Models of Economic Dynamics: Proceedings of a Workshop held at the IMA, University of Minnesota, Minneapolis, USA, October 24–28, 1983

219 Pages·1986·5.456 MB·English

by  Donald G. Saari (auth.), Prof. Hugo F. Sonnenschein (eds.)

Lecture Notes
in Economics and
Mathematical Systems
Managing Editors: M. Beckmann and W. Krelle
Subseries: Institute for Mathematics and its Applications, Minneapolis
Advisers: H. Weinberger and G.R. Sell
264
Models of Economic Dynamics
Proceedings of a Workshop held at the IMA,
University of Minnesota, Minneapolis, USA
October 24-28, 1983
Edited by Hugo F. Sonnenschein
Springer-Verlag
Berlin Heidelberg New York Tokyo
Editorial Board
H. Albach M. Beckmann (Managing Editor) P. Ohrymes
G. Fandel J. Green W Hildenbrand W Krelle (Managing Editor) H.P. KQnzi
G.L. Nemhauser K. Ritter R. Sato U. Schittko P. SchOnfeld R. Selten
Managing Editors
Prof. Dr. M. Beckmann
Brown University
Providence, RI 02912, USA
Prof. Dr. W. Krelle
Institut fOr Gesellschafts-und Wirtschaftswissenschaften
der Universitat Bonn
Adenauerallee 24-42, 0-5300 Bonn, FRG
Editor
Prof. Hugo F. Sonnenschein
Department of Economics, Princeton University
Princeton, NJ 08544, USA
ISBN 978-3-540-16098-4 ISBN 978-3-642-51645-0 (eBook)
DOI 10.10071978-3-642-51645-0
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© by Springer-Verlag Berlin Heidelberg 1986
214213140-543210
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PREFACE
During the week of October 24-28, 1983, a group of mathematicians and eco
nomists met at the Institute for Mathematics and it Applications at the
University of Minnesota. The workshop dealt with economic models in which time
plays an essential role, and both the description of adjustment to a static
equilibrium and the description of equilibrium paths were considered. From a
mathematical point of view, discrete dynamical systems and the dynamics of ordi
nary and partial differential equations played a major role.
The conference consisted of lectures by economists and by mathematicians
which treated some of the principal ideas of economic dynamics. Donald Saari pro
vided some discrete dynamical systems background for a paper by Jean-Michel
Grandmont on business cycles; the Grandmont paper was a major focus of the
Workshop. Daniel Goroff, Jose Scheinkman, Christopher Sims. Neil Wallace. and
Michael Woodford discussed the Grandmont paper after its presentation. The ideas
of tatonnement were introduced by Leonid Hurwicz and extended by Andreu MasColell
and H. Jerome Keisler. Four papers on economic dynamics follow (W.A. Brock, Truman
Bewley. W.A. Brock and M. Rothschild and Yieh-Hei Wan). The remaining papers are
devoted to issues of quantity and/or price adjustment (William Novshek and Hugo
Sonnenschein. Phillipe Artzner. Carl Simon and Hugo Sonnenschein). equilibrium
with a continuum of commodities (Larry Jones), and the adjustment of expectations
(Lawrence Blume and James Jordan).
A distinctive feature of the Workshop was the substantial interaction between
mathematicians and economists. many of whom had not been previously exposed to the
ideas that were presented. This volume is intended to introduce a wider audience
to the contents of these lectures and to stimulate further dialogue between econo
mists and mathematicians.
ACKNOWLEDGEMENTS
The organizers of the Workshop thank Debbie Bradley and Kaye Smith for their
careful typing of this volume.
