p.1271

Values of Markets with Satiation or Fixed Prices

Robert J. Aumann
Jacques H. Dreze

Abstract

In markets with satiation, competitive equilibria may fail to exist, because no matter what the prices are, the satiation points of some traders may be in the interiors of their budget sets. Thus some traders will be using less than the maximum budget available to them, creating a total budget excess. This suggests a revision of the equilibrium concept that allows the budget excess to be divided among all the traders, as dividends. Each trader's budget is then the sum of his dividend and the market value of his endowment. A given system of dividends and prices defines a dividend equilibrium if it generates equal supply and demand. This in itself is not satisfactory because it is too broad: Every Pareto optimal allocation is sustained by some system of dividends and prices. However, the Shapely value yields much more specific information. We prove that, when there are many individually insignificant agents, every Shapely value allocation is generated by a system of dividends and prices in which all dividends are nonnegative and depend only on the net trade sets of the agents, not on their utilities. Moreover, the dependence is monotonic; the larger the net trade set, the higher the dividend. The same result holds for markets with fixed prices, which can be analyzed formally as a special case of markets with satiation. On a more technical level, our analysis has some unusual features. We use a finite-type asymptotic model, rather than a nonatomic continuum. Surprisingly, the results are qualitatively different. (The continuum is too rough a tool for our problem, and leads to inconclusive results.) Also, small coalitions play a critical role in our analysis. (We are led to equations in which the first-order terms cancel; the second-order terms, which take events of small probability into account, become decisive.)