Productivity Accounting in Vertically (Hyper-)integrated Terms: Bridging the Gap between Theory and Empirics

PRODUCTIVITY ACCOUNTING IN VERTICALLY(HYPER-)INTEGRATED TERMS: BRIDGING THE GAP

BETWEEN THEORY AND EMPIRICS

Nadia Garbellini and Ariel Luis Wirkierman*Università degli studi di Pavia and Università Cattolica del Sacro Cuore

(April 2013; revised August 2013)

ABSTRACT

The present paper is a methodological contribution introducing a disaggregated physical productivityaccounting framework in vertically (hyper-)integrated terms, establishing a direct correspondencebetween Supply–Use Tables and Pasinetti’s (1973, 1988) theoretical magnitudes. As an empiricalapplication, we computed productivity indicators and indexes of direction of technical change at thesubsystem level for the case of Italy during 1999–2007. Our findings suggest that: (a) only 60 per centof productivity growth accrued to real wages, (b) the degree of mechanization increased, (c) the mostdynamic subsystems correspond to consolidated sectors, and (d) technical change has almost alwaysbeen capital intensity increasing.

1. INTRODUCTION

Productivity analysis within the Classical tradition has frequently departedfrom the insight that partial industry measures cannot adequately deal withthe interdependent character of technical change, while system measuresinvolving total (direct and indirect) input requirements per unit of net outputdo fulfil this essential pre-requisite (see, for example, Gupta and Steedman,1971). The idea that observing only direct labour-saving trends could be

* The present paper has been written within the framework of the Research Group PRIN 2009Structural Change and Growth, of the Italian Ministry of Instruction, University and Research(MIUR). We thank all the group participants for discussions. A previous version of this paperwas the output of participating to the International School of Input–Output Analysis, 2012Edition: we thank our supervisors for their advice. We wish to thank two anonymous referees fortheir helpful and valuable comments and suggestions. The usual disclaimers apply. Specialacknowledgement goes to Prof. Luigi Pasinetti, for his continuous encouragement and support.We would finally like to spare a thought in memory of our dear late friend Angelo Reati.

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Metroeconomica 65:1 (2014) 154–190doi: 10.1111/meca.12036

© 2013 John Wiley & Sons Ltd

misleading has been early recognized (Leontief, 1953, pp. 38–40) and explicitattempts to recursively measure input requirements from every supportingindustry to produce each component of the net product have been conceivedat an early stage as well (see, for example, Vincent, 1962).1

But only the explicit notion of a net output subsystem (Sraffa, 1960,Appendix A) provided a precise alternative description of the technique inuse, which allowed for a disaggregated analysis of technical change in physi-cal terms (see, for example, Pasinetti, 1963; Gossling, 1972). The conceptualapportionment of gross outputs, means of production and quantities oflabour into (relatively) autonomous parts—each reproducing system-wideinterdependence due to circularity—allowed to overcome the problem ofaggregation, which usually characterizes sectoral measures of productivitychanges.2

In this sense, the compact representation of self-replacing subsystems interms of vertically integrated sectors (Pasinetti, 1973) can operationalize theperiod-by-period mapping between industries and subsystems, shifting thedisaggregated unit of analysis for the purpose of quantifying the overalleffects of technical change.3

The use of vertically integrated sectors within productivity studies has beenwidespread (see, for example, Rampa, 1981; Ochoa, 1986; Seyfried, 1988;Elmslie and Milberg, 1996; De Juan and Febrero, 2000). To our knowledge,however, there are, at least, four key issues which have not been dealt with inthe literature:

(1) the explicit recognition that observed empirical structures contain agrowth/decay element, so that self-replacement and expansion require-ments in Pasinetti (1973) have to be accurately specified,4

(2) the need to conceptually and empirically distinguish betweendepreciation (an income side magnitude) and replacement needs

1 For example, in discussing the problem of the influence of compositional changes in finaldemand on the measurement of technical change, Leontief thought that the solution would bethat of ‘computing for each one of the different structural situations a complete set of A’s [inputcoefficients], each showing the dependence of one particular output on one individual componentof the final demand’ (Leontief, 1953, p. 40, italics added).2 See, for example, Steedman (1983).3 In fact, it should always be kept in mind that ‘the analytical device of vertical integration is notmeant to catch the detailed and localized sources of technical change; on the contrary, it is meantto synthesize the overall effect of technical change’ (Pasinetti, 1990, p. 258).4 Exceptions can be found in Lager (1997, 2000), although these contributions do not deal withproductivity accounting.

Productivity Accounting in Vertically (Hyper-)integrated Terms 155


(a physical notion) in the construction of subsystems including fixedcapital inputs,5

(3) the need to provide an empirical counterpart to growing subsystems orvertically hyper-integrated sectors, as formulated by Pasinetti (1988), and

(4) the necessity to assess the consequences of shifting from a single productsystem towards a pure joint products framework, compatible with a setof Supply–Use Tables.6

As regards the first point, data limitations may render impossible toperform an empirical separation between what re-enters the circular flow toreplace productive capacity and what contributes to the expansion/contraction of the economy, whereas an explicit formulation in terms ofmeasurable empirical magnitudes is required. As to the second issue, a dis-tinction between depreciation and physical replacements is necessary, giventhat productivity accounting relies on the expenditure side of the system, andnot on the income (or value-added) side. Third, the empirical formulationof growing subsystems can operationalize the device of vertical hyper-integration, which is of particular interest in a dynamic setting.

Finally, the shift towards a Supply–Use framework needs to be consideredin some more detail. Usually, contributions in this field start from a single-product model, derive theoretical indicators, and eventually adapt empiricaldata to fit (with varying accuracy) the theoretical concepts. This paper pro-ceeds differently. We depart from an empirical Supply–Use scheme, corre-sponding to a square commodity × industry system, and with this frameworkwe directly construct subsystems and productivity indicators, graduallyestablishing a mapping from empirical magnitudes to theoretical concepts.In this way, all complexities involving joint products, valuation schemes,imported commodities and activity levels, among others, are dealt withimmediately. Hence, for example, instead of adopting an Input–Output tech-nology assumption to do away with joint production, we deal with theproblems that joint products bring into the picture (e.g. the empirical non-fulfilment of the all-productiveness property).

Essentially, the set of Supply–Use Tables of the System of NationalAccounts (SNA, hereinafter) allows to make a precise separation between(commodity) prices and volume changes, which is a crucial pre-requisite for

5 Instead of concentrating on physical replacements, applied studies usually focus on deprecia-tion, even to assess physical productivity changes. See, for example, Gupta and Steedman (1971);Ochoa (1986); De Juan and Febrero (2000); Flaschel et al. (2013).6 See Soklis (2011) in this direction, although concerning the empirical computation of wage–profit curves, and involving circulating capital only.

156 Nadia Garbellini and Ariel Luis Wirkierman


setting-up an accurate disaggregated physical productivity accountingscheme.7 But taking the SNA as an explicit point of departure intends toconvey a further message. From a Classical standpoint, the evolution of thetheoretical position of the SNA should raise worrying awareness. While theSNA-1968 (partially drafted by Richard Stone) contained a set of produc-tivity indicators deeply connected to the Classical tradition (UN, 1968,pp. 66–77), the latest SNA-2008 has completely done away any formulationwhich does not comply with the logic of Neoclassical Multi-Factor Produc-tivity (MFP) (UN, 2009, p. 412). In this light, formulating a Classical pro-ductivity accounting scheme using the empirical elements provided by theSNA-2008 seems to be utterly justified.

The aim of the present paper is therefore to set-up a framework forphysical productivity accounting in terms of vertically integrated and hyper-integrated sectors, introducing disaggregated (subsystem-specific) measuresof productivity changes and of the direction of technical change that do notdepend on relative prices (and thus, income distribution) and on the compo-sition of the net product. To illustrate the use of the proposed framework,we provide an application for the Italian economy during 1999–2007. Itshould be kept in mind, however, that the focus of this paper is primarilymethodological.

After this introduction, we proceed by formulating vertically integratedand hyper-integrated sectors in terms of the empirical categories emergingfrom a set of Supply–Use Tables (section 2). Measures of productivitychanges and of the direction of technical change are introduced in section 3.Section 4 reports and discusses the empirical results. Finally, section 5 sum-marizes and concludes.

2. ALTERNATIVE DESCRIPTIONS OF THE TECHNIQUE IN USE

In order to give an analytical formulation of vertically integrated and hyper-integrated sectors in terms of the empirical categories of a set of Supply–UseTables, we depart from commodity balances by source of demand for domes-tically produced items, i.e. the physical counterpart to the expenditure side of

7 This is not possible by adopting any of the main Input–Output technology assumptions, as isargued in detail below. For an early discussion of this issue as regards the ‘industry’ and‘commodity’ technology assumptions, see Flaschel (1980). In fact, ‘[m]easures of sectoral andtotal labour productivity should be based on technological data as much as possible (subject toan unavoidable degree of aggregation) and they should not definitionally depend on pricevariables’ (Flaschel et al., 2013, p. 380).



an Input–Output system. Gross outputs in physical terms are identicallyequal to8

q V e U e F e c≡ ≡ + +q q kq (1)

where c contains all sources of expenditure that do not re-enter the circularflow of re-production as productive capacity.9

2.1 Vertically integrated sectors

To establish a correspondence between commodity balances and the theo-retical magnitudes appearing in Pasinetti’s (1973) article on vertical integra-tion we should distinguish self-replacement from expansion requirements, forboth fixed and circulating capital inputs.

