Coding economic dynamics to represent regime dynamics. A teach-yourself exercise

Coding economic dynamics to represent regime

dynamics. A teach-yourself exercise

Juan G. Brida a,*, Martin Puchet Anyul b,1, LionelloF. Punzo a,*

a Department of Political Economy, University of Siena, Piazza S. Francesco 7, 53100 Siena, Italyb Faculty of Economics, UNAM, 18 Ciudad Universitaria, C. de Mexico, D.F. 04510, Mexico

Received 1 December 2001; received in revised form 1 November 2002; accepted 1 December 2002

Abstract

Much of the recent economic history of various countries, regions and/or sectors in the

world economy can usefully be reconstructed as sequences of repeated, basically endogenously

induced, changes of growth regimes. Coded dynamics (CD) is proposed hereafter as the

appropriate tool for the analysis of such multi-regime dynamics, i.e. dynamics where switches

between growth regimes represent structural changes, or in other words, abrupt alterations in

an economy’s qualitative dynamics . On a theoretical tone, one of our arguments in favor of the

adoption of a CD approach derives from the lesson that can be drawn from the complex

dynamics literature. Often, some form of regularity, while it cannot be found in the punctual

analysis of motion across states, can be recovered from the system’s dynamics over a partition

of its state space. This dynamics can be represented by strings of symbols (instead of real

numbers), or symbolic trajectories . The 2-fold purpose of this paper is to introduce the

formalism and terminology of multi-regime dynamics, and to try our hand with the technique

of coding through a set of simple exercises. In fact, we consider only cases with two and three

regimes, instead of the six of the Framework Space introduced in the Preface. Via such

exercises, we also trace the origins of the multi-regime framework in the tradition of classical

macrodynamics.

# 2003 Elsevier Science B.V. All rights reserved.

JEL classifications: C60; C61; E30; E32

* Corresponding authors. Tel.: �/39-057-723-2780; fax: �/39-057-723-2661.

E-mail addresses: [email protected] (J.G. Brida), [email protected]( L.F. Punzo) , [email protected]

(M.P. Anyul).1 Tel.: �/55-56-222-341; fax: �/55-56-160-834.

Structural Change and Economic Dynamics

14 (2003) 133�/157 www.elsevier.com/locate/econbase

0954-349X/03/$ - see front matter # 2003 Elsevier Science B.V. All rights reserved.

doi:10.1016/S0954-349X(03)00003-1

mailto:[email protected]



Keywords: Economic regime; Multi-regime dynamics; Coded dynamics; Symbolic dynamics

1. Introduction

One of the discoveries of the recent macroeconomic literature is that even very

simple dynamic models can display a great variety of surprising behaviour. Thus, it

has become standard to consider models with, e.g. multiple and/or indeterminate

equilibria, endogenous cycles and irregular fluctuations, and the like. In many cases,

this modeling innovation indicates an effort to introduce formally a notion (that of

dynamic regime), which already belongs to the jargon of the more empirically

oriented economists.

And, in fact, they are often more interested in highlighting only certain qualitative

features of an economy, i.e. in capturing its development pattern , to compare it with

other economies and/or to reconstruct episodes of abrupt change. The latter are

taken to indicate structural changes, i.e. changes in the underlying generating

mechanism or growth model . We believe a useful way to address this tightly knitted

set of issues is by resorting to the twin notions of dynamic regime and of regime

switch, as defined in this paper.

In the first part (Sections 3 and 4), we discuss how to formalize a fully-fledged

multi-regime structure and the implied dynamics. This notion was introduced as the

theoretical intuition of a fact-finding research on comparative structural dynamics in

a number of papers by Bohm and Punzo (2001). (Day (1994, 2000) and Day (1995)

formulates a fundamentally similar framework, multi-phase dynamics).

We build upon this previous work but we also take it one crucial step further. The

basic innovation introduced in this paper is that, keeping track of a system’s time

evolution, we just focus upon its qualitative behaviour, i.e. its dynamics as a

sequence of regimes. This leads to the symbolic technique. A multi-regime approach

has really something new to say when an economy’s path traverses (it ‘switches’)

once or repeatedly from one regime to another, thus depicting a more or less

complex history of relatively smooth dynamics with intermittently sudden and large

jumps. The latter can be the result of exogenous shocks and/or endogenously

accumulated forces of change. One of the tasks still ahead is to construct an ability to

discriminate between the two (if one such program makes sense at all). As it will be

seen, we support an endogenous interpretation, according to which structural change

is basically the result of regime instability.

In this light, conventional economic dynamics can be seen to rest upon the implicit

assumption that events of endogenous regime switches are rare, or else they are

irrelevant transients. It is on the other hand in this same light that one can appreciate

the increasing relevance recently assumed in less conventional approaches by such

notions as evolution seen as a sequence of unequally spaced regime switches. With

Day, we believe this framework is one way to capture this kind of issues, among

others.

Our definition of regime is meant to pin down what we believe to be essential of

the many available definitions. The corresponding macro-dynamics is formalized by

J.G. Brida et al. / Structural Change and Economic Dynamics 14 (2003) 133�/157134

a mathematical model built upon the twofold idea of within - and cross -regime

dynamics. If we were interested in the composition of these two , we would represent

the former by, e.g. a continuous time-continuous space model (a ‘dynamical model

written in conventional form’). Then, cross regime dynamics could be captured by

superimposing a switching mechanism, generating time-discrete jumps across

regimes. In this way, our model would technically be a hybrid system .2 Instead,

as the logical innovation mentioned above, we choose to re-define states as

discrete events (i.e. ‘belonging to a given regime’, ‘switching to a different one’),

so that the ensuing dynamics is of the sort studied by automata theory. Thus,

regime dynamics can be represented in a synthetic way by a labeling technique

to obtain a coded dynamics . CD is related to (and partly overlaps with)

symbolic dynamics, at the interface between mathematics, computer and information

theory, and linguistics. (For an introduction to symbolic dynamics, see

Adler, 1998; Lind and Marcus, 1995; for its relation to topological dynamics,

Robinson, 1995.) When our models happen to satisfy certain restrictive assumptions,

such proximity permits us the use of well-established formal techniques. An

economist’s motivation for coding however may rather lay in her need of handling

and extracting relevant information from noisy time series, often whole sets

of them.

While the idea of coding may be new to economists, the notion of a dynamics that

may take across regimes is not. It belongs to classical Macrodynamics, the theory of

cycles and growth initiated by Frisch (Frisch, 1933), and then vigorously

developed by Harrod, Hicks and Goodwin, among others, in path-breaking works

of the 1940s and 1950s.3 There, regimes are identified with the phases of the

business cycle, an idea that has recently re-surfaced in some important non-linear

econometric work. In the second part of this paper, (from Section 5 on) we illustrate

our approach with examples taken from such tradition. Only two phases or

three regimes are considered, as is typical of business cycle (BC) theories,

and, thus, our detour into the history of economic analysis is also an

introduction to the more complex framework of the framework space (FS ) where

growth as well as regular and irregular oscillations may take place at the same

time.

On top of expressing the issues of dynamically coupled regimes in a more

appropriate language, thus as an additional bonus, the coding technique appears to

simplify the treatment of a theoretical problem, which more conventional analytical

tools left unsolved, though already intuitively well understood.

