SFB 303 Discussion Paper No. A - 344

Author: Brown, Donald, and Jan Werner
Title: Arbitrage and Existence of Equilibrium in Infinite Asset Markets
Abstract: This paper develops a model for the general equilibrium analysis of markets with infinitely many assets. A partial equilibrium model of large asset markets, with prices exogenously given, has been analyzed by Chamberlain and Rothschild (1983) and Chamberlain (1983) in their investigation of the Arbitrage Pricing Theory of Ross (1976). A complete presentation of the APT requires an infinite number of assets, optimizing investors, and an endogenous determination of equilibrium prices as a consequence of the lack of arbitrage opportunities. Our paper provides such a model.

We combine two recent developments in equilibrium theory: the existence of equilibrium in economies with infinite dimensional commodity spaces, and the existence of equilibria in economies with finite dimensional commodity spaces where the consumption sets are not bounded below. In the literature on the existence of equilibria in infinite dimensional spaces (see Aliprantis, Brown, and Burkinshaw (1989), and Mas-Colell and Zame (1990)) it is typically assumed that the consumption set is the positive orthant of the commodity space, which is bounded below. This assumption is quite plausible for equilibrium models of consumption but does not apply to models of asset markets. The prototypical equilibrium model of finite asset markets is due to Hart (1974). The existence of equilibrium result for an arbitrage-free economy has been extended to more general (finite dimensional) setting by Werner (1987) and Nielsen (1989). It has long been recognized in the literature on asset markets that the concept of the absence of arbitrage opportunities as developed for finite markets is far too weak for infinite markets (see Kreps (1981)). In Section 3 we provide a discussion of concepts of an arbitrage opportunity and develop a notion suitable for equilibrium analysis of infinite markets.

Our principal result (Section 4) is that if the set of arbitrage-free prices is nonempty then there exists an equilibrium. Our proof follows closely the proof of the existence of equilibrium theorem of Mas-Colell, as given by Aliprantis, Brown, and Burkinshaw (1989). In Section 5 we present an example of Hart's (1974) securities market model with infinitely many securities and apply to it our main result of Section 4.
Creation-Date: June 1991
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