Cycles and Chaos in Economic Equilibrium 9780691225210, 0691042497, 0691003920

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CYCLES AND CHAOS IN ECONOMIC EQUILIBRIUM

CYCLES AND CHAOS IN ECONOMIC EQUILIBRIUM

Edited by Jess Benhabib

P R I N C E T O N

U N I V E R S I T Y

P R E S S

P R I N C E T O N ,

N E W

J E R S E Y

Copyright © 1992 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, Oxford All Rights Reserved Library of Congress Cataloging-in-Publication-Data Cycles and chaos in economic equilibrium / edited by Jess Benhabib. p. cm. Includes bibliographical references. ISBN 0-691-04249-7 — ISBN 0-691-00392-0 (pbk.) 1. Business cycles—Mathematical models. 2. Equilibrium (Economics)— Mathematical models. I. Benhabib, Jess, 1948HB3711.C9 1991 339.5—dc20 91-28251 CIP This book has been composed in Linotron Times Roman Princeton University Press books are printed on acid-free paper, and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources Printed in the United States of America 10

9 8 7 6 5 4 3 2 1

(Pbk.) 10 9 8 7 6 5 4 3 2 1

For Madeline, Michael, and Nicki

Contents

Introduction

3

1. Equilibrium Models Displaying Endogenous Fluctuations and Chaos: A Survey Michele Boldrin and Michael Woodford

8

2. Periodic and Aperiodic Behaviour in Discrete One-Dimensional Dynamical Systems Jean-Michel Grandmont

44

3. A Characterization of Erratic Dynamics in the Overlapping Generations Model Jess Benhabib and Richard H. Day

64

4. On Endogenous Competitive Business Cycles Jean-Michel Grandmont

82

5. Competitive Business Cycles in an Overlapping Generations Economy with Productive Investment Bruno Jullien

138

6. Endogenous Fluctuations in a Two-Sector Overlapping Generations Economy Pietro Reichlin

158

7. Recent Theories of the Business Cycle: The Role of Speculative Inventories Guy Laroque

180

8. Endogenous Cycles with Uncertain Lifespans in Continuous Time William Whitesell

199

9. The Hopf Bifurcation and the Existence and Stability of Closed Orbits in Multisector Models of Optimal Economic Growth Jess Benhabib and Kazuo Nishimura

206

10. Sources of Complex Dynamics in Two-Sector Growth Models Michele Boldrin and Raymond J. Deneckere

228

11. Imperfect Financial Intermediation and Complex Dynamics Michael Woodford

253

Viii

CONTENTS

12. Dynamical Systems that Solve Continuous-Time Concave Optimization Problems: Anything Goes Luigi Montrucchio

111

13. Stochastic Equilibrium Oscillations Jess Benhabib and Kazuo Nishimura

289

14. Cyclical and Chaotic Behavior in a Dynamic Equilibrium Model, with Implications for Fiscal Policy Raymond J. Deneckere and Kenneth L. Judd

308

15. Endogenous Business Cycles with Self-Fulfilling Optimism: A Model with Entry Allan Drazen

330

16. Keynesian Chaos Richard H. Day and Wayne Shafer

339

17. Feedback Between R&D and Productivity Growth: A Chaos Model William J. Baumol and Edward N. Wolff

355

18. Is the Business Cycle Characterized by Deterministic Chaos? William A. Brock and Chera L. Sayers

374

19. The Statistical Properties of Dimension Calculations Using Small Data Sets: Some Economic Applications James B. Ramsey, Chera L. Sayers, and Philip Rothman

394

20. Some Evidence on the Non-Linearity of Economic Time Series: 1890-1981 Bruce McNevin and Salih Neftqi

429

21. Nonlinear Dynamics and Stock Returns Jose A. Scheinkman and Blake LeBaron

446

List of Contributors

475

CYCLES AND CHAOS IN ECONOMIC EQUILIBRIUM

Introduction

A RECURRENT theme in economic literature is the self-correcting nature of the economic system. Shortages create incentives for increased supply; dire necessities give rise to inventions as the invisible hand guides the allocation of resources. An equally recurrent theme (especially in the literature on business cycles) is that of instability: the multiplier interacts with the accelerator, leading to explosive or implosive investment expenditures; self-fulfilling expectations give rise to bubbles and crashes. In combination, these two themes suggest a nonlinear system, somewhat unstable at the core, effectively contained (by nonlinearities) further out. Yet at first blush the relationship between these two themes seems uneasy—cyclic and chaotic dynamics do not sit well with the idea of strict economic equilibrium. The standard wisdom seems to suggest that deterministic patterns and the associated regularities in economic time series, even if contaminated by some random noise, should be eliminated by intertemporal arbitrage, especially if they occur in price series. But in recent years a large number of studies have shown that the logic of this intuitive view is questionable. Apparently, cycles and chaos are perfectly compatible with a wide variety of standard equilibrium models that incorporate the assumption of stationarity of preferences and technology. It may then be tempting to swing to another extreme and conclude that deterministic cycles by themselves can account for most economic fluctuations, that deterministic chaos can explain our inability to make accurate forecasts, and that stochastic elements are not of essence in economic modeling, Yet this also is not a very defensible conclusion. It is more helpful to consider endogenously oscillatory dynamics as complementary to the role of stochastic elements in accounting for economic fluctuations. After all, it does not really make a big difference if endogenous mechanisms by themselves generate regular or irregular persistent oscillations or whether they give rise to damped oscillations that are sustained by stochastic shocks. On the other hand, the recognition of the role of oscillatory dynamics may diminish our reliance on unrealistically large shocks to explain economic data, for example, in business cycle theory. One of the main purposes of this volume is to collect and to draw attention to the wide variety of oscillatory "generating mechanisms" that have recently been discovered in the study of nonlinear models of economic equilibrium. It is hoped that this will elicit further empirical work in this area. For the most part, the papers have been selected to provide the reader with an overview of the various economic mechanisms that generate cyclic or chaotic dynamics in

4

INTRODUCTION

equilibrium. While some of the original contributions are unavoidably technical, others contain sufficient exposition and discussion to be helpful to the reader who is not inclined to plough through mathematical proofs. The first paper by Boldrin and Woodford provides a nontechnical survey of the economic literature on endogenous cycles, while the second paper by Grandmont is a technical introduction to the mathematics of chaotic dynamics. The next two groups of papers contain contributions to the literature on oscillatory equilibria in models with overlapping generations and with infinitely lived representative agents. The last group focuses on empirical methods to study and detect nonlinear relationships and cyclic and chaotic dynamics in time series data. In the first group, papers by Benhabib and Day and by Grandmont describe the emergence of cycles and chaotic dynamics in overlapping generations' models of exchange. Their results rely on the existence of strong income effects (although the inferiority of consumption at different ages is not required). Jullien's model is used to study the possibilities of cycles with a role for productive capital as well as money [see also Reichlin (1986); Farmer (1986); and Benhabib and Laroque (1988)].1 Reichlin establishes the role of factor intensities in generating cyclic equilibrium behavior in a two-sector economy. Laroque discusses the role of inventories and their destabilizing properties in simple overlapping generation economies. Finally, Whitesell shows the possibility of cycles in the continuous-time overlapping generations model of Blanchard (1985), which can be viewed as a bridge between the overlapping generations model and the model with infinitely lived representative agents. Here again the source of cycles seems to be an elastic labor supply and a low elasticity of substitution in production. The second group of papers is confined to growth models with infinitely lived representative agents, for which many turnpike theorems exist. Benhabib and Nishimura show how cycles can arise from the structure of technology, in particular because of the structure of factor intensities [see also Benhabib and Nishimura (1985)] .2 Boldrin and Deneckere explore how factorintensity reversals in a two-sector model can lead to chaotic dynamics. In the paper by Woodford, cycles arise because of imperfect financial markets with the consequence that investment, for the most part, is financed out of profits. 1 See Reichlin's "Endogenous Fluctuations in a Two-Sector Overlapping Generations Economy," in this volume, for a discussion. In these works, the existence of cycles is related either to strong income effects or to a wage elastic labor supply coupled with either a low elasticity of substitution in production, or with negative outside money. 2 The examples in Benhabib and Nishimura use large discount rates. A misinterpretation of turnpike theorems has led some to conclude that low discounting is incompatible with persistent cycles. Benhabib and Rustichini (1990) explicitly show that cycles with infinitely lived representative agents are possible for any discount rate for large and robust families of standard nonjoint Cobb-Douglas technologies.

INTRODUCTION

5

The paper by Montrucchio extends the earlier results of Boldrin and Montrucchio to continuous time [see Boldrin and Montrucchio (1986)], and shows that any smooth differential equation system, including those that have cyclic or "turbulent" trajectories, can be generated as a solution to an infinite horizon concave dynamic programming problem with a sufficiently high discount rate. This indicates that some of the economic problems that can be formulated as concave dynamic programming problems may have cyclic or chaotic solutions. Benhabib and Nishimura explore how the concepts of cycles and oscillations can be extended to a stochastic framework for a two-sector growth model, in order to study some of the empirical regularities of the business cycle. Finally, Deneckere and Judd study the emergence of cycles and chaos in a model with market imperfections, where monopoly rents are obtained from innovations until imitators drive the rents and the investments down before a new cycle begins. While the next three papers are not in the general equilibrium framework of overlapping generations or optimal growth models, they provide interesting ' 'generating mechanisms'' for endogenous oscillations. The paper by Drazen3 demonstrates how the presence of search externalities can give rise to persistent cycles [see also Diamond and Fudenberg (1989)]. Day and Shaffer show the possibility of chaos in an IS-LM model where investment depends on income (induced investment) as well as the interest rate. [A longer version of this paper contains an excellent discussion of ergodicity for a simple chaotic system. See Day and Shafer (1986).] Finally, the paper by Baumol and Wolff explores the possibilities of cycles and chaos associated with R&D expenditures. Much of the recent work on the empirical methods to study time series data generated by complex nonlinear models has focused on the possibilities of chaos. Some of these methods and their applications to economic time series are described in the papers by Brock and Sayers and Scheinkman and LeBaron [see also Barnett and Chen (1988)]. Unlike the hundreds of thousands of observations that can be generated by controlled experiments in the natural sciences (for which many of these methods have been designed), economic time series are typically quite short. These empirical methods must therefore face considerable theoretical difficulties, as discussed in this volume in the paper by Ramsey, Sayers, and Rothman. One promising approach is the use of the nonlinearity tests constructed by Brock, Dechert, Scheinkman, and LeBaron (1988). These tests are designed to identify data generated not only by simple (or cyclic) nonlinear systems but also data generated by chaotic systems, which a simpler test may judge to be random. A fruitful direction suggested by theoretical models is the use of such tests on disaggregated sectoral or industry data, whose components, because of resource constraints or other scar3

This is a revised version of his paper that appeared in American Economic Review (1988).

6

INTRODUCTION

cities, can move in ways that partially offset one another's cyclic or irregular movements. The empirical work of McNevin and Neftgi in this volume [see also Neftgi (1984); Sichel (1989a), (1989b); among others] seeks to ferret out nonlinearities by looking for asymmetries in the upswings and downswings of sectoral data. There is also other work, not included in this volume, by Hal White (1988) and John Geweke (1988) that designs alternative empirical methods to identify nonlinear or chaotic structures. Empirical and estimation issues aside, and quite apart from cyclic or oscillatory dynamics, a theoretical question concerning chaos still remains. "Strange attractors" or "observable chaos" refer to chaotic systems for which the set of points consisting of bounded aperiodic trajectories, together with the points that these trajectories attract, are in some sense large. Theoretically, the degree of prevalence of such chaotic systems has not yet been fully settled. For applied economics, however, one may include in the category of complex dynamics stable orbits of long periodicity, which for all practical purposes will appear aperiodic over relatively short time intervals. With such an expanded definition, systems that give rise to "complex dynamics" may be sufficiently common. On the other hand, over sufficiently long intervals the economy may be plagued by nonstationarities, giving rise to additional difficulties. In any case, given the complicated nature of economic systems with a large number of interrelated variables and nonlinear relationships, the possibility of complex dynamics in economics, and its implications for statistical estimation and forecasting, should not be dismissed lightly. The literature on nonlinearities in economics has been growing. Many excellent and original papers which should have been included in this volume had to be left out for lack of space. In particular, the literature on the relation between sunspots and cycles [see, for instance, Azariadis and Guesnerie (1986)], or on learning and dynamic instabilities [see Grandmont and Laroque (1989)] are not represented. Many excellent papers in Nonlinear Economic Dynamics (Academic Press 1988), edited by J. M. Grandmont, were not reproduced since that volume is readily available. Apologies are due to many authors whose excellent papers are left out in order to keep the proportions of this book manageable (at least manageable for this editor). I of course have innumerable debts to many who helped me organize my thoughts and compile this book. I am particularly indebted, however, to my colleagues Will Baumol, Kazuo Nishimura, and Aldo Rustichini, and to the economics editor at Princeton University Press, Jack Repcheck.

References Azariadis, C , andR. Guesnerie. 1986. "Sunspots and Cycles." Review of Economics Studies 53: 725-37.

INTRODUCTION

7

Barnett, W., and P. Chen. 1988. "The Aggregation Theoretic Monetary Aggregates are Chaotic and have Strange Attractors." In Dynamic Econometric Modelling, ed. W. Barnett, E. Bernt, and H. White. Cambridge: Cambridge University Press. Benhabib, J., and G. Laroque. 1988. "On Competitive Cycles in Productive Economies." Journal of Economic Theory 45: 145-170. Benhabib, H., and K. Nishimura. 1985. "Competitive Equilibrium Cycles." Journal of Economic Theory 35: 287-306. Benhabib, J., and A. Rustichini. 1990. "Equilibrium Cycling with Small Discounting." Journal of Economic Theory 52: 423^32. Blanchard, O. J. 1985. "Debt, Deficits and Finite Horizons." Journal of Political Economy 93: 223-42. Boldrin, M., and L. Montrucchio. 1986. "On the Indeterminacy of Capital Accumulation Paths." Journal of Economic Theory 40: 26-39. Brock, W. A., W. D. Dechert, J. A. Sheinkman, and B. LeBaron. 1988. "A Test for Independence Based Upon the Correlation Dimension." Manuscript, University of Wisconsin-Madison. Day, R. H., and W. Shafer. 1986. "Ergodic Fluctuations in Deterministic Economic Models." MRG working paper 8631, University of Southern California. Diamond, P. and D. Fudenberg. 1989. "Rational Expectations Business Cycles in Search Equilibrium." Journal of Political Economy 97: 606-19. Drazen, A. 1988. "Self-Fulfilling Optimism in a Trade-Friction Model of the Business Cycle." American Economic Review, Papers and Proceedings 78 (May): 369-72. Farmer, R. 1986. "Inference and Forecasting for Chaotic Nonlinear Time Series." Journal of Economic Theory 40: 77-88. Geweke, J. 1988. "Inference and Forecasting for Nonlinear Time Series." Working paper, Duke University. Grandmont, J. M., andG. Laroque. 1989. "Economic Dynamics with Learning: Some Instability Examples." Working paper 9007, Cepremap, Paris. Neftci, S. 1984. "Are Economic Time Series Asymmetric Over the Business Cycle?" Journal of Political Economy 92: 307-28. Reichlin, P. 1986. "Equilibrium Cycles and Stabilization Policies in an Overlapping Generations Economy with Production." Journal of Economic Theory 40: 89-103. Sichel, D. E. 1989a. "Business Cycle Asymmetry: A Deeper Look." Working Paper 98, Federal Reserve Board of Governors, Division of Research and Statistics. Sichel, D. E. 1989b. "Are Business Cycles Symmetric? A Correction." Journal of Political Economy 97: 1255-60. White, H. 1988. "Economic Prediction Using Neural Networks: The Case of IBM Daily Stock Returns." Discussion paper, U.C. San Diego, no. 88-20.

Equilibrium Models Displaying Endogenous Fluctuations and Chaos: A Survey* MICHELE BOLDRIN AND MICHAEL WOODFORD

1. Introduction The idea that market mechanisms are inherently dynamically unstable has played a minor role in studies of aggregate fluctuations over the past quarter century. Instead, the dominant strategy, both in equilibrium business cycle theory and in econometric modeling of aggregate fluctuations, has been to assume model specifications for which equilibrium is determinate and intrinsically stable, so that in the absence of continuing exogenous shocks, the economy would tend toward a steady state growth path. Recent work, however, has seen a revival of interest in the hypothesis that aggregate fluctuations might represent an endogenous phenomenon that would persist even in the absence of stochastic "shocks" to the economy. The endogenous cycle hypothesis is not new. Indeed, the earliest formal models of business cycles were largely of this type, including most notably the business cycle models proposed by John Hicks, Nicholas Kaldor, and Richard Goodwin.1 By the late 1950s, however, this method of attempting to model aggregate fluctuations had largely fallen out of favor, the dominant approach having become instead the Slutsky-Frisch-Tinbergen methodology of exogenous stochastic "impulses" that are transformed into a characteristic pattern of oscillations through the filtering properties of the economy's "propagation mechanism." There were probably three main reasons for the overwhelming popularity of the latter methodology, apart from whatever comfort may have been provided by a vision of the market process as fundamentally self-stabilizing. First, endogenous cycle models are essentially nonlinear, while the linear specifications that were possible in the case of the exogenous shock models were extremely convenient, both from the analytical point of view and from the point of view of empirical testing. * Originally published in Journal of Monetary Economics, 25: 189-222, 1990. Reprinted with the permission of Elsevier Science Publishers B.V. (North-Holland). 1 For overviews of the early (nonoptimizing, nonequilibrium) literature, see, e.g., Blatt (1983) or Lorenz (1989). For recent extensions of Goodwin's model, see Goodwin and Punzo (1988).

FLUCTUATIONS AND CHAOS: A SURVEY

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Second, the endogenous cycle hypothesis was thought to have been empirically refuted. It was easily shown that actual business cycles are far from being regular periodic motions.2 For example, the spectrum of an aggregate time series typically exhibits no pronounced peaks, let alone actual spikes as one would expect in the case of deterministic cycles. And econometric models were estimated that, when simulated with repeated exogenous stochastic shocks, produced data that looked like actual business cycles, but that, when simulated without the exogenous shocks, converged to a steady state (Adelman and Adelman 1959). Demonstrations of this kind appeared to show that the true structural relations implied an intrinsically stable economy. These types of considerations have less force today than they must have seemed to possess around 1960. We have come to understand that the simple empirical "refutations" of the endogenous cycle hypothesis do not prove as much as they might have seemed to. It is now understood that deterministic dynamical systems can generate chaotic dynamics, which can look very irregular and can have autocorrelation functions and spectra that exactly mimic those of a "stable" linear stochastic model, such as a stationary AR(1) (Sakai and Tokumaru 1980). Furthermore, it is now recognized that the fact that a stable model gives the best fit within the class of models considered is no proof that the true (or more accurate) model may not be an unstable one that generates endogenous cycles. John Blatt (1978) showed that when a linear autoregression was fit to periodic data from a simulation of the Hicks cycle model, the parameter estimates implied a stable second-order autoregressive process for output, of the kind that is in fact obtained from autoregressions of actual GNP data. Techniques that can, in principle, distinguish certain stochastic fluctuations from deterministic or even "noisy" chaotic data have been developed in the natural sciences, especially among physicists [see the excellent discussion in Eckmann and Ruelle (1985)]. They have been refined, improved, and recently applied to economic data by Brock, Scheinkman, and their coauthors. We will not review this literature here, but instead refer the reader to Brock (1988) and Scheinkman (1990). In general, the types of nonparametric tests for nonlinearity and endogenous instability that have been proposed seem to require quite large samples if reliable results are to be obtained [on this point, see Ramsey and Yuan (1989), and Ramsey, Sayers, and Rothman (1988)], and this may well mean that definitive conclusions will not be possible in the case of economic time series, proceeding in this fashion. The question of the relative empirical validity of the exogenous and endogenous cycle hypotheses is likely to be decided only by comparing the predictions of theoretical models from both classes, whose 2

Sir John Hicks, in private communication, has indicated that this was the reason for his loss of interest in endogenous cycle models.

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MICHELE BOLDRIN & MICHAEL WOODFORD

parameters are either econometrically estimated or' 'calibrated'' after the fashion of Kydland and Prescott (1982). Thus far, no tests of this kind have been performed using endogenous cycle models (and admittedly, none of the available theoretical models appear likely to fare well under such a test—some reasons for which are discussed below). The development of theoretical models that could be tested in this way should be a major object of further research. Another reason for the decline from favor of the endogenous cycle hypothesis concerns the inadequate behavioral foundations of the early models of this kind. The stability results obtained for many simple equilibrium models based upon optimizing behavior—in particular the celebrated "turnpike theorems" for optimal growth models (discussed below)—doubtless led many economists to suppose that the endogenous cycle models were not only lacking in explicit foundations in terms of optimizing behavior, but depended upon behavioral assumptions that were necessarily inconsistent with optimization. This latter issue is the central focus of the present paper. We survey the literature that shows that endogenous fluctuations (either periodic or chaotic) can persist in the absence of exogenous shocks, in rigorously formulated equilibrium models in which agents optimize with perfect foresight. We find it useful to divide the known examples into two categories. On the one hand (sections 2, 3, and 4), there exist models with a unique perfect foresight equilibrium which involves perpetual fluctuations for most initial conditions. In such cases, it is clear how the forces that bring about a competitive equilibrium also require the economy to exhibit endogenous fluctuations. On the other hand (sections 5 and 6), we find models in which perfect foresight equilibrium is indeterminate, and among the large set of possible equilibria are those in which the state of the economy oscillates forever. In cases of this sort, the forces that bring about competitive equilibrium do not require that perpetual fluctuations occur. While we regard the indeterminacy in these types of cases to indicate a type of instability of the competitive process—in that arbitrary events can determine which equilibrium occurs—it is of qualitatively a different sort than in the case of models of the first type, which are the primary focus of the present survey. We begin (section 2) with a brief discussion of an early nonoptimizing model of complex economic dynamics, because it allows us to raise some issues regarding the consistency of the postulated behavior with optimization that are then resolved in the more recent literature on optimizing models. We also discuss in the context of this simple example some of the techniques that can be used to demonstrate the existence of endogenous fluctuations. For reasons of brevity we have minimized the number of mathematical definitions and theorems used. The reader who wishes to know more about the mathematics of endogenous cycles and chaos may wish to consult such standard references as Collet and Eckmann (1980), Devaney (1986), Guckenheimer

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and Holmes (1983), Chow and Hale (1982), Lasota and Mackey (1985), and Ruelle (1989).

2. A Simple Example of Complex Economic Dynamics3 Day (1982) considers a one-sector, neoclassical growth model in which the dynamics of capital accumulation has the form: k,+ i =

s(k,)-f(k.) .

,

x

=

KK)

(2.1)

where s is the saving function, / the production function, and \ > 0 is the exogenous population growth rate. This is a discrete-time version of the famous Solow (1956) growth model. In the discrete-time form (2.1), Solow's assumption of a constant, exogenous saving rate and of a neoclassical, concave production function give rise to a map h(k,) which is monotonically increasing and has one and only one interior steady state k* = h{k*). A typical case is represented in Figure 1.1a. For example, in the case of a constant saving ratio a and a Cobb-Douglas form for/, (2.1) becomes k,+ 1 = °° such that h'n(xn) —» x. The collection of all such x's is called the non-wandering set and is denoted by fi(h). For monotonic maps, the set £l(h) turns out to be extremely simple. In fact, if /z: / —*• / is monotonic increasing, then Cl(h) = Fix(h), and if it is monotonic decreasing, then il(h) = {Fix(ft) U Per2(/z)}. This should be obvious from graphical inspection, plus the fact that when h is monotonic decreasing, h2 is monotonic increasing. This also implies that we need h to be nonmonotonic in order to get trajectories more complicated than Per2(A). Even if h(x) = \AJC(1 —X) is not monotonic, it is not difficult to see that for (X < 3, £l(h) = Fix(/i), while at \L = 3 a first cycle of period 2 emerges that is attractive for 3 < (JL < 3.449499. At fx = 3.449499, a cycle of period 4 emerges. Similarly, attracting cycles of period 2" emerge (following the Sarkovskii's order!) at the values JJL^, where (UL3 = 3.549090, \x4 = 3.564407, |x5 = 3.568759, JJU6 = 3.569692, . . . . We later describe the "bifurcation" through which a period 2" cycle emerges from one of period 2"~'. But cycles of higher order are not the most complicated kind of possible asymptotic behavior. Intuitively we may say that a trajectory x,+ l = h(x,) is "complicated" if: (a) it does not converge to any point in Fix(/i) or Per(/t); (b) 6

Moreover, there are a few simple, qualitative properties that guarantee the existence of a period 3. In fact, let h: I —* I be continuous, with / an interval; if there exist distinct, disjoint subintervals /i C / and I2 C I such that h(I,) D [2 and h(I2) D (I\ U /2) then there is a period 3 for h. See Devaney (1986) for more details.

FLUCTUATIONS AND CHAOS: A SURVEY

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it does not explode either to ± °° (i.e., it stays within a bounded interval that we may identify with / = [0,1]); (c) trajectories x't+l = h(x?) starting from x'o "nearby" x0 tend to "move away" from xt+1 = h(x,) after enough iterations. This is only an intuitive description to which one may attach different formal "interpretations." A simple one is: Definition 2. Let h: I—* I define a dynamical system as in (2.4). We say that h exhibits topological chaos if: (i) for every period N, there exist points xNe.I such that h?{xN) = xN; (ii) there exists a non-denumerable set S C / and an e > 0 such that for every pair x and y in 5 with x i= y: lim^o. sup \h"(x) - h"(y)\ > e, lim^« inf \hn(x) - h"(y)\ = 0 and for every y 6E Per(fc) and A GE S: linv,,. sup \h"{x) — h"(y)\ a e.

Li and Yorke (1975) prove that when h has a period-3 point, it will have topological chaos in the sense just defined. Nevertheless, this kind of chaos may not be very interesting to economists. The reason is that the scrambled set, S, may turn out to be of (Lebesgue) measure zero, i.e., the probability of starting in S is zero, while any initial condition outside 5 results in an orbit converging to a cycle of finite period. For the quadratic map this occurs, for example, at JJL, = 3.828427, where topological chaos exists but almost all initial conditions lead asymptotically to a period-3 cycle. In the technical literature, a notion of "observable chaos" or "ergodic chaos" has emerged. Ergodic chaos (loosely speaking) means: (a) S has positive (Lebesgue) measure (it has full measure, for example, for the quadratic map with (x = 4) so that aperiodic trajectories are in fact observable and (b) asymptotically the sequence {x}?= 0 obtained by iterating h(x,) approximates an ergodic and absolutely continuous distribution which is invariant under h and which summarizes the limiting statistical properties of the (deterministic) chaotic trajectories. Again, for the quadratic map with |A = 4, an invariant distribution for (juc(l —x) exists, namely/(x) = {ir[x(l -x)] 1 ' 2 }" 1 . Unfortunately, proving that "ergodic chaos" exists is not a trivial task. In particular, no standard all-purpose theorems currently exist. A relatively simple result, in the one-dimensional case, is the following: Theorem [Lasota and Yorke (1973)]. Let h: I —» / be piecewise C2, and expansive, i.e., such that inf,eI | h! (x) \ > 1. Then h has an absolutely continuous invariant measure, whose support accordingly has positive Lebesgue measure. Moreover, if h is unimodal, then the measure is ergodic, so that for almost all initial conditions this measure describes the long-run frequency with which different neighborhoods are visited. An example, indeed a classical one, of a map satisfying the conditions of

16

MICHELE BOLDRIN & MICHAEL WOODFORD

the Lasota-Yorke theorem is given by the so-called "tent map" (Figure 1.2), which has the general form: 1)

h(x) =

(2.5)

1 - \ In this case it is easy to understand why orbits starting from a generic point in [0,1] cannot converge to any cycle of a finite period: the slope of h is everywhere larger than one and therefore no fixed point of W(x), for any JV S: 1, can possibly be attractive. On the other hand, any orbit has to stay bounded forever in [0,1], so that it has to "wander around" the unit interval and, eventually, visit every neighborhood of it, hence the ergodic behavior. The quadratic map for (JL = 4 also exhibits this behavior, since under a change of variables it can be shown to be equivalent to a tent map with X. = 2. A (technical) word of caution should be added here. Point (c) in our "intuitive" definition of chaos could be expressed, in the jargon of the field, as "sensitive dependence on initial conditions." For many mathematicians, this property (technically, the existence of "positive Liapunov exponents") defines "chaos." We will not treat this difficult technical concept here; the curious reader is invited to consult Eckmann and Ruelle (1985). It should be pointed out, though, that neither "topological" nor "ergodic" chaos as de-

Fig. 1.2

t+l

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fined here necessarily implies the existence of a positive Liapunov exponent. The map Ax(\ -x) is, however, chaotic also in this third sense, as it has a Liapunov exponent equal to Iog2. To complete our survey of discrete-time dynamical systems, we need to say a few words about "bifurcation theory." Consider a continuous family of systems of the form (2.4) indexed by a parameter ( i £ i , i.e., x,+ l = h^ix,). Assume that for, say, (x < a, the system h^ has an attractor A of a certain kind (e.g., a stable fixed point). Assume that at |x = a the attractor A "loses stability" (nearby orbits will not converge to A any more) and that a "new attractor" (e.g., a periodic cycle) A' ¥= A appears which is stable for (x £ [a, a + e], where e > 0, and that orbits near A now converge to A'. Then |x = a is a bifurcation point for the system and, in the jargon, we say that A' bifurcates from A at |x = a. In the case of the family hjjc) = (JUC(1 — JC), (x = 3 is a first bifurcation point, as explained before. Obviously there are many different types of bifurcations, even in the case of simple one-dimensional maps; each different bifurcation produces a different new attractor A' from a given preexisting attractor A. The theory is in fact very general; it applies in high dimensions, to both continuous- and discrete-time dynamical systems, and to the study of singularities as well. A general reference is Chow and Hale (1982); simpler discussions may be found in the references given at the end of section 1. A special, albeit rather pervasive, type of bifurcation for one-dimensional maps has often been used in economics; this is the flip or period-doubling bifurcation. Loosely speaking (see, e.g., Devaney [1986] for a precise statement), it says that if JC* = hjx*), dhjx*)/dx = - 1, and d(dh,,.(x*))/dx/diL =/= 0 at p, = jx, then (given a generically valid condition on higher derivatives) a stable cycle of period 2 will bifurcate from x* for JJL in a (right or left) neighborhood of |1. Notice that replacing h with h" in the former one gets a period 2N cycle out of one of period N. The period-2 orbit emerging at (x = 3 for |xx(l —x) is created by means of a flip bifurcation, as are all the other subsequent ones of period 2" at the parameter values |xn listed before. A cascade of period-doubling bifurcations (following along the "power of two" portion of the Sarkovskii ordering) is a frequent (albeit not the unique) pattern observed for unimodal maps in the "transition to chaos." Once again, |xx(l -x) perfectly respects this pattern. It is now time to go back to economics. Day's examples generate families of unimodal maps that illustrate these propositions. Thus, they show that extremely simple behavioral hypotheses and model structures can produce very complicated dynamics. However, one may nevertheless question whether the sort of behavior assumed is in fact consistent with optimization within the assumed environment. For example, the assumption of a constant saving ratio was often used in the early "descrip-

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tive" growth models and can indeed be derived from intertemporal utility maximization under certain hypotheses, but it becomes especially implausible when a production function of the type embodied in (2.2) is proposed. Why should a maximizing agent ever save up to the point at which marginal returns to capital are negative if he can obtain the same output level with much less capital stock? Although it is less obvious, Day's case of a variable saving ratio and a monotonic production function [i.e., equation (2.3)] is equally inconsistent with intertemporal utility maximization at least in the case of a representative consumer model. This was pointed out (in a general form) in Dechert (1984). If the consumer-producer maximizes 2™_0 u(c,)b', where u is concave and 8 is in (0,1), even if the production function is not concave, the optimal program {ko,kuk2, . . .} can be expressed by a policy function kt+l = T (k,), which is monotonically increasing. The economic prediction is, accordingly, that such a society will asymptotically converge to some stationary position. The latter is unique when/is concave. From this we have to conclude that the chaotic examples derived from a one-sector growth model would not pass the rationality critique. Such a critique turns out to be rather special itself, as it holds true only for the special type of growth model considered above. This will be illustrated in the next two sections.

