Production structure and economic fluctuationslowast

Tommaso Ciarlidagger Marco ValenteDagger

November 18 2006

Abstract

We aim at contributing to the debate on the mechanisms and prop-erties of economic fluctuations We consider a crucial aspect amongmany thought to influence this ubiquitous and extremely relevant phe-nomenon the interaction structure that characterises the organisationof production that is the production relation among sectors of a sys-tem

We build mdash and simulate mdash a very simple model representing aninputndashoutput system where sectorsfirms adapt production and desiredlevels of stocks Their output serves both an exogenous final demandand the intermediate demand solicited by the other sectors of the sys-tem Series of simulation runs allow to derive relevant and nonndashobviousconclusions concerning the levels and more importantly the volatilityof economic activity as an outcome of the same inherent economicstructure

We claim that the results that we obtain through the highly ab-stract representation we use provide useful intuitions on the workingof economic cycles to be later integrated by further studies

As a byndashproduct of our analysis we also suggest that the method-ology we adopt can provide valuable insights by allowing a detailedanalysis of the time path generated in the artificial systems and there-fore assessing with precisions the same mechanisms that affect realndashworld systems The natural following step left for further research isto investigate how those mechanisms are empirically generated

Keywords Production structure micro- and macro-volatility sim-ulation models

JEL E32 E37 C63 C67 D57

lowastThe paper has benefited from comments received during presentations in the followingvenues 12th ACSEG Conference (ldquoConnectionist Approaches in Economics and Manage-mentrdquo) Marseille Wehia 2006 (1st International Conference on Economic Sciences withHeterogeneous Interacting Agents) Bologna 11th ISS Conference (International Schum-peter Society) Nice But we are the only responsible for the outcome

daggerFaculty of Economics University of LrsquoAquila amp University of Bolognatommasociarli[at]uniboit

DaggerFaculty of Economics University of LrsquoAquila valente[at]ecunivaqit

1

1 Introduction

Business cycles have always been a major concern for economists becauseof they are a phenomenon of great impact on societiesrsquo welfare and theyhave shown to be one of the most constant feature of economies across timeSuch is the importance of business cycles that economists seem to have spentmore attention in providing normative tools to control them than in pro-viding a detail explanation of their origin and nature1 In broad terms thegeneral agreement is that business cycles stem from the interaction of twofeatures comovement of economic variables (including actorsrsquo decisions) onthe one hand and exogenous events usually random (at micro andor macrolevel) on the other hand Typically a model meant to explain cycles studyhow specific stochastic events (ie a flow of random shocks) can perturb aneconomy from its equilibrium state In some cases for example economistssuggests that micro-level shocks partially cancel out at aggregate levels withthe fluctuations being due to stochastic excesses not absorbed Converselyother works suggest that micro-shocks tend to reinforce each other gener-ating aggregate fluctuations larger than the shocks that originated themA recent strand of the literature on business cycle has correctly pointed tothe central relevance of the production structure Depending on the way inwhich the entities composing the supply side of the system interact they willreinforce or smooth away random shocks generating or dumping aggregatefluctuations

In this work we contribute to the debate by removing a number of draw-backs that curtail in our opinion the analytical power of most of the litera-ture on this subject First of all most of the literature considers jointly theeffects of stochastic events and structural features of the economic systemsIndeed real systems do face a constant flow of non predictable changesimpacting on interacting economic agents However realistic the joint con-sideration of the two features of an economic system (structure and stochas-ticity) prevents the rigorous assessment of their separate contribution to theaggregate phenomenon of fluctuations We propose to take a different routewe make the highly unrealistic assumption that there is no exogenous flowof shocks and concentrate on the role of the production structure Giventhe simplification obtained by non considering the noise of random eventswe will be able to make a detailed assessment of the properties of a real-istic representation of an economic system in respect of fluctuations Forsimilar reasons we also ignore other aspects proposed as relevant in gener-ating fluctuations like technological development price adjustments lumpyinvestment financial constraints etc Our approach does not deny the rel-evance of these factors we sustain that they are logically if not practically

1A similar claim is at the base of the research originated in Dosi Fagiolo and Roventini(2006)

2

separated Their contribution must be studied in order to produce a robusttheoretical understanding of the business cycle phenomenon Indeed thephysical structure through which any micro change (or reaction to a change)is transmitted in the system is the production structure And the way inwhich shocks propagate is at least as relevant as the sources of propagationIn the words of Zarnovitz (1977) we investigate a lsquotheoretical possibilityrsquo ofbusiness fluctuations that for the very nature of production systems is alsoan lsquoexplanationrsquo (while not an assessment) The aim of this paper is thento analyse transmission properties of multisectoral production systems

The structure of the paper is the following Next section briefly reviewsthe major contributions of the literature on the subject highlighting theworks more closely related to our approach In the third section we high-light the main elements that characterise the transmission mechanisms ofa production structure The fourth section describes a very simple modelrepresenting a dynamical production system comprised of an input-outputmatrix and a few simple behavioural rules governing the actions of economicsectors Section five then discusses the major results of the model under afew parametrisations Finally the last section will draw the conclusions andsuggest directions for further research

2 Economic fluctuations

A wide number of reasons may explain why economic aggregates fluctuateand a wide number of theories attempt to explain short run and long runwaves in economic growth For example large shocks that affect an en-tire economic system are likely to fully displace it changing its long termpattern Though while such shocks may be plausible for long run wavesthey are less likely to be the cause of short run cycles given that they donot occur with the same frequency Moreover aggregate shocks may affectthe various economic entities differently making it quite difficult to statethe final result of a complex combination of reactions Conversely shocksat more disaggregate levels of the economy are undoubtedly more frequentThis itself renders the study of the influence of micro shocks on aggregatefluctuations an important piece of analysis and eventual understanding ofbusiness cycles

It is selfndashevident that being economic entities interconnected to eachother they may absorb linearly transmit or reinforce the quantitativechanges that hit on them mdash induced by neighbouring entities mdash to differentextents This can be observed at different levels of aggregation An eco-nomic crisis in one country causes shocks in related countries (eg BackusKehoe and Kydland 1995 Head 1995 Kraay and Ventura 1998) a crisisin the financial system causes readjustment in the proximate systems inboth supply and demand (eg Stiglitz and Greenwald 2003 Delli Gatti

3

Di Guilmi Gaffeo Giulioni Gallegati and Palestrini 2004)2 the failure ofa large company induces readjustments in the same and related sectors andson on

Besides we expect that the more we disaggregate the economic units ofanalysis the more shocksrsquo intensity is likely to reduce and symmetric shockstend to cancel out The application of the law of large numbers would sug-gest that an economy with normally distributed entities (at the same level ofaggregation) mdash or even better with representative entities would not showaggregate fluctuations in the presence of uncorrelated disaggregated shocks(Lucas 1977) This has led to focus most theoretical explanations (or rep-resentations) of short business cycles on aggregate shocks (Horvath 2000)Although they are undoubtedly relevant and have an impact also on mi-cro changes aggregate interpretations are quite limited in their explanatorypower While having to assume exogenous origin of shocks they do notprovide an understanding of how macro shocks rebound on the economicentities (consumption change production shift productivity shifts invest-ments etc)

Indeed in order for micro shocks to generate aggregate fluctuations twoconditions are necessary a propagating structure mdash a connection mdash be-tween micro economic entities and a sluggish absorption of changes Inthis respect production structures made up of sectors connected by inputndashoutput relations represent a good candidate a part from being the knownstructure of a basic industrial economy Two main class of models havebeen developed to analyse the effect of sectoral shocks on aggregate fluc-tuations Real Business Cycles (RBC) and Avalanche (Av) models Bothtype of models make attempt to introduce mechanisms that allow for thepersistence of micro shocks In this respect both classes are based either onquite unrealistic behavioural assumption which determine the conditions ofthe system or threshold behaviors which impose the conditions for shockspersistence We briefly review them below

21 Mechanisms of shocks propagation in IndashO structures

A number of contributions study the phenomenon of generation and prop-agation of cycles independently from the sectoral structure of the economyHowever the most recent contributions point in the direction of giving thisaspect of economic system a strong relevance Moreover the two approachesare not incompatible so that we focus on the literature that explicitly makesuse of the IndashO structure In this section we briefly review three strands ofthis literature highlighting the elements that inspire our work

2Both cases are easily shown with evidence from the Latin American crisis during theeighties rooted in the crisis of the Mexican financial system the financial crisis in Asiaduring mid nineties the recent Argentinian financial crisis etc

4

Plain IndashO structure

The pioneering RBC model that analyses the diffusion of sectoral shocks isdue to Long and Plosser (1983) They use a general equilibrium multi sectormodel with fully rational infinitely lived perfectly informed homogeneousetc individuals In such a system by assuming a perfectly maximising be-haviour (via allocation of production and consumption of resources) andthat individuals prefer to smooth savings across time and goods it is pos-sible to generate persistence of continuous albeit uncorrelated shocks inproduction capacity across sectors In the words of Long and Plosser [p67] ldquoAt constant relative prices this [the assumed maximising behavior]suggests that business-cycle features like persistence and comovement arecharacteristics of desired consumption plansrdquo

The necessity to maintain the assumption of perfectly rational agentsforces these strand of models to attribute the origin of cycles to externalfactors only like exogenous shocks on technology As such they seem un-likely to provide a consistent explanation (or representation) of aggregatefluctuations Dupor (1999) shows that under quite general conditions theavailable multindashsector RBC models produce the same results in terms ofvariance convergence as their respective single sector models This is shownalso under particular conditions of input matrix or high coefficient of spe-cific inputs Dupor concludes that [p 405] ldquoresearchers who wish to useindependent sector shocks as a source of aggregate fluctuations must appealto sector shocks that are large relative to their single sector counterparts orelse consider a different set of models than those discussed in this paperrdquoIn doing so the author though suggests that another mechanism must beintroduced somehow putting the desired results ahead of the economic issueanalysed

The criticism to this kind of literature suggests that in order to allowthe IndashO structure to play a role in the explanation of cycles we need agentswith less then perfect foresight and capacity of adaptation to unndashexpectedconditions As we will see below agents that need time to both realizethe economic changes of their time and to introduce the necessary mod-ifications are sufficient conditions to observe aggregate fluctuations froma single change in one part of the economy Though being ldquorationalrdquo inthe sense that they make the right decisions they are limited by lags inthe information flow and physical constraints to the changes that they canapply

Input matrix incompleteness

Using a similar approach to RBC models Horvath (1998) and Horvath(2000) focus the attention on the properties stemming from peculiar produc-tion structures as represented by characteristics of the input-output matrix

5

The author shows that sectoral shocks will tend to cancel out in case theeconomic system is sufficiently distributed that is when the IndashO matrixcontains evenly distributed values over all the cells Conversely sectoralshocks are not absorbed and can be reinforced in case many cells of thematrix are empty This case indicate that there will be few sectors affect-ing many other sectors preventing the possibility of compensating certainshocks

These conclusions are potentially interesting because point directly toan easy to observe aspect of economic system for example providing thepossibility to test the prediction of the analysis It is worth to note that thisapproach draws conclusions that are not neutral to the aggregation levelused In fact a highly disaggregated matrix is more likely to contain manyempty cells than the matrix for the same system obtained aggregatingsectors

Dupor (1999) also considers the case of inputndashoutput matrices with dif-ferent input coefficients finding them irrelevant for the results But he doesnot include matrices with empty cells

The cited contributions and those inspired to the same approach sug-gest that the type of interaction as expressed by input-output coefficientscan be a relevant factor in determining the existence and dimension of fluc-tuations However this literature is not able yet to draw conclusive results

Limited interactions and production constraints

A different kind of mechanisms are assumed and shown to play a role in theAv models they add an analysis on firms production (technological) possi-bilities to the production structure They are inspired by and similar to Jo-vanovic (1987) model in which the limitation in the number of interactionsupon which each playerrsquos decision depends generates shock cascades andaggregate persistence Bak Chen Scheinkman and Woodford (1993) andScheinkman and Woodford (1994) show that when the interaction structurebetween sectors is constrained to a completely rigid system coupled withnon convexities in the production function independent shocks to differentsectors do not cancel in the aggregate In other words the authors place aset of restrictions to the set of possible actions of sectors First structuringthe production networks with a fixed invariable lattice produces a rigiditythat do not allow producers to change (or add) inputs (or suppliers) orincrease the produced quantity This is combined with the structure of theavalanche that propagates through the unchanged network Second a pro-duction function that is maximised only when a discrete number of unitsof a good is produced coupled with a fixed (discrete) maximum amount ofinventories that can be stored by the representative firm Such a structuredetermines production oscillation and shocks avalanches due to the factthat firms are constrained in their production opportunities

6

Using a very similar production structure with limited interactionsWeisbuch and Battiston (2005) play on the bankruptcies generated on down-stream firms by production failures occurring in upstream firms (reducingthe flow of inputs) Stochastically generated failures to produce in a firmcauses production constraints in its downstream clients hence lower invest-ment and eventually bankruptcies3 A lag between the period in whicha firm goes bankrupt and a new firm appears generates a lack of supplywhich propagates on the downstream sectors

Due to the original literature on avalanches these contributions sharetwo common features that are ill-adapted to economic systems First theyassume very peculiar interaction structures of one-directional productionrelations between sectors which amount to assume one half of the input-output matrix empty and are not applicable to the general case with cyclicalproduction structure or full IndashO matrices Second the strongly rely on quiterigid thresholds determining a mechanicist behaviour by agents Basicallyagents have only two options available (ie produce or not produce) andthe choice is made on the basis of crude decisional mechanism4

Notwithstanding the limitations this approach provides a quite sensiblemechanism of the origin and transmission of shocks through a system Ourproposal can be interpreted as an extension of this approach by relaxing thestrong assumptions on the interaction structure and allowing agents for amore fluid decisional process as we will see in the next section

3 The basic elements of a production structure

Our overall hypothesis is that the very production structure of an economymay generate persistent fluctuations even in the absence of continuous un-related shocks and rigid thresholds that determine non linearities The aimis to provide a generalised interpretative framework to explain the occur-rence of short term fluctuations in the economy simply as an outcome ofthe division of labour in a number of sectors related through trade In itsextreme interpretation any production system which is not fully integratedis bound to produce business cycles As we will show a number of factorsinfluence their extent

Quantitative adjustments in production in any one sector require ad-justments in related sectors When the structure of the economy is trulyInputndashOutput (not a directed linear supply chain as in the Av models) in-put flows are cyclical and adjustment to long run production equilibrium islikely to take a long mdash when not infinite mdash time Our work departs from

3A similar model by Battiston Delli Gatti Gallegati Greenwald and Stiglitz (2005)relates bankruptcies to price changes buyers payment timing and the credit market (acentral bank)

4Nirei (2005) is an attempt to generalise such models though maintaining the assump-tion of threshold behavior

7

both RBC and Av models with respect to structure modelling criteriaanalysis and methodological approach Indeed our model may be easilyreconduced (or restricted) to both We are much in line with a recent se-ries of papers by Helbing Witt Lammer and Brenner (2004) that addressthe same question although with a number of differences that are discussedlater on5

We aim to analyse to what extent the structure of inputndashoutput relationscauses aggregate fluctuations by itself The main idea is to show that ag-gregate fluctuations may derive from simple micro imbalances without theneed for continuous exogenous shocks not even at the firm or sectoral levelIn other words normal production activities including lags technologicaladjustments information asymmetries circularity of the inputndashoutput sys-tem and so on may be enough to explain part of the macro volatility in aclosed system6

Our assumptions draw on the existing literature but try to avoid themajor difficulties highlighted above Firstly we avoid to impose either mar-ket clearance assumptions or crude threshold behaviours Rather we adoptan assumption close to a Simonian approach agents in an economy do theirbest within the informational and physical constraint they are subject toWe represent decision makers as responding to a single piece of informationthey obtain the quantities demanded by their clients Crucially their ac-tions are inspired to a conservative approach an increase in the demandgenerates a smaller (desired see below) sudden increment of productionThis assumption stems from the fact that decision makers are aware of theinstability of their environment and therefore want to avoid getting perma-nently caught with over production following a temporary spike of demandIn other words firms attempt to smooth business cycle generating a morestable (or less uncertain) environment as the economic literature would sug-gest Notice that this representation in line with a routinized representationof organizations (Nelson and Winter 1982) does not contradict market equi-librium Rather in a stable environment such decisional procedure generatesan asymptotic pattern to the equilibrium level in the absence of feedndashbacks

A second assumption concerns the technological possibilities available tothe firms When firms realise the need for a change in production levels saya desired increase of production they face a complex and costly organisa-tional transformation of the production capacity risky investments an soon As Bresnahan and Ramey (1994 p 622) suggest inducing from a de-tailed analysis of the automobile industry in the US ldquoadjusting production

5We thank Matteo Richiardi for pointing out these similar attempts See also Helbingand Lammer (2005) and Helbing Lammer Witt and Brenner (2004)

6An open system entails many other factors including price factors of both inputsand outputs mdash terms of trade labour cost external investment etc As with exogenousshocks our analysis does not deny the importance of these factors but simply separatethem from the structural explanation of cycles

8

is a more complicated process than simply ldquochanging Qrdquo or choosing themix of capital and laborrdquo And this is not only a matter of non convexitiesin production technologies The inconsistency of time reversibility in thetechnological choice is accompanied by the need to operate on the lsquoavail-able technological frontierrsquo (David 1975) If a firm plans a given productionactivity for a short period of one week even though the next day there is anunexpected fall in the demand it is unlikely that it is followed by a similardownturn in production The case is evident when we think at productionplanning that involve hiring of workers an capital investments (both in-volving sunk costs)7 Therefore any desired change generates a pattern ofsmall modifications approaching slowly the target level Moreover thesechanges are irreversible in the short term since only a prolonged change ofthe environment triggers the modification of previous desired levels

Resting on those simple conjectures we thus analyse a model in whichwe only allow for sluggish adaptation of production due to conservative(adaptive) behaviour of firms and the physical constraints faced when vary-ing the production levels Notice that these assumptions are contrary tothe generation of instability of a system In fact a firm behaving as de-scribed above reduces if not eliminates any disturbance affecting its ownstate if acting in isolation Other factors like new technologies extremelylong term perspective price changes etc are likely to be sources of volatil-ity therefore contributing to generate aggregate system fluctuations (as forexample in Acemoglu and Scott 1997) Our neglect of these and other fac-tors however unrealistic serves the purpose of testing whether a volatilityreducing microndashbehaviour turns into volatility generating aggregate resulttwo widely observed dynamics Quite interestingly (and encouraging) DosiFagiolo and Roventini (2006) have concomitantly proposed a model whichincorporates detailed representation of both sluggishness robustly based onempirical evidence Heterogeneous firms which undergo lumpy investmentand slowly adapt their expectation on changes of the future demand inducecomovement of macroeconomic variables qualitatively similar to the onesobserved in the real world To pursue our line of research we first needto concentrate on how sectorsfirms interact in the system in the effort tocomplement those promising results

Therefore the second element we consider is meant to represent a gen-eralized economic system We allow for the circularity of inputndashoutput sys-tems a crucial feature in determining feedback mechanisms and a principal

7See also the discussion in Acemoglu and Scott (1997) where the authors argue thatldquo[w]hile the presence of fixed costs can account for the discreteness of economic turningpoints it does not naturally lead to persistence because once an individual undertakes anaction they are less likely to do so in the near future [] [A]lthough the presence of fixedcosts leads to increasing returns these are intratemporal the full extent of economiesof scale arising from fixed costs can be exploited within a periodrdquo [p 502 authorsrsquoemphasis]

9

component of business cycle It may be true that most backward linkagesconcern capital goods (Weisbuch and Battiston 2005) but a rapid glanceto an inputndashoutput matrix would show that technical coefficients are nonnull in both directions and ldquomost commodities are inputs to the produc-tion process of other commoditiesrdquo (Horvath 2000 p 70) The higher thesectoral aggregation mdash observed or assumed mdash the more the relevance ofintermediate supply holds true (as mentioned above with respect to ma-trix incompleteness) We claim that the existence of backward and forwardlinkages are a crucial element to determine the persistence of shocks in theeconomy A perfectly linear (one directional) system needs much less adap-tation of firms and sectors in order to stabilise the economy Nonethelesscircularity does not imply that inputs are perfectly substitutable amongthem in order to determine the maximising allocation even at high level ofdisaggregation

By using a fixed coefficients production function we both maintain themid term rigidity of production processes and we do not impose limitedinteractions We claim that the circular property of the production structureis by itself sufficient to produce shock persistence even when the inputmatrix is full (although with heterogeneous input coefficients) mdash all sectorsshop in the remaining n minus 1 sectors

In conclusion we represent a model for an economic system made onlyof i) production level decisions and ii) input-output relations Clearly sucha model lacks many features present in real economic systems likely we areconvinced to affect the propensity to generate cycles As such we cannotaim at reproducing realistic data so our results cannot be tested comparingtime series generated by the model with real ones Instead given the paucityof the elements comprising the model we can investigate in detail to whichextent the (few) features present in our model affect the cyclical behaviour ofthe system As we will see we will be able to show quite a number of resultin which aggregate fluctuations are generated whereas the individual microstructure of the model aims at monotonous adaptation Before discussingthe results next section describes in detail the implementation of the simplemodel

4 The model

We model an economic system in which a number of sectors i = (1 N)are potentially connected via (intermediate) market relations A N times NInputndashOutput matrix determines the mij units of each input j = (1 N)mdash bought from the N minus 1 residual sectors mdash necessary to produce oneunit of output i8 For simplicity given the focus of this paper on the IndashOstructure as a generator of aggregate fluctuation we abstract from market

8IndashO coefficients are therefore physical coefficients

10

interactions within sectors each sector is a production unit In fact forour purposes it is not relevant which firm produces the sectoral outcomebut only the total amount produced by each sector9 Moreover besidesmaking the analysis more neat this assumption allows also to enjoy a largefreedom in determining the aggregation level since we donrsquot need to assessthe effects of between sectors competitiveness which for example is likelyto increase for increasing levels of disaggregation Our economic systemis therefore composed of N production units each of which may buy andprovide inputs from and to any other sector depending on the IndashO structureof the economy

The demand for each sector i is determined by a constant consumersdemand Di (E) and the input needs of other sectors

Di = Di (E) +sum

j 6=iisinCi

mjiQj (1)

where Ci is the set of j intermediate clients of i (sum

j isin C lt N) mji is thetechnical coefficient of firm j for input i ie the amount of i required toproduce one unit of j Qj is the actual output produced by sector j

Concerning the simultaneous production level of each sector in time twe make two lsquorealisticrsquo assumptions First from the technical viewpoint afirm like a tanker cannot abruptly change its course Therefore changesto the production level can be introduced only gradually through capital(dis)investment and job h(f)iring (see discussion in Section 3) Second weassume that firms do not decide the production level directly on the basisof the observed demand level but conscious of the uncertainty of demandplan their production in order to maintain invariant a desired level of stocksand avoid stockout10

Quantity produced by sector i in period t is computed as follows

Qit = αiQtminus1 + (1 minus αi)(

Slowastitminus1 minus Sitminus1

)

(2)

where αi measures the physical constraint in adapting the production capac-ity to changes in the required output Slowast

itminus1are firmrsquos i desired stocks and

Sitminus1 its actual stocks Notice that both variables related to stocks appearwith a lag This is due to two reasons First it is a modelling necessity inorder to allow the simultaneous determination of production levels for allsectors Secondly as frequently happens modelling necessities reveal sen-sible if not generally considered aspects of the modelled system In factproduction decisions concern management in the production plant while

9Adding heterogeneous firms within sectors would definitely enrich the causal expla-nation put forward and parametrised in this paper (see below) For example to analysethe effect of firms constraints It is left over for future research as in the present work weopt for the analysis of the main structural parameters that affect aggregate fluctuations

10See also discussion in Schuh (1996)

11

demand through sales is observed by the commercial staff It is thereforelogical that the information from demand reaches plant managers with somelag

The actual stock level is trivially computed subtracting from the previousheap of stock the amount sold and adding the new production

Sit = Sitminus1 + Qit minus Dit (3)

Concerning the desired level of stocks instead we introduce the conser-vative behaviour mentioned above Firms ideally would keep an amountof stocks proportional to the demand they receive say a multiple σ ofthat amount However when demand varies they adjust the desired levelsmoothly preventing sudden changes to the variable representing the goalof the firm

Slowastit = Slowast

itminus1 + αsi

(

σDit minus Slowastitminus1

)

(4)

where αsi measures the adaptation to demand changes

In summary the dynamics of the model consists of only the equationsabove Firstly at the beginning of a time step firms determine the quantityto produce as a function of past imbalances in stock levels and the pastproduction level These levels define the demand for all sectors determinedby the technical coefficients applied to all produced quantities Finallyactual and desired stocks can be updated Figure 1 represents visually thisdynamics highlighting the decisional components of the model in respect ofthe ldquomechanicsrdquo of trade

5 Results production structure and economic fluc-

tuations

In this section we describe some of the most relevant results concerningthe effects of the production structure to determine aggregate fluctuationsGiven the unusual analytical instrument chosen (at least concerning thistopic) we start with a brief methodological introduction meant to clarifythe nature of the results we claim to obtain Next we describe a generalsetting for the model which is the (extremely) limited area of the potentialparametersrsquo space of the model that we will explore In the rest of thesection we discuss the results from the model

51 Methodological considerations

Real economic systems are constantly subject to a continuous flow of shocksexternal to the production system besides changes within the system itselfIn order to appreciate the systemic properties we need to abstract from all

12

Compute the quantityto produce

F (Q(t-1) S(t-1) s(t-1) alpha)

Communicate thequantity of inputsrequired to each

supplier

Determine the overalldemand

F(D(E) D(I))

Revise the past decisionon stock plans

(desired stocks)F(D S(t-1) sigma alpha(s))

Compute the final levelof stocks for next period

F(s(t-1) Q D)

KEY

ModelDirection

Variablecomputed

at t-1(t-1)

Step includingfirm behavior

Figure 1 One period dynamics

the nonndashrelevant disturbances and concentrate on how the system endoge-nously contributes to the observed dynamics Therefore we will generatehighly abstract patterns whose analysis will provide insights on the systemproperties that would be otherwise lost in the noise generated by all theelements of a real system

We are not trying to fit the model to a specific empirical data set nor weare interested in studying the complete behaviour of the model for areas ofthe parameterrsquos space that make no economic sense (eg negative or infiniteproduction levels) Therefore the values of initial settings and results matteronly in relative and not in absolute terms

We are going to describe a simulation model with which we will produceseveral simulation runs and we claim obtain results relevant for the debatediscussed above Rigorously speaking we may claim only certainty for thevalidation of the results that is that the model does actually generate theresults presented and for the reasons we explain We limit this part to thepresentation of graphs and verbal explanations that we believe are ratheruncontroversial11 Concerning the relevance of our studies for real systems

11Interested readers can request the model code and and simulation data for replicationand extensions of the results We can guarantee that such possibility does not require

13

(verification) we simply avoid to pretend that our model is a quantitativeapproximation of real economies (which in which period) Our opinionis that the study of the model allows us to make general considerationsnon-obvious and relevant for applied debates supported by strong logicalarguments which can be integrated but not reverted by arguments con-cerning more and more elements of the economic world

52 Model setup

We go through the model properties analysing an economic system madeup of ten sectors (N = 10)12 Each sector shops inputs from the remain-ing N minus 1 and sells to them part of its own production as intermediategoods This means that the input matrix is complete except for the diag-onal (mij gt 0 forallj 6= i mii = 0) The sum of all N minus 1 coefficients for asingle sector is given by Mi and its benchmark value is kept at a plausiblemedium level (unless differently specified) to avoid distortions from highlydemanding sectors Each input coefficient is randomly drawn from a uniformdistribution

Sectors start at an equilibrium level their supply matches intermediateand final demand and in each period the same quantity of output mustbe produced to compensate for the used stocks and keep the desired stocksunchanged In thier turn stocks must be ten times the value of total sectoraldemand (σ) Eventually technological adjustment of production is ratherslow and producers are quiet refractory to follow sudden changes in thedemand in general a smoothing behaviour strongly prevails (α and αs)

Under those general preconditions in the following sections we anal-yse the dynamics of the economic system in relation to model parametersWe mainly focus on the cyclical behaviour of systems and their speed ofconvergence to the asymptotic equilibrium13

The very abstract assumption and the homogeneous initialisation donot allow for the multiple equilibria that are likely to rise in complex systemswith feedbacks Once more such simplification allows to have a betterunderstanding of the crucial impacts of the production structure on businessfluctuations Departing from the related literature discussed above14 weconsider the effects of a single shock rather then a continuous flow whichin our model is superfluous in order to obtain fluctuation persistence15 To

extensive programming or statistical skills besides the usual education of economists12You can think of a high level of aggregation in the statistical observation The com-

plete initialisation and parameters values are available in table 1 in the Appendix13Notice that sectors only approach a final equilibrium level of production which can

be prendashdetermined under static conditions14Included the more proximate work of Helbing Witt Lammer and Brenner (2004)15An exception are the last results presented in Section 58 which we see more as an

opportunity for further research than a closing of the paper

14

start with next section (53) shows the aggregate cyclical behaviour of astandard setting

53 Single shock analysis

We generate a simulated history following the dynamics of the modelrsquos vari-ables when the system is set to an equilibrium level and a single shockmodifies one of the exogenous elements namely the external demand forsectors Modifications across sectors are random and not correlated andmay increase or reduce external demand by at most 10

100 225 350 475 600

193894

195861

197828

199795

201762 Industrial Production

Time

Figure 2 Total industrial production fluctuations following a shock fromexternal demand

Figure 2 shows the dynamics generated by the total industrial productionof the system mdash that is also a proxy for the systemrsquos GDP fluctuations Thenew external demand changes the equilibrium level but the system cannotreach the new equilibrium with a monotonous pattern and converges to thenew level with a fluctuating pattern Each change in the external demandinduces also modifications of the intermediate demand and sectoral adjust-ments in the quantity produced typically overshoot the aggregate patternAs it can be appreciated from the figure it takes time before the system isable to converge to the new static equilibrium conditions

We study two features of this pattern absolute level of the new equilib-rium and volatility induced by the shock

54 Technical coefficients and industrial output

Production level

The equilibrium levels of industrial output depend on two factors only Ob-viously the levels of external demand determine how much of the total pro-

15

duction needs to be supplied to consumers exogenous from the productionsystem Second the technical coefficients also influence strongly the level ofactivity of the system defining the intermediate demand to each sector

180

200

220

240

260

280

300

01 02 03 04 05 06 07 08 09

Out

put l

evel

(In

dust

rial p

rodu

ctio

n)

Sum of total input coefficients (M)

Total industrial production for different input coefficients levels

Figure 3 The relevance of input coefficients on output level

Figure 3 shows the equilibrium level of industrial output reached bysystems with identical external demand where we fixed the sum of inputcoefficients for each sector of the economy (Mi) to different values16 Thehigher the sum the higher is the total production sustained by the system

This result depends on the very production structure represented by theinputndashoutput matrix In fact the higher is the level of production requiredby intermediate sectors the higher is the total production given a constantlevel of external demand

Notice that the level of production does not depend on the distribution ofthe technical coefficients As long as the sum of the coefficients is identicalthe same production level is obtained irrespective of their distribution (otherthings being equal)

Output shock absorption

Input coefficients have a relevant role also on the output stability of thesystem Starting from an output equilibrium level we hit the system witha single shock on the external demand for all sectors uncorrelated amongsectors17 modifing (positively or negatively) the demand for all sectors by

16Within this limit the actual value of the coefficients are randomly drawn17In each sector the external demand changes by plusmnX where X is a uniformly dis-

tributed percentage between 1 and 10

16

at most10 As show in Figure 2 the economic systems produces a lenghtlysmoothing oscillatory behaviour We compute the relative deviation of theaggregate production (industrial output) as the ratio between its standarddeviation and its mean δ = σY microY where Y =

sum

Qi Figure 4 shows thebehaviour of δ as a function of the sum of input coefficients for each sector80 periods after the shock As time goes by the level of δ reduces for allMi but the relation with the sum of input coefficient is unchanged

00055

0006

00065

0007

00075

0008

00085

0009

01 02 03 04 05 06 07 08 09

Rel

ativ

e va

rianc

e of

tota

l out

put (

delta

)

Sum of total input coefficients

Aggregate output deviations for different technical coefficients

Figure 4 The relevance of input coefficients on output stability

In sum input coefficients determine the extent of overshooting behaviourThe higher the coefficients the higher is the backlash even when shockscould cancel out through sectors

55 Shock persistence and adjustment coefficients

We now consider the effect of micro smoothing behaviours on the volatilityof the production system In particular we are interested in testing theeffects of two parameters of the model on the volatility of the productionsystem First we consider the effects of the αrsquos representing the ldquostickinessrdquoof the production technology to desired changes of Q aimed at filling thegap between actual and desired stocks The higher is α the slower is thereaction to stock unbalances Concerning its effect referring to the technicalpossibilities available ie production capital and organization we may putforward two opposite hypothesis relevant to systemrsquos volatility

Hypothesis 1 Slower production adjustment (high α) reduces volatilitybecause unbalances in demand of one sector being compensated slowly slow

17

down the diffusion of shocks to other sectors

Hypothesis 2 Slower production adjustment (high α) increase volatil-ity because the longer lasts the mismatch between actual and equilibriumproduction level the deeper will become the mismatch between actual anddesired stocks

20 25 30 35 40 45 50 55 60 65

Cyclesrsquo width

07 075

08 085

09alpha 0005

001 0015

002 0025

003 0035

004 0045

005

alphaS

20 25 30 35 40 45 50 55 60 65

Figure 5 Maximum width recorded between highest and lowest peaks fortotal production for different values of α and αS

Secondly we consider the parameters αS representing the speed of ad-justment of desired stock to observed demand The higher this parame-ter the faster sectors revise the desired levels of stocks to current levelsof observed demand This parameter concerns sectorsrsquo behavioral decisionprocess Higher αS represent a stronger belief that the most recently ob-served demand level is a reliable indicator of future demand levels whilelower levels of the parameter represent a more conservative approach wheremanagers require a long observation of demand levels different from the pastones before adjusting their expectations Also concerning the effect of thisparameter on the volatility induced by an exogenous shock we may maketwo hypothesis

Hypothesis 1 Faster adaptation of desired stocks to new levels of de-mand (higher αS) reduce volatility because changes in demand are rapidlytranslated in adjusted levels of production

Hypothesis 2 Faster adaptation of desired stocks to new levels of demand(higher αS) increase volatility because temporary unbalances of demand in

18

one sector are transmitted faster to other sector reinforcing the feedbackmechanism of fluctuations

To test which of these hypotheses are confirmed by the model we runsimulation runs with identical systems (eg same technical coefficients samelevels of demand same shocks) but for the values of the αrsquos and αS rsquos testingall combinations of these parameters over a range of values (each run usedthe same parameters for all the sectors of the system) For each simulation(ie each couple of values of the parameters) we computed two indicatorsof volatility One measures the maximum distance of total production be-tween the highest and the lowest peak that is the maximum width of theoscillations registered in the production time series The second indicatormeasures the relative deviation δ = σY microY Figures 5 and 6 report theseresults

0004 0006 0008 001 0012 0014 0016 0018 002

Relative deviation

07 075

08 085

09alpha 0005

001 0015

002 0025

003 0035

004 0045

005

alphaS

0004 0006 0008 001

0012 0014 0016 0018 002

Figure 6 Relative deviation of total production for different values of α andαS

The results confirms unequivocally both hypotheses 2 and reject hy-potheses 1 In a sense these hypotheses confirm the nature of the economicsystem represented in the model as a complex system where the interactionsamong sectors play a far more important role than the individual sector be-haviour In fact in both cases we may interpret the results that the strongerthe efforts to mitigate the effects of shocks (ie slow productionrsquos and quickexpectationsrsquo adjustment) the opposite result is actually obtained the fluc-tuations become even more accentuated It is also worth to note that thestronger impact is generated by the behavioural parameters rather than thetechnical one as indicated by the stronger effects of the αS in respect of theα This result seems to suggest that ceteris paribus information mattersmore than technical constraints Though the model is by any means in-

19

adequate to discuss normative aspects because of the naive ways strategicdecision making is represented it can anyway suggest the effects of errorsin management decisions (ie wrong desired levels of stocks) as opposed tothe technical constraints preventing sudden adjustment of production lev-els Stretching a bit the interpretation of these results we may suggests thatefforts to reduce volatility should better be oriented to improve the trans-mission of information in order to coordinate future production plans (ieidentifying quickly the desired levels of stocks and therefore the new equilib-rium production level) rather than improving the flexibility of productionmethods (reducing α)

56 Volatility and stock levels

In our model stock levels are maintained to buffer a sectorrsquos sale againstunexpected changes in demand Therefore a system made of risk adversemanagers (keeping higher levels of stocks) may be thought to lead to asmoother absorption of shocks in respect of a system composed of firmskeeping a short supply to cushion unexpected events To test this hypothesiswe tested the behaviour of the model for different values of σ the levels ofstock represented as a multiple of demand For each level of σ we run asimulation using all other initialization identical as usual representing aneconomic system ldquoalmostrdquo at equilibrium but for a single small shock indemand uncorrelated across sectors18

00044

00046

00048

0005

00052

00054

00056

00058

0006

00062

5 55 6 65 7 75 8 85 9 95 10

Rel

ativ

e va

riatio

n of

tota

l out

put (

delta

)

Ratio of stocked goods over production (sigma)

Stocks and production fluctuations

Figure 7 Fluctuations persistence on systems with different stock strategies(σ)

18Demand changes by plusmn1-10

20

Figure 7 shows the value of the relative deviation of output production(δ) generated by heterogeneous uncorrelated shocks in the external demandon identical systems where firms store a different multiple of their demand asstocks We obtain actually the opposite result that we may have expectedfluctuation become larger and more persistent the higher the level of stocksIn some way a strategy that for the individual firm is meant to reducethe fluctuations (using stocks as buffers) ends up by actually increasing thevolatility at aggregate levels

The result is though explainable both at the theoretical and empiricallevel In fact if no stocks were maintained production would need to adjustfor changes in demand only in case of a shock That is the firm may movemonotonically from the old to the new production levels sending consistentsignals to the rest of the system (ie its own suppliers) Instead if thesame shock affects a firm maintaining large stocks of output over the sameperiod production needs to adjust to the new demand and and to alignstocks with the new demand This generates besides the adjustment to thedemand a potentially inconsistent signal to the suppliers since the firm willhave a temporary level of production not meant to continue in equilibrium(adjusting stocks)

From the empirical viewpoint the last few decades have seen the almostuniversal diffusion of production system meant to keep stocks to a minimumif at all Lean production methods justndashinndashtime etc are solutions tothe problems of the costs necessary to maintain stocks Over the sameperiod economic fluctuations have been strongly reduced and few empiricalassessments have pointed to the relations between the two phenomena Forexample concerning the US ldquochanges in inventory behavior have playeda direct role in reducing real output volatilityrdquo (Kahn McConnell andPerez-Quiros 2002 p 183)19

57 Irrelevance of technical coefficients distribution

Horvath (1998) and Horvath (2000) suggest that systems with sparse ma-trices ie inputndashoutput matrices with many empty cells tend to be morevolatile in respect of systems with more evenly distributed matrices Noticethat this claim risk if not adequately qualified to generate an illogical resultIn fact matrices sparseness increases as a function of sectoral disaggregationand comparing the same systemrsquos fluctuations using inputndashoutput matricesat different levels of aggregation needs to produce the same type of volatil-ity Indeed this is not the case if all remaining parameters stand equaland do not adjust to the different level of analysis considered If aggregatepatterns of adjustment speed are maintained unchanged the higher the sec-toral disaggregation the higher the systemrsquos volatility ceteris paribus But

19Similar results are found in Blanchard and Simon (2001)

21

for example in our model reducing the aggregation also requires a changein the adaptation coefficients the more micro we undergo the analysis thehigher is the speed of adjustment of economic objects If for example weconsider sectors we are evaluating the adaptation of an entire set of firmswhich pose questions on how to aggregate their dynamics Things changewhen we consider the response of a single firm

Going back to Horvatsrsquo argument highly disaggregated matrices arelikely to contain many empty cells so that we may forecast different levelsof volatility depending on the aggregation used for the same system whichas mentioned is not caused by sparseness but to a failed adjustment inreaction mechanisms Results from our model20 show that volatility is com-pletely unaffected by input coefficients distribution Given their sectoralsum mdash through rows of the matrix mdash the value of each coefficient is irrele-vant The same occurs in the case of very skewed distributions that presenta high number of empty cells (zero input coefficients)

58 Microndash and Macrondashvolatility

As we have seen above one of the major discussion in the literature concernsthe relations between microndash and macrondashvolatility Put it simply one sideof the literature considers that uncorrelated microndashshocks cancel each otherout reducing their effects at macrondashlevel Conversely other researchersconsider that microndashshocks are multiplied at macrondashlevel generating highermacrondashvolatility than that we can observe at microndashlevel The debate isof obvious importance to determine the sources of business cycles to pre-dict their effects and determine the more effective policies to mitigate theirnegative effects

Our model is only a partial representation of realndashworld economic sys-tem and is also overly simplistic However based on these limitations italso offers very robust results allowing reliable conclusions on the (limited)elements included in the model In particular we consider our model a re-liable representation of the basic structures of the production interactionsamong sectors or firms Thus we can use the model to provide an answerto the debate concerning the contributions to volatility from the productioninteractions only Other contributions (eg from financial markets priceadjustments technological innovation etc) may re-inforce or counter theforces we analyse but they need to be considered as additional elements(and we opine also generally less relevant) not alternative to our analysisWe consider a necessity in order to apply a rigorous methodological ap-proach to be able to identify with certainty the results provided by partialstudies in order to eventually obtain by gradual extensions of the analy-sis more and more detailed representations of realistic economic systems

20Available from the authors

22

The results we obtain in this paper concerning this microndashmacro debate aremeant as a first relevant step to a more extended analysis

0

002

004

006

008

01

012

014

016

018

02

022

05 1 15 2 25 3

Var Demand

Macro (Output) and micro (Demand) volatility

delta Outputdelta Demand

Figure 8 Values of δ (relative deviation) for aggregate demand and ag-gregate output in respect of different levels of variance for sector specificuncorrelated external demand shocks

The flows of shocks affecting the firms of a system can be distinguishedalong two dimensions strength of the shock (eg amplitude of the changesin demand) and frequency of the shock (rare or frequent) Concerning thefirst aspect we initialized as usual a system to equilibrium levels (eg pro-duction quantities equal to external and intermediate demand for all sectorsconstant level of stocks equal to the desired level) We then ran the simu-lation generating the external demand for all sectors as a random variablewith constant (equilibrium) mean level and varying values of variance

Figure 8 shows unequivocally that as far as our model has propertiessimilar to real systems the hypothesis that macrondashfluctuations have largervolatility than their sectoral level shocks is fully confirmed At any levelof variance the aggregate volatility of demand as captured by the relativedeviation index is sensibly lower than the volatility shown by aggregateoutput that is the total production of the system This result is not sur-prising throughout all tests presented in the previous sections every resultsuggested that the system lsquooverndashreactsrsquo to any individual microndashlevel changeeven in the case these changes were meant to contrast disturbances There-fore when we move from considering one single shock to a flow of shockswe are not surprised to obtain that the system shows higher volatility thanthat implied by the microndashlevel disturbances

We now consider the second dimension defining a flow of shocks theirfrequency The question is whether for the same level of demand shocks

23

0 002 004 006 008 01 012 014 016

Difference of relative deviation for Output and Demand

0 5 10 15 20 25 30 35 40 45 50 55

Shock frequency 05

1

15

2

25

3

Var Demand

0 002 004 006 008 01

012 014 016

Figure 9 Differences of values of δ for aggregate output minus those foraggregate demand in respect of different levels of variance of demand and offrequency of the shocks

(ie same variance) more frequent changes trigger a higher or lower systemvolatility than rare disturbances The results shown in figure 9 suggest arather elaborated answer which partly vindicates (though in a particularsense) the arguments in favour of compensation of micro shocks The fig-ure shows the difference between aggregate output volatility and aggregatedemand volatility The independent variables are as before the volatilityof demand at microndashlevel21 and the time intervals in between two shocksAggregate demand volatility does not change for the frequency of shocksat least when computed over a long period But output volatility doesin a nonndashmonotonous way For high frequency of shocks the difference isstill positive and increasing for higher variance indicating that the ldquooverndashreactionrdquo argument is valid even in these cases but it is rather low Whilethe frequency of shocks slows down aggregate volatility of output increasesshowing that time is needed for the microndashshock to unfold their full effectsat the macrondashlevel However after a certain threshold less and less frequentshocks generate a lower volatility of the aggregate level of output

This result is quite sensible though not obvious at first sight and onceagain shows the power of such a simple model to generate and explain com-plex properties For frenetic levels of demand modifications the system hasno time to adjust but part of the job to align actual and desired productionfor constantly changing demand is performed by the varying demand itselfIn other terms the shocks cancel out through time if not through their quan-

21Beware of the scale of variance For reasons of visibility the scale is inverted

24

titative effects on the system While frequency decreases the disturbancesof the shocks not compensated by other (unlikely) shocks have the time tofilter through the production system generating higher and higher volatil-ity In a sense any system has a particular frequency of shocks that make itmaximally ldquoresonatingrdquo with the amplification of microndashshocks When therate of shocks gets even rarer the system can enjoy periods of longer andlonger quietness after each shock is absorbed and before a new one startsagain a cycle of initial local disturbance propagation of the disturbances toother sectors and settlement to the new equilibrium

6 Discussion and extensions

We presented a simple model of production composed by interacting sectorThe main (quit realistic) assumption of the model is that the productionprocesses in any sector of the economy make use of input from any othersector Moreover we assumed that sectors (represented as a single firm)adjust slowly both the desired level of production (smoothed revision of ex-pectation) and the actual level (smoothed modification of plant utilization)The model voluntarily leaves aside a large number of aspects of real-worldeconomic systems (eg financial markets price adjustments growth etc)The reason is that the model can be easily and reliably tested in order toextract its properties The possibility to transfer the (certain) model resultto a real context depend of course on an ample ceteris paribus clause How-ever we are confident that the reported results concerning the productionstructure play a relevant role in real world systems Therefore our resultsand particularly their motivations being accessible because of the relativesimplicity of the model will not be diminished by the integration of ourconsiderations with other aspects of economic systems

Our model is in essence a representation of an input-output table inter-preted in physical terms that is as the amount of inputs required for oneunit of output Besides this we include two straightforward assumptionsFirstly production levels cannot be suddenly changed but require time toscale up or down the utilization rate that is the levels of production Stocksare meant to make up for the difference between sales and production Thesecond assumption concerns the intentional behaviour of producers We im-plement this aspect by assuming that exists a level of stocks desired for eachlevel of demand However a change of demand does not translate suddenlyin a new level of desired stock but again only a long time at a different levelof demand will convince managers to update the desired levels of stocks

We show that the model generate under a wide number of settings oscil-latory patterns converging toward the equilibrium level where each sectorgenerate exactly the quantity demanded by final consumers and by othersectors as input The oscillations are provoked by over- and under-shooting

25

generated by the difficulty of coordination among sectors that by assump-tion ldquocommunicaterdquo only via variation of demand to their direct input sup-pliers We also show that the total production levels generated by sectors(sum of the physical units) produced at equilibrium depends (for constantfinal demand) on the total sum of the technical coefficients per sector irre-spective of their distribution This result is motivated by the impossibilityof input substitution and offers interesting suggestions for the analysis ofthe study of input-output tables of real-world systems

We then consider systems at equilibrium facing a single small change ofthe external demand for sectors These simulations allow to study the waythe system reacts to shocks We see that the sum of the coefficients affectsbesides the levels also the volatility of aggregate output the larger a systemthe stronger is the impact of a given shock Less obviously we find thatfor systems with the same dimension (ie same sums of coefficients) theaggregate volatility increases with both the inertia to change of productionlevel and the speed of adjustment of desired stocks This result is somewhatcounter-intuitive since both the cited variables are associated to a moreconservative behaviour that may be supposed should smooth away spikesin the pattern from an old to a new equilibrium However we show thatthe opposite actually holds given the complex interactions among sectorindividual-sector attitude intended to smooth shocks actually increase theiraggregate impact

In the same vein we show that systems maintaining high levels of stocksare more volatile that systems with reduced amounts of stocks Again thisis a counter-intuitive result reverting the goal of individual producers andthe aggregate result Interestingly this result finds strong support from theevidence of the recent changes in production methods (ie low or no stocks)and diminished cycles

In relation to specific issues discussed in the literature we find no supportto the suggestion that particular distributions of input-output coefficientsmodify the volatility of a system As said this depends in case on thesums of these coefficients but not on their distribution Further we canprovide an answer to a long and hotly debated questions whether microndashlevels volatility generates stronger or lower aggregate volatility Our modelshows unequivocally that microndashvolatility (eg variance of final demand)persists and generates stronger volatility at aggregate level Again we canidentify the reason on the complex production structure that exalts micro-level shocks by generating lsquowrongrsquo messages among producers Moreoverwe can also point to a usually neglected aspect of volatility we show thatthe frequency of shocks is a very relevant aspect besides their dimensionin determining the volatility of aggregate output

Our work can be developed along two complementary directions Firstlywe can continue to study the properties of the model by extending the ele-ments considered (eg pricing) analysing the effects on the present results

26

and generating more analytical evidence Of particular relevance is in ouropinion the problem of aggregation Representations of the same systemswith different codification systems (eg different number of SIC digits)must not modify overall system properties for example the volatility ob-served Given that our result show that the sums of coefficients matter forthe levels and volatility of the system and that these sum do depend on theaggregation chosen it should be possible to induce the reaction coefficientsthat we know affect the volatility only

Secondly we can find an application of our results to real world evidenceIn fact there are many data sets available for the data we deal with basicallyinput-output tables Our model is readily adapt to be parametrized alongany number of sector and coefficient necessary so that we may replicate pastseries in order to adapt the unobservable parameters and use the resultingcomplete model to explain business fluctuations and provide intuitions onfuture patterns

27

References

Acemoglu D and A Scott (1997) ldquoAsymmetric Business CyclesTheory and TimendashSeries Evidencerdquo Journal of Monetary Economics 40501ndash533

Backus D P Kehoe and F Kydland (1995) ldquoInternational BusinessCycles Theory and Evidencerdquo in Frontiers of Business Cycle Researched by T Cooley Princeton University Press Princeton

Bak P K Chen J Scheinkman and M Woodford (1993) ldquoAg-gregate Fluctuations from Independent Sectoral Shocks SelfndashOrganizedCritically in a Model of Production and Inventory Dynamicsrdquo RicercheEconomiche 47(1) 3ndash30

Battiston S D Delli Gatti M Gallegati B Greenwald and

J Stiglitz (2005) ldquoCredit Chains and Bankruptcies Avalanches in Sup-ply Networksrdquo in Annual Workshop on Economic Heterogeneous Inter-acting Agents (WEHIA 2005) University of Essex UK

Blanchard O and J Simon (2001) ldquoThe Long and Large Declinein US Output Volatilityrdquo Brookings Papers on Economic Activity 1135ndash174

Bresnahan T F and V A Ramey (1994) ldquoOutput Fluctuations atthe Plant Levelrdquo The Quarterly Journal of Economics 109(3) 593ndash924

David P A (1975) Technical choice innovation and economic growth essays on American and British experience in the nineteenth centuryCambridge University Press London

Delli Gatti D C Di Guilmi E Gaffeo G Giulioni M Galle-

gati and A Palestrini (2004) ldquoA New Approach to Business Fluc-tuations Heterogeneous Interacting Agents Scaling Laws and FinancialFragilityrdquo Journal of Economic Behavior amp Organization 56 489ndash512

Dosi G G Fagiolo and A Roventini (2006) ldquoAn EvolutionaryModel of Endogenous Business Cyclesrdquo Computational Economics 273ndash34

Dupor B (1999) ldquoAggregation and the Irrelevance in MultindashSector Mod-elsrdquo Journal of Monetary Economics 43 391ndash409

Head A C (1995) ldquoCountry Size Aggregate Fluctuations and Interna-tional Risk Sharingrdquo The Canadian Journal of Economics 28(4b) 1096ndash1119

28

Helbing D and S Lammer (2005) ldquoSupply and Production NetworksFrom the Bullwhip Effect to Business Cyclesrdquo in Networks of Interactingmachines Production Organization in Complex Industrial Systems andBiological Cells ed by D Armbruster K Kaneko and A S Mikhailovvol 3 of World Scientific Lecture Notes in Complex Systems World Sci-entific

Helbing D S Lammer U Witt and T Brenner (2004) ldquoNetworkndashInduced Oscillatory Behavior in Material Flow Networks and IrregularBusiness Cyclesrdquo Phisical Review E 70(056118) 1ndash6

Helbing D U Witt S Lammer and T Brenner (2004) ldquoNetwork-Induced Oscillatory Behaviour of Macroeconomic Output Flowsrdquo in In-dustry and Labour Dynamics II The Agent-based Computational Eco-nomics Approach Proceedings of the wildace 2004 conference ed byR Leombruni and M Richiardi no 5 LABORatorio R Revelli

Horvath M (1998) ldquoCyclically and Sectoral Linkages Aggregate Fluctu-ations from Independent Sectoral Shocksrdquo Review of Economic Dynam-ics 1 781ndash808

(2000) ldquoSectoral Shocks and Aggregate Fluctuationsrdquo Journal ofMonetary Economics 45(1) 69ndash106

Jovanovic B (1987) ldquoMicro Shocks and Aggregate Riskrdquo The QuarterlyJournal of Economics 102(2) 395ndash410

Kahn J A M M McConnell and G Perez-Quiros (2002) ldquoOnthe Causes of the Increased Stability of the US Economyrdquo FRBNY Eco-nomic Policy Review 8(1) 183ndash202

Kraay A and J Ventura (1998) ldquoComparative Advantage and theCrossndashSection of Business Cyclesrdquo World Bank Woorking paper Series1948 World Bank

Long J B and C I Plosser (1983) ldquoReal Business Cyclesrdquo Journalof Political Economy 31(1) 39ndash69

Lucas R E (1977) ldquoUnderstanding Business Cyclesrdquo CarnegiendashRochester Conference Series on Public Policy (special issue of Journalof Monetary Economics) 5 7ndash29

Nelson R R and S G Winter (1982) An Evolutionary Theory ofEconomic Change Harvard University Press Cambridge MA

Nirei M (2005) ldquoThreshold Behavior and Aggregate Fluctuationsrdquo Jour-nal of Economic Theory in press

29

Scheinkman J and M Woodford (1994) ldquoSelfndashOrganized Criticallyand Economic Fluctuationsrdquo American Economic Review 84(2 Papersand Proceedings of teh AEA) 417ndash421

Schuh S (1996) ldquoEvidence of the Link Between FirmndashLevel and Ag-gregate Inventory Behaviorrdquo Finance and Economics Discussion Series1996-46 Federal Reserve Board

Stiglitz J and B Greenwald (2003) Towards a New Paradigm inMonetary Economics Raffaele Mattioli Lectures Cambridge UniversityPress Cambridge

Weisbuch G and S Battiston (2005) ldquoProduction Networks andFailure Avalanchesrdquo Working Paper mimeo Ecole Normale SupErieure

Zarnovitz V (1977) ldquoBusiness Cycles Observed and Assessed Why andHow they Matterrdquo NBER Working Paper Series 6230 National Bureauof Economic Research

30

A Tables

Table 1 Parameters valuesParametera Description Value

N Number of sectorsunits i of production 10Ci Number of j buyer sectors for sector i (IO matrix sparseness) N minus 1Di (E) Value of the external demand 10mij Physical input coefficient of input j for sector i sim U(0 1)Mi Sum of the mij input coefficients j for the production of i 05αi friction to changes of production levels Degree of lockndashin on

production technology085

αsi Speed of adaptation to demand changes 004

σ Desired multiple of production to stock 10

aInto parenthesis the number of lags of the initial value for lagged variables

31

separated Their contribution must be studied in order to produce a robusttheoretical understanding of the business cycle phenomenon Indeed thephysical structure through which any micro change (or reaction to a change)is transmitted in the system is the production structure And the way inwhich shocks propagate is at least as relevant as the sources of propagationIn the words of Zarnovitz (1977) we investigate a lsquotheoretical possibilityrsquo ofbusiness fluctuations that for the very nature of production systems is alsoan lsquoexplanationrsquo (while not an assessment) The aim of this paper is thento analyse transmission properties of multisectoral production systems

The structure of the paper is the following Next section briefly reviewsthe major contributions of the literature on the subject highlighting theworks more closely related to our approach In the third section we high-light the main elements that characterise the transmission mechanisms ofa production structure The fourth section describes a very simple modelrepresenting a dynamical production system comprised of an input-outputmatrix and a few simple behavioural rules governing the actions of economicsectors Section five then discusses the major results of the model under afew parametrisations Finally the last section will draw the conclusions andsuggest directions for further research

2 Economic fluctuations

A wide number of reasons may explain why economic aggregates fluctuateand a wide number of theories attempt to explain short run and long runwaves in economic growth For example large shocks that affect an en-tire economic system are likely to fully displace it changing its long termpattern Though while such shocks may be plausible for long run wavesthey are less likely to be the cause of short run cycles given that they donot occur with the same frequency Moreover aggregate shocks may affectthe various economic entities differently making it quite difficult to statethe final result of a complex combination of reactions Conversely shocksat more disaggregate levels of the economy are undoubtedly more frequentThis itself renders the study of the influence of micro shocks on aggregatefluctuations an important piece of analysis and eventual understanding ofbusiness cycles

It is selfndashevident that being economic entities interconnected to eachother they may absorb linearly transmit or reinforce the quantitativechanges that hit on them mdash induced by neighbouring entities mdash to differentextents This can be observed at different levels of aggregation An eco-nomic crisis in one country causes shocks in related countries (eg BackusKehoe and Kydland 1995 Head 1995 Kraay and Ventura 1998) a crisisin the financial system causes readjustment in the proximate systems inboth supply and demand (eg Stiglitz and Greenwald 2003 Delli Gatti

3

Di Guilmi Gaffeo Giulioni Gallegati and Palestrini 2004)2 the failure ofa large company induces readjustments in the same and related sectors andson on

Besides we expect that the more we disaggregate the economic units ofanalysis the more shocksrsquo intensity is likely to reduce and symmetric shockstend to cancel out The application of the law of large numbers would sug-gest that an economy with normally distributed entities (at the same level ofaggregation) mdash or even better with representative entities would not showaggregate fluctuations in the presence of uncorrelated disaggregated shocks(Lucas 1977) This has led to focus most theoretical explanations (or rep-resentations) of short business cycles on aggregate shocks (Horvath 2000)Although they are undoubtedly relevant and have an impact also on mi-cro changes aggregate interpretations are quite limited in their explanatorypower While having to assume exogenous origin of shocks they do notprovide an understanding of how macro shocks rebound on the economicentities (consumption change production shift productivity shifts invest-ments etc)

Indeed in order for micro shocks to generate aggregate fluctuations twoconditions are necessary a propagating structure mdash a connection mdash be-tween micro economic entities and a sluggish absorption of changes Inthis respect production structures made up of sectors connected by inputndashoutput relations represent a good candidate a part from being the knownstructure of a basic industrial economy Two main class of models havebeen developed to analyse the effect of sectoral shocks on aggregate fluc-tuations Real Business Cycles (RBC) and Avalanche (Av) models Bothtype of models make attempt to introduce mechanisms that allow for thepersistence of micro shocks In this respect both classes are based either onquite unrealistic behavioural assumption which determine the conditions ofthe system or threshold behaviors which impose the conditions for shockspersistence We briefly review them below

21 Mechanisms of shocks propagation in IndashO structures

A number of contributions study the phenomenon of generation and prop-agation of cycles independently from the sectoral structure of the economyHowever the most recent contributions point in the direction of giving thisaspect of economic system a strong relevance Moreover the two approachesare not incompatible so that we focus on the literature that explicitly makesuse of the IndashO structure In this section we briefly review three strands ofthis literature highlighting the elements that inspire our work

2Both cases are easily shown with evidence from the Latin American crisis during theeighties rooted in the crisis of the Mexican financial system the financial crisis in Asiaduring mid nineties the recent Argentinian financial crisis etc

4

Plain IndashO structure

The pioneering RBC model that analyses the diffusion of sectoral shocks isdue to Long and Plosser (1983) They use a general equilibrium multi sectormodel with fully rational infinitely lived perfectly informed homogeneousetc individuals In such a system by assuming a perfectly maximising be-haviour (via allocation of production and consumption of resources) andthat individuals prefer to smooth savings across time and goods it is pos-sible to generate persistence of continuous albeit uncorrelated shocks inproduction capacity across sectors In the words of Long and Plosser [p67] ldquoAt constant relative prices this [the assumed maximising behavior]suggests that business-cycle features like persistence and comovement arecharacteristics of desired consumption plansrdquo

The necessity to maintain the assumption of perfectly rational agentsforces these strand of models to attribute the origin of cycles to externalfactors only like exogenous shocks on technology As such they seem un-likely to provide a consistent explanation (or representation) of aggregatefluctuations Dupor (1999) shows that under quite general conditions theavailable multindashsector RBC models produce the same results in terms ofvariance convergence as their respective single sector models This is shownalso under particular conditions of input matrix or high coefficient of spe-cific inputs Dupor concludes that [p 405] ldquoresearchers who wish to useindependent sector shocks as a source of aggregate fluctuations must appealto sector shocks that are large relative to their single sector counterparts orelse consider a different set of models than those discussed in this paperrdquoIn doing so the author though suggests that another mechanism must beintroduced somehow putting the desired results ahead of the economic issueanalysed

The criticism to this kind of literature suggests that in order to allowthe IndashO structure to play a role in the explanation of cycles we need agentswith less then perfect foresight and capacity of adaptation to unndashexpectedconditions As we will see below agents that need time to both realizethe economic changes of their time and to introduce the necessary mod-ifications are sufficient conditions to observe aggregate fluctuations froma single change in one part of the economy Though being ldquorationalrdquo inthe sense that they make the right decisions they are limited by lags inthe information flow and physical constraints to the changes that they canapply

Input matrix incompleteness

Using a similar approach to RBC models Horvath (1998) and Horvath(2000) focus the attention on the properties stemming from peculiar produc-tion structures as represented by characteristics of the input-output matrix

5

The author shows that sectoral shocks will tend to cancel out in case theeconomic system is sufficiently distributed that is when the IndashO matrixcontains evenly distributed values over all the cells Conversely sectoralshocks are not absorbed and can be reinforced in case many cells of thematrix are empty This case indicate that there will be few sectors affect-ing many other sectors preventing the possibility of compensating certainshocks

These conclusions are potentially interesting because point directly toan easy to observe aspect of economic system for example providing thepossibility to test the prediction of the analysis It is worth to note that thisapproach draws conclusions that are not neutral to the aggregation levelused In fact a highly disaggregated matrix is more likely to contain manyempty cells than the matrix for the same system obtained aggregatingsectors

Dupor (1999) also considers the case of inputndashoutput matrices with dif-ferent input coefficients finding them irrelevant for the results But he doesnot include matrices with empty cells

The cited contributions and those inspired to the same approach sug-gest that the type of interaction as expressed by input-output coefficientscan be a relevant factor in determining the existence and dimension of fluc-tuations However this literature is not able yet to draw conclusive results

Limited interactions and production constraints

A different kind of mechanisms are assumed and shown to play a role in theAv models they add an analysis on firms production (technological) possi-bilities to the production structure They are inspired by and similar to Jo-vanovic (1987) model in which the limitation in the number of interactionsupon which each playerrsquos decision depends generates shock cascades andaggregate persistence Bak Chen Scheinkman and Woodford (1993) andScheinkman and Woodford (1994) show that when the interaction structurebetween sectors is constrained to a completely rigid system coupled withnon convexities in the production function independent shocks to differentsectors do not cancel in the aggregate In other words the authors place aset of restrictions to the set of possible actions of sectors First structuringthe production networks with a fixed invariable lattice produces a rigiditythat do not allow producers to change (or add) inputs (or suppliers) orincrease the produced quantity This is combined with the structure of theavalanche that propagates through the unchanged network Second a pro-duction function that is maximised only when a discrete number of unitsof a good is produced coupled with a fixed (discrete) maximum amount ofinventories that can be stored by the representative firm Such a structuredetermines production oscillation and shocks avalanches due to the factthat firms are constrained in their production opportunities

6

Using a very similar production structure with limited interactionsWeisbuch and Battiston (2005) play on the bankruptcies generated on down-stream firms by production failures occurring in upstream firms (reducingthe flow of inputs) Stochastically generated failures to produce in a firmcauses production constraints in its downstream clients hence lower invest-ment and eventually bankruptcies3 A lag between the period in whicha firm goes bankrupt and a new firm appears generates a lack of supplywhich propagates on the downstream sectors

Due to the original literature on avalanches these contributions sharetwo common features that are ill-adapted to economic systems First theyassume very peculiar interaction structures of one-directional productionrelations between sectors which amount to assume one half of the input-output matrix empty and are not applicable to the general case with cyclicalproduction structure or full IndashO matrices Second the strongly rely on quiterigid thresholds determining a mechanicist behaviour by agents Basicallyagents have only two options available (ie produce or not produce) andthe choice is made on the basis of crude decisional mechanism4

Notwithstanding the limitations this approach provides a quite sensiblemechanism of the origin and transmission of shocks through a system Ourproposal can be interpreted as an extension of this approach by relaxing thestrong assumptions on the interaction structure and allowing agents for amore fluid decisional process as we will see in the next section

3 The basic elements of a production structure

Our overall hypothesis is that the very production structure of an economymay generate persistent fluctuations even in the absence of continuous un-related shocks and rigid thresholds that determine non linearities The aimis to provide a generalised interpretative framework to explain the occur-rence of short term fluctuations in the economy simply as an outcome ofthe division of labour in a number of sectors related through trade In itsextreme interpretation any production system which is not fully integratedis bound to produce business cycles As we will show a number of factorsinfluence their extent

Quantitative adjustments in production in any one sector require ad-justments in related sectors When the structure of the economy is trulyInputndashOutput (not a directed linear supply chain as in the Av models) in-put flows are cyclical and adjustment to long run production equilibrium islikely to take a long mdash when not infinite mdash time Our work departs from

3A similar model by Battiston Delli Gatti Gallegati Greenwald and Stiglitz (2005)relates bankruptcies to price changes buyers payment timing and the credit market (acentral bank)

4Nirei (2005) is an attempt to generalise such models though maintaining the assump-tion of threshold behavior

7

both RBC and Av models with respect to structure modelling criteriaanalysis and methodological approach Indeed our model may be easilyreconduced (or restricted) to both We are much in line with a recent se-ries of papers by Helbing Witt Lammer and Brenner (2004) that addressthe same question although with a number of differences that are discussedlater on5

We aim to analyse to what extent the structure of inputndashoutput relationscauses aggregate fluctuations by itself The main idea is to show that ag-gregate fluctuations may derive from simple micro imbalances without theneed for continuous exogenous shocks not even at the firm or sectoral levelIn other words normal production activities including lags technologicaladjustments information asymmetries circularity of the inputndashoutput sys-tem and so on may be enough to explain part of the macro volatility in aclosed system6

Our assumptions draw on the existing literature but try to avoid themajor difficulties highlighted above Firstly we avoid to impose either mar-ket clearance assumptions or crude threshold behaviours Rather we adoptan assumption close to a Simonian approach agents in an economy do theirbest within the informational and physical constraint they are subject toWe represent decision makers as responding to a single piece of informationthey obtain the quantities demanded by their clients Crucially their ac-tions are inspired to a conservative approach an increase in the demandgenerates a smaller (desired see below) sudden increment of productionThis assumption stems from the fact that decision makers are aware of theinstability of their environment and therefore want to avoid getting perma-nently caught with over production following a temporary spike of demandIn other words firms attempt to smooth business cycle generating a morestable (or less uncertain) environment as the economic literature would sug-gest Notice that this representation in line with a routinized representationof organizations (Nelson and Winter 1982) does not contradict market equi-librium Rather in a stable environment such decisional procedure generatesan asymptotic pattern to the equilibrium level in the absence of feedndashbacks

A second assumption concerns the technological possibilities available tothe firms When firms realise the need for a change in production levels saya desired increase of production they face a complex and costly organisa-tional transformation of the production capacity risky investments an soon As Bresnahan and Ramey (1994 p 622) suggest inducing from a de-tailed analysis of the automobile industry in the US ldquoadjusting production

5We thank Matteo Richiardi for pointing out these similar attempts See also Helbingand Lammer (2005) and Helbing Lammer Witt and Brenner (2004)

6An open system entails many other factors including price factors of both inputsand outputs mdash terms of trade labour cost external investment etc As with exogenousshocks our analysis does not deny the importance of these factors but simply separatethem from the structural explanation of cycles

8

is a more complicated process than simply ldquochanging Qrdquo or choosing themix of capital and laborrdquo And this is not only a matter of non convexitiesin production technologies The inconsistency of time reversibility in thetechnological choice is accompanied by the need to operate on the lsquoavail-able technological frontierrsquo (David 1975) If a firm plans a given productionactivity for a short period of one week even though the next day there is anunexpected fall in the demand it is unlikely that it is followed by a similardownturn in production The case is evident when we think at productionplanning that involve hiring of workers an capital investments (both in-volving sunk costs)7 Therefore any desired change generates a pattern ofsmall modifications approaching slowly the target level Moreover thesechanges are irreversible in the short term since only a prolonged change ofthe environment triggers the modification of previous desired levels

Resting on those simple conjectures we thus analyse a model in whichwe only allow for sluggish adaptation of production due to conservative(adaptive) behaviour of firms and the physical constraints faced when vary-ing the production levels Notice that these assumptions are contrary tothe generation of instability of a system In fact a firm behaving as de-scribed above reduces if not eliminates any disturbance affecting its ownstate if acting in isolation Other factors like new technologies extremelylong term perspective price changes etc are likely to be sources of volatil-ity therefore contributing to generate aggregate system fluctuations (as forexample in Acemoglu and Scott 1997) Our neglect of these and other fac-tors however unrealistic serves the purpose of testing whether a volatilityreducing microndashbehaviour turns into volatility generating aggregate resulttwo widely observed dynamics Quite interestingly (and encouraging) DosiFagiolo and Roventini (2006) have concomitantly proposed a model whichincorporates detailed representation of both sluggishness robustly based onempirical evidence Heterogeneous firms which undergo lumpy investmentand slowly adapt their expectation on changes of the future demand inducecomovement of macroeconomic variables qualitatively similar to the onesobserved in the real world To pursue our line of research we first needto concentrate on how sectorsfirms interact in the system in the effort tocomplement those promising results

Therefore the second element we consider is meant to represent a gen-eralized economic system We allow for the circularity of inputndashoutput sys-tems a crucial feature in determining feedback mechanisms and a principal

7See also the discussion in Acemoglu and Scott (1997) where the authors argue thatldquo[w]hile the presence of fixed costs can account for the discreteness of economic turningpoints it does not naturally lead to persistence because once an individual undertakes anaction they are less likely to do so in the near future [] [A]lthough the presence of fixedcosts leads to increasing returns these are intratemporal the full extent of economiesof scale arising from fixed costs can be exploited within a periodrdquo [p 502 authorsrsquoemphasis]

9

component of business cycle It may be true that most backward linkagesconcern capital goods (Weisbuch and Battiston 2005) but a rapid glanceto an inputndashoutput matrix would show that technical coefficients are nonnull in both directions and ldquomost commodities are inputs to the produc-tion process of other commoditiesrdquo (Horvath 2000 p 70) The higher thesectoral aggregation mdash observed or assumed mdash the more the relevance ofintermediate supply holds true (as mentioned above with respect to ma-trix incompleteness) We claim that the existence of backward and forwardlinkages are a crucial element to determine the persistence of shocks in theeconomy A perfectly linear (one directional) system needs much less adap-tation of firms and sectors in order to stabilise the economy Nonethelesscircularity does not imply that inputs are perfectly substitutable amongthem in order to determine the maximising allocation even at high level ofdisaggregation

By using a fixed coefficients production function we both maintain themid term rigidity of production processes and we do not impose limitedinteractions We claim that the circular property of the production structureis by itself sufficient to produce shock persistence even when the inputmatrix is full (although with heterogeneous input coefficients) mdash all sectorsshop in the remaining n minus 1 sectors

In conclusion we represent a model for an economic system made onlyof i) production level decisions and ii) input-output relations Clearly sucha model lacks many features present in real economic systems likely we areconvinced to affect the propensity to generate cycles As such we cannotaim at reproducing realistic data so our results cannot be tested comparingtime series generated by the model with real ones Instead given the paucityof the elements comprising the model we can investigate in detail to whichextent the (few) features present in our model affect the cyclical behaviour ofthe system As we will see we will be able to show quite a number of resultin which aggregate fluctuations are generated whereas the individual microstructure of the model aims at monotonous adaptation Before discussingthe results next section describes in detail the implementation of the simplemodel

4 The model

We model an economic system in which a number of sectors i = (1 N)are potentially connected via (intermediate) market relations A N times NInputndashOutput matrix determines the mij units of each input j = (1 N)mdash bought from the N minus 1 residual sectors mdash necessary to produce oneunit of output i8 For simplicity given the focus of this paper on the IndashOstructure as a generator of aggregate fluctuation we abstract from market

8IndashO coefficients are therefore physical coefficients

10

interactions within sectors each sector is a production unit In fact forour purposes it is not relevant which firm produces the sectoral outcomebut only the total amount produced by each sector9 Moreover besidesmaking the analysis more neat this assumption allows also to enjoy a largefreedom in determining the aggregation level since we donrsquot need to assessthe effects of between sectors competitiveness which for example is likelyto increase for increasing levels of disaggregation Our economic systemis therefore composed of N production units each of which may buy andprovide inputs from and to any other sector depending on the IndashO structureof the economy

The demand for each sector i is determined by a constant consumersdemand Di (E) and the input needs of other sectors

Di = Di (E) +sum

j 6=iisinCi

mjiQj (1)

where Ci is the set of j intermediate clients of i (sum

j isin C lt N) mji is thetechnical coefficient of firm j for input i ie the amount of i required toproduce one unit of j Qj is the actual output produced by sector j

Concerning the simultaneous production level of each sector in time twe make two lsquorealisticrsquo assumptions First from the technical viewpoint afirm like a tanker cannot abruptly change its course Therefore changesto the production level can be introduced only gradually through capital(dis)investment and job h(f)iring (see discussion in Section 3) Second weassume that firms do not decide the production level directly on the basisof the observed demand level but conscious of the uncertainty of demandplan their production in order to maintain invariant a desired level of stocksand avoid stockout10

Quantity produced by sector i in period t is computed as follows

Qit = αiQtminus1 + (1 minus αi)(

Slowastitminus1 minus Sitminus1

)

(2)

where αi measures the physical constraint in adapting the production capac-ity to changes in the required output Slowast

itminus1are firmrsquos i desired stocks and

Sitminus1 its actual stocks Notice that both variables related to stocks appearwith a lag This is due to two reasons First it is a modelling necessity inorder to allow the simultaneous determination of production levels for allsectors Secondly as frequently happens modelling necessities reveal sen-sible if not generally considered aspects of the modelled system In factproduction decisions concern management in the production plant while

9Adding heterogeneous firms within sectors would definitely enrich the causal expla-nation put forward and parametrised in this paper (see below) For example to analysethe effect of firms constraints It is left over for future research as in the present work weopt for the analysis of the main structural parameters that affect aggregate fluctuations

10See also discussion in Schuh (1996)

11

demand through sales is observed by the commercial staff It is thereforelogical that the information from demand reaches plant managers with somelag

The actual stock level is trivially computed subtracting from the previousheap of stock the amount sold and adding the new production

Sit = Sitminus1 + Qit minus Dit (3)

Concerning the desired level of stocks instead we introduce the conser-vative behaviour mentioned above Firms ideally would keep an amountof stocks proportional to the demand they receive say a multiple σ ofthat amount However when demand varies they adjust the desired levelsmoothly preventing sudden changes to the variable representing the goalof the firm

Slowastit = Slowast

itminus1 + αsi

(

σDit minus Slowastitminus1

)

(4)

where αsi measures the adaptation to demand changes

In summary the dynamics of the model consists of only the equationsabove Firstly at the beginning of a time step firms determine the quantityto produce as a function of past imbalances in stock levels and the pastproduction level These levels define the demand for all sectors determinedby the technical coefficients applied to all produced quantities Finallyactual and desired stocks can be updated Figure 1 represents visually thisdynamics highlighting the decisional components of the model in respect ofthe ldquomechanicsrdquo of trade

5 Results production structure and economic fluc-

tuations

In this section we describe some of the most relevant results concerningthe effects of the production structure to determine aggregate fluctuationsGiven the unusual analytical instrument chosen (at least concerning thistopic) we start with a brief methodological introduction meant to clarifythe nature of the results we claim to obtain Next we describe a generalsetting for the model which is the (extremely) limited area of the potentialparametersrsquo space of the model that we will explore In the rest of thesection we discuss the results from the model

51 Methodological considerations

Real economic systems are constantly subject to a continuous flow of shocksexternal to the production system besides changes within the system itselfIn order to appreciate the systemic properties we need to abstract from all

12

Compute the quantityto produce

F (Q(t-1) S(t-1) s(t-1) alpha)

Communicate thequantity of inputsrequired to each

supplier

Determine the overalldemand

F(D(E) D(I))

Revise the past decisionon stock plans

(desired stocks)F(D S(t-1) sigma alpha(s))

Compute the final levelof stocks for next period

F(s(t-1) Q D)

KEY

ModelDirection

Variablecomputed

at t-1(t-1)

Step includingfirm behavior

Figure 1 One period dynamics

the nonndashrelevant disturbances and concentrate on how the system endoge-nously contributes to the observed dynamics Therefore we will generatehighly abstract patterns whose analysis will provide insights on the systemproperties that would be otherwise lost in the noise generated by all theelements of a real system

We are not trying to fit the model to a specific empirical data set nor weare interested in studying the complete behaviour of the model for areas ofthe parameterrsquos space that make no economic sense (eg negative or infiniteproduction levels) Therefore the values of initial settings and results matteronly in relative and not in absolute terms

We are going to describe a simulation model with which we will produceseveral simulation runs and we claim obtain results relevant for the debatediscussed above Rigorously speaking we may claim only certainty for thevalidation of the results that is that the model does actually generate theresults presented and for the reasons we explain We limit this part to thepresentation of graphs and verbal explanations that we believe are ratheruncontroversial11 Concerning the relevance of our studies for real systems

11Interested readers can request the model code and and simulation data for replicationand extensions of the results We can guarantee that such possibility does not require

13

(verification) we simply avoid to pretend that our model is a quantitativeapproximation of real economies (which in which period) Our opinionis that the study of the model allows us to make general considerationsnon-obvious and relevant for applied debates supported by strong logicalarguments which can be integrated but not reverted by arguments con-cerning more and more elements of the economic world

52 Model setup

We go through the model properties analysing an economic system madeup of ten sectors (N = 10)12 Each sector shops inputs from the remain-ing N minus 1 and sells to them part of its own production as intermediategoods This means that the input matrix is complete except for the diag-onal (mij gt 0 forallj 6= i mii = 0) The sum of all N minus 1 coefficients for asingle sector is given by Mi and its benchmark value is kept at a plausiblemedium level (unless differently specified) to avoid distortions from highlydemanding sectors Each input coefficient is randomly drawn from a uniformdistribution

Sectors start at an equilibrium level their supply matches intermediateand final demand and in each period the same quantity of output mustbe produced to compensate for the used stocks and keep the desired stocksunchanged In thier turn stocks must be ten times the value of total sectoraldemand (σ) Eventually technological adjustment of production is ratherslow and producers are quiet refractory to follow sudden changes in thedemand in general a smoothing behaviour strongly prevails (α and αs)

Under those general preconditions in the following sections we anal-yse the dynamics of the economic system in relation to model parametersWe mainly focus on the cyclical behaviour of systems and their speed ofconvergence to the asymptotic equilibrium13

The very abstract assumption and the homogeneous initialisation donot allow for the multiple equilibria that are likely to rise in complex systemswith feedbacks Once more such simplification allows to have a betterunderstanding of the crucial impacts of the production structure on businessfluctuations Departing from the related literature discussed above14 weconsider the effects of a single shock rather then a continuous flow whichin our model is superfluous in order to obtain fluctuation persistence15 To

extensive programming or statistical skills besides the usual education of economists12You can think of a high level of aggregation in the statistical observation The com-

plete initialisation and parameters values are available in table 1 in the Appendix13Notice that sectors only approach a final equilibrium level of production which can

be prendashdetermined under static conditions14Included the more proximate work of Helbing Witt Lammer and Brenner (2004)15An exception are the last results presented in Section 58 which we see more as an

opportunity for further research than a closing of the paper

14

start with next section (53) shows the aggregate cyclical behaviour of astandard setting

53 Single shock analysis

We generate a simulated history following the dynamics of the modelrsquos vari-ables when the system is set to an equilibrium level and a single shockmodifies one of the exogenous elements namely the external demand forsectors Modifications across sectors are random and not correlated andmay increase or reduce external demand by at most 10

100 225 350 475 600

193894

195861

197828

199795

201762 Industrial Production

Time

Figure 2 Total industrial production fluctuations following a shock fromexternal demand

Figure 2 shows the dynamics generated by the total industrial productionof the system mdash that is also a proxy for the systemrsquos GDP fluctuations Thenew external demand changes the equilibrium level but the system cannotreach the new equilibrium with a monotonous pattern and converges to thenew level with a fluctuating pattern Each change in the external demandinduces also modifications of the intermediate demand and sectoral adjust-ments in the quantity produced typically overshoot the aggregate patternAs it can be appreciated from the figure it takes time before the system isable to converge to the new static equilibrium conditions

We study two features of this pattern absolute level of the new equilib-rium and volatility induced by the shock

54 Technical coefficients and industrial output

Production level

The equilibrium levels of industrial output depend on two factors only Ob-viously the levels of external demand determine how much of the total pro-

15

duction needs to be supplied to consumers exogenous from the productionsystem Second the technical coefficients also influence strongly the level ofactivity of the system defining the intermediate demand to each sector

180

200

220

240

260

280

300

01 02 03 04 05 06 07 08 09

Out

put l

evel

(In

dust

rial p

rodu

ctio

n)

Sum of total input coefficients (M)

Total industrial production for different input coefficients levels

Figure 3 The relevance of input coefficients on output level

Figure 3 shows the equilibrium level of industrial output reached bysystems with identical external demand where we fixed the sum of inputcoefficients for each sector of the economy (Mi) to different values16 Thehigher the sum the higher is the total production sustained by the system

This result depends on the very production structure represented by theinputndashoutput matrix In fact the higher is the level of production requiredby intermediate sectors the higher is the total production given a constantlevel of external demand

Notice that the level of production does not depend on the distribution ofthe technical coefficients As long as the sum of the coefficients is identicalthe same production level is obtained irrespective of their distribution (otherthings being equal)

Output shock absorption

Input coefficients have a relevant role also on the output stability of thesystem Starting from an output equilibrium level we hit the system witha single shock on the external demand for all sectors uncorrelated amongsectors17 modifing (positively or negatively) the demand for all sectors by

16Within this limit the actual value of the coefficients are randomly drawn17In each sector the external demand changes by plusmnX where X is a uniformly dis-

tributed percentage between 1 and 10

16

at most10 As show in Figure 2 the economic systems produces a lenghtlysmoothing oscillatory behaviour We compute the relative deviation of theaggregate production (industrial output) as the ratio between its standarddeviation and its mean δ = σY microY where Y =

sum

Qi Figure 4 shows thebehaviour of δ as a function of the sum of input coefficients for each sector80 periods after the shock As time goes by the level of δ reduces for allMi but the relation with the sum of input coefficient is unchanged

00055

0006

00065

0007

00075

0008

00085

0009

01 02 03 04 05 06 07 08 09

Rel

ativ

e va

rianc

e of

tota

l out

put (

delta

)

Sum of total input coefficients

Aggregate output deviations for different technical coefficients

Figure 4 The relevance of input coefficients on output stability

In sum input coefficients determine the extent of overshooting behaviourThe higher the coefficients the higher is the backlash even when shockscould cancel out through sectors

55 Shock persistence and adjustment coefficients

We now consider the effect of micro smoothing behaviours on the volatilityof the production system In particular we are interested in testing theeffects of two parameters of the model on the volatility of the productionsystem First we consider the effects of the αrsquos representing the ldquostickinessrdquoof the production technology to desired changes of Q aimed at filling thegap between actual and desired stocks The higher is α the slower is thereaction to stock unbalances Concerning its effect referring to the technicalpossibilities available ie production capital and organization we may putforward two opposite hypothesis relevant to systemrsquos volatility

Hypothesis 1 Slower production adjustment (high α) reduces volatilitybecause unbalances in demand of one sector being compensated slowly slow

17

down the diffusion of shocks to other sectors

Hypothesis 2 Slower production adjustment (high α) increase volatil-ity because the longer lasts the mismatch between actual and equilibriumproduction level the deeper will become the mismatch between actual anddesired stocks

20 25 30 35 40 45 50 55 60 65

Cyclesrsquo width

07 075

08 085

09alpha 0005

001 0015

002 0025

003 0035

004 0045

005

alphaS

20 25 30 35 40 45 50 55 60 65

Figure 5 Maximum width recorded between highest and lowest peaks fortotal production for different values of α and αS

Secondly we consider the parameters αS representing the speed of ad-justment of desired stock to observed demand The higher this parame-ter the faster sectors revise the desired levels of stocks to current levelsof observed demand This parameter concerns sectorsrsquo behavioral decisionprocess Higher αS represent a stronger belief that the most recently ob-served demand level is a reliable indicator of future demand levels whilelower levels of the parameter represent a more conservative approach wheremanagers require a long observation of demand levels different from the pastones before adjusting their expectations Also concerning the effect of thisparameter on the volatility induced by an exogenous shock we may maketwo hypothesis

Hypothesis 1 Faster adaptation of desired stocks to new levels of de-mand (higher αS) reduce volatility because changes in demand are rapidlytranslated in adjusted levels of production

Hypothesis 2 Faster adaptation of desired stocks to new levels of demand(higher αS) increase volatility because temporary unbalances of demand in

18

one sector are transmitted faster to other sector reinforcing the feedbackmechanism of fluctuations

To test which of these hypotheses are confirmed by the model we runsimulation runs with identical systems (eg same technical coefficients samelevels of demand same shocks) but for the values of the αrsquos and αS rsquos testingall combinations of these parameters over a range of values (each run usedthe same parameters for all the sectors of the system) For each simulation(ie each couple of values of the parameters) we computed two indicatorsof volatility One measures the maximum distance of total production be-tween the highest and the lowest peak that is the maximum width of theoscillations registered in the production time series The second indicatormeasures the relative deviation δ = σY microY Figures 5 and 6 report theseresults

0004 0006 0008 001 0012 0014 0016 0018 002

Relative deviation

07 075

08 085

09alpha 0005

001 0015

002 0025

003 0035

004 0045

005

alphaS

0004 0006 0008 001

0012 0014 0016 0018 002

Figure 6 Relative deviation of total production for different values of α andαS

The results confirms unequivocally both hypotheses 2 and reject hy-potheses 1 In a sense these hypotheses confirm the nature of the economicsystem represented in the model as a complex system where the interactionsamong sectors play a far more important role than the individual sector be-haviour In fact in both cases we may interpret the results that the strongerthe efforts to mitigate the effects of shocks (ie slow productionrsquos and quickexpectationsrsquo adjustment) the opposite result is actually obtained the fluc-tuations become even more accentuated It is also worth to note that thestronger impact is generated by the behavioural parameters rather than thetechnical one as indicated by the stronger effects of the αS in respect of theα This result seems to suggest that ceteris paribus information mattersmore than technical constraints Though the model is by any means in-

19

adequate to discuss normative aspects because of the naive ways strategicdecision making is represented it can anyway suggest the effects of errorsin management decisions (ie wrong desired levels of stocks) as opposed tothe technical constraints preventing sudden adjustment of production lev-els Stretching a bit the interpretation of these results we may suggests thatefforts to reduce volatility should better be oriented to improve the trans-mission of information in order to coordinate future production plans (ieidentifying quickly the desired levels of stocks and therefore the new equilib-rium production level) rather than improving the flexibility of productionmethods (reducing α)

56 Volatility and stock levels

In our model stock levels are maintained to buffer a sectorrsquos sale againstunexpected changes in demand Therefore a system made of risk adversemanagers (keeping higher levels of stocks) may be thought to lead to asmoother absorption of shocks in respect of a system composed of firmskeeping a short supply to cushion unexpected events To test this hypothesiswe tested the behaviour of the model for different values of σ the levels ofstock represented as a multiple of demand For each level of σ we run asimulation using all other initialization identical as usual representing aneconomic system ldquoalmostrdquo at equilibrium but for a single small shock indemand uncorrelated across sectors18

00044

00046

00048

0005

00052

00054

00056

00058

0006

00062

5 55 6 65 7 75 8 85 9 95 10

Rel

ativ

e va

riatio

n of

tota

l out

put (

delta

)

Ratio of stocked goods over production (sigma)

Stocks and production fluctuations

Figure 7 Fluctuations persistence on systems with different stock strategies(σ)

18Demand changes by plusmn1-10

20

Figure 7 shows the value of the relative deviation of output production(δ) generated by heterogeneous uncorrelated shocks in the external demandon identical systems where firms store a different multiple of their demand asstocks We obtain actually the opposite result that we may have expectedfluctuation become larger and more persistent the higher the level of stocksIn some way a strategy that for the individual firm is meant to reducethe fluctuations (using stocks as buffers) ends up by actually increasing thevolatility at aggregate levels

The result is though explainable both at the theoretical and empiricallevel In fact if no stocks were maintained production would need to adjustfor changes in demand only in case of a shock That is the firm may movemonotonically from the old to the new production levels sending consistentsignals to the rest of the system (ie its own suppliers) Instead if thesame shock affects a firm maintaining large stocks of output over the sameperiod production needs to adjust to the new demand and and to alignstocks with the new demand This generates besides the adjustment to thedemand a potentially inconsistent signal to the suppliers since the firm willhave a temporary level of production not meant to continue in equilibrium(adjusting stocks)

From the empirical viewpoint the last few decades have seen the almostuniversal diffusion of production system meant to keep stocks to a minimumif at all Lean production methods justndashinndashtime etc are solutions tothe problems of the costs necessary to maintain stocks Over the sameperiod economic fluctuations have been strongly reduced and few empiricalassessments have pointed to the relations between the two phenomena Forexample concerning the US ldquochanges in inventory behavior have playeda direct role in reducing real output volatilityrdquo (Kahn McConnell andPerez-Quiros 2002 p 183)19

57 Irrelevance of technical coefficients distribution

Horvath (1998) and Horvath (2000) suggest that systems with sparse ma-trices ie inputndashoutput matrices with many empty cells tend to be morevolatile in respect of systems with more evenly distributed matrices Noticethat this claim risk if not adequately qualified to generate an illogical resultIn fact matrices sparseness increases as a function of sectoral disaggregationand comparing the same systemrsquos fluctuations using inputndashoutput matricesat different levels of aggregation needs to produce the same type of volatil-ity Indeed this is not the case if all remaining parameters stand equaland do not adjust to the different level of analysis considered If aggregatepatterns of adjustment speed are maintained unchanged the higher the sec-toral disaggregation the higher the systemrsquos volatility ceteris paribus But

19Similar results are found in Blanchard and Simon (2001)

21

for example in our model reducing the aggregation also requires a changein the adaptation coefficients the more micro we undergo the analysis thehigher is the speed of adjustment of economic objects If for example weconsider sectors we are evaluating the adaptation of an entire set of firmswhich pose questions on how to aggregate their dynamics Things changewhen we consider the response of a single firm

Going back to Horvatsrsquo argument highly disaggregated matrices arelikely to contain many empty cells so that we may forecast different levelsof volatility depending on the aggregation used for the same system whichas mentioned is not caused by sparseness but to a failed adjustment inreaction mechanisms Results from our model20 show that volatility is com-pletely unaffected by input coefficients distribution Given their sectoralsum mdash through rows of the matrix mdash the value of each coefficient is irrele-vant The same occurs in the case of very skewed distributions that presenta high number of empty cells (zero input coefficients)

58 Microndash and Macrondashvolatility

As we have seen above one of the major discussion in the literature concernsthe relations between microndash and macrondashvolatility Put it simply one sideof the literature considers that uncorrelated microndashshocks cancel each otherout reducing their effects at macrondashlevel Conversely other researchersconsider that microndashshocks are multiplied at macrondashlevel generating highermacrondashvolatility than that we can observe at microndashlevel The debate isof obvious importance to determine the sources of business cycles to pre-dict their effects and determine the more effective policies to mitigate theirnegative effects

Our model is only a partial representation of realndashworld economic sys-tem and is also overly simplistic However based on these limitations italso offers very robust results allowing reliable conclusions on the (limited)elements included in the model In particular we consider our model a re-liable representation of the basic structures of the production interactionsamong sectors or firms Thus we can use the model to provide an answerto the debate concerning the contributions to volatility from the productioninteractions only Other contributions (eg from financial markets priceadjustments technological innovation etc) may re-inforce or counter theforces we analyse but they need to be considered as additional elements(and we opine also generally less relevant) not alternative to our analysisWe consider a necessity in order to apply a rigorous methodological ap-proach to be able to identify with certainty the results provided by partialstudies in order to eventually obtain by gradual extensions of the analy-sis more and more detailed representations of realistic economic systems

20Available from the authors

22

The results we obtain in this paper concerning this microndashmacro debate aremeant as a first relevant step to a more extended analysis

0

002

004

006

008

01

012

014

016

018

02

022

05 1 15 2 25 3

Var Demand

Macro (Output) and micro (Demand) volatility

delta Outputdelta Demand

Figure 8 Values of δ (relative deviation) for aggregate demand and ag-gregate output in respect of different levels of variance for sector specificuncorrelated external demand shocks

The flows of shocks affecting the firms of a system can be distinguishedalong two dimensions strength of the shock (eg amplitude of the changesin demand) and frequency of the shock (rare or frequent) Concerning thefirst aspect we initialized as usual a system to equilibrium levels (eg pro-duction quantities equal to external and intermediate demand for all sectorsconstant level of stocks equal to the desired level) We then ran the simu-lation generating the external demand for all sectors as a random variablewith constant (equilibrium) mean level and varying values of variance

Figure 8 shows unequivocally that as far as our model has propertiessimilar to real systems the hypothesis that macrondashfluctuations have largervolatility than their sectoral level shocks is fully confirmed At any levelof variance the aggregate volatility of demand as captured by the relativedeviation index is sensibly lower than the volatility shown by aggregateoutput that is the total production of the system This result is not sur-prising throughout all tests presented in the previous sections every resultsuggested that the system lsquooverndashreactsrsquo to any individual microndashlevel changeeven in the case these changes were meant to contrast disturbances There-fore when we move from considering one single shock to a flow of shockswe are not surprised to obtain that the system shows higher volatility thanthat implied by the microndashlevel disturbances

We now consider the second dimension defining a flow of shocks theirfrequency The question is whether for the same level of demand shocks

23

0 002 004 006 008 01 012 014 016

Difference of relative deviation for Output and Demand

0 5 10 15 20 25 30 35 40 45 50 55

Shock frequency 05

1

15

2

25

3

Var Demand

0 002 004 006 008 01

012 014 016

Figure 9 Differences of values of δ for aggregate output minus those foraggregate demand in respect of different levels of variance of demand and offrequency of the shocks

(ie same variance) more frequent changes trigger a higher or lower systemvolatility than rare disturbances The results shown in figure 9 suggest arather elaborated answer which partly vindicates (though in a particularsense) the arguments in favour of compensation of micro shocks The fig-ure shows the difference between aggregate output volatility and aggregatedemand volatility The independent variables are as before the volatilityof demand at microndashlevel21 and the time intervals in between two shocksAggregate demand volatility does not change for the frequency of shocksat least when computed over a long period But output volatility doesin a nonndashmonotonous way For high frequency of shocks the difference isstill positive and increasing for higher variance indicating that the ldquooverndashreactionrdquo argument is valid even in these cases but it is rather low Whilethe frequency of shocks slows down aggregate volatility of output increasesshowing that time is needed for the microndashshock to unfold their full effectsat the macrondashlevel However after a certain threshold less and less frequentshocks generate a lower volatility of the aggregate level of output

This result is quite sensible though not obvious at first sight and onceagain shows the power of such a simple model to generate and explain com-plex properties For frenetic levels of demand modifications the system hasno time to adjust but part of the job to align actual and desired productionfor constantly changing demand is performed by the varying demand itselfIn other terms the shocks cancel out through time if not through their quan-

21Beware of the scale of variance For reasons of visibility the scale is inverted

24

titative effects on the system While frequency decreases the disturbancesof the shocks not compensated by other (unlikely) shocks have the time tofilter through the production system generating higher and higher volatil-ity In a sense any system has a particular frequency of shocks that make itmaximally ldquoresonatingrdquo with the amplification of microndashshocks When therate of shocks gets even rarer the system can enjoy periods of longer andlonger quietness after each shock is absorbed and before a new one startsagain a cycle of initial local disturbance propagation of the disturbances toother sectors and settlement to the new equilibrium