National accountants compute gross stocks of fixed capital by accumulat-ing successive vintages of gross investment flows valued at a common pricesystem. In each year, to every past flow corresponding to each type of capitalgood a survival probability is applied, in order to identify those durableinstruments of production that have not been discarded. Hence, the differ-ence between current and previous period stocks (valued at the same pricesystem) will include both current gross fixed capital formation and retire-ments of capital items which (statistically) could not survive to this date.10

Formally, for the current time period we have

K K F Rq q k kq q∗ − ∗ = ∗ − ∗

−( )1

But given that Kq∗, Kq( )−

∗1 and Fkq

∗ are those magnitudes usually reported,fixed capital retirements are obtained as a residual:

R K K Fk q q kq q∗ = − ∗ − ∗ − ∗

−( )( )1

8 Appendix A specifies matrix notation for all those symbols which are not explicitly introducedin the main text. Notation has been chosen in order to keep the present formulation as close aspossible to symbols frequently used in empirical Supply–Use schemes coming from the SNA.9 These sources include final private and government consumption as well as exports (even ofintermediates or capital goods). Hence, in matrix notation: f p c f f fc s c g xp= = + +ˆ .10 See OECD (2009, pp. 38–40) for details.



Moreover, assume that durable capital inputs have a gestation period of one‘year’.11 Hence, by subtracting retirements from gross fixed capital formationwe are implicitly defining the concept of new investments, i.e. current expen-diture on capital goods devoted to the expansion of productive capacity,which will be ready for operation in the following period. Formally, we have

J F Rk k kq q q∗ = ∗ − ∗ (2)

The concept and meaning of new investments cast lights on two points. First,it allows to objectively think of retirements as replacement needs, i.e. anartificial apportionment of investment expenditure aimed at reconstitutingproductive capacity in place.12 Second, it allows to make a crucial distinctionwith respect to the traditional concept of net investment, i.e. gross investmentminus depreciation.

In fact, the reader might be puzzled as to why we have focused on replace-ment needs instead of depreciation to identify self-replacement requirements.Depreciation is a book-keeping concept, belonging to the value-added side ofthe economy. As such, it does not have a physical counterpart. According tobook-keeping (linear) depreciation schemes, when a new machine is boughtand enters productive capacity, in order not to alter the cost/profits relationin the corresponding accounting period, an estimate of its life-time is made.The value of the machine is then split into as many parts as its estimatedlife-time, and thus spread over the whole period, in order to smooth theassociated increase in production costs. This has nothing to do with thepurely physical concept of replacements, which instead pertains exclusivelyto the expenditure side of the system in physical terms.13

11 From a theoretical standpoint, this may seem an arbitrary assumption. However, note thatthe column of gross fixed capital formation in a typical Use Table includes only acquisition lessdisposal of fixed assets, while work-in-progress (which constitute production during the currentaccounting period in need of further processing to be saleable) are instead included in the vectorof changes in inventories (for details, see EUROSTAT, 2008, pp. 154–6). This means thatcurrent gross investments consist entirely of finished capital goods, which add to productivecapacity.12 In principle, investment decisions clearly depend on a multiplicity of factors. However, giventhat our aim is to device an objective accounting framework, it is our contention that this shouldbe done without recourse to behavioural hypotheses. As such, an apportionment of gross flowsinto a self-replacement and expansion component based on empirically given retirements seemsto be an acceptable strategy.13 In his 1959 paper, Pasinetti had already noticed that ‘[a]ll quantities could be interpreted in netterms but, since depreciation allowances always contain elements of arbitrariness, it is better towork with gross quantities’ (Pasinetti, 1959, p. 275). In fact, as regards physical productivityaccounting, we couldn’t agree more with Steindl reporting Leontief’s view: ‘I remember a lectureby Leontief in which he said that depreciation is a concept used by the tax administration, it isnot an economic concept at all’ (Steindl, 1993, p. 121).



Additionally, national accounts generally do not render available grossstock and flow fixed capital matrices (of dimension ‘commodity oforigin × industry of destination’) separating domestic from imported sources,but only domestic and imported column vectors of gross investments bycommodity ( fkq and fk

mq, respectively). Given that this separation is necessary

for productivity accounting, we assumed, for all industries and for each fixedcapital input, the same proportion of imported to domestic demand.Computationally:14

F F R R f f f f fk q k k q k q k k k k p km

q q q q q q q q q= ∗ = ∗ = ∗ ∗ = +−ˆ , ˆ , ˆ ˆ (ˆ ) , ˆq q q e1 ,, ˆ ˆ ˆe p s

ms= −p p 1

As regards circulating capital inputs, the analytical separation between self-replacement and expansion requires to compute, under the technique cur-rently in use, those activity levels allowing each industry to reproduce grossoutputs from the previous period. In matrix terms:15

x V q q V e( ) ( ) ( ) ( ),−−

− − −= =11

1 1 1q q

R U xu qq = −ˆ ( )1

R U xu qq∗ = ∗

−ˆ ( )1

where x(−1) is the activity level vector applied to current domestic and totalUse matrices (Uq and Uq

∗, respectively) to obtain domestic and total circu-lating capital replacements (Ruq and Ruq

∗ , respectively).16 Thus, also in thecase of circulating capital it is possible to define (total) expansion require-ments as

14 Note that the vector of total fixed capital inputs by commodity ( )fkq∗ depends on the diagonal

matrix of terms of trade by commodity e p( ). Hence, the estimated separation between domes-tically produced and imported gross investments qq( ) depends on given base-period terms oftrade. See Appendix B for details on the relationship between physical and nominal magnitudes.15 Note that it is necessary to assume that the Make matrix Vq is non-singular.16 This procedure evinces a crucial difference between circulating and fixed capital. Whereas inthe first case Ruq is productively consumed during the current period—so the consequences oft − 1 activity levels are exhausted in t—this is not so with fixed capital. Investment consequenceswill probably carry on for many periods to come (which also means that current productionconditions are influenced by lagged investment flows of several past periods). Hence, an artificialseparation between self-replacement and expansion based only in activity levels of t and t − 1 isout of place. Instead, we have proceeded by using empirical period-by-period gross stocks andflows of durable means of production. Alternatively, it could have been possible to discard allstock magnitudes and, for a given (final commodity-specific) steady growth path, reconstructreplacement and expansion requirements from flow magnitudes (see, for example, Lager, 1997).



J U Ru q uq q∗ = ∗ − ∗ (3)

At this point, introduce domestic fixed and circulating capital replacementsinto commodity balances (1)

V e U e F e c R e R e R e R eq q k u u k kq q q q q= + + + − + −( ) ( )

and re-order conveniently

V e R R e U R e F R e cq u k q u k kq q q q q= + + − + − +( ) ( ) ( )

Proceeding as in (2) and (3), domestic expansion requirements for fixed andcirculating capital may be defined as J F Rk k kq q q= − and J U Ru q uq q= − ,respectively, obtaining

V e R R e J J e cq u k u kq q q q= + + + +( ) ( )

By further defining R R Rq u kq q≡ + as the matrix of (domestic) replacementneeds and J J Jq u kq q≡ + as the (domestic) expansion matrix, we finally have

( )V R e yq q− = (4)

where y = Jqe + c is the vertically integrated net product.17

We can now compare expression (4) with the original system in Pasinetti(1973, equation (2.1), p. 4):

( ) ( ) ( )I A X Y− =� t t (5)

with A A A� = +( ) ( )C F d , A = A(C) + A(F).The first apparent difference is a consequence of introducing pure joint

products. Specifically, the original identity matrix I in (5) is here replaced bythe Make matrix Vq in (4). Second, matrix A�, ‘representing that part of theinitial stocks of [circulating and fixed] capital goods that are actually used upeach year by the production process’ (Pasinetti, 1973, p. 4) is given, in ourformulation, by the matrix of replacement needs Rq.18

17 Note that net product vector y differs from the traditional Input–Output vector of final demanddomestically produced. Besides their coincidence as regards truly final uses (private and govern-ment consumption as well as exports), the latter includes gross fixed capital formation, whereas theformer includes expansion requirements only, for both circulating and fixed capital inputs.18 Note that A F( )d is a matrix of depreciation charges obtained by applying given a prioricommodity-specific (row-wise) depreciation quotas d to stock matrix A(F). Instead, workingwith physical replacement needs avoids looking at fixed capital flows from the value-added side.



The key difference, however, is that the original vector of gross outputs,X(t) in (5), is here replaced by the observed unitary vector of activity levels, ein (4). While in system (5) there is a clear-cut separation between currentgross output levels and the technique in use, this is not possible in (4). Inempirical Use Tables, any separation between activity levels and techniquesinvolves an element of arbitrariness, given that current period matricescontain implicit growth (or decay) components, allowing for extended repro-duction in following period(s).19

As a consequence, changes in empirical transaction matrices may be due tovarying activity levels or ongoing technical progress; the analytical distinc-tion between growth and technical change need not be unique. Given theaim of building an objective scheme for productivity accounting, while we donot separate unitary input requirements from the level of operation of eachindustry, we do consider replacement and expansion components separately.

Turning back to (4), provided (Vq − Rq) is non-singular, observed unitaryactivity levels e may be expressed as

e V R y= − −( )q q1

while activity level indexes for each vertically integrated sector i = 1, . . . , nare given by

x V R yν( ) ( )( )i

q qi= − −1

(6)

with y(i) = eiyi, y y( )ii∑ = and x eν

( )ii∑ = .

