Profits, Cycles, and Chaos

Some time ago, Goodwin (1967) offered an elegant and influential model to represent part of Marx’s thinking on business cycles. In that model he was able to show how the interaction of the reserve army of labor and the process of capital accumulation could produce self-sustaining oscillatory behavior. Increases in the real wage cause decreases in the rate of growth of the capital stock, since all wages are consumed and all profits invested. The declining rate of accumulation in turn causes a decline in the employment rate, which eventually causes the wage rate to decline. The eventual expansion in the growth rate of the capital stock begins the process over again. This behavior was described by fitting a model of one good economy into the Lotka-Volterra equations, the solution to which is well known. While it has proved extremely fruitful, this model also has some well-known limitations. It is, first of all, a center, so that no limit cycle produced by the model is stable. Second, it takes a rather asocial approach to the creation of the labor force, assuming that it is governed exclusively by an exogenously given rate of population growth. Also, the model assumes that all technical change occurs at a constant, autonomously given rate, and allows for no induced components.

In what follows, some minor alterations to the Goodwin model are shown to introduce interesting new behavior. By making technical change depend on economic and social phenomena, and by assuming that the labor force grows at least in part in response to social phenomena, it is easy to show that the model will now generate stable limit cycles. When the model is changed still further, to allow for systematic periodic influences—such as those an economy might experience as a result of seasonal changes in labor force participation or productivity—somewhat more dramatic dynamic behavior follows. Under certain conditions, the resulting behavior is more business cycle–like because it is irregular. But at the same time, the existence chaos implies difficulties for empirical "deseasonalization" of data.

The possibility of chaos introduces some questions for the study of business cycles. One is whether it is possible to discriminate between economic phenomena that are induced by chaos-generating nonlinearities and those that are introduced by stochastic shocks to some underlying nonlinear system. The model is used to illustrate this problem, and to show how an existing technique for testing for chaos—the calculation of Lyapunov exponents—is able to handle it.