6 Discussion and extensions

We presented a simple model of production composed by interacting sectorThe main (quit realistic) assumption of the model is that the productionprocesses in any sector of the economy make use of input from any othersector Moreover we assumed that sectors (represented as a single firm)adjust slowly both the desired level of production (smoothed revision of ex-pectation) and the actual level (smoothed modification of plant utilization)The model voluntarily leaves aside a large number of aspects of real-worldeconomic systems (eg financial markets price adjustments growth etc)The reason is that the model can be easily and reliably tested in order toextract its properties The possibility to transfer the (certain) model resultto a real context depend of course on an ample ceteris paribus clause How-ever we are confident that the reported results concerning the productionstructure play a relevant role in real world systems Therefore our resultsand particularly their motivations being accessible because of the relativesimplicity of the model will not be diminished by the integration of ourconsiderations with other aspects of economic systems

Our model is in essence a representation of an input-output table inter-preted in physical terms that is as the amount of inputs required for oneunit of output Besides this we include two straightforward assumptionsFirstly production levels cannot be suddenly changed but require time toscale up or down the utilization rate that is the levels of production Stocksare meant to make up for the difference between sales and production Thesecond assumption concerns the intentional behaviour of producers We im-plement this aspect by assuming that exists a level of stocks desired for eachlevel of demand However a change of demand does not translate suddenlyin a new level of desired stock but again only a long time at a different levelof demand will convince managers to update the desired levels of stocks

We show that the model generate under a wide number of settings oscil-latory patterns converging toward the equilibrium level where each sectorgenerate exactly the quantity demanded by final consumers and by othersectors as input The oscillations are provoked by over- and under-shooting

25

generated by the difficulty of coordination among sectors that by assump-tion ldquocommunicaterdquo only via variation of demand to their direct input sup-pliers We also show that the total production levels generated by sectors(sum of the physical units) produced at equilibrium depends (for constantfinal demand) on the total sum of the technical coefficients per sector irre-spective of their distribution This result is motivated by the impossibilityof input substitution and offers interesting suggestions for the analysis ofthe study of input-output tables of real-world systems

We then consider systems at equilibrium facing a single small change ofthe external demand for sectors These simulations allow to study the waythe system reacts to shocks We see that the sum of the coefficients affectsbesides the levels also the volatility of aggregate output the larger a systemthe stronger is the impact of a given shock Less obviously we find thatfor systems with the same dimension (ie same sums of coefficients) theaggregate volatility increases with both the inertia to change of productionlevel and the speed of adjustment of desired stocks This result is somewhatcounter-intuitive since both the cited variables are associated to a moreconservative behaviour that may be supposed should smooth away spikesin the pattern from an old to a new equilibrium However we show thatthe opposite actually holds given the complex interactions among sectorindividual-sector attitude intended to smooth shocks actually increase theiraggregate impact

In the same vein we show that systems maintaining high levels of stocksare more volatile that systems with reduced amounts of stocks Again thisis a counter-intuitive result reverting the goal of individual producers andthe aggregate result Interestingly this result finds strong support from theevidence of the recent changes in production methods (ie low or no stocks)and diminished cycles

In relation to specific issues discussed in the literature we find no supportto the suggestion that particular distributions of input-output coefficientsmodify the volatility of a system As said this depends in case on thesums of these coefficients but not on their distribution Further we canprovide an answer to a long and hotly debated questions whether microndashlevels volatility generates stronger or lower aggregate volatility Our modelshows unequivocally that microndashvolatility (eg variance of final demand)persists and generates stronger volatility at aggregate level Again we canidentify the reason on the complex production structure that exalts micro-level shocks by generating lsquowrongrsquo messages among producers Moreoverwe can also point to a usually neglected aspect of volatility we show thatthe frequency of shocks is a very relevant aspect besides their dimensionin determining the volatility of aggregate output

Our work can be developed along two complementary directions Firstlywe can continue to study the properties of the model by extending the ele-ments considered (eg pricing) analysing the effects on the present results

26

and generating more analytical evidence Of particular relevance is in ouropinion the problem of aggregation Representations of the same systemswith different codification systems (eg different number of SIC digits)must not modify overall system properties for example the volatility ob-served Given that our result show that the sums of coefficients matter forthe levels and volatility of the system and that these sum do depend on theaggregation chosen it should be possible to induce the reaction coefficientsthat we know affect the volatility only

Secondly we can find an application of our results to real world evidenceIn fact there are many data sets available for the data we deal with basicallyinput-output tables Our model is readily adapt to be parametrized alongany number of sector and coefficient necessary so that we may replicate pastseries in order to adapt the unobservable parameters and use the resultingcomplete model to explain business fluctuations and provide intuitions onfuture patterns

27

References

Acemoglu D and A Scott (1997) ldquoAsymmetric Business CyclesTheory and TimendashSeries Evidencerdquo Journal of Monetary Economics 40501ndash533

Backus D P Kehoe and F Kydland (1995) ldquoInternational BusinessCycles Theory and Evidencerdquo in Frontiers of Business Cycle Researched by T Cooley Princeton University Press Princeton

Bak P K Chen J Scheinkman and M Woodford (1993) ldquoAg-gregate Fluctuations from Independent Sectoral Shocks SelfndashOrganizedCritically in a Model of Production and Inventory Dynamicsrdquo RicercheEconomiche 47(1) 3ndash30

Battiston S D Delli Gatti M Gallegati B Greenwald and

J Stiglitz (2005) ldquoCredit Chains and Bankruptcies Avalanches in Sup-ply Networksrdquo in Annual Workshop on Economic Heterogeneous Inter-acting Agents (WEHIA 2005) University of Essex UK

Blanchard O and J Simon (2001) ldquoThe Long and Large Declinein US Output Volatilityrdquo Brookings Papers on Economic Activity 1135ndash174

Bresnahan T F and V A Ramey (1994) ldquoOutput Fluctuations atthe Plant Levelrdquo The Quarterly Journal of Economics 109(3) 593ndash924

David P A (1975) Technical choice innovation and economic growth essays on American and British experience in the nineteenth centuryCambridge University Press London

Delli Gatti D C Di Guilmi E Gaffeo G Giulioni M Galle-

gati and A Palestrini (2004) ldquoA New Approach to Business Fluc-tuations Heterogeneous Interacting Agents Scaling Laws and FinancialFragilityrdquo Journal of Economic Behavior amp Organization 56 489ndash512

Dosi G G Fagiolo and A Roventini (2006) ldquoAn EvolutionaryModel of Endogenous Business Cyclesrdquo Computational Economics 273ndash34

Dupor B (1999) ldquoAggregation and the Irrelevance in MultindashSector Mod-elsrdquo Journal of Monetary Economics 43 391ndash409

Head A C (1995) ldquoCountry Size Aggregate Fluctuations and Interna-tional Risk Sharingrdquo The Canadian Journal of Economics 28(4b) 1096ndash1119

28

Helbing D and S Lammer (2005) ldquoSupply and Production NetworksFrom the Bullwhip Effect to Business Cyclesrdquo in Networks of Interactingmachines Production Organization in Complex Industrial Systems andBiological Cells ed by D Armbruster K Kaneko and A S Mikhailovvol 3 of World Scientific Lecture Notes in Complex Systems World Sci-entific

Helbing D S Lammer U Witt and T Brenner (2004) ldquoNetworkndashInduced Oscillatory Behavior in Material Flow Networks and IrregularBusiness Cyclesrdquo Phisical Review E 70(056118) 1ndash6

Helbing D U Witt S Lammer and T Brenner (2004) ldquoNetwork-Induced Oscillatory Behaviour of Macroeconomic Output Flowsrdquo in In-dustry and Labour Dynamics II The Agent-based Computational Eco-nomics Approach Proceedings of the wildace 2004 conference ed byR Leombruni and M Richiardi no 5 LABORatorio R Revelli

Horvath M (1998) ldquoCyclically and Sectoral Linkages Aggregate Fluctu-ations from Independent Sectoral Shocksrdquo Review of Economic Dynam-ics 1 781ndash808

(2000) ldquoSectoral Shocks and Aggregate Fluctuationsrdquo Journal ofMonetary Economics 45(1) 69ndash106

Jovanovic B (1987) ldquoMicro Shocks and Aggregate Riskrdquo The QuarterlyJournal of Economics 102(2) 395ndash410

Kahn J A M M McConnell and G Perez-Quiros (2002) ldquoOnthe Causes of the Increased Stability of the US Economyrdquo FRBNY Eco-nomic Policy Review 8(1) 183ndash202

Kraay A and J Ventura (1998) ldquoComparative Advantage and theCrossndashSection of Business Cyclesrdquo World Bank Woorking paper Series1948 World Bank

Long J B and C I Plosser (1983) ldquoReal Business Cyclesrdquo Journalof Political Economy 31(1) 39ndash69

Lucas R E (1977) ldquoUnderstanding Business Cyclesrdquo CarnegiendashRochester Conference Series on Public Policy (special issue of Journalof Monetary Economics) 5 7ndash29

Nelson R R and S G Winter (1982) An Evolutionary Theory ofEconomic Change Harvard University Press Cambridge MA

Nirei M (2005) ldquoThreshold Behavior and Aggregate Fluctuationsrdquo Jour-nal of Economic Theory in press

29

Scheinkman J and M Woodford (1994) ldquoSelfndashOrganized Criticallyand Economic Fluctuationsrdquo American Economic Review 84(2 Papersand Proceedings of teh AEA) 417ndash421

Schuh S (1996) ldquoEvidence of the Link Between FirmndashLevel and Ag-gregate Inventory Behaviorrdquo Finance and Economics Discussion Series1996-46 Federal Reserve Board

Stiglitz J and B Greenwald (2003) Towards a New Paradigm inMonetary Economics Raffaele Mattioli Lectures Cambridge UniversityPress Cambridge

Weisbuch G and S Battiston (2005) ldquoProduction Networks andFailure Avalanchesrdquo Working Paper mimeo Ecole Normale SupErieure

Zarnovitz V (1977) ldquoBusiness Cycles Observed and Assessed Why andHow they Matterrdquo NBER Working Paper Series 6230 National Bureauof Economic Research

30

A Tables

Table 1 Parameters valuesParametera Description Value

N Number of sectorsunits i of production 10Ci Number of j buyer sectors for sector i (IO matrix sparseness) N minus 1Di (E) Value of the external demand 10mij Physical input coefficient of input j for sector i sim U(0 1)Mi Sum of the mij input coefficients j for the production of i 05αi friction to changes of production levels Degree of lockndashin on

production technology085

αsi Speed of adaptation to demand changes 004

σ Desired multiple of production to stock 10

aInto parenthesis the number of lags of the initial value for lagged variables

31

Di Guilmi Gaffeo Giulioni Gallegati and Palestrini 2004)2 the failure ofa large company induces readjustments in the same and related sectors andson on

Besides we expect that the more we disaggregate the economic units ofanalysis the more shocksrsquo intensity is likely to reduce and symmetric shockstend to cancel out The application of the law of large numbers would sug-gest that an economy with normally distributed entities (at the same level ofaggregation) mdash or even better with representative entities would not showaggregate fluctuations in the presence of uncorrelated disaggregated shocks(Lucas 1977) This has led to focus most theoretical explanations (or rep-resentations) of short business cycles on aggregate shocks (Horvath 2000)Although they are undoubtedly relevant and have an impact also on mi-cro changes aggregate interpretations are quite limited in their explanatorypower While having to assume exogenous origin of shocks they do notprovide an understanding of how macro shocks rebound on the economicentities (consumption change production shift productivity shifts invest-ments etc)

Indeed in order for micro shocks to generate aggregate fluctuations twoconditions are necessary a propagating structure mdash a connection mdash be-tween micro economic entities and a sluggish absorption of changes Inthis respect production structures made up of sectors connected by inputndashoutput relations represent a good candidate a part from being the knownstructure of a basic industrial economy Two main class of models havebeen developed to analyse the effect of sectoral shocks on aggregate fluc-tuations Real Business Cycles (RBC) and Avalanche (Av) models Bothtype of models make attempt to introduce mechanisms that allow for thepersistence of micro shocks In this respect both classes are based either onquite unrealistic behavioural assumption which determine the conditions ofthe system or threshold behaviors which impose the conditions for shockspersistence We briefly review them below

21 Mechanisms of shocks propagation in IndashO structures

A number of contributions study the phenomenon of generation and prop-agation of cycles independently from the sectoral structure of the economyHowever the most recent contributions point in the direction of giving thisaspect of economic system a strong relevance Moreover the two approachesare not incompatible so that we focus on the literature that explicitly makesuse of the IndashO structure In this section we briefly review three strands ofthis literature highlighting the elements that inspire our work

2Both cases are easily shown with evidence from the Latin American crisis during theeighties rooted in the crisis of the Mexican financial system the financial crisis in Asiaduring mid nineties the recent Argentinian financial crisis etc

4

Plain IndashO structure

The pioneering RBC model that analyses the diffusion of sectoral shocks isdue to Long and Plosser (1983) They use a general equilibrium multi sectormodel with fully rational infinitely lived perfectly informed homogeneousetc individuals In such a system by assuming a perfectly maximising be-haviour (via allocation of production and consumption of resources) andthat individuals prefer to smooth savings across time and goods it is pos-sible to generate persistence of continuous albeit uncorrelated shocks inproduction capacity across sectors In the words of Long and Plosser [p67] ldquoAt constant relative prices this [the assumed maximising behavior]suggests that business-cycle features like persistence and comovement arecharacteristics of desired consumption plansrdquo

The necessity to maintain the assumption of perfectly rational agentsforces these strand of models to attribute the origin of cycles to externalfactors only like exogenous shocks on technology As such they seem un-likely to provide a consistent explanation (or representation) of aggregatefluctuations Dupor (1999) shows that under quite general conditions theavailable multindashsector RBC models produce the same results in terms ofvariance convergence as their respective single sector models This is shownalso under particular conditions of input matrix or high coefficient of spe-cific inputs Dupor concludes that [p 405] ldquoresearchers who wish to useindependent sector shocks as a source of aggregate fluctuations must appealto sector shocks that are large relative to their single sector counterparts orelse consider a different set of models than those discussed in this paperrdquoIn doing so the author though suggests that another mechanism must beintroduced somehow putting the desired results ahead of the economic issueanalysed

The criticism to this kind of literature suggests that in order to allowthe IndashO structure to play a role in the explanation of cycles we need agentswith less then perfect foresight and capacity of adaptation to unndashexpectedconditions As we will see below agents that need time to both realizethe economic changes of their time and to introduce the necessary mod-ifications are sufficient conditions to observe aggregate fluctuations froma single change in one part of the economy Though being ldquorationalrdquo inthe sense that they make the right decisions they are limited by lags inthe information flow and physical constraints to the changes that they canapply

Input matrix incompleteness

Using a similar approach to RBC models Horvath (1998) and Horvath(2000) focus the attention on the properties stemming from peculiar produc-tion structures as represented by characteristics of the input-output matrix

5

The author shows that sectoral shocks will tend to cancel out in case theeconomic system is sufficiently distributed that is when the IndashO matrixcontains evenly distributed values over all the cells Conversely sectoralshocks are not absorbed and can be reinforced in case many cells of thematrix are empty This case indicate that there will be few sectors affect-ing many other sectors preventing the possibility of compensating certainshocks

These conclusions are potentially interesting because point directly toan easy to observe aspect of economic system for example providing thepossibility to test the prediction of the analysis It is worth to note that thisapproach draws conclusions that are not neutral to the aggregation levelused In fact a highly disaggregated matrix is more likely to contain manyempty cells than the matrix for the same system obtained aggregatingsectors

Dupor (1999) also considers the case of inputndashoutput matrices with dif-ferent input coefficients finding them irrelevant for the results But he doesnot include matrices with empty cells

The cited contributions and those inspired to the same approach sug-gest that the type of interaction as expressed by input-output coefficientscan be a relevant factor in determining the existence and dimension of fluc-tuations However this literature is not able yet to draw conclusive results

Limited interactions and production constraints

A different kind of mechanisms are assumed and shown to play a role in theAv models they add an analysis on firms production (technological) possi-bilities to the production structure They are inspired by and similar to Jo-vanovic (1987) model in which the limitation in the number of interactionsupon which each playerrsquos decision depends generates shock cascades andaggregate persistence Bak Chen Scheinkman and Woodford (1993) andScheinkman and Woodford (1994) show that when the interaction structurebetween sectors is constrained to a completely rigid system coupled withnon convexities in the production function independent shocks to differentsectors do not cancel in the aggregate In other words the authors place aset of restrictions to the set of possible actions of sectors First structuringthe production networks with a fixed invariable lattice produces a rigiditythat do not allow producers to change (or add) inputs (or suppliers) orincrease the produced quantity This is combined with the structure of theavalanche that propagates through the unchanged network Second a pro-duction function that is maximised only when a discrete number of unitsof a good is produced coupled with a fixed (discrete) maximum amount ofinventories that can be stored by the representative firm Such a structuredetermines production oscillation and shocks avalanches due to the factthat firms are constrained in their production opportunities

6

Using a very similar production structure with limited interactionsWeisbuch and Battiston (2005) play on the bankruptcies generated on down-stream firms by production failures occurring in upstream firms (reducingthe flow of inputs) Stochastically generated failures to produce in a firmcauses production constraints in its downstream clients hence lower invest-ment and eventually bankruptcies3 A lag between the period in whicha firm goes bankrupt and a new firm appears generates a lack of supplywhich propagates on the downstream sectors

Due to the original literature on avalanches these contributions sharetwo common features that are ill-adapted to economic systems First theyassume very peculiar interaction structures of one-directional productionrelations between sectors which amount to assume one half of the input-output matrix empty and are not applicable to the general case with cyclicalproduction structure or full IndashO matrices Second the strongly rely on quiterigid thresholds determining a mechanicist behaviour by agents Basicallyagents have only two options available (ie produce or not produce) andthe choice is made on the basis of crude decisional mechanism4

Notwithstanding the limitations this approach provides a quite sensiblemechanism of the origin and transmission of shocks through a system Ourproposal can be interpreted as an extension of this approach by relaxing thestrong assumptions on the interaction structure and allowing agents for amore fluid decisional process as we will see in the next section

3 The basic elements of a production structure

Our overall hypothesis is that the very production structure of an economymay generate persistent fluctuations even in the absence of continuous un-related shocks and rigid thresholds that determine non linearities The aimis to provide a generalised interpretative framework to explain the occur-rence of short term fluctuations in the economy simply as an outcome ofthe division of labour in a number of sectors related through trade In itsextreme interpretation any production system which is not fully integratedis bound to produce business cycles As we will show a number of factorsinfluence their extent

Quantitative adjustments in production in any one sector require ad-justments in related sectors When the structure of the economy is trulyInputndashOutput (not a directed linear supply chain as in the Av models) in-put flows are cyclical and adjustment to long run production equilibrium islikely to take a long mdash when not infinite mdash time Our work departs from

3A similar model by Battiston Delli Gatti Gallegati Greenwald and Stiglitz (2005)relates bankruptcies to price changes buyers payment timing and the credit market (acentral bank)

4Nirei (2005) is an attempt to generalise such models though maintaining the assump-tion of threshold behavior

7

both RBC and Av models with respect to structure modelling criteriaanalysis and methodological approach Indeed our model may be easilyreconduced (or restricted) to both We are much in line with a recent se-ries of papers by Helbing Witt Lammer and Brenner (2004) that addressthe same question although with a number of differences that are discussedlater on5

We aim to analyse to what extent the structure of inputndashoutput relationscauses aggregate fluctuations by itself The main idea is to show that ag-gregate fluctuations may derive from simple micro imbalances without theneed for continuous exogenous shocks not even at the firm or sectoral levelIn other words normal production activities including lags technologicaladjustments information asymmetries circularity of the inputndashoutput sys-tem and so on may be enough to explain part of the macro volatility in aclosed system6

Our assumptions draw on the existing literature but try to avoid themajor difficulties highlighted above Firstly we avoid to impose either mar-ket clearance assumptions or crude threshold behaviours Rather we adoptan assumption close to a Simonian approach agents in an economy do theirbest within the informational and physical constraint they are subject toWe represent decision makers as responding to a single piece of informationthey obtain the quantities demanded by their clients Crucially their ac-tions are inspired to a conservative approach an increase in the demandgenerates a smaller (desired see below) sudden increment of productionThis assumption stems from the fact that decision makers are aware of theinstability of their environment and therefore want to avoid getting perma-nently caught with over production following a temporary spike of demandIn other words firms attempt to smooth business cycle generating a morestable (or less uncertain) environment as the economic literature would sug-gest Notice that this representation in line with a routinized representationof organizations (Nelson and Winter 1982) does not contradict market equi-librium Rather in a stable environment such decisional procedure generatesan asymptotic pattern to the equilibrium level in the absence of feedndashbacks

A second assumption concerns the technological possibilities available tothe firms When firms realise the need for a change in production levels saya desired increase of production they face a complex and costly organisa-tional transformation of the production capacity risky investments an soon As Bresnahan and Ramey (1994 p 622) suggest inducing from a de-tailed analysis of the automobile industry in the US ldquoadjusting production

5We thank Matteo Richiardi for pointing out these similar attempts See also Helbingand Lammer (2005) and Helbing Lammer Witt and Brenner (2004)

6An open system entails many other factors including price factors of both inputsand outputs mdash terms of trade labour cost external investment etc As with exogenousshocks our analysis does not deny the importance of these factors but simply separatethem from the structural explanation of cycles

8

is a more complicated process than simply ldquochanging Qrdquo or choosing themix of capital and laborrdquo And this is not only a matter of non convexitiesin production technologies The inconsistency of time reversibility in thetechnological choice is accompanied by the need to operate on the lsquoavail-able technological frontierrsquo (David 1975) If a firm plans a given productionactivity for a short period of one week even though the next day there is anunexpected fall in the demand it is unlikely that it is followed by a similardownturn in production The case is evident when we think at productionplanning that involve hiring of workers an capital investments (both in-volving sunk costs)7 Therefore any desired change generates a pattern ofsmall modifications approaching slowly the target level Moreover thesechanges are irreversible in the short term since only a prolonged change ofthe environment triggers the modification of previous desired levels

Resting on those simple conjectures we thus analyse a model in whichwe only allow for sluggish adaptation of production due to conservative(adaptive) behaviour of firms and the physical constraints faced when vary-ing the production levels Notice that these assumptions are contrary tothe generation of instability of a system In fact a firm behaving as de-scribed above reduces if not eliminates any disturbance affecting its ownstate if acting in isolation Other factors like new technologies extremelylong term perspective price changes etc are likely to be sources of volatil-ity therefore contributing to generate aggregate system fluctuations (as forexample in Acemoglu and Scott 1997) Our neglect of these and other fac-tors however unrealistic serves the purpose of testing whether a volatilityreducing microndashbehaviour turns into volatility generating aggregate resulttwo widely observed dynamics Quite interestingly (and encouraging) DosiFagiolo and Roventini (2006) have concomitantly proposed a model whichincorporates detailed representation of both sluggishness robustly based onempirical evidence Heterogeneous firms which undergo lumpy investmentand slowly adapt their expectation on changes of the future demand inducecomovement of macroeconomic variables qualitatively similar to the onesobserved in the real world To pursue our line of research we first needto concentrate on how sectorsfirms interact in the system in the effort tocomplement those promising results

Therefore the second element we consider is meant to represent a gen-eralized economic system We allow for the circularity of inputndashoutput sys-tems a crucial feature in determining feedback mechanisms and a principal

7See also the discussion in Acemoglu and Scott (1997) where the authors argue thatldquo[w]hile the presence of fixed costs can account for the discreteness of economic turningpoints it does not naturally lead to persistence because once an individual undertakes anaction they are less likely to do so in the near future [] [A]lthough the presence of fixedcosts leads to increasing returns these are intratemporal the full extent of economiesof scale arising from fixed costs can be exploited within a periodrdquo [p 502 authorsrsquoemphasis]

9

component of business cycle It may be true that most backward linkagesconcern capital goods (Weisbuch and Battiston 2005) but a rapid glanceto an inputndashoutput matrix would show that technical coefficients are nonnull in both directions and ldquomost commodities are inputs to the produc-tion process of other commoditiesrdquo (Horvath 2000 p 70) The higher thesectoral aggregation mdash observed or assumed mdash the more the relevance ofintermediate supply holds true (as mentioned above with respect to ma-trix incompleteness) We claim that the existence of backward and forwardlinkages are a crucial element to determine the persistence of shocks in theeconomy A perfectly linear (one directional) system needs much less adap-tation of firms and sectors in order to stabilise the economy Nonethelesscircularity does not imply that inputs are perfectly substitutable amongthem in order to determine the maximising allocation even at high level ofdisaggregation

By using a fixed coefficients production function we both maintain themid term rigidity of production processes and we do not impose limitedinteractions We claim that the circular property of the production structureis by itself sufficient to produce shock persistence even when the inputmatrix is full (although with heterogeneous input coefficients) mdash all sectorsshop in the remaining n minus 1 sectors

In conclusion we represent a model for an economic system made onlyof i) production level decisions and ii) input-output relations Clearly sucha model lacks many features present in real economic systems likely we areconvinced to affect the propensity to generate cycles As such we cannotaim at reproducing realistic data so our results cannot be tested comparingtime series generated by the model with real ones Instead given the paucityof the elements comprising the model we can investigate in detail to whichextent the (few) features present in our model affect the cyclical behaviour ofthe system As we will see we will be able to show quite a number of resultin which aggregate fluctuations are generated whereas the individual microstructure of the model aims at monotonous adaptation Before discussingthe results next section describes in detail the implementation of the simplemodel

4 The model

We model an economic system in which a number of sectors i = (1 N)are potentially connected via (intermediate) market relations A N times NInputndashOutput matrix determines the mij units of each input j = (1 N)mdash bought from the N minus 1 residual sectors mdash necessary to produce oneunit of output i8 For simplicity given the focus of this paper on the IndashOstructure as a generator of aggregate fluctuation we abstract from market

8IndashO coefficients are therefore physical coefficients

10

interactions within sectors each sector is a production unit In fact forour purposes it is not relevant which firm produces the sectoral outcomebut only the total amount produced by each sector9 Moreover besidesmaking the analysis more neat this assumption allows also to enjoy a largefreedom in determining the aggregation level since we donrsquot need to assessthe effects of between sectors competitiveness which for example is likelyto increase for increasing levels of disaggregation Our economic systemis therefore composed of N production units each of which may buy andprovide inputs from and to any other sector depending on the IndashO structureof the economy

The demand for each sector i is determined by a constant consumersdemand Di (E) and the input needs of other sectors

Di = Di (E) +sum

j 6=iisinCi

mjiQj (1)

where Ci is the set of j intermediate clients of i (sum

j isin C lt N) mji is thetechnical coefficient of firm j for input i ie the amount of i required toproduce one unit of j Qj is the actual output produced by sector j

Concerning the simultaneous production level of each sector in time twe make two lsquorealisticrsquo assumptions First from the technical viewpoint afirm like a tanker cannot abruptly change its course Therefore changesto the production level can be introduced only gradually through capital(dis)investment and job h(f)iring (see discussion in Section 3) Second weassume that firms do not decide the production level directly on the basisof the observed demand level but conscious of the uncertainty of demandplan their production in order to maintain invariant a desired level of stocksand avoid stockout10

Quantity produced by sector i in period t is computed as follows

Qit = αiQtminus1 + (1 minus αi)(

Slowastitminus1 minus Sitminus1

)

(2)

where αi measures the physical constraint in adapting the production capac-ity to changes in the required output Slowast

itminus1are firmrsquos i desired stocks and

Sitminus1 its actual stocks Notice that both variables related to stocks appearwith a lag This is due to two reasons First it is a modelling necessity inorder to allow the simultaneous determination of production levels for allsectors Secondly as frequently happens modelling necessities reveal sen-sible if not generally considered aspects of the modelled system In factproduction decisions concern management in the production plant while

9Adding heterogeneous firms within sectors would definitely enrich the causal expla-nation put forward and parametrised in this paper (see below) For example to analysethe effect of firms constraints It is left over for future research as in the present work weopt for the analysis of the main structural parameters that affect aggregate fluctuations

10See also discussion in Schuh (1996)

11

demand through sales is observed by the commercial staff It is thereforelogical that the information from demand reaches plant managers with somelag

The actual stock level is trivially computed subtracting from the previousheap of stock the amount sold and adding the new production

Sit = Sitminus1 + Qit minus Dit (3)

Concerning the desired level of stocks instead we introduce the conser-vative behaviour mentioned above Firms ideally would keep an amountof stocks proportional to the demand they receive say a multiple σ ofthat amount However when demand varies they adjust the desired levelsmoothly preventing sudden changes to the variable representing the goalof the firm

Slowastit = Slowast

itminus1 + αsi

(

σDit minus Slowastitminus1

)

(4)

where αsi measures the adaptation to demand changes

In summary the dynamics of the model consists of only the equationsabove Firstly at the beginning of a time step firms determine the quantityto produce as a function of past imbalances in stock levels and the pastproduction level These levels define the demand for all sectors determinedby the technical coefficients applied to all produced quantities Finallyactual and desired stocks can be updated Figure 1 represents visually thisdynamics highlighting the decisional components of the model in respect ofthe ldquomechanicsrdquo of trade

5 Results production structure and economic fluc-

tuations

In this section we describe some of the most relevant results concerningthe effects of the production structure to determine aggregate fluctuationsGiven the unusual analytical instrument chosen (at least concerning thistopic) we start with a brief methodological introduction meant to clarifythe nature of the results we claim to obtain Next we describe a generalsetting for the model which is the (extremely) limited area of the potentialparametersrsquo space of the model that we will explore In the rest of thesection we discuss the results from the model

51 Methodological considerations

Real economic systems are constantly subject to a continuous flow of shocksexternal to the production system besides changes within the system itselfIn order to appreciate the systemic properties we need to abstract from all

12

Compute the quantityto produce

F (Q(t-1) S(t-1) s(t-1) alpha)

Communicate thequantity of inputsrequired to each

supplier

Determine the overalldemand

F(D(E) D(I))

Revise the past decisionon stock plans

(desired stocks)F(D S(t-1) sigma alpha(s))

Compute the final levelof stocks for next period

F(s(t-1) Q D)

KEY

ModelDirection

Variablecomputed

at t-1(t-1)

Step includingfirm behavior

Figure 1 One period dynamics

the nonndashrelevant disturbances and concentrate on how the system endoge-nously contributes to the observed dynamics Therefore we will generatehighly abstract patterns whose analysis will provide insights on the systemproperties that would be otherwise lost in the noise generated by all theelements of a real system

We are not trying to fit the model to a specific empirical data set nor weare interested in studying the complete behaviour of the model for areas ofthe parameterrsquos space that make no economic sense (eg negative or infiniteproduction levels) Therefore the values of initial settings and results matteronly in relative and not in absolute terms

We are going to describe a simulation model with which we will produceseveral simulation runs and we claim obtain results relevant for the debatediscussed above Rigorously speaking we may claim only certainty for thevalidation of the results that is that the model does actually generate theresults presented and for the reasons we explain We limit this part to thepresentation of graphs and verbal explanations that we believe are ratheruncontroversial11 Concerning the relevance of our studies for real systems

11Interested readers can request the model code and and simulation data for replicationand extensions of the results We can guarantee that such possibility does not require

13

(verification) we simply avoid to pretend that our model is a quantitativeapproximation of real economies (which in which period) Our opinionis that the study of the model allows us to make general considerationsnon-obvious and relevant for applied debates supported by strong logicalarguments which can be integrated but not reverted by arguments con-cerning more and more elements of the economic world

52 Model setup

We go through the model properties analysing an economic system madeup of ten sectors (N = 10)12 Each sector shops inputs from the remain-ing N minus 1 and sells to them part of its own production as intermediategoods This means that the input matrix is complete except for the diag-onal (mij gt 0 forallj 6= i mii = 0) The sum of all N minus 1 coefficients for asingle sector is given by Mi and its benchmark value is kept at a plausiblemedium level (unless differently specified) to avoid distortions from highlydemanding sectors Each input coefficient is randomly drawn from a uniformdistribution

Sectors start at an equilibrium level their supply matches intermediateand final demand and in each period the same quantity of output mustbe produced to compensate for the used stocks and keep the desired stocksunchanged In thier turn stocks must be ten times the value of total sectoraldemand (σ) Eventually technological adjustment of production is ratherslow and producers are quiet refractory to follow sudden changes in thedemand in general a smoothing behaviour strongly prevails (α and αs)

Under those general preconditions in the following sections we anal-yse the dynamics of the economic system in relation to model parametersWe mainly focus on the cyclical behaviour of systems and their speed ofconvergence to the asymptotic equilibrium13

The very abstract assumption and the homogeneous initialisation donot allow for the multiple equilibria that are likely to rise in complex systemswith feedbacks Once more such simplification allows to have a betterunderstanding of the crucial impacts of the production structure on businessfluctuations Departing from the related literature discussed above14 weconsider the effects of a single shock rather then a continuous flow whichin our model is superfluous in order to obtain fluctuation persistence15 To

extensive programming or statistical skills besides the usual education of economists12You can think of a high level of aggregation in the statistical observation The com-

plete initialisation and parameters values are available in table 1 in the Appendix13Notice that sectors only approach a final equilibrium level of production which can

be prendashdetermined under static conditions14Included the more proximate work of Helbing Witt Lammer and Brenner (2004)15An exception are the last results presented in Section 58 which we see more as an

opportunity for further research than a closing of the paper

14

start with next section (53) shows the aggregate cyclical behaviour of astandard setting

53 Single shock analysis

We generate a simulated history following the dynamics of the modelrsquos vari-ables when the system is set to an equilibrium level and a single shockmodifies one of the exogenous elements namely the external demand forsectors Modifications across sectors are random and not correlated andmay increase or reduce external demand by at most 10

100 225 350 475 600

193894

195861

197828

199795

201762 Industrial Production

Time

Figure 2 Total industrial production fluctuations following a shock fromexternal demand

Figure 2 shows the dynamics generated by the total industrial productionof the system mdash that is also a proxy for the systemrsquos GDP fluctuations Thenew external demand changes the equilibrium level but the system cannotreach the new equilibrium with a monotonous pattern and converges to thenew level with a fluctuating pattern Each change in the external demandinduces also modifications of the intermediate demand and sectoral adjust-ments in the quantity produced typically overshoot the aggregate patternAs it can be appreciated from the figure it takes time before the system isable to converge to the new static equilibrium conditions

We study two features of this pattern absolute level of the new equilib-rium and volatility induced by the shock

54 Technical coefficients and industrial output

Production level

The equilibrium levels of industrial output depend on two factors only Ob-viously the levels of external demand determine how much of the total pro-

15

duction needs to be supplied to consumers exogenous from the productionsystem Second the technical coefficients also influence strongly the level ofactivity of the system defining the intermediate demand to each sector

180

200

220

240

260

280

300

01 02 03 04 05 06 07 08 09

Out

put l

evel

(In

dust

rial p

rodu

ctio

n)

Sum of total input coefficients (M)

Total industrial production for different input coefficients levels

Figure 3 The relevance of input coefficients on output level

Figure 3 shows the equilibrium level of industrial output reached bysystems with identical external demand where we fixed the sum of inputcoefficients for each sector of the economy (Mi) to different values16 Thehigher the sum the higher is the total production sustained by the system

This result depends on the very production structure represented by theinputndashoutput matrix In fact the higher is the level of production requiredby intermediate sectors the higher is the total production given a constantlevel of external demand

Notice that the level of production does not depend on the distribution ofthe technical coefficients As long as the sum of the coefficients is identicalthe same production level is obtained irrespective of their distribution (otherthings being equal)

Output shock absorption

Input coefficients have a relevant role also on the output stability of thesystem Starting from an output equilibrium level we hit the system witha single shock on the external demand for all sectors uncorrelated amongsectors17 modifing (positively or negatively) the demand for all sectors by

16Within this limit the actual value of the coefficients are randomly drawn17In each sector the external demand changes by plusmnX where X is a uniformly dis-

tributed percentage between 1 and 10

16

at most10 As show in Figure 2 the economic systems produces a lenghtlysmoothing oscillatory behaviour We compute the relative deviation of theaggregate production (industrial output) as the ratio between its standarddeviation and its mean δ = σY microY where Y =

sum

Qi Figure 4 shows thebehaviour of δ as a function of the sum of input coefficients for each sector80 periods after the shock As time goes by the level of δ reduces for allMi but the relation with the sum of input coefficient is unchanged

00055

0006

00065

0007

00075

0008

00085

0009

01 02 03 04 05 06 07 08 09

Rel

ativ

e va

rianc

e of

tota

l out

put (

delta

)

Sum of total input coefficients

Aggregate output deviations for different technical coefficients

Figure 4 The relevance of input coefficients on output stability

In sum input coefficients determine the extent of overshooting behaviourThe higher the coefficients the higher is the backlash even when shockscould cancel out through sectors

55 Shock persistence and adjustment coefficients

We now consider the effect of micro smoothing behaviours on the volatilityof the production system In particular we are interested in testing theeffects of two parameters of the model on the volatility of the productionsystem First we consider the effects of the αrsquos representing the ldquostickinessrdquoof the production technology to desired changes of Q aimed at filling thegap between actual and desired stocks The higher is α the slower is thereaction to stock unbalances Concerning its effect referring to the technicalpossibilities available ie production capital and organization we may putforward two opposite hypothesis relevant to systemrsquos volatility

Hypothesis 1 Slower production adjustment (high α) reduces volatilitybecause unbalances in demand of one sector being compensated slowly slow

17

down the diffusion of shocks to other sectors

Hypothesis 2 Slower production adjustment (high α) increase volatil-ity because the longer lasts the mismatch between actual and equilibriumproduction level the deeper will become the mismatch between actual anddesired stocks

20 25 30 35 40 45 50 55 60 65

Cyclesrsquo width

07 075

08 085

09alpha 0005

001 0015

002 0025

003 0035

004 0045

005

alphaS

20 25 30 35 40 45 50 55 60 65

Figure 5 Maximum width recorded between highest and lowest peaks fortotal production for different values of α and αS

Secondly we consider the parameters αS representing the speed of ad-justment of desired stock to observed demand The higher this parame-ter the faster sectors revise the desired levels of stocks to current levelsof observed demand This parameter concerns sectorsrsquo behavioral decisionprocess Higher αS represent a stronger belief that the most recently ob-served demand level is a reliable indicator of future demand levels whilelower levels of the parameter represent a more conservative approach wheremanagers require a long observation of demand levels different from the pastones before adjusting their expectations Also concerning the effect of thisparameter on the volatility induced by an exogenous shock we may maketwo hypothesis

Hypothesis 1 Faster adaptation of desired stocks to new levels of de-mand (higher αS) reduce volatility because changes in demand are rapidlytranslated in adjusted levels of production

Hypothesis 2 Faster adaptation of desired stocks to new levels of demand(higher αS) increase volatility because temporary unbalances of demand in

18

one sector are transmitted faster to other sector reinforcing the feedbackmechanism of fluctuations

To test which of these hypotheses are confirmed by the model we runsimulation runs with identical systems (eg same technical coefficients samelevels of demand same shocks) but for the values of the αrsquos and αS rsquos testingall combinations of these parameters over a range of values (each run usedthe same parameters for all the sectors of the system) For each simulation(ie each couple of values of the parameters) we computed two indicatorsof volatility One measures the maximum distance of total production be-tween the highest and the lowest peak that is the maximum width of theoscillations registered in the production time series The second indicatormeasures the relative deviation δ = σY microY Figures 5 and 6 report theseresults

0004 0006 0008 001 0012 0014 0016 0018 002

Relative deviation

07 075

08 085

09alpha 0005

001 0015

002 0025

003 0035

004 0045

005

alphaS

0004 0006 0008 001

0012 0014 0016 0018 002

Figure 6 Relative deviation of total production for different values of α andαS

The results confirms unequivocally both hypotheses 2 and reject hy-potheses 1 In a sense these hypotheses confirm the nature of the economicsystem represented in the model as a complex system where the interactionsamong sectors play a far more important role than the individual sector be-haviour In fact in both cases we may interpret the results that the strongerthe efforts to mitigate the effects of shocks (ie slow productionrsquos and quickexpectationsrsquo adjustment) the opposite result is actually obtained the fluc-tuations become even more accentuated It is also worth to note that thestronger impact is generated by the behavioural parameters rather than thetechnical one as indicated by the stronger effects of the αS in respect of theα This result seems to suggest that ceteris paribus information mattersmore than technical constraints Though the model is by any means in-

19

adequate to discuss normative aspects because of the naive ways strategicdecision making is represented it can anyway suggest the effects of errorsin management decisions (ie wrong desired levels of stocks) as opposed tothe technical constraints preventing sudden adjustment of production lev-els Stretching a bit the interpretation of these results we may suggests thatefforts to reduce volatility should better be oriented to improve the trans-mission of information in order to coordinate future production plans (ieidentifying quickly the desired levels of stocks and therefore the new equilib-rium production level) rather than improving the flexibility of productionmethods (reducing α)

56 Volatility and stock levels

In our model stock levels are maintained to buffer a sectorrsquos sale againstunexpected changes in demand Therefore a system made of risk adversemanagers (keeping higher levels of stocks) may be thought to lead to asmoother absorption of shocks in respect of a system composed of firmskeeping a short supply to cushion unexpected events To test this hypothesiswe tested the behaviour of the model for different values of σ the levels ofstock represented as a multiple of demand For each level of σ we run asimulation using all other initialization identical as usual representing aneconomic system ldquoalmostrdquo at equilibrium but for a single small shock indemand uncorrelated across sectors18

00044

00046

00048

0005

00052

00054

00056

00058

0006

00062

5 55 6 65 7 75 8 85 9 95 10

Rel

ativ

e va

riatio

n of

tota

l out

put (

delta

)

Ratio of stocked goods over production (sigma)

Stocks and production fluctuations

Figure 7 Fluctuations persistence on systems with different stock strategies(σ)

18Demand changes by plusmn1-10

20

Figure 7 shows the value of the relative deviation of output production(δ) generated by heterogeneous uncorrelated shocks in the external demandon identical systems where firms store a different multiple of their demand asstocks We obtain actually the opposite result that we may have expectedfluctuation become larger and more persistent the higher the level of stocksIn some way a strategy that for the individual firm is meant to reducethe fluctuations (using stocks as buffers) ends up by actually increasing thevolatility at aggregate levels

The result is though explainable both at the theoretical and empiricallevel In fact if no stocks were maintained production would need to adjustfor changes in demand only in case of a shock That is the firm may movemonotonically from the old to the new production levels sending consistentsignals to the rest of the system (ie its own suppliers) Instead if thesame shock affects a firm maintaining large stocks of output over the sameperiod production needs to adjust to the new demand and and to alignstocks with the new demand This generates besides the adjustment to thedemand a potentially inconsistent signal to the suppliers since the firm willhave a temporary level of production not meant to continue in equilibrium(adjusting stocks)

From the empirical viewpoint the last few decades have seen the almostuniversal diffusion of production system meant to keep stocks to a minimumif at all Lean production methods justndashinndashtime etc are solutions tothe problems of the costs necessary to maintain stocks Over the sameperiod economic fluctuations have been strongly reduced and few empiricalassessments have pointed to the relations between the two phenomena Forexample concerning the US ldquochanges in inventory behavior have playeda direct role in reducing real output volatilityrdquo (Kahn McConnell andPerez-Quiros 2002 p 183)19

57 Irrelevance of technical coefficients distribution

Horvath (1998) and Horvath (2000) suggest that systems with sparse ma-trices ie inputndashoutput matrices with many empty cells tend to be morevolatile in respect of systems with more evenly distributed matrices Noticethat this claim risk if not adequately qualified to generate an illogical resultIn fact matrices sparseness increases as a function of sectoral disaggregationand comparing the same systemrsquos fluctuations using inputndashoutput matricesat different levels of aggregation needs to produce the same type of volatil-ity Indeed this is not the case if all remaining parameters stand equaland do not adjust to the different level of analysis considered If aggregatepatterns of adjustment speed are maintained unchanged the higher the sec-toral disaggregation the higher the systemrsquos volatility ceteris paribus But

19Similar results are found in Blanchard and Simon (2001)

21

for example in our model reducing the aggregation also requires a changein the adaptation coefficients the more micro we undergo the analysis thehigher is the speed of adjustment of economic objects If for example weconsider sectors we are evaluating the adaptation of an entire set of firmswhich pose questions on how to aggregate their dynamics Things changewhen we consider the response of a single firm

Going back to Horvatsrsquo argument highly disaggregated matrices arelikely to contain many empty cells so that we may forecast different levelsof volatility depending on the aggregation used for the same system whichas mentioned is not caused by sparseness but to a failed adjustment inreaction mechanisms Results from our model20 show that volatility is com-pletely unaffected by input coefficients distribution Given their sectoralsum mdash through rows of the matrix mdash the value of each coefficient is irrele-vant The same occurs in the case of very skewed distributions that presenta high number of empty cells (zero input coefficients)

58 Microndash and Macrondashvolatility

As we have seen above one of the major discussion in the literature concernsthe relations between microndash and macrondashvolatility Put it simply one sideof the literature considers that uncorrelated microndashshocks cancel each otherout reducing their effects at macrondashlevel Conversely other researchersconsider that microndashshocks are multiplied at macrondashlevel generating highermacrondashvolatility than that we can observe at microndashlevel The debate isof obvious importance to determine the sources of business cycles to pre-dict their effects and determine the more effective policies to mitigate theirnegative effects

Our model is only a partial representation of realndashworld economic sys-tem and is also overly simplistic However based on these limitations italso offers very robust results allowing reliable conclusions on the (limited)elements included in the model In particular we consider our model a re-liable representation of the basic structures of the production interactionsamong sectors or firms Thus we can use the model to provide an answerto the debate concerning the contributions to volatility from the productioninteractions only Other contributions (eg from financial markets priceadjustments technological innovation etc) may re-inforce or counter theforces we analyse but they need to be considered as additional elements(and we opine also generally less relevant) not alternative to our analysisWe consider a necessity in order to apply a rigorous methodological ap-proach to be able to identify with certainty the results provided by partialstudies in order to eventually obtain by gradual extensions of the analy-sis more and more detailed representations of realistic economic systems

20Available from the authors

22

The results we obtain in this paper concerning this microndashmacro debate aremeant as a first relevant step to a more extended analysis

0

002

004

006

008

01

012

014

016

018

02

022

05 1 15 2 25 3

Var Demand

Macro (Output) and micro (Demand) volatility

delta Outputdelta Demand

Figure 8 Values of δ (relative deviation) for aggregate demand and ag-gregate output in respect of different levels of variance for sector specificuncorrelated external demand shocks

The flows of shocks affecting the firms of a system can be distinguishedalong two dimensions strength of the shock (eg amplitude of the changesin demand) and frequency of the shock (rare or frequent) Concerning thefirst aspect we initialized as usual a system to equilibrium levels (eg pro-duction quantities equal to external and intermediate demand for all sectorsconstant level of stocks equal to the desired level) We then ran the simu-lation generating the external demand for all sectors as a random variablewith constant (equilibrium) mean level and varying values of variance

Figure 8 shows unequivocally that as far as our model has propertiessimilar to real systems the hypothesis that macrondashfluctuations have largervolatility than their sectoral level shocks is fully confirmed At any levelof variance the aggregate volatility of demand as captured by the relativedeviation index is sensibly lower than the volatility shown by aggregateoutput that is the total production of the system This result is not sur-prising throughout all tests presented in the previous sections every resultsuggested that the system lsquooverndashreactsrsquo to any individual microndashlevel changeeven in the case these changes were meant to contrast disturbances There-fore when we move from considering one single shock to a flow of shockswe are not surprised to obtain that the system shows higher volatility thanthat implied by the microndashlevel disturbances

We now consider the second dimension defining a flow of shocks theirfrequency The question is whether for the same level of demand shocks

23

0 002 004 006 008 01 012 014 016

Difference of relative deviation for Output and Demand

0 5 10 15 20 25 30 35 40 45 50 55

Shock frequency 05

1

15

2

25

3

Var Demand

0 002 004 006 008 01

012 014 016

Figure 9 Differences of values of δ for aggregate output minus those foraggregate demand in respect of different levels of variance of demand and offrequency of the shocks

(ie same variance) more frequent changes trigger a higher or lower systemvolatility than rare disturbances The results shown in figure 9 suggest arather elaborated answer which partly vindicates (though in a particularsense) the arguments in favour of compensation of micro shocks The fig-ure shows the difference between aggregate output volatility and aggregatedemand volatility The independent variables are as before the volatilityof demand at microndashlevel21 and the time intervals in between two shocksAggregate demand volatility does not change for the frequency of shocksat least when computed over a long period But output volatility doesin a nonndashmonotonous way For high frequency of shocks the difference isstill positive and increasing for higher variance indicating that the ldquooverndashreactionrdquo argument is valid even in these cases but it is rather low Whilethe frequency of shocks slows down aggregate volatility of output increasesshowing that time is needed for the microndashshock to unfold their full effectsat the macrondashlevel However after a certain threshold less and less frequentshocks generate a lower volatility of the aggregate level of output

This result is quite sensible though not obvious at first sight and onceagain shows the power of such a simple model to generate and explain com-plex properties For frenetic levels of demand modifications the system hasno time to adjust but part of the job to align actual and desired productionfor constantly changing demand is performed by the varying demand itselfIn other terms the shocks cancel out through time if not through their quan-

21Beware of the scale of variance For reasons of visibility the scale is inverted

24

titative effects on the system While frequency decreases the disturbancesof the shocks not compensated by other (unlikely) shocks have the time tofilter through the production system generating higher and higher volatil-ity In a sense any system has a particular frequency of shocks that make itmaximally ldquoresonatingrdquo with the amplification of microndashshocks When therate of shocks gets even rarer the system can enjoy periods of longer andlonger quietness after each shock is absorbed and before a new one startsagain a cycle of initial local disturbance propagation of the disturbances toother sectors and settlement to the new equilibrium

6 Discussion and extensions

We presented a simple model of production composed by interacting sectorThe main (quit realistic) assumption of the model is that the productionprocesses in any sector of the economy make use of input from any othersector Moreover we assumed that sectors (represented as a single firm)adjust slowly both the desired level of production (smoothed revision of ex-pectation) and the actual level (smoothed modification of plant utilization)The model voluntarily leaves aside a large number of aspects of real-worldeconomic systems (eg financial markets price adjustments growth etc)The reason is that the model can be easily and reliably tested in order toextract its properties The possibility to transfer the (certain) model resultto a real context depend of course on an ample ceteris paribus clause How-ever we are confident that the reported results concerning the productionstructure play a relevant role in real world systems Therefore our resultsand particularly their motivations being accessible because of the relativesimplicity of the model will not be diminished by the integration of ourconsiderations with other aspects of economic systems

Our model is in essence a representation of an input-output table inter-preted in physical terms that is as the amount of inputs required for oneunit of output Besides this we include two straightforward assumptionsFirstly production levels cannot be suddenly changed but require time toscale up or down the utilization rate that is the levels of production Stocksare meant to make up for the difference between sales and production Thesecond assumption concerns the intentional behaviour of producers We im-plement this aspect by assuming that exists a level of stocks desired for eachlevel of demand However a change of demand does not translate suddenlyin a new level of desired stock but again only a long time at a different levelof demand will convince managers to update the desired levels of stocks

We show that the model generate under a wide number of settings oscil-latory patterns converging toward the equilibrium level where each sectorgenerate exactly the quantity demanded by final consumers and by othersectors as input The oscillations are provoked by over- and under-shooting

25

generated by the difficulty of coordination among sectors that by assump-tion ldquocommunicaterdquo only via variation of demand to their direct input sup-pliers We also show that the total production levels generated by sectors(sum of the physical units) produced at equilibrium depends (for constantfinal demand) on the total sum of the technical coefficients per sector irre-spective of their distribution This result is motivated by the impossibilityof input substitution and offers interesting suggestions for the analysis ofthe study of input-output tables of real-world systems

We then consider systems at equilibrium facing a single small change ofthe external demand for sectors These simulations allow to study the waythe system reacts to shocks We see that the sum of the coefficients affectsbesides the levels also the volatility of aggregate output the larger a systemthe stronger is the impact of a given shock Less obviously we find thatfor systems with the same dimension (ie same sums of coefficients) theaggregate volatility increases with both the inertia to change of productionlevel and the speed of adjustment of desired stocks This result is somewhatcounter-intuitive since both the cited variables are associated to a moreconservative behaviour that may be supposed should smooth away spikesin the pattern from an old to a new equilibrium However we show thatthe opposite actually holds given the complex interactions among sectorindividual-sector attitude intended to smooth shocks actually increase theiraggregate impact

In the same vein we show that systems maintaining high levels of stocksare more volatile that systems with reduced amounts of stocks Again thisis a counter-intuitive result reverting the goal of individual producers andthe aggregate result Interestingly this result finds strong support from theevidence of the recent changes in production methods (ie low or no stocks)and diminished cycles

In relation to specific issues discussed in the literature we find no supportto the suggestion that particular distributions of input-output coefficientsmodify the volatility of a system As said this depends in case on thesums of these coefficients but not on their distribution Further we canprovide an answer to a long and hotly debated questions whether microndashlevels volatility generates stronger or lower aggregate volatility Our modelshows unequivocally that microndashvolatility (eg variance of final demand)persists and generates stronger volatility at aggregate level Again we canidentify the reason on the complex production structure that exalts micro-level shocks by generating lsquowrongrsquo messages among producers Moreoverwe can also point to a usually neglected aspect of volatility we show thatthe frequency of shocks is a very relevant aspect besides their dimensionin determining the volatility of aggregate output

Our work can be developed along two complementary directions Firstlywe can continue to study the properties of the model by extending the ele-ments considered (eg pricing) analysing the effects on the present results

26

and generating more analytical evidence Of particular relevance is in ouropinion the problem of aggregation Representations of the same systemswith different codification systems (eg different number of SIC digits)must not modify overall system properties for example the volatility ob-served Given that our result show that the sums of coefficients matter forthe levels and volatility of the system and that these sum do depend on theaggregation chosen it should be possible to induce the reaction coefficientsthat we know affect the volatility only

Secondly we can find an application of our results to real world evidenceIn fact there are many data sets available for the data we deal with basicallyinput-output tables Our model is readily adapt to be parametrized alongany number of sector and coefficient necessary so that we may replicate pastseries in order to adapt the unobservable parameters and use the resultingcomplete model to explain business fluctuations and provide intuitions onfuture patterns

27

References

Acemoglu D and A Scott (1997) ldquoAsymmetric Business CyclesTheory and TimendashSeries Evidencerdquo Journal of Monetary Economics 40501ndash533

Backus D P Kehoe and F Kydland (1995) ldquoInternational BusinessCycles Theory and Evidencerdquo in Frontiers of Business Cycle Researched by T Cooley Princeton University Press Princeton

Bak P K Chen J Scheinkman and M Woodford (1993) ldquoAg-gregate Fluctuations from Independent Sectoral Shocks SelfndashOrganizedCritically in a Model of Production and Inventory Dynamicsrdquo RicercheEconomiche 47(1) 3ndash30

Battiston S D Delli Gatti M Gallegati B Greenwald and

J Stiglitz (2005) ldquoCredit Chains and Bankruptcies Avalanches in Sup-ply Networksrdquo in Annual Workshop on Economic Heterogeneous Inter-acting Agents (WEHIA 2005) University of Essex UK

Blanchard O and J Simon (2001) ldquoThe Long and Large Declinein US Output Volatilityrdquo Brookings Papers on Economic Activity 1135ndash174

Bresnahan T F and V A Ramey (1994) ldquoOutput Fluctuations atthe Plant Levelrdquo The Quarterly Journal of Economics 109(3) 593ndash924

David P A (1975) Technical choice innovation and economic growth essays on American and British experience in the nineteenth centuryCambridge University Press London

Delli Gatti D C Di Guilmi E Gaffeo G Giulioni M Galle-

gati and A Palestrini (2004) ldquoA New Approach to Business Fluc-tuations Heterogeneous Interacting Agents Scaling Laws and FinancialFragilityrdquo Journal of Economic Behavior amp Organization 56 489ndash512

Dosi G G Fagiolo and A Roventini (2006) ldquoAn EvolutionaryModel of Endogenous Business Cyclesrdquo Computational Economics 273ndash34

Dupor B (1999) ldquoAggregation and the Irrelevance in MultindashSector Mod-elsrdquo Journal of Monetary Economics 43 391ndash409

Head A C (1995) ldquoCountry Size Aggregate Fluctuations and Interna-tional Risk Sharingrdquo The Canadian Journal of Economics 28(4b) 1096ndash1119

28

Helbing D and S Lammer (2005) ldquoSupply and Production NetworksFrom the Bullwhip Effect to Business Cyclesrdquo in Networks of Interactingmachines Production Organization in Complex Industrial Systems andBiological Cells ed by D Armbruster K Kaneko and A S Mikhailovvol 3 of World Scientific Lecture Notes in Complex Systems World Sci-entific

Helbing D S Lammer U Witt and T Brenner (2004) ldquoNetworkndashInduced Oscillatory Behavior in Material Flow Networks and IrregularBusiness Cyclesrdquo Phisical Review E 70(056118) 1ndash6

Helbing D U Witt S Lammer and T Brenner (2004) ldquoNetwork-Induced Oscillatory Behaviour of Macroeconomic Output Flowsrdquo in In-dustry and Labour Dynamics II The Agent-based Computational Eco-nomics Approach Proceedings of the wildace 2004 conference ed byR Leombruni and M Richiardi no 5 LABORatorio R Revelli

Horvath M (1998) ldquoCyclically and Sectoral Linkages Aggregate Fluctu-ations from Independent Sectoral Shocksrdquo Review of Economic Dynam-ics 1 781ndash808

(2000) ldquoSectoral Shocks and Aggregate Fluctuationsrdquo Journal ofMonetary Economics 45(1) 69ndash106

Jovanovic B (1987) ldquoMicro Shocks and Aggregate Riskrdquo The QuarterlyJournal of Economics 102(2) 395ndash410

Kahn J A M M McConnell and G Perez-Quiros (2002) ldquoOnthe Causes of the Increased Stability of the US Economyrdquo FRBNY Eco-nomic Policy Review 8(1) 183ndash202

Kraay A and J Ventura (1998) ldquoComparative Advantage and theCrossndashSection of Business Cyclesrdquo World Bank Woorking paper Series1948 World Bank

Long J B and C I Plosser (1983) ldquoReal Business Cyclesrdquo Journalof Political Economy 31(1) 39ndash69

Lucas R E (1977) ldquoUnderstanding Business Cyclesrdquo CarnegiendashRochester Conference Series on Public Policy (special issue of Journalof Monetary Economics) 5 7ndash29

Nelson R R and S G Winter (1982) An Evolutionary Theory ofEconomic Change Harvard University Press Cambridge MA

Nirei M (2005) ldquoThreshold Behavior and Aggregate Fluctuationsrdquo Jour-nal of Economic Theory in press

29

Scheinkman J and M Woodford (1994) ldquoSelfndashOrganized Criticallyand Economic Fluctuationsrdquo American Economic Review 84(2 Papersand Proceedings of teh AEA) 417ndash421

Schuh S (1996) ldquoEvidence of the Link Between FirmndashLevel and Ag-gregate Inventory Behaviorrdquo Finance and Economics Discussion Series1996-46 Federal Reserve Board

Stiglitz J and B Greenwald (2003) Towards a New Paradigm inMonetary Economics Raffaele Mattioli Lectures Cambridge UniversityPress Cambridge

Weisbuch G and S Battiston (2005) ldquoProduction Networks andFailure Avalanchesrdquo Working Paper mimeo Ecole Normale SupErieure

Zarnovitz V (1977) ldquoBusiness Cycles Observed and Assessed Why andHow they Matterrdquo NBER Working Paper Series 6230 National Bureauof Economic Research

30

A Tables

Table 1 Parameters valuesParametera Description Value

N Number of sectorsunits i of production 10Ci Number of j buyer sectors for sector i (IO matrix sparseness) N minus 1Di (E) Value of the external demand 10mij Physical input coefficient of input j for sector i sim U(0 1)Mi Sum of the mij input coefficients j for the production of i 05αi friction to changes of production levels Degree of lockndashin on

production technology085

αsi Speed of adaptation to demand changes 004

σ Desired multiple of production to stock 10

aInto parenthesis the number of lags of the initial value for lagged variables

31

Plain IndashO structure

The pioneering RBC model that analyses the diffusion of sectoral shocks isdue to Long and Plosser (1983) They use a general equilibrium multi sectormodel with fully rational infinitely lived perfectly informed homogeneousetc individuals In such a system by assuming a perfectly maximising be-haviour (via allocation of production and consumption of resources) andthat individuals prefer to smooth savings across time and goods it is pos-sible to generate persistence of continuous albeit uncorrelated shocks inproduction capacity across sectors In the words of Long and Plosser [p67] ldquoAt constant relative prices this [the assumed maximising behavior]suggests that business-cycle features like persistence and comovement arecharacteristics of desired consumption plansrdquo

The necessity to maintain the assumption of perfectly rational agentsforces these strand of models to attribute the origin of cycles to externalfactors only like exogenous shocks on technology As such they seem un-likely to provide a consistent explanation (or representation) of aggregatefluctuations Dupor (1999) shows that under quite general conditions theavailable multindashsector RBC models produce the same results in terms ofvariance convergence as their respective single sector models This is shownalso under particular conditions of input matrix or high coefficient of spe-cific inputs Dupor concludes that [p 405] ldquoresearchers who wish to useindependent sector shocks as a source of aggregate fluctuations must appealto sector shocks that are large relative to their single sector counterparts orelse consider a different set of models than those discussed in this paperrdquoIn doing so the author though suggests that another mechanism must beintroduced somehow putting the desired results ahead of the economic issueanalysed

The criticism to this kind of literature suggests that in order to allowthe IndashO structure to play a role in the explanation of cycles we need agentswith less then perfect foresight and capacity of adaptation to unndashexpectedconditions As we will see below agents that need time to both realizethe economic changes of their time and to introduce the necessary mod-ifications are sufficient conditions to observe aggregate fluctuations froma single change in one part of the economy Though being ldquorationalrdquo inthe sense that they make the right decisions they are limited by lags inthe information flow and physical constraints to the changes that they canapply

Input matrix incompleteness

Using a similar approach to RBC models Horvath (1998) and Horvath(2000) focus the attention on the properties stemming from peculiar produc-tion structures as represented by characteristics of the input-output matrix

5

The author shows that sectoral shocks will tend to cancel out in case theeconomic system is sufficiently distributed that is when the IndashO matrixcontains evenly distributed values over all the cells Conversely sectoralshocks are not absorbed and can be reinforced in case many cells of thematrix are empty This case indicate that there will be few sectors affect-ing many other sectors preventing the possibility of compensating certainshocks

These conclusions are potentially interesting because point directly toan easy to observe aspect of economic system for example providing thepossibility to test the prediction of the analysis It is worth to note that thisapproach draws conclusions that are not neutral to the aggregation levelused In fact a highly disaggregated matrix is more likely to contain manyempty cells than the matrix for the same system obtained aggregatingsectors

Dupor (1999) also considers the case of inputndashoutput matrices with dif-ferent input coefficients finding them irrelevant for the results But he doesnot include matrices with empty cells

The cited contributions and those inspired to the same approach sug-gest that the type of interaction as expressed by input-output coefficientscan be a relevant factor in determining the existence and dimension of fluc-tuations However this literature is not able yet to draw conclusive results

Limited interactions and production constraints

A different kind of mechanisms are assumed and shown to play a role in theAv models they add an analysis on firms production (technological) possi-bilities to the production structure They are inspired by and similar to Jo-vanovic (1987) model in which the limitation in the number of interactionsupon which each playerrsquos decision depends generates shock cascades andaggregate persistence Bak Chen Scheinkman and Woodford (1993) andScheinkman and Woodford (1994) show that when the interaction structurebetween sectors is constrained to a completely rigid system coupled withnon convexities in the production function independent shocks to differentsectors do not cancel in the aggregate In other words the authors place aset of restrictions to the set of possible actions of sectors First structuringthe production networks with a fixed invariable lattice produces a rigiditythat do not allow producers to change (or add) inputs (or suppliers) orincrease the produced quantity This is combined with the structure of theavalanche that propagates through the unchanged network Second a pro-duction function that is maximised only when a discrete number of unitsof a good is produced coupled with a fixed (discrete) maximum amount ofinventories that can be stored by the representative firm Such a structuredetermines production oscillation and shocks avalanches due to the factthat firms are constrained in their production opportunities

6

Using a very similar production structure with limited interactionsWeisbuch and Battiston (2005) play on the bankruptcies generated on down-stream firms by production failures occurring in upstream firms (reducingthe flow of inputs) Stochastically generated failures to produce in a firmcauses production constraints in its downstream clients hence lower invest-ment and eventually bankruptcies3 A lag between the period in whicha firm goes bankrupt and a new firm appears generates a lack of supplywhich propagates on the downstream sectors

Due to the original literature on avalanches these contributions sharetwo common features that are ill-adapted to economic systems First theyassume very peculiar interaction structures of one-directional productionrelations between sectors which amount to assume one half of the input-output matrix empty and are not applicable to the general case with cyclicalproduction structure or full IndashO matrices Second the strongly rely on quiterigid thresholds determining a mechanicist behaviour by agents Basicallyagents have only two options available (ie produce or not produce) andthe choice is made on the basis of crude decisional mechanism4

Notwithstanding the limitations this approach provides a quite sensiblemechanism of the origin and transmission of shocks through a system Ourproposal can be interpreted as an extension of this approach by relaxing thestrong assumptions on the interaction structure and allowing agents for amore fluid decisional process as we will see in the next section

3 The basic elements of a production structure

Our overall hypothesis is that the very production structure of an economymay generate persistent fluctuations even in the absence of continuous un-related shocks and rigid thresholds that determine non linearities The aimis to provide a generalised interpretative framework to explain the occur-rence of short term fluctuations in the economy simply as an outcome ofthe division of labour in a number of sectors related through trade In itsextreme interpretation any production system which is not fully integratedis bound to produce business cycles As we will show a number of factorsinfluence their extent

Quantitative adjustments in production in any one sector require ad-justments in related sectors When the structure of the economy is trulyInputndashOutput (not a directed linear supply chain as in the Av models) in-put flows are cyclical and adjustment to long run production equilibrium islikely to take a long mdash when not infinite mdash time Our work departs from

3A similar model by Battiston Delli Gatti Gallegati Greenwald and Stiglitz (2005)relates bankruptcies to price changes buyers payment timing and the credit market (acentral bank)

4Nirei (2005) is an attempt to generalise such models though maintaining the assump-tion of threshold behavior

7

both RBC and Av models with respect to structure modelling criteriaanalysis and methodological approach Indeed our model may be easilyreconduced (or restricted) to both We are much in line with a recent se-ries of papers by Helbing Witt Lammer and Brenner (2004) that addressthe same question although with a number of differences that are discussedlater on5

We aim to analyse to what extent the structure of inputndashoutput relationscauses aggregate fluctuations by itself The main idea is to show that ag-gregate fluctuations may derive from simple micro imbalances without theneed for continuous exogenous shocks not even at the firm or sectoral levelIn other words normal production activities including lags technologicaladjustments information asymmetries circularity of the inputndashoutput sys-tem and so on may be enough to explain part of the macro volatility in aclosed system6

Our assumptions draw on the existing literature but try to avoid themajor difficulties highlighted above Firstly we avoid to impose either mar-ket clearance assumptions or crude threshold behaviours Rather we adoptan assumption close to a Simonian approach agents in an economy do theirbest within the informational and physical constraint they are subject toWe represent decision makers as responding to a single piece of informationthey obtain the quantities demanded by their clients Crucially their ac-tions are inspired to a conservative approach an increase in the demandgenerates a smaller (desired see below) sudden increment of productionThis assumption stems from the fact that decision makers are aware of theinstability of their environment and therefore want to avoid getting perma-nently caught with over production following a temporary spike of demandIn other words firms attempt to smooth business cycle generating a morestable (or less uncertain) environment as the economic literature would sug-gest Notice that this representation in line with a routinized representationof organizations (Nelson and Winter 1982) does not contradict market equi-librium Rather in a stable environment such decisional procedure generatesan asymptotic pattern to the equilibrium level in the absence of feedndashbacks

A second assumption concerns the technological possibilities available tothe firms When firms realise the need for a change in production levels saya desired increase of production they face a complex and costly organisa-tional transformation of the production capacity risky investments an soon As Bresnahan and Ramey (1994 p 622) suggest inducing from a de-tailed analysis of the automobile industry in the US ldquoadjusting production

5We thank Matteo Richiardi for pointing out these similar attempts See also Helbingand Lammer (2005) and Helbing Lammer Witt and Brenner (2004)

6An open system entails many other factors including price factors of both inputsand outputs mdash terms of trade labour cost external investment etc As with exogenousshocks our analysis does not deny the importance of these factors but simply separatethem from the structural explanation of cycles

8

is a more complicated process than simply ldquochanging Qrdquo or choosing themix of capital and laborrdquo And this is not only a matter of non convexitiesin production technologies The inconsistency of time reversibility in thetechnological choice is accompanied by the need to operate on the lsquoavail-able technological frontierrsquo (David 1975) If a firm plans a given productionactivity for a short period of one week even though the next day there is anunexpected fall in the demand it is unlikely that it is followed by a similardownturn in production The case is evident when we think at productionplanning that involve hiring of workers an capital investments (both in-volving sunk costs)7 Therefore any desired change generates a pattern ofsmall modifications approaching slowly the target level Moreover thesechanges are irreversible in the short term since only a prolonged change ofthe environment triggers the modification of previous desired levels

Resting on those simple conjectures we thus analyse a model in whichwe only allow for sluggish adaptation of production due to conservative(adaptive) behaviour of firms and the physical constraints faced when vary-ing the production levels Notice that these assumptions are contrary tothe generation of instability of a system In fact a firm behaving as de-scribed above reduces if not eliminates any disturbance affecting its ownstate if acting in isolation Other factors like new technologies extremelylong term perspective price changes etc are likely to be sources of volatil-ity therefore contributing to generate aggregate system fluctuations (as forexample in Acemoglu and Scott 1997) Our neglect of these and other fac-tors however unrealistic serves the purpose of testing whether a volatilityreducing microndashbehaviour turns into volatility generating aggregate resulttwo widely observed dynamics Quite interestingly (and encouraging) DosiFagiolo and Roventini (2006) have concomitantly proposed a model whichincorporates detailed representation of both sluggishness robustly based onempirical evidence Heterogeneous firms which undergo lumpy investmentand slowly adapt their expectation on changes of the future demand inducecomovement of macroeconomic variables qualitatively similar to the onesobserved in the real world To pursue our line of research we first needto concentrate on how sectorsfirms interact in the system in the effort tocomplement those promising results

Therefore the second element we consider is meant to represent a gen-eralized economic system We allow for the circularity of inputndashoutput sys-tems a crucial feature in determining feedback mechanisms and a principal

7See also the discussion in Acemoglu and Scott (1997) where the authors argue thatldquo[w]hile the presence of fixed costs can account for the discreteness of economic turningpoints it does not naturally lead to persistence because once an individual undertakes anaction they are less likely to do so in the near future [] [A]lthough the presence of fixedcosts leads to increasing returns these are intratemporal the full extent of economiesof scale arising from fixed costs can be exploited within a periodrdquo [p 502 authorsrsquoemphasis]

9

component of business cycle It may be true that most backward linkagesconcern capital goods (Weisbuch and Battiston 2005) but a rapid glanceto an inputndashoutput matrix would show that technical coefficients are nonnull in both directions and ldquomost commodities are inputs to the produc-tion process of other commoditiesrdquo (Horvath 2000 p 70) The higher thesectoral aggregation mdash observed or assumed mdash the more the relevance ofintermediate supply holds true (as mentioned above with respect to ma-trix incompleteness) We claim that the existence of backward and forwardlinkages are a crucial element to determine the persistence of shocks in theeconomy A perfectly linear (one directional) system needs much less adap-tation of firms and sectors in order to stabilise the economy Nonethelesscircularity does not imply that inputs are perfectly substitutable amongthem in order to determine the maximising allocation even at high level ofdisaggregation

By using a fixed coefficients production function we both maintain themid term rigidity of production processes and we do not impose limitedinteractions We claim that the circular property of the production structureis by itself sufficient to produce shock persistence even when the inputmatrix is full (although with heterogeneous input coefficients) mdash all sectorsshop in the remaining n minus 1 sectors

In conclusion we represent a model for an economic system made onlyof i) production level decisions and ii) input-output relations Clearly sucha model lacks many features present in real economic systems likely we areconvinced to affect the propensity to generate cycles As such we cannotaim at reproducing realistic data so our results cannot be tested comparingtime series generated by the model with real ones Instead given the paucityof the elements comprising the model we can investigate in detail to whichextent the (few) features present in our model affect the cyclical behaviour ofthe system As we will see we will be able to show quite a number of resultin which aggregate fluctuations are generated whereas the individual microstructure of the model aims at monotonous adaptation Before discussingthe results next section describes in detail the implementation of the simplemodel

4 The model

We model an economic system in which a number of sectors i = (1 N)are potentially connected via (intermediate) market relations A N times NInputndashOutput matrix determines the mij units of each input j = (1 N)mdash bought from the N minus 1 residual sectors mdash necessary to produce oneunit of output i8 For simplicity given the focus of this paper on the IndashOstructure as a generator of aggregate fluctuation we abstract from market

8IndashO coefficients are therefore physical coefficients

10

interactions within sectors each sector is a production unit In fact forour purposes it is not relevant which firm produces the sectoral outcomebut only the total amount produced by each sector9 Moreover besidesmaking the analysis more neat this assumption allows also to enjoy a largefreedom in determining the aggregation level since we donrsquot need to assessthe effects of between sectors competitiveness which for example is likelyto increase for increasing levels of disaggregation Our economic systemis therefore composed of N production units each of which may buy andprovide inputs from and to any other sector depending on the IndashO structureof the economy

The demand for each sector i is determined by a constant consumersdemand Di (E) and the input needs of other sectors

Di = Di (E) +sum

j 6=iisinCi

mjiQj (1)

where Ci is the set of j intermediate clients of i (sum

j isin C lt N) mji is thetechnical coefficient of firm j for input i ie the amount of i required toproduce one unit of j Qj is the actual output produced by sector j

Concerning the simultaneous production level of each sector in time twe make two lsquorealisticrsquo assumptions First from the technical viewpoint afirm like a tanker cannot abruptly change its course Therefore changesto the production level can be introduced only gradually through capital(dis)investment and job h(f)iring (see discussion in Section 3) Second weassume that firms do not decide the production level directly on the basisof the observed demand level but conscious of the uncertainty of demandplan their production in order to maintain invariant a desired level of stocksand avoid stockout10

Quantity produced by sector i in period t is computed as follows

Qit = αiQtminus1 + (1 minus αi)(

Slowastitminus1 minus Sitminus1

)

(2)

where αi measures the physical constraint in adapting the production capac-ity to changes in the required output Slowast

itminus1are firmrsquos i desired stocks and

Sitminus1 its actual stocks Notice that both variables related to stocks appearwith a lag This is due to two reasons First it is a modelling necessity inorder to allow the simultaneous determination of production levels for allsectors Secondly as frequently happens modelling necessities reveal sen-sible if not generally considered aspects of the modelled system In factproduction decisions concern management in the production plant while

9Adding heterogeneous firms within sectors would definitely enrich the causal expla-nation put forward and parametrised in this paper (see below) For example to analysethe effect of firms constraints It is left over for future research as in the present work weopt for the analysis of the main structural parameters that affect aggregate fluctuations

10See also discussion in Schuh (1996)

11

demand through sales is observed by the commercial staff It is thereforelogical that the information from demand reaches plant managers with somelag

The actual stock level is trivially computed subtracting from the previousheap of stock the amount sold and adding the new production

Sit = Sitminus1 + Qit minus Dit (3)

Concerning the desired level of stocks instead we introduce the conser-vative behaviour mentioned above Firms ideally would keep an amountof stocks proportional to the demand they receive say a multiple σ ofthat amount However when demand varies they adjust the desired levelsmoothly preventing sudden changes to the variable representing the goalof the firm

Slowastit = Slowast

itminus1 + αsi

(

σDit minus Slowastitminus1

)

(4)

where αsi measures the adaptation to demand changes

In summary the dynamics of the model consists of only the equationsabove Firstly at the beginning of a time step firms determine the quantityto produce as a function of past imbalances in stock levels and the pastproduction level These levels define the demand for all sectors determinedby the technical coefficients applied to all produced quantities Finallyactual and desired stocks can be updated Figure 1 represents visually thisdynamics highlighting the decisional components of the model in respect ofthe ldquomechanicsrdquo of trade

5 Results production structure and economic fluc-

tuations

In this section we describe some of the most relevant results concerningthe effects of the production structure to determine aggregate fluctuationsGiven the unusual analytical instrument chosen (at least concerning thistopic) we start with a brief methodological introduction meant to clarifythe nature of the results we claim to obtain Next we describe a generalsetting for the model which is the (extremely) limited area of the potentialparametersrsquo space of the model that we will explore In the rest of thesection we discuss the results from the model

51 Methodological considerations

Real economic systems are constantly subject to a continuous flow of shocksexternal to the production system besides changes within the system itselfIn order to appreciate the systemic properties we need to abstract from all

12

Compute the quantityto produce

F (Q(t-1) S(t-1) s(t-1) alpha)

Communicate thequantity of inputsrequired to each

supplier

Determine the overalldemand

F(D(E) D(I))

Revise the past decisionon stock plans

(desired stocks)F(D S(t-1) sigma alpha(s))

Compute the final levelof stocks for next period

F(s(t-1) Q D)

KEY

ModelDirection

Variablecomputed

at t-1(t-1)

Step includingfirm behavior

Figure 1 One period dynamics

the nonndashrelevant disturbances and concentrate on how the system endoge-nously contributes to the observed dynamics Therefore we will generatehighly abstract patterns whose analysis will provide insights on the systemproperties that would be otherwise lost in the noise generated by all theelements of a real system

We are not trying to fit the model to a specific empirical data set nor weare interested in studying the complete behaviour of the model for areas ofthe parameterrsquos space that make no economic sense (eg negative or infiniteproduction levels) Therefore the values of initial settings and results matteronly in relative and not in absolute terms

We are going to describe a simulation model with which we will produceseveral simulation runs and we claim obtain results relevant for the debatediscussed above Rigorously speaking we may claim only certainty for thevalidation of the results that is that the model does actually generate theresults presented and for the reasons we explain We limit this part to thepresentation of graphs and verbal explanations that we believe are ratheruncontroversial11 Concerning the relevance of our studies for real systems

11Interested readers can request the model code and and simulation data for replicationand extensions of the results We can guarantee that such possibility does not require

13

(verification) we simply avoid to pretend that our model is a quantitativeapproximation of real economies (which in which period) Our opinionis that the study of the model allows us to make general considerationsnon-obvious and relevant for applied debates supported by strong logicalarguments which can be integrated but not reverted by arguments con-cerning more and more elements of the economic world

52 Model setup

We go through the model properties analysing an economic system madeup of ten sectors (N = 10)12 Each sector shops inputs from the remain-ing N minus 1 and sells to them part of its own production as intermediategoods This means that the input matrix is complete except for the diag-onal (mij gt 0 forallj 6= i mii = 0) The sum of all N minus 1 coefficients for asingle sector is given by Mi and its benchmark value is kept at a plausiblemedium level (unless differently specified) to avoid distortions from highlydemanding sectors Each input coefficient is randomly drawn from a uniformdistribution

Sectors start at an equilibrium level their supply matches intermediateand final demand and in each period the same quantity of output mustbe produced to compensate for the used stocks and keep the desired stocksunchanged In thier turn stocks must be ten times the value of total sectoraldemand (σ) Eventually technological adjustment of production is ratherslow and producers are quiet refractory to follow sudden changes in thedemand in general a smoothing behaviour strongly prevails (α and αs)

Under those general preconditions in the following sections we anal-yse the dynamics of the economic system in relation to model parametersWe mainly focus on the cyclical behaviour of systems and their speed ofconvergence to the asymptotic equilibrium13

The very abstract assumption and the homogeneous initialisation donot allow for the multiple equilibria that are likely to rise in complex systemswith feedbacks Once more such simplification allows to have a betterunderstanding of the crucial impacts of the production structure on businessfluctuations Departing from the related literature discussed above14 weconsider the effects of a single shock rather then a continuous flow whichin our model is superfluous in order to obtain fluctuation persistence15 To

extensive programming or statistical skills besides the usual education of economists12You can think of a high level of aggregation in the statistical observation The com-

plete initialisation and parameters values are available in table 1 in the Appendix13Notice that sectors only approach a final equilibrium level of production which can

be prendashdetermined under static conditions14Included the more proximate work of Helbing Witt Lammer and Brenner (2004)15An exception are the last results presented in Section 58 which we see more as an

opportunity for further research than a closing of the paper

14

start with next section (53) shows the aggregate cyclical behaviour of astandard setting

53 Single shock analysis

We generate a simulated history following the dynamics of the modelrsquos vari-ables when the system is set to an equilibrium level and a single shockmodifies one of the exogenous elements namely the external demand forsectors Modifications across sectors are random and not correlated andmay increase or reduce external demand by at most 10

100 225 350 475 600

193894

195861

197828

199795

201762 Industrial Production

Time

Figure 2 Total industrial production fluctuations following a shock fromexternal demand

Figure 2 shows the dynamics generated by the total industrial productionof the system mdash that is also a proxy for the systemrsquos GDP fluctuations Thenew external demand changes the equilibrium level but the system cannotreach the new equilibrium with a monotonous pattern and converges to thenew level with a fluctuating pattern Each change in the external demandinduces also modifications of the intermediate demand and sectoral adjust-ments in the quantity produced typically overshoot the aggregate patternAs it can be appreciated from the figure it takes time before the system isable to converge to the new static equilibrium conditions

We study two features of this pattern absolute level of the new equilib-rium and volatility induced by the shock

54 Technical coefficients and industrial output

Production level

The equilibrium levels of industrial output depend on two factors only Ob-viously the levels of external demand determine how much of the total pro-

15

duction needs to be supplied to consumers exogenous from the productionsystem Second the technical coefficients also influence strongly the level ofactivity of the system defining the intermediate demand to each sector

180

200

220

240

260

280

300

01 02 03 04 05 06 07 08 09

Out

put l

evel

(In

dust

rial p

rodu

ctio

n)

Sum of total input coefficients (M)

Total industrial production for different input coefficients levels

Figure 3 The relevance of input coefficients on output level

Figure 3 shows the equilibrium level of industrial output reached bysystems with identical external demand where we fixed the sum of inputcoefficients for each sector of the economy (Mi) to different values16 Thehigher the sum the higher is the total production sustained by the system

This result depends on the very production structure represented by theinputndashoutput matrix In fact the higher is the level of production requiredby intermediate sectors the higher is the total production given a constantlevel of external demand

Notice that the level of production does not depend on the distribution ofthe technical coefficients As long as the sum of the coefficients is identicalthe same production level is obtained irrespective of their distribution (otherthings being equal)

Output shock absorption

Input coefficients have a relevant role also on the output stability of thesystem Starting from an output equilibrium level we hit the system witha single shock on the external demand for all sectors uncorrelated amongsectors17 modifing (positively or negatively) the demand for all sectors by

16Within this limit the actual value of the coefficients are randomly drawn17In each sector the external demand changes by plusmnX where X is a uniformly dis-

tributed percentage between 1 and 10

16

at most10 As show in Figure 2 the economic systems produces a lenghtlysmoothing oscillatory behaviour We compute the relative deviation of theaggregate production (industrial output) as the ratio between its standarddeviation and its mean δ = σY microY where Y =

sum

Qi Figure 4 shows thebehaviour of δ as a function of the sum of input coefficients for each sector80 periods after the shock As time goes by the level of δ reduces for allMi but the relation with the sum of input coefficient is unchanged

00055

0006

00065

0007

00075

0008

00085

0009

01 02 03 04 05 06 07 08 09

Rel

ativ

e va

rianc

e of

tota

l out

put (

delta

)

Sum of total input coefficients

Aggregate output deviations for different technical coefficients

Figure 4 The relevance of input coefficients on output stability

In sum input coefficients determine the extent of overshooting behaviourThe higher the coefficients the higher is the backlash even when shockscould cancel out through sectors

55 Shock persistence and adjustment coefficients

We now consider the effect of micro smoothing behaviours on the volatilityof the production system In particular we are interested in testing theeffects of two parameters of the model on the volatility of the productionsystem First we consider the effects of the αrsquos representing the ldquostickinessrdquoof the production technology to desired changes of Q aimed at filling thegap between actual and desired stocks The higher is α the slower is thereaction to stock unbalances Concerning its effect referring to the technicalpossibilities available ie production capital and organization we may putforward two opposite hypothesis relevant to systemrsquos volatility

Hypothesis 1 Slower production adjustment (high α) reduces volatilitybecause unbalances in demand of one sector being compensated slowly slow

17

down the diffusion of shocks to other sectors

Hypothesis 2 Slower production adjustment (high α) increase volatil-ity because the longer lasts the mismatch between actual and equilibriumproduction level the deeper will become the mismatch between actual anddesired stocks

20 25 30 35 40 45 50 55 60 65

Cyclesrsquo width

07 075

08 085

09alpha 0005

001 0015

002 0025

003 0035

004 0045

005

alphaS

20 25 30 35 40 45 50 55 60 65

Figure 5 Maximum width recorded between highest and lowest peaks fortotal production for different values of α and αS

Secondly we consider the parameters αS representing the speed of ad-justment of desired stock to observed demand The higher this parame-ter the faster sectors revise the desired levels of stocks to current levelsof observed demand This parameter concerns sectorsrsquo behavioral decisionprocess Higher αS represent a stronger belief that the most recently ob-served demand level is a reliable indicator of future demand levels whilelower levels of the parameter represent a more conservative approach wheremanagers require a long observation of demand levels different from the pastones before adjusting their expectations Also concerning the effect of thisparameter on the volatility induced by an exogenous shock we may maketwo hypothesis

Hypothesis 1 Faster adaptation of desired stocks to new levels of de-mand (higher αS) reduce volatility because changes in demand are rapidlytranslated in adjusted levels of production

Hypothesis 2 Faster adaptation of desired stocks to new levels of demand(higher αS) increase volatility because temporary unbalances of demand in

18

one sector are transmitted faster to other sector reinforcing the feedbackmechanism of fluctuations

To test which of these hypotheses are confirmed by the model we runsimulation runs with identical systems (eg same technical coefficients samelevels of demand same shocks) but for the values of the αrsquos and αS rsquos testingall combinations of these parameters over a range of values (each run usedthe same parameters for all the sectors of the system) For each simulation(ie each couple of values of the parameters) we computed two indicatorsof volatility One measures the maximum distance of total production be-tween the highest and the lowest peak that is the maximum width of theoscillations registered in the production time series The second indicatormeasures the relative deviation δ = σY microY Figures 5 and 6 report theseresults

0004 0006 0008 001 0012 0014 0016 0018 002

Relative deviation

07 075

08 085

09alpha 0005

001 0015

002 0025

003 0035

004 0045

005

alphaS

0004 0006 0008 001

0012 0014 0016 0018 002

Figure 6 Relative deviation of total production for different values of α andαS

The results confirms unequivocally both hypotheses 2 and reject hy-potheses 1 In a sense these hypotheses confirm the nature of the economicsystem represented in the model as a complex system where the interactionsamong sectors play a far more important role than the individual sector be-haviour In fact in both cases we may interpret the results that the strongerthe efforts to mitigate the effects of shocks (ie slow productionrsquos and quickexpectationsrsquo adjustment) the opposite result is actually obtained the fluc-tuations become even more accentuated It is also worth to note that thestronger impact is generated by the behavioural parameters rather than thetechnical one as indicated by the stronger effects of the αS in respect of theα This result seems to suggest that ceteris paribus information mattersmore than technical constraints Though the model is by any means in-

19

adequate to discuss normative aspects because of the naive ways strategicdecision making is represented it can anyway suggest the effects of errorsin management decisions (ie wrong desired levels of stocks) as opposed tothe technical constraints preventing sudden adjustment of production lev-els Stretching a bit the interpretation of these results we may suggests thatefforts to reduce volatility should better be oriented to improve the trans-mission of information in order to coordinate future production plans (ieidentifying quickly the desired levels of stocks and therefore the new equilib-rium production level) rather than improving the flexibility of productionmethods (reducing α)

56 Volatility and stock levels

In our model stock levels are maintained to buffer a sectorrsquos sale againstunexpected changes in demand Therefore a system made of risk adversemanagers (keeping higher levels of stocks) may be thought to lead to asmoother absorption of shocks in respect of a system composed of firmskeeping a short supply to cushion unexpected events To test this hypothesiswe tested the behaviour of the model for different values of σ the levels ofstock represented as a multiple of demand For each level of σ we run asimulation using all other initialization identical as usual representing aneconomic system ldquoalmostrdquo at equilibrium but for a single small shock indemand uncorrelated across sectors18

00044

00046

00048

0005

00052

00054

00056

00058

0006

00062

5 55 6 65 7 75 8 85 9 95 10

Rel

ativ

e va

riatio

n of

tota

l out

put (

delta

)

Ratio of stocked goods over production (sigma)

Stocks and production fluctuations

Figure 7 Fluctuations persistence on systems with different stock strategies(σ)

18Demand changes by plusmn1-10

20

Figure 7 shows the value of the relative deviation of output production(δ) generated by heterogeneous uncorrelated shocks in the external demandon identical systems where firms store a different multiple of their demand asstocks We obtain actually the opposite result that we may have expectedfluctuation become larger and more persistent the higher the level of stocksIn some way a strategy that for the individual firm is meant to reducethe fluctuations (using stocks as buffers) ends up by actually increasing thevolatility at aggregate levels

The result is though explainable both at the theoretical and empiricallevel In fact if no stocks were maintained production would need to adjustfor changes in demand only in case of a shock That is the firm may movemonotonically from the old to the new production levels sending consistentsignals to the rest of the system (ie its own suppliers) Instead if thesame shock affects a firm maintaining large stocks of output over the sameperiod production needs to adjust to the new demand and and to alignstocks with the new demand This generates besides the adjustment to thedemand a potentially inconsistent signal to the suppliers since the firm willhave a temporary level of production not meant to continue in equilibrium(adjusting stocks)

From the empirical viewpoint the last few decades have seen the almostuniversal diffusion of production system meant to keep stocks to a minimumif at all Lean production methods justndashinndashtime etc are solutions tothe problems of the costs necessary to maintain stocks Over the sameperiod economic fluctuations have been strongly reduced and few empiricalassessments have pointed to the relations between the two phenomena Forexample concerning the US ldquochanges in inventory behavior have playeda direct role in reducing real output volatilityrdquo (Kahn McConnell andPerez-Quiros 2002 p 183)19

57 Irrelevance of technical coefficients distribution

Horvath (1998) and Horvath (2000) suggest that systems with sparse ma-trices ie inputndashoutput matrices with many empty cells tend to be morevolatile in respect of systems with more evenly distributed matrices Noticethat this claim risk if not adequately qualified to generate an illogical resultIn fact matrices sparseness increases as a function of sectoral disaggregationand comparing the same systemrsquos fluctuations using inputndashoutput matricesat different levels of aggregation needs to produce the same type of volatil-ity Indeed this is not the case if all remaining parameters stand equaland do not adjust to the different level of analysis considered If aggregatepatterns of adjustment speed are maintained unchanged the higher the sec-toral disaggregation the higher the systemrsquos volatility ceteris paribus But

19Similar results are found in Blanchard and Simon (2001)

21

for example in our model reducing the aggregation also requires a changein the adaptation coefficients the more micro we undergo the analysis thehigher is the speed of adjustment of economic objects If for example weconsider sectors we are evaluating the adaptation of an entire set of firmswhich pose questions on how to aggregate their dynamics Things changewhen we consider the response of a single firm

Going back to Horvatsrsquo argument highly disaggregated matrices arelikely to contain many empty cells so that we may forecast different levelsof volatility depending on the aggregation used for the same system whichas mentioned is not caused by sparseness but to a failed adjustment inreaction mechanisms Results from our model20 show that volatility is com-pletely unaffected by input coefficients distribution Given their sectoralsum mdash through rows of the matrix mdash the value of each coefficient is irrele-vant The same occurs in the case of very skewed distributions that presenta high number of empty cells (zero input coefficients)

58 Microndash and Macrondashvolatility

As we have seen above one of the major discussion in the literature concernsthe relations between microndash and macrondashvolatility Put it simply one sideof the literature considers that uncorrelated microndashshocks cancel each otherout reducing their effects at macrondashlevel Conversely other researchersconsider that microndashshocks are multiplied at macrondashlevel generating highermacrondashvolatility than that we can observe at microndashlevel The debate isof obvious importance to determine the sources of business cycles to pre-dict their effects and determine the more effective policies to mitigate theirnegative effects

Our model is only a partial representation of realndashworld economic sys-tem and is also overly simplistic However based on these limitations italso offers very robust results allowing reliable conclusions on the (limited)elements included in the model In particular we consider our model a re-liable representation of the basic structures of the production interactionsamong sectors or firms Thus we can use the model to provide an answerto the debate concerning the contributions to volatility from the productioninteractions only Other contributions (eg from financial markets priceadjustments technological innovation etc) may re-inforce or counter theforces we analyse but they need to be considered as additional elements(and we opine also generally less relevant) not alternative to our analysisWe consider a necessity in order to apply a rigorous methodological ap-proach to be able to identify with certainty the results provided by partialstudies in order to eventually obtain by gradual extensions of the analy-sis more and more detailed representations of realistic economic systems

20Available from the authors

22

The results we obtain in this paper concerning this microndashmacro debate aremeant as a first relevant step to a more extended analysis

0

002

004

006

008

01

012

014

016

018

02

022

05 1 15 2 25 3

Var Demand

Macro (Output) and micro (Demand) volatility

delta Outputdelta Demand

Figure 8 Values of δ (relative deviation) for aggregate demand and ag-gregate output in respect of different levels of variance for sector specificuncorrelated external demand shocks

The flows of shocks affecting the firms of a system can be distinguishedalong two dimensions strength of the shock (eg amplitude of the changesin demand) and frequency of the shock (rare or frequent) Concerning thefirst aspect we initialized as usual a system to equilibrium levels (eg pro-duction quantities equal to external and intermediate demand for all sectorsconstant level of stocks equal to the desired level) We then ran the simu-lation generating the external demand for all sectors as a random variablewith constant (equilibrium) mean level and varying values of variance

Figure 8 shows unequivocally that as far as our model has propertiessimilar to real systems the hypothesis that macrondashfluctuations have largervolatility than their sectoral level shocks is fully confirmed At any levelof variance the aggregate volatility of demand as captured by the relativedeviation index is sensibly lower than the volatility shown by aggregateoutput that is the total production of the system This result is not sur-prising throughout all tests presented in the previous sections every resultsuggested that the system lsquooverndashreactsrsquo to any individual microndashlevel changeeven in the case these changes were meant to contrast disturbances There-fore when we move from considering one single shock to a flow of shockswe are not surprised to obtain that the system shows higher volatility thanthat implied by the microndashlevel disturbances

We now consider the second dimension defining a flow of shocks theirfrequency The question is whether for the same level of demand shocks

23

0 002 004 006 008 01 012 014 016

Difference of relative deviation for Output and Demand

0 5 10 15 20 25 30 35 40 45 50 55

Shock frequency 05

1

15

2

25

3

Var Demand

0 002 004 006 008 01

012 014 016

Figure 9 Differences of values of δ for aggregate output minus those foraggregate demand in respect of different levels of variance of demand and offrequency of the shocks

(ie same variance) more frequent changes trigger a higher or lower systemvolatility than rare disturbances The results shown in figure 9 suggest arather elaborated answer which partly vindicates (though in a particularsense) the arguments in favour of compensation of micro shocks The fig-ure shows the difference between aggregate output volatility and aggregatedemand volatility The independent variables are as before the volatilityof demand at microndashlevel21 and the time intervals in between two shocksAggregate demand volatility does not change for the frequency of shocksat least when computed over a long period But output volatility doesin a nonndashmonotonous way For high frequency of shocks the difference isstill positive and increasing for higher variance indicating that the ldquooverndashreactionrdquo argument is valid even in these cases but it is rather low Whilethe frequency of shocks slows down aggregate volatility of output increasesshowing that time is needed for the microndashshock to unfold their full effectsat the macrondashlevel However after a certain threshold less and less frequentshocks generate a lower volatility of the aggregate level of output

This result is quite sensible though not obvious at first sight and onceagain shows the power of such a simple model to generate and explain com-plex properties For frenetic levels of demand modifications the system hasno time to adjust but part of the job to align actual and desired productionfor constantly changing demand is performed by the varying demand itselfIn other terms the shocks cancel out through time if not through their quan-

21Beware of the scale of variance For reasons of visibility the scale is inverted

24

titative effects on the system While frequency decreases the disturbancesof the shocks not compensated by other (unlikely) shocks have the time tofilter through the production system generating higher and higher volatil-ity In a sense any system has a particular frequency of shocks that make itmaximally ldquoresonatingrdquo with the amplification of microndashshocks When therate of shocks gets even rarer the system can enjoy periods of longer andlonger quietness after each shock is absorbed and before a new one startsagain a cycle of initial local disturbance propagation of the disturbances toother sectors and settlement to the new equilibrium

