This material has been published in *Journal of Mathematical Economics* 34, (2000), pages 439-469, the only definitive repository of the content that has been certified and accepted after peer review.

PII: S0304-4068(00)00052-5

Copyright © 2000 Elsevier Science S.A. All rights reserved.

Frank H. Page Jr. ,a, Myrna H. Wooders, b, and Paulo K. Monteiro, c.

a Department of Finance, University of Alabama, Tuscaloosa, AL 35487, USA

b Department of Economics, University of Warwick, Coventry CV4 7AL, UK

c EPGE/FGV, Praia de Botafogo 190 sala 1125, 22253-900 Rio de Janeiro, RJ, Brazil

Received 4 April 1999; revised 27 June 2000; accepted 7 July 2000. Available online 10 November 2000.

##### Abstract

We introduce the concept of inconsequential arbitrage and, in the context of a model allowing short-sales and half-lines in indifference surfaces, prove that inconsequential arbitrage is sufficient for existence of equilibrium. Moreover, with a slightly stronger condition of nonsatiation than that required for existence of equilibrium and with a mild uniformity condition on arbitrage opportunities, we show that inconsequential arbitrage, the existence of a Pareto optimal allocation, and compactness of the set of utility possibilities are equivalent. Thus, when all equilibria are Pareto optimal - for example, when local nonsatiation holds - inconsequential arbitrage is necessary and sufficient for existence of an equilibrium. By further

strengthening our nonsatiation condition, we obtain a second welfare theorem for exchange economies allowing short sales. Finally, we compare inconsequential arbitrage to the conditions limiting arbitrage of Hart [Hart, O.D., 1974. J. Econ. Theory 9, 293-311], Werner [Werner,J., 1987. Econometrica 55, abs 1403-1418], Dana et al. [Dana, R.A., Le Van, C., Magnien, F.,

1999. J. Econ. Theory 87, 169-193] and Allouch [Allouch,N., 1999. Equilibrium and no market arbitrage. CERMSEM, Universite de Paris I]. For example, we show that the condition of Hart (translated to a general equilibrium setting) and the condition of Werner are equivalent. We then show that the Hart/Werner conditions imply inconsequential arbitrage. To highlight the extent to which we extend Hart and Werner, we construct an example of an exchange economy in which inconsequential arbitrage holds (and is necessary and sufficient for existence), while the Hart/Werner conditions do not hold.

Author Keywords: Arbitrage; Pareto optimality; Recession cones, Increasing cones, Existence of equilibrium,

##### Article Outline

1. Introduction

2. An economy with short sales

3. Inconsequential arbitrage

4. Implications of inconsequential arbitrage

4.1. Increasing cones and exhaustible arbitrages

4.2. Compactness of the set of utility possibilities

5. Main existence result

6. Pareto optimality, inconsequential arbitrage, and existence of equilibrium

6.1. Two equivalence results

6.2. A second welfare theorem

7. Examples

8. Other conditions limiting arbitrage: some comparisons

8.1. A Comparison to Hart andWerner

8.1.1. The equivalence of Hart's condition and Werner's condition

8.1.2. Hart/Werner conditions and inconsequential arbitrage

8.1.3. A summary of results

8.1.4. Examples

8.2. A comparison to Dana etal. and Allouch

8.2.1. Dana et al

8.2.2. Allouch

8.3. A fundamental equivalence result

9. Proofs

9.1. Bounded economies

9.2. A lemma

9.3. Proofs of Theorems

9.3.1. Proof of Theorem 1 (inconsequentiality implies exhaustibility)

9.3.2. Proof of Theorem 2 (inconsequentiality and boundedness of utilities)

9.3.3. Proof of Theorem 3 (inconsequential arbitrage implies existenceof a quasi-equilibrium)

9.3.4. Proof of Theorem 4 (equivalence of existence of a Pareto optimal allocation, inconsequential arbitrage, and

compactness of utility possibilities)

9.3.5. Proof of Theorem 6 (second welfare theorem)

9.3.6. Proof of Theorem 7 (equivalence of Hart and Werner)

9.3.7. Proof of Theorem 8 (Hart/Werner imply inconsequential)

Acknowledgements

References