2.2 Vertically hyper-integrated sectors

Recovering data for computing vertically hyper-integrated sectors is morestraightforward, in empirical terms, than for vertically integrated ones.Building growing subsystems requires first a redefinition of the concept of net

19 In fact, matrix A(C) of circulating capital inputs per unit of gross output cannot be identifiedwith Uq in (1), as the Use Table collected by statistical institutes ‘includes all non-durable goodsand services with an expected life of less than one year which are used up in the process ofproduction by industries’ (EUROSTAT, 2008, p. 146) not only to reproduce past output levels,but to expand (or contract) productive capacity. While A(C) describes a technique in use, Uq

includes both a given technique and effects from changing activity levels. Crucially, given thatproduction takes time, current inputs are met from past outputs, so observed matrices depend onactivity levels of several periods. For a clarification of the relation between empirical magnitudesand theoretical concepts in dynamic Input–Output schemes, see Lager (2000, especially,pp. 248–51).



output, which includes demand for final commodities only, as given by vectorc in our set-up. This implies that gross investments, and not only self-replacement requirements, are part of the means of production, and thereforethere is no need to distinguish between replacement needs and new invest-ments as we did in section 2.1. The key concept behind hyper-integration isthe induced character of (fixed and circulating) investment expenditures.20

In order to compute vertically hyper-integrated sectors, we depart fromcommodity balances (1), re-ordering terms to obtain

( )V U F e cq q kq− − = (7)

Pasinetti’s (1988) original formulation of growing subsystems is given by21

BX AX AX A X C( ) ( ) ( ) ( ) ( )( )t t g t r t tii− − − =∑ (8)

To connect (7) and (8), note first Pasinetti’s (1988) treatment of fixed capitalas a joint product, in the tradition of Sraffa (1960). Instead, we adopt anempirically more tractable procedure, working with gross fixed capital flowand stock matrices, though incorporating pure joint products.

Also in this case, the vector of operation intensities X(t) in (8) is replacedby the unitary vector e in (7), while BX(t) performs a similar role than Vqe.22

The most apparent difference is represented by the fact that it is not possibleto find a one-to-one correspondence between matrices A, gA, A Xr ti

iˆ ( )( )∑and matrices Uq, Fkq; rather, ( )U F eq kq+ performs in empirical system (7) therole that AX AX A X( ) ( ) ( )( )t g t r ti

i+ + ∑ has in (8).23 Finally, net output C(t)in (8) corresponds to vector c in (7).

There is an additional subtle but crucial point in comparing (7) and (8).The subsystem-specific expansion component A Xr ti

i( ) ( )∑ implies that activ-ity levels have to be solved separately for each vertically hyper-integratedsector in the theoretical formulation. This is so because the analysis evaluates

20 Precisely, ‘[i]t is this derived demand aspect of investment goods, due to their being used asmeans of production, that is new and typical of production systems’ (Pasinetti, 1981, p. 176).21 See Pasinetti (1989, expression (2.1a), p. 479). We have slightly re-ordered terms in thispresentation.22 Note, however, that output matrix B also includes ‘old’ durable instruments of productionwhile Make matrix Vq includes only ‘new’ finished products.23 Two reasons motivate this: (i) while AX(t) are self-replacement requirements andg t r ti

iAX A X( ) ( )( )+ ∑ correspond to expansion components, both of circulating and fixed capitalwithout distinction, Uq and Fkq separate between circulating and fixed capital inputs, respec-tively, but without splitting into replacement and expansion requirements; and (ii) while in thetheoretical scheme there is a neat separation between technique (matrix A) and activity levels, inempirical structures this is not attempted.



gross output requirements to satisfy a counter-factual (or normative) steadygrowth path in final consumption, in which specific demand expansion com-ponents (ri) are given data. Instead, as our aim is to set up a productivityaccounting framework based on actually observed period-by-period changes,it need not be true that intermediate transactions will have followed acommodity-specific steady growth path, conforming to the dynamics of netoutput.

Hence, in this empirical reconstruction of growing subsystems, we takechanging activity levels for what they have been, building hyper-integratedsectors by re-proportioning observed unitary operation intensities in corre-spondence to each single component of net output vector c.

To sum up, by comparing the load of assumptions and computationsrequired to arrive at system (4) with respect to system (7), it emerges thatobserved statistical outlays are much more suitable for vertically hyper-integrated analyses than for vertically integrated ones. The key issue is thatthe separation between replacements and new investments in vertically inte-grated analyses—which is always subject to quite a high degree ofarbitrariness—is not at all necessary in setting up growing subsystems, wherematrices Uq and Fkq include both self-replacement and expansion compo-nents. This purely computational advantage has, however, a deeper concep-tual foundation:

[w]ith technical change going on, each machine is never replaced by an exact similarphysical machine, and this makes it impossible to say what is that is replaced andkept intact. Measurement in terms of units of [hyper-integrated] productive capacityovercomes this possibility. (Pasinetti, 1981, p. 178)

Turning back to (7), provided ( )V U Fq q kq− − is non-singular, observedunitary activity levels e may be expressed as

e V U F c= − − −( )q q kq1

while activity level indexes for each vertically hyper-integrated sector i = 1,. . . , n are given by

x V U F cη( ) ( )( )i

q q ki

q= − − −1(9)

with c(i) = eici, c c( )ii∑ = and x eη

( )ii∑ = .

Obtaining (net output, activity level indexes) tuples at the vertically inte-grated and hyper-integrated levels—(y(i), xν

( )i ) from (6) and (c(i), xη( )i ) from (9),

respectively—is our departure point for setting up a productivity accountingscheme in disaggregated physical terms.



2.3 Direct, integrated and hyper-integrated labour requirements andproductive capacities

In view of devising a framework exclusively based on observable magnitudes,the technical conditions of production may be summarized into labourrequirements and stocks of fixed and circulating capital goods (i.e. produc-tive capacity). In formal terms:

( , ) ( , )l S l K RT Tqq u= ∗ + ∗ (10)

( , ) ( , )L Ts l e Se= (11)

( , ) ( , ) ( , )L Li i i i i iTs l e Se s= = × ×1 1 (12)

where lT stands for the industry employment vector (in man-hours), and Scontains fixed and circulating capital stocks (both domestically produced andimported) required to support the production of gross outputs q, with eachindustry operating at observed unitary intensities e. In (12), it is explicitthat each industry’s employment (Li) and commodity requirements (si) areexpressed per unitary operation intensity.

As has been pointed out by Kurz and Salvadori (1995, pp. 168–9), verti-cally integrated (and hyper-integrated, we may add) coefficients offer alter-native descriptions of the technique in use. These alternatives, by shifting thedisaggregated unit of analysis from the industry to the commodity subsys-tem, allow to perform a true price–volume separation, which is not possibleotherwise.24

Conceptually, measuring the change in technical requirements for thereproduction of a given net output allows to account for circularity in pro-duction within each subsystem, while keeping a disaggregated physicaldimension of technical progress. To see this point, consider the expressionscorresponding to (10)–(12) at the vertically integrated and hyper-integratedlevels, respectively:

( , ) ( ( ) , ( ) )1 1nT Tq q q qH l V R S V R= − −− − (13)

( ) ( )L T, ,s y Hy= n (14)

24 Note, in fact, that in the presence of pure joint products, any disaggregated industry magni-tude (e.g. labour requirements) is usually computed with respect to gross output by industry,given by: e p VT

s qˆ , which necessarily involves a price aggregator (e.g. ps). This is not so when theanalysis is kept at the commodity level.



( , ) ( , ) ( , ) ( , )( ) ( ) ( ) ( )L y y y yi i i ii i i i i i i i

T Tν ν νs y Hy e He h= = =n n (15)

where νT is the vector of vertically integrated labour coefficients, and H thematrix of vertically integrated productive capacity,25 and

( , ) ( ( ) , ( ) )1 1hT Tq q k q q kM l V U F S V U F= − − − −− − (16)

( , ) ( )L Ts c Mc= h , (17)

( , ) ( , ) ( , ) ( , )( ) ( ) ( ) ( )L c c c ci i i ii i i i i i i i

T Tη η ηs c Mc e Me m= = =h h (18)

where ηT is the vector of vertically hyper-integrated labour coefficients, andM the matrix of vertically hyper-integrated productive capacity.26

Expressions (10), (13) and (16) provide synthetic alternative descriptions ofthe technique in use, each referring to a different concept of output—unitaryoperation intensities in (10), consumption-cum-expansion requirements in(13), and consumption requirements only in (16)—although at the same timeexhausting aggregate labour force and commodity requirements (L, s), as canbe seen from (11), (14) and (17).27

The key point of this comparison lies in industry or commodity-levelexpressions (12), (15) and (18). Productivity analysis should focus on mea-suring changing technical requirements per physical unit of net output, whilefor industry magnitudes (12) the separation between technique and unitaryobserved activity levels is not unique and, even if attempted, would assesslabour and commodity requirements to reproduce gross output. Instead, thelogical operation of vertical (hyper-)integration, as reflected in (15) and (18),allows to establish a clear-cut separation between technical requirements andnet output, even at a disaggregated level.

