Cycles and Chaos in Economic Equilibrium.

This book contains 21 papers on the subject of nonlinear determinism and chaos in economics. Fifteen of the 21 have appeared in print in one venue or another, and a number of these would have to be considered among the more important articles on nonlinear determinism in economics. The remaining papers elaborate on or extend themes developed in the published work.

The papers in this book utilize tools originally developed in the natural sciences for the study of complex deterministic dynamical systems. Most economists have had some exposure to deterministic dynamics at one time or another during their graduate studies, perhaps taking a course from a book like Kamien and Schwartz's Dynamic Optimization |2~ or Stokey and Lucas's Recursive Methods in Economic Dynamics |3~. The usual approach is to develop conditions under which the underlying dynamic optimization problem has a unique solution. Typically, one then studies the asymptotic behavior of a general system by linearizing within a neighborhood of the equilibrium and studying the stability properties of the corresponding linear system. Linear systems are easy to work with, and exhibit only a limited range of behaviors: they are stable or explosive, periodic oscillatory or monotonic. The student takes away from the course the subliminal message that economically relevant deterministic dynamical systems are too simple to be empirically interesting. Deterministic systems are quickly discarded in favor of stochastic systems, which appear to generate time series that more closely resemble economic data sets.

The French mathematician Henri Poincare, writing around the turn of the century, is generally credited with the discovery that nonlinear deterministic dynamical systems can exhibit strange, erratic behaviors. These results were mainly of intellectual interest until the '60s and '70s, when researchers in the hard sciences discovered that systems in meteorology, biology, and physics exhibited complex behaviors that could be usefully modeled with simple, nonlinear systems. Where physicists go, economists soon follow, and by the mid '80s came a flood of papers on nonlinear deterministic models of economic systems.

Two features of nonlinear systems captured the attention of economists working in the area. First, it is easy to design a simple nonlinear system that can generate aperiodic, irregular cycles. Second, there are some nice examples of nonlinear systems that appear to be random to many popular linear statistical tests. Among economists studying the business cycle, these tools are used to address questions such as whether cyclical movements in macroeconomic time series are the damped oscillations of a stable system reacting to exogenous, stochastic shocks, or are the result of endogenous cycles in a nonlinear system. The possibility of determinism masquerading as randomness is of interest to financial economists, who use nonlinear methods to test market efficiency by looking for subtle patterns in financial asset return data.

The majority of the papers included in this volume (17 of 21) are theoretical papers illustrating various ways one can obtain complex dynamics from the standard optimizing models used by macro economists. A common approach is to derive equilibrium trajectories to dynamic systems, and show how varying key parameters can lead to the period-doubling route to chaos. The papers illustrate how features of preferences, technology, or economic institutions can be exploited in this manner. The lesson one draws from these exercises is that it appears to be surprisingly easy to get complex dynamics out of the usual macro models.

The skeptic might argue that showing that complex dynamics are a logical possibility is not the same as showing that they are economically relevant. There is little effort in these papers to calibrate the models to show that the erratic behavior is generated in economically relevant regions of the parameter space. Nonetheless, the careful empirical researcher who estimates a sufficiently complex dynamic model should probably simulate the behavior of the corresponding deterministic system at the estimated parameter values to check for signs of nonlinear instabilities.

The four empirical papers in this volume are not focused on estimating model parameters, but instead on searching for the signature of nonlinearity in economic data. Two of the four empirical papers use techniques which are inspired by or borrowed from work in the physical sciences. The papers by Brock and Sayers and by Scheinkman and LeBaron are both widely cited, and while the statistical methods have been refined (by, for example, Brock and Baek |1~), these papers illustrate a very useful approach to testing for nonlinearity. Briefly, the approach is to estimate a "null hypothesis" model on a data set, and then look for additional evidence of dependence in the estimated residuals of the null model. If the dependence is significant, then one can reject the null model as a data generator. The editor also includes a paper by Ramsey, Sayers, and Rothman providing a review of the empirical methodology which details the problems associated with its uncritical application.

On balance, this book serves the useful purpose of collecting in one volume a number of important papers on chaos in economics. The papers in this volume are most suitable for those looking for an entry into a rapidly expanding field.

References

1. Brock, W. A. and E. G. Baek, "Some Theory of Statistical Inference for Nonlinear Science." Review of Economic Studies 58, 1991, 697-716.

2. Kamien, M. I., and N. L. Schwartz, Dynamic Optimization: The Calculus of Variations and Optimal Control in Economics and Management. New York: North-Holland, 1981.

3. Stokey, N. L. and R. E. Lucas. Recursive Methods in Economic Dynamics. Cambridge, Mass.: Harvard University Press, 1989.