Journal Article Unscrambling the Concept of Chaos Through Thick and Thin: Reply Get access Richard H. Day Richard H. Day University of Southern California Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Economics, Volume 101, Issue 2, May 1986, Pages 425–426, https://doi.org/10.2307/1891124 Published: 01 May 1986

This paper shows how fluctuations of an erratic and unstable nature can emerge from the classical, deterministic economic growth process. Implications of the analysis would appear to be at least two. First, pronounced changes in the way an economy behaves need not cause us to reject our understanding of how it works. Second, we need not seek in exogenous forces an explanation as to why behavioral patterns change and why it may be so difficult to anticipate future events from the profile of past experience.

Introduction, 302. — The learning model, 304. — Demand, cost, and environmental feedback, 307. — Satisficing's convergence to marginalism, 308. — A simple example, 310.

This paper uses the familiar, neoclassical theory of capital accumulation to show how complex behavior can emerge from quite simple economic structures. Indeed, when sufficient nonlinearities and a production lag are present, the interaction alone of the propensity to save and the productivity of capital can lead to growth cycles that exhibit a wandering, sawtooth pattern not unlike those observed in reality. These fluctuations need not converge to a cycle of any regular periodicity so they are not quasi periodic. Because such trajectories are unstable, errors of estimation in parameters or initial conditions, however tiny, will accumulate rapidly into substantial errors of forecast. Moreover, periods of erratic cycling can be interspersed with periods of more or less stable growth. Evidently, the future behavior of a model solution cannot be anticipated from its patterns in the past, a situation that seems to mimic experience. Apparent structural change and unpredictability is explained in the present theory by a deterministic, single equation model. Random shocks play no role. So the reader can visualize just what it is we are are talking about, a noteworthy simulation is presented in Figure 1 for GNP in a growth model that is described below in Section III. A period of relatively rapid growth is followed by a period of cycles. Then, remarkably, for a considerable time (about twenty periods) apparently steady-state growth occurs. Wandering cycles, however, emerge. Another brief period close to the steady state appears again toward the end of the series. I establish conditions of savings and productivity that lead to results of this kind. This analysis makes use of the mathematical theory of which, in the form exploited here, originated in the work of Edward Lorenz. A formal definition of chaos and sufficient conditions for chaotic trajectories were provided in a seminal paper by T-Y Li and James Yorke. A survey of these related contributions is found in Yorke and Evelyn Yorke. This theory was introduced into economics by Jess Benhabib and myself (1981), where we showed that sequences of rational choices can be erratic when preferences depend on experience in a certain way; by Michael Stutzer, who provides a detailed analysis of Trygve Haavelmo's growth model; by Benhabib and myself (1980) in an application of the overlapping generations model; and in my forthcoming study of the classical growth model.

In this paper decision systems are structured so that topological concepts can be applied to formulate and help solve existence problems. Existence of stationary states and orbits is established. The analysis is then applied to programs, a special class of decision systems in which the decision operator is a mathematical program. The theory is extended to of many decision makers with rolling schedules of future actions. RECURSIVE DECISION SYSTEMS (RDS's) are dynamic systems based on discrete time that represent the positive behavior of decision makers. They have been put to three basic uses, (1) to describe the behavior of various economic sectors, (2) to show how indirect policies can in some particular way improve the performance of the economic system under investigation, and (3) to formulate and analyze a variety of dynamic economic theories. In this paper we define RDS's so that topological concepts and theorems can be used to study existence questions. Existence theorems are then given for stationary states and compact orbit sets. Special attention is given to the class of RDS's called programs (RP) of which various recursive program- ming models are special cases. The paper concludes with some brief comments on the assumptions used in the analysis. Before proceeding to the formal definitions, we briefly review in a nontechnical manner the basic concepts underlying RDS's and their use in economic research. RDS's as defined here are mathematical of socioeconomic processes having two basic components: (1) a decision operator that describes the manner in which final decisions or actions are derived from a given amount of information about the decision maker's environment ;2 and (2) afeedback operator that describes how decisions once acted on, or once scheduled for the future, interact with the decision maker's environment to produce new information upon which succeeding plans can be based.3 A given decision operator may represent the decision process not only of a single decision maker, but also of a group of decision makers who make their decisions independently-or collusively-during the same time period. Further- more, the decision at a given time may represent not only an immediate choice,