In fact, total subsystem labour—L yii iν ν( ) = in (15) and L ci

i iη η( ) = in (18)—isa scalar magnitude that, nevertheless, captures the whole intricate networkof inter-industry relations through coefficients νi and ηi. This is clearly notthe case with industry-level magnitudes. Moreover, when capital goods aremeasured in units of subsystem-specific productive capacity—i.e. in terms ofcomposite commodities hi in (15) and mi in (18)—it is possible to distinguish,

25 C.f. expressions (4.2b) and (4.3b) and the related interpretation of vertically integratedcoefficients in Pasinetti (1973, p. 6).26 C.f. expressions (2.9) and (2.10) and the related interpretation of vertically hyper-integratedcoefficients in Pasinetti (1988, pp. 127–8).27 Note that each of these expressions provides an alternative decomposition of the sameaggregates.



period-by-period, the required pace of capital accumulation (given by thedynamics of net output) from the changing physical composition of produc-tive capacities (given by the evolution of vectors hi and mi).28

Clearly, National Accounts do not provide physical, but nominal data.However, it can be shown29 that, since period-by-period magnitudes may beexpressed at constant (or current and past-year) prices, computing variationsthrough time of commodity-level variables in a pure joint products frame-work makes price effects to vanish. By proceeding in this way we obtain purevolume changes, which remain an essential pre-requisite for productivityaccounting in physical terms.

3. MEASURES OF CHANGES IN PRODUCTIVITY AND DIRECTION OFTECHNICAL CHANGE

Based on the alternative descriptions of the technique in use provided insection 2.3, we compute vertically integrated and hyper-integrated measuresof productivity changes and indexes of direction of technical change, both atthe sectoral and at the aggregate level.30

3.1 Productivity changes

Departing from (15), total labour productivity in each vertically integratedsector i is given by the ratio of net product to total labour requirements in thecorresponding self-replacing subsystem:

αν νν

ν

( )( ) 1

=1 1

( )i i

ii

i i i q q i

yL

yy T

= = =− −l V R e (19)

Note that αν( )i is not a ‘partial’ measure as, besides labour inputs, it takes into

account changes in circulating and fixed capital requirements for self-replacement.

Correspondingly, total labour productivity in each vertically hyper-integrated sector i is given by the corresponding ratio of net product tosubsystem labour requirements:

28 See Pasinetti (1973, pp. 28–9) for a reflection on this latter point.29 See Appendix B for details.30 In order to ease exposition, we present all indicators in levels. As shown in Appendix B,computing changes of commodity-level magnitudes valued using the same price system cancelsout the influence of prices.



αη ηη

η

( )( ) 1

1 1i ii

i

i i i q q kq i

cL

cc T

= = = =− − −l V U F e( ) (20)

But what is the rationale for formulating vertically hyper-integrated mea-sures with respect to vertically integrated ones? As pointed out by Garbellini(2010, pp. 48–9), the key issue lies in the switch from static to dynamicanalysis. The vertically integrated sector is an essentially static construct.Being new investments included in the net output of each vertically integratedsector, the part of it consisting in capital goods needs to be exchangedbetween (or redistributed among) subsystems for them to expand (or con-tract) their productive capacity—composite commodity hi in (15). As a con-sequence, as soon as we consider the evolution of subsystems through time,they cease to be completely autonomous. Allowing for a true separationbetween changes in technique and dynamics of net output requires grossinvestments to be included among the means of production, as we did in (20)following Pasinetti (1988). Hence, hyper-integrated productivity measuresreflect comprehensive though disaggregated surplus generating capacity inphysical terms, within a set of subsystem-specific expanding circular flows.

To see this point, we may decompose changes in total labour requirementsat the integrated and hyper-integrated levels from (19) and (20),respectively:31

Δ Δ Δ% % %( ) ( )L yii

iν να= − (21)

Δ Δ Δ% % %( ) ( )L cii

iη ηα= − (22)

Expression (21) is a ‘spurious’ decomposition: changes in yi are due tochanges in final demand for both consumption commodities and new invest-ment goods. But the process of reproduction of capital goods is itself subjectto technical change, so that changes in vertically integrated net output arealso influenced by changes in productivity, i.e. by the second addendum ofthe decomposition. On the contrary, expression (22) correctly separates theeffect of changes in the composition of effective demand for final uses fromthe effects of technical progress, thereby separating what does from whatdoes not re-enter the circular flow.32 Hence, it is Δ% ( )L i

η that displays thestructural dynamics of employment as intended by Pasinetti (1981, pp. 94–7).

31 By Δ%x we denote the rate of change between t − 1 and t of variable x.32 See also Pasinetti (1986, p. 7) on the role of subsystems as ‘an analytical device that allows usto separate in an unambiguous way what pertains to the surplus from what pertains to thecircular process’.



On a first thought, the fact that vertically hyper-integrated labour produc-tivity depends on the pattern of accumulation may be seen as a disadvantage,because productivity usually reflects the surplus generating capacity of aneconomy, independently of the use made of this social surplus (as new invest-ment demand, for example).

However, if we adopt the view that economic systems are continuouslyundergoing growth (or decay), considering new investment as part ofthe means of production has the advantage of providing us with a notionof surplus that captures the truly final effects of technical progress. Itsaggregate expected outcome should be increasing private consumption percapita in a closed economy without government, and not necessarily higherconsumption-cum-investment per head, as in the vertically integrated case.

It must be noticed that, ceteris paribus, higher growth rates of final effectivedemand imply lower vertically hyper-integrated productivity levels. On thispotential source of criticism, two points should be made.33 First, if the exer-cise performed is one of comparative dynamics, the accumulation rates them-selves are part of the data known or assumed. So productivity levels may becomputed for every feasible growth path. When analysing productivity dif-ferences in a single economy this should suffice to discard the critique,because the analysis is conditional upon every feasible set of data. Second,the proviso ceteris paribus is surely not very realistic. A fast accumulatingsociety embodying technical progress in new machines will not have the samedirect labour input or material input requirements as a stagnant society withalmost no new investment.

When performing instead an empirical exercise without assuming any apriori set of growth rates, total labour productivity differences may beexplained by multiple underlying determinants. In this case, acceleratedgrowth may imply a tendency towards increasing hyper-integrated labourcontent of commodities, as employment would be expected to rise. So it willbe up to the decreasing nature of technical coefficients—either embodied inthe capital inputs introduced through new investment or through the inten-sification of the division of labour—to act as counter-tendencies for anoverall decreasing trend of labour content to manifest itself. Therefore, thefinal effect of these two countervailing forces is only reflected in a hyper-integrated productivity measure, not in a vertically integrated one.

Besides looking at the dynamics of labour productivity as a single magni-tude, it is also informative to decompose each vertically hyper-integratedlabour coefficient into its direct and (hyper-)indirect component. In order to

33 We would like to thank one anonymous referee for calling our attention on the need toconsider more carefully the usefulness of hyper-integration for measuring productivity changes.



do so, from the definition of ηT in (18), for each final commodity i we maywrite

ηi i q i q q k q q k iT T T

q q= = + + − −− − −h e l V e l V U F V U F e1 1 1( )( ) (23)

where each addendum in the right-hand side of the equation may bedefined as

η ηdir hyp( ) ( ), ( )( )i

q ii

q q k q q k iT T

q q= = + − −− − −l V e l V U F V U F e1 1 1

The first addendum (ηdir( )i ) represents the direct labour employed by all indus-

tries producing commodity i, while the second (ηhyp( )i ) stands for the indirect

and hyper-indirect labour requirements to reproduce a unit of commodity ifor final uses. Accordingly, by defining ω η ηη,

( )d

ii= dir / as the share of direct

labour in vertically hyper-integrated labour per unit of net output, it ispossible to assess the (relative) degree of interdependence of each subsystem(which will be given by 1 − ωη,d).

Finally, to obtain a synthetic measure of total productivity changes at anaggregate level, we compute Pasinetti’s (1981) standard rate of growth ofproductivity:34

ρα

αη η

η

ηη*

%%

( ) ( )

( )

( )( )= ==

==

∑∑

∑L

L

LL

i ii

n

ii

n

ii

i

nΔΔ1

11

(24)

The standard rate of productivity growth consists of a weighted averageof total labour productivity changes in every hyper-integrated sector, theweights being the ratio of total labour requirements in the correspondinggrowing subsystem to aggregate employment.

3.2 Direction of technical change

Replying to Solow (1957), Pasinetti (1959) advanced a methodological pro-posal to compute productivity changes and the direction of technical change

34 Note that Pasinetti (1981) advanced this measure in the context of a simplified description ofthe technique in use (in which there were no inter-industry relations). Instead, formula (24)provides a computable generalization within an empirical Supply–Use framework with purejoint products, explicitly acknowledging its vertically hyper-integrated character.



in which the reproducible character of produced means of production wasexplicitly taken into account.35

In particular, Pasinetti (1959) focused his attention on the evolution of tworatios: Q/L and C/N, where Q is the quantity of final consumption commod-ity actually produced, C is the productive capacity necessary for reproducingQ, L is the (direct) labour employed in its production and N ‘can be inter-preted as the quantity of labor which would be necessary for reproducing theexistent capacity, with the technique available at the time observations aremade’ (Pasinetti, 1959, p. 273). While Q/L is the labour productivity in theproduction of net-output, C/N measures labour productivity in the repro-duction of capacity.36 Hence:

A change through time of Q/L can be assumed by itself to be an indication of changein productivity only if C/N changes in the same proportion. If C/N does not changein the same proportion at least two parts of the change have to be distinguished—aneutral effect equal to the proportional change of that ratio which has changed theleast, and a labor saving effect—or alternatively a capital saving effect—given by theexcess of the proportional change of Q/L over C/N—or alternatively of C/N overQ/L. (Pasinetti, 1959, p. 273)

Thus, by defining

β = Q LC N

//

(25)

we may assess the direction of technical change, according to the movementof β.

