Arbitrage, Equilibrium, and Nonsatiation

Abstract

In his seminal paper on arbitrage and competitive equilibrium in unbounded exchange economies, Werner (Econometrica, 1987) proved the existence of a competitive equilibrium, under a price noarbitrage condition, without assuming either local or global nonsatiation. Werner's existence result contrasts sharply with classical A substantial part of the paper was written while Allouch w as at CERMSEM, University o f P aris 1, and at the Department of Finance at the University of Alabama. Allouch thanks CERMSEM, Paris 1, and the Department of Finance at the University of Alabama for their generous support.y This paper was begun while Le Van was visiting the Department of Finance at the University of Alabama. Le Van gratefully acknowledges the support and hospitality o f the Department of Finance at the University of Alabama.z The current v ersion of the paper was completed while Page was visiting CERMSEM at Paris 1 and the Department of Economics at Warwick. Page gratefully acknowledges the support and hospitality of CERMSEM, Paris 1, and the Department of Economics at Warwick. Page also acknowledges generous support from the Department o f F i n a n c e at the University of Alabama. 1 existence results for bounded exchange economies which require, at minimum, global nonsatiation at rational allocations. Why do unbounded exchange economies admit existence without local or global nonsatiation? This question is the focus of our paper. We m a k e t wo main contributions to the theory of arbitrage and competitive equilibrium. First, w e s h o w that, in general, in unbounded exchange economies (for example, asset exchange economies allowing short sales), even if some agents' preferences are satiated, the absence of arbitrage is su cient for the existence of competitive equilibria, as long as each agent who is satiated has a nonempty set of useful net trades -that is, as long as agents' preferences satisfy weak nonsatiation. Second, w e provide a new approach t o p r o ving existence in unbounded exchange economies. The key step in our new approach is to transform the original economy to an economy satisfying global nonsatiation such that all equilibria of the transformed economy a r e equilibria of the original economy. What our approach m a k es clear is that it is precisely the condition of weak nonsatiation -a condition considerably weaker than local or global nonsatiation -that makes possible this transformation. Moreover, as we show via examples, without weak nonsatiation, existence fails.