2 A point made by one of the anonymous referees.3 The relevant references appear later in the text.

J.G. Brida et al. / Structural Change and Economic Dynamics 14 (2003) 133�/157 135

2. The intuition

Intuitively speaking, a regime is a qualitative behaviour usefully distinguished

from other dynamical behaviours. The interesting case, of course, is when

there is a whole set of such regimes, which may be mutually excluding alternatives,

or else potential components of a sort of dynamical menu. The issue, then,

becomes how regimes get stringed together into a path, as the history of a system

(be it an economy or a sector, or else a firm) may now display qualitative

change.

Hereafter, we focus upon the possibility of a system displaying overtime dynamic

behaviours that are qualitatively different in the sense precisely defined in the next

section. One image taken from the phase portrait of one such system is therefore a

path taking across trajectories in principle belonging to different regimes. Therefore,

each path can be seen as constructed piecewise, by stringing together pieces of

trajectories predicted by different models . Each model would be a kind of local

representation of an overall dynamics, thus accounting for only part of the

economy’s history.

Put differently, an actual or simulated history is interpreted as one specific

realization of a collection of already available regimes , a time sequence of

part-trajectories through regimes with their own timing and duration in some

agreed clock . A given system can go through the whole collection (regularly or with

varying degrees of irregularity) or through only a subset. Any change of

regime naturally signals some form of structural change, the sequential

ordering of visited regimes and other parameters of the time dimension

giving information relevant to understand which form. In fact, history may display

an identifiable pattern of regimes, i.e. a repetitive and stable sequence; more

often, we will see a varying sequence and/or an irregular ticking of the clock.

History may turn up to be more or less similar across economies, sectors or

firms.

This representation of dynamics, which keeps track of structural change in

the sense just defined, is called regime, or multi-regime, dynamics. Regime

dynamics is therefore best conceived as a dynamical system over a set of models,

and such set as a kind of menu of qualitative behaviours available to an

economy. Formalizing such dynamics requires introducing the notion of a state

space over models , which is naturally discrete. Now we can focus upon

regime switches as discontinuous changes or discrete jumps. It is to isolate this

kind of dynamics that regimes are assigned distinct symbols from an

adequate alphabet, and our description goes symbolic . The use of such technique

is the fundamental difference of our approach from the conventional one where state

variables are real numbers.


3. Defining regimes for multi-regime dynamics

Following Day, a regime is defined as a pair (8j , Xj ) where Xj is a set in a

partition4 of the system’s state space X such that X�//@nj�1/Xj , and 8j is the rule (i.e.

the dynamic law) over the set Xj ; i.e. 8j : Xj0/X . Often, the rule is improperly

identified with the model, a convenient shortcut we will also be using sometimes.

Also mostly for convenience, we will consider only rules in discrete time.

Of course, when X1�/X , we are in the standard, single regime situation, and the

regime and the model that describes it identify with one another. To be interesting,

therefore, the partition should slice the state space into two or more nonempty sets

Xj , j�/1, 2, . . ., n , or regime domains , and it needs to cover the whole state space. On

the other hand, the partition member Xj is associated with a given dynamic law fj as

its domain. In the present application, such law may be thought of as experimental

or an economic hypothesis interpreting certain chosen state variables and thus

restricting their acceptable values.

Clearly, with multiple regimes a variety of dynamical behaviours become available

to construct a system’s history. The latter displays, in principle, a twofold dynamics:

one within a given regime and one across regimes . Dynamics that we call

conventional for easy reference focuses upon the former overlooking the latter.

Coded dynamics we introduce later, is meant to focus upon the latter. Thus, the two

approaches are in principle complementary to one another. Their intermingling can

produce any kind of complex, erratic-like behaviour we may be looking for.

The key to our definition above is that the partition of the state space into a finite

number of regime domains goes together with multiple dynamic models in a specific

way. Regime dynamics is defined over the set of (8j , Xj ), j�/1, 2, . . ., n . We can think

of each of them as the mathematical representation of a specific model in the

economist’s sense. Regime dynamics, on the other hand, is defined over a space of

dynamical models (which are therefore naturally local ), and in this way its definition

goes beyond other similar definitions already available in the literature.

Often, however, economists use the notion of regime in much looser, though

intuitive, sense and with a number of different meanings that reflect alternative

criteria in defining qualitatively different behaviours. Correspondingly, there are

distinct procedures to introduce the notion, briefly reviewed in the next section.

4 As usual in Symbolic Dynamics theory*/see e.g. Alligood et al. (1997), p. 125�/126*/in this paper we

will abuse terminology in calling a ‘partition’ of the state space X , a collection of subsets of X , which have

pairwise disjoint interiors and whose union is X . E.g. a partition of an interval I is a collection of

subintervals whose union is I and which are pairwise disjoint except perhaps at the endpoints. For a well

defined dynamic rule, at a point x : x �/Xi S/Xj,8i (x )�/8j (x ) must hold, of course.


4. Regimes as mutually exclusive alternatives

Basically, the distinction in dynamic regimes can be based upon three criteria:

modal, bifurcation, and equilibrium. Ours is a fourth, distinct criterion, which can be

shown to have a logical connection with the last one of the list.

Regime is often used as a synonym for certain specific dynamic behaviours, i.e. it

stands for a dynamic mode . Typical is its use to identify growth (trend) from

oscillation. A model may exhibit qualitatively different dynamics (hence, regimes in

the sense above) depending upon the values of one or more of its parameters. With a

linear formulation this is quite easily seen, for parameters enter the coefficients of a

characteristic equation, hence determining its roots, which are the system’s dynamic

modes. For a textbook example, in a well-known model (Samuelson, 1939) of the

interaction between multiplier and accelerator, depending upon where the vector of

coefficients (the average propensity to consume c and the capital�/output ratio k ) lies

in the rectangular open region C�/K in R2, we get either growth or oscillations.

Hence, two modal regimes are generated because the parameter space can be

partitioned into sets IC and IR, IC the set of vectors yielding complex, IR real roots,

respectively, and neither of them is empty.

The co-existence or emergence of different dynamic modes can thus be ascertained

through an analysis of the structure of the parameter space, so that the modal and

the bifurcation criterion in defining regimes are basically two ways of looking at the

same thing. The bifurcation approach, actually, can be thought as a specific and

well-understood procedure to get a modal classification by parameterizing the

dynamical law (i.e. the model). In state space, on the other hand, these two regimes

are mutually exclusive alternatives, in the sense that a system would be locked into

one or the other unless parameters were to change exogenously . Which regime is

going to emerge, when and how, and when it will change, all this does not depend

upon where in the system state space dynamics is taking place. Each model is local in

the parameter space, while its predicted dynamics is global over the state space.

The above definition of regime, in other words, does not rely upon a partition of

the state space as required by our definition, though of course one such partition

may not be excluded for particular purposes. Still, the definition can only serve

purposes different from ours. Formally put, just like in the case where a model does

not contemplate the possibility, hence implicitly assuming a unique regime, system

state space and its partition are one and the same: X1�/X , and in state space they

behave as single regime models.

