Day, Richard H., "Dynamical Systems Theory and Complicated Economic Behavior", 1983

DYNAMICAL SYSTEMS THEORY AND
COMPLICATED ECONOMIC BEHAVIOR

RICHARD H, DAY

MRG WORKING PAPER #8215

Abstract

Recent developments in mathematics show that more-or-less random
behavior and spontaneously evolving structures .can be given analytical
and deterministic representations. Both empirical simulation and theo-
retical models have been developed in economics that have similar capac~
ities. This suggests that we are entering a new period when structural
change and inherently unpredictable events can be explained or under-

stood in terms of endogenous economic forces.

July 1982

DYNAMICAL SYSTEMS THEORY AND COMPLICATED ECONOMIC BEHAVIOR

by
Richard H. Day

This paper outlines. several important related developments in
dynamical systems theory and in mathematical economics. It seems that
these developments may provide an improved understanding of how econo-

mies work.

I, BASIS FOR A NEW PERSPECTIVE

Classical Mechanics

Analytical dynamics begins with Newton to whom we owe the calculus
and the idea of differential equations. All of the problems presented
by his theory of celestial motion in its general form have not been
completely solved. They challenge mathematicians even now.

Of special interst in the present context is Poincaré's perception
that solutions of the N-body problem when N.2 3 can be complicated, far
more so than the periodic orbits that had been the focus of attention to
that time. In the 1930's, Birkhof demonstrated that there could be
periodic orbits of every periodicity, all existing simultaneously.
Subsequent investigators established the existence of trajectories that

were not attracted even to quasi-stationary orbits but which wandered in
a more-or-less random or quasi-random fashion. Since such trajectories
are difficult to characterize attention has focussed on their statis~
tical properties, the subject of ergodic theory. A review of the liter-

ature and key results will be found in Moser [1973].

Dynamical Systems
Central to the analysis that has emerged from classical mechanics

is the view that it is not just the property of particular solutions of
differential equations that are of interest. One can never be sure
where a given process is at any point in time; its state can never be
measured with complete accuracy. Therefore, one wants to study solu-
tions whose initial states are neighbors. This means studying the
sensitivity of solutions or their stability.

Moreover, one can't specify all aspects of a real system in one's
model equations which may be thought of as subject to perturbations.
One wants therefore to stiidy how a model's solutions are influenced by
these perturbations. This is the study of structural stability.

The study of the behavior of solutions (1) whose initial condi-
tions are perturbed and (2) whose parameters are perturbed, is, in
economics called "comparative dynamics." In the mathematical literature
it was given an elegant statement along with a more-or-less comprehen=
sive survey of results (up to that time) in Smale's path breaking and
highly sophisticated article on “Differentiable Dynamical Systems"
[1967].

‘An extremely important idea central to the analysis of dynamical
systems is the idea of an attractor. Such an object is a set of states

in the state space of the system which attracts all trajectories emanat-

ing from neighboring points. The attractor does not describe change
over time. Indeed, even if it is a regular body behavior on or near it
may be complicated. Still it expresses something essential about the
long-run behavior of the system.

For many parameter values of a given dynamical system its attrag-
tors -- if they exist -- are regular objects described by closed curves
or surfaces. For other parameter values they may have an extremely
complex structure that cannot be described in any simple way. Such
attractors. are called strange, a term coined by Ruelle and Takins

[1971]. Their existence was already suspected by Poincaré.

Nonperiodic Behavior
In a series of seminal papers Edvard Lorenz in the early 60's

examined various "forced dissipative systems" approximated by certain
quadratic differential equations. His work, which was motivated by an
effort to use hydrodynamic theory to explain meteorological variables,
suggested the existence of wandering behavior of a highly complex type
which he called nonperiodic since it was neither periodic or quasi-
periodic. Figure 1 reproduces a projection onto 2-space of a trajec-
tory, simulated by computer, for a fourteen equation model of "vacilla~
tion." (Lorenz [1963])

Lorenz's work is noteworthy in both its analysis and in the
thoughtfulness with which its author contemplated its implications. It
follows a reductionist approach in which a “naturally motivated” or
"realistic" model is successively simplified so as to obtain precise
results while at the same time retaining salient qualitative features

both of the model and of the empirical phenomena to be explained. It
Figure 2:

YKo

WANDERING TRAJECTORY FOR LORENZ'S VACILLATION EQUATIONS
Source: Lorenz [1963b, p. 459].

also emphasized comparative dynamics in both its senses of stability and
structural stability.