3. Optimal Growth Models The kind of strong characterization given above of the qualitative properties of equilibrium dynamics turns out not to be possible even for more general optimal growth models, perfectly competitive dynamic economies in which all agents are identical and optimize over the entire infinite horizon of the economy.7 We will fully describe only a discrete-time class of such economies, though we also discuss a continuous-time version of the same model; the translation should be immediate.8 In every period t = 0,1,2, . . . , a representative agent derives satisfaction from a consumption vector c, G U1, according to a utility function u(c,) which is taken to be increasing, concave, and as smooth as needed. The state of the world is described by a vector x, €E Ui of stocks and by a feasible set F C R2+" x IRm composed of all the triples of today's stocks, today's consumption, 7 See Bewley (1982) and the literature quoted therein for a reconciliation of the abstraction of a single representative agent that controls both consumption and production decisions with the case of many independent consumers and producers. 8 The reader is referred to Cass and Shell (1976), Bewley (1982), Becker and Majumdar (1989), and especially McKenzie (1986) and (1987) for more complete treatments.

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and tomorrow's stocks that are technologically compatible, i.e., a point in F has the form (xt, c,, x,+ l). Now define: V(x,y) = max u(c) s.t. (x,c,y) e F

(3.1)

and let D C !R2+ be the projection of F along the c coordinates. Then V, which is called the short-run or instantaneous return function, will give the maximum utility achievable at time t if the state is x and we have chosen to go into state v by tomorrow. It should be easy to see that to maximize the discounted sum 27 = 0 u(c,)b' s.t. (x,, c,, x,+ i) G F is equivalent to max 27 = 0 V(x,,xl+l)h' s.t. (x,, x,+,) G D. (Here the parameter 8 indicates the rate at which future utilities are discounted from today's standpoint (impatience), and takes values in [0,1].) It is mathematically simpler to consider the problem in the latter (reduced) form. Standard neoclassical assumptions on u and F would typically yield a pair V and D satisfying the following properties. The return function V(x,y) is strictly concave, increasing in x and decreasing in y. The technology set D C X x X C M2? is convex and compact. X is the feasible set which is also convex and compact. The initial state x0 is given. The optimization problems we are facing can be equivalently described as one of dynamic programming [see Stokey et al. (1989) for the details of this derivation]. The value function for such a problem is defined by the Bellman equation: W(x) = Maxy{V(x,y) + bW(y), s.t.(x,y) X describing the optimal sequence of states {xo,xux2, . . .} as a dynamical system x,+ l = T&(X,) on X. The time evolution described by T8 contains all the relevant information about the dynamic behavior of our model economy. In particular, the price vectors p, of the stocks x, that realize the optimal program as a competitive equilibrium over time follow a dynamic process that (when the solution {x} is interior to X and V is smooth) is homeomorphic to the one for the stocks. In other words: pl+l = 8(pt) with 6 = 8W • T • ( W 8 ) 1 , where W is the first derivative of the value function. Accordingly, the question that concerns us is: what are the predictions of the theory about the asymptotic behavior of the dynamical system T8? In particular, do competitive equilibrium and perfect foresight imply convergence to a stationary state, as in the one-sector growth model? A first, remarkable answer is given by the following: Turnpike Theorem (Discrete Time). Under the maintained assumptions, for given V and D there exists a level 8 of the discount factor such that for all 8 in the non-empty interval [8,1], the function T8that solves (3.2) has a unique

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globally attractive fixed point x* = T8(;C*). Such an x* is also interior to X under additional mild restrictions. This means that under a set of fairly general hypotheses, we are able to predict that if people are not "too impatient" relative to the given V and D, then they should move toward a stationary state where history repeats itself indefinitely and no surprises ever occur.9 As remarkable as it is, the turnpike property requires rather special conditions. In particular, how close must 8 be to one, and what happens when 8 is smaller than 8? These are important questions. It is hard to rely heavily on a property that may depend critically on such a volatile and not directly observable factor as "society's average degree of impatience." It turns out in fact that as the discount factor moves away from 8 toward zero, Tsmay become "practically anything." This was proved in Boldrin and Montrucchio (1986b) [but see also Boldrin and Montrucchio (1984) and (1986a), and Montrucchio (1986) for additional results]: Theorem (Anti-turnpike). Let 9: X —» X be any C2 map describing a dynamical system on the compact, convex setX C U". Then there exist a technology set D, a return function V, and a discount factor 8 E (0,1) satisfying the maintained hypotheses and such that 6 is the policy function Tsthat solves (3.2) for the given D, V, and 8. The proof is of a constructive type, so that one may effectively compute a fictitious economy for any desired dynamics. This "general possibility theorem'' makes clear that any kind of strange dynamic behavior is fully compatible with competitive markets, perfect foresight, decreasing returns, etc. Independently, Deneckere and Pelikan (1986) also presented some onedimensional examples of models satisfying our assumptions and having the quadratic map 4x(l - x) as their optimal policy function for appropriately selected values of 8. Also, Neumann et al. (1988) provide a technical improvement on the Boldrin-Montrucchio construction that, after a slight modification, enables one to derive the classical chaotic map J&(X) = 4x(l - x) for 8 = .25, which is 25 times larger than the initial estimates (see Boldrin and Montrucchio [1989, chap. 4] for this and other examples). These results may contrast with the common intuition according to which infinite-horizon concave programming problems should yield optimal policies that are "dynamically regular" as concavity typically penalizes oscillations. In order to see the economic reasons for this, it is worth looking at some specific, simple, examples. Let us begin with the one-sector model we briefly introduced at the end of section 2, and which was used by Dechert to prove 9

In the form given here, the turnpike theorem is due to Scheinkman (1976). McKenzie (1976) and Rockafellar (1976) proved it for the continuous-time version (on this point see the discussion below), and Bewley (1982) and Yano (1984) generalized it to the many-agents case (but see McKenzie [1986] for a more careful attribution of credit).

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that cycles and chaos are not optimal in that framework. This is a special case of the general model we are considering, with V(x,, x,+1) = u\f(x,) —x,+,] and D = {{x,, xl+1) s.t. 0 ^ xt+1 ^/(*,)}. For that model, the turnpike theorem holds independently of the discount factor as T8 is always monotonic increasing. Unfortunately, such a nice feature does not persist even if the simplest generalization of the one-sector model is taken into account. This was first proved by Benhabib and Nishimura (1985). They considered a model with two goods—consumption and capital—which are produced by two different sectors by means of capital and labor. Given the two concave and homogeneous of degree-one production functions, one can then define a Production Possibility Frontier (PPF) T(x,, xt+1) = ct, that gives the producible amount of consumption when the aggregated capital stock is x, (a scalar), labor is efficiently and fully employed, and tomorrow's stock must be xt+1. The return function is now V(x,, xt+1) = u[T(x,, JC,+ 1 )] andD = {(x,, xt+l) s.t. 0 < xt+l ^ F(x,,l)} where F is the production function of the capital good sector and labor has been normalized to one. In such a case, T8 is not always upwardsloping. If the consumption sector uses a capital/labor ratio higher than the one used by the capital sector, it will be downward-sloping. Let x* be the (unique) interior fixed point (i.e., TS(X*) = x*). This is the candidate for the turnpike. Assume, for simplicity, that T8is differentiable in a neighborhood of x*. The derivative T8(JC*) at the steady state changes as 8 moves in (0,1), everything else equal. Benhabib and Nishimura show that it may take the value - 1 for admissible 8's in such a way that the conditions for a flip (perioddoubling) bifurcation are realized. In this case an optimal cycle of period-2 will exist which also is attractive: no more turnpike! One may provide examples of this phenomenon showing that such an outcome is by no means due to "pathological" technologies and preferences. One may go even further and show that cycles of every period as well as chaos (in the sense of topological chaos) may arise in the same class of twosector optimal growth models. A theoretical analysis is provided in Boldrin (1986). It is proved that the policy function xl+1 = T&(X,) is unimodal when (for example) factor-intensity reversal occurs between the two sectors (this is not strictly necessary). Suppose that there is a level, say k*, of the aggregate capital stock such that when k, is in [0,k*), the capital sector has a higher capital-labor ratio, whereas the opposite is true when k, is in (k*,k], where k is the maximum level of capital that the economy can sustain. This technological feature provides the underlying reason for the unimodal shape for T8 (i.e., TS is as in Figure 1.1b). Variations in the level of the discount factor 8 then can produce a cascade of period-doubling bifurcations that (technicalities aside) leads to period-3 orbits and chaos. By means of a simple example where the two production functions are respectively CES (consumption sector) and Leontief (investment sector), it can be shown that even very standard technologies allow for period-doubling bifurcations at certain parameter values. This

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MICHELE BOLDRIN & MICHAEL WOODFORD

example is fully worked out in Boldrin and Deneckere (1987) for the case in which the production function for the consumption sector is Cobb-Douglas.10 Assuming, in order to simplify the algebra, that the utility function is linear, the short-run return function V(k,,k,+ l) may be written as (1 — kl+1 + [ik,)a • (k,{\ + 7(A) - ykt+,)' ~", where a is the Cobb-Douglas coefficient, 0 < 7 < 1 is the capital/labor ratio in the investment sector, and (1 — jx) is the capital depreciation rate. The value k, = 7 corresponds to the critical point of the policy function; for all k, < 7, the investment sector is more capital/labor intensive and, therefore, any increase in the capital stock today will make it optimal to further increase it tomorrow (TS is upward sloping); for k, > 7 the opposite is true and the substitution effect along the production possibility frontier makes it optimal to have smaller stocks of capital k1+1 associated to larger stocks k, (T8 is downward sloping). The reader may notice that the basic logic behind this is the same as with the Rybczynski theorem. Once properly parameterized the economy displays various types of dynamic behavior, from the simple convergence to a stationary state, to cycles of different finite periods, to "chaos." In particular, for any given level of discounting 8, a stable period 2 can always appear for values of a, 7, and |x in the unit interval. The same in fact seems to be true for cycles of orders 2". It should also be noticed that the technological parameter values at which this occurs are rather extreme, and they become even more so when chaos is obtained. A typical chaotic triple would have a = .03, 7 = .09, 8 — .1 ((JL = 0 here, but similar sets of parameters work for (x next to one). If ones goes back to the general CES formulation, things improve only slightly. For levels of the elasticity of substitution in production that are not too extreme, aperiodic motions appear when the discount factor is in the range (.2,.3) (this still implies an interest rate around 400%). When the elasticity of substitution becomes extremely small (i.e., in the range .01, .02), then chaos is present also for less unreasonable levels of discounting, such as .7 or .8. Cycles are not special to the discrete-time version of such models. In a very early work, Magill (1979) had pointed out that cyclical (albeit converging to a steady state) motions were possible for solutions to undiscounted continuous-time optimization problems. He was able to show that the origin of oscillations along optimal trajectories is directly related to the existence of asymmetries in the Hessian function of the short-run maximand evaluated at the steady state. The use of a model without discounting prevented him from making these oscillatory motions persistent and from proving the existence of limit cycles. This was achieved in Benhabib and Nishimura (1979). The two authors use bifurcation theory for ordinary differential equations to prove that limit cycles can occur [consult again Chow and Hale (1982) under "Hopf 10

Ivar Ekeland and Jos6 Scheinkman had conjectured earlier that such a parametric form may lead to irregular trajectories (Scheinkman 1984).

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bifurcation" for the details]. Let us show very briefly how this can happen. In continuous time, we face an optimal control problem of the form: max \ V(x,x) exp( - pt)dt s.t. (x,x) 6 D , x(0) given. o

(3.3)

Here, x(t) is a vector depending on time, x is its time derivative, D, again the convex feasible set, and p, the discount factor in [0,°°]. (Note that p = 0 is equivalent to 8 = 1 in discrete time.) Using the Maximum Principle, one defines a Hamiltonian H(x,q) =

max{V(x,x) + (q,x),

x

s.t. (x,x) 0, and that it is (3-convex in 11

It remains true, and global convergence is assured, in the special case in which d H(x,q)/d2q = [-d2H(x,q)/S2x]T. This was proved in Magill and Scheinkman (1979). When this symmetry condition does not hold, there is room for oscillations. 2

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MICHELE BOLDRIN & MICHAEL WOODFORD

q if H(x,q) - (J/2 \\q\\2 is convex in q on its domain for all admissible x and (5 > 0. Then one has: Turnpike Theorem (Continuous Time). Suppose the Hamiltonian given in (3.4) is a-concave and (J-convex in a convex neighborhood of (x,q) G lRn X Un, where (x,q) is a rest point for (3.5) (i.e., dH(x,q)/dq = 0 and -dH(x,q)/dx + pq = 0). Assume that the discount rate satisfies p2 < 4a(3. Then (under a few additional technical conditions), for every initial condition (xo,qo), the unique solution [x(t),q(t)] to (3.5) that maximizes (3.3) converges to (x,q) asf-» +oo. This version of the turnpike property is useful because it relates the level of discounting to the "curvature" of the Hamiltonian, which in turn depends (albeit in a very complicated way) on the curvatures of the technology and the preferences. The more concave-convex is H, the higher is the level of impatience compatible with regular dynamic behaviors. But as Benhabib and Nishimura (1979) showed, when p grows for fixed a and p, a pair (or more than a pair) of eigenvalues may change the sign of their real part by crossing the imaginary axis. In such a case (care taken for the technical details), a Hopf bifurcation occurs. The limit cycle associated with it turns out to be an attractor for the system (3.5). Once again the turnpike property is lost as people become more impatient. Some characteristics of the oscillatory or chaotic paths so obtained in both versions of the optimal growth model need to be stressed. First of all, they are realized as equilibrium paths, in the sense that all markets are continuously clearing at each point in time, prices adjust completely, and no productive resource is unemployed. Moreover, they are Pareto efficient in the sense that it is impossible to modify the allocation of resources that they imply, in order to increase the welfare of some agent without making somebody else worse off. The economic policy implications of these facts are obvious and we do not intend to elaborate further on them. Second, oscillations here are strictly market-driven: it is the existence of certain factor-intensity relations across sectors that makes it profitable for the producers (and the consumers alike) to invest, produce (and consume) in an oscillatory form. Even if all the prices are the "right ones" (i.e., no conditions for profitable arbitrage exist), the pure seeking of individual profits will bring about cyclic behavior. We should point out here that an "anti-turnpike" theorem is available also for continuous-time models. This was proved in Montrucchio (1987). The proof proceeds essentially as in the discrete case and therefore permits the construction of fictitious economies that optimally evolve according to any prescribed law of motion x = f(x). The extension to the continuous-time case is particularly useful in clarifying the extent to which a high rate of time discount is required for the existence of complex dynamics. In both Boldrin and Montrucchio (1986a, 1986b), Boldrin

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(1986), and Deneckere and Pelikan (1986), all of the parametric examples of chaotic optimal accumulation paths require very small discount factors, or equivalently high rates of time preference. Some have therefore concluded that chaotic oscillations are mere mathematical curiosa (at least in optimal growth models) and not a plausible line of research in business cycle theory. In fact, it is clear that appropriate choice of the technology and the singleperiod utility function can allow chaos to exist for as low a rate of time preference as one likes. This is clearest in the case of the continuous-time examples. In such examples, arbitrary rescaling of the time unit gives economies in which the rate of time preference (in, say, percent per year) can be arbitrarily low. Suppose that the economy characterized by V, p, and D exhibits endogenous cycles or chaos. Then consider a new economy with an objective function V(x,x) = V(x,e-'x), a discount factor p = ep, and a feasible set D = {(x,x)\(x, e~ J i) £ D}. This will also be a standard optimal growth model, and if the equilibrium law of motion for the original economy was x = f(x), the law of motion for the new economy will be x = f(x) = ef(x). The new economy will also exhibit cycles or chaos, since the dynamical systems are equivalent, but by choosing e « 1, we can make p arbitrarily small. It is straightforward to conclude that any kind of dynamics can be made optimal at any level of discounting, no matter how small the latter is [Boldrin and Montrucchio (1989, chap. 3)]. This does not contradict the turnpike theorem of Rockafellar quoted above, because the Hamiltonian of the new economy is given by H(x,q) = H(x,eq), so that if H was a-concave and @-convex, H is a-concave but only Pe2 —convex. Thus, as e is made small, the degree of convexity of the Hamiltonian becomes small as well. Before it is possible to conclude that empirically realistic values for the rate of time preference are "too low" to allow endogenous cycles, one must discuss what is an empirically realistic degree of curvature for the Hamiltonian.12

4. Models with Market Imperfections and Determinate Equilibrium Dynamics In the event that markets are incomplete, imperfectly competitive, or otherwise less than fully efficient, the conditions under which endogenous equilibrium fluctuations can occur are less stringent. There is no general "turnpike" 12

We are indebted to David Levine for discussion of this point. A similar point is made by Benhabib and Rustichini (1990), who show that for any discount rate p greater than the rate of depreciation \x, of capital stocks, one can find parameter values for a three-sector Cobb-Douglas production technology that results in an optimal growth path attracted to a limit cycle. Their construction assumes a linear utility function, which helps to keep the curvature of the Hamiltonian low.

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theorem for this case; in fact, it is possible to construct economies in which borrowing constraints permit one to make consumers' rate of time preference arbitrarily small while continuing to have endogenous cycles or chaos. Nor is the kind of inter-sectoral relations in the production technology considered earlier necessary in order for endogenous fluctuations to occur; for example, endogenous cycles and chaos can occur even in the case of a one-sector production technology. A simple type of market imperfection is an assumption that agents are unable to borrow against all types of future incomes. The first demonstration that borrowing constraints could make endogenous cycles possible even in the case of a finite number of infinite lived consumer types and a one-sector production technology was due to Bewley (1986).13 Bewley showed how borrowing constraints could result in equilibrium dynamics in such a model formally analogous to the capital accumulation paths that could occur in the overlapping generations model of Diamond (1965). Bewley's result depends upon specifications that typically also imply indeterminacy of perfect foresight equilibrium. Hence, further discussion of this example is deferred to section 6. Another model based on borrowing constraints is examined by Woodford (1988b). In this economy, there are two types of infinite-lived consumers— workers, who supply labor inputs to the production process, and entrepreneurs, who own the capital stock and organize production, and hence who make the investment decisions. Workers are assumed to be unable to save by accumulating physical capital and organizing production themselves. There is also a limitation upon the extent to which workers can indirectly invest in productive capital by lending to entrepreneurs. In the simplest case, loan contracts are assumed to be completely unenforceable. In this case, in each period, workers must consume exactly the wage bill, and entrepreneurs must finance investment entirely out of retained earnings from that period. The capital stock in each period will then be equal to the previous period's gross returns to capital, times the fraction of their wealth that entrepreneurs do not wish to consume. This can easily result in a unimodal map kl+l = f(kt) of the form shown in Figure 1. lb of section 2, since gross returns to capital will be a decreasing function of the capital stock if labor supply is sufficiently inelastic at high levels of labor supply, and capital is not too easily substituted for labor. This is the case when, for example, the single period utility function of the entrepreneurs is logarithmic in consumption. Then entrepreneurs will consume a constant fraction ( 1 - 8 ) (8 being again the discount factor) of their wealth in each period. If the preferences of workers are additively separable between periods, equilibrium labor supply will depend only upon the current real wage, so that it can be represented by a function s(w(). Assume that the 13

A somewhat similar type of borrowing constraint is considered in Scheinkman-Weiss (1986), but there, exogenous stochastic shocks play a major role in sustaining oscillations.

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latter is monotonically increasing. With constant returns to scale production function Y, = F(k,,lt), the equilibrium real wage w, will be determined by the current capital stock k, as the unique solution w(k,) to the relation F,(k,,s(w,)) = w,. The unique equilibrium solution for the following period's capital stock will then be: k,+ l = f(k.) = pkMKXMK)))

(4.1)

The function f(k) in (4.1) can easily be unimodal. For example, if workers' preferences are such that the labor supply function is linear s(w) = mw, and, in addition, the technology is Leontief, with a > 0 units of output being produced per unit of capital using b > 0 units of labor, then (4.1) becomes *(+1 = Pfat, - (P/m)kfi 2

2

(4.2)

Setting |JL = $a m/b , (4.2) becomes the quadratic map discussed earlier in section 2, and chaos, both "topological'' and "ergodic," will occur for (x G [3.57,4]. (Note that for appropriate a, b, and m, chaos may persist as (3 is made to approach 1. Hence, there is no "turnpike" property.) The existence of ergodic chaotic dynamics can be assured for open sets of parameter values by constructing an example in which the map/(&) in (4.1) is everywhere expanding. This requires a kink in f(k) at the peak, but this can easily come about if, for example, the elasticity of labor supply is discontinuous at this point. If the elasticity of labor supply falls sufficiently greatly at the kink, it is possible for f(k) to be sharply increasing before the peak and sharply decreasing thereafter. Endogenous cycles and chaos in this type of economy do not depend upon the assumption of a Leontief technology, but it is important that the substitutability between capital and labor not be too great. For the fall in kt+1 for large values of kt obviously depends upon total gross returns to capital being a decreasing function of the size of the current capital stock at that point, which is only possible if it is not possible to easily substitute capital for labor when the real wage rises. In fact, Hernandez (1988) proves a "turnpike" theorem for a class of economies in which consumers cannot borrow against future labor income, under the assumption that the production technology allows sufficient substitutability between factors for total returns to capital to be a monotonically increasing function of the capital stock. It is not clear how far this result can be generalized, but it suggests that low factor substitutability may be important for the existence of endogenous instability even in more complicated examples. See also the discussion by Becker and Foias (1987). Similar dynamics are also possible in the case of a much weaker restriction upon financial intermediation, namely, in the case that workers can lend to entrepreneurs, but debt contracts contingent upon firm-specific technology shocks are unenforceable. In such a case, if there is a continuum of firms with independent realizations of the technology shock, and the technology shock

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MICHELE BOLDRIN & MICHAEL WOODFORD

takes an appropriate form, the equilibrium real wage and aggregate production are deterministic functions of the aggregate capital stock despite the existence of a stochastic technology for each individual entrepreneur/firm. If entrepreneurs' preferences are homothetic, each entrepreneur will choose levels of consumption, investment, and borrowing that are proportional to the amount by which his or her gross returns in the current period exceed his or her debt commitment. As a result, the deterministic (perfect foresight) dynamics of the aggregate capital stock are independent of the distribution of capital holdings across entrepreneurs with different histories of technology shocks. In this model, if the technology shock occurring in the worst state that cannot be insured against is sufficiently bad, a low level of gross returns to capital will result in a low capital stock in the following period, even though total current income is high. Thus, k,+ 1 can again be a decreasing function of kt, for high values of kt, and all of the phenomena discussed above can occur. The distribution of income continues to have a significant effect upon capital accumulation, despite the existence of a competitive market for (noncontingent) loans. Financial constraints are not the only kind of market imperfections that can give rise to endogenous fluctuations even with an arbitrarily low rate of time discount. Deneckere and Judd (1986) demonstrate the possibility of endogenous fluctuations in the rate of introduction of new products in an economy in which the creation of a new product involves a one-time fixed cost, and allows the innovator a one-period monopoly of production of the new product. The model is a simplified discrete-time variant of that of Judd (1985). Each period, new products are introduced to the point where the monopoly rents from the production of each new product are no greater than the fixed cost of creation of a new product. The total number of products produced in a given period (N,) is a determinate function of the number of old (nonmonpolized) products in existence, and hence ofN,-x. Deneckere and Judd show that for a certain parametric class of preferences, N, is a function of JV,_! similar to the one shown in Figure 1.2, only turned upside down. Like the Woodford example, this one in no way relies upon high rates of time preference. Producers' decisions about whether to introduce new products do not depend upon the rate at which profits are discounted, because profits are obtained only for a single period and the fixed cost is paid in that same period. The level of monopoly rents depends only upon consumers' elasticity of substitution between different products within a given period, not upon their preferences regarding present as opposed to future consumption. Neither of these examples can easily be parametrized for comparison with actual data, since each makes a number of very special assumptions in order to obtain equilibrium dynamics that can be described by a first-order nonlinear difference equation for a single state variable. This is, as of now, the only case in which a reasonably thorough analytical characterization of the types of pos-

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sible asymptotic dynamics is available. Further progress in evaluation of the empirical relevance of the endogenous cycle hypothesis will doubtless depend upon the use of numerical simulations to determine whether chaotic fluctuations occur for realistic parameter values in more complicated models.