A key point of this formulation lies in measuring capital goods in units ofcapacity C currently required to reproduce a given (hyper-integrated) net-output Q. In fact, by adopting this unit of measurement for capital goods,C = Q in every period and β = N/L.

Clearly, β was originally conceived in the context of an economy produc-ing a single final commodity, without taking into account the complexities ofinter-industry relations. However, as soon as general interdependence isaccounted for, such an aggregate index could never be purely ‘technical’, asit would depend on compositional changes in the vector of final uses. At best,the original measure could be conceptually thought of as a subsystem-specific

35 A detailed presentation of this debate is beyond the scope of this paper. The discussionbetween Pasinetti and Solow on this specific topic has re-emerged after the posthumous publi-cation of a research paper by Stone (1998[1960]), which gave rise to further exchanges (Pasinetti,1998; Solow, 1998).36 It should be clear that C/N measures a counter-factual, as N corresponds to a measure ofcurrent and co-existing labour, no reference at all being made to series of dated labour quantities.



index. Moreover, this indicator should mirror the evolution of the capital/net-output ratio, reflecting the overall capital intensity of the system, in thesense of Harrod (1948).

To translate the logic of Pasinetti’s (1959) index β into the formulation forgrowing subsystems in (16)–(18), we need to establish a correspondence withQ, L, C and N. When considered at the level of the single hyper-integratedsector, L may be associated with L i

η( ) in (18), while Q and C correspond to ci.

Finally, we may define

N cii i

Tη( ) = h m (26)

i.e. the quantity of co-existing vertically hyper-integrated labour that wouldbe necessary for the reproduction of the existing productive capacity with thetechnique actually in use.

In this way, the disaggregated index for the direction of technical change ineach vertically hyper-integrated sector i may be written as

βη η

η

η

η

η

( )( )

( )

( )

( )i i

i

ii

i

ii i

i i

i

i

c Lc N

NL

cc

T T

= = = =//

h hm m(27)

while the economy-wide index of capital intensity is given by

βη

η*

( )

( )= = = =∑

∑Q LC N

NL

N

L

ii

ii

T

T

//

hh

Mcc

(28)

Note that the series of subsystem-specific indexes β(i) as well as the aggregateindex β* are ‘pure numbers’.37 Moreover, it is worth stressing that while β*depends on the composition of final consumption c (its movement throughtime thus depending on compositional changes in net-output), sectoral indexesβ(i) are intrinsically ‘technical’, since they are independent of the structure offinal uses.38 The intrinsically technical character of subsystem magnitudes withan overall average that depends on the composition of final demand is alsopresent in Δ% ( )αη

i and ρ* (expressions (20) and (24) above, respectively).

37 In fact, it is straightforward to show that the absolute level of both measures can be computedstarting from nominal magnitudes, since the effect of prices cancels out

hh

hh

Ts s s s

Ts s

T

T

ˆ ˆ ˆ ˆˆ ˆ

p p Mp p cp p c

Mcc

− −

− = =1 1

1β*

h hTs s i i

i i

Ti

i

ipp

ˆ ˆ ( )p p m m− −

− = =1 1

1η ηβ

38 In this sense, while β(i) adequately reflects the direction of technical change in each growingsubsystem, the interpretation of β* as indicating the ‘type’ of technical change at an aggregatelevel is not warranted. See Pasinetti (1981, p. 214).



4. AN EMPIRICAL EXPLORATION

In this section we present and discuss the results of computing sectoral andaggregate measures of productivity increase and direction of technicalchange, introduced in section 3, for the Italian case throughout 1999–2007.Yearly series of square 30 × 30 (commodity × industry) Supply–Use Tablesat the two-digit NACE Rev. 1 level, as well as gross fixed capital stock andflow matrices and labour input data have been obtained from the ItalianNational Institute of Statistics (ISTAT).39

4.1 Effect of pure joint products

A distinctive feature of the present study consists in taking a set of squarecommodity × industry Supply–Use Tables as the point of departure, insteadof adopting an Input–Output technology assumption to obtain an ‘industry’or ‘commodity’ model.40 This is done not only for theoretical consistencewith Classical production models,41 but also because none of the main tech-nology assumptions allows for both a genuine price–volume separation andan assured semi-positivity of the direct input requirements matrix.42

However, adopting a joint products framework prevents from obtaining‘well-behaved’ systems so neatly as with single-product schemes. Theoreticalmodels may overcome this limitation by assuming that the economy is ‘all-productive’, meaning that every commodity in the net-output is separatelyproducible at non-negative activity levels43 or, for the case of growing econo-mies (at a uniform rate g), that the system is ‘g-all-productive’.44

As regards vertical integration, ‘all-productiveness’ holds if and only if(Vq − Rq)−1 is semi-positive. For the hyper-integrated case, we have notassumed uniform steady growth, so ready-made theoretical conditions do notstrictly apply. However, ‘g-all-productiveness’ need hold if ( )V U Fq q kq− − −1

is semi-positive.

39 As regards particular characteristics of the data set, as well as data preparation and estimationprocedures, please refer to Wirkierman (2012, Appendix C).40 See EUROSTAT (2008, Chapter 11) for an exhaustive presentation of the four main Input–Output technology assumptions/transformation models: (a) commodity technology, (b) industrytechnology, (c) fixed industry sales structure, and (d) fixed product sales structure.41 On theoretical grounds, Input–Output technology assumptions neglect joint products byre-allocating secondary production according to pre-defined criteria; see Lager (2011) for adetailed analysis.42 See Wirkierman (2012, Appendix B) for a formal analysis.43 Alternatively, ‘[a] system is all-productive if and only if all its subsystems have non-negativeactivity levels’ (Schefold, 1989, p. 61).44 See Bidard (1996, p. 330) for sufficient conditions to guarantee ‘g-all-productiveness’.



Unfortunately, as can be read from table 1, the Italian economy (during1999–2007) is neither all-productive nor g-all-productive, given that bothinverse matrices contain some negative elements. But the fact of not possess-ing the (g-)all-productiveness property does not mean that empirical analysesare not meaningful. First, the fact that both νT and ηT (i.e. the vectors ofvertically integrated and hyper-integrated labour coefficients, respectively)do not contain any negatives is reassuring, in view of computing total labourproductivity changes. Second, the few negative elements in matrix M ofvertically hyper-integrated productive capacity correspond only to theMining-Energy industry,45 thus, sectoral indexes for the direction of technicalchange may be computed for all but one growing subsystem.46

45 Specifically, in its interaction with Construction and Non-metallic minerals, and only between2000 and 2002.46 Moreover, it can be informative to assess which commodities are separately producible at thevertically integrated or hyper-integrated level. Systems which are all-productive with respect toa subset of commodities are called ‘partially all-productive’ (Schefold, 1989, p. 196). In our case,it is noticeable that by just removing the negative entries of the Health services row in matrix( )V U Fq q kq− − −1, 14 out of 30 commodities would be separately producible at the hyper-integrated level. Note that negative entries never account for more than 4.7 per cent of totalyearly transactions.

Table 1. Negative elements in alternative descriptions of the technique in use

(In number of negative entries; yearly total transactions are 302 = 900)

1999 2000 2001 2002 2003 2004 2005 2006 2007

Inverse matrices(Vq − Rq)−1 n.a. 25 28 22 28 24 43 42 43( )V U Fq q kq− − −1 20 23 20 19 20 22 38 39 43

Productive capacitiesH n.a. 1 1 1 1 1 2 2 2M n.a. 2 2 1 0 0 0 0 0

Labour contentνT n.a. 0 0 0 0 0 0 0 0ηT 0 0 0 0 0 0 0 0 0

Source: Own computation based on Supply–Use Tables (SUT) and National Accounts Data,ISTAT.Notes: ‘n.a.’ stands for not available, given that unavailable previous year data are required forcurrent year computations.



4.2 Distribution of the fruits of technical progress

Turning now to productivity accounting, table 2 reports the aggregatedynamics of employment (Δ%L), average real wage rate Δ%( )w cp/ ∗ , produc-tivity (ρ*) and overall capital intensity (β*).

A first remark concerns the extent to which productivity increases accruedto real wages. For the whole 1999–2007 period, ρ* has exceeded Δ%( *)w cp/ bya yearly average of 0.25 p.p., although it is interesting to notice that whenproductivity is falling (2000–2003), the real wage decreases to a lesser extent(their yearly average difference is −0.53 p.p.). Hence, productivity movementsamplify those of the real wage rate in both directions; however, the overalltrend suggests that, on average, only 60 per cent of productivity growth hasaccrued to wages; leaving real wages lagging behind productivity dynamics.

4.3 Substitution and degree of mechanization

The link between increasing amounts of fixed capital inputs and technologi-cal unemployment is subtle and should be treated with care.47 In the first

47 This is clearly not the case in marginalist analyses of technical change, where the substitutionmechanism conceives capital as a homogeneous quantity with a ‘factor price’ (the rate of profits),and suggests an inverse monotonic relation between the capital labour ratio and relative ‘factor’prices (the ratio of the rate of profits to the real wage rate). See Pasinetti (1977) for a critique ofthe marginalist mechanism of ‘factor input substitution’.

Table 2. Aggregate dynamics of employment, real wage, productivity andcapital intensity in Italy (1999–2007)

(Rates of change in p.p., mean values in average yearly p.p.)