Lorenz's work inspired a number of contributions by others and the
50 called Lorenz attractor has become an intensive object of study by
specialists in nonlinear dynamical systems theory. Well known review ,
articles summarize much of this work so I will not survey it further
here. But a further aspect of Lorenz's work has a direct bearing on
what is to follow and illustrates in the simplest way the basic idea of

bifurcation theory.

Bifurcation Theory
Suppose we have the equation

(1) Xtaz = mp CL~ x), Xq © (0, 11, me (0, 4].

(Lorenz used a slightly different but equivalent form.) When O<m<1
all trajectories converge to x = 0 which is globally asymptotically
stable. When m> 1 but < 3 all trajectories move away from 0 and con-
verge to the asymptotically stable stationary state X= 1- 1/m. When m
> 3 but < 1+ J, X becomes unstable and a stable, two period cycle
emerges.

The points m, = 1, my = 3, are bifurcation points where the quali-
tative behavior implied by (1) changes. They are points of structural
instability. As it turns out, my = 1 + ¥® is also a bifurcation point
and in fact as m increases cycles of even, period doubling order
emerge. Lorenz also found that as m approaches 4 behavior of such

great complexity emerged that no periodic or quasi-periodic motion
approximated it; the past did not repeat itself and solutions were
highly sensitive to small perturbations in both initial conditions and
in the parameter m, exactly the qualitative properties he had found in

higher dimension, continuous-time models.

Self-Organization
The capacity of systems to evolve strikingly different qualitative

patterns of behavior as a parameter is varied became the basis for
Prigogine's celebrated work on bifurcation and self-organization.
Imagine a dynamic system characterized by states, some of which move
relatively rapidly, described by "variables", and some of which move
relatively slowly, described by “parameters.” Fixing the slow moving
parameters one gets reduced dynamical systems whose variables may con-
verge to some kind of an attractor. Now vary the parameters. The size
and shape of the attractor may change gradually as a parameter or sev-
eral parameters are varied until, when a bifurcation point in the para~
meter space is passed, the attractor suddenly changes its form altoget-
her. If we imagined that the variables represented particles in space,
(atoms, molecules, etc.) what we would observe is the appearance of
spontaneous reorganization of the particles, much like a marching band
changing formation at the half-time of a football game.

If the parameters ("slow variables) are subject to random shocks,
then, whether or not a system reorganizes in the sense just described is
partly a matter of chance, Moreover, once a chance reorganization takes
place the change may be irreversible or nearly so. This phenomena is

illustrated by the equation

@) koa bx tex? - ax, a,b,c,d>0,

where "a" is a parameter that we may regard as increasing slowly but
with small positive or negative random shocks superimposed. See
Figure 2.

Suppose the system is initially near state S,. This state will be
observed to move gradually to the right. As time passes it will exhibit.
smal] random fluctuations around a gradually increasing state. However,
if at any time a positive shock occurs such as to move the system from
the general situation in curve 1 to that of curve 2, the system will
rapidly evolve to the stable stationary state S,. Even if secular
change in the parameter "a" terminates so that most of the time the
phase diagram resumes its initial qualitative form (curve 1) the system,
instead of returning to $, will remain at Sp.

In a suggestive rendition of language Prigogine refers to this kind
of phenomenon as the "self-organization from bifurcation through fluc
tuations." Obviously, this idea would be of little importance if it
were only applicable to situations as simple as that shown in the dia-
gram. But when there are several state variables the change in form can
be much more cap lteaede than the jump from one stationary state to
another. Then bifurcation can involve altogether different geometric

attractors.

Castastrophe .