To see this, we introduce here the coded dynamics notation that will prove really

useful only later. From now onwards, symbols such as A, B will stand for regimes,

i.e. pairs (8j , Xj ), while Xj will indicate the domain of 8j . (Quite often, ‘domain’ will

also mean the ‘regime’.)

Let us have two regimes defined on the basis of whatever pre-determined criterion,

e.g. in this case the modal criterion. Thus, calling B the regime of oscillations, A that

of monotonic growth, trajectories can be re-written as infinite strings of symbols by

inserting a ‘A’ or else a ‘B’ whenever a path is monotonic or, respectively, oscillatory.

This convention is here a preliminary step to coded dynamics . For the moment it only


implies a re-coding of paths described in real number coordinates: from the infinite

sequence {. . ., x0, x1,. . ., xk ,. . .} through an initial state x0 (where any xj is a vector

of real numbers), to a symbolic string of As and Bs.

A coded sequence, therefore, records a path over the given set of regimes. Thus, in

the case of modal regimes, we can observe either an infinite string of A: A.AA. . .�/

(A)�, or else of B.BB. . .�/(B)�, but no strings of mixed symbols . Moreover, as said,

(A)� will be associated with parameter vectors in IR, while (B)� for vectors in IC,

(sets IC, IR defined above). In other words, let the two modal regimes be A�/

(8R, X1), and B�/(8C, X2), functions 8C, 8R with parameters appropriately chosen:

their domains cannot be distinguished from one another: X1�/X2�/X .

That this be so, is essential to our definition above. It implies that continuous

dynamics can be separated from discontinuous or structural change, and that initial

conditions and/or the sequence of states (i.e. ‘history’) in general do matter in

determining current and future dynamics. In the multi-regime framework, if

parameter values were to change so as to bring about also a regime change, they

would be driven, in principle, by some state-dependent rule. In theory , regime

selection is in other words endogenous, without this excluding the possibility of

combined stochastic forces in reality . However, coding dynamics has not much use

whenever dynamics takes place within one and the same regime, as in the

conventional approach and in the example above. In both of them, there is only

dynamics within as we call it.

An amended version of the example above, which can be attributed to Harrod

(1936),5 though it still yields coded dynamics of the same kind, highlights the role of

state-space dimension in defining regimes.6

With the average propensity to save s�/ (1�/c ), and k the capital/output ratio,

Harrod’s model predicts the monotonic trajectory for output

Y t��(1�s=k)tY0 (1)

as the unique solution (from initial level of output Y0) to

Yt� (1�s=k)Yt�1�f (Yt�1) (2)

which, in its turn, is the reduced form of the well known structural model

St�sYt�1

It�k(Yt�Yt�1)�kDYt

It�St

8

<

:

(3)

5 Notice that, here as later, we are not referring to the better known article by Harrod (1939)!!!

Attributing Eqs. (1) and (2) to Harrod leads to a discussion of the relation between Samuelson’s and

Harrod’s models, which we do not want to pursue here. Their use is purely illustrative, and therefore the

naming is introduced for easy reference.6 We are using textbook symbols: c , s , k stand for the parameters propensity to consume and to save,

and capital output ratio; Y for income; S and I for savings and gross investment, respectively.


Whatever the parameter vector (s , k ), now as opposed to before, only monotonic

growth can arise with s /k/�/gw the so-called ‘warranted rate of growth’.7 In other

words, the parameter space cannot be partitioned in terms of the modal criterion.

Coding is trivial because there is only one regime, the equilibrium regime . In our

notation, A�/(8, R�), f being defined by Eq. (2).

We need to remove either assumption, linearity and/or dimension one, to get more

interesting coded dynamics. Hereafter, we remove the latter. In fact, the modal

criterion that here fails in partitioning the state half-line R�, performs better in

higher dimension. We can indeed get distinct dynamic modes associated with a

partition of a X⁄Rn state space, n]2. It can easily be shown that for a Leontief-

like model dynamized with a simple lag, after setting the impulse function identically

equal to zero, the matrix equation

xt�Zxt�1 (4)

(where Z is a nonnegative matrix of order n and xt is a vector of activity levels), can

be considered mathematically identical to Harrod’s Eq. (2) except for dimension. In

fact, growth factor (1�/s/k ) as a one-dimensional matrix Z , is also its (unique)

eigenvalue, nonnegative due to the economic restrictions upon the two structural

parameters, s and k. In the n -dimensional case, the eigenvalues of Z are akin to

generalized warranted rates of expansion (contraction, respectively), but, obviously,

some may very likely turn up complex.8

This, of course, implies a decomposition of the state space consistent with the

modal criterion for regimes, but wrongly assimilated to the one introduced by our

definition. Let L be the diagonal canonical form of Z above, i.e. L�/ (P�1ZP ), P

being the modal matrix of Z . We now have two regimes in the modal sense, namely

B�/(LC, XC) and A�/(LR, XR), where each regime is obtained from the original ?

restricted on initial conditions lying in XC, or XR, respectively. Thus, using principal

coordinate axes the n-dimensional state space, Rn� , gets decomposed into the direct

sum Rn� �/XR�/XC, where XR is the linear subspace of monotonic and XC, the linear

subspace of oscillatory dynamics. Both subspaces are generally non-empty, and by

selecting initial conditions appropriately, we can generate monotonic expansion

alone (contraction, when this is the case),9 or alternatively, only oscillations. This is,

in a sense, the modal regime criterion fully blown : for, both regimes emerge for any

generic dynamic matrix Z .

Thus, provided state space dimension be great enough, regimes of oscillation and

growth are no longer mutually exclusive, and even linear models appear to be able to

7 Recall that while in Samuelson’s model the accelerator yields a second order difference equation,

Harrod’s yields only a first order equation. It is for this reason, as is well known, that dynamics in the

former case may exhibit also oscillatory motions that cannot arise in the latter.8 The set of theorems associated with the names of Perron and Frobenius apply here, for we have taken

the input-output coefficient matrix to be also the system’s dynamic matrix. Of course, the unique

eigenvalue of a scalar matrix is its dominant eigenvalue.9 Corresponding to dominant eigenvalues that are negative or larger than unity, depending on the

dynamic formulation in continuous or discrete time.


display different qualitative behaviours over different slices of their state space.

Here, modal regimes hold one key property (dependence upon initial conditions) of

regime dynamics as we define it, though in a sort of weaker form.