5. Indeterminacy and Endogenous Fluctuations in Overlapping Generations Models Endogenous cycles and chaos again occur as equilibrium phenomena in the presence of complete and perfectly competitive intertemporal markets, also in the case of an economy made up of overlapping generations of finite lived consumers. In this case, as noted earlier, the equilibria involving perpetual deterministic fluctuations are only some members of a large set of rational expectations equilibria, which also includes equilibria converging to a stationary state. Nonetheless, this class of examples has been crucial for the development of modern interest in the endogenous cycle hypothesis, since it provided the first general equilibrium examples of the possibility of chaotic economic dynamics, through the work of a Benhabib and Day (1982) and Grandmont(1985). Consider the simple overlapping generations model treated by Gale (1973). The economy consists of a sequence of generations of two period lived consumers, each identical in number. There is a single perishable consumption good each period, of which each consumer has an endowment wt in the first period of life and w2 in the second period of life. Each consumer born in period t seeks to maximize U(cll,c2,+ i), where c,,is consumption in period t by consumers in the yth period of life, and where U is a concave function increasing in both arguments. Finally, there exists a single asset, fiat money, in constant supply M > 0, all of which is initially held by a group of consumers who are already in their final period of life in the first period of the model. Let p, be the price of the consumption good in terms of money in period t. A perfect foresight equilibrium price sequence {p,} must solve the difference equation / U\wx \

M M\ , w2 + = Pi

Pi+W

( U2 w, \

M M\ p, ,w2 + -^Pi

Pt+U

P«+i

(5.1)

The relation between M/p, and M/p,+ 1 required by equation (5.1) can be graphed as in Figure 1.3. When both first and second period consumption are normal goods, there will be a unique positive solution for M/p, for each positive value for M/p,+ l, that we may write M/p, = f(M/p,+ 1). However, as indicated by the figure, the function/need not be invertible. The sort of unimodal function shown occurs if preferences are such that desired saving in

30

MICHELE BOLDRIN & MICHAEL WOODFORD

Fig. 1.3

youth is a decreasing function of the expected real return on money, for high enough levels of that return. It is apparent that, when the unimodal map is steep enough (which Grandmont [1985] shows to require simply that the marginal utility of second period consumption fall sharply enough with increases in second period consumption, near the level of consumption that occurs in the stationary monetary equilibrium), the (backward) perfect foresight trajectories so traced can involve complex oscillations. As was first noted by Gale, deterministic cycles are possible. Cass, Okuno, and Zilcha (1979) showed that the deterministic cycles could be of arbitrary period, and Grandmont discusses in detail the order in which cycles of various periods occur as the map is made progressively steeper, applying the theory of unimodal maps set out in Collet and Eckmann (1980). All such cycles obviously represent possible equilibria of the forward perfect foresight dynamics as well; i.e., they can be extended indefinitely into the future as well as indefinitely into the past. Grandmont also discusses conditions under which the backward perfect

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foresight dynamics would exhibit topological chaos. This result is of less obvious significance for the forward perfect foresight dynamics, since the "chaotic" property of a trajectory can be denned only in terms of its asymptotic behavior as it is continued indefinitely, and the existence of such a property for trajectories extended backward indefinitely from a date T does not necessarily imply anything about the kind of trajectories that exist going forward from a date T. Furthermore, it is clear that not just chaotic but genuinely random perfect foresight trajectories do exist for the forward dynamics, albeit for reasons that do not require the use of the theory of nonlinear maps of the interval. For it is evident from Figure 1.3 that for many values of p,, there would be two different values of p,+ 1, expectation of either of which would result in a market clearing price of pt. So we can construct a (forward) perfect foresight equilibrium trajectory by starting at some arbitrary p0 > P_ [where M/P = sup p>0 /(M/p)], and proceeding recursively, for each value of t, choosing p,+ i, givenp,, so that (5.1) is satisfied, and also so thatp t+l > F . It is clear that this iteration can be continued forever. Let us suppose furthermore that the map/is so steep that/(M/P) w2), while the endowment pattern of type B consumers is exactly the reverse. Suppose furthermore that debt contracts are unenforceable, so that neither consumer type is able to borrow against future income. Consumers can save only by holding fiat money, which exists in a fixed positive supply M > 0. An equilibrium is possible in which the entire money supply is held at the end of each period by consumers of the type with endowment H^ if the price level sequence \p,} satisfies the following sequence of conditions: w,

= Pi I

I M\ u'[w2 + —) > \

Pi)

8«' w2 + \

8M'

I w, \

\-2-

Pi+i/ Pi+i

M\ p, -^-

Pi+\) Pi+i

(6.1)

(6.2)

Condition (6.1) shows how pt is determined in period t as a function of expectations regarding the value oip,+ l. This equation is of the form (5.1), and again cyclic or chaotic trajectories are among its solutions if the function u is sufficiently concave. It is necessary, however, in the present case also to check that the fluctuating solutions never violate the bound (6.2). This will necessarily be satisfied (given 8 < 1) in the case that {p,} fluctuates over a sufficiently small range, but large price fluctuations resulting in large variations in the marginal utility of consumption (as must occur in the case of chaotic dynamics) are consistent with (6.2) only if 8 is significantly less than one. If consumers smooth their consumption path by accumulating capital (rather than fiat money) in their high-endowment periods, and one interprets the endowments as being of labor rather than of the consumption good, a similar model of infinite lived consumers subject to borrowing constraints can mimic the dynamics of capital accumulation in the Diamond (1965) overlapping generations model. In this model, endogenous equilibrium cycles are possible in the case that aggregate savings are a backward-bending function of the expected real return. In this way, Bewley (1986) shows that endogenous cycles are possible. This example again depends upon a high rate of time discount (8 = .5). Woodford (1988a) shows that cash-in-advance constraints can result in dynamics similar to those of an overlapping generations model with short lifetimes even when all agents are infinite lived. Consider, for example, the cashin-advance model studied by Wilson (1979), in which the infinite lived representative consumer seeks to maximize - nt)

36

MICHELE BOLDRIN & MICHAEL WOODFORD

where c, is consumption in period t and n, is output supplied, 0 < 8 < 1, and U is increasing and concave in both arguments, subject to the sequence of budget constraints p,c, < M,

(6.3a)

M, + 1 = M, + p,(n, - c,)

(6.3b)

where M, is money balances carried into period t. In the case of a constant money supply M > 0, perfect foresight equilibria in which the cash-in-advance constraint (6.3a) always binds correspond to price sequences {p} satisfying the conditions (M

M\

IM

\P,

Ptj

\P,+ i

(M U\-,n \P,

M\

U1[-,n--\=WA

,n

M\

p, M2-

P,+ i/P,+ i

M M\ > [ / , - , « - P.I \Pt P.I

(6.4)

(6.5)

in all periods. Condition (6.4) indicates how the equilibrium price level p, is determined by expectations regarding pt+i. It is of the same general form as (5.1), and again cyclic or chaotic trajectories are among its solutions if U is a sufficiently concave function of its first argument. But, as in the case just discussed, it is also necessary to check that the fluctuating solutions never violate the bound (6.5). Again, this is necessarily satisfied (given 8 < 1) if {p} fluctuates over a sufficiently small range, but 8 much less than one is required in order for large fluctuations in the price level to be consistent with this condition. Similar results are also possible in representative consumer monetary models of the Sidrausky-Brock type, in which services from real money balances are directly an argument of the utility function. Indeed, the Lucas-Stokey model is known to be formally analogous to a particular case of such a model. Matsuyama (1989a, 1989b, 1989c) demonstrates the possibility of deterministic equilibrium cycles, chaotic equilibrium trajectories, and "sunspot" equilibria in a parametric class of models of this type. Woodford (1986) also shows that equilibrium dynamics similar to those occurring in an overlapping generations model with production can result in a model with infinite lived workers and entrepreneurs, from the existence of a cash-in-advance constraint upon workers' consumption purchases, together with a constraint of the sort discussed in section 4, according to which entrepreneurs must finance investment entirely out of the returns to capital. In this case, the equilibrium dynamics are very close to those characteristic of the Reichlin (1986) model, and as a result endogenous cycles are possible under circumstances similar to those discussed by Reichlin. In particular, in this case

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(unlike the Bewley example of a production economy), endogenous cycles can occur even when savings are an increasing function of the expected return. These examples show that it is not entirely fair to criticize the examples of endogenous fluctuations in economies with overlapping generations of twoperiod lived consumers as involving mechanisms that could generate cycles only at very low frequencies. The same sort of intertemporal substitution can also generate endogenous fluctuations in models where consumers are represented as infinite lived, and in which the period of the fluctuations bears no relation to any time scale relating to human biology. Indeed, the "periods" in the models with financial constraints just described must surely be interpreted as relatively short, so that if anything, these models should be criticized for only generating cycles of frequency too high to correspond to actual' 'business cycles." 15 Reichlin's overlapping generations model yields cycles that are approximately six "periods" in length, or three times the lifetime of consumers, in the life cycle interpretation of his model. This is a time scale much longer than that on which "business cycles" occur. But in the Woodford reinterpretation of this model, six "periods" is still too short for a "business cycle," under the most reasonable interpretation of the length of a ' 'period'' in a cashin-advance model.16 Hence, construction of examples that allow endogenous cycles at "business cycle" frequencies in the case of empirically realistic parameter specifications remains an important challenge for this line of research.17

References Adelman, I., and F. L. Adelman. 1959. "The Dynamic Properties of the Klein-Goldberger Model." Econometrica 27: 596—625. 15 The possibility in these models of equilibrium cycles of arbitrarily long periodicity, as discussed by Grandmont (1985), does not imply the possibility of "long" business cycles in the sense that would usually be given this term. In any model that is formally analogous to the Grandmont model, real money balances can never increase for more than one consecutive "period," so that fluctuations are necessarily high frequency even if they do not exactly repeat for a very long time. 16 However, Woodford (1988a) shows that if one considers stationary "sunspot" equilibria rather than only the limiting case of purely deterministic cycles, it is possible to obtain cycles that last for several quarters on average, regardless of how short' 'periods" are taken to be. 17 Perfect foresight equilibrium is also indeterminate in many other kinds of models with infinite lived agents in the case of various market imperfections that need not involve financial constraints. In many such cases, endogenous cycles or "chaos" are among the possible types of equilibrium dynamics [Schleifer (1986), Murphy et al. (1988), Diamond and Fudenberg (1989), Howitt and McAfee (1988), Hammour (1988)]. Perhaps some of these mechanisms can produce cycles at ' 'business cycle" frequencies, although none of the papers listed discuss this. The Deneckere and Judd (1986) cycles could be at "business cycle" frequencies, if firms that introduce new products can exploit their monopoly for something like a two-year period.

38

MICHELE BOLDRIN & MICHAEL WOODFORD

Aiyagari, S. R. 1989. "Can There Be Short-Period Deterministic Cycles When People Are Long Lived?" Quarterly Journal of Economics CIV: 163-85. Azariadis, Costas. 1983. "Self-Fulfilling Prophecies." Journal of Economic Theory 25: 380-96. Azariadis, Costas, and Roger Guesnerie. 1986. "Sunspots and Cycles." Review of Economic Studies 53: 725-36. Baumol, W., and R. Quandt. 1985. "Chaos Models and Their Implications for Forecasting." Eastern Economic Journal 11: 3-15. Becker, R. A., and C. Foias. 1987. "A Characterization of Ramsey Equilibrium." Journal of Economic Theory 41: 173-84. Becker, R. A., and M. Majumdar. 1989. "Optimality and Decentralization in Infinite Horizon Economies." In Joan Robinson and Modern Economic Theory, ed. G. R. Feiwel. London: MacMillan Press. Benhabib, J., and R. Day. 1980. "Erratic Accumulation." Economic Letters 6: 113— 17. . 1981. "Rational Choice and Erratic Behavior." Review of Economic Studies XLVIII: 459-71. . 1982. "A Characterization of Erratic Dynamics in the Overlapping Generations Model," Journal of Economic Dynamics and Control 4: 37-55. Benhabib, J., and G. Laroque. 1988. "On Competitive Cycles in Productive Economies." Journal of Economic Theory 45: 145-70. Benhabib, J., and K. Nishimura. 1979. "The Hopf Bifurcation and the Existence and Stability of Closed Orbits in Multisector Models of Optimal Economic Growth." Journal of Economic Theory 21: 421—44. . 1985. "Competitive Equilibrium Cycles." Journal of Economic Theory 35: 284-306. . 1986. "Endogenous Fluctuations in the Barro-Becker Theory of Fertility." New York University: mimeo, October. Benhabib, J., and A. Rustichini. 1990. "Equilibrium Cycling with Small Discounting." Journal of Economic Theory 52: 423-432. Bewley, T. 1980. "The Optimum Quantity of Money." In Models of Monetary Economics, eds. J. H. Kareken and N. Wallace. Minneapolis: Federal Reserve Bank of Minneapolis. . 1982. "An Integration of Equilibrium Theory and Turnpike Theory." Journal of Mathematical Economics 10: 233-68. . 1986. "Dynamic Implications of the Form of the Budget Constraint." In Models of Economic Dynamics, ed. H. F. Sonnenschein. Berlin, New York: Springer-Verlag. Blatt, J. M. 1978. "On the Econometric Approach to Business-Cycle Analysis." Oxford Economic Papers 30: 292-300. . 1983. Dynamic Economic Systems. Armonk, NY: M. G. Sharpe. Boldrin, M. 1983. "Applying Bifurcation Theory: Some Simple Results on the Keynesian Business Cycle." Note di Lavoro #8403. Univ. di Venezia: Dip. di Scienze Economiche. . 1986. "Paths of Optimal Accumulation in Two-Sector Models." In Economic Complexity: Chaos, Sunspots, Bubbles and Nonlinearity, eds. W. Barnett, J. Geweke, and K. Shell. Cambridge: Cambridge University Press.

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Boldrin, M., and R. Deneckere. 1987. "Simple Macroeconomic Models with a Very Complicated Behavior." University of Chicago and Northwestern University, mimeo. Boldrin, M., and L. Montrucchio. 1984. "The Emergence of Dynamic Complexities in Models of Optimal Growth: The Role of Impatience." Working paper # 7 , Rochester Center for Economic Research: University of Rochester. . 1986a. "Cyclic and Chaotic Behavior in Intertemporal Optimization Models." Mathematical Modelling 8: 627-700. . 1986b. "On the Indeterminacy of Capital Accumulation Paths." Journal of Economic Theory 40: 26-39. . 1989. Dynamic Complexities of Intertemporal Competitive Equilibria. London, New York: Oxford University Press. Forthcoming. Brock, W. 1988. "Nonlinearities and Complex Dynamics in Economics and Finance." In The Economy as an Evolving Complex System, eds. P. Anderson, K. Arrow, and D. Pines, NY: Addison Wesley. Cass, D. 1965. "Optimum Growth in an Aggregative Model of Capital Accumulation." Review of Economic Studies 32: 233-40. Cass, D., M. Okuno, and I. Zilcha. 1979. "The Role of Money in Supporting the Pareto Optimality of Competitive Equilibrium in Consumption-Loan Models." Journal of Economic Theory 20: 41-80. Cass, D., andK. Shell, eds. 1976. The Hamiltonian Approach to Dynamic Economics. New York: Academic Press. Chang, W. W., and D. J. Smith. 1971. "The Existence and Persistence of Cycles in a Nonlinear Model: Kaldor's 1940 Model Re-examined." Review of Economic Studies 38: 3 7 ^ 4 . Chow, S. N., and J. K. Hale .1982. Methods ofBifurcation Theory. Berlin, New York: Springer-Verlag. Collet, P., and J. P. Eckmann. 1980. Iterated Maps on the Interval as Dynamical Systems. Boston: Birkhauser. Cugno, F., and L. Montrucchio. 1982a. "Stability and Instability in a Two-Dimensional Dynamical System: A Mathematical Approach to Kaldor's Theory of the Trade Cycle." In New Quantitative Techniques for Economics, ed. G. P. Szego. New York: Academic Press. . 1982b. "Cyclical Growth and Inflation: A Qualitative Approach to Goodwin's Model With Money Prices." Economic Notes 3: 93-107. . 1983. "Disequilibrium Dynamics in a Multidimensional Macroeconomic Model: A Bifurcational Approach." Ricerche Economiche XXXVII: 3-21. Dana, R. A., and P. Malgrange. 1984. "The Dynamics of a Discrete Version of a Growth Cycle Model." In Analyzing the Structure of Econometric Models, ed. J. P. Ancot. Amsterdam: M. Nijhoff. Dana, R. A., and L. Montrucchio. 1986. "Dynamic Complexity in Duopoly Games." Journal of Economic Theory 40: 40-56. Day, R. 1982. "Irregular Growth Cycles." American Economic Review 72(3): 40614. . 1983. "The Emergence of Chaos From Classical Economic Growth." Quarterly Journal of Economics 98: 201-13.

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Day, R., and J. W. Shafer. 1985. "Keynesian Chaos." Journal ofMacroeconomics 1: 277-95. . 1987. "Ergodic Fluctuations in Deterministic Economic Models." Journal of Economic Behavior and Organization 8: 339-61. Dechert, W. D. 1984. "Does Optimal Growth Preclude Chaos? A Theorem on Monotonicity." Zeitschrift fur Nationalokonomie 44: 57-61. Deneckere, R., and K. L. Judd. 1986. "Cyclical and Chaotic Behavior in a Dynamic Equilibrium Model With Implications for Fiscal Policy." Northwestern University, mimeo, June. Deneckere, R., and S. Pelikan. 1986. "Competitive Chaos." Journal of Economic Theory 40: 13-25. Devaney, R. L. 1986. An Introduction to Chaotic Dynamical Systems. Menlo Park, CA: Benjamin Cummings Publishing Co. Diamond, P. 1965. "National Debt in a Neoclassical Growth Model." American Economic Review 55: 1026—50. Diamond, P., and D. Fudenberg. 1989. "An Example of Rational Expectations Business Cycles in Search Equilibrium." Journal of Political Economy 97: 606-19. Drazen, A. 1988. "Self-Fulfilling Optimism in a Trade-Friction Model of the Business Cycle." American Economic Review Papers and Proceedings 78: 369—72. Eckmann, J. P., and D. Ruelle. 1985. "Ergodic Theory of Chaos and Strange Attractors." Reviews of Modern Physics 57 (3) Pt. I: 617-56. Farmer, R. A. 1986. "Deficits and Cycles." Journal of Economic Theory 40: 77-88. Farmer, R. A., and M. Woodford. 1984. "Self-Fulfilling Prophecies and the Business Cycle." CARESS W.P. #84-12, University of Pennsylvania, April. Feichtinger, G., and G. Sorger. 1986. "Optimal Oscillations in Control Models: How Can Constant Demand Lead to Cyclical Production?" Operations Research Letters 5:277-81. Feichtinger, G., R. Harte, G. Sorger, and A. Steindl. "On the Optimality of Cyclical Employment Policies." Journal of Economic Dynamics and Control 10: 457-66. Gale, D. 1973. "Pure Exchange Equilibrium of Dynamic Economic Models." Journal of Economic Theory 6: 12—36.

Goodwin, R. M. 1982. Essays in Economic Dynamics. London: Macmillan Press Ltd. Goodwin, R. M., and L. Punzo. 1988. The Dynamics of a Capitalist Economy. Boulder, CO: Westview Press. J. M. Grandmont, 1985. "On Endogenous Competitive Business Cycles." Econometrica 53: 995-1046. Grandmont, J. M., ed. 1987. Nonlinear Economic Dynamics. New York: Academic Press. Grandmont, J. M., and G. Laroque. 1986. "Stability of Cycles and Expectations." Journal of Economic Theory 40: 138-51. Guckenheimer, J., and P. Holmes. 1983. Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields. New York Springer-Verlag. Hammour, M. 1988. "Increasing Returns and Endogenous Business Cycles." M.I.T., mimeo, October. Hernandez, A. D. 1988. "The Dynamics of Competitive Equilibrium Allocations with Borrowing Constraints." chap. Ill of a Ph.D. dissertation. Rochester, NY: University of Rochester.

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Howitt, P., andR. P. McAfee, 1988. "Animal Spirits." mimeo, University of Western Ontario. Judd, K. 1985. "On Performance of Patents." Econometrica 53: 567-86. Jullien, B. 1988. "Competitive Business Cycles in an Overlapping Generations Economy With Productive Investment." Journal of Economic Theory 46: 45-65. Kehoe, T., D. Levine, A. Mas-Colell, andM. Woodford. 1986. "Gross Substitutability in Large-Square Economies." mimeo, May. Kydland, F., and E. C. Prescott. 1982. "Time to Build and Aggregate Fluctuations." Econometrica 50: 1345-70. Lasota, A., and J. A. Yorke. 1973. "On the Existence of Invariant Measures for Piecewise Monotonic Transformation." Transactions of the American Mathematical

Society 186:481-88.

Lasota, A., and M. C. Mackey. 1985. Probabilistic Properties of Deterministic Systems. Cambridge: Cambridge University Press. Li, T. Y., and J. A. Yorke. 1975. "Period Three Implies Chaos." American Mathematical Monthly 82: 985-92. Lorenz, H. W. 1989. Nonlinear Dynamical Economics and Chaotic Motions. Berlin, NY: Springer-Verlag. Magill, M.J.P. 1979. "The Origin of Cyclical Motions in Dynamic Economic Models." Journal of Economic Dynamics and Control 1: 199—218. Magill, M.J.P., and J. A. Scheinkman. 1979. "Stability of Regular Equilibria and the Correspondence Principle for Symmetric Variational Problems." International Economic Review 20: 297-313. Matsuyama, K. 1989a. "Endogenous Price Fluctuations in an Optimizing Model of a Monetary Economy." CMSEMS Disc, paper #825, Northwestern University, March. . 1989b. "Serial Correlation of Sunspot Equilibria (Rational Bubbles) in Two Popular Models of Monetary Economies." CMSEMS Disc, paper #827, Northwestern University, March. . 1989c. "Comparative Statics and Complicated Topological Structure of the Set of Equilibrium Prices." Unpublished manuscript, Northwestern University, July. McKenzie, L. W. 1976. "Turnpike Theory." Econometrica 44: 841-55. . 1986. "Optimal Economic Growth, Turnpike Theorems and Comparative Dynamics." In Handbook of Mathematical Economics, vol. Ill, eds. K. J. Arrow and M. D. Intriligator. Amsterdam, NY: North-Holland. . 1987. "Turnpike Theory." In The New Palgrave. NY: Stockton Press. Medio, A. 1984. "Synergetics and Dynamic Economic Models." In. Non-Linear Models of Fluctuating Growth, eds. R. M. Goodwin, M. Kruger, and A. Vercelli. Berlin and NY: Springer-Verlag. Montrucchio, L. 1982. "Some Mathematical Aspects of the Political Business Cycle." Journal of Optimization Theory and Applications 36: 251-75. . 1986. "Optimal Decisions Over Time and Strange Attractors: An Analysis by the Bellman Principle." Mathematical Modelling 7: 341-52. . 1987. "Dynamical Systems That Solve Continuous Time Infinite Horizon Optimization Problems: Anything Goes." mimeo, Politecnico di Torino, October.

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Neumann, D., T. O'Brien, J. Hoag, and H. Kim. 1988. "Policy Functions for Capital Accumulation Paths." Journal of Economic Theory 46: 205-14. Pohjola, M. T. 1981. "Stable, Cyclic and Chaotic Growth: The Dynamics of a Discrete-Time Version of Goodwin's Growth Cycle Model." Zeitschrift fur Nationalokonomie 41: 27-38. Ramsey, J. B., C. L. Sayers, and P. Rothman. 1988. "The Statistical Properties of Dimension Calculations Using Small Data Sets: Some Economic Applications." C. V. Starr Center w.p. #88-10, New York University. Ramsey, J. B., and H. J. Yuan. 1989. "Bias and Error Bars in Dimension Calculations and Their Evaluations in Some Simple Models." Physics Letters A 134: 287-97. Rand, D. 1978. "Exotic Phenomena in Games and Duopoly Models." Journal of Mathematical Economics 5: 173-84. Reichlin, P. 1986. "Equilibrium Cycles in an Overlapping Generations Economy With Production." Journal of Economic Theory 40: 89-102. . 1987. "Endogenous Fluctuations in a Two-Sector Overlapping Generations Economy." Working paper No. 87/264, Florence: European University Institute. . 1989. "Endogenous Cycles with Long Lived Agents." unpublished manuscript, Univ. di Roma, March. Rockafellar, T. R. 1976. "Saddle Points of Hamiltonian Systems in Convex Lagrange Problems Having a Non-Zero Discount Rate." Journal of Economic Theory 12: 71— 113. Ruelle, D. 1989. Elements of Differentiate Dynamics and Bifurcation Theory. New York: Academic Press. Rustichini, A. 1983. "Equilibrium Points and 'Strange Trajectories', in Keynesian Dynamic Models." Economic Notes 3: 161-79. Sakai, H., and H. Tokumaru. 1980. "Autocorrelations of a Certain Chaos." IEEE Transactions in Acoustics, Speech, and Signal Processing 28: 588—90. Sargent, T. J. 1987. DynamicMacroeconomic Theory. Cambridge: Harvard University Press. Sarkovskii, A. N. 1964. "Coexistence of Cycles of a Continuous Map of the Line Into Itself." Ukrain. Matem. Zhur. 16: 61-71. Scheinkman, J. A. 1976. "On Optimal Steady State of n-Sector Growth Models When Utility is Discounted." Journal of Economic Theory 12: 11-30. . 1984. "General Equilibrium Models of Economic Fluctuations: A Survey." mimeo, Dept. of Economics, University of Chicago, September. . 1990. "Nonlinearities in Economic Dynamics." Economic Journal. Forthcoming. Scheinkman, J. A., and L. Weiss. 1986. "Borrowing Constraints and Aggregate Economic Activity." Econometrica 54: 23-46. Shell, K. 1977. "Monnaie et Allocation Intertemporelle." mimeo, Seminaire de E. Malinvaud, Paris. Shleifer, A. 1986. "Implementation Cycles." Journal of Political Economy 94: 116390. Sims, C. A. 1986. "Comments." In Models of Economic Dynamics, ed. H. F. Sonnenschein. New York: Springer-Verlag. Solow, R. 1956. "A Contribution to the Theory of Economic Growth." Quarterly Journal of Economics 70: 65-94.

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Stokey, N., and R. E. Lucas, Jr. 1989. Recursive Methods in Economic Dynamics. Cambridge: Harvard University Press. Stutzer, M. 1980. "Chaotic Dynamics and Bifurcation in a Macro Model." Journal of Economic Dynamics and Control 1: 377-93. Torre, V. 1977. "Existence of Limit Cycles and Control in Complete Keynesian Systems by Theory of Bifurcations." Econometrica 45: 1457-66. Whitesell, W. 1986. "Endogenous Cycles With Uncertain Lifespans in Continuous Time." Economic Letters 22: 153-58. Wilson, C. 1979. "An Infinite Horizon Model With Money." In General Equilibrium Growth and Trade, Essays in Honor of Lionel McKenzie, eds. J. R. Green and J. A. Scheinkman. New York: Academic Press. Woodford, M. 1984. "Indeterminacy of Equilibrium in the Overlapping Generations Model: A Survey." unpublished manuscript, Columbia University, May. . 1986. "Stationary Sunspot Equilibria in a Finance Constrained Economy." Journal of Economic Theory 40: 128—37. . 1988a. "Expectations, Finance Constraints, and Aggregate Instability." In Finance Constraints, Expectations, and Macroeconomics, eds. M. Kohn and S. C. Tsiang. New York: Oxford University Press. . 1988b. "Imperfect Financial Intermediation and Complex Dynamics." In Economic Complexity: Chaos, Sunspots, Bubbles and Nonlinearity, eds. W. Barnett, J. Geweke, and K. Shell. Cambridge: Cambridge University Press. Yano, M. 1984. "The Turnpike of Dynamic General Equilibrium Paths and Its Insensitivity to Initial Conditions." Journal of Mathematical Economics 13: 235-54.

Periodic and Aperiodic Behaviour in Discrete One-Dimensional Dynamical Systems JEAN-MICHEL GRANDMONT*

1. Introduction The theory of one-dimensional non-linear difference equations underwent considerable progress in recent years, as the result of the efforts of theorists from several fields—in particular from physics—to get a better understanding, by making use of the notion of the "bifurcation" of a dynamical system, of the appearance of cycles and of the transition to aperiodic or "chaotic" behaviour in physical, biological, or ecological systems. These new developments seem to be potentially very useful for the study of periodic and aperiodic phenomena in economics. Parts of this theory have indeed been used already in economic or game theory by Benhabib and Day (1981, 1982), Dana and Malgrange (1984), Day (1982, 1983), Grandmont (1985), Jensen and Urban (1982), Rand (1978). The aim of this paper is to present some of these new developments in a compact form which will be, it is hoped, useable by economic theorists. The emphasis will be on the mathematical results of the theory, rather than on its possible applications.' Our basic reference will be Collet and Eckmann's book (1980)—thereafter denoted "CE." In order to simplify the presentation, we shall use in a few places stronger assumptions than in CE's book, which means that the reader interested in the more general (but more complicated) case and who wishes to look for complements, will have to go back to their book. The definitions and the statements of the results will be self-contained. However, in the proofs of * Originally published in Contributions to Mathematical Economics, ed. W. Hildenbrand and A. Mas Collel [Copyright © 1986, pp. 227-65 by Elsevier Science Publishers B.V. (North-Holland)]. The financial support of the U.S. Office of Naval Research under contract ONR-R0001479-C-0685 at the IMSSS at Stanford University, of the French Commissariat General du Plan and the University of Lausanne is gratefully acknowledged. I wish to thank very much Rose-Anne Dana and Pierre Malgrange, who introduced me to the mathematics of the subject. I also had very useful conversations with Philippe Aghion, Pierre Collet, John Geanakopoulos, and Dominique Levy. 1 For applications to economics, see the references cited above. For an excellent review of the applications in other fields, see May (1976).

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45

a few facts, we shall use freely the concepts introduced by CE, but we shall indicate where to find the appropriate definitions in that book.2

2. One-dimensional non-linear difference equations We are concerned thereafter with the difference equation xj+l = /(x,), in which/is a continuous function that maps the interval [a, b] into itself. The object of the theory is the study of the existence (and the stability) of periodic solutions of this difference equation. To this effect, one defines recursively the iterates o f / b y / 0 (x) = x for all x (f° is the identity map),/ 1 = / a n d / ' = f° / ' " ' . The orbit ofx is then the set {x, f(x),f2(x), . . .}, which is composed of all iterates ofx. We say that A: is a periodic point of/ (with period k) if (1) x is a fixed point of/*, i.e., x = fk(x), and (2) k is the smallest integer having this property, i.e., x + f'(x) for all i = 1, . . . , k — 1. The corresponding cycle (or periodic orbit) is the set {x,, . . . , xk} of all the iterates x, = / ' ~ ' (x) ofx, for i = 1, . . . , k. Of course, if x is a periodic point of/with period k, any element x, of the corresponding cycle { * , , . . . , xk} is also a periodic point of /, with the same period. Let us first consider the case depicted in Figures 2.1a and 2. lb. The m a p / has then a unique fixed point x, which is globally stable. The dynamics induced by/are particularly simple, which is presumably why economists focus attention typically on this case. Consider now Figure 2.2a, in which/is assumed to be continuously differentiable. The map/has then two fixed points a and x, but they are both unstable since/'(a) > 1 and/'(x) < — 1. Economists tend usually to avoid such a configuration. It gives rise, however, to an interesting phenomenon, since there exists then a cycle of period 2. According to the definition above, a cycle of period 2 is identified with a fixed point of/2 that differs from a and from x. Of course a and x are fixed points of/2, and by applying the chain rule of differentiation, the derivative of/2, i.e., D/2(x), evaluated atx = a or (x = x) is equal to \f'(a)]2 (or \f'(x)]2) and is thus greater than 1. By continuity,/ 2 has a fixed point in (a, x). One can pursue further this matter by looking at the case described in Figure 2.2b, in which / i s assumed again continuously differentiable. As in the previous case,/has a cycle of period 2. The interesting point is that/has also a cycle of period 3. Such a cycle is characterized by a fixed point x of/ 3 that differs from a and fromx. Now, a is clearly afixedpoint of/3, and by applying again the chain rule of differentiation. 2 More information on the behaviour of non-linear dynamical systems in more than one dimension—which is much richer but also less understood—is provided by the recent book by Guckenheimer and Holmes (1983).

Fig. 2.1a

Fig. 2.1b

Fig. 2.2a

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D/ 3 (a) = [/'(a)] 3 > 1. Figure 2.2b postulates the existence of x* < x such that/3 (x*) < x*. Thus by continuity,/ 3 has a fixed point in (a, x*). The central feature of the example is that/displays an important "hump" so that/(jc*) is "large" and/ 2 (x*) is "small."

3. Sarkovskii's theorem The above arguments imply that cycles of different periods should typically coexist. The following beautiful and striking result, which is due to Sarkovskii (1964) [see also Stefan (1977)], provides very useful information about this coexistence. Theorem 1 (Sarkovskii).

Consider the ordering of the integers

3>5>7- • • >2-3>2-5>2-7>- • • > 2" • 3 > 2" • 5 > 2" • 1 > • • • > • • • > 2m>

• •

- > 8 > 4 > 2 > 1 .

That is, first the odd integers greater than or equal to 3 forward, then the powers of two times these odd integers, and then the powers of 2 backward. If/is continuous and has a cycle with period k, then it has a cycle of period k' for every integer k' such that k> k'. Proof. This is Sarkovskii's theorem. Note that the theorem is stated in CE, Theorem II.3.10, p. 91, only for a "unimodal" map, i.e., a map that is first increasing, reaches its maximum at some point** of (a, b) and then decreases. Sarkovskii's theorem, however, is valid for an arbitrary continuous map. • Sarkovskii's theorem implies that the map represented in Figure 2.2b, which looks rather simple, may yield very tricky dynamics since it possesses an infinity of different cycles! In order to assess more precisely this issue, however, one needs to study which of these cycles, if any, is stable. Indeed, Sarkovskii's theorem does not say anything about stability.