2000 2001 2002 2003 2004 2005 2006 2007 mean

Employment and real wageΔ%L 1.82 1.78 1.27 0.62 0.37 0.16 1.54 0.96 1.07Δ%( *)w cp/ −0.24 0.17 −0.31 −0.40 1.26 1.31 1.06 0.10 0.37

Productivity: standard rate and machinery subsystemsρ* 2.54 −0.22 −1.23 −0.59 1.76 1.22 0.95 0.84 0.66Δ% ( )αη

DK 6.78 1.05 −3.99 0.35 6.02 1.37 4.19 1.04 2.10Δ% ( )αη

DL 7.24 −0.44 −1.06 −2.26 5.89 4.48 2.54 0.73 2.14

Overall capital intensity levelβ* 6.37 6.43 6.58 6.68 6.66 6.80 6.77 6.88 6.65

Source: Own computation based on Supply–Use Tables (SUT) and National Accounts Data,ISTAT.



place, as noted by Pasinetti (1981, Chapter IX), the ratio of fixed capital tolabour should not be considered as an index of capital intensity but as anindicator of the degree of mechanization of the system. Second, the substi-tution of capital for labour is not the consequence of changing relative‘factor’ prices triggering movements along an isoquant, but is instead adynamic process intimately connected to the hyper-integrated productivitygrowth of subsystems producing machinery with respect to the standard rateof productivity growth and the dynamics of the real wage rate. In fact:

if, in any production process, at a certain point of time, machines are substituted forlabour, the reason simply is that productivity in the machine producing sector isincreasing faster than the over-all wage rate. (Pasinetti, 1981, p. 217)

In his assertion, Pasinetti (1981) is thinking in terms of a ‘natural economicsystem’, in which the average real wage increases precisely at rate ρ*.However, the key point is the comparison between labour saving trends insubsystems producing fixed capital (mainly machinery) and overall produc-tivity growth (as measured by ρ*), even if productivity increases do not fullyaccrue to wages (as has actually occurred in Italy between 1999 and 2007).

Intuitively, when productivity is increasing faster in the machinery subsys-tems than in the overall economy, there is a comprehensive saving of labourin operating new machines with fewer workers with respect to using olddurable instruments under current employment conditions.48

Hence, the frequently observed pattern of increasing degree of mechani-zation reflects these labour saving trends, which may be only partiallycounter-balanced by an increase in effective demand for final commodities.49

By following the dynamics of hyper-subsystems DK and DL (correspond-ing to electrical and mechanical machinery products) in table 2, an increasingdegree of mechanization in Italy is confirmed by the higher rate of produc-tivity growth of the machinery complex (2.10 and 2.14 p.p.) with respect toboth ρ* and the real wage (0.66 and 0.37 p.p., respectively), on average.

4.4 Identification of dynamic subsystems

Table 3 reports sectoral output, employment and productivity changes at thedirect, vertically integrated and hyper-integrated levels.

48 Additionally, note that movements in the average wage rate will affect all subsystems, notonly those producing final commodities; see Pasinetti (1981, p. 216, n. 28).49 An increase which, in turn, depends on the extent to which productivity increases accrue toreal wages, allowing for higher real incomes and further final consumption.



Tab

le3.

Dyn

amic

sof

outp

ut,e

mpl

oym

ent

and

labo

urpr

oduc

tivi

tyin

Ital

y(1

999–

2007

)

(Mea

nva

lues

for

peri

od19

99–2

007:

rate

sof

chan

gein

year

lyav

erag

epe

rcen

tage

poin

ts,l

evel

sin

%)

Dir

ect

Inte

grat

edH

yper

-int

egra

ted

Δ%q i

Δ%L

jΔ%

y iΔ

%(

)L

i νΔ

%(

)α

νiΔ%

c iΔ

%(

)L

i ηΔ

%(

)α

ηi%

()

Li η

ωη,(

) diβ(i

)

(01)

(02)

(03)

(04)

(05)

(06)

(07)

(08)

(09)

(10)

(11)

Dyn

amic

subs

yste

ms:

Δ%

()

αρ

ηi>

*an

dΔ

%(

)L

i η>

0D

G:C

hem

ical

s0.

26−0

.45

2.48

0.33

2.31

3.59

0.43

3.40

1.43

27.4

09.

38D

L:E

lect

r.M

achi

nery

2.19

1.13

0.28

−0.5

60.

862.

890.

712.

141.

5446

.02

5.42

DA

:Foo

d-T

obac

co1.

170.

191.

411.

030.

582.

160.

042.

145.

7218

.60

6.81

DK

:Mac

hine

ryn.

e.c.

3.37

1.53

2.57

1.82

0.80

4.18

2.05

2.10

3.87

36.1

06.

52D

H:P

last

ics

−0.0

7−1

.34

1.38

−0.0

71.

472.

730.

771.

960.

6740

.11

7.51

JJ:F

inan

ce3.

700.

872.

801.

601.

103.

651.

791.

791.

5850

.00

6.63

II:T

rans

port

-Com

m.

3.76

1.32

1.91

0.85

1.08

2.91

1.36

1.55

5.42

41.8

46.

95D

M:T

rans

port

Equ

ip.

0.92

−0.3

4−0

.31

−0.9

30.

602.

090.

481.

541.

8426

.89

7.49

DE

:Pap

er-P

rint

ing

1.52

−0.5

30.

530.

390.

131.

810.

591.

240.

9837

.70

7.13

DJ:

Met

als

2.45

1.47

5.58

4.79

0.65

6.75

5.76

0.91

1.78

42.4

76.

56N

N:H

ealt

h2.

351.

082.

481.

810.

662.

561.

770.

797.

9172

.17

2.83



Tab

le3.

Con

tinu

ed

Dir

ect

Inte

grat

edH

yper

-int

egra

ted

Δ%q i

Δ%L

jΔ%

y iΔ

%(

)L

i νΔ

%(

)α

νiΔ%

c iΔ

%(

)L

i ηΔ

%(

)α

ηi%

()

Li η

ωη,(

) diβ(i

)

(01)

(02)

(03)

(04)

(05)

(06)

(07)

(08)

(09)

(10)

(11)

Dyn

amic

prod

ucti

vity

/labo

urex

pelli

ngsu

bsys

tem

s:Δ

%(

)α

ρηi

>*

and

Δ%

()

Li η

<0

DD

:Woo

d0.

25−1

.47

1.29

−1.0

51.

780.

09−2

.80

2.93

0.19

47.7

25.

91D

C:L

eath

er−0

.96

−2.7

8−3

.34

−3.2

80.

05−1

.53

−3.5

92.

161.

3238

.05

5.69

DB

:Tex

tile

s−1

.32

−2.4

1−2

.00

−1.8

6−0

.04

−0.5

8−2

.36

1.87

3.75

46.2

85.

81D

N:M

anuf

actu

ren.

e.c.

0.19

−0.5

0−2

.23

−1.5

1−0

.60

−1.1

5−2

.53

1.43

1.92

41.5

65.

80L

L:P

ublic

Adm

in.

1.24

−0.8

51.

22−0

.34

1.58

1.25

−0.1

71.

439.

2062

.96

10.5

7D

I:N

on-m

et.m

iner

als

1.96

0.39

−1.5

3−1

.60

0.05

0.06

−0.9

61.

080.

7038

.25

7.25

AA

:Agr

icul

ture

−0.3

4−1

.66

−1.2

9−3

.29

0.47

0.82

−0.0

50.

881.

9270

.89

5.45

Pro

duct

ivit

yla

ggin

gsu

bsys

tem

s:Δ

%(

)α

ρηi

<*

CB

:Min

ing

non-

ener

gy−0

.23

−1.8

50.

783.

68−2

.32

2.21

1.97

0.55

0.03

40.0

39.

40M

M:E

duca

tion

0.58

0.47

0.66

0.27

0.39

0.58

0.11

0.47

6.76

88.7

91.

86G

G:T

rade

1.70

0.69

0.76

0.85

−0.0

91.

110.

990.

1316

.03

50.9

95.

48P

P:H

ouse

hold

Serv

ices

2.93

2.92

3.09

3.09

0.00

2.93

2.92

0.01

3.28

100.

000.

00F

F:C

onst

ruct

ion

2.40

3.03

1.33

2.12

−0.7

6−0

.61

−0.5

5−0

.06

0.63

50.6

74.

35H

H:H

otel

-Res

taur

ant

1.68

2.56

0.90

2.40

−1.3

51.

982.

38−0

.37

7.87

57.3

84.

69E

E:E

nerg

y2.

02−1

.38

1.12

1.72

−0.6

21.

041.

71−0

.46

0.73

20.8

915

.80

BB

:Fis

hing

−1.1

3−0

.36

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23−3

.39

−0.7

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.80

0.19

81.6

42.

80O

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erso

nalS

ervi

ces

0.20

2.01

1.14

2.42

−1.1

11.

503.

10−1

.46

3.64

58.0

85.

21K

K:B

usin

ess

Serv

ices

2.41

3.79

0.62

2.55

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422.

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DF

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etro

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0.03

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0.29

18.4

411

.55

Sou

rce:

Ow

nco

mpu

tati

onba

sed

onSu

pply

–Use

Tab

les

(SU

T)

and

Nat

iona

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ount

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ata,

IST

AT

.



Given that productivity movements crucially depend on employmenttrends, it is useful to draw conclusions by looking at the joint dynamics ofhyper-integrated labour productivity and total labour requirements, Δ% ( )αη

i

and Δ% ( )L iη , respectively. In fact, we have classified subsystems according to

whether:

(i) Productivity growth is faster than average and subsystem labour isincreasing (Δ% ( )α ρη

i > * and Δ% ( )L iη > 0, respectively).

