Catastrophe Theory

Abstract

The theory of general equilibrium defines equilibrium prices p as the solutions in the commodity space of the vector equation defined by equality of supply and demand, namely z(p) = 0, where z denotes aggregate excess demand. This formulation leads to a purely mathematical problem, namely the study of the properties of the solutions of the equation z(p) = 0. The first problem to come into the picture is that of existence. Its positive solution leads to new issues such as the determinateness of the solutions or their number. The fact that these problems cannot be solved uniformly with exactly the same answer for every economy necessitates the introduction of suitable parameters in terms of which the properties of the solutions of the equilibrium equation can be properly described. Let ω denote this parameter chosen in some suitable vector space Ω. This means that the aggregate demand function z can be viewed as depending on ω∈Ω which we now denote by z(·,ω) and the goal of equilibrium theory becomes one of relating the properties of the solutions to z(p, ω) = 0 with the parameter ω. In practice, one chooses for ω the initial endowments of every consumer, the equilibrium model simply describing a pure exchange economy.