Consumer Preferences, Linear Demand Functions and Aggregation in Competitive Asset Markets

Abstract

In Arrow (1970) and Debreu (1959) it was shown that one could obtain a well-defined economic theory with uncertainty over states of the world, which was formally equivalent to the certainty theory of value. Although the Arrow-Debreu framework is a useful starting-point for introducing uncertainty, its complete market assumption is patently unrealistic, so that most subsequent research has been directed toward incomplete market models. Compared with the elegant generality of the original Arrow-Debreu formulation, subsequent models abound with restrictive assumptions upon consumer preferences and probability distributions.' Since the pioneering work of Arrow and Pratt (1964) on measures of risk-aversion, many writers have applied these measures, or their derivative utility functions, to problems under uncertainty. Furthermore, there has developed a literature in Finance Theory, which has evolved the closely related HARA class2 of utility functions, from which one can deduce a number of interesting aggregation theorems. What is not widely understood, is that much of this recent work is intimately related to the theory of aggregation which has a comparatively long history in economic theory. By making explicit the relationship between these two bodies of knowledge, one can gain important insights and generalizations of these recent results in uncertainty economics. In essence, the HARA class (which is in fact a generalization of the older Bergson class of utility function) generates linear affine Engel curves for assets. Of course, linear Engel curves considerably simplify an analysis of the consumer's problem. Furthermore, it is well known that if agents have linear Engel curves with the same slope, then this is a necessary and sufficient condition for aggregation over the agents. Given these two results, it's easy to see a simple geometric interpretation of a whole literature in uncertainty theory; and to see its advantages and limitations. The paper has been divided into six sections; Section 1 surveys additive utility functions with linear Engel curves; Section 2 summarizes the Gorman, representative-consumer theorem; Section 3 translates the theorems of the previous sections into the Arrow-Debreu complete market model and obtains particular cases for the more general results; Section 4 considers a two period incomplete market model and the aggregation theorem; Section 5 extends the analysis to many periods; and Section 6 provides some general concluding comments.