TABLE OF CONTENTS
Dynamical Systems and Mathematical Economics 1
Donald G. Saari
On Endogenous Competitive Business Cycles 25
Jean-Michel Grandmont
COlll11entaries on the Grandmont Paper "Endogenous Competitive Business Cycles" 35
Daniel Goroff •••• 35
Jose A. Scheinkman
36
Christopher Sims 37
Neil Wa 11 ace 40
Michael Woodford 41
On the Stability of the Tatonnement Approach to Competitive Equilibrium 45
Leon i d Hu rwi cz
Notes on Price and Quantity Tatonnement Dynamics •• 49
Andreu Mas-Colell
A Price Adjustment Model with Infinitesimal Traders 69
H. Jerome Keisler
A Revised Version of Samuelson's Correspondence Principle:
Applications of Recent Results on the Asymptotic Stability of
Optimal Control to the Problem of Comparing Long-Run Equilibrium 86
W.A. Brock
Dynamic Implications of the Form of the Budget Constraint 117
Truman F. Bewley
Comparative Statics for Multidimensional Optimal Stopping Problems 124
W.A. Brock and M. Rothschild
The Cournot Problem with Bounded Memory Strategies •••• 139
Yieh-Hei Wan
Quantity Adjustment in an Arrow-Debreu-McKenzie Type Model 148
William Novshek and Hugo Sonnenschein
Convergence of Myopic Firms to Long-Run Equilibrium via the Method of
Characteristics ••••••••••••••••••••••••• 157
Philippe Artzner, Carl P. Simon, and Hugo Sonnenschein
Special Problems Arising in the Study of Economies with Infinitely Many
Convnodi t ; es •••••• • • • • • • • • • • • • • • • • • • • 184
Larry E. Jones
Introduction to Expectations Equilibrium. • • • • • • • • • • • • • • • • •• 206
Lawrence Blume and James Jordan
DYNAMICAL SYSTEMS AND MATHEMATICAL ECONOMICS
Donald G. Saari
Department of Mathematics
Northwestern University
Evanston, Illinois 60201
1. INTRODUCTION
The modern mathematical economics literature is permeated with dynamics.
This starts with a simple tatonnement story of how prices adjust according to
supply and demand, and it continues with the more sophisticated price adjustment
models which involve speculation, etc. Dynamics arise from the Euler, or the
Bellman equations, to define the optimal paths in growth models, as well as in
other optimization problems. Arguments based upon dynamics are advanced to
justify various forms of equilibria; here we find issues such as the accessihility
of pareto points or the comparison of different bargaining solution concepts. In
recent years, as manifested by several of the papers presented at this conference,
dynamics has been used to explain non-stationary behavior such as business cycles.
It became clear in the 1970's that mathematical economics is blessed with
almost an overly rich supply of dynamical systems. This is the period when the
work initiated by H. Sonnenschein and advanced by Mantel, Debreu, and ~as Colell
culminated in the conclusion that just about any differential equation, or vector
field on the price simplex, can be viewed as being the aggregate excess demand
function for some economy. (A good reference would be the expository paper hy
Shafer and Sonnenschein (1].) Other topics from mathematical economics also admit
a wealth of dynamical systems with the accompanying varying behavior of the solu-
tions. For example, in optimal growth models or overlapping generations models, a
large class of systems can be generated by altering the production sets, the uti
lity functions, and the discount rates. In what follow, I will indicate how these
simple changes can lead to all sorts of unexpected orbit behavior.
Because the modelling admits an abundance of dynamical systems, mathematical
economics can benefit from most advances in Dynamical Systems. Consequently, it
is fortunate for economics that during this same period of the late 60's and the
2
70's, mathematicians made significant advances in the geometric study of these
systems. Part of the earlier intuition concerning solutions resulted from the
behavior of dynamical systems in the plane. Here, under some general conditions
(known as the Poincare - Bendixson Theorem), it is known that all bounded solu
tions asymptotically approach either a fixed point or a periodic orbit. But it
has been known for some time that the solution behavior isn't as nicely described
in a higher dimensional space. Important progress has been made'on the difficult
problem of describing the "general" or "generic" behavior of solutions: an earlier
exposition of this is in [2J while a more recent one is [3J.
We now know that the flexibility acquired by increasing the dimension of the
phase space from two to three admits orbits which only a decade ago would have
been viewed as being pathological. For example, it is natural to think of a
periodic orbit as being a distortion of a simple loop. However, in a three dimen
sional space, there are periodic orbits for differential equations which define
almost any conceivable knot one can tie. That is, tie a piece of string into a
complicated knot. Once this is done, attach the loose ends of the string
together. It turns out that there are differential equations where a periodic
orbit ties this knot! (An excellent, recent paper by Franks and Williams [4J
shows that an infinite number of different knots occur in a system which satisfy a
certain "topological entropy" condition.) Furthermore, we are learning that the
limiting sets of orbits can assume complicated and strange structures: such sets
are descri bed by the now standard term "strange attractors".
Although this orbit behavior may seem to be bizarre, it can occur in models
from economics! For example, as soon as four commodities are admitted into an
economy, the tatonnement story is modelled by a dynamic ina three dimensional
price simplex. The Sonnenschein result asserts that the defining system can be
quite arbitrary, so all of the above orbit behavior is admitted. Thus, the story
of the auctioneer admits some knotty consequences.