The rapid jump from one kind of motion to another through the
bifurcation of parameters or slowly moving variables as exemplified by
(1) or (2) have been called "catastrophes," a term coined by René Thom
who exploited the idea as a means of characterizing biological morpho-

genesis; that is, the spontaneous ("endogenous") evolution of new living
x

Figure 2:

Perturbation increases a to a!

Curve 2

x
NR . 1N
Curve 1

A positive perturbation in "a" sends the
system from smal] fluctuations in the
neighborhood of S; to small fluctuations
in the neighborhood of S».

forms from old ones. The idea seems to be essentially the same as that
of Prigogine's "self-organization".- Thom, however, was able to give an
exhaustive characterization of all possible types of catastrophes for
dynamical systems with three slowly moving parameters. These have been

so frequently described that I will not do so again here.

Chaos

Before turning to economics explicitly, let us linger a bit longer
with the kind of nonlinearities underlying all of the complications
outlined so far. Lorenz's work established the existence of extremely
complex dynamics for a simple difference equation (1). Using techniques
developed by Smale, Li and Yorke [1975] discoverd a sufficient nonlin-
earity or overshoot condition that analytically established the exis-
tence of cycles of all orders and a scrambled set in the state space of
a single variable system in which all: trajectories were non-periodic,
and asymptotically unstable, i.e., they moved away from cycles of any
order and were sensitive to changes in initial conditions. They refer-

red to trajectories with this character as chaotic, (their paper was

entitled "Period 3 Implies Chaos.") The existence theorem was extended
to n-dimension discrete dynamical systems (n > 2) by Diamond [1976].

The significance of this work lies in its lack of dependence on
smoothness of the dynamical system in question. Instead it relies only
on continuous and topological properties. This feature has made it
possible to develop chaos existence theory for a variety of economic
models where non-differentiability is typical.

In Figure 3 the picture shows a continuous but non-smooth, (non-

differentiable) mapping on the interval [0, 1]. The points d <a <b <
10

(x)

Figure 4: The sufficient overshoot

conditions. Points in B map into

zero, points in A or A‘ map into
and so on.

1

c satisfy the Li-Yorke existence condition and imply the existence of an

interval X, say with the properties that

(3) XO F(X) =

and XM £(X) € F(F(X))

One sees how in the process of successsive iterations of the map f
(i.e., with the passage of discrete time intervals) states in the set X
get mixed and scattered much like the shuffling of cards or molecules of

dough in croissants.

Summary

What has emerged from theoretical developments in dynamical systems
theory is a gradually improving analytical understanding of complex
dynamics. By the latter term I mean (1) unstable nonperiodic (wander-
‘ing, chaotic, erratic, etc.) behavior and (2) evolving regimes of quali-
tatively different behavior. Behavior with these characteristics would
seem to be nowhere more evident than in economic phenomenon. It may
still come as a surprise however, that such possibilities might be
generic in the sense that large classes of dynamic processes repre
senting economic behavior can be shown to possess these properties. The

next section reviews evidence in support of this assertion.

II. COMPLEX ECONOMIC DYNAMICS

Erratic “Fluctuations

Do we need reminding that economic behavior appears to be compli-

cated in the sense just defined? Figure 4 shows some typical macro-
(a) Average Prime Rate i

~
(c) . Stock Prices

(d) Unemployment Rate

“ve

{e) Industrial Production

(f) Rate of Capacity Utilization, Manufacturing

DARN

(g) Change in Business Inventories

inner NA

Figure 4: Various Economic Indexes, 1955-1980
Source; Business Conditions Bigest, June 1980

13

economic series. These data are a little out of date now and more
recent figures would add some further drama to the general picture, but
perhaps they are sufficient to establish the empirical relevance of
wandering, erratic fluctuations in economics.

Traditionally, such patterns have been explained by the super=*
imposition of random shocks on what is (usually) assumed to be a stable,
deterministic, linear procéss. See Sargent [1979, p. 215ff]. That such
patterns might be deterministically generated is a novel idea in econo-

mics (though one anticipated by Georgescu-Roegen [1954]).