4. Stable cycles It is important to know how many stable cycles—if any—the map/possesses. It is only recently that a real breakthrough was achieved on this matter by

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Singer (1978), who discovered that a unimodal map with a negative "Schwarzian derivative" could have at most one stable cycle.3 Let us first define stability. Given the map/from [a, b] into itself, consider a periodic orbit {x,, . . . , xk}. Since x, is a fixed point of/*, we may say that this periodic orbit is (locally) stable if there exists an open neighborhood U of x, such that for every x in U,fkj(x) stays in U for ally s l and lim^^/^Xx) = Xi. When/is continuous, this implies that/^(/'^'(•*)) converges to x, as well for every i = 2, . . . , k. If/ is continuously differentiable, this means that the derivative of/* at xx has a modulus less than 1, i.e., |D/*(x,)| < 1. Of course, in order to make any sense, this definition should not depend upon the point chosen on the periodic orbit. As a matter of fact, we have by the chain rule of differentiation D/*

(*,)

=

/'(**) D / * - ' (Xx) =

• • • = f'(xk)

• • •/'(*,)

When/is continuously differentiable, we may therefore say that the cycle {x,, . . . , x j is stable if |D/*(x,)| < 1. The cycle will be said to be weakly stable if iD/^x^l Si 1 (this definition allows for "one-sided" stability only).4 Finally, it will be said to be superstable if Dfk(xx) = 0. This means that a critical point of/, i.e., a point x such that/'(x) = 0, belongs to the periodic orbit. We define next the notion of a Schwarzian derivative. Assume that/is thrice continuously differentiable. The Schwarzian derivative of / at x, denoted S/(x), is defined by fix)

2 [_/'(*)_

whenever/'(x) + 0. Direct computation shows that S/ = - 2 |/'|" 2 D 2 [ | / ' | - " 2 ] . So the condition that' '/has a negative Schwarzian derivative'' [S/< 0 at every x such that/'(x) + 0] means that [/'|~1/2 is convex on every interval of monotony of/. It will be satisfied in particular if \f'\ (or log \f'\) is concave on such intervals. But these sufficient conditions are by no means necessary. Finally the reader will note that the concavity of/is neither necessary nor sufficient to guarantee S / < 0. The following result gives sufficient conditons on the map/that guarantees the existence of at most one weakly stable cycle. In order to simplify the ex3

Singer's result is actually more general, since he showed that the number of stable cycles of an arbitrary map with a negative Schwarzian derivative is bounded above by the number of its critical points. 4 CE use "stable" to denote what we call "weakly stable."

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position, we shall concentrate on maps that are unimodal, from now on. More precisely, we say that/is unimodal if 1. / i s continuous; 2. there exists x* in (a, b) such that/is increasing on [a, x*] — i.e.,/(jc) > fix') for all x, x' in [a, x*] such that x > x' — and decreasing on [x*, b\\ 3. /(**) = b.

We shall say that/is C'-unimodal if in addition 4. /is once continuously differentiable and/'(*) + 0 when* + x*.5 Note that the assumption/(x*) = b involves no loss of generality as soon as/(x*) > x*, since we can then restrict ourselves to the invariant interval [a, f(x*)]. On the other hand, the case/(x*) ^ x* does not generate interesting phenomena, since it is similar to the case described in Figure 2.1a. Consider next the following conditions: 5.1. / i s C'-unimodal. 5.2. / i s thrice continuously differentiable. 5.3. S/(x) < 0 for all x in [a, b], x + x*. Then we have: Theorem 2. Assume that/ specifies S. 1, S. 2, and S. 3, f(x) > x for all x in (a, x*), and/'(a) > 1 whenever/(a) = a. Then: 1. The map/has at most one weakly stable periodic orbit. This periodic orbit lies in the interval \f(b), b]. 2. If/has a weakly stable periodic orbit, it attracts the critical point x*, that is, it coincides with the set of accumulation points of the sequence (/?(**)).

Proof. We may note incidentally that under S.I, S.2, and S.3, one has/O) > x for all x in (a, x*) whenever/' (a) > 1. This follows from the fact that since S/< 0 , / ' cannot have a positive local minimum on that interval (see step 3 of the proof of Theorem II.4.1 in CE, p. 97). Indeed, if there existed x in {a, x*) such that/(x) Si x, then by the mean value theorem there would be yu y2, with a < yt ^ x ^ y2 < x* such t h a t / ' ^ ) 2= 1 x for all x in (a, x*) implies that i. /maps the interval [fib), b] into itself [onto if and only if fib) s x*]; ii. for every JC in (a,/(fc)), there exists^' such that/(;c) E. [fib), b]. 5

CE require that a = —l,x* = 0,b= l. However, none of their arguments depend upon that specification and they are valid for the case at hand. We shall use that fact repeatedly without any further explicit reference.

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This follows from elementary considerations that are left to the reader. This shows that all periodic orbits—with the possible exception of an unstable fixed point of/at x = a—must lie in \f(b), b]. In particular, any weakly stable cycle belongs to that interval. Corollary II.4.2 in CE implies therefore that the statements of Theorem 2 are valid provided that/satisfies the additional condition: S.4. /maps \f(b), b] onto itself. However, a closer look at CE's proof of this corollary shows that it is still valid if S. 4 is replaced by the weaker condition: S.4'. /maps the interval \f(b), b] into itself. But we have seen that this condition was implied by the assumptions of Theorem 2. The proof is complete. • We shall note for further reference: S.4". (i)/(x) > x for all x in (a, x*), and (ii)/'(a) > 1 when/(a) = a. As we have seen, if/is unimodal, then S.4" implies S.4', while it implies S.4 if and only iff(b) S= x*. The maps represented in Figures 2.2a and 2.2b are unimodal and satisfy S.4". The foregoing result is important, not only because it gives conditions ensuring the existence of at most one weakly stable cycle, but also because it provides us with a nice experimental way to discover it (whenever it exists) in situations in which an analytical treatment would be untractable, as for instance in Figure 2.2b. It suffices indeed to iterate the critical point x* on a computer, to check whether or not the iterates converge and to verify that the limit cycle is indeed weakly stable. These operations are easy to perform with the help of modern computers. Of course, since the iterations must be stopped after a finite time in practice, this experimental way of proceeding will be unable to distinguish between the absence of any weakly stable periodic orbit and the presence of a weakly stable cycle that has a long period or that is only weakly attracting. One can read Theorem 2 from the opposite viewpoint, since it states that a given map/that satisfies S.I, S.2, S.3, and S.4", for which the iterates/>'(**) do not converge or converge to an unstable cycle, has no weakly stable periodic orbit. It is clear that such maps do exist [think of the case in which/(a) = a and/(ft) = a for instance], and they are good candidates for portraying ' 'turbulent'' behaviour, since for any initial point that does not belong to a periodic orbit, the associated trajectory will be aperiodic even when one waits very long. The next statement provides a condition involving the trajectory of the critical point x* off only, that ensures the existence of a (unique) weakly stable

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JEAN-MICHEL GRANDMONT

cycle. To this effect, we introduce some notation. Given a unimodal map/, for every x in [a, b], the extended itinerary of x describes how the iterates/>'(*) behave qualitatively, i.e., whether or not they fall on the right or on the left of the critical point x*. More precisely, this extended itinerary IE(x) is an infinite sequence of R's, of L's, and of C's obeying the following rule. If [IE(x)]j denotes thejth element of IE(x) forj = 0 , 1 , . . . , then f/E(x)]y = R if/(x) > x*, [IE(x)]j = C if fi(x) = x* and [IE(x)]j = L if/(x) < x*. We shall say that IB(x) is periodic with (primitive) period k if [IE(x)]j+k = [IE(x)]j for ally and if k is the smallest integer having this property. Proposition 3.

Assume that/satisfies S.I, S.2, S.3, S.4", and

S.5. f'(x*) < 0. Then/has a (unique) weakly stable cycle P if and only if the extended itinerary of the endpoint b, i.e., IE(b), is periodic. If the period of IE(b) is k, the period of Pis k or 2k. Proof. Assume that IE(b) has period k. Iff(b) = S x*, then S.4 is satisfied, and from the "if" part of CE, Proposition II.6.2, / h a s a weakly stable cycle in \f(b), b]. Iff(b) > x*, then/(fc) S/(x*) for all; S l. But it is then easy to verify that the restriction of/to (f(b), b) has a sink in the sense of CE, p. 107. Therefore from CE, Lemma II.5.1,/has a weakly stable periodic orbit in \f(b), b] in that case too (one can alternatively prove directly that/ 1 has a weakly stable fixed point \f(b), b], see the proof of Proposition 4). In all cases the weakly stable cycle is unique from Theorem 2. Finally, the fact that its period is k or 2k is an immediate consequence of CE, Lemma II.3.2. Assume conversely that/has a (unique) weakly stable cycle P of period k. It must lie in \f(b), b]. We wish to apply the "only if" part of CE, Proposition II.6.2. A close look at their argument shows that their result is valid if S.4 is replaced by S.4'—and thus under S.4"—but that it is correct only when the rightmost point of P, say x, satisfies x & x*—which is the case under S. 1, S. 2, S.3, and S. 4', if and only if k S 2 or when the periodic orbit is a fixed point in (x*, b). The "only if" part of CE, Proposition II.6.2, is not correct however under their assumptions if P is a weakly stable fixed point x of/such that x < x* (counterexamples are provided by making symmetric the cases 1^4 of Figure II.8 in CE, p. 102).6 The latter circumstance is ruled out however under S.4", so the "only if" part of CE, Proposition II.6.2, is valid under our assumptions. Thus IE(b) is periodic, and from CE, Lemma II.3.2, its period is k

ork/2.

•

The concept of (weak) stability that we have used is only local. It is thus important to know how large is the basin of attraction of a given weakly stable 6

These facts have been confirmed to me privately by Pierre Collet.

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cycle. The next result states that under the conditions of Proposition 3, if there exists a weakly stable periodic orbit, which is then unique, the set of points that are not attracted to it is "exceptional." Proposition 4. Assume that/satisfies S.I, S.2, S.3, S.4", and S.5 and that it has a weakly stable cycle P. Let E be the set of points x in [a,b] such that f(x) does not tend to P. Then E has Lebesgue measure 0. Proof. Iff(b) ^ x*, S.4 is satisfied. Then from CE, Proposition II.5.7, the set of Ef of points in [f(b), b] that are not attracted to the weakly stable periodic orbit P, has Lebesgue measure 0.7 Let E'f be the set of points x in [a, f{b)) such that fj(x) G Effor some/ Since/is increasing on [a, x*), the Lebesgue measure of E'f is also 0. The set of points of [a, b] that are not attracted to P is Ef U E'f, to which one must add the endpoint a whenever/(a) = a, which shows the result in that case. The case in which x* 0 for all x in A, x * x*. Furthermore,/ 2 has a negative Schwarzian derivative on (x*, b] and has finitely many fixed points in [x*, b]—see steps 2 and 4 of the proof of Theorem II. 4.1 in CE, pp. 97-98. Consider first the case in which the period of P is 1. Then since/^x*) = f(b) > x*,f(b) < b and Df{x) S 1, one must have/^x) > x for all x in [x*, x) and/2(x) < x for all x in (x, b], otherwise there would be another weakly stable periodic orbit (of period 2). Thus/^x), and thus/^x), converges to x as j tends to + °° for all x in [x*, b] [see Figure 2.3a]. The other case in which the period of P is 2 is dealt with similarly. Let xx and x2 be the two points of P. They satisfy x* < X; < x < x2 < b. From the uniqueness of the weakly stable cycle, we have D/^x) > 1 and in fact/2(x) > x for all x in (x*, x,) or (x, x2), and/2(x) < x for every x in (x,, x) or (x2, b) [see Figure 2.3b], Thus/^Xx), and thus/(x), converges to P as j tends to + °° for all x in [x*, b] except x = x. Thus if the period of P is 1, it attracts the whole interval [a, b], except a iff (a) = a. If the period of P is 2, it attracts again the whole interval [a, b], with the exception of the preimages of x, i.e., of all points x of [a, x*) such that/(x) = x for s o m e / and of the endpoint a when/(a) = a. In the two cases, the exceptional set is finite or countable, which completes the proof. • 7

To be precise, Proposition II.5.7 in CE is correct under assumptions S.I, S.2, S.3, S.4, and

S.5, provided that f{b) is not a fixed point of f satisfying f (f(b)) = 1 (this fact has also been confirmed to me privately by Pierre Collet). This circumstance is however ruled out by S.4". We may therefore apply their Proposition II.5.7 when/(ft) 3= x*.

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JEAN-MICHEL GRANDMONT

f2(b)

Fig. 2.3a

Remark. Proposition 4 shows that some claims according to which ' 'period 3 implies chaos" are not always warranted. For instance, a consequence of the results of Li and Yorke (1975) is that if/is unimodal and if there exists a cycle of period 3, then there is an uncountable set Sin [a, b] and an e > 0 such that for every x and y in S, lim sup

- fKy) I ^ e,

and liminf I

- f'{y)\ =

0.

Thus trajectories with initial points in S—which may be called the "chaotic" set—come arbitrarily close and then noticeably separated infinitely often. Some theorists have used this result (or a variant of it) to claim that the existence of a cycle of period 3 was an indication of chaotic behaviour [see in particular in economics Benhabib and Day (1981, 1982), Day (1982, 1983)]. Proposition 4 shows that such a claim is unwarranted, for if there is a stable cycle, then the "chaotic" set S may be of Lebesgue measure 0 (think of a Cantor set) and thus essentially unobservable.

55

PERIODIC AND APERIODIC DYNAMICS

f 2 (b)

Fig. 2.3b

5. Aperiodic dynamics As we said, maps/that have no weakly stable cycles appear to be good candidates to describe turbulent or "chaotic" behaviour in one-dimensional dynamical systems. In that case, for any initial point x that does not belong to a cycle, the orbit of x is aperiodic, even if one iterates it long enough. Among the class of such aperiodic maps, of special interest are those which possess a unique invariant probability measure which is absolutely continuous with respect to the Lebesgue measure and which is ergodic. The probability measure v on [a, b] (endowed with its Borel o--algebra) is said to be invariant with respect to/if v(f~'(A)) = v(A) for any Borel set. It is absolutely continuous with respect to the Lebesgue measure X (absolutely continuous for short) if for any Borel set A, X(A) = 0 implies v(A) = 0 (v has then a X-integrable density with respect to X). Finally, v is said to be ergodic if for any v-integrable real-valued function g,

-Y nj=\

gdv,

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JEAN-MICHEL GRANDMONT

as n tends to + °°, for v-almost every x. This implies in particular that if one considers for each x and every n, the empirical distribution vn(x) that is generated by the iterates f(x) for j = 0, . . . , n — 1, which assigns probability 1/n to each/>(x), then the sequence vn(x) converges weakly to v for v-almost every x.8 Thus if v is absolutely continuous and ergodic, although a given trajectory may look somewhat erratic since the iterates fill up eventually the support of the limit distribution v, empirical distributions and time averages become ultimately fairly stable for v-almost every initial point. The next results give a sufficient condition for the existence of a unique absolutely continuous invariant measure, which is ergodic. Theorem 5. Assume that/satisfies S.I, S.2, S.3, and S.5, that it has no weakly stable periodic orbit, and that there exists an open neighbourhood V of x* such that/^x*) £ V for j is 1. Then/has a unique absolutely continuous invariant probability measure. It is ergodic. Proof. Note first that if all cycles of/are unstable, S. 1, S.2, and S.3 imply S.4", otherwise/would have a weakly stable fixed point in [a, x*]. Second, one must have/(fc) Si x*, so that S.4 is satisfied, otherwise/ would have a weakly stable cycle in [x*, b\. Thus we may apply Theorem III.8.3 in

CE.

•

Corollary 6. If/satisfies S. 1, S.2, S.3, S.4", and S.5 and if the iterates/(x*) of the critical point converge to an unstable cycle, then/has a unique absolutely continuous invariant probability measure. It is ergodic. Proof. In view of Theorem 2, / h a s no weakly stable cycle and the iterates of x* stay at a finite distance of x*. Thus Theorem 5 applies. • Remark. The foregoing results go in the direction of showing that aperiodic maps (having only unstable cycles) may display strong statistical regularities after all. Another direction of research has been to show that some (but not all) aperiodic maps may generate trajectories that are very sensitive to a small variation of initial conditions, thereby exhibiting the kind of phenomena that are observed, e.g., in turbulent flows (maps that have a unique weakly stable periodic orbit as in Theorem 2 do not have such a sensitivity to initial conditions). For an aperiodic and sensitive map, a small error of measurement of the initial state, for instance, may result in very large prediction errors (relatively speaking) for future dates, even if the forecaster knows very well the law of motion of the system (the map/). For such maps, there will exist a set of initial points, with positive Lebesgue measure, such that any trajectory starting from it looks "chaotic"—e.g., its spectrum closely resembles the spectrum that would be generated by a stationary stochastic process (random 8

See, e.g. Parthasarathy (1967, theorem 9.1). That book also contains the definition of the weak convergence of probability measures.

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noise). Such maps are actually used by theorists of various scientific disciplines to explain phenomena that look apparently random by a deterministic non-linear dynamic system. An example of such a map isf(x) = 4x(l - x), with x in [0,1], which has been employed for quite some time to generate . . . "random" numbers! For various definitions of the concept of sensitivity and a discussion of their implications, see CE, pp. 15-22, 30-35, and Section II.7.

6. Topological conjugacy There is nothing intrinsic in the representation of a one-dimensional dynamical system by a particular difference equation xJ+1 = /(*,•), since one can always make a change of coordinates. We investigate now what happens when one makes a change of variable y = h(x), in which h is an homeomorphism that maps [a, b] onto [a', b'], i.e., h is one to one and onto, continuous and h~x is also continuous. With the new variable, the dynamical system is represented by a new function g which maps [a1, b'] into itself and satisfies g(y) = h{f(h~](y))]. Thus g = h ° / ° h~l, we say then that/and g are topological conjugates. The maps / and g describe the same dynamics since the iterates of/ and g are linked by g> = h °fi °h~l for ally =£ 0. In particular {.*;,, . . . , x j is a cycle of/if and only if {h(xx), . . . , h(xk)} is a cycle of g, and stability or unstability of a periodic orbit is topologically invariant. If h is a C'-diffeomorphism, i.e., h is r times continuously differentiable and h'(x) ± 0 for all x in [a, b], with r g 1, we say that/and g are O-conjugates. In that case, one gets by differentiations, for all x and k, Dgk(h(x))h'(x) =

h'(f(x))Df(x).

In particular for k = 1, this shows that the sign of derivatives is unchanged. If {*!, . . . , xk} is a cycle of/, one has Dgk(h(xt)) = D/*(x,) at any point of the periodic orbit. It is clear from the above discussion that one can always assume that h preserves orientation, i.e., that h is increasing—or h'(x) > 0 for all x if h is continuously differentiable—since one can always go back to that case by making the additional change of variable z = —y without altering the dynamics of the system. It is now immediate that S.I is topologically invariant, in the sense that/ satisfies this condition if and only if g does, when h preserves orientation. S.2 is unaltered if h is thrice continuously differentiable. Conditions like/(x) > x are also topologically invariant, as well as S.4, S.4', or S.4", again if h preserves orientation. Finally, it is easily seen that the condition/'(x*) < 0 is unchanged through the change of variable provided one assumes h to be an

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JEAN-MICHEL GRANDMONT

increasing C2-diffeomorphism. Differentiating g(h(x)) = h(f(x)) twice and evaluating the expressions at x = x* yields indeed g"(h(x*))[h'(x*)]2 =

f"(x*)h'(f(x*)),

if one takes into account the fact that g'(h(x*)) = f'(x*) = 0. However, the condition S.3—which says in effect that D 2 |/'(x)|~ 1/2 is positive for all x in [a,ft],x + x*—is not generally invariant (like any convexity statement) through a (non-linear) change of variable. The point of this discussion is that even when a particular map f does not satisfy S.3, the foregoing results, i.e., Theorem 2 through Corollary 6, are still valid provided that one of the topological conjugates g of the original map f satisfies the assumptions made in anyone of these statements.

7. Bifurcations: Period doubling and the transition to turbulence Numerical experimentation with one-dimensional non-linear dynamical systems yields remarkable regularities that do not appear to depend much upon the maps under consideration. More precisely, consider a family of one-dimensional unimodal maps/ x that depend upon some real number X, that may be thought as indexing one of the characteristics of the system (the parameter may be for instance under the control of some outside observer in a physical experiment). If we look back at Theorem 1, we should expect that the fashion in which cycles appear when X is varying, should display some degree of conformity with Sarkovskii's ordering of the integers. Namely, we should expect cycles having a period that is a power of 2 to appear first. Numerical experimentation shows that this is indeed the case. In fact, this is true for (weakly) stable cycles. Let us assume that for each X, we iterate the critical point x^ of / x on a computer. If each/ x satisfies the conditions of Theorem 2 and has in particular a negative Schwarzian derivative, we know that this procedure discloses (weakly) stable cycles that have a small period and that are attracting enough. Suppose now that we put X on an horizontal axis and that above each value of X we plot vertically the values taken by the iteratesf'x(x*x) for, say, t = 200 to 300. Computer simulations of this type yield typically a very neat "bifurcation diagram" which displays first a whole interval in which period doubling bifurcations occur more and more rapidly, a stable fixed point giving rise to a stable cycle of period 2, which yields then a stable cycle of period 4 and so on (see Figure 2.4). The values of X for which such period doubling bifurcations occur tend to some limit value X*, beyond which one enters the "chaotic" region: for X > X t , one often observes a ' 'mess''—meaning that one has either an aperiodic ("chaotic") map or a stable cycle with a very long period—in the middle of which windows may appear that show stable cycles with low

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Fig. 2.4 periods like 3, 5, 6, or 7 (that depends of course on the degree of resolution of the diagram).9 The results that follow explain why such an outcome should be typically observed. Formally, we consider a one-parameter family of maps / x in which X belongs to [0,1]. For each X in that interval,/x maps the interval [ax, Z?J into itself, is C'-unimodal with a unique critical point x% in (ax, frx) and/x(jc^) = bx. We assume that ax and bx depend continuously on X, as well as/ x and its derivatives. More precisely, for any sequence Xn that tends to X. in [0,1], then a = a n \n a n d bn = bKn tend to ax and bx, respectively, while for any sequence xn E [an, bn] that converges to x E [ax, fcj, the sequences fXn(xn) and f'Xn(xn) converge tofK(x) and/ x (x), respectively. We shall say that the family is/«// if 1. for \ = 0, one has fo(bo) > x% . In that case, as one can easily verify, all iterates/if*?) = fh'KK) belong to the interval \fo{ba), b0] for; a 1; 2. for \ = 1, one has/?(*"{) < xf and/](x1) < x*u Then we have: Theorem 7. Consider a full one-parameter family of C'-unimodal maps indexed by X in [0, 1]. Then: 9 Diagrams of this type are numerous in the literature. See CE, p. 26, and May (1976). Such bifurcation diagrams have been obtained in economic models by Dana and Malgrange (1984), Grandmont (1985), Jensen and Urban (1982).

60

JEAN-MICHEL GRANDMONT

1. Given an arbitrary t g 2 , the set of parameters X for which the map/ x has a superstable cycle of period k is closed and non-empty. Given such a \ , there is an open interval around A. such that/ v has a stable cycle of period k for all V in this interval. 2. Let X* be the first value of the parameters X for which a superstable cycle of period 2> obtains for; & 1. Then the sequence Xf increases with j and converges to some value k% < 1 asj tends to + = °. For each X in [0, k%), all cycles of the map/ x have a period that is a power of 2 or are fixed points. The critical point x% of fk is attracted to one of these. 3. If superstable cycles of periods 2> and 2'' withy' > j + 1 occur respectively for the values X and X' in [0, X*), then a superstable cycle of period 2' with/ > i > j must appear for some value in the open interval determined by X and X'.

Proof. As a preliminary remark, CE require that ax = — 1, bK = 1, JC^ = 0 for all X, but the proofs of the results we shall use employ only simple continuity arguments that do not depend upon these specific assumptions. Second, our assumptions imply that the itinerary of b0, denoted K(f0), is i?°°, while that of bu denotedK(ft), staitsRLL . . . (itineraries are defined in CE, p. 64). 1. According to CE, Theorem III. 1.1, every maximal admissible sequence A satisfying K(f0) < A < K(ft) occurs as the itinerary K(fJ of bx for some X in (0,1) (admissible sequences are defined in CE, p. 64, the ordering between admissible sequences is defined in CE, pp. 65-66, while maximal sequences are defined in CE, p. 71). In fact, it follows from the proof of this theorem (see CE, p. 175) that the set of such X's is non-empty and closed provided that A * {BRT and A = t (BL)°°. Choose now an integer ks^2, and consider a maximal sequence BC in which the sequence B contains k - 1 elements, such that R™ • • • > RLR'~3C > {RLRl-2Y > RLRixC > • • • > • • • > R * RLRi 3C >R* (RLR'-2)- >R* RLR>~lC > • • • > • •• > R'" * RLR'lC > R'" * (RLR'-2r > R'" * RLR'~'C > • • • > . . . >R*(>» + »*RC>R* to a period 2J+', then (kj — \ , _ l)/(kj+1 - Kj) tends very rapidly, as y diverges to + oo, to some number 8 = 4.66920. . . , that seem independent of the family fK under consideration. For a discussion of this and related points, and a theorem that gives a partial mathematical explanation of this "empirical" phenomenon, see CE, Section 1.6 and III.3. For extension to families of maps on Um, with m § 2, see CE, Section III.4.

References Benhabib, J. and R.H. Day, 1981, Rational choice and erratic behaviour, Review of Economic Studies 48, 459-472. Benhabib, J. and R.H. Day, 1982, A characterisation of erratic dynamics in the overlapping generations model, Journal of Economic Dynamics and Control 4, 37-55. Collet, P. and J.-P. Eckmann, 1980. Iterated Maps on the Interval as Dynamical Systems (Birkhauser, Boston MA). Dana, R.A. and P. Malgrange, 1984, The dynamics of a discrete version of a growth cycle model, in: J.P. Ancot, ed., Analysing the Structure of Econometric Models (M. Nijhoff, Amsterdam). Day, R.H., 1982, Irregular growth cycles, American Economic Review 72, 406-414. Day, R.H., 1983, The emergence of chaos from classical economic growth, Quarterly Journal of Economics 98; 201-213. Grandmont, J.M., 1985, On endogenous competitive business cycles, Econometrica 53: 995-1045. Guckenheimer, J. and P. Holmes, 1983, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Applied Mathematical Sciences Series no. 42 (Springer: New York). Jakobson, M.V., 1981, Absolutely continuous invariant measures for one parameter families of one-dimensional maps, Comm. Math. Phys. 81, 39-88. Jensen, R.U. and R. Urban, 1982, Chaotic price behaviour in a nonlinear Cobweb model, Mimeo. (Yale University, New Haven CT). Li, T. and J.A. Yorke, 1975, Period three implies chaos, American Mathematical Monthly 82, 985-992. May, R.B., 1976, Simple mathematical models with very complicated dynamics, Nature 261, 459-467. Parthasarathy, K.R., 1967, Probability Measures on Metric Spaces (Academic Press, New York). Rand, D., 1978, Exotic phenomena in games and duopoly models, Journal of Mathematical Economics, 5, 173-184. Sarkovskii, A.N., 1964, Coexistence of cycles of a continuous map of the line into itself. Ukr. Mat. Z. 16, 61-71. Singer, D., 1978, Stable orbits and bifurcations of maps of the interval. SIAM Journal of Applied Mathematics 35, 260. Stefan, P., 1977, A theorem of Sarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line. Comm. Math. Phys. 54, 237-248.

A Characterization of Erratic Dynamics in the Overlapping Generations Model JESS BENHABIB AND RICHARD H. DAY*

1. Introduction The pure "consumption-loan" model of Samuelson (1958) has been one of the most widely discussed subjects in the contemporary economic literature [see Meckling (1960), Cass and Yaari (1966, 1967), Diamond (1965), Thompson (1967), Shell (1971), Starret (1972), Gale (1973), Cass, Okuno, and Zilcha (1979), Brock and Scheinkman (1980)]. Much of the discussion centered around the efficiency of the allocation (in the Pareto-sense) that results from overlapping generations. But even when efficiency is not at issue, the model has much to offer. In contrast to models which have a finite horizon (a built-in doomsday), the horizon in the consumption-loan model extends indefinitely. Individuals, however, have a finite horizon and plan their lifetime consumption-saving path over their finite lifetime.1 As pointed out by Cass and Shell (1980), this allows the life of the economic system to extend indefinitely while not forcing the doomsday problem onto the agents in the economy or onto some representative individual. And in contrast to other infinite horizon models, it does not require economic agents to have perfect foresight up to infinity. In the standard Samuelson version, the heterogeneity of consumers is restricted to the simultaneous existence of the young and the old. Recently, Cass, Okuno, and Zilcha (1979) introduced heterogeneity within generations and obtained illuminating results on the existence and Pareto-efficiency of equilibrium trajectories. It is the possibility of oscillitary trajectories on which we focus our attention in the present paper. In particular we examine non-stationary paths where net trades between generations remain positive and feasible but exhibit an erratic * Originally published in the Journal of Economic Dynamics and Control 4, 37-55, 1982. Reprinted with the permission of Elsevier Science publishers B.V. (North-Holland). The illuminating and helpful discussions with Professors W.A. Brock and D. Cass are happily acknowledged. 1 The sequential generations models of Leontief (1958), Day (1969), and Day and Fan (1976) also share this property. Their mathematical structure is similar to that of the overlapping generations model.