(ii) Productivity growth is faster than average but labour is being expelledfrom the subsystem (Δ% ( )α ρη

i > * and Δ% ( )L iη < 0, respectively).

(iii) Productivity growth is slower than average Δ% ( )α ρηi < *.

Group (i) includes dynamic subsystems, and mainly involves consolidatedItalian manufacturing sectors such as machinery and processed food prod-ucts (DA, DL, DK, DM), diffused intermediates such as plastics and metalproducts (DJ, DH), the pharma-health complex (DG, NN), together withlogistics and financial services (II, JJ).

Group (ii) includes labour-expelling subsystems with faster than averageproductivity growth and mainly involves sectors producing inputs to theConstruction industry—such as Wood (DD), Furniture (the most importantindustry in DN) and Non-metallic minerals (DN, mainly cement)—andsectors in which international off-shoring and strong price competition hastaken place (such as Leather and Textiles). Note that in all these sectorshyper-integrated net-output dynamics—Δ%ci in column (06)—has beennegative or almost nil.

Group (iii) is composed of subsystems lagging behind overall productivitygrowth (ρ*), mainly including the energy complex (CB, DF, EE) and twotypes of core service products: trade, accommodation, restaurants and busi-ness services (GG, HH, KK) on the one hand, and education and personalservices (OO, PP, MM), on the other.

To have a quantitative idea of the relative weight of these subsystems in theeconomy, summing over column (09) within each category gives that groups(i), (ii) and (iii) represent, on average, about 32.74 per cent, 19 per cent and48.26 per cent of total labour requirements, respectively. Columns (06), (07),(08) capture the structural dynamics of employment (as obtained in expression(22) above). Labour expelled by subsystems in group (ii) has been, to a greatextent, absorbed by service subsystems in group (iii)—HH, KK, OO, PP—aswell as by mechanical machinery (DK), metal products (DJ), health services(NN), logistics (II) and finance (JJ) in group (i). While in the latter casedemand dynamics—column (06)—has more than counter-balanced laboursaving trends, the opposite has occurred in sectors belonging to group (ii).



Note that a shift in the disaggregated unit of analysis from the industry tothe growing subsystem leads to a complete change in some results. Forexample, from a comparison between Δ%Lj in column (02) and Δ% ( )L i

η incolumn (07), we have that while Chemicals, Plastics, Transport Equip. andPaper-Printing industries have expelled employment, the growing subsystemsassociated to their main product have instead absorbed labour. Interestingly,the dynamics associated to the relatively low share of direct in totallabour—ωη,

( )d

i in column (10)—has been offset by indirect-cum-hyper-indirectdynamics.

However, a thorough inspection of columns (08) and (10) shows that nomechanical relationship can be established between the degree of interdepen-dence of a subsystem (given by 1− ωη,

( )d

i ) and its productivity performance( % )( )Δ αη

i . The same holds for the subsystem level of capital intensity—as‘proxied’ by β(i) in column (11). In fact, note that some of the most capitalintensive sectors produce Chemicals (β(i) = 9.38), Public Administration(β(i) = 10.57) and Business Services (β(i) = 16.02), each of them belonging to adifferent group among (i)–(iii).

4.5 Direction of technical change and capital intensity

Focusing on the sectoral direction of technical change, i.e. the dynamics of β(i)

in (27), we have that if Δ%β(i) > 0, then Δ Δ%( ) %( )( ) ( )c L c Nii

ii/ /η η> , implying

that total labour productivity increases faster than the reduction in labourcontent required to reproduce subsystem’s i productive capacity. In terms ofPasinetti (1981, p. 209), this pattern corresponds to ‘capital intensity increas-ing’ technical progress. In the case under study, it results from our compu-tations that all growing subsystems but Education (MM) and BusinessServices (KK) follow this upward trend.50

By looking at column (11) in table 3 the reader might wonder why weincluded average levels of β(i), instead of their rate of change. Being a purenumber, β(i) might be directly compared across subsystems. In fact, in section3.2 we claimed that β(i) should mirror the capital/net-output ratio, i.e. thecapital intensity, of each growing subsystem. In terms of statistically given(basic) prices ps, capital intensity in sector i is given by

κ ( )( )

( )i s

i

si

s i i

s i

s i

s

T

T

T

i

T

i

cp c p

= = =p Mcp c

p m p m(29)

50 The values for Δ%β(MM) and Δ%β(KK) are −0.14 per cent and −0.63 per cent, on a yearly averagebasis, respectively.



whereas the economy-wide index is

κ = =p Mcp c

p See f

s

s

s

c

T

T

T

T (30)

A quick comparison of (29) and (30) with respect to (27) and (28), respec-tively, shows that the key difference lies in the use of a different set of weightsto aggregate physical quantities: ps

T in (29) and (30) as compared with ηT

in (27) and (28). There is no a priori reason to expect that κ(i) is reflectedin β(i) or, to control for the upward general trend in β(i), that β(i)/β* mirrorsκ(i)/κ.

Figure 1 plots β(i)/β* (x-axis) against κ(i)/κ (y-axis) for the year 2007.51

Points (almost) lying on the dashed 45-degree line plot those subsystemswhose deviation from the economy-wide capital intensity is (almost) cor-rectly predicted by β(i)/β* (examples are PP, MM, NN, II, LL; which are allservice products). Below (above) the 45-degree line, β(i)/β* overestimates(underestimates) κ(i)/κ. With the exception of some outliers (subsystems AA,BB, KK, EE, DF, DG, JJ), conditional prediction of κ(i)/κ by β(i)/β* is rela-tively accurate.52

5. CONCLUDING REMARKS

The aim of this paper has been to set up a physical productivity accountingscheme at a disaggregated level. To do so, we have relied on the notions ofself-replacing and growing subsystems, rendered operational by means ofvertical integration and hyper-integration, respectively. In particular, weestablished a precise correspondence between empirical categories of a set ofSupply–Use Tables and the theoretical notions introduced by Pasinetti (1973,1988). This allowed us to obtain alternative descriptions of the technique inuse, and derive productivity indicators as well as indexes of direction oftechnical change.

While vertically integrated sectors have been frequently used in the lit-erature, this is not the case for hyper-integrated sectors. It has been shown

51 The main conclusions reached do not change by considering any other year between 1999 and2006.52 Estimating a linear projection of β(i)/β* on κ(i)/κ conditional on subsystem and year gives amultiple R2 of 0.9923, in which the unconditional mean (the intercept), β(i)/β* and subsystemdummies are statistically significant (while year dummies are not). Results are available uponrequest.



that each of these two notions requires a different concept of net product(none of which strictly coincides with the traditional Input–Output conceptof final demand). Moreover, it has been argued that to accurately applyvertical integration, self-replacing requirements should be singled out,even though actual data include both self-replacement and expansion/contraction components. On the contrary, computation of vertically hyper-integrated sectors precisely requires these actual data. Hence, while growingsubsystems are straightforward to obtain, self-replacing subsystems involveadditional assumptions to empirically separate self-replacement fromgrowth (or decay).

Figure 1. Deviation of capital intensity indexes (β(i), κ(i)) with respect to their economy-wideaverages (β*, κ), Italy, 2007. Circle size represents the weight of the hyper-subsystem in total

employment (L Liη( ) / ).

Source: Own computation based on Supply–Use Tables (SUT) and National AccountsData, ISTAT.



A crucial difference between this paper and other studies dealing with fixedcapital inputs and vertical integration is that while in the latter depreciationmatrices are considered a valid measure of self-replacement, we argue that adistinction between depreciation and physical replacements is essential.Depreciation pertains to the income (value-added) side of an Input–Outputscheme, while physical replacements concern the expenditure side. Given thatproductivity accounting in physical terms should be always carried outdeparting from the set of commodity balances by source of demand (i.e. thephysical counterpart to the expenditure side), physical replacements, notdepreciation, is the adequate magnitude to build subsystems for the measure-ment of technical change.

The fact of having directly departed from a set of Supply–Use Tablesand gradually arrived at theoretical concepts allowed us to see that: (a) theseparation between activity levels and techniques in empirically givenstructures involves an element of arbitrariness, and (b) a genuine price/volume-change separation cannot be obtained by applying any of the mainInput–Output technology assumptions (while at the same time keeping asemi-positive direct input coefficients matrix). These two insights haveimportant implications in the specification of a productivity accountingframework.

Empirically, the paper explored the case of the Italian economy during1999–2007. By means of applying the set-up devised, it emerged that:

(a) only 60 per cent of productivity growth has accrued to real wages, onaverage;

(b) the degree of mechanization did increase;(c) the most dynamic subsystems correspond to consolidated Italian sectors

such as machinery and processed food products, diffused intermediatessuch as plastics and metal products, the pharma-health complex,together with logistics and financial services;

(d) technical change at the sectoral level has been (almost always) ‘capitalintensity increasing’ in the sense of Pasinetti (1981, pp. 208–9).

Future research efforts should concentrate on a wider empirical applica-tion of the measures introduced (e.g. ρ*, Δ% ( )αη

i , β(i), β*), in order to seehow they perform in a long-period setting and across countries. Clearly,data availability restricts the realm of application, in particular due to ageneral lack of fixed capital stock and flow matrices, Use matrices fordomestic output at basic prices, as well as the general difficulty of obtainingdata in constant prices. But it is our firm conviction that only through thespecification of an accurate theory of measurement it will be possible to



obtain from statistical offices the elements required for the measurementof theory.