6 Discussion and extensions

We presented a simple model of production composed by interacting sectorThe main (quit realistic) assumption of the model is that the productionprocesses in any sector of the economy make use of input from any othersector Moreover we assumed that sectors (represented as a single firm)adjust slowly both the desired level of production (smoothed revision of ex-pectation) and the actual level (smoothed modification of plant utilization)The model voluntarily leaves aside a large number of aspects of real-worldeconomic systems (eg financial markets price adjustments growth etc)The reason is that the model can be easily and reliably tested in order toextract its properties The possibility to transfer the (certain) model resultto a real context depend of course on an ample ceteris paribus clause How-ever we are confident that the reported results concerning the productionstructure play a relevant role in real world systems Therefore our resultsand particularly their motivations being accessible because of the relativesimplicity of the model will not be diminished by the integration of ourconsiderations with other aspects of economic systems

Our model is in essence a representation of an input-output table inter-preted in physical terms that is as the amount of inputs required for oneunit of output Besides this we include two straightforward assumptionsFirstly production levels cannot be suddenly changed but require time toscale up or down the utilization rate that is the levels of production Stocksare meant to make up for the difference between sales and production Thesecond assumption concerns the intentional behaviour of producers We im-plement this aspect by assuming that exists a level of stocks desired for eachlevel of demand However a change of demand does not translate suddenlyin a new level of desired stock but again only a long time at a different levelof demand will convince managers to update the desired levels of stocks

We show that the model generate under a wide number of settings oscil-latory patterns converging toward the equilibrium level where each sectorgenerate exactly the quantity demanded by final consumers and by othersectors as input The oscillations are provoked by over- and under-shooting

25

generated by the difficulty of coordination among sectors that by assump-tion ldquocommunicaterdquo only via variation of demand to their direct input sup-pliers We also show that the total production levels generated by sectors(sum of the physical units) produced at equilibrium depends (for constantfinal demand) on the total sum of the technical coefficients per sector irre-spective of their distribution This result is motivated by the impossibilityof input substitution and offers interesting suggestions for the analysis ofthe study of input-output tables of real-world systems

We then consider systems at equilibrium facing a single small change ofthe external demand for sectors These simulations allow to study the waythe system reacts to shocks We see that the sum of the coefficients affectsbesides the levels also the volatility of aggregate output the larger a systemthe stronger is the impact of a given shock Less obviously we find thatfor systems with the same dimension (ie same sums of coefficients) theaggregate volatility increases with both the inertia to change of productionlevel and the speed of adjustment of desired stocks This result is somewhatcounter-intuitive since both the cited variables are associated to a moreconservative behaviour that may be supposed should smooth away spikesin the pattern from an old to a new equilibrium However we show thatthe opposite actually holds given the complex interactions among sectorindividual-sector attitude intended to smooth shocks actually increase theiraggregate impact

In the same vein we show that systems maintaining high levels of stocksare more volatile that systems with reduced amounts of stocks Again thisis a counter-intuitive result reverting the goal of individual producers andthe aggregate result Interestingly this result finds strong support from theevidence of the recent changes in production methods (ie low or no stocks)and diminished cycles

In relation to specific issues discussed in the literature we find no supportto the suggestion that particular distributions of input-output coefficientsmodify the volatility of a system As said this depends in case on thesums of these coefficients but not on their distribution Further we canprovide an answer to a long and hotly debated questions whether microndashlevels volatility generates stronger or lower aggregate volatility Our modelshows unequivocally that microndashvolatility (eg variance of final demand)persists and generates stronger volatility at aggregate level Again we canidentify the reason on the complex production structure that exalts micro-level shocks by generating lsquowrongrsquo messages among producers Moreoverwe can also point to a usually neglected aspect of volatility we show thatthe frequency of shocks is a very relevant aspect besides their dimensionin determining the volatility of aggregate output

Our work can be developed along two complementary directions Firstlywe can continue to study the properties of the model by extending the ele-ments considered (eg pricing) analysing the effects on the present results

26

and generating more analytical evidence Of particular relevance is in ouropinion the problem of aggregation Representations of the same systemswith different codification systems (eg different number of SIC digits)must not modify overall system properties for example the volatility ob-served Given that our result show that the sums of coefficients matter forthe levels and volatility of the system and that these sum do depend on theaggregation chosen it should be possible to induce the reaction coefficientsthat we know affect the volatility only

Secondly we can find an application of our results to real world evidenceIn fact there are many data sets available for the data we deal with basicallyinput-output tables Our model is readily adapt to be parametrized alongany number of sector and coefficient necessary so that we may replicate pastseries in order to adapt the unobservable parameters and use the resultingcomplete model to explain business fluctuations and provide intuitions onfuture patterns

27

References

Acemoglu D and A Scott (1997) ldquoAsymmetric Business CyclesTheory and TimendashSeries Evidencerdquo Journal of Monetary Economics 40501ndash533

Backus D P Kehoe and F Kydland (1995) ldquoInternational BusinessCycles Theory and Evidencerdquo in Frontiers of Business Cycle Researched by T Cooley Princeton University Press Princeton

Bak P K Chen J Scheinkman and M Woodford (1993) ldquoAg-gregate Fluctuations from Independent Sectoral Shocks SelfndashOrganizedCritically in a Model of Production and Inventory Dynamicsrdquo RicercheEconomiche 47(1) 3ndash30

Battiston S D Delli Gatti M Gallegati B Greenwald and

J Stiglitz (2005) ldquoCredit Chains and Bankruptcies Avalanches in Sup-ply Networksrdquo in Annual Workshop on Economic Heterogeneous Inter-acting Agents (WEHIA 2005) University of Essex UK

Blanchard O and J Simon (2001) ldquoThe Long and Large Declinein US Output Volatilityrdquo Brookings Papers on Economic Activity 1135ndash174

Bresnahan T F and V A Ramey (1994) ldquoOutput Fluctuations atthe Plant Levelrdquo The Quarterly Journal of Economics 109(3) 593ndash924

David P A (1975) Technical choice innovation and economic growth essays on American and British experience in the nineteenth centuryCambridge University Press London

Delli Gatti D C Di Guilmi E Gaffeo G Giulioni M Galle-

gati and A Palestrini (2004) ldquoA New Approach to Business Fluc-tuations Heterogeneous Interacting Agents Scaling Laws and FinancialFragilityrdquo Journal of Economic Behavior amp Organization 56 489ndash512

Dosi G G Fagiolo and A Roventini (2006) ldquoAn EvolutionaryModel of Endogenous Business Cyclesrdquo Computational Economics 273ndash34

Dupor B (1999) ldquoAggregation and the Irrelevance in MultindashSector Mod-elsrdquo Journal of Monetary Economics 43 391ndash409

Head A C (1995) ldquoCountry Size Aggregate Fluctuations and Interna-tional Risk Sharingrdquo The Canadian Journal of Economics 28(4b) 1096ndash1119

28

Helbing D and S Lammer (2005) ldquoSupply and Production NetworksFrom the Bullwhip Effect to Business Cyclesrdquo in Networks of Interactingmachines Production Organization in Complex Industrial Systems andBiological Cells ed by D Armbruster K Kaneko and A S Mikhailovvol 3 of World Scientific Lecture Notes in Complex Systems World Sci-entific

Helbing D S Lammer U Witt and T Brenner (2004) ldquoNetworkndashInduced Oscillatory Behavior in Material Flow Networks and IrregularBusiness Cyclesrdquo Phisical Review E 70(056118) 1ndash6

Helbing D U Witt S Lammer and T Brenner (2004) ldquoNetwork-Induced Oscillatory Behaviour of Macroeconomic Output Flowsrdquo in In-dustry and Labour Dynamics II The Agent-based Computational Eco-nomics Approach Proceedings of the wildace 2004 conference ed byR Leombruni and M Richiardi no 5 LABORatorio R Revelli

Horvath M (1998) ldquoCyclically and Sectoral Linkages Aggregate Fluctu-ations from Independent Sectoral Shocksrdquo Review of Economic Dynam-ics 1 781ndash808

(2000) ldquoSectoral Shocks and Aggregate Fluctuationsrdquo Journal ofMonetary Economics 45(1) 69ndash106

Jovanovic B (1987) ldquoMicro Shocks and Aggregate Riskrdquo The QuarterlyJournal of Economics 102(2) 395ndash410

Kahn J A M M McConnell and G Perez-Quiros (2002) ldquoOnthe Causes of the Increased Stability of the US Economyrdquo FRBNY Eco-nomic Policy Review 8(1) 183ndash202

Kraay A and J Ventura (1998) ldquoComparative Advantage and theCrossndashSection of Business Cyclesrdquo World Bank Woorking paper Series1948 World Bank

Long J B and C I Plosser (1983) ldquoReal Business Cyclesrdquo Journalof Political Economy 31(1) 39ndash69

Lucas R E (1977) ldquoUnderstanding Business Cyclesrdquo CarnegiendashRochester Conference Series on Public Policy (special issue of Journalof Monetary Economics) 5 7ndash29

Nelson R R and S G Winter (1982) An Evolutionary Theory ofEconomic Change Harvard University Press Cambridge MA

Nirei M (2005) ldquoThreshold Behavior and Aggregate Fluctuationsrdquo Jour-nal of Economic Theory in press

29

Scheinkman J and M Woodford (1994) ldquoSelfndashOrganized Criticallyand Economic Fluctuationsrdquo American Economic Review 84(2 Papersand Proceedings of teh AEA) 417ndash421

Schuh S (1996) ldquoEvidence of the Link Between FirmndashLevel and Ag-gregate Inventory Behaviorrdquo Finance and Economics Discussion Series1996-46 Federal Reserve Board

Stiglitz J and B Greenwald (2003) Towards a New Paradigm inMonetary Economics Raffaele Mattioli Lectures Cambridge UniversityPress Cambridge

Weisbuch G and S Battiston (2005) ldquoProduction Networks andFailure Avalanchesrdquo Working Paper mimeo Ecole Normale SupErieure

Zarnovitz V (1977) ldquoBusiness Cycles Observed and Assessed Why andHow they Matterrdquo NBER Working Paper Series 6230 National Bureauof Economic Research

30

A Tables

Table 1 Parameters valuesParametera Description Value

N Number of sectorsunits i of production 10Ci Number of j buyer sectors for sector i (IO matrix sparseness) N minus 1Di (E) Value of the external demand 10mij Physical input coefficient of input j for sector i sim U(0 1)Mi Sum of the mij input coefficients j for the production of i 05αi friction to changes of production levels Degree of lockndashin on

production technology085

αsi Speed of adaptation to demand changes 004

σ Desired multiple of production to stock 10

aInto parenthesis the number of lags of the initial value for lagged variables

31

The author shows that sectoral shocks will tend to cancel out in case theeconomic system is sufficiently distributed that is when the IndashO matrixcontains evenly distributed values over all the cells Conversely sectoralshocks are not absorbed and can be reinforced in case many cells of thematrix are empty This case indicate that there will be few sectors affect-ing many other sectors preventing the possibility of compensating certainshocks

These conclusions are potentially interesting because point directly toan easy to observe aspect of economic system for example providing thepossibility to test the prediction of the analysis It is worth to note that thisapproach draws conclusions that are not neutral to the aggregation levelused In fact a highly disaggregated matrix is more likely to contain manyempty cells than the matrix for the same system obtained aggregatingsectors

Dupor (1999) also considers the case of inputndashoutput matrices with dif-ferent input coefficients finding them irrelevant for the results But he doesnot include matrices with empty cells

The cited contributions and those inspired to the same approach sug-gest that the type of interaction as expressed by input-output coefficientscan be a relevant factor in determining the existence and dimension of fluc-tuations However this literature is not able yet to draw conclusive results

Limited interactions and production constraints

A different kind of mechanisms are assumed and shown to play a role in theAv models they add an analysis on firms production (technological) possi-bilities to the production structure They are inspired by and similar to Jo-vanovic (1987) model in which the limitation in the number of interactionsupon which each playerrsquos decision depends generates shock cascades andaggregate persistence Bak Chen Scheinkman and Woodford (1993) andScheinkman and Woodford (1994) show that when the interaction structurebetween sectors is constrained to a completely rigid system coupled withnon convexities in the production function independent shocks to differentsectors do not cancel in the aggregate In other words the authors place aset of restrictions to the set of possible actions of sectors First structuringthe production networks with a fixed invariable lattice produces a rigiditythat do not allow producers to change (or add) inputs (or suppliers) orincrease the produced quantity This is combined with the structure of theavalanche that propagates through the unchanged network Second a pro-duction function that is maximised only when a discrete number of unitsof a good is produced coupled with a fixed (discrete) maximum amount ofinventories that can be stored by the representative firm Such a structuredetermines production oscillation and shocks avalanches due to the factthat firms are constrained in their production opportunities

6

Using a very similar production structure with limited interactionsWeisbuch and Battiston (2005) play on the bankruptcies generated on down-stream firms by production failures occurring in upstream firms (reducingthe flow of inputs) Stochastically generated failures to produce in a firmcauses production constraints in its downstream clients hence lower invest-ment and eventually bankruptcies3 A lag between the period in whicha firm goes bankrupt and a new firm appears generates a lack of supplywhich propagates on the downstream sectors

Due to the original literature on avalanches these contributions sharetwo common features that are ill-adapted to economic systems First theyassume very peculiar interaction structures of one-directional productionrelations between sectors which amount to assume one half of the input-output matrix empty and are not applicable to the general case with cyclicalproduction structure or full IndashO matrices Second the strongly rely on quiterigid thresholds determining a mechanicist behaviour by agents Basicallyagents have only two options available (ie produce or not produce) andthe choice is made on the basis of crude decisional mechanism4

Notwithstanding the limitations this approach provides a quite sensiblemechanism of the origin and transmission of shocks through a system Ourproposal can be interpreted as an extension of this approach by relaxing thestrong assumptions on the interaction structure and allowing agents for amore fluid decisional process as we will see in the next section

3 The basic elements of a production structure

Our overall hypothesis is that the very production structure of an economymay generate persistent fluctuations even in the absence of continuous un-related shocks and rigid thresholds that determine non linearities The aimis to provide a generalised interpretative framework to explain the occur-rence of short term fluctuations in the economy simply as an outcome ofthe division of labour in a number of sectors related through trade In itsextreme interpretation any production system which is not fully integratedis bound to produce business cycles As we will show a number of factorsinfluence their extent

Quantitative adjustments in production in any one sector require ad-justments in related sectors When the structure of the economy is trulyInputndashOutput (not a directed linear supply chain as in the Av models) in-put flows are cyclical and adjustment to long run production equilibrium islikely to take a long mdash when not infinite mdash time Our work departs from

3A similar model by Battiston Delli Gatti Gallegati Greenwald and Stiglitz (2005)relates bankruptcies to price changes buyers payment timing and the credit market (acentral bank)

4Nirei (2005) is an attempt to generalise such models though maintaining the assump-tion of threshold behavior

7

both RBC and Av models with respect to structure modelling criteriaanalysis and methodological approach Indeed our model may be easilyreconduced (or restricted) to both We are much in line with a recent se-ries of papers by Helbing Witt Lammer and Brenner (2004) that addressthe same question although with a number of differences that are discussedlater on5

We aim to analyse to what extent the structure of inputndashoutput relationscauses aggregate fluctuations by itself The main idea is to show that ag-gregate fluctuations may derive from simple micro imbalances without theneed for continuous exogenous shocks not even at the firm or sectoral levelIn other words normal production activities including lags technologicaladjustments information asymmetries circularity of the inputndashoutput sys-tem and so on may be enough to explain part of the macro volatility in aclosed system6

Our assumptions draw on the existing literature but try to avoid themajor difficulties highlighted above Firstly we avoid to impose either mar-ket clearance assumptions or crude threshold behaviours Rather we adoptan assumption close to a Simonian approach agents in an economy do theirbest within the informational and physical constraint they are subject toWe represent decision makers as responding to a single piece of informationthey obtain the quantities demanded by their clients Crucially their ac-tions are inspired to a conservative approach an increase in the demandgenerates a smaller (desired see below) sudden increment of productionThis assumption stems from the fact that decision makers are aware of theinstability of their environment and therefore want to avoid getting perma-nently caught with over production following a temporary spike of demandIn other words firms attempt to smooth business cycle generating a morestable (or less uncertain) environment as the economic literature would sug-gest Notice that this representation in line with a routinized representationof organizations (Nelson and Winter 1982) does not contradict market equi-librium Rather in a stable environment such decisional procedure generatesan asymptotic pattern to the equilibrium level in the absence of feedndashbacks

A second assumption concerns the technological possibilities available tothe firms When firms realise the need for a change in production levels saya desired increase of production they face a complex and costly organisa-tional transformation of the production capacity risky investments an soon As Bresnahan and Ramey (1994 p 622) suggest inducing from a de-tailed analysis of the automobile industry in the US ldquoadjusting production

5We thank Matteo Richiardi for pointing out these similar attempts See also Helbingand Lammer (2005) and Helbing Lammer Witt and Brenner (2004)

6An open system entails many other factors including price factors of both inputsand outputs mdash terms of trade labour cost external investment etc As with exogenousshocks our analysis does not deny the importance of these factors but simply separatethem from the structural explanation of cycles

8

is a more complicated process than simply ldquochanging Qrdquo or choosing themix of capital and laborrdquo And this is not only a matter of non convexitiesin production technologies The inconsistency of time reversibility in thetechnological choice is accompanied by the need to operate on the lsquoavail-able technological frontierrsquo (David 1975) If a firm plans a given productionactivity for a short period of one week even though the next day there is anunexpected fall in the demand it is unlikely that it is followed by a similardownturn in production The case is evident when we think at productionplanning that involve hiring of workers an capital investments (both in-volving sunk costs)7 Therefore any desired change generates a pattern ofsmall modifications approaching slowly the target level Moreover thesechanges are irreversible in the short term since only a prolonged change ofthe environment triggers the modification of previous desired levels

Resting on those simple conjectures we thus analyse a model in whichwe only allow for sluggish adaptation of production due to conservative(adaptive) behaviour of firms and the physical constraints faced when vary-ing the production levels Notice that these assumptions are contrary tothe generation of instability of a system In fact a firm behaving as de-scribed above reduces if not eliminates any disturbance affecting its ownstate if acting in isolation Other factors like new technologies extremelylong term perspective price changes etc are likely to be sources of volatil-ity therefore contributing to generate aggregate system fluctuations (as forexample in Acemoglu and Scott 1997) Our neglect of these and other fac-tors however unrealistic serves the purpose of testing whether a volatilityreducing microndashbehaviour turns into volatility generating aggregate resulttwo widely observed dynamics Quite interestingly (and encouraging) DosiFagiolo and Roventini (2006) have concomitantly proposed a model whichincorporates detailed representation of both sluggishness robustly based onempirical evidence Heterogeneous firms which undergo lumpy investmentand slowly adapt their expectation on changes of the future demand inducecomovement of macroeconomic variables qualitatively similar to the onesobserved in the real world To pursue our line of research we first needto concentrate on how sectorsfirms interact in the system in the effort tocomplement those promising results

Therefore the second element we consider is meant to represent a gen-eralized economic system We allow for the circularity of inputndashoutput sys-tems a crucial feature in determining feedback mechanisms and a principal

7See also the discussion in Acemoglu and Scott (1997) where the authors argue thatldquo[w]hile the presence of fixed costs can account for the discreteness of economic turningpoints it does not naturally lead to persistence because once an individual undertakes anaction they are less likely to do so in the near future [] [A]lthough the presence of fixedcosts leads to increasing returns these are intratemporal the full extent of economiesof scale arising from fixed costs can be exploited within a periodrdquo [p 502 authorsrsquoemphasis]

9

component of business cycle It may be true that most backward linkagesconcern capital goods (Weisbuch and Battiston 2005) but a rapid glanceto an inputndashoutput matrix would show that technical coefficients are nonnull in both directions and ldquomost commodities are inputs to the produc-tion process of other commoditiesrdquo (Horvath 2000 p 70) The higher thesectoral aggregation mdash observed or assumed mdash the more the relevance ofintermediate supply holds true (as mentioned above with respect to ma-trix incompleteness) We claim that the existence of backward and forwardlinkages are a crucial element to determine the persistence of shocks in theeconomy A perfectly linear (one directional) system needs much less adap-tation of firms and sectors in order to stabilise the economy Nonethelesscircularity does not imply that inputs are perfectly substitutable amongthem in order to determine the maximising allocation even at high level ofdisaggregation

By using a fixed coefficients production function we both maintain themid term rigidity of production processes and we do not impose limitedinteractions We claim that the circular property of the production structureis by itself sufficient to produce shock persistence even when the inputmatrix is full (although with heterogeneous input coefficients) mdash all sectorsshop in the remaining n minus 1 sectors

In conclusion we represent a model for an economic system made onlyof i) production level decisions and ii) input-output relations Clearly sucha model lacks many features present in real economic systems likely we areconvinced to affect the propensity to generate cycles As such we cannotaim at reproducing realistic data so our results cannot be tested comparingtime series generated by the model with real ones Instead given the paucityof the elements comprising the model we can investigate in detail to whichextent the (few) features present in our model affect the cyclical behaviour ofthe system As we will see we will be able to show quite a number of resultin which aggregate fluctuations are generated whereas the individual microstructure of the model aims at monotonous adaptation Before discussingthe results next section describes in detail the implementation of the simplemodel

4 The model

We model an economic system in which a number of sectors i = (1 N)are potentially connected via (intermediate) market relations A N times NInputndashOutput matrix determines the mij units of each input j = (1 N)mdash bought from the N minus 1 residual sectors mdash necessary to produce oneunit of output i8 For simplicity given the focus of this paper on the IndashOstructure as a generator of aggregate fluctuation we abstract from market

8IndashO coefficients are therefore physical coefficients

10

interactions within sectors each sector is a production unit In fact forour purposes it is not relevant which firm produces the sectoral outcomebut only the total amount produced by each sector9 Moreover besidesmaking the analysis more neat this assumption allows also to enjoy a largefreedom in determining the aggregation level since we donrsquot need to assessthe effects of between sectors competitiveness which for example is likelyto increase for increasing levels of disaggregation Our economic systemis therefore composed of N production units each of which may buy andprovide inputs from and to any other sector depending on the IndashO structureof the economy

The demand for each sector i is determined by a constant consumersdemand Di (E) and the input needs of other sectors

Di = Di (E) +sum

j 6=iisinCi

mjiQj (1)

where Ci is the set of j intermediate clients of i (sum

j isin C lt N) mji is thetechnical coefficient of firm j for input i ie the amount of i required toproduce one unit of j Qj is the actual output produced by sector j

Concerning the simultaneous production level of each sector in time twe make two lsquorealisticrsquo assumptions First from the technical viewpoint afirm like a tanker cannot abruptly change its course Therefore changesto the production level can be introduced only gradually through capital(dis)investment and job h(f)iring (see discussion in Section 3) Second weassume that firms do not decide the production level directly on the basisof the observed demand level but conscious of the uncertainty of demandplan their production in order to maintain invariant a desired level of stocksand avoid stockout10

Quantity produced by sector i in period t is computed as follows

Qit = αiQtminus1 + (1 minus αi)(

Slowastitminus1 minus Sitminus1

)

(2)

where αi measures the physical constraint in adapting the production capac-ity to changes in the required output Slowast

itminus1are firmrsquos i desired stocks and

Sitminus1 its actual stocks Notice that both variables related to stocks appearwith a lag This is due to two reasons First it is a modelling necessity inorder to allow the simultaneous determination of production levels for allsectors Secondly as frequently happens modelling necessities reveal sen-sible if not generally considered aspects of the modelled system In factproduction decisions concern management in the production plant while

9Adding heterogeneous firms within sectors would definitely enrich the causal expla-nation put forward and parametrised in this paper (see below) For example to analysethe effect of firms constraints It is left over for future research as in the present work weopt for the analysis of the main structural parameters that affect aggregate fluctuations

10See also discussion in Schuh (1996)

11

demand through sales is observed by the commercial staff It is thereforelogical that the information from demand reaches plant managers with somelag

The actual stock level is trivially computed subtracting from the previousheap of stock the amount sold and adding the new production

Sit = Sitminus1 + Qit minus Dit (3)

Concerning the desired level of stocks instead we introduce the conser-vative behaviour mentioned above Firms ideally would keep an amountof stocks proportional to the demand they receive say a multiple σ ofthat amount However when demand varies they adjust the desired levelsmoothly preventing sudden changes to the variable representing the goalof the firm

Slowastit = Slowast

itminus1 + αsi

(

σDit minus Slowastitminus1

)

(4)

where αsi measures the adaptation to demand changes

In summary the dynamics of the model consists of only the equationsabove Firstly at the beginning of a time step firms determine the quantityto produce as a function of past imbalances in stock levels and the pastproduction level These levels define the demand for all sectors determinedby the technical coefficients applied to all produced quantities Finallyactual and desired stocks can be updated Figure 1 represents visually thisdynamics highlighting the decisional components of the model in respect ofthe ldquomechanicsrdquo of trade

5 Results production structure and economic fluc-

tuations

In this section we describe some of the most relevant results concerningthe effects of the production structure to determine aggregate fluctuationsGiven the unusual analytical instrument chosen (at least concerning thistopic) we start with a brief methodological introduction meant to clarifythe nature of the results we claim to obtain Next we describe a generalsetting for the model which is the (extremely) limited area of the potentialparametersrsquo space of the model that we will explore In the rest of thesection we discuss the results from the model

51 Methodological considerations

Real economic systems are constantly subject to a continuous flow of shocksexternal to the production system besides changes within the system itselfIn order to appreciate the systemic properties we need to abstract from all

12

Compute the quantityto produce

F (Q(t-1) S(t-1) s(t-1) alpha)

Communicate thequantity of inputsrequired to each

supplier

Determine the overalldemand

F(D(E) D(I))

Revise the past decisionon stock plans

(desired stocks)F(D S(t-1) sigma alpha(s))

Compute the final levelof stocks for next period

F(s(t-1) Q D)

KEY

ModelDirection

Variablecomputed

at t-1(t-1)

Step includingfirm behavior

Figure 1 One period dynamics

the nonndashrelevant disturbances and concentrate on how the system endoge-nously contributes to the observed dynamics Therefore we will generatehighly abstract patterns whose analysis will provide insights on the systemproperties that would be otherwise lost in the noise generated by all theelements of a real system

We are not trying to fit the model to a specific empirical data set nor weare interested in studying the complete behaviour of the model for areas ofthe parameterrsquos space that make no economic sense (eg negative or infiniteproduction levels) Therefore the values of initial settings and results matteronly in relative and not in absolute terms

We are going to describe a simulation model with which we will produceseveral simulation runs and we claim obtain results relevant for the debatediscussed above Rigorously speaking we may claim only certainty for thevalidation of the results that is that the model does actually generate theresults presented and for the reasons we explain We limit this part to thepresentation of graphs and verbal explanations that we believe are ratheruncontroversial11 Concerning the relevance of our studies for real systems

11Interested readers can request the model code and and simulation data for replicationand extensions of the results We can guarantee that such possibility does not require

13

(verification) we simply avoid to pretend that our model is a quantitativeapproximation of real economies (which in which period) Our opinionis that the study of the model allows us to make general considerationsnon-obvious and relevant for applied debates supported by strong logicalarguments which can be integrated but not reverted by arguments con-cerning more and more elements of the economic world

52 Model setup

We go through the model properties analysing an economic system madeup of ten sectors (N = 10)12 Each sector shops inputs from the remain-ing N minus 1 and sells to them part of its own production as intermediategoods This means that the input matrix is complete except for the diag-onal (mij gt 0 forallj 6= i mii = 0) The sum of all N minus 1 coefficients for asingle sector is given by Mi and its benchmark value is kept at a plausiblemedium level (unless differently specified) to avoid distortions from highlydemanding sectors Each input coefficient is randomly drawn from a uniformdistribution

Sectors start at an equilibrium level their supply matches intermediateand final demand and in each period the same quantity of output mustbe produced to compensate for the used stocks and keep the desired stocksunchanged In thier turn stocks must be ten times the value of total sectoraldemand (σ) Eventually technological adjustment of production is ratherslow and producers are quiet refractory to follow sudden changes in thedemand in general a smoothing behaviour strongly prevails (α and αs)

Under those general preconditions in the following sections we anal-yse the dynamics of the economic system in relation to model parametersWe mainly focus on the cyclical behaviour of systems and their speed ofconvergence to the asymptotic equilibrium13

The very abstract assumption and the homogeneous initialisation donot allow for the multiple equilibria that are likely to rise in complex systemswith feedbacks Once more such simplification allows to have a betterunderstanding of the crucial impacts of the production structure on businessfluctuations Departing from the related literature discussed above14 weconsider the effects of a single shock rather then a continuous flow whichin our model is superfluous in order to obtain fluctuation persistence15 To

extensive programming or statistical skills besides the usual education of economists12You can think of a high level of aggregation in the statistical observation The com-

plete initialisation and parameters values are available in table 1 in the Appendix13Notice that sectors only approach a final equilibrium level of production which can

be prendashdetermined under static conditions14Included the more proximate work of Helbing Witt Lammer and Brenner (2004)15An exception are the last results presented in Section 58 which we see more as an

opportunity for further research than a closing of the paper

14

start with next section (53) shows the aggregate cyclical behaviour of astandard setting

53 Single shock analysis

We generate a simulated history following the dynamics of the modelrsquos vari-ables when the system is set to an equilibrium level and a single shockmodifies one of the exogenous elements namely the external demand forsectors Modifications across sectors are random and not correlated andmay increase or reduce external demand by at most 10

100 225 350 475 600

193894

195861

197828

199795

201762 Industrial Production

Time

Figure 2 Total industrial production fluctuations following a shock fromexternal demand

Figure 2 shows the dynamics generated by the total industrial productionof the system mdash that is also a proxy for the systemrsquos GDP fluctuations Thenew external demand changes the equilibrium level but the system cannotreach the new equilibrium with a monotonous pattern and converges to thenew level with a fluctuating pattern Each change in the external demandinduces also modifications of the intermediate demand and sectoral adjust-ments in the quantity produced typically overshoot the aggregate patternAs it can be appreciated from the figure it takes time before the system isable to converge to the new static equilibrium conditions

We study two features of this pattern absolute level of the new equilib-rium and volatility induced by the shock

54 Technical coefficients and industrial output

Production level

The equilibrium levels of industrial output depend on two factors only Ob-viously the levels of external demand determine how much of the total pro-

15

duction needs to be supplied to consumers exogenous from the productionsystem Second the technical coefficients also influence strongly the level ofactivity of the system defining the intermediate demand to each sector

180

200

220

240

260

280

300

01 02 03 04 05 06 07 08 09

Out

put l

evel

(In

dust

rial p

rodu

ctio

n)

Sum of total input coefficients (M)

Total industrial production for different input coefficients levels

Figure 3 The relevance of input coefficients on output level

Figure 3 shows the equilibrium level of industrial output reached bysystems with identical external demand where we fixed the sum of inputcoefficients for each sector of the economy (Mi) to different values16 Thehigher the sum the higher is the total production sustained by the system

This result depends on the very production structure represented by theinputndashoutput matrix In fact the higher is the level of production requiredby intermediate sectors the higher is the total production given a constantlevel of external demand

Notice that the level of production does not depend on the distribution ofthe technical coefficients As long as the sum of the coefficients is identicalthe same production level is obtained irrespective of their distribution (otherthings being equal)

Output shock absorption

Input coefficients have a relevant role also on the output stability of thesystem Starting from an output equilibrium level we hit the system witha single shock on the external demand for all sectors uncorrelated amongsectors17 modifing (positively or negatively) the demand for all sectors by

16Within this limit the actual value of the coefficients are randomly drawn17In each sector the external demand changes by plusmnX where X is a uniformly dis-

tributed percentage between 1 and 10

16

at most10 As show in Figure 2 the economic systems produces a lenghtlysmoothing oscillatory behaviour We compute the relative deviation of theaggregate production (industrial output) as the ratio between its standarddeviation and its mean δ = σY microY where Y =

sum

Qi Figure 4 shows thebehaviour of δ as a function of the sum of input coefficients for each sector80 periods after the shock As time goes by the level of δ reduces for allMi but the relation with the sum of input coefficient is unchanged

00055

0006

00065

0007

00075

0008

00085

0009

01 02 03 04 05 06 07 08 09

Rel

ativ

e va

rianc

e of

tota

l out

put (

delta

)

Sum of total input coefficients

Aggregate output deviations for different technical coefficients

Figure 4 The relevance of input coefficients on output stability

In sum input coefficients determine the extent of overshooting behaviourThe higher the coefficients the higher is the backlash even when shockscould cancel out through sectors

55 Shock persistence and adjustment coefficients

We now consider the effect of micro smoothing behaviours on the volatilityof the production system In particular we are interested in testing theeffects of two parameters of the model on the volatility of the productionsystem First we consider the effects of the αrsquos representing the ldquostickinessrdquoof the production technology to desired changes of Q aimed at filling thegap between actual and desired stocks The higher is α the slower is thereaction to stock unbalances Concerning its effect referring to the technicalpossibilities available ie production capital and organization we may putforward two opposite hypothesis relevant to systemrsquos volatility

Hypothesis 1 Slower production adjustment (high α) reduces volatilitybecause unbalances in demand of one sector being compensated slowly slow

17

down the diffusion of shocks to other sectors

Hypothesis 2 Slower production adjustment (high α) increase volatil-ity because the longer lasts the mismatch between actual and equilibriumproduction level the deeper will become the mismatch between actual anddesired stocks

20 25 30 35 40 45 50 55 60 65

Cyclesrsquo width

07 075

08 085

09alpha 0005

001 0015

002 0025

003 0035

004 0045

005

alphaS

20 25 30 35 40 45 50 55 60 65

Figure 5 Maximum width recorded between highest and lowest peaks fortotal production for different values of α and αS

Secondly we consider the parameters αS representing the speed of ad-justment of desired stock to observed demand The higher this parame-ter the faster sectors revise the desired levels of stocks to current levelsof observed demand This parameter concerns sectorsrsquo behavioral decisionprocess Higher αS represent a stronger belief that the most recently ob-served demand level is a reliable indicator of future demand levels whilelower levels of the parameter represent a more conservative approach wheremanagers require a long observation of demand levels different from the pastones before adjusting their expectations Also concerning the effect of thisparameter on the volatility induced by an exogenous shock we may maketwo hypothesis

Hypothesis 1 Faster adaptation of desired stocks to new levels of de-mand (higher αS) reduce volatility because changes in demand are rapidlytranslated in adjusted levels of production

Hypothesis 2 Faster adaptation of desired stocks to new levels of demand(higher αS) increase volatility because temporary unbalances of demand in

18

one sector are transmitted faster to other sector reinforcing the feedbackmechanism of fluctuations

To test which of these hypotheses are confirmed by the model we runsimulation runs with identical systems (eg same technical coefficients samelevels of demand same shocks) but for the values of the αrsquos and αS rsquos testingall combinations of these parameters over a range of values (each run usedthe same parameters for all the sectors of the system) For each simulation(ie each couple of values of the parameters) we computed two indicatorsof volatility One measures the maximum distance of total production be-tween the highest and the lowest peak that is the maximum width of theoscillations registered in the production time series The second indicatormeasures the relative deviation δ = σY microY Figures 5 and 6 report theseresults

0004 0006 0008 001 0012 0014 0016 0018 002

Relative deviation

07 075

08 085

09alpha 0005

001 0015

002 0025

003 0035

004 0045

005

alphaS

0004 0006 0008 001

0012 0014 0016 0018 002

Figure 6 Relative deviation of total production for different values of α andαS

The results confirms unequivocally both hypotheses 2 and reject hy-potheses 1 In a sense these hypotheses confirm the nature of the economicsystem represented in the model as a complex system where the interactionsamong sectors play a far more important role than the individual sector be-haviour In fact in both cases we may interpret the results that the strongerthe efforts to mitigate the effects of shocks (ie slow productionrsquos and quickexpectationsrsquo adjustment) the opposite result is actually obtained the fluc-tuations become even more accentuated It is also worth to note that thestronger impact is generated by the behavioural parameters rather than thetechnical one as indicated by the stronger effects of the αS in respect of theα This result seems to suggest that ceteris paribus information mattersmore than technical constraints Though the model is by any means in-

19

adequate to discuss normative aspects because of the naive ways strategicdecision making is represented it can anyway suggest the effects of errorsin management decisions (ie wrong desired levels of stocks) as opposed tothe technical constraints preventing sudden adjustment of production lev-els Stretching a bit the interpretation of these results we may suggests thatefforts to reduce volatility should better be oriented to improve the trans-mission of information in order to coordinate future production plans (ieidentifying quickly the desired levels of stocks and therefore the new equilib-rium production level) rather than improving the flexibility of productionmethods (reducing α)

56 Volatility and stock levels

In our model stock levels are maintained to buffer a sectorrsquos sale againstunexpected changes in demand Therefore a system made of risk adversemanagers (keeping higher levels of stocks) may be thought to lead to asmoother absorption of shocks in respect of a system composed of firmskeeping a short supply to cushion unexpected events To test this hypothesiswe tested the behaviour of the model for different values of σ the levels ofstock represented as a multiple of demand For each level of σ we run asimulation using all other initialization identical as usual representing aneconomic system ldquoalmostrdquo at equilibrium but for a single small shock indemand uncorrelated across sectors18

00044

00046

00048

0005

00052

00054

00056

00058

0006

00062

5 55 6 65 7 75 8 85 9 95 10

Rel

ativ

e va

riatio

n of

tota

l out

put (

delta

)

Ratio of stocked goods over production (sigma)

Stocks and production fluctuations

Figure 7 Fluctuations persistence on systems with different stock strategies(σ)

18Demand changes by plusmn1-10

20

Figure 7 shows the value of the relative deviation of output production(δ) generated by heterogeneous uncorrelated shocks in the external demandon identical systems where firms store a different multiple of their demand asstocks We obtain actually the opposite result that we may have expectedfluctuation become larger and more persistent the higher the level of stocksIn some way a strategy that for the individual firm is meant to reducethe fluctuations (using stocks as buffers) ends up by actually increasing thevolatility at aggregate levels

The result is though explainable both at the theoretical and empiricallevel In fact if no stocks were maintained production would need to adjustfor changes in demand only in case of a shock That is the firm may movemonotonically from the old to the new production levels sending consistentsignals to the rest of the system (ie its own suppliers) Instead if thesame shock affects a firm maintaining large stocks of output over the sameperiod production needs to adjust to the new demand and and to alignstocks with the new demand This generates besides the adjustment to thedemand a potentially inconsistent signal to the suppliers since the firm willhave a temporary level of production not meant to continue in equilibrium(adjusting stocks)

From the empirical viewpoint the last few decades have seen the almostuniversal diffusion of production system meant to keep stocks to a minimumif at all Lean production methods justndashinndashtime etc are solutions tothe problems of the costs necessary to maintain stocks Over the sameperiod economic fluctuations have been strongly reduced and few empiricalassessments have pointed to the relations between the two phenomena Forexample concerning the US ldquochanges in inventory behavior have playeda direct role in reducing real output volatilityrdquo (Kahn McConnell andPerez-Quiros 2002 p 183)19

57 Irrelevance of technical coefficients distribution

Horvath (1998) and Horvath (2000) suggest that systems with sparse ma-trices ie inputndashoutput matrices with many empty cells tend to be morevolatile in respect of systems with more evenly distributed matrices Noticethat this claim risk if not adequately qualified to generate an illogical resultIn fact matrices sparseness increases as a function of sectoral disaggregationand comparing the same systemrsquos fluctuations using inputndashoutput matricesat different levels of aggregation needs to produce the same type of volatil-ity Indeed this is not the case if all remaining parameters stand equaland do not adjust to the different level of analysis considered If aggregatepatterns of adjustment speed are maintained unchanged the higher the sec-toral disaggregation the higher the systemrsquos volatility ceteris paribus But

19Similar results are found in Blanchard and Simon (2001)

21

for example in our model reducing the aggregation also requires a changein the adaptation coefficients the more micro we undergo the analysis thehigher is the speed of adjustment of economic objects If for example weconsider sectors we are evaluating the adaptation of an entire set of firmswhich pose questions on how to aggregate their dynamics Things changewhen we consider the response of a single firm

Going back to Horvatsrsquo argument highly disaggregated matrices arelikely to contain many empty cells so that we may forecast different levelsof volatility depending on the aggregation used for the same system whichas mentioned is not caused by sparseness but to a failed adjustment inreaction mechanisms Results from our model20 show that volatility is com-pletely unaffected by input coefficients distribution Given their sectoralsum mdash through rows of the matrix mdash the value of each coefficient is irrele-vant The same occurs in the case of very skewed distributions that presenta high number of empty cells (zero input coefficients)

58 Microndash and Macrondashvolatility

As we have seen above one of the major discussion in the literature concernsthe relations between microndash and macrondashvolatility Put it simply one sideof the literature considers that uncorrelated microndashshocks cancel each otherout reducing their effects at macrondashlevel Conversely other researchersconsider that microndashshocks are multiplied at macrondashlevel generating highermacrondashvolatility than that we can observe at microndashlevel The debate isof obvious importance to determine the sources of business cycles to pre-dict their effects and determine the more effective policies to mitigate theirnegative effects

Our model is only a partial representation of realndashworld economic sys-tem and is also overly simplistic However based on these limitations italso offers very robust results allowing reliable conclusions on the (limited)elements included in the model In particular we consider our model a re-liable representation of the basic structures of the production interactionsamong sectors or firms Thus we can use the model to provide an answerto the debate concerning the contributions to volatility from the productioninteractions only Other contributions (eg from financial markets priceadjustments technological innovation etc) may re-inforce or counter theforces we analyse but they need to be considered as additional elements(and we opine also generally less relevant) not alternative to our analysisWe consider a necessity in order to apply a rigorous methodological ap-proach to be able to identify with certainty the results provided by partialstudies in order to eventually obtain by gradual extensions of the analy-sis more and more detailed representations of realistic economic systems

20Available from the authors

22

The results we obtain in this paper concerning this microndashmacro debate aremeant as a first relevant step to a more extended analysis

0

002

004

006

008

01

012

014

016

018

02

022

05 1 15 2 25 3

Var Demand

Macro (Output) and micro (Demand) volatility

delta Outputdelta Demand

Figure 8 Values of δ (relative deviation) for aggregate demand and ag-gregate output in respect of different levels of variance for sector specificuncorrelated external demand shocks

The flows of shocks affecting the firms of a system can be distinguishedalong two dimensions strength of the shock (eg amplitude of the changesin demand) and frequency of the shock (rare or frequent) Concerning thefirst aspect we initialized as usual a system to equilibrium levels (eg pro-duction quantities equal to external and intermediate demand for all sectorsconstant level of stocks equal to the desired level) We then ran the simu-lation generating the external demand for all sectors as a random variablewith constant (equilibrium) mean level and varying values of variance

Figure 8 shows unequivocally that as far as our model has propertiessimilar to real systems the hypothesis that macrondashfluctuations have largervolatility than their sectoral level shocks is fully confirmed At any levelof variance the aggregate volatility of demand as captured by the relativedeviation index is sensibly lower than the volatility shown by aggregateoutput that is the total production of the system This result is not sur-prising throughout all tests presented in the previous sections every resultsuggested that the system lsquooverndashreactsrsquo to any individual microndashlevel changeeven in the case these changes were meant to contrast disturbances There-fore when we move from considering one single shock to a flow of shockswe are not surprised to obtain that the system shows higher volatility thanthat implied by the microndashlevel disturbances

We now consider the second dimension defining a flow of shocks theirfrequency The question is whether for the same level of demand shocks

23

0 002 004 006 008 01 012 014 016

Difference of relative deviation for Output and Demand

0 5 10 15 20 25 30 35 40 45 50 55

Shock frequency 05

1

15

2

25

3

Var Demand

0 002 004 006 008 01

012 014 016

Figure 9 Differences of values of δ for aggregate output minus those foraggregate demand in respect of different levels of variance of demand and offrequency of the shocks

(ie same variance) more frequent changes trigger a higher or lower systemvolatility than rare disturbances The results shown in figure 9 suggest arather elaborated answer which partly vindicates (though in a particularsense) the arguments in favour of compensation of micro shocks The fig-ure shows the difference between aggregate output volatility and aggregatedemand volatility The independent variables are as before the volatilityof demand at microndashlevel21 and the time intervals in between two shocksAggregate demand volatility does not change for the frequency of shocksat least when computed over a long period But output volatility doesin a nonndashmonotonous way For high frequency of shocks the difference isstill positive and increasing for higher variance indicating that the ldquooverndashreactionrdquo argument is valid even in these cases but it is rather low Whilethe frequency of shocks slows down aggregate volatility of output increasesshowing that time is needed for the microndashshock to unfold their full effectsat the macrondashlevel However after a certain threshold less and less frequentshocks generate a lower volatility of the aggregate level of output

This result is quite sensible though not obvious at first sight and onceagain shows the power of such a simple model to generate and explain com-plex properties For frenetic levels of demand modifications the system hasno time to adjust but part of the job to align actual and desired productionfor constantly changing demand is performed by the varying demand itselfIn other terms the shocks cancel out through time if not through their quan-

21Beware of the scale of variance For reasons of visibility the scale is inverted

24

titative effects on the system While frequency decreases the disturbancesof the shocks not compensated by other (unlikely) shocks have the time tofilter through the production system generating higher and higher volatil-ity In a sense any system has a particular frequency of shocks that make itmaximally ldquoresonatingrdquo with the amplification of microndashshocks When therate of shocks gets even rarer the system can enjoy periods of longer andlonger quietness after each shock is absorbed and before a new one startsagain a cycle of initial local disturbance propagation of the disturbances toother sectors and settlement to the new equilibrium

6 Discussion and extensions

We presented a simple model of production composed by interacting sectorThe main (quit realistic) assumption of the model is that the productionprocesses in any sector of the economy make use of input from any othersector Moreover we assumed that sectors (represented as a single firm)adjust slowly both the desired level of production (smoothed revision of ex-pectation) and the actual level (smoothed modification of plant utilization)The model voluntarily leaves aside a large number of aspects of real-worldeconomic systems (eg financial markets price adjustments growth etc)The reason is that the model can be easily and reliably tested in order toextract its properties The possibility to transfer the (certain) model resultto a real context depend of course on an ample ceteris paribus clause How-ever we are confident that the reported results concerning the productionstructure play a relevant role in real world systems Therefore our resultsand particularly their motivations being accessible because of the relativesimplicity of the model will not be diminished by the integration of ourconsiderations with other aspects of economic systems

Our model is in essence a representation of an input-output table inter-preted in physical terms that is as the amount of inputs required for oneunit of output Besides this we include two straightforward assumptionsFirstly production levels cannot be suddenly changed but require time toscale up or down the utilization rate that is the levels of production Stocksare meant to make up for the difference between sales and production Thesecond assumption concerns the intentional behaviour of producers We im-plement this aspect by assuming that exists a level of stocks desired for eachlevel of demand However a change of demand does not translate suddenlyin a new level of desired stock but again only a long time at a different levelof demand will convince managers to update the desired levels of stocks

We show that the model generate under a wide number of settings oscil-latory patterns converging toward the equilibrium level where each sectorgenerate exactly the quantity demanded by final consumers and by othersectors as input The oscillations are provoked by over- and under-shooting

25

generated by the difficulty of coordination among sectors that by assump-tion ldquocommunicaterdquo only via variation of demand to their direct input sup-pliers We also show that the total production levels generated by sectors(sum of the physical units) produced at equilibrium depends (for constantfinal demand) on the total sum of the technical coefficients per sector irre-spective of their distribution This result is motivated by the impossibilityof input substitution and offers interesting suggestions for the analysis ofthe study of input-output tables of real-world systems

We then consider systems at equilibrium facing a single small change ofthe external demand for sectors These simulations allow to study the waythe system reacts to shocks We see that the sum of the coefficients affectsbesides the levels also the volatility of aggregate output the larger a systemthe stronger is the impact of a given shock Less obviously we find thatfor systems with the same dimension (ie same sums of coefficients) theaggregate volatility increases with both the inertia to change of productionlevel and the speed of adjustment of desired stocks This result is somewhatcounter-intuitive since both the cited variables are associated to a moreconservative behaviour that may be supposed should smooth away spikesin the pattern from an old to a new equilibrium However we show thatthe opposite actually holds given the complex interactions among sectorindividual-sector attitude intended to smooth shocks actually increase theiraggregate impact

In the same vein we show that systems maintaining high levels of stocksare more volatile that systems with reduced amounts of stocks Again thisis a counter-intuitive result reverting the goal of individual producers andthe aggregate result Interestingly this result finds strong support from theevidence of the recent changes in production methods (ie low or no stocks)and diminished cycles

In relation to specific issues discussed in the literature we find no supportto the suggestion that particular distributions of input-output coefficientsmodify the volatility of a system As said this depends in case on thesums of these coefficients but not on their distribution Further we canprovide an answer to a long and hotly debated questions whether microndashlevels volatility generates stronger or lower aggregate volatility Our modelshows unequivocally that microndashvolatility (eg variance of final demand)persists and generates stronger volatility at aggregate level Again we canidentify the reason on the complex production structure that exalts micro-level shocks by generating lsquowrongrsquo messages among producers Moreoverwe can also point to a usually neglected aspect of volatility we show thatthe frequency of shocks is a very relevant aspect besides their dimensionin determining the volatility of aggregate output

Our work can be developed along two complementary directions Firstlywe can continue to study the properties of the model by extending the ele-ments considered (eg pricing) analysing the effects on the present results

26

and generating more analytical evidence Of particular relevance is in ouropinion the problem of aggregation Representations of the same systemswith different codification systems (eg different number of SIC digits)must not modify overall system properties for example the volatility ob-served Given that our result show that the sums of coefficients matter forthe levels and volatility of the system and that these sum do depend on theaggregation chosen it should be possible to induce the reaction coefficientsthat we know affect the volatility only

Secondly we can find an application of our results to real world evidenceIn fact there are many data sets available for the data we deal with basicallyinput-output tables Our model is readily adapt to be parametrized alongany number of sector and coefficient necessary so that we may replicate pastseries in order to adapt the unobservable parameters and use the resultingcomplete model to explain business fluctuations and provide intuitions onfuture patterns

27

References

Acemoglu D and A Scott (1997) ldquoAsymmetric Business CyclesTheory and TimendashSeries Evidencerdquo Journal of Monetary Economics 40501ndash533

Backus D P Kehoe and F Kydland (1995) ldquoInternational BusinessCycles Theory and Evidencerdquo in Frontiers of Business Cycle Researched by T Cooley Princeton University Press Princeton

Bak P K Chen J Scheinkman and M Woodford (1993) ldquoAg-gregate Fluctuations from Independent Sectoral Shocks SelfndashOrganizedCritically in a Model of Production and Inventory Dynamicsrdquo RicercheEconomiche 47(1) 3ndash30

Battiston S D Delli Gatti M Gallegati B Greenwald and

J Stiglitz (2005) ldquoCredit Chains and Bankruptcies Avalanches in Sup-ply Networksrdquo in Annual Workshop on Economic Heterogeneous Inter-acting Agents (WEHIA 2005) University of Essex UK

Blanchard O and J Simon (2001) ldquoThe Long and Large Declinein US Output Volatilityrdquo Brookings Papers on Economic Activity 1135ndash174

Bresnahan T F and V A Ramey (1994) ldquoOutput Fluctuations atthe Plant Levelrdquo The Quarterly Journal of Economics 109(3) 593ndash924

David P A (1975) Technical choice innovation and economic growth essays on American and British experience in the nineteenth centuryCambridge University Press London

Delli Gatti D C Di Guilmi E Gaffeo G Giulioni M Galle-

gati and A Palestrini (2004) ldquoA New Approach to Business Fluc-tuations Heterogeneous Interacting Agents Scaling Laws and FinancialFragilityrdquo Journal of Economic Behavior amp Organization 56 489ndash512

Dosi G G Fagiolo and A Roventini (2006) ldquoAn EvolutionaryModel of Endogenous Business Cyclesrdquo Computational Economics 273ndash34

Dupor B (1999) ldquoAggregation and the Irrelevance in MultindashSector Mod-elsrdquo Journal of Monetary Economics 43 391ndash409

Head A C (1995) ldquoCountry Size Aggregate Fluctuations and Interna-tional Risk Sharingrdquo The Canadian Journal of Economics 28(4b) 1096ndash1119

28

Helbing D and S Lammer (2005) ldquoSupply and Production NetworksFrom the Bullwhip Effect to Business Cyclesrdquo in Networks of Interactingmachines Production Organization in Complex Industrial Systems andBiological Cells ed by D Armbruster K Kaneko and A S Mikhailovvol 3 of World Scientific Lecture Notes in Complex Systems World Sci-entific

Helbing D S Lammer U Witt and T Brenner (2004) ldquoNetworkndashInduced Oscillatory Behavior in Material Flow Networks and IrregularBusiness Cyclesrdquo Phisical Review E 70(056118) 1ndash6

Helbing D U Witt S Lammer and T Brenner (2004) ldquoNetwork-Induced Oscillatory Behaviour of Macroeconomic Output Flowsrdquo in In-dustry and Labour Dynamics II The Agent-based Computational Eco-nomics Approach Proceedings of the wildace 2004 conference ed byR Leombruni and M Richiardi no 5 LABORatorio R Revelli

Horvath M (1998) ldquoCyclically and Sectoral Linkages Aggregate Fluctu-ations from Independent Sectoral Shocksrdquo Review of Economic Dynam-ics 1 781ndash808

(2000) ldquoSectoral Shocks and Aggregate Fluctuationsrdquo Journal ofMonetary Economics 45(1) 69ndash106

Jovanovic B (1987) ldquoMicro Shocks and Aggregate Riskrdquo The QuarterlyJournal of Economics 102(2) 395ndash410

Kahn J A M M McConnell and G Perez-Quiros (2002) ldquoOnthe Causes of the Increased Stability of the US Economyrdquo FRBNY Eco-nomic Policy Review 8(1) 183ndash202

Kraay A and J Ventura (1998) ldquoComparative Advantage and theCrossndashSection of Business Cyclesrdquo World Bank Woorking paper Series1948 World Bank

Long J B and C I Plosser (1983) ldquoReal Business Cyclesrdquo Journalof Political Economy 31(1) 39ndash69

Lucas R E (1977) ldquoUnderstanding Business Cyclesrdquo CarnegiendashRochester Conference Series on Public Policy (special issue of Journalof Monetary Economics) 5 7ndash29

Nelson R R and S G Winter (1982) An Evolutionary Theory ofEconomic Change Harvard University Press Cambridge MA

Nirei M (2005) ldquoThreshold Behavior and Aggregate Fluctuationsrdquo Jour-nal of Economic Theory in press

29

Scheinkman J and M Woodford (1994) ldquoSelfndashOrganized Criticallyand Economic Fluctuationsrdquo American Economic Review 84(2 Papersand Proceedings of teh AEA) 417ndash421

Schuh S (1996) ldquoEvidence of the Link Between FirmndashLevel and Ag-gregate Inventory Behaviorrdquo Finance and Economics Discussion Series1996-46 Federal Reserve Board

Stiglitz J and B Greenwald (2003) Towards a New Paradigm inMonetary Economics Raffaele Mattioli Lectures Cambridge UniversityPress Cambridge

Weisbuch G and S Battiston (2005) ldquoProduction Networks andFailure Avalanchesrdquo Working Paper mimeo Ecole Normale SupErieure

Zarnovitz V (1977) ldquoBusiness Cycles Observed and Assessed Why andHow they Matterrdquo NBER Working Paper Series 6230 National Bureauof Economic Research

30

A Tables

Table 1 Parameters valuesParametera Description Value

N Number of sectorsunits i of production 10Ci Number of j buyer sectors for sector i (IO matrix sparseness) N minus 1Di (E) Value of the external demand 10mij Physical input coefficient of input j for sector i sim U(0 1)Mi Sum of the mij input coefficients j for the production of i 05αi friction to changes of production levels Degree of lockndashin on

production technology085

αsi Speed of adaptation to demand changes 004

σ Desired multiple of production to stock 10

aInto parenthesis the number of lags of the initial value for lagged variables

31

Using a very similar production structure with limited interactionsWeisbuch and Battiston (2005) play on the bankruptcies generated on down-stream firms by production failures occurring in upstream firms (reducingthe flow of inputs) Stochastically generated failures to produce in a firmcauses production constraints in its downstream clients hence lower invest-ment and eventually bankruptcies3 A lag between the period in whicha firm goes bankrupt and a new firm appears generates a lack of supplywhich propagates on the downstream sectors

Due to the original literature on avalanches these contributions sharetwo common features that are ill-adapted to economic systems First theyassume very peculiar interaction structures of one-directional productionrelations between sectors which amount to assume one half of the input-output matrix empty and are not applicable to the general case with cyclicalproduction structure or full IndashO matrices Second the strongly rely on quiterigid thresholds determining a mechanicist behaviour by agents Basicallyagents have only two options available (ie produce or not produce) andthe choice is made on the basis of crude decisional mechanism4

Notwithstanding the limitations this approach provides a quite sensiblemechanism of the origin and transmission of shocks through a system Ourproposal can be interpreted as an extension of this approach by relaxing thestrong assumptions on the interaction structure and allowing agents for amore fluid decisional process as we will see in the next section

3 The basic elements of a production structure

Our overall hypothesis is that the very production structure of an economymay generate persistent fluctuations even in the absence of continuous un-related shocks and rigid thresholds that determine non linearities The aimis to provide a generalised interpretative framework to explain the occur-rence of short term fluctuations in the economy simply as an outcome ofthe division of labour in a number of sectors related through trade In itsextreme interpretation any production system which is not fully integratedis bound to produce business cycles As we will show a number of factorsinfluence their extent

Quantitative adjustments in production in any one sector require ad-justments in related sectors When the structure of the economy is trulyInputndashOutput (not a directed linear supply chain as in the Av models) in-put flows are cyclical and adjustment to long run production equilibrium islikely to take a long mdash when not infinite mdash time Our work departs from

3A similar model by Battiston Delli Gatti Gallegati Greenwald and Stiglitz (2005)relates bankruptcies to price changes buyers payment timing and the credit market (acentral bank)

4Nirei (2005) is an attempt to generalise such models though maintaining the assump-tion of threshold behavior

7

both RBC and Av models with respect to structure modelling criteriaanalysis and methodological approach Indeed our model may be easilyreconduced (or restricted) to both We are much in line with a recent se-ries of papers by Helbing Witt Lammer and Brenner (2004) that addressthe same question although with a number of differences that are discussedlater on5

We aim to analyse to what extent the structure of inputndashoutput relationscauses aggregate fluctuations by itself The main idea is to show that ag-gregate fluctuations may derive from simple micro imbalances without theneed for continuous exogenous shocks not even at the firm or sectoral levelIn other words normal production activities including lags technologicaladjustments information asymmetries circularity of the inputndashoutput sys-tem and so on may be enough to explain part of the macro volatility in aclosed system6

Our assumptions draw on the existing literature but try to avoid themajor difficulties highlighted above Firstly we avoid to impose either mar-ket clearance assumptions or crude threshold behaviours Rather we adoptan assumption close to a Simonian approach agents in an economy do theirbest within the informational and physical constraint they are subject toWe represent decision makers as responding to a single piece of informationthey obtain the quantities demanded by their clients Crucially their ac-tions are inspired to a conservative approach an increase in the demandgenerates a smaller (desired see below) sudden increment of productionThis assumption stems from the fact that decision makers are aware of theinstability of their environment and therefore want to avoid getting perma-nently caught with over production following a temporary spike of demandIn other words firms attempt to smooth business cycle generating a morestable (or less uncertain) environment as the economic literature would sug-gest Notice that this representation in line with a routinized representationof organizations (Nelson and Winter 1982) does not contradict market equi-librium Rather in a stable environment such decisional procedure generatesan asymptotic pattern to the equilibrium level in the absence of feedndashbacks

A second assumption concerns the technological possibilities available tothe firms When firms realise the need for a change in production levels saya desired increase of production they face a complex and costly organisa-tional transformation of the production capacity risky investments an soon As Bresnahan and Ramey (1994 p 622) suggest inducing from a de-tailed analysis of the automobile industry in the US ldquoadjusting production

5We thank Matteo Richiardi for pointing out these similar attempts See also Helbingand Lammer (2005) and Helbing Lammer Witt and Brenner (2004)

6An open system entails many other factors including price factors of both inputsand outputs mdash terms of trade labour cost external investment etc As with exogenousshocks our analysis does not deny the importance of these factors but simply separatethem from the structural explanation of cycles

8

is a more complicated process than simply ldquochanging Qrdquo or choosing themix of capital and laborrdquo And this is not only a matter of non convexitiesin production technologies The inconsistency of time reversibility in thetechnological choice is accompanied by the need to operate on the lsquoavail-able technological frontierrsquo (David 1975) If a firm plans a given productionactivity for a short period of one week even though the next day there is anunexpected fall in the demand it is unlikely that it is followed by a similardownturn in production The case is evident when we think at productionplanning that involve hiring of workers an capital investments (both in-volving sunk costs)7 Therefore any desired change generates a pattern ofsmall modifications approaching slowly the target level Moreover thesechanges are irreversible in the short term since only a prolonged change ofthe environment triggers the modification of previous desired levels

Resting on those simple conjectures we thus analyse a model in whichwe only allow for sluggish adaptation of production due to conservative(adaptive) behaviour of firms and the physical constraints faced when vary-ing the production levels Notice that these assumptions are contrary tothe generation of instability of a system In fact a firm behaving as de-scribed above reduces if not eliminates any disturbance affecting its ownstate if acting in isolation Other factors like new technologies extremelylong term perspective price changes etc are likely to be sources of volatil-ity therefore contributing to generate aggregate system fluctuations (as forexample in Acemoglu and Scott 1997) Our neglect of these and other fac-tors however unrealistic serves the purpose of testing whether a volatilityreducing microndashbehaviour turns into volatility generating aggregate resulttwo widely observed dynamics Quite interestingly (and encouraging) DosiFagiolo and Roventini (2006) have concomitantly proposed a model whichincorporates detailed representation of both sluggishness robustly based onempirical evidence Heterogeneous firms which undergo lumpy investmentand slowly adapt their expectation on changes of the future demand inducecomovement of macroeconomic variables qualitatively similar to the onesobserved in the real world To pursue our line of research we first needto concentrate on how sectorsfirms interact in the system in the effort tocomplement those promising results

Therefore the second element we consider is meant to represent a gen-eralized economic system We allow for the circularity of inputndashoutput sys-tems a crucial feature in determining feedback mechanisms and a principal

7See also the discussion in Acemoglu and Scott (1997) where the authors argue thatldquo[w]hile the presence of fixed costs can account for the discreteness of economic turningpoints it does not naturally lead to persistence because once an individual undertakes anaction they are less likely to do so in the near future [] [A]lthough the presence of fixedcosts leads to increasing returns these are intratemporal the full extent of economiesof scale arising from fixed costs can be exploited within a periodrdquo [p 502 authorsrsquoemphasis]

9

component of business cycle It may be true that most backward linkagesconcern capital goods (Weisbuch and Battiston 2005) but a rapid glanceto an inputndashoutput matrix would show that technical coefficients are nonnull in both directions and ldquomost commodities are inputs to the produc-tion process of other commoditiesrdquo (Horvath 2000 p 70) The higher thesectoral aggregation mdash observed or assumed mdash the more the relevance ofintermediate supply holds true (as mentioned above with respect to ma-trix incompleteness) We claim that the existence of backward and forwardlinkages are a crucial element to determine the persistence of shocks in theeconomy A perfectly linear (one directional) system needs much less adap-tation of firms and sectors in order to stabilise the economy Nonethelesscircularity does not imply that inputs are perfectly substitutable amongthem in order to determine the maximising allocation even at high level ofdisaggregation

By using a fixed coefficients production function we both maintain themid term rigidity of production processes and we do not impose limitedinteractions We claim that the circular property of the production structureis by itself sufficient to produce shock persistence even when the inputmatrix is full (although with heterogeneous input coefficients) mdash all sectorsshop in the remaining n minus 1 sectors

In conclusion we represent a model for an economic system made onlyof i) production level decisions and ii) input-output relations Clearly sucha model lacks many features present in real economic systems likely we areconvinced to affect the propensity to generate cycles As such we cannotaim at reproducing realistic data so our results cannot be tested comparingtime series generated by the model with real ones Instead given the paucityof the elements comprising the model we can investigate in detail to whichextent the (few) features present in our model affect the cyclical behaviour ofthe system As we will see we will be able to show quite a number of resultin which aggregate fluctuations are generated whereas the individual microstructure of the model aims at monotonous adaptation Before discussingthe results next section describes in detail the implementation of the simplemodel

4 The model

We model an economic system in which a number of sectors i = (1 N)are potentially connected via (intermediate) market relations A N times NInputndashOutput matrix determines the mij units of each input j = (1 N)mdash bought from the N minus 1 residual sectors mdash necessary to produce oneunit of output i8 For simplicity given the focus of this paper on the IndashOstructure as a generator of aggregate fluctuation we abstract from market

8IndashO coefficients are therefore physical coefficients

10

interactions within sectors each sector is a production unit In fact forour purposes it is not relevant which firm produces the sectoral outcomebut only the total amount produced by each sector9 Moreover besidesmaking the analysis more neat this assumption allows also to enjoy a largefreedom in determining the aggregation level since we donrsquot need to assessthe effects of between sectors competitiveness which for example is likelyto increase for increasing levels of disaggregation Our economic systemis therefore composed of N production units each of which may buy andprovide inputs from and to any other sector depending on the IndashO structureof the economy

The demand for each sector i is determined by a constant consumersdemand Di (E) and the input needs of other sectors

Di = Di (E) +sum

j 6=iisinCi

mjiQj (1)

where Ci is the set of j intermediate clients of i (sum

j isin C lt N) mji is thetechnical coefficient of firm j for input i ie the amount of i required toproduce one unit of j Qj is the actual output produced by sector j

Concerning the simultaneous production level of each sector in time twe make two lsquorealisticrsquo assumptions First from the technical viewpoint afirm like a tanker cannot abruptly change its course Therefore changesto the production level can be introduced only gradually through capital(dis)investment and job h(f)iring (see discussion in Section 3) Second weassume that firms do not decide the production level directly on the basisof the observed demand level but conscious of the uncertainty of demandplan their production in order to maintain invariant a desired level of stocksand avoid stockout10

Quantity produced by sector i in period t is computed as follows

Qit = αiQtminus1 + (1 minus αi)(

Slowastitminus1 minus Sitminus1

)

(2)

where αi measures the physical constraint in adapting the production capac-ity to changes in the required output Slowast

itminus1are firmrsquos i desired stocks and

Sitminus1 its actual stocks Notice that both variables related to stocks appearwith a lag This is due to two reasons First it is a modelling necessity inorder to allow the simultaneous determination of production levels for allsectors Secondly as frequently happens modelling necessities reveal sen-sible if not generally considered aspects of the modelled system In factproduction decisions concern management in the production plant while

9Adding heterogeneous firms within sectors would definitely enrich the causal expla-nation put forward and parametrised in this paper (see below) For example to analysethe effect of firms constraints It is left over for future research as in the present work weopt for the analysis of the main structural parameters that affect aggregate fluctuations

10See also discussion in Schuh (1996)

11

demand through sales is observed by the commercial staff It is thereforelogical that the information from demand reaches plant managers with somelag

The actual stock level is trivially computed subtracting from the previousheap of stock the amount sold and adding the new production

Sit = Sitminus1 + Qit minus Dit (3)

Concerning the desired level of stocks instead we introduce the conser-vative behaviour mentioned above Firms ideally would keep an amountof stocks proportional to the demand they receive say a multiple σ ofthat amount However when demand varies they adjust the desired levelsmoothly preventing sudden changes to the variable representing the goalof the firm

Slowastit = Slowast

itminus1 + αsi

(

σDit minus Slowastitminus1

)

(4)

where αsi measures the adaptation to demand changes

In summary the dynamics of the model consists of only the equationsabove Firstly at the beginning of a time step firms determine the quantityto produce as a function of past imbalances in stock levels and the pastproduction level These levels define the demand for all sectors determinedby the technical coefficients applied to all produced quantities Finallyactual and desired stocks can be updated Figure 1 represents visually thisdynamics highlighting the decisional components of the model in respect ofthe ldquomechanicsrdquo of trade

5 Results production structure and economic fluc-

tuations

In this section we describe some of the most relevant results concerningthe effects of the production structure to determine aggregate fluctuationsGiven the unusual analytical instrument chosen (at least concerning thistopic) we start with a brief methodological introduction meant to clarifythe nature of the results we claim to obtain Next we describe a generalsetting for the model which is the (extremely) limited area of the potentialparametersrsquo space of the model that we will explore In the rest of thesection we discuss the results from the model

51 Methodological considerations

Real economic systems are constantly subject to a continuous flow of shocksexternal to the production system besides changes within the system itselfIn order to appreciate the systemic properties we need to abstract from all

12

Compute the quantityto produce

F (Q(t-1) S(t-1) s(t-1) alpha)

Communicate thequantity of inputsrequired to each

supplier

Determine the overalldemand

F(D(E) D(I))

Revise the past decisionon stock plans

(desired stocks)F(D S(t-1) sigma alpha(s))

Compute the final levelof stocks for next period

F(s(t-1) Q D)

KEY

ModelDirection

Variablecomputed

at t-1(t-1)

Step includingfirm behavior

Figure 1 One period dynamics

the nonndashrelevant disturbances and concentrate on how the system endoge-nously contributes to the observed dynamics Therefore we will generatehighly abstract patterns whose analysis will provide insights on the systemproperties that would be otherwise lost in the noise generated by all theelements of a real system

We are not trying to fit the model to a specific empirical data set nor weare interested in studying the complete behaviour of the model for areas ofthe parameterrsquos space that make no economic sense (eg negative or infiniteproduction levels) Therefore the values of initial settings and results matteronly in relative and not in absolute terms

We are going to describe a simulation model with which we will produceseveral simulation runs and we claim obtain results relevant for the debatediscussed above Rigorously speaking we may claim only certainty for thevalidation of the results that is that the model does actually generate theresults presented and for the reasons we explain We limit this part to thepresentation of graphs and verbal explanations that we believe are ratheruncontroversial11 Concerning the relevance of our studies for real systems

11Interested readers can request the model code and and simulation data for replicationand extensions of the results We can guarantee that such possibility does not require

13

(verification) we simply avoid to pretend that our model is a quantitativeapproximation of real economies (which in which period) Our opinionis that the study of the model allows us to make general considerationsnon-obvious and relevant for applied debates supported by strong logicalarguments which can be integrated but not reverted by arguments con-cerning more and more elements of the economic world

52 Model setup

We go through the model properties analysing an economic system madeup of ten sectors (N = 10)12 Each sector shops inputs from the remain-ing N minus 1 and sells to them part of its own production as intermediategoods This means that the input matrix is complete except for the diag-onal (mij gt 0 forallj 6= i mii = 0) The sum of all N minus 1 coefficients for asingle sector is given by Mi and its benchmark value is kept at a plausiblemedium level (unless differently specified) to avoid distortions from highlydemanding sectors Each input coefficient is randomly drawn from a uniformdistribution

Sectors start at an equilibrium level their supply matches intermediateand final demand and in each period the same quantity of output mustbe produced to compensate for the used stocks and keep the desired stocksunchanged In thier turn stocks must be ten times the value of total sectoraldemand (σ) Eventually technological adjustment of production is ratherslow and producers are quiet refractory to follow sudden changes in thedemand in general a smoothing behaviour strongly prevails (α and αs)

Under those general preconditions in the following sections we anal-yse the dynamics of the economic system in relation to model parametersWe mainly focus on the cyclical behaviour of systems and their speed ofconvergence to the asymptotic equilibrium13

The very abstract assumption and the homogeneous initialisation donot allow for the multiple equilibria that are likely to rise in complex systemswith feedbacks Once more such simplification allows to have a betterunderstanding of the crucial impacts of the production structure on businessfluctuations Departing from the related literature discussed above14 weconsider the effects of a single shock rather then a continuous flow whichin our model is superfluous in order to obtain fluctuation persistence15 To

extensive programming or statistical skills besides the usual education of economists12You can think of a high level of aggregation in the statistical observation The com-

plete initialisation and parameters values are available in table 1 in the Appendix13Notice that sectors only approach a final equilibrium level of production which can

be prendashdetermined under static conditions14Included the more proximate work of Helbing Witt Lammer and Brenner (2004)15An exception are the last results presented in Section 58 which we see more as an

opportunity for further research than a closing of the paper

14

start with next section (53) shows the aggregate cyclical behaviour of astandard setting

53 Single shock analysis

We generate a simulated history following the dynamics of the modelrsquos vari-ables when the system is set to an equilibrium level and a single shockmodifies one of the exogenous elements namely the external demand forsectors Modifications across sectors are random and not correlated andmay increase or reduce external demand by at most 10

100 225 350 475 600

193894

195861

197828

199795

201762 Industrial Production

Time

Figure 2 Total industrial production fluctuations following a shock fromexternal demand

Figure 2 shows the dynamics generated by the total industrial productionof the system mdash that is also a proxy for the systemrsquos GDP fluctuations Thenew external demand changes the equilibrium level but the system cannotreach the new equilibrium with a monotonous pattern and converges to thenew level with a fluctuating pattern Each change in the external demandinduces also modifications of the intermediate demand and sectoral adjust-ments in the quantity produced typically overshoot the aggregate patternAs it can be appreciated from the figure it takes time before the system isable to converge to the new static equilibrium conditions

We study two features of this pattern absolute level of the new equilib-rium and volatility induced by the shock

54 Technical coefficients and industrial output

Production level

The equilibrium levels of industrial output depend on two factors only Ob-viously the levels of external demand determine how much of the total pro-

15

duction needs to be supplied to consumers exogenous from the productionsystem Second the technical coefficients also influence strongly the level ofactivity of the system defining the intermediate demand to each sector

180

200

220

240

260

280

300

01 02 03 04 05 06 07 08 09

Out

put l

evel

(In

dust

rial p

rodu

ctio

n)

Sum of total input coefficients (M)

Total industrial production for different input coefficients levels

Figure 3 The relevance of input coefficients on output level

Figure 3 shows the equilibrium level of industrial output reached bysystems with identical external demand where we fixed the sum of inputcoefficients for each sector of the economy (Mi) to different values16 Thehigher the sum the higher is the total production sustained by the system

This result depends on the very production structure represented by theinputndashoutput matrix In fact the higher is the level of production requiredby intermediate sectors the higher is the total production given a constantlevel of external demand

Notice that the level of production does not depend on the distribution ofthe technical coefficients As long as the sum of the coefficients is identicalthe same production level is obtained irrespective of their distribution (otherthings being equal)

Output shock absorption

Input coefficients have a relevant role also on the output stability of thesystem Starting from an output equilibrium level we hit the system witha single shock on the external demand for all sectors uncorrelated amongsectors17 modifing (positively or negatively) the demand for all sectors by

16Within this limit the actual value of the coefficients are randomly drawn17In each sector the external demand changes by plusmnX where X is a uniformly dis-

tributed percentage between 1 and 10

16

at most10 As show in Figure 2 the economic systems produces a lenghtlysmoothing oscillatory behaviour We compute the relative deviation of theaggregate production (industrial output) as the ratio between its standarddeviation and its mean δ = σY microY where Y =

sum

Qi Figure 4 shows thebehaviour of δ as a function of the sum of input coefficients for each sector80 periods after the shock As time goes by the level of δ reduces for allMi but the relation with the sum of input coefficient is unchanged

00055

0006

00065

0007

00075

0008

00085

0009

01 02 03 04 05 06 07 08 09

Rel

ativ

e va

rianc

e of

tota

l out

put (

delta

)

Sum of total input coefficients

Aggregate output deviations for different technical coefficients

Figure 4 The relevance of input coefficients on output stability

In sum input coefficients determine the extent of overshooting behaviourThe higher the coefficients the higher is the backlash even when shockscould cancel out through sectors

55 Shock persistence and adjustment coefficients

We now consider the effect of micro smoothing behaviours on the volatilityof the production system In particular we are interested in testing theeffects of two parameters of the model on the volatility of the productionsystem First we consider the effects of the αrsquos representing the ldquostickinessrdquoof the production technology to desired changes of Q aimed at filling thegap between actual and desired stocks The higher is α the slower is thereaction to stock unbalances Concerning its effect referring to the technicalpossibilities available ie production capital and organization we may putforward two opposite hypothesis relevant to systemrsquos volatility

Hypothesis 1 Slower production adjustment (high α) reduces volatilitybecause unbalances in demand of one sector being compensated slowly slow

17

down the diffusion of shocks to other sectors

Hypothesis 2 Slower production adjustment (high α) increase volatil-ity because the longer lasts the mismatch between actual and equilibriumproduction level the deeper will become the mismatch between actual anddesired stocks

20 25 30 35 40 45 50 55 60 65

Cyclesrsquo width

07 075

08 085

09alpha 0005

001 0015

002 0025

003 0035

004 0045

005

alphaS

20 25 30 35 40 45 50 55 60 65

Figure 5 Maximum width recorded between highest and lowest peaks fortotal production for different values of α and αS

Secondly we consider the parameters αS representing the speed of ad-justment of desired stock to observed demand The higher this parame-ter the faster sectors revise the desired levels of stocks to current levelsof observed demand This parameter concerns sectorsrsquo behavioral decisionprocess Higher αS represent a stronger belief that the most recently ob-served demand level is a reliable indicator of future demand levels whilelower levels of the parameter represent a more conservative approach wheremanagers require a long observation of demand levels different from the pastones before adjusting their expectations Also concerning the effect of thisparameter on the volatility induced by an exogenous shock we may maketwo hypothesis

Hypothesis 1 Faster adaptation of desired stocks to new levels of de-mand (higher αS) reduce volatility because changes in demand are rapidlytranslated in adjusted levels of production

Hypothesis 2 Faster adaptation of desired stocks to new levels of demand(higher αS) increase volatility because temporary unbalances of demand in

18

one sector are transmitted faster to other sector reinforcing the feedbackmechanism of fluctuations

To test which of these hypotheses are confirmed by the model we runsimulation runs with identical systems (eg same technical coefficients samelevels of demand same shocks) but for the values of the αrsquos and αS rsquos testingall combinations of these parameters over a range of values (each run usedthe same parameters for all the sectors of the system) For each simulation(ie each couple of values of the parameters) we computed two indicatorsof volatility One measures the maximum distance of total production be-tween the highest and the lowest peak that is the maximum width of theoscillations registered in the production time series The second indicatormeasures the relative deviation δ = σY microY Figures 5 and 6 report theseresults

0004 0006 0008 001 0012 0014 0016 0018 002

Relative deviation

07 075

08 085

09alpha 0005

001 0015

002 0025

003 0035

004 0045

005

alphaS

0004 0006 0008 001

0012 0014 0016 0018 002

Figure 6 Relative deviation of total production for different values of α andαS

The results confirms unequivocally both hypotheses 2 and reject hy-potheses 1 In a sense these hypotheses confirm the nature of the economicsystem represented in the model as a complex system where the interactionsamong sectors play a far more important role than the individual sector be-haviour In fact in both cases we may interpret the results that the strongerthe efforts to mitigate the effects of shocks (ie slow productionrsquos and quickexpectationsrsquo adjustment) the opposite result is actually obtained the fluc-tuations become even more accentuated It is also worth to note that thestronger impact is generated by the behavioural parameters rather than thetechnical one as indicated by the stronger effects of the αS in respect of theα This result seems to suggest that ceteris paribus information mattersmore than technical constraints Though the model is by any means in-

19

adequate to discuss normative aspects because of the naive ways strategicdecision making is represented it can anyway suggest the effects of errorsin management decisions (ie wrong desired levels of stocks) as opposed tothe technical constraints preventing sudden adjustment of production lev-els Stretching a bit the interpretation of these results we may suggests thatefforts to reduce volatility should better be oriented to improve the trans-mission of information in order to coordinate future production plans (ieidentifying quickly the desired levels of stocks and therefore the new equilib-rium production level) rather than improving the flexibility of productionmethods (reducing α)

56 Volatility and stock levels

In our model stock levels are maintained to buffer a sectorrsquos sale againstunexpected changes in demand Therefore a system made of risk adversemanagers (keeping higher levels of stocks) may be thought to lead to asmoother absorption of shocks in respect of a system composed of firmskeeping a short supply to cushion unexpected events To test this hypothesiswe tested the behaviour of the model for different values of σ the levels ofstock represented as a multiple of demand For each level of σ we run asimulation using all other initialization identical as usual representing aneconomic system ldquoalmostrdquo at equilibrium but for a single small shock indemand uncorrelated across sectors18

00044

00046

00048

0005

00052

00054

00056

00058

0006

00062

5 55 6 65 7 75 8 85 9 95 10

Rel

ativ

e va

riatio

n of

tota

l out

put (

delta

)

Ratio of stocked goods over production (sigma)

Stocks and production fluctuations

Figure 7 Fluctuations persistence on systems with different stock strategies(σ)

18Demand changes by plusmn1-10

20

Figure 7 shows the value of the relative deviation of output production(δ) generated by heterogeneous uncorrelated shocks in the external demandon identical systems where firms store a different multiple of their demand asstocks We obtain actually the opposite result that we may have expectedfluctuation become larger and more persistent the higher the level of stocksIn some way a strategy that for the individual firm is meant to reducethe fluctuations (using stocks as buffers) ends up by actually increasing thevolatility at aggregate levels

The result is though explainable both at the theoretical and empiricallevel In fact if no stocks were maintained production would need to adjustfor changes in demand only in case of a shock That is the firm may movemonotonically from the old to the new production levels sending consistentsignals to the rest of the system (ie its own suppliers) Instead if thesame shock affects a firm maintaining large stocks of output over the sameperiod production needs to adjust to the new demand and and to alignstocks with the new demand This generates besides the adjustment to thedemand a potentially inconsistent signal to the suppliers since the firm willhave a temporary level of production not meant to continue in equilibrium(adjusting stocks)

From the empirical viewpoint the last few decades have seen the almostuniversal diffusion of production system meant to keep stocks to a minimumif at all Lean production methods justndashinndashtime etc are solutions tothe problems of the costs necessary to maintain stocks Over the sameperiod economic fluctuations have been strongly reduced and few empiricalassessments have pointed to the relations between the two phenomena Forexample concerning the US ldquochanges in inventory behavior have playeda direct role in reducing real output volatilityrdquo (Kahn McConnell andPerez-Quiros 2002 p 183)19

57 Irrelevance of technical coefficients distribution

Horvath (1998) and Horvath (2000) suggest that systems with sparse ma-trices ie inputndashoutput matrices with many empty cells tend to be morevolatile in respect of systems with more evenly distributed matrices Noticethat this claim risk if not adequately qualified to generate an illogical resultIn fact matrices sparseness increases as a function of sectoral disaggregationand comparing the same systemrsquos fluctuations using inputndashoutput matricesat different levels of aggregation needs to produce the same type of volatil-ity Indeed this is not the case if all remaining parameters stand equaland do not adjust to the different level of analysis considered If aggregatepatterns of adjustment speed are maintained unchanged the higher the sec-toral disaggregation the higher the systemrsquos volatility ceteris paribus But

19Similar results are found in Blanchard and Simon (2001)

21

for example in our model reducing the aggregation also requires a changein the adaptation coefficients the more micro we undergo the analysis thehigher is the speed of adjustment of economic objects If for example weconsider sectors we are evaluating the adaptation of an entire set of firmswhich pose questions on how to aggregate their dynamics Things changewhen we consider the response of a single firm

Going back to Horvatsrsquo argument highly disaggregated matrices arelikely to contain many empty cells so that we may forecast different levelsof volatility depending on the aggregation used for the same system whichas mentioned is not caused by sparseness but to a failed adjustment inreaction mechanisms Results from our model20 show that volatility is com-pletely unaffected by input coefficients distribution Given their sectoralsum mdash through rows of the matrix mdash the value of each coefficient is irrele-vant The same occurs in the case of very skewed distributions that presenta high number of empty cells (zero input coefficients)

58 Microndash and Macrondashvolatility

As we have seen above one of the major discussion in the literature concernsthe relations between microndash and macrondashvolatility Put it simply one sideof the literature considers that uncorrelated microndashshocks cancel each otherout reducing their effects at macrondashlevel Conversely other researchersconsider that microndashshocks are multiplied at macrondashlevel generating highermacrondashvolatility than that we can observe at microndashlevel The debate isof obvious importance to determine the sources of business cycles to pre-dict their effects and determine the more effective policies to mitigate theirnegative effects

Our model is only a partial representation of realndashworld economic sys-tem and is also overly simplistic However based on these limitations italso offers very robust results allowing reliable conclusions on the (limited)elements included in the model In particular we consider our model a re-liable representation of the basic structures of the production interactionsamong sectors or firms Thus we can use the model to provide an answerto the debate concerning the contributions to volatility from the productioninteractions only Other contributions (eg from financial markets priceadjustments technological innovation etc) may re-inforce or counter theforces we analyse but they need to be considered as additional elements(and we opine also generally less relevant) not alternative to our analysisWe consider a necessity in order to apply a rigorous methodological ap-proach to be able to identify with certainty the results provided by partialstudies in order to eventually obtain by gradual extensions of the analy-sis more and more detailed representations of realistic economic systems

20Available from the authors

22

The results we obtain in this paper concerning this microndashmacro debate aremeant as a first relevant step to a more extended analysis

0

002

004

006

008

01

012

014

016

018

02

022

05 1 15 2 25 3

Var Demand

Macro (Output) and micro (Demand) volatility

delta Outputdelta Demand

Figure 8 Values of δ (relative deviation) for aggregate demand and ag-gregate output in respect of different levels of variance for sector specificuncorrelated external demand shocks

The flows of shocks affecting the firms of a system can be distinguishedalong two dimensions strength of the shock (eg amplitude of the changesin demand) and frequency of the shock (rare or frequent) Concerning thefirst aspect we initialized as usual a system to equilibrium levels (eg pro-duction quantities equal to external and intermediate demand for all sectorsconstant level of stocks equal to the desired level) We then ran the simu-lation generating the external demand for all sectors as a random variablewith constant (equilibrium) mean level and varying values of variance

Figure 8 shows unequivocally that as far as our model has propertiessimilar to real systems the hypothesis that macrondashfluctuations have largervolatility than their sectoral level shocks is fully confirmed At any levelof variance the aggregate volatility of demand as captured by the relativedeviation index is sensibly lower than the volatility shown by aggregateoutput that is the total production of the system This result is not sur-prising throughout all tests presented in the previous sections every resultsuggested that the system lsquooverndashreactsrsquo to any individual microndashlevel changeeven in the case these changes were meant to contrast disturbances There-fore when we move from considering one single shock to a flow of shockswe are not surprised to obtain that the system shows higher volatility thanthat implied by the microndashlevel disturbances

We now consider the second dimension defining a flow of shocks theirfrequency The question is whether for the same level of demand shocks

23

0 002 004 006 008 01 012 014 016

Difference of relative deviation for Output and Demand

0 5 10 15 20 25 30 35 40 45 50 55

Shock frequency 05

1

15

2

25

3

Var Demand

0 002 004 006 008 01

012 014 016

Figure 9 Differences of values of δ for aggregate output minus those foraggregate demand in respect of different levels of variance of demand and offrequency of the shocks

(ie same variance) more frequent changes trigger a higher or lower systemvolatility than rare disturbances The results shown in figure 9 suggest arather elaborated answer which partly vindicates (though in a particularsense) the arguments in favour of compensation of micro shocks The fig-ure shows the difference between aggregate output volatility and aggregatedemand volatility The independent variables are as before the volatilityof demand at microndashlevel21 and the time intervals in between two shocksAggregate demand volatility does not change for the frequency of shocksat least when computed over a long period But output volatility doesin a nonndashmonotonous way For high frequency of shocks the difference isstill positive and increasing for higher variance indicating that the ldquooverndashreactionrdquo argument is valid even in these cases but it is rather low Whilethe frequency of shocks slows down aggregate volatility of output increasesshowing that time is needed for the microndashshock to unfold their full effectsat the macrondashlevel However after a certain threshold less and less frequentshocks generate a lower volatility of the aggregate level of output

This result is quite sensible though not obvious at first sight and onceagain shows the power of such a simple model to generate and explain com-plex properties For frenetic levels of demand modifications the system hasno time to adjust but part of the job to align actual and desired productionfor constantly changing demand is performed by the varying demand itselfIn other terms the shocks cancel out through time if not through their quan-

21Beware of the scale of variance For reasons of visibility the scale is inverted

24

titative effects on the system While frequency decreases the disturbancesof the shocks not compensated by other (unlikely) shocks have the time tofilter through the production system generating higher and higher volatil-ity In a sense any system has a particular frequency of shocks that make itmaximally ldquoresonatingrdquo with the amplification of microndashshocks When therate of shocks gets even rarer the system can enjoy periods of longer andlonger quietness after each shock is absorbed and before a new one startsagain a cycle of initial local disturbance propagation of the disturbances toother sectors and settlement to the new equilibrium

6 Discussion and extensions

We presented a simple model of production composed by interacting sectorThe main (quit realistic) assumption of the model is that the productionprocesses in any sector of the economy make use of input from any othersector Moreover we assumed that sectors (represented as a single firm)adjust slowly both the desired level of production (smoothed revision of ex-pectation) and the actual level (smoothed modification of plant utilization)The model voluntarily leaves aside a large number of aspects of real-worldeconomic systems (eg financial markets price adjustments growth etc)The reason is that the model can be easily and reliably tested in order toextract its properties The possibility to transfer the (certain) model resultto a real context depend of course on an ample ceteris paribus clause How-ever we are confident that the reported results concerning the productionstructure play a relevant role in real world systems Therefore our resultsand particularly their motivations being accessible because of the relativesimplicity of the model will not be diminished by the integration of ourconsiderations with other aspects of economic systems

Our model is in essence a representation of an input-output table inter-preted in physical terms that is as the amount of inputs required for oneunit of output Besides this we include two straightforward assumptionsFirstly production levels cannot be suddenly changed but require time toscale up or down the utilization rate that is the levels of production Stocksare meant to make up for the difference between sales and production Thesecond assumption concerns the intentional behaviour of producers We im-plement this aspect by assuming that exists a level of stocks desired for eachlevel of demand However a change of demand does not translate suddenlyin a new level of desired stock but again only a long time at a different levelof demand will convince managers to update the desired levels of stocks

We show that the model generate under a wide number of settings oscil-latory patterns converging toward the equilibrium level where each sectorgenerate exactly the quantity demanded by final consumers and by othersectors as input The oscillations are provoked by over- and under-shooting

25

generated by the difficulty of coordination among sectors that by assump-tion ldquocommunicaterdquo only via variation of demand to their direct input sup-pliers We also show that the total production levels generated by sectors(sum of the physical units) produced at equilibrium depends (for constantfinal demand) on the total sum of the technical coefficients per sector irre-spective of their distribution This result is motivated by the impossibilityof input substitution and offers interesting suggestions for the analysis ofthe study of input-output tables of real-world systems

We then consider systems at equilibrium facing a single small change ofthe external demand for sectors These simulations allow to study the waythe system reacts to shocks We see that the sum of the coefficients affectsbesides the levels also the volatility of aggregate output the larger a systemthe stronger is the impact of a given shock Less obviously we find thatfor systems with the same dimension (ie same sums of coefficients) theaggregate volatility increases with both the inertia to change of productionlevel and the speed of adjustment of desired stocks This result is somewhatcounter-intuitive since both the cited variables are associated to a moreconservative behaviour that may be supposed should smooth away spikesin the pattern from an old to a new equilibrium However we show thatthe opposite actually holds given the complex interactions among sectorindividual-sector attitude intended to smooth shocks actually increase theiraggregate impact

In the same vein we show that systems maintaining high levels of stocksare more volatile that systems with reduced amounts of stocks Again thisis a counter-intuitive result reverting the goal of individual producers andthe aggregate result Interestingly this result finds strong support from theevidence of the recent changes in production methods (ie low or no stocks)and diminished cycles

In relation to specific issues discussed in the literature we find no supportto the suggestion that particular distributions of input-output coefficientsmodify the volatility of a system As said this depends in case on thesums of these coefficients but not on their distribution Further we canprovide an answer to a long and hotly debated questions whether microndashlevels volatility generates stronger or lower aggregate volatility Our modelshows unequivocally that microndashvolatility (eg variance of final demand)persists and generates stronger volatility at aggregate level Again we canidentify the reason on the complex production structure that exalts micro-level shocks by generating lsquowrongrsquo messages among producers Moreoverwe can also point to a usually neglected aspect of volatility we show thatthe frequency of shocks is a very relevant aspect besides their dimensionin determining the volatility of aggregate output

Our work can be developed along two complementary directions Firstlywe can continue to study the properties of the model by extending the ele-ments considered (eg pricing) analysing the effects on the present results

26

and generating more analytical evidence Of particular relevance is in ouropinion the problem of aggregation Representations of the same systemswith different codification systems (eg different number of SIC digits)must not modify overall system properties for example the volatility ob-served Given that our result show that the sums of coefficients matter forthe levels and volatility of the system and that these sum do depend on theaggregation chosen it should be possible to induce the reaction coefficientsthat we know affect the volatility only

Secondly we can find an application of our results to real world evidenceIn fact there are many data sets available for the data we deal with basicallyinput-output tables Our model is readily adapt to be parametrized alongany number of sector and coefficient necessary so that we may replicate pastseries in order to adapt the unobservable parameters and use the resultingcomplete model to explain business fluctuations and provide intuitions onfuture patterns

27

References

Acemoglu D and A Scott (1997) ldquoAsymmetric Business CyclesTheory and TimendashSeries Evidencerdquo Journal of Monetary Economics 40501ndash533

Backus D P Kehoe and F Kydland (1995) ldquoInternational BusinessCycles Theory and Evidencerdquo in Frontiers of Business Cycle Researched by T Cooley Princeton University Press Princeton

Bak P K Chen J Scheinkman and M Woodford (1993) ldquoAg-gregate Fluctuations from Independent Sectoral Shocks SelfndashOrganizedCritically in a Model of Production and Inventory Dynamicsrdquo RicercheEconomiche 47(1) 3ndash30

Battiston S D Delli Gatti M Gallegati B Greenwald and

J Stiglitz (2005) ldquoCredit Chains and Bankruptcies Avalanches in Sup-ply Networksrdquo in Annual Workshop on Economic Heterogeneous Inter-acting Agents (WEHIA 2005) University of Essex UK

Blanchard O and J Simon (2001) ldquoThe Long and Large Declinein US Output Volatilityrdquo Brookings Papers on Economic Activity 1135ndash174

Bresnahan T F and V A Ramey (1994) ldquoOutput Fluctuations atthe Plant Levelrdquo The Quarterly Journal of Economics 109(3) 593ndash924

David P A (1975) Technical choice innovation and economic growth essays on American and British experience in the nineteenth centuryCambridge University Press London

Delli Gatti D C Di Guilmi E Gaffeo G Giulioni M Galle-

gati and A Palestrini (2004) ldquoA New Approach to Business Fluc-tuations Heterogeneous Interacting Agents Scaling Laws and FinancialFragilityrdquo Journal of Economic Behavior amp Organization 56 489ndash512

Dosi G G Fagiolo and A Roventini (2006) ldquoAn EvolutionaryModel of Endogenous Business Cyclesrdquo Computational Economics 273ndash34

Dupor B (1999) ldquoAggregation and the Irrelevance in MultindashSector Mod-elsrdquo Journal of Monetary Economics 43 391ndash409

Head A C (1995) ldquoCountry Size Aggregate Fluctuations and Interna-tional Risk Sharingrdquo The Canadian Journal of Economics 28(4b) 1096ndash1119

28

Helbing D and S Lammer (2005) ldquoSupply and Production NetworksFrom the Bullwhip Effect to Business Cyclesrdquo in Networks of Interactingmachines Production Organization in Complex Industrial Systems andBiological Cells ed by D Armbruster K Kaneko and A S Mikhailovvol 3 of World Scientific Lecture Notes in Complex Systems World Sci-entific

Helbing D S Lammer U Witt and T Brenner (2004) ldquoNetworkndashInduced Oscillatory Behavior in Material Flow Networks and IrregularBusiness Cyclesrdquo Phisical Review E 70(056118) 1ndash6

Helbing D U Witt S Lammer and T Brenner (2004) ldquoNetwork-Induced Oscillatory Behaviour of Macroeconomic Output Flowsrdquo in In-dustry and Labour Dynamics II The Agent-based Computational Eco-nomics Approach Proceedings of the wildace 2004 conference ed byR Leombruni and M Richiardi no 5 LABORatorio R Revelli

Horvath M (1998) ldquoCyclically and Sectoral Linkages Aggregate Fluctu-ations from Independent Sectoral Shocksrdquo Review of Economic Dynam-ics 1 781ndash808

(2000) ldquoSectoral Shocks and Aggregate Fluctuationsrdquo Journal ofMonetary Economics 45(1) 69ndash106

Jovanovic B (1987) ldquoMicro Shocks and Aggregate Riskrdquo The QuarterlyJournal of Economics 102(2) 395ndash410

Kahn J A M M McConnell and G Perez-Quiros (2002) ldquoOnthe Causes of the Increased Stability of the US Economyrdquo FRBNY Eco-nomic Policy Review 8(1) 183ndash202

