Computing Economic Chaos

Olegs Pavlovs

Computing Economic Chaos

2004, Computational Economics

Abstract

Existence theory in economics is usually in real domains such as the findingsof chaotic trajectories in models of economic growth, tâtonnement, oroverlapping generations models. Computational examples, however, sometimesconverge rapidly to cyclic orbits when in theory they should be nonperiodicalmost surely. We explain this anomaly as the result of digital approximationand conclude that both theoretical and numerical behavior can still illuminateessential features of the real data.

The possibility of cycles and chaos arising from nonlinear dynamics in economics emerged in the literature in the 1980s, and it came as a surprise. 1 The possibility of deterministic cycles in economic models had been noted before, for example in the well-known multiplier-accelerator models, but not in equilibrium models with complete markets, no frictions, and full intertemporal arbitrage. 2 The reason for the surprise was understandable: deterministic fluctuations in equilibrium models involve predictable changes in relative prices which should be ruled out by intertemporal arbitrage. In models of overlapping generations, however, finite lives can restrict complete arbitrage over time. As a result, some people thought, and still think, that cycles that are shorter than the agents' postulated lifespans would not be possible in equilibrium models, and therefore are irrelevant for business-cycle analysis. This view is clearly wrong, and of course ignores the extensive literature on cycles and chaos in optimal growth models with infinitely lived agents. In such models deterministic cycles in relative prices occur easily, but the amplitudes of the cycles remain within bounds of the discount rate. 3 It is not difficult to show in the context of multisector growth models, say with Cobb-Douglas production functions, that for any positive discount rate there is a large class of technologies for which cycles occur. (See Benhabib and Rustichini [1990].) Getting chaos, however, is harder. Recent works by Sorger (1992), by Mitra (1995), and by Nishimura and Yano (1995) give lower bounds for the discount rate, below which chaos is ruled out for one-sector models of optimal growth. Yet even in that context, going to a multisector framework may considerably lower the bounds on the discount rate thus far established. A second reason for the attention that chaotic dynamics received in the economics literature regards prediction. The common wisdom has been that economic fluctuations are driven by exogenous shocks. Chaotic dynamics not only supplied an alternative explanation for at least some part of economic fluctuations, but also provided an excuse for economists' difficulties with forecasting. Sometimes, however, an important feature of chaotic dynamics that makes forecasting difficult, namely, sensitive dependence on initial conditions, is used in a cavalier way to explain short-run dynamics, forgetting that the effect of sensitive dependence becomes significant only after some periods, but not in the very short run. When it became obvious that very-standard equilibrium models could easily generate cycles and chaos, the attention in the literature naturally turned to the empirical plausibility of such dynamics. The most interesting approach, inspired by developments in natural sciences and mathematics, was also atheoretical, and reminiscent of VAR methods of time series. 4 The idea was to try to infer whether a particular economic time series was generated by a deterministic, low (at most four-or five-) dimensional system that was chaotic, or whether it came from a simple (linear) stochastic system. It is not difficult to see that such inferences are hard to make when the time-series data is short, as is the case with most economic series, with the exception of financial data. It is not surprising, then, that many applications of this approach are in the area of finance, but even there, where we have very high-frequency data, it is hard to pick up fluctuations that may occur at lower

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