Still, also the definition of regime(s) on the basis of a principal axes decomposition

(Punzo, 1995), where local models are obtained through restrictions on initial

conditions instead of parameter values, does not meet the requirements of the one

proposed in Section 3. First of all, it is really appropriate only to linear models. More

crucial is that paths starting in a linear subspace cannot cross over to another one, as

initial conditions get them trapped into the starting regime. There is a multiplicity of

regimes but no interesting dynamics. Coded orbits can be constructed as infinite

strings inserting B for any dated state xt � /XC, A for xt � /XR. Here, if a sequence

starts from an initial state in XC (alternatively, from a x0 � /XR), it will show the same

symbol forever. And, finally, from generic initial conditions, dynamics will display

both oscillations and monotonic behaviour (Goodwin, 1949; Goodwin and Punzo,

1987), and therefore, such regime dynamics cannot be effectively coded.10

The example above is, therefore, meant to illustrate the dynamic variety (in terms

of modal regimes) that can be obtained by stretching a linear model as much as

possible (Punzo, 1988, 1995). For multi-regime dynamics,

¯

dimension plays no

essential role; the key to it is non-linearity. It is to explain this issue that we continue

reviewing elementary models in the classical tradition of Macrodynamics. They

provide a familiar benchmark and a common environment. Moreover, it is true that

the idea of dynamics as a sequence of regime can be found already in Hicks’

approach to the theory of business cycle (Hicks, 1950). The historically subsequent

move to Business Cycles models formulated in continuous time has obscured the

original intuition of an economic dynamics built up of a finite set of local

behaviours, the primitive idea of regime.

5. Towards a simple scheme of regime coupling

Linear models are global models, and they chart dynamics in an un-structured

state space. In our jargon, they yield single-regime dynamics. Only models that are

constructed as local representations, possibly of a globally non-linear model, may

show paths across (two or more) regimes. If they are parameterized, their parameter

values must depend upon state location. This is the key form of non-linearity that

may explain (the possibility of) regime switches, in other words the discontinuous

dynamics across regimes implying, in our definition, some form of structural change.

We can build a road to this kind of models starting from the single-state variable

equation above and introducing two co-existing regimes. The peculiarity of

dynamics described by Eq. (1), is in that the solution path connects a monotonic

sequence that coincides with the state space, the half-line R� of Yt , to the right of

the given Y0. A coded path can only be of the type AAA. . .�/(A)�. As we start from

10 In other words, in general dynamics cannot take place on disjoint sets of the state space.


an initial state by definition already in A and can only stay within A forever, a very

simple dynamics within A can issue. The mathematical simplicity of this model lies in

this coincidence of three logically distinct notions: the regime domain R�, the system

state space and the equilibrium set. To understand its conceptual complexity we have

to embed it into a larger space.

A simple graphical representation may be useful to focus upon the point we want

to make here. The equilibrium dynamics can be projected into the origin of the state

space, so that all Harrodian-like growth paths are zoomed into it regardless of their

different initial conditions. This is done by taking the deviation from the equilibrium

path(s): yt�/Yt�/Yt* as state variable, where Yt� is the mobile equilibrium solution

to the original equation. Whence, the new equation in deviation from the reference

equilibrium is

yt�(1�s=k)yt�1�8(yt�1): (5)

where, of course, yt"/0, provided yt"/0, some t5/t .11

An equilibrium for this equation is a fixed point of map 8 ; i.e. a value yt such that

yt�/yt�1. Linearity of 8 implies that there is only one such fixed point, i.e. yt�/

yt�1�/0, while elsewhere in R /{0} a nonzero deviation is monotonically increasing

to 9/�. Therefore, the unique fixed point is an unstable equilibrium. Following

Harrod (1936), we now introduce the distinction between an ‘equilibrium regime’

and a ‘disequilibrium regime’. And in fact, having embedded path (1) into a larger

space, a distinct rule can be associated with two slices of the state space. On the one

hand, we have the rule describing the equilibrium path of the values of the levels of

Y : this is the 8 defined in Eq. (2) over domain X�/R�. On the other hand, in

deviations y from equilibrium, the state space becomes the whole of R . Therefore,

for clarity, we will indicate the 8 in Eq. (5)) restricted to the single point domain I1�/

{0}, with symbol 81, and we will assign 82 to govern the disequilibrium behaviour

expressed in deviation, where 82 is the restriction of the same 8 to the domain: I2�/

R /{0}. Notice that, in compliance with our definition, the partition of the state space

satisfies the condition: X/�/R�/I1@ /I2. The possibility of having simultaneously two

rules on the same state space, working on different parts of it, could not appear in

the original Harrod’s model because there was no distinction between regimes

(though it was hinted at by Harrod himself (see Punzo, 1988)).

As is well known, any path starting in equilibrium (hence, in the equilibrium

regime) will stick to it, just as before. The novelty is that now, starting or after having

been shocked out of the equilibrium regime, the economy will never return to it, and

one can see that the Harrodian equilibrium or growth regime is not an attractor.12

As before, after coding, growth regime A and Harrod’s equilibrium path coincide,

while symbol B is associated with the set of disequilibrium states as a distinct regime.

11 We are assuming that the deviation from equilibrium is induced by a single shock at some time t�/0,

which propagates at the same rate thereafter. (This logical consequence of the linearity assumption was

dismissed by Harrod himself.)12 This is another way of expressing, in terms of regimes, the known knife-edge or (in-)stability

property of Harrodian paths.


On the other hand, the partition P of X : X�/I1@ /I2, can be further refined with

I2�/I2�@ /I2� to obtain a mathematically more convenient construction. Looking at

the dynamics as an ordered sequence of regimes visited and coding with the

technique introduced above, we obtain only two of the possible infinite strings that

can be constructed with two regimes: i.e. either (A)� or, alternatively, (B)�,

depending on the initial regime. We still cannot construct strings with anything like

AB or BA as part components, in other words with traverse orbits leading from one

to the other regime.

The previous re-formulation, while confirming well-known features of the

standard version of Harrod’s model, adds something new. One realizes that the

economy cannot stay forever in the disequilibrium regime, as the distance from

equilibrium would increase without bounds were the system following a rule 8 like

the one in Eq. (5).13 In other words, such regime is made up of transient states only,

and thus it cannot display any systematic (‘long run’) behaviour. While its presence

does not alter coded dynamics, it helps us to understand the importance of the

outward properties of a regime in determining (cross-) regime dynamics.

In fact, to produce a proper regime, the rule applicable to interval I2 has to be

different from Eq. (5), as it must allow for the possibility of ‘turning inwards’, or

admit an equilibrium inside the domain, or both. We slightly modify the basic model

above, to see how this alters dynamics within in a relevant way.

6. Going non-linear

Two features are peculiar in the previous model. First, regime A collapses into its

unique equilibrium, while a regime normally should comprises also a set of transient,

or off equilibrium states, if not additional equilibria. By contrast, regime B has no

equilibrium at all, only transients. In the economist’s intuition, with regimes

basically being the same as models, regimes must comprise at least one equilibrium,

the basis for prediction. This peculiar situation can be easily amended, as it stems

from the linearity assumption. In fact, under linearity, states that are not in the basin

of attraction of the (unique) equilibrium can only span a null regime, as is the case

here. If, instead, they are in its basin of attraction, they cannot span a distinct regime

domain. The full decomposition into distinct or uncoupled regimes in the model

above is somehow artificial. In any case, it is not very interesting.

From the fact that the domain of regime A is a point set , stems also the ‘odd’

situation that paths from almost all initial conditions in X fly away from

equilibrium, the economy is practically always in the null regime B. In other words,

equilibrium (i.e. the regime A) is rarely observed . This profound intuition in

Harrod’s own argument becomes, we believe, evident in our multi-regime environ-

ment.