Kraay A and J Ventura (1998) ldquoComparative Advantage and theCrossndashSection of Business Cyclesrdquo World Bank Woorking paper Series1948 World Bank

Long J B and C I Plosser (1983) ldquoReal Business Cyclesrdquo Journalof Political Economy 31(1) 39ndash69

Lucas R E (1977) ldquoUnderstanding Business Cyclesrdquo CarnegiendashRochester Conference Series on Public Policy (special issue of Journalof Monetary Economics) 5 7ndash29

Nelson R R and S G Winter (1982) An Evolutionary Theory ofEconomic Change Harvard University Press Cambridge MA

Nirei M (2005) ldquoThreshold Behavior and Aggregate Fluctuationsrdquo Jour-nal of Economic Theory in press

29

Scheinkman J and M Woodford (1994) ldquoSelfndashOrganized Criticallyand Economic Fluctuationsrdquo American Economic Review 84(2 Papersand Proceedings of teh AEA) 417ndash421

Schuh S (1996) ldquoEvidence of the Link Between FirmndashLevel and Ag-gregate Inventory Behaviorrdquo Finance and Economics Discussion Series1996-46 Federal Reserve Board

Stiglitz J and B Greenwald (2003) Towards a New Paradigm inMonetary Economics Raffaele Mattioli Lectures Cambridge UniversityPress Cambridge

Weisbuch G and S Battiston (2005) ldquoProduction Networks andFailure Avalanchesrdquo Working Paper mimeo Ecole Normale SupErieure

Zarnovitz V (1977) ldquoBusiness Cycles Observed and Assessed Why andHow they Matterrdquo NBER Working Paper Series 6230 National Bureauof Economic Research

30

A Tables

Table 1 Parameters valuesParametera Description Value

N Number of sectorsunits i of production 10Ci Number of j buyer sectors for sector i (IO matrix sparseness) N minus 1Di (E) Value of the external demand 10mij Physical input coefficient of input j for sector i sim U(0 1)Mi Sum of the mij input coefficients j for the production of i 05αi friction to changes of production levels Degree of lockndashin on

production technology085

αsi Speed of adaptation to demand changes 004

σ Desired multiple of production to stock 10

aInto parenthesis the number of lags of the initial value for lagged variables

31

both RBC and Av models with respect to structure modelling criteriaanalysis and methodological approach Indeed our model may be easilyreconduced (or restricted) to both We are much in line with a recent se-ries of papers by Helbing Witt Lammer and Brenner (2004) that addressthe same question although with a number of differences that are discussedlater on5

We aim to analyse to what extent the structure of inputndashoutput relationscauses aggregate fluctuations by itself The main idea is to show that ag-gregate fluctuations may derive from simple micro imbalances without theneed for continuous exogenous shocks not even at the firm or sectoral levelIn other words normal production activities including lags technologicaladjustments information asymmetries circularity of the inputndashoutput sys-tem and so on may be enough to explain part of the macro volatility in aclosed system6

Our assumptions draw on the existing literature but try to avoid themajor difficulties highlighted above Firstly we avoid to impose either mar-ket clearance assumptions or crude threshold behaviours Rather we adoptan assumption close to a Simonian approach agents in an economy do theirbest within the informational and physical constraint they are subject toWe represent decision makers as responding to a single piece of informationthey obtain the quantities demanded by their clients Crucially their ac-tions are inspired to a conservative approach an increase in the demandgenerates a smaller (desired see below) sudden increment of productionThis assumption stems from the fact that decision makers are aware of theinstability of their environment and therefore want to avoid getting perma-nently caught with over production following a temporary spike of demandIn other words firms attempt to smooth business cycle generating a morestable (or less uncertain) environment as the economic literature would sug-gest Notice that this representation in line with a routinized representationof organizations (Nelson and Winter 1982) does not contradict market equi-librium Rather in a stable environment such decisional procedure generatesan asymptotic pattern to the equilibrium level in the absence of feedndashbacks

A second assumption concerns the technological possibilities available tothe firms When firms realise the need for a change in production levels saya desired increase of production they face a complex and costly organisa-tional transformation of the production capacity risky investments an soon As Bresnahan and Ramey (1994 p 622) suggest inducing from a de-tailed analysis of the automobile industry in the US ldquoadjusting production

5We thank Matteo Richiardi for pointing out these similar attempts See also Helbingand Lammer (2005) and Helbing Lammer Witt and Brenner (2004)

6An open system entails many other factors including price factors of both inputsand outputs mdash terms of trade labour cost external investment etc As with exogenousshocks our analysis does not deny the importance of these factors but simply separatethem from the structural explanation of cycles

8

is a more complicated process than simply ldquochanging Qrdquo or choosing themix of capital and laborrdquo And this is not only a matter of non convexitiesin production technologies The inconsistency of time reversibility in thetechnological choice is accompanied by the need to operate on the lsquoavail-able technological frontierrsquo (David 1975) If a firm plans a given productionactivity for a short period of one week even though the next day there is anunexpected fall in the demand it is unlikely that it is followed by a similardownturn in production The case is evident when we think at productionplanning that involve hiring of workers an capital investments (both in-volving sunk costs)7 Therefore any desired change generates a pattern ofsmall modifications approaching slowly the target level Moreover thesechanges are irreversible in the short term since only a prolonged change ofthe environment triggers the modification of previous desired levels

Resting on those simple conjectures we thus analyse a model in whichwe only allow for sluggish adaptation of production due to conservative(adaptive) behaviour of firms and the physical constraints faced when vary-ing the production levels Notice that these assumptions are contrary tothe generation of instability of a system In fact a firm behaving as de-scribed above reduces if not eliminates any disturbance affecting its ownstate if acting in isolation Other factors like new technologies extremelylong term perspective price changes etc are likely to be sources of volatil-ity therefore contributing to generate aggregate system fluctuations (as forexample in Acemoglu and Scott 1997) Our neglect of these and other fac-tors however unrealistic serves the purpose of testing whether a volatilityreducing microndashbehaviour turns into volatility generating aggregate resulttwo widely observed dynamics Quite interestingly (and encouraging) DosiFagiolo and Roventini (2006) have concomitantly proposed a model whichincorporates detailed representation of both sluggishness robustly based onempirical evidence Heterogeneous firms which undergo lumpy investmentand slowly adapt their expectation on changes of the future demand inducecomovement of macroeconomic variables qualitatively similar to the onesobserved in the real world To pursue our line of research we first needto concentrate on how sectorsfirms interact in the system in the effort tocomplement those promising results

Therefore the second element we consider is meant to represent a gen-eralized economic system We allow for the circularity of inputndashoutput sys-tems a crucial feature in determining feedback mechanisms and a principal

7See also the discussion in Acemoglu and Scott (1997) where the authors argue thatldquo[w]hile the presence of fixed costs can account for the discreteness of economic turningpoints it does not naturally lead to persistence because once an individual undertakes anaction they are less likely to do so in the near future [] [A]lthough the presence of fixedcosts leads to increasing returns these are intratemporal the full extent of economiesof scale arising from fixed costs can be exploited within a periodrdquo [p 502 authorsrsquoemphasis]

9

component of business cycle It may be true that most backward linkagesconcern capital goods (Weisbuch and Battiston 2005) but a rapid glanceto an inputndashoutput matrix would show that technical coefficients are nonnull in both directions and ldquomost commodities are inputs to the produc-tion process of other commoditiesrdquo (Horvath 2000 p 70) The higher thesectoral aggregation mdash observed or assumed mdash the more the relevance ofintermediate supply holds true (as mentioned above with respect to ma-trix incompleteness) We claim that the existence of backward and forwardlinkages are a crucial element to determine the persistence of shocks in theeconomy A perfectly linear (one directional) system needs much less adap-tation of firms and sectors in order to stabilise the economy Nonethelesscircularity does not imply that inputs are perfectly substitutable amongthem in order to determine the maximising allocation even at high level ofdisaggregation

By using a fixed coefficients production function we both maintain themid term rigidity of production processes and we do not impose limitedinteractions We claim that the circular property of the production structureis by itself sufficient to produce shock persistence even when the inputmatrix is full (although with heterogeneous input coefficients) mdash all sectorsshop in the remaining n minus 1 sectors

In conclusion we represent a model for an economic system made onlyof i) production level decisions and ii) input-output relations Clearly sucha model lacks many features present in real economic systems likely we areconvinced to affect the propensity to generate cycles As such we cannotaim at reproducing realistic data so our results cannot be tested comparingtime series generated by the model with real ones Instead given the paucityof the elements comprising the model we can investigate in detail to whichextent the (few) features present in our model affect the cyclical behaviour ofthe system As we will see we will be able to show quite a number of resultin which aggregate fluctuations are generated whereas the individual microstructure of the model aims at monotonous adaptation Before discussingthe results next section describes in detail the implementation of the simplemodel

4 The model

We model an economic system in which a number of sectors i = (1 N)are potentially connected via (intermediate) market relations A N times NInputndashOutput matrix determines the mij units of each input j = (1 N)mdash bought from the N minus 1 residual sectors mdash necessary to produce oneunit of output i8 For simplicity given the focus of this paper on the IndashOstructure as a generator of aggregate fluctuation we abstract from market

8IndashO coefficients are therefore physical coefficients

10

interactions within sectors each sector is a production unit In fact forour purposes it is not relevant which firm produces the sectoral outcomebut only the total amount produced by each sector9 Moreover besidesmaking the analysis more neat this assumption allows also to enjoy a largefreedom in determining the aggregation level since we donrsquot need to assessthe effects of between sectors competitiveness which for example is likelyto increase for increasing levels of disaggregation Our economic systemis therefore composed of N production units each of which may buy andprovide inputs from and to any other sector depending on the IndashO structureof the economy

The demand for each sector i is determined by a constant consumersdemand Di (E) and the input needs of other sectors

Di = Di (E) +sum

j 6=iisinCi

mjiQj (1)

where Ci is the set of j intermediate clients of i (sum

j isin C lt N) mji is thetechnical coefficient of firm j for input i ie the amount of i required toproduce one unit of j Qj is the actual output produced by sector j

Concerning the simultaneous production level of each sector in time twe make two lsquorealisticrsquo assumptions First from the technical viewpoint afirm like a tanker cannot abruptly change its course Therefore changesto the production level can be introduced only gradually through capital(dis)investment and job h(f)iring (see discussion in Section 3) Second weassume that firms do not decide the production level directly on the basisof the observed demand level but conscious of the uncertainty of demandplan their production in order to maintain invariant a desired level of stocksand avoid stockout10

Quantity produced by sector i in period t is computed as follows

Qit = αiQtminus1 + (1 minus αi)(

Slowastitminus1 minus Sitminus1

)

(2)

where αi measures the physical constraint in adapting the production capac-ity to changes in the required output Slowast

itminus1are firmrsquos i desired stocks and

Sitminus1 its actual stocks Notice that both variables related to stocks appearwith a lag This is due to two reasons First it is a modelling necessity inorder to allow the simultaneous determination of production levels for allsectors Secondly as frequently happens modelling necessities reveal sen-sible if not generally considered aspects of the modelled system In factproduction decisions concern management in the production plant while

9Adding heterogeneous firms within sectors would definitely enrich the causal expla-nation put forward and parametrised in this paper (see below) For example to analysethe effect of firms constraints It is left over for future research as in the present work weopt for the analysis of the main structural parameters that affect aggregate fluctuations

10See also discussion in Schuh (1996)

11

demand through sales is observed by the commercial staff It is thereforelogical that the information from demand reaches plant managers with somelag

The actual stock level is trivially computed subtracting from the previousheap of stock the amount sold and adding the new production

Sit = Sitminus1 + Qit minus Dit (3)

Concerning the desired level of stocks instead we introduce the conser-vative behaviour mentioned above Firms ideally would keep an amountof stocks proportional to the demand they receive say a multiple σ ofthat amount However when demand varies they adjust the desired levelsmoothly preventing sudden changes to the variable representing the goalof the firm

Slowastit = Slowast

itminus1 + αsi

(

σDit minus Slowastitminus1

)

(4)

where αsi measures the adaptation to demand changes

In summary the dynamics of the model consists of only the equationsabove Firstly at the beginning of a time step firms determine the quantityto produce as a function of past imbalances in stock levels and the pastproduction level These levels define the demand for all sectors determinedby the technical coefficients applied to all produced quantities Finallyactual and desired stocks can be updated Figure 1 represents visually thisdynamics highlighting the decisional components of the model in respect ofthe ldquomechanicsrdquo of trade

5 Results production structure and economic fluc-

tuations

In this section we describe some of the most relevant results concerningthe effects of the production structure to determine aggregate fluctuationsGiven the unusual analytical instrument chosen (at least concerning thistopic) we start with a brief methodological introduction meant to clarifythe nature of the results we claim to obtain Next we describe a generalsetting for the model which is the (extremely) limited area of the potentialparametersrsquo space of the model that we will explore In the rest of thesection we discuss the results from the model

51 Methodological considerations

Real economic systems are constantly subject to a continuous flow of shocksexternal to the production system besides changes within the system itselfIn order to appreciate the systemic properties we need to abstract from all

12

Compute the quantityto produce

F (Q(t-1) S(t-1) s(t-1) alpha)

Communicate thequantity of inputsrequired to each

supplier

Determine the overalldemand

F(D(E) D(I))

Revise the past decisionon stock plans

(desired stocks)F(D S(t-1) sigma alpha(s))

Compute the final levelof stocks for next period

F(s(t-1) Q D)

KEY

ModelDirection

Variablecomputed

at t-1(t-1)

Step includingfirm behavior

Figure 1 One period dynamics

the nonndashrelevant disturbances and concentrate on how the system endoge-nously contributes to the observed dynamics Therefore we will generatehighly abstract patterns whose analysis will provide insights on the systemproperties that would be otherwise lost in the noise generated by all theelements of a real system

We are not trying to fit the model to a specific empirical data set nor weare interested in studying the complete behaviour of the model for areas ofthe parameterrsquos space that make no economic sense (eg negative or infiniteproduction levels) Therefore the values of initial settings and results matteronly in relative and not in absolute terms

We are going to describe a simulation model with which we will produceseveral simulation runs and we claim obtain results relevant for the debatediscussed above Rigorously speaking we may claim only certainty for thevalidation of the results that is that the model does actually generate theresults presented and for the reasons we explain We limit this part to thepresentation of graphs and verbal explanations that we believe are ratheruncontroversial11 Concerning the relevance of our studies for real systems

11Interested readers can request the model code and and simulation data for replicationand extensions of the results We can guarantee that such possibility does not require

13

(verification) we simply avoid to pretend that our model is a quantitativeapproximation of real economies (which in which period) Our opinionis that the study of the model allows us to make general considerationsnon-obvious and relevant for applied debates supported by strong logicalarguments which can be integrated but not reverted by arguments con-cerning more and more elements of the economic world

52 Model setup

We go through the model properties analysing an economic system madeup of ten sectors (N = 10)12 Each sector shops inputs from the remain-ing N minus 1 and sells to them part of its own production as intermediategoods This means that the input matrix is complete except for the diag-onal (mij gt 0 forallj 6= i mii = 0) The sum of all N minus 1 coefficients for asingle sector is given by Mi and its benchmark value is kept at a plausiblemedium level (unless differently specified) to avoid distortions from highlydemanding sectors Each input coefficient is randomly drawn from a uniformdistribution

Sectors start at an equilibrium level their supply matches intermediateand final demand and in each period the same quantity of output mustbe produced to compensate for the used stocks and keep the desired stocksunchanged In thier turn stocks must be ten times the value of total sectoraldemand (σ) Eventually technological adjustment of production is ratherslow and producers are quiet refractory to follow sudden changes in thedemand in general a smoothing behaviour strongly prevails (α and αs)

Under those general preconditions in the following sections we anal-yse the dynamics of the economic system in relation to model parametersWe mainly focus on the cyclical behaviour of systems and their speed ofconvergence to the asymptotic equilibrium13

The very abstract assumption and the homogeneous initialisation donot allow for the multiple equilibria that are likely to rise in complex systemswith feedbacks Once more such simplification allows to have a betterunderstanding of the crucial impacts of the production structure on businessfluctuations Departing from the related literature discussed above14 weconsider the effects of a single shock rather then a continuous flow whichin our model is superfluous in order to obtain fluctuation persistence15 To

extensive programming or statistical skills besides the usual education of economists12You can think of a high level of aggregation in the statistical observation The com-

plete initialisation and parameters values are available in table 1 in the Appendix13Notice that sectors only approach a final equilibrium level of production which can

be prendashdetermined under static conditions14Included the more proximate work of Helbing Witt Lammer and Brenner (2004)15An exception are the last results presented in Section 58 which we see more as an

opportunity for further research than a closing of the paper

14

start with next section (53) shows the aggregate cyclical behaviour of astandard setting

53 Single shock analysis

We generate a simulated history following the dynamics of the modelrsquos vari-ables when the system is set to an equilibrium level and a single shockmodifies one of the exogenous elements namely the external demand forsectors Modifications across sectors are random and not correlated andmay increase or reduce external demand by at most 10

100 225 350 475 600

193894

195861

197828

199795

201762 Industrial Production

Time

Figure 2 Total industrial production fluctuations following a shock fromexternal demand

Figure 2 shows the dynamics generated by the total industrial productionof the system mdash that is also a proxy for the systemrsquos GDP fluctuations Thenew external demand changes the equilibrium level but the system cannotreach the new equilibrium with a monotonous pattern and converges to thenew level with a fluctuating pattern Each change in the external demandinduces also modifications of the intermediate demand and sectoral adjust-ments in the quantity produced typically overshoot the aggregate patternAs it can be appreciated from the figure it takes time before the system isable to converge to the new static equilibrium conditions

We study two features of this pattern absolute level of the new equilib-rium and volatility induced by the shock

54 Technical coefficients and industrial output

Production level

The equilibrium levels of industrial output depend on two factors only Ob-viously the levels of external demand determine how much of the total pro-

15

duction needs to be supplied to consumers exogenous from the productionsystem Second the technical coefficients also influence strongly the level ofactivity of the system defining the intermediate demand to each sector

180

200

220

240

260

280

300

01 02 03 04 05 06 07 08 09

Out

put l

evel

(In

dust

rial p

rodu

ctio

n)

Sum of total input coefficients (M)

Total industrial production for different input coefficients levels

Figure 3 The relevance of input coefficients on output level

Figure 3 shows the equilibrium level of industrial output reached bysystems with identical external demand where we fixed the sum of inputcoefficients for each sector of the economy (Mi) to different values16 Thehigher the sum the higher is the total production sustained by the system

This result depends on the very production structure represented by theinputndashoutput matrix In fact the higher is the level of production requiredby intermediate sectors the higher is the total production given a constantlevel of external demand

Notice that the level of production does not depend on the distribution ofthe technical coefficients As long as the sum of the coefficients is identicalthe same production level is obtained irrespective of their distribution (otherthings being equal)

Output shock absorption

Input coefficients have a relevant role also on the output stability of thesystem Starting from an output equilibrium level we hit the system witha single shock on the external demand for all sectors uncorrelated amongsectors17 modifing (positively or negatively) the demand for all sectors by

16Within this limit the actual value of the coefficients are randomly drawn17In each sector the external demand changes by plusmnX where X is a uniformly dis-

tributed percentage between 1 and 10

16

at most10 As show in Figure 2 the economic systems produces a lenghtlysmoothing oscillatory behaviour We compute the relative deviation of theaggregate production (industrial output) as the ratio between its standarddeviation and its mean δ = σY microY where Y =

sum

Qi Figure 4 shows thebehaviour of δ as a function of the sum of input coefficients for each sector80 periods after the shock As time goes by the level of δ reduces for allMi but the relation with the sum of input coefficient is unchanged

00055

0006

00065

0007

00075

0008

00085

0009

01 02 03 04 05 06 07 08 09

Rel

ativ

e va

rianc

e of

tota

l out

put (

delta

)

Sum of total input coefficients

Aggregate output deviations for different technical coefficients

Figure 4 The relevance of input coefficients on output stability

In sum input coefficients determine the extent of overshooting behaviourThe higher the coefficients the higher is the backlash even when shockscould cancel out through sectors

55 Shock persistence and adjustment coefficients

We now consider the effect of micro smoothing behaviours on the volatilityof the production system In particular we are interested in testing theeffects of two parameters of the model on the volatility of the productionsystem First we consider the effects of the αrsquos representing the ldquostickinessrdquoof the production technology to desired changes of Q aimed at filling thegap between actual and desired stocks The higher is α the slower is thereaction to stock unbalances Concerning its effect referring to the technicalpossibilities available ie production capital and organization we may putforward two opposite hypothesis relevant to systemrsquos volatility

Hypothesis 1 Slower production adjustment (high α) reduces volatilitybecause unbalances in demand of one sector being compensated slowly slow

17

down the diffusion of shocks to other sectors

Hypothesis 2 Slower production adjustment (high α) increase volatil-ity because the longer lasts the mismatch between actual and equilibriumproduction level the deeper will become the mismatch between actual anddesired stocks

20 25 30 35 40 45 50 55 60 65

Cyclesrsquo width

07 075

08 085

09alpha 0005

001 0015

002 0025

003 0035

004 0045

005

alphaS

20 25 30 35 40 45 50 55 60 65

Figure 5 Maximum width recorded between highest and lowest peaks fortotal production for different values of α and αS

Secondly we consider the parameters αS representing the speed of ad-justment of desired stock to observed demand The higher this parame-ter the faster sectors revise the desired levels of stocks to current levelsof observed demand This parameter concerns sectorsrsquo behavioral decisionprocess Higher αS represent a stronger belief that the most recently ob-served demand level is a reliable indicator of future demand levels whilelower levels of the parameter represent a more conservative approach wheremanagers require a long observation of demand levels different from the pastones before adjusting their expectations Also concerning the effect of thisparameter on the volatility induced by an exogenous shock we may maketwo hypothesis

Hypothesis 1 Faster adaptation of desired stocks to new levels of de-mand (higher αS) reduce volatility because changes in demand are rapidlytranslated in adjusted levels of production

Hypothesis 2 Faster adaptation of desired stocks to new levels of demand(higher αS) increase volatility because temporary unbalances of demand in

18

one sector are transmitted faster to other sector reinforcing the feedbackmechanism of fluctuations

To test which of these hypotheses are confirmed by the model we runsimulation runs with identical systems (eg same technical coefficients samelevels of demand same shocks) but for the values of the αrsquos and αS rsquos testingall combinations of these parameters over a range of values (each run usedthe same parameters for all the sectors of the system) For each simulation(ie each couple of values of the parameters) we computed two indicatorsof volatility One measures the maximum distance of total production be-tween the highest and the lowest peak that is the maximum width of theoscillations registered in the production time series The second indicatormeasures the relative deviation δ = σY microY Figures 5 and 6 report theseresults

0004 0006 0008 001 0012 0014 0016 0018 002

Relative deviation

07 075

08 085

09alpha 0005

001 0015

002 0025

003 0035

004 0045

005

alphaS

0004 0006 0008 001

0012 0014 0016 0018 002

Figure 6 Relative deviation of total production for different values of α andαS

The results confirms unequivocally both hypotheses 2 and reject hy-potheses 1 In a sense these hypotheses confirm the nature of the economicsystem represented in the model as a complex system where the interactionsamong sectors play a far more important role than the individual sector be-haviour In fact in both cases we may interpret the results that the strongerthe efforts to mitigate the effects of shocks (ie slow productionrsquos and quickexpectationsrsquo adjustment) the opposite result is actually obtained the fluc-tuations become even more accentuated It is also worth to note that thestronger impact is generated by the behavioural parameters rather than thetechnical one as indicated by the stronger effects of the αS in respect of theα This result seems to suggest that ceteris paribus information mattersmore than technical constraints Though the model is by any means in-

19

adequate to discuss normative aspects because of the naive ways strategicdecision making is represented it can anyway suggest the effects of errorsin management decisions (ie wrong desired levels of stocks) as opposed tothe technical constraints preventing sudden adjustment of production lev-els Stretching a bit the interpretation of these results we may suggests thatefforts to reduce volatility should better be oriented to improve the trans-mission of information in order to coordinate future production plans (ieidentifying quickly the desired levels of stocks and therefore the new equilib-rium production level) rather than improving the flexibility of productionmethods (reducing α)

56 Volatility and stock levels

In our model stock levels are maintained to buffer a sectorrsquos sale againstunexpected changes in demand Therefore a system made of risk adversemanagers (keeping higher levels of stocks) may be thought to lead to asmoother absorption of shocks in respect of a system composed of firmskeeping a short supply to cushion unexpected events To test this hypothesiswe tested the behaviour of the model for different values of σ the levels ofstock represented as a multiple of demand For each level of σ we run asimulation using all other initialization identical as usual representing aneconomic system ldquoalmostrdquo at equilibrium but for a single small shock indemand uncorrelated across sectors18

00044

00046

00048

0005

00052

00054

00056

00058

0006

00062

5 55 6 65 7 75 8 85 9 95 10

Rel

ativ

e va

riatio

n of

tota

l out

put (

delta

)

Ratio of stocked goods over production (sigma)

Stocks and production fluctuations

Figure 7 Fluctuations persistence on systems with different stock strategies(σ)

18Demand changes by plusmn1-10

20

Figure 7 shows the value of the relative deviation of output production(δ) generated by heterogeneous uncorrelated shocks in the external demandon identical systems where firms store a different multiple of their demand asstocks We obtain actually the opposite result that we may have expectedfluctuation become larger and more persistent the higher the level of stocksIn some way a strategy that for the individual firm is meant to reducethe fluctuations (using stocks as buffers) ends up by actually increasing thevolatility at aggregate levels

The result is though explainable both at the theoretical and empiricallevel In fact if no stocks were maintained production would need to adjustfor changes in demand only in case of a shock That is the firm may movemonotonically from the old to the new production levels sending consistentsignals to the rest of the system (ie its own suppliers) Instead if thesame shock affects a firm maintaining large stocks of output over the sameperiod production needs to adjust to the new demand and and to alignstocks with the new demand This generates besides the adjustment to thedemand a potentially inconsistent signal to the suppliers since the firm willhave a temporary level of production not meant to continue in equilibrium(adjusting stocks)

From the empirical viewpoint the last few decades have seen the almostuniversal diffusion of production system meant to keep stocks to a minimumif at all Lean production methods justndashinndashtime etc are solutions tothe problems of the costs necessary to maintain stocks Over the sameperiod economic fluctuations have been strongly reduced and few empiricalassessments have pointed to the relations between the two phenomena Forexample concerning the US ldquochanges in inventory behavior have playeda direct role in reducing real output volatilityrdquo (Kahn McConnell andPerez-Quiros 2002 p 183)19

57 Irrelevance of technical coefficients distribution

Horvath (1998) and Horvath (2000) suggest that systems with sparse ma-trices ie inputndashoutput matrices with many empty cells tend to be morevolatile in respect of systems with more evenly distributed matrices Noticethat this claim risk if not adequately qualified to generate an illogical resultIn fact matrices sparseness increases as a function of sectoral disaggregationand comparing the same systemrsquos fluctuations using inputndashoutput matricesat different levels of aggregation needs to produce the same type of volatil-ity Indeed this is not the case if all remaining parameters stand equaland do not adjust to the different level of analysis considered If aggregatepatterns of adjustment speed are maintained unchanged the higher the sec-toral disaggregation the higher the systemrsquos volatility ceteris paribus But

19Similar results are found in Blanchard and Simon (2001)

21

for example in our model reducing the aggregation also requires a changein the adaptation coefficients the more micro we undergo the analysis thehigher is the speed of adjustment of economic objects If for example weconsider sectors we are evaluating the adaptation of an entire set of firmswhich pose questions on how to aggregate their dynamics Things changewhen we consider the response of a single firm

Going back to Horvatsrsquo argument highly disaggregated matrices arelikely to contain many empty cells so that we may forecast different levelsof volatility depending on the aggregation used for the same system whichas mentioned is not caused by sparseness but to a failed adjustment inreaction mechanisms Results from our model20 show that volatility is com-pletely unaffected by input coefficients distribution Given their sectoralsum mdash through rows of the matrix mdash the value of each coefficient is irrele-vant The same occurs in the case of very skewed distributions that presenta high number of empty cells (zero input coefficients)

58 Microndash and Macrondashvolatility

As we have seen above one of the major discussion in the literature concernsthe relations between microndash and macrondashvolatility Put it simply one sideof the literature considers that uncorrelated microndashshocks cancel each otherout reducing their effects at macrondashlevel Conversely other researchersconsider that microndashshocks are multiplied at macrondashlevel generating highermacrondashvolatility than that we can observe at microndashlevel The debate isof obvious importance to determine the sources of business cycles to pre-dict their effects and determine the more effective policies to mitigate theirnegative effects

Our model is only a partial representation of realndashworld economic sys-tem and is also overly simplistic However based on these limitations italso offers very robust results allowing reliable conclusions on the (limited)elements included in the model In particular we consider our model a re-liable representation of the basic structures of the production interactionsamong sectors or firms Thus we can use the model to provide an answerto the debate concerning the contributions to volatility from the productioninteractions only Other contributions (eg from financial markets priceadjustments technological innovation etc) may re-inforce or counter theforces we analyse but they need to be considered as additional elements(and we opine also generally less relevant) not alternative to our analysisWe consider a necessity in order to apply a rigorous methodological ap-proach to be able to identify with certainty the results provided by partialstudies in order to eventually obtain by gradual extensions of the analy-sis more and more detailed representations of realistic economic systems

20Available from the authors

22

The results we obtain in this paper concerning this microndashmacro debate aremeant as a first relevant step to a more extended analysis

0

002

004

006

008

01

012

014

016

018

02

022

05 1 15 2 25 3

Var Demand

Macro (Output) and micro (Demand) volatility

delta Outputdelta Demand

Figure 8 Values of δ (relative deviation) for aggregate demand and ag-gregate output in respect of different levels of variance for sector specificuncorrelated external demand shocks

The flows of shocks affecting the firms of a system can be distinguishedalong two dimensions strength of the shock (eg amplitude of the changesin demand) and frequency of the shock (rare or frequent) Concerning thefirst aspect we initialized as usual a system to equilibrium levels (eg pro-duction quantities equal to external and intermediate demand for all sectorsconstant level of stocks equal to the desired level) We then ran the simu-lation generating the external demand for all sectors as a random variablewith constant (equilibrium) mean level and varying values of variance

Figure 8 shows unequivocally that as far as our model has propertiessimilar to real systems the hypothesis that macrondashfluctuations have largervolatility than their sectoral level shocks is fully confirmed At any levelof variance the aggregate volatility of demand as captured by the relativedeviation index is sensibly lower than the volatility shown by aggregateoutput that is the total production of the system This result is not sur-prising throughout all tests presented in the previous sections every resultsuggested that the system lsquooverndashreactsrsquo to any individual microndashlevel changeeven in the case these changes were meant to contrast disturbances There-fore when we move from considering one single shock to a flow of shockswe are not surprised to obtain that the system shows higher volatility thanthat implied by the microndashlevel disturbances

We now consider the second dimension defining a flow of shocks theirfrequency The question is whether for the same level of demand shocks

23

0 002 004 006 008 01 012 014 016

Difference of relative deviation for Output and Demand

0 5 10 15 20 25 30 35 40 45 50 55

Shock frequency 05

1

15

2

25

3

Var Demand

0 002 004 006 008 01

012 014 016

Figure 9 Differences of values of δ for aggregate output minus those foraggregate demand in respect of different levels of variance of demand and offrequency of the shocks

(ie same variance) more frequent changes trigger a higher or lower systemvolatility than rare disturbances The results shown in figure 9 suggest arather elaborated answer which partly vindicates (though in a particularsense) the arguments in favour of compensation of micro shocks The fig-ure shows the difference between aggregate output volatility and aggregatedemand volatility The independent variables are as before the volatilityof demand at microndashlevel21 and the time intervals in between two shocksAggregate demand volatility does not change for the frequency of shocksat least when computed over a long period But output volatility doesin a nonndashmonotonous way For high frequency of shocks the difference isstill positive and increasing for higher variance indicating that the ldquooverndashreactionrdquo argument is valid even in these cases but it is rather low Whilethe frequency of shocks slows down aggregate volatility of output increasesshowing that time is needed for the microndashshock to unfold their full effectsat the macrondashlevel However after a certain threshold less and less frequentshocks generate a lower volatility of the aggregate level of output

This result is quite sensible though not obvious at first sight and onceagain shows the power of such a simple model to generate and explain com-plex properties For frenetic levels of demand modifications the system hasno time to adjust but part of the job to align actual and desired productionfor constantly changing demand is performed by the varying demand itselfIn other terms the shocks cancel out through time if not through their quan-

21Beware of the scale of variance For reasons of visibility the scale is inverted

24

titative effects on the system While frequency decreases the disturbancesof the shocks not compensated by other (unlikely) shocks have the time tofilter through the production system generating higher and higher volatil-ity In a sense any system has a particular frequency of shocks that make itmaximally ldquoresonatingrdquo with the amplification of microndashshocks When therate of shocks gets even rarer the system can enjoy periods of longer andlonger quietness after each shock is absorbed and before a new one startsagain a cycle of initial local disturbance propagation of the disturbances toother sectors and settlement to the new equilibrium

6 Discussion and extensions

We presented a simple model of production composed by interacting sectorThe main (quit realistic) assumption of the model is that the productionprocesses in any sector of the economy make use of input from any othersector Moreover we assumed that sectors (represented as a single firm)adjust slowly both the desired level of production (smoothed revision of ex-pectation) and the actual level (smoothed modification of plant utilization)The model voluntarily leaves aside a large number of aspects of real-worldeconomic systems (eg financial markets price adjustments growth etc)The reason is that the model can be easily and reliably tested in order toextract its properties The possibility to transfer the (certain) model resultto a real context depend of course on an ample ceteris paribus clause How-ever we are confident that the reported results concerning the productionstructure play a relevant role in real world systems Therefore our resultsand particularly their motivations being accessible because of the relativesimplicity of the model will not be diminished by the integration of ourconsiderations with other aspects of economic systems

Our model is in essence a representation of an input-output table inter-preted in physical terms that is as the amount of inputs required for oneunit of output Besides this we include two straightforward assumptionsFirstly production levels cannot be suddenly changed but require time toscale up or down the utilization rate that is the levels of production Stocksare meant to make up for the difference between sales and production Thesecond assumption concerns the intentional behaviour of producers We im-plement this aspect by assuming that exists a level of stocks desired for eachlevel of demand However a change of demand does not translate suddenlyin a new level of desired stock but again only a long time at a different levelof demand will convince managers to update the desired levels of stocks

We show that the model generate under a wide number of settings oscil-latory patterns converging toward the equilibrium level where each sectorgenerate exactly the quantity demanded by final consumers and by othersectors as input The oscillations are provoked by over- and under-shooting

25

generated by the difficulty of coordination among sectors that by assump-tion ldquocommunicaterdquo only via variation of demand to their direct input sup-pliers We also show that the total production levels generated by sectors(sum of the physical units) produced at equilibrium depends (for constantfinal demand) on the total sum of the technical coefficients per sector irre-spective of their distribution This result is motivated by the impossibilityof input substitution and offers interesting suggestions for the analysis ofthe study of input-output tables of real-world systems

We then consider systems at equilibrium facing a single small change ofthe external demand for sectors These simulations allow to study the waythe system reacts to shocks We see that the sum of the coefficients affectsbesides the levels also the volatility of aggregate output the larger a systemthe stronger is the impact of a given shock Less obviously we find thatfor systems with the same dimension (ie same sums of coefficients) theaggregate volatility increases with both the inertia to change of productionlevel and the speed of adjustment of desired stocks This result is somewhatcounter-intuitive since both the cited variables are associated to a moreconservative behaviour that may be supposed should smooth away spikesin the pattern from an old to a new equilibrium However we show thatthe opposite actually holds given the complex interactions among sectorindividual-sector attitude intended to smooth shocks actually increase theiraggregate impact

In the same vein we show that systems maintaining high levels of stocksare more volatile that systems with reduced amounts of stocks Again thisis a counter-intuitive result reverting the goal of individual producers andthe aggregate result Interestingly this result finds strong support from theevidence of the recent changes in production methods (ie low or no stocks)and diminished cycles

In relation to specific issues discussed in the literature we find no supportto the suggestion that particular distributions of input-output coefficientsmodify the volatility of a system As said this depends in case on thesums of these coefficients but not on their distribution Further we canprovide an answer to a long and hotly debated questions whether microndashlevels volatility generates stronger or lower aggregate volatility Our modelshows unequivocally that microndashvolatility (eg variance of final demand)persists and generates stronger volatility at aggregate level Again we canidentify the reason on the complex production structure that exalts micro-level shocks by generating lsquowrongrsquo messages among producers Moreoverwe can also point to a usually neglected aspect of volatility we show thatthe frequency of shocks is a very relevant aspect besides their dimensionin determining the volatility of aggregate output

Our work can be developed along two complementary directions Firstlywe can continue to study the properties of the model by extending the ele-ments considered (eg pricing) analysing the effects on the present results

26

and generating more analytical evidence Of particular relevance is in ouropinion the problem of aggregation Representations of the same systemswith different codification systems (eg different number of SIC digits)must not modify overall system properties for example the volatility ob-served Given that our result show that the sums of coefficients matter forthe levels and volatility of the system and that these sum do depend on theaggregation chosen it should be possible to induce the reaction coefficientsthat we know affect the volatility only

Secondly we can find an application of our results to real world evidenceIn fact there are many data sets available for the data we deal with basicallyinput-output tables Our model is readily adapt to be parametrized alongany number of sector and coefficient necessary so that we may replicate pastseries in order to adapt the unobservable parameters and use the resultingcomplete model to explain business fluctuations and provide intuitions onfuture patterns

27

References

Acemoglu D and A Scott (1997) ldquoAsymmetric Business CyclesTheory and TimendashSeries Evidencerdquo Journal of Monetary Economics 40501ndash533

Backus D P Kehoe and F Kydland (1995) ldquoInternational BusinessCycles Theory and Evidencerdquo in Frontiers of Business Cycle Researched by T Cooley Princeton University Press Princeton

Bak P K Chen J Scheinkman and M Woodford (1993) ldquoAg-gregate Fluctuations from Independent Sectoral Shocks SelfndashOrganizedCritically in a Model of Production and Inventory Dynamicsrdquo RicercheEconomiche 47(1) 3ndash30

Battiston S D Delli Gatti M Gallegati B Greenwald and

J Stiglitz (2005) ldquoCredit Chains and Bankruptcies Avalanches in Sup-ply Networksrdquo in Annual Workshop on Economic Heterogeneous Inter-acting Agents (WEHIA 2005) University of Essex UK

Blanchard O and J Simon (2001) ldquoThe Long and Large Declinein US Output Volatilityrdquo Brookings Papers on Economic Activity 1135ndash174

Bresnahan T F and V A Ramey (1994) ldquoOutput Fluctuations atthe Plant Levelrdquo The Quarterly Journal of Economics 109(3) 593ndash924

David P A (1975) Technical choice innovation and economic growth essays on American and British experience in the nineteenth centuryCambridge University Press London

Delli Gatti D C Di Guilmi E Gaffeo G Giulioni M Galle-

gati and A Palestrini (2004) ldquoA New Approach to Business Fluc-tuations Heterogeneous Interacting Agents Scaling Laws and FinancialFragilityrdquo Journal of Economic Behavior amp Organization 56 489ndash512

Dosi G G Fagiolo and A Roventini (2006) ldquoAn EvolutionaryModel of Endogenous Business Cyclesrdquo Computational Economics 273ndash34

Dupor B (1999) ldquoAggregation and the Irrelevance in MultindashSector Mod-elsrdquo Journal of Monetary Economics 43 391ndash409

Head A C (1995) ldquoCountry Size Aggregate Fluctuations and Interna-tional Risk Sharingrdquo The Canadian Journal of Economics 28(4b) 1096ndash1119

28

Helbing D and S Lammer (2005) ldquoSupply and Production NetworksFrom the Bullwhip Effect to Business Cyclesrdquo in Networks of Interactingmachines Production Organization in Complex Industrial Systems andBiological Cells ed by D Armbruster K Kaneko and A S Mikhailovvol 3 of World Scientific Lecture Notes in Complex Systems World Sci-entific

Helbing D S Lammer U Witt and T Brenner (2004) ldquoNetworkndashInduced Oscillatory Behavior in Material Flow Networks and IrregularBusiness Cyclesrdquo Phisical Review E 70(056118) 1ndash6

Helbing D U Witt S Lammer and T Brenner (2004) ldquoNetwork-Induced Oscillatory Behaviour of Macroeconomic Output Flowsrdquo in In-dustry and Labour Dynamics II The Agent-based Computational Eco-nomics Approach Proceedings of the wildace 2004 conference ed byR Leombruni and M Richiardi no 5 LABORatorio R Revelli

Horvath M (1998) ldquoCyclically and Sectoral Linkages Aggregate Fluctu-ations from Independent Sectoral Shocksrdquo Review of Economic Dynam-ics 1 781ndash808

(2000) ldquoSectoral Shocks and Aggregate Fluctuationsrdquo Journal ofMonetary Economics 45(1) 69ndash106

Jovanovic B (1987) ldquoMicro Shocks and Aggregate Riskrdquo The QuarterlyJournal of Economics 102(2) 395ndash410

Kahn J A M M McConnell and G Perez-Quiros (2002) ldquoOnthe Causes of the Increased Stability of the US Economyrdquo FRBNY Eco-nomic Policy Review 8(1) 183ndash202

Kraay A and J Ventura (1998) ldquoComparative Advantage and theCrossndashSection of Business Cyclesrdquo World Bank Woorking paper Series1948 World Bank

Long J B and C I Plosser (1983) ldquoReal Business Cyclesrdquo Journalof Political Economy 31(1) 39ndash69

Lucas R E (1977) ldquoUnderstanding Business Cyclesrdquo CarnegiendashRochester Conference Series on Public Policy (special issue of Journalof Monetary Economics) 5 7ndash29

Nelson R R and S G Winter (1982) An Evolutionary Theory ofEconomic Change Harvard University Press Cambridge MA

Nirei M (2005) ldquoThreshold Behavior and Aggregate Fluctuationsrdquo Jour-nal of Economic Theory in press

29

Scheinkman J and M Woodford (1994) ldquoSelfndashOrganized Criticallyand Economic Fluctuationsrdquo American Economic Review 84(2 Papersand Proceedings of teh AEA) 417ndash421

Schuh S (1996) ldquoEvidence of the Link Between FirmndashLevel and Ag-gregate Inventory Behaviorrdquo Finance and Economics Discussion Series1996-46 Federal Reserve Board

Stiglitz J and B Greenwald (2003) Towards a New Paradigm inMonetary Economics Raffaele Mattioli Lectures Cambridge UniversityPress Cambridge

Weisbuch G and S Battiston (2005) ldquoProduction Networks andFailure Avalanchesrdquo Working Paper mimeo Ecole Normale SupErieure

Zarnovitz V (1977) ldquoBusiness Cycles Observed and Assessed Why andHow they Matterrdquo NBER Working Paper Series 6230 National Bureauof Economic Research

30

A Tables

Table 1 Parameters valuesParametera Description Value

N Number of sectorsunits i of production 10Ci Number of j buyer sectors for sector i (IO matrix sparseness) N minus 1Di (E) Value of the external demand 10mij Physical input coefficient of input j for sector i sim U(0 1)Mi Sum of the mij input coefficients j for the production of i 05αi friction to changes of production levels Degree of lockndashin on

production technology085

αsi Speed of adaptation to demand changes 004

σ Desired multiple of production to stock 10

aInto parenthesis the number of lags of the initial value for lagged variables

31

is a more complicated process than simply ldquochanging Qrdquo or choosing themix of capital and laborrdquo And this is not only a matter of non convexitiesin production technologies The inconsistency of time reversibility in thetechnological choice is accompanied by the need to operate on the lsquoavail-able technological frontierrsquo (David 1975) If a firm plans a given productionactivity for a short period of one week even though the next day there is anunexpected fall in the demand it is unlikely that it is followed by a similardownturn in production The case is evident when we think at productionplanning that involve hiring of workers an capital investments (both in-volving sunk costs)7 Therefore any desired change generates a pattern ofsmall modifications approaching slowly the target level Moreover thesechanges are irreversible in the short term since only a prolonged change ofthe environment triggers the modification of previous desired levels

Resting on those simple conjectures we thus analyse a model in whichwe only allow for sluggish adaptation of production due to conservative(adaptive) behaviour of firms and the physical constraints faced when vary-ing the production levels Notice that these assumptions are contrary tothe generation of instability of a system In fact a firm behaving as de-scribed above reduces if not eliminates any disturbance affecting its ownstate if acting in isolation Other factors like new technologies extremelylong term perspective price changes etc are likely to be sources of volatil-ity therefore contributing to generate aggregate system fluctuations (as forexample in Acemoglu and Scott 1997) Our neglect of these and other fac-tors however unrealistic serves the purpose of testing whether a volatilityreducing microndashbehaviour turns into volatility generating aggregate resulttwo widely observed dynamics Quite interestingly (and encouraging) DosiFagiolo and Roventini (2006) have concomitantly proposed a model whichincorporates detailed representation of both sluggishness robustly based onempirical evidence Heterogeneous firms which undergo lumpy investmentand slowly adapt their expectation on changes of the future demand inducecomovement of macroeconomic variables qualitatively similar to the onesobserved in the real world To pursue our line of research we first needto concentrate on how sectorsfirms interact in the system in the effort tocomplement those promising results

Therefore the second element we consider is meant to represent a gen-eralized economic system We allow for the circularity of inputndashoutput sys-tems a crucial feature in determining feedback mechanisms and a principal

7See also the discussion in Acemoglu and Scott (1997) where the authors argue thatldquo[w]hile the presence of fixed costs can account for the discreteness of economic turningpoints it does not naturally lead to persistence because once an individual undertakes anaction they are less likely to do so in the near future [] [A]lthough the presence of fixedcosts leads to increasing returns these are intratemporal the full extent of economiesof scale arising from fixed costs can be exploited within a periodrdquo [p 502 authorsrsquoemphasis]

9

component of business cycle It may be true that most backward linkagesconcern capital goods (Weisbuch and Battiston 2005) but a rapid glanceto an inputndashoutput matrix would show that technical coefficients are nonnull in both directions and ldquomost commodities are inputs to the produc-tion process of other commoditiesrdquo (Horvath 2000 p 70) The higher thesectoral aggregation mdash observed or assumed mdash the more the relevance ofintermediate supply holds true (as mentioned above with respect to ma-trix incompleteness) We claim that the existence of backward and forwardlinkages are a crucial element to determine the persistence of shocks in theeconomy A perfectly linear (one directional) system needs much less adap-tation of firms and sectors in order to stabilise the economy Nonethelesscircularity does not imply that inputs are perfectly substitutable amongthem in order to determine the maximising allocation even at high level ofdisaggregation

By using a fixed coefficients production function we both maintain themid term rigidity of production processes and we do not impose limitedinteractions We claim that the circular property of the production structureis by itself sufficient to produce shock persistence even when the inputmatrix is full (although with heterogeneous input coefficients) mdash all sectorsshop in the remaining n minus 1 sectors

In conclusion we represent a model for an economic system made onlyof i) production level decisions and ii) input-output relations Clearly sucha model lacks many features present in real economic systems likely we areconvinced to affect the propensity to generate cycles As such we cannotaim at reproducing realistic data so our results cannot be tested comparingtime series generated by the model with real ones Instead given the paucityof the elements comprising the model we can investigate in detail to whichextent the (few) features present in our model affect the cyclical behaviour ofthe system As we will see we will be able to show quite a number of resultin which aggregate fluctuations are generated whereas the individual microstructure of the model aims at monotonous adaptation Before discussingthe results next section describes in detail the implementation of the simplemodel

4 The model

We model an economic system in which a number of sectors i = (1 N)are potentially connected via (intermediate) market relations A N times NInputndashOutput matrix determines the mij units of each input j = (1 N)mdash bought from the N minus 1 residual sectors mdash necessary to produce oneunit of output i8 For simplicity given the focus of this paper on the IndashOstructure as a generator of aggregate fluctuation we abstract from market

8IndashO coefficients are therefore physical coefficients

10

interactions within sectors each sector is a production unit In fact forour purposes it is not relevant which firm produces the sectoral outcomebut only the total amount produced by each sector9 Moreover besidesmaking the analysis more neat this assumption allows also to enjoy a largefreedom in determining the aggregation level since we donrsquot need to assessthe effects of between sectors competitiveness which for example is likelyto increase for increasing levels of disaggregation Our economic systemis therefore composed of N production units each of which may buy andprovide inputs from and to any other sector depending on the IndashO structureof the economy

The demand for each sector i is determined by a constant consumersdemand Di (E) and the input needs of other sectors

Di = Di (E) +sum

j 6=iisinCi

mjiQj (1)

where Ci is the set of j intermediate clients of i (sum

j isin C lt N) mji is thetechnical coefficient of firm j for input i ie the amount of i required toproduce one unit of j Qj is the actual output produced by sector j

Concerning the simultaneous production level of each sector in time twe make two lsquorealisticrsquo assumptions First from the technical viewpoint afirm like a tanker cannot abruptly change its course Therefore changesto the production level can be introduced only gradually through capital(dis)investment and job h(f)iring (see discussion in Section 3) Second weassume that firms do not decide the production level directly on the basisof the observed demand level but conscious of the uncertainty of demandplan their production in order to maintain invariant a desired level of stocksand avoid stockout10

Quantity produced by sector i in period t is computed as follows

Qit = αiQtminus1 + (1 minus αi)(

Slowastitminus1 minus Sitminus1

)

(2)

where αi measures the physical constraint in adapting the production capac-ity to changes in the required output Slowast

itminus1are firmrsquos i desired stocks and

Sitminus1 its actual stocks Notice that both variables related to stocks appearwith a lag This is due to two reasons First it is a modelling necessity inorder to allow the simultaneous determination of production levels for allsectors Secondly as frequently happens modelling necessities reveal sen-sible if not generally considered aspects of the modelled system In factproduction decisions concern management in the production plant while

9Adding heterogeneous firms within sectors would definitely enrich the causal expla-nation put forward and parametrised in this paper (see below) For example to analysethe effect of firms constraints It is left over for future research as in the present work weopt for the analysis of the main structural parameters that affect aggregate fluctuations

10See also discussion in Schuh (1996)

11

demand through sales is observed by the commercial staff It is thereforelogical that the information from demand reaches plant managers with somelag

The actual stock level is trivially computed subtracting from the previousheap of stock the amount sold and adding the new production

Sit = Sitminus1 + Qit minus Dit (3)

Concerning the desired level of stocks instead we introduce the conser-vative behaviour mentioned above Firms ideally would keep an amountof stocks proportional to the demand they receive say a multiple σ ofthat amount However when demand varies they adjust the desired levelsmoothly preventing sudden changes to the variable representing the goalof the firm

Slowastit = Slowast

itminus1 + αsi

(

σDit minus Slowastitminus1

)

(4)

where αsi measures the adaptation to demand changes

In summary the dynamics of the model consists of only the equationsabove Firstly at the beginning of a time step firms determine the quantityto produce as a function of past imbalances in stock levels and the pastproduction level These levels define the demand for all sectors determinedby the technical coefficients applied to all produced quantities Finallyactual and desired stocks can be updated Figure 1 represents visually thisdynamics highlighting the decisional components of the model in respect ofthe ldquomechanicsrdquo of trade

5 Results production structure and economic fluc-

tuations

In this section we describe some of the most relevant results concerningthe effects of the production structure to determine aggregate fluctuationsGiven the unusual analytical instrument chosen (at least concerning thistopic) we start with a brief methodological introduction meant to clarifythe nature of the results we claim to obtain Next we describe a generalsetting for the model which is the (extremely) limited area of the potentialparametersrsquo space of the model that we will explore In the rest of thesection we discuss the results from the model

51 Methodological considerations

Real economic systems are constantly subject to a continuous flow of shocksexternal to the production system besides changes within the system itselfIn order to appreciate the systemic properties we need to abstract from all

12

Compute the quantityto produce

F (Q(t-1) S(t-1) s(t-1) alpha)

Communicate thequantity of inputsrequired to each

supplier

Determine the overalldemand

F(D(E) D(I))

Revise the past decisionon stock plans

(desired stocks)F(D S(t-1) sigma alpha(s))

Compute the final levelof stocks for next period

F(s(t-1) Q D)

KEY

ModelDirection

Variablecomputed

at t-1(t-1)

Step includingfirm behavior

Figure 1 One period dynamics

the nonndashrelevant disturbances and concentrate on how the system endoge-nously contributes to the observed dynamics Therefore we will generatehighly abstract patterns whose analysis will provide insights on the systemproperties that would be otherwise lost in the noise generated by all theelements of a real system

We are not trying to fit the model to a specific empirical data set nor weare interested in studying the complete behaviour of the model for areas ofthe parameterrsquos space that make no economic sense (eg negative or infiniteproduction levels) Therefore the values of initial settings and results matteronly in relative and not in absolute terms

We are going to describe a simulation model with which we will produceseveral simulation runs and we claim obtain results relevant for the debatediscussed above Rigorously speaking we may claim only certainty for thevalidation of the results that is that the model does actually generate theresults presented and for the reasons we explain We limit this part to thepresentation of graphs and verbal explanations that we believe are ratheruncontroversial11 Concerning the relevance of our studies for real systems

11Interested readers can request the model code and and simulation data for replicationand extensions of the results We can guarantee that such possibility does not require

13

(verification) we simply avoid to pretend that our model is a quantitativeapproximation of real economies (which in which period) Our opinionis that the study of the model allows us to make general considerationsnon-obvious and relevant for applied debates supported by strong logicalarguments which can be integrated but not reverted by arguments con-cerning more and more elements of the economic world

52 Model setup

We go through the model properties analysing an economic system madeup of ten sectors (N = 10)12 Each sector shops inputs from the remain-ing N minus 1 and sells to them part of its own production as intermediategoods This means that the input matrix is complete except for the diag-onal (mij gt 0 forallj 6= i mii = 0) The sum of all N minus 1 coefficients for asingle sector is given by Mi and its benchmark value is kept at a plausiblemedium level (unless differently specified) to avoid distortions from highlydemanding sectors Each input coefficient is randomly drawn from a uniformdistribution

Sectors start at an equilibrium level their supply matches intermediateand final demand and in each period the same quantity of output mustbe produced to compensate for the used stocks and keep the desired stocksunchanged In thier turn stocks must be ten times the value of total sectoraldemand (σ) Eventually technological adjustment of production is ratherslow and producers are quiet refractory to follow sudden changes in thedemand in general a smoothing behaviour strongly prevails (α and αs)

Under those general preconditions in the following sections we anal-yse the dynamics of the economic system in relation to model parametersWe mainly focus on the cyclical behaviour of systems and their speed ofconvergence to the asymptotic equilibrium13

The very abstract assumption and the homogeneous initialisation donot allow for the multiple equilibria that are likely to rise in complex systemswith feedbacks Once more such simplification allows to have a betterunderstanding of the crucial impacts of the production structure on businessfluctuations Departing from the related literature discussed above14 weconsider the effects of a single shock rather then a continuous flow whichin our model is superfluous in order to obtain fluctuation persistence15 To

extensive programming or statistical skills besides the usual education of economists12You can think of a high level of aggregation in the statistical observation The com-

plete initialisation and parameters values are available in table 1 in the Appendix13Notice that sectors only approach a final equilibrium level of production which can

be prendashdetermined under static conditions14Included the more proximate work of Helbing Witt Lammer and Brenner (2004)15An exception are the last results presented in Section 58 which we see more as an

opportunity for further research than a closing of the paper

14

start with next section (53) shows the aggregate cyclical behaviour of astandard setting

53 Single shock analysis

We generate a simulated history following the dynamics of the modelrsquos vari-ables when the system is set to an equilibrium level and a single shockmodifies one of the exogenous elements namely the external demand forsectors Modifications across sectors are random and not correlated andmay increase or reduce external demand by at most 10

100 225 350 475 600

193894

195861

197828

199795

201762 Industrial Production

Time

Figure 2 Total industrial production fluctuations following a shock fromexternal demand

Figure 2 shows the dynamics generated by the total industrial productionof the system mdash that is also a proxy for the systemrsquos GDP fluctuations Thenew external demand changes the equilibrium level but the system cannotreach the new equilibrium with a monotonous pattern and converges to thenew level with a fluctuating pattern Each change in the external demandinduces also modifications of the intermediate demand and sectoral adjust-ments in the quantity produced typically overshoot the aggregate patternAs it can be appreciated from the figure it takes time before the system isable to converge to the new static equilibrium conditions

We study two features of this pattern absolute level of the new equilib-rium and volatility induced by the shock

54 Technical coefficients and industrial output

Production level

The equilibrium levels of industrial output depend on two factors only Ob-viously the levels of external demand determine how much of the total pro-

15

duction needs to be supplied to consumers exogenous from the productionsystem Second the technical coefficients also influence strongly the level ofactivity of the system defining the intermediate demand to each sector

180

200

220

240

260

280

300

01 02 03 04 05 06 07 08 09

Out

put l

evel

(In

dust

rial p

rodu

ctio

n)

Sum of total input coefficients (M)

Total industrial production for different input coefficients levels

Figure 3 The relevance of input coefficients on output level

Figure 3 shows the equilibrium level of industrial output reached bysystems with identical external demand where we fixed the sum of inputcoefficients for each sector of the economy (Mi) to different values16 Thehigher the sum the higher is the total production sustained by the system

This result depends on the very production structure represented by theinputndashoutput matrix In fact the higher is the level of production requiredby intermediate sectors the higher is the total production given a constantlevel of external demand

Notice that the level of production does not depend on the distribution ofthe technical coefficients As long as the sum of the coefficients is identicalthe same production level is obtained irrespective of their distribution (otherthings being equal)

Output shock absorption

Input coefficients have a relevant role also on the output stability of thesystem Starting from an output equilibrium level we hit the system witha single shock on the external demand for all sectors uncorrelated amongsectors17 modifing (positively or negatively) the demand for all sectors by

16Within this limit the actual value of the coefficients are randomly drawn17In each sector the external demand changes by plusmnX where X is a uniformly dis-

tributed percentage between 1 and 10

16

at most10 As show in Figure 2 the economic systems produces a lenghtlysmoothing oscillatory behaviour We compute the relative deviation of theaggregate production (industrial output) as the ratio between its standarddeviation and its mean δ = σY microY where Y =

sum

Qi Figure 4 shows thebehaviour of δ as a function of the sum of input coefficients for each sector80 periods after the shock As time goes by the level of δ reduces for allMi but the relation with the sum of input coefficient is unchanged

00055

0006

00065

0007

00075

0008

00085

0009

01 02 03 04 05 06 07 08 09

Rel

ativ

e va

rianc

e of

tota

l out

put (

delta

)

Sum of total input coefficients

Aggregate output deviations for different technical coefficients

Figure 4 The relevance of input coefficients on output stability

In sum input coefficients determine the extent of overshooting behaviourThe higher the coefficients the higher is the backlash even when shockscould cancel out through sectors

55 Shock persistence and adjustment coefficients

We now consider the effect of micro smoothing behaviours on the volatilityof the production system In particular we are interested in testing theeffects of two parameters of the model on the volatility of the productionsystem First we consider the effects of the αrsquos representing the ldquostickinessrdquoof the production technology to desired changes of Q aimed at filling thegap between actual and desired stocks The higher is α the slower is thereaction to stock unbalances Concerning its effect referring to the technicalpossibilities available ie production capital and organization we may putforward two opposite hypothesis relevant to systemrsquos volatility

Hypothesis 1 Slower production adjustment (high α) reduces volatilitybecause unbalances in demand of one sector being compensated slowly slow

17

down the diffusion of shocks to other sectors

Hypothesis 2 Slower production adjustment (high α) increase volatil-ity because the longer lasts the mismatch between actual and equilibriumproduction level the deeper will become the mismatch between actual anddesired stocks

20 25 30 35 40 45 50 55 60 65

Cyclesrsquo width

07 075

08 085

09alpha 0005

001 0015

002 0025

003 0035

004 0045

005

alphaS

20 25 30 35 40 45 50 55 60 65

Figure 5 Maximum width recorded between highest and lowest peaks fortotal production for different values of α and αS

Secondly we consider the parameters αS representing the speed of ad-justment of desired stock to observed demand The higher this parame-ter the faster sectors revise the desired levels of stocks to current levelsof observed demand This parameter concerns sectorsrsquo behavioral decisionprocess Higher αS represent a stronger belief that the most recently ob-served demand level is a reliable indicator of future demand levels whilelower levels of the parameter represent a more conservative approach wheremanagers require a long observation of demand levels different from the pastones before adjusting their expectations Also concerning the effect of thisparameter on the volatility induced by an exogenous shock we may maketwo hypothesis

Hypothesis 1 Faster adaptation of desired stocks to new levels of de-mand (higher αS) reduce volatility because changes in demand are rapidlytranslated in adjusted levels of production

Hypothesis 2 Faster adaptation of desired stocks to new levels of demand(higher αS) increase volatility because temporary unbalances of demand in

18

one sector are transmitted faster to other sector reinforcing the feedbackmechanism of fluctuations

To test which of these hypotheses are confirmed by the model we runsimulation runs with identical systems (eg same technical coefficients samelevels of demand same shocks) but for the values of the αrsquos and αS rsquos testingall combinations of these parameters over a range of values (each run usedthe same parameters for all the sectors of the system) For each simulation(ie each couple of values of the parameters) we computed two indicatorsof volatility One measures the maximum distance of total production be-tween the highest and the lowest peak that is the maximum width of theoscillations registered in the production time series The second indicatormeasures the relative deviation δ = σY microY Figures 5 and 6 report theseresults

0004 0006 0008 001 0012 0014 0016 0018 002

Relative deviation

07 075

08 085

09alpha 0005

001 0015

002 0025

003 0035

004 0045

005

alphaS

0004 0006 0008 001

0012 0014 0016 0018 002

Figure 6 Relative deviation of total production for different values of α andαS

The results confirms unequivocally both hypotheses 2 and reject hy-potheses 1 In a sense these hypotheses confirm the nature of the economicsystem represented in the model as a complex system where the interactionsamong sectors play a far more important role than the individual sector be-haviour In fact in both cases we may interpret the results that the strongerthe efforts to mitigate the effects of shocks (ie slow productionrsquos and quickexpectationsrsquo adjustment) the opposite result is actually obtained the fluc-tuations become even more accentuated It is also worth to note that thestronger impact is generated by the behavioural parameters rather than thetechnical one as indicated by the stronger effects of the αS in respect of theα This result seems to suggest that ceteris paribus information mattersmore than technical constraints Though the model is by any means in-

19

adequate to discuss normative aspects because of the naive ways strategicdecision making is represented it can anyway suggest the effects of errorsin management decisions (ie wrong desired levels of stocks) as opposed tothe technical constraints preventing sudden adjustment of production lev-els Stretching a bit the interpretation of these results we may suggests thatefforts to reduce volatility should better be oriented to improve the trans-mission of information in order to coordinate future production plans (ieidentifying quickly the desired levels of stocks and therefore the new equilib-rium production level) rather than improving the flexibility of productionmethods (reducing α)

56 Volatility and stock levels

In our model stock levels are maintained to buffer a sectorrsquos sale againstunexpected changes in demand Therefore a system made of risk adversemanagers (keeping higher levels of stocks) may be thought to lead to asmoother absorption of shocks in respect of a system composed of firmskeeping a short supply to cushion unexpected events To test this hypothesiswe tested the behaviour of the model for different values of σ the levels ofstock represented as a multiple of demand For each level of σ we run asimulation using all other initialization identical as usual representing aneconomic system ldquoalmostrdquo at equilibrium but for a single small shock indemand uncorrelated across sectors18

00044

00046

00048

0005

00052

00054

00056

00058

0006

00062

5 55 6 65 7 75 8 85 9 95 10

Rel

ativ

e va

riatio

n of

tota

l out

put (

delta

)

Ratio of stocked goods over production (sigma)

Stocks and production fluctuations

Figure 7 Fluctuations persistence on systems with different stock strategies(σ)

18Demand changes by plusmn1-10

20

Figure 7 shows the value of the relative deviation of output production(δ) generated by heterogeneous uncorrelated shocks in the external demandon identical systems where firms store a different multiple of their demand asstocks We obtain actually the opposite result that we may have expectedfluctuation become larger and more persistent the higher the level of stocksIn some way a strategy that for the individual firm is meant to reducethe fluctuations (using stocks as buffers) ends up by actually increasing thevolatility at aggregate levels

The result is though explainable both at the theoretical and empiricallevel In fact if no stocks were maintained production would need to adjustfor changes in demand only in case of a shock That is the firm may movemonotonically from the old to the new production levels sending consistentsignals to the rest of the system (ie its own suppliers) Instead if thesame shock affects a firm maintaining large stocks of output over the sameperiod production needs to adjust to the new demand and and to alignstocks with the new demand This generates besides the adjustment to thedemand a potentially inconsistent signal to the suppliers since the firm willhave a temporary level of production not meant to continue in equilibrium(adjusting stocks)

From the empirical viewpoint the last few decades have seen the almostuniversal diffusion of production system meant to keep stocks to a minimumif at all Lean production methods justndashinndashtime etc are solutions tothe problems of the costs necessary to maintain stocks Over the sameperiod economic fluctuations have been strongly reduced and few empiricalassessments have pointed to the relations between the two phenomena Forexample concerning the US ldquochanges in inventory behavior have playeda direct role in reducing real output volatilityrdquo (Kahn McConnell andPerez-Quiros 2002 p 183)19

57 Irrelevance of technical coefficients distribution

Horvath (1998) and Horvath (2000) suggest that systems with sparse ma-trices ie inputndashoutput matrices with many empty cells tend to be morevolatile in respect of systems with more evenly distributed matrices Noticethat this claim risk if not adequately qualified to generate an illogical resultIn fact matrices sparseness increases as a function of sectoral disaggregationand comparing the same systemrsquos fluctuations using inputndashoutput matricesat different levels of aggregation needs to produce the same type of volatil-ity Indeed this is not the case if all remaining parameters stand equaland do not adjust to the different level of analysis considered If aggregatepatterns of adjustment speed are maintained unchanged the higher the sec-toral disaggregation the higher the systemrsquos volatility ceteris paribus But

19Similar results are found in Blanchard and Simon (2001)

21

for example in our model reducing the aggregation also requires a changein the adaptation coefficients the more micro we undergo the analysis thehigher is the speed of adjustment of economic objects If for example weconsider sectors we are evaluating the adaptation of an entire set of firmswhich pose questions on how to aggregate their dynamics Things changewhen we consider the response of a single firm

Going back to Horvatsrsquo argument highly disaggregated matrices arelikely to contain many empty cells so that we may forecast different levelsof volatility depending on the aggregation used for the same system whichas mentioned is not caused by sparseness but to a failed adjustment inreaction mechanisms Results from our model20 show that volatility is com-pletely unaffected by input coefficients distribution Given their sectoralsum mdash through rows of the matrix mdash the value of each coefficient is irrele-vant The same occurs in the case of very skewed distributions that presenta high number of empty cells (zero input coefficients)

58 Microndash and Macrondashvolatility

As we have seen above one of the major discussion in the literature concernsthe relations between microndash and macrondashvolatility Put it simply one sideof the literature considers that uncorrelated microndashshocks cancel each otherout reducing their effects at macrondashlevel Conversely other researchersconsider that microndashshocks are multiplied at macrondashlevel generating highermacrondashvolatility than that we can observe at microndashlevel The debate isof obvious importance to determine the sources of business cycles to pre-dict their effects and determine the more effective policies to mitigate theirnegative effects

Our model is only a partial representation of realndashworld economic sys-tem and is also overly simplistic However based on these limitations italso offers very robust results allowing reliable conclusions on the (limited)elements included in the model In particular we consider our model a re-liable representation of the basic structures of the production interactionsamong sectors or firms Thus we can use the model to provide an answerto the debate concerning the contributions to volatility from the productioninteractions only Other contributions (eg from financial markets priceadjustments technological innovation etc) may re-inforce or counter theforces we analyse but they need to be considered as additional elements(and we opine also generally less relevant) not alternative to our analysisWe consider a necessity in order to apply a rigorous methodological ap-proach to be able to identify with certainty the results provided by partialstudies in order to eventually obtain by gradual extensions of the analy-sis more and more detailed representations of realistic economic systems

20Available from the authors

22

The results we obtain in this paper concerning this microndashmacro debate aremeant as a first relevant step to a more extended analysis

0

002

004

006

008

01

012

014

016

018

02

022

05 1 15 2 25 3

Var Demand

Macro (Output) and micro (Demand) volatility

delta Outputdelta Demand

Figure 8 Values of δ (relative deviation) for aggregate demand and ag-gregate output in respect of different levels of variance for sector specificuncorrelated external demand shocks

The flows of shocks affecting the firms of a system can be distinguishedalong two dimensions strength of the shock (eg amplitude of the changesin demand) and frequency of the shock (rare or frequent) Concerning thefirst aspect we initialized as usual a system to equilibrium levels (eg pro-duction quantities equal to external and intermediate demand for all sectorsconstant level of stocks equal to the desired level) We then ran the simu-lation generating the external demand for all sectors as a random variablewith constant (equilibrium) mean level and varying values of variance

Figure 8 shows unequivocally that as far as our model has propertiessimilar to real systems the hypothesis that macrondashfluctuations have largervolatility than their sectoral level shocks is fully confirmed At any levelof variance the aggregate volatility of demand as captured by the relativedeviation index is sensibly lower than the volatility shown by aggregateoutput that is the total production of the system This result is not sur-prising throughout all tests presented in the previous sections every resultsuggested that the system lsquooverndashreactsrsquo to any individual microndashlevel changeeven in the case these changes were meant to contrast disturbances There-fore when we move from considering one single shock to a flow of shockswe are not surprised to obtain that the system shows higher volatility thanthat implied by the microndashlevel disturbances

We now consider the second dimension defining a flow of shocks theirfrequency The question is whether for the same level of demand shocks

23

0 002 004 006 008 01 012 014 016

Difference of relative deviation for Output and Demand

0 5 10 15 20 25 30 35 40 45 50 55

Shock frequency 05

1

15

2

25

3

Var Demand

0 002 004 006 008 01

012 014 016

Figure 9 Differences of values of δ for aggregate output minus those foraggregate demand in respect of different levels of variance of demand and offrequency of the shocks

(ie same variance) more frequent changes trigger a higher or lower systemvolatility than rare disturbances The results shown in figure 9 suggest arather elaborated answer which partly vindicates (though in a particularsense) the arguments in favour of compensation of micro shocks The fig-ure shows the difference between aggregate output volatility and aggregatedemand volatility The independent variables are as before the volatilityof demand at microndashlevel21 and the time intervals in between two shocksAggregate demand volatility does not change for the frequency of shocksat least when computed over a long period But output volatility doesin a nonndashmonotonous way For high frequency of shocks the difference isstill positive and increasing for higher variance indicating that the ldquooverndashreactionrdquo argument is valid even in these cases but it is rather low Whilethe frequency of shocks slows down aggregate volatility of output increasesshowing that time is needed for the microndashshock to unfold their full effectsat the macrondashlevel However after a certain threshold less and less frequentshocks generate a lower volatility of the aggregate level of output

This result is quite sensible though not obvious at first sight and onceagain shows the power of such a simple model to generate and explain com-plex properties For frenetic levels of demand modifications the system hasno time to adjust but part of the job to align actual and desired productionfor constantly changing demand is performed by the varying demand itselfIn other terms the shocks cancel out through time if not through their quan-

21Beware of the scale of variance For reasons of visibility the scale is inverted

24

titative effects on the system While frequency decreases the disturbancesof the shocks not compensated by other (unlikely) shocks have the time tofilter through the production system generating higher and higher volatil-ity In a sense any system has a particular frequency of shocks that make itmaximally ldquoresonatingrdquo with the amplification of microndashshocks When therate of shocks gets even rarer the system can enjoy periods of longer andlonger quietness after each shock is absorbed and before a new one startsagain a cycle of initial local disturbance propagation of the disturbances toother sectors and settlement to the new equilibrium

6 Discussion and extensions

We presented a simple model of production composed by interacting sectorThe main (quit realistic) assumption of the model is that the productionprocesses in any sector of the economy make use of input from any othersector Moreover we assumed that sectors (represented as a single firm)adjust slowly both the desired level of production (smoothed revision of ex-pectation) and the actual level (smoothed modification of plant utilization)The model voluntarily leaves aside a large number of aspects of real-worldeconomic systems (eg financial markets price adjustments growth etc)The reason is that the model can be easily and reliably tested in order toextract its properties The possibility to transfer the (certain) model resultto a real context depend of course on an ample ceteris paribus clause How-ever we are confident that the reported results concerning the productionstructure play a relevant role in real world systems Therefore our resultsand particularly their motivations being accessible because of the relativesimplicity of the model will not be diminished by the integration of ourconsiderations with other aspects of economic systems

Our model is in essence a representation of an input-output table inter-preted in physical terms that is as the amount of inputs required for oneunit of output Besides this we include two straightforward assumptionsFirstly production levels cannot be suddenly changed but require time toscale up or down the utilization rate that is the levels of production Stocksare meant to make up for the difference between sales and production Thesecond assumption concerns the intentional behaviour of producers We im-plement this aspect by assuming that exists a level of stocks desired for eachlevel of demand However a change of demand does not translate suddenlyin a new level of desired stock but again only a long time at a different levelof demand will convince managers to update the desired levels of stocks

We show that the model generate under a wide number of settings oscil-latory patterns converging toward the equilibrium level where each sectorgenerate exactly the quantity demanded by final consumers and by othersectors as input The oscillations are provoked by over- and under-shooting

25

generated by the difficulty of coordination among sectors that by assump-tion ldquocommunicaterdquo only via variation of demand to their direct input sup-pliers We also show that the total production levels generated by sectors(sum of the physical units) produced at equilibrium depends (for constantfinal demand) on the total sum of the technical coefficients per sector irre-spective of their distribution This result is motivated by the impossibilityof input substitution and offers interesting suggestions for the analysis ofthe study of input-output tables of real-world systems

We then consider systems at equilibrium facing a single small change ofthe external demand for sectors These simulations allow to study the waythe system reacts to shocks We see that the sum of the coefficients affectsbesides the levels also the volatility of aggregate output the larger a systemthe stronger is the impact of a given shock Less obviously we find thatfor systems with the same dimension (ie same sums of coefficients) theaggregate volatility increases with both the inertia to change of productionlevel and the speed of adjustment of desired stocks This result is somewhatcounter-intuitive since both the cited variables are associated to a moreconservative behaviour that may be supposed should smooth away spikesin the pattern from an old to a new equilibrium However we show thatthe opposite actually holds given the complex interactions among sectorindividual-sector attitude intended to smooth shocks actually increase theiraggregate impact

In the same vein we show that systems maintaining high levels of stocksare more volatile that systems with reduced amounts of stocks Again thisis a counter-intuitive result reverting the goal of individual producers andthe aggregate result Interestingly this result finds strong support from theevidence of the recent changes in production methods (ie low or no stocks)and diminished cycles

In relation to specific issues discussed in the literature we find no supportto the suggestion that particular distributions of input-output coefficientsmodify the volatility of a system As said this depends in case on thesums of these coefficients but not on their distribution Further we canprovide an answer to a long and hotly debated questions whether microndashlevels volatility generates stronger or lower aggregate volatility Our modelshows unequivocally that microndashvolatility (eg variance of final demand)persists and generates stronger volatility at aggregate level Again we canidentify the reason on the complex production structure that exalts micro-level shocks by generating lsquowrongrsquo messages among producers Moreoverwe can also point to a usually neglected aspect of volatility we show thatthe frequency of shocks is a very relevant aspect besides their dimensionin determining the volatility of aggregate output

Our work can be developed along two complementary directions Firstlywe can continue to study the properties of the model by extending the ele-ments considered (eg pricing) analysing the effects on the present results

26

and generating more analytical evidence Of particular relevance is in ouropinion the problem of aggregation Representations of the same systemswith different codification systems (eg different number of SIC digits)must not modify overall system properties for example the volatility ob-served Given that our result show that the sums of coefficients matter forthe levels and volatility of the system and that these sum do depend on theaggregation chosen it should be possible to induce the reaction coefficientsthat we know affect the volatility only

Secondly we can find an application of our results to real world evidenceIn fact there are many data sets available for the data we deal with basicallyinput-output tables Our model is readily adapt to be parametrized alongany number of sector and coefficient necessary so that we may replicate pastseries in order to adapt the unobservable parameters and use the resultingcomplete model to explain business fluctuations and provide intuitions onfuture patterns

27

References

Acemoglu D and A Scott (1997) ldquoAsymmetric Business CyclesTheory and TimendashSeries Evidencerdquo Journal of Monetary Economics 40501ndash533

Backus D P Kehoe and F Kydland (1995) ldquoInternational BusinessCycles Theory and Evidencerdquo in Frontiers of Business Cycle Researched by T Cooley Princeton University Press Princeton

Bak P K Chen J Scheinkman and M Woodford (1993) ldquoAg-gregate Fluctuations from Independent Sectoral Shocks SelfndashOrganizedCritically in a Model of Production and Inventory Dynamicsrdquo RicercheEconomiche 47(1) 3ndash30

Battiston S D Delli Gatti M Gallegati B Greenwald and

J Stiglitz (2005) ldquoCredit Chains and Bankruptcies Avalanches in Sup-ply Networksrdquo in Annual Workshop on Economic Heterogeneous Inter-acting Agents (WEHIA 2005) University of Essex UK

Blanchard O and J Simon (2001) ldquoThe Long and Large Declinein US Output Volatilityrdquo Brookings Papers on Economic Activity 1135ndash174

Bresnahan T F and V A Ramey (1994) ldquoOutput Fluctuations atthe Plant Levelrdquo The Quarterly Journal of Economics 109(3) 593ndash924

David P A (1975) Technical choice innovation and economic growth essays on American and British experience in the nineteenth centuryCambridge University Press London

Delli Gatti D C Di Guilmi E Gaffeo G Giulioni M Galle-

gati and A Palestrini (2004) ldquoA New Approach to Business Fluc-tuations Heterogeneous Interacting Agents Scaling Laws and FinancialFragilityrdquo Journal of Economic Behavior amp Organization 56 489ndash512

Dosi G G Fagiolo and A Roventini (2006) ldquoAn EvolutionaryModel of Endogenous Business Cyclesrdquo Computational Economics 273ndash34

Dupor B (1999) ldquoAggregation and the Irrelevance in MultindashSector Mod-elsrdquo Journal of Monetary Economics 43 391ndash409

Head A C (1995) ldquoCountry Size Aggregate Fluctuations and Interna-tional Risk Sharingrdquo The Canadian Journal of Economics 28(4b) 1096ndash1119

28

Helbing D and S Lammer (2005) ldquoSupply and Production NetworksFrom the Bullwhip Effect to Business Cyclesrdquo in Networks of Interactingmachines Production Organization in Complex Industrial Systems andBiological Cells ed by D Armbruster K Kaneko and A S Mikhailovvol 3 of World Scientific Lecture Notes in Complex Systems World Sci-entific

Helbing D S Lammer U Witt and T Brenner (2004) ldquoNetworkndashInduced Oscillatory Behavior in Material Flow Networks and IrregularBusiness Cyclesrdquo Phisical Review E 70(056118) 1ndash6

Helbing D U Witt S Lammer and T Brenner (2004) ldquoNetwork-Induced Oscillatory Behaviour of Macroeconomic Output Flowsrdquo in In-dustry and Labour Dynamics II The Agent-based Computational Eco-nomics Approach Proceedings of the wildace 2004 conference ed byR Leombruni and M Richiardi no 5 LABORatorio R Revelli

Horvath M (1998) ldquoCyclically and Sectoral Linkages Aggregate Fluctu-ations from Independent Sectoral Shocksrdquo Review of Economic Dynam-ics 1 781ndash808

(2000) ldquoSectoral Shocks and Aggregate Fluctuationsrdquo Journal ofMonetary Economics 45(1) 69ndash106

Jovanovic B (1987) ldquoMicro Shocks and Aggregate Riskrdquo The QuarterlyJournal of Economics 102(2) 395ndash410

Kahn J A M M McConnell and G Perez-Quiros (2002) ldquoOnthe Causes of the Increased Stability of the US Economyrdquo FRBNY Eco-nomic Policy Review 8(1) 183ndash202

Kraay A and J Ventura (1998) ldquoComparative Advantage and theCrossndashSection of Business Cyclesrdquo World Bank Woorking paper Series1948 World Bank

Long J B and C I Plosser (1983) ldquoReal Business Cyclesrdquo Journalof Political Economy 31(1) 39ndash69

Lucas R E (1977) ldquoUnderstanding Business Cyclesrdquo CarnegiendashRochester Conference Series on Public Policy (special issue of Journalof Monetary Economics) 5 7ndash29

Nelson R R and S G Winter (1982) An Evolutionary Theory ofEconomic Change Harvard University Press Cambridge MA

Nirei M (2005) ldquoThreshold Behavior and Aggregate Fluctuationsrdquo Jour-nal of Economic Theory in press

29

Scheinkman J and M Woodford (1994) ldquoSelfndashOrganized Criticallyand Economic Fluctuationsrdquo American Economic Review 84(2 Papersand Proceedings of teh AEA) 417ndash421

Schuh S (1996) ldquoEvidence of the Link Between FirmndashLevel and Ag-gregate Inventory Behaviorrdquo Finance and Economics Discussion Series1996-46 Federal Reserve Board

Stiglitz J and B Greenwald (2003) Towards a New Paradigm inMonetary Economics Raffaele Mattioli Lectures Cambridge UniversityPress Cambridge

Weisbuch G and S Battiston (2005) ldquoProduction Networks andFailure Avalanchesrdquo Working Paper mimeo Ecole Normale SupErieure

Zarnovitz V (1977) ldquoBusiness Cycles Observed and Assessed Why andHow they Matterrdquo NBER Working Paper Series 6230 National Bureauof Economic Research

30

A Tables

Table 1 Parameters valuesParametera Description Value

N Number of sectorsunits i of production 10Ci Number of j buyer sectors for sector i (IO matrix sparseness) N minus 1Di (E) Value of the external demand 10mij Physical input coefficient of input j for sector i sim U(0 1)Mi Sum of the mij input coefficients j for the production of i 05αi friction to changes of production levels Degree of lockndashin on

production technology085

αsi Speed of adaptation to demand changes 004

σ Desired multiple of production to stock 10

aInto parenthesis the number of lags of the initial value for lagged variables

31

component of business cycle It may be true that most backward linkagesconcern capital goods (Weisbuch and Battiston 2005) but a rapid glanceto an inputndashoutput matrix would show that technical coefficients are nonnull in both directions and ldquomost commodities are inputs to the produc-tion process of other commoditiesrdquo (Horvath 2000 p 70) The higher thesectoral aggregation mdash observed or assumed mdash the more the relevance ofintermediate supply holds true (as mentioned above with respect to ma-trix incompleteness) We claim that the existence of backward and forwardlinkages are a crucial element to determine the persistence of shocks in theeconomy A perfectly linear (one directional) system needs much less adap-tation of firms and sectors in order to stabilise the economy Nonethelesscircularity does not imply that inputs are perfectly substitutable amongthem in order to determine the maximising allocation even at high level ofdisaggregation

By using a fixed coefficients production function we both maintain themid term rigidity of production processes and we do not impose limitedinteractions We claim that the circular property of the production structureis by itself sufficient to produce shock persistence even when the inputmatrix is full (although with heterogeneous input coefficients) mdash all sectorsshop in the remaining n minus 1 sectors

In conclusion we represent a model for an economic system made onlyof i) production level decisions and ii) input-output relations Clearly sucha model lacks many features present in real economic systems likely we areconvinced to affect the propensity to generate cycles As such we cannotaim at reproducing realistic data so our results cannot be tested comparingtime series generated by the model with real ones Instead given the paucityof the elements comprising the model we can investigate in detail to whichextent the (few) features present in our model affect the cyclical behaviour ofthe system As we will see we will be able to show quite a number of resultin which aggregate fluctuations are generated whereas the individual microstructure of the model aims at monotonous adaptation Before discussingthe results next section describes in detail the implementation of the simplemodel

4 The model

We model an economic system in which a number of sectors i = (1 N)are potentially connected via (intermediate) market relations A N times NInputndashOutput matrix determines the mij units of each input j = (1 N)mdash bought from the N minus 1 residual sectors mdash necessary to produce oneunit of output i8 For simplicity given the focus of this paper on the IndashOstructure as a generator of aggregate fluctuation we abstract from market

8IndashO coefficients are therefore physical coefficients

10

interactions within sectors each sector is a production unit In fact forour purposes it is not relevant which firm produces the sectoral outcomebut only the total amount produced by each sector9 Moreover besidesmaking the analysis more neat this assumption allows also to enjoy a largefreedom in determining the aggregation level since we donrsquot need to assessthe effects of between sectors competitiveness which for example is likelyto increase for increasing levels of disaggregation Our economic systemis therefore composed of N production units each of which may buy andprovide inputs from and to any other sector depending on the IndashO structureof the economy

The demand for each sector i is determined by a constant consumersdemand Di (E) and the input needs of other sectors

Di = Di (E) +sum

j 6=iisinCi

mjiQj (1)

where Ci is the set of j intermediate clients of i (sum

j isin C lt N) mji is thetechnical coefficient of firm j for input i ie the amount of i required toproduce one unit of j Qj is the actual output produced by sector j

Concerning the simultaneous production level of each sector in time twe make two lsquorealisticrsquo assumptions First from the technical viewpoint afirm like a tanker cannot abruptly change its course Therefore changesto the production level can be introduced only gradually through capital(dis)investment and job h(f)iring (see discussion in Section 3) Second weassume that firms do not decide the production level directly on the basisof the observed demand level but conscious of the uncertainty of demandplan their production in order to maintain invariant a desired level of stocksand avoid stockout10

Quantity produced by sector i in period t is computed as follows

Qit = αiQtminus1 + (1 minus αi)(

Slowastitminus1 minus Sitminus1

)

(2)

where αi measures the physical constraint in adapting the production capac-ity to changes in the required output Slowast

itminus1are firmrsquos i desired stocks and

Sitminus1 its actual stocks Notice that both variables related to stocks appearwith a lag This is due to two reasons First it is a modelling necessity inorder to allow the simultaneous determination of production levels for allsectors Secondly as frequently happens modelling necessities reveal sen-sible if not generally considered aspects of the modelled system In factproduction decisions concern management in the production plant while

9Adding heterogeneous firms within sectors would definitely enrich the causal expla-nation put forward and parametrised in this paper (see below) For example to analysethe effect of firms constraints It is left over for future research as in the present work weopt for the analysis of the main structural parameters that affect aggregate fluctuations

10See also discussion in Schuh (1996)

11

demand through sales is observed by the commercial staff It is thereforelogical that the information from demand reaches plant managers with somelag

The actual stock level is trivially computed subtracting from the previousheap of stock the amount sold and adding the new production

Sit = Sitminus1 + Qit minus Dit (3)

Concerning the desired level of stocks instead we introduce the conser-vative behaviour mentioned above Firms ideally would keep an amountof stocks proportional to the demand they receive say a multiple σ ofthat amount However when demand varies they adjust the desired levelsmoothly preventing sudden changes to the variable representing the goalof the firm

Slowastit = Slowast

itminus1 + αsi

(

σDit minus Slowastitminus1

)

(4)

where αsi measures the adaptation to demand changes

In summary the dynamics of the model consists of only the equationsabove Firstly at the beginning of a time step firms determine the quantityto produce as a function of past imbalances in stock levels and the pastproduction level These levels define the demand for all sectors determinedby the technical coefficients applied to all produced quantities Finallyactual and desired stocks can be updated Figure 1 represents visually thisdynamics highlighting the decisional components of the model in respect ofthe ldquomechanicsrdquo of trade

5 Results production structure and economic fluc-

tuations

In this section we describe some of the most relevant results concerningthe effects of the production structure to determine aggregate fluctuationsGiven the unusual analytical instrument chosen (at least concerning thistopic) we start with a brief methodological introduction meant to clarifythe nature of the results we claim to obtain Next we describe a generalsetting for the model which is the (extremely) limited area of the potentialparametersrsquo space of the model that we will explore In the rest of thesection we discuss the results from the model

51 Methodological considerations

Real economic systems are constantly subject to a continuous flow of shocksexternal to the production system besides changes within the system itselfIn order to appreciate the systemic properties we need to abstract from all

12

Compute the quantityto produce

F (Q(t-1) S(t-1) s(t-1) alpha)

Communicate thequantity of inputsrequired to each

supplier

Determine the overalldemand

F(D(E) D(I))

Revise the past decisionon stock plans

(desired stocks)F(D S(t-1) sigma alpha(s))

Compute the final levelof stocks for next period

F(s(t-1) Q D)

KEY

ModelDirection

Variablecomputed

at t-1(t-1)

Step includingfirm behavior

Figure 1 One period dynamics

the nonndashrelevant disturbances and concentrate on how the system endoge-nously contributes to the observed dynamics Therefore we will generatehighly abstract patterns whose analysis will provide insights on the systemproperties that would be otherwise lost in the noise generated by all theelements of a real system

We are not trying to fit the model to a specific empirical data set nor weare interested in studying the complete behaviour of the model for areas ofthe parameterrsquos space that make no economic sense (eg negative or infiniteproduction levels) Therefore the values of initial settings and results matteronly in relative and not in absolute terms

We are going to describe a simulation model with which we will produceseveral simulation runs and we claim obtain results relevant for the debatediscussed above Rigorously speaking we may claim only certainty for thevalidation of the results that is that the model does actually generate theresults presented and for the reasons we explain We limit this part to thepresentation of graphs and verbal explanations that we believe are ratheruncontroversial11 Concerning the relevance of our studies for real systems

11Interested readers can request the model code and and simulation data for replicationand extensions of the results We can guarantee that such possibility does not require

13

(verification) we simply avoid to pretend that our model is a quantitativeapproximation of real economies (which in which period) Our opinionis that the study of the model allows us to make general considerationsnon-obvious and relevant for applied debates supported by strong logicalarguments which can be integrated but not reverted by arguments con-cerning more and more elements of the economic world

52 Model setup

We go through the model properties analysing an economic system madeup of ten sectors (N = 10)12 Each sector shops inputs from the remain-ing N minus 1 and sells to them part of its own production as intermediategoods This means that the input matrix is complete except for the diag-onal (mij gt 0 forallj 6= i mii = 0) The sum of all N minus 1 coefficients for asingle sector is given by Mi and its benchmark value is kept at a plausiblemedium level (unless differently specified) to avoid distortions from highlydemanding sectors Each input coefficient is randomly drawn from a uniformdistribution

Sectors start at an equilibrium level their supply matches intermediateand final demand and in each period the same quantity of output mustbe produced to compensate for the used stocks and keep the desired stocksunchanged In thier turn stocks must be ten times the value of total sectoraldemand (σ) Eventually technological adjustment of production is ratherslow and producers are quiet refractory to follow sudden changes in thedemand in general a smoothing behaviour strongly prevails (α and αs)

Under those general preconditions in the following sections we anal-yse the dynamics of the economic system in relation to model parametersWe mainly focus on the cyclical behaviour of systems and their speed ofconvergence to the asymptotic equilibrium13

The very abstract assumption and the homogeneous initialisation donot allow for the multiple equilibria that are likely to rise in complex systemswith feedbacks Once more such simplification allows to have a betterunderstanding of the crucial impacts of the production structure on businessfluctuations Departing from the related literature discussed above14 weconsider the effects of a single shock rather then a continuous flow whichin our model is superfluous in order to obtain fluctuation persistence15 To

extensive programming or statistical skills besides the usual education of economists12You can think of a high level of aggregation in the statistical observation The com-

plete initialisation and parameters values are available in table 1 in the Appendix13Notice that sectors only approach a final equilibrium level of production which can

be prendashdetermined under static conditions14Included the more proximate work of Helbing Witt Lammer and Brenner (2004)15An exception are the last results presented in Section 58 which we see more as an

opportunity for further research than a closing of the paper

14

start with next section (53) shows the aggregate cyclical behaviour of astandard setting

53 Single shock analysis

We generate a simulated history following the dynamics of the modelrsquos vari-ables when the system is set to an equilibrium level and a single shockmodifies one of the exogenous elements namely the external demand forsectors Modifications across sectors are random and not correlated andmay increase or reduce external demand by at most 10

100 225 350 475 600

193894

195861

197828

199795

201762 Industrial Production

Time

Figure 2 Total industrial production fluctuations following a shock fromexternal demand

Figure 2 shows the dynamics generated by the total industrial productionof the system mdash that is also a proxy for the systemrsquos GDP fluctuations Thenew external demand changes the equilibrium level but the system cannotreach the new equilibrium with a monotonous pattern and converges to thenew level with a fluctuating pattern Each change in the external demandinduces also modifications of the intermediate demand and sectoral adjust-ments in the quantity produced typically overshoot the aggregate patternAs it can be appreciated from the figure it takes time before the system isable to converge to the new static equilibrium conditions

We study two features of this pattern absolute level of the new equilib-rium and volatility induced by the shock

54 Technical coefficients and industrial output

Production level

The equilibrium levels of industrial output depend on two factors only Ob-viously the levels of external demand determine how much of the total pro-

15

duction needs to be supplied to consumers exogenous from the productionsystem Second the technical coefficients also influence strongly the level ofactivity of the system defining the intermediate demand to each sector

180

200

220

240

260

280

300

01 02 03 04 05 06 07 08 09

Out

put l

evel

(In

dust

rial p

rodu

ctio

n)

Sum of total input coefficients (M)

Total industrial production for different input coefficients levels

Figure 3 The relevance of input coefficients on output level

Figure 3 shows the equilibrium level of industrial output reached bysystems with identical external demand where we fixed the sum of inputcoefficients for each sector of the economy (Mi) to different values16 Thehigher the sum the higher is the total production sustained by the system

This result depends on the very production structure represented by theinputndashoutput matrix In fact the higher is the level of production requiredby intermediate sectors the higher is the total production given a constantlevel of external demand

Notice that the level of production does not depend on the distribution ofthe technical coefficients As long as the sum of the coefficients is identicalthe same production level is obtained irrespective of their distribution (otherthings being equal)

Output shock absorption

Input coefficients have a relevant role also on the output stability of thesystem Starting from an output equilibrium level we hit the system witha single shock on the external demand for all sectors uncorrelated amongsectors17 modifing (positively or negatively) the demand for all sectors by

16Within this limit the actual value of the coefficients are randomly drawn17In each sector the external demand changes by plusmnX where X is a uniformly dis-

tributed percentage between 1 and 10

16

at most10 As show in Figure 2 the economic systems produces a lenghtlysmoothing oscillatory behaviour We compute the relative deviation of theaggregate production (industrial output) as the ratio between its standarddeviation and its mean δ = σY microY where Y =

sum

Qi Figure 4 shows thebehaviour of δ as a function of the sum of input coefficients for each sector80 periods after the shock As time goes by the level of δ reduces for allMi but the relation with the sum of input coefficient is unchanged

00055

0006

00065

0007

00075

0008

00085

0009

01 02 03 04 05 06 07 08 09

Rel

ativ

e va

rianc

e of

tota

l out

put (

delta

)

Sum of total input coefficients

Aggregate output deviations for different technical coefficients

Figure 4 The relevance of input coefficients on output stability

In sum input coefficients determine the extent of overshooting behaviourThe higher the coefficients the higher is the backlash even when shockscould cancel out through sectors

55 Shock persistence and adjustment coefficients

We now consider the effect of micro smoothing behaviours on the volatilityof the production system In particular we are interested in testing theeffects of two parameters of the model on the volatility of the productionsystem First we consider the effects of the αrsquos representing the ldquostickinessrdquoof the production technology to desired changes of Q aimed at filling thegap between actual and desired stocks The higher is α the slower is thereaction to stock unbalances Concerning its effect referring to the technicalpossibilities available ie production capital and organization we may putforward two opposite hypothesis relevant to systemrsquos volatility

Hypothesis 1 Slower production adjustment (high α) reduces volatilitybecause unbalances in demand of one sector being compensated slowly slow

17

down the diffusion of shocks to other sectors

Hypothesis 2 Slower production adjustment (high α) increase volatil-ity because the longer lasts the mismatch between actual and equilibriumproduction level the deeper will become the mismatch between actual anddesired stocks

20 25 30 35 40 45 50 55 60 65

Cyclesrsquo width

07 075

08 085

09alpha 0005

001 0015

002 0025

003 0035

004 0045

005

alphaS

20 25 30 35 40 45 50 55 60 65

Figure 5 Maximum width recorded between highest and lowest peaks fortotal production for different values of α and αS

Secondly we consider the parameters αS representing the speed of ad-justment of desired stock to observed demand The higher this parame-ter the faster sectors revise the desired levels of stocks to current levelsof observed demand This parameter concerns sectorsrsquo behavioral decisionprocess Higher αS represent a stronger belief that the most recently ob-served demand level is a reliable indicator of future demand levels whilelower levels of the parameter represent a more conservative approach wheremanagers require a long observation of demand levels different from the pastones before adjusting their expectations Also concerning the effect of thisparameter on the volatility induced by an exogenous shock we may maketwo hypothesis

