SFB 303 Discussion Paper No. B - 50

Author: Werner, Jan
Title: Asset Prices and Real Indeterminacy in Equilibrium with Financial Markets
Abstract: One of the features of the Arrow-Debreu model of a competitive economy that makes it an inadequate tool for representing the world we live in is the assumption of a complete set of markets. Over the past decade, there have been significant attempts to modify the original model in order to reflect the incomplete and sequential aspect of real world trading. Much of the interest have been focused in the modelling of futures markets. Futures markets can be thought of as being of two kinds: markets for contingent deliveries of commodities and markets for financial assets, i.e. for contingent deliveries of units of account.

In the present paper we study an exchange economy extending over two periods with incomplete futures markets for financial assets. A general equilibrium in such an economy has been previously studied by Werner (1985), Cass (1984), and Duffie (1985). The first part of the paper is devoted to equilibrium prices of assets. We show that every asset price vector that admits no arbitrage opportunity (i.e. such that there is no portfolio of assets that yields a positive and non-zero return stream, and can be purchased at zero cost) is an equilibrium asset price vector. More precisely, for every such an asset price vector there exists a vector of prices of commodities such that consumers’ optimal consumption plans and portfolios of assets are market clearing. This result suggests a large indeterminacy of equilibrium allocations generated by the indeterminacy of asset prices in equilibrium.

In the second part of the paper we study the degree of indeterminacy of equilibrium allocations. We are interested in real indeterminacy, i.e. affecting the allocation of commodities. We show that, if there are less assets than states of nature, then, generically, distinct vectors of asset prices lead to distinct equilibrium allocations of commodities. It turns out, however, that even with a fixed asset price vector there is an indeterminacy of equilibrium allocations. We find that, generically, there are as many dimensions of real indeterminacy with fixed asset prices as the number of states of nature minus the number of assets. Since the dimension of the set of different (normalized) non-arbitrage asset price vectors is the number of assets minus one, it follows that the dimension of real indeterminacy with variable asset prices is the number of states of nature minus one.
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Creation-Date: October 1986
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