Limited Arbitrage Is Necessary and Sufficient for the Existence of a Competitive Equilibrium
A condition of limited arbitrage is defined on the endowments and the preferences of the traders in an Arrow-Debreu economy. It bounds the diversity of the traders in the economy, and the gains from trade which they can afford from initial endowments. Theorem 1 shows that limited arbitrage is necessary and sufficient for the existence of a competitive equilibrium, when consumption sets are either positive orthants or the whole euclidean space. The results apply therefore to market economies with or without bounds on short sales. Theorem 6 establishes that an Arrow-Debreu economy has a competitive equilibrium if and only if every subeconomy with N + 1 traders does, where N is the number of commodities. Limited arbitrage has been shown elsewhere to be equivalent to the contractibility of spaces of preferences, and therefore, by the results of Chichilnisky and Heal , to be necessary and sufficient for the existence of social choice rules defined on individual preferences over allocations, rules which are continuous, anonymous and respect unanimity.
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