Hypothesis 1 Faster adaptation of desired stocks to new levels of de-mand (higher αS) reduce volatility because changes in demand are rapidlytranslated in adjusted levels of production

Hypothesis 2 Faster adaptation of desired stocks to new levels of demand(higher αS) increase volatility because temporary unbalances of demand in

18

one sector are transmitted faster to other sector reinforcing the feedbackmechanism of fluctuations

To test which of these hypotheses are confirmed by the model we runsimulation runs with identical systems (eg same technical coefficients samelevels of demand same shocks) but for the values of the αrsquos and αS rsquos testingall combinations of these parameters over a range of values (each run usedthe same parameters for all the sectors of the system) For each simulation(ie each couple of values of the parameters) we computed two indicatorsof volatility One measures the maximum distance of total production be-tween the highest and the lowest peak that is the maximum width of theoscillations registered in the production time series The second indicatormeasures the relative deviation δ = σY microY Figures 5 and 6 report theseresults

0004 0006 0008 001 0012 0014 0016 0018 002

Relative deviation

07 075

08 085

09alpha 0005

001 0015

002 0025

003 0035

004 0045

005

alphaS

0004 0006 0008 001

0012 0014 0016 0018 002

Figure 6 Relative deviation of total production for different values of α andαS

The results confirms unequivocally both hypotheses 2 and reject hy-potheses 1 In a sense these hypotheses confirm the nature of the economicsystem represented in the model as a complex system where the interactionsamong sectors play a far more important role than the individual sector be-haviour In fact in both cases we may interpret the results that the strongerthe efforts to mitigate the effects of shocks (ie slow productionrsquos and quickexpectationsrsquo adjustment) the opposite result is actually obtained the fluc-tuations become even more accentuated It is also worth to note that thestronger impact is generated by the behavioural parameters rather than thetechnical one as indicated by the stronger effects of the αS in respect of theα This result seems to suggest that ceteris paribus information mattersmore than technical constraints Though the model is by any means in-

19

adequate to discuss normative aspects because of the naive ways strategicdecision making is represented it can anyway suggest the effects of errorsin management decisions (ie wrong desired levels of stocks) as opposed tothe technical constraints preventing sudden adjustment of production lev-els Stretching a bit the interpretation of these results we may suggests thatefforts to reduce volatility should better be oriented to improve the trans-mission of information in order to coordinate future production plans (ieidentifying quickly the desired levels of stocks and therefore the new equilib-rium production level) rather than improving the flexibility of productionmethods (reducing α)

56 Volatility and stock levels

In our model stock levels are maintained to buffer a sectorrsquos sale againstunexpected changes in demand Therefore a system made of risk adversemanagers (keeping higher levels of stocks) may be thought to lead to asmoother absorption of shocks in respect of a system composed of firmskeeping a short supply to cushion unexpected events To test this hypothesiswe tested the behaviour of the model for different values of σ the levels ofstock represented as a multiple of demand For each level of σ we run asimulation using all other initialization identical as usual representing aneconomic system ldquoalmostrdquo at equilibrium but for a single small shock indemand uncorrelated across sectors18

00044

00046

00048

0005

00052

00054

00056

00058

0006

00062

5 55 6 65 7 75 8 85 9 95 10

Rel

ativ

e va

riatio

n of

tota

l out

put (

delta

)

Ratio of stocked goods over production (sigma)

Stocks and production fluctuations

Figure 7 Fluctuations persistence on systems with different stock strategies(σ)

18Demand changes by plusmn1-10

20

Figure 7 shows the value of the relative deviation of output production(δ) generated by heterogeneous uncorrelated shocks in the external demandon identical systems where firms store a different multiple of their demand asstocks We obtain actually the opposite result that we may have expectedfluctuation become larger and more persistent the higher the level of stocksIn some way a strategy that for the individual firm is meant to reducethe fluctuations (using stocks as buffers) ends up by actually increasing thevolatility at aggregate levels

The result is though explainable both at the theoretical and empiricallevel In fact if no stocks were maintained production would need to adjustfor changes in demand only in case of a shock That is the firm may movemonotonically from the old to the new production levels sending consistentsignals to the rest of the system (ie its own suppliers) Instead if thesame shock affects a firm maintaining large stocks of output over the sameperiod production needs to adjust to the new demand and and to alignstocks with the new demand This generates besides the adjustment to thedemand a potentially inconsistent signal to the suppliers since the firm willhave a temporary level of production not meant to continue in equilibrium(adjusting stocks)

From the empirical viewpoint the last few decades have seen the almostuniversal diffusion of production system meant to keep stocks to a minimumif at all Lean production methods justndashinndashtime etc are solutions tothe problems of the costs necessary to maintain stocks Over the sameperiod economic fluctuations have been strongly reduced and few empiricalassessments have pointed to the relations between the two phenomena Forexample concerning the US ldquochanges in inventory behavior have playeda direct role in reducing real output volatilityrdquo (Kahn McConnell andPerez-Quiros 2002 p 183)19

57 Irrelevance of technical coefficients distribution

Horvath (1998) and Horvath (2000) suggest that systems with sparse ma-trices ie inputndashoutput matrices with many empty cells tend to be morevolatile in respect of systems with more evenly distributed matrices Noticethat this claim risk if not adequately qualified to generate an illogical resultIn fact matrices sparseness increases as a function of sectoral disaggregationand comparing the same systemrsquos fluctuations using inputndashoutput matricesat different levels of aggregation needs to produce the same type of volatil-ity Indeed this is not the case if all remaining parameters stand equaland do not adjust to the different level of analysis considered If aggregatepatterns of adjustment speed are maintained unchanged the higher the sec-toral disaggregation the higher the systemrsquos volatility ceteris paribus But

19Similar results are found in Blanchard and Simon (2001)

21

for example in our model reducing the aggregation also requires a changein the adaptation coefficients the more micro we undergo the analysis thehigher is the speed of adjustment of economic objects If for example weconsider sectors we are evaluating the adaptation of an entire set of firmswhich pose questions on how to aggregate their dynamics Things changewhen we consider the response of a single firm

Going back to Horvatsrsquo argument highly disaggregated matrices arelikely to contain many empty cells so that we may forecast different levelsof volatility depending on the aggregation used for the same system whichas mentioned is not caused by sparseness but to a failed adjustment inreaction mechanisms Results from our model20 show that volatility is com-pletely unaffected by input coefficients distribution Given their sectoralsum mdash through rows of the matrix mdash the value of each coefficient is irrele-vant The same occurs in the case of very skewed distributions that presenta high number of empty cells (zero input coefficients)

58 Microndash and Macrondashvolatility

As we have seen above one of the major discussion in the literature concernsthe relations between microndash and macrondashvolatility Put it simply one sideof the literature considers that uncorrelated microndashshocks cancel each otherout reducing their effects at macrondashlevel Conversely other researchersconsider that microndashshocks are multiplied at macrondashlevel generating highermacrondashvolatility than that we can observe at microndashlevel The debate isof obvious importance to determine the sources of business cycles to pre-dict their effects and determine the more effective policies to mitigate theirnegative effects

Our model is only a partial representation of realndashworld economic sys-tem and is also overly simplistic However based on these limitations italso offers very robust results allowing reliable conclusions on the (limited)elements included in the model In particular we consider our model a re-liable representation of the basic structures of the production interactionsamong sectors or firms Thus we can use the model to provide an answerto the debate concerning the contributions to volatility from the productioninteractions only Other contributions (eg from financial markets priceadjustments technological innovation etc) may re-inforce or counter theforces we analyse but they need to be considered as additional elements(and we opine also generally less relevant) not alternative to our analysisWe consider a necessity in order to apply a rigorous methodological ap-proach to be able to identify with certainty the results provided by partialstudies in order to eventually obtain by gradual extensions of the analy-sis more and more detailed representations of realistic economic systems

20Available from the authors

22

The results we obtain in this paper concerning this microndashmacro debate aremeant as a first relevant step to a more extended analysis

0

002

004

006

008

01

012

014

016

018

02

022

05 1 15 2 25 3

Var Demand

Macro (Output) and micro (Demand) volatility

delta Outputdelta Demand

Figure 8 Values of δ (relative deviation) for aggregate demand and ag-gregate output in respect of different levels of variance for sector specificuncorrelated external demand shocks

The flows of shocks affecting the firms of a system can be distinguishedalong two dimensions strength of the shock (eg amplitude of the changesin demand) and frequency of the shock (rare or frequent) Concerning thefirst aspect we initialized as usual a system to equilibrium levels (eg pro-duction quantities equal to external and intermediate demand for all sectorsconstant level of stocks equal to the desired level) We then ran the simu-lation generating the external demand for all sectors as a random variablewith constant (equilibrium) mean level and varying values of variance

Figure 8 shows unequivocally that as far as our model has propertiessimilar to real systems the hypothesis that macrondashfluctuations have largervolatility than their sectoral level shocks is fully confirmed At any levelof variance the aggregate volatility of demand as captured by the relativedeviation index is sensibly lower than the volatility shown by aggregateoutput that is the total production of the system This result is not sur-prising throughout all tests presented in the previous sections every resultsuggested that the system lsquooverndashreactsrsquo to any individual microndashlevel changeeven in the case these changes were meant to contrast disturbances There-fore when we move from considering one single shock to a flow of shockswe are not surprised to obtain that the system shows higher volatility thanthat implied by the microndashlevel disturbances

We now consider the second dimension defining a flow of shocks theirfrequency The question is whether for the same level of demand shocks

23

0 002 004 006 008 01 012 014 016

Difference of relative deviation for Output and Demand

0 5 10 15 20 25 30 35 40 45 50 55

Shock frequency 05

1

15

2

25

3

Var Demand

0 002 004 006 008 01

012 014 016

Figure 9 Differences of values of δ for aggregate output minus those foraggregate demand in respect of different levels of variance of demand and offrequency of the shocks

(ie same variance) more frequent changes trigger a higher or lower systemvolatility than rare disturbances The results shown in figure 9 suggest arather elaborated answer which partly vindicates (though in a particularsense) the arguments in favour of compensation of micro shocks The fig-ure shows the difference between aggregate output volatility and aggregatedemand volatility The independent variables are as before the volatilityof demand at microndashlevel21 and the time intervals in between two shocksAggregate demand volatility does not change for the frequency of shocksat least when computed over a long period But output volatility doesin a nonndashmonotonous way For high frequency of shocks the difference isstill positive and increasing for higher variance indicating that the ldquooverndashreactionrdquo argument is valid even in these cases but it is rather low Whilethe frequency of shocks slows down aggregate volatility of output increasesshowing that time is needed for the microndashshock to unfold their full effectsat the macrondashlevel However after a certain threshold less and less frequentshocks generate a lower volatility of the aggregate level of output

This result is quite sensible though not obvious at first sight and onceagain shows the power of such a simple model to generate and explain com-plex properties For frenetic levels of demand modifications the system hasno time to adjust but part of the job to align actual and desired productionfor constantly changing demand is performed by the varying demand itselfIn other terms the shocks cancel out through time if not through their quan-

21Beware of the scale of variance For reasons of visibility the scale is inverted

24

titative effects on the system While frequency decreases the disturbancesof the shocks not compensated by other (unlikely) shocks have the time tofilter through the production system generating higher and higher volatil-ity In a sense any system has a particular frequency of shocks that make itmaximally ldquoresonatingrdquo with the amplification of microndashshocks When therate of shocks gets even rarer the system can enjoy periods of longer andlonger quietness after each shock is absorbed and before a new one startsagain a cycle of initial local disturbance propagation of the disturbances toother sectors and settlement to the new equilibrium

6 Discussion and extensions

We presented a simple model of production composed by interacting sectorThe main (quit realistic) assumption of the model is that the productionprocesses in any sector of the economy make use of input from any othersector Moreover we assumed that sectors (represented as a single firm)adjust slowly both the desired level of production (smoothed revision of ex-pectation) and the actual level (smoothed modification of plant utilization)The model voluntarily leaves aside a large number of aspects of real-worldeconomic systems (eg financial markets price adjustments growth etc)The reason is that the model can be easily and reliably tested in order toextract its properties The possibility to transfer the (certain) model resultto a real context depend of course on an ample ceteris paribus clause How-ever we are confident that the reported results concerning the productionstructure play a relevant role in real world systems Therefore our resultsand particularly their motivations being accessible because of the relativesimplicity of the model will not be diminished by the integration of ourconsiderations with other aspects of economic systems

Our model is in essence a representation of an input-output table inter-preted in physical terms that is as the amount of inputs required for oneunit of output Besides this we include two straightforward assumptionsFirstly production levels cannot be suddenly changed but require time toscale up or down the utilization rate that is the levels of production Stocksare meant to make up for the difference between sales and production Thesecond assumption concerns the intentional behaviour of producers We im-plement this aspect by assuming that exists a level of stocks desired for eachlevel of demand However a change of demand does not translate suddenlyin a new level of desired stock but again only a long time at a different levelof demand will convince managers to update the desired levels of stocks

We show that the model generate under a wide number of settings oscil-latory patterns converging toward the equilibrium level where each sectorgenerate exactly the quantity demanded by final consumers and by othersectors as input The oscillations are provoked by over- and under-shooting

25

generated by the difficulty of coordination among sectors that by assump-tion ldquocommunicaterdquo only via variation of demand to their direct input sup-pliers We also show that the total production levels generated by sectors(sum of the physical units) produced at equilibrium depends (for constantfinal demand) on the total sum of the technical coefficients per sector irre-spective of their distribution This result is motivated by the impossibilityof input substitution and offers interesting suggestions for the analysis ofthe study of input-output tables of real-world systems

We then consider systems at equilibrium facing a single small change ofthe external demand for sectors These simulations allow to study the waythe system reacts to shocks We see that the sum of the coefficients affectsbesides the levels also the volatility of aggregate output the larger a systemthe stronger is the impact of a given shock Less obviously we find thatfor systems with the same dimension (ie same sums of coefficients) theaggregate volatility increases with both the inertia to change of productionlevel and the speed of adjustment of desired stocks This result is somewhatcounter-intuitive since both the cited variables are associated to a moreconservative behaviour that may be supposed should smooth away spikesin the pattern from an old to a new equilibrium However we show thatthe opposite actually holds given the complex interactions among sectorindividual-sector attitude intended to smooth shocks actually increase theiraggregate impact

In the same vein we show that systems maintaining high levels of stocksare more volatile that systems with reduced amounts of stocks Again thisis a counter-intuitive result reverting the goal of individual producers andthe aggregate result Interestingly this result finds strong support from theevidence of the recent changes in production methods (ie low or no stocks)and diminished cycles

In relation to specific issues discussed in the literature we find no supportto the suggestion that particular distributions of input-output coefficientsmodify the volatility of a system As said this depends in case on thesums of these coefficients but not on their distribution Further we canprovide an answer to a long and hotly debated questions whether microndashlevels volatility generates stronger or lower aggregate volatility Our modelshows unequivocally that microndashvolatility (eg variance of final demand)persists and generates stronger volatility at aggregate level Again we canidentify the reason on the complex production structure that exalts micro-level shocks by generating lsquowrongrsquo messages among producers Moreoverwe can also point to a usually neglected aspect of volatility we show thatthe frequency of shocks is a very relevant aspect besides their dimensionin determining the volatility of aggregate output

Our work can be developed along two complementary directions Firstlywe can continue to study the properties of the model by extending the ele-ments considered (eg pricing) analysing the effects on the present results

26

and generating more analytical evidence Of particular relevance is in ouropinion the problem of aggregation Representations of the same systemswith different codification systems (eg different number of SIC digits)must not modify overall system properties for example the volatility ob-served Given that our result show that the sums of coefficients matter forthe levels and volatility of the system and that these sum do depend on theaggregation chosen it should be possible to induce the reaction coefficientsthat we know affect the volatility only

Secondly we can find an application of our results to real world evidenceIn fact there are many data sets available for the data we deal with basicallyinput-output tables Our model is readily adapt to be parametrized alongany number of sector and coefficient necessary so that we may replicate pastseries in order to adapt the unobservable parameters and use the resultingcomplete model to explain business fluctuations and provide intuitions onfuture patterns

27

References

Acemoglu D and A Scott (1997) ldquoAsymmetric Business CyclesTheory and TimendashSeries Evidencerdquo Journal of Monetary Economics 40501ndash533

Backus D P Kehoe and F Kydland (1995) ldquoInternational BusinessCycles Theory and Evidencerdquo in Frontiers of Business Cycle Researched by T Cooley Princeton University Press Princeton

Bak P K Chen J Scheinkman and M Woodford (1993) ldquoAg-gregate Fluctuations from Independent Sectoral Shocks SelfndashOrganizedCritically in a Model of Production and Inventory Dynamicsrdquo RicercheEconomiche 47(1) 3ndash30

Battiston S D Delli Gatti M Gallegati B Greenwald and

J Stiglitz (2005) ldquoCredit Chains and Bankruptcies Avalanches in Sup-ply Networksrdquo in Annual Workshop on Economic Heterogeneous Inter-acting Agents (WEHIA 2005) University of Essex UK

Blanchard O and J Simon (2001) ldquoThe Long and Large Declinein US Output Volatilityrdquo Brookings Papers on Economic Activity 1135ndash174

Bresnahan T F and V A Ramey (1994) ldquoOutput Fluctuations atthe Plant Levelrdquo The Quarterly Journal of Economics 109(3) 593ndash924

David P A (1975) Technical choice innovation and economic growth essays on American and British experience in the nineteenth centuryCambridge University Press London

Delli Gatti D C Di Guilmi E Gaffeo G Giulioni M Galle-

gati and A Palestrini (2004) ldquoA New Approach to Business Fluc-tuations Heterogeneous Interacting Agents Scaling Laws and FinancialFragilityrdquo Journal of Economic Behavior amp Organization 56 489ndash512

Dosi G G Fagiolo and A Roventini (2006) ldquoAn EvolutionaryModel of Endogenous Business Cyclesrdquo Computational Economics 273ndash34

Dupor B (1999) ldquoAggregation and the Irrelevance in MultindashSector Mod-elsrdquo Journal of Monetary Economics 43 391ndash409

Head A C (1995) ldquoCountry Size Aggregate Fluctuations and Interna-tional Risk Sharingrdquo The Canadian Journal of Economics 28(4b) 1096ndash1119

28

Helbing D and S Lammer (2005) ldquoSupply and Production NetworksFrom the Bullwhip Effect to Business Cyclesrdquo in Networks of Interactingmachines Production Organization in Complex Industrial Systems andBiological Cells ed by D Armbruster K Kaneko and A S Mikhailovvol 3 of World Scientific Lecture Notes in Complex Systems World Sci-entific

Helbing D S Lammer U Witt and T Brenner (2004) ldquoNetworkndashInduced Oscillatory Behavior in Material Flow Networks and IrregularBusiness Cyclesrdquo Phisical Review E 70(056118) 1ndash6

Helbing D U Witt S Lammer and T Brenner (2004) ldquoNetwork-Induced Oscillatory Behaviour of Macroeconomic Output Flowsrdquo in In-dustry and Labour Dynamics II The Agent-based Computational Eco-nomics Approach Proceedings of the wildace 2004 conference ed byR Leombruni and M Richiardi no 5 LABORatorio R Revelli

Horvath M (1998) ldquoCyclically and Sectoral Linkages Aggregate Fluctu-ations from Independent Sectoral Shocksrdquo Review of Economic Dynam-ics 1 781ndash808

(2000) ldquoSectoral Shocks and Aggregate Fluctuationsrdquo Journal ofMonetary Economics 45(1) 69ndash106

Jovanovic B (1987) ldquoMicro Shocks and Aggregate Riskrdquo The QuarterlyJournal of Economics 102(2) 395ndash410

Kahn J A M M McConnell and G Perez-Quiros (2002) ldquoOnthe Causes of the Increased Stability of the US Economyrdquo FRBNY Eco-nomic Policy Review 8(1) 183ndash202

Kraay A and J Ventura (1998) ldquoComparative Advantage and theCrossndashSection of Business Cyclesrdquo World Bank Woorking paper Series1948 World Bank

Long J B and C I Plosser (1983) ldquoReal Business Cyclesrdquo Journalof Political Economy 31(1) 39ndash69

Lucas R E (1977) ldquoUnderstanding Business Cyclesrdquo CarnegiendashRochester Conference Series on Public Policy (special issue of Journalof Monetary Economics) 5 7ndash29

Nelson R R and S G Winter (1982) An Evolutionary Theory ofEconomic Change Harvard University Press Cambridge MA

Nirei M (2005) ldquoThreshold Behavior and Aggregate Fluctuationsrdquo Jour-nal of Economic Theory in press

29

Scheinkman J and M Woodford (1994) ldquoSelfndashOrganized Criticallyand Economic Fluctuationsrdquo American Economic Review 84(2 Papersand Proceedings of teh AEA) 417ndash421

Schuh S (1996) ldquoEvidence of the Link Between FirmndashLevel and Ag-gregate Inventory Behaviorrdquo Finance and Economics Discussion Series1996-46 Federal Reserve Board

Stiglitz J and B Greenwald (2003) Towards a New Paradigm inMonetary Economics Raffaele Mattioli Lectures Cambridge UniversityPress Cambridge

Weisbuch G and S Battiston (2005) ldquoProduction Networks andFailure Avalanchesrdquo Working Paper mimeo Ecole Normale SupErieure

Zarnovitz V (1977) ldquoBusiness Cycles Observed and Assessed Why andHow they Matterrdquo NBER Working Paper Series 6230 National Bureauof Economic Research

30

A Tables

Table 1 Parameters valuesParametera Description Value

N Number of sectorsunits i of production 10Ci Number of j buyer sectors for sector i (IO matrix sparseness) N minus 1Di (E) Value of the external demand 10mij Physical input coefficient of input j for sector i sim U(0 1)Mi Sum of the mij input coefficients j for the production of i 05αi friction to changes of production levels Degree of lockndashin on

production technology085

αsi Speed of adaptation to demand changes 004

σ Desired multiple of production to stock 10

aInto parenthesis the number of lags of the initial value for lagged variables

31

interactions within sectors each sector is a production unit In fact forour purposes it is not relevant which firm produces the sectoral outcomebut only the total amount produced by each sector9 Moreover besidesmaking the analysis more neat this assumption allows also to enjoy a largefreedom in determining the aggregation level since we donrsquot need to assessthe effects of between sectors competitiveness which for example is likelyto increase for increasing levels of disaggregation Our economic systemis therefore composed of N production units each of which may buy andprovide inputs from and to any other sector depending on the IndashO structureof the economy

The demand for each sector i is determined by a constant consumersdemand Di (E) and the input needs of other sectors

Di = Di (E) +sum

j 6=iisinCi

mjiQj (1)

where Ci is the set of j intermediate clients of i (sum

j isin C lt N) mji is thetechnical coefficient of firm j for input i ie the amount of i required toproduce one unit of j Qj is the actual output produced by sector j

Concerning the simultaneous production level of each sector in time twe make two lsquorealisticrsquo assumptions First from the technical viewpoint afirm like a tanker cannot abruptly change its course Therefore changesto the production level can be introduced only gradually through capital(dis)investment and job h(f)iring (see discussion in Section 3) Second weassume that firms do not decide the production level directly on the basisof the observed demand level but conscious of the uncertainty of demandplan their production in order to maintain invariant a desired level of stocksand avoid stockout10

Quantity produced by sector i in period t is computed as follows

Qit = αiQtminus1 + (1 minus αi)(

Slowastitminus1 minus Sitminus1

)

(2)

where αi measures the physical constraint in adapting the production capac-ity to changes in the required output Slowast

itminus1are firmrsquos i desired stocks and

Sitminus1 its actual stocks Notice that both variables related to stocks appearwith a lag This is due to two reasons First it is a modelling necessity inorder to allow the simultaneous determination of production levels for allsectors Secondly as frequently happens modelling necessities reveal sen-sible if not generally considered aspects of the modelled system In factproduction decisions concern management in the production plant while

9Adding heterogeneous firms within sectors would definitely enrich the causal expla-nation put forward and parametrised in this paper (see below) For example to analysethe effect of firms constraints It is left over for future research as in the present work weopt for the analysis of the main structural parameters that affect aggregate fluctuations

10See also discussion in Schuh (1996)

11

demand through sales is observed by the commercial staff It is thereforelogical that the information from demand reaches plant managers with somelag

The actual stock level is trivially computed subtracting from the previousheap of stock the amount sold and adding the new production

Sit = Sitminus1 + Qit minus Dit (3)

Concerning the desired level of stocks instead we introduce the conser-vative behaviour mentioned above Firms ideally would keep an amountof stocks proportional to the demand they receive say a multiple σ ofthat amount However when demand varies they adjust the desired levelsmoothly preventing sudden changes to the variable representing the goalof the firm

Slowastit = Slowast

itminus1 + αsi

(

σDit minus Slowastitminus1

)

(4)

where αsi measures the adaptation to demand changes

In summary the dynamics of the model consists of only the equationsabove Firstly at the beginning of a time step firms determine the quantityto produce as a function of past imbalances in stock levels and the pastproduction level These levels define the demand for all sectors determinedby the technical coefficients applied to all produced quantities Finallyactual and desired stocks can be updated Figure 1 represents visually thisdynamics highlighting the decisional components of the model in respect ofthe ldquomechanicsrdquo of trade

5 Results production structure and economic fluc-

tuations

In this section we describe some of the most relevant results concerningthe effects of the production structure to determine aggregate fluctuationsGiven the unusual analytical instrument chosen (at least concerning thistopic) we start with a brief methodological introduction meant to clarifythe nature of the results we claim to obtain Next we describe a generalsetting for the model which is the (extremely) limited area of the potentialparametersrsquo space of the model that we will explore In the rest of thesection we discuss the results from the model

51 Methodological considerations

Real economic systems are constantly subject to a continuous flow of shocksexternal to the production system besides changes within the system itselfIn order to appreciate the systemic properties we need to abstract from all

12

Compute the quantityto produce

F (Q(t-1) S(t-1) s(t-1) alpha)

Communicate thequantity of inputsrequired to each

supplier

Determine the overalldemand

F(D(E) D(I))

Revise the past decisionon stock plans

(desired stocks)F(D S(t-1) sigma alpha(s))

Compute the final levelof stocks for next period

F(s(t-1) Q D)

KEY

ModelDirection

Variablecomputed

at t-1(t-1)

Step includingfirm behavior

Figure 1 One period dynamics

the nonndashrelevant disturbances and concentrate on how the system endoge-nously contributes to the observed dynamics Therefore we will generatehighly abstract patterns whose analysis will provide insights on the systemproperties that would be otherwise lost in the noise generated by all theelements of a real system

We are not trying to fit the model to a specific empirical data set nor weare interested in studying the complete behaviour of the model for areas ofthe parameterrsquos space that make no economic sense (eg negative or infiniteproduction levels) Therefore the values of initial settings and results matteronly in relative and not in absolute terms

We are going to describe a simulation model with which we will produceseveral simulation runs and we claim obtain results relevant for the debatediscussed above Rigorously speaking we may claim only certainty for thevalidation of the results that is that the model does actually generate theresults presented and for the reasons we explain We limit this part to thepresentation of graphs and verbal explanations that we believe are ratheruncontroversial11 Concerning the relevance of our studies for real systems

11Interested readers can request the model code and and simulation data for replicationand extensions of the results We can guarantee that such possibility does not require

13

(verification) we simply avoid to pretend that our model is a quantitativeapproximation of real economies (which in which period) Our opinionis that the study of the model allows us to make general considerationsnon-obvious and relevant for applied debates supported by strong logicalarguments which can be integrated but not reverted by arguments con-cerning more and more elements of the economic world

52 Model setup

We go through the model properties analysing an economic system madeup of ten sectors (N = 10)12 Each sector shops inputs from the remain-ing N minus 1 and sells to them part of its own production as intermediategoods This means that the input matrix is complete except for the diag-onal (mij gt 0 forallj 6= i mii = 0) The sum of all N minus 1 coefficients for asingle sector is given by Mi and its benchmark value is kept at a plausiblemedium level (unless differently specified) to avoid distortions from highlydemanding sectors Each input coefficient is randomly drawn from a uniformdistribution

Sectors start at an equilibrium level their supply matches intermediateand final demand and in each period the same quantity of output mustbe produced to compensate for the used stocks and keep the desired stocksunchanged In thier turn stocks must be ten times the value of total sectoraldemand (σ) Eventually technological adjustment of production is ratherslow and producers are quiet refractory to follow sudden changes in thedemand in general a smoothing behaviour strongly prevails (α and αs)

Under those general preconditions in the following sections we anal-yse the dynamics of the economic system in relation to model parametersWe mainly focus on the cyclical behaviour of systems and their speed ofconvergence to the asymptotic equilibrium13

The very abstract assumption and the homogeneous initialisation donot allow for the multiple equilibria that are likely to rise in complex systemswith feedbacks Once more such simplification allows to have a betterunderstanding of the crucial impacts of the production structure on businessfluctuations Departing from the related literature discussed above14 weconsider the effects of a single shock rather then a continuous flow whichin our model is superfluous in order to obtain fluctuation persistence15 To

extensive programming or statistical skills besides the usual education of economists12You can think of a high level of aggregation in the statistical observation The com-

plete initialisation and parameters values are available in table 1 in the Appendix13Notice that sectors only approach a final equilibrium level of production which can

be prendashdetermined under static conditions14Included the more proximate work of Helbing Witt Lammer and Brenner (2004)15An exception are the last results presented in Section 58 which we see more as an

opportunity for further research than a closing of the paper

14

start with next section (53) shows the aggregate cyclical behaviour of astandard setting

53 Single shock analysis

We generate a simulated history following the dynamics of the modelrsquos vari-ables when the system is set to an equilibrium level and a single shockmodifies one of the exogenous elements namely the external demand forsectors Modifications across sectors are random and not correlated andmay increase or reduce external demand by at most 10

100 225 350 475 600

193894

195861

197828

199795

201762 Industrial Production

Time

Figure 2 Total industrial production fluctuations following a shock fromexternal demand

Figure 2 shows the dynamics generated by the total industrial productionof the system mdash that is also a proxy for the systemrsquos GDP fluctuations Thenew external demand changes the equilibrium level but the system cannotreach the new equilibrium with a monotonous pattern and converges to thenew level with a fluctuating pattern Each change in the external demandinduces also modifications of the intermediate demand and sectoral adjust-ments in the quantity produced typically overshoot the aggregate patternAs it can be appreciated from the figure it takes time before the system isable to converge to the new static equilibrium conditions

We study two features of this pattern absolute level of the new equilib-rium and volatility induced by the shock

54 Technical coefficients and industrial output

Production level

The equilibrium levels of industrial output depend on two factors only Ob-viously the levels of external demand determine how much of the total pro-

15

duction needs to be supplied to consumers exogenous from the productionsystem Second the technical coefficients also influence strongly the level ofactivity of the system defining the intermediate demand to each sector

180

200

220

240

260

280

300

01 02 03 04 05 06 07 08 09

Out

put l

evel

(In

dust

rial p

rodu

ctio

n)

Sum of total input coefficients (M)

Total industrial production for different input coefficients levels

Figure 3 The relevance of input coefficients on output level

Figure 3 shows the equilibrium level of industrial output reached bysystems with identical external demand where we fixed the sum of inputcoefficients for each sector of the economy (Mi) to different values16 Thehigher the sum the higher is the total production sustained by the system

This result depends on the very production structure represented by theinputndashoutput matrix In fact the higher is the level of production requiredby intermediate sectors the higher is the total production given a constantlevel of external demand

Notice that the level of production does not depend on the distribution ofthe technical coefficients As long as the sum of the coefficients is identicalthe same production level is obtained irrespective of their distribution (otherthings being equal)

Output shock absorption

Input coefficients have a relevant role also on the output stability of thesystem Starting from an output equilibrium level we hit the system witha single shock on the external demand for all sectors uncorrelated amongsectors17 modifing (positively or negatively) the demand for all sectors by

16Within this limit the actual value of the coefficients are randomly drawn17In each sector the external demand changes by plusmnX where X is a uniformly dis-

tributed percentage between 1 and 10

16

at most10 As show in Figure 2 the economic systems produces a lenghtlysmoothing oscillatory behaviour We compute the relative deviation of theaggregate production (industrial output) as the ratio between its standarddeviation and its mean δ = σY microY where Y =

sum

Qi Figure 4 shows thebehaviour of δ as a function of the sum of input coefficients for each sector80 periods after the shock As time goes by the level of δ reduces for allMi but the relation with the sum of input coefficient is unchanged

00055

0006

00065

0007

00075

0008

00085

0009

01 02 03 04 05 06 07 08 09

Rel

ativ

e va

rianc

e of

tota

l out

put (

delta

)

Sum of total input coefficients

Aggregate output deviations for different technical coefficients

Figure 4 The relevance of input coefficients on output stability

In sum input coefficients determine the extent of overshooting behaviourThe higher the coefficients the higher is the backlash even when shockscould cancel out through sectors

55 Shock persistence and adjustment coefficients

We now consider the effect of micro smoothing behaviours on the volatilityof the production system In particular we are interested in testing theeffects of two parameters of the model on the volatility of the productionsystem First we consider the effects of the αrsquos representing the ldquostickinessrdquoof the production technology to desired changes of Q aimed at filling thegap between actual and desired stocks The higher is α the slower is thereaction to stock unbalances Concerning its effect referring to the technicalpossibilities available ie production capital and organization we may putforward two opposite hypothesis relevant to systemrsquos volatility

Hypothesis 1 Slower production adjustment (high α) reduces volatilitybecause unbalances in demand of one sector being compensated slowly slow

17

down the diffusion of shocks to other sectors

Hypothesis 2 Slower production adjustment (high α) increase volatil-ity because the longer lasts the mismatch between actual and equilibriumproduction level the deeper will become the mismatch between actual anddesired stocks

20 25 30 35 40 45 50 55 60 65

Cyclesrsquo width

07 075

08 085

09alpha 0005

001 0015

002 0025

003 0035

004 0045

005

alphaS

20 25 30 35 40 45 50 55 60 65

Figure 5 Maximum width recorded between highest and lowest peaks fortotal production for different values of α and αS

Secondly we consider the parameters αS representing the speed of ad-justment of desired stock to observed demand The higher this parame-ter the faster sectors revise the desired levels of stocks to current levelsof observed demand This parameter concerns sectorsrsquo behavioral decisionprocess Higher αS represent a stronger belief that the most recently ob-served demand level is a reliable indicator of future demand levels whilelower levels of the parameter represent a more conservative approach wheremanagers require a long observation of demand levels different from the pastones before adjusting their expectations Also concerning the effect of thisparameter on the volatility induced by an exogenous shock we may maketwo hypothesis

Hypothesis 1 Faster adaptation of desired stocks to new levels of de-mand (higher αS) reduce volatility because changes in demand are rapidlytranslated in adjusted levels of production

Hypothesis 2 Faster adaptation of desired stocks to new levels of demand(higher αS) increase volatility because temporary unbalances of demand in

18

one sector are transmitted faster to other sector reinforcing the feedbackmechanism of fluctuations

To test which of these hypotheses are confirmed by the model we runsimulation runs with identical systems (eg same technical coefficients samelevels of demand same shocks) but for the values of the αrsquos and αS rsquos testingall combinations of these parameters over a range of values (each run usedthe same parameters for all the sectors of the system) For each simulation(ie each couple of values of the parameters) we computed two indicatorsof volatility One measures the maximum distance of total production be-tween the highest and the lowest peak that is the maximum width of theoscillations registered in the production time series The second indicatormeasures the relative deviation δ = σY microY Figures 5 and 6 report theseresults

0004 0006 0008 001 0012 0014 0016 0018 002

Relative deviation

07 075

08 085

09alpha 0005

001 0015

002 0025

003 0035

004 0045

005

alphaS

0004 0006 0008 001

0012 0014 0016 0018 002

Figure 6 Relative deviation of total production for different values of α andαS

The results confirms unequivocally both hypotheses 2 and reject hy-potheses 1 In a sense these hypotheses confirm the nature of the economicsystem represented in the model as a complex system where the interactionsamong sectors play a far more important role than the individual sector be-haviour In fact in both cases we may interpret the results that the strongerthe efforts to mitigate the effects of shocks (ie slow productionrsquos and quickexpectationsrsquo adjustment) the opposite result is actually obtained the fluc-tuations become even more accentuated It is also worth to note that thestronger impact is generated by the behavioural parameters rather than thetechnical one as indicated by the stronger effects of the αS in respect of theα This result seems to suggest that ceteris paribus information mattersmore than technical constraints Though the model is by any means in-

19

adequate to discuss normative aspects because of the naive ways strategicdecision making is represented it can anyway suggest the effects of errorsin management decisions (ie wrong desired levels of stocks) as opposed tothe technical constraints preventing sudden adjustment of production lev-els Stretching a bit the interpretation of these results we may suggests thatefforts to reduce volatility should better be oriented to improve the trans-mission of information in order to coordinate future production plans (ieidentifying quickly the desired levels of stocks and therefore the new equilib-rium production level) rather than improving the flexibility of productionmethods (reducing α)

56 Volatility and stock levels

In our model stock levels are maintained to buffer a sectorrsquos sale againstunexpected changes in demand Therefore a system made of risk adversemanagers (keeping higher levels of stocks) may be thought to lead to asmoother absorption of shocks in respect of a system composed of firmskeeping a short supply to cushion unexpected events To test this hypothesiswe tested the behaviour of the model for different values of σ the levels ofstock represented as a multiple of demand For each level of σ we run asimulation using all other initialization identical as usual representing aneconomic system ldquoalmostrdquo at equilibrium but for a single small shock indemand uncorrelated across sectors18

00044

00046

00048

0005

00052

00054

00056

00058

0006

00062

5 55 6 65 7 75 8 85 9 95 10

Rel

ativ

e va

riatio

n of

tota

l out

put (

delta

)

Ratio of stocked goods over production (sigma)

Stocks and production fluctuations

Figure 7 Fluctuations persistence on systems with different stock strategies(σ)

18Demand changes by plusmn1-10

20

Figure 7 shows the value of the relative deviation of output production(δ) generated by heterogeneous uncorrelated shocks in the external demandon identical systems where firms store a different multiple of their demand asstocks We obtain actually the opposite result that we may have expectedfluctuation become larger and more persistent the higher the level of stocksIn some way a strategy that for the individual firm is meant to reducethe fluctuations (using stocks as buffers) ends up by actually increasing thevolatility at aggregate levels

The result is though explainable both at the theoretical and empiricallevel In fact if no stocks were maintained production would need to adjustfor changes in demand only in case of a shock That is the firm may movemonotonically from the old to the new production levels sending consistentsignals to the rest of the system (ie its own suppliers) Instead if thesame shock affects a firm maintaining large stocks of output over the sameperiod production needs to adjust to the new demand and and to alignstocks with the new demand This generates besides the adjustment to thedemand a potentially inconsistent signal to the suppliers since the firm willhave a temporary level of production not meant to continue in equilibrium(adjusting stocks)

From the empirical viewpoint the last few decades have seen the almostuniversal diffusion of production system meant to keep stocks to a minimumif at all Lean production methods justndashinndashtime etc are solutions tothe problems of the costs necessary to maintain stocks Over the sameperiod economic fluctuations have been strongly reduced and few empiricalassessments have pointed to the relations between the two phenomena Forexample concerning the US ldquochanges in inventory behavior have playeda direct role in reducing real output volatilityrdquo (Kahn McConnell andPerez-Quiros 2002 p 183)19

57 Irrelevance of technical coefficients distribution

Horvath (1998) and Horvath (2000) suggest that systems with sparse ma-trices ie inputndashoutput matrices with many empty cells tend to be morevolatile in respect of systems with more evenly distributed matrices Noticethat this claim risk if not adequately qualified to generate an illogical resultIn fact matrices sparseness increases as a function of sectoral disaggregationand comparing the same systemrsquos fluctuations using inputndashoutput matricesat different levels of aggregation needs to produce the same type of volatil-ity Indeed this is not the case if all remaining parameters stand equaland do not adjust to the different level of analysis considered If aggregatepatterns of adjustment speed are maintained unchanged the higher the sec-toral disaggregation the higher the systemrsquos volatility ceteris paribus But

19Similar results are found in Blanchard and Simon (2001)

21

for example in our model reducing the aggregation also requires a changein the adaptation coefficients the more micro we undergo the analysis thehigher is the speed of adjustment of economic objects If for example weconsider sectors we are evaluating the adaptation of an entire set of firmswhich pose questions on how to aggregate their dynamics Things changewhen we consider the response of a single firm

Going back to Horvatsrsquo argument highly disaggregated matrices arelikely to contain many empty cells so that we may forecast different levelsof volatility depending on the aggregation used for the same system whichas mentioned is not caused by sparseness but to a failed adjustment inreaction mechanisms Results from our model20 show that volatility is com-pletely unaffected by input coefficients distribution Given their sectoralsum mdash through rows of the matrix mdash the value of each coefficient is irrele-vant The same occurs in the case of very skewed distributions that presenta high number of empty cells (zero input coefficients)

58 Microndash and Macrondashvolatility

As we have seen above one of the major discussion in the literature concernsthe relations between microndash and macrondashvolatility Put it simply one sideof the literature considers that uncorrelated microndashshocks cancel each otherout reducing their effects at macrondashlevel Conversely other researchersconsider that microndashshocks are multiplied at macrondashlevel generating highermacrondashvolatility than that we can observe at microndashlevel The debate isof obvious importance to determine the sources of business cycles to pre-dict their effects and determine the more effective policies to mitigate theirnegative effects

Our model is only a partial representation of realndashworld economic sys-tem and is also overly simplistic However based on these limitations italso offers very robust results allowing reliable conclusions on the (limited)elements included in the model In particular we consider our model a re-liable representation of the basic structures of the production interactionsamong sectors or firms Thus we can use the model to provide an answerto the debate concerning the contributions to volatility from the productioninteractions only Other contributions (eg from financial markets priceadjustments technological innovation etc) may re-inforce or counter theforces we analyse but they need to be considered as additional elements(and we opine also generally less relevant) not alternative to our analysisWe consider a necessity in order to apply a rigorous methodological ap-proach to be able to identify with certainty the results provided by partialstudies in order to eventually obtain by gradual extensions of the analy-sis more and more detailed representations of realistic economic systems

20Available from the authors

22

The results we obtain in this paper concerning this microndashmacro debate aremeant as a first relevant step to a more extended analysis

0

002

004

006

008

01

012

014

016

018

02

022

05 1 15 2 25 3

Var Demand

Macro (Output) and micro (Demand) volatility

delta Outputdelta Demand

Figure 8 Values of δ (relative deviation) for aggregate demand and ag-gregate output in respect of different levels of variance for sector specificuncorrelated external demand shocks

The flows of shocks affecting the firms of a system can be distinguishedalong two dimensions strength of the shock (eg amplitude of the changesin demand) and frequency of the shock (rare or frequent) Concerning thefirst aspect we initialized as usual a system to equilibrium levels (eg pro-duction quantities equal to external and intermediate demand for all sectorsconstant level of stocks equal to the desired level) We then ran the simu-lation generating the external demand for all sectors as a random variablewith constant (equilibrium) mean level and varying values of variance

Figure 8 shows unequivocally that as far as our model has propertiessimilar to real systems the hypothesis that macrondashfluctuations have largervolatility than their sectoral level shocks is fully confirmed At any levelof variance the aggregate volatility of demand as captured by the relativedeviation index is sensibly lower than the volatility shown by aggregateoutput that is the total production of the system This result is not sur-prising throughout all tests presented in the previous sections every resultsuggested that the system lsquooverndashreactsrsquo to any individual microndashlevel changeeven in the case these changes were meant to contrast disturbances There-fore when we move from considering one single shock to a flow of shockswe are not surprised to obtain that the system shows higher volatility thanthat implied by the microndashlevel disturbances

We now consider the second dimension defining a flow of shocks theirfrequency The question is whether for the same level of demand shocks

23

0 002 004 006 008 01 012 014 016

Difference of relative deviation for Output and Demand

0 5 10 15 20 25 30 35 40 45 50 55

Shock frequency 05

1

15

2

25

3

Var Demand

0 002 004 006 008 01

012 014 016

Figure 9 Differences of values of δ for aggregate output minus those foraggregate demand in respect of different levels of variance of demand and offrequency of the shocks

(ie same variance) more frequent changes trigger a higher or lower systemvolatility than rare disturbances The results shown in figure 9 suggest arather elaborated answer which partly vindicates (though in a particularsense) the arguments in favour of compensation of micro shocks The fig-ure shows the difference between aggregate output volatility and aggregatedemand volatility The independent variables are as before the volatilityof demand at microndashlevel21 and the time intervals in between two shocksAggregate demand volatility does not change for the frequency of shocksat least when computed over a long period But output volatility doesin a nonndashmonotonous way For high frequency of shocks the difference isstill positive and increasing for higher variance indicating that the ldquooverndashreactionrdquo argument is valid even in these cases but it is rather low Whilethe frequency of shocks slows down aggregate volatility of output increasesshowing that time is needed for the microndashshock to unfold their full effectsat the macrondashlevel However after a certain threshold less and less frequentshocks generate a lower volatility of the aggregate level of output

This result is quite sensible though not obvious at first sight and onceagain shows the power of such a simple model to generate and explain com-plex properties For frenetic levels of demand modifications the system hasno time to adjust but part of the job to align actual and desired productionfor constantly changing demand is performed by the varying demand itselfIn other terms the shocks cancel out through time if not through their quan-

21Beware of the scale of variance For reasons of visibility the scale is inverted

24

titative effects on the system While frequency decreases the disturbancesof the shocks not compensated by other (unlikely) shocks have the time tofilter through the production system generating higher and higher volatil-ity In a sense any system has a particular frequency of shocks that make itmaximally ldquoresonatingrdquo with the amplification of microndashshocks When therate of shocks gets even rarer the system can enjoy periods of longer andlonger quietness after each shock is absorbed and before a new one startsagain a cycle of initial local disturbance propagation of the disturbances toother sectors and settlement to the new equilibrium

6 Discussion and extensions

We presented a simple model of production composed by interacting sectorThe main (quit realistic) assumption of the model is that the productionprocesses in any sector of the economy make use of input from any othersector Moreover we assumed that sectors (represented as a single firm)adjust slowly both the desired level of production (smoothed revision of ex-pectation) and the actual level (smoothed modification of plant utilization)The model voluntarily leaves aside a large number of aspects of real-worldeconomic systems (eg financial markets price adjustments growth etc)The reason is that the model can be easily and reliably tested in order toextract its properties The possibility to transfer the (certain) model resultto a real context depend of course on an ample ceteris paribus clause How-ever we are confident that the reported results concerning the productionstructure play a relevant role in real world systems Therefore our resultsand particularly their motivations being accessible because of the relativesimplicity of the model will not be diminished by the integration of ourconsiderations with other aspects of economic systems

Our model is in essence a representation of an input-output table inter-preted in physical terms that is as the amount of inputs required for oneunit of output Besides this we include two straightforward assumptionsFirstly production levels cannot be suddenly changed but require time toscale up or down the utilization rate that is the levels of production Stocksare meant to make up for the difference between sales and production Thesecond assumption concerns the intentional behaviour of producers We im-plement this aspect by assuming that exists a level of stocks desired for eachlevel of demand However a change of demand does not translate suddenlyin a new level of desired stock but again only a long time at a different levelof demand will convince managers to update the desired levels of stocks

We show that the model generate under a wide number of settings oscil-latory patterns converging toward the equilibrium level where each sectorgenerate exactly the quantity demanded by final consumers and by othersectors as input The oscillations are provoked by over- and under-shooting

25

generated by the difficulty of coordination among sectors that by assump-tion ldquocommunicaterdquo only via variation of demand to their direct input sup-pliers We also show that the total production levels generated by sectors(sum of the physical units) produced at equilibrium depends (for constantfinal demand) on the total sum of the technical coefficients per sector irre-spective of their distribution This result is motivated by the impossibilityof input substitution and offers interesting suggestions for the analysis ofthe study of input-output tables of real-world systems

We then consider systems at equilibrium facing a single small change ofthe external demand for sectors These simulations allow to study the waythe system reacts to shocks We see that the sum of the coefficients affectsbesides the levels also the volatility of aggregate output the larger a systemthe stronger is the impact of a given shock Less obviously we find thatfor systems with the same dimension (ie same sums of coefficients) theaggregate volatility increases with both the inertia to change of productionlevel and the speed of adjustment of desired stocks This result is somewhatcounter-intuitive since both the cited variables are associated to a moreconservative behaviour that may be supposed should smooth away spikesin the pattern from an old to a new equilibrium However we show thatthe opposite actually holds given the complex interactions among sectorindividual-sector attitude intended to smooth shocks actually increase theiraggregate impact

In the same vein we show that systems maintaining high levels of stocksare more volatile that systems with reduced amounts of stocks Again thisis a counter-intuitive result reverting the goal of individual producers andthe aggregate result Interestingly this result finds strong support from theevidence of the recent changes in production methods (ie low or no stocks)and diminished cycles

In relation to specific issues discussed in the literature we find no supportto the suggestion that particular distributions of input-output coefficientsmodify the volatility of a system As said this depends in case on thesums of these coefficients but not on their distribution Further we canprovide an answer to a long and hotly debated questions whether microndashlevels volatility generates stronger or lower aggregate volatility Our modelshows unequivocally that microndashvolatility (eg variance of final demand)persists and generates stronger volatility at aggregate level Again we canidentify the reason on the complex production structure that exalts micro-level shocks by generating lsquowrongrsquo messages among producers Moreoverwe can also point to a usually neglected aspect of volatility we show thatthe frequency of shocks is a very relevant aspect besides their dimensionin determining the volatility of aggregate output

Our work can be developed along two complementary directions Firstlywe can continue to study the properties of the model by extending the ele-ments considered (eg pricing) analysing the effects on the present results

26

and generating more analytical evidence Of particular relevance is in ouropinion the problem of aggregation Representations of the same systemswith different codification systems (eg different number of SIC digits)must not modify overall system properties for example the volatility ob-served Given that our result show that the sums of coefficients matter forthe levels and volatility of the system and that these sum do depend on theaggregation chosen it should be possible to induce the reaction coefficientsthat we know affect the volatility only

Secondly we can find an application of our results to real world evidenceIn fact there are many data sets available for the data we deal with basicallyinput-output tables Our model is readily adapt to be parametrized alongany number of sector and coefficient necessary so that we may replicate pastseries in order to adapt the unobservable parameters and use the resultingcomplete model to explain business fluctuations and provide intuitions onfuture patterns

27

References

Acemoglu D and A Scott (1997) ldquoAsymmetric Business CyclesTheory and TimendashSeries Evidencerdquo Journal of Monetary Economics 40501ndash533

Backus D P Kehoe and F Kydland (1995) ldquoInternational BusinessCycles Theory and Evidencerdquo in Frontiers of Business Cycle Researched by T Cooley Princeton University Press Princeton

Bak P K Chen J Scheinkman and M Woodford (1993) ldquoAg-gregate Fluctuations from Independent Sectoral Shocks SelfndashOrganizedCritically in a Model of Production and Inventory Dynamicsrdquo RicercheEconomiche 47(1) 3ndash30

Battiston S D Delli Gatti M Gallegati B Greenwald and

J Stiglitz (2005) ldquoCredit Chains and Bankruptcies Avalanches in Sup-ply Networksrdquo in Annual Workshop on Economic Heterogeneous Inter-acting Agents (WEHIA 2005) University of Essex UK

Blanchard O and J Simon (2001) ldquoThe Long and Large Declinein US Output Volatilityrdquo Brookings Papers on Economic Activity 1135ndash174

Bresnahan T F and V A Ramey (1994) ldquoOutput Fluctuations atthe Plant Levelrdquo The Quarterly Journal of Economics 109(3) 593ndash924

David P A (1975) Technical choice innovation and economic growth essays on American and British experience in the nineteenth centuryCambridge University Press London

Delli Gatti D C Di Guilmi E Gaffeo G Giulioni M Galle-

gati and A Palestrini (2004) ldquoA New Approach to Business Fluc-tuations Heterogeneous Interacting Agents Scaling Laws and FinancialFragilityrdquo Journal of Economic Behavior amp Organization 56 489ndash512

Dosi G G Fagiolo and A Roventini (2006) ldquoAn EvolutionaryModel of Endogenous Business Cyclesrdquo Computational Economics 273ndash34

Dupor B (1999) ldquoAggregation and the Irrelevance in MultindashSector Mod-elsrdquo Journal of Monetary Economics 43 391ndash409

Head A C (1995) ldquoCountry Size Aggregate Fluctuations and Interna-tional Risk Sharingrdquo The Canadian Journal of Economics 28(4b) 1096ndash1119

28

Helbing D and S Lammer (2005) ldquoSupply and Production NetworksFrom the Bullwhip Effect to Business Cyclesrdquo in Networks of Interactingmachines Production Organization in Complex Industrial Systems andBiological Cells ed by D Armbruster K Kaneko and A S Mikhailovvol 3 of World Scientific Lecture Notes in Complex Systems World Sci-entific

Helbing D S Lammer U Witt and T Brenner (2004) ldquoNetworkndashInduced Oscillatory Behavior in Material Flow Networks and IrregularBusiness Cyclesrdquo Phisical Review E 70(056118) 1ndash6

Helbing D U Witt S Lammer and T Brenner (2004) ldquoNetwork-Induced Oscillatory Behaviour of Macroeconomic Output Flowsrdquo in In-dustry and Labour Dynamics II The Agent-based Computational Eco-nomics Approach Proceedings of the wildace 2004 conference ed byR Leombruni and M Richiardi no 5 LABORatorio R Revelli

Horvath M (1998) ldquoCyclically and Sectoral Linkages Aggregate Fluctu-ations from Independent Sectoral Shocksrdquo Review of Economic Dynam-ics 1 781ndash808

(2000) ldquoSectoral Shocks and Aggregate Fluctuationsrdquo Journal ofMonetary Economics 45(1) 69ndash106

Jovanovic B (1987) ldquoMicro Shocks and Aggregate Riskrdquo The QuarterlyJournal of Economics 102(2) 395ndash410

Kahn J A M M McConnell and G Perez-Quiros (2002) ldquoOnthe Causes of the Increased Stability of the US Economyrdquo FRBNY Eco-nomic Policy Review 8(1) 183ndash202

Kraay A and J Ventura (1998) ldquoComparative Advantage and theCrossndashSection of Business Cyclesrdquo World Bank Woorking paper Series1948 World Bank

Long J B and C I Plosser (1983) ldquoReal Business Cyclesrdquo Journalof Political Economy 31(1) 39ndash69

Lucas R E (1977) ldquoUnderstanding Business Cyclesrdquo CarnegiendashRochester Conference Series on Public Policy (special issue of Journalof Monetary Economics) 5 7ndash29

Nelson R R and S G Winter (1982) An Evolutionary Theory ofEconomic Change Harvard University Press Cambridge MA

Nirei M (2005) ldquoThreshold Behavior and Aggregate Fluctuationsrdquo Jour-nal of Economic Theory in press

29

Scheinkman J and M Woodford (1994) ldquoSelfndashOrganized Criticallyand Economic Fluctuationsrdquo American Economic Review 84(2 Papersand Proceedings of teh AEA) 417ndash421

Schuh S (1996) ldquoEvidence of the Link Between FirmndashLevel and Ag-gregate Inventory Behaviorrdquo Finance and Economics Discussion Series1996-46 Federal Reserve Board

Stiglitz J and B Greenwald (2003) Towards a New Paradigm inMonetary Economics Raffaele Mattioli Lectures Cambridge UniversityPress Cambridge

Weisbuch G and S Battiston (2005) ldquoProduction Networks andFailure Avalanchesrdquo Working Paper mimeo Ecole Normale SupErieure

Zarnovitz V (1977) ldquoBusiness Cycles Observed and Assessed Why andHow they Matterrdquo NBER Working Paper Series 6230 National Bureauof Economic Research

30

A Tables

Table 1 Parameters valuesParametera Description Value

N Number of sectorsunits i of production 10Ci Number of j buyer sectors for sector i (IO matrix sparseness) N minus 1Di (E) Value of the external demand 10mij Physical input coefficient of input j for sector i sim U(0 1)Mi Sum of the mij input coefficients j for the production of i 05αi friction to changes of production levels Degree of lockndashin on

production technology085

αsi Speed of adaptation to demand changes 004

σ Desired multiple of production to stock 10

aInto parenthesis the number of lags of the initial value for lagged variables

31

demand through sales is observed by the commercial staff It is thereforelogical that the information from demand reaches plant managers with somelag

The actual stock level is trivially computed subtracting from the previousheap of stock the amount sold and adding the new production

Sit = Sitminus1 + Qit minus Dit (3)

Concerning the desired level of stocks instead we introduce the conser-vative behaviour mentioned above Firms ideally would keep an amountof stocks proportional to the demand they receive say a multiple σ ofthat amount However when demand varies they adjust the desired levelsmoothly preventing sudden changes to the variable representing the goalof the firm

Slowastit = Slowast

itminus1 + αsi

(

σDit minus Slowastitminus1

)

(4)

where αsi measures the adaptation to demand changes

In summary the dynamics of the model consists of only the equationsabove Firstly at the beginning of a time step firms determine the quantityto produce as a function of past imbalances in stock levels and the pastproduction level These levels define the demand for all sectors determinedby the technical coefficients applied to all produced quantities Finallyactual and desired stocks can be updated Figure 1 represents visually thisdynamics highlighting the decisional components of the model in respect ofthe ldquomechanicsrdquo of trade

5 Results production structure and economic fluc-

tuations

In this section we describe some of the most relevant results concerningthe effects of the production structure to determine aggregate fluctuationsGiven the unusual analytical instrument chosen (at least concerning thistopic) we start with a brief methodological introduction meant to clarifythe nature of the results we claim to obtain Next we describe a generalsetting for the model which is the (extremely) limited area of the potentialparametersrsquo space of the model that we will explore In the rest of thesection we discuss the results from the model

51 Methodological considerations

Real economic systems are constantly subject to a continuous flow of shocksexternal to the production system besides changes within the system itselfIn order to appreciate the systemic properties we need to abstract from all

12

Compute the quantityto produce

F (Q(t-1) S(t-1) s(t-1) alpha)

Communicate thequantity of inputsrequired to each

supplier

Determine the overalldemand

F(D(E) D(I))

Revise the past decisionon stock plans

(desired stocks)F(D S(t-1) sigma alpha(s))

Compute the final levelof stocks for next period

F(s(t-1) Q D)

KEY

ModelDirection

Variablecomputed

at t-1(t-1)

Step includingfirm behavior

Figure 1 One period dynamics

the nonndashrelevant disturbances and concentrate on how the system endoge-nously contributes to the observed dynamics Therefore we will generatehighly abstract patterns whose analysis will provide insights on the systemproperties that would be otherwise lost in the noise generated by all theelements of a real system

We are not trying to fit the model to a specific empirical data set nor weare interested in studying the complete behaviour of the model for areas ofthe parameterrsquos space that make no economic sense (eg negative or infiniteproduction levels) Therefore the values of initial settings and results matteronly in relative and not in absolute terms

We are going to describe a simulation model with which we will produceseveral simulation runs and we claim obtain results relevant for the debatediscussed above Rigorously speaking we may claim only certainty for thevalidation of the results that is that the model does actually generate theresults presented and for the reasons we explain We limit this part to thepresentation of graphs and verbal explanations that we believe are ratheruncontroversial11 Concerning the relevance of our studies for real systems

11Interested readers can request the model code and and simulation data for replicationand extensions of the results We can guarantee that such possibility does not require

13

(verification) we simply avoid to pretend that our model is a quantitativeapproximation of real economies (which in which period) Our opinionis that the study of the model allows us to make general considerationsnon-obvious and relevant for applied debates supported by strong logicalarguments which can be integrated but not reverted by arguments con-cerning more and more elements of the economic world

52 Model setup

We go through the model properties analysing an economic system madeup of ten sectors (N = 10)12 Each sector shops inputs from the remain-ing N minus 1 and sells to them part of its own production as intermediategoods This means that the input matrix is complete except for the diag-onal (mij gt 0 forallj 6= i mii = 0) The sum of all N minus 1 coefficients for asingle sector is given by Mi and its benchmark value is kept at a plausiblemedium level (unless differently specified) to avoid distortions from highlydemanding sectors Each input coefficient is randomly drawn from a uniformdistribution

Sectors start at an equilibrium level their supply matches intermediateand final demand and in each period the same quantity of output mustbe produced to compensate for the used stocks and keep the desired stocksunchanged In thier turn stocks must be ten times the value of total sectoraldemand (σ) Eventually technological adjustment of production is ratherslow and producers are quiet refractory to follow sudden changes in thedemand in general a smoothing behaviour strongly prevails (α and αs)

Under those general preconditions in the following sections we anal-yse the dynamics of the economic system in relation to model parametersWe mainly focus on the cyclical behaviour of systems and their speed ofconvergence to the asymptotic equilibrium13

The very abstract assumption and the homogeneous initialisation donot allow for the multiple equilibria that are likely to rise in complex systemswith feedbacks Once more such simplification allows to have a betterunderstanding of the crucial impacts of the production structure on businessfluctuations Departing from the related literature discussed above14 weconsider the effects of a single shock rather then a continuous flow whichin our model is superfluous in order to obtain fluctuation persistence15 To

extensive programming or statistical skills besides the usual education of economists12You can think of a high level of aggregation in the statistical observation The com-

plete initialisation and parameters values are available in table 1 in the Appendix13Notice that sectors only approach a final equilibrium level of production which can

be prendashdetermined under static conditions14Included the more proximate work of Helbing Witt Lammer and Brenner (2004)15An exception are the last results presented in Section 58 which we see more as an

opportunity for further research than a closing of the paper

14

start with next section (53) shows the aggregate cyclical behaviour of astandard setting

53 Single shock analysis

We generate a simulated history following the dynamics of the modelrsquos vari-ables when the system is set to an equilibrium level and a single shockmodifies one of the exogenous elements namely the external demand forsectors Modifications across sectors are random and not correlated andmay increase or reduce external demand by at most 10

100 225 350 475 600

193894

195861

197828

199795

201762 Industrial Production

Time

Figure 2 Total industrial production fluctuations following a shock fromexternal demand

Figure 2 shows the dynamics generated by the total industrial productionof the system mdash that is also a proxy for the systemrsquos GDP fluctuations Thenew external demand changes the equilibrium level but the system cannotreach the new equilibrium with a monotonous pattern and converges to thenew level with a fluctuating pattern Each change in the external demandinduces also modifications of the intermediate demand and sectoral adjust-ments in the quantity produced typically overshoot the aggregate patternAs it can be appreciated from the figure it takes time before the system isable to converge to the new static equilibrium conditions

We study two features of this pattern absolute level of the new equilib-rium and volatility induced by the shock

54 Technical coefficients and industrial output

Production level

The equilibrium levels of industrial output depend on two factors only Ob-viously the levels of external demand determine how much of the total pro-

15

duction needs to be supplied to consumers exogenous from the productionsystem Second the technical coefficients also influence strongly the level ofactivity of the system defining the intermediate demand to each sector

180

200

220

240

260

280

300

01 02 03 04 05 06 07 08 09

Out

put l

evel

(In

dust

rial p

rodu

ctio

n)

Sum of total input coefficients (M)

Total industrial production for different input coefficients levels

Figure 3 The relevance of input coefficients on output level

Figure 3 shows the equilibrium level of industrial output reached bysystems with identical external demand where we fixed the sum of inputcoefficients for each sector of the economy (Mi) to different values16 Thehigher the sum the higher is the total production sustained by the system

This result depends on the very production structure represented by theinputndashoutput matrix In fact the higher is the level of production requiredby intermediate sectors the higher is the total production given a constantlevel of external demand

Notice that the level of production does not depend on the distribution ofthe technical coefficients As long as the sum of the coefficients is identicalthe same production level is obtained irrespective of their distribution (otherthings being equal)

Output shock absorption

Input coefficients have a relevant role also on the output stability of thesystem Starting from an output equilibrium level we hit the system witha single shock on the external demand for all sectors uncorrelated amongsectors17 modifing (positively or negatively) the demand for all sectors by

16Within this limit the actual value of the coefficients are randomly drawn17In each sector the external demand changes by plusmnX where X is a uniformly dis-

tributed percentage between 1 and 10

16

at most10 As show in Figure 2 the economic systems produces a lenghtlysmoothing oscillatory behaviour We compute the relative deviation of theaggregate production (industrial output) as the ratio between its standarddeviation and its mean δ = σY microY where Y =

sum

Qi Figure 4 shows thebehaviour of δ as a function of the sum of input coefficients for each sector80 periods after the shock As time goes by the level of δ reduces for allMi but the relation with the sum of input coefficient is unchanged

00055

0006

00065

0007

00075

0008

00085

0009

01 02 03 04 05 06 07 08 09

Rel

ativ

e va

rianc

e of

tota

l out

put (

delta

)

Sum of total input coefficients

Aggregate output deviations for different technical coefficients

Figure 4 The relevance of input coefficients on output stability

In sum input coefficients determine the extent of overshooting behaviourThe higher the coefficients the higher is the backlash even when shockscould cancel out through sectors

55 Shock persistence and adjustment coefficients

We now consider the effect of micro smoothing behaviours on the volatilityof the production system In particular we are interested in testing theeffects of two parameters of the model on the volatility of the productionsystem First we consider the effects of the αrsquos representing the ldquostickinessrdquoof the production technology to desired changes of Q aimed at filling thegap between actual and desired stocks The higher is α the slower is thereaction to stock unbalances Concerning its effect referring to the technicalpossibilities available ie production capital and organization we may putforward two opposite hypothesis relevant to systemrsquos volatility

Hypothesis 1 Slower production adjustment (high α) reduces volatilitybecause unbalances in demand of one sector being compensated slowly slow

17

down the diffusion of shocks to other sectors

Hypothesis 2 Slower production adjustment (high α) increase volatil-ity because the longer lasts the mismatch between actual and equilibriumproduction level the deeper will become the mismatch between actual anddesired stocks

20 25 30 35 40 45 50 55 60 65

Cyclesrsquo width

07 075

08 085

09alpha 0005

001 0015

002 0025

003 0035

004 0045

005

alphaS

20 25 30 35 40 45 50 55 60 65

Figure 5 Maximum width recorded between highest and lowest peaks fortotal production for different values of α and αS

Secondly we consider the parameters αS representing the speed of ad-justment of desired stock to observed demand The higher this parame-ter the faster sectors revise the desired levels of stocks to current levelsof observed demand This parameter concerns sectorsrsquo behavioral decisionprocess Higher αS represent a stronger belief that the most recently ob-served demand level is a reliable indicator of future demand levels whilelower levels of the parameter represent a more conservative approach wheremanagers require a long observation of demand levels different from the pastones before adjusting their expectations Also concerning the effect of thisparameter on the volatility induced by an exogenous shock we may maketwo hypothesis

Hypothesis 1 Faster adaptation of desired stocks to new levels of de-mand (higher αS) reduce volatility because changes in demand are rapidlytranslated in adjusted levels of production

Hypothesis 2 Faster adaptation of desired stocks to new levels of demand(higher αS) increase volatility because temporary unbalances of demand in

18

one sector are transmitted faster to other sector reinforcing the feedbackmechanism of fluctuations

To test which of these hypotheses are confirmed by the model we runsimulation runs with identical systems (eg same technical coefficients samelevels of demand same shocks) but for the values of the αrsquos and αS rsquos testingall combinations of these parameters over a range of values (each run usedthe same parameters for all the sectors of the system) For each simulation(ie each couple of values of the parameters) we computed two indicatorsof volatility One measures the maximum distance of total production be-tween the highest and the lowest peak that is the maximum width of theoscillations registered in the production time series The second indicatormeasures the relative deviation δ = σY microY Figures 5 and 6 report theseresults

0004 0006 0008 001 0012 0014 0016 0018 002

Relative deviation

07 075

08 085

09alpha 0005

001 0015

002 0025

003 0035

004 0045

005

alphaS

0004 0006 0008 001

0012 0014 0016 0018 002

Figure 6 Relative deviation of total production for different values of α andαS

The results confirms unequivocally both hypotheses 2 and reject hy-potheses 1 In a sense these hypotheses confirm the nature of the economicsystem represented in the model as a complex system where the interactionsamong sectors play a far more important role than the individual sector be-haviour In fact in both cases we may interpret the results that the strongerthe efforts to mitigate the effects of shocks (ie slow productionrsquos and quickexpectationsrsquo adjustment) the opposite result is actually obtained the fluc-tuations become even more accentuated It is also worth to note that thestronger impact is generated by the behavioural parameters rather than thetechnical one as indicated by the stronger effects of the αS in respect of theα This result seems to suggest that ceteris paribus information mattersmore than technical constraints Though the model is by any means in-

19

adequate to discuss normative aspects because of the naive ways strategicdecision making is represented it can anyway suggest the effects of errorsin management decisions (ie wrong desired levels of stocks) as opposed tothe technical constraints preventing sudden adjustment of production lev-els Stretching a bit the interpretation of these results we may suggests thatefforts to reduce volatility should better be oriented to improve the trans-mission of information in order to coordinate future production plans (ieidentifying quickly the desired levels of stocks and therefore the new equilib-rium production level) rather than improving the flexibility of productionmethods (reducing α)

56 Volatility and stock levels

In our model stock levels are maintained to buffer a sectorrsquos sale againstunexpected changes in demand Therefore a system made of risk adversemanagers (keeping higher levels of stocks) may be thought to lead to asmoother absorption of shocks in respect of a system composed of firmskeeping a short supply to cushion unexpected events To test this hypothesiswe tested the behaviour of the model for different values of σ the levels ofstock represented as a multiple of demand For each level of σ we run asimulation using all other initialization identical as usual representing aneconomic system ldquoalmostrdquo at equilibrium but for a single small shock indemand uncorrelated across sectors18

00044

00046

00048

0005

00052

00054

00056

00058

0006

00062

5 55 6 65 7 75 8 85 9 95 10

Rel

ativ

e va

riatio

n of

tota

l out

put (

delta

)

Ratio of stocked goods over production (sigma)

Stocks and production fluctuations

Figure 7 Fluctuations persistence on systems with different stock strategies(σ)

18Demand changes by plusmn1-10

20

Figure 7 shows the value of the relative deviation of output production(δ) generated by heterogeneous uncorrelated shocks in the external demandon identical systems where firms store a different multiple of their demand asstocks We obtain actually the opposite result that we may have expectedfluctuation become larger and more persistent the higher the level of stocksIn some way a strategy that for the individual firm is meant to reducethe fluctuations (using stocks as buffers) ends up by actually increasing thevolatility at aggregate levels

The result is though explainable both at the theoretical and empiricallevel In fact if no stocks were maintained production would need to adjustfor changes in demand only in case of a shock That is the firm may movemonotonically from the old to the new production levels sending consistentsignals to the rest of the system (ie its own suppliers) Instead if thesame shock affects a firm maintaining large stocks of output over the sameperiod production needs to adjust to the new demand and and to alignstocks with the new demand This generates besides the adjustment to thedemand a potentially inconsistent signal to the suppliers since the firm willhave a temporary level of production not meant to continue in equilibrium(adjusting stocks)

From the empirical viewpoint the last few decades have seen the almostuniversal diffusion of production system meant to keep stocks to a minimumif at all Lean production methods justndashinndashtime etc are solutions tothe problems of the costs necessary to maintain stocks Over the sameperiod economic fluctuations have been strongly reduced and few empiricalassessments have pointed to the relations between the two phenomena Forexample concerning the US ldquochanges in inventory behavior have playeda direct role in reducing real output volatilityrdquo (Kahn McConnell andPerez-Quiros 2002 p 183)19

57 Irrelevance of technical coefficients distribution

Horvath (1998) and Horvath (2000) suggest that systems with sparse ma-trices ie inputndashoutput matrices with many empty cells tend to be morevolatile in respect of systems with more evenly distributed matrices Noticethat this claim risk if not adequately qualified to generate an illogical resultIn fact matrices sparseness increases as a function of sectoral disaggregationand comparing the same systemrsquos fluctuations using inputndashoutput matricesat different levels of aggregation needs to produce the same type of volatil-ity Indeed this is not the case if all remaining parameters stand equaland do not adjust to the different level of analysis considered If aggregatepatterns of adjustment speed are maintained unchanged the higher the sec-toral disaggregation the higher the systemrsquos volatility ceteris paribus But

19Similar results are found in Blanchard and Simon (2001)

21

for example in our model reducing the aggregation also requires a changein the adaptation coefficients the more micro we undergo the analysis thehigher is the speed of adjustment of economic objects If for example weconsider sectors we are evaluating the adaptation of an entire set of firmswhich pose questions on how to aggregate their dynamics Things changewhen we consider the response of a single firm

Going back to Horvatsrsquo argument highly disaggregated matrices arelikely to contain many empty cells so that we may forecast different levelsof volatility depending on the aggregation used for the same system whichas mentioned is not caused by sparseness but to a failed adjustment inreaction mechanisms Results from our model20 show that volatility is com-pletely unaffected by input coefficients distribution Given their sectoralsum mdash through rows of the matrix mdash the value of each coefficient is irrele-vant The same occurs in the case of very skewed distributions that presenta high number of empty cells (zero input coefficients)

58 Microndash and Macrondashvolatility

As we have seen above one of the major discussion in the literature concernsthe relations between microndash and macrondashvolatility Put it simply one sideof the literature considers that uncorrelated microndashshocks cancel each otherout reducing their effects at macrondashlevel Conversely other researchersconsider that microndashshocks are multiplied at macrondashlevel generating highermacrondashvolatility than that we can observe at microndashlevel The debate isof obvious importance to determine the sources of business cycles to pre-dict their effects and determine the more effective policies to mitigate theirnegative effects

Our model is only a partial representation of realndashworld economic sys-tem and is also overly simplistic However based on these limitations italso offers very robust results allowing reliable conclusions on the (limited)elements included in the model In particular we consider our model a re-liable representation of the basic structures of the production interactionsamong sectors or firms Thus we can use the model to provide an answerto the debate concerning the contributions to volatility from the productioninteractions only Other contributions (eg from financial markets priceadjustments technological innovation etc) may re-inforce or counter theforces we analyse but they need to be considered as additional elements(and we opine also generally less relevant) not alternative to our analysisWe consider a necessity in order to apply a rigorous methodological ap-proach to be able to identify with certainty the results provided by partialstudies in order to eventually obtain by gradual extensions of the analy-sis more and more detailed representations of realistic economic systems

20Available from the authors

22

The results we obtain in this paper concerning this microndashmacro debate aremeant as a first relevant step to a more extended analysis

0

002

004

006

008

01

012

014

016

018

02

022

05 1 15 2 25 3

Var Demand

Macro (Output) and micro (Demand) volatility

delta Outputdelta Demand

Figure 8 Values of δ (relative deviation) for aggregate demand and ag-gregate output in respect of different levels of variance for sector specificuncorrelated external demand shocks

The flows of shocks affecting the firms of a system can be distinguishedalong two dimensions strength of the shock (eg amplitude of the changesin demand) and frequency of the shock (rare or frequent) Concerning thefirst aspect we initialized as usual a system to equilibrium levels (eg pro-duction quantities equal to external and intermediate demand for all sectorsconstant level of stocks equal to the desired level) We then ran the simu-lation generating the external demand for all sectors as a random variablewith constant (equilibrium) mean level and varying values of variance

Figure 8 shows unequivocally that as far as our model has propertiessimilar to real systems the hypothesis that macrondashfluctuations have largervolatility than their sectoral level shocks is fully confirmed At any levelof variance the aggregate volatility of demand as captured by the relativedeviation index is sensibly lower than the volatility shown by aggregateoutput that is the total production of the system This result is not sur-prising throughout all tests presented in the previous sections every resultsuggested that the system lsquooverndashreactsrsquo to any individual microndashlevel changeeven in the case these changes were meant to contrast disturbances There-fore when we move from considering one single shock to a flow of shockswe are not surprised to obtain that the system shows higher volatility thanthat implied by the microndashlevel disturbances

We now consider the second dimension defining a flow of shocks theirfrequency The question is whether for the same level of demand shocks

23

0 002 004 006 008 01 012 014 016

Difference of relative deviation for Output and Demand

0 5 10 15 20 25 30 35 40 45 50 55

Shock frequency 05

1

15

2

25

3

Var Demand

0 002 004 006 008 01

012 014 016

Figure 9 Differences of values of δ for aggregate output minus those foraggregate demand in respect of different levels of variance of demand and offrequency of the shocks

(ie same variance) more frequent changes trigger a higher or lower systemvolatility than rare disturbances The results shown in figure 9 suggest arather elaborated answer which partly vindicates (though in a particularsense) the arguments in favour of compensation of micro shocks The fig-ure shows the difference between aggregate output volatility and aggregatedemand volatility The independent variables are as before the volatilityof demand at microndashlevel21 and the time intervals in between two shocksAggregate demand volatility does not change for the frequency of shocksat least when computed over a long period But output volatility doesin a nonndashmonotonous way For high frequency of shocks the difference isstill positive and increasing for higher variance indicating that the ldquooverndashreactionrdquo argument is valid even in these cases but it is rather low Whilethe frequency of shocks slows down aggregate volatility of output increasesshowing that time is needed for the microndashshock to unfold their full effectsat the macrondashlevel However after a certain threshold less and less frequentshocks generate a lower volatility of the aggregate level of output

This result is quite sensible though not obvious at first sight and onceagain shows the power of such a simple model to generate and explain com-plex properties For frenetic levels of demand modifications the system hasno time to adjust but part of the job to align actual and desired productionfor constantly changing demand is performed by the varying demand itselfIn other terms the shocks cancel out through time if not through their quan-

21Beware of the scale of variance For reasons of visibility the scale is inverted

24

titative effects on the system While frequency decreases the disturbancesof the shocks not compensated by other (unlikely) shocks have the time tofilter through the production system generating higher and higher volatil-ity In a sense any system has a particular frequency of shocks that make itmaximally ldquoresonatingrdquo with the amplification of microndashshocks When therate of shocks gets even rarer the system can enjoy periods of longer andlonger quietness after each shock is absorbed and before a new one startsagain a cycle of initial local disturbance propagation of the disturbances toother sectors and settlement to the new equilibrium

6 Discussion and extensions

We presented a simple model of production composed by interacting sectorThe main (quit realistic) assumption of the model is that the productionprocesses in any sector of the economy make use of input from any othersector Moreover we assumed that sectors (represented as a single firm)adjust slowly both the desired level of production (smoothed revision of ex-pectation) and the actual level (smoothed modification of plant utilization)The model voluntarily leaves aside a large number of aspects of real-worldeconomic systems (eg financial markets price adjustments growth etc)The reason is that the model can be easily and reliably tested in order toextract its properties The possibility to transfer the (certain) model resultto a real context depend of course on an ample ceteris paribus clause How-ever we are confident that the reported results concerning the productionstructure play a relevant role in real world systems Therefore our resultsand particularly their motivations being accessible because of the relativesimplicity of the model will not be diminished by the integration of ourconsiderations with other aspects of economic systems

Our model is in essence a representation of an input-output table inter-preted in physical terms that is as the amount of inputs required for oneunit of output Besides this we include two straightforward assumptionsFirstly production levels cannot be suddenly changed but require time toscale up or down the utilization rate that is the levels of production Stocksare meant to make up for the difference between sales and production Thesecond assumption concerns the intentional behaviour of producers We im-plement this aspect by assuming that exists a level of stocks desired for eachlevel of demand However a change of demand does not translate suddenlyin a new level of desired stock but again only a long time at a different levelof demand will convince managers to update the desired levels of stocks

We show that the model generate under a wide number of settings oscil-latory patterns converging toward the equilibrium level where each sectorgenerate exactly the quantity demanded by final consumers and by othersectors as input The oscillations are provoked by over- and under-shooting

25

generated by the difficulty of coordination among sectors that by assump-tion ldquocommunicaterdquo only via variation of demand to their direct input sup-pliers We also show that the total production levels generated by sectors(sum of the physical units) produced at equilibrium depends (for constantfinal demand) on the total sum of the technical coefficients per sector irre-spective of their distribution This result is motivated by the impossibilityof input substitution and offers interesting suggestions for the analysis ofthe study of input-output tables of real-world systems

We then consider systems at equilibrium facing a single small change ofthe external demand for sectors These simulations allow to study the waythe system reacts to shocks We see that the sum of the coefficients affectsbesides the levels also the volatility of aggregate output the larger a systemthe stronger is the impact of a given shock Less obviously we find thatfor systems with the same dimension (ie same sums of coefficients) theaggregate volatility increases with both the inertia to change of productionlevel and the speed of adjustment of desired stocks This result is somewhatcounter-intuitive since both the cited variables are associated to a moreconservative behaviour that may be supposed should smooth away spikesin the pattern from an old to a new equilibrium However we show thatthe opposite actually holds given the complex interactions among sectorindividual-sector attitude intended to smooth shocks actually increase theiraggregate impact

In the same vein we show that systems maintaining high levels of stocksare more volatile that systems with reduced amounts of stocks Again thisis a counter-intuitive result reverting the goal of individual producers andthe aggregate result Interestingly this result finds strong support from theevidence of the recent changes in production methods (ie low or no stocks)and diminished cycles

In relation to specific issues discussed in the literature we find no supportto the suggestion that particular distributions of input-output coefficientsmodify the volatility of a system As said this depends in case on thesums of these coefficients but not on their distribution Further we canprovide an answer to a long and hotly debated questions whether microndashlevels volatility generates stronger or lower aggregate volatility Our modelshows unequivocally that microndashvolatility (eg variance of final demand)persists and generates stronger volatility at aggregate level Again we canidentify the reason on the complex production structure that exalts micro-level shocks by generating lsquowrongrsquo messages among producers Moreoverwe can also point to a usually neglected aspect of volatility we show thatthe frequency of shocks is a very relevant aspect besides their dimensionin determining the volatility of aggregate output

Our work can be developed along two complementary directions Firstlywe can continue to study the properties of the model by extending the ele-ments considered (eg pricing) analysing the effects on the present results

26

and generating more analytical evidence Of particular relevance is in ouropinion the problem of aggregation Representations of the same systemswith different codification systems (eg different number of SIC digits)must not modify overall system properties for example the volatility ob-served Given that our result show that the sums of coefficients matter forthe levels and volatility of the system and that these sum do depend on theaggregation chosen it should be possible to induce the reaction coefficientsthat we know affect the volatility only

Secondly we can find an application of our results to real world evidenceIn fact there are many data sets available for the data we deal with basicallyinput-output tables Our model is readily adapt to be parametrized alongany number of sector and coefficient necessary so that we may replicate pastseries in order to adapt the unobservable parameters and use the resultingcomplete model to explain business fluctuations and provide intuitions onfuture patterns

27

References

Acemoglu D and A Scott (1997) ldquoAsymmetric Business CyclesTheory and TimendashSeries Evidencerdquo Journal of Monetary Economics 40501ndash533

Backus D P Kehoe and F Kydland (1995) ldquoInternational BusinessCycles Theory and Evidencerdquo in Frontiers of Business Cycle Researched by T Cooley Princeton University Press Princeton

Bak P K Chen J Scheinkman and M Woodford (1993) ldquoAg-gregate Fluctuations from Independent Sectoral Shocks SelfndashOrganizedCritically in a Model of Production and Inventory Dynamicsrdquo RicercheEconomiche 47(1) 3ndash30

Battiston S D Delli Gatti M Gallegati B Greenwald and

J Stiglitz (2005) ldquoCredit Chains and Bankruptcies Avalanches in Sup-ply Networksrdquo in Annual Workshop on Economic Heterogeneous Inter-acting Agents (WEHIA 2005) University of Essex UK

Blanchard O and J Simon (2001) ldquoThe Long and Large Declinein US Output Volatilityrdquo Brookings Papers on Economic Activity 1135ndash174

Bresnahan T F and V A Ramey (1994) ldquoOutput Fluctuations atthe Plant Levelrdquo The Quarterly Journal of Economics 109(3) 593ndash924

David P A (1975) Technical choice innovation and economic growth essays on American and British experience in the nineteenth centuryCambridge University Press London

Delli Gatti D C Di Guilmi E Gaffeo G Giulioni M Galle-

gati and A Palestrini (2004) ldquoA New Approach to Business Fluc-tuations Heterogeneous Interacting Agents Scaling Laws and FinancialFragilityrdquo Journal of Economic Behavior amp Organization 56 489ndash512

Dosi G G Fagiolo and A Roventini (2006) ldquoAn EvolutionaryModel of Endogenous Business Cyclesrdquo Computational Economics 273ndash34

Dupor B (1999) ldquoAggregation and the Irrelevance in MultindashSector Mod-elsrdquo Journal of Monetary Economics 43 391ndash409

Head A C (1995) ldquoCountry Size Aggregate Fluctuations and Interna-tional Risk Sharingrdquo The Canadian Journal of Economics 28(4b) 1096ndash1119

28

Helbing D and S Lammer (2005) ldquoSupply and Production NetworksFrom the Bullwhip Effect to Business Cyclesrdquo in Networks of Interactingmachines Production Organization in Complex Industrial Systems andBiological Cells ed by D Armbruster K Kaneko and A S Mikhailovvol 3 of World Scientific Lecture Notes in Complex Systems World Sci-entific

Helbing D S Lammer U Witt and T Brenner (2004) ldquoNetworkndashInduced Oscillatory Behavior in Material Flow Networks and IrregularBusiness Cyclesrdquo Phisical Review E 70(056118) 1ndash6

Helbing D U Witt S Lammer and T Brenner (2004) ldquoNetwork-Induced Oscillatory Behaviour of Macroeconomic Output Flowsrdquo in In-dustry and Labour Dynamics II The Agent-based Computational Eco-nomics Approach Proceedings of the wildace 2004 conference ed byR Leombruni and M Richiardi no 5 LABORatorio R Revelli

Horvath M (1998) ldquoCyclically and Sectoral Linkages Aggregate Fluctu-ations from Independent Sectoral Shocksrdquo Review of Economic Dynam-ics 1 781ndash808

(2000) ldquoSectoral Shocks and Aggregate Fluctuationsrdquo Journal ofMonetary Economics 45(1) 69ndash106

Jovanovic B (1987) ldquoMicro Shocks and Aggregate Riskrdquo The QuarterlyJournal of Economics 102(2) 395ndash410

Kahn J A M M McConnell and G Perez-Quiros (2002) ldquoOnthe Causes of the Increased Stability of the US Economyrdquo FRBNY Eco-nomic Policy Review 8(1) 183ndash202

Kraay A and J Ventura (1998) ldquoComparative Advantage and theCrossndashSection of Business Cyclesrdquo World Bank Woorking paper Series1948 World Bank

Long J B and C I Plosser (1983) ldquoReal Business Cyclesrdquo Journalof Political Economy 31(1) 39ndash69

Lucas R E (1977) ldquoUnderstanding Business Cyclesrdquo CarnegiendashRochester Conference Series on Public Policy (special issue of Journalof Monetary Economics) 5 7ndash29

Nelson R R and S G Winter (1982) An Evolutionary Theory ofEconomic Change Harvard University Press Cambridge MA

Nirei M (2005) ldquoThreshold Behavior and Aggregate Fluctuationsrdquo Jour-nal of Economic Theory in press

29

Scheinkman J and M Woodford (1994) ldquoSelfndashOrganized Criticallyand Economic Fluctuationsrdquo American Economic Review 84(2 Papersand Proceedings of teh AEA) 417ndash421

Schuh S (1996) ldquoEvidence of the Link Between FirmndashLevel and Ag-gregate Inventory Behaviorrdquo Finance and Economics Discussion Series1996-46 Federal Reserve Board

Stiglitz J and B Greenwald (2003) Towards a New Paradigm inMonetary Economics Raffaele Mattioli Lectures Cambridge UniversityPress Cambridge

Weisbuch G and S Battiston (2005) ldquoProduction Networks andFailure Avalanchesrdquo Working Paper mimeo Ecole Normale SupErieure

Zarnovitz V (1977) ldquoBusiness Cycles Observed and Assessed Why andHow they Matterrdquo NBER Working Paper Series 6230 National Bureauof Economic Research

30

A Tables

Table 1 Parameters valuesParametera Description Value

N Number of sectorsunits i of production 10Ci Number of j buyer sectors for sector i (IO matrix sparseness) N minus 1Di (E) Value of the external demand 10mij Physical input coefficient of input j for sector i sim U(0 1)Mi Sum of the mij input coefficients j for the production of i 05αi friction to changes of production levels Degree of lockndashin on

production technology085

αsi Speed of adaptation to demand changes 004

σ Desired multiple of production to stock 10

aInto parenthesis the number of lags of the initial value for lagged variables

31

Compute the quantityto produce

F (Q(t-1) S(t-1) s(t-1) alpha)

Communicate thequantity of inputsrequired to each

supplier

Determine the overalldemand

F(D(E) D(I))

Revise the past decisionon stock plans

(desired stocks)F(D S(t-1) sigma alpha(s))

Compute the final levelof stocks for next period

F(s(t-1) Q D)

KEY

ModelDirection

Variablecomputed

at t-1(t-1)

Step includingfirm behavior

Figure 1 One period dynamics

the nonndashrelevant disturbances and concentrate on how the system endoge-nously contributes to the observed dynamics Therefore we will generatehighly abstract patterns whose analysis will provide insights on the systemproperties that would be otherwise lost in the noise generated by all theelements of a real system

We are not trying to fit the model to a specific empirical data set nor weare interested in studying the complete behaviour of the model for areas ofthe parameterrsquos space that make no economic sense (eg negative or infiniteproduction levels) Therefore the values of initial settings and results matteronly in relative and not in absolute terms

We are going to describe a simulation model with which we will produceseveral simulation runs and we claim obtain results relevant for the debatediscussed above Rigorously speaking we may claim only certainty for thevalidation of the results that is that the model does actually generate theresults presented and for the reasons we explain We limit this part to thepresentation of graphs and verbal explanations that we believe are ratheruncontroversial11 Concerning the relevance of our studies for real systems

11Interested readers can request the model code and and simulation data for replicationand extensions of the results We can guarantee that such possibility does not require

13

(verification) we simply avoid to pretend that our model is a quantitativeapproximation of real economies (which in which period) Our opinionis that the study of the model allows us to make general considerationsnon-obvious and relevant for applied debates supported by strong logicalarguments which can be integrated but not reverted by arguments con-cerning more and more elements of the economic world

52 Model setup

We go through the model properties analysing an economic system madeup of ten sectors (N = 10)12 Each sector shops inputs from the remain-ing N minus 1 and sells to them part of its own production as intermediategoods This means that the input matrix is complete except for the diag-onal (mij gt 0 forallj 6= i mii = 0) The sum of all N minus 1 coefficients for asingle sector is given by Mi and its benchmark value is kept at a plausiblemedium level (unless differently specified) to avoid distortions from highlydemanding sectors Each input coefficient is randomly drawn from a uniformdistribution

Sectors start at an equilibrium level their supply matches intermediateand final demand and in each period the same quantity of output mustbe produced to compensate for the used stocks and keep the desired stocksunchanged In thier turn stocks must be ten times the value of total sectoraldemand (σ) Eventually technological adjustment of production is ratherslow and producers are quiet refractory to follow sudden changes in thedemand in general a smoothing behaviour strongly prevails (α and αs)

Under those general preconditions in the following sections we anal-yse the dynamics of the economic system in relation to model parametersWe mainly focus on the cyclical behaviour of systems and their speed ofconvergence to the asymptotic equilibrium13

The very abstract assumption and the homogeneous initialisation donot allow for the multiple equilibria that are likely to rise in complex systemswith feedbacks Once more such simplification allows to have a betterunderstanding of the crucial impacts of the production structure on businessfluctuations Departing from the related literature discussed above14 weconsider the effects of a single shock rather then a continuous flow whichin our model is superfluous in order to obtain fluctuation persistence15 To

extensive programming or statistical skills besides the usual education of economists12You can think of a high level of aggregation in the statistical observation The com-

plete initialisation and parameters values are available in table 1 in the Appendix13Notice that sectors only approach a final equilibrium level of production which can

be prendashdetermined under static conditions14Included the more proximate work of Helbing Witt Lammer and Brenner (2004)15An exception are the last results presented in Section 58 which we see more as an

opportunity for further research than a closing of the paper

14

start with next section (53) shows the aggregate cyclical behaviour of astandard setting

53 Single shock analysis

We generate a simulated history following the dynamics of the modelrsquos vari-ables when the system is set to an equilibrium level and a single shockmodifies one of the exogenous elements namely the external demand forsectors Modifications across sectors are random and not correlated andmay increase or reduce external demand by at most 10

100 225 350 475 600

193894

195861

197828

199795

201762 Industrial Production

Time

Figure 2 Total industrial production fluctuations following a shock fromexternal demand

Figure 2 shows the dynamics generated by the total industrial productionof the system mdash that is also a proxy for the systemrsquos GDP fluctuations Thenew external demand changes the equilibrium level but the system cannotreach the new equilibrium with a monotonous pattern and converges to thenew level with a fluctuating pattern Each change in the external demandinduces also modifications of the intermediate demand and sectoral adjust-ments in the quantity produced typically overshoot the aggregate patternAs it can be appreciated from the figure it takes time before the system isable to converge to the new static equilibrium conditions

We study two features of this pattern absolute level of the new equilib-rium and volatility induced by the shock

54 Technical coefficients and industrial output

Production level

The equilibrium levels of industrial output depend on two factors only Ob-viously the levels of external demand determine how much of the total pro-

15

duction needs to be supplied to consumers exogenous from the productionsystem Second the technical coefficients also influence strongly the level ofactivity of the system defining the intermediate demand to each sector

180

200

220

240

260

280

300

01 02 03 04 05 06 07 08 09

Out

put l

evel

(In

dust

rial p

rodu

ctio

n)

Sum of total input coefficients (M)

Total industrial production for different input coefficients levels

Figure 3 The relevance of input coefficients on output level

Figure 3 shows the equilibrium level of industrial output reached bysystems with identical external demand where we fixed the sum of inputcoefficients for each sector of the economy (Mi) to different values16 Thehigher the sum the higher is the total production sustained by the system

This result depends on the very production structure represented by theinputndashoutput matrix In fact the higher is the level of production requiredby intermediate sectors the higher is the total production given a constantlevel of external demand

Notice that the level of production does not depend on the distribution ofthe technical coefficients As long as the sum of the coefficients is identicalthe same production level is obtained irrespective of their distribution (otherthings being equal)

Output shock absorption

Input coefficients have a relevant role also on the output stability of thesystem Starting from an output equilibrium level we hit the system witha single shock on the external demand for all sectors uncorrelated amongsectors17 modifing (positively or negatively) the demand for all sectors by

16Within this limit the actual value of the coefficients are randomly drawn17In each sector the external demand changes by plusmnX where X is a uniformly dis-

tributed percentage between 1 and 10

16

at most10 As show in Figure 2 the economic systems produces a lenghtlysmoothing oscillatory behaviour We compute the relative deviation of theaggregate production (industrial output) as the ratio between its standarddeviation and its mean δ = σY microY where Y =

sum

Qi Figure 4 shows thebehaviour of δ as a function of the sum of input coefficients for each sector80 periods after the shock As time goes by the level of δ reduces for allMi but the relation with the sum of input coefficient is unchanged

00055

0006

00065

0007

00075

0008

00085

0009

01 02 03 04 05 06 07 08 09

Rel

ativ

e va

rianc

e of

tota

l out

put (

delta

)

Sum of total input coefficients

Aggregate output deviations for different technical coefficients

Figure 4 The relevance of input coefficients on output stability

In sum input coefficients determine the extent of overshooting behaviourThe higher the coefficients the higher is the backlash even when shockscould cancel out through sectors

55 Shock persistence and adjustment coefficients

We now consider the effect of micro smoothing behaviours on the volatilityof the production system In particular we are interested in testing theeffects of two parameters of the model on the volatility of the productionsystem First we consider the effects of the αrsquos representing the ldquostickinessrdquoof the production technology to desired changes of Q aimed at filling thegap between actual and desired stocks The higher is α the slower is thereaction to stock unbalances Concerning its effect referring to the technicalpossibilities available ie production capital and organization we may putforward two opposite hypothesis relevant to systemrsquos volatility

Hypothesis 1 Slower production adjustment (high α) reduces volatilitybecause unbalances in demand of one sector being compensated slowly slow

17

down the diffusion of shocks to other sectors

Hypothesis 2 Slower production adjustment (high α) increase volatil-ity because the longer lasts the mismatch between actual and equilibriumproduction level the deeper will become the mismatch between actual anddesired stocks

20 25 30 35 40 45 50 55 60 65

Cyclesrsquo width

07 075

08 085

09alpha 0005

001 0015

002 0025

003 0035

004 0045

005

alphaS

20 25 30 35 40 45 50 55 60 65

Figure 5 Maximum width recorded between highest and lowest peaks fortotal production for different values of α and αS

Secondly we consider the parameters αS representing the speed of ad-justment of desired stock to observed demand The higher this parame-ter the faster sectors revise the desired levels of stocks to current levelsof observed demand This parameter concerns sectorsrsquo behavioral decisionprocess Higher αS represent a stronger belief that the most recently ob-served demand level is a reliable indicator of future demand levels whilelower levels of the parameter represent a more conservative approach wheremanagers require a long observation of demand levels different from the pastones before adjusting their expectations Also concerning the effect of thisparameter on the volatility induced by an exogenous shock we may maketwo hypothesis