Evolving Regimes
The notion of evolving regimes, that is, successive periods of

distinguishably different qualitative behavior, is likewise a common-
place observation to those acquainted with a little economic history or
those even modestly acquainted with current events. A few specific
examples are always i1luminating, however.

Consider energy supplies and prices. After over half a century of
nearly exponential growth in energy supplies and monotonic decline in
energy costs we have entered a period of fluctuating supplies and
prices; after a long period of ranking among the safest of the invest-
ment grade corporations, the utilities have become financially insecure,
many threatened with bankruptcy. The same may be said for numerous
sectors (steel, autos, and so on).

In the realm of individual technologies, we see overlapping waves
as a given technique is innovated, then expands first gradually replac~
ing then driving out competitors at an accelerated rate, only to enter
an obsolescent phase as another new technique that will eventually

replace it altogether enters the picture.
4

At the level of societies as a whole we see socio-economic ways-of-
life gradually grow to prominence, dominate large regions, perhaps
rising to such great importance as to be identified as "civilizations,"

then go into a decline, perhaps precipitous, eventually dying out alto~

gether, some leaving few traces, some leaving vast monuments and widely .

scattered artifacts to attest to their once grand but mysteriously
vanished power.

To many observers the present time seems to be one of rapidly
changing futures, trend reversals, newly emerging problems and oppor-
tunities, rapidly decaying viability of recently successful economic
activity and so on (Forrester [1972]. )

"catastrophe" may be too melodramatic a term to apply to periods of
rapid qualitative change, but the existence of such periods can scarcely
be argued away. That such periods of transition from one distinct
regime to another might be explained by an endogenous theory, while not
a novel idea, is at least one that has received little attention within
standard or orthodox economics, where, instead, exogenous, ad hoc,
explanations are more common. (For example, the innovator or entre-
preneur of Schumpeterian theory or the "random shock model" of econo-

metrics.)

Deterministic Dynamic Simulation Models
The possibility that patterns of behavior of the kind briefly

described could be given an analytical, theoretical representation
occurred to me as the result of experience with a class of simulation
models designed to simulate production, investment and technological
change in various industries and agricultural regions. Early examples

of some of this work including studies of the American and Japanese

15

steel industries, petroleum refining, coal mining and agricultural areas
in Brazil, West Germany and the Indian Pinjab will be found in Day and
Cigno [1978].

Many of these models are quite detailed involving several conmodi-
ties, a variety of alternative technologies for producing each one and a,
considerable number of intermediate products, capital goods and
resources, Much in the spirit of Lorenz's reductionist approach, sim-
plified versions of these models were then specified that retained
salient features of the large-scale, “realistic" models but which could
be analyzed, or failing that, could be studied using computer simulation
at low cost.

Three examples will illustrate some typical model behaviors.
Figure 5a shows the mode? generated values for a “corn-hog" model.
Illustrated are market and anticipated prices, pork supply and invest-
ment in buildings, hog feeding and breeding stocks. Note especially the
time profile for investment in buildings. After half a decade a five
period cycle seems to appear, being approximately reproduced three
times. After period 21 however the time path moves away from this cycle
in an irregular oscillation. The effect of an addition of a trend in
the demand for pork is shown in Figure 5b. The irregularity of invest~
ment is seen again.

These results were obtained from a model specified in the fall of
1968 at Gottingen University by myself, the late T. Heidhues and Garriet
Muller now at the University of Frankfurt, using realistic data from
West Germany but with the purpose of gaining a better understanding of

the rcursive programming modelling approach.
aut piece ean 5
PR age ye

Figure 5:

(a) Stationary Demand

nec 885. and ep. pice
won,

(b) Caponential Demand Shifter

SIMULATIONS OF THE GENERALIZED COBWEB MODEL.
Miller and Day [1978, pp. 242, 247].

v7

More recently two colleagues (Robert Boyd and Scott Moreland) using
a similar type of model produced experimental simulations of the elec
tric power industry that was designed to explain investment in two
alternative technologies (capital intensive and capital saving) in the
presence of growing demand but under alternative inflationary condi
tions. We see in Figure 6(a) and (b) two examples. “Cap.I" is capital
intensive investment represented by the shaded area. The unshaded area
represents investment in the capital saving technique. Total investment
is the upper line. The increasing irregularity of both total investment
and its composition in response to inflation should be noted. In this
mode1 we see the economic effect of a shift in a "parameter" (the infla~
tion rate). It results in extreme shifts to capital saving caused by
severe financial strains and changes in the cost of capital.

