A New Approach to the Uniqueness of Equilibrium

Abstract

A problem of major concern to economists is that of knowing whether certain phenomena are likely or not. For example, the econometrician would doubtless be happy if it were true that Cobb-Douglas production functions are very likely in the sense that any production function could be closely approximated by a function of the Cobb-Douglas type. Such results would provide excellent protection against those critics who claim that such functions are only special cases . The lack of these results is due largely to the idea of being likely having been used in too general and intuitive a way. In showing here that economies with a unique equilibrium are likely we will incidentally show that if being likely is interpreted carefully, within the context of a particular problem, then a surprising number of apparently improbable phenomena may in fact rigorously, be shown to be likely. We will now examine the fundamental problem of establishing whether uniqueness of equilibrium is likely (see Arrow and Hahn [1, Chap. 9]). The standard way to demonstrate the likeliness of some property is to show that it holds generically, i.e. on an open dense set with respect to some meaningful topology. We show that for a carefully chosen topology on the space of all economies, which we call the y topology just such a statement is true for the uniqueness of equilibrium.3 A suggestion that such a statement might be true can be inferred from Dierker's result [4] that the number of equilibrium prices is generically odd. He also gives an indication of the significance of establishing the uniqueness of equilibrium. Following Debreu [3] we define an economy with a commodity space R' as a tuple (f1, *--, fm1~l ..., COm). Here m denotes the number of consumers involved, fi (resp. woi E R') denotes the C' demand function 4 (resp. the strictly positive resources) of the ith consumer. An equilibrium for this economy is a tuple (p, x1, ..., x.) of prices p E R', zPh = 1, and commodity bundles xi e R' , such that lixi = Sioi and xi = fi(p, p cioi) for each consumer i. In the following all topological notions are with respect to the y topology. The definition of this topology, however, is somewhat technical, so it is relegated to the appendix, as one of the proofs of the theorems. We now state our main result.