APPENDIX A: MATRIX NOTATION

All throughout the paper, vectors are indicated by lower case boldface char-acters (e.g. z), and have to be intended as column vectors unless explicitlytransposed (e.g. zT); matrices are indicated by upper case boldface characters(e.g. X), except for lower case characters with a hat (e.g. z ), indicatingdiagonal matrices with the vector elements on the main diagonal. Moreover,e = [1 . . . 1]T is the sum vector and ei = [0 . . . 1 . . . 0]T, with 1 in the ithposition, is a column selector vector.

In order not to make notation too heavy, we are going to distinguish aphysical-quantity matrix from the corresponding one in nominal termssimply by adding the subscript q; moreover, in general, almost all magnitudeswill refer to time period t; in case of exceptions, the time lag i with respect totime period t will be indicated with the subscript ±i.53 In the same way, allmagnitudes without any special superscript will be intended as domesticallyproduced ones; in all cases in which it will be necessary to refer to theimported component or to the sum of both domestically produced andimported components of a variable, we will do it by means of superscripts m

and *, respectively.The list of symbols composing the accounting framework introduced in

section 2 is reported in table A1 below.

53 So, for example, while matrix V will denote the make matrix, evaluated at current basic prices,in time period t, matrix Vq(−1) will denote the make matrix, in physical terms, referring to timeperiod t − 1.



Table A1. Symbols used throughout the paper

Symbol Dimensions Description

U n × n (commodity × activity) Use matrix of circulating capitalat basic prices

V n × n (commodity × activity) Make matrix of gross outputs atbasic prices

Fk n × n (commodity × activity) Gross fixed capital formation atbasic prices

fc n × 1 (commodity × 1) Hyper-integrated net output atbasic prices

fcp n × 1 (commodity × 1) Final private consumption atbasic prices

fg n × 1 (commodity × 1) Government consumptionexpenditure at basic prices

fx n × 1 (commodity × 1) Exports at basic FOB pricesy n × 1 (commodity × 1) Vertically integrated net output

in physical termsc n × 1 (commodity × 1) Hyper-integrated net output in

physical termsq n × 1 (commodity × 1) Gross outputs in physical termsx n × 1 (activity × 1) Activity levelsl n × 1 (activity × 1) Employment by industry (in

number of persons engaged)K n × n (commodity × activity) Gross fixed capital stocks at

basic pricesJk n × n (commodity × activity) Fixed capital new investments at

basic pricesJu n × n (commodity × activity) Circulating capital expansion at

basic pricesRk n × n (commodity × activity) Retirements of fixed capital

inputs at basic pricesRu n × n (commodity × activity) Circulating capital Replacement

needs at basic pricesps n × 1 (commodity × 1) Statistical basic prices of

domestically produced comm.ps

m n × 1 (commodity × 1) Statistical prices of importedcommodities



APPENDIX B: PRICE–VOLUME SEPARATION IN EMPIRICALSUPPLY–USE SCHEMES

Consider a set of square n × n Supply–Use Tables, together with comple-mentary gross fixed capital stock and flow matrices. In order to neatlyseparate price and volume components, we adopt the following workingassumptions:

(1) There is a uniform statistical basic price54 for each domestically pro-duced commodity (given by ps), as well as for each imported product(given by ps

m).(2) The terms of trade ˆ ˆ ˆe p s

ms= −p p 1 of the base-period are given.

Assumption (1) means that statistical basic prices used to constructSupply–Use Tables can be represented by diagonal matrices. This is justifiedin that, once taxes on products and trade and transport margins have beendeducted, the price of a unit at the factory is equal for every unit produced,during the current accounting period.

Assumption (2) is required to work with total (domestically produced-cum-imported) magnitudes. As domestic and imported commodities have differentprice systems, in order to re-express nominal magnitudes with a different pricebase, it is necessary to assume that the proportion between domestic andimported prices bears a given fixed relation, during the current accountingperiod.

Under these assumptions, table B1 reports the price–volume decomposi-tion of magnitudes composing the Supply–Use framework used throughoutthe analysis.

Note that by expressing previous period magnitudes in current prices (e.g.ˆ ( )p qs −1 ), activity level vector x(−1) does not depend on relative prices. More-

over, every domestic-cum-imported magnitude (identified by superscript *)depends on the diagonal matrix of terms of trade e p.

54 The sequence of prices in the SNA can be described as follows (UN, 2009, p. 103): Basicprices + Taxes on products excluding invoiced VAT—Subsidies on products = Producers’prices + VAT not deductible by the purchaser + Separately invoiced transport charges+ Wholesalers’ and retailers’ margins = Purchasers’ prices.



Table B1. Price–volume decomposition

Nominal = Price × volume Nominal = Price × volume

z = p qs K* = ˆ ˆ ( ˆ )p K p K Ks q s q p qm∗ = + e

V = p Vs q Fk∗ = ˆ ˆ ( ˆ )p F p F Fs k s k p k

mq q q∗ = + e

U = p Us q Fk = ˆ ˆ ˆ ˆq qF p F p Fk s q k s kq q∗ = ∗ =

fc = p cs Rk∗ = ˆ ˆ ( ˆ )p R p R Rs k s k p k

mq q q

∗ = + efk∗ = ˆ ˆ ( ˆ )p f p f fs k s k p k

mq q q∗ = + e Rk = ˆ ˆ ˆ ˆq qR p R p Rk s q k s kq q

∗ = ∗ =fk = p fs kq Jk

∗ = ˆ ( )p F Rs k kq q∗ − ∗

q = ˆ ˆ ˆp ps q sq −1 Jk = ˆ ˆ ˆ ( )q qJ p F Rk s q k kq q∗ = ∗ − ∗

U* = ˆ ˆ ( ˆ )p U p U Us q s q p qm∗ = + e Ru

∗ = U x p U x p R*ˆ ˆ ˆ ˆ( ) ( )− −= ∗ = ∗1 1s q s uq

x(−1) = ˆ ˆp V p q V qs q s q( ) =−−( )

−−( )

11

11 Ru = Ux p U x p Rˆ ˆ ˆ ˆ( ) ( )− −= =1 1s q s uq

Hence, absolute magnitudes corresponding to the vertically integrateddescription of the technique in use that we can actually compute depend onstatistical basic prices:

l V R R l p V p R p R

l V R p

T Tq q

T

u k s q s u s k

q q s

− −( ) = − −( ) =

= −( ) =

− −

− −

1 1

1 1

ˆ ˆ ˆ

ˆ

== −nTsp 1

for vertically integrated labour coefficients νT, and

K R V R R p K p R p V p R p R* + ∗( ) − −( ) = ∗ + ∗( ) − −( )− −u u k s q s u s q s u s kq q q

1 ˆ ˆ ˆ ˆ ˆ 11

1 1

1 1

=

= ∗ + ∗( ) − −( ) =

= −( ) =

=

− −

− −

ˆ ˆ

ˆ ˆ

p K R V R R p

p S V R p

s q u q u k s

s q q s

q q q

ˆ ˆp Hps s−1

for vertically integrated productive capacity matrix H, as defined in (13).However, this is not so for variations through time of vector νT and for

changes in the units of vertically integrated productive capacity—i.e. in thecolumns of matrix H. By expressing all magnitudes in a common statisticalprice system and computing rates of change, the effect of prices vanishes:

n n n n n nT T T Ts s sˆ ˆ ˆ ˆ ˆ( ) ( ) ( ) ( )p p p−

−−

−−

− −−−( ) ( ) = −( )( )1

11

11

1 11

p p pj j s s j j s j j j jˆ ˆ ˆ ˆ ˆ

( ) ( ) ( )h p p h p h h h−− − −

−−

−−( ) −( ) = ( ) −1

1 1 11

11

1hh j ( )−( )1



Proceeding in an analogous way for the vertically hyper-integrated case:

l V U F l p V p U p F

l V U F p

T Tq

T

k s q s q s k

q k s

− −( ) = − −( ) =

= − −( ) =

− −

− −

1 1

1 1

ˆ ˆ ˆ

ˆ

== −hTsp 1

for vertically hyper-integrated labour coefficients ηT, and

K R V U F p K p R p V p U p F* + ∗( ) − −( ) = ∗ + ∗( ) − −( ) =− −u k s q s u s q s q s kq q

1 1ˆ ˆ ˆ ˆ ˆ

== ∗ + ∗( ) − −( ) =

=

− −

−

ˆ ˆ

ˆ

p K R V U F p

p Mp

s q u q q k s

s s

q q

1 1

1

for vertically hyper-integrated productive capacity matrix M, as definedin (16).

Also in this case, computing changes through time allows us to get rid ofprice effects:

h h h h h hT T T Ts s sˆ ˆ ˆ ˆ ˆ( ) ( ) ( ) ( )p p p−

−−

−−

− −−−( ) ( ) = −( )( )1

11

11

1 11

p p pj j s s j j s j j j jˆ ˆ ˆ ˆ ˆ( ) ( ) ( )m p p m p m m m−− − −

−−

−−( ) −( ) = ( ) −1

1 1 11

11

1 mm j ( )−( )1

where mj is the jth column of matrix M.

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Nadia GarbelliniFacoltà di EconomiaUniversità degli studi di PaviaPaviaItalyE-mail: [email protected]

Ariel Luis WirkiermanITEMQUniversità Cattolica del Sacro CuoreVia Necchi 520123 Milano (MI)ItalyE-mail: [email protected]



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Productivity Accounting in Vertically (Hyper-)integrated Terms: Bridging the Gap between Theory and Empirics