Here again our effort was directed at representing economizing
behavior in a "realistic" way but with Jittle idea in advance as to what
might turn up. Yet, the model portrays the shift from a regime of
secular expansion and economic health to one of financial crisis and
sudden sefcching: of ‘investment strategies in a more-or-less realistic
fashion.

Finally, consider Figure 7, which shows the results of another
deterministic computer simulation of a recursive programming model of
the same general class. The underlying model was designed to represent
economic devetopment in a highly populated, open economy initially
dominated by agriculture but with an infant industry just beginning to
expand. Of special interest is the “overlapping-wave" character of
technology shown in Figure 7(a) and the cusp-like switch from growth to

decay in fibre and the export crop in Figure 7(b). These drastic
1000 MW

1000 Mw

|
|

(2 \
Rete
prea

(b) 12% Inflation

FIGURE 6: SIMULATIONS OF THE EDEM MODEL
Economic Dynamics [1980]

re

fe) TRADMONAL ano MODERN
GRAIN FARMING

fe) 2-000 te) mvt SIMCWT AND CAaPrTAL STOCK

Figure 7: A Dynamic Model of Urban-Regional Development
20

"structural" changes are occurring while aggregate capital stock accumu-
lates according to the smooth, classic, sigmoid pattern of Figure 7(d).
The z-good charted in Figure 7(c) represents the allocation of labor to

service activities in the urban sector. Here, as in reality, the

advanced stages of growth are accompanied by a vast expansion of the,

tertiary econony.

Certainly in a conference on System Dynamics one would be remiss in
failing to cite simulation models constructed according to the Forres-
terian paradigm which likewise emphasizes nonlinearity in feedback
systems. A search of the diagrams of Forrester's original treatise
[1961] or of many of the studies of his followers would reveal numerous
additional examples of the phenomenon we are talking about in this

Paper.

The Robertson-Williams Cobweb Model

The characteristics of the simulation models just reviewed that are
responsible for fluctuations in prices, output and investment are (1)
the dependence of revenue through feedback on a denand function with
variable elasticity, (2) the presence of a financial constraint that
depends on revenue, and (3) the independence of pricing from the pro-

duction and investment decision: either prices or outputs are determined

by purely competitive markets. If one begins with sufficiently small
initial endowments of working capital there is an inital period of
growth. Eventually output levels reach the inelastic portion of demand;
consequently, revenues fall. This reduces working capital and borrowing
ability for the subsequent period. Production and/or investment must be

reduced, or a shift to money-saving production and investment alterna-

21

tives effected. Later, because market supplies are reduced, demand
"recovers," prices increase and output can expand once more. (In the
developing context these fluctuations are avoided temporarily by the
continued growth in demand. ) ,

D.H. Robertson's observation, made in the context of a discussion.
of the Keynesian multiplier, that currect expenditures are made fron
previous incone can serve as the basis of a model of extrene simplicity,
yet one that can produce many of the features of complex dynamics that
we have just been talking about.

Robertson's basic equation

Current Expenditure depends on Lagged Income becomes, in the con-
text of the firm Current Production Costs are limited by Lagged

Revenue,

a statement that reflects John Burr Williams (1967) “current assets
mechanism." A sales maximizing hypothesis (Baumol [1959]) coupled with
a financial constraint Yeads to such an equation (Day 1967). Given
appropriate aggregation conditions and setting cX equal to total pro-

duction costs (where “c" is unit cost and X industry output" and D(X,

a) equal to demand (where "a" is a parameter measuring the size or
“extent of the market" to use Adam Smith's phrase we arrive at the equa~

tion

) Keen = AX DOK) 5 m=afe>0,

and where for simplicity we have D(-, a) = aD(-). The parameter "m" is

therefore a measure of the extent of demand in “efficiency units".
22

Now it is easy to see that if the revenue function XD(X) has the
usual "bel1" or “single-humped" shape, m can be increased, revealing a
sequence of bifurcation points at which successively higher order cycles

emerge accumulating to a critical value, m., say, at which the Li-Yorke

Theorem is satisfied so that cycles of all orders and unstable chaotic,

trajectories exist!