Hypothesis 1 Faster adaptation of desired stocks to new levels of de-mand (higher αS) reduce volatility because changes in demand are rapidlytranslated in adjusted levels of production

Hypothesis 2 Faster adaptation of desired stocks to new levels of demand(higher αS) increase volatility because temporary unbalances of demand in

18

one sector are transmitted faster to other sector reinforcing the feedbackmechanism of fluctuations

To test which of these hypotheses are confirmed by the model we runsimulation runs with identical systems (eg same technical coefficients samelevels of demand same shocks) but for the values of the αrsquos and αS rsquos testingall combinations of these parameters over a range of values (each run usedthe same parameters for all the sectors of the system) For each simulation(ie each couple of values of the parameters) we computed two indicatorsof volatility One measures the maximum distance of total production be-tween the highest and the lowest peak that is the maximum width of theoscillations registered in the production time series The second indicatormeasures the relative deviation δ = σY microY Figures 5 and 6 report theseresults

0004 0006 0008 001 0012 0014 0016 0018 002

Relative deviation

07 075

08 085

09alpha 0005

001 0015

002 0025

003 0035

004 0045

005

alphaS

0004 0006 0008 001

0012 0014 0016 0018 002

Figure 6 Relative deviation of total production for different values of α andαS

The results confirms unequivocally both hypotheses 2 and reject hy-potheses 1 In a sense these hypotheses confirm the nature of the economicsystem represented in the model as a complex system where the interactionsamong sectors play a far more important role than the individual sector be-haviour In fact in both cases we may interpret the results that the strongerthe efforts to mitigate the effects of shocks (ie slow productionrsquos and quickexpectationsrsquo adjustment) the opposite result is actually obtained the fluc-tuations become even more accentuated It is also worth to note that thestronger impact is generated by the behavioural parameters rather than thetechnical one as indicated by the stronger effects of the αS in respect of theα This result seems to suggest that ceteris paribus information mattersmore than technical constraints Though the model is by any means in-

19

adequate to discuss normative aspects because of the naive ways strategicdecision making is represented it can anyway suggest the effects of errorsin management decisions (ie wrong desired levels of stocks) as opposed tothe technical constraints preventing sudden adjustment of production lev-els Stretching a bit the interpretation of these results we may suggests thatefforts to reduce volatility should better be oriented to improve the trans-mission of information in order to coordinate future production plans (ieidentifying quickly the desired levels of stocks and therefore the new equilib-rium production level) rather than improving the flexibility of productionmethods (reducing α)

56 Volatility and stock levels

In our model stock levels are maintained to buffer a sectorrsquos sale againstunexpected changes in demand Therefore a system made of risk adversemanagers (keeping higher levels of stocks) may be thought to lead to asmoother absorption of shocks in respect of a system composed of firmskeeping a short supply to cushion unexpected events To test this hypothesiswe tested the behaviour of the model for different values of σ the levels ofstock represented as a multiple of demand For each level of σ we run asimulation using all other initialization identical as usual representing aneconomic system ldquoalmostrdquo at equilibrium but for a single small shock indemand uncorrelated across sectors18

00044

00046

00048

0005

00052

00054

00056

00058

0006

00062

5 55 6 65 7 75 8 85 9 95 10

Rel

ativ

e va

riatio

n of

tota

l out

put (

delta

)

Ratio of stocked goods over production (sigma)

Stocks and production fluctuations

Figure 7 Fluctuations persistence on systems with different stock strategies(σ)

18Demand changes by plusmn1-10

20

Figure 7 shows the value of the relative deviation of output production(δ) generated by heterogeneous uncorrelated shocks in the external demandon identical systems where firms store a different multiple of their demand asstocks We obtain actually the opposite result that we may have expectedfluctuation become larger and more persistent the higher the level of stocksIn some way a strategy that for the individual firm is meant to reducethe fluctuations (using stocks as buffers) ends up by actually increasing thevolatility at aggregate levels

The result is though explainable both at the theoretical and empiricallevel In fact if no stocks were maintained production would need to adjustfor changes in demand only in case of a shock That is the firm may movemonotonically from the old to the new production levels sending consistentsignals to the rest of the system (ie its own suppliers) Instead if thesame shock affects a firm maintaining large stocks of output over the sameperiod production needs to adjust to the new demand and and to alignstocks with the new demand This generates besides the adjustment to thedemand a potentially inconsistent signal to the suppliers since the firm willhave a temporary level of production not meant to continue in equilibrium(adjusting stocks)

From the empirical viewpoint the last few decades have seen the almostuniversal diffusion of production system meant to keep stocks to a minimumif at all Lean production methods justndashinndashtime etc are solutions tothe problems of the costs necessary to maintain stocks Over the sameperiod economic fluctuations have been strongly reduced and few empiricalassessments have pointed to the relations between the two phenomena Forexample concerning the US ldquochanges in inventory behavior have playeda direct role in reducing real output volatilityrdquo (Kahn McConnell andPerez-Quiros 2002 p 183)19

57 Irrelevance of technical coefficients distribution

Horvath (1998) and Horvath (2000) suggest that systems with sparse ma-trices ie inputndashoutput matrices with many empty cells tend to be morevolatile in respect of systems with more evenly distributed matrices Noticethat this claim risk if not adequately qualified to generate an illogical resultIn fact matrices sparseness increases as a function of sectoral disaggregationand comparing the same systemrsquos fluctuations using inputndashoutput matricesat different levels of aggregation needs to produce the same type of volatil-ity Indeed this is not the case if all remaining parameters stand equaland do not adjust to the different level of analysis considered If aggregatepatterns of adjustment speed are maintained unchanged the higher the sec-toral disaggregation the higher the systemrsquos volatility ceteris paribus But

19Similar results are found in Blanchard and Simon (2001)

21

for example in our model reducing the aggregation also requires a changein the adaptation coefficients the more micro we undergo the analysis thehigher is the speed of adjustment of economic objects If for example weconsider sectors we are evaluating the adaptation of an entire set of firmswhich pose questions on how to aggregate their dynamics Things changewhen we consider the response of a single firm

Going back to Horvatsrsquo argument highly disaggregated matrices arelikely to contain many empty cells so that we may forecast different levelsof volatility depending on the aggregation used for the same system whichas mentioned is not caused by sparseness but to a failed adjustment inreaction mechanisms Results from our model20 show that volatility is com-pletely unaffected by input coefficients distribution Given their sectoralsum mdash through rows of the matrix mdash the value of each coefficient is irrele-vant The same occurs in the case of very skewed distributions that presenta high number of empty cells (zero input coefficients)

58 Microndash and Macrondashvolatility

As we have seen above one of the major discussion in the literature concernsthe relations between microndash and macrondashvolatility Put it simply one sideof the literature considers that uncorrelated microndashshocks cancel each otherout reducing their effects at macrondashlevel Conversely other researchersconsider that microndashshocks are multiplied at macrondashlevel generating highermacrondashvolatility than that we can observe at microndashlevel The debate isof obvious importance to determine the sources of business cycles to pre-dict their effects and determine the more effective policies to mitigate theirnegative effects

Our model is only a partial representation of realndashworld economic sys-tem and is also overly simplistic However based on these limitations italso offers very robust results allowing reliable conclusions on the (limited)elements included in the model In particular we consider our model a re-liable representation of the basic structures of the production interactionsamong sectors or firms Thus we can use the model to provide an answerto the debate concerning the contributions to volatility from the productioninteractions only Other contributions (eg from financial markets priceadjustments technological innovation etc) may re-inforce or counter theforces we analyse but they need to be considered as additional elements(and we opine also generally less relevant) not alternative to our analysisWe consider a necessity in order to apply a rigorous methodological ap-proach to be able to identify with certainty the results provided by partialstudies in order to eventually obtain by gradual extensions of the analy-sis more and more detailed representations of realistic economic systems

20Available from the authors

22

The results we obtain in this paper concerning this microndashmacro debate aremeant as a first relevant step to a more extended analysis

0

002

004

006

008

01

012

014

016

018

02

022

05 1 15 2 25 3

Var Demand

Macro (Output) and micro (Demand) volatility

delta Outputdelta Demand

Figure 8 Values of δ (relative deviation) for aggregate demand and ag-gregate output in respect of different levels of variance for sector specificuncorrelated external demand shocks

The flows of shocks affecting the firms of a system can be distinguishedalong two dimensions strength of the shock (eg amplitude of the changesin demand) and frequency of the shock (rare or frequent) Concerning thefirst aspect we initialized as usual a system to equilibrium levels (eg pro-duction quantities equal to external and intermediate demand for all sectorsconstant level of stocks equal to the desired level) We then ran the simu-lation generating the external demand for all sectors as a random variablewith constant (equilibrium) mean level and varying values of variance

Figure 8 shows unequivocally that as far as our model has propertiessimilar to real systems the hypothesis that macrondashfluctuations have largervolatility than their sectoral level shocks is fully confirmed At any levelof variance the aggregate volatility of demand as captured by the relativedeviation index is sensibly lower than the volatility shown by aggregateoutput that is the total production of the system This result is not sur-prising throughout all tests presented in the previous sections every resultsuggested that the system lsquooverndashreactsrsquo to any individual microndashlevel changeeven in the case these changes were meant to contrast disturbances There-fore when we move from considering one single shock to a flow of shockswe are not surprised to obtain that the system shows higher volatility thanthat implied by the microndashlevel disturbances

We now consider the second dimension defining a flow of shocks theirfrequency The question is whether for the same level of demand shocks

23

0 002 004 006 008 01 012 014 016

Difference of relative deviation for Output and Demand

0 5 10 15 20 25 30 35 40 45 50 55

Shock frequency 05

1

15

2

25

3

Var Demand

0 002 004 006 008 01

012 014 016

Figure 9 Differences of values of δ for aggregate output minus those foraggregate demand in respect of different levels of variance of demand and offrequency of the shocks

(ie same variance) more frequent changes trigger a higher or lower systemvolatility than rare disturbances The results shown in figure 9 suggest arather elaborated answer which partly vindicates (though in a particularsense) the arguments in favour of compensation of micro shocks The fig-ure shows the difference between aggregate output volatility and aggregatedemand volatility The independent variables are as before the volatilityof demand at microndashlevel21 and the time intervals in between two shocksAggregate demand volatility does not change for the frequency of shocksat least when computed over a long period But output volatility doesin a nonndashmonotonous way For high frequency of shocks the difference isstill positive and increasing for higher variance indicating that the ldquooverndashreactionrdquo argument is valid even in these cases but it is rather low Whilethe frequency of shocks slows down aggregate volatility of output increasesshowing that time is needed for the microndashshock to unfold their full effectsat the macrondashlevel However after a certain threshold less and less frequentshocks generate a lower volatility of the aggregate level of output

This result is quite sensible though not obvious at first sight and onceagain shows the power of such a simple model to generate and explain com-plex properties For frenetic levels of demand modifications the system hasno time to adjust but part of the job to align actual and desired productionfor constantly changing demand is performed by the varying demand itselfIn other terms the shocks cancel out through time if not through their quan-

21Beware of the scale of variance For reasons of visibility the scale is inverted

24

titative effects on the system While frequency decreases the disturbancesof the shocks not compensated by other (unlikely) shocks have the time tofilter through the production system generating higher and higher volatil-ity In a sense any system has a particular frequency of shocks that make itmaximally ldquoresonatingrdquo with the amplification of microndashshocks When therate of shocks gets even rarer the system can enjoy periods of longer andlonger quietness after each shock is absorbed and before a new one startsagain a cycle of initial local disturbance propagation of the disturbances toother sectors and settlement to the new equilibrium

6 Discussion and extensions

We presented a simple model of production composed by interacting sectorThe main (quit realistic) assumption of the model is that the productionprocesses in any sector of the economy make use of input from any othersector Moreover we assumed that sectors (represented as a single firm)adjust slowly both the desired level of production (smoothed revision of ex-pectation) and the actual level (smoothed modification of plant utilization)The model voluntarily leaves aside a large number of aspects of real-worldeconomic systems (eg financial markets price adjustments growth etc)The reason is that the model can be easily and reliably tested in order toextract its properties The possibility to transfer the (certain) model resultto a real context depend of course on an ample ceteris paribus clause How-ever we are confident that the reported results concerning the productionstructure play a relevant role in real world systems Therefore our resultsand particularly their motivations being accessible because of the relativesimplicity of the model will not be diminished by the integration of ourconsiderations with other aspects of economic systems

Our model is in essence a representation of an input-output table inter-preted in physical terms that is as the amount of inputs required for oneunit of output Besides this we include two straightforward assumptionsFirstly production levels cannot be suddenly changed but require time toscale up or down the utilization rate that is the levels of production Stocksare meant to make up for the difference between sales and production Thesecond assumption concerns the intentional behaviour of producers We im-plement this aspect by assuming that exists a level of stocks desired for eachlevel of demand However a change of demand does not translate suddenlyin a new level of desired stock but again only a long time at a different levelof demand will convince managers to update the desired levels of stocks

We show that the model generate under a wide number of settings oscil-latory patterns converging toward the equilibrium level where each sectorgenerate exactly the quantity demanded by final consumers and by othersectors as input The oscillations are provoked by over- and under-shooting

25

generated by the difficulty of coordination among sectors that by assump-tion ldquocommunicaterdquo only via variation of demand to their direct input sup-pliers We also show that the total production levels generated by sectors(sum of the physical units) produced at equilibrium depends (for constantfinal demand) on the total sum of the technical coefficients per sector irre-spective of their distribution This result is motivated by the impossibilityof input substitution and offers interesting suggestions for the analysis ofthe study of input-output tables of real-world systems

We then consider systems at equilibrium facing a single small change ofthe external demand for sectors These simulations allow to study the waythe system reacts to shocks We see that the sum of the coefficients affectsbesides the levels also the volatility of aggregate output the larger a systemthe stronger is the impact of a given shock Less obviously we find thatfor systems with the same dimension (ie same sums of coefficients) theaggregate volatility increases with both the inertia to change of productionlevel and the speed of adjustment of desired stocks This result is somewhatcounter-intuitive since both the cited variables are associated to a moreconservative behaviour that may be supposed should smooth away spikesin the pattern from an old to a new equilibrium However we show thatthe opposite actually holds given the complex interactions among sectorindividual-sector attitude intended to smooth shocks actually increase theiraggregate impact

In the same vein we show that systems maintaining high levels of stocksare more volatile that systems with reduced amounts of stocks Again thisis a counter-intuitive result reverting the goal of individual producers andthe aggregate result Interestingly this result finds strong support from theevidence of the recent changes in production methods (ie low or no stocks)and diminished cycles

In relation to specific issues discussed in the literature we find no supportto the suggestion that particular distributions of input-output coefficientsmodify the volatility of a system As said this depends in case on thesums of these coefficients but not on their distribution Further we canprovide an answer to a long and hotly debated questions whether microndashlevels volatility generates stronger or lower aggregate volatility Our modelshows unequivocally that microndashvolatility (eg variance of final demand)persists and generates stronger volatility at aggregate level Again we canidentify the reason on the complex production structure that exalts micro-level shocks by generating lsquowrongrsquo messages among producers Moreoverwe can also point to a usually neglected aspect of volatility we show thatthe frequency of shocks is a very relevant aspect besides their dimensionin determining the volatility of aggregate output

Our work can be developed along two complementary directions Firstlywe can continue to study the properties of the model by extending the ele-ments considered (eg pricing) analysing the effects on the present results

26

and generating more analytical evidence Of particular relevance is in ouropinion the problem of aggregation Representations of the same systemswith different codification systems (eg different number of SIC digits)must not modify overall system properties for example the volatility ob-served Given that our result show that the sums of coefficients matter forthe levels and volatility of the system and that these sum do depend on theaggregation chosen it should be possible to induce the reaction coefficientsthat we know affect the volatility only

Secondly we can find an application of our results to real world evidenceIn fact there are many data sets available for the data we deal with basicallyinput-output tables Our model is readily adapt to be parametrized alongany number of sector and coefficient necessary so that we may replicate pastseries in order to adapt the unobservable parameters and use the resultingcomplete model to explain business fluctuations and provide intuitions onfuture patterns

27

References

Acemoglu D and A Scott (1997) ldquoAsymmetric Business CyclesTheory and TimendashSeries Evidencerdquo Journal of Monetary Economics 40501ndash533

Backus D P Kehoe and F Kydland (1995) ldquoInternational BusinessCycles Theory and Evidencerdquo in Frontiers of Business Cycle Researched by T Cooley Princeton University Press Princeton

Bak P K Chen J Scheinkman and M Woodford (1993) ldquoAg-gregate Fluctuations from Independent Sectoral Shocks SelfndashOrganizedCritically in a Model of Production and Inventory Dynamicsrdquo RicercheEconomiche 47(1) 3ndash30

Battiston S D Delli Gatti M Gallegati B Greenwald and

J Stiglitz (2005) ldquoCredit Chains and Bankruptcies Avalanches in Sup-ply Networksrdquo in Annual Workshop on Economic Heterogeneous Inter-acting Agents (WEHIA 2005) University of Essex UK

Blanchard O and J Simon (2001) ldquoThe Long and Large Declinein US Output Volatilityrdquo Brookings Papers on Economic Activity 1135ndash174

Bresnahan T F and V A Ramey (1994) ldquoOutput Fluctuations atthe Plant Levelrdquo The Quarterly Journal of Economics 109(3) 593ndash924

David P A (1975) Technical choice innovation and economic growth essays on American and British experience in the nineteenth centuryCambridge University Press London

Delli Gatti D C Di Guilmi E Gaffeo G Giulioni M Galle-

gati and A Palestrini (2004) ldquoA New Approach to Business Fluc-tuations Heterogeneous Interacting Agents Scaling Laws and FinancialFragilityrdquo Journal of Economic Behavior amp Organization 56 489ndash512

Dosi G G Fagiolo and A Roventini (2006) ldquoAn EvolutionaryModel of Endogenous Business Cyclesrdquo Computational Economics 273ndash34

Dupor B (1999) ldquoAggregation and the Irrelevance in MultindashSector Mod-elsrdquo Journal of Monetary Economics 43 391ndash409

Head A C (1995) ldquoCountry Size Aggregate Fluctuations and Interna-tional Risk Sharingrdquo The Canadian Journal of Economics 28(4b) 1096ndash1119

28

Helbing D and S Lammer (2005) ldquoSupply and Production NetworksFrom the Bullwhip Effect to Business Cyclesrdquo in Networks of Interactingmachines Production Organization in Complex Industrial Systems andBiological Cells ed by D Armbruster K Kaneko and A S Mikhailovvol 3 of World Scientific Lecture Notes in Complex Systems World Sci-entific

Helbing D S Lammer U Witt and T Brenner (2004) ldquoNetworkndashInduced Oscillatory Behavior in Material Flow Networks and IrregularBusiness Cyclesrdquo Phisical Review E 70(056118) 1ndash6

Helbing D U Witt S Lammer and T Brenner (2004) ldquoNetwork-Induced Oscillatory Behaviour of Macroeconomic Output Flowsrdquo in In-dustry and Labour Dynamics II The Agent-based Computational Eco-nomics Approach Proceedings of the wildace 2004 conference ed byR Leombruni and M Richiardi no 5 LABORatorio R Revelli

Horvath M (1998) ldquoCyclically and Sectoral Linkages Aggregate Fluctu-ations from Independent Sectoral Shocksrdquo Review of Economic Dynam-ics 1 781ndash808

(2000) ldquoSectoral Shocks and Aggregate Fluctuationsrdquo Journal ofMonetary Economics 45(1) 69ndash106

Jovanovic B (1987) ldquoMicro Shocks and Aggregate Riskrdquo The QuarterlyJournal of Economics 102(2) 395ndash410

Kahn J A M M McConnell and G Perez-Quiros (2002) ldquoOnthe Causes of the Increased Stability of the US Economyrdquo FRBNY Eco-nomic Policy Review 8(1) 183ndash202

Kraay A and J Ventura (1998) ldquoComparative Advantage and theCrossndashSection of Business Cyclesrdquo World Bank Woorking paper Series1948 World Bank

Long J B and C I Plosser (1983) ldquoReal Business Cyclesrdquo Journalof Political Economy 31(1) 39ndash69

Lucas R E (1977) ldquoUnderstanding Business Cyclesrdquo CarnegiendashRochester Conference Series on Public Policy (special issue of Journalof Monetary Economics) 5 7ndash29

Nelson R R and S G Winter (1982) An Evolutionary Theory ofEconomic Change Harvard University Press Cambridge MA

Nirei M (2005) ldquoThreshold Behavior and Aggregate Fluctuationsrdquo Jour-nal of Economic Theory in press

29

Scheinkman J and M Woodford (1994) ldquoSelfndashOrganized Criticallyand Economic Fluctuationsrdquo American Economic Review 84(2 Papersand Proceedings of teh AEA) 417ndash421

Schuh S (1996) ldquoEvidence of the Link Between FirmndashLevel and Ag-gregate Inventory Behaviorrdquo Finance and Economics Discussion Series1996-46 Federal Reserve Board

Stiglitz J and B Greenwald (2003) Towards a New Paradigm inMonetary Economics Raffaele Mattioli Lectures Cambridge UniversityPress Cambridge

Weisbuch G and S Battiston (2005) ldquoProduction Networks andFailure Avalanchesrdquo Working Paper mimeo Ecole Normale SupErieure

Zarnovitz V (1977) ldquoBusiness Cycles Observed and Assessed Why andHow they Matterrdquo NBER Working Paper Series 6230 National Bureauof Economic Research

30

A Tables

Table 1 Parameters valuesParametera Description Value

N Number of sectorsunits i of production 10Ci Number of j buyer sectors for sector i (IO matrix sparseness) N minus 1Di (E) Value of the external demand 10mij Physical input coefficient of input j for sector i sim U(0 1)Mi Sum of the mij input coefficients j for the production of i 05αi friction to changes of production levels Degree of lockndashin on

production technology085

αsi Speed of adaptation to demand changes 004

σ Desired multiple of production to stock 10

aInto parenthesis the number of lags of the initial value for lagged variables

31

(verification) we simply avoid to pretend that our model is a quantitativeapproximation of real economies (which in which period) Our opinionis that the study of the model allows us to make general considerationsnon-obvious and relevant for applied debates supported by strong logicalarguments which can be integrated but not reverted by arguments con-cerning more and more elements of the economic world

52 Model setup

We go through the model properties analysing an economic system madeup of ten sectors (N = 10)12 Each sector shops inputs from the remain-ing N minus 1 and sells to them part of its own production as intermediategoods This means that the input matrix is complete except for the diag-onal (mij gt 0 forallj 6= i mii = 0) The sum of all N minus 1 coefficients for asingle sector is given by Mi and its benchmark value is kept at a plausiblemedium level (unless differently specified) to avoid distortions from highlydemanding sectors Each input coefficient is randomly drawn from a uniformdistribution

Sectors start at an equilibrium level their supply matches intermediateand final demand and in each period the same quantity of output mustbe produced to compensate for the used stocks and keep the desired stocksunchanged In thier turn stocks must be ten times the value of total sectoraldemand (σ) Eventually technological adjustment of production is ratherslow and producers are quiet refractory to follow sudden changes in thedemand in general a smoothing behaviour strongly prevails (α and αs)

Under those general preconditions in the following sections we anal-yse the dynamics of the economic system in relation to model parametersWe mainly focus on the cyclical behaviour of systems and their speed ofconvergence to the asymptotic equilibrium13

The very abstract assumption and the homogeneous initialisation donot allow for the multiple equilibria that are likely to rise in complex systemswith feedbacks Once more such simplification allows to have a betterunderstanding of the crucial impacts of the production structure on businessfluctuations Departing from the related literature discussed above14 weconsider the effects of a single shock rather then a continuous flow whichin our model is superfluous in order to obtain fluctuation persistence15 To

extensive programming or statistical skills besides the usual education of economists12You can think of a high level of aggregation in the statistical observation The com-

plete initialisation and parameters values are available in table 1 in the Appendix13Notice that sectors only approach a final equilibrium level of production which can

be prendashdetermined under static conditions14Included the more proximate work of Helbing Witt Lammer and Brenner (2004)15An exception are the last results presented in Section 58 which we see more as an

opportunity for further research than a closing of the paper

14

start with next section (53) shows the aggregate cyclical behaviour of astandard setting

53 Single shock analysis

We generate a simulated history following the dynamics of the modelrsquos vari-ables when the system is set to an equilibrium level and a single shockmodifies one of the exogenous elements namely the external demand forsectors Modifications across sectors are random and not correlated andmay increase or reduce external demand by at most 10

100 225 350 475 600

193894

195861

197828

199795

201762 Industrial Production

Time

Figure 2 Total industrial production fluctuations following a shock fromexternal demand

Figure 2 shows the dynamics generated by the total industrial productionof the system mdash that is also a proxy for the systemrsquos GDP fluctuations Thenew external demand changes the equilibrium level but the system cannotreach the new equilibrium with a monotonous pattern and converges to thenew level with a fluctuating pattern Each change in the external demandinduces also modifications of the intermediate demand and sectoral adjust-ments in the quantity produced typically overshoot the aggregate patternAs it can be appreciated from the figure it takes time before the system isable to converge to the new static equilibrium conditions

We study two features of this pattern absolute level of the new equilib-rium and volatility induced by the shock

54 Technical coefficients and industrial output

Production level

The equilibrium levels of industrial output depend on two factors only Ob-viously the levels of external demand determine how much of the total pro-

15

duction needs to be supplied to consumers exogenous from the productionsystem Second the technical coefficients also influence strongly the level ofactivity of the system defining the intermediate demand to each sector

180

200

220

240

260

280

300

01 02 03 04 05 06 07 08 09

Out

put l

evel

(In

dust

rial p

rodu

ctio

n)

Sum of total input coefficients (M)

Total industrial production for different input coefficients levels

Figure 3 The relevance of input coefficients on output level

Figure 3 shows the equilibrium level of industrial output reached bysystems with identical external demand where we fixed the sum of inputcoefficients for each sector of the economy (Mi) to different values16 Thehigher the sum the higher is the total production sustained by the system

This result depends on the very production structure represented by theinputndashoutput matrix In fact the higher is the level of production requiredby intermediate sectors the higher is the total production given a constantlevel of external demand

Notice that the level of production does not depend on the distribution ofthe technical coefficients As long as the sum of the coefficients is identicalthe same production level is obtained irrespective of their distribution (otherthings being equal)

Output shock absorption

Input coefficients have a relevant role also on the output stability of thesystem Starting from an output equilibrium level we hit the system witha single shock on the external demand for all sectors uncorrelated amongsectors17 modifing (positively or negatively) the demand for all sectors by

16Within this limit the actual value of the coefficients are randomly drawn17In each sector the external demand changes by plusmnX where X is a uniformly dis-

tributed percentage between 1 and 10

16

at most10 As show in Figure 2 the economic systems produces a lenghtlysmoothing oscillatory behaviour We compute the relative deviation of theaggregate production (industrial output) as the ratio between its standarddeviation and its mean δ = σY microY where Y =

sum

Qi Figure 4 shows thebehaviour of δ as a function of the sum of input coefficients for each sector80 periods after the shock As time goes by the level of δ reduces for allMi but the relation with the sum of input coefficient is unchanged

00055

0006

00065

0007

00075

0008

00085

0009

01 02 03 04 05 06 07 08 09

Rel

ativ

e va

rianc

e of

tota

l out

put (

delta

)

Sum of total input coefficients

Aggregate output deviations for different technical coefficients

Figure 4 The relevance of input coefficients on output stability

In sum input coefficients determine the extent of overshooting behaviourThe higher the coefficients the higher is the backlash even when shockscould cancel out through sectors

55 Shock persistence and adjustment coefficients

We now consider the effect of micro smoothing behaviours on the volatilityof the production system In particular we are interested in testing theeffects of two parameters of the model on the volatility of the productionsystem First we consider the effects of the αrsquos representing the ldquostickinessrdquoof the production technology to desired changes of Q aimed at filling thegap between actual and desired stocks The higher is α the slower is thereaction to stock unbalances Concerning its effect referring to the technicalpossibilities available ie production capital and organization we may putforward two opposite hypothesis relevant to systemrsquos volatility

Hypothesis 1 Slower production adjustment (high α) reduces volatilitybecause unbalances in demand of one sector being compensated slowly slow

17

down the diffusion of shocks to other sectors

Hypothesis 2 Slower production adjustment (high α) increase volatil-ity because the longer lasts the mismatch between actual and equilibriumproduction level the deeper will become the mismatch between actual anddesired stocks

20 25 30 35 40 45 50 55 60 65

Cyclesrsquo width

07 075

08 085

09alpha 0005

001 0015

002 0025

003 0035

004 0045

005

alphaS

20 25 30 35 40 45 50 55 60 65

Figure 5 Maximum width recorded between highest and lowest peaks fortotal production for different values of α and αS

Secondly we consider the parameters αS representing the speed of ad-justment of desired stock to observed demand The higher this parame-ter the faster sectors revise the desired levels of stocks to current levelsof observed demand This parameter concerns sectorsrsquo behavioral decisionprocess Higher αS represent a stronger belief that the most recently ob-served demand level is a reliable indicator of future demand levels whilelower levels of the parameter represent a more conservative approach wheremanagers require a long observation of demand levels different from the pastones before adjusting their expectations Also concerning the effect of thisparameter on the volatility induced by an exogenous shock we may maketwo hypothesis

Hypothesis 1 Faster adaptation of desired stocks to new levels of de-mand (higher αS) reduce volatility because changes in demand are rapidlytranslated in adjusted levels of production

Hypothesis 2 Faster adaptation of desired stocks to new levels of demand(higher αS) increase volatility because temporary unbalances of demand in

18

one sector are transmitted faster to other sector reinforcing the feedbackmechanism of fluctuations

To test which of these hypotheses are confirmed by the model we runsimulation runs with identical systems (eg same technical coefficients samelevels of demand same shocks) but for the values of the αrsquos and αS rsquos testingall combinations of these parameters over a range of values (each run usedthe same parameters for all the sectors of the system) For each simulation(ie each couple of values of the parameters) we computed two indicatorsof volatility One measures the maximum distance of total production be-tween the highest and the lowest peak that is the maximum width of theoscillations registered in the production time series The second indicatormeasures the relative deviation δ = σY microY Figures 5 and 6 report theseresults

0004 0006 0008 001 0012 0014 0016 0018 002

Relative deviation

07 075

08 085

09alpha 0005

001 0015

002 0025

003 0035

004 0045

005

alphaS

0004 0006 0008 001

0012 0014 0016 0018 002

Figure 6 Relative deviation of total production for different values of α andαS

The results confirms unequivocally both hypotheses 2 and reject hy-potheses 1 In a sense these hypotheses confirm the nature of the economicsystem represented in the model as a complex system where the interactionsamong sectors play a far more important role than the individual sector be-haviour In fact in both cases we may interpret the results that the strongerthe efforts to mitigate the effects of shocks (ie slow productionrsquos and quickexpectationsrsquo adjustment) the opposite result is actually obtained the fluc-tuations become even more accentuated It is also worth to note that thestronger impact is generated by the behavioural parameters rather than thetechnical one as indicated by the stronger effects of the αS in respect of theα This result seems to suggest that ceteris paribus information mattersmore than technical constraints Though the model is by any means in-

19

adequate to discuss normative aspects because of the naive ways strategicdecision making is represented it can anyway suggest the effects of errorsin management decisions (ie wrong desired levels of stocks) as opposed tothe technical constraints preventing sudden adjustment of production lev-els Stretching a bit the interpretation of these results we may suggests thatefforts to reduce volatility should better be oriented to improve the trans-mission of information in order to coordinate future production plans (ieidentifying quickly the desired levels of stocks and therefore the new equilib-rium production level) rather than improving the flexibility of productionmethods (reducing α)

56 Volatility and stock levels

In our model stock levels are maintained to buffer a sectorrsquos sale againstunexpected changes in demand Therefore a system made of risk adversemanagers (keeping higher levels of stocks) may be thought to lead to asmoother absorption of shocks in respect of a system composed of firmskeeping a short supply to cushion unexpected events To test this hypothesiswe tested the behaviour of the model for different values of σ the levels ofstock represented as a multiple of demand For each level of σ we run asimulation using all other initialization identical as usual representing aneconomic system ldquoalmostrdquo at equilibrium but for a single small shock indemand uncorrelated across sectors18

00044

00046

00048

0005

00052

00054

00056

00058

0006

00062

5 55 6 65 7 75 8 85 9 95 10

Rel

ativ

e va

riatio

n of

tota

l out

put (

delta

)

Ratio of stocked goods over production (sigma)

Stocks and production fluctuations

Figure 7 Fluctuations persistence on systems with different stock strategies(σ)

18Demand changes by plusmn1-10

20

Figure 7 shows the value of the relative deviation of output production(δ) generated by heterogeneous uncorrelated shocks in the external demandon identical systems where firms store a different multiple of their demand asstocks We obtain actually the opposite result that we may have expectedfluctuation become larger and more persistent the higher the level of stocksIn some way a strategy that for the individual firm is meant to reducethe fluctuations (using stocks as buffers) ends up by actually increasing thevolatility at aggregate levels

The result is though explainable both at the theoretical and empiricallevel In fact if no stocks were maintained production would need to adjustfor changes in demand only in case of a shock That is the firm may movemonotonically from the old to the new production levels sending consistentsignals to the rest of the system (ie its own suppliers) Instead if thesame shock affects a firm maintaining large stocks of output over the sameperiod production needs to adjust to the new demand and and to alignstocks with the new demand This generates besides the adjustment to thedemand a potentially inconsistent signal to the suppliers since the firm willhave a temporary level of production not meant to continue in equilibrium(adjusting stocks)

From the empirical viewpoint the last few decades have seen the almostuniversal diffusion of production system meant to keep stocks to a minimumif at all Lean production methods justndashinndashtime etc are solutions tothe problems of the costs necessary to maintain stocks Over the sameperiod economic fluctuations have been strongly reduced and few empiricalassessments have pointed to the relations between the two phenomena Forexample concerning the US ldquochanges in inventory behavior have playeda direct role in reducing real output volatilityrdquo (Kahn McConnell andPerez-Quiros 2002 p 183)19

57 Irrelevance of technical coefficients distribution

Horvath (1998) and Horvath (2000) suggest that systems with sparse ma-trices ie inputndashoutput matrices with many empty cells tend to be morevolatile in respect of systems with more evenly distributed matrices Noticethat this claim risk if not adequately qualified to generate an illogical resultIn fact matrices sparseness increases as a function of sectoral disaggregationand comparing the same systemrsquos fluctuations using inputndashoutput matricesat different levels of aggregation needs to produce the same type of volatil-ity Indeed this is not the case if all remaining parameters stand equaland do not adjust to the different level of analysis considered If aggregatepatterns of adjustment speed are maintained unchanged the higher the sec-toral disaggregation the higher the systemrsquos volatility ceteris paribus But

19Similar results are found in Blanchard and Simon (2001)

21

for example in our model reducing the aggregation also requires a changein the adaptation coefficients the more micro we undergo the analysis thehigher is the speed of adjustment of economic objects If for example weconsider sectors we are evaluating the adaptation of an entire set of firmswhich pose questions on how to aggregate their dynamics Things changewhen we consider the response of a single firm

Going back to Horvatsrsquo argument highly disaggregated matrices arelikely to contain many empty cells so that we may forecast different levelsof volatility depending on the aggregation used for the same system whichas mentioned is not caused by sparseness but to a failed adjustment inreaction mechanisms Results from our model20 show that volatility is com-pletely unaffected by input coefficients distribution Given their sectoralsum mdash through rows of the matrix mdash the value of each coefficient is irrele-vant The same occurs in the case of very skewed distributions that presenta high number of empty cells (zero input coefficients)

58 Microndash and Macrondashvolatility

As we have seen above one of the major discussion in the literature concernsthe relations between microndash and macrondashvolatility Put it simply one sideof the literature considers that uncorrelated microndashshocks cancel each otherout reducing their effects at macrondashlevel Conversely other researchersconsider that microndashshocks are multiplied at macrondashlevel generating highermacrondashvolatility than that we can observe at microndashlevel The debate isof obvious importance to determine the sources of business cycles to pre-dict their effects and determine the more effective policies to mitigate theirnegative effects

Our model is only a partial representation of realndashworld economic sys-tem and is also overly simplistic However based on these limitations italso offers very robust results allowing reliable conclusions on the (limited)elements included in the model In particular we consider our model a re-liable representation of the basic structures of the production interactionsamong sectors or firms Thus we can use the model to provide an answerto the debate concerning the contributions to volatility from the productioninteractions only Other contributions (eg from financial markets priceadjustments technological innovation etc) may re-inforce or counter theforces we analyse but they need to be considered as additional elements(and we opine also generally less relevant) not alternative to our analysisWe consider a necessity in order to apply a rigorous methodological ap-proach to be able to identify with certainty the results provided by partialstudies in order to eventually obtain by gradual extensions of the analy-sis more and more detailed representations of realistic economic systems

20Available from the authors

22

The results we obtain in this paper concerning this microndashmacro debate aremeant as a first relevant step to a more extended analysis

0

002

004

006

008

01

012

014

016

018

02

022

05 1 15 2 25 3

Var Demand

Macro (Output) and micro (Demand) volatility

delta Outputdelta Demand

Figure 8 Values of δ (relative deviation) for aggregate demand and ag-gregate output in respect of different levels of variance for sector specificuncorrelated external demand shocks

The flows of shocks affecting the firms of a system can be distinguishedalong two dimensions strength of the shock (eg amplitude of the changesin demand) and frequency of the shock (rare or frequent) Concerning thefirst aspect we initialized as usual a system to equilibrium levels (eg pro-duction quantities equal to external and intermediate demand for all sectorsconstant level of stocks equal to the desired level) We then ran the simu-lation generating the external demand for all sectors as a random variablewith constant (equilibrium) mean level and varying values of variance

Figure 8 shows unequivocally that as far as our model has propertiessimilar to real systems the hypothesis that macrondashfluctuations have largervolatility than their sectoral level shocks is fully confirmed At any levelof variance the aggregate volatility of demand as captured by the relativedeviation index is sensibly lower than the volatility shown by aggregateoutput that is the total production of the system This result is not sur-prising throughout all tests presented in the previous sections every resultsuggested that the system lsquooverndashreactsrsquo to any individual microndashlevel changeeven in the case these changes were meant to contrast disturbances There-fore when we move from considering one single shock to a flow of shockswe are not surprised to obtain that the system shows higher volatility thanthat implied by the microndashlevel disturbances

We now consider the second dimension defining a flow of shocks theirfrequency The question is whether for the same level of demand shocks

23

0 002 004 006 008 01 012 014 016

Difference of relative deviation for Output and Demand

0 5 10 15 20 25 30 35 40 45 50 55

Shock frequency 05

1

15

2

25

3

Var Demand

0 002 004 006 008 01

012 014 016

Figure 9 Differences of values of δ for aggregate output minus those foraggregate demand in respect of different levels of variance of demand and offrequency of the shocks

(ie same variance) more frequent changes trigger a higher or lower systemvolatility than rare disturbances The results shown in figure 9 suggest arather elaborated answer which partly vindicates (though in a particularsense) the arguments in favour of compensation of micro shocks The fig-ure shows the difference between aggregate output volatility and aggregatedemand volatility The independent variables are as before the volatilityof demand at microndashlevel21 and the time intervals in between two shocksAggregate demand volatility does not change for the frequency of shocksat least when computed over a long period But output volatility doesin a nonndashmonotonous way For high frequency of shocks the difference isstill positive and increasing for higher variance indicating that the ldquooverndashreactionrdquo argument is valid even in these cases but it is rather low Whilethe frequency of shocks slows down aggregate volatility of output increasesshowing that time is needed for the microndashshock to unfold their full effectsat the macrondashlevel However after a certain threshold less and less frequentshocks generate a lower volatility of the aggregate level of output

This result is quite sensible though not obvious at first sight and onceagain shows the power of such a simple model to generate and explain com-plex properties For frenetic levels of demand modifications the system hasno time to adjust but part of the job to align actual and desired productionfor constantly changing demand is performed by the varying demand itselfIn other terms the shocks cancel out through time if not through their quan-

21Beware of the scale of variance For reasons of visibility the scale is inverted

24

titative effects on the system While frequency decreases the disturbancesof the shocks not compensated by other (unlikely) shocks have the time tofilter through the production system generating higher and higher volatil-ity In a sense any system has a particular frequency of shocks that make itmaximally ldquoresonatingrdquo with the amplification of microndashshocks When therate of shocks gets even rarer the system can enjoy periods of longer andlonger quietness after each shock is absorbed and before a new one startsagain a cycle of initial local disturbance propagation of the disturbances toother sectors and settlement to the new equilibrium

6 Discussion and extensions

We presented a simple model of production composed by interacting sectorThe main (quit realistic) assumption of the model is that the productionprocesses in any sector of the economy make use of input from any othersector Moreover we assumed that sectors (represented as a single firm)adjust slowly both the desired level of production (smoothed revision of ex-pectation) and the actual level (smoothed modification of plant utilization)The model voluntarily leaves aside a large number of aspects of real-worldeconomic systems (eg financial markets price adjustments growth etc)The reason is that the model can be easily and reliably tested in order toextract its properties The possibility to transfer the (certain) model resultto a real context depend of course on an ample ceteris paribus clause How-ever we are confident that the reported results concerning the productionstructure play a relevant role in real world systems Therefore our resultsand particularly their motivations being accessible because of the relativesimplicity of the model will not be diminished by the integration of ourconsiderations with other aspects of economic systems

Our model is in essence a representation of an input-output table inter-preted in physical terms that is as the amount of inputs required for oneunit of output Besides this we include two straightforward assumptionsFirstly production levels cannot be suddenly changed but require time toscale up or down the utilization rate that is the levels of production Stocksare meant to make up for the difference between sales and production Thesecond assumption concerns the intentional behaviour of producers We im-plement this aspect by assuming that exists a level of stocks desired for eachlevel of demand However a change of demand does not translate suddenlyin a new level of desired stock but again only a long time at a different levelof demand will convince managers to update the desired levels of stocks

We show that the model generate under a wide number of settings oscil-latory patterns converging toward the equilibrium level where each sectorgenerate exactly the quantity demanded by final consumers and by othersectors as input The oscillations are provoked by over- and under-shooting

25

generated by the difficulty of coordination among sectors that by assump-tion ldquocommunicaterdquo only via variation of demand to their direct input sup-pliers We also show that the total production levels generated by sectors(sum of the physical units) produced at equilibrium depends (for constantfinal demand) on the total sum of the technical coefficients per sector irre-spective of their distribution This result is motivated by the impossibilityof input substitution and offers interesting suggestions for the analysis ofthe study of input-output tables of real-world systems

We then consider systems at equilibrium facing a single small change ofthe external demand for sectors These simulations allow to study the waythe system reacts to shocks We see that the sum of the coefficients affectsbesides the levels also the volatility of aggregate output the larger a systemthe stronger is the impact of a given shock Less obviously we find thatfor systems with the same dimension (ie same sums of coefficients) theaggregate volatility increases with both the inertia to change of productionlevel and the speed of adjustment of desired stocks This result is somewhatcounter-intuitive since both the cited variables are associated to a moreconservative behaviour that may be supposed should smooth away spikesin the pattern from an old to a new equilibrium However we show thatthe opposite actually holds given the complex interactions among sectorindividual-sector attitude intended to smooth shocks actually increase theiraggregate impact

In the same vein we show that systems maintaining high levels of stocksare more volatile that systems with reduced amounts of stocks Again thisis a counter-intuitive result reverting the goal of individual producers andthe aggregate result Interestingly this result finds strong support from theevidence of the recent changes in production methods (ie low or no stocks)and diminished cycles

In relation to specific issues discussed in the literature we find no supportto the suggestion that particular distributions of input-output coefficientsmodify the volatility of a system As said this depends in case on thesums of these coefficients but not on their distribution Further we canprovide an answer to a long and hotly debated questions whether microndashlevels volatility generates stronger or lower aggregate volatility Our modelshows unequivocally that microndashvolatility (eg variance of final demand)persists and generates stronger volatility at aggregate level Again we canidentify the reason on the complex production structure that exalts micro-level shocks by generating lsquowrongrsquo messages among producers Moreoverwe can also point to a usually neglected aspect of volatility we show thatthe frequency of shocks is a very relevant aspect besides their dimensionin determining the volatility of aggregate output

Our work can be developed along two complementary directions Firstlywe can continue to study the properties of the model by extending the ele-ments considered (eg pricing) analysing the effects on the present results

26

and generating more analytical evidence Of particular relevance is in ouropinion the problem of aggregation Representations of the same systemswith different codification systems (eg different number of SIC digits)must not modify overall system properties for example the volatility ob-served Given that our result show that the sums of coefficients matter forthe levels and volatility of the system and that these sum do depend on theaggregation chosen it should be possible to induce the reaction coefficientsthat we know affect the volatility only

Secondly we can find an application of our results to real world evidenceIn fact there are many data sets available for the data we deal with basicallyinput-output tables Our model is readily adapt to be parametrized alongany number of sector and coefficient necessary so that we may replicate pastseries in order to adapt the unobservable parameters and use the resultingcomplete model to explain business fluctuations and provide intuitions onfuture patterns

27

References

Acemoglu D and A Scott (1997) ldquoAsymmetric Business CyclesTheory and TimendashSeries Evidencerdquo Journal of Monetary Economics 40501ndash533

Backus D P Kehoe and F Kydland (1995) ldquoInternational BusinessCycles Theory and Evidencerdquo in Frontiers of Business Cycle Researched by T Cooley Princeton University Press Princeton

Bak P K Chen J Scheinkman and M Woodford (1993) ldquoAg-gregate Fluctuations from Independent Sectoral Shocks SelfndashOrganizedCritically in a Model of Production and Inventory Dynamicsrdquo RicercheEconomiche 47(1) 3ndash30

Battiston S D Delli Gatti M Gallegati B Greenwald and

J Stiglitz (2005) ldquoCredit Chains and Bankruptcies Avalanches in Sup-ply Networksrdquo in Annual Workshop on Economic Heterogeneous Inter-acting Agents (WEHIA 2005) University of Essex UK

Blanchard O and J Simon (2001) ldquoThe Long and Large Declinein US Output Volatilityrdquo Brookings Papers on Economic Activity 1135ndash174

Bresnahan T F and V A Ramey (1994) ldquoOutput Fluctuations atthe Plant Levelrdquo The Quarterly Journal of Economics 109(3) 593ndash924

David P A (1975) Technical choice innovation and economic growth essays on American and British experience in the nineteenth centuryCambridge University Press London

Delli Gatti D C Di Guilmi E Gaffeo G Giulioni M Galle-

gati and A Palestrini (2004) ldquoA New Approach to Business Fluc-tuations Heterogeneous Interacting Agents Scaling Laws and FinancialFragilityrdquo Journal of Economic Behavior amp Organization 56 489ndash512

Dosi G G Fagiolo and A Roventini (2006) ldquoAn EvolutionaryModel of Endogenous Business Cyclesrdquo Computational Economics 273ndash34

Dupor B (1999) ldquoAggregation and the Irrelevance in MultindashSector Mod-elsrdquo Journal of Monetary Economics 43 391ndash409

Head A C (1995) ldquoCountry Size Aggregate Fluctuations and Interna-tional Risk Sharingrdquo The Canadian Journal of Economics 28(4b) 1096ndash1119

28

Helbing D and S Lammer (2005) ldquoSupply and Production NetworksFrom the Bullwhip Effect to Business Cyclesrdquo in Networks of Interactingmachines Production Organization in Complex Industrial Systems andBiological Cells ed by D Armbruster K Kaneko and A S Mikhailovvol 3 of World Scientific Lecture Notes in Complex Systems World Sci-entific

Helbing D S Lammer U Witt and T Brenner (2004) ldquoNetworkndashInduced Oscillatory Behavior in Material Flow Networks and IrregularBusiness Cyclesrdquo Phisical Review E 70(056118) 1ndash6

Helbing D U Witt S Lammer and T Brenner (2004) ldquoNetwork-Induced Oscillatory Behaviour of Macroeconomic Output Flowsrdquo in In-dustry and Labour Dynamics II The Agent-based Computational Eco-nomics Approach Proceedings of the wildace 2004 conference ed byR Leombruni and M Richiardi no 5 LABORatorio R Revelli

Horvath M (1998) ldquoCyclically and Sectoral Linkages Aggregate Fluctu-ations from Independent Sectoral Shocksrdquo Review of Economic Dynam-ics 1 781ndash808

(2000) ldquoSectoral Shocks and Aggregate Fluctuationsrdquo Journal ofMonetary Economics 45(1) 69ndash106

Jovanovic B (1987) ldquoMicro Shocks and Aggregate Riskrdquo The QuarterlyJournal of Economics 102(2) 395ndash410

Kahn J A M M McConnell and G Perez-Quiros (2002) ldquoOnthe Causes of the Increased Stability of the US Economyrdquo FRBNY Eco-nomic Policy Review 8(1) 183ndash202

Kraay A and J Ventura (1998) ldquoComparative Advantage and theCrossndashSection of Business Cyclesrdquo World Bank Woorking paper Series1948 World Bank

Long J B and C I Plosser (1983) ldquoReal Business Cyclesrdquo Journalof Political Economy 31(1) 39ndash69

Lucas R E (1977) ldquoUnderstanding Business Cyclesrdquo CarnegiendashRochester Conference Series on Public Policy (special issue of Journalof Monetary Economics) 5 7ndash29

Nelson R R and S G Winter (1982) An Evolutionary Theory ofEconomic Change Harvard University Press Cambridge MA

Nirei M (2005) ldquoThreshold Behavior and Aggregate Fluctuationsrdquo Jour-nal of Economic Theory in press

29

Scheinkman J and M Woodford (1994) ldquoSelfndashOrganized Criticallyand Economic Fluctuationsrdquo American Economic Review 84(2 Papersand Proceedings of teh AEA) 417ndash421

Schuh S (1996) ldquoEvidence of the Link Between FirmndashLevel and Ag-gregate Inventory Behaviorrdquo Finance and Economics Discussion Series1996-46 Federal Reserve Board

Stiglitz J and B Greenwald (2003) Towards a New Paradigm inMonetary Economics Raffaele Mattioli Lectures Cambridge UniversityPress Cambridge

Weisbuch G and S Battiston (2005) ldquoProduction Networks andFailure Avalanchesrdquo Working Paper mimeo Ecole Normale SupErieure

Zarnovitz V (1977) ldquoBusiness Cycles Observed and Assessed Why andHow they Matterrdquo NBER Working Paper Series 6230 National Bureauof Economic Research

30

A Tables

Table 1 Parameters valuesParametera Description Value

N Number of sectorsunits i of production 10Ci Number of j buyer sectors for sector i (IO matrix sparseness) N minus 1Di (E) Value of the external demand 10mij Physical input coefficient of input j for sector i sim U(0 1)Mi Sum of the mij input coefficients j for the production of i 05αi friction to changes of production levels Degree of lockndashin on

production technology085

αsi Speed of adaptation to demand changes 004

σ Desired multiple of production to stock 10

aInto parenthesis the number of lags of the initial value for lagged variables

31

start with next section (53) shows the aggregate cyclical behaviour of astandard setting

53 Single shock analysis

We generate a simulated history following the dynamics of the modelrsquos vari-ables when the system is set to an equilibrium level and a single shockmodifies one of the exogenous elements namely the external demand forsectors Modifications across sectors are random and not correlated andmay increase or reduce external demand by at most 10

100 225 350 475 600

193894

195861

197828

199795

201762 Industrial Production

Time

Figure 2 Total industrial production fluctuations following a shock fromexternal demand

Figure 2 shows the dynamics generated by the total industrial productionof the system mdash that is also a proxy for the systemrsquos GDP fluctuations Thenew external demand changes the equilibrium level but the system cannotreach the new equilibrium with a monotonous pattern and converges to thenew level with a fluctuating pattern Each change in the external demandinduces also modifications of the intermediate demand and sectoral adjust-ments in the quantity produced typically overshoot the aggregate patternAs it can be appreciated from the figure it takes time before the system isable to converge to the new static equilibrium conditions

We study two features of this pattern absolute level of the new equilib-rium and volatility induced by the shock

54 Technical coefficients and industrial output

Production level

The equilibrium levels of industrial output depend on two factors only Ob-viously the levels of external demand determine how much of the total pro-

15

duction needs to be supplied to consumers exogenous from the productionsystem Second the technical coefficients also influence strongly the level ofactivity of the system defining the intermediate demand to each sector

180

200

220

240

260

280

300

01 02 03 04 05 06 07 08 09

Out

put l

evel

(In

dust

rial p

rodu

ctio

n)

Sum of total input coefficients (M)

Total industrial production for different input coefficients levels

Figure 3 The relevance of input coefficients on output level

Figure 3 shows the equilibrium level of industrial output reached bysystems with identical external demand where we fixed the sum of inputcoefficients for each sector of the economy (Mi) to different values16 Thehigher the sum the higher is the total production sustained by the system

This result depends on the very production structure represented by theinputndashoutput matrix In fact the higher is the level of production requiredby intermediate sectors the higher is the total production given a constantlevel of external demand

Notice that the level of production does not depend on the distribution ofthe technical coefficients As long as the sum of the coefficients is identicalthe same production level is obtained irrespective of their distribution (otherthings being equal)

Output shock absorption

Input coefficients have a relevant role also on the output stability of thesystem Starting from an output equilibrium level we hit the system witha single shock on the external demand for all sectors uncorrelated amongsectors17 modifing (positively or negatively) the demand for all sectors by

16Within this limit the actual value of the coefficients are randomly drawn17In each sector the external demand changes by plusmnX where X is a uniformly dis-

tributed percentage between 1 and 10

16

at most10 As show in Figure 2 the economic systems produces a lenghtlysmoothing oscillatory behaviour We compute the relative deviation of theaggregate production (industrial output) as the ratio between its standarddeviation and its mean δ = σY microY where Y =

sum

Qi Figure 4 shows thebehaviour of δ as a function of the sum of input coefficients for each sector80 periods after the shock As time goes by the level of δ reduces for allMi but the relation with the sum of input coefficient is unchanged

00055

0006

00065

0007

00075

0008

00085

0009

01 02 03 04 05 06 07 08 09

Rel

ativ

e va

rianc

e of

tota

l out

put (

delta

)

Sum of total input coefficients

Aggregate output deviations for different technical coefficients

Figure 4 The relevance of input coefficients on output stability

In sum input coefficients determine the extent of overshooting behaviourThe higher the coefficients the higher is the backlash even when shockscould cancel out through sectors

55 Shock persistence and adjustment coefficients

We now consider the effect of micro smoothing behaviours on the volatilityof the production system In particular we are interested in testing theeffects of two parameters of the model on the volatility of the productionsystem First we consider the effects of the αrsquos representing the ldquostickinessrdquoof the production technology to desired changes of Q aimed at filling thegap between actual and desired stocks The higher is α the slower is thereaction to stock unbalances Concerning its effect referring to the technicalpossibilities available ie production capital and organization we may putforward two opposite hypothesis relevant to systemrsquos volatility

Hypothesis 1 Slower production adjustment (high α) reduces volatilitybecause unbalances in demand of one sector being compensated slowly slow

17

down the diffusion of shocks to other sectors

Hypothesis 2 Slower production adjustment (high α) increase volatil-ity because the longer lasts the mismatch between actual and equilibriumproduction level the deeper will become the mismatch between actual anddesired stocks

20 25 30 35 40 45 50 55 60 65

Cyclesrsquo width

07 075

08 085

09alpha 0005

001 0015

002 0025

003 0035

004 0045

005

alphaS

20 25 30 35 40 45 50 55 60 65

Figure 5 Maximum width recorded between highest and lowest peaks fortotal production for different values of α and αS

Secondly we consider the parameters αS representing the speed of ad-justment of desired stock to observed demand The higher this parame-ter the faster sectors revise the desired levels of stocks to current levelsof observed demand This parameter concerns sectorsrsquo behavioral decisionprocess Higher αS represent a stronger belief that the most recently ob-served demand level is a reliable indicator of future demand levels whilelower levels of the parameter represent a more conservative approach wheremanagers require a long observation of demand levels different from the pastones before adjusting their expectations Also concerning the effect of thisparameter on the volatility induced by an exogenous shock we may maketwo hypothesis

Hypothesis 1 Faster adaptation of desired stocks to new levels of de-mand (higher αS) reduce volatility because changes in demand are rapidlytranslated in adjusted levels of production

Hypothesis 2 Faster adaptation of desired stocks to new levels of demand(higher αS) increase volatility because temporary unbalances of demand in

18

one sector are transmitted faster to other sector reinforcing the feedbackmechanism of fluctuations

To test which of these hypotheses are confirmed by the model we runsimulation runs with identical systems (eg same technical coefficients samelevels of demand same shocks) but for the values of the αrsquos and αS rsquos testingall combinations of these parameters over a range of values (each run usedthe same parameters for all the sectors of the system) For each simulation(ie each couple of values of the parameters) we computed two indicatorsof volatility One measures the maximum distance of total production be-tween the highest and the lowest peak that is the maximum width of theoscillations registered in the production time series The second indicatormeasures the relative deviation δ = σY microY Figures 5 and 6 report theseresults

0004 0006 0008 001 0012 0014 0016 0018 002

Relative deviation

07 075

08 085

09alpha 0005

001 0015

002 0025

003 0035

004 0045

005

alphaS

0004 0006 0008 001

0012 0014 0016 0018 002

Figure 6 Relative deviation of total production for different values of α andαS

The results confirms unequivocally both hypotheses 2 and reject hy-potheses 1 In a sense these hypotheses confirm the nature of the economicsystem represented in the model as a complex system where the interactionsamong sectors play a far more important role than the individual sector be-haviour In fact in both cases we may interpret the results that the strongerthe efforts to mitigate the effects of shocks (ie slow productionrsquos and quickexpectationsrsquo adjustment) the opposite result is actually obtained the fluc-tuations become even more accentuated It is also worth to note that thestronger impact is generated by the behavioural parameters rather than thetechnical one as indicated by the stronger effects of the αS in respect of theα This result seems to suggest that ceteris paribus information mattersmore than technical constraints Though the model is by any means in-

19

adequate to discuss normative aspects because of the naive ways strategicdecision making is represented it can anyway suggest the effects of errorsin management decisions (ie wrong desired levels of stocks) as opposed tothe technical constraints preventing sudden adjustment of production lev-els Stretching a bit the interpretation of these results we may suggests thatefforts to reduce volatility should better be oriented to improve the trans-mission of information in order to coordinate future production plans (ieidentifying quickly the desired levels of stocks and therefore the new equilib-rium production level) rather than improving the flexibility of productionmethods (reducing α)

56 Volatility and stock levels

In our model stock levels are maintained to buffer a sectorrsquos sale againstunexpected changes in demand Therefore a system made of risk adversemanagers (keeping higher levels of stocks) may be thought to lead to asmoother absorption of shocks in respect of a system composed of firmskeeping a short supply to cushion unexpected events To test this hypothesiswe tested the behaviour of the model for different values of σ the levels ofstock represented as a multiple of demand For each level of σ we run asimulation using all other initialization identical as usual representing aneconomic system ldquoalmostrdquo at equilibrium but for a single small shock indemand uncorrelated across sectors18

00044

00046

00048

0005

00052

00054

00056

00058

0006

00062

5 55 6 65 7 75 8 85 9 95 10

Rel

ativ

e va

riatio

n of

tota

l out

put (

delta

)

Ratio of stocked goods over production (sigma)

Stocks and production fluctuations

Figure 7 Fluctuations persistence on systems with different stock strategies(σ)

18Demand changes by plusmn1-10

20

Figure 7 shows the value of the relative deviation of output production(δ) generated by heterogeneous uncorrelated shocks in the external demandon identical systems where firms store a different multiple of their demand asstocks We obtain actually the opposite result that we may have expectedfluctuation become larger and more persistent the higher the level of stocksIn some way a strategy that for the individual firm is meant to reducethe fluctuations (using stocks as buffers) ends up by actually increasing thevolatility at aggregate levels

The result is though explainable both at the theoretical and empiricallevel In fact if no stocks were maintained production would need to adjustfor changes in demand only in case of a shock That is the firm may movemonotonically from the old to the new production levels sending consistentsignals to the rest of the system (ie its own suppliers) Instead if thesame shock affects a firm maintaining large stocks of output over the sameperiod production needs to adjust to the new demand and and to alignstocks with the new demand This generates besides the adjustment to thedemand a potentially inconsistent signal to the suppliers since the firm willhave a temporary level of production not meant to continue in equilibrium(adjusting stocks)

From the empirical viewpoint the last few decades have seen the almostuniversal diffusion of production system meant to keep stocks to a minimumif at all Lean production methods justndashinndashtime etc are solutions tothe problems of the costs necessary to maintain stocks Over the sameperiod economic fluctuations have been strongly reduced and few empiricalassessments have pointed to the relations between the two phenomena Forexample concerning the US ldquochanges in inventory behavior have playeda direct role in reducing real output volatilityrdquo (Kahn McConnell andPerez-Quiros 2002 p 183)19

57 Irrelevance of technical coefficients distribution

Horvath (1998) and Horvath (2000) suggest that systems with sparse ma-trices ie inputndashoutput matrices with many empty cells tend to be morevolatile in respect of systems with more evenly distributed matrices Noticethat this claim risk if not adequately qualified to generate an illogical resultIn fact matrices sparseness increases as a function of sectoral disaggregationand comparing the same systemrsquos fluctuations using inputndashoutput matricesat different levels of aggregation needs to produce the same type of volatil-ity Indeed this is not the case if all remaining parameters stand equaland do not adjust to the different level of analysis considered If aggregatepatterns of adjustment speed are maintained unchanged the higher the sec-toral disaggregation the higher the systemrsquos volatility ceteris paribus But

19Similar results are found in Blanchard and Simon (2001)

21

for example in our model reducing the aggregation also requires a changein the adaptation coefficients the more micro we undergo the analysis thehigher is the speed of adjustment of economic objects If for example weconsider sectors we are evaluating the adaptation of an entire set of firmswhich pose questions on how to aggregate their dynamics Things changewhen we consider the response of a single firm

Going back to Horvatsrsquo argument highly disaggregated matrices arelikely to contain many empty cells so that we may forecast different levelsof volatility depending on the aggregation used for the same system whichas mentioned is not caused by sparseness but to a failed adjustment inreaction mechanisms Results from our model20 show that volatility is com-pletely unaffected by input coefficients distribution Given their sectoralsum mdash through rows of the matrix mdash the value of each coefficient is irrele-vant The same occurs in the case of very skewed distributions that presenta high number of empty cells (zero input coefficients)

58 Microndash and Macrondashvolatility

As we have seen above one of the major discussion in the literature concernsthe relations between microndash and macrondashvolatility Put it simply one sideof the literature considers that uncorrelated microndashshocks cancel each otherout reducing their effects at macrondashlevel Conversely other researchersconsider that microndashshocks are multiplied at macrondashlevel generating highermacrondashvolatility than that we can observe at microndashlevel The debate isof obvious importance to determine the sources of business cycles to pre-dict their effects and determine the more effective policies to mitigate theirnegative effects

Our model is only a partial representation of realndashworld economic sys-tem and is also overly simplistic However based on these limitations italso offers very robust results allowing reliable conclusions on the (limited)elements included in the model In particular we consider our model a re-liable representation of the basic structures of the production interactionsamong sectors or firms Thus we can use the model to provide an answerto the debate concerning the contributions to volatility from the productioninteractions only Other contributions (eg from financial markets priceadjustments technological innovation etc) may re-inforce or counter theforces we analyse but they need to be considered as additional elements(and we opine also generally less relevant) not alternative to our analysisWe consider a necessity in order to apply a rigorous methodological ap-proach to be able to identify with certainty the results provided by partialstudies in order to eventually obtain by gradual extensions of the analy-sis more and more detailed representations of realistic economic systems

20Available from the authors

22

The results we obtain in this paper concerning this microndashmacro debate aremeant as a first relevant step to a more extended analysis

0

002

004

006

008

01

012

014

016

018

02

022

05 1 15 2 25 3

Var Demand

Macro (Output) and micro (Demand) volatility

delta Outputdelta Demand

Figure 8 Values of δ (relative deviation) for aggregate demand and ag-gregate output in respect of different levels of variance for sector specificuncorrelated external demand shocks

The flows of shocks affecting the firms of a system can be distinguishedalong two dimensions strength of the shock (eg amplitude of the changesin demand) and frequency of the shock (rare or frequent) Concerning thefirst aspect we initialized as usual a system to equilibrium levels (eg pro-duction quantities equal to external and intermediate demand for all sectorsconstant level of stocks equal to the desired level) We then ran the simu-lation generating the external demand for all sectors as a random variablewith constant (equilibrium) mean level and varying values of variance

Figure 8 shows unequivocally that as far as our model has propertiessimilar to real systems the hypothesis that macrondashfluctuations have largervolatility than their sectoral level shocks is fully confirmed At any levelof variance the aggregate volatility of demand as captured by the relativedeviation index is sensibly lower than the volatility shown by aggregateoutput that is the total production of the system This result is not sur-prising throughout all tests presented in the previous sections every resultsuggested that the system lsquooverndashreactsrsquo to any individual microndashlevel changeeven in the case these changes were meant to contrast disturbances There-fore when we move from considering one single shock to a flow of shockswe are not surprised to obtain that the system shows higher volatility thanthat implied by the microndashlevel disturbances

We now consider the second dimension defining a flow of shocks theirfrequency The question is whether for the same level of demand shocks

23

0 002 004 006 008 01 012 014 016

Difference of relative deviation for Output and Demand

0 5 10 15 20 25 30 35 40 45 50 55

Shock frequency 05

1

15

2

25

3

Var Demand

0 002 004 006 008 01

012 014 016

Figure 9 Differences of values of δ for aggregate output minus those foraggregate demand in respect of different levels of variance of demand and offrequency of the shocks

(ie same variance) more frequent changes trigger a higher or lower systemvolatility than rare disturbances The results shown in figure 9 suggest arather elaborated answer which partly vindicates (though in a particularsense) the arguments in favour of compensation of micro shocks The fig-ure shows the difference between aggregate output volatility and aggregatedemand volatility The independent variables are as before the volatilityof demand at microndashlevel21 and the time intervals in between two shocksAggregate demand volatility does not change for the frequency of shocksat least when computed over a long period But output volatility doesin a nonndashmonotonous way For high frequency of shocks the difference isstill positive and increasing for higher variance indicating that the ldquooverndashreactionrdquo argument is valid even in these cases but it is rather low Whilethe frequency of shocks slows down aggregate volatility of output increasesshowing that time is needed for the microndashshock to unfold their full effectsat the macrondashlevel However after a certain threshold less and less frequentshocks generate a lower volatility of the aggregate level of output

This result is quite sensible though not obvious at first sight and onceagain shows the power of such a simple model to generate and explain com-plex properties For frenetic levels of demand modifications the system hasno time to adjust but part of the job to align actual and desired productionfor constantly changing demand is performed by the varying demand itselfIn other terms the shocks cancel out through time if not through their quan-

21Beware of the scale of variance For reasons of visibility the scale is inverted

24

titative effects on the system While frequency decreases the disturbancesof the shocks not compensated by other (unlikely) shocks have the time tofilter through the production system generating higher and higher volatil-ity In a sense any system has a particular frequency of shocks that make itmaximally ldquoresonatingrdquo with the amplification of microndashshocks When therate of shocks gets even rarer the system can enjoy periods of longer andlonger quietness after each shock is absorbed and before a new one startsagain a cycle of initial local disturbance propagation of the disturbances toother sectors and settlement to the new equilibrium

6 Discussion and extensions

We presented a simple model of production composed by interacting sectorThe main (quit realistic) assumption of the model is that the productionprocesses in any sector of the economy make use of input from any othersector Moreover we assumed that sectors (represented as a single firm)adjust slowly both the desired level of production (smoothed revision of ex-pectation) and the actual level (smoothed modification of plant utilization)The model voluntarily leaves aside a large number of aspects of real-worldeconomic systems (eg financial markets price adjustments growth etc)The reason is that the model can be easily and reliably tested in order toextract its properties The possibility to transfer the (certain) model resultto a real context depend of course on an ample ceteris paribus clause How-ever we are confident that the reported results concerning the productionstructure play a relevant role in real world systems Therefore our resultsand particularly their motivations being accessible because of the relativesimplicity of the model will not be diminished by the integration of ourconsiderations with other aspects of economic systems

Our model is in essence a representation of an input-output table inter-preted in physical terms that is as the amount of inputs required for oneunit of output Besides this we include two straightforward assumptionsFirstly production levels cannot be suddenly changed but require time toscale up or down the utilization rate that is the levels of production Stocksare meant to make up for the difference between sales and production Thesecond assumption concerns the intentional behaviour of producers We im-plement this aspect by assuming that exists a level of stocks desired for eachlevel of demand However a change of demand does not translate suddenlyin a new level of desired stock but again only a long time at a different levelof demand will convince managers to update the desired levels of stocks

We show that the model generate under a wide number of settings oscil-latory patterns converging toward the equilibrium level where each sectorgenerate exactly the quantity demanded by final consumers and by othersectors as input The oscillations are provoked by over- and under-shooting

25

generated by the difficulty of coordination among sectors that by assump-tion ldquocommunicaterdquo only via variation of demand to their direct input sup-pliers We also show that the total production levels generated by sectors(sum of the physical units) produced at equilibrium depends (for constantfinal demand) on the total sum of the technical coefficients per sector irre-spective of their distribution This result is motivated by the impossibilityof input substitution and offers interesting suggestions for the analysis ofthe study of input-output tables of real-world systems

We then consider systems at equilibrium facing a single small change ofthe external demand for sectors These simulations allow to study the waythe system reacts to shocks We see that the sum of the coefficients affectsbesides the levels also the volatility of aggregate output the larger a systemthe stronger is the impact of a given shock Less obviously we find thatfor systems with the same dimension (ie same sums of coefficients) theaggregate volatility increases with both the inertia to change of productionlevel and the speed of adjustment of desired stocks This result is somewhatcounter-intuitive since both the cited variables are associated to a moreconservative behaviour that may be supposed should smooth away spikesin the pattern from an old to a new equilibrium However we show thatthe opposite actually holds given the complex interactions among sectorindividual-sector attitude intended to smooth shocks actually increase theiraggregate impact

In the same vein we show that systems maintaining high levels of stocksare more volatile that systems with reduced amounts of stocks Again thisis a counter-intuitive result reverting the goal of individual producers andthe aggregate result Interestingly this result finds strong support from theevidence of the recent changes in production methods (ie low or no stocks)and diminished cycles

In relation to specific issues discussed in the literature we find no supportto the suggestion that particular distributions of input-output coefficientsmodify the volatility of a system As said this depends in case on thesums of these coefficients but not on their distribution Further we canprovide an answer to a long and hotly debated questions whether microndashlevels volatility generates stronger or lower aggregate volatility Our modelshows unequivocally that microndashvolatility (eg variance of final demand)persists and generates stronger volatility at aggregate level Again we canidentify the reason on the complex production structure that exalts micro-level shocks by generating lsquowrongrsquo messages among producers Moreoverwe can also point to a usually neglected aspect of volatility we show thatthe frequency of shocks is a very relevant aspect besides their dimensionin determining the volatility of aggregate output

Our work can be developed along two complementary directions Firstlywe can continue to study the properties of the model by extending the ele-ments considered (eg pricing) analysing the effects on the present results

26

and generating more analytical evidence Of particular relevance is in ouropinion the problem of aggregation Representations of the same systemswith different codification systems (eg different number of SIC digits)must not modify overall system properties for example the volatility ob-served Given that our result show that the sums of coefficients matter forthe levels and volatility of the system and that these sum do depend on theaggregation chosen it should be possible to induce the reaction coefficientsthat we know affect the volatility only

Secondly we can find an application of our results to real world evidenceIn fact there are many data sets available for the data we deal with basicallyinput-output tables Our model is readily adapt to be parametrized alongany number of sector and coefficient necessary so that we may replicate pastseries in order to adapt the unobservable parameters and use the resultingcomplete model to explain business fluctuations and provide intuitions onfuture patterns

27

References

Acemoglu D and A Scott (1997) ldquoAsymmetric Business CyclesTheory and TimendashSeries Evidencerdquo Journal of Monetary Economics 40501ndash533

Backus D P Kehoe and F Kydland (1995) ldquoInternational BusinessCycles Theory and Evidencerdquo in Frontiers of Business Cycle Researched by T Cooley Princeton University Press Princeton

Bak P K Chen J Scheinkman and M Woodford (1993) ldquoAg-gregate Fluctuations from Independent Sectoral Shocks SelfndashOrganizedCritically in a Model of Production and Inventory Dynamicsrdquo RicercheEconomiche 47(1) 3ndash30

Battiston S D Delli Gatti M Gallegati B Greenwald and

J Stiglitz (2005) ldquoCredit Chains and Bankruptcies Avalanches in Sup-ply Networksrdquo in Annual Workshop on Economic Heterogeneous Inter-acting Agents (WEHIA 2005) University of Essex UK

Blanchard O and J Simon (2001) ldquoThe Long and Large Declinein US Output Volatilityrdquo Brookings Papers on Economic Activity 1135ndash174

Bresnahan T F and V A Ramey (1994) ldquoOutput Fluctuations atthe Plant Levelrdquo The Quarterly Journal of Economics 109(3) 593ndash924

David P A (1975) Technical choice innovation and economic growth essays on American and British experience in the nineteenth centuryCambridge University Press London

Delli Gatti D C Di Guilmi E Gaffeo G Giulioni M Galle-

gati and A Palestrini (2004) ldquoA New Approach to Business Fluc-tuations Heterogeneous Interacting Agents Scaling Laws and FinancialFragilityrdquo Journal of Economic Behavior amp Organization 56 489ndash512

Dosi G G Fagiolo and A Roventini (2006) ldquoAn EvolutionaryModel of Endogenous Business Cyclesrdquo Computational Economics 273ndash34

Dupor B (1999) ldquoAggregation and the Irrelevance in MultindashSector Mod-elsrdquo Journal of Monetary Economics 43 391ndash409

Head A C (1995) ldquoCountry Size Aggregate Fluctuations and Interna-tional Risk Sharingrdquo The Canadian Journal of Economics 28(4b) 1096ndash1119

28

Helbing D and S Lammer (2005) ldquoSupply and Production NetworksFrom the Bullwhip Effect to Business Cyclesrdquo in Networks of Interactingmachines Production Organization in Complex Industrial Systems andBiological Cells ed by D Armbruster K Kaneko and A S Mikhailovvol 3 of World Scientific Lecture Notes in Complex Systems World Sci-entific

Helbing D S Lammer U Witt and T Brenner (2004) ldquoNetworkndashInduced Oscillatory Behavior in Material Flow Networks and IrregularBusiness Cyclesrdquo Phisical Review E 70(056118) 1ndash6

Helbing D U Witt S Lammer and T Brenner (2004) ldquoNetwork-Induced Oscillatory Behaviour of Macroeconomic Output Flowsrdquo in In-dustry and Labour Dynamics II The Agent-based Computational Eco-nomics Approach Proceedings of the wildace 2004 conference ed byR Leombruni and M Richiardi no 5 LABORatorio R Revelli

Horvath M (1998) ldquoCyclically and Sectoral Linkages Aggregate Fluctu-ations from Independent Sectoral Shocksrdquo Review of Economic Dynam-ics 1 781ndash808

(2000) ldquoSectoral Shocks and Aggregate Fluctuationsrdquo Journal ofMonetary Economics 45(1) 69ndash106

Jovanovic B (1987) ldquoMicro Shocks and Aggregate Riskrdquo The QuarterlyJournal of Economics 102(2) 395ndash410

Kahn J A M M McConnell and G Perez-Quiros (2002) ldquoOnthe Causes of the Increased Stability of the US Economyrdquo FRBNY Eco-nomic Policy Review 8(1) 183ndash202

Kraay A and J Ventura (1998) ldquoComparative Advantage and theCrossndashSection of Business Cyclesrdquo World Bank Woorking paper Series1948 World Bank

Long J B and C I Plosser (1983) ldquoReal Business Cyclesrdquo Journalof Political Economy 31(1) 39ndash69

Lucas R E (1977) ldquoUnderstanding Business Cyclesrdquo CarnegiendashRochester Conference Series on Public Policy (special issue of Journalof Monetary Economics) 5 7ndash29

Nelson R R and S G Winter (1982) An Evolutionary Theory ofEconomic Change Harvard University Press Cambridge MA

Nirei M (2005) ldquoThreshold Behavior and Aggregate Fluctuationsrdquo Jour-nal of Economic Theory in press

29

Scheinkman J and M Woodford (1994) ldquoSelfndashOrganized Criticallyand Economic Fluctuationsrdquo American Economic Review 84(2 Papersand Proceedings of teh AEA) 417ndash421

Schuh S (1996) ldquoEvidence of the Link Between FirmndashLevel and Ag-gregate Inventory Behaviorrdquo Finance and Economics Discussion Series1996-46 Federal Reserve Board

Stiglitz J and B Greenwald (2003) Towards a New Paradigm inMonetary Economics Raffaele Mattioli Lectures Cambridge UniversityPress Cambridge

Weisbuch G and S Battiston (2005) ldquoProduction Networks andFailure Avalanchesrdquo Working Paper mimeo Ecole Normale SupErieure

Zarnovitz V (1977) ldquoBusiness Cycles Observed and Assessed Why andHow they Matterrdquo NBER Working Paper Series 6230 National Bureauof Economic Research

30

A Tables

Table 1 Parameters valuesParametera Description Value

N Number of sectorsunits i of production 10Ci Number of j buyer sectors for sector i (IO matrix sparseness) N minus 1Di (E) Value of the external demand 10mij Physical input coefficient of input j for sector i sim U(0 1)Mi Sum of the mij input coefficients j for the production of i 05αi friction to changes of production levels Degree of lockndashin on

production technology085

αsi Speed of adaptation to demand changes 004

σ Desired multiple of production to stock 10

aInto parenthesis the number of lags of the initial value for lagged variables

31

duction needs to be supplied to consumers exogenous from the productionsystem Second the technical coefficients also influence strongly the level ofactivity of the system defining the intermediate demand to each sector

180

200

220

240

260

280

300

01 02 03 04 05 06 07 08 09

Out

put l

evel

(In

dust

rial p

rodu

ctio

n)

Sum of total input coefficients (M)

Total industrial production for different input coefficients levels

Figure 3 The relevance of input coefficients on output level

Figure 3 shows the equilibrium level of industrial output reached bysystems with identical external demand where we fixed the sum of inputcoefficients for each sector of the economy (Mi) to different values16 Thehigher the sum the higher is the total production sustained by the system

This result depends on the very production structure represented by theinputndashoutput matrix In fact the higher is the level of production requiredby intermediate sectors the higher is the total production given a constantlevel of external demand

Notice that the level of production does not depend on the distribution ofthe technical coefficients As long as the sum of the coefficients is identicalthe same production level is obtained irrespective of their distribution (otherthings being equal)

Output shock absorption

Input coefficients have a relevant role also on the output stability of thesystem Starting from an output equilibrium level we hit the system witha single shock on the external demand for all sectors uncorrelated amongsectors17 modifing (positively or negatively) the demand for all sectors by

16Within this limit the actual value of the coefficients are randomly drawn17In each sector the external demand changes by plusmnX where X is a uniformly dis-

tributed percentage between 1 and 10

16

at most10 As show in Figure 2 the economic systems produces a lenghtlysmoothing oscillatory behaviour We compute the relative deviation of theaggregate production (industrial output) as the ratio between its standarddeviation and its mean δ = σY microY where Y =

sum

Qi Figure 4 shows thebehaviour of δ as a function of the sum of input coefficients for each sector80 periods after the shock As time goes by the level of δ reduces for allMi but the relation with the sum of input coefficient is unchanged

00055

0006

00065

0007

00075

0008

00085

0009

01 02 03 04 05 06 07 08 09

Rel

ativ

e va

rianc

e of

tota

l out

put (

delta

)

Sum of total input coefficients

Aggregate output deviations for different technical coefficients

Figure 4 The relevance of input coefficients on output stability

In sum input coefficients determine the extent of overshooting behaviourThe higher the coefficients the higher is the backlash even when shockscould cancel out through sectors

55 Shock persistence and adjustment coefficients

We now consider the effect of micro smoothing behaviours on the volatilityof the production system In particular we are interested in testing theeffects of two parameters of the model on the volatility of the productionsystem First we consider the effects of the αrsquos representing the ldquostickinessrdquoof the production technology to desired changes of Q aimed at filling thegap between actual and desired stocks The higher is α the slower is thereaction to stock unbalances Concerning its effect referring to the technicalpossibilities available ie production capital and organization we may putforward two opposite hypothesis relevant to systemrsquos volatility

Hypothesis 1 Slower production adjustment (high α) reduces volatilitybecause unbalances in demand of one sector being compensated slowly slow

17

down the diffusion of shocks to other sectors

Hypothesis 2 Slower production adjustment (high α) increase volatil-ity because the longer lasts the mismatch between actual and equilibriumproduction level the deeper will become the mismatch between actual anddesired stocks

20 25 30 35 40 45 50 55 60 65

Cyclesrsquo width

07 075

08 085

09alpha 0005

001 0015

002 0025

003 0035

004 0045

005

alphaS

20 25 30 35 40 45 50 55 60 65

Figure 5 Maximum width recorded between highest and lowest peaks fortotal production for different values of α and αS

Secondly we consider the parameters αS representing the speed of ad-justment of desired stock to observed demand The higher this parame-ter the faster sectors revise the desired levels of stocks to current levelsof observed demand This parameter concerns sectorsrsquo behavioral decisionprocess Higher αS represent a stronger belief that the most recently ob-served demand level is a reliable indicator of future demand levels whilelower levels of the parameter represent a more conservative approach wheremanagers require a long observation of demand levels different from the pastones before adjusting their expectations Also concerning the effect of thisparameter on the volatility induced by an exogenous shock we may maketwo hypothesis

Hypothesis 1 Faster adaptation of desired stocks to new levels of de-mand (higher αS) reduce volatility because changes in demand are rapidlytranslated in adjusted levels of production

Hypothesis 2 Faster adaptation of desired stocks to new levels of demand(higher αS) increase volatility because temporary unbalances of demand in

18

one sector are transmitted faster to other sector reinforcing the feedbackmechanism of fluctuations

To test which of these hypotheses are confirmed by the model we runsimulation runs with identical systems (eg same technical coefficients samelevels of demand same shocks) but for the values of the αrsquos and αS rsquos testingall combinations of these parameters over a range of values (each run usedthe same parameters for all the sectors of the system) For each simulation(ie each couple of values of the parameters) we computed two indicatorsof volatility One measures the maximum distance of total production be-tween the highest and the lowest peak that is the maximum width of theoscillations registered in the production time series The second indicatormeasures the relative deviation δ = σY microY Figures 5 and 6 report theseresults

0004 0006 0008 001 0012 0014 0016 0018 002

Relative deviation

07 075

08 085

09alpha 0005

001 0015

002 0025

003 0035

004 0045

005

alphaS

0004 0006 0008 001

0012 0014 0016 0018 002

Figure 6 Relative deviation of total production for different values of α andαS

The results confirms unequivocally both hypotheses 2 and reject hy-potheses 1 In a sense these hypotheses confirm the nature of the economicsystem represented in the model as a complex system where the interactionsamong sectors play a far more important role than the individual sector be-haviour In fact in both cases we may interpret the results that the strongerthe efforts to mitigate the effects of shocks (ie slow productionrsquos and quickexpectationsrsquo adjustment) the opposite result is actually obtained the fluc-tuations become even more accentuated It is also worth to note that thestronger impact is generated by the behavioural parameters rather than thetechnical one as indicated by the stronger effects of the αS in respect of theα This result seems to suggest that ceteris paribus information mattersmore than technical constraints Though the model is by any means in-

19

adequate to discuss normative aspects because of the naive ways strategicdecision making is represented it can anyway suggest the effects of errorsin management decisions (ie wrong desired levels of stocks) as opposed tothe technical constraints preventing sudden adjustment of production lev-els Stretching a bit the interpretation of these results we may suggests thatefforts to reduce volatility should better be oriented to improve the trans-mission of information in order to coordinate future production plans (ieidentifying quickly the desired levels of stocks and therefore the new equilib-rium production level) rather than improving the flexibility of productionmethods (reducing α)

56 Volatility and stock levels

In our model stock levels are maintained to buffer a sectorrsquos sale againstunexpected changes in demand Therefore a system made of risk adversemanagers (keeping higher levels of stocks) may be thought to lead to asmoother absorption of shocks in respect of a system composed of firmskeeping a short supply to cushion unexpected events To test this hypothesiswe tested the behaviour of the model for different values of σ the levels ofstock represented as a multiple of demand For each level of σ we run asimulation using all other initialization identical as usual representing aneconomic system ldquoalmostrdquo at equilibrium but for a single small shock indemand uncorrelated across sectors18

00044

00046

00048

0005

00052

00054

00056

00058

0006

00062

5 55 6 65 7 75 8 85 9 95 10

Rel

ativ

e va

riatio

n of

tota

l out

put (

delta

)

Ratio of stocked goods over production (sigma)

Stocks and production fluctuations

Figure 7 Fluctuations persistence on systems with different stock strategies(σ)

18Demand changes by plusmn1-10

20

Figure 7 shows the value of the relative deviation of output production(δ) generated by heterogeneous uncorrelated shocks in the external demandon identical systems where firms store a different multiple of their demand asstocks We obtain actually the opposite result that we may have expectedfluctuation become larger and more persistent the higher the level of stocksIn some way a strategy that for the individual firm is meant to reducethe fluctuations (using stocks as buffers) ends up by actually increasing thevolatility at aggregate levels

The result is though explainable both at the theoretical and empiricallevel In fact if no stocks were maintained production would need to adjustfor changes in demand only in case of a shock That is the firm may movemonotonically from the old to the new production levels sending consistentsignals to the rest of the system (ie its own suppliers) Instead if thesame shock affects a firm maintaining large stocks of output over the sameperiod production needs to adjust to the new demand and and to alignstocks with the new demand This generates besides the adjustment to thedemand a potentially inconsistent signal to the suppliers since the firm willhave a temporary level of production not meant to continue in equilibrium(adjusting stocks)

From the empirical viewpoint the last few decades have seen the almostuniversal diffusion of production system meant to keep stocks to a minimumif at all Lean production methods justndashinndashtime etc are solutions tothe problems of the costs necessary to maintain stocks Over the sameperiod economic fluctuations have been strongly reduced and few empiricalassessments have pointed to the relations between the two phenomena Forexample concerning the US ldquochanges in inventory behavior have playeda direct role in reducing real output volatilityrdquo (Kahn McConnell andPerez-Quiros 2002 p 183)19

57 Irrelevance of technical coefficients distribution

Horvath (1998) and Horvath (2000) suggest that systems with sparse ma-trices ie inputndashoutput matrices with many empty cells tend to be morevolatile in respect of systems with more evenly distributed matrices Noticethat this claim risk if not adequately qualified to generate an illogical resultIn fact matrices sparseness increases as a function of sectoral disaggregationand comparing the same systemrsquos fluctuations using inputndashoutput matricesat different levels of aggregation needs to produce the same type of volatil-ity Indeed this is not the case if all remaining parameters stand equaland do not adjust to the different level of analysis considered If aggregatepatterns of adjustment speed are maintained unchanged the higher the sec-toral disaggregation the higher the systemrsquos volatility ceteris paribus But

19Similar results are found in Blanchard and Simon (2001)

21

for example in our model reducing the aggregation also requires a changein the adaptation coefficients the more micro we undergo the analysis thehigher is the speed of adjustment of economic objects If for example weconsider sectors we are evaluating the adaptation of an entire set of firmswhich pose questions on how to aggregate their dynamics Things changewhen we consider the response of a single firm

Going back to Horvatsrsquo argument highly disaggregated matrices arelikely to contain many empty cells so that we may forecast different levelsof volatility depending on the aggregation used for the same system whichas mentioned is not caused by sparseness but to a failed adjustment inreaction mechanisms Results from our model20 show that volatility is com-pletely unaffected by input coefficients distribution Given their sectoralsum mdash through rows of the matrix mdash the value of each coefficient is irrele-vant The same occurs in the case of very skewed distributions that presenta high number of empty cells (zero input coefficients)

58 Microndash and Macrondashvolatility

As we have seen above one of the major discussion in the literature concernsthe relations between microndash and macrondashvolatility Put it simply one sideof the literature considers that uncorrelated microndashshocks cancel each otherout reducing their effects at macrondashlevel Conversely other researchersconsider that microndashshocks are multiplied at macrondashlevel generating highermacrondashvolatility than that we can observe at microndashlevel The debate isof obvious importance to determine the sources of business cycles to pre-dict their effects and determine the more effective policies to mitigate theirnegative effects

Our model is only a partial representation of realndashworld economic sys-tem and is also overly simplistic However based on these limitations italso offers very robust results allowing reliable conclusions on the (limited)elements included in the model In particular we consider our model a re-liable representation of the basic structures of the production interactionsamong sectors or firms Thus we can use the model to provide an answerto the debate concerning the contributions to volatility from the productioninteractions only Other contributions (eg from financial markets priceadjustments technological innovation etc) may re-inforce or counter theforces we analyse but they need to be considered as additional elements(and we opine also generally less relevant) not alternative to our analysisWe consider a necessity in order to apply a rigorous methodological ap-proach to be able to identify with certainty the results provided by partialstudies in order to eventually obtain by gradual extensions of the analy-sis more and more detailed representations of realistic economic systems

20Available from the authors

22

The results we obtain in this paper concerning this microndashmacro debate aremeant as a first relevant step to a more extended analysis

0

002

004

006

008

01

012

014

016

018

02

022

05 1 15 2 25 3

Var Demand

Macro (Output) and micro (Demand) volatility

delta Outputdelta Demand

Figure 8 Values of δ (relative deviation) for aggregate demand and ag-gregate output in respect of different levels of variance for sector specificuncorrelated external demand shocks

The flows of shocks affecting the firms of a system can be distinguishedalong two dimensions strength of the shock (eg amplitude of the changesin demand) and frequency of the shock (rare or frequent) Concerning thefirst aspect we initialized as usual a system to equilibrium levels (eg pro-duction quantities equal to external and intermediate demand for all sectorsconstant level of stocks equal to the desired level) We then ran the simu-lation generating the external demand for all sectors as a random variablewith constant (equilibrium) mean level and varying values of variance

Figure 8 shows unequivocally that as far as our model has propertiessimilar to real systems the hypothesis that macrondashfluctuations have largervolatility than their sectoral level shocks is fully confirmed At any levelof variance the aggregate volatility of demand as captured by the relativedeviation index is sensibly lower than the volatility shown by aggregateoutput that is the total production of the system This result is not sur-prising throughout all tests presented in the previous sections every resultsuggested that the system lsquooverndashreactsrsquo to any individual microndashlevel changeeven in the case these changes were meant to contrast disturbances There-fore when we move from considering one single shock to a flow of shockswe are not surprised to obtain that the system shows higher volatility thanthat implied by the microndashlevel disturbances

We now consider the second dimension defining a flow of shocks theirfrequency The question is whether for the same level of demand shocks

23

0 002 004 006 008 01 012 014 016

Difference of relative deviation for Output and Demand

0 5 10 15 20 25 30 35 40 45 50 55

Shock frequency 05

1

15

2

25

3

Var Demand

0 002 004 006 008 01

012 014 016

Figure 9 Differences of values of δ for aggregate output minus those foraggregate demand in respect of different levels of variance of demand and offrequency of the shocks

(ie same variance) more frequent changes trigger a higher or lower systemvolatility than rare disturbances The results shown in figure 9 suggest arather elaborated answer which partly vindicates (though in a particularsense) the arguments in favour of compensation of micro shocks The fig-ure shows the difference between aggregate output volatility and aggregatedemand volatility The independent variables are as before the volatilityof demand at microndashlevel21 and the time intervals in between two shocksAggregate demand volatility does not change for the frequency of shocksat least when computed over a long period But output volatility doesin a nonndashmonotonous way For high frequency of shocks the difference isstill positive and increasing for higher variance indicating that the ldquooverndashreactionrdquo argument is valid even in these cases but it is rather low Whilethe frequency of shocks slows down aggregate volatility of output increasesshowing that time is needed for the microndashshock to unfold their full effectsat the macrondashlevel However after a certain threshold less and less frequentshocks generate a lower volatility of the aggregate level of output

This result is quite sensible though not obvious at first sight and onceagain shows the power of such a simple model to generate and explain com-plex properties For frenetic levels of demand modifications the system hasno time to adjust but part of the job to align actual and desired productionfor constantly changing demand is performed by the varying demand itselfIn other terms the shocks cancel out through time if not through their quan-

21Beware of the scale of variance For reasons of visibility the scale is inverted

24

titative effects on the system While frequency decreases the disturbancesof the shocks not compensated by other (unlikely) shocks have the time tofilter through the production system generating higher and higher volatil-ity In a sense any system has a particular frequency of shocks that make itmaximally ldquoresonatingrdquo with the amplification of microndashshocks When therate of shocks gets even rarer the system can enjoy periods of longer andlonger quietness after each shock is absorbed and before a new one startsagain a cycle of initial local disturbance propagation of the disturbances toother sectors and settlement to the new equilibrium

6 Discussion and extensions

We presented a simple model of production composed by interacting sectorThe main (quit realistic) assumption of the model is that the productionprocesses in any sector of the economy make use of input from any othersector Moreover we assumed that sectors (represented as a single firm)adjust slowly both the desired level of production (smoothed revision of ex-pectation) and the actual level (smoothed modification of plant utilization)The model voluntarily leaves aside a large number of aspects of real-worldeconomic systems (eg financial markets price adjustments growth etc)The reason is that the model can be easily and reliably tested in order toextract its properties The possibility to transfer the (certain) model resultto a real context depend of course on an ample ceteris paribus clause How-ever we are confident that the reported results concerning the productionstructure play a relevant role in real world systems Therefore our resultsand particularly their motivations being accessible because of the relativesimplicity of the model will not be diminished by the integration of ourconsiderations with other aspects of economic systems

Our model is in essence a representation of an input-output table inter-preted in physical terms that is as the amount of inputs required for oneunit of output Besides this we include two straightforward assumptionsFirstly production levels cannot be suddenly changed but require time toscale up or down the utilization rate that is the levels of production Stocksare meant to make up for the difference between sales and production Thesecond assumption concerns the intentional behaviour of producers We im-plement this aspect by assuming that exists a level of stocks desired for eachlevel of demand However a change of demand does not translate suddenlyin a new level of desired stock but again only a long time at a different levelof demand will convince managers to update the desired levels of stocks

We show that the model generate under a wide number of settings oscil-latory patterns converging toward the equilibrium level where each sectorgenerate exactly the quantity demanded by final consumers and by othersectors as input The oscillations are provoked by over- and under-shooting

25

generated by the difficulty of coordination among sectors that by assump-tion ldquocommunicaterdquo only via variation of demand to their direct input sup-pliers We also show that the total production levels generated by sectors(sum of the physical units) produced at equilibrium depends (for constantfinal demand) on the total sum of the technical coefficients per sector irre-spective of their distribution This result is motivated by the impossibilityof input substitution and offers interesting suggestions for the analysis ofthe study of input-output tables of real-world systems

We then consider systems at equilibrium facing a single small change ofthe external demand for sectors These simulations allow to study the waythe system reacts to shocks We see that the sum of the coefficients affectsbesides the levels also the volatility of aggregate output the larger a systemthe stronger is the impact of a given shock Less obviously we find thatfor systems with the same dimension (ie same sums of coefficients) theaggregate volatility increases with both the inertia to change of productionlevel and the speed of adjustment of desired stocks This result is somewhatcounter-intuitive since both the cited variables are associated to a moreconservative behaviour that may be supposed should smooth away spikesin the pattern from an old to a new equilibrium However we show thatthe opposite actually holds given the complex interactions among sectorindividual-sector attitude intended to smooth shocks actually increase theiraggregate impact

In the same vein we show that systems maintaining high levels of stocksare more volatile that systems with reduced amounts of stocks Again thisis a counter-intuitive result reverting the goal of individual producers andthe aggregate result Interestingly this result finds strong support from theevidence of the recent changes in production methods (ie low or no stocks)and diminished cycles

In relation to specific issues discussed in the literature we find no supportto the suggestion that particular distributions of input-output coefficientsmodify the volatility of a system As said this depends in case on thesums of these coefficients but not on their distribution Further we canprovide an answer to a long and hotly debated questions whether microndashlevels volatility generates stronger or lower aggregate volatility Our modelshows unequivocally that microndashvolatility (eg variance of final demand)persists and generates stronger volatility at aggregate level Again we canidentify the reason on the complex production structure that exalts micro-level shocks by generating lsquowrongrsquo messages among producers Moreoverwe can also point to a usually neglected aspect of volatility we show thatthe frequency of shocks is a very relevant aspect besides their dimensionin determining the volatility of aggregate output

Our work can be developed along two complementary directions Firstlywe can continue to study the properties of the model by extending the ele-ments considered (eg pricing) analysing the effects on the present results

26

and generating more analytical evidence Of particular relevance is in ouropinion the problem of aggregation Representations of the same systemswith different codification systems (eg different number of SIC digits)must not modify overall system properties for example the volatility ob-served Given that our result show that the sums of coefficients matter forthe levels and volatility of the system and that these sum do depend on theaggregation chosen it should be possible to induce the reaction coefficientsthat we know affect the volatility only