13 Regimes where the system behaves in this way, are called null regimes in Day (see e.g. his (1994)).


Let us consider a slightly generalized version of the model above, with a ceiling to

the investment function

It�kDYt; for DYtBI�=kI�; for DYt]I�=k

�

(6)

where I* is a positive value for investment. This can be determined, e.g. by the

constraint equation: I�/N (1�/c ), where N is the full employment level of income

(assuming unit productivity of labour) and the right hand side is the corresponding

level of savings given consumption habits. Then,14 as long as ytB/N , we get the path

of monotonic expansion yt�/(1�/s /k )ty0. While, once N has been reached, thereafter

the economy switches onto the stationary path yt�/N�/I*/s .

Correspondingly, in the dynamic equation: yt�/8(yt�1), 8 becomes piecewise

linear with one kink at y�//y�Nk=k�s; i.e.

8 (yt)�8 1(yt�1)�

�

1�s

k

�

yt�1 for yt�1B y

8 2(yt�1)�N for y5yt�15N

8

<

:

(7)

and in the (yt�1 , yt )�/plane, the situation looks like Fig. 1.

Here, regime A is the pair (81, I1), while regime B is (82, I2), with corresponding

domains I1�/(�/�, y]; I2�/[/y; N ], respectively, and of course X�/(�/�, N ]�/I1@ /

I2, while the two rules 8js correspond to restrictions of 8 to their respective domains.

Therefore, the trivial and unstable equilibrium y�/0 together with Harrod’s growth

path defined above, still span regime A, but the introduction of one non-linearity has

14 Recall that the reference equilibrium value of Y was set into the origin of the state space, the Real

line. Therefore all variables and constants are measured in deviations.

Fig. 1. Harrod’s piecewise linear model.


added a new equilibrium path N that lies in a distinct regime B and acts as a global

attractor. The internal structure of regime B is again very simple. But, now, in the re-

defined domain I1 of A, any sequence starting at y � /I1 with y�/0, will fly away

towards the single path N � /I2, thus entering regime B after a finite number of

iterations. Still, paths may stay in I1 (or in I2) forever if they have as initial condition

a y5/0 (or any value y : y5y5N; respectively).The corresponding coded dynamics is slightly altered by this modification: in

addition to the strings (A)� and (B)� produced by the previous model, the

emergence of a traverse from regime A to regime B (when we start at the right of the

unique equilibrium in I2) introduces a new type of coded sequences:

A.A. . .ABB. . .BB. . .�/A.A. . .A(B)�; i.e. a finite string of As followed by an infinite

sequence of Bs.

7. Garden-variety regime dynamics

Fig. 2 below illustrates a simple graphical way of representing coded dynamics for

map 8 by using finite directed graphs, when we have the covering partition15 above:

P�/{I1, I2}. Graph G, called the transition graph for the partition P, is constructed

using a covering rule : for example, an edge from A to B is drawn if and only if the

image 8 (I1) of I1 contains the interval I2. Thus, if we have been able to construct,

from our economic models, a covering partition P of the domain of the

corresponding dynamical system, its associated coded dynamics can be effectively

represented by a transition graph. Another way to represent the same dynamics is via

the transition matrix associated with the graph. This is a square {0, 1} matrix

whereby there is a 1 (alternatively, a 0) in the ij -entry whenever there is an arrow in

the transition graph leading from the i-th to the j-th vertex-regime.

This and the previous sections illustrate elementary regime dynamics that can arise

with a 2-regime menu . Clearly, a symbolic string does not give enough information

to uniquely recover the model that has generated it. The scenarios lying behind

15 P is a covering partition for the map 8 if every element of P is mapped by 8 onto a union of

elements of P . (See, e.g. Alligood et al., 1997.) The choice of using such mathematically convenient,

though restrictive, partition does not generally contradict the economic intuition behind our definition of

specific regimes. However, we will resort to a less demanding version of it.

Fig. 2. Directed graph G and adjacency matrix T representing coded dynamics for the two versions of the

basic (Harrod’s) model. For the nonlinear version, G has a new arrow from A to B and T , a 1 (instead of

0) as the upper-right entry.


simple strings may be far richer and interesting. But the representation in the coarser

regime dynamics highlights the properties of fully decomposable and unilaterally

coupled dynamics of these two cases. The arrow from A to A and the one from B to

B indicate that any path starting in a given regime may remain trapped there forever,

the two regimes not mixing up with one another to generate traverse paths.

However, in the latter case, the economy is (more ) likely to switch from regime A to

regime B than to stay put in the former. While it does not have the capability of

switching back, still a tendency to change regime (to undergo structural change, in

our definition) is in-built into its dynamics, so to say. In other words, structural

change can be the result of endogenous forces, possibly coupled with or triggered by

exogenous shocks.16

In this paper, we focus upon endogenous conditions for the emergence of such

traverse paths, this being in a sense the purpose and essence of multi-regime

dynamics. These conditions can obtain as the result of the dynamical coupling of

regimes and the latter can follow a variety of (basically non-linear) schemes.17 With

two regimes, one can already anticipate the possibility of more interesting cases. E.g.

we may immediately envisage regime dynamics that, once coded, look like

A.BABAB. . .�/(AB)�, or B.ABABA. . .�/(BA)�, of course. More complex strings

may contain a block or sub-string of symbols in a repeating pattern after an initial

transient lingering in a different regime. The simplest instance is, indeed,

AA. . ..AABABAB. . .�/AA. . ..A(AB)�, the bracket identifying a repeating 2-regime

block, while outside there is a transient in A that may take up any length of time to

dissipate. Thus, already with two regimes there is a whole variety of regime dynamics

to take into account; and accordingly one should construct a multi-regime

framework able to reproduce it.

Preparing for the next simplest case, let us adapt conventional terminology so as

to be appropriate to coded dynamics . Keeping to the one-dimensional state space X/⁄/

R , the selected partition criterion must be able to generate a finite set of pairs (8j ,

Xj ), j�/1, 2,. . .,n Xj being intervals on the real line satisfying the union criterion of

our definition. Each such pair (or regime) can be coded with one of j symbols, say

the first 1 to n letters of the Latin alphabet. An infinite symbolic string is a regime

trajectory , sometimes called orbit, provided an initial regime be specified and,

afterwards, only regimes be recorded in terms of the alphabet, in place of actual

states along the R(eal)-axis. Thus, an infinite string displaying one and the same

symbol will naturally represent an equilibrium in the regime sense ; likewise, an

infinite sequence of the type (AB)� is a period-two cycle in terms of regimes. The

presence of an initial set of As (or Bs) indicates a transient dynamics within that

regime, which eventually may settle down to a regular or repetitive behaviour in

16 An exogenous shock is needed to displace an economy in the Harrod regime A into its corresponding

null regime B, when as in the linear model above no connecting arrow exists between them. This is a simple

exogenous interpretation of why regime changes take place.17 The natural reference here is to Goodwin (1947).


regime sense, a regime attractor or else a regime cycle which is therefore akin to a

limit cycle.18

In this way, we have recovered by analogy notions, like that of regular dynamics

as represented by equilibria, point attractors, and regular cycles, which are typical of

conventional analysis. They are now re-expressed in terms of regime dynamics. We

are going to discuss how a regular cycle can be generated. However, we should keep

in mind that, beyond such garden-variety regime dynamics, more complicated

examples can easily be thought of (and discovered ‘in reality’). They are investigated

in the companion papers in this issue. The study of these latter more complex cases

is, in fact, the motivation for going coded and/or using other techniques that are

amenable to translation into the regime framework.

