Outcome Functions Yielding Walrasian and Lindahl Allocations at Nash Equilibrium Points

Abstract

In this paper we shall present balanced outcome functions . These are functions which, for any given set of messages by individual agents in the economy, determine as their outcome an economic allocation satisfying the resource balance constraints of the economyi.e. the constraints that, for any good, total demand must not exceed total supply. Now, it has been known for a long time that outcome functions realizing the Walrasian (or Lindahl) equilibrium correspondence can easily be constructed provided an auctioneer is permitted and balance not required. In unpublished work, Schmeidler has presented a balanced outcome function without an auctioneer which realizes the Walrasian correspondence when there are at least three traders; a later variant (in a forthcoming paper by the Author) appears also to work for two traders. As he himself noted, his outcome function is not individually feasible. It is also discontinuous. Groves and Ledyard (1977) proposed balance outcome functions whose equilibrium correspondences yield allocations which are Pareto optimal but not individually rational (in the sense that traders can be worse off than if they refused to trade at all), and hence not Lindahl allocations. Here again it is required that there be at least three traders. The outcome functions used by Groves and Ledyard are quadratic, and so smooth. For the pure exchange economy with at least three traders Hurwicz (1976) has constructed analogous balanced outcome functions which are Pareto optimal but not individually rational, as well as non-balanced outcome functions which are Pareto optimal (but not individually rational) when there are two traders. Shapley, Shubik, Schmeidler, Pazner and Postlewaite have also constructed feasible outcome functions that can yield allocations outside the Pareto optimal set. Either the outcome functions or the message spaces depend, however, on the initial endowment vectors, so that agents have to know one anothers' endowments. Thus, these outcome functions are not informationally decentralized. In the present paper we show balanced outcome functions that are smooth and, without employing an auctioneer, yield equivalence between Nash and Walrasian (or, where appropriate, Lindahl) allocations, for three or more traders. Our outcome functions are smooth, but not individually feasible away from equilibrium.