Competitive Equilibrium in Sobolev Spaces without Bounds on Short Sales

Abstract

Following Chichilnisky and Chichilnisky-Kalman we establish existence and optimality of competitive equilibrium when commodity spaces are infinite dimensional Sobolov spaces, including Hilbert spaces such as weighted L2 which have L∞, as dense subspaces. We allow general consumption sets with or without lower bounds, thus including securities markets with infinitely many assets and unbounded short sales, and economies with production. We give non-arbitrage conditions on endowments and preferences which suffice for the existence of an equilibrium. Prices are in the same space as commodities. Equilibrium allocations are approximated by allocations in other frequently used spaces such as C(R) and L∞. © 1993 Academic Press. All rights reserved.