Secondly we can find an application of our results to real world evidenceIn fact there are many data sets available for the data we deal with basicallyinput-output tables Our model is readily adapt to be parametrized alongany number of sector and coefficient necessary so that we may replicate pastseries in order to adapt the unobservable parameters and use the resultingcomplete model to explain business fluctuations and provide intuitions onfuture patterns

27

References

Acemoglu D and A Scott (1997) ldquoAsymmetric Business CyclesTheory and TimendashSeries Evidencerdquo Journal of Monetary Economics 40501ndash533

Backus D P Kehoe and F Kydland (1995) ldquoInternational BusinessCycles Theory and Evidencerdquo in Frontiers of Business Cycle Researched by T Cooley Princeton University Press Princeton

Bak P K Chen J Scheinkman and M Woodford (1993) ldquoAg-gregate Fluctuations from Independent Sectoral Shocks SelfndashOrganizedCritically in a Model of Production and Inventory Dynamicsrdquo RicercheEconomiche 47(1) 3ndash30

Battiston S D Delli Gatti M Gallegati B Greenwald and

J Stiglitz (2005) ldquoCredit Chains and Bankruptcies Avalanches in Sup-ply Networksrdquo in Annual Workshop on Economic Heterogeneous Inter-acting Agents (WEHIA 2005) University of Essex UK

Blanchard O and J Simon (2001) ldquoThe Long and Large Declinein US Output Volatilityrdquo Brookings Papers on Economic Activity 1135ndash174

Bresnahan T F and V A Ramey (1994) ldquoOutput Fluctuations atthe Plant Levelrdquo The Quarterly Journal of Economics 109(3) 593ndash924

David P A (1975) Technical choice innovation and economic growth essays on American and British experience in the nineteenth centuryCambridge University Press London

Delli Gatti D C Di Guilmi E Gaffeo G Giulioni M Galle-

gati and A Palestrini (2004) ldquoA New Approach to Business Fluc-tuations Heterogeneous Interacting Agents Scaling Laws and FinancialFragilityrdquo Journal of Economic Behavior amp Organization 56 489ndash512

Dosi G G Fagiolo and A Roventini (2006) ldquoAn EvolutionaryModel of Endogenous Business Cyclesrdquo Computational Economics 273ndash34

Dupor B (1999) ldquoAggregation and the Irrelevance in MultindashSector Mod-elsrdquo Journal of Monetary Economics 43 391ndash409

Head A C (1995) ldquoCountry Size Aggregate Fluctuations and Interna-tional Risk Sharingrdquo The Canadian Journal of Economics 28(4b) 1096ndash1119

28

Helbing D and S Lammer (2005) ldquoSupply and Production NetworksFrom the Bullwhip Effect to Business Cyclesrdquo in Networks of Interactingmachines Production Organization in Complex Industrial Systems andBiological Cells ed by D Armbruster K Kaneko and A S Mikhailovvol 3 of World Scientific Lecture Notes in Complex Systems World Sci-entific

Helbing D S Lammer U Witt and T Brenner (2004) ldquoNetworkndashInduced Oscillatory Behavior in Material Flow Networks and IrregularBusiness Cyclesrdquo Phisical Review E 70(056118) 1ndash6

Helbing D U Witt S Lammer and T Brenner (2004) ldquoNetwork-Induced Oscillatory Behaviour of Macroeconomic Output Flowsrdquo in In-dustry and Labour Dynamics II The Agent-based Computational Eco-nomics Approach Proceedings of the wildace 2004 conference ed byR Leombruni and M Richiardi no 5 LABORatorio R Revelli

Horvath M (1998) ldquoCyclically and Sectoral Linkages Aggregate Fluctu-ations from Independent Sectoral Shocksrdquo Review of Economic Dynam-ics 1 781ndash808

(2000) ldquoSectoral Shocks and Aggregate Fluctuationsrdquo Journal ofMonetary Economics 45(1) 69ndash106

Jovanovic B (1987) ldquoMicro Shocks and Aggregate Riskrdquo The QuarterlyJournal of Economics 102(2) 395ndash410

Kahn J A M M McConnell and G Perez-Quiros (2002) ldquoOnthe Causes of the Increased Stability of the US Economyrdquo FRBNY Eco-nomic Policy Review 8(1) 183ndash202

Kraay A and J Ventura (1998) ldquoComparative Advantage and theCrossndashSection of Business Cyclesrdquo World Bank Woorking paper Series1948 World Bank

Long J B and C I Plosser (1983) ldquoReal Business Cyclesrdquo Journalof Political Economy 31(1) 39ndash69

Lucas R E (1977) ldquoUnderstanding Business Cyclesrdquo CarnegiendashRochester Conference Series on Public Policy (special issue of Journalof Monetary Economics) 5 7ndash29

Nelson R R and S G Winter (1982) An Evolutionary Theory ofEconomic Change Harvard University Press Cambridge MA

Nirei M (2005) ldquoThreshold Behavior and Aggregate Fluctuationsrdquo Jour-nal of Economic Theory in press

29

Scheinkman J and M Woodford (1994) ldquoSelfndashOrganized Criticallyand Economic Fluctuationsrdquo American Economic Review 84(2 Papersand Proceedings of teh AEA) 417ndash421

Schuh S (1996) ldquoEvidence of the Link Between FirmndashLevel and Ag-gregate Inventory Behaviorrdquo Finance and Economics Discussion Series1996-46 Federal Reserve Board

Stiglitz J and B Greenwald (2003) Towards a New Paradigm inMonetary Economics Raffaele Mattioli Lectures Cambridge UniversityPress Cambridge

Weisbuch G and S Battiston (2005) ldquoProduction Networks andFailure Avalanchesrdquo Working Paper mimeo Ecole Normale SupErieure

Zarnovitz V (1977) ldquoBusiness Cycles Observed and Assessed Why andHow they Matterrdquo NBER Working Paper Series 6230 National Bureauof Economic Research

30

A Tables

Table 1 Parameters valuesParametera Description Value

N Number of sectorsunits i of production 10Ci Number of j buyer sectors for sector i (IO matrix sparseness) N minus 1Di (E) Value of the external demand 10mij Physical input coefficient of input j for sector i sim U(0 1)Mi Sum of the mij input coefficients j for the production of i 05αi friction to changes of production levels Degree of lockndashin on

production technology085

αsi Speed of adaptation to demand changes 004

σ Desired multiple of production to stock 10

aInto parenthesis the number of lags of the initial value for lagged variables

31

at most10 As show in Figure 2 the economic systems produces a lenghtlysmoothing oscillatory behaviour We compute the relative deviation of theaggregate production (industrial output) as the ratio between its standarddeviation and its mean δ = σY microY where Y =

sum

Qi Figure 4 shows thebehaviour of δ as a function of the sum of input coefficients for each sector80 periods after the shock As time goes by the level of δ reduces for allMi but the relation with the sum of input coefficient is unchanged

00055

0006

00065

0007

00075

0008

00085

0009

01 02 03 04 05 06 07 08 09

Rel

ativ

e va

rianc

e of

tota

l out

put (

delta

)

Sum of total input coefficients

Aggregate output deviations for different technical coefficients

Figure 4 The relevance of input coefficients on output stability

In sum input coefficients determine the extent of overshooting behaviourThe higher the coefficients the higher is the backlash even when shockscould cancel out through sectors

55 Shock persistence and adjustment coefficients

We now consider the effect of micro smoothing behaviours on the volatilityof the production system In particular we are interested in testing theeffects of two parameters of the model on the volatility of the productionsystem First we consider the effects of the αrsquos representing the ldquostickinessrdquoof the production technology to desired changes of Q aimed at filling thegap between actual and desired stocks The higher is α the slower is thereaction to stock unbalances Concerning its effect referring to the technicalpossibilities available ie production capital and organization we may putforward two opposite hypothesis relevant to systemrsquos volatility

Hypothesis 1 Slower production adjustment (high α) reduces volatilitybecause unbalances in demand of one sector being compensated slowly slow

17

down the diffusion of shocks to other sectors

Hypothesis 2 Slower production adjustment (high α) increase volatil-ity because the longer lasts the mismatch between actual and equilibriumproduction level the deeper will become the mismatch between actual anddesired stocks

20 25 30 35 40 45 50 55 60 65

Cyclesrsquo width

07 075

08 085

09alpha 0005

001 0015

002 0025

003 0035

004 0045

005

alphaS

20 25 30 35 40 45 50 55 60 65

Figure 5 Maximum width recorded between highest and lowest peaks fortotal production for different values of α and αS

Secondly we consider the parameters αS representing the speed of ad-justment of desired stock to observed demand The higher this parame-ter the faster sectors revise the desired levels of stocks to current levelsof observed demand This parameter concerns sectorsrsquo behavioral decisionprocess Higher αS represent a stronger belief that the most recently ob-served demand level is a reliable indicator of future demand levels whilelower levels of the parameter represent a more conservative approach wheremanagers require a long observation of demand levels different from the pastones before adjusting their expectations Also concerning the effect of thisparameter on the volatility induced by an exogenous shock we may maketwo hypothesis

Hypothesis 1 Faster adaptation of desired stocks to new levels of de-mand (higher αS) reduce volatility because changes in demand are rapidlytranslated in adjusted levels of production

Hypothesis 2 Faster adaptation of desired stocks to new levels of demand(higher αS) increase volatility because temporary unbalances of demand in

18

one sector are transmitted faster to other sector reinforcing the feedbackmechanism of fluctuations

To test which of these hypotheses are confirmed by the model we runsimulation runs with identical systems (eg same technical coefficients samelevels of demand same shocks) but for the values of the αrsquos and αS rsquos testingall combinations of these parameters over a range of values (each run usedthe same parameters for all the sectors of the system) For each simulation(ie each couple of values of the parameters) we computed two indicatorsof volatility One measures the maximum distance of total production be-tween the highest and the lowest peak that is the maximum width of theoscillations registered in the production time series The second indicatormeasures the relative deviation δ = σY microY Figures 5 and 6 report theseresults

0004 0006 0008 001 0012 0014 0016 0018 002

Relative deviation

07 075

08 085

09alpha 0005

001 0015

002 0025

003 0035

004 0045

005

alphaS

0004 0006 0008 001

0012 0014 0016 0018 002

Figure 6 Relative deviation of total production for different values of α andαS

The results confirms unequivocally both hypotheses 2 and reject hy-potheses 1 In a sense these hypotheses confirm the nature of the economicsystem represented in the model as a complex system where the interactionsamong sectors play a far more important role than the individual sector be-haviour In fact in both cases we may interpret the results that the strongerthe efforts to mitigate the effects of shocks (ie slow productionrsquos and quickexpectationsrsquo adjustment) the opposite result is actually obtained the fluc-tuations become even more accentuated It is also worth to note that thestronger impact is generated by the behavioural parameters rather than thetechnical one as indicated by the stronger effects of the αS in respect of theα This result seems to suggest that ceteris paribus information mattersmore than technical constraints Though the model is by any means in-

19

adequate to discuss normative aspects because of the naive ways strategicdecision making is represented it can anyway suggest the effects of errorsin management decisions (ie wrong desired levels of stocks) as opposed tothe technical constraints preventing sudden adjustment of production lev-els Stretching a bit the interpretation of these results we may suggests thatefforts to reduce volatility should better be oriented to improve the trans-mission of information in order to coordinate future production plans (ieidentifying quickly the desired levels of stocks and therefore the new equilib-rium production level) rather than improving the flexibility of productionmethods (reducing α)

56 Volatility and stock levels

In our model stock levels are maintained to buffer a sectorrsquos sale againstunexpected changes in demand Therefore a system made of risk adversemanagers (keeping higher levels of stocks) may be thought to lead to asmoother absorption of shocks in respect of a system composed of firmskeeping a short supply to cushion unexpected events To test this hypothesiswe tested the behaviour of the model for different values of σ the levels ofstock represented as a multiple of demand For each level of σ we run asimulation using all other initialization identical as usual representing aneconomic system ldquoalmostrdquo at equilibrium but for a single small shock indemand uncorrelated across sectors18

00044

00046

00048

0005

00052

00054

00056

00058

0006

00062

5 55 6 65 7 75 8 85 9 95 10

Rel

ativ

e va

riatio

n of

tota

l out

put (

delta

)

Ratio of stocked goods over production (sigma)

Stocks and production fluctuations

Figure 7 Fluctuations persistence on systems with different stock strategies(σ)

18Demand changes by plusmn1-10

20

Figure 7 shows the value of the relative deviation of output production(δ) generated by heterogeneous uncorrelated shocks in the external demandon identical systems where firms store a different multiple of their demand asstocks We obtain actually the opposite result that we may have expectedfluctuation become larger and more persistent the higher the level of stocksIn some way a strategy that for the individual firm is meant to reducethe fluctuations (using stocks as buffers) ends up by actually increasing thevolatility at aggregate levels

The result is though explainable both at the theoretical and empiricallevel In fact if no stocks were maintained production would need to adjustfor changes in demand only in case of a shock That is the firm may movemonotonically from the old to the new production levels sending consistentsignals to the rest of the system (ie its own suppliers) Instead if thesame shock affects a firm maintaining large stocks of output over the sameperiod production needs to adjust to the new demand and and to alignstocks with the new demand This generates besides the adjustment to thedemand a potentially inconsistent signal to the suppliers since the firm willhave a temporary level of production not meant to continue in equilibrium(adjusting stocks)

From the empirical viewpoint the last few decades have seen the almostuniversal diffusion of production system meant to keep stocks to a minimumif at all Lean production methods justndashinndashtime etc are solutions tothe problems of the costs necessary to maintain stocks Over the sameperiod economic fluctuations have been strongly reduced and few empiricalassessments have pointed to the relations between the two phenomena Forexample concerning the US ldquochanges in inventory behavior have playeda direct role in reducing real output volatilityrdquo (Kahn McConnell andPerez-Quiros 2002 p 183)19

57 Irrelevance of technical coefficients distribution

Horvath (1998) and Horvath (2000) suggest that systems with sparse ma-trices ie inputndashoutput matrices with many empty cells tend to be morevolatile in respect of systems with more evenly distributed matrices Noticethat this claim risk if not adequately qualified to generate an illogical resultIn fact matrices sparseness increases as a function of sectoral disaggregationand comparing the same systemrsquos fluctuations using inputndashoutput matricesat different levels of aggregation needs to produce the same type of volatil-ity Indeed this is not the case if all remaining parameters stand equaland do not adjust to the different level of analysis considered If aggregatepatterns of adjustment speed are maintained unchanged the higher the sec-toral disaggregation the higher the systemrsquos volatility ceteris paribus But

19Similar results are found in Blanchard and Simon (2001)

21

for example in our model reducing the aggregation also requires a changein the adaptation coefficients the more micro we undergo the analysis thehigher is the speed of adjustment of economic objects If for example weconsider sectors we are evaluating the adaptation of an entire set of firmswhich pose questions on how to aggregate their dynamics Things changewhen we consider the response of a single firm

Going back to Horvatsrsquo argument highly disaggregated matrices arelikely to contain many empty cells so that we may forecast different levelsof volatility depending on the aggregation used for the same system whichas mentioned is not caused by sparseness but to a failed adjustment inreaction mechanisms Results from our model20 show that volatility is com-pletely unaffected by input coefficients distribution Given their sectoralsum mdash through rows of the matrix mdash the value of each coefficient is irrele-vant The same occurs in the case of very skewed distributions that presenta high number of empty cells (zero input coefficients)

58 Microndash and Macrondashvolatility

As we have seen above one of the major discussion in the literature concernsthe relations between microndash and macrondashvolatility Put it simply one sideof the literature considers that uncorrelated microndashshocks cancel each otherout reducing their effects at macrondashlevel Conversely other researchersconsider that microndashshocks are multiplied at macrondashlevel generating highermacrondashvolatility than that we can observe at microndashlevel The debate isof obvious importance to determine the sources of business cycles to pre-dict their effects and determine the more effective policies to mitigate theirnegative effects

Our model is only a partial representation of realndashworld economic sys-tem and is also overly simplistic However based on these limitations italso offers very robust results allowing reliable conclusions on the (limited)elements included in the model In particular we consider our model a re-liable representation of the basic structures of the production interactionsamong sectors or firms Thus we can use the model to provide an answerto the debate concerning the contributions to volatility from the productioninteractions only Other contributions (eg from financial markets priceadjustments technological innovation etc) may re-inforce or counter theforces we analyse but they need to be considered as additional elements(and we opine also generally less relevant) not alternative to our analysisWe consider a necessity in order to apply a rigorous methodological ap-proach to be able to identify with certainty the results provided by partialstudies in order to eventually obtain by gradual extensions of the analy-sis more and more detailed representations of realistic economic systems

20Available from the authors

22

The results we obtain in this paper concerning this microndashmacro debate aremeant as a first relevant step to a more extended analysis

0

002

004

006

008

01

012

014

016

018

02

022

05 1 15 2 25 3

Var Demand

Macro (Output) and micro (Demand) volatility

delta Outputdelta Demand

Figure 8 Values of δ (relative deviation) for aggregate demand and ag-gregate output in respect of different levels of variance for sector specificuncorrelated external demand shocks

The flows of shocks affecting the firms of a system can be distinguishedalong two dimensions strength of the shock (eg amplitude of the changesin demand) and frequency of the shock (rare or frequent) Concerning thefirst aspect we initialized as usual a system to equilibrium levels (eg pro-duction quantities equal to external and intermediate demand for all sectorsconstant level of stocks equal to the desired level) We then ran the simu-lation generating the external demand for all sectors as a random variablewith constant (equilibrium) mean level and varying values of variance

Figure 8 shows unequivocally that as far as our model has propertiessimilar to real systems the hypothesis that macrondashfluctuations have largervolatility than their sectoral level shocks is fully confirmed At any levelof variance the aggregate volatility of demand as captured by the relativedeviation index is sensibly lower than the volatility shown by aggregateoutput that is the total production of the system This result is not sur-prising throughout all tests presented in the previous sections every resultsuggested that the system lsquooverndashreactsrsquo to any individual microndashlevel changeeven in the case these changes were meant to contrast disturbances There-fore when we move from considering one single shock to a flow of shockswe are not surprised to obtain that the system shows higher volatility thanthat implied by the microndashlevel disturbances

We now consider the second dimension defining a flow of shocks theirfrequency The question is whether for the same level of demand shocks

23

0 002 004 006 008 01 012 014 016

Difference of relative deviation for Output and Demand

0 5 10 15 20 25 30 35 40 45 50 55

Shock frequency 05

1

15

2

25

3

Var Demand

0 002 004 006 008 01

012 014 016

Figure 9 Differences of values of δ for aggregate output minus those foraggregate demand in respect of different levels of variance of demand and offrequency of the shocks

(ie same variance) more frequent changes trigger a higher or lower systemvolatility than rare disturbances The results shown in figure 9 suggest arather elaborated answer which partly vindicates (though in a particularsense) the arguments in favour of compensation of micro shocks The fig-ure shows the difference between aggregate output volatility and aggregatedemand volatility The independent variables are as before the volatilityof demand at microndashlevel21 and the time intervals in between two shocksAggregate demand volatility does not change for the frequency of shocksat least when computed over a long period But output volatility doesin a nonndashmonotonous way For high frequency of shocks the difference isstill positive and increasing for higher variance indicating that the ldquooverndashreactionrdquo argument is valid even in these cases but it is rather low Whilethe frequency of shocks slows down aggregate volatility of output increasesshowing that time is needed for the microndashshock to unfold their full effectsat the macrondashlevel However after a certain threshold less and less frequentshocks generate a lower volatility of the aggregate level of output

This result is quite sensible though not obvious at first sight and onceagain shows the power of such a simple model to generate and explain com-plex properties For frenetic levels of demand modifications the system hasno time to adjust but part of the job to align actual and desired productionfor constantly changing demand is performed by the varying demand itselfIn other terms the shocks cancel out through time if not through their quan-

21Beware of the scale of variance For reasons of visibility the scale is inverted

24

titative effects on the system While frequency decreases the disturbancesof the shocks not compensated by other (unlikely) shocks have the time tofilter through the production system generating higher and higher volatil-ity In a sense any system has a particular frequency of shocks that make itmaximally ldquoresonatingrdquo with the amplification of microndashshocks When therate of shocks gets even rarer the system can enjoy periods of longer andlonger quietness after each shock is absorbed and before a new one startsagain a cycle of initial local disturbance propagation of the disturbances toother sectors and settlement to the new equilibrium

6 Discussion and extensions

We presented a simple model of production composed by interacting sectorThe main (quit realistic) assumption of the model is that the productionprocesses in any sector of the economy make use of input from any othersector Moreover we assumed that sectors (represented as a single firm)adjust slowly both the desired level of production (smoothed revision of ex-pectation) and the actual level (smoothed modification of plant utilization)The model voluntarily leaves aside a large number of aspects of real-worldeconomic systems (eg financial markets price adjustments growth etc)The reason is that the model can be easily and reliably tested in order toextract its properties The possibility to transfer the (certain) model resultto a real context depend of course on an ample ceteris paribus clause How-ever we are confident that the reported results concerning the productionstructure play a relevant role in real world systems Therefore our resultsand particularly their motivations being accessible because of the relativesimplicity of the model will not be diminished by the integration of ourconsiderations with other aspects of economic systems

Our model is in essence a representation of an input-output table inter-preted in physical terms that is as the amount of inputs required for oneunit of output Besides this we include two straightforward assumptionsFirstly production levels cannot be suddenly changed but require time toscale up or down the utilization rate that is the levels of production Stocksare meant to make up for the difference between sales and production Thesecond assumption concerns the intentional behaviour of producers We im-plement this aspect by assuming that exists a level of stocks desired for eachlevel of demand However a change of demand does not translate suddenlyin a new level of desired stock but again only a long time at a different levelof demand will convince managers to update the desired levels of stocks

We show that the model generate under a wide number of settings oscil-latory patterns converging toward the equilibrium level where each sectorgenerate exactly the quantity demanded by final consumers and by othersectors as input The oscillations are provoked by over- and under-shooting

25

generated by the difficulty of coordination among sectors that by assump-tion ldquocommunicaterdquo only via variation of demand to their direct input sup-pliers We also show that the total production levels generated by sectors(sum of the physical units) produced at equilibrium depends (for constantfinal demand) on the total sum of the technical coefficients per sector irre-spective of their distribution This result is motivated by the impossibilityof input substitution and offers interesting suggestions for the analysis ofthe study of input-output tables of real-world systems

We then consider systems at equilibrium facing a single small change ofthe external demand for sectors These simulations allow to study the waythe system reacts to shocks We see that the sum of the coefficients affectsbesides the levels also the volatility of aggregate output the larger a systemthe stronger is the impact of a given shock Less obviously we find thatfor systems with the same dimension (ie same sums of coefficients) theaggregate volatility increases with both the inertia to change of productionlevel and the speed of adjustment of desired stocks This result is somewhatcounter-intuitive since both the cited variables are associated to a moreconservative behaviour that may be supposed should smooth away spikesin the pattern from an old to a new equilibrium However we show thatthe opposite actually holds given the complex interactions among sectorindividual-sector attitude intended to smooth shocks actually increase theiraggregate impact

In the same vein we show that systems maintaining high levels of stocksare more volatile that systems with reduced amounts of stocks Again thisis a counter-intuitive result reverting the goal of individual producers andthe aggregate result Interestingly this result finds strong support from theevidence of the recent changes in production methods (ie low or no stocks)and diminished cycles

In relation to specific issues discussed in the literature we find no supportto the suggestion that particular distributions of input-output coefficientsmodify the volatility of a system As said this depends in case on thesums of these coefficients but not on their distribution Further we canprovide an answer to a long and hotly debated questions whether microndashlevels volatility generates stronger or lower aggregate volatility Our modelshows unequivocally that microndashvolatility (eg variance of final demand)persists and generates stronger volatility at aggregate level Again we canidentify the reason on the complex production structure that exalts micro-level shocks by generating lsquowrongrsquo messages among producers Moreoverwe can also point to a usually neglected aspect of volatility we show thatthe frequency of shocks is a very relevant aspect besides their dimensionin determining the volatility of aggregate output

Our work can be developed along two complementary directions Firstlywe can continue to study the properties of the model by extending the ele-ments considered (eg pricing) analysing the effects on the present results

26

and generating more analytical evidence Of particular relevance is in ouropinion the problem of aggregation Representations of the same systemswith different codification systems (eg different number of SIC digits)must not modify overall system properties for example the volatility ob-served Given that our result show that the sums of coefficients matter forthe levels and volatility of the system and that these sum do depend on theaggregation chosen it should be possible to induce the reaction coefficientsthat we know affect the volatility only

Secondly we can find an application of our results to real world evidenceIn fact there are many data sets available for the data we deal with basicallyinput-output tables Our model is readily adapt to be parametrized alongany number of sector and coefficient necessary so that we may replicate pastseries in order to adapt the unobservable parameters and use the resultingcomplete model to explain business fluctuations and provide intuitions onfuture patterns

27

References

Acemoglu D and A Scott (1997) ldquoAsymmetric Business CyclesTheory and TimendashSeries Evidencerdquo Journal of Monetary Economics 40501ndash533

Backus D P Kehoe and F Kydland (1995) ldquoInternational BusinessCycles Theory and Evidencerdquo in Frontiers of Business Cycle Researched by T Cooley Princeton University Press Princeton

Bak P K Chen J Scheinkman and M Woodford (1993) ldquoAg-gregate Fluctuations from Independent Sectoral Shocks SelfndashOrganizedCritically in a Model of Production and Inventory Dynamicsrdquo RicercheEconomiche 47(1) 3ndash30

Battiston S D Delli Gatti M Gallegati B Greenwald and

J Stiglitz (2005) ldquoCredit Chains and Bankruptcies Avalanches in Sup-ply Networksrdquo in Annual Workshop on Economic Heterogeneous Inter-acting Agents (WEHIA 2005) University of Essex UK

Blanchard O and J Simon (2001) ldquoThe Long and Large Declinein US Output Volatilityrdquo Brookings Papers on Economic Activity 1135ndash174

Bresnahan T F and V A Ramey (1994) ldquoOutput Fluctuations atthe Plant Levelrdquo The Quarterly Journal of Economics 109(3) 593ndash924

David P A (1975) Technical choice innovation and economic growth essays on American and British experience in the nineteenth centuryCambridge University Press London

Delli Gatti D C Di Guilmi E Gaffeo G Giulioni M Galle-

gati and A Palestrini (2004) ldquoA New Approach to Business Fluc-tuations Heterogeneous Interacting Agents Scaling Laws and FinancialFragilityrdquo Journal of Economic Behavior amp Organization 56 489ndash512

Dosi G G Fagiolo and A Roventini (2006) ldquoAn EvolutionaryModel of Endogenous Business Cyclesrdquo Computational Economics 273ndash34

Dupor B (1999) ldquoAggregation and the Irrelevance in MultindashSector Mod-elsrdquo Journal of Monetary Economics 43 391ndash409

Head A C (1995) ldquoCountry Size Aggregate Fluctuations and Interna-tional Risk Sharingrdquo The Canadian Journal of Economics 28(4b) 1096ndash1119

28

Helbing D and S Lammer (2005) ldquoSupply and Production NetworksFrom the Bullwhip Effect to Business Cyclesrdquo in Networks of Interactingmachines Production Organization in Complex Industrial Systems andBiological Cells ed by D Armbruster K Kaneko and A S Mikhailovvol 3 of World Scientific Lecture Notes in Complex Systems World Sci-entific

Helbing D S Lammer U Witt and T Brenner (2004) ldquoNetworkndashInduced Oscillatory Behavior in Material Flow Networks and IrregularBusiness Cyclesrdquo Phisical Review E 70(056118) 1ndash6

Helbing D U Witt S Lammer and T Brenner (2004) ldquoNetwork-Induced Oscillatory Behaviour of Macroeconomic Output Flowsrdquo in In-dustry and Labour Dynamics II The Agent-based Computational Eco-nomics Approach Proceedings of the wildace 2004 conference ed byR Leombruni and M Richiardi no 5 LABORatorio R Revelli

Horvath M (1998) ldquoCyclically and Sectoral Linkages Aggregate Fluctu-ations from Independent Sectoral Shocksrdquo Review of Economic Dynam-ics 1 781ndash808

(2000) ldquoSectoral Shocks and Aggregate Fluctuationsrdquo Journal ofMonetary Economics 45(1) 69ndash106

Jovanovic B (1987) ldquoMicro Shocks and Aggregate Riskrdquo The QuarterlyJournal of Economics 102(2) 395ndash410

Kahn J A M M McConnell and G Perez-Quiros (2002) ldquoOnthe Causes of the Increased Stability of the US Economyrdquo FRBNY Eco-nomic Policy Review 8(1) 183ndash202

Kraay A and J Ventura (1998) ldquoComparative Advantage and theCrossndashSection of Business Cyclesrdquo World Bank Woorking paper Series1948 World Bank

Long J B and C I Plosser (1983) ldquoReal Business Cyclesrdquo Journalof Political Economy 31(1) 39ndash69

Lucas R E (1977) ldquoUnderstanding Business Cyclesrdquo CarnegiendashRochester Conference Series on Public Policy (special issue of Journalof Monetary Economics) 5 7ndash29

Nelson R R and S G Winter (1982) An Evolutionary Theory ofEconomic Change Harvard University Press Cambridge MA

Nirei M (2005) ldquoThreshold Behavior and Aggregate Fluctuationsrdquo Jour-nal of Economic Theory in press

29

Scheinkman J and M Woodford (1994) ldquoSelfndashOrganized Criticallyand Economic Fluctuationsrdquo American Economic Review 84(2 Papersand Proceedings of teh AEA) 417ndash421

Schuh S (1996) ldquoEvidence of the Link Between FirmndashLevel and Ag-gregate Inventory Behaviorrdquo Finance and Economics Discussion Series1996-46 Federal Reserve Board

Stiglitz J and B Greenwald (2003) Towards a New Paradigm inMonetary Economics Raffaele Mattioli Lectures Cambridge UniversityPress Cambridge

Weisbuch G and S Battiston (2005) ldquoProduction Networks andFailure Avalanchesrdquo Working Paper mimeo Ecole Normale SupErieure

Zarnovitz V (1977) ldquoBusiness Cycles Observed and Assessed Why andHow they Matterrdquo NBER Working Paper Series 6230 National Bureauof Economic Research

30

A Tables

Table 1 Parameters valuesParametera Description Value

N Number of sectorsunits i of production 10Ci Number of j buyer sectors for sector i (IO matrix sparseness) N minus 1Di (E) Value of the external demand 10mij Physical input coefficient of input j for sector i sim U(0 1)Mi Sum of the mij input coefficients j for the production of i 05αi friction to changes of production levels Degree of lockndashin on

production technology085

αsi Speed of adaptation to demand changes 004

σ Desired multiple of production to stock 10

aInto parenthesis the number of lags of the initial value for lagged variables

31

down the diffusion of shocks to other sectors

Hypothesis 2 Slower production adjustment (high α) increase volatil-ity because the longer lasts the mismatch between actual and equilibriumproduction level the deeper will become the mismatch between actual anddesired stocks

20 25 30 35 40 45 50 55 60 65

Cyclesrsquo width

07 075

08 085

09alpha 0005

001 0015

002 0025

003 0035

004 0045

005

alphaS

20 25 30 35 40 45 50 55 60 65

Figure 5 Maximum width recorded between highest and lowest peaks fortotal production for different values of α and αS

Secondly we consider the parameters αS representing the speed of ad-justment of desired stock to observed demand The higher this parame-ter the faster sectors revise the desired levels of stocks to current levelsof observed demand This parameter concerns sectorsrsquo behavioral decisionprocess Higher αS represent a stronger belief that the most recently ob-served demand level is a reliable indicator of future demand levels whilelower levels of the parameter represent a more conservative approach wheremanagers require a long observation of demand levels different from the pastones before adjusting their expectations Also concerning the effect of thisparameter on the volatility induced by an exogenous shock we may maketwo hypothesis

Hypothesis 1 Faster adaptation of desired stocks to new levels of de-mand (higher αS) reduce volatility because changes in demand are rapidlytranslated in adjusted levels of production

Hypothesis 2 Faster adaptation of desired stocks to new levels of demand(higher αS) increase volatility because temporary unbalances of demand in

18

one sector are transmitted faster to other sector reinforcing the feedbackmechanism of fluctuations

To test which of these hypotheses are confirmed by the model we runsimulation runs with identical systems (eg same technical coefficients samelevels of demand same shocks) but for the values of the αrsquos and αS rsquos testingall combinations of these parameters over a range of values (each run usedthe same parameters for all the sectors of the system) For each simulation(ie each couple of values of the parameters) we computed two indicatorsof volatility One measures the maximum distance of total production be-tween the highest and the lowest peak that is the maximum width of theoscillations registered in the production time series The second indicatormeasures the relative deviation δ = σY microY Figures 5 and 6 report theseresults

0004 0006 0008 001 0012 0014 0016 0018 002

Relative deviation

07 075

08 085

09alpha 0005

001 0015

002 0025

003 0035

004 0045

005

alphaS

0004 0006 0008 001

0012 0014 0016 0018 002

Figure 6 Relative deviation of total production for different values of α andαS

The results confirms unequivocally both hypotheses 2 and reject hy-potheses 1 In a sense these hypotheses confirm the nature of the economicsystem represented in the model as a complex system where the interactionsamong sectors play a far more important role than the individual sector be-haviour In fact in both cases we may interpret the results that the strongerthe efforts to mitigate the effects of shocks (ie slow productionrsquos and quickexpectationsrsquo adjustment) the opposite result is actually obtained the fluc-tuations become even more accentuated It is also worth to note that thestronger impact is generated by the behavioural parameters rather than thetechnical one as indicated by the stronger effects of the αS in respect of theα This result seems to suggest that ceteris paribus information mattersmore than technical constraints Though the model is by any means in-

19

adequate to discuss normative aspects because of the naive ways strategicdecision making is represented it can anyway suggest the effects of errorsin management decisions (ie wrong desired levels of stocks) as opposed tothe technical constraints preventing sudden adjustment of production lev-els Stretching a bit the interpretation of these results we may suggests thatefforts to reduce volatility should better be oriented to improve the trans-mission of information in order to coordinate future production plans (ieidentifying quickly the desired levels of stocks and therefore the new equilib-rium production level) rather than improving the flexibility of productionmethods (reducing α)

56 Volatility and stock levels

In our model stock levels are maintained to buffer a sectorrsquos sale againstunexpected changes in demand Therefore a system made of risk adversemanagers (keeping higher levels of stocks) may be thought to lead to asmoother absorption of shocks in respect of a system composed of firmskeeping a short supply to cushion unexpected events To test this hypothesiswe tested the behaviour of the model for different values of σ the levels ofstock represented as a multiple of demand For each level of σ we run asimulation using all other initialization identical as usual representing aneconomic system ldquoalmostrdquo at equilibrium but for a single small shock indemand uncorrelated across sectors18

00044

00046

00048

0005

00052

00054

00056

00058

0006

00062

5 55 6 65 7 75 8 85 9 95 10

Rel

ativ

e va

riatio

n of

tota

l out

put (

delta

)

Ratio of stocked goods over production (sigma)

Stocks and production fluctuations

Figure 7 Fluctuations persistence on systems with different stock strategies(σ)

18Demand changes by plusmn1-10

20

Figure 7 shows the value of the relative deviation of output production(δ) generated by heterogeneous uncorrelated shocks in the external demandon identical systems where firms store a different multiple of their demand asstocks We obtain actually the opposite result that we may have expectedfluctuation become larger and more persistent the higher the level of stocksIn some way a strategy that for the individual firm is meant to reducethe fluctuations (using stocks as buffers) ends up by actually increasing thevolatility at aggregate levels

The result is though explainable both at the theoretical and empiricallevel In fact if no stocks were maintained production would need to adjustfor changes in demand only in case of a shock That is the firm may movemonotonically from the old to the new production levels sending consistentsignals to the rest of the system (ie its own suppliers) Instead if thesame shock affects a firm maintaining large stocks of output over the sameperiod production needs to adjust to the new demand and and to alignstocks with the new demand This generates besides the adjustment to thedemand a potentially inconsistent signal to the suppliers since the firm willhave a temporary level of production not meant to continue in equilibrium(adjusting stocks)

From the empirical viewpoint the last few decades have seen the almostuniversal diffusion of production system meant to keep stocks to a minimumif at all Lean production methods justndashinndashtime etc are solutions tothe problems of the costs necessary to maintain stocks Over the sameperiod economic fluctuations have been strongly reduced and few empiricalassessments have pointed to the relations between the two phenomena Forexample concerning the US ldquochanges in inventory behavior have playeda direct role in reducing real output volatilityrdquo (Kahn McConnell andPerez-Quiros 2002 p 183)19

57 Irrelevance of technical coefficients distribution

Horvath (1998) and Horvath (2000) suggest that systems with sparse ma-trices ie inputndashoutput matrices with many empty cells tend to be morevolatile in respect of systems with more evenly distributed matrices Noticethat this claim risk if not adequately qualified to generate an illogical resultIn fact matrices sparseness increases as a function of sectoral disaggregationand comparing the same systemrsquos fluctuations using inputndashoutput matricesat different levels of aggregation needs to produce the same type of volatil-ity Indeed this is not the case if all remaining parameters stand equaland do not adjust to the different level of analysis considered If aggregatepatterns of adjustment speed are maintained unchanged the higher the sec-toral disaggregation the higher the systemrsquos volatility ceteris paribus But

19Similar results are found in Blanchard and Simon (2001)

21

for example in our model reducing the aggregation also requires a changein the adaptation coefficients the more micro we undergo the analysis thehigher is the speed of adjustment of economic objects If for example weconsider sectors we are evaluating the adaptation of an entire set of firmswhich pose questions on how to aggregate their dynamics Things changewhen we consider the response of a single firm

Going back to Horvatsrsquo argument highly disaggregated matrices arelikely to contain many empty cells so that we may forecast different levelsof volatility depending on the aggregation used for the same system whichas mentioned is not caused by sparseness but to a failed adjustment inreaction mechanisms Results from our model20 show that volatility is com-pletely unaffected by input coefficients distribution Given their sectoralsum mdash through rows of the matrix mdash the value of each coefficient is irrele-vant The same occurs in the case of very skewed distributions that presenta high number of empty cells (zero input coefficients)

58 Microndash and Macrondashvolatility

As we have seen above one of the major discussion in the literature concernsthe relations between microndash and macrondashvolatility Put it simply one sideof the literature considers that uncorrelated microndashshocks cancel each otherout reducing their effects at macrondashlevel Conversely other researchersconsider that microndashshocks are multiplied at macrondashlevel generating highermacrondashvolatility than that we can observe at microndashlevel The debate isof obvious importance to determine the sources of business cycles to pre-dict their effects and determine the more effective policies to mitigate theirnegative effects

Our model is only a partial representation of realndashworld economic sys-tem and is also overly simplistic However based on these limitations italso offers very robust results allowing reliable conclusions on the (limited)elements included in the model In particular we consider our model a re-liable representation of the basic structures of the production interactionsamong sectors or firms Thus we can use the model to provide an answerto the debate concerning the contributions to volatility from the productioninteractions only Other contributions (eg from financial markets priceadjustments technological innovation etc) may re-inforce or counter theforces we analyse but they need to be considered as additional elements(and we opine also generally less relevant) not alternative to our analysisWe consider a necessity in order to apply a rigorous methodological ap-proach to be able to identify with certainty the results provided by partialstudies in order to eventually obtain by gradual extensions of the analy-sis more and more detailed representations of realistic economic systems

20Available from the authors

22

The results we obtain in this paper concerning this microndashmacro debate aremeant as a first relevant step to a more extended analysis

0

002

004

006

008

01

012

014

016

018

02

022

05 1 15 2 25 3

Var Demand

Macro (Output) and micro (Demand) volatility

delta Outputdelta Demand

Figure 8 Values of δ (relative deviation) for aggregate demand and ag-gregate output in respect of different levels of variance for sector specificuncorrelated external demand shocks

The flows of shocks affecting the firms of a system can be distinguishedalong two dimensions strength of the shock (eg amplitude of the changesin demand) and frequency of the shock (rare or frequent) Concerning thefirst aspect we initialized as usual a system to equilibrium levels (eg pro-duction quantities equal to external and intermediate demand for all sectorsconstant level of stocks equal to the desired level) We then ran the simu-lation generating the external demand for all sectors as a random variablewith constant (equilibrium) mean level and varying values of variance

Figure 8 shows unequivocally that as far as our model has propertiessimilar to real systems the hypothesis that macrondashfluctuations have largervolatility than their sectoral level shocks is fully confirmed At any levelof variance the aggregate volatility of demand as captured by the relativedeviation index is sensibly lower than the volatility shown by aggregateoutput that is the total production of the system This result is not sur-prising throughout all tests presented in the previous sections every resultsuggested that the system lsquooverndashreactsrsquo to any individual microndashlevel changeeven in the case these changes were meant to contrast disturbances There-fore when we move from considering one single shock to a flow of shockswe are not surprised to obtain that the system shows higher volatility thanthat implied by the microndashlevel disturbances

We now consider the second dimension defining a flow of shocks theirfrequency The question is whether for the same level of demand shocks

23

0 002 004 006 008 01 012 014 016

Difference of relative deviation for Output and Demand

0 5 10 15 20 25 30 35 40 45 50 55

Shock frequency 05

1

15

2

25

3

Var Demand

0 002 004 006 008 01

012 014 016

Figure 9 Differences of values of δ for aggregate output minus those foraggregate demand in respect of different levels of variance of demand and offrequency of the shocks

(ie same variance) more frequent changes trigger a higher or lower systemvolatility than rare disturbances The results shown in figure 9 suggest arather elaborated answer which partly vindicates (though in a particularsense) the arguments in favour of compensation of micro shocks The fig-ure shows the difference between aggregate output volatility and aggregatedemand volatility The independent variables are as before the volatilityof demand at microndashlevel21 and the time intervals in between two shocksAggregate demand volatility does not change for the frequency of shocksat least when computed over a long period But output volatility doesin a nonndashmonotonous way For high frequency of shocks the difference isstill positive and increasing for higher variance indicating that the ldquooverndashreactionrdquo argument is valid even in these cases but it is rather low Whilethe frequency of shocks slows down aggregate volatility of output increasesshowing that time is needed for the microndashshock to unfold their full effectsat the macrondashlevel However after a certain threshold less and less frequentshocks generate a lower volatility of the aggregate level of output

This result is quite sensible though not obvious at first sight and onceagain shows the power of such a simple model to generate and explain com-plex properties For frenetic levels of demand modifications the system hasno time to adjust but part of the job to align actual and desired productionfor constantly changing demand is performed by the varying demand itselfIn other terms the shocks cancel out through time if not through their quan-

21Beware of the scale of variance For reasons of visibility the scale is inverted

24

titative effects on the system While frequency decreases the disturbancesof the shocks not compensated by other (unlikely) shocks have the time tofilter through the production system generating higher and higher volatil-ity In a sense any system has a particular frequency of shocks that make itmaximally ldquoresonatingrdquo with the amplification of microndashshocks When therate of shocks gets even rarer the system can enjoy periods of longer andlonger quietness after each shock is absorbed and before a new one startsagain a cycle of initial local disturbance propagation of the disturbances toother sectors and settlement to the new equilibrium

6 Discussion and extensions

We presented a simple model of production composed by interacting sectorThe main (quit realistic) assumption of the model is that the productionprocesses in any sector of the economy make use of input from any othersector Moreover we assumed that sectors (represented as a single firm)adjust slowly both the desired level of production (smoothed revision of ex-pectation) and the actual level (smoothed modification of plant utilization)The model voluntarily leaves aside a large number of aspects of real-worldeconomic systems (eg financial markets price adjustments growth etc)The reason is that the model can be easily and reliably tested in order toextract its properties The possibility to transfer the (certain) model resultto a real context depend of course on an ample ceteris paribus clause How-ever we are confident that the reported results concerning the productionstructure play a relevant role in real world systems Therefore our resultsand particularly their motivations being accessible because of the relativesimplicity of the model will not be diminished by the integration of ourconsiderations with other aspects of economic systems

Our model is in essence a representation of an input-output table inter-preted in physical terms that is as the amount of inputs required for oneunit of output Besides this we include two straightforward assumptionsFirstly production levels cannot be suddenly changed but require time toscale up or down the utilization rate that is the levels of production Stocksare meant to make up for the difference between sales and production Thesecond assumption concerns the intentional behaviour of producers We im-plement this aspect by assuming that exists a level of stocks desired for eachlevel of demand However a change of demand does not translate suddenlyin a new level of desired stock but again only a long time at a different levelof demand will convince managers to update the desired levels of stocks

We show that the model generate under a wide number of settings oscil-latory patterns converging toward the equilibrium level where each sectorgenerate exactly the quantity demanded by final consumers and by othersectors as input The oscillations are provoked by over- and under-shooting

25

generated by the difficulty of coordination among sectors that by assump-tion ldquocommunicaterdquo only via variation of demand to their direct input sup-pliers We also show that the total production levels generated by sectors(sum of the physical units) produced at equilibrium depends (for constantfinal demand) on the total sum of the technical coefficients per sector irre-spective of their distribution This result is motivated by the impossibilityof input substitution and offers interesting suggestions for the analysis ofthe study of input-output tables of real-world systems

We then consider systems at equilibrium facing a single small change ofthe external demand for sectors These simulations allow to study the waythe system reacts to shocks We see that the sum of the coefficients affectsbesides the levels also the volatility of aggregate output the larger a systemthe stronger is the impact of a given shock Less obviously we find thatfor systems with the same dimension (ie same sums of coefficients) theaggregate volatility increases with both the inertia to change of productionlevel and the speed of adjustment of desired stocks This result is somewhatcounter-intuitive since both the cited variables are associated to a moreconservative behaviour that may be supposed should smooth away spikesin the pattern from an old to a new equilibrium However we show thatthe opposite actually holds given the complex interactions among sectorindividual-sector attitude intended to smooth shocks actually increase theiraggregate impact

In the same vein we show that systems maintaining high levels of stocksare more volatile that systems with reduced amounts of stocks Again thisis a counter-intuitive result reverting the goal of individual producers andthe aggregate result Interestingly this result finds strong support from theevidence of the recent changes in production methods (ie low or no stocks)and diminished cycles

In relation to specific issues discussed in the literature we find no supportto the suggestion that particular distributions of input-output coefficientsmodify the volatility of a system As said this depends in case on thesums of these coefficients but not on their distribution Further we canprovide an answer to a long and hotly debated questions whether microndashlevels volatility generates stronger or lower aggregate volatility Our modelshows unequivocally that microndashvolatility (eg variance of final demand)persists and generates stronger volatility at aggregate level Again we canidentify the reason on the complex production structure that exalts micro-level shocks by generating lsquowrongrsquo messages among producers Moreoverwe can also point to a usually neglected aspect of volatility we show thatthe frequency of shocks is a very relevant aspect besides their dimensionin determining the volatility of aggregate output

Our work can be developed along two complementary directions Firstlywe can continue to study the properties of the model by extending the ele-ments considered (eg pricing) analysing the effects on the present results

26

and generating more analytical evidence Of particular relevance is in ouropinion the problem of aggregation Representations of the same systemswith different codification systems (eg different number of SIC digits)must not modify overall system properties for example the volatility ob-served Given that our result show that the sums of coefficients matter forthe levels and volatility of the system and that these sum do depend on theaggregation chosen it should be possible to induce the reaction coefficientsthat we know affect the volatility only

Secondly we can find an application of our results to real world evidenceIn fact there are many data sets available for the data we deal with basicallyinput-output tables Our model is readily adapt to be parametrized alongany number of sector and coefficient necessary so that we may replicate pastseries in order to adapt the unobservable parameters and use the resultingcomplete model to explain business fluctuations and provide intuitions onfuture patterns

27

References

Acemoglu D and A Scott (1997) ldquoAsymmetric Business CyclesTheory and TimendashSeries Evidencerdquo Journal of Monetary Economics 40501ndash533

Backus D P Kehoe and F Kydland (1995) ldquoInternational BusinessCycles Theory and Evidencerdquo in Frontiers of Business Cycle Researched by T Cooley Princeton University Press Princeton

Bak P K Chen J Scheinkman and M Woodford (1993) ldquoAg-gregate Fluctuations from Independent Sectoral Shocks SelfndashOrganizedCritically in a Model of Production and Inventory Dynamicsrdquo RicercheEconomiche 47(1) 3ndash30

Battiston S D Delli Gatti M Gallegati B Greenwald and

J Stiglitz (2005) ldquoCredit Chains and Bankruptcies Avalanches in Sup-ply Networksrdquo in Annual Workshop on Economic Heterogeneous Inter-acting Agents (WEHIA 2005) University of Essex UK

Blanchard O and J Simon (2001) ldquoThe Long and Large Declinein US Output Volatilityrdquo Brookings Papers on Economic Activity 1135ndash174

Bresnahan T F and V A Ramey (1994) ldquoOutput Fluctuations atthe Plant Levelrdquo The Quarterly Journal of Economics 109(3) 593ndash924

David P A (1975) Technical choice innovation and economic growth essays on American and British experience in the nineteenth centuryCambridge University Press London

Delli Gatti D C Di Guilmi E Gaffeo G Giulioni M Galle-

gati and A Palestrini (2004) ldquoA New Approach to Business Fluc-tuations Heterogeneous Interacting Agents Scaling Laws and FinancialFragilityrdquo Journal of Economic Behavior amp Organization 56 489ndash512

Dosi G G Fagiolo and A Roventini (2006) ldquoAn EvolutionaryModel of Endogenous Business Cyclesrdquo Computational Economics 273ndash34

Dupor B (1999) ldquoAggregation and the Irrelevance in MultindashSector Mod-elsrdquo Journal of Monetary Economics 43 391ndash409

Head A C (1995) ldquoCountry Size Aggregate Fluctuations and Interna-tional Risk Sharingrdquo The Canadian Journal of Economics 28(4b) 1096ndash1119

28

Helbing D and S Lammer (2005) ldquoSupply and Production NetworksFrom the Bullwhip Effect to Business Cyclesrdquo in Networks of Interactingmachines Production Organization in Complex Industrial Systems andBiological Cells ed by D Armbruster K Kaneko and A S Mikhailovvol 3 of World Scientific Lecture Notes in Complex Systems World Sci-entific

Helbing D S Lammer U Witt and T Brenner (2004) ldquoNetworkndashInduced Oscillatory Behavior in Material Flow Networks and IrregularBusiness Cyclesrdquo Phisical Review E 70(056118) 1ndash6

Helbing D U Witt S Lammer and T Brenner (2004) ldquoNetwork-Induced Oscillatory Behaviour of Macroeconomic Output Flowsrdquo in In-dustry and Labour Dynamics II The Agent-based Computational Eco-nomics Approach Proceedings of the wildace 2004 conference ed byR Leombruni and M Richiardi no 5 LABORatorio R Revelli

Horvath M (1998) ldquoCyclically and Sectoral Linkages Aggregate Fluctu-ations from Independent Sectoral Shocksrdquo Review of Economic Dynam-ics 1 781ndash808

(2000) ldquoSectoral Shocks and Aggregate Fluctuationsrdquo Journal ofMonetary Economics 45(1) 69ndash106

Jovanovic B (1987) ldquoMicro Shocks and Aggregate Riskrdquo The QuarterlyJournal of Economics 102(2) 395ndash410

Kahn J A M M McConnell and G Perez-Quiros (2002) ldquoOnthe Causes of the Increased Stability of the US Economyrdquo FRBNY Eco-nomic Policy Review 8(1) 183ndash202

Kraay A and J Ventura (1998) ldquoComparative Advantage and theCrossndashSection of Business Cyclesrdquo World Bank Woorking paper Series1948 World Bank

Long J B and C I Plosser (1983) ldquoReal Business Cyclesrdquo Journalof Political Economy 31(1) 39ndash69

Lucas R E (1977) ldquoUnderstanding Business Cyclesrdquo CarnegiendashRochester Conference Series on Public Policy (special issue of Journalof Monetary Economics) 5 7ndash29

Nelson R R and S G Winter (1982) An Evolutionary Theory ofEconomic Change Harvard University Press Cambridge MA

Nirei M (2005) ldquoThreshold Behavior and Aggregate Fluctuationsrdquo Jour-nal of Economic Theory in press

29

Scheinkman J and M Woodford (1994) ldquoSelfndashOrganized Criticallyand Economic Fluctuationsrdquo American Economic Review 84(2 Papersand Proceedings of teh AEA) 417ndash421

Schuh S (1996) ldquoEvidence of the Link Between FirmndashLevel and Ag-gregate Inventory Behaviorrdquo Finance and Economics Discussion Series1996-46 Federal Reserve Board

Stiglitz J and B Greenwald (2003) Towards a New Paradigm inMonetary Economics Raffaele Mattioli Lectures Cambridge UniversityPress Cambridge

Weisbuch G and S Battiston (2005) ldquoProduction Networks andFailure Avalanchesrdquo Working Paper mimeo Ecole Normale SupErieure

Zarnovitz V (1977) ldquoBusiness Cycles Observed and Assessed Why andHow they Matterrdquo NBER Working Paper Series 6230 National Bureauof Economic Research

30

A Tables

Table 1 Parameters valuesParametera Description Value

N Number of sectorsunits i of production 10Ci Number of j buyer sectors for sector i (IO matrix sparseness) N minus 1Di (E) Value of the external demand 10mij Physical input coefficient of input j for sector i sim U(0 1)Mi Sum of the mij input coefficients j for the production of i 05αi friction to changes of production levels Degree of lockndashin on

production technology085

αsi Speed of adaptation to demand changes 004

σ Desired multiple of production to stock 10

aInto parenthesis the number of lags of the initial value for lagged variables

31

one sector are transmitted faster to other sector reinforcing the feedbackmechanism of fluctuations

To test which of these hypotheses are confirmed by the model we runsimulation runs with identical systems (eg same technical coefficients samelevels of demand same shocks) but for the values of the αrsquos and αS rsquos testingall combinations of these parameters over a range of values (each run usedthe same parameters for all the sectors of the system) For each simulation(ie each couple of values of the parameters) we computed two indicatorsof volatility One measures the maximum distance of total production be-tween the highest and the lowest peak that is the maximum width of theoscillations registered in the production time series The second indicatormeasures the relative deviation δ = σY microY Figures 5 and 6 report theseresults

0004 0006 0008 001 0012 0014 0016 0018 002

Relative deviation

07 075

08 085

09alpha 0005

001 0015

002 0025

003 0035

004 0045

005

alphaS

0004 0006 0008 001

0012 0014 0016 0018 002

Figure 6 Relative deviation of total production for different values of α andαS

The results confirms unequivocally both hypotheses 2 and reject hy-potheses 1 In a sense these hypotheses confirm the nature of the economicsystem represented in the model as a complex system where the interactionsamong sectors play a far more important role than the individual sector be-haviour In fact in both cases we may interpret the results that the strongerthe efforts to mitigate the effects of shocks (ie slow productionrsquos and quickexpectationsrsquo adjustment) the opposite result is actually obtained the fluc-tuations become even more accentuated It is also worth to note that thestronger impact is generated by the behavioural parameters rather than thetechnical one as indicated by the stronger effects of the αS in respect of theα This result seems to suggest that ceteris paribus information mattersmore than technical constraints Though the model is by any means in-

19

adequate to discuss normative aspects because of the naive ways strategicdecision making is represented it can anyway suggest the effects of errorsin management decisions (ie wrong desired levels of stocks) as opposed tothe technical constraints preventing sudden adjustment of production lev-els Stretching a bit the interpretation of these results we may suggests thatefforts to reduce volatility should better be oriented to improve the trans-mission of information in order to coordinate future production plans (ieidentifying quickly the desired levels of stocks and therefore the new equilib-rium production level) rather than improving the flexibility of productionmethods (reducing α)

56 Volatility and stock levels

In our model stock levels are maintained to buffer a sectorrsquos sale againstunexpected changes in demand Therefore a system made of risk adversemanagers (keeping higher levels of stocks) may be thought to lead to asmoother absorption of shocks in respect of a system composed of firmskeeping a short supply to cushion unexpected events To test this hypothesiswe tested the behaviour of the model for different values of σ the levels ofstock represented as a multiple of demand For each level of σ we run asimulation using all other initialization identical as usual representing aneconomic system ldquoalmostrdquo at equilibrium but for a single small shock indemand uncorrelated across sectors18

00044

00046

00048

0005

00052

00054

00056

00058

0006

00062

5 55 6 65 7 75 8 85 9 95 10

Rel

ativ

e va

riatio

n of

tota

l out

put (

delta

)

Ratio of stocked goods over production (sigma)

Stocks and production fluctuations

Figure 7 Fluctuations persistence on systems with different stock strategies(σ)

18Demand changes by plusmn1-10

20

Figure 7 shows the value of the relative deviation of output production(δ) generated by heterogeneous uncorrelated shocks in the external demandon identical systems where firms store a different multiple of their demand asstocks We obtain actually the opposite result that we may have expectedfluctuation become larger and more persistent the higher the level of stocksIn some way a strategy that for the individual firm is meant to reducethe fluctuations (using stocks as buffers) ends up by actually increasing thevolatility at aggregate levels

The result is though explainable both at the theoretical and empiricallevel In fact if no stocks were maintained production would need to adjustfor changes in demand only in case of a shock That is the firm may movemonotonically from the old to the new production levels sending consistentsignals to the rest of the system (ie its own suppliers) Instead if thesame shock affects a firm maintaining large stocks of output over the sameperiod production needs to adjust to the new demand and and to alignstocks with the new demand This generates besides the adjustment to thedemand a potentially inconsistent signal to the suppliers since the firm willhave a temporary level of production not meant to continue in equilibrium(adjusting stocks)

From the empirical viewpoint the last few decades have seen the almostuniversal diffusion of production system meant to keep stocks to a minimumif at all Lean production methods justndashinndashtime etc are solutions tothe problems of the costs necessary to maintain stocks Over the sameperiod economic fluctuations have been strongly reduced and few empiricalassessments have pointed to the relations between the two phenomena Forexample concerning the US ldquochanges in inventory behavior have playeda direct role in reducing real output volatilityrdquo (Kahn McConnell andPerez-Quiros 2002 p 183)19

57 Irrelevance of technical coefficients distribution

Horvath (1998) and Horvath (2000) suggest that systems with sparse ma-trices ie inputndashoutput matrices with many empty cells tend to be morevolatile in respect of systems with more evenly distributed matrices Noticethat this claim risk if not adequately qualified to generate an illogical resultIn fact matrices sparseness increases as a function of sectoral disaggregationand comparing the same systemrsquos fluctuations using inputndashoutput matricesat different levels of aggregation needs to produce the same type of volatil-ity Indeed this is not the case if all remaining parameters stand equaland do not adjust to the different level of analysis considered If aggregatepatterns of adjustment speed are maintained unchanged the higher the sec-toral disaggregation the higher the systemrsquos volatility ceteris paribus But

19Similar results are found in Blanchard and Simon (2001)

21

for example in our model reducing the aggregation also requires a changein the adaptation coefficients the more micro we undergo the analysis thehigher is the speed of adjustment of economic objects If for example weconsider sectors we are evaluating the adaptation of an entire set of firmswhich pose questions on how to aggregate their dynamics Things changewhen we consider the response of a single firm

Going back to Horvatsrsquo argument highly disaggregated matrices arelikely to contain many empty cells so that we may forecast different levelsof volatility depending on the aggregation used for the same system whichas mentioned is not caused by sparseness but to a failed adjustment inreaction mechanisms Results from our model20 show that volatility is com-pletely unaffected by input coefficients distribution Given their sectoralsum mdash through rows of the matrix mdash the value of each coefficient is irrele-vant The same occurs in the case of very skewed distributions that presenta high number of empty cells (zero input coefficients)

58 Microndash and Macrondashvolatility

As we have seen above one of the major discussion in the literature concernsthe relations between microndash and macrondashvolatility Put it simply one sideof the literature considers that uncorrelated microndashshocks cancel each otherout reducing their effects at macrondashlevel Conversely other researchersconsider that microndashshocks are multiplied at macrondashlevel generating highermacrondashvolatility than that we can observe at microndashlevel The debate isof obvious importance to determine the sources of business cycles to pre-dict their effects and determine the more effective policies to mitigate theirnegative effects

Our model is only a partial representation of realndashworld economic sys-tem and is also overly simplistic However based on these limitations italso offers very robust results allowing reliable conclusions on the (limited)elements included in the model In particular we consider our model a re-liable representation of the basic structures of the production interactionsamong sectors or firms Thus we can use the model to provide an answerto the debate concerning the contributions to volatility from the productioninteractions only Other contributions (eg from financial markets priceadjustments technological innovation etc) may re-inforce or counter theforces we analyse but they need to be considered as additional elements(and we opine also generally less relevant) not alternative to our analysisWe consider a necessity in order to apply a rigorous methodological ap-proach to be able to identify with certainty the results provided by partialstudies in order to eventually obtain by gradual extensions of the analy-sis more and more detailed representations of realistic economic systems

20Available from the authors

22

The results we obtain in this paper concerning this microndashmacro debate aremeant as a first relevant step to a more extended analysis

0

002

004

006

008

01

012

014

016

018

02

022

05 1 15 2 25 3

Var Demand

Macro (Output) and micro (Demand) volatility

delta Outputdelta Demand

Figure 8 Values of δ (relative deviation) for aggregate demand and ag-gregate output in respect of different levels of variance for sector specificuncorrelated external demand shocks

The flows of shocks affecting the firms of a system can be distinguishedalong two dimensions strength of the shock (eg amplitude of the changesin demand) and frequency of the shock (rare or frequent) Concerning thefirst aspect we initialized as usual a system to equilibrium levels (eg pro-duction quantities equal to external and intermediate demand for all sectorsconstant level of stocks equal to the desired level) We then ran the simu-lation generating the external demand for all sectors as a random variablewith constant (equilibrium) mean level and varying values of variance

Figure 8 shows unequivocally that as far as our model has propertiessimilar to real systems the hypothesis that macrondashfluctuations have largervolatility than their sectoral level shocks is fully confirmed At any levelof variance the aggregate volatility of demand as captured by the relativedeviation index is sensibly lower than the volatility shown by aggregateoutput that is the total production of the system This result is not sur-prising throughout all tests presented in the previous sections every resultsuggested that the system lsquooverndashreactsrsquo to any individual microndashlevel changeeven in the case these changes were meant to contrast disturbances There-fore when we move from considering one single shock to a flow of shockswe are not surprised to obtain that the system shows higher volatility thanthat implied by the microndashlevel disturbances

We now consider the second dimension defining a flow of shocks theirfrequency The question is whether for the same level of demand shocks

23

0 002 004 006 008 01 012 014 016

Difference of relative deviation for Output and Demand

0 5 10 15 20 25 30 35 40 45 50 55

Shock frequency 05

1

15

2

25

3

Var Demand

0 002 004 006 008 01

012 014 016

Figure 9 Differences of values of δ for aggregate output minus those foraggregate demand in respect of different levels of variance of demand and offrequency of the shocks

(ie same variance) more frequent changes trigger a higher or lower systemvolatility than rare disturbances The results shown in figure 9 suggest arather elaborated answer which partly vindicates (though in a particularsense) the arguments in favour of compensation of micro shocks The fig-ure shows the difference between aggregate output volatility and aggregatedemand volatility The independent variables are as before the volatilityof demand at microndashlevel21 and the time intervals in between two shocksAggregate demand volatility does not change for the frequency of shocksat least when computed over a long period But output volatility doesin a nonndashmonotonous way For high frequency of shocks the difference isstill positive and increasing for higher variance indicating that the ldquooverndashreactionrdquo argument is valid even in these cases but it is rather low Whilethe frequency of shocks slows down aggregate volatility of output increasesshowing that time is needed for the microndashshock to unfold their full effectsat the macrondashlevel However after a certain threshold less and less frequentshocks generate a lower volatility of the aggregate level of output

This result is quite sensible though not obvious at first sight and onceagain shows the power of such a simple model to generate and explain com-plex properties For frenetic levels of demand modifications the system hasno time to adjust but part of the job to align actual and desired productionfor constantly changing demand is performed by the varying demand itselfIn other terms the shocks cancel out through time if not through their quan-

21Beware of the scale of variance For reasons of visibility the scale is inverted

24

titative effects on the system While frequency decreases the disturbancesof the shocks not compensated by other (unlikely) shocks have the time tofilter through the production system generating higher and higher volatil-ity In a sense any system has a particular frequency of shocks that make itmaximally ldquoresonatingrdquo with the amplification of microndashshocks When therate of shocks gets even rarer the system can enjoy periods of longer andlonger quietness after each shock is absorbed and before a new one startsagain a cycle of initial local disturbance propagation of the disturbances toother sectors and settlement to the new equilibrium

6 Discussion and extensions

We presented a simple model of production composed by interacting sectorThe main (quit realistic) assumption of the model is that the productionprocesses in any sector of the economy make use of input from any othersector Moreover we assumed that sectors (represented as a single firm)adjust slowly both the desired level of production (smoothed revision of ex-pectation) and the actual level (smoothed modification of plant utilization)The model voluntarily leaves aside a large number of aspects of real-worldeconomic systems (eg financial markets price adjustments growth etc)The reason is that the model can be easily and reliably tested in order toextract its properties The possibility to transfer the (certain) model resultto a real context depend of course on an ample ceteris paribus clause How-ever we are confident that the reported results concerning the productionstructure play a relevant role in real world systems Therefore our resultsand particularly their motivations being accessible because of the relativesimplicity of the model will not be diminished by the integration of ourconsiderations with other aspects of economic systems

Our model is in essence a representation of an input-output table inter-preted in physical terms that is as the amount of inputs required for oneunit of output Besides this we include two straightforward assumptionsFirstly production levels cannot be suddenly changed but require time toscale up or down the utilization rate that is the levels of production Stocksare meant to make up for the difference between sales and production Thesecond assumption concerns the intentional behaviour of producers We im-plement this aspect by assuming that exists a level of stocks desired for eachlevel of demand However a change of demand does not translate suddenlyin a new level of desired stock but again only a long time at a different levelof demand will convince managers to update the desired levels of stocks

We show that the model generate under a wide number of settings oscil-latory patterns converging toward the equilibrium level where each sectorgenerate exactly the quantity demanded by final consumers and by othersectors as input The oscillations are provoked by over- and under-shooting

25

generated by the difficulty of coordination among sectors that by assump-tion ldquocommunicaterdquo only via variation of demand to their direct input sup-pliers We also show that the total production levels generated by sectors(sum of the physical units) produced at equilibrium depends (for constantfinal demand) on the total sum of the technical coefficients per sector irre-spective of their distribution This result is motivated by the impossibilityof input substitution and offers interesting suggestions for the analysis ofthe study of input-output tables of real-world systems

We then consider systems at equilibrium facing a single small change ofthe external demand for sectors These simulations allow to study the waythe system reacts to shocks We see that the sum of the coefficients affectsbesides the levels also the volatility of aggregate output the larger a systemthe stronger is the impact of a given shock Less obviously we find thatfor systems with the same dimension (ie same sums of coefficients) theaggregate volatility increases with both the inertia to change of productionlevel and the speed of adjustment of desired stocks This result is somewhatcounter-intuitive since both the cited variables are associated to a moreconservative behaviour that may be supposed should smooth away spikesin the pattern from an old to a new equilibrium However we show thatthe opposite actually holds given the complex interactions among sectorindividual-sector attitude intended to smooth shocks actually increase theiraggregate impact

In the same vein we show that systems maintaining high levels of stocksare more volatile that systems with reduced amounts of stocks Again thisis a counter-intuitive result reverting the goal of individual producers andthe aggregate result Interestingly this result finds strong support from theevidence of the recent changes in production methods (ie low or no stocks)and diminished cycles

In relation to specific issues discussed in the literature we find no supportto the suggestion that particular distributions of input-output coefficientsmodify the volatility of a system As said this depends in case on thesums of these coefficients but not on their distribution Further we canprovide an answer to a long and hotly debated questions whether microndashlevels volatility generates stronger or lower aggregate volatility Our modelshows unequivocally that microndashvolatility (eg variance of final demand)persists and generates stronger volatility at aggregate level Again we canidentify the reason on the complex production structure that exalts micro-level shocks by generating lsquowrongrsquo messages among producers Moreoverwe can also point to a usually neglected aspect of volatility we show thatthe frequency of shocks is a very relevant aspect besides their dimensionin determining the volatility of aggregate output

Our work can be developed along two complementary directions Firstlywe can continue to study the properties of the model by extending the ele-ments considered (eg pricing) analysing the effects on the present results

26

and generating more analytical evidence Of particular relevance is in ouropinion the problem of aggregation Representations of the same systemswith different codification systems (eg different number of SIC digits)must not modify overall system properties for example the volatility ob-served Given that our result show that the sums of coefficients matter forthe levels and volatility of the system and that these sum do depend on theaggregation chosen it should be possible to induce the reaction coefficientsthat we know affect the volatility only

Secondly we can find an application of our results to real world evidenceIn fact there are many data sets available for the data we deal with basicallyinput-output tables Our model is readily adapt to be parametrized alongany number of sector and coefficient necessary so that we may replicate pastseries in order to adapt the unobservable parameters and use the resultingcomplete model to explain business fluctuations and provide intuitions onfuture patterns

27

References

Acemoglu D and A Scott (1997) ldquoAsymmetric Business CyclesTheory and TimendashSeries Evidencerdquo Journal of Monetary Economics 40501ndash533

Backus D P Kehoe and F Kydland (1995) ldquoInternational BusinessCycles Theory and Evidencerdquo in Frontiers of Business Cycle Researched by T Cooley Princeton University Press Princeton

Bak P K Chen J Scheinkman and M Woodford (1993) ldquoAg-gregate Fluctuations from Independent Sectoral Shocks SelfndashOrganizedCritically in a Model of Production and Inventory Dynamicsrdquo RicercheEconomiche 47(1) 3ndash30

Battiston S D Delli Gatti M Gallegati B Greenwald and

J Stiglitz (2005) ldquoCredit Chains and Bankruptcies Avalanches in Sup-ply Networksrdquo in Annual Workshop on Economic Heterogeneous Inter-acting Agents (WEHIA 2005) University of Essex UK

Blanchard O and J Simon (2001) ldquoThe Long and Large Declinein US Output Volatilityrdquo Brookings Papers on Economic Activity 1135ndash174

Bresnahan T F and V A Ramey (1994) ldquoOutput Fluctuations atthe Plant Levelrdquo The Quarterly Journal of Economics 109(3) 593ndash924

David P A (1975) Technical choice innovation and economic growth essays on American and British experience in the nineteenth centuryCambridge University Press London

Delli Gatti D C Di Guilmi E Gaffeo G Giulioni M Galle-

gati and A Palestrini (2004) ldquoA New Approach to Business Fluc-tuations Heterogeneous Interacting Agents Scaling Laws and FinancialFragilityrdquo Journal of Economic Behavior amp Organization 56 489ndash512

Dosi G G Fagiolo and A Roventini (2006) ldquoAn EvolutionaryModel of Endogenous Business Cyclesrdquo Computational Economics 273ndash34

Dupor B (1999) ldquoAggregation and the Irrelevance in MultindashSector Mod-elsrdquo Journal of Monetary Economics 43 391ndash409

Head A C (1995) ldquoCountry Size Aggregate Fluctuations and Interna-tional Risk Sharingrdquo The Canadian Journal of Economics 28(4b) 1096ndash1119

28

Helbing D and S Lammer (2005) ldquoSupply and Production NetworksFrom the Bullwhip Effect to Business Cyclesrdquo in Networks of Interactingmachines Production Organization in Complex Industrial Systems andBiological Cells ed by D Armbruster K Kaneko and A S Mikhailovvol 3 of World Scientific Lecture Notes in Complex Systems World Sci-entific

Helbing D S Lammer U Witt and T Brenner (2004) ldquoNetworkndashInduced Oscillatory Behavior in Material Flow Networks and IrregularBusiness Cyclesrdquo Phisical Review E 70(056118) 1ndash6

Helbing D U Witt S Lammer and T Brenner (2004) ldquoNetwork-Induced Oscillatory Behaviour of Macroeconomic Output Flowsrdquo in In-dustry and Labour Dynamics II The Agent-based Computational Eco-nomics Approach Proceedings of the wildace 2004 conference ed byR Leombruni and M Richiardi no 5 LABORatorio R Revelli

Horvath M (1998) ldquoCyclically and Sectoral Linkages Aggregate Fluctu-ations from Independent Sectoral Shocksrdquo Review of Economic Dynam-ics 1 781ndash808

(2000) ldquoSectoral Shocks and Aggregate Fluctuationsrdquo Journal ofMonetary Economics 45(1) 69ndash106

Jovanovic B (1987) ldquoMicro Shocks and Aggregate Riskrdquo The QuarterlyJournal of Economics 102(2) 395ndash410

Kahn J A M M McConnell and G Perez-Quiros (2002) ldquoOnthe Causes of the Increased Stability of the US Economyrdquo FRBNY Eco-nomic Policy Review 8(1) 183ndash202

Kraay A and J Ventura (1998) ldquoComparative Advantage and theCrossndashSection of Business Cyclesrdquo World Bank Woorking paper Series1948 World Bank

Long J B and C I Plosser (1983) ldquoReal Business Cyclesrdquo Journalof Political Economy 31(1) 39ndash69

Lucas R E (1977) ldquoUnderstanding Business Cyclesrdquo CarnegiendashRochester Conference Series on Public Policy (special issue of Journalof Monetary Economics) 5 7ndash29

Nelson R R and S G Winter (1982) An Evolutionary Theory ofEconomic Change Harvard University Press Cambridge MA

Nirei M (2005) ldquoThreshold Behavior and Aggregate Fluctuationsrdquo Jour-nal of Economic Theory in press

29

Scheinkman J and M Woodford (1994) ldquoSelfndashOrganized Criticallyand Economic Fluctuationsrdquo American Economic Review 84(2 Papersand Proceedings of teh AEA) 417ndash421

Schuh S (1996) ldquoEvidence of the Link Between FirmndashLevel and Ag-gregate Inventory Behaviorrdquo Finance and Economics Discussion Series1996-46 Federal Reserve Board

Stiglitz J and B Greenwald (2003) Towards a New Paradigm inMonetary Economics Raffaele Mattioli Lectures Cambridge UniversityPress Cambridge

Weisbuch G and S Battiston (2005) ldquoProduction Networks andFailure Avalanchesrdquo Working Paper mimeo Ecole Normale SupErieure

Zarnovitz V (1977) ldquoBusiness Cycles Observed and Assessed Why andHow they Matterrdquo NBER Working Paper Series 6230 National Bureauof Economic Research

30

A Tables

Table 1 Parameters valuesParametera Description Value

N Number of sectorsunits i of production 10Ci Number of j buyer sectors for sector i (IO matrix sparseness) N minus 1Di (E) Value of the external demand 10mij Physical input coefficient of input j for sector i sim U(0 1)Mi Sum of the mij input coefficients j for the production of i 05αi friction to changes of production levels Degree of lockndashin on

production technology085

αsi Speed of adaptation to demand changes 004

σ Desired multiple of production to stock 10

aInto parenthesis the number of lags of the initial value for lagged variables

31

adequate to discuss normative aspects because of the naive ways strategicdecision making is represented it can anyway suggest the effects of errorsin management decisions (ie wrong desired levels of stocks) as opposed tothe technical constraints preventing sudden adjustment of production lev-els Stretching a bit the interpretation of these results we may suggests thatefforts to reduce volatility should better be oriented to improve the trans-mission of information in order to coordinate future production plans (ieidentifying quickly the desired levels of stocks and therefore the new equilib-rium production level) rather than improving the flexibility of productionmethods (reducing α)

56 Volatility and stock levels

In our model stock levels are maintained to buffer a sectorrsquos sale againstunexpected changes in demand Therefore a system made of risk adversemanagers (keeping higher levels of stocks) may be thought to lead to asmoother absorption of shocks in respect of a system composed of firmskeeping a short supply to cushion unexpected events To test this hypothesiswe tested the behaviour of the model for different values of σ the levels ofstock represented as a multiple of demand For each level of σ we run asimulation using all other initialization identical as usual representing aneconomic system ldquoalmostrdquo at equilibrium but for a single small shock indemand uncorrelated across sectors18

00044

00046

00048

0005

00052

00054

00056

00058

0006

00062

5 55 6 65 7 75 8 85 9 95 10

Rel

ativ

e va

riatio

n of

tota

l out

put (

delta

)

Ratio of stocked goods over production (sigma)

Stocks and production fluctuations

Figure 7 Fluctuations persistence on systems with different stock strategies(σ)

18Demand changes by plusmn1-10

20

Figure 7 shows the value of the relative deviation of output production(δ) generated by heterogeneous uncorrelated shocks in the external demandon identical systems where firms store a different multiple of their demand asstocks We obtain actually the opposite result that we may have expectedfluctuation become larger and more persistent the higher the level of stocksIn some way a strategy that for the individual firm is meant to reducethe fluctuations (using stocks as buffers) ends up by actually increasing thevolatility at aggregate levels

The result is though explainable both at the theoretical and empiricallevel In fact if no stocks were maintained production would need to adjustfor changes in demand only in case of a shock That is the firm may movemonotonically from the old to the new production levels sending consistentsignals to the rest of the system (ie its own suppliers) Instead if thesame shock affects a firm maintaining large stocks of output over the sameperiod production needs to adjust to the new demand and and to alignstocks with the new demand This generates besides the adjustment to thedemand a potentially inconsistent signal to the suppliers since the firm willhave a temporary level of production not meant to continue in equilibrium(adjusting stocks)

From the empirical viewpoint the last few decades have seen the almostuniversal diffusion of production system meant to keep stocks to a minimumif at all Lean production methods justndashinndashtime etc are solutions tothe problems of the costs necessary to maintain stocks Over the sameperiod economic fluctuations have been strongly reduced and few empiricalassessments have pointed to the relations between the two phenomena Forexample concerning the US ldquochanges in inventory behavior have playeda direct role in reducing real output volatilityrdquo (Kahn McConnell andPerez-Quiros 2002 p 183)19

57 Irrelevance of technical coefficients distribution

Horvath (1998) and Horvath (2000) suggest that systems with sparse ma-trices ie inputndashoutput matrices with many empty cells tend to be morevolatile in respect of systems with more evenly distributed matrices Noticethat this claim risk if not adequately qualified to generate an illogical resultIn fact matrices sparseness increases as a function of sectoral disaggregationand comparing the same systemrsquos fluctuations using inputndashoutput matricesat different levels of aggregation needs to produce the same type of volatil-ity Indeed this is not the case if all remaining parameters stand equaland do not adjust to the different level of analysis considered If aggregatepatterns of adjustment speed are maintained unchanged the higher the sec-toral disaggregation the higher the systemrsquos volatility ceteris paribus But

19Similar results are found in Blanchard and Simon (2001)

21

for example in our model reducing the aggregation also requires a changein the adaptation coefficients the more micro we undergo the analysis thehigher is the speed of adjustment of economic objects If for example weconsider sectors we are evaluating the adaptation of an entire set of firmswhich pose questions on how to aggregate their dynamics Things changewhen we consider the response of a single firm

Going back to Horvatsrsquo argument highly disaggregated matrices arelikely to contain many empty cells so that we may forecast different levelsof volatility depending on the aggregation used for the same system whichas mentioned is not caused by sparseness but to a failed adjustment inreaction mechanisms Results from our model20 show that volatility is com-pletely unaffected by input coefficients distribution Given their sectoralsum mdash through rows of the matrix mdash the value of each coefficient is irrele-vant The same occurs in the case of very skewed distributions that presenta high number of empty cells (zero input coefficients)

58 Microndash and Macrondashvolatility

As we have seen above one of the major discussion in the literature concernsthe relations between microndash and macrondashvolatility Put it simply one sideof the literature considers that uncorrelated microndashshocks cancel each otherout reducing their effects at macrondashlevel Conversely other researchersconsider that microndashshocks are multiplied at macrondashlevel generating highermacrondashvolatility than that we can observe at microndashlevel The debate isof obvious importance to determine the sources of business cycles to pre-dict their effects and determine the more effective policies to mitigate theirnegative effects

Our model is only a partial representation of realndashworld economic sys-tem and is also overly simplistic However based on these limitations italso offers very robust results allowing reliable conclusions on the (limited)elements included in the model In particular we consider our model a re-liable representation of the basic structures of the production interactionsamong sectors or firms Thus we can use the model to provide an answerto the debate concerning the contributions to volatility from the productioninteractions only Other contributions (eg from financial markets priceadjustments technological innovation etc) may re-inforce or counter theforces we analyse but they need to be considered as additional elements(and we opine also generally less relevant) not alternative to our analysisWe consider a necessity in order to apply a rigorous methodological ap-proach to be able to identify with certainty the results provided by partialstudies in order to eventually obtain by gradual extensions of the analy-sis more and more detailed representations of realistic economic systems

20Available from the authors

22

The results we obtain in this paper concerning this microndashmacro debate aremeant as a first relevant step to a more extended analysis

0

002

004

006

008

01

012

014

016

018

02

022

05 1 15 2 25 3

Var Demand

Macro (Output) and micro (Demand) volatility

delta Outputdelta Demand

Figure 8 Values of δ (relative deviation) for aggregate demand and ag-gregate output in respect of different levels of variance for sector specificuncorrelated external demand shocks

The flows of shocks affecting the firms of a system can be distinguishedalong two dimensions strength of the shock (eg amplitude of the changesin demand) and frequency of the shock (rare or frequent) Concerning thefirst aspect we initialized as usual a system to equilibrium levels (eg pro-duction quantities equal to external and intermediate demand for all sectorsconstant level of stocks equal to the desired level) We then ran the simu-lation generating the external demand for all sectors as a random variablewith constant (equilibrium) mean level and varying values of variance

Figure 8 shows unequivocally that as far as our model has propertiessimilar to real systems the hypothesis that macrondashfluctuations have largervolatility than their sectoral level shocks is fully confirmed At any levelof variance the aggregate volatility of demand as captured by the relativedeviation index is sensibly lower than the volatility shown by aggregateoutput that is the total production of the system This result is not sur-prising throughout all tests presented in the previous sections every resultsuggested that the system lsquooverndashreactsrsquo to any individual microndashlevel changeeven in the case these changes were meant to contrast disturbances There-fore when we move from considering one single shock to a flow of shockswe are not surprised to obtain that the system shows higher volatility thanthat implied by the microndashlevel disturbances

We now consider the second dimension defining a flow of shocks theirfrequency The question is whether for the same level of demand shocks

23

0 002 004 006 008 01 012 014 016

Difference of relative deviation for Output and Demand

0 5 10 15 20 25 30 35 40 45 50 55

Shock frequency 05

1

15

2

25

3

Var Demand

0 002 004 006 008 01

012 014 016

Figure 9 Differences of values of δ for aggregate output minus those foraggregate demand in respect of different levels of variance of demand and offrequency of the shocks

(ie same variance) more frequent changes trigger a higher or lower systemvolatility than rare disturbances The results shown in figure 9 suggest arather elaborated answer which partly vindicates (though in a particularsense) the arguments in favour of compensation of micro shocks The fig-ure shows the difference between aggregate output volatility and aggregatedemand volatility The independent variables are as before the volatilityof demand at microndashlevel21 and the time intervals in between two shocksAggregate demand volatility does not change for the frequency of shocksat least when computed over a long period But output volatility doesin a nonndashmonotonous way For high frequency of shocks the difference isstill positive and increasing for higher variance indicating that the ldquooverndashreactionrdquo argument is valid even in these cases but it is rather low Whilethe frequency of shocks slows down aggregate volatility of output increasesshowing that time is needed for the microndashshock to unfold their full effectsat the macrondashlevel However after a certain threshold less and less frequentshocks generate a lower volatility of the aggregate level of output

This result is quite sensible though not obvious at first sight and onceagain shows the power of such a simple model to generate and explain com-plex properties For frenetic levels of demand modifications the system hasno time to adjust but part of the job to align actual and desired productionfor constantly changing demand is performed by the varying demand itselfIn other terms the shocks cancel out through time if not through their quan-

21Beware of the scale of variance For reasons of visibility the scale is inverted

24

titative effects on the system While frequency decreases the disturbancesof the shocks not compensated by other (unlikely) shocks have the time tofilter through the production system generating higher and higher volatil-ity In a sense any system has a particular frequency of shocks that make itmaximally ldquoresonatingrdquo with the amplification of microndashshocks When therate of shocks gets even rarer the system can enjoy periods of longer andlonger quietness after each shock is absorbed and before a new one startsagain a cycle of initial local disturbance propagation of the disturbances toother sectors and settlement to the new equilibrium

6 Discussion and extensions

We presented a simple model of production composed by interacting sectorThe main (quit realistic) assumption of the model is that the productionprocesses in any sector of the economy make use of input from any othersector Moreover we assumed that sectors (represented as a single firm)adjust slowly both the desired level of production (smoothed revision of ex-pectation) and the actual level (smoothed modification of plant utilization)The model voluntarily leaves aside a large number of aspects of real-worldeconomic systems (eg financial markets price adjustments growth etc)The reason is that the model can be easily and reliably tested in order toextract its properties The possibility to transfer the (certain) model resultto a real context depend of course on an ample ceteris paribus clause How-ever we are confident that the reported results concerning the productionstructure play a relevant role in real world systems Therefore our resultsand particularly their motivations being accessible because of the relativesimplicity of the model will not be diminished by the integration of ourconsiderations with other aspects of economic systems

Our model is in essence a representation of an input-output table inter-preted in physical terms that is as the amount of inputs required for oneunit of output Besides this we include two straightforward assumptionsFirstly production levels cannot be suddenly changed but require time toscale up or down the utilization rate that is the levels of production Stocksare meant to make up for the difference between sales and production Thesecond assumption concerns the intentional behaviour of producers We im-plement this aspect by assuming that exists a level of stocks desired for eachlevel of demand However a change of demand does not translate suddenlyin a new level of desired stock but again only a long time at a different levelof demand will convince managers to update the desired levels of stocks

We show that the model generate under a wide number of settings oscil-latory patterns converging toward the equilibrium level where each sectorgenerate exactly the quantity demanded by final consumers and by othersectors as input The oscillations are provoked by over- and under-shooting

25

generated by the difficulty of coordination among sectors that by assump-tion ldquocommunicaterdquo only via variation of demand to their direct input sup-pliers We also show that the total production levels generated by sectors(sum of the physical units) produced at equilibrium depends (for constantfinal demand) on the total sum of the technical coefficients per sector irre-spective of their distribution This result is motivated by the impossibilityof input substitution and offers interesting suggestions for the analysis ofthe study of input-output tables of real-world systems

We then consider systems at equilibrium facing a single small change ofthe external demand for sectors These simulations allow to study the waythe system reacts to shocks We see that the sum of the coefficients affectsbesides the levels also the volatility of aggregate output the larger a systemthe stronger is the impact of a given shock Less obviously we find thatfor systems with the same dimension (ie same sums of coefficients) theaggregate volatility increases with both the inertia to change of productionlevel and the speed of adjustment of desired stocks This result is somewhatcounter-intuitive since both the cited variables are associated to a moreconservative behaviour that may be supposed should smooth away spikesin the pattern from an old to a new equilibrium However we show thatthe opposite actually holds given the complex interactions among sectorindividual-sector attitude intended to smooth shocks actually increase theiraggregate impact

In the same vein we show that systems maintaining high levels of stocksare more volatile that systems with reduced amounts of stocks Again thisis a counter-intuitive result reverting the goal of individual producers andthe aggregate result Interestingly this result finds strong support from theevidence of the recent changes in production methods (ie low or no stocks)and diminished cycles

In relation to specific issues discussed in the literature we find no supportto the suggestion that particular distributions of input-output coefficientsmodify the volatility of a system As said this depends in case on thesums of these coefficients but not on their distribution Further we canprovide an answer to a long and hotly debated questions whether microndashlevels volatility generates stronger or lower aggregate volatility Our modelshows unequivocally that microndashvolatility (eg variance of final demand)persists and generates stronger volatility at aggregate level Again we canidentify the reason on the complex production structure that exalts micro-level shocks by generating lsquowrongrsquo messages among producers Moreoverwe can also point to a usually neglected aspect of volatility we show thatthe frequency of shocks is a very relevant aspect besides their dimensionin determining the volatility of aggregate output

Our work can be developed along two complementary directions Firstlywe can continue to study the properties of the model by extending the ele-ments considered (eg pricing) analysing the effects on the present results

26

and generating more analytical evidence Of particular relevance is in ouropinion the problem of aggregation Representations of the same systemswith different codification systems (eg different number of SIC digits)must not modify overall system properties for example the volatility ob-served Given that our result show that the sums of coefficients matter forthe levels and volatility of the system and that these sum do depend on theaggregation chosen it should be possible to induce the reaction coefficientsthat we know affect the volatility only

Secondly we can find an application of our results to real world evidenceIn fact there are many data sets available for the data we deal with basicallyinput-output tables Our model is readily adapt to be parametrized alongany number of sector and coefficient necessary so that we may replicate pastseries in order to adapt the unobservable parameters and use the resultingcomplete model to explain business fluctuations and provide intuitions onfuture patterns

27

References

Acemoglu D and A Scott (1997) ldquoAsymmetric Business CyclesTheory and TimendashSeries Evidencerdquo Journal of Monetary Economics 40501ndash533

Backus D P Kehoe and F Kydland (1995) ldquoInternational BusinessCycles Theory and Evidencerdquo in Frontiers of Business Cycle Researched by T Cooley Princeton University Press Princeton

Bak P K Chen J Scheinkman and M Woodford (1993) ldquoAg-gregate Fluctuations from Independent Sectoral Shocks SelfndashOrganizedCritically in a Model of Production and Inventory Dynamicsrdquo RicercheEconomiche 47(1) 3ndash30

Battiston S D Delli Gatti M Gallegati B Greenwald and

J Stiglitz (2005) ldquoCredit Chains and Bankruptcies Avalanches in Sup-ply Networksrdquo in Annual Workshop on Economic Heterogeneous Inter-acting Agents (WEHIA 2005) University of Essex UK

Blanchard O and J Simon (2001) ldquoThe Long and Large Declinein US Output Volatilityrdquo Brookings Papers on Economic Activity 1135ndash174

Bresnahan T F and V A Ramey (1994) ldquoOutput Fluctuations atthe Plant Levelrdquo The Quarterly Journal of Economics 109(3) 593ndash924

David P A (1975) Technical choice innovation and economic growth essays on American and British experience in the nineteenth centuryCambridge University Press London

Delli Gatti D C Di Guilmi E Gaffeo G Giulioni M Galle-

gati and A Palestrini (2004) ldquoA New Approach to Business Fluc-tuations Heterogeneous Interacting Agents Scaling Laws and FinancialFragilityrdquo Journal of Economic Behavior amp Organization 56 489ndash512

Dosi G G Fagiolo and A Roventini (2006) ldquoAn EvolutionaryModel of Endogenous Business Cyclesrdquo Computational Economics 273ndash34

Dupor B (1999) ldquoAggregation and the Irrelevance in MultindashSector Mod-elsrdquo Journal of Monetary Economics 43 391ndash409

Head A C (1995) ldquoCountry Size Aggregate Fluctuations and Interna-tional Risk Sharingrdquo The Canadian Journal of Economics 28(4b) 1096ndash1119

28

Helbing D and S Lammer (2005) ldquoSupply and Production NetworksFrom the Bullwhip Effect to Business Cyclesrdquo in Networks of Interactingmachines Production Organization in Complex Industrial Systems andBiological Cells ed by D Armbruster K Kaneko and A S Mikhailovvol 3 of World Scientific Lecture Notes in Complex Systems World Sci-entific

Helbing D S Lammer U Witt and T Brenner (2004) ldquoNetworkndashInduced Oscillatory Behavior in Material Flow Networks and IrregularBusiness Cyclesrdquo Phisical Review E 70(056118) 1ndash6

Helbing D U Witt S Lammer and T Brenner (2004) ldquoNetwork-Induced Oscillatory Behaviour of Macroeconomic Output Flowsrdquo in In-dustry and Labour Dynamics II The Agent-based Computational Eco-nomics Approach Proceedings of the wildace 2004 conference ed byR Leombruni and M Richiardi no 5 LABORatorio R Revelli

Horvath M (1998) ldquoCyclically and Sectoral Linkages Aggregate Fluctu-ations from Independent Sectoral Shocksrdquo Review of Economic Dynam-ics 1 781ndash808

(2000) ldquoSectoral Shocks and Aggregate Fluctuationsrdquo Journal ofMonetary Economics 45(1) 69ndash106

Jovanovic B (1987) ldquoMicro Shocks and Aggregate Riskrdquo The QuarterlyJournal of Economics 102(2) 395ndash410

Kahn J A M M McConnell and G Perez-Quiros (2002) ldquoOnthe Causes of the Increased Stability of the US Economyrdquo FRBNY Eco-nomic Policy Review 8(1) 183ndash202

Kraay A and J Ventura (1998) ldquoComparative Advantage and theCrossndashSection of Business Cyclesrdquo World Bank Woorking paper Series1948 World Bank

Long J B and C I Plosser (1983) ldquoReal Business Cyclesrdquo Journalof Political Economy 31(1) 39ndash69

Lucas R E (1977) ldquoUnderstanding Business Cyclesrdquo CarnegiendashRochester Conference Series on Public Policy (special issue of Journalof Monetary Economics) 5 7ndash29

Nelson R R and S G Winter (1982) An Evolutionary Theory ofEconomic Change Harvard University Press Cambridge MA

Nirei M (2005) ldquoThreshold Behavior and Aggregate Fluctuationsrdquo Jour-nal of Economic Theory in press

29

Scheinkman J and M Woodford (1994) ldquoSelfndashOrganized Criticallyand Economic Fluctuationsrdquo American Economic Review 84(2 Papersand Proceedings of teh AEA) 417ndash421

Schuh S (1996) ldquoEvidence of the Link Between FirmndashLevel and Ag-gregate Inventory Behaviorrdquo Finance and Economics Discussion Series1996-46 Federal Reserve Board

Stiglitz J and B Greenwald (2003) Towards a New Paradigm inMonetary Economics Raffaele Mattioli Lectures Cambridge UniversityPress Cambridge

Weisbuch G and S Battiston (2005) ldquoProduction Networks andFailure Avalanchesrdquo Working Paper mimeo Ecole Normale SupErieure

Zarnovitz V (1977) ldquoBusiness Cycles Observed and Assessed Why andHow they Matterrdquo NBER Working Paper Series 6230 National Bureauof Economic Research

30

A Tables

Table 1 Parameters valuesParametera Description Value

N Number of sectorsunits i of production 10Ci Number of j buyer sectors for sector i (IO matrix sparseness) N minus 1Di (E) Value of the external demand 10mij Physical input coefficient of input j for sector i sim U(0 1)Mi Sum of the mij input coefficients j for the production of i 05αi friction to changes of production levels Degree of lockndashin on

production technology085

αsi Speed of adaptation to demand changes 004

σ Desired multiple of production to stock 10

aInto parenthesis the number of lags of the initial value for lagged variables

31

Figure 7 shows the value of the relative deviation of output production(δ) generated by heterogeneous uncorrelated shocks in the external demandon identical systems where firms store a different multiple of their demand asstocks We obtain actually the opposite result that we may have expectedfluctuation become larger and more persistent the higher the level of stocksIn some way a strategy that for the individual firm is meant to reducethe fluctuations (using stocks as buffers) ends up by actually increasing thevolatility at aggregate levels

The result is though explainable both at the theoretical and empiricallevel In fact if no stocks were maintained production would need to adjustfor changes in demand only in case of a shock That is the firm may movemonotonically from the old to the new production levels sending consistentsignals to the rest of the system (ie its own suppliers) Instead if thesame shock affects a firm maintaining large stocks of output over the sameperiod production needs to adjust to the new demand and and to alignstocks with the new demand This generates besides the adjustment to thedemand a potentially inconsistent signal to the suppliers since the firm willhave a temporary level of production not meant to continue in equilibrium(adjusting stocks)

From the empirical viewpoint the last few decades have seen the almostuniversal diffusion of production system meant to keep stocks to a minimumif at all Lean production methods justndashinndashtime etc are solutions tothe problems of the costs necessary to maintain stocks Over the sameperiod economic fluctuations have been strongly reduced and few empiricalassessments have pointed to the relations between the two phenomena Forexample concerning the US ldquochanges in inventory behavior have playeda direct role in reducing real output volatilityrdquo (Kahn McConnell andPerez-Quiros 2002 p 183)19

57 Irrelevance of technical coefficients distribution

Horvath (1998) and Horvath (2000) suggest that systems with sparse ma-trices ie inputndashoutput matrices with many empty cells tend to be morevolatile in respect of systems with more evenly distributed matrices Noticethat this claim risk if not adequately qualified to generate an illogical resultIn fact matrices sparseness increases as a function of sectoral disaggregationand comparing the same systemrsquos fluctuations using inputndashoutput matricesat different levels of aggregation needs to produce the same type of volatil-ity Indeed this is not the case if all remaining parameters stand equaland do not adjust to the different level of analysis considered If aggregatepatterns of adjustment speed are maintained unchanged the higher the sec-toral disaggregation the higher the systemrsquos volatility ceteris paribus But

19Similar results are found in Blanchard and Simon (2001)

