Existence of competitive equilibrium in a non-standard exchange economy

NONSTANDARD EXCHANGE ECONOMIES and the associated nonstandard concepts of the core and competitive equilibrium were first defined in Brown-Robinson. In that paper they proved the equivalence between the nonstandard core and the set of nonstandard competitive equilibria. In a second paper, using the equivalence theorem for nonstandard exchange economies, they showed that core allocations in large standard exchange economies were approximately competitive equilibria.
In this paper, we shall modify slightly their definition of a competitive equilibrium and then prove the existence of this nonstandard competitive equilibrium. In a future paper, we hope to show that this existence theorem implies the existence of approximate or "ε"-equilibria in large standard exchange economies.
Our proof is based, in part, on Schmeidler's elegant proof of the existence of a competitive equilibrium for a continuous economy. As in the continuous case we do not assume convexity of preferences, since we use a nonstandard analogue of the Lyapunov theorem, on the convexity of vector-valued measures, due to P. Loeb. We have also benefited a great deal from Auman's seminal papers, and Debreu's paper on the integration of correspondences.

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