If what one wishes to do is to examine the structural stability of a particular pattern of bifurcation, or perhaps more specifically to compare the topological characteristics of two distinct patterns of discontinuities in economics, then proper catastrophe theory is clearly the most appropriate method to use for sufficiently low dimensional systems with gradient dynamics derived from a potential function. If, however, what one is interested in doing is simply to model dynamic discontinuities within economic processes, other alternatives certainly exist, most of which do not rely on the specific set of mathematical assumptions that frequently do not hold in specific situations. The use of alternative approaches is especially indicated if one of the control variables in the process under study is time.
Among modern complexity theorists a variety of methods have emerged that can produce phase transitions or dynamic discontinuities of one sort or another in models with heterogeneous, interacting agents. Interacting particle models from statistical mechanics in physics has been the origin of some of these (Föllmer, 1974), with the mean field method one that provides distinct bifurcations that describe phase transitions between different forms of organization of a system (Brock, 1993). Another arises in cases of multiple equilibria when the basins of attraction boundaries are fractally interwoven with each other as in Lorenz (1992). Yet another involves self-organizing criticality wherein small exogenous shocks can trigger much larger endogenous reactions (Bak, Chen, Scheinkman, and Woodford, 1993). Still another uses the idea of synergetics, especially as involving the use of the master equation approach (Weidlich and Braun, 1992). In its emphasis upon distinguishing between slowly changing control variables and more rapidly changing slaved variables, the synergetics approach has a much stronger superficial similarity to catastrophe theory.
Finally we must note the increasing spread throughout economics of models that posit multiple equilibria. In many cases these models generate possible equilibrium surfaces that have similarities to the equilibrium manifolds of catastrophe theory, although they often fail to fulfill all of the mathematical characteristics of true catastrophe theory. Nevertheless, these models can produce dynamic discontinuities as control parameters are varied in ways that cause the system to cross bifurcation points that separate one equilibrium zone from a discretely different equilibrium zone. Such models are so widespread and ubiquitous that it is not worth even beginning to list them, although they have been knows as long as economists have been aware of the possibility of multiple equilibria. This is now approaching a century and a half (von Mangoldt, 1863; Walras, 1874; Marshall, 1890). When such models generate equilibrium surfaces that resemble those of true catastrophe theory, we are dealing with something that is rather like a close sibling of catastrophe theory. This sibling now sits ever taller in the high chairs of the house of economics, and its close relationship with catastrophe theory might as well be more generally recognized.
An increasingly popular example of such an approach is that using Skiba points (or regions or surfaces), originally studied for convex-concave production functions in optimal growth models (Skiba ,1978; Dechert and Nishimura, 1983). The Skiba point separates the basins of attraction of the distinct equilibria and for this model was used to explain dualistic growth outcomes along the lines seen using endogenous growth models. More recently this has been applied to a wide variety of problems (Deissenberg, Feichtinger, Semmler, and Wirl, 2003). An especially striking model is due to Wagener (2003) of multiple equilibria in an ecologic-economic model of pollutants in a lake system (Brock, Carpenter, and Ludwig, 1999; Dechert and Brock, 2000). Catastrophe theory is never mentioned in this paper explicitly, but when it comes to determining the conditions under which a Skiba point exists for this lake system, Wagener finds that a sufficient condition is for the existence of a Hamiltonian cusp bifurcation as described in Thom’s book (1972, p. 62). Catastrophe theory may be all but dead, but in the guise of the study of Skiba points and related phenomena, it lives again.
Catastrophe theory experienced one of the most dramatic intellectual bubbles ever seen. After a gradual development over many decades, it burst onto the intellectual scene in the early and mid-1970s following the publicizing of the work of René Thom and Christopher Zeeman. One can readily speculate that part of the reason for its faddishness at that time was the condition of the socio-cultural environment. Radical political movements abounded, and dramatic changes in the world economy were happening such as the extreme shocks to food and oil prices in the early 1970s. The idea that huge, sudden, and revolutionary changes might happen had considerable widespread appeal, especially among more dissident intellectuals. But widespread applications of the theory that were inappropriate either theoretically or methodologically undermined its credibility. A counterattack came in the late 1970s, and as the 1980s wore on, fewer and fewer applications of catastrophe theory were seen, especially in economics, although catastrophe theory always retained more respectability among mathematicians as a special case of bifurcation theory. Nevertheless, there were many applications of catastrophe theory in economics that were properly done before the counterattack’s influence was fully felt.
Criticisms of applications of catastrophe theory included that it involved excessive reliance on qualitative methods, that many applications involved spurious quantizations or improper statistical methods, and the general failure of many models to fulfill certain mathematical conditions such as possessing a true potential function or by including time as an independent variable in the analysis. Also, in response to those suggesting some kind of universal applicability of catastrophe theory it was noted that the elementary catastrophes were only a small set of the more general set of bifurcations and singularities. Nevertherless, empirical methods such as multi-modal models exist that can be used for estimating catastrophe theory models, with these approaches having been used surprisingly little in economics.
The general critics of catastrophe theory also subjected Zeeman’s (1974) of financial market dynamics to harsh criticism. However, a careful reevaluation suggests that some of these criticisms were misplaced and misguided. To the extent that economists have shied away from using catastrophe theory because of those critiques, they should no longer do so.
With the decline of catastrophe theory, a variety of alternative methods of modeling dynamic discontinuities in economic models have appeared, although some of them have been around for much longer than catastrophe theory has, and some have close connections with catastrophe theory, especially the analysis of Skiba points in multiple equilibria dynamical systems.
In sum, it would appear that indeed the baby of catastrophe theory was largely thrown out with the bathwater of its inappropriate applications to a large extent. Although there are serious limits to its proper application in economics, there remain many potential such proper applications. Economists should no longer shy away from its use and should include it with the family of other methods for studying dynamic discontinuity. It should be revalued from its currently low state on the intellectual bourse and right the wrong of its excessive devaluation, while avoiding any return to the hype and overvaluation that occurred during the 1970s. A reasonable middle ground can and should be found.