Existence of Approximate Equilibria and Cores

IT IS WELL KNOWN that for a finite exchange economy, where preferences are not assumed to be convex, there may be no price equilibrium or even the core may be empty. For this reason it was proposed to enlarge the set of price equilibria and the core by introducing the concepts of "approximate equilibrium" and "approximate core." The existence of approximate equilibria for exchange economies, where preferences are not assumed to be convex, has been investigated by R. Starr [6]. He showed that there exists a "quasi-equilibrium," provided the number of participants is large enough and there is a bound on the "degree of non-convexity" [6, p. 30, Assumption D]. In this note we shall show the existence of "approximate equilibria" (a stronger concept than the one considered by Starr [6, p. 31]) for large economies where the preferences are neither assumed to be convex nor complete. To obtain our result we shall assume that the preferences and the endowments of all participating agents belong to a compact set. In [5] Shapley and Shubik proved that, for a large replica of a given economy with transferable utility, the e-core is nonempty. We shall generalize this result to large economies without transferable utility by using the concept of ?-core as introduced by Kannai [2]. The nonemptiness of the e-core follows easily from the existence of approximate equilibria and a relationship between the set of approximate equilibria and e-core. The existence of c-core for large economies (with a fixed number of types) can also be deduced from Kannai's Theorem C' [2] in its stronger form (Theorem C" in [3]).