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Saturday, November 28, 2009

"Catastrophe Theory and the Business Cycle"

As a follow up to the recent post on non-linear dynamics that continued the discussion on what's wrong with modern macroeconomics, here is a paper written many years ago by Hal Varian that extends the Goodwin-Kaldor model of business cycles. It is old-fashioned macro, but the interesting part is the wealth effect causing the difference between recessions and depressions. In particular, the results of the paper imply that shocks to wealth that change savings propensities -- as we are seeing now -- can cause recoveries that "may take a very long time, and differ quite substantially from the recovery pattern of a [typical] recession."

Here are a few selections from the paper:

Catastrophe Theory and the Business Cycle, by Hal Varian: In this paper we examine a variation on Kaldor's (1940) model of the business cycle using some of the methods of catastrophe theory. (Thom (1975), Zeeman (1977)). The development proceeds in several stages. Section I provides a brief outline of catastrophe theory, while Section II applies some of these techniques to a simple macroeconomic model. This model yields, as a special case, Kaldor's business cycles. ... In Section III, we describe a generalization of Kaldor's model that allows not only for cyclical recessions, but also allows for long term depressions. Section IV presents a brief review and summary.

This paper is frankly speculative. It presents, in my opinion, some interesting models concerning important macroeconomic phenomena. However, the hypotheses of the models are neither derived from microeconomic models of maximizing behavior, nor are they subjected to serious empirical testing. The hypotheses are not without economic plausibility, but they are far from being established truths. Hence, this paper can only be said to present some interesting stories of macroeconomic instability. Whether these stories have any empirical basis is an important, and much more difficult, question. ...

Applied catastrophe theory is not without its detractors (Sussman and Zahler (1978)). Some of the applied work in catastrophe theory has been criticized for being ad hoc, unscientific, and oversimplified. As with any new approach to established subjects, catastrophe theory has been to some extent oversold. In some cases, applications of the techniques may have been overly hasty. Nevertheless, the basic approach of the subject seems, to this author at least, potentially fruitful. Catastrophe theory may provide some descriptive models and some hypotheses which, when coupled with serious empirical work, may help to explain real phenomena. ...

[I]t makes sense to model the system as if the state variables adjust immediately to some "short run" equilibrium, and then the parameters adjust in some "long run" manner. In the parlance of catastrophe theory, the state variables are referred to as "fast" variables, and the "parameters" are referred to as "slow" variables. This distinction is, of course, common in economic modeling. For example, when we model short run macroeconomic processes we take certain variables, such as the capital stock, as fixed at some predetermined level. Then when we wish to examine long run macroeconomic growth processes, we imagine that economy instantaneously adjusts to a short run equilibrium, and focus exclusively on the
long run adjustment process.

Catastrophe theory is concerned with the interactions between the short run equilibria and the long term dynamic process. To be more explicit, catastrophe theory studies the movements of short run equilibria as the long run variables evolve. A particularly interesting kind of movement is when a short run equilibrium jumps from one region of the state space to another. Such jumps are known as catastrophes. Under certain assumptions catastrophes can be classified into a small number of distinct qualitative types. ... In the economic model that follows we will only utilize the two simplest catastrophes, the fold and the cusp. In these low dimensional cases, there are no restrictions on the nature of dynamical systems involved. ...

[One result from the macroeconomic model] is the case considered by Kaldor (1940) and, more rigorously, by Chang and Smyth (1972). It has been shown by Chang and Smyth that when the speed of adjustment parameter is large enough, and certain technical conditions are met, there must exist a limit cycle in the phase space. In the appendix I prove a slightly simplified and modified version of this result.

This "business cycle" proposition is clearly the result intuited by Kaldor thirty years ago. However, the existence of a regular, periodic business cycle causes certain theoretical and empirical difficulties. Recent theoretical work involving rational expectations (Lucas (1975)) and empirical work on business cycles (McCullough (1975), (1977), Savin (1977)) have argued that (1) regular cycles seem to be incompatible with rational economic behavior, and (2) there is little statistically significant evidence for a business cycle anyway.

However, there does seem to be some evidence for a kind of "cyclic behavior" in the economy. It is commonplace to hear descriptions of how exogenous shocks may send the economy spiraling into a recession, from which it sooner or later recovers. Leijonhufvud (1973) has suggested that economies operate as if there is a kind of "corridor of stability": that is, there is a local stability of equilibrium, but a global instability. Small shocks are dampened out, but large shocks may be amplified. ...

Such a story seems to me to be a reasonable description of the functioning of the commonly described "inventory recession." [L]et us, for the sake of argument, accept such a story as providing a possible explanation of the "cyclic" behavior of an economy. Then there is yet another puzzle. Each recession in this model will behave rather similarly: First some kind of shock, then a rapid fall, followed by a slow change in some stock variables with, eventually, a rapid recovery. Although this story seems to be descriptive of some recessions, it does not describe all types of fluctuations of income. Sometimes the economy experiences depressions. That is, sometimes the return from a crash is very gradual and drawn out. ...

Here is the interesting feature of the model. Suppose as before, that there is some kind of perturbation in one of the stock variables. For definiteness let us suppose [there is] some kind of shock (a stock market crash?).... If the shock is relatively small, we have much the same story as with the inventory recession... If on the other hand the shock is relatively large, wealth may decrease so much as to significantly affect the propensity to save. In this case,... national income will remain at a relatively low level rather than experience a jump return. Eventually the gradual increase in wealth due to the increased savings will move the system slowly back towards the long run equilibrium. ...

According to this story the major difference between a recession and a depression is in the effect on consumption. If a shock affects wealth so much as to change savings propensities, recovery may take a very long time, and differ quite substantially from the recovery pattern of a recession. This explanation does not seem to be in contradiction with observed behavior, but as I have mentioned earlier, it rests on unproven (but not implausible) assumptions about savings and investment behavior.

1.4 Review and summary

We have shown how nonlinearities in investment behavior can give rise to cyclic or cycle like behavior in a simple dynamic macroeconomic model.

This behavior shares some features with empirically observed behavior. If savings behavior also exhibits nonlinearities of a plausible sort, the model can allow for both rapid recoveries which characterize recessions, as well as extended recoveries typical of a depression.

    Posted by on Saturday, November 28, 2009 at 12:33 PM in Economics, Macroeconomics, Methodology | Permalink  TrackBack (0)  Comments (6)

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