CHAOS IN OVERLAPPING GENERATIONS

65

character. We first characterize a class of utility functions, assumed to be unchanging over the generations, which give rise to such erratic trajectories. These trajectories essentially are paths that do not converge to periodic orbits or stationary points but remain bounded. Co-existing with such trajectories are also periodic orbits of arbitrarily large periods, which for all practical purposes are indistinguishable from the aperiodic ones. What is surprising is that such trajectories arise from a very wide class of utility functions that are robust with respect to perturbations in the parameters of the system (see section 6). In section 4 we discuss the specific mechanism by which intergenerational exchanges can take place [on this see also Gale (1973, sect. 5.C)]. This requires the introduction of credit (say in the form of checking accounts) which is extended to the agents by a central authority. The expansion of credit is controlled by this central authority via a nominal rate of interest. We show that the path of intergenerational trades is insensitive to this interest rate. A nominal rate fixed at a given level, for instance, results in a constant rate of expansion of the nominal value of credit. Yet, we show that the real value of credit can fluctuate erratically. What of the Pareto efficiency of intergenerational exchange in a chaotic regime? Using results of Balasko and Shell (1980) and Okuno and Zilcha (1980), it is possible to establish efficiency for all periodic orbits and erratic trajectories. This is done in section 5 of this paper. Our discussion concludes with a review of some of the technical aspects of chaotic dynamics and some possible extensions of our work. Of particular interest is the potential use of ergodic theory to obtain a statistical description of the deterministic trajectories of key variables (intergenerational trades, prices, the real value of credit) in the chaotic region.

2. Intergenerational exchange 2.1. The overlapping generations model Following Gale (1973) we consider exchange between a population of overlapping generations growing at a rate 7. A representative individual lives for two periods, and, when young, determines a (non-negative) consumption for his youth co(t) and for his old age cx{t + 1). His preferences are represented by a utility function U(co(t), ct(t + 1)), and he receives an endowment w0 in his youth and wt in his old age. The interest factor at time t, p, (defining the exchange rate between present and future consumption) determines the representative individual's budget constraint, c , ( t + 1 ) = w , + p , [ w 0 - c o ( t ) ] , co(t) a 0 , c , ( ? + ! ) & 0 .

(1)

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JESS BENHABIB & RICHARD DAY

Since by assumption the aggregate endowment grows at the rate 7, the marketclearing equilibrium condition for the economy as a whole is (1 + 7)1*0 - co(t)] +

Wl

- c,(0 = 0.

(2)

This materials balance constraint, together with the budget constraint (1), defines the set of feasible programs for this economy. Define (c%(t), cx(t + 1)) to be the consumption vector maximizing the utility of the fth generation subject to its budget constraint given by eq. (1). Let U(c%(t), c*t(t + 1); pr, w0, w,) = C/*(p,, w0, w,). Then a dynamic, pure exchange equilibrium consists of price and consumption sequences in which the latter are individually optimal and consistent: Definition 1. A pure exchange equilibrium trajectory is a sequence of vectors (pr, co(t), c,(0)r= i s u c n that, for all t, U(co(t),Cl(t + 1); p,, w0, w,) = U*(p,, w0, w,),

(3)

and the materials balance constraint (2) holds. Members of each generation may either save or borrow in their youth, thereby carrying claims or debts into their old age which they then settle with members of the new generation. They either claim their savings plus interest or pay their debts, which correspond, in the same order, to the new generation's savings or borrowings. The equilibrium price or interest factor sequence assures that the claims (debts) of the old generation are equal to the savings (borrowings) of the young generation. The institutional elements to assure the settlement of claims can range from a social security system to a central clearing house dealing in I.O.Us [see Gale (1973) and section 4 below]. Alternatively, the existence of a non-perishable and negotiable asset can also serve as a convenient medium of exchange and a store of value. Later on we will explore a model where we introduce 'credit' to serve as a medium of intergenerational exchange. Gale named the case where the young exhibit impatience and borrow from the old, "classical," and the case where the young save and lend to the old, "Samuelson." Which of the two cases obtains depends on the utility functions and the endowments. In either case, whether the "classical" or the "Samuelson" case is feasible, there exists a no-trade equilibrium associated with a constant price sequence: if in period zero there is no exchange, then there will be no exchange in future periods. The classical no-trade equilibrium is locally unstable: for any amount of borrowing by the young in period zero, no matter how small, the market will not return to the no-trade equilibrium. Conversely, Samuelson's no-trade equilibrium is locally stable [see Gale (1973, theorem 4)]. This has led Gale to conclude that while the "Samuelson" case is logically consistent, the case relevant to the real world (and to the theory of interest) is the "classical" one.

CHAOS IN OVERLAPPING GENERATIONS

67

The issue of how and why any trade should take place between the young and the old has been discussed by many authors. In period zero, why should the young lend to the old who will not be around to repay their debts in the next period? And why should the old lend to the young if they will not be around to collect? In the Samuelson case, where the young are the lenders, they could be compensated when they become old by the forthcoming young generation, which in turn could be compensated when they become old, etc. That time never ends must be taken as axiomatic for infinitely farsighted individuals or they will not lend when young since, in an economy of finite duration, the "last" young generation will have no compensation for their savings. In the classical case the old repay debts of their youth. But this leads to an infinite regress, possibly to exchange among the prehominids. Let us, therefore, consider an initial situation in which, for whatever reason (government taxes or transfers in the first period, altruism, small perturbations which destabilize the no-trade equilibrium) that trade takes place.

2.2. The dynamics of the classical case In order to characterize the dynamics of the pure exchange equilibrium trajectory the following assumptions are invoked: 1. The utility function for the representative generation is strictly concave, twice differentiable, increasing in its arguments and either separable or homothetic.

ASSUMPTION

2. For the prices along the equilibrium path, the solution to the utility maximization problem of each generation is interior, i.e., co(t), c,(f + 1) > 0 for t = 0, 1, . . . . (This would hold if the utility function satisfied appropriate Inada conditions.)

ASSUMPTION

Under Assumption 2 the first-order conditions for the utility maximization problem (3) reduces to P

'

=

uo(co(t),Cl{t + l ) ) £/,(co(0,c,(f + 1))'

in which Uo and £/, are the partial derivatives of U. Substituting this into the budget constraint (1) we get U0(c0(t),Cl(t + 1))

U^cM^t + D)

=

w, -

Cl(t

+ 1)

co(t) - w0

Our problem now is to reduce (5) to a difference equation in co(t) [or c,(f)]. To obtain a well-defined difference equation we confine our attention to the

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JESS BENHABIB & RICHARD DAY

classical case. (For the problems presented by the "Samuelson" case, see the next session.) Gale (1973, theorem 5) provides the prerequisite result: Lemma 1. In the classical case and given Assumptions 1 and 2 above, the function (5) can be solved uniquely for cx(t + 1). Call this function c,(7 + 1) =

G(co(t); w0, w,).

(6)

Now define the constrained marginal rate of substitution (CMRS) function, ... . , , V(co(t); w0, w,): =

U0(c0(.t),c,(t + 1)] — , £/,[co(O,Ci(f + 1)]

(7)

where use is made of (6) to eliminate c,(f + 1). The function V describes the marginal rate of substitution of present for future consumption for individually optimal and feasible programs, i.e., programs which satisfy the individuals' budget constraints. Combining (7) with the equilibrium condition (2) we obtain the difference equation2 co(t + 1) = w0 + -—— V(co(t); wB, w,)(co(O - w0): = /(co(r)).

(8)

Once trade begins in the classical case the trajectory must remain classical. This result is readily established from (8) by observing the fact that pr = Uo/U1 > 0, for all co(f),Ci(f + 1) > 0 and using induction. Hence, the difference equation (8) characterizes the pure exchange equilibrium trajectories when co(0) > w0. Gale (1973, p. 23) asserted that in the case of cycles the trajectory would converge either to the stationary state or to a limit cycle. However, we show that cyclical trajectories may oscillate without ever converging to a cycle of any order. Before establishing this result we will briefly consider the dynamics of the Samuelson case. 2.3. The Samuelson case So far we have only treated the case where the young people borrow. This is because we have been able to unambiguously define the dynamics of exchange only for co(t) > w0 (see Lemma 1). This is not simply a technical difficulty because cyclical behavior implies an ambiguity in the dynamical system when young people lend to the old. This is illustrated in fig. 3.1 [see also Cass, Okuno, and Zilcha (1979)]. The line XY is the materials balance constraint, AF is the offer curve, A is the stationary no-trade equilibrium, and D is the 2

Solve (2) for c(t + 1), substitute into (1) and cancel to obtain c(t + 1) = (p,/l + y)(w0 co(0). Then (8) follows from (4) and (7).

69

CHAOS IN OVERLAPPING GENERATIONS

Bl

co(t) Fig. 3.1

stationary exchange equilibrium. Note that to have non-monotonic trajectories, AF must cut XY at D with a positive slope. But this implies that for a given co(t),Ci(t + 1) and therefore co(t + 1) is not uniquely determined. Only if an additional selection criterion is imposed can we discuss the dynamics of exchange and prices. One possibility, for example, is random selection. On the other hand, we could assume that when multiple equilibria are possible, the one yielding the highest utility to the young obtains, at the expense of the yet unborn. Assume that the youth of each generation can obtain the equilibrium price yielding the highest utility and that the offer curve is as shown (with EF steep) in fig. 3.1. At some point along the equilibrium path the young want to lend AB in return for BF in their old age. But to the youth of the following generation such a sacrifice is not acceptable at any price. Therefore, if we insist on equilibrium, the feasible exchange for the present young is AB for BC, where BC is substantially less than BF. From such a point on, however, if equilibrium and feasibility are to hold, the system must converge to the no-trade equilibrium point A.

3. Erratic exchange equilibria 3.1. The chaos theorem By erratic exchange equilibria we mean trajectories (p,, co(t), cx(t + l))o whose components do not converge to a stationary state or limit cycle but

70

JESS BENHABIB & RICHARD DAY

remain bounded. Under conditions sufficient to generate such behavior limit cycles of all orders will also exist. All this is made precise in a theorem proved by Li and Yorke in their paper "Period Three Implies Chaos." We first define the following: Definition 2. The iterated mzpfk(x) is defined as/*(*) = f{fk~\x)) where f°(x) = x. A point x is /^-periodic under/iffk(x) = x andfJ(x) + x for 0 < j < k. The basic theorem which we will apply to the consumption loan model is the following [Li and Yorke (1975)]: Chaos Theorem. tion

Let J be an interval in R and consider the difference equa*,+ i = / ( * . ) ,

(9)

in which/is a continuous mapping ofJ—>J. Suppose there exists a point x £ J such that / ' ( * ) £ * < / ( * ) 0, t—*co

lim mf\f'(x)-f'(y)\ =

0;

I—»co

(b) for all period points x and all points y £ S, lim sup |/'(x) - f'iy) I > 0 . We note that statement (ii-a) of the theorem implies that trajectories in the set S starting from different points eventually get arbitrarily close but then diverge again. Definition 3. If/ is a continuous mapping of an interval J into itself and there exists some initial x for which (10) is satisfied, then the difference eq. (9) will be said to be chaotic on J. Trajectories in the chaotic set S of the chaos theorem will be said to be erratic. [For an illuminating discussion of chaotic difference equations, see Guckenheimer, Oster, and Ipaktchi (1977).] Letting z, — co(t) and defining/as in eq. (8), we can apply the chaos theorem to characterize erratic exchange equilibria. The question now to be ex-

CHAOS IN OVERLAPPING GENERATIONS

71

amined is: "Since the qualitative behavior of trajectories of (8) depend on the utility function £/(•,•), what characteristics of the utility function are sufficient to generate chaos in the sense of Li and Yorke?'' 3.2. The sufficient substitutability condition We will characterize chaotic trajectories in terms of the constrained marginal rate of substitution function, V(-), defined in (7). Indeed, the sufficient condition about to be stated essentially requires the isoquants of the utility map to have enough curvature so that the marginal rate of substitution can vary sufficiently: Sufficient Substitutability Condition. The utility function is said to satisfy the sufficient substitutability condition if there exists a c > w0 such that (i)

a, = ( — — ) V(c) > 1

(resp. < 1)

(ii)

a2 =

(resp. < 1)

—;

(iii) 0 < a3 = a,a2

V(al(? + (1 - a,)w0) > 1

VCa^c + (1 - a,a2)w0) § 1 (resp. s 1) \1 + 7/ As before, 7 is the rate of population growth. • It is easily shown that the functions satisfying the "sufficient substitutability" condition form a very large class. Note that the condition requires (i), (ii), and (iii) to hold for some c, not all c. Condition (i) simply requires that V(c) > (1 + 7). The function V(c) then defines a n a b which together with c, w0, and 7 define a second point at which V(c) has to be evaluated. Condition (ii) puts a restriction on V(c) at that point; it also must be greater than (1 + 7). Note that c0 > w0 implies that this new point is greater than c. Finally, condition (iii) uses the value of this second point to define a third point at which to evaluate V(c). This point falls to the right on the second point and (iii) requires V(c), evaluated at this third point, to be less than or equal to (1 + 7)/ot,a 2 . Fig. 3.2 illustrates the "sufficient substitutability" condition. It is obvious from inspection that the construction of such functions is very easy. Essentially it requires V(c) to be steep at first and then to taper off sufficiently. [For example, let vv0 = 0, 7 = 0, c = 1. Then choose V(c) such that V(l) = 2, V{2) = 1.5, and V(3) = 0.3. Such a V(c) will satisfy the condition given above.] When the utility function is separable and linear in old age consumption,

72

JESS BENHABIB & RICHARD DAY

Fig. 3.2

«1 14.765). For example, let r = 100 and x0 = 0.01 on eq. (7). Simple calculations showx 3 0,

and let the endowment vector be (wo,w,) = (0,w) with w > a/b and let the population be stationary. Then the difference equation describing the dynamics is given by co(t + 1) = aco(t){\ - (b/a)co(t)).

(14)

Note that co(t) G [0,a/b] for all co(O) £ [0, a/b], provided a S 4. In fact, co(t + 1) attains a maximum for co(t) = a/2b. Eq. (14) can be shown to satisfy 3

The utility derived from first-period consumption is described by the standard, widely used functional form exhibiting constant risk aversion. 4 Since co(t) ~ w0 and c,(f) are positive along the erratic paths, Assumptions 1 and 2 are clearly satisfied.

CHAOS IN OVERLAPPING GENERATIONS

75

the requirements of our theorem for a E [3.53,4], b = a. For a = 3.83, for example, (14) has (approximately) a 3-period cycle for c0 = 0.1561 and where F{c) = 0.5096 and F2(c0) = 0.9579. [For further discussions, see Marotto (1978, example 4.1) or Hoppensteadt and Hyman (1977).] Note that for c0 = a/b utility saturates. But the erratic trajectories and those with period greater than one never attain b/a since if c(t) = b/a, c(t + 0 = 0 for alli = 1,2, (iii) For a final example let the concave utility function be U(co,Cl) =

\(c + by-"

- 1 - 05 — + c, 1 — a

a ^ o, a + 1, \ > 0, b a 0.

(15)

This leads to the difference equation co(t + 1) - w0 = T - — — — ,_,_.. (co(0 - w0), (16) 1 + 7 ((co(O - w0) + ky where k=b + w0 and 7 is again the rate of population growth. For large values of \ (= S 50) and a ( s 5) again chaos emerges [see Hassel, Lawton, and May (1976) and May and Oster (1976, table 1)].

4. Consumption loans with the social contrivance of "credit" So far we have not discussed the mechanism by which trades between generations take place. It turns out that this is a delicate matter. In the "Samuelson'' case young people sell part of their endowment for a negotiable asset which they in turn use to purchase goods in their old age. This negotiable asset performs one of the functions of money: it serves as a store of value. In the "classical" case the young borrow and the old settle their debts. This requires an intermediary or a central authority who extends credit to the young (mortgage loans, for instance) who wish to borrow, say in the form of checking accounts. The young then buy goods (for example, houses) from the old with checks; the old in turn deposit these checks and settle their debts. The checking accounts demanded by the young and provided by the central authority as credit then serve as a medium of exchange between generations.5 The central authority can also try to regulate the amount of credit by stipulating a nominal interest on credit. We will see that erratic equilibrium trajectories in the real value of credit may arise even though nominal credit expands at a constant rate. The representative young consumer maximizes his two-period utility function, U(cQ(t), ct(t + 1)), subject to 5 In general the young may on balance either borrow from or lend to the old and a more complete analysis would require a model where generations live longer than two periods. For a formal treatment of this, see Gale (1973, sect 6).

76

JESS BENHABIB & RICHARD DAY p(t)co(t) =

(17)

p(t)w0 + m(f),

p(t + l)c,(/ + 1) =

p{t + l)w, - (1 + 0 for any erratic path. This is also obviously true for any periodic orbit with x(0) + 0 as well. Since l/p(t) = co(f) — w0 for all t as shown above, the theorem is proved. •

6. Final remarks We conclude by briefly pointing out certain technical aspects of erratic or chaotic dynamics since this is a relatively new subject: 1. Erratic (or chaotic) trajectories seem to be a ubiquitous feature of nonlinear difference equations. That such dynamics can be robust (structurally stable) with respect to small perturbations of the map defining the difference equation has been shown by Butler and Pianigiani (1978) and by way of example by Smale and Williams (1976). See also Kloeden (1976). 2. A characterization of chaotic behavior in higher order systems has been given by Phil Diamond (1976) as well as Marotto (1978). 3. Chaotic trajectories arise in differential equations as well. A number of studies have exhibited such possibilities (called "strange attractors") in differential equation systems of order higher than three [see Lorentz (1963) and Ruelle and Tackens (1971)]. 4. The dynamics within erratic sets may often be indistinguishable from stochastic processes, sometimes even from simple Bernoulli processes [see Guckenheimer, Oster, and Ipaktchi (1977) or May and Oster (1976)]. This has led to the use of statistical techniques and ergodic theory to investigate the average behavior of erratic trajectories. One can denote the fraction of iterates {x,F(x), . . . , F"~i(x)} that are in an interval [aua2] (or take a Markov partition of the set to be studied). The fraction can be represented by ip(x,n,[a1,a2]) and one can take the limit as n—»°° if such a limit exists. The subject of ergodic theory is the existence of limiting fractions for intervals [aua2]. Such limiting co(t + 1) - Wo =

V((co(t) - w0) + wo)(co(O - w0),

(29')

1 + 7 and (27) becomes l/ir(/ + 1) = V(co(f))(l/TT(/))-

(27')

Normalizing such prices that TT(0) = co(O) — w0 and combining (27') and (29') we obtain = (1/(1 + 7)')(c (O - w ) or l/p(t) = (1 + •y)'/ir(O = c (t) - w . Since the theorem of Okuno ando Zilcha iso given in terms of 1 /p(t), the proof of oTheorem0 2 above applies to the case of population growth as well.

CHAOS IN OVERLAPPING GENERATIONS

79

fractions are of particular interest if they are independent of the initial point x. For the model discussed in the previous section such results would allow us to characterize a limiting long-run average of net trades and real balances even though such quantities fluctuate erratically from period to period. For a given initial value of net trades it is sometimes possible to show that the limiting function (f>(x,N(aua2)) exists. From Theorem 1 we know that the function defining net trades F(co(t); w0) maps the interval J = [w0; wj(l + 7)] into itself. We can then apply the Kriloff-Bougolioubof Theorem8 to deduce the existence of an invariant measure on J. Then the Birkhoff Ergodic Theorem (1936) [see also Halmos (1956)] asserts the existence of the limiting fractions for "all initial x £ J except for a subset of measure zero." The trouble is that the invariant measure defined on J, since it is not continuous, may be supported on a very small set or even on a point. In such a case "all initial x E J except for a subset of measure zero" may be just one point. When erratic paths (as defined in Theorem 1) exist, however, it may be possible to show the existence of continuous invariant measures (assigning zero measure to points) on J [see Lasota and Yorke (1977) and Pianigiani (1979a)]. Then the set of initial values for which time averages exist cannot be supported on isolated points. In certain cases the limiting fractions can be independent of the initial value.9 In example (ii) of section 3.4, for the value of " a " = 4, b = 1 [see Stein and Ulam (1974), or for " a " = 3.6785735 . . . see Ruelle (1977)], such independence can be obtained. In fact for this difference equation it can be shown that there are an infinite number of such values of " a " [see Pianigiani (1979b)].

References Balasko, Y. and K. Shell, 1980, The overlapping-generations model, I: The case of pure exchange without money, Journal of Economic Theory 23, 281-306. Birkhoff, A.D., 1936, Proof of the ergodic theorem, Proceedings of the National Academy of Sciences 17, 656-660. Brock, W.A. and J.A. Scheinkman, 1980, The optimum mean and variance of the monetary growth rate in an overlapping generations model, in: J.H. Kareken and N. Wallace, eds., Models of monetary economy (Federal Research Bank of Minneapolis, MN) 211-232. Butler, G.J. and A. Pianigiani, 1978, Periodic points and chaotic functions in the unit interval, Bulletin of the Australian Mathematical Society 18, 255-265. 8

For an exposition, see Pianigiani (1979a). Birkhoff s Erogodic Theorem shows what is needed for independence from initial values (i.e., ergodicity) is "metrical transitivity." "Metrical transitivity" requires that under the transformation considered there do not exist two disjoint invariant sets of positive measure. In practice, this is extremely hard to check for. 9

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Cass, D. and K. Shell, 1980, In defense of a basic approach, in: J.H. Kareken and N. Wallace, eds., Models of monetary economy (Federal Research Bank of Minneapolis, MN) 251-260. Cass, D. and M. Yaari, 1966, A re-examination of the pure consumption loans model, Journal of Political Economy 74, 353-367. Cass, D. and M. Yaari, 1967, Individual savings, aggregate capital accumulation, and efficient growth, in: K. Shell, ed., Essays on the theory of optimal economic growth (MIT Press, Cambridge, MA) 233-269. Cass, D., M. Okuno, and I. Zilcha, 1979, The role of money in supporting the Pareto optimality of competitive equilibrium in consumption-loan models, Journal of Economic Theory 20, 41-80. Day, R.H., 1969, Flexible utility and myopic expectations in economic growth, Oxford Economic Papers 21, 299-311. Day, R.H. and Y.K. Fan, 1976, Myopic optimizing, economic growth and the Golden Rule, Hong Kong Economic Papers 10, 12-20. Diamond, P.A., 1965, National debt in a neoclassical growth model, American Economic Review 55, 1126-1150. Diamond, Phil, 1976, Chaotic behavior of systems of difference equations, International Journal of Systems Science 7, 953-956. Gale, D., 1973, Pure exchange equilibrium of dynamic economic model, Journal of Economic Theory 6, 12-36. Guckenheimer, J., G. Oster and A. Ipaktchi, 1977, Dynamics of density dependent population models, Journal of Mathematical Biology 4, 101-147. Hassel, M., J. Lawton, and R.M. May, 1976, Patterns of dynamical behavior in single-species populations, Journal of Animal Ecology 45, 471^4-86. Halmos, P.R., 1956, Lectures on erogodic theory (The Mathematical Society of Japan, Tokyo). Hoppensteadt, F.C. and J.M. Hyman, 1977, Periodic solutions of a logistic difference equation, SIAM Journal of Applied Mathematics 32, 73-81. Kloeden, P.E., 1976, Chaotic difference equations are dense, Bulletin of the Australian Mathematical Society 15, 371-379. Lasota, A. and J.A. Yorke, 1977, On the existence of invariant measure of transformations with strictly turbulent trajectories, Bulletin del'Academie Polonaise des Sciences Serie des Sciences Mathematiques Astronomiques et Physique 25, 233-238. Leontief, W.W., 1958, Theoretical note on time-preference, productivity of capital, stagnation and economic growth, American Economic Review 48, 105-111. Li, T.Y. and J.A. Yorke, 1975, Period three implies chaos, American Mathematical Monthly 82, 985-992. Lorentz, E.N., 1963, Deterministic nonperiodic flow, Journal of Atmospheric Science 20, 130-141. Marotto, F.R., 1978, Snapback repellers imply chaos in R", Journal of Mathematical Analysis and Applications 63, 199-223. May, R.M., 1976, Simple mathematical models with very complicated dynamics, Nature 261, 459-467. May, R.M. and G.F. Oster, 1976, Bifurcations and dynamic complexity in simple ecological models, The American Naturalist 110, 573-595.

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Meckling, W.H., 1960, An exact consumption-loan model of interest: A comment, Journal of Political Economy 68, 72-75. Okuno, M. and I. Zilcha, 1980, On the efficiency of a competitive equilibrium in infinite horizon monetary models, Review of Economic Studies 47, 797-808. Pianigiani, A., 1979a, On the existence of invariant measures, in: V. Lankshmikanthan, ed., Applied nonlinear analysis (Academic Press, New York) 299-307. Pianigiani A., 1979b, Absolutely invariant measures for the process xn +, =Axn(\ — xn), Bolletino Unione della Matematica Italiana (5), 16-A, 374-378. Ruelle, D., 1977, Applications conservant une mesure absolument continue par rapport a dx sur [0,1], Communications and Mathematical Physics 55, 47-51. Ruelle, D. and F. Takens, 1971, On the nature of turbulence, Communications and Mathematical Physics, 167-192. Samuelson, P. A., 1958, An exact consumption-loan model of interest with or without the social contrivance of money, Journal of Political Economy 66, 467^-82. Shell, K., 1971, Notes on the economics of infinity, Journal of Political Economy 79, 1002-1011. Smale, S. and R.W. Williams, 1976, The qualitative analysis of a difference equation of growth, Journal of Mathematical Biology 3, 1-5. Starret, D., 1972, On golden rules, the "biological theory of interest" and competitive inefficiency, Journal of Political Economy 80, 276-291. Stein, P.R. and S.M. Ulam, 1974, Nonlinear transformations on the electronic computers, in: W.A. Beyer, J. Myeidlski, and A.C. Rota, eds., Stanislaw Ulam: Set, numbers and universes: Selected papers (MIT Press, Cambridge, MA) 401^-84. Thompson, E.A., 1967, Debt instruments in macroeconomic and capital theory, American Economic Review 57, 1196-1210.

On Endogenous Competitive Business Cycles* JEAN-MICHEL GRANDMONT1

Introduction The belief that the long run equilibrium of a competitive monetary economy that does not experience any exogenous shocks—whether originating from the external environment or from policy—should be modeled as a state that is stationary or perhaps growing at a constant rate seems to be deeply rooted in the mind of some economists. The most outspoken believers in the market's invisible hand go indeed as far as claiming that any departure from a long run Walrasian equilibrium should be regarded as purely transitory and that accordingly the basic tendencies of a competitive economy may be represented adequately by such a * Originally published in Econometrica 22, pp. 995-1037, 1985. Reprinted with the permission of the Econometric Society. 1 The material of this paper has been presented as part of the Walras-Bowley lecture at the North American Summer Meetings of the Econometric Society, Stanford, June, 1984. I wish to thank very much the members of the ensuing discussion panel, K. Arrow, R. Hall, R. Lucas, C. Sims, and J. Stiglitz for their perceptive comments. An earlier version was presented at the Workshop on Price Adjustment, Quantity Adjustment, and Business Cycles, in October, 1983, at the Institute for Mathematics and Its Applications of the University of Minnesota, organized by Hugo Sonnenschein. Special thanks are due to the discussants of the paper at the Workshop, Daniel Goroff, Jose Scheinkman, Christopher Sims, Neil Wallace, and Mike Woodford, for their valuable comments and constructive criticisms. I wish to thank very much Rose Anne Dana and Pierre Malgrange who introduced me to the mathematics of the subject: without them, this research could not have been completed. Christian Gourieroux deserves also a special mention for his decisive advice in connection with the stability issue of Section 3. Rose Anne Dana and Dominique Levy ran the computer experiment reported in Section 4. Their help is gratefully acknowledged. Earlier simulations by Alain Morineau were also very valuable. I had stimulating conversations with and received useful comments from many colleagues at various stages of this work. I wish to thank them all, and especially Philippe Aghion, Albert Ando, Jean-Pascal Benassy, Robert Boyer, David Cass, Jean-Pierre Danthine, Frank Hahn, Guy Laroque, and Yves Younes. Last, but not least, I wish to express my gratitude to the four referees, in particular to Margaret Bray, for their very professional reports and their suggestions, which led to significant improvements of the paper. Financial support from the French Commissariat General du Plan, the University of Lausanne, and the U.S. Office of Naval Research under Contract ONR-R00014-79-C0685 at the Institute for Mathematical Studies in the Social Sciences at Stanford University is also gratefully acknowledged.