It is clear that dynamics leading from one regime to another can be generated only

if one or both of them are unstable and at the same time it is not a null regime (in

other words, if it ‘points inwards’, as said before). Then, transients having died away,

some long run behaviour in the regime sense will emerge. With two regimes

available, if both of them point inwards, we may get a regular cycle. In other words,

counting the equilibria, three of such ‘long run’ strings can be generated.

These symbolic trajectories can be generated also by models with any number k]/

2 of regimes (and symbols). Thus, we will say that a multi-regime model is a

candidate theory for a given regime path if it can reproduce that coded string with a

minimal number of assumed regimes. We illustrate this idea continuing our detour ,

and introducing a classic model with three regimes, which is also an interesting piece

of history of economic analysis.

8. Simple indecomposable regime dynamics

For any given partition of the state space in regime domains , a rich dynamics can

be constructed by concatenation of local models (phase structures, in the language of

Day). Dynamical variety can thus be generated by varying the coupling scheme. This

is the idea to be illustrated in the sequel. As said, it was Hicks (1950) who first tried

to formalize the notion of a concatenated three-regime dynamics as the explanation

of the business cycle, though Goodwin (1947) had already explored it in a general, n -

dimensional setting. Hicks’ argument can be re-cast into the terminology of regime

dynamics.

To recall: a model was obtained before, expressed in deviations from the long run

equilibrium that is represented by an exogenously-driven growth path. Locally, such

a trend is unstable and any small deviation gets amplified over time via the working

of the interaction between the multiplier and a standard, linear accelerator

mechanisms. This latter applies to deviations that are not too large, symmetrically

on either side of the equilibrium path. This identifies the domain for a middle regime

18 One such cycle is an instance (and the simplest instance) of the structural cycle discussed by Boehm

and Punzo, in e.g. (2001), and in the IDEE project, mentioned in the Preface to this issue.


with its own (linear) model. Such explosive dynamics, however, is checked by the

presence of the two threshold values for DY, a ceiling and a floor . There, the

investment function will switch to a different form, and the system’s dynamic rule 8

will do so, accordingly. Hence, a three-regime structure is implied.

Thus, also here, the investment function It is a non-linear function of the rate of

change of income DYt but now with two non-linearities (see appendix for the

derivation) and in the dynamic Eq. (7), the specific function 8 will be

8 (y)�8 1�m1y�n1; for y5y1��n1(m2�m1)

�1

8 2�m2y; for y15y5y2�n3(m2�m3)�1

8 3�m3y�n3; for y25y

8

<

:

(8)

with m1B/c , 1B/m2, m3B/c , 0B/n1 and 0B/n3.

Our basic model has again a first-order non-linear difference equation.19

However, the new map f is piecewise linear with two kinks at y1 and y2, defining

the borderlines of the three regime domains. According to our definition, Eq. (8)

defines three regimes, one for each branch 8j , j�/1, 2, 3, of 8 together with its own

domain. Dynamics, of course, depends upon the values of the five parameters: m1,

m2, m3, n1, and n3.

Map 8 has still a fixed point at y�/0 which is repelling (unstable) as m2�/1. On

the other hand, compared with our version of Harrod’s model of Section 6, it has

two more fixed points p1B/0 and p3�/0 (as m1B/1 and m3B/1). And pi is stable if

and only if mi�/�/1.

As long as both m1, m3�/�/1, regime dynamics is as simple as before. Any initial

y0"/0 goes either to p1 or to p3, as t goes to infinity. This reproduces Harrod’s

situation of a decomposable dynamics, but now in a fuller way: for, both non-zero

fixed points are attractors within their own regimes. Regime dynamics is the simplest

as it decomposes into two local and uncoupled behaviours, each with its own

attractor, while the middle domain splits up in two: the left-hand and the right-hand

sides becoming transients with respect to the adjacent stable regimes. To generate a

regular (regime) cycle corresponding to Hicks’ result, the real novel result with

respect to Harrod, both regimes have to be unstable in a specific way.

One can think of such result as obtained by smoothly changing slopes mi , hence

decreasing them to less than�/1 till they cross their stability bifurcation values. (This

would be akin to a flip or period-doubling bifurcation.) We could, in other words,

resort to bifurcation techniques and investigate the changes in behaviour of the one-

parameter family of functions {8(y ); m1}, the parameter m1 being allowed to change

to the left of the critical value �/1. (An analogous exercise could be carried on with

the other parameters, to obtain a complete picture for a family of maps {8(y); m}

parameterized by a vector m .) More heuristically, we are going to investigate what

regime dynamics is likely to be for a sample of values of a selected parameter, in its

19 In Hicks, a second-order lag hypothesis is introduced, i.e. that DYt -1 �/ (Yt -1- Yt -2), but we prefer to

stick to DYt : that the global reduced form model is non-linear replaces, in fact, such rather arbitrary

hypothesis in generating the sought oscillations.


Fig. 3. Hicks’ model. (A) Graph of f for parameter values m1�/�/2, m2�/2, m3�/�/4, n1�/4 and n3�/6. The interval I�/[�/2, 2] is invariant under 8 and all

interesting dynamics lies in I . The latter has been partitioned in four regime domains, such that: I1�/[�/2,�/1], I20�/[�/1, 0], I21�/[0, 1] and I3�/[1, 2]. Under

such partition: I1@/I20�/8(I1), I1@/I20�/8(I20), I21@/I3�/8(I21) and I1@/I20�/8(I3). (B) Graph of 8 for m1�/�/3, m2�/2, m3�/�/4, n1�/4 and n3�/6. Now, the

regime partition is such that: I1@/I20@/I21�/8(I1), I1@/I20�/8(I20), I21@/I3�/8(I21) but again: I1@/I20�/8(I3). (C) Graph of 8 for m1�/�/4, m2�/2, m3�/�/4,

n1�/4 and n3�/6. In this case, the regime partition I�/I1@/I20@/I21@/I3�/8(I1), verifies I1@/I20�/8(I20), I21@/I3�/8(I21). While as before, I1@/I20�/8(I3).

J.G.Brid

aet

al./Stru

cturalChangeandEconomic

Dynamics

14(2003)133�/157

149

instability interval. The exercise is meant to show that regime dynamics from

concatenation via instability is quite rich, in fact richer than Hicks’ himself saw.

With m3�/�/4, m2�/2, x2�/1, x1�/�/1, n3�/6, as n1�/2�/m1, map 8 depends

solely on parameter m1 and it has three branches again, as shown by the formula

8 (y)�

8 1�m1y�m1�2; if y5�1

8 2�2y; if �15y51

8 3��4y�6; if 15y

; where m1B�1:

8

<

:

(9)

There are three interesting cases to look at: for m1�/�/2, m1�/�/3 and m1�/�/4,

respectively. Fig. 3 shows graphs of 8 in these cases.