Noting that when D(X) = 1~- X (which is equivalent to a - bX under
a suitable transformation) we have the classic equation of Lorenz. The
simulation of Figure 8 can be thought of as representing a highly vola~
tile industry (hoola hoop or skate boards) that enjoys periods of growth
and sporadic booms followed by collapses of greater or less magnitude.
Here we have an existence theorem for the irregular fluctuations of an
unstable nature displayed in the simulation models.

When I first analyzed this model back in the early 60's, sometime
after going to the University of Wisconsin, I was not aware of Lorenz's
work on analogous dynamical systems, nor was the Li-Yorke theorem or any
of the other work summarized in Part I of this paper available. I was
puzzted by the simulations but dropped the bifurcation analysis after
obtaining the first several bifurcation points for m Day [1967]. It
was only at the suggestion of the mathematician Kenneth Cooke that I
looked into Lorenz's papers in 1978. Then in collaboration with Jess
Benhabib of USC (now at NYU) who had come across the same literature in
connection with quite different work on growth theory, a way was found
to use the Li-Yorke theorem to prove existence for a variety of economic
models.

The key point in the present model (4) is that the type of non-
linear feedback induced by the financial constraint is going to be

generic in economic models that treat financial resources as well as

23

ea EE ne

a) Chaos (1/c = 4.0)

Figure 8
24

prices. Even assuming linear cost and demand functions a quadratic

feedback occurs. The moral is that an economic world in which money

enters in a non-trivial way can be highly complex in its behavior in
theory just as in reality! .

In addition to various papers by Benhabib and myself other studies
have begun to appear of economic chaos in a wide variety of settings and
I expect this to be one of the most interesting and rewarding areas of

theoretical research in economics in the coming few years ahead.

Self-Organization and Catastrophes in Economics
Prigogine's application of bifurcation theory to forced dissipative

systems to obtain an explanation of "self-organization" has been given
an imaginative application to urban-regional science by Peter Allen et
al. (undated) whose results have been obtained using computer simula-
tion. They have shown how the introduction of a single railroad or
highway can induce a switch in regime and impel] an economy onto a path
leading to a new distribution of activity, to bring about self-organiza~
tion or self-reorganization as it were, Allen (1981) has also contri-
buted a thoughtful exegesis of the general idea as well as its economic
application.

As indicated above I think the basic insights of Prigogine and Thom
are quite similar, even if different terminology has been used. Cer-
tainly they have both stimulated a great deal of imaginative theorizing
in a wide variety of fields in social science.

Unfortunately, many of these applications have been "poorly moti-
vated" in the sense that the underlying equations used to explain catas~

trophes or self-organization in the field in question are not derived

25

from compelling empirical hypotheses or carefully posed theoretical
axioms or propositions. Indeed, criticism by mathematicians has been
strong. See for example Sussman and Zahler [1977]. But as with chaos
theory, it has proven possible to use catastrophe theory to illuminate
dynamic properties of well established theories and models.

In economics "well-established" cannot be taken to mean "accepted"
or "non-controversial." Rather I use it to mean that emminent scholars
have taken an idea seriously and that it has played an important role in
advancing our understanding of important phenomena. Such an example is
the paper of Varian [1978] who applied Thom's theory to a clever exege-
sis of Kaldor's business cycle model, in this way deriving logical
consequences of certain behavioral assumptions that were certainly
unknown before. Varian forthrightly emphasized the lack of micro-
theoretical foundations to the Kaldor macro-model, and this must be
candidly admitted as a flaw. Nonetheless, a lack of micro-foundations
is not in itself sufficient to justify a charge of “ad hocery", not if
the macro-assumptions have sufficient plausibility to invite serious
attention, and one does not casually reject an argument set forth by so

keen an observer as Kaldor.