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"Classical" stationary equilibrium. The most recent reformulation of the Classical approach has been to model economic fluctuations by adding random shocks to the deterministic stationary state and to underscore the role of incomplete (and asymmetric) information in the influence of economic policy on real equilibrium variables. The outcome of this reformulation is a model that preserves very cleverly stationarity while incorporating in the analysis something that looks like business cycles (Barro [5], Kydland and Prescott [40], Lucas [42, 43, 44, 45, 46], McCallum [47], Sargent and Wallace [50]).2 An important implication of many, but not all, of these models is that the systematic (deterministic) component of economic policy can have no real effect whenever it is anticipated by the private sector. The arguments put forward by the opposing (Keynesian) school appear often, by contrast, almost exclusively defensive. Proponents of this school seem to accept in effect the theoretical validity of the claim according to which the long run equilibrium positions of a competitive economy may be described by deterministic stationary states. They tend to question primarily the practical relevance, for the description of short run and medium run phenomena, of the mere notion of a long run stationary equilibrium and of its underlying assumptions. The list is long: prices cannot move fast enough to clear markets, anticipations adjust only slowly, New Classical macroeconomic models rely upon extremely specific assumptions concerning the distribution of information, the Classical stationary state may be unstable or convergence to it may be so slow that it becomes practically irrelevant in calendar time, and so on.3 The purpose of this work is to demonstrate that, by contrast to currently accepted views, a competitive monetary economy of which the environment is stationary may undergo persistent and large deterministic fluctuations under laissez faire; that these cyclical fluctuations may display moreover the sort of correlations that recent Classical macroeconomic models have sought to incorporate, even under complete information, without having to make the ad hoc assumption that cycles are due to exogenous shocks; and finally, that the Government, in the face of such autonomous deterministic fluctuations, has indeed in principle the power to stabilize the economy by implementing simple deterministic—and publicly known—countercyclical policies. Although one of the goals of present work is to develop concepts and methods that can be applied, it is hoped, to a larger class of situations, the analysis will proceed by studying a particular example, i.e., an overlapping generations model very much like the model developed by R. J. Lucas in his seminal paper [42], with the noticeable difference that we shall assume that the economy is not subjected to any shock of any sort. Business deterministic cycles will be shown to appear in a purely endogenous fashion under laissez faire. 2

See also Grandmont and Hildenbrand [35]. For an excellent account of the long feud between Keynesian and (Old, Neo- or New) Classical economists, see, e.g., Tobin [58]. 3

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Markets will be assumed to clear in the Walrasian sense at every date, and traders will have perfect foresight along the cycles. The origin of these endogenous deterministic cycles will be seen to be the potential conflict between the wealth effect and the intertemporal substitution effect that are associated with real interest rate movements. Business cycles will emerge in particular when the degree of concavity of a trader's utility function—which we shall measure, although there is no uncertainty in the model, by the so-called Arrow-Pratt "relative degree of risk aversion"—is sufficiently higher for old agents than for younger ones.4 An important outcome of the analysis will be that cycles of different periods will typically coexist—in some cases, there may be an infinity of these. The techniques employed to study the occurrence and the stability of such business cycles will be borrowed partly from recent mathematical theories that have been constructed by using the notion of the ' 'bifurcation'' of a dynamical system in order to explain the emergence of cycles and the transition to turbulent ("chaotic") behavior in physical, biological, or ecological systems.5 The equilibrium output level will be shown to be negatively related to the equilibrium level of the real interest rate. A similar relation exists (but in the opposite direction) between equilibrium real money balances and real interest rates. These relations hold both in the long run, i.e., along business cycles, and in the short run, i.e., on the transition path, and whether movements of the real interest rate are anticipated or not. The basic ingredient there will be the condition that older agents have a higher marginal propensity to consume leisure. Finally, monetary policy by means of nominal interest payments will be shown to be extremely effective. A permanent change of the rate of growth of the money supply by these means will be seen to be superneutral. Yet, it will 4 The idea that endogenous deterministic cycles may emerge in an overlapping generations model is already present in the literature. For instance David Gale [30] presents a numerical example of a cycle of period 2, while David Cass [12], Brock and Scheinkman [11] discuss the possibility of their occurrence. Moreover, the deterministic business cycles that are the subject of this paper may be assimilated to what has been called recently "sunspots" equilibria. The analysis of sunspot equilibria has been started by Karl Shell [51] and later developed by C. Azariadis [2], D. Cass and K. Shell [13,14], and Spear [53]. Independently of the present work, C. Azariadis and R. Guesnerie [4] establish in a model similar to ours that there are sunspots equilibria if and only if deterministic cycles exist. A recent paper by P. Diamond and D. Fudenberg [20] provides an example of an endogenous rational expectations business cycle in a search equilibrium model, in an otherwise stationary environment. Finally, Benhabib and Nishimura [8,9] investigate the occurrence of endogenous deterministic cycles in a neoclassical growth model. 5 One important mathematical reference in this field is Collet and Eckmann [16]. For a stimulating review of various applications of the theory, see May [48]. Part of this theory has been already applied in economics or game theory, in particular by Benhabib and Day [6,7], Dana and Malgrange [17], Day [18,19], Jensen and Urban [39], Rand [49]. Some results of this theory that seemed (to me) relevant and useable by economic theorists are reviewed in Grandmont [34].

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be shown that there exists a very simple deterministic counter-cyclical policy that enables monetary authorities to stabilize completely business cycles and to force the economy back to the unique (Golden rule) stationary state. Due to the nonlinearity of the model, such a policy affects not only the variances of real equilibrium magnitudes but also their means. The central point here is that there are typically many long run periodic equilibria that coexist under laissez faire, and that policies may be designed which force the economy to settle at only one of these—here the stationary state.6 The paper is organized as follows. We specify in Section 1 the structure of the model and study there the traders' microeconomic behavior. Long run periodic equilibria with perfect foresight are defined in Section 2. The issue of the stability of these periodic equilibria is partly analyzed in Section 3. The existence, the multiplicity, and the bifurcations of periodic competitive equilibria are investigated systematically in Section 4. The long run and short run relationships between equilibrium output or real balances, and anticipated or unanticipated real interest rates are established in Section 5. Finally, the impact of monetary policy through deterministic proportional money transfers is dealt with in Section 6. A few concluding remarks are given in Section 7, while some proofs are gathered in a separate Appendix. Section 4 is the most technical, and although it is in some respects the most interesting one, the nonmathematically oriented reader may skip it on a first reading. Section 5 and Section 6 can be read right after the first two sections of the paper.

1. Behavioral Assumptions We shall use the simple structure of an overlapping generation model, with a constant population and without bequests, in which agents live two periods only. For simplicity we shall assume that all agents are identical, or equivalently that there is a single member in each generation. There will be accordingly two agents in every period, one "young" and one "old." The model involves one perishable consumption good, which is produced from the labor that is supplied by consumers. There is no production lag, and producing one unit of output requires one unit of labor. Young consumers have the opportunity to save part of their income in each period by holding a nonnegative money balance. For the most part of the paper, the money stock will be assumed to be constant over time. It will be denoted M > 0. 6 The idea that there may be a large number of "rational expectations" or perfect foresight equilibria in a monetary economy and that accordingly, one possible role of policy is to select one of these, is also already present in the literature. In particular, this fact has been well known by theorists who worked with the overlapping generations model; see, e.g., Geanakoplos and Polemarchakis [31]. The point has been most forcefully reiterated recently by F. Hahn [38].

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At each date t, there are competitive spot markets for the consumption good, for labor, and for money. The money price of the good and the money wage rate need not be distinguished, however, since profit maximization requires that the equilibrium real wage be equal to unity. An agent's intertemporal characteristics may then be described as follows. Consumption cT in each period T of his life (T = 1,2) must be nonnegative. On the other hand, it is assumed that the agent has a labor endowment in each period of his life, I*, and that his labor supply ZT, or equivalently his consumption of leisure I* - ZT, must satisfy 0 < Z* - ZT < I* for T = 1,2. The agent's intertemporal tastes are represented by the separable utility function £/,((;!,Z*! - Z,) + U2(c2, l*2 - l2) which is denned on the set of cT and Z* ZT that satisfy the foregoing feasibility constraints. We shall assume: ASSUMPTION (l.a): 1% > 0 and I* > 0.

(l.b): E/T(cT, Z* - ZT) is continuous, increasing in each argument, and strictly concave for T = 1,2.

ASSUMPTION

We consider now the decision problem that a young agent has to solve at an arbitrary date. Let/? > 0 be the money price of the good that he observes in the current period and let pe > 0 be the price that he expects for the next date (current and expected money wages are taken to be equal to p and p" respectively). The agent's problem is then to choose his current consumption c, > 0, his current labor supply 0 < Z, < I*, his demand for nominal money m > 0, and to plan for the next date his future consumption c 2 s 0 and labor supply 0 < Z2 < 1%, so as to maximize his intertemporal utility function subject to the current and expected budget constraints (1.1)

p{ci - /,) + m =

0

and

p'(c2 - l2) =

m.

Under Assumptions ( l . a ) , (l.b), the problem has a unique solution. The optimum value of (cT - ZT) depends only on 9 = p/pe, or equivalently on the consumer's expected real interest rate 0 — 1 , and is denoted zT(0), for T = 1,2 (absence of money illusion). Since one unit of labor yields one unit of good, these values can be interpreted as the trader's current and expected excess demands for the good. On the other hand, the agent's demand for money md(p, pe) is given by (1.2)

m"(p, p') =

-pz,(9) =

P'z2(Q)

which implies of course (1.3)

8zi(6) + z2(9) =

0

for every 9 > 0.

It is convenient to decompose the trader's decision problem into two subproblems as follows. First, given aT a 0, the agent maximizes UT(cr, Z* ZT) subject to

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87

cT + (/* - /T) =

aT.

Under Assumptions (l.a), (l.b), this problem has a unique solution which determines the agent's consumption cT(a^) and his labor supply /T(aT) in each period of his life as a function of his "real wealth" aT, for T = 1,2. Let VT(aT) be the maximum of UT subject to (1.4) and consider the problem of maximizing V,(a,) + V2(a2) with respect to ax > 0, a2 > 0, m > 0 subject to (1.5)

/?«[ + m = p/f

and p*^ = p 8 ^ + m.

By comparing (1.4) and (1.5) with (1.1), it is clear that the optimum values of aT - I* and of m that result from this last problem coincide indeed with zT(6) and md(p,pe). The argument shows that considering a variable production model like this one is formally equivalent to looking at an exchange economy (i.e., without production) in which every agent is endowed with the quantity I* of the good in each period of his life. The procedure will also allow us to state our assumptions more compactly on the indirect utility functions VT instead of deriving them for the original function UT. Under Assumptions (1 .a), (1 .b), each VT is continuous, strictly concave and increasing. We shall make in fact the stronger assumption: (l.c): For each T = 1,2, the indirect utility function VT is continuous on [0, + °°) and twice continuously differentiable on (0, +°°), with K ( O > 0, l i m ^ o K K ) = += o, y ' X ) < o.

ASSUMPTION

We end up this section with a brief analysis of a few elementary facts about the excess demand functions z, and z2 that will be used repeatedly in the sequel. Lemma 1.1. Assume (l.a) and (l.c) and let 6 = V[(l*)/V2(l*2). Then z,(9) and z2(0) are continuous on the open interval (0, + °°). Moreover, (i) z,(9) = z2(9) = 0 whenever 9 < 0, and - /* < Zl(6) < 0, z2(0) > 0 whenever 9 > 9; (ii) for every 9 & 9, one has (1.6)

V'm + zi(0)) = evm

+ z2(9)).

We shall not give a formal proof of this statement, which follows from elementary considerations. Note that in view of (1.3), the relation (1.6) may take the equivalent form (1.7)

-Zl(8)V;(/t + Z!(9)) = z2(Q)V[{l% + z2(6))

for every 6 > 6.

The above lemma implies, in view of (1.2), that a consumer has a positive demand for money if and only if 0 > 9. We shall be concerned in the sequel with the case where the agents have a positive demand for money when the price of the good is constant over time. This corresponds to the Samuelson

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case, according to the terminology of David Gale, in his excellent paper [30] on the overlapping generations model. Formally, we have the following assumption. ASSUMPTION

(l.d): 6 = V\Q*^/V'2{1\) is less than 1.

The next fact gives more information about the variation of Zj and z2 with 6. Lemma 1.2. Assume (1 .a) and (1 .c). Then the restrictions of the excess demand functions z,(9) and z2(6) to the interval [9, + °°) are continuously differentiable. For every 6 > 9, z!(6) = [V2(/*2 + z2(9)) + z2(8)V2'(/*2 + z2(9))]/A, z2(6) =

-\V\{l*i

+ zi(6)) +

in which A = V[{l*x + z,(9)) + 02V2(/*2 + z2(9)) < 0. In particular z2(9) > 0 for every 8 ^ 9 . Moreover z2(9) diverges to + °° whenever 9 tends to + °°. The first part of this statement is obvious by differentiating (1.6) or (1.7) and by using (1.3). On the other hand, the fact that z2(9) diverges to +°° whenever 6 increases without bound is not difficult to verify. Indeed, we get from (1.3) that when 9 tends to + °°, z,(9) must tend to 0 if z2(9) remains bounded. But in that case, the left hand side of (1.6) is bounded while the right hand side diverges to + °°. This is a contradiction which shows that lim^ + =oZ2(9) = + 00.

The foregoing analysis gives some insight about the consequences upon a trader's behavior of a variation of 9 (or of his "expected real interest rate" which is given by 9 - 1). A change of 9 generates an intertemporal substitution as well as income or wealth effects. A rise of 6 induces always an increase of z2(6) because intertemporal substitution and income effects work in the same direction for future consumption. On the other hand, the induced variation of z,(9) is ambiguous. We know indeed from Lemma 1.2 that — z2(9)/ z[(S) = 9. Therefore z',(6) < 0, and thus by continuity z[(Q) < 0 if 0 is larger than but close enough to 0. However, the sign of z',(9) is a priori indeterminate for large values of 6, because income and substitution effects are working in opposite directions for current consumption. As a matter of fact the origin of the business cycles that are going to be analyzed in the present paper is precisely this conflict between intertemporal substitution and wealth effects. Examination of the expression of z',(0) and z2(9) suggests that an important role in this regard should be played by the so-called Arrow-Pratt relative degrees of risk aversion which are well defined whenever aT > 0. These expressions measure in effect the degree of concavity (curvature) of each VT. We shall use the following assumption:

ENDOGENOUS BUSINESS CYCLES

ASSUMPTION

89

(l.e): R2(a2) is a nondecreasing function of a2 for every a2 > 0.

For a justification of such an assumption (in a context involving uncertainty), see Arrow [1, Ch. 3]. It is then easy to get the following fact. Lemma 1.3. Assume (l.a) and (l.c). Then for every 0 > 0, z[(d) < 0 if and only if R2(l% + z2(9)) < (l\ + z2(9))/z2 (9).7 Accordingly, (i) if R2{a2) < 1 for all a2 > 0, then z',(9) < 0 for every 0 > 9; (ii) if (l.e) holds and if there exists a2 > 0 such that/?2(a2) > 1, then there exists a unique 9* > 0 such that z[(Q) < 0 for every 9 < 0 < 0*, z',(0*) = 0 and zj(0) > 0 for every 0 > 9*. The claim that z',(0) < 0 if and only if R2(l*2 + z2(0)) < (If + z2(G))/z2(9) can be verified immediately by looking at the expression of zj(0) in Lemma 1.2. Then if R2(a2) ^ 1 for all a2 > 0, the left-hand side of this inequality never exceeds 1, while the right-hand side is always greater than 1 whenever 6 > 6 , which shows (i). If Assumption (l.e) holds, the left-hand side of this inequality is a nondecreasing function of 0. When 9 tends to + °°, z2(0) tends also to + °° and thus under the assumptions of (ii), R2(l*2 + z2(0)) exceeds 1 for 0 large enough. On the other hand, (/* + z2(0))/z2(0) decreases from +°° to 1 when 9 rises from 0 to + °°. There is thus, by continuity, a unique 9* > 9 such that R2(l% + z2(9*)) = (l*2 + z2(8*))/z2(0*) and it is clear that z',(9) < 0 whenever 0 < 0 < 0*, z',(9*) = 0 and z',(0) > 0 when 9 > 0*. It may be useful to illustrate our findings by drawing in the plane (a,, a2) a consumer's offer curve, that is, the locus of all points of coordinates a, = l\ + z,(0), a2 = /*, + z2(0) when 0 varies. The result is shown in Figure 4.1. In view of (1.3), finding the point of the offer curve that corresponds to 0 amounts to looking at its intersection with the intertemporal budget line of equation 0(a, - IX) + (a2 - l*2) = 0. According to the previous lemmas, the offer curve is smooth and goes through the endowment point A = (I*, l\)—this corresponds to 0 ^ 9. Its normal there is the vector (9, 1). The curve lies below the 45° line AB when 9 < 9 < 1, and above when 9 > 1. Figures 4.1a and 4. lb are drawn under the assumption that 9 < 1. Of course, if 0 > 1, the curve would lie entirely above AB. Figure 4.1a corresponds to the case considered in (i) of Lemma 1.3, in which R2{a2) ^ 1 for all a2 > 1. The curve is then "monotone," i.e., it has no critical point. Figure 4.1b corresponds to the case in which (1 .e) holds and in which R2(a2) > 1 for some a2 > 0. The offer curve has then a unique critical point corresponding to the value 0 = 0*. 7

We allow the value + °° when 9 = 6 .

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JEAN-MICHEL GRANDMONT

Fig. 4.1a

When 6 < 6* the substitution effect dominates, while the income effect is prominent for 9 > 6*.8

2. Periodic Competitive Equilibria We now turn our attention to the definition of periodic equilibria with perfect foresight. The evolution of the economy is described by a sequence of temporary competitive equilibria. Since an old trader's excess demand for the good at any date is equal to the real value of his money stock, equilibrium at date t for the good and for money is given by (2.1) (2.2)

) + M/p, = m\p,,

pf+i) =

0,

M,

8 W. Brock brought to my attention after the completion of this work that an earlier paper by Brock and Scheinkman [11] contains a discussion of the slope of a trader's offer curve in relationship to the coefficients of relative risk aversion.

91

ENDOGENOUS BUSINESS CYCLES

a2'

e.

/* Fig. 4.1b in which pt is the current price and pe,+, the price that is expected to prevail at the next date.9 The two equations are* of course equivalent by Walras's Law; see (1.2). A sequence of competitive equilibria with perfect foresight is a sequence (/?,) that is a solution of (2.1) or (2.2) with/??+, = p,+, for all t. It is a straightforward matter to see that under Assumptions (l.a), (l.c), such a sequence is a solution of the following system (and conversely): (2.3)

z,(6,) + z2(6,_,) = 0,

(2.4)

P.+1*2(0.) =

M,

in which 6, = p,/pt+l > 6 for all t. Indeed, in view of (1.2), (2.4) is nothing else than (2.2) with/??+1 = p,+ l. This relation tells us that an old trader's excess demand for the good at an arbitrary date t, i.e., M/p,, is under perfect 9

Equilibrium on the labor market has already been taken into account by setting the real wage equal to one and by assuming that output is equal to the traders' labor supply.

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JEAN-MICHEL GRANDMONT

foresight what he planned to do when he was young, i.e., z2(9,_ ,)• Replacing M/p, by z2(Q,_1) in (2.1), with/?f+1 = p,+ 1, yields (2.3). The fact that M is positive implies z2(0,) > 0 and thus 0, > 0. The system (2.3), (2.4) describes the dynamics of the economy under perfect foresight.10 The important property of the system is that it dichotomizes. The good market equation (2.3) determines the equilibrium sequence of 0,— or equivalently of real interest rates 0, — 1—and indeed all real equilibrium quantities, independently of (2.4), i.e., of the level of the money stock M. The level of equilibrium prices is in turn determined by the money equation, i.e., through p,+ 1 = M/z2{Q,).n In fact equilibrium prices are, under perfect foresight, proportional to the money stock M. This is the traditional Quantity Theory {neutrality of money). A periodic {monetary) equilibrium with perfect foresight is defined as a periodic sequence {p,) that satisfies (2.1) and (2.2) with/?f+1 = pl+i for all t, alternatively (2.3) and (2.4). From the above discussion, such a periodic sequence {p,) can be identified with a periodic sequence (0,) that fulfills (2.3) with 0, > 0 for all t, through the relation pt+l = M/z2(0,). Such a periodic sequence (p,) or (0,), say with period k, may be described by the k consecutive values that it takes, i.e., with its periodic orbit {p%, . . . , p*k) or (9*,, . . . , 0*,). The system (2.3), (2.4) induces a well defined (but fictitious) backward perfect foresight (b.p.f.) dynamics. Under Assumptions (l.a), (l.c), the restriction of z2 to the interval [0, + oo) is increasing and maps that interval onto [0, +°°) (Lemma 1.2). It has therefore an inverse. Then (2.3) and (2.4) give rise to very simple difference equations of the form (2.5)

6,_, =

zf'C-

(2.6)

p, = pl+1z2-'(M/p,+

1)

= *(/>,+1)-

The b.p.f map . Of course the b.p.f. dynamics associated to

/*, and thus there could not exist a 0, such that Zl (e,) + z2(e,_,) = 0. A forward perfect foresight dynamics can be defined nevertheless locally in a neighborhood of a periodic orbit (p*t, . . . , p*k) provided that D 1 that are not of the form j = nk — 1 for some integer n s 1. All other derivatives \\ij withy = nk — 1 ^ 1 are nonnegative.

ASSUMPTION

It is instructive to see the implications of (3.h) in the light of Assumption (3.g), in the limiting case where e = 0. Consider a sequence of prices that has period k (p,, . . . , p,-T). Then for any (even aperiodic) sequence (p,, . . . , p,-T) in its neighborhood, a trader's expected price is approximately given by T

ty(p,, . . . , p,-T) ~ ty(Pt, • • • , P,-T) + 2 (P,-j - P,-j)tyj(,Pn • • • , P,-T) j-0

or T

(Pt, • • • , Pt-r) =

2

j-0

(P,-j¥j(Pt,

• • • • P,-T)

since i|/(Xjor, . . . , \p,- r ) is homogeneous of degree 1 with respect to X under Assumption (3.g). On the other hand, this assumption implies that the ex-

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JEAN-MICHEL GRANDMONT

pected price i|; is unchanged for each integer 0 < j < k — 2 whenever the constant sequence (p,_ ; , p,-j-k, • • • >Pt-j-nk> • • •) is multiplied by an arbitrary real number X > 0, and that \\i changes proportionately to X when j = k — 1. One gets thus by differentiation with respect to X in each case 2

>!>;+Kt-l(P>, • • •

»I4-I =

1

Then condition (3.h) ,P,-T),

the first term of this sum being 0 when k = 1. The trader's forecast is locally some sort of "average," with nonnegative weights, of the relevant past prices p,-k+1,pt-2k+i> a n d so on (the current price being given no weight when k = 1). If e is "small," the trader's behavior is close to the one we just described, which does not sound implausible. Then we have the following proposition:21 Proposition 3.2. Assume (1.a), (1.c), and (3.f). Under Assumption (3.g), a cycle (/?*], . . . , p*k) with perfect foresight with period k is a periodic solution of the dynamics with learning (3.6) with the same period and vice versa. Under the additional Assumption (3.h), stability of (p%, . . . , p\) under the backward perfect foresight dynamics ( 0. The phenomenon will be seen to appear in particular, in the constant elas23

One important mathematical reference in this field is Collet and Eckmann [16], For an excellent review of various applications of the theory, see May [48]. Part of this theory has been applied already in economics or game theory, in particular Benhabib and Day [6,7], Dana and Malgrange [17], Day [18,19], Jensen and Urban [39], Rand [49]. Some parts of this theory that seemed (to me) relevant and useable by economic theorists are reviewed in Grandmont [34].

ENDOGENOUS BUSINESS CYCLES

103

ticity case, when old consumers have utility functions that are sufficiently more concave than those of the young traders. A most interesting feature of this model is the coexistence of cycles with different periods. An instance of the phenomenon was already given in Figure 4.2, in which a cycle with period 2 coexisted with the unique monetary steady state. Much more can be said in fact. If the set of positive integers is ordered in a specific way, which we may call the "Sarkovskii's ordering" from the name of the mathematician who discovered it, and if the map

0 when 0 < 0 < 9*, 0*. In that case, the conditions cp'(l) < 0 o r 9 * < 1 or 9* < 0 of ip3 that differs from 1—the corresponding orbit is then described by (0, ip(0), 92(0)). Now the function cp3 maps [0, +oo) into [0, 0) and is continuously differentiable. One has 1 and 1, the function v2 has a unique maximum which occurs at the critical point JJL* of x = uf 1 ° v2. The condition |x* < xCl^*) means in effect thati>2(|A*) > D^JJL*). Since V, is increasing, the inequality |x* < XCM-*) will thus obtain if and only if there exists ji & (JL* such thatv2(|j,*) > v,(|i). In this case we have JJL* < | i < x(M-*) a n d since X'(MO < 0 for all (x > |x* we get 27 The condition is also necessary. For otherwise the orbit (8t, 8$, 8?) of the cycle would satisfy 8* > 1 and thus 8f > 81 > 6f > ip(61), a contradiction. The same reasoning shows that under (l.a), (l.c), (l.d), a necessary and sufficient condition for the existence of a cycle of period 3 is that ip3(8) > 8 for some 8 > 1. The reader will note incidentally that the map

0, 28 we have

The condition x3(lx*) — H-* will obtain accordingly (in fact with a strict inequality sign) if or equivalently if (4.12)

u 2 (|i) — 'ui(9|JL*)-

Since the curve representing the function vl must lie above its tangent at the origin [see (4.8)], we havev,(9(x*) >V[(l*i) 9(x*, and thus (4.12) will automatically be satisfied if (4.13)

v2((L) — Vi'(/f)8(jL*.

To sum up, we have obtained Proposition 4.4:29 28

This follows directly from (4.4) and the fact that z^"1 (|x) > 8 whenever (x > 0. One can also argue directly that the curve x is isometric to the trader's offer curve. 29 The conditions that we get for the existence of a cycle of period 3 are closely related to the

111

ENDOGENOUS BUSINESS CYCLES

Fig. 4.3b Proposition 4.4. Assume (1. a), (l.e), (l.d), (l.e), and a2 = supR2(a2)> 1. Let Vi and v2 be defined by (4.7) and let |x* be the unique maximum of v2. Assume that there exists (i > JJL* such thatu2(n,*) > i>i(|i), that satisfies v2(\L) s D^SIX*) or the stronger condition v2(jx) ^ V[(l*{)Q^*- Then one has30 X2(|x*) < x3d-L*) < M.* < x(M-*)

and there exists a cycle of period 3. Apart from providing a criterion to verify whether a cycle of period 3 exists, the foregoing Proposition gives a way to "generate" utility functions that entail the appearance of a cycle of period 3. Choose (JL* and |x such that |x* < | i < 1%. We then keep fixed all the characteristics of the model, except the old trader's utility function V2 (oru2) which we shall subject to the restriction that (l.e) holds, and the v2 reaches its maximum at JJL*. Choose a value of V'2(l*2 + |x*j that is sufficiently high to satisfy \x.*V'2{l%

> p. V,'(/1f -

conditions obtained by Benhabib and Day [7] in their study of the occurrence of "chaotic" behavior in an overlapping generations model. 30 The same statement could have been made of course by using

0. Assume 0 < l\ < 1 and keep fixed all the characteristics of the model except the old trader's relative risk aversion a 2 that is free to vary. Then if /*, + l%> (l/l*2), the assumptions of Proposition 4.4 are verified, and a cycle of period 3 thus exists, when a 2 is large enough. The proof of this assertion uses only elementary algebra. In the particular case under consideration, we have

aa

VJCIA) =

|j/r°" a n d v 2 O , a2) =

ii.(l*t al

p(l*2 + ( x ) « 2 .

a

We have then 6(a2) = (l%) /(l\) * which is a decreasing function of a 2 and goes to 0 as a2 tends to infinity, since l% 1 - l*2, and that v2(l - l*2, a2) = 1 - l\ for all ct2. On the other hand, when a 2 > 1, the maximum of v2 with respect to |x occurs at |x*(a2) = l\K (1/1%), then /*, + 1% > 1 since 1% < 1. Choose now an arbitrary |i such that 1 - 1% < |1< t\. We have clearly |x*(a2) < |i and D2((x*(a2),a2) > v,(|x) for a 2 large enough. On the other hand, (4.13) will be satisfied if one can choose (i so that for a 2 large enough. But it is not difficult to verify that the left-hand member of this inequality goes actually to 0 when a 2 tends to + °o if |1 + l%> (1/1%). Thus if we pick up (1 such that 1 — 1% < |i< 1% and p- + 1% > (1//*), which is always possible when V\ + 1% > (1/1%), then the assumptions of Proposition 4.4 are fulfilled when a 2 is large, as claimed.

Uniqueness ofy-stable Cycles Theorem 4.3 above shows that cycles of very different periods will typically coexist. We now present a condition which ensures that there exists at most

ENDOGENOUS BUSINESS CYCLES

113

one ip-stable cycle. The condition is essentially that the map 1, and that the critical point of cp satisfies 0* < 1. Consider the functions v, and v2 defined in (4.7) and assume that SU^JJL) 2: 0 for all |x in [0, l\) and 5v2((x) < 0 on the interval [0, x(n-*)L Then S\ < 0 and [0, x(n*)l and Theorem 4.5 applies. In particular, the foregoing condition on Vt and v2 obtains in the constant elasticity case—VT(aT) = a'1-*•*>/( 1 - aT) and thus/?T(aT) = aT >0—if 0 < a, < 1 and a2 > 2. The proof of this statement uses the fact which may be verified by direct computation that the Schwarzian derivative of the composition of two (thrice continuously differentiable) maps/and g is given by (4.15)

S(f-gXx) =

Sf(g(x))[g'(xW + Sg(x).

Applying (4.15) to u, ° x = u2, we get for all |x > 0, The first part of the lemma is then immediate. In the case of a constant relative risk aversion, one has Ui(|x) = p,(^* 012 JJL)-°I and v2((x) = ^(1% + M-)" - Since x>;(|x) > 0 for all jx in [0, I*,), Su, is defined everywhere on that interval. Direct computation shows that SVi is equal to (up to an everywhere positive factor) (1 - a,)[(l - a,)(2 - COM.2 -4/= ?(2 - a,)(x + 6(/= ?)2].

The expression between the brackets is easily seen to be a monotone function of (x on the interval [0, l\), which is positive on this interval. Thus Svx > 0 on [0, 1%) if and only if a t < 1. As for v2, Sv2 is defined for every JJL + (x* on [0, + oo) and the same sort of computation shows that it has the same sign as -( 0 when a 2 > 2, which establishes the Lemma. Remark. Under the assumptions of Theorem 4.5, it is possible to show that if the dynamical system ip has a weakly stable periodic orbit, then the set of points that are not attracted to it is "exceptional," i.e., has Lebesgue measured zero (this follows from Collet and Eckmann [16, Proposition II.5.7]). We do not insist on this otherwise nice result for the backward perfect foresight dynamics has no clear meaning in the present context, beyond the fact that cp-stability implies stability in the dynamics with learning under Assumption (3.h). We mention the result nonetheless since it shows that some claims that "period three implies chaos" are generally unwarranted. In particular, Benhabib and Day [6,7], Day [18,19] use a result of Li and Yorke [41], or a variant of it, to exhibit, under the assumption that there is a cycle of period 3,

116

JEAN-MICHEL GRANDMONT

a "chaotic set," that is a set such that any perfect foresight trajectory starting from it becomes eventually erratic. This discussion shows the limits of such a statement. For if there is a weakly stable periodic orbit, the "chaotic set"— of which the existence is rightly asserted—may well be of Lebesgue measure zero, and erratic behavior may thus be essentially unobservable—see also the remarks in Collet and Eckmann [16, p. 20].

