Optimization and Chaos

A dynamical system is described by a pair (X, f) where X is a nonempty set (called the state space) and f is a function (called the law of motion) from X into X. Thus, if x _t is the state of the system in period t, then is the state of the system in period t + 1.

Dynamic optimization models and methods are currently in use in a number of different areas in economics, to address a wide variety of issues. The purpose of this chapter is to provide an introduction to the subject of dynamic optimization theory which should be particularly useful in economic applications. The current section provides a brief history of our subject, to put this survey in proper perspective. It then provides an overview of the topics covered in the various sections of this review.

In this chapter, we examine a discrete-time aggregative model of discounted dynamic optimization where the felicity function depends on both consumption and capital stock. The need for studying such a model has been stressed in the theory of optimal growth and also in the economics of natural resources.

This chapter presents a new characterization of chaotic optimal capital accumulation by which a chaotic optimal path can be constructed in a simple systematic manner. In the existing literature, it has been demonstrated that optimal capital accumulation may be chaotic in the sense of Li and Yorke (1975); see Boldrin and Montrucchio (1986b) and Deneckere and Pelikan (1986).¹ As Scheinkman’s survey (1990) discusses, this finding indicates that the deterministic equilibrium model of a dynamic economy may explain various complex dynamic behaviors of economic variables. And, in fact, the search for such explanations has already begun.² In the existing literature, however, not much as been revealed with respect to the circumstances under which optimal accumulation exhibits complex nonlinear dynamics.³

The hypothesis that aggregate fluctuations may represent an endogenous feature of dynamic competitive economies with incomplete markets has been advanced in several papers.¹ The role of incomplete markets seems essential for the appearance of cycles in one sector models.² Becker and Foias (1987) and Woodford (1988a) have pointed out that the elasticity of substitution of the production function plays a fundamental role in the existence of cyclic equilibrium paths. In those papers cycles are generated if the substitutability between capital and labor is not too great. Heterogeneity of households is also a crucial component of their fluctuation theories.

In this chapter we demonstrate the possibility of periodicity and indeterminacy of perfect foresight equilibrium under borrowing constraints in a dynamic one-sector economy with infinitely-lived heterogeneous households. As a consequence of indeterminacy we can also prove the existence of sunspot equilibria in this economy. The Ramsey model we consider is basically the same as the one used by Becker (1980), Becker and Foias (1987, 1994), and Becker et al. (1991). It differs from standard neoclassical growth models in two aspects, namely, that households are heterogeneous and that markets are incomplete which prevents households from using their discounted future income to finance present consumption.

where the function h satisfies the Li-Yorke condition for “chaotic” or “complex” behavior. Besides the relative abundance of examples of chaos, yet another theme has rightly been stressed: quite simple models of economic theory may lead to such examples.

In the recent literature, it has been demonstrated that optimal capital accumulation may be chaotic; see Boldrin and Montrucchio (1986) , and Deneckere and Pelikan (1986). ¹ This finding indicates, as Scheinkman (1990) discusses, that the deterministic equilibrium model of a dynamic economy may explain various complex dynamic behaviors of economic variables, and, in fact, search for such explanations has already begun (see Brock, 1986, and Scheinkman and LeBaron, 1989, for example) . In the existing literature, however, not much has yet been revealed with respect to the circumstances under which optimal accumulation exhibits complex non-linear dynamics. In particular, it has not yet been known whether or not chaotic optimal accumulation may appear in the case in which future utilities are discounted not so strongly.²

Whether or not erratic economic behavior can be explained by models of infinitely-lived rational agents has been discussed in the economic literature for some years. A positive answer to this question was provided first by Deneckere and Pelikan (1986) and Boldrin and Montrucchio (1986); that is, optimal paths of capital accumulation may behave chaotically. Until recently, however, the possibility of chaotic optimal dynamics has been established only for the case in which future utilities are discounted extremely heavily. This fact is in line with the intuition obtained from the turnpike literature, namely, that weak myopia tends to simplify the dynamic behavior of optimal growth paths (see, e.g, Brock and Scheinkman (1976), Cass and Shell (1976), McKenzie (1976, 1983), Scheinkman (1976), Yano (1990)). In Sorger (1992a,b, 1995) this intuition is further strengthened by the minimum impatience theorems which demonstrate that for any given dynamical system generating complicated dynamics there is an upper bound for the set of those discount factors with which that system is the optimal policy function of an optimal growth model. These results, however, do not exclude the possibility of chaotic optimal dynamics for arbitrary low myopia.

The occurrence of complex dynamics in economic models has received wide attention by the economic profession during the last decade (see Boldrin and Woodford (1990) for a recent survey). One of the most interesting and surprising results obtained so far is the indeterminacy theorem of Boldrin and Montrucchio (1986b), which implies that virtually every dynamical behavior is fully compatible with the standard assumptions of decreasing returns, competitive markets, and perfect foresight. Deneckere and Pelikan (1986) have used a related approach to derive similar results for the special case of one-dimensional dynamics. Whereas the analysis in Boldrin and Montrucchio (1986b) and Deneckere and Pelikan (1986) is restricted to optimal growth models formulated in discrete time, an analogous result holds also for the continuous time case (see Montrucchio (1988) and Sorger (1990)). All of these indeterminacy theorems have been proved by a constructive approach which requires a sufficiently high rate of impatience on the side of the decision maker. In particular, to construct optimal growth models exhibiting some well known chaotic maps (like the logistic map, or the Henon map) as optimal policy functions, one needs time preference rates of more than 400%.

Consider a standard aggregative dynamic optimization framework (Ω, u, δ), where Ω is the transition possibility (technology) set, u is a (reduced form) utility function defined on this set, and 0 < δ < 1 a discount factor. Can an optimal program in this framework exhibit a period-three cycle?

In this chapter, we derive, in the standard class of optimal p growth models, the least upper bound of discount factors of future utilities for which a cyclical optimal path of period 3 may emerges.1 On the one hand, Ni s h imur a and Yano (1992) and Ni s himura, Sorger and Yano (Chapter 9) construct examples in which a cyclical optimal path of period 3 emerges for discount factors around 0.36. On the other hand, Sorger (1992, 1994) demonstrates that if such a path emerges in an optimal growth model of the standard class, the model’s discount factor cannot exceed 0.55. These results imply that the least upper bound of discount factors that can give rise to cyclical optimal paths of period 3 must lie between 0.36 and 0.55.2 We demonstrate that the least upper bound is \(\hat p = \left( {3 - \sqrt 5 } \right)/2\).