1992 Reports

# 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 [12], 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.

## Subjects

## Files

- econ_9293_650.pdf application/pdf 1.91 MB Download File

## More About This Work

- Academic Units
- Economics
- Publisher
- Department of Economics, Columbia University
- Series
- Department of Economics Discussion Papers, 650
- Published Here
- February 17, 2011

## Notes

December 1992.