Irregular Growth Cycles

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.