As we shall see, critics of catastrophe theory have argued that many applications of it in many fields were either done in violation of necessary assumptions or were carried out in other ways that are questionable for one reason or another. Let us list a few examples of applications in economics, most of which in this author’s view were done in a reasonable manner.5
The earliest published application was due to Zeeman (1974) and was an effort to model bubbles and crashes in stock markets. This example has been much criticized (Zahler and Sussman, 1977; Weintraub,1983). We shall return later to discuss the criticisms of this particular model after we discuss the question of the broader criticisms of catastrophe theory and the debate that arose around it.
Although he did not use catastrophe theory directly, Debreu (1970) set the stage for doing so in regard to general equilibrium theory with his distinction between regular and critical economies, the latter containing equilibria that are singularities. Discontinuous structural transformations of general equilibria in response to slow and continuous variation of control variables can occur at such equilibria. Analysis of this possible phenomenon was carried out using catastrophe theory by Rand (1976)6 and Balasko (1978).7 Rand in particular derives such a case when at least one trader in a pure exchange economy has non-convex preferences, as depicted in Figure 3.
[insert Figure 3]
Bonanno (1987) studied a model of monopoly in which there were non-monotonic marginal revenue curves due to market segmentation. Multiple equilibria can arise with smoothly shifting cost curves, which he analyzed using catastrophe theory.
Perhaps the most influential application of catastrophe theory in economics was to the analysis of business cycles in a paper by Varian (1979). He adopted a nonlinear investment function of Kaldor (1940) as modified by Chang and Smyth (1971) to construct the following model.
dy/dt = s(C(y)) + I(y,k) – y (5)
dk/dtg = I(y,k) – I0 (6)
C(y) = cy + D, (7)
with y as national income, k as capital stock measured against a long-run trend, C(y) as the consumption function, I(y,k) as the gross investment function with I0 an autonomous level of replacement investment, and s a speed of adjustment parameter assumed to be rapid relative to the movements of the capital stock. The nonlinear investment function was assumed to have a sigmoid shape and would shift with the capital stock as depicted in Figure 4, with S = I being the equilibrium condition.
[insert Figure 4]
Within this model a hysteresis cycle with discontinuities can arise as the investment function shifts back and forth during the course of a business cycle, as depicted in Figure 5.
[insert Figure 5]
Varian then extended this model by allowing the consumption function to include wealth, w, as a control variable as follows:
C(y,w) = c(w)y + D(w), (8)
with c’(w) > 0 and D’(w) > 0. This formulation allows for a tilting of the savings function such that there are no longer any multiple equilibria. This allowed Varian to distinguish between simple recessions and longer term depressions. This was depicted by a cusp catastrophe in which wealth is the splitting factor, as depicted in Figure 6.
[insert Figure 6]
One of the few efforts to empirically estimate a catastrophe theory model in economics was of a model of inflationary hysteresis involving a presumably shifting Phillips Curve. This was due to Fischer and Jammernegg (1986). The method they used was a multi-modal density function due to Cobb (1978, 1981).8 For U.S. data for the period of June 1957 to June 1983, they found a cusp point in the space of the unemployment rate and inflationary expectations of about 7 percent for each variable. This drew on an ad hoc model suggested by Woodcock and Davis (1978), and in effect argued that this system could be viewed as a cusp catastrophe, with the economy jumping to the “higher” sheet of the equilibrium manifold during 1973-74 and then back down again, but then jumping up again at the end of 1977 only to gradually come back down (by going around the cusp point after 1980.
Drawing on models due to Bruno (1967) and Magill (1977), Rosser (1983) analyzed dynamic discontinuities in an optimal control theoretic growth theory model that contained capital theoretic paradoxes. Ho and Saunders (1980) developed a catastrophe theory model of bank failure when risk factors go beyond critical levels.
The areas of urban and regional economics saw especially large numbers of applications of catastrophe theory, including the use of catastrophe theory models of higher dimensionality than the three dimensional cusp catastrophe seen above, although some of this work was done by geographers rather than economists. Amson (1975) initiated the formal use of it with a cusp catastrophe model urban density as a function of rent and “opulence.” Mees (1975) modeled the revival of cities in medieval Europe using the five dimensional “butterfly” catastrophe. Wilson (1976) studied modal transportation choice as a fold catastrophe, Dendrinos (1979) modeled the formation of urban slums using the six dimensional parabolic or “mushroom” catastrophe. Structural change in regional trading systems was analyzed using the five dimensional hyperbolic and elliptic umbilic catastrophes by Puu (1979, 1981a, b)9 and by Beckmann and Puu (1985). Andersson (1986) modeled “logistical revolutions” in interurban transportation and communications relations and patterns as a function of long run technological change using a fold catastrophe. Some of these applications were somewhat ad hoc, although the ones by Puu and by Beckmann and Puu especially stand out as fulfilling all the mathematical conditions for proper application of catastrophe theory.
Within ecologic-economic systems considerable focus has been paid to systems in which there are discontinuous changes in biological populations, including collapses to extinction as a result of interaction with human activities. The multiple equilibria model of fishery dynamics in the case of backward-bending supply curves was initially studied by Copes (1970), and Clark (1976) examined it in the context of catastrophe theory. The basic pattern is depicted in Figure 7 in which outward shifts of the demand curve due to rising incomes or preferences for fish can lead to discontinuous changes in equilibria. A somewhat similar model with improved fishing technology as a control variable was due to Jones and Walters (1976).
[insert Figure 7]
Another vein of argumentation drew on models of predator-prey dynamics, such as the spruce budworm dynamics in forests modeled by Ludwig, Jones, and Holling (1978). Most notably, Walters (1986) examined a fold catastrophe model of Great Lakes trout dynamics using such a predator-prey model to study how yields could be maximized while avoiding a catastrophic collapse by using a so-called “surfing” strategy. This example refutes the widespread argument heard that catastrophe theory had no practical application. Unsurprisingly there has continued to be much more interest in catastrophe theory models, or variations on them, in ecologic-economic modeling, although often these are models with multiple equilibria in fold or cusp patterns that are not identified with catastrophe theory explicitly (Wagener, 2003). Of course this may well reflect the current disrepute in which catastrophe theory is so widely held and the desire not to be tainted with it.
In international finance, George (1981) studied foreign currency speculation in a model with non-convex risk preferences. This used a cusp catastrophe and essentially followed the approach used in the Rand (1976) general equilibrium model. Although he did not put it formally into a catastrophe theory framework, Krugman’s (1984) of multiple equilibria in the demand for foreign currencies could rather easily be put into such a framework following along the lines of the Varian (1979) approach.10 Of course many models are now studied of multiple equilibria in foreign exchange rate models, with many of these taken very seriously given the numerous foreign exchange crises that have occurred in recent years.