The rise and fall of catastrophe theory applications in economics: was the baby thrown out with the bathwater?

Criticism of an Economic Application

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Criticism of an Economic Application

Of course it is not much of a defense of catastrophe theory to point out that criticisms made of specific models happen to have been misguided. But in regard to economic applications, there was relatively little specific discussion during the main debates in the late 1970s. The main economic application that was discussed was probably the first one ever made, the model of stock market crashes due to Zeeman (1974). As we shall now see, much of the criticism directed at this model was seriously misguided. That these criticisms have somehow fed into the current negative attitude towards applying catastrophe theory in economics therefore calls for correction.

Zeeman models stock market dynamics as reflecting the interactions of two different kinds of agents, fundamentalists who know what the true value of an asset is and who buy when the asset is below that true value and sell when it is above that value, and chartists who chase trends, who buy as price rises and sell as price falls. The formulation is somewhat different from most economic models in that what is modeled is the rate of change of price rather than the level of price. This rate of change of price is J, the state variable. It is modeled as determined by the excess demands of the two groups, F for the excess demand of the fundamentalists and C for the excess demand of the chartists. These two are the control variables then for a cusp catastrophe in which F is the normal factor and C is the splitting factor, as shown in Figure 8. If all agents are fundamentalists, then the market will be well-behaved and stable, with a unique equilibrium, keeping in mind that this equilibrium is actually a rate of change of price, although if the equilibrium for the midpoint equals zero, then that will essentially coincide with the random walk model. As C increases and the cusp point is passed, the possibility of instability appears and of discontinuous changes in J. Zeeman’s original story involved C rising as the price accelerated until there was a crash, at which point C would decline as chastened investors reverted to more cautious fundamentalist behavior.

[insert Figure 8]

The more widely publicized critique of this model came in the Zahler and Sussman (1977) general assault on applications of catastrophe theory, with their critique of this model being their one salvo specifically aimed at economics applications. They declared the model to be “unscientific” on grounds that the chartist agents would not have rational expectations. In retrospect such a charge seems quite ridiculous, although it is now largely forgotten among economists that this was the basis of their rejection of Zeeman’s model. The model may well be open to criticism on other grounds, such as its making J equal the rate of change of price rather than the level of price. But to reject this model because it does not assume rational expectations for all its agents, and to use this as a basis for more broadly criticizing catastrophe theory, looks like not only throwing out the baby with the bathwater, but in fact throwing out the baby when the water it was bathing in was reasonably clean and not even in need of being thrown out.

How could they make such a silly argument? It must be remembered that 1977 was pretty near the peak of the intellectual bubble of rational expectations, although that idea certainly has considerable value and continues to be used by many macroeconomists in many models. Financial economists in 1977 were especially enamored of efficient markets models of rational, homogeneous agents, arguably even more so than most other economists. It only began to become intellectually respectable to allow for heterogeneous agents in financial market models after Fischer Black (1986) gave his famous presidential speech to the American Finance Association in which he suggested the possibility that “noise traders” might exist. This idea became suddenly much more respectable after the Dow-Jones Average for the U.S. stock market declined 22 percent on October 19, 1987. More recent experience with obvious stock market bubbles and crashes has made such ideas even more acceptable among financial economists, and today such models are commonplace, especially given the broader shift to using heterogeneous agent models that has been inspired by the complexity modelers of the Santa Fe Institute and their associates in the econophysics movement.16 Interestingly, although few of these models use catastrophe theory explicitly, the Zeeman model is now increasingly cited in these papers and its essential insight has been supported, that the stability or instability of financial market reflects the balance between traders who behave more like the traditional fundamentalists and those who behave more like the traditional chartists as posited by Zeeman. Despite Zahler and Sussman’s earlier dismissal, the Zeeman model is back to some degree.17

A less well-known criticism was made later by Weintraub (1983), although he did not criticize using catastrophe theory in economics in general. Coming from a discussion of the tâtonnement process in general equilibrium theory and the stability conditions for that process, he concluded that Zeeman’s model implied that chartist traders must have upward-sloping demand curves. He then cited Stigler (1948) who argued that there never has been a true Giffen good with an upward-sloping demand curve. Regarding the idea that a stock market participant might believe that tomorrow’s price will depend on what the price has been, Weintraub declared in italics (1983, p. 80), “There is no evidence whatsoever to support such an hypothesis,” citing Malkiel (1975) and the random walk model. Although Weintraub was writing after Zahler and Sussman, this was still prior to the speech by Black (1986) and the dramatic events in the stock market in 1987, a period the random walk model was more widely accepted than it is today.

As regards the claim that a chartist speculator must have an upward-sloping demand curve, which cannot exist because George Stigler said so in 1948, what is involved is not a static demand curve, but a situation where the demand curve shifts outwards as the price rises (or accelerates). Weintraub might well reasonably respond that in this case it is meaningless to describe the surface in Figure 8 as an equilibrium manifold. But, as he himself points out, it is not a proper general equilibrium manifold anyway because J is a rate of change of price rather than price itself. It is thus a different kind of equilibrium entirely, one about a pattern of shifts in demand curves rather than about movements along fixed demand curves. Stigler’s argument is simply irrelevant.

The Zeeman model may have its oddities, but many of the criticisms of it were fallacious. The idea that these criticisms somehow constituted a case for not using catastrophe theory in economics is simply absurd. Although the current models of financial market dynamics do not generally use catastrophe theory per se, the Zeeman model has regained its respectability and is now recognized as a source of useful insights for understanding such dynamics. A historical wrong has been righted.

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