Arbitrage, duality and asset equilibria

Abstract

In finite dimensional economies, it was proven by Werner [Werner, J., 1987. Arbitrage and the existence of competitive equilibrium. Econometrica 55, 1403–1418.], that if there exists a no-arbitrage price (equivalently, under standard assumptions on agents' utilities, if aggregate demand exists for some price), then there exists an equilibrium. This result does not generalize to the infinite dimension. The purpose of this paper is to propose a “utility weight” interpretation of the notion of “of no-arbitrage price”. We define “fair utility weight vectors” as utility weight vectors for which the representative agent problem has a unique solution. They correspond to no-arbitrage prices. The assumption that there exists a Pareto-optimum, can be viewed as the equivalent of the assumption of existence of aggregate demand. We may then define in the space of utility weight vector, the excess utility correspondence, which has the properties of an excess demand correspondence. We use a generalized version of Gale–Nikaido–Debreu's lemma to prove the existence of an equilibrium.