Bifurcation ofcp-stable Periodic Equilibria We consider now the following experiment. Suppose that we take a "onedimensional" family of economies. That is, we index the characteristics of the economy by a real number and then move this parameter over the real line. Sarkovskii's theorem suggests that the emergence of cycles as the parameter moves on should display some regularities. We show below that it is indeed the case for y-stable cycles, and that these regularities are in particular very strong for those cp-stable cycles that should "appear first" according to the Sarkovskii's ordering (4.11), namely cycles that have a period equal to a power of 2. Finally, we shall report on a computer experiment that was done in the case of a constant relative risk aversion, in which the old traders' risk aversion is varied, in fact increased. The result of this experiment is a "period doubling" bifurcation scheme very much like the diagrams that are traditional in the analysis of nonlinear one-dimensional dynamical systems. In order to go on we must make precise what we mean by a (one-parameter) family of economies. Let us index the characteristics of the economy (the endowments /* and the indirect utility functions VT) by some parameter X, which will be taken as a real number that belongs to say, the interval [0,1]. The result of this indexation is denoted /*x and VTX(aT), T = 1,2. We assume of course that for each X the characteristics of the economy satisfy Assumptions (l.a), (l.c), (l.d), and in order to make the problem nontrivial, also Assumption (1 .e) as well as a2x = supa2 R2\(a2) > 1 • The corresponding backward perfect foresight map cpx has then a unique critical point for each X, say 9t, and we shall postulate that 9 t < 1 for all X (again, this condition is necessary in view of Lemma 4.2 to get nondegenerate cycles). We shall say finally that the family is continuous if in addition to these assumptions, the endowments /*x depend continuously on X and if Kx(aT) as well as V^(aT) axe. jointly continuous in (aT, X) for T = 1,2. In order to state the results most clearly, it will be convenient to consider a family of economies that is "rich" enough. We shall say accordingly that the family is full if it satisfies the two following conditions: 1. When X = 0, the second iterate of the critical point Q*o is such that 6% < %). In other words, since from Lemma 4. l(iv) we have already

ENDOGENOUS BUSINESS CYCLES

117

1 < cpo(6*). a n d since ip0 is unimodal, all points of the trajectory of the critical point 6% are greater than 0% and oscillate around the stationary equilibrium 0 = 1. It can be shown that under this condition, then if the map 0, in which the parameter indexing the family has been taken to be the old traders' relative risk aversion a2. Then we know from Lemma 4.6 that if c^ < 1 and a2 > 2, Theorem 4.5 applies. In that case, it suffices to iterate the critical point 8* sufficiently long to discover the unique weakly stable cycle whenever it exists. This procedure was applied in fact to the map x that describes the backward perfect foresight dynamics of equilibrium real balances. The parameter a 2 was made to vary between 2 and 16 by steps of 0.05. For each such value of a 2 on the horizontal axis, the corresponding map x was iterated 300 times, the initial point being the critical point of the map, 35

That is X* = m i n M 0, and expects the price pe > 0 to prevail as well as the nominal interest rate r* > - 1 to be paid in the future. lfxe= 1 + r > 0, his problem is to maximize V^a,) + V2(a2) with respect to a, > 0, a2 s 0, m > 0 subject to (6.1)

pat + m = pt\ andp e a 2 = Pel* + "ix"-

Comparison of this decision problem with (1.5) shows that the optimum excess demands for the good at each age are z,(8), in which 0 = pxe'/pe is again one plus the expected real interest rate, while money demand is 37 The picture is even worse, since there are many two-states Markov "sunspots" equilibria in the present model as soon as there exists a cycle of period 2 with perfect foresight, as shown by Azariadis and Guesnerie [4]. 38 A cycle that is unstable in the b.p.f. dynamics, we recall, is stable in the local forward perfect foresight dynamics; see Section 3.

124

(6.2)

JEAN-MICHEL GRANDMONT

m"{p, p*/x*) = - p z , (9) = p'z 2 (6)A«.

Competitive equilibrium at date t describes again the equality of supply and demand, that is (6.3)

zi(p,*f+i/pf+i) + Mjp, =

(6.4)

m"(p,, puJxf+l) =

0,

M,,

in which p, is the current price, pet+1 the expected price, and rf+1 = xf+1 — 1 > — 1 the nominal rate on money balances that is expected for the next date, and M, is the money supply in period t. As usual, the two equations are equivalent by Walras's Law. The sequence of money supplies M, is given by (6.5)

M, = x,M,-i,

Mo given,

in which M,_x > 0 is the beginning of the period (pretransfer) money stock, and r, = x, — 1 > — 1 the nominal interest rate actually paid by the Government at t on money balances. In that set up, the variables actually controlled by the Government are the values of x,.

Perfect Foresight Given a sequence of transfers (x,), aperiodic equilibrium with perfect foresight is a sequence of equilibrium prices (p,) satisfying (6.3), (6.4), (6.5) with Pi+i = Pt+i and xf+i = x,+ 1 for all t, such that real equilibrium variables (e.g., real interest rates) are periodic. By the same argument that was used to derive (2.3) and (2.4) it is easily seen that such an equilibrium is a periodic solution of the following system: (6.6) (6.7)

Zl(9()

+ z2(e,_,) = 0, Mjp, = z2(e,_,),

in which 9, = ppc,+ Jp,+ l for all t, and the sequence of money supplies M, satisfies (6.5). The equation (6.6) is identical to (2.3) and thus generates the same backward perfect foresight dynamics on real interest rates as in the case of laissezfaire, i.e., 6 t _, = 0 for all v, then the left-hand side of (6.10) is, givenp_ x and Mo, a decreasing function of p0, which tends to + °° when/?0 goes to 0 and becomes negative for p0 large enough. Thus (6.10) has a unique solution in p0, which yields 90 =

i(po/P-i)-

Given 60 in/ = (1 — e, 1 + e), a sequence 9, = £,(p,/p,-,) verifies (6.11) for all t > 1 if and only if it satisfies 6,_, = 2 and from 2 to m + 1 when k = 1). Our purpose, we recall, is to show that ^'(p*,) . . . ^ ( p t ) ! < 1 implies that all eigenvalues of DWk(q*x) have a modulus less than 1. These eigenvalues are the solutions of the characteristic equation det(DW*(«1) - X/) = 0 in which / is this time the unit (T, T) matrix. It is known that given the specific form of the Jacobian matrix DWk(q*f) shown above, this characteristic equation may be written equivalently det (B(X)) = 0, in which B(X) is the (k, k) matrix that is equal to45

S(X) = Xm/ - J£ V-'A, (here / denotes the (k, k) unit matrix). If k = 1, the characteristic equation becomes m

(A.4) 0 = \m - 2 >",,„ + , V - \ Iffca 2, we obtain, by employing the expressions of the matrices An given above, Xm

0

0

-Vi(X)

X"1

0

0

-V 2 (X)

\»

0

0

0

0

0

0

B(X) =

•• •

...

0

0

0

0

X-

0

with, for i = 1, . . . , k, winkm

and

The expansion of the determinant yields det (B(X)) = If we recall that wkm+, =

X"k -

• • • 2>,(X)(H>tlXm +

0 whenever i > 2, we see that det (B(X)) =

45

fct.,(\)

See Wilkinson [59, pp. 33-34].

X"1* - Xfc,(X) • • • bk(\)

X"1

.

133

ENDOGENOUS BUSINESS CYCLES

when t > 2 ; the characteristics equation becomes therefore (A.5)

o = x- - x 2

The last step of the proof is to show that ^ ' ( p * ) • • • * ' ( p * ) | < 1 implies that all solutions of (A.4) when k = 1, or of (A.5) when l > 2 have a modulus less than 1. Consider any complex number X with |X| a 1. If k = 1, let p* be the stationary price and q* = (p*, p*, . . .). Since all w, „ + , = W'n(q*) have the same sign and sum to w e obtain where u, l*2 + (JL

Differentiating again, v'2(^*) = V2(l*2 + |x*)p'(n-*) < 0 since p(n-*) = 0 and p'(px) < 0 under Assumption (l.e).

•

References [1] Arrow, K. J.: Essays in the Theory of Risk Bearing. London: North-Holland, 1970. [2] Azariadis, C : "Self-Fulfilling Prophecies," Journal of Economic Theory, 25(1981), 380-396. [3] Azariadis, C , and Y. Balasko: "The Pigou Effect and the Phillips Correspondence," mimeo, University of Pennsylvania, 1983. [4] Azariadis, C , and R. Guesnerie: "Sunspots and Cycles," CARESS Working Paper 8322, University of Pennsylvania at Philadelphia, 1983. [5] Barro, R. J.: "The Equilibrium Approach to Business Cycles," Chapter 2 in Money Expectations, and Business Cycles. New York: Academic Press, 1981. [6] Benhabib, J., and R. H. Day: "Rational Choice and Erratic Behaviour," Review of Economic Studies, 48(1981), 459^72.

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[7] : "A Characterisation of Erratic Dynamics in the Overlapping Generations Model," Journal of Economic Dynamics and Control, 4(1982), 37-55. [8] Benhabib, J., and K. Nishimura: "The Hopf Bifurcation and the Existence and Stability of Closed Orbits in Multisector Models of Optimum Economic Growth," Journal of Economic Theory, 21(1979), 421^44. [9] : "Competitive Equilibrium Cycles," C.U. Starr Center RP 83-30, revised January, 1984. [10] Bray, M.: "Learning, Estimation and the Stability of Rational Expectations," Journal of Economic Theory, 26(1982), 318-339. [11] Brock, W. A., and J. A. Scheinkman: '' Some Remarks on Monetary Policy in an Overlapping Generations Model," in Models of Monetary Economies, ed. by J. H. Kareken and N. Wallace. Minneapolis: Federal Reserve Bank of Minneapolis, 1980. [12] Cass, D.: "Money in Consumption-Loan Type Models: An Addendum," inModels of Monetary Economies, ed. by J. H. Kareken and N. Wallace. Minneapolis: Federal Reserve Bank of Minneapolis, 1980. [13] Cass, D., and K. Shell: "Do Sunspots Matter?" CARESS Working Paper 8009 R, University of Pennsylvania, 1981. Also appearing in French as "Les taches solaires ont-elles de 1'importance?" Cahiers du Seminaire d'Econometrie, 24(1982), 93-127. [14] : "Do Sunspots Matter?" Journal of Political Economy, 91(1983), 193227. [15] Champsaur, P.: "On the Stability of Rational Expectations Equilibria," CORE Discussion Paper 8324, University Catholique de Louvain, 1983. [16] Collet, P., and J.-P. Eckmann: Iterated Maps on the Interval as Dynamical Systems. Boston: Birkhaiiser, 1980. [17] Dana, R. A., and P. Malgrange: "The Dynamics of a Discrete Version of a Growth Cycle Model," in Analysing the Structure of Econometric Models, ed. by J. P. Ancot. Amsterdam: M. Nijhoff, 1984. [18] Day, R. H.: "Irregular Growth Cycles," American Economic Review, 72(1982), 406-^14. [19] : "The Emergence of Chaos from Classical Economic Growth," Quarterly Journal of Economics, 98(1983), 201-213. [20] Diamond, P., and D. Fudenberg: "An Example of Rational Expectations Business Cycles in Search Equilibrium," mimeo, M.I.T., 1983. [21] Fisher, S.: "Anticipations and the Nonneutrality of Money," Journal of Political Economy, 87(1979), 225-252. [22] Friedman, B.: "Optimal Expectations and the Extreme Information Assumptions of 'Rational Expectations' Macromodels," Journal of Monetary Economics, 5(1979), 23-^1. [23] Friedman, M.: The Optimum Quantity of Money and Other Essays. Chicago: Aldine, 1969. [24] Fuchs, G.: "Asymptotic Stability of Stationary Temporary Equilibria and Changes in Expectations," Journal of Economic Theory, 12(1976), 201-216. [25] : "Dynamic Role and Evolution of Expectations," in Systemes Dynamiques et Modeles Economiques, Proceedings of a C. N. R. S. International Meeting, 259(1977), 183.

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[26] : ' 'Formation of Expectations. A Model in Temporary General Equilibrium Theory," Journal of Mathematical Economics, 4(1977), 167-188. [27] : "Dynamics of Expectations in Temporary General Equilibrium Theory," Journal of Mathematical Economics, 6(1979), 229-252. [28] : "Are Error Learning Behaviours Stabilizing?", Journal of Economic Theory, 3(1979), 300-317. [29] Fuchs, G., and G. Laroque: "Dynamics of Temporary Equilibria and Expectations," Econometrica, 44(1976), 1157-1178. [30] Gale, D.: "Pure Exchange Equilibrium of Dynamic Economic Models," Journal of Economic Theory, 6(1973), 12-36. [31] Geanakoplos, J. D., and H. M. Polemarchakis: "Walrasian Indeterminacy and Dynamic Macroeconomic Equilibrium: The Case of Certainty," mimeo, Yale University, 1983. [32] Grandmont, J. M.: "Temporary General Equilibrium Theory," Econometrica, 45(1977), 535-572. [33] : Money and Value, Econometric Society Series. Cambridge: Cambridge University Press, 1983. [34] : "Periodic and Aperiodic Behaviour in Discrete One-dimensional Dynamical Systems," CEPREMAP D.P. No. 8317. Also available as a Technical Report of IMSSS, Economics, Stanford University and a Technical Report of EHEC, University of Lausanne, 1983. [35] Grandmont, J. M., and W. Hildenbrand: "Stochastic Processes of Temporary Equilibria," Journal of Mathematical Economics, 1(1974), 247-277. [36] Grandmont, J. M., and G. Laroque: "Money in the Pure Consumption Loan Model," Journal of Economic Theory, 6(1973), 382-395. [37] Grandmont, J. M., and Y. Younes: "On the Efficiency of a Monetary Equilibrium," Review of Economic Studies, 40(1973), 149-165. [38] Hahn, F. H.: Money and Inflation. Oxford: Basil Blackwell, 1982. [39] Jensen, R. U., and R. Urban: "Chaotic Price Behaviour in a Nonlinear Cobweb Model," mimeo, Yale University, 1982. [40] Kydland, F. E., andE. C. Prescott: "Time to Build and Aggregate Fluctuations," Econometrica, 50(1982), 1345-1370. [41] Li, T., and J. A. Yorke: "Period Three Implies Chaos," American Mathematical Monthly, 82(1975), 985-992. [42] Lucas, R. E., Jr.: "Expectations and the Neutrality of Money," Journal of Economic Theory, 4(1972), 103-124. [43] : "An Equilibrium Model of the Business Cycle," Journal of Political Economy, 83(1975), 1113-1144. [44] : "Understanding Business Cycles," Journal of Monetary Economics, 3 (Supplement) (1977), 7-30. [45] : "Methods and Problems in Business Cycle Theory," Journal of Money, Credit and Banking, 12(1980), 696-715. [46] : "Tobin and Monetarism: A Review Article," Journal of Economic Literature, 19(1981), 558-567. [47] McCallum, B. T.: "Rational Expectations and Macroeconomic Stabilization Policy," Journal of Money, Credit and Banking, 12(1980), 716-746.

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[48] May, R. B.: "Simple Mathematical Models with Very Complicated Dynamics," Nature, 261(1976), 459-467. [49] Rand, D.: "Exotic Phenomena in Games and Duopoly Models," Journal of Mathematical Economics, 5(1978), 173-184. [50] Sargent, T.J., and N. Wallace: "Rational Expectations, the Optimal Monetary Instrument, and the Optimal Money Supply Rule," Journal of Political Economy, 83(1975), 241-254. [51] Shell, K.: "Monnaie et allocation intertemporelle," CNRS Seminaire d'Econometrie Roy-Malinvaud, mimeo, Paris, 1977. [52] Singer, D.: "Stable Orbits and Bifurcations of Maps of the Interval," SI AM Journal of Applied Mathematics, 35(1978), 260. [53] Spear, S. E.: "Sufficient Conditions for the Existence of Sunspot Equilibria," mimeo, Carnegie-Mellon University, revised December, 1983. [54] Stiglitz, J. E.: "Recurrence of Techniques in a Dynamic Economy," in Models of Economic Growth, ed. by J. A. Mirlees and N. H. Stern. New York: Macmillan, 1973. [55] Stiglitz, J. E.: "On the Relevance or Irrelevance of Public Financial Policy: Indexation, Price Rigidities and Optimal Monetary Policy," NBER Working Paper 1106, 1983. [56] Tillmann, G.: "Stability in a Simple Pure Consumption Loan Model," Journal of Economic Theory, 30(1983), 315-329. [57] Tirole, J.: "Asset Bubbles and Overlapping Generations," Econometrica, 53(1985), 1071-1100. [58] Tobin, J.: Asset Accumulation and Economic Activity, Yrjo Jahnsson Lectures. Oxford: Basil Blackwell, 1980. [59] Wilkinson, J. H.: The Algebraic Eigenvalue Problem. Oxford: Clarendon Press, 1965. [60] Woodford, M.: "Indeterminacy of Equilibrium in the Overlapping Generations Model: A Survey," mimeo. IMSSS, Stanford, 1984.

Competitive Business Cycles in an Overlapping Generations Economy with Productive Investment* BRUNO JULLIEN**

Introduction Since D. Gale [13] pointed out that equilibrium cycles may arise in overlapping generations (OLG) models, it is known that a perfectly competitive economy can exhibit persistent fluctuations under "laissez-faire." The emergence of cycles in an OLG model with no capital accumulation was studied recently by Grandmont [15]. The object of the present paper is to study the dynamics of a two-period OLG model when the technology requires both capital and labor. (See Woodford [19] for a good survey on OLG models.) The existing literature on the subject studies the appearance of cycles near the steady state through bifurcation theory. Both Farmer [12] and Benhabib and Laroque [4] show that fluctuations can occur if savings decrease with the interest rate or outside money is negative. Reichlin [17] shows how there can be cycles when there is no outside money but enough complementarities in the technology. Due to the nature of the bifurcation theory, all the results are local: fluctuations are nearby the steady state and occur when the parameters of the model are close to some "critical value." In this paper an alternative approach is proposed. We will provide a global analysis of the dynamics of the model. Under an assumption of substitutability of the production function, we will show that the long-term behavior of the economy is captured by a onedimensional dynamical system, although the original problem is two-dimensional (prices and capital). The advantage is that global results are available on the cyclical behavior of one-dimensional maps. The theory developed is based on the saving-investment relation in the context of a monetary economy. It will be shown that if the response of savings to the interest rate is nonmonotonic, the dynamics of the economy may induce * Originally published in Journal of Economic Theory 46, pp. 45-65 (Copyright © 1988 by Academic Press, Inc.). ** I am very indebted to Jean-Michel Grandmont for introducing me to the subject and for the constant interest he has shown in my work. I am grateful to Philippe Aghion, Patrick Bolton, Bernard Caillaud, Andreu Mas-Colell, Eric Maskin, and Jean Tirole for useful discussions and comments.

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excessive movements of investment and periodic equilibria may exist. The phenomenon described is monetary: the existence of some nominal asset is necessary to relax the link between investment and aggregate saving, and generate cycles through self-fulfilling expectations on returns. In the present paper this will be done through inflation with a fixed quantity of money, but the approach is valid for any kind of aggregate bubble. The paper is organised as follows. Sections 1 and 2 expose the model and define the equilibria. The model used is very close to Diamond's original OLG model [11], and to its version developed by Tirole [18] for the study of asset bubbles. Section 3 examines the dynamics and shows how to reduce the dimension of the problem. In particular it exhibits an invariant curve to which all periodic equilibria must belong and studies its properties. Section 4 studies the emergence and the nature of cycles.

1. The Model The model is an extension of Diamond's version [11] of the OLG model. At each date t (t goes from 0 to infinity) a single good is produced. This good can be consumed during the period or stored as an input (capital good) for future production. Each generation lives two periods and reproduces identically. The young generation sells one unit of labor inelastically at a real wage wt, consumes the quantity Cu of the good in the first period, and saves the real quantity S, for next period consumption by holding money and capital. The old generation spends all its savings from the previous period. A typical consumer is characterized by his utility function U(Cu,C2t) and faces the following intertemporal maximization problem: Max U(CU, C2I) s.t. C , + 5, < w, C2,SRI+1

• S,

C,, > 0 ,

= i

l , 2 .

R,+1 is referred to as the real rate of interest between t and t + 1. Notice that we have assumed away labor substitution. (For an analysis of intertemporal labor substitution, see Barro and King [1].) Under standard assumptions, the consumer's decision problem has a unique solution characterized by the savings function S(w,,R,+ 1). We assume that the savings function S is derived from a nicely behaved utility function and verifies: ASSUMPTION (l.a).

0 < S(w, R) < w,

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S is continuously differentiable, increasing with w, and R • S(w,R) is increasing with/?, lim S(w, R) = +00, R

lim RS(w, R) = +00.

Production is made through a neoclassical constant return to scale technology. Output per capita is a function of capital intensity, y, = /(&,), where/is a gross production function including depreciated capital. ASSUMPTION (l.b).

/ i s increasing, strictly concave on R + , and C2 on R*+, lim /'(*) e [0,l[,

- » + «>

lim f(k) - kf'{k) = + °°,

*-

lim /(jfc) - */'(*) = 0,

&f'(£) is non-decreasing. The first conditions are standard.1 The last one will be used to insure some regularity of the dynamical system (namely it will be characterized by a monotonic map).2 In our model it means that profits do not decrease when the amount of investment increases. This seems reasonably to assume at an aggregate level, and it requires that there are not too many complementarities in production.3 For example, it is verified by the Cobb-Douglas production functions and by the CES production functions with an elasticity of substitution larger than 1. The competitive behavior of firms leads to the equalization of the marginal productivity of each factor to its cost: w, = f(k,) - k,f'(k,) = W(k,). We will refer to W(k) as the wage function. It is well known that under Assumption (l.b) this function is C1 on R*+, increasing, and maps R + into itself. There is a fixed nominal amount of money M available in the economy. The capital assets are assumed to be sold at their market fundamental value.4 1 The assumptions on the limit properties of the wage and the rate of interest are mainly technical. We could drop them and the main results of the paper would remain the same, but the mathematics would become much more complicated. 2 This assumption could be relaxed but would become less intuitive. We want the revenue from monetary savings to increase with the rate of interest ( 0 , m, + k,+ l = S(W(k,),f'(k,+ 1))

(2.1)

m,+ l = /'(*,+,)m,,

(2.2)

k, > 0, m, > 0, k0 given.

Equation (2.2) expresses that the interest rate on money pjp,+ y is equal to the interest rate on capital f'(kt+,). Equation (2.1) then equalizes the demand and supply of assets, given that the labor market is in equilibrium. We call an equilibrium non-monetary or monetary according to m0 = 0 or m0 > 0. It appears from the system above that, when the savings function is not monotonic, the equilibria cannot be characterized by using a forward dynamic map (i.e., the equilibrium values at date t + 1 are not functions of the equilibrium values at date t or less). On the contrary, (2.1), (2.2) induce a welldefined backward dynamics. Combining the two equations, we can replace (2.1) by m,+ , + k,+ lf'(k,+ l) = S(W(kt),f'(k,+ 1)) -f'(k,+ 1).

(2.3)

Under Assumptions (La) and (l.b), the right-hand side of (2.3) increases from 0 to infinity with k,. So we can invert the relation and express the current capital stock as a function of the future capital stock and the future real quantity of money. We denote this relation k, = g(kt+l, m,+ 1 ).

The function g is defined for kt+ x > 0 and m,+, a 0. Lemma (2.1). g is C1, increasing in each of its arguments, and g(k, m) tends to 0 (resp. infinity) when k goes to 0 (resp. infinity) with m fixed. Proof. S(W(k,),f'(k,+,)) • /(*,+ ,) ~ mt+1 - kl+, f'(kt+,) is C1, increasing with k,, and decreasing with k,+ l, ml+1, so the first part follows from the implicit function theorem. When kl+l goes to 0, S(W(k,),f'(kt+i)) •f'(k,+ 1) stays bounded, which is only possible if k, tends to 0. W(kt) > S> kt+l implies that k, tends to infinity when k,+ x goes to infinity. • model just expresses that there is an aggregate bubble in the economy. For a discussion of bubbles, seeTirole [18].

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Now define the C1, increasing map F as F(k, m) = (g(k, m), m/f'{k))

for

k> 0

and

m > 0.

Then a PF equilibrium is a sequence (kt, /n,),a0 such that 12 0

(*,,/»,) = F(kt+1, mt+1) k, > 0, k0 given m0 > 0 if monetary equilibrium m0 = 0 if non-monetary equilibrium.

(2.4)

2.1 Steady States We make no assumption about the non-monetary economy contrary to Diamond [11] and Tirole [18]. We assume that a monetary steady state exists (the Samuelson case in the terminology of Gale [13]). We refer to this steady state by using the notation X* = (k*, m*). It is uniquely defined by /'(**) = 1

m* = S(W(k*), 1) - k*.

(2.5) (2.6)

From Assumption (l.b), k* is well defined. In order for a monetary steady state to exist we need to assume: ASSUMPTION (2.a). S(W(k*),

l)>k*.

Lemma 2.2. Under Assumptions (l.a), (l.b), and (2.a) there exist a unique monetary steady state X* and at least one inefficient non-monetary steady state

(k > k*).

Proof. The first point has already been shown. A non-monetary steady state is a level k of capital stock which verifies S(W(k),f'(k)) = k. It is inefficient if k > k*. It is straightforward to show that W(k) < k for k large enough. So S(W(k),f'(k)) < W(k) < k, for k large enough. The result follows by continuity

of S(W(k),f(k)).

•

Among all the inefficient non-monetary steady states, the less capital intensive one will be of special interest. We will refer to it as ks = inf{k > k*/g(k, 0) = *}.

3. Backward Dynamics and Cycles A periodic PF equilibrium (cycle) is a periodic sequence (k,, m,)ts0 which verifies (2.4). The function g(k, 0) being increasing, the only periodic non-monetary PF equilibria are the steady states. The set of cycles is in bijection with the set of periodic orbits of the map F: if (k,, m,),s0 is a cycle with period p,

OLG CYCLES WITH INVESTMENT

143

{(k,+p_u nit+p-i), . . . , (kt, m,)} is a periodic orbit of F with period p. The problem of the existence of periodic orbits of two-dimensional map is in general studied by using the bifurcation theory.5 The drawback is that the theory only provides a local analysis. The strategy adopted in the present paper is to exploit the monotonicity of F to reduce the dimension to one. This is the purpose of the present section. We will show that all the periodic orbits must belong to a C1 invariant curve. As a consequence, we will be able in the subsequent section to focus on the dynamics reduced to the curve and to derive a global analysis of the cycles. Define F"(X) = (kn(X), mn(X)), where X = (k, m). Notice that, from the budget constraint, the image of F is bounded by the relation W(g(k,m)) > m/f'(k) + k, which expresses that the young generation saves less than its wage income. It follows that W(k{(X)) > m{{X).

(3.1)

This property, along with the monotonicity of F, allows us to derive the results summarized in the following diagram (fig. 5.1): The real quantity of money increases to infinity along the orbit of a point greater than (k*, m*), and decreases to 0 along the orbit of a smaller point (this is because the interest rate remains smaller than 1 in the former case and larger than 1 in the latter case). Between these two behaviours it is possible to exhibit a set of points with an orbit bounded away from 0 and infinity. This set is a C1 curve. Theorem 3.1. There exist a compact set K C R*+ 2 and a function h, decreasing and C , from R*+ to R*+, such that if we define the sets T = {(*, m) G R*+2A = h(m)} T+ = {(*, m) e R*+2A > Km)} r_ = {(*, m) e R*+2A < Km)} then {F,r + ,F~} is an F-invariant partition of R*+2, and Xer

+

«

X e L «

lim mn(X) = +oo lim mn(X) = 0

X e T O F"(X) e K

for n large enough.

Proof. See Appendix. 5

A bifurcation occurs when the stability of a family of maps changes for some value of the parameters. It is associated with the emergence of an invariant set nearby the steady state. A Hopf bifurcation is characterized by the emergence of an invariant circle and happens when the eigenvalues at the steady state are complex and cross the unit circle. A Flip bifurcation is characterized by the emergence of a period 2 cycle and happens when one eigenvalue crosses — 1. (See Guckenheimer and Holmes [16] for an exposition of the theory, and Benhabib and Nishimura [5,6] and Dana and Malgrange [8] for earlier applications to economics.)

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W(k)

m

m*

Fig. 5.1

Modified versions of the theorem can be found for different sets of assumptions. We give here two of them, skipping the proof for simplicity. It can be obtained by slight modifications of the original proof. If/' (k) remains bounded when k goes to 0, the function h is defined for m in a bounded interval. If W(k) remains bounded when k goes to infinity, F"(X) will eventually not exist forX

er + .

The useful and robust result is that all cycles belong to the invariant curve. In fact, for cyclical points, we can find some additional information on the derivative of h. • Proposition 3.1. Let X = (k, m) be a cyclical point of order p. Then X belongs to F and (1, h'(m)) is the eigenvector of DFP{X) associated to its smallest eigenvalue (it has two distinct real eigenvalues). Proof. Let X = (k, m) be a cyclical point of order p (p is the smallest integer such that FP{X) — X). Then X belongs to F because its orbit is bounded. Define V = (1, h\m)), V is tangent to F at X. Since F is C1 and F-invariant, DF"(X) • V is also tangent to F at X. Therefore there exists a that DFP(X) • V = aV. V is an eigenvector of DFP(X). But DFP(X) has all its elements positive. It has two distinct real eigenvalues. The coordinates of the eigenvector associated to the largest eigenvalue have the same sign. The coordinates of the eigenvector associated to the smallest eigenvalue are of opposite signs. h'{m) is negative, so a must be the smallest eigenvalue. D The next step is to characterize the limit properties of h(-). Since the equi-

OLG CYCLES WITH INVESTMENT

145

libria are bounded by the budget constraint relation (3.1), we are only interested in the behaviour of h(m) when m becomes small. Proposition 3.2.

h(m) tends to ks when m goes to 0.

Proof. Let k0 be the limit of h(m) as m goes to 0 ( + °° allowed). Then ko> k* = h{m*). Suppose first that k0 < ks. As ks is the smallest k> k* such that g(k, 0) = k and, by Assumption (2.a), g(k*, 0) < k*, Vk E [k*, ks[, g(k, 0) < k. There exists n such that g"(k0, 0) < k* and therefore Fn(k0, 0) < (k*, m*). By continuity of F, for m small enough, F"(h(m), m) < (k*, m*), which is impossible. Suppose now that ko> ks, then for m small enough (h(m), m) > (ks, 0). This is true for all the iterates of (h(m), m), which implies Vn > 0

mn(h(m), m) > m/(f'(ks)y.

