Consistent Solutions in Atomless Economies

Abstract

We consider the problem of allocating a bundle of commodities among a group of agents who are collectively entitled to them. We prove: For an atomless economy with possibly satiated preferences, any solution that is efficient, equitable, and consistent must select allocations that are supported by equal-budget Walrasian equilibria with slack. WE CONSIDER THE PROBLEM of allocating a bundle of commodities among a group of agents who are collectively entitled to them. We search for systematic methods, or solutions, of resolving this problem. For a solution to be considered desirable, we propose that it should satisfy several requirements. First, it should be efficient. Second, it should be equitable. A variety of distributional require- ments have been proposed in the literature. The requirement we will impose here is that it select allocations at which no agent would prefer the equal division of the social endowment to his own consumption. In addition to these two basic efficiency and equity conditions, we require that the solution be consistent: the recommendations it makes for any economy are never contra- dicted by the recommendations it makes for any reduced economy obtained by allowing some subgroup of agents to leave the scene with their allotted consumptions. The requirement of consistency is very natural. Here is a motivation for it, offered by Thomson (1988): consider all the resources available on the planet earth; one would like not only that they be distributed fairly when the planet is considered as a whole but also that whatever resources end up in each continent, country, city,... be fairly distributed among the members of that subpopulation, when considered in isolation. In this paper we study economies with a large number of agents modelled as a complete atomless measure space. Our main result is a characterization of the solutions that satisfy all three requirements of efficiency, equity, and con- sistency. It follows from our result that, under standard assumptions on preferences, any such solution must select allocations that are supported by equal-income Walrasian equilibria. However, our characterization holds for more general economies since it applies to economies in which preferences are possibly satiated. Mas-Colell (1988) extended traditional general equilibrium theory to the class of such economies by introducing the notion of Walrasian