It is natural to question whether this unusual orbit behavior can be
dismissed on the grounds that it only occurs either in isolated, or in complicated
examples. This is not the case. For example, in the 60's Lorenz introduced the
3
following differential equation as a model for turbulence.
x' a o x
y' b -1 o y
z' o o -c z
Here, a, b, and c are positive constants where the original values are a=10,
b=28, and c=8/3. The linear part of this equation can be studied with a standard
eigenvalue analysis. Because the equation is only a quadratic system, it might
appear that the full system shouldn't be unduly difficult to analyze. The fact
is, in spite of the efforts of several very good mathematicians, this equation is
yet to be completely understood. It is known that this quadratic equation, admits
solutions such as the knotty periodic orbits described above, and that there are
other, more complicated trajectories. (A recent book by Guckenheimer and Holmes
[5] discusses this system in great detail.) The important point, which is why
this example is introduced here, is that although ~ nonlinear system Y2.. elementary
~ form, 2! can admit ~ very complicated orbit structure! (For an example of a
simple, nonlinear, second order scalar equation with a periodic forcing function
which has complicated solutions, I refer you to Mark levi's seminal work on the
van der Pol equation [6].)
When the modelling of an economic setting involves discrete units of time,
the analysis yields an iterative, discrete dynamic. (Indeed, often the intended
mOdel is a discrete one, but a continuous model is used to approximate and to
simplify the analysis; e.g., a differential equation is the limiting case of the
desired iterative system.) Since discrete, iterative dynamics playa vital role
in economics, it is important to determine whether their orbit structures can be
as complicated as above. They can, and these complexities already appear in one
dimensional systems!
In this paper, I will provide an elementary introduction to selective aspects
of this area. The explanation of the causes for "chaos" in discrete dynamics are
easier if only because they occur for one-dimensional systems. This permits the
4
ideas to be described in terms of graphs of functions. For this reason, this is
the topic I'll emphasize. But, because Dynamical Systems is such an exciting and
rapidly growing field of mathematics, any introduction cannot be complete. So, my
approach is to present those ideas which I suspect are of immediate interest and
applications to mathematical economics. My emphasis will be to indicate why cer
tain types of orbit behavior occur, and to describe the identifying features of
the defining system. These properties will be illustrated by suggesting how this
relates to models in economics such as cycles in growth problems and price dyna
mics. (The details for some of these models will be given elsewhere.) A selection
of the appropriate references for the mathematics will be cited for the reader
interested in more detailed information.
2. DISCRETE DYNAMICS AND THE TATONNEMENT PROCEDURE
In this section, I will discuss a simple dynamical system represented by a
mapping G from an interval I back into itself. To be specific, suppose the
graph of G is given in Figure 1 where the iterative dynamic is given by
(2.1)
/
Figure 1
/
5
Geometrically, this can be described in the following way. The value of
xN+l = G(XN) ,which is found from the graph of Gover xN' provides the next
value for x. To determine the location of xN+1 on the x-axis, move horizon
tally from the graph of G until you reach the diagonal y=x, then move ver-
tically to the x-axis. The next iterate, xN+2' is given by the value on the
graph of G over this point. An equivalent description of this iteration, which
omits the vertical projection, is to move horizontally from the graph of G to the
line y=x and then move vertically to the graph of G. The process is con
tinued, and it is depicted by the directed lines in the figure.
It is obvious from the above that the intersection of the graph of G with
the diagonal y=x determines a fixed point p of the map; G(p)=p. By use of
the mean value theorem, it is an elementary exercise to show that a fixed point is
stable at x=p if IG'(p)1 < 1 and unstable if IG'(p)1 > 1. This is because
IxN+1-pl = IG(xN) - G(p)1 = IG'(c)llxN-pl for some c between xN and p. Thus
if IG'(p) I < I, then once an iterate xN is sufficiently close to p, all future
iterates are closer. If IG'(p) I > I, then the future iterates are pushed away
from p. These conditions are the basis for the usual analysis of fixed points in
mathematical economics.
This system admits much more interesting activity than that captured by an
equilibrium analysis; for instance, it admits a highly random, orbit behavior in
the three regions labelled A, B, and C. More specifically, select a region in
which you wish the initial point to be, say region C. Next, specify the interval
in which you wish the first iterate to land, say A. Continuing in this fashion, a
sequence of intervals is specified, say (C,A,A,B, ••• ), where the (N+l)th term in
the sequence identifies the interval in which you wish the Nth iterate to be.
What I will show is that for any specified sequence, there exist points for which
their trajectories have this future! In other words, for any random defined
future, there are orbits of this deterministic system which will obey it.
To show this, all of the points which satisfy a finite portion of this
future will be found. Let C(C,A) be the set of points in interval C which
are mapped to the closed interval A. The crucial factor in describing this set