Default Risk in a General Equilibrium Asset Economy with Incomplete Markets

Abstract

IN THE ARROW [1], Debreu [2] economy with contingent commodities, or derivable primitive securities (i. e., a security that pays one unit of account if a certain state of the world occurs), the analytical apparatus of certainty general equilibrium theory can be applied directly with suitable reinterpretation. Nevertheless, many writers (e. g., Arrow [1], Radner [11]) have criticized the complete market model as unrealistic, in requiring an embarrassingly rich set of forward insurance markets. Arrow has argued that transaction costs and moral hazard problems would be sufficient to destroy the existence of complete markets and introduce an incomplete market theory.2 There are some simplified incomplete market models in existence. One of the first formulations was based upon the mean-variance formulation of portfolio theory. This model was developed by Sharpe [12], Lintner [7] and other writers, and is the basis for much subsequent work in finance theory. 3 Another model was introduced by Diamond [4], who used the contingent commodity framework, but restricted trades to linear combinations or contingent commodities. Diamond's model is more general for analytical purposes than the mean-variance formulation, because it does not restrict preferences of subjective probability distributions. In these types of models there has been some confusion over the role of default risk, when agents borrow or short-sell. Diamond assumes no-bankruptcy, but does not include this assumption explicitly in his analysis, nor does he explore its implications. A similar situation applies for the mean-variance formulation.4 In this paper we will attempt to clarify the default risk issue, and at the same time generalize the Diamond model in a number of directions. Because the model has the same properties as the Debreu model (except for the possibility of unboundedness below for consumption sets when there are short sales) it is easy to show the existence of an equilibrium for a suitably restricted asset economy. The paper is divided into two sections: Section 1 provides the formal model and proofs; Section 2 is concerned with interpretations, extensions and limita-