Notice that in all cases the interval I�/[�/2, 2] is invariant under the map 8

defined in Eq. (9) and therefore all interesting dynamics lies in I . We further

partition it into four intervals: I1�/[�/2,�/1], I20�/[�/1, 0], I21�/[0, 1] and I3�/[1, 2].

Such partition reflects, to an extent, the regime classification implied by the

economic model we are using, but it is in fact also finer for, again, it also adheres

to a criterion of mathematical convenience (something we have done before

introducing the notion of the covering partition). In fact, while I1 and I3 do

correspond to the outer regimes as defined in Hicks’ model, the domain of the

middle regime is here treated as the union of two intervals: I20@ /I21.20 However, one

can easily check that the two (economic and mathematical) criteria are not in

contradiction with one another, as splitting the middle domain does not introduce a

new regime: as the rule f2 remains the same over both I20 and I21, regime B is split

into B� and B� only in order to identify two intervals in its domain.

For the three cases associated with the different values of m1 and the chosen

partition of I, the following relations hold under map 8:

I1⁄8(I1); I20⁄8(I1);

I1@I20�8(I20);

I21@I3�8(I21);

I1@I20�8(I3):

Moreover, we have I1@ /I20�/8(I1) for m1�/�/2, I1@ /I20@ /I21�/8(I1) for m1�/�/3,

and I1@ /I20@ /I21@ /I3�/8(I1) for m1�/�/4. Thus, for the three selected values of m1,

P�/{I1, I20, I21, I3} is a very convenient covering partition. Let A�/(81, I1), B��/

(82, I20), B��/(82, I21), and C�/(83, I3). Then, for one such partition and the

associated maps 8j , coded dynamics is represented by the directed graphs and

transition matrices shown in Fig. 4.

20 In fact, distinguishing the middle region into I2 and I3 is useful also from the point of view of the

theory when fixed points in I1 and/or I4 are attractors, as is in the previous case (and in our non-linear

version of Harrod).


The structurally stable feature of the dynamics associated with this three-regime

structure, therefore, appears to be the presence of two, period�/two mathematically

distinct cycles, (AB�)� and (B�C)�. In fact, for all three sample values of the

selected parameter m1, two minor cycles arise: one within what may be called for

short, the ‘positive piece’ of I , i.e. I21@ /I3, the other within the ‘negative piece’, i.e. in

the set I1@ /I20.

To understand this situation, first notice that adjacent sets in the partition share a

frontier state that therefore carries two tags. Let us now look at the structure of the

outer domains, I1and I3. In I1 (similarly, in I3), there is a unique fixed point that is a

repulsor; hence all dynamics surging out of the fixed point will point outwards. Thus,

there will be a direction for a representative point to travel such as it will get

eventually lost (the left hand side in I1, the right hand side in I3, with respect to the

corresponding fixed point). In a finer partition: I1/{p1}�/I1�@ /I1� (with I1�, I1�the right- and the left-hand side open intervals, p1 the fixed point),21 while I1� acts

locally as the domain of a null phase regime, I1� enters into the basin of attraction of

the inner regime, i.e. I20@ /I21. An analogous argument applies for the outer-regime

I3: representative points are flown towards I1� (from I20) and I3� (from I3,

respectively). This explains the presence of the minor cycles, coded by the strings

(AB�)� and (B�C)�, mentioned above.

For sample value m1�/�/2, on the other hand, there are also paths leading from

the positive to the negative piece, but not in the opposite direction, so there is no

cycle across regimes . For m1�/�/3, we may go from I1 to I21 and then a new path

from A to B� emerges introducing the possibility of crossing from the negative to

21 The same can be done for I3.

Fig. 4. Transition graph and matrix Tm1for the partition P�/{I1, I20, I21, I3} and map 8 corresponding to

the three parameter values: m1�/�/2; m1�/�/3; m1�/�/4.


the positive piece, thus entering into a distinct cycle within . Only for m1�/�/4, can

one also go from I1 to I3 so that a cycle (AC) arises between the two outer regimes.

As for the sought major, regime cycle A�/C, which was the objective of the whole

exercise, we have to call on intuition to help understanding the implied dynamics.

This cycle does emerge but it gets interlocked with the two minor cycles, and stylized

period-two cycles mix up with cycles with different periods up to at least 6.

This is a Hicksian dynamics, which, set in regime format, looks quite rich, in fact

richer that it needed to be for the purpose. And this is what Goodwin thought of it.

9. But two regimes are sufficient for a regular cycle

Goodwin’s (1951) intuition was that the minimal number of regimes that need to

be coupled to generate a regular cycle is two, instead of the three of the Hicksian

approach. This can be easily seen from a version of the basic model, which is in a

sense simplified to allow for one nonlinearity only.22 This version of Goodwin’s non-

linear accelerator-multiplier model in discrete time can be still represented by the

same first order difference, where (see Appendix A) the function 8 is a piecewise

linear map with two, instead of three, branches of the type

8 (y)�8 1�ay; if y5y0�d(a�b)�1

8 2�by�d; if y0By

�

(10)

where 1B/a , bB/cB/1 and 0B/d ).

The map has again two fixed points: 0 and p1 and the value y0 lies in between, i.e.

0B/y0B/p1. Given the conditions that must be verified by parameters a, b and d , the

stability properties of the fixed points depend only on the values of parameter b : 0 is

a repulsor whatever the value of b , and p1 is stable for �/1B/bB/1, unstable for b5/

�/1.

We distinguish two regimes, one for each monotone branch of 8. Correspond-

ingly, we label I1 for the left and I2 for the right interval in a partition of the phase

space at the threshold value y0 obtaining regimes A�/(81, I1) and B�/(82, I2),

respectively. Here too, mathematical convenience suggests to further split I1 into:

I1��/ (�/�, 0), I1��/ (0, y0] and point set {0}, relative to 0 as the fixed point,

because the dynamics there is of the following form:

a) paths starting at 0, remain there forever;

b) any path starting in I1�, goes away to infinity and the system is self-destructing;

c) any path starting in I1�, goes to the stable fixed point if �/1B/b and when �/

1�/b it goes to I2 to remain there.

22 See Goodwin (1950, 1951); see also Punzo and Velupillai (1997) on Goodwin’s use of Occam’s razor .

On dynamical systems with a single non-linearity, see Le Corbeiller’s paper reprinted in Goodwin (1982)

(Le Corbeiller, 1960).


From this we can deduce that, as long as �/15/b , the symbolic sequences for

regime dynamics are: AAA. . .�/(A)�, BAAA. . .�/B(A)�, BAA. . .ABBB. . .�/

BAA. . .A(B)�, AA. . .ABBB. . .�/AA. . .A(B)�, and BBB. . .�/(B)�. Thus, basically

the same situation seen in the Harrodian generalized model above appears again and

no cycle is present. To look for this result we need to consider the case where bB/�/

1. In fact, for bB/�/1, there is a zero of 8 at y1�/�/d /b and all the interesting

dynamics occurs in the interval [0, y1]. (See Fig. 5 for a representative graph of 8 for

bB/�/1).