As f'(ks) 5> 7> •• • > 2 •3

>2-5>2-7>---

> 2" • 3 > 2" • 5 > 2" • 7 >

> 2m > • • • > 4 > 2 > 1.

Theorem 4.1. If a cycle of order p exists, then there exists a cycle of order n for any positive integer n such that/? > n in the above order. This theorem has important implications. First it shows that a necessary condition for the existence of some cycle is the existence of a period 2 cycle. Thus period 2 cycles are of particular interest and will be analysed in greater detail. Second, except when there is no cycle, multiple periodic equilibria coexist. In particular a sufficient condition for the existence of cycles of all orders is the existence of a period 3 cycle. Between these two limiting cases, the model is able to present any configuration of periodic allowed by Sarkovskii's order. Let us mention that the existence of a period 3 cycle is also associated with chaotic behavior of some trajectories. This will not be analysed in the paper. (For studies of erratic behaviors in different economic set-up, see Benhabib and Day [2, 3], Day [9, 10].)

4.2 Cycles of Order 2 We know that for m sufficiently small 4>2(TW) is greater than m. Hence if the derivative of 4>2 evaluated at m* is greater than one there will exist a period 2 cycle.

Lemma (4.1).

§'(m*) is the smallest eigenvalue of DF(k*, m*).

Proof. verifies V/n > 0, (h(§(m)), (m)) = F(h(m), m). By differentiating with respect to m, we obtain '(m*) is less than 1 (it is a positive matrix with 1 as a diagonal term). Going back to the savings and production functions, the following condition of existence is obtained:

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Theorem (4.2). is that

A sufficient condition for the existence of a cycle of order 2

S(W(k*),l) - k* + 2Sg(W(k*),l) + 2k*S^.W(k*),l) - 2/f"(k*) < 0. (4.2) Proof. A direct calculus shows that Eq. (4.2) is equivalent to h'^k) > 2, m* tends to 1 when a goes to infinity (notice that x* > 2). We choose the initial point such that mjco = 1. The successive iterates are then *, = expft, + 1 - x + *o[(l + mo)/2]"), k2 =

exp(3 • k i — x),

m, = 1, m2 =

+ * , ) - * + * 2 [(1 + jfcO/2]"),

ku

m 3 = t , exp(3 • kx -

x).

When x > x*, by choosing k0 close enough to x — 1 but larger, the iterates will verify 1 < k0
ra0 = l/£ 0 >

lim m3.

D

Conclusion We have shown how endogenous fluctuations may occur in an OLG economy with investment and money. By choosing a one-sector model with an inelastic supply of labor and a well-behaved production function we emphasized the role of the financial sector. It is clear from the results that what is really determinant is the demand for unproductive assets (money). This demand results jointly from savings and investment decisions. This suggests that it may be fruitful to relax the assumptions made on the productive sector. In particular what happens in a multisector model is an open question. The main technical tool was the possibility to reduce the problem from two dimensions to one. For that purpose, we used a result close to the stable manifold theorem. The main difference is that it is global and applies even when both eigenvalues are unstable. This result relies mainly on the monotonicity of the map and its range of applicability should go much beyond the framework of the model studied.

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Appendix Proof of Theorem (3.1) When there will be no ambiguity we will note F"(X) = (kn, mn). Define IS+ = {X > X*}, IS- = {X < X*}, IS = IS+ UIS., CC = ISC. LemmaA.l.

XSIS+

(resp. IS.) = > mn(X)—>^+co + oo (resp. 0). IS + and IS _ are F-invariant.

Proof. Let X £ / J t , then for all n > 0, F"(X) > X*, so belongs to /5 + , mn + l = mn/f'(kn) > mn, by kn > k*. So the sequence mn is increasing. Suppose it has afinitelimit m > m*, then/'(&,,) must tend to 1 and kn to k*. This implies that g(k*, m) = k* and so m = m*, which contradicts m > m*. By the same argument, if X m* and kn+, s fe*, we have seen that F" + '(X) £ /H. (b) Suppose that mn+1 < m*, kn+1 > ^*. If F"(X) G *T1 then m n+ , > m*/f'(V/-\m*)) and /tn+1 < g(k*, W(k*)), so F"+1(X) G K2. If F"(X) G #2\{X*} then m n+1 > mn. This implies that kn+1 is strictly less than kn. If it was not the case the sequence (F"+I(X))S should be increasing in both arguments so that finally it should lie outside CC. So we obtain m* > mn + 1 > mn a: m*/f'(W-\m*)) and £* < ifcn+1 < *„ < g(/t*, W(/t*)). F"+1(X) G K2.

OLG CYCLES WITH INVESTMENT

Lemma A3. CC.

153 n

If X > X', it is impossible that for all n > 0 , F"(X) and F (X') belong to

Proof. From Lemma A.2 and the fact that F is increasing we can restrict to (X, X') €E K2. LetX > X' andF"(X) e CC andF"(X') e CC, for all n. When n > 0, F"(X) §> F"(A") and both belong to A". Then mn + l/m'n+1 =

{mJm'n)(f{K)/f'(kn))

> mjm'n > mjm\ =

II > 1.

Let 8 be the minimum on {(k, m) £ K, (k', m') e K, m/m' > II} of g(k', m) - g(k', m'): &„+1 — ^+1 a g(^, mn) - g(K, m'n) > 8 > 0 . Let 8, be the minimum on {(k, m) e K, (k', m') 1 and mn + 1/rn'n+1 > h^mjm'n). This ratio must tend to infinity, which is impossible since K is bounded. •

Existence ofh We construct h(m) as a limit of a functional sequence. Let the sequence (pn)n be given by pn: R* ^ R* g(pn(m), m) = p n _,tm//'(p n (/n))],

po(m) =

This defines a sequence of C1, decreasing functions (for n > 0) which has the property k = pn(m) O £„(£, m) = k*. Let the sequence (/„)„ be given by /„: R* -* R* As before we have defined a sequence of C , decreasing functions which this time has the property m = /„(&) mn{k, m) = m*. It is straightforward to derive that lim lo{k) = +oo, either lim^ + 0O ln(k) = 0, or lim^ +co /„(*:) = (lim^ + xln-,(k)) • (limk^ + xf'(k)). The minimum of /„ tends to 0 when n goes to infinity. For m positive, pn(m) and (/n)"'(m) exist for n large enough. Lemma A.4.

For n large enough and .s non-negative. mi[pn{m), /"'(m)] < A,+S(m), t+'s(m) < sup [pn(m),

^(m)].

Proof. lfk< inf[pn(m), t ' ( m ) ] , then (k, m) < (pn(m), m) and (k, m) < (k, ln(k)). This implies that kn(k, m) < k* and mn(k, m) < m* so thatF"(£, m) e IS. and neither fcn+s(jfe, /n) can be k* nor mn+s(k, m) be m*. The reasoning is symmetric for the other inequality. • This implies first that the sequencespn(m) and l^l{m) are bounded. Let k be an adherence value of any of them. If pn s k s /„"' then kn(k, m) > k* and m,,(£, m) :£ m*. If /-1 < /t < p n then kn{k, m) < A;* and mn(k, m)>m*. So for all n > 0, F"(A, m) e CC.

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Lemma A.3 shows that for a fixed m there can be at most one such k. Therefore the two sequences converge to the same limit. We define h{m) =

lim pn(m) =

lim /^'(m).

The proof of Lemma A.4 shows also that the orbit of (k, m) will remain in CC, end up in IS + or IS _, according to k = , > , of 0, and N be such that sup[a t] | Ifj \m) — pN(m)\ £ e. Then Vn a: A', sup \h(m) - pn(m)\ < sup \l^\ni) - pn(m)\ [a,b]

[a,b]

< sup \lN\m) - pN(m)\ < e.

•

[ 0. But then inf(p,C« 0, X, £ 75f/AT. But A: has been built such that if k, < £* then X, e /Sf/Zf. We have just shown that if (k, 0) is adherence value of X,, k, > k* for f large and k a fes. It follows that m, is decreasing and converges to 0. If k,+ l = g(k,+2, m,+2) > k, = g(k,+ 1, m,+ 1 ), then*;,+2 > fcI+1 (since m,+2 < m,+ 1 ).fc,is either non-increasing or increasing after some time. In both cases it converges to k. So (X,), converges to X, which implies that X is a fixed point. The theorem results directly from Lemmas A.6, A.8, and A.9. •

References 1. R. Barro and R. King, Time-separable preference and intertemporal substitution models of business cycles, Quart J. Econom. 99 (1984), 817-840. 2. J. Benhabib and R. H. Day, Rational choice and erratic behavior, Rev. Econ. Stud. 48 (1981), 459-^72. 3. J. Benhabib and R. H. Day, A characterisation of erratic dynamics in the overlapping generations model, J. Econom. Dynamics Control 4 (1982), 37-55. 4. J. Benhabib and G. Laroque, On competitive cycles in productive economies, mimeo, 1986. 5. J. Benhabib and K. Nishimura, The Hopf bifurcation and the existence and stability of closed orbits in multi-sector models of optimum economic growth, J. Econom. Theory 21 (1979), 421-^44. 6. J. Benhabib and K. Nishimura, Competitive equilibrium cycles, J. Econom. Theory 35 (1985), 284-306. 7. P. Collet and J. P. Eckmann, "Iterated Maps on the Interval as Dynamical System," Birkhauser, Boston, 1980. 8. R. A. Dana and P. Malgrange, The dynamics of a discrete version of a growth cycle model, in "Analysing the Structure of Econometric Models" (J. P. Ancot, Ed.), Nijhoff, Amsterdam, 1984. 9. R. H. Day, Irregular growth cycles, Amer. Econom. Rev. 72 (1982), 406-414. 10. R. H. Day, The emergence of chaos from the classical economic growth, Quart J. Econom. 98 (1983), 201-213. 11. P. Diamond, National debt in a neoclassical growth model, Amer. Econom. Rev. 55(1965), 1126-1150.

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12. R. Farmer, Deficits and cycles, J. Econom. Theory 40 (1986), 77-89. 13. D. Gale, Pure exchange equilibrium of dynamic economic models, J. Econom. Theory 6 (1973), 12-36. 14. J. M. Grandmont, Periodic and aperiodic behavior in discrete one-dimensional dynamical system, in "Contribution to Mathematical Economics" (W. Hildenbrand and A. Mas-Colell, Eds.), North-Holland, Amsterdam, 1986. 15. J. M. Grandmont, On endogenous competitive business cycles, Econometrica 53 (1985), 995-1045. 16. J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields," Springer-Verlag, New York, 1983. 17. P. Reichlin, Equilibrium business cycle and stabilization policies in an overlapping generations economy with production, J. Econom. Theory 40 (1986), 89-103. 18. J. Tirole, Asset bubbles and overlapping generations, Econometrica 53 (1985), 1071-1100. 19. M. Woodford, Indeterminacy of equilibrium in the overlapping generations model: A survey, mimeo, IMSS, Stanford University, 1984.

Endogenous Fluctuations in a Two-Sector Overlapping Generations Economy* PIETRO REICHLIN

Introduction It is widely known by now that competitive equilibrium models may generate endogenous fluctuations and that these fluctuations are entirely consistent with complete markets and perfect foresight. Most of the literature on the subject has focused on the overlapping generations (OG) model. In fact, it is generally recognized that this is the only neoclassical model requiring sequential trading (i.e., having a truly dynamic structure) in the absence of market imperfections. OG economies may have a very complicated dynamics when agents live for several periods and/or one allows for the existence of a multiplicity of goods. In particular, it has been shown [see Kehoe and Levine (1985)] that when the number of consumers in each generation is sufficiently high, the dynamic properties of the OG model are completely "generic" up to very mild restrictions. This is a consequence of the Sonnenschein-Mantel-Debreu Theorem, since the dynamics of equilibrium trajectories of OG models is completely specified in terms of aggregate excess demand functions. Therefore, one may not be surprised that the existence of equilibrium fluctuations is a possible outcome of dynamic disaggregative economies. The existing literature, however, has shown that complicated dynamic phenomena may exist in OG models with a very simple structure. In other words, one does not need to rely on the heterogeneity of agents' preferences and the multiplicity of goods in order to prove the existence of endogenous fluctuations. From the studies of Benhabib and Day (1982) and Grandmont (1985), we know that a one-good, pure exchange OG model in which identical agents live for two periods may generate periodic equilibrium trajectories and also "chaotic" dynamics around the golden rule (GR) stationary state if the agents' excess demand functions are characterized by a significant income effect.' * This is a revised version of a paper presented at the Workshop in Mathematical Economics, San Miniato, September 1986. 1 It may be noticed that the same phenomenon which is responsible for the cycles in pure

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Following these findings, the natural question has arisen whether the income effect is the only potential cause of endogenous fluctuations in aggregative OG models. In order to investigate this question, one has to look at economies whose structure is not completely characterized by agents' preferences. For this reason, some work has been focused on OG models with production in the setting developed by Diamond (1965). As it turns out, by introducing productive capital into the OG economy, one can prove the existence of endogenous cycles without assuming a strong income effect. Farmer (1986) has shown that when the GR steady state involves negative outside money, a Hopf bifurcation may occur, giving rise to the emergence of invariant circles. The same phenomenon may be observed in the vicinity of a nonmonetary stationary state if the agents' labor supply is wageelastic and there is enough complementarity in the production function [see Reichlin(1986)]. A study by Benhabib and Laroque (1986) has confirmed and extended these results in the framework of Diamond's model with wage-elastic labor supply and outside money. The study is an attempt to characterize different sets of economies in terms of their dynamic behavior by looking at some relevant parameters, mainly, the elasticity of substitution between factors in the production function (ESP) and the interest rate elasticity of saving (IES). The following statements can be derived: a. Cycles are either associated with a negative IES or a low ESP. However, an increased ESP will always reduce the scope for cycles. b. When the IES is positive, a nonpositive outside money is necessary for a low ESP to be associated with endogenous fluctuations.

Thus, the role of a low ESP in generating cycles is not clearly separated from the existence of an evolving debt of the private sector and/or a labor supply response to wage-rate fluctuations. Moreover, there seems to be a tight relation between the type of cycles and the characteristics of the economies. In particular, the existence of chaotic dynamics is only associated with economies where agents' excess demand functions are characterized by a very strong income effect. On the other hand, the existence of trajectories lying on a closed curve around the stationary state exchange economies, i.e., the strong income effect, is generally responsible for the indeterminacy of stationary equilibria. This is not surprising because the emergence of cycles in these models is always detected using bifurcation theory, i.e., they arise when, under a perturbation of the economy, some eigenvalue of the associated dynamical system crosses the unit circle at a stationary state. As noted by Woodford (1984), examples characterized by indeterminate Pareto optimal steady states must always involve excess demand functions displaying significant income effects even within more general pure exchange OG models. Therefore, it may be conjectured that the existence of cycles in the vicinity of a GR equilibrium are associated to the income effect even in more disaggregative models.

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PIETRO REICHLIN

has only been proved in economies where private agents' net wealth is negative and evolving over time [Farmer (1986)] and/or labor supply is wage-elastic [Reichlin (1986)]. In the present paper, I will slightly modify Diamond's framework in order to evaluate the generality of the above statements. The only departure from the models considered so far concerns the production side. In particular, I will assume the existence of a two-sector technology producing a consumption and a capital good. Moreover, in the attempt to simplify the exposition and confine the analysis to the questions at hand, I will also assume an exogenously fixed labor supply and a zero ESP in both sectors.2 In this way, the role of complementarity in production is isolated from any other potential cause of endogenous fluctuations. Notice also that a zero ESP rules out any factor intensity reversal between sectors, which can be proved to be a potential cause of dynamic complexities in optimal growth models with infinite lived agents [see Benhabib and Nishimura (1985)]. The following analysis shows that: a. periodic and chaotic dynamics are possible with a positive IES; b. invariant circles around golden rule stationary states are not necessarily associated with a wage-elastic labor supply or a nonpositive outside money.

Two parameters of the model are crucial in proving these results: the IES and a purely technological parameter 6 whose magnitude depends on the relative factor intensities in the two sectors of production and the rate of depreciation of the capital stock. The paper is organized in the following way. The next section briefly describes the technology of the model. Section 2 studies the behavior of an autarkic economy, i.e., an economy where no outside money is allowed and private wealth equals the aggregate stock of capital. This economy is shown to possess a wide range of potential dynamic behaviors: from saddle path stability to chaotic dynamics, even for a fixed positive IES. In section 3, agents are allowed to hold outside money besides capital. The possibility of generating invariant circles around a GR steady state involving positive outside money is investigated. This dynamic phenomenon is shown to imply a negative IES. The model that I am going to analyze—the two-sector OG economy—was studied for the first time by Gale in 1972. In his formulation, the young generation consumes a constant fraction of income. Therefore, anticipations about next period variables do not affect agents' saving decisions. It may be worth emphasizing that this assumption is enough to rule out any endogenous fluctuations in my specification of the model. 2

This same model has been used by Calvo (1978) to provide examples of indeterminate perfect foresight equilibria in OG economies.

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1. The Technology The economy that I am going to describe produces a consumption and a capital good, denoted by c and /, respectively, using labor and capital in fixed proportionality. Thus, the technology is represented by the following technical coefficients: a0, b0 = the labor-output ratios in sectors c and /; ax,bx = the capital-output ratios in c and /. Following Diamond (1965), I will assume that the stock of capital K is productive with a lag of one period. Therefore, assuming that the total supply of labor is equal to 1, the equilibrium condition in the factor markets reads: 1 a aoc, + bol, K, isa lC , + 1 + V , + 1

(la)

Now let 8 e [0,1] be the depreciation rate of capital. Then, the latter evolves according to: K, = (1 - 8)tf,_, + /,

(2)

Perfect competition in this economy implies that production activity does not allow positive profits to be earned. Assuming that firms take into consideration capital gains and losses in evaluating profits and letting the price of the consumption good be the numeraire, we have: 1 + (1 - 8)a^, ^ aow, + citR.q,-!

(3a)

q,[l + (1 - b)bx] § bow, + b,R,q,.x

(3b)

where q is the relative price of the capital good, w is the real wage, and R is the real interest factor. An obvious interpretation of equations (3) is the following. Firms are able to finance their production plan from t to / + 1 (i.e., to buy the total stock qj(,) by issuing a bond whose relative price is \/R,+ l. Among the set of equilibria compatible with the present technology, I will only study the ones in which there is no excess supply of capital and labor and there is strictly positive production of c and /. Defining A = axb0 — aobu it is shown in the appendix that these equilibria are associated with prices q and capital stocks K such that ? £ / } = [(.bjaj, (bo/ao)] if A > 0; q*Pq =

[(bo/ao), (fe./a,)] if A < 0;

Ktl\ =

{{bjbo), (fl./oo)) if A > 0,

K e I{ = ((a,/a 0 ), (*i/*o)) if A < 0.

In particular, the existence of an optimal production plan with full employment and positive production implies that q and K belong to one of the above intervals. At the same production plan (3) is satisfied with equality and:

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PIETRO REICHLIN

w, g 0, [R.q.-i - (1 - h)q,] & 0 l

Now define 9 = A[a0 - (1 - b)A]- ,x, =

Then, it follows that:

for all t.

R^.-i-

K, = ( fll /A) - ( l / e ) * , . , w, = (8/A)[l + bx (1 - 8) - a,*,]

(4) (5) (6)

for A * 0, 6 = £ 0, and K, = a,/oo w, = (I/a,) + (fc./a.Xl - 8 - /?,)

for A = 6 = 0. The parameters A and 0 play a central role in the dynamics described in (4)(6). The first may be thought of as a measure of the curvature of the factor price frontier, and its sign depends on the relative capital intensity of the two sectors. The latter crucially depends on the magnitude of 8 and A and affects the dynamics of K and the relation between the interest factor x and the other prices. In order to avoid pathological situations, I will assume: (Al) (1 - 8)A * a0. At a stationary state (w*,q*,R*), prices are linked by the following relationships: q* = (&o/ao)[l + 6(1 - 8)(1 + e/J*)]- 1 w* = (1A,)[1 " b,(R* - (1 - 8))] - 1/8

if 8 g 0 if 8 < 0.

As far as the capital stock is concerned, a stationary state is readily found to be: K* = ai(a0 + 8A)- 1

In order to ensure that the stationary stock of capital is compatible with a full-employment equilibrium, I assume: (A2) bb, < 1 if A > 0; [1 + ha^Ja^

> 8i, > 1 if A < 0.

In fact, K* e Vk (i = 1,2) implies (A2). Equations (5) and (6) show that there is a one-to-one relation between xt, w,, and q,. Thus, from the dynamics of a sequence {x,\ t = 1,2, . . .} we can infer the dynamics of the sequence {w,,q,,Rt+i; t = 1,2, . . .}, whereas sta-

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163

tionary values of w, q, and R correspond to a stationary value of x and vice versa. For this reason, in order to study the evolution of prices, I can focus on x with no loss of information. Now, qel'q(i = 1,2) implies: x e Q = [*0(l - 8)/oo, (I/a,) + b,{\ - 8)/aJ if A s= 0, xePx = [(I/a,) + fc,(l - 8)/a,, &0(l - 8)/a 0 ] if A =S 0.

Notice that, by the above assumptions, if 0 < 0, then A < 0. In fact, let 0 < 0 and A > 0. Then, we have (8/A) < 0. By (5), in order for wages to be nonnegative, we have to impose x = £ [(I/a,) + b,(l — SVaJ, i.e., x e Px. However, this is a contradiction, since A > 0 implies x e l\.

2. A Wealth-Capital Economy As was mentioned in the introduction, the model that I intend to set up is a general equilibrium model based on Diamond's (1965) classical paper. In particular, I will assume that there are two types of identical agents in any t, young and old. Population and preferences are constant over time and the economic activity is started by a generation of old people whose exclusive role is to initiate the production process with a given stock of capital Ko. All consumers live for two periods and supply labor inelastically only when they are young. For simplicity, no bequests are allowed, so that, as in Diamond's original formulation, old people consume all of their wealth and young agents start life with an initial wealth of zero. Finally, I will also assume that young agents are endowed with a given amount E of the consumption good. This assumption, whose importance will become clearer later, essentially widens the set of economies for which there exist equilibrium trajectories with strictly positive private wealth. In the last section it was shown that a competitive production activity implies the zero profit condition (3). Therefore, at full employment we have: c, - w, =

/f, ^,_i Jf,_, -

q,K,.

Letting W, be the private sector's total wealth in t, one can combine the above equation with the consumers' budget constraint to get: /?,W,_, - W, = R.q.-.K,^

- q,K,.

Therefore, a steady state equilibrium (W*, K*, R*, q*) satisfies the following condition: (R* -

1)W* =

(R* -

\)q*K*

and we see that it must be one of two types: wealth-capital (WC): with W* = q*K* golden rule (GR): with R* = 1.

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PIETRO REICHLIN

This classification is used by Gale (1972). The term wealth-capital simply refers to the fact that the corresponding equilibria imply the equality between the aggregate stock of capital and total private wealth. On the other hand, in a GR equilibrium this equality will not hold in generic cases. Notice, however, that, as opposed to pure exchange economies, where non-GR stationary states are called autarkic [see Gale (1973)], both types of equilibria imply the existence of intergenerational trade. In this section I will consider economies (WC economies) in which the only asset available to private agents is productive capital. In this setting, all competitive equilibria imply W, = qji, and R* will be different from 1 in generic cases. In the next section, this assumption will be relaxed by allowing people to hold an alternative asset. The problem now arises of proving that full employment equilibria with positive wealth are feasible. Formally, we need to show that there are nonnegative values of q,, w,, and K, satisfying: q,K, g w, + E In what follows, I will focus on equilibria with K, in small neighborhoods of K*. Thus, consider the constraint: q,K* K*[x, - (1 - 8fc,)/a,]. Now, it is immediate from the definition of l'x that the right-hand side of the above inequality is bounded for all the equilibria that I am going to consider. Therefore, for any neighborhood NofK*, we can clearly find a value E > 0 at whichqK^w + £forany*e/i (i = 1,2),KeN. Without formalizing the representative agent's decision problem, I will simply impose the existence of an aggregate saving function S(w,R), derived from the solution of a standard problem of preference maximization. For simplicity, the preference structure is assumed to be such that: (A3) S(w,,Rt+ , ) i s a C ( r g l ) function with Sw e (0, l),SR±0 = 1,2).

for all x e lx (i

Now, denoting (5) and (6) by w(x) and q(x) respectively, we can define a perfect foresight equilibrium for the economy as a sequence {x,,Kt_ „; t = 1,2, . . .} satisfying: q(x,)K, = S[w(x,), xt+l/q(x,)] KI+1 = [(i + eye]*-* K0 = k with x, e /i and K, e I'k (i = 1 or 2) for all t.

(9a)

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165

Notice that the dynamics of capital is totally independent of the dynamics of x. This implies that the existence of a nonstationary perfect foresight full employment equilibrium from any t on requires a very simple assumption, i.e., the stability of equation (9b). More precisely: (A4) Either 6 = 0or|8| > 1. From (A4) it follows that: Prop. 1.

If K* > 0, 9 * 0 and (A4) holds, then A is positive and 0 > 1.

Proof. 8 < - 1 implies 8A < - a0, which in turn implies the negativity of K*.lfQ> l , t h e n A > a o [ l + (1 - 8 ) ] - 1 > 0 . • From now on I will only consider parameter values at which K* > 0 and (A4) holds. Then, either the model has a linear factor price frontier, or the consumption sector is more capital intensive. Assume now that system (9) has at least one stationary equilibrium (x*,K*), then the latter satisfies: S[w(x*),R*] = q(x*)K*

where R* = x*/q(x*). Now, by (A3) it follows that there exists a neighborhood of x* where a unique function x,+ r = F(x,,Kt) is implicitly defined by: q(x,)K, =

Differentiating with respect to x at the stationary state we get: Fx(x*,K*) = (R*/e*)[S%(\ + 6) - 9(1 + e*)]

where e* is the elasticity of saving with respect to the interest rate and 5*, the marginal propensity to save out of wage income both evaluated at (x* ,K*). The expression for Fx{x*,K*) shows that the local dynamic behavior of equilibrium trajectories depends on the magnitude as well as on the sign of a set of parameters (e*,S\,%) describing the preferences and the technology of the model. Among these parameters I will single out the influence of 8 on the dynamics of (9). Notice that: Prop. 2. If Fx(x*,K*) + 1 for some 8', then there exists a continuous and differentiable function x* = x*(Q) for 8 in a neighborhood of 8'. Proof. Letting/(x,8) = S[w(x),x/q(x)] - q(x)K*, we have: fx = (e/R)q(x)K* [1 - Fx(x,K*)]. Thus, if the above expression is different from 0 at (x*,8'), by the implicit • function theorem, we get the proposition.

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PIETRO REICHLIN

Let 9° be defined as the parameter value at which the present economy is such that: FX[X*(B°),K*(Q°)] =

-l.

Under this circumstance, one of the following relations is satisfied: e* = [5*(1 + 9°) - e°]fl* (9°fl* - 1)-' S*(l + 9°) = 9°

if/?* + 1/9° HR* = 1/6°

Now, using bifurcation theory, it may be possible to prove the existence of periodic solutions generated by models in which one of the above equalities "almost" hold. By the flip bifurcation theorem,3 we get the following proposition: Prop. 3.

Assume that:

a. the 9-derivative of Fx[x*(Q),K*(Q)] at 9° is nonzero; b. (l/2)[d2F(x,K*(Q))/dx2]2 + [d3F(x,K*(Q))/dx3]/3 i= 0 at (**,80);

then, (9) generates a two-period cycle for 0 in a neighborhood of 9°. Proof. The Jacobian of (9) at (x*,K*) has the following eigenvalues: X, = Fx(x*,K*),

\=2

- 1/9.

Since a solution (x,,Kt) of (9) through (x*,K*) lies on the curve K — h(x) = K*, the latter is an invariant manifold for the dynamical system. When ki = - 1, h(x) = K* is the center manifold of (9) since h(x*) = K* and h'(x*) = 0. Now consider the equation: xl+1 =

F(x,,K*).

(10)

By the center manifold theorem: i. equation (10) contains all the necessary information needed to determine the asymptotic behavior of local trajectories of (9); ii. all bifurcation phenomena take place on the invariant manifold h(x), i.e., they are exhibited by the trajectories obtained from (10).

Therefore, applying to equation (10) the flip bifurcation theorem4 for onedimensional systems, we get the proposition. • In generic cases, conditions (a) and (b) are clearly verified in the present model. Therefore, "flip" cycles may arise for any 9 > 1 belonging to an interval containing 6°. These cycles are clearly possible even in the absence of an inverse relation between saving and the interest rate as long as 9 + 0. This implies that the existence of more than one sector in the economy and the 3 4

See Guckenheimer and Holmes (1983), p. 158. See Guckenheimer and Holmes (1983), p. 158.

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167

nonlinearity of the factor-price frontier is essential for having cycles with e > 0. In particular, a two-period cycle in which the interest elasticity of saving is positive may occur provided that either: 5* & 8°/(l + 9% R* S l/e ° 6°), R* § 1/6°. Clearly, 0 plays a central role in the dynamics of (9) despite the restriction imposed by the stability assumption (A4). A flip bifurcation may generate a stable or an unstable periodic orbit around the steady state according to the magnitude of the second and third derivatives of F at (x*,K*,Q°). It is worth emphasizing, however, that the existing cycles are only in prices, interest rate, and consumption, and they are associated with a stationary level of the capital stock. This is a consequence of having assumed a particular type of technology, i.e., a production function characterized by a fixed proportionality between factors of production in both sectors. For more general technologies, one would expect to find also a cycling behavior of A". Notice also that the trajectories on the orbits alternate from one side of the steady state to the other along the line K = K*. The following may be an intuitive explanation of the reason why, even when saving is increasing in the interest rate, non-monotonic trajectories may be easily generated by system (9). Suppose that, with the system initially at the stationary point, x, falls. Then, both q, and w, have to increase. While the value of capital will unambiguously rise, the effect on saving may go in either direction. In particular, we have to evaluate the relative strength of the substitution and the income effect and the magnitude of the technological parameters. Because R,+ 1 = x,+ ,/