Multiple Phase Dynamics in Recursive Programs and Games
The simulation models whose results I briefly described at the

beginning of this part belong to a general class of recursive programs
and games that explicitly represent economizing but which can incorpor-
ate other assumptions of an essentially behavioral nature as well. It
is difficult to prove theorems for other than special cases of such

models and indeed as far as I know the stationary state and compact
26

orbit theorems of Day and Kennedy [1970] and the Chaos Existence Theo-
rem, Day [1981], are the only "general theorems" that are known.
A central feature of these models is that they incorporate inequal-

‘ity constraints so that at each discrete, decision-making period, econo-

mizing can be thought of as selecting a set of positive or active activ-,

ities and a set of binding constraints. For this reason, although the
dynamical system derived for an RP model is set-valued, it possesses
solutions that satisfy in a piecewise fashion, sets of difference equa~
tions. At each point in time a given set of difference equations pre-
vail but from time to time this set’ switches so that the solution as a
whole can be characterized by an evolving sequence of endogenously
generated phase structures, Day [1963], Day and Cigno, [ibid, Chapter 3]
and Day [1982].

The result is the appearance of such behaviors as trend reversals,
oscillation emerging out of growth and: various types of catastrophe or
self-organization as illustrated in the examples of Figures 5-7. In the
"RP models", however, each phase structure in a given sequence is
derived directly from a specific set of economic choices and a specific

set of scarce and abundant resources and other limiting factors.

Differential Inclusions

I spoke of the difficulty of mathematically analyzing general
recursive programs and games. This seems to be the result of their
discrete time, non-differentiable nature. By going to continuous time,
adding strong regularity properties and boundary conditions great pro-
gress has been made in closely related economic models that lead to

differential inclusions where instead of a rate being determined by an

27

equation, the rate is indeterminant but constrained by a set. Thus, one

writes
& © A(x)

where A. is a set-valued map or correspondence.

Most of the work in this area has so far involved monotone solu-
tions of a highly regular type (see Dreze and Valle) and questions of
instability, catastrophe, self-organization and evolving phase struc~
tures have not yet been posed in this context Tet alone answered, but a
rich analysis has nontheless been developed for such systems leading to
‘the important new book by Aubin and Celina [undated]. Aubin in fact has
shown how a decentralized price system obeying a generalized Walras Law
can lead an economy to-a generally improving performance, while Aubin
and Day [1980] have shown how an adaptive economics theory ala Simon,
Cyert and March can be formalized using differential inclusions and can

be shown to possess improving economic evolutions.

Differentiable Dynamics
Although economics naturally leads to non-differentiable, set-

valued dynamical processes, much progress in dynamical systems theory
has exploited differentiable, single-valued systems. This includes
virtually all of the work mentioned in section 2 except the Li-Yorke
existence theorem. In order to make progress in analytical understand-
ing of complex economic dynamics it would seem worth giving up some of
the realism of discrete time, nondifferentiable, set-valued structures

and to study instead dynamic economic models without these complications
28

but that retain the essential nonlinearities that are there in reality
and that lead to complicated behavior of a qualitatively realistic type.
This would make many of the techniques developed for nonlinear systems

applicable to economics and might accelerate progress just as the calcu-

lus helped establish neoclassical theory on a rigorous footing before ,

modern convex analysis and topological techniques were innovated.

EPILOG

Recent developments in mathematics show that more or less random
behavior and spontaneously evolving structures can be given analytical
and deterministic representations. When applied to a specific field of
scientific inquiry they provide possibilities for endogenous theories of
complicated dynamics, that is, theories which explain irregular fluctua-
tions and evolving structures by underlying material, mental and social
forces rather than by “random shocks," “great men", “the weather" or
other unexplained outside or "exogenous" events.

Nowhere do we observe complicated behavior more frequently than in
economics. We would therefore seem to be standing at a threshold across
which lies a new intellectual domain in which events may be recognized
as more or less unpredictable, but nonetheless understandable. What
this may mean both for the further progress of scientific method and for

practical policy one can only guess.

29

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