In this case, map 8 is akin to the well known tent map. In particular, when

8(y0)�/y1 the partition {[0, y0], [y0, y1]} of interval [0, y1] is covering and it verifies

8([0; y0])� [0; y0]@ [y0; y1] and 8([y0; y1])� [0; y0]@ [y0; y1]:

Thus, by the covering rule the corresponding regime dynamics can be represented

by the transition graph and the transition matrix shown in Fig. 6 below.

Fig. 5. The graph of map 8 for bB/�/1. We have also drawn the graph of the identity map and the square

[0, 8(y0)]2. The fixed points 0 and p1 are repulsors. All interesting dynamics is in the interval [0, y1]. If

8(y0)5/y1, like in this figure, the interval [0, 8(y0)] is invariant.

Fig. 6. Transition graph and transition matrix T for the partition P�/{I1, I2} and map 8 that represents

Goodwin’s model when bB/�/1 and 8(y0)�/y1.


The remarkable feature of regime dynamics in this case is the arising of a pure

period-two regime cycle AB, which is what we wanted to show: if properly

dynamically coupled, two (rather than three) regimes are already sufficient to yield

a regular cycle. Actually, they may yield much more than that, for the tent-like map

above (to which the dynamic law has been reduced) is known to be capable of a

much wider variety of dynamics. This is clearly made apparent by the all-ones

transition matrix, including unpredictable behaviour of the worst type.23 Never-

theless, Goodwin’s intuition that two regimes may be so coupled as to yield periodic

dynamics, is vindicated.

10. Summing up and looking ahead

The worth of using a complex scheme such as multi-regime dynamics can really be

appreciated only when we deal with multivariable systems, i.e. systems with a large

number of interacting components and many state variables. From this point of

view, this paper is no more than an introduction to a particular treatment of this vast

and expanding topic. In this primer to coding and similar techniques, our strategy

has been dictated by a principle of economy.

We have investigated alternative patterns of multi-regime dynamics that can be

obtained via coupling models of local dynamics (i.e. dynamics within a given regime )

to one another through non-linear schemes in order to explain cross regime

dynamics . It is important to stress that all models in our exercise were linear, and

that they were chosen from the history of Business Cycle theory. Non-linear, local

models can also used, and, actually, in principle they perform better, producing more

interesting dynamics. To the best of our knowledge, no one has tried this yet, and

this is the challenge ahead. We have tried to offer a large-scale chart of what can be

obtained from a coding technique applied to a multi-regime approach.

There is good reason to try and go ahead. Basically, as they are defined here,

regimes coincide with local models in the sense in which economists understand

them, and our coded dynamics is a sort of sub-field of the better-established

symbolic dynamics. This shows up in particular when a mathematically more

appropriate (symbolic) partition can be used that is finer than the regime partition

induced by pure economic reasoning. To generate the former, we had to resort to a

cross product of economic and the mathematical criteria.

In fact, while our chosen definition of regime implies a partition of the system’s

state space, the latter may be introduced without paying any attention to its

economic significance. Still, a regime classification on the basis of some specific

economic motivation can be a reasonable starting point to construct a mathema-

tically useful partition retaining key dynamical features. Thus, although it is a

conceptual construction of the economist, coded dynamics is related with the

Symbolic Dynamics of the mathematicians. Sometimes, the former can fully avail

23 This, of course, descends also from our choice of a time-discrete formulation.


itself of the symbolic methods. Broadly speaking, symbolic dynamic techniques can

be effectively used to understand dynamics within a multi-regime framework

whenever the partitions associated with the latter are included in those demanded

by the former. This happens whenever the regime partition satisfies the requirement

of being, e.g. a covering partition . At this stage of our understanding, this appears a

reasonable justification for an economist’s exploratory use of these mathematical

techniques. We were lucky to be able to construct such partitions in all our exercises

above without doing much violence to an economist’s understanding. This might not

be as easy in general, though.

Partitions like the one behind the six regimes of the Framework Space (see

Preface), for instance, most likely fall into the latter category. Therefore, we ought to

think of our coding on the basis of an economic understanding, as a technique that

generates symbolic strings only some of which are well understood mathematically.

Whenever this is not the case, we have to resort to other techniques. And, in fact,

even within the simple settings of our previous exercises, we have seen that

mathematics often has to yield to statistical and other, more qualitative techniques,

if we want to try to account for the diversity and irregularity of observed regime

dynamics. This, however, is to be taken as a further reminder that coded and

symbolic dynamics are not the same thing.

Acknowledgements

Comments by two anonymous referees were particularly useful in improving the

paper, and they are gratefully acknowledged, with the usual caveats. ‘‘This research

was carried on with the financial support of the program PAR UNSI 1999, and as a

MIUR Progetto di Interesse Nazionale, 2000-20002, prot.MM 13308994.’’

Appendix A

A.1. Derivation of Eq. (8)

Let us have

It�k1DYt�a1 if DYt5a1(k2�k1)

�1

k2DYt if a1(k2�k1)�1

5DYt5a3(k2�k3)�1

k3DYt�a3 if a3(k2�k3)�1

5DYt

8

<

:

where: 0B/k1B/1, 1B/k2, 0B/k3B/1, a1B/0 and 0B/a3, while the consumption

function at time t , is as usual:

Ct�cYt�1; where 0BcB1

Replacing Ct and It in the equilibrium equation, and after making some

rearrangements, we have the equation (a modified Hicks model)


yt�(c�k1)(1�k1)

�1yt�1�a1(1�k1)�1; if yt�15a1(1�k2)[(k1�k2)(1�c)]�1

(c�k2)(1�k2)�1yt�1; if a1(1�k2)[(k1�k2)(1�c)]�1

5yt�15a3(1�k2)[(k3�k2)(1�c)]�1

(c�k3)(1�k3)�1yt�1�a3(1�k3)

�1; if a3(1�k2)[(k3�k2)(1�c)]�15 yt�1

8

<

:

Letting: m1�/(c�/k1)(1�/k1)�1, m2�/(c�/k2)(1�/k2)

�1, m1�/(c�/k3)(1�/k3)�1, n1�/

�/a1(1�/k1)�1, n3�/a3(1�/k3)

�1, then the conditions 0B/k1B/1, 1B/k2, 0B/k3B/1,

a1B/0 and 0B/a3 imply that m1B/c , 1B/m2, m3B/c , 0B/n1 and 0B/n3. In

correspondence with the dynamic equation yt�/8 (yt�1), function 8 is the one of

Eq. (8).

A.2. Derivation of Eq. (10)

If the investment function is:

It�k2DYt if DYt5a3(k2�k3)

�1(k2�1)

k3DYt�a3 if a3(k2�k3)�1

5DYt(0Bk3B1; a3�0)

�

we get the reduced form equation

yt�(c�k2)(1�k2)

�1yt�1 if yt�15a3(1�k2)[(k3�k2)s]�1

(c�k3)(1�k3)�1yt�1�a3(1�k3)

�1 if a3(1�k2)[(k3�k2)s]�1

5yt�1

�

Let us rename the coefficients: a�/(c�/k2)(1�/k2)�1, b�/(c�/k3)(1�/k3)

�1 and

d�/a3(1�/k3)�1. Then, we have: 1B/a , bB/cB/1 and 0B/d and corresponding with

the dynamic equation yt�/8 ( yt�1), function 8 is the one of Eq. (10).

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