21

for example in our model reducing the aggregation also requires a changein the adaptation coefficients the more micro we undergo the analysis thehigher is the speed of adjustment of economic objects If for example weconsider sectors we are evaluating the adaptation of an entire set of firmswhich pose questions on how to aggregate their dynamics Things changewhen we consider the response of a single firm

Going back to Horvatsrsquo argument highly disaggregated matrices arelikely to contain many empty cells so that we may forecast different levelsof volatility depending on the aggregation used for the same system whichas mentioned is not caused by sparseness but to a failed adjustment inreaction mechanisms Results from our model20 show that volatility is com-pletely unaffected by input coefficients distribution Given their sectoralsum mdash through rows of the matrix mdash the value of each coefficient is irrele-vant The same occurs in the case of very skewed distributions that presenta high number of empty cells (zero input coefficients)

58 Microndash and Macrondashvolatility

As we have seen above one of the major discussion in the literature concernsthe relations between microndash and macrondashvolatility Put it simply one sideof the literature considers that uncorrelated microndashshocks cancel each otherout reducing their effects at macrondashlevel Conversely other researchersconsider that microndashshocks are multiplied at macrondashlevel generating highermacrondashvolatility than that we can observe at microndashlevel The debate isof obvious importance to determine the sources of business cycles to pre-dict their effects and determine the more effective policies to mitigate theirnegative effects

Our model is only a partial representation of realndashworld economic sys-tem and is also overly simplistic However based on these limitations italso offers very robust results allowing reliable conclusions on the (limited)elements included in the model In particular we consider our model a re-liable representation of the basic structures of the production interactionsamong sectors or firms Thus we can use the model to provide an answerto the debate concerning the contributions to volatility from the productioninteractions only Other contributions (eg from financial markets priceadjustments technological innovation etc) may re-inforce or counter theforces we analyse but they need to be considered as additional elements(and we opine also generally less relevant) not alternative to our analysisWe consider a necessity in order to apply a rigorous methodological ap-proach to be able to identify with certainty the results provided by partialstudies in order to eventually obtain by gradual extensions of the analy-sis more and more detailed representations of realistic economic systems

20Available from the authors

22

The results we obtain in this paper concerning this microndashmacro debate aremeant as a first relevant step to a more extended analysis

0

002

004

006

008

01

012

014

016

018

02

022

05 1 15 2 25 3

Var Demand

Macro (Output) and micro (Demand) volatility

delta Outputdelta Demand

Figure 8 Values of δ (relative deviation) for aggregate demand and ag-gregate output in respect of different levels of variance for sector specificuncorrelated external demand shocks

The flows of shocks affecting the firms of a system can be distinguishedalong two dimensions strength of the shock (eg amplitude of the changesin demand) and frequency of the shock (rare or frequent) Concerning thefirst aspect we initialized as usual a system to equilibrium levels (eg pro-duction quantities equal to external and intermediate demand for all sectorsconstant level of stocks equal to the desired level) We then ran the simu-lation generating the external demand for all sectors as a random variablewith constant (equilibrium) mean level and varying values of variance

Figure 8 shows unequivocally that as far as our model has propertiessimilar to real systems the hypothesis that macrondashfluctuations have largervolatility than their sectoral level shocks is fully confirmed At any levelof variance the aggregate volatility of demand as captured by the relativedeviation index is sensibly lower than the volatility shown by aggregateoutput that is the total production of the system This result is not sur-prising throughout all tests presented in the previous sections every resultsuggested that the system lsquooverndashreactsrsquo to any individual microndashlevel changeeven in the case these changes were meant to contrast disturbances There-fore when we move from considering one single shock to a flow of shockswe are not surprised to obtain that the system shows higher volatility thanthat implied by the microndashlevel disturbances

We now consider the second dimension defining a flow of shocks theirfrequency The question is whether for the same level of demand shocks

23

0 002 004 006 008 01 012 014 016

Difference of relative deviation for Output and Demand

0 5 10 15 20 25 30 35 40 45 50 55

Shock frequency 05

1

15

2

25

3

Var Demand

0 002 004 006 008 01

012 014 016

Figure 9 Differences of values of δ for aggregate output minus those foraggregate demand in respect of different levels of variance of demand and offrequency of the shocks

(ie same variance) more frequent changes trigger a higher or lower systemvolatility than rare disturbances The results shown in figure 9 suggest arather elaborated answer which partly vindicates (though in a particularsense) the arguments in favour of compensation of micro shocks The fig-ure shows the difference between aggregate output volatility and aggregatedemand volatility The independent variables are as before the volatilityof demand at microndashlevel21 and the time intervals in between two shocksAggregate demand volatility does not change for the frequency of shocksat least when computed over a long period But output volatility doesin a nonndashmonotonous way For high frequency of shocks the difference isstill positive and increasing for higher variance indicating that the ldquooverndashreactionrdquo argument is valid even in these cases but it is rather low Whilethe frequency of shocks slows down aggregate volatility of output increasesshowing that time is needed for the microndashshock to unfold their full effectsat the macrondashlevel However after a certain threshold less and less frequentshocks generate a lower volatility of the aggregate level of output

This result is quite sensible though not obvious at first sight and onceagain shows the power of such a simple model to generate and explain com-plex properties For frenetic levels of demand modifications the system hasno time to adjust but part of the job to align actual and desired productionfor constantly changing demand is performed by the varying demand itselfIn other terms the shocks cancel out through time if not through their quan-

21Beware of the scale of variance For reasons of visibility the scale is inverted

24

titative effects on the system While frequency decreases the disturbancesof the shocks not compensated by other (unlikely) shocks have the time tofilter through the production system generating higher and higher volatil-ity In a sense any system has a particular frequency of shocks that make itmaximally ldquoresonatingrdquo with the amplification of microndashshocks When therate of shocks gets even rarer the system can enjoy periods of longer andlonger quietness after each shock is absorbed and before a new one startsagain a cycle of initial local disturbance propagation of the disturbances toother sectors and settlement to the new equilibrium

6 Discussion and extensions

We presented a simple model of production composed by interacting sectorThe main (quit realistic) assumption of the model is that the productionprocesses in any sector of the economy make use of input from any othersector Moreover we assumed that sectors (represented as a single firm)adjust slowly both the desired level of production (smoothed revision of ex-pectation) and the actual level (smoothed modification of plant utilization)The model voluntarily leaves aside a large number of aspects of real-worldeconomic systems (eg financial markets price adjustments growth etc)The reason is that the model can be easily and reliably tested in order toextract its properties The possibility to transfer the (certain) model resultto a real context depend of course on an ample ceteris paribus clause How-ever we are confident that the reported results concerning the productionstructure play a relevant role in real world systems Therefore our resultsand particularly their motivations being accessible because of the relativesimplicity of the model will not be diminished by the integration of ourconsiderations with other aspects of economic systems

Our model is in essence a representation of an input-output table inter-preted in physical terms that is as the amount of inputs required for oneunit of output Besides this we include two straightforward assumptionsFirstly production levels cannot be suddenly changed but require time toscale up or down the utilization rate that is the levels of production Stocksare meant to make up for the difference between sales and production Thesecond assumption concerns the intentional behaviour of producers We im-plement this aspect by assuming that exists a level of stocks desired for eachlevel of demand However a change of demand does not translate suddenlyin a new level of desired stock but again only a long time at a different levelof demand will convince managers to update the desired levels of stocks

We show that the model generate under a wide number of settings oscil-latory patterns converging toward the equilibrium level where each sectorgenerate exactly the quantity demanded by final consumers and by othersectors as input The oscillations are provoked by over- and under-shooting

25

generated by the difficulty of coordination among sectors that by assump-tion ldquocommunicaterdquo only via variation of demand to their direct input sup-pliers We also show that the total production levels generated by sectors(sum of the physical units) produced at equilibrium depends (for constantfinal demand) on the total sum of the technical coefficients per sector irre-spective of their distribution This result is motivated by the impossibilityof input substitution and offers interesting suggestions for the analysis ofthe study of input-output tables of real-world systems

We then consider systems at equilibrium facing a single small change ofthe external demand for sectors These simulations allow to study the waythe system reacts to shocks We see that the sum of the coefficients affectsbesides the levels also the volatility of aggregate output the larger a systemthe stronger is the impact of a given shock Less obviously we find thatfor systems with the same dimension (ie same sums of coefficients) theaggregate volatility increases with both the inertia to change of productionlevel and the speed of adjustment of desired stocks This result is somewhatcounter-intuitive since both the cited variables are associated to a moreconservative behaviour that may be supposed should smooth away spikesin the pattern from an old to a new equilibrium However we show thatthe opposite actually holds given the complex interactions among sectorindividual-sector attitude intended to smooth shocks actually increase theiraggregate impact

In the same vein we show that systems maintaining high levels of stocksare more volatile that systems with reduced amounts of stocks Again thisis a counter-intuitive result reverting the goal of individual producers andthe aggregate result Interestingly this result finds strong support from theevidence of the recent changes in production methods (ie low or no stocks)and diminished cycles

In relation to specific issues discussed in the literature we find no supportto the suggestion that particular distributions of input-output coefficientsmodify the volatility of a system As said this depends in case on thesums of these coefficients but not on their distribution Further we canprovide an answer to a long and hotly debated questions whether microndashlevels volatility generates stronger or lower aggregate volatility Our modelshows unequivocally that microndashvolatility (eg variance of final demand)persists and generates stronger volatility at aggregate level Again we canidentify the reason on the complex production structure that exalts micro-level shocks by generating lsquowrongrsquo messages among producers Moreoverwe can also point to a usually neglected aspect of volatility we show thatthe frequency of shocks is a very relevant aspect besides their dimensionin determining the volatility of aggregate output

Our work can be developed along two complementary directions Firstlywe can continue to study the properties of the model by extending the ele-ments considered (eg pricing) analysing the effects on the present results

26

and generating more analytical evidence Of particular relevance is in ouropinion the problem of aggregation Representations of the same systemswith different codification systems (eg different number of SIC digits)must not modify overall system properties for example the volatility ob-served Given that our result show that the sums of coefficients matter forthe levels and volatility of the system and that these sum do depend on theaggregation chosen it should be possible to induce the reaction coefficientsthat we know affect the volatility only

Secondly we can find an application of our results to real world evidenceIn fact there are many data sets available for the data we deal with basicallyinput-output tables Our model is readily adapt to be parametrized alongany number of sector and coefficient necessary so that we may replicate pastseries in order to adapt the unobservable parameters and use the resultingcomplete model to explain business fluctuations and provide intuitions onfuture patterns

27

References

Acemoglu D and A Scott (1997) ldquoAsymmetric Business CyclesTheory and TimendashSeries Evidencerdquo Journal of Monetary Economics 40501ndash533

Backus D P Kehoe and F Kydland (1995) ldquoInternational BusinessCycles Theory and Evidencerdquo in Frontiers of Business Cycle Researched by T Cooley Princeton University Press Princeton

Bak P K Chen J Scheinkman and M Woodford (1993) ldquoAg-gregate Fluctuations from Independent Sectoral Shocks SelfndashOrganizedCritically in a Model of Production and Inventory Dynamicsrdquo RicercheEconomiche 47(1) 3ndash30

Battiston S D Delli Gatti M Gallegati B Greenwald and

J Stiglitz (2005) ldquoCredit Chains and Bankruptcies Avalanches in Sup-ply Networksrdquo in Annual Workshop on Economic Heterogeneous Inter-acting Agents (WEHIA 2005) University of Essex UK

Blanchard O and J Simon (2001) ldquoThe Long and Large Declinein US Output Volatilityrdquo Brookings Papers on Economic Activity 1135ndash174

Bresnahan T F and V A Ramey (1994) ldquoOutput Fluctuations atthe Plant Levelrdquo The Quarterly Journal of Economics 109(3) 593ndash924

David P A (1975) Technical choice innovation and economic growth essays on American and British experience in the nineteenth centuryCambridge University Press London

Delli Gatti D C Di Guilmi E Gaffeo G Giulioni M Galle-

gati and A Palestrini (2004) ldquoA New Approach to Business Fluc-tuations Heterogeneous Interacting Agents Scaling Laws and FinancialFragilityrdquo Journal of Economic Behavior amp Organization 56 489ndash512

Dosi G G Fagiolo and A Roventini (2006) ldquoAn EvolutionaryModel of Endogenous Business Cyclesrdquo Computational Economics 273ndash34

Dupor B (1999) ldquoAggregation and the Irrelevance in MultindashSector Mod-elsrdquo Journal of Monetary Economics 43 391ndash409

Head A C (1995) ldquoCountry Size Aggregate Fluctuations and Interna-tional Risk Sharingrdquo The Canadian Journal of Economics 28(4b) 1096ndash1119

28

Helbing D and S Lammer (2005) ldquoSupply and Production NetworksFrom the Bullwhip Effect to Business Cyclesrdquo in Networks of Interactingmachines Production Organization in Complex Industrial Systems andBiological Cells ed by D Armbruster K Kaneko and A S Mikhailovvol 3 of World Scientific Lecture Notes in Complex Systems World Sci-entific

Helbing D S Lammer U Witt and T Brenner (2004) ldquoNetworkndashInduced Oscillatory Behavior in Material Flow Networks and IrregularBusiness Cyclesrdquo Phisical Review E 70(056118) 1ndash6

Helbing D U Witt S Lammer and T Brenner (2004) ldquoNetwork-Induced Oscillatory Behaviour of Macroeconomic Output Flowsrdquo in In-dustry and Labour Dynamics II The Agent-based Computational Eco-nomics Approach Proceedings of the wildace 2004 conference ed byR Leombruni and M Richiardi no 5 LABORatorio R Revelli

Horvath M (1998) ldquoCyclically and Sectoral Linkages Aggregate Fluctu-ations from Independent Sectoral Shocksrdquo Review of Economic Dynam-ics 1 781ndash808

(2000) ldquoSectoral Shocks and Aggregate Fluctuationsrdquo Journal ofMonetary Economics 45(1) 69ndash106

Jovanovic B (1987) ldquoMicro Shocks and Aggregate Riskrdquo The QuarterlyJournal of Economics 102(2) 395ndash410

Kahn J A M M McConnell and G Perez-Quiros (2002) ldquoOnthe Causes of the Increased Stability of the US Economyrdquo FRBNY Eco-nomic Policy Review 8(1) 183ndash202

Kraay A and J Ventura (1998) ldquoComparative Advantage and theCrossndashSection of Business Cyclesrdquo World Bank Woorking paper Series1948 World Bank

Long J B and C I Plosser (1983) ldquoReal Business Cyclesrdquo Journalof Political Economy 31(1) 39ndash69

Lucas R E (1977) ldquoUnderstanding Business Cyclesrdquo CarnegiendashRochester Conference Series on Public Policy (special issue of Journalof Monetary Economics) 5 7ndash29

Nelson R R and S G Winter (1982) An Evolutionary Theory ofEconomic Change Harvard University Press Cambridge MA

Nirei M (2005) ldquoThreshold Behavior and Aggregate Fluctuationsrdquo Jour-nal of Economic Theory in press

29

Scheinkman J and M Woodford (1994) ldquoSelfndashOrganized Criticallyand Economic Fluctuationsrdquo American Economic Review 84(2 Papersand Proceedings of teh AEA) 417ndash421

Schuh S (1996) ldquoEvidence of the Link Between FirmndashLevel and Ag-gregate Inventory Behaviorrdquo Finance and Economics Discussion Series1996-46 Federal Reserve Board

Stiglitz J and B Greenwald (2003) Towards a New Paradigm inMonetary Economics Raffaele Mattioli Lectures Cambridge UniversityPress Cambridge

Weisbuch G and S Battiston (2005) ldquoProduction Networks andFailure Avalanchesrdquo Working Paper mimeo Ecole Normale SupErieure

Zarnovitz V (1977) ldquoBusiness Cycles Observed and Assessed Why andHow they Matterrdquo NBER Working Paper Series 6230 National Bureauof Economic Research

30

A Tables

Table 1 Parameters valuesParametera Description Value

N Number of sectorsunits i of production 10Ci Number of j buyer sectors for sector i (IO matrix sparseness) N minus 1Di (E) Value of the external demand 10mij Physical input coefficient of input j for sector i sim U(0 1)Mi Sum of the mij input coefficients j for the production of i 05αi friction to changes of production levels Degree of lockndashin on

production technology085

αsi Speed of adaptation to demand changes 004

σ Desired multiple of production to stock 10

aInto parenthesis the number of lags of the initial value for lagged variables

31

for example in our model reducing the aggregation also requires a changein the adaptation coefficients the more micro we undergo the analysis thehigher is the speed of adjustment of economic objects If for example weconsider sectors we are evaluating the adaptation of an entire set of firmswhich pose questions on how to aggregate their dynamics Things changewhen we consider the response of a single firm

Going back to Horvatsrsquo argument highly disaggregated matrices arelikely to contain many empty cells so that we may forecast different levelsof volatility depending on the aggregation used for the same system whichas mentioned is not caused by sparseness but to a failed adjustment inreaction mechanisms Results from our model20 show that volatility is com-pletely unaffected by input coefficients distribution Given their sectoralsum mdash through rows of the matrix mdash the value of each coefficient is irrele-vant The same occurs in the case of very skewed distributions that presenta high number of empty cells (zero input coefficients)

58 Microndash and Macrondashvolatility

As we have seen above one of the major discussion in the literature concernsthe relations between microndash and macrondashvolatility Put it simply one sideof the literature considers that uncorrelated microndashshocks cancel each otherout reducing their effects at macrondashlevel Conversely other researchersconsider that microndashshocks are multiplied at macrondashlevel generating highermacrondashvolatility than that we can observe at microndashlevel The debate isof obvious importance to determine the sources of business cycles to pre-dict their effects and determine the more effective policies to mitigate theirnegative effects

Our model is only a partial representation of realndashworld economic sys-tem and is also overly simplistic However based on these limitations italso offers very robust results allowing reliable conclusions on the (limited)elements included in the model In particular we consider our model a re-liable representation of the basic structures of the production interactionsamong sectors or firms Thus we can use the model to provide an answerto the debate concerning the contributions to volatility from the productioninteractions only Other contributions (eg from financial markets priceadjustments technological innovation etc) may re-inforce or counter theforces we analyse but they need to be considered as additional elements(and we opine also generally less relevant) not alternative to our analysisWe consider a necessity in order to apply a rigorous methodological ap-proach to be able to identify with certainty the results provided by partialstudies in order to eventually obtain by gradual extensions of the analy-sis more and more detailed representations of realistic economic systems

20Available from the authors

22

The results we obtain in this paper concerning this microndashmacro debate aremeant as a first relevant step to a more extended analysis

0

002

004

006

008

01

012

014

016

018

02

022

05 1 15 2 25 3

Var Demand

Macro (Output) and micro (Demand) volatility

delta Outputdelta Demand

Figure 8 Values of δ (relative deviation) for aggregate demand and ag-gregate output in respect of different levels of variance for sector specificuncorrelated external demand shocks

The flows of shocks affecting the firms of a system can be distinguishedalong two dimensions strength of the shock (eg amplitude of the changesin demand) and frequency of the shock (rare or frequent) Concerning thefirst aspect we initialized as usual a system to equilibrium levels (eg pro-duction quantities equal to external and intermediate demand for all sectorsconstant level of stocks equal to the desired level) We then ran the simu-lation generating the external demand for all sectors as a random variablewith constant (equilibrium) mean level and varying values of variance

Figure 8 shows unequivocally that as far as our model has propertiessimilar to real systems the hypothesis that macrondashfluctuations have largervolatility than their sectoral level shocks is fully confirmed At any levelof variance the aggregate volatility of demand as captured by the relativedeviation index is sensibly lower than the volatility shown by aggregateoutput that is the total production of the system This result is not sur-prising throughout all tests presented in the previous sections every resultsuggested that the system lsquooverndashreactsrsquo to any individual microndashlevel changeeven in the case these changes were meant to contrast disturbances There-fore when we move from considering one single shock to a flow of shockswe are not surprised to obtain that the system shows higher volatility thanthat implied by the microndashlevel disturbances

We now consider the second dimension defining a flow of shocks theirfrequency The question is whether for the same level of demand shocks

23

0 002 004 006 008 01 012 014 016

Difference of relative deviation for Output and Demand

0 5 10 15 20 25 30 35 40 45 50 55

Shock frequency 05

1

15

2

25

3

Var Demand

0 002 004 006 008 01

012 014 016

Figure 9 Differences of values of δ for aggregate output minus those foraggregate demand in respect of different levels of variance of demand and offrequency of the shocks

(ie same variance) more frequent changes trigger a higher or lower systemvolatility than rare disturbances The results shown in figure 9 suggest arather elaborated answer which partly vindicates (though in a particularsense) the arguments in favour of compensation of micro shocks The fig-ure shows the difference between aggregate output volatility and aggregatedemand volatility The independent variables are as before the volatilityof demand at microndashlevel21 and the time intervals in between two shocksAggregate demand volatility does not change for the frequency of shocksat least when computed over a long period But output volatility doesin a nonndashmonotonous way For high frequency of shocks the difference isstill positive and increasing for higher variance indicating that the ldquooverndashreactionrdquo argument is valid even in these cases but it is rather low Whilethe frequency of shocks slows down aggregate volatility of output increasesshowing that time is needed for the microndashshock to unfold their full effectsat the macrondashlevel However after a certain threshold less and less frequentshocks generate a lower volatility of the aggregate level of output

This result is quite sensible though not obvious at first sight and onceagain shows the power of such a simple model to generate and explain com-plex properties For frenetic levels of demand modifications the system hasno time to adjust but part of the job to align actual and desired productionfor constantly changing demand is performed by the varying demand itselfIn other terms the shocks cancel out through time if not through their quan-

21Beware of the scale of variance For reasons of visibility the scale is inverted

24

titative effects on the system While frequency decreases the disturbancesof the shocks not compensated by other (unlikely) shocks have the time tofilter through the production system generating higher and higher volatil-ity In a sense any system has a particular frequency of shocks that make itmaximally ldquoresonatingrdquo with the amplification of microndashshocks When therate of shocks gets even rarer the system can enjoy periods of longer andlonger quietness after each shock is absorbed and before a new one startsagain a cycle of initial local disturbance propagation of the disturbances toother sectors and settlement to the new equilibrium

6 Discussion and extensions

We presented a simple model of production composed by interacting sectorThe main (quit realistic) assumption of the model is that the productionprocesses in any sector of the economy make use of input from any othersector Moreover we assumed that sectors (represented as a single firm)adjust slowly both the desired level of production (smoothed revision of ex-pectation) and the actual level (smoothed modification of plant utilization)The model voluntarily leaves aside a large number of aspects of real-worldeconomic systems (eg financial markets price adjustments growth etc)The reason is that the model can be easily and reliably tested in order toextract its properties The possibility to transfer the (certain) model resultto a real context depend of course on an ample ceteris paribus clause How-ever we are confident that the reported results concerning the productionstructure play a relevant role in real world systems Therefore our resultsand particularly their motivations being accessible because of the relativesimplicity of the model will not be diminished by the integration of ourconsiderations with other aspects of economic systems

Our model is in essence a representation of an input-output table inter-preted in physical terms that is as the amount of inputs required for oneunit of output Besides this we include two straightforward assumptionsFirstly production levels cannot be suddenly changed but require time toscale up or down the utilization rate that is the levels of production Stocksare meant to make up for the difference between sales and production Thesecond assumption concerns the intentional behaviour of producers We im-plement this aspect by assuming that exists a level of stocks desired for eachlevel of demand However a change of demand does not translate suddenlyin a new level of desired stock but again only a long time at a different levelof demand will convince managers to update the desired levels of stocks

We show that the model generate under a wide number of settings oscil-latory patterns converging toward the equilibrium level where each sectorgenerate exactly the quantity demanded by final consumers and by othersectors as input The oscillations are provoked by over- and under-shooting

25

generated by the difficulty of coordination among sectors that by assump-tion ldquocommunicaterdquo only via variation of demand to their direct input sup-pliers We also show that the total production levels generated by sectors(sum of the physical units) produced at equilibrium depends (for constantfinal demand) on the total sum of the technical coefficients per sector irre-spective of their distribution This result is motivated by the impossibilityof input substitution and offers interesting suggestions for the analysis ofthe study of input-output tables of real-world systems

We then consider systems at equilibrium facing a single small change ofthe external demand for sectors These simulations allow to study the waythe system reacts to shocks We see that the sum of the coefficients affectsbesides the levels also the volatility of aggregate output the larger a systemthe stronger is the impact of a given shock Less obviously we find thatfor systems with the same dimension (ie same sums of coefficients) theaggregate volatility increases with both the inertia to change of productionlevel and the speed of adjustment of desired stocks This result is somewhatcounter-intuitive since both the cited variables are associated to a moreconservative behaviour that may be supposed should smooth away spikesin the pattern from an old to a new equilibrium However we show thatthe opposite actually holds given the complex interactions among sectorindividual-sector attitude intended to smooth shocks actually increase theiraggregate impact

In the same vein we show that systems maintaining high levels of stocksare more volatile that systems with reduced amounts of stocks Again thisis a counter-intuitive result reverting the goal of individual producers andthe aggregate result Interestingly this result finds strong support from theevidence of the recent changes in production methods (ie low or no stocks)and diminished cycles

In relation to specific issues discussed in the literature we find no supportto the suggestion that particular distributions of input-output coefficientsmodify the volatility of a system As said this depends in case on thesums of these coefficients but not on their distribution Further we canprovide an answer to a long and hotly debated questions whether microndashlevels volatility generates stronger or lower aggregate volatility Our modelshows unequivocally that microndashvolatility (eg variance of final demand)persists and generates stronger volatility at aggregate level Again we canidentify the reason on the complex production structure that exalts micro-level shocks by generating lsquowrongrsquo messages among producers Moreoverwe can also point to a usually neglected aspect of volatility we show thatthe frequency of shocks is a very relevant aspect besides their dimensionin determining the volatility of aggregate output

Our work can be developed along two complementary directions Firstlywe can continue to study the properties of the model by extending the ele-ments considered (eg pricing) analysing the effects on the present results

26

and generating more analytical evidence Of particular relevance is in ouropinion the problem of aggregation Representations of the same systemswith different codification systems (eg different number of SIC digits)must not modify overall system properties for example the volatility ob-served Given that our result show that the sums of coefficients matter forthe levels and volatility of the system and that these sum do depend on theaggregation chosen it should be possible to induce the reaction coefficientsthat we know affect the volatility only

Secondly we can find an application of our results to real world evidenceIn fact there are many data sets available for the data we deal with basicallyinput-output tables Our model is readily adapt to be parametrized alongany number of sector and coefficient necessary so that we may replicate pastseries in order to adapt the unobservable parameters and use the resultingcomplete model to explain business fluctuations and provide intuitions onfuture patterns

27

References

Acemoglu D and A Scott (1997) ldquoAsymmetric Business CyclesTheory and TimendashSeries Evidencerdquo Journal of Monetary Economics 40501ndash533

Backus D P Kehoe and F Kydland (1995) ldquoInternational BusinessCycles Theory and Evidencerdquo in Frontiers of Business Cycle Researched by T Cooley Princeton University Press Princeton

Bak P K Chen J Scheinkman and M Woodford (1993) ldquoAg-gregate Fluctuations from Independent Sectoral Shocks SelfndashOrganizedCritically in a Model of Production and Inventory Dynamicsrdquo RicercheEconomiche 47(1) 3ndash30

Battiston S D Delli Gatti M Gallegati B Greenwald and

J Stiglitz (2005) ldquoCredit Chains and Bankruptcies Avalanches in Sup-ply Networksrdquo in Annual Workshop on Economic Heterogeneous Inter-acting Agents (WEHIA 2005) University of Essex UK

Blanchard O and J Simon (2001) ldquoThe Long and Large Declinein US Output Volatilityrdquo Brookings Papers on Economic Activity 1135ndash174

Bresnahan T F and V A Ramey (1994) ldquoOutput Fluctuations atthe Plant Levelrdquo The Quarterly Journal of Economics 109(3) 593ndash924

David P A (1975) Technical choice innovation and economic growth essays on American and British experience in the nineteenth centuryCambridge University Press London

Delli Gatti D C Di Guilmi E Gaffeo G Giulioni M Galle-

gati and A Palestrini (2004) ldquoA New Approach to Business Fluc-tuations Heterogeneous Interacting Agents Scaling Laws and FinancialFragilityrdquo Journal of Economic Behavior amp Organization 56 489ndash512

Dosi G G Fagiolo and A Roventini (2006) ldquoAn EvolutionaryModel of Endogenous Business Cyclesrdquo Computational Economics 273ndash34

Dupor B (1999) ldquoAggregation and the Irrelevance in MultindashSector Mod-elsrdquo Journal of Monetary Economics 43 391ndash409

Head A C (1995) ldquoCountry Size Aggregate Fluctuations and Interna-tional Risk Sharingrdquo The Canadian Journal of Economics 28(4b) 1096ndash1119

28

Helbing D and S Lammer (2005) ldquoSupply and Production NetworksFrom the Bullwhip Effect to Business Cyclesrdquo in Networks of Interactingmachines Production Organization in Complex Industrial Systems andBiological Cells ed by D Armbruster K Kaneko and A S Mikhailovvol 3 of World Scientific Lecture Notes in Complex Systems World Sci-entific

Helbing D S Lammer U Witt and T Brenner (2004) ldquoNetworkndashInduced Oscillatory Behavior in Material Flow Networks and IrregularBusiness Cyclesrdquo Phisical Review E 70(056118) 1ndash6

Helbing D U Witt S Lammer and T Brenner (2004) ldquoNetwork-Induced Oscillatory Behaviour of Macroeconomic Output Flowsrdquo in In-dustry and Labour Dynamics II The Agent-based Computational Eco-nomics Approach Proceedings of the wildace 2004 conference ed byR Leombruni and M Richiardi no 5 LABORatorio R Revelli

Horvath M (1998) ldquoCyclically and Sectoral Linkages Aggregate Fluctu-ations from Independent Sectoral Shocksrdquo Review of Economic Dynam-ics 1 781ndash808

(2000) ldquoSectoral Shocks and Aggregate Fluctuationsrdquo Journal ofMonetary Economics 45(1) 69ndash106

Jovanovic B (1987) ldquoMicro Shocks and Aggregate Riskrdquo The QuarterlyJournal of Economics 102(2) 395ndash410

Kahn J A M M McConnell and G Perez-Quiros (2002) ldquoOnthe Causes of the Increased Stability of the US Economyrdquo FRBNY Eco-nomic Policy Review 8(1) 183ndash202

Kraay A and J Ventura (1998) ldquoComparative Advantage and theCrossndashSection of Business Cyclesrdquo World Bank Woorking paper Series1948 World Bank

Long J B and C I Plosser (1983) ldquoReal Business Cyclesrdquo Journalof Political Economy 31(1) 39ndash69

Lucas R E (1977) ldquoUnderstanding Business Cyclesrdquo CarnegiendashRochester Conference Series on Public Policy (special issue of Journalof Monetary Economics) 5 7ndash29

Nelson R R and S G Winter (1982) An Evolutionary Theory ofEconomic Change Harvard University Press Cambridge MA

Nirei M (2005) ldquoThreshold Behavior and Aggregate Fluctuationsrdquo Jour-nal of Economic Theory in press

29

Scheinkman J and M Woodford (1994) ldquoSelfndashOrganized Criticallyand Economic Fluctuationsrdquo American Economic Review 84(2 Papersand Proceedings of teh AEA) 417ndash421

Schuh S (1996) ldquoEvidence of the Link Between FirmndashLevel and Ag-gregate Inventory Behaviorrdquo Finance and Economics Discussion Series1996-46 Federal Reserve Board

Stiglitz J and B Greenwald (2003) Towards a New Paradigm inMonetary Economics Raffaele Mattioli Lectures Cambridge UniversityPress Cambridge

Weisbuch G and S Battiston (2005) ldquoProduction Networks andFailure Avalanchesrdquo Working Paper mimeo Ecole Normale SupErieure

Zarnovitz V (1977) ldquoBusiness Cycles Observed and Assessed Why andHow they Matterrdquo NBER Working Paper Series 6230 National Bureauof Economic Research

30

A Tables

Table 1 Parameters valuesParametera Description Value

N Number of sectorsunits i of production 10Ci Number of j buyer sectors for sector i (IO matrix sparseness) N minus 1Di (E) Value of the external demand 10mij Physical input coefficient of input j for sector i sim U(0 1)Mi Sum of the mij input coefficients j for the production of i 05αi friction to changes of production levels Degree of lockndashin on

production technology085

αsi Speed of adaptation to demand changes 004

σ Desired multiple of production to stock 10

aInto parenthesis the number of lags of the initial value for lagged variables

31

The results we obtain in this paper concerning this microndashmacro debate aremeant as a first relevant step to a more extended analysis

0

002

004

006

008

01

012

014

016

018

02

022

05 1 15 2 25 3

Var Demand

Macro (Output) and micro (Demand) volatility

delta Outputdelta Demand

Figure 8 Values of δ (relative deviation) for aggregate demand and ag-gregate output in respect of different levels of variance for sector specificuncorrelated external demand shocks

The flows of shocks affecting the firms of a system can be distinguishedalong two dimensions strength of the shock (eg amplitude of the changesin demand) and frequency of the shock (rare or frequent) Concerning thefirst aspect we initialized as usual a system to equilibrium levels (eg pro-duction quantities equal to external and intermediate demand for all sectorsconstant level of stocks equal to the desired level) We then ran the simu-lation generating the external demand for all sectors as a random variablewith constant (equilibrium) mean level and varying values of variance

Figure 8 shows unequivocally that as far as our model has propertiessimilar to real systems the hypothesis that macrondashfluctuations have largervolatility than their sectoral level shocks is fully confirmed At any levelof variance the aggregate volatility of demand as captured by the relativedeviation index is sensibly lower than the volatility shown by aggregateoutput that is the total production of the system This result is not sur-prising throughout all tests presented in the previous sections every resultsuggested that the system lsquooverndashreactsrsquo to any individual microndashlevel changeeven in the case these changes were meant to contrast disturbances There-fore when we move from considering one single shock to a flow of shockswe are not surprised to obtain that the system shows higher volatility thanthat implied by the microndashlevel disturbances

We now consider the second dimension defining a flow of shocks theirfrequency The question is whether for the same level of demand shocks

23

0 002 004 006 008 01 012 014 016

Difference of relative deviation for Output and Demand

0 5 10 15 20 25 30 35 40 45 50 55

Shock frequency 05

1

15

2

25

3

Var Demand

0 002 004 006 008 01

012 014 016

Figure 9 Differences of values of δ for aggregate output minus those foraggregate demand in respect of different levels of variance of demand and offrequency of the shocks

(ie same variance) more frequent changes trigger a higher or lower systemvolatility than rare disturbances The results shown in figure 9 suggest arather elaborated answer which partly vindicates (though in a particularsense) the arguments in favour of compensation of micro shocks The fig-ure shows the difference between aggregate output volatility and aggregatedemand volatility The independent variables are as before the volatilityof demand at microndashlevel21 and the time intervals in between two shocksAggregate demand volatility does not change for the frequency of shocksat least when computed over a long period But output volatility doesin a nonndashmonotonous way For high frequency of shocks the difference isstill positive and increasing for higher variance indicating that the ldquooverndashreactionrdquo argument is valid even in these cases but it is rather low Whilethe frequency of shocks slows down aggregate volatility of output increasesshowing that time is needed for the microndashshock to unfold their full effectsat the macrondashlevel However after a certain threshold less and less frequentshocks generate a lower volatility of the aggregate level of output

This result is quite sensible though not obvious at first sight and onceagain shows the power of such a simple model to generate and explain com-plex properties For frenetic levels of demand modifications the system hasno time to adjust but part of the job to align actual and desired productionfor constantly changing demand is performed by the varying demand itselfIn other terms the shocks cancel out through time if not through their quan-

21Beware of the scale of variance For reasons of visibility the scale is inverted

24

titative effects on the system While frequency decreases the disturbancesof the shocks not compensated by other (unlikely) shocks have the time tofilter through the production system generating higher and higher volatil-ity In a sense any system has a particular frequency of shocks that make itmaximally ldquoresonatingrdquo with the amplification of microndashshocks When therate of shocks gets even rarer the system can enjoy periods of longer andlonger quietness after each shock is absorbed and before a new one startsagain a cycle of initial local disturbance propagation of the disturbances toother sectors and settlement to the new equilibrium

6 Discussion and extensions

We presented a simple model of production composed by interacting sectorThe main (quit realistic) assumption of the model is that the productionprocesses in any sector of the economy make use of input from any othersector Moreover we assumed that sectors (represented as a single firm)adjust slowly both the desired level of production (smoothed revision of ex-pectation) and the actual level (smoothed modification of plant utilization)The model voluntarily leaves aside a large number of aspects of real-worldeconomic systems (eg financial markets price adjustments growth etc)The reason is that the model can be easily and reliably tested in order toextract its properties The possibility to transfer the (certain) model resultto a real context depend of course on an ample ceteris paribus clause How-ever we are confident that the reported results concerning the productionstructure play a relevant role in real world systems Therefore our resultsand particularly their motivations being accessible because of the relativesimplicity of the model will not be diminished by the integration of ourconsiderations with other aspects of economic systems

Our model is in essence a representation of an input-output table inter-preted in physical terms that is as the amount of inputs required for oneunit of output Besides this we include two straightforward assumptionsFirstly production levels cannot be suddenly changed but require time toscale up or down the utilization rate that is the levels of production Stocksare meant to make up for the difference between sales and production Thesecond assumption concerns the intentional behaviour of producers We im-plement this aspect by assuming that exists a level of stocks desired for eachlevel of demand However a change of demand does not translate suddenlyin a new level of desired stock but again only a long time at a different levelof demand will convince managers to update the desired levels of stocks

We show that the model generate under a wide number of settings oscil-latory patterns converging toward the equilibrium level where each sectorgenerate exactly the quantity demanded by final consumers and by othersectors as input The oscillations are provoked by over- and under-shooting

25

generated by the difficulty of coordination among sectors that by assump-tion ldquocommunicaterdquo only via variation of demand to their direct input sup-pliers We also show that the total production levels generated by sectors(sum of the physical units) produced at equilibrium depends (for constantfinal demand) on the total sum of the technical coefficients per sector irre-spective of their distribution This result is motivated by the impossibilityof input substitution and offers interesting suggestions for the analysis ofthe study of input-output tables of real-world systems

We then consider systems at equilibrium facing a single small change ofthe external demand for sectors These simulations allow to study the waythe system reacts to shocks We see that the sum of the coefficients affectsbesides the levels also the volatility of aggregate output the larger a systemthe stronger is the impact of a given shock Less obviously we find thatfor systems with the same dimension (ie same sums of coefficients) theaggregate volatility increases with both the inertia to change of productionlevel and the speed of adjustment of desired stocks This result is somewhatcounter-intuitive since both the cited variables are associated to a moreconservative behaviour that may be supposed should smooth away spikesin the pattern from an old to a new equilibrium However we show thatthe opposite actually holds given the complex interactions among sectorindividual-sector attitude intended to smooth shocks actually increase theiraggregate impact

In the same vein we show that systems maintaining high levels of stocksare more volatile that systems with reduced amounts of stocks Again thisis a counter-intuitive result reverting the goal of individual producers andthe aggregate result Interestingly this result finds strong support from theevidence of the recent changes in production methods (ie low or no stocks)and diminished cycles

In relation to specific issues discussed in the literature we find no supportto the suggestion that particular distributions of input-output coefficientsmodify the volatility of a system As said this depends in case on thesums of these coefficients but not on their distribution Further we canprovide an answer to a long and hotly debated questions whether microndashlevels volatility generates stronger or lower aggregate volatility Our modelshows unequivocally that microndashvolatility (eg variance of final demand)persists and generates stronger volatility at aggregate level Again we canidentify the reason on the complex production structure that exalts micro-level shocks by generating lsquowrongrsquo messages among producers Moreoverwe can also point to a usually neglected aspect of volatility we show thatthe frequency of shocks is a very relevant aspect besides their dimensionin determining the volatility of aggregate output

Our work can be developed along two complementary directions Firstlywe can continue to study the properties of the model by extending the ele-ments considered (eg pricing) analysing the effects on the present results

26

and generating more analytical evidence Of particular relevance is in ouropinion the problem of aggregation Representations of the same systemswith different codification systems (eg different number of SIC digits)must not modify overall system properties for example the volatility ob-served Given that our result show that the sums of coefficients matter forthe levels and volatility of the system and that these sum do depend on theaggregation chosen it should be possible to induce the reaction coefficientsthat we know affect the volatility only

Secondly we can find an application of our results to real world evidenceIn fact there are many data sets available for the data we deal with basicallyinput-output tables Our model is readily adapt to be parametrized alongany number of sector and coefficient necessary so that we may replicate pastseries in order to adapt the unobservable parameters and use the resultingcomplete model to explain business fluctuations and provide intuitions onfuture patterns

27

References

Acemoglu D and A Scott (1997) ldquoAsymmetric Business CyclesTheory and TimendashSeries Evidencerdquo Journal of Monetary Economics 40501ndash533

Backus D P Kehoe and F Kydland (1995) ldquoInternational BusinessCycles Theory and Evidencerdquo in Frontiers of Business Cycle Researched by T Cooley Princeton University Press Princeton

Bak P K Chen J Scheinkman and M Woodford (1993) ldquoAg-gregate Fluctuations from Independent Sectoral Shocks SelfndashOrganizedCritically in a Model of Production and Inventory Dynamicsrdquo RicercheEconomiche 47(1) 3ndash30

Battiston S D Delli Gatti M Gallegati B Greenwald and

J Stiglitz (2005) ldquoCredit Chains and Bankruptcies Avalanches in Sup-ply Networksrdquo in Annual Workshop on Economic Heterogeneous Inter-acting Agents (WEHIA 2005) University of Essex UK

Blanchard O and J Simon (2001) ldquoThe Long and Large Declinein US Output Volatilityrdquo Brookings Papers on Economic Activity 1135ndash174

Bresnahan T F and V A Ramey (1994) ldquoOutput Fluctuations atthe Plant Levelrdquo The Quarterly Journal of Economics 109(3) 593ndash924

David P A (1975) Technical choice innovation and economic growth essays on American and British experience in the nineteenth centuryCambridge University Press London

Delli Gatti D C Di Guilmi E Gaffeo G Giulioni M Galle-

gati and A Palestrini (2004) ldquoA New Approach to Business Fluc-tuations Heterogeneous Interacting Agents Scaling Laws and FinancialFragilityrdquo Journal of Economic Behavior amp Organization 56 489ndash512

Dosi G G Fagiolo and A Roventini (2006) ldquoAn EvolutionaryModel of Endogenous Business Cyclesrdquo Computational Economics 273ndash34

Dupor B (1999) ldquoAggregation and the Irrelevance in MultindashSector Mod-elsrdquo Journal of Monetary Economics 43 391ndash409

Head A C (1995) ldquoCountry Size Aggregate Fluctuations and Interna-tional Risk Sharingrdquo The Canadian Journal of Economics 28(4b) 1096ndash1119

28

Helbing D and S Lammer (2005) ldquoSupply and Production NetworksFrom the Bullwhip Effect to Business Cyclesrdquo in Networks of Interactingmachines Production Organization in Complex Industrial Systems andBiological Cells ed by D Armbruster K Kaneko and A S Mikhailovvol 3 of World Scientific Lecture Notes in Complex Systems World Sci-entific

Helbing D S Lammer U Witt and T Brenner (2004) ldquoNetworkndashInduced Oscillatory Behavior in Material Flow Networks and IrregularBusiness Cyclesrdquo Phisical Review E 70(056118) 1ndash6

Helbing D U Witt S Lammer and T Brenner (2004) ldquoNetwork-Induced Oscillatory Behaviour of Macroeconomic Output Flowsrdquo in In-dustry and Labour Dynamics II The Agent-based Computational Eco-nomics Approach Proceedings of the wildace 2004 conference ed byR Leombruni and M Richiardi no 5 LABORatorio R Revelli

Horvath M (1998) ldquoCyclically and Sectoral Linkages Aggregate Fluctu-ations from Independent Sectoral Shocksrdquo Review of Economic Dynam-ics 1 781ndash808

(2000) ldquoSectoral Shocks and Aggregate Fluctuationsrdquo Journal ofMonetary Economics 45(1) 69ndash106

Jovanovic B (1987) ldquoMicro Shocks and Aggregate Riskrdquo The QuarterlyJournal of Economics 102(2) 395ndash410

Kahn J A M M McConnell and G Perez-Quiros (2002) ldquoOnthe Causes of the Increased Stability of the US Economyrdquo FRBNY Eco-nomic Policy Review 8(1) 183ndash202

Kraay A and J Ventura (1998) ldquoComparative Advantage and theCrossndashSection of Business Cyclesrdquo World Bank Woorking paper Series1948 World Bank

Long J B and C I Plosser (1983) ldquoReal Business Cyclesrdquo Journalof Political Economy 31(1) 39ndash69

Lucas R E (1977) ldquoUnderstanding Business Cyclesrdquo CarnegiendashRochester Conference Series on Public Policy (special issue of Journalof Monetary Economics) 5 7ndash29

Nelson R R and S G Winter (1982) An Evolutionary Theory ofEconomic Change Harvard University Press Cambridge MA

Nirei M (2005) ldquoThreshold Behavior and Aggregate Fluctuationsrdquo Jour-nal of Economic Theory in press

29

Scheinkman J and M Woodford (1994) ldquoSelfndashOrganized Criticallyand Economic Fluctuationsrdquo American Economic Review 84(2 Papersand Proceedings of teh AEA) 417ndash421

Schuh S (1996) ldquoEvidence of the Link Between FirmndashLevel and Ag-gregate Inventory Behaviorrdquo Finance and Economics Discussion Series1996-46 Federal Reserve Board

Stiglitz J and B Greenwald (2003) Towards a New Paradigm inMonetary Economics Raffaele Mattioli Lectures Cambridge UniversityPress Cambridge

Weisbuch G and S Battiston (2005) ldquoProduction Networks andFailure Avalanchesrdquo Working Paper mimeo Ecole Normale SupErieure

Zarnovitz V (1977) ldquoBusiness Cycles Observed and Assessed Why andHow they Matterrdquo NBER Working Paper Series 6230 National Bureauof Economic Research

30

A Tables

Table 1 Parameters valuesParametera Description Value

N Number of sectorsunits i of production 10Ci Number of j buyer sectors for sector i (IO matrix sparseness) N minus 1Di (E) Value of the external demand 10mij Physical input coefficient of input j for sector i sim U(0 1)Mi Sum of the mij input coefficients j for the production of i 05αi friction to changes of production levels Degree of lockndashin on

production technology085

αsi Speed of adaptation to demand changes 004

σ Desired multiple of production to stock 10

aInto parenthesis the number of lags of the initial value for lagged variables

31

0 002 004 006 008 01 012 014 016

Difference of relative deviation for Output and Demand

0 5 10 15 20 25 30 35 40 45 50 55

Shock frequency 05

1

15

2

25

3

Var Demand

0 002 004 006 008 01

012 014 016

Figure 9 Differences of values of δ for aggregate output minus those foraggregate demand in respect of different levels of variance of demand and offrequency of the shocks

(ie same variance) more frequent changes trigger a higher or lower systemvolatility than rare disturbances The results shown in figure 9 suggest arather elaborated answer which partly vindicates (though in a particularsense) the arguments in favour of compensation of micro shocks The fig-ure shows the difference between aggregate output volatility and aggregatedemand volatility The independent variables are as before the volatilityof demand at microndashlevel21 and the time intervals in between two shocksAggregate demand volatility does not change for the frequency of shocksat least when computed over a long period But output volatility doesin a nonndashmonotonous way For high frequency of shocks the difference isstill positive and increasing for higher variance indicating that the ldquooverndashreactionrdquo argument is valid even in these cases but it is rather low Whilethe frequency of shocks slows down aggregate volatility of output increasesshowing that time is needed for the microndashshock to unfold their full effectsat the macrondashlevel However after a certain threshold less and less frequentshocks generate a lower volatility of the aggregate level of output

This result is quite sensible though not obvious at first sight and onceagain shows the power of such a simple model to generate and explain com-plex properties For frenetic levels of demand modifications the system hasno time to adjust but part of the job to align actual and desired productionfor constantly changing demand is performed by the varying demand itselfIn other terms the shocks cancel out through time if not through their quan-

21Beware of the scale of variance For reasons of visibility the scale is inverted

24

titative effects on the system While frequency decreases the disturbancesof the shocks not compensated by other (unlikely) shocks have the time tofilter through the production system generating higher and higher volatil-ity In a sense any system has a particular frequency of shocks that make itmaximally ldquoresonatingrdquo with the amplification of microndashshocks When therate of shocks gets even rarer the system can enjoy periods of longer andlonger quietness after each shock is absorbed and before a new one startsagain a cycle of initial local disturbance propagation of the disturbances toother sectors and settlement to the new equilibrium

6 Discussion and extensions

We presented a simple model of production composed by interacting sectorThe main (quit realistic) assumption of the model is that the productionprocesses in any sector of the economy make use of input from any othersector Moreover we assumed that sectors (represented as a single firm)adjust slowly both the desired level of production (smoothed revision of ex-pectation) and the actual level (smoothed modification of plant utilization)The model voluntarily leaves aside a large number of aspects of real-worldeconomic systems (eg financial markets price adjustments growth etc)The reason is that the model can be easily and reliably tested in order toextract its properties The possibility to transfer the (certain) model resultto a real context depend of course on an ample ceteris paribus clause How-ever we are confident that the reported results concerning the productionstructure play a relevant role in real world systems Therefore our resultsand particularly their motivations being accessible because of the relativesimplicity of the model will not be diminished by the integration of ourconsiderations with other aspects of economic systems

Our model is in essence a representation of an input-output table inter-preted in physical terms that is as the amount of inputs required for oneunit of output Besides this we include two straightforward assumptionsFirstly production levels cannot be suddenly changed but require time toscale up or down the utilization rate that is the levels of production Stocksare meant to make up for the difference between sales and production Thesecond assumption concerns the intentional behaviour of producers We im-plement this aspect by assuming that exists a level of stocks desired for eachlevel of demand However a change of demand does not translate suddenlyin a new level of desired stock but again only a long time at a different levelof demand will convince managers to update the desired levels of stocks

We show that the model generate under a wide number of settings oscil-latory patterns converging toward the equilibrium level where each sectorgenerate exactly the quantity demanded by final consumers and by othersectors as input The oscillations are provoked by over- and under-shooting

25

generated by the difficulty of coordination among sectors that by assump-tion ldquocommunicaterdquo only via variation of demand to their direct input sup-pliers We also show that the total production levels generated by sectors(sum of the physical units) produced at equilibrium depends (for constantfinal demand) on the total sum of the technical coefficients per sector irre-spective of their distribution This result is motivated by the impossibilityof input substitution and offers interesting suggestions for the analysis ofthe study of input-output tables of real-world systems

We then consider systems at equilibrium facing a single small change ofthe external demand for sectors These simulations allow to study the waythe system reacts to shocks We see that the sum of the coefficients affectsbesides the levels also the volatility of aggregate output the larger a systemthe stronger is the impact of a given shock Less obviously we find thatfor systems with the same dimension (ie same sums of coefficients) theaggregate volatility increases with both the inertia to change of productionlevel and the speed of adjustment of desired stocks This result is somewhatcounter-intuitive since both the cited variables are associated to a moreconservative behaviour that may be supposed should smooth away spikesin the pattern from an old to a new equilibrium However we show thatthe opposite actually holds given the complex interactions among sectorindividual-sector attitude intended to smooth shocks actually increase theiraggregate impact

In the same vein we show that systems maintaining high levels of stocksare more volatile that systems with reduced amounts of stocks Again thisis a counter-intuitive result reverting the goal of individual producers andthe aggregate result Interestingly this result finds strong support from theevidence of the recent changes in production methods (ie low or no stocks)and diminished cycles

In relation to specific issues discussed in the literature we find no supportto the suggestion that particular distributions of input-output coefficientsmodify the volatility of a system As said this depends in case on thesums of these coefficients but not on their distribution Further we canprovide an answer to a long and hotly debated questions whether microndashlevels volatility generates stronger or lower aggregate volatility Our modelshows unequivocally that microndashvolatility (eg variance of final demand)persists and generates stronger volatility at aggregate level Again we canidentify the reason on the complex production structure that exalts micro-level shocks by generating lsquowrongrsquo messages among producers Moreoverwe can also point to a usually neglected aspect of volatility we show thatthe frequency of shocks is a very relevant aspect besides their dimensionin determining the volatility of aggregate output

Our work can be developed along two complementary directions Firstlywe can continue to study the properties of the model by extending the ele-ments considered (eg pricing) analysing the effects on the present results

26

and generating more analytical evidence Of particular relevance is in ouropinion the problem of aggregation Representations of the same systemswith different codification systems (eg different number of SIC digits)must not modify overall system properties for example the volatility ob-served Given that our result show that the sums of coefficients matter forthe levels and volatility of the system and that these sum do depend on theaggregation chosen it should be possible to induce the reaction coefficientsthat we know affect the volatility only

Secondly we can find an application of our results to real world evidenceIn fact there are many data sets available for the data we deal with basicallyinput-output tables Our model is readily adapt to be parametrized alongany number of sector and coefficient necessary so that we may replicate pastseries in order to adapt the unobservable parameters and use the resultingcomplete model to explain business fluctuations and provide intuitions onfuture patterns

27

References

Acemoglu D and A Scott (1997) ldquoAsymmetric Business CyclesTheory and TimendashSeries Evidencerdquo Journal of Monetary Economics 40501ndash533

Backus D P Kehoe and F Kydland (1995) ldquoInternational BusinessCycles Theory and Evidencerdquo in Frontiers of Business Cycle Researched by T Cooley Princeton University Press Princeton

Bak P K Chen J Scheinkman and M Woodford (1993) ldquoAg-gregate Fluctuations from Independent Sectoral Shocks SelfndashOrganizedCritically in a Model of Production and Inventory Dynamicsrdquo RicercheEconomiche 47(1) 3ndash30

Battiston S D Delli Gatti M Gallegati B Greenwald and

J Stiglitz (2005) ldquoCredit Chains and Bankruptcies Avalanches in Sup-ply Networksrdquo in Annual Workshop on Economic Heterogeneous Inter-acting Agents (WEHIA 2005) University of Essex UK

Blanchard O and J Simon (2001) ldquoThe Long and Large Declinein US Output Volatilityrdquo Brookings Papers on Economic Activity 1135ndash174

Bresnahan T F and V A Ramey (1994) ldquoOutput Fluctuations atthe Plant Levelrdquo The Quarterly Journal of Economics 109(3) 593ndash924

David P A (1975) Technical choice innovation and economic growth essays on American and British experience in the nineteenth centuryCambridge University Press London

Delli Gatti D C Di Guilmi E Gaffeo G Giulioni M Galle-

gati and A Palestrini (2004) ldquoA New Approach to Business Fluc-tuations Heterogeneous Interacting Agents Scaling Laws and FinancialFragilityrdquo Journal of Economic Behavior amp Organization 56 489ndash512

Dosi G G Fagiolo and A Roventini (2006) ldquoAn EvolutionaryModel of Endogenous Business Cyclesrdquo Computational Economics 273ndash34

Dupor B (1999) ldquoAggregation and the Irrelevance in MultindashSector Mod-elsrdquo Journal of Monetary Economics 43 391ndash409

Head A C (1995) ldquoCountry Size Aggregate Fluctuations and Interna-tional Risk Sharingrdquo The Canadian Journal of Economics 28(4b) 1096ndash1119

28

Helbing D and S Lammer (2005) ldquoSupply and Production NetworksFrom the Bullwhip Effect to Business Cyclesrdquo in Networks of Interactingmachines Production Organization in Complex Industrial Systems andBiological Cells ed by D Armbruster K Kaneko and A S Mikhailovvol 3 of World Scientific Lecture Notes in Complex Systems World Sci-entific

Helbing D S Lammer U Witt and T Brenner (2004) ldquoNetworkndashInduced Oscillatory Behavior in Material Flow Networks and IrregularBusiness Cyclesrdquo Phisical Review E 70(056118) 1ndash6

Helbing D U Witt S Lammer and T Brenner (2004) ldquoNetwork-Induced Oscillatory Behaviour of Macroeconomic Output Flowsrdquo in In-dustry and Labour Dynamics II The Agent-based Computational Eco-nomics Approach Proceedings of the wildace 2004 conference ed byR Leombruni and M Richiardi no 5 LABORatorio R Revelli

Horvath M (1998) ldquoCyclically and Sectoral Linkages Aggregate Fluctu-ations from Independent Sectoral Shocksrdquo Review of Economic Dynam-ics 1 781ndash808

(2000) ldquoSectoral Shocks and Aggregate Fluctuationsrdquo Journal ofMonetary Economics 45(1) 69ndash106

Jovanovic B (1987) ldquoMicro Shocks and Aggregate Riskrdquo The QuarterlyJournal of Economics 102(2) 395ndash410

Kahn J A M M McConnell and G Perez-Quiros (2002) ldquoOnthe Causes of the Increased Stability of the US Economyrdquo FRBNY Eco-nomic Policy Review 8(1) 183ndash202

Kraay A and J Ventura (1998) ldquoComparative Advantage and theCrossndashSection of Business Cyclesrdquo World Bank Woorking paper Series1948 World Bank

Long J B and C I Plosser (1983) ldquoReal Business Cyclesrdquo Journalof Political Economy 31(1) 39ndash69

Lucas R E (1977) ldquoUnderstanding Business Cyclesrdquo CarnegiendashRochester Conference Series on Public Policy (special issue of Journalof Monetary Economics) 5 7ndash29

Nelson R R and S G Winter (1982) An Evolutionary Theory ofEconomic Change Harvard University Press Cambridge MA

Nirei M (2005) ldquoThreshold Behavior and Aggregate Fluctuationsrdquo Jour-nal of Economic Theory in press

29

Scheinkman J and M Woodford (1994) ldquoSelfndashOrganized Criticallyand Economic Fluctuationsrdquo American Economic Review 84(2 Papersand Proceedings of teh AEA) 417ndash421

Schuh S (1996) ldquoEvidence of the Link Between FirmndashLevel and Ag-gregate Inventory Behaviorrdquo Finance and Economics Discussion Series1996-46 Federal Reserve Board

Stiglitz J and B Greenwald (2003) Towards a New Paradigm inMonetary Economics Raffaele Mattioli Lectures Cambridge UniversityPress Cambridge

Weisbuch G and S Battiston (2005) ldquoProduction Networks andFailure Avalanchesrdquo Working Paper mimeo Ecole Normale SupErieure

Zarnovitz V (1977) ldquoBusiness Cycles Observed and Assessed Why andHow they Matterrdquo NBER Working Paper Series 6230 National Bureauof Economic Research

30

A Tables

Table 1 Parameters valuesParametera Description Value

N Number of sectorsunits i of production 10Ci Number of j buyer sectors for sector i (IO matrix sparseness) N minus 1Di (E) Value of the external demand 10mij Physical input coefficient of input j for sector i sim U(0 1)Mi Sum of the mij input coefficients j for the production of i 05αi friction to changes of production levels Degree of lockndashin on

production technology085

αsi Speed of adaptation to demand changes 004

σ Desired multiple of production to stock 10

aInto parenthesis the number of lags of the initial value for lagged variables

31

titative effects on the system While frequency decreases the disturbancesof the shocks not compensated by other (unlikely) shocks have the time tofilter through the production system generating higher and higher volatil-ity In a sense any system has a particular frequency of shocks that make itmaximally ldquoresonatingrdquo with the amplification of microndashshocks When therate of shocks gets even rarer the system can enjoy periods of longer andlonger quietness after each shock is absorbed and before a new one startsagain a cycle of initial local disturbance propagation of the disturbances toother sectors and settlement to the new equilibrium

6 Discussion and extensions

We presented a simple model of production composed by interacting sectorThe main (quit realistic) assumption of the model is that the productionprocesses in any sector of the economy make use of input from any othersector Moreover we assumed that sectors (represented as a single firm)adjust slowly both the desired level of production (smoothed revision of ex-pectation) and the actual level (smoothed modification of plant utilization)The model voluntarily leaves aside a large number of aspects of real-worldeconomic systems (eg financial markets price adjustments growth etc)The reason is that the model can be easily and reliably tested in order toextract its properties The possibility to transfer the (certain) model resultto a real context depend of course on an ample ceteris paribus clause How-ever we are confident that the reported results concerning the productionstructure play a relevant role in real world systems Therefore our resultsand particularly their motivations being accessible because of the relativesimplicity of the model will not be diminished by the integration of ourconsiderations with other aspects of economic systems

Our model is in essence a representation of an input-output table inter-preted in physical terms that is as the amount of inputs required for oneunit of output Besides this we include two straightforward assumptionsFirstly production levels cannot be suddenly changed but require time toscale up or down the utilization rate that is the levels of production Stocksare meant to make up for the difference between sales and production Thesecond assumption concerns the intentional behaviour of producers We im-plement this aspect by assuming that exists a level of stocks desired for eachlevel of demand However a change of demand does not translate suddenlyin a new level of desired stock but again only a long time at a different levelof demand will convince managers to update the desired levels of stocks

We show that the model generate under a wide number of settings oscil-latory patterns converging toward the equilibrium level where each sectorgenerate exactly the quantity demanded by final consumers and by othersectors as input The oscillations are provoked by over- and under-shooting

25

generated by the difficulty of coordination among sectors that by assump-tion ldquocommunicaterdquo only via variation of demand to their direct input sup-pliers We also show that the total production levels generated by sectors(sum of the physical units) produced at equilibrium depends (for constantfinal demand) on the total sum of the technical coefficients per sector irre-spective of their distribution This result is motivated by the impossibilityof input substitution and offers interesting suggestions for the analysis ofthe study of input-output tables of real-world systems

We then consider systems at equilibrium facing a single small change ofthe external demand for sectors These simulations allow to study the waythe system reacts to shocks We see that the sum of the coefficients affectsbesides the levels also the volatility of aggregate output the larger a systemthe stronger is the impact of a given shock Less obviously we find thatfor systems with the same dimension (ie same sums of coefficients) theaggregate volatility increases with both the inertia to change of productionlevel and the speed of adjustment of desired stocks This result is somewhatcounter-intuitive since both the cited variables are associated to a moreconservative behaviour that may be supposed should smooth away spikesin the pattern from an old to a new equilibrium However we show thatthe opposite actually holds given the complex interactions among sectorindividual-sector attitude intended to smooth shocks actually increase theiraggregate impact

In the same vein we show that systems maintaining high levels of stocksare more volatile that systems with reduced amounts of stocks Again thisis a counter-intuitive result reverting the goal of individual producers andthe aggregate result Interestingly this result finds strong support from theevidence of the recent changes in production methods (ie low or no stocks)and diminished cycles

In relation to specific issues discussed in the literature we find no supportto the suggestion that particular distributions of input-output coefficientsmodify the volatility of a system As said this depends in case on thesums of these coefficients but not on their distribution Further we canprovide an answer to a long and hotly debated questions whether microndashlevels volatility generates stronger or lower aggregate volatility Our modelshows unequivocally that microndashvolatility (eg variance of final demand)persists and generates stronger volatility at aggregate level Again we canidentify the reason on the complex production structure that exalts micro-level shocks by generating lsquowrongrsquo messages among producers Moreoverwe can also point to a usually neglected aspect of volatility we show thatthe frequency of shocks is a very relevant aspect besides their dimensionin determining the volatility of aggregate output

Our work can be developed along two complementary directions Firstlywe can continue to study the properties of the model by extending the ele-ments considered (eg pricing) analysing the effects on the present results

26

and generating more analytical evidence Of particular relevance is in ouropinion the problem of aggregation Representations of the same systemswith different codification systems (eg different number of SIC digits)must not modify overall system properties for example the volatility ob-served Given that our result show that the sums of coefficients matter forthe levels and volatility of the system and that these sum do depend on theaggregation chosen it should be possible to induce the reaction coefficientsthat we know affect the volatility only

Secondly we can find an application of our results to real world evidenceIn fact there are many data sets available for the data we deal with basicallyinput-output tables Our model is readily adapt to be parametrized alongany number of sector and coefficient necessary so that we may replicate pastseries in order to adapt the unobservable parameters and use the resultingcomplete model to explain business fluctuations and provide intuitions onfuture patterns

27

References

Acemoglu D and A Scott (1997) ldquoAsymmetric Business CyclesTheory and TimendashSeries Evidencerdquo Journal of Monetary Economics 40501ndash533

Backus D P Kehoe and F Kydland (1995) ldquoInternational BusinessCycles Theory and Evidencerdquo in Frontiers of Business Cycle Researched by T Cooley Princeton University Press Princeton

Bak P K Chen J Scheinkman and M Woodford (1993) ldquoAg-gregate Fluctuations from Independent Sectoral Shocks SelfndashOrganizedCritically in a Model of Production and Inventory Dynamicsrdquo RicercheEconomiche 47(1) 3ndash30

Battiston S D Delli Gatti M Gallegati B Greenwald and

J Stiglitz (2005) ldquoCredit Chains and Bankruptcies Avalanches in Sup-ply Networksrdquo in Annual Workshop on Economic Heterogeneous Inter-acting Agents (WEHIA 2005) University of Essex UK

Blanchard O and J Simon (2001) ldquoThe Long and Large Declinein US Output Volatilityrdquo Brookings Papers on Economic Activity 1135ndash174

Bresnahan T F and V A Ramey (1994) ldquoOutput Fluctuations atthe Plant Levelrdquo The Quarterly Journal of Economics 109(3) 593ndash924

David P A (1975) Technical choice innovation and economic growth essays on American and British experience in the nineteenth centuryCambridge University Press London

Delli Gatti D C Di Guilmi E Gaffeo G Giulioni M Galle-

gati and A Palestrini (2004) ldquoA New Approach to Business Fluc-tuations Heterogeneous Interacting Agents Scaling Laws and FinancialFragilityrdquo Journal of Economic Behavior amp Organization 56 489ndash512

Dosi G G Fagiolo and A Roventini (2006) ldquoAn EvolutionaryModel of Endogenous Business Cyclesrdquo Computational Economics 273ndash34

Dupor B (1999) ldquoAggregation and the Irrelevance in MultindashSector Mod-elsrdquo Journal of Monetary Economics 43 391ndash409

Head A C (1995) ldquoCountry Size Aggregate Fluctuations and Interna-tional Risk Sharingrdquo The Canadian Journal of Economics 28(4b) 1096ndash1119

28

Helbing D and S Lammer (2005) ldquoSupply and Production NetworksFrom the Bullwhip Effect to Business Cyclesrdquo in Networks of Interactingmachines Production Organization in Complex Industrial Systems andBiological Cells ed by D Armbruster K Kaneko and A S Mikhailovvol 3 of World Scientific Lecture Notes in Complex Systems World Sci-entific

Helbing D S Lammer U Witt and T Brenner (2004) ldquoNetworkndashInduced Oscillatory Behavior in Material Flow Networks and IrregularBusiness Cyclesrdquo Phisical Review E 70(056118) 1ndash6

Helbing D U Witt S Lammer and T Brenner (2004) ldquoNetwork-Induced Oscillatory Behaviour of Macroeconomic Output Flowsrdquo in In-dustry and Labour Dynamics II The Agent-based Computational Eco-nomics Approach Proceedings of the wildace 2004 conference ed byR Leombruni and M Richiardi no 5 LABORatorio R Revelli

Horvath M (1998) ldquoCyclically and Sectoral Linkages Aggregate Fluctu-ations from Independent Sectoral Shocksrdquo Review of Economic Dynam-ics 1 781ndash808

(2000) ldquoSectoral Shocks and Aggregate Fluctuationsrdquo Journal ofMonetary Economics 45(1) 69ndash106

Jovanovic B (1987) ldquoMicro Shocks and Aggregate Riskrdquo The QuarterlyJournal of Economics 102(2) 395ndash410

Kahn J A M M McConnell and G Perez-Quiros (2002) ldquoOnthe Causes of the Increased Stability of the US Economyrdquo FRBNY Eco-nomic Policy Review 8(1) 183ndash202

Kraay A and J Ventura (1998) ldquoComparative Advantage and theCrossndashSection of Business Cyclesrdquo World Bank Woorking paper Series1948 World Bank

Long J B and C I Plosser (1983) ldquoReal Business Cyclesrdquo Journalof Political Economy 31(1) 39ndash69

Lucas R E (1977) ldquoUnderstanding Business Cyclesrdquo CarnegiendashRochester Conference Series on Public Policy (special issue of Journalof Monetary Economics) 5 7ndash29

Nelson R R and S G Winter (1982) An Evolutionary Theory ofEconomic Change Harvard University Press Cambridge MA

Nirei M (2005) ldquoThreshold Behavior and Aggregate Fluctuationsrdquo Jour-nal of Economic Theory in press

29

Scheinkman J and M Woodford (1994) ldquoSelfndashOrganized Criticallyand Economic Fluctuationsrdquo American Economic Review 84(2 Papersand Proceedings of teh AEA) 417ndash421

Schuh S (1996) ldquoEvidence of the Link Between FirmndashLevel and Ag-gregate Inventory Behaviorrdquo Finance and Economics Discussion Series1996-46 Federal Reserve Board

Stiglitz J and B Greenwald (2003) Towards a New Paradigm inMonetary Economics Raffaele Mattioli Lectures Cambridge UniversityPress Cambridge

Weisbuch G and S Battiston (2005) ldquoProduction Networks andFailure Avalanchesrdquo Working Paper mimeo Ecole Normale SupErieure

Zarnovitz V (1977) ldquoBusiness Cycles Observed and Assessed Why andHow they Matterrdquo NBER Working Paper Series 6230 National Bureauof Economic Research

30

A Tables

Table 1 Parameters valuesParametera Description Value

N Number of sectorsunits i of production 10Ci Number of j buyer sectors for sector i (IO matrix sparseness) N minus 1Di (E) Value of the external demand 10mij Physical input coefficient of input j for sector i sim U(0 1)Mi Sum of the mij input coefficients j for the production of i 05αi friction to changes of production levels Degree of lockndashin on

production technology085

αsi Speed of adaptation to demand changes 004

σ Desired multiple of production to stock 10

aInto parenthesis the number of lags of the initial value for lagged variables

31

generated by the difficulty of coordination among sectors that by assump-tion ldquocommunicaterdquo only via variation of demand to their direct input sup-pliers We also show that the total production levels generated by sectors(sum of the physical units) produced at equilibrium depends (for constantfinal demand) on the total sum of the technical coefficients per sector irre-spective of their distribution This result is motivated by the impossibilityof input substitution and offers interesting suggestions for the analysis ofthe study of input-output tables of real-world systems

We then consider systems at equilibrium facing a single small change ofthe external demand for sectors These simulations allow to study the waythe system reacts to shocks We see that the sum of the coefficients affectsbesides the levels also the volatility of aggregate output the larger a systemthe stronger is the impact of a given shock Less obviously we find thatfor systems with the same dimension (ie same sums of coefficients) theaggregate volatility increases with both the inertia to change of productionlevel and the speed of adjustment of desired stocks This result is somewhatcounter-intuitive since both the cited variables are associated to a moreconservative behaviour that may be supposed should smooth away spikesin the pattern from an old to a new equilibrium However we show thatthe opposite actually holds given the complex interactions among sectorindividual-sector attitude intended to smooth shocks actually increase theiraggregate impact

In the same vein we show that systems maintaining high levels of stocksare more volatile that systems with reduced amounts of stocks Again thisis a counter-intuitive result reverting the goal of individual producers andthe aggregate result Interestingly this result finds strong support from theevidence of the recent changes in production methods (ie low or no stocks)and diminished cycles

In relation to specific issues discussed in the literature we find no supportto the suggestion that particular distributions of input-output coefficientsmodify the volatility of a system As said this depends in case on thesums of these coefficients but not on their distribution Further we canprovide an answer to a long and hotly debated questions whether microndashlevels volatility generates stronger or lower aggregate volatility Our modelshows unequivocally that microndashvolatility (eg variance of final demand)persists and generates stronger volatility at aggregate level Again we canidentify the reason on the complex production structure that exalts micro-level shocks by generating lsquowrongrsquo messages among producers Moreoverwe can also point to a usually neglected aspect of volatility we show thatthe frequency of shocks is a very relevant aspect besides their dimensionin determining the volatility of aggregate output

Our work can be developed along two complementary directions Firstlywe can continue to study the properties of the model by extending the ele-ments considered (eg pricing) analysing the effects on the present results

26

and generating more analytical evidence Of particular relevance is in ouropinion the problem of aggregation Representations of the same systemswith different codification systems (eg different number of SIC digits)must not modify overall system properties for example the volatility ob-served Given that our result show that the sums of coefficients matter forthe levels and volatility of the system and that these sum do depend on theaggregation chosen it should be possible to induce the reaction coefficientsthat we know affect the volatility only

Secondly we can find an application of our results to real world evidenceIn fact there are many data sets available for the data we deal with basicallyinput-output tables Our model is readily adapt to be parametrized alongany number of sector and coefficient necessary so that we may replicate pastseries in order to adapt the unobservable parameters and use the resultingcomplete model to explain business fluctuations and provide intuitions onfuture patterns

27

References

Acemoglu D and A Scott (1997) ldquoAsymmetric Business CyclesTheory and TimendashSeries Evidencerdquo Journal of Monetary Economics 40501ndash533

Backus D P Kehoe and F Kydland (1995) ldquoInternational BusinessCycles Theory and Evidencerdquo in Frontiers of Business Cycle Researched by T Cooley Princeton University Press Princeton

Bak P K Chen J Scheinkman and M Woodford (1993) ldquoAg-gregate Fluctuations from Independent Sectoral Shocks SelfndashOrganizedCritically in a Model of Production and Inventory Dynamicsrdquo RicercheEconomiche 47(1) 3ndash30

Battiston S D Delli Gatti M Gallegati B Greenwald and

J Stiglitz (2005) ldquoCredit Chains and Bankruptcies Avalanches in Sup-ply Networksrdquo in Annual Workshop on Economic Heterogeneous Inter-acting Agents (WEHIA 2005) University of Essex UK

Blanchard O and J Simon (2001) ldquoThe Long and Large Declinein US Output Volatilityrdquo Brookings Papers on Economic Activity 1135ndash174

Bresnahan T F and V A Ramey (1994) ldquoOutput Fluctuations atthe Plant Levelrdquo The Quarterly Journal of Economics 109(3) 593ndash924

David P A (1975) Technical choice innovation and economic growth essays on American and British experience in the nineteenth centuryCambridge University Press London

Delli Gatti D C Di Guilmi E Gaffeo G Giulioni M Galle-

gati and A Palestrini (2004) ldquoA New Approach to Business Fluc-tuations Heterogeneous Interacting Agents Scaling Laws and FinancialFragilityrdquo Journal of Economic Behavior amp Organization 56 489ndash512

Dosi G G Fagiolo and A Roventini (2006) ldquoAn EvolutionaryModel of Endogenous Business Cyclesrdquo Computational Economics 273ndash34

Dupor B (1999) ldquoAggregation and the Irrelevance in MultindashSector Mod-elsrdquo Journal of Monetary Economics 43 391ndash409

Head A C (1995) ldquoCountry Size Aggregate Fluctuations and Interna-tional Risk Sharingrdquo The Canadian Journal of Economics 28(4b) 1096ndash1119

28

Helbing D and S Lammer (2005) ldquoSupply and Production NetworksFrom the Bullwhip Effect to Business Cyclesrdquo in Networks of Interactingmachines Production Organization in Complex Industrial Systems andBiological Cells ed by D Armbruster K Kaneko and A S Mikhailovvol 3 of World Scientific Lecture Notes in Complex Systems World Sci-entific

Helbing D S Lammer U Witt and T Brenner (2004) ldquoNetworkndashInduced Oscillatory Behavior in Material Flow Networks and IrregularBusiness Cyclesrdquo Phisical Review E 70(056118) 1ndash6

Helbing D U Witt S Lammer and T Brenner (2004) ldquoNetwork-Induced Oscillatory Behaviour of Macroeconomic Output Flowsrdquo in In-dustry and Labour Dynamics II The Agent-based Computational Eco-nomics Approach Proceedings of the wildace 2004 conference ed byR Leombruni and M Richiardi no 5 LABORatorio R Revelli

Horvath M (1998) ldquoCyclically and Sectoral Linkages Aggregate Fluctu-ations from Independent Sectoral Shocksrdquo Review of Economic Dynam-ics 1 781ndash808

(2000) ldquoSectoral Shocks and Aggregate Fluctuationsrdquo Journal ofMonetary Economics 45(1) 69ndash106

Jovanovic B (1987) ldquoMicro Shocks and Aggregate Riskrdquo The QuarterlyJournal of Economics 102(2) 395ndash410

Kahn J A M M McConnell and G Perez-Quiros (2002) ldquoOnthe Causes of the Increased Stability of the US Economyrdquo FRBNY Eco-nomic Policy Review 8(1) 183ndash202

Kraay A and J Ventura (1998) ldquoComparative Advantage and theCrossndashSection of Business Cyclesrdquo World Bank Woorking paper Series1948 World Bank

Long J B and C I Plosser (1983) ldquoReal Business Cyclesrdquo Journalof Political Economy 31(1) 39ndash69

Lucas R E (1977) ldquoUnderstanding Business Cyclesrdquo CarnegiendashRochester Conference Series on Public Policy (special issue of Journalof Monetary Economics) 5 7ndash29

Nelson R R and S G Winter (1982) An Evolutionary Theory ofEconomic Change Harvard University Press Cambridge MA

Nirei M (2005) ldquoThreshold Behavior and Aggregate Fluctuationsrdquo Jour-nal of Economic Theory in press

29

Scheinkman J and M Woodford (1994) ldquoSelfndashOrganized Criticallyand Economic Fluctuationsrdquo American Economic Review 84(2 Papersand Proceedings of teh AEA) 417ndash421

Schuh S (1996) ldquoEvidence of the Link Between FirmndashLevel and Ag-gregate Inventory Behaviorrdquo Finance and Economics Discussion Series1996-46 Federal Reserve Board

Stiglitz J and B Greenwald (2003) Towards a New Paradigm inMonetary Economics Raffaele Mattioli Lectures Cambridge UniversityPress Cambridge

Weisbuch G and S Battiston (2005) ldquoProduction Networks andFailure Avalanchesrdquo Working Paper mimeo Ecole Normale SupErieure

Zarnovitz V (1977) ldquoBusiness Cycles Observed and Assessed Why andHow they Matterrdquo NBER Working Paper Series 6230 National Bureauof Economic Research

30

A Tables

Table 1 Parameters valuesParametera Description Value

N Number of sectorsunits i of production 10Ci Number of j buyer sectors for sector i (IO matrix sparseness) N minus 1Di (E) Value of the external demand 10mij Physical input coefficient of input j for sector i sim U(0 1)Mi Sum of the mij input coefficients j for the production of i 05αi friction to changes of production levels Degree of lockndashin on

production technology085

αsi Speed of adaptation to demand changes 004

σ Desired multiple of production to stock 10

aInto parenthesis the number of lags of the initial value for lagged variables

31

and generating more analytical evidence Of particular relevance is in ouropinion the problem of aggregation Representations of the same systemswith different codification systems (eg different number of SIC digits)must not modify overall system properties for example the volatility ob-served Given that our result show that the sums of coefficients matter forthe levels and volatility of the system and that these sum do depend on theaggregation chosen it should be possible to induce the reaction coefficientsthat we know affect the volatility only

Secondly we can find an application of our results to real world evidenceIn fact there are many data sets available for the data we deal with basicallyinput-output tables Our model is readily adapt to be parametrized alongany number of sector and coefficient necessary so that we may replicate pastseries in order to adapt the unobservable parameters and use the resultingcomplete model to explain business fluctuations and provide intuitions onfuture patterns

27

References

Acemoglu D and A Scott (1997) ldquoAsymmetric Business CyclesTheory and TimendashSeries Evidencerdquo Journal of Monetary Economics 40501ndash533

Backus D P Kehoe and F Kydland (1995) ldquoInternational BusinessCycles Theory and Evidencerdquo in Frontiers of Business Cycle Researched by T Cooley Princeton University Press Princeton

Bak P K Chen J Scheinkman and M Woodford (1993) ldquoAg-gregate Fluctuations from Independent Sectoral Shocks SelfndashOrganizedCritically in a Model of Production and Inventory Dynamicsrdquo RicercheEconomiche 47(1) 3ndash30

Battiston S D Delli Gatti M Gallegati B Greenwald and

J Stiglitz (2005) ldquoCredit Chains and Bankruptcies Avalanches in Sup-ply Networksrdquo in Annual Workshop on Economic Heterogeneous Inter-acting Agents (WEHIA 2005) University of Essex UK

Blanchard O and J Simon (2001) ldquoThe Long and Large Declinein US Output Volatilityrdquo Brookings Papers on Economic Activity 1135ndash174

Bresnahan T F and V A Ramey (1994) ldquoOutput Fluctuations atthe Plant Levelrdquo The Quarterly Journal of Economics 109(3) 593ndash924

David P A (1975) Technical choice innovation and economic growth essays on American and British experience in the nineteenth centuryCambridge University Press London

Delli Gatti D C Di Guilmi E Gaffeo G Giulioni M Galle-

gati and A Palestrini (2004) ldquoA New Approach to Business Fluc-tuations Heterogeneous Interacting Agents Scaling Laws and FinancialFragilityrdquo Journal of Economic Behavior amp Organization 56 489ndash512

Dosi G G Fagiolo and A Roventini (2006) ldquoAn EvolutionaryModel of Endogenous Business Cyclesrdquo Computational Economics 273ndash34

Dupor B (1999) ldquoAggregation and the Irrelevance in MultindashSector Mod-elsrdquo Journal of Monetary Economics 43 391ndash409

Head A C (1995) ldquoCountry Size Aggregate Fluctuations and Interna-tional Risk Sharingrdquo The Canadian Journal of Economics 28(4b) 1096ndash1119

28

Helbing D and S Lammer (2005) ldquoSupply and Production NetworksFrom the Bullwhip Effect to Business Cyclesrdquo in Networks of Interactingmachines Production Organization in Complex Industrial Systems andBiological Cells ed by D Armbruster K Kaneko and A S Mikhailovvol 3 of World Scientific Lecture Notes in Complex Systems World Sci-entific

Helbing D S Lammer U Witt and T Brenner (2004) ldquoNetworkndashInduced Oscillatory Behavior in Material Flow Networks and IrregularBusiness Cyclesrdquo Phisical Review E 70(056118) 1ndash6

Helbing D U Witt S Lammer and T Brenner (2004) ldquoNetwork-Induced Oscillatory Behaviour of Macroeconomic Output Flowsrdquo in In-dustry and Labour Dynamics II The Agent-based Computational Eco-nomics Approach Proceedings of the wildace 2004 conference ed byR Leombruni and M Richiardi no 5 LABORatorio R Revelli

Horvath M (1998) ldquoCyclically and Sectoral Linkages Aggregate Fluctu-ations from Independent Sectoral Shocksrdquo Review of Economic Dynam-ics 1 781ndash808

(2000) ldquoSectoral Shocks and Aggregate Fluctuationsrdquo Journal ofMonetary Economics 45(1) 69ndash106

Jovanovic B (1987) ldquoMicro Shocks and Aggregate Riskrdquo The QuarterlyJournal of Economics 102(2) 395ndash410

Kahn J A M M McConnell and G Perez-Quiros (2002) ldquoOnthe Causes of the Increased Stability of the US Economyrdquo FRBNY Eco-nomic Policy Review 8(1) 183ndash202

Kraay A and J Ventura (1998) ldquoComparative Advantage and theCrossndashSection of Business Cyclesrdquo World Bank Woorking paper Series1948 World Bank

Long J B and C I Plosser (1983) ldquoReal Business Cyclesrdquo Journalof Political Economy 31(1) 39ndash69

Lucas R E (1977) ldquoUnderstanding Business Cyclesrdquo CarnegiendashRochester Conference Series on Public Policy (special issue of Journalof Monetary Economics) 5 7ndash29

Nelson R R and S G Winter (1982) An Evolutionary Theory ofEconomic Change Harvard University Press Cambridge MA

Nirei M (2005) ldquoThreshold Behavior and Aggregate Fluctuationsrdquo Jour-nal of Economic Theory in press

29

Scheinkman J and M Woodford (1994) ldquoSelfndashOrganized Criticallyand Economic Fluctuationsrdquo American Economic Review 84(2 Papersand Proceedings of teh AEA) 417ndash421

Schuh S (1996) ldquoEvidence of the Link Between FirmndashLevel and Ag-gregate Inventory Behaviorrdquo Finance and Economics Discussion Series1996-46 Federal Reserve Board

Stiglitz J and B Greenwald (2003) Towards a New Paradigm inMonetary Economics Raffaele Mattioli Lectures Cambridge UniversityPress Cambridge

Weisbuch G and S Battiston (2005) ldquoProduction Networks andFailure Avalanchesrdquo Working Paper mimeo Ecole Normale SupErieure

Zarnovitz V (1977) ldquoBusiness Cycles Observed and Assessed Why andHow they Matterrdquo NBER Working Paper Series 6230 National Bureauof Economic Research

30

A Tables

Table 1 Parameters valuesParametera Description Value

N Number of sectorsunits i of production 10Ci Number of j buyer sectors for sector i (IO matrix sparseness) N minus 1Di (E) Value of the external demand 10mij Physical input coefficient of input j for sector i sim U(0 1)Mi Sum of the mij input coefficients j for the production of i 05αi friction to changes of production levels Degree of lockndashin on

production technology085

αsi Speed of adaptation to demand changes 004

σ Desired multiple of production to stock 10

aInto parenthesis the number of lags of the initial value for lagged variables

31

References

Acemoglu D and A Scott (1997) ldquoAsymmetric Business CyclesTheory and TimendashSeries Evidencerdquo Journal of Monetary Economics 40501ndash533

Backus D P Kehoe and F Kydland (1995) ldquoInternational BusinessCycles Theory and Evidencerdquo in Frontiers of Business Cycle Researched by T Cooley Princeton University Press Princeton

Bak P K Chen J Scheinkman and M Woodford (1993) ldquoAg-gregate Fluctuations from Independent Sectoral Shocks SelfndashOrganizedCritically in a Model of Production and Inventory Dynamicsrdquo RicercheEconomiche 47(1) 3ndash30

Battiston S D Delli Gatti M Gallegati B Greenwald and

J Stiglitz (2005) ldquoCredit Chains and Bankruptcies Avalanches in Sup-ply Networksrdquo in Annual Workshop on Economic Heterogeneous Inter-acting Agents (WEHIA 2005) University of Essex UK

Blanchard O and J Simon (2001) ldquoThe Long and Large Declinein US Output Volatilityrdquo Brookings Papers on Economic Activity 1135ndash174

Bresnahan T F and V A Ramey (1994) ldquoOutput Fluctuations atthe Plant Levelrdquo The Quarterly Journal of Economics 109(3) 593ndash924

David P A (1975) Technical choice innovation and economic growth essays on American and British experience in the nineteenth centuryCambridge University Press London

Delli Gatti D C Di Guilmi E Gaffeo G Giulioni M Galle-

gati and A Palestrini (2004) ldquoA New Approach to Business Fluc-tuations Heterogeneous Interacting Agents Scaling Laws and FinancialFragilityrdquo Journal of Economic Behavior amp Organization 56 489ndash512

Dosi G G Fagiolo and A Roventini (2006) ldquoAn EvolutionaryModel of Endogenous Business Cyclesrdquo Computational Economics 273ndash34

Dupor B (1999) ldquoAggregation and the Irrelevance in MultindashSector Mod-elsrdquo Journal of Monetary Economics 43 391ndash409

Head A C (1995) ldquoCountry Size Aggregate Fluctuations and Interna-tional Risk Sharingrdquo The Canadian Journal of Economics 28(4b) 1096ndash1119

28

Helbing D and S Lammer (2005) ldquoSupply and Production NetworksFrom the Bullwhip Effect to Business Cyclesrdquo in Networks of Interactingmachines Production Organization in Complex Industrial Systems andBiological Cells ed by D Armbruster K Kaneko and A S Mikhailovvol 3 of World Scientific Lecture Notes in Complex Systems World Sci-entific

Helbing D S Lammer U Witt and T Brenner (2004) ldquoNetworkndashInduced Oscillatory Behavior in Material Flow Networks and IrregularBusiness Cyclesrdquo Phisical Review E 70(056118) 1ndash6

Helbing D U Witt S Lammer and T Brenner (2004) ldquoNetwork-Induced Oscillatory Behaviour of Macroeconomic Output Flowsrdquo in In-dustry and Labour Dynamics II The Agent-based Computational Eco-nomics Approach Proceedings of the wildace 2004 conference ed byR Leombruni and M Richiardi no 5 LABORatorio R Revelli

Horvath M (1998) ldquoCyclically and Sectoral Linkages Aggregate Fluctu-ations from Independent Sectoral Shocksrdquo Review of Economic Dynam-ics 1 781ndash808

(2000) ldquoSectoral Shocks and Aggregate Fluctuationsrdquo Journal ofMonetary Economics 45(1) 69ndash106

Jovanovic B (1987) ldquoMicro Shocks and Aggregate Riskrdquo The QuarterlyJournal of Economics 102(2) 395ndash410

Kahn J A M M McConnell and G Perez-Quiros (2002) ldquoOnthe Causes of the Increased Stability of the US Economyrdquo FRBNY Eco-nomic Policy Review 8(1) 183ndash202

Kraay A and J Ventura (1998) ldquoComparative Advantage and theCrossndashSection of Business Cyclesrdquo World Bank Woorking paper Series1948 World Bank

Long J B and C I Plosser (1983) ldquoReal Business Cyclesrdquo Journalof Political Economy 31(1) 39ndash69

Lucas R E (1977) ldquoUnderstanding Business Cyclesrdquo CarnegiendashRochester Conference Series on Public Policy (special issue of Journalof Monetary Economics) 5 7ndash29

Nelson R R and S G Winter (1982) An Evolutionary Theory ofEconomic Change Harvard University Press Cambridge MA

Nirei M (2005) ldquoThreshold Behavior and Aggregate Fluctuationsrdquo Jour-nal of Economic Theory in press

29

Scheinkman J and M Woodford (1994) ldquoSelfndashOrganized Criticallyand Economic Fluctuationsrdquo American Economic Review 84(2 Papersand Proceedings of teh AEA) 417ndash421

Schuh S (1996) ldquoEvidence of the Link Between FirmndashLevel and Ag-gregate Inventory Behaviorrdquo Finance and Economics Discussion Series1996-46 Federal Reserve Board

Stiglitz J and B Greenwald (2003) Towards a New Paradigm inMonetary Economics Raffaele Mattioli Lectures Cambridge UniversityPress Cambridge

Weisbuch G and S Battiston (2005) ldquoProduction Networks andFailure Avalanchesrdquo Working Paper mimeo Ecole Normale SupErieure

Zarnovitz V (1977) ldquoBusiness Cycles Observed and Assessed Why andHow they Matterrdquo NBER Working Paper Series 6230 National Bureauof Economic Research

30

A Tables

Table 1 Parameters valuesParametera Description Value

N Number of sectorsunits i of production 10Ci Number of j buyer sectors for sector i (IO matrix sparseness) N minus 1Di (E) Value of the external demand 10mij Physical input coefficient of input j for sector i sim U(0 1)Mi Sum of the mij input coefficients j for the production of i 05αi friction to changes of production levels Degree of lockndashin on

production technology085

αsi Speed of adaptation to demand changes 004

σ Desired multiple of production to stock 10

aInto parenthesis the number of lags of the initial value for lagged variables

31

Helbing D and S Lammer (2005) ldquoSupply and Production NetworksFrom the Bullwhip Effect to Business Cyclesrdquo in Networks of Interactingmachines Production Organization in Complex Industrial Systems andBiological Cells ed by D Armbruster K Kaneko and A S Mikhailovvol 3 of World Scientific Lecture Notes in Complex Systems World Sci-entific

Helbing D S Lammer U Witt and T Brenner (2004) ldquoNetworkndashInduced Oscillatory Behavior in Material Flow Networks and IrregularBusiness Cyclesrdquo Phisical Review E 70(056118) 1ndash6

Helbing D U Witt S Lammer and T Brenner (2004) ldquoNetwork-Induced Oscillatory Behaviour of Macroeconomic Output Flowsrdquo in In-dustry and Labour Dynamics II The Agent-based Computational Eco-nomics Approach Proceedings of the wildace 2004 conference ed byR Leombruni and M Richiardi no 5 LABORatorio R Revelli

Horvath M (1998) ldquoCyclically and Sectoral Linkages Aggregate Fluctu-ations from Independent Sectoral Shocksrdquo Review of Economic Dynam-ics 1 781ndash808

(2000) ldquoSectoral Shocks and Aggregate Fluctuationsrdquo Journal ofMonetary Economics 45(1) 69ndash106

Jovanovic B (1987) ldquoMicro Shocks and Aggregate Riskrdquo The QuarterlyJournal of Economics 102(2) 395ndash410

Kahn J A M M McConnell and G Perez-Quiros (2002) ldquoOnthe Causes of the Increased Stability of the US Economyrdquo FRBNY Eco-nomic Policy Review 8(1) 183ndash202

Kraay A and J Ventura (1998) ldquoComparative Advantage and theCrossndashSection of Business Cyclesrdquo World Bank Woorking paper Series1948 World Bank

Long J B and C I Plosser (1983) ldquoReal Business Cyclesrdquo Journalof Political Economy 31(1) 39ndash69

Lucas R E (1977) ldquoUnderstanding Business Cyclesrdquo CarnegiendashRochester Conference Series on Public Policy (special issue of Journalof Monetary Economics) 5 7ndash29

Nelson R R and S G Winter (1982) An Evolutionary Theory ofEconomic Change Harvard University Press Cambridge MA

Nirei M (2005) ldquoThreshold Behavior and Aggregate Fluctuationsrdquo Jour-nal of Economic Theory in press

29

Scheinkman J and M Woodford (1994) ldquoSelfndashOrganized Criticallyand Economic Fluctuationsrdquo American Economic Review 84(2 Papersand Proceedings of teh AEA) 417ndash421

Schuh S (1996) ldquoEvidence of the Link Between FirmndashLevel and Ag-gregate Inventory Behaviorrdquo Finance and Economics Discussion Series1996-46 Federal Reserve Board

Stiglitz J and B Greenwald (2003) Towards a New Paradigm inMonetary Economics Raffaele Mattioli Lectures Cambridge UniversityPress Cambridge

Weisbuch G and S Battiston (2005) ldquoProduction Networks andFailure Avalanchesrdquo Working Paper mimeo Ecole Normale SupErieure

Zarnovitz V (1977) ldquoBusiness Cycles Observed and Assessed Why andHow they Matterrdquo NBER Working Paper Series 6230 National Bureauof Economic Research

30

A Tables

Table 1 Parameters valuesParametera Description Value

N Number of sectorsunits i of production 10Ci Number of j buyer sectors for sector i (IO matrix sparseness) N minus 1Di (E) Value of the external demand 10mij Physical input coefficient of input j for sector i sim U(0 1)Mi Sum of the mij input coefficients j for the production of i 05αi friction to changes of production levels Degree of lockndashin on

production technology085

αsi Speed of adaptation to demand changes 004

σ Desired multiple of production to stock 10

aInto parenthesis the number of lags of the initial value for lagged variables

31

Scheinkman J and M Woodford (1994) ldquoSelfndashOrganized Criticallyand Economic Fluctuationsrdquo American Economic Review 84(2 Papersand Proceedings of teh AEA) 417ndash421

Schuh S (1996) ldquoEvidence of the Link Between FirmndashLevel and Ag-gregate Inventory Behaviorrdquo Finance and Economics Discussion Series1996-46 Federal Reserve Board

Stiglitz J and B Greenwald (2003) Towards a New Paradigm inMonetary Economics Raffaele Mattioli Lectures Cambridge UniversityPress Cambridge

Weisbuch G and S Battiston (2005) ldquoProduction Networks andFailure Avalanchesrdquo Working Paper mimeo Ecole Normale SupErieure

Zarnovitz V (1977) ldquoBusiness Cycles Observed and Assessed Why andHow they Matterrdquo NBER Working Paper Series 6230 National Bureauof Economic Research

30

A Tables

Table 1 Parameters valuesParametera Description Value

N Number of sectorsunits i of production 10Ci Number of j buyer sectors for sector i (IO matrix sparseness) N minus 1Di (E) Value of the external demand 10mij Physical input coefficient of input j for sector i sim U(0 1)Mi Sum of the mij input coefficients j for the production of i 05αi friction to changes of production levels Degree of lockndashin on

production technology085

αsi Speed of adaptation to demand changes 004

σ Desired multiple of production to stock 10

aInto parenthesis the number of lags of the initial value for lagged variables

31

A Tables

Table 1 Parameters valuesParametera Description Value

N Number of sectorsunits i of production 10Ci Number of j buyer sectors for sector i (IO matrix sparseness) N minus 1Di (E) Value of the external demand 10mij Physical input coefficient of input j for sector i sim U(0 1)Mi Sum of the mij input coefficients j for the production of i 05αi friction to changes of production levels Degree of lockndashin on

production technology085

αsi Speed of adaptation to demand changes 004

σ Desired multiple of production to stock 10

aInto parenthesis the number of lags of the initial value for lagged variables

31