Herbert Scarf - Major Works - 4. Core and Competitive Equilibrium Equivalence

4. Core and Competitive Equilibrium Equivalence

Consider an economic system composed of many self-interested individuals each of whom is endowed with a bundle of goods, has preferences over the available bundles and wishes to achieve a maximal satisfaction by exchanging his/her own goods with others. The system requires every individual to respect the private ownership and the voluntary and non-coercive trade rule. Given this system, what will be a natural outcome of chaotic and countless independent actions of these self-interested agents? Adam Smith in his book ``The Wealth of Nations” (1776) first recognized how the invisible hand - a competitive market mechanism - can reconcile the complicated and conflicting forces of self-interested agents and guides the system to an equilibrium. The equilibrium is a state in which there exists a system of prices (i.e., market-clearing prices) at which every agent gets a best bundle of goods under his/her budget constraint and the supply of each good meets its demand. The list of the bundles obtained by all agents in the equilibrium state is called a competitive equilibrium allocation and is a redistribution of all agents’ initial endowments of goods. Wald (1936), Arrow and Debreu (1954), and McKenzie (1959) among many others established fundamental results on the existence of competitive equilibrium. The assumption of perfect competition or price-taking behaviour is crucial in these analyses. It essentially requires that the influence of every agent in the system should be negligible.

Another equally appealing and natural outcome of the economic system was first proposed by Francis Edgeworth in his book ``Mathematical Psychics’’ (1881), and is now known as the core allocation (in the case of two goods, it is any point in the contract curve of the Edgeworth box). Formally, a redistribution of all agents’ initial endowments of goods among all agents in the system is a core allocation if no group of agents can redistribute their own initial endowments among themselves so as to improve the satisfaction of someone in the group without impairing that of any other in the group. Clearly, a core allocation is Pareto efficient in the sense that there is no way to make some agent better off without making any other worse off. It is now well known that every competitive equilibrium allocation must be a core allocation but a core allocation need not be a competitive equilibrium allocation. Edgeworth worked with an economic system consisting of only two agents and two goods, and then replicated the economy many times. What he found is that as the replication tends to infinity, the set of core allocations converges to the set of competitive equilibrium allocations. This result provides a perfect justification of price-taking behaviour but in a very specific setting. However, Edgeworth’s approach is based on the geometrical picture of the Edgeworth box and cannot be applied to the general case involving more than two agents and more than two types of goods. The general case is known as Edgeworth conjecture and remained widely open for many several decades.

Based on the earlier paper of Scarf (1962), Debreu and Scarf (1963) resolved the outstanding theoretical problem in a brilliant and elegant manner. They started with a general economy consisting of any finitely many agents and a finite number of goods and proved that if one replicates the economy infinitely many times, then the set of core allocations coincides with the set of competitive equilibrium allocations. This offers an impeccable validation of perfect competition in a most general and most natural setting. This study has spawned a large body of literature on the relationship between the core and the set of competitive equilibrium allocations. One of the most significant contributions to this literature is the paper of Aumann (1964). Having heard Scarf’s discussion on his original 1962 paper at a conference at Princeton in 1962, Aumann established a model of pure exchange economy with a continuum of agents in which the core and the set of competitive equilibrium allocations are the same.

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