Singularities and Catastrophes in Economics: Historical Perspectives and Future Directions
Abstract
Economic theory is a mathematically rich field in which there are opportunities for the formal analysis of singularities and catastrophes. This article looks at the historical context of singularities through the work of two eminent Frenchmen around the late 1960s and 1970s. René Thom (19232002) was an acclaimed mathematician having received the Fields Medal in 1958, whereas Gérard Debreu (19212004) would receive the Nobel Prize in economics in 1983. Both were highly influential within their fields and given the fundamental nature of their work, the potential for crossfertilisation would seem to be quite promising. This was not to be the case: Debreu knew of Thom's work and cited it in the analysis of his own work, but despite this and other applied mathematicians taking catastrophe theory to economics, the theory never achieved a lasting following and relatively few results were published. This article reviews Debreu's analysis of the so called ${\it regular}$ and ${\it crtitical}$ economies in order to draw some insights into the economic perspective of singularities before moving to how singularities arise naturally in the Nash equilibria of game theory. Finally a modern treatment of stochastic game theory is covered through recent work on the quantal response equilibrium. In this view the Nash equilibrium is to the quantal response equilibrium what deterministic catastrophe theory is to stochastic catastrophe theory, with some caveats regarding when this analogy breaks down discussed at the end.
 Publication:

arXiv eprints
 Pub Date:
 July 2019
 arXiv:
 arXiv:1907.05582
 Bibcode:
 2019arXiv190705582H
 Keywords:

 Economics  General Economics;
 91A15;
 91A80;
 91A05;
 91B50;
 91B06
 EPrint:
 23 pages, 5 figures