Aggregation and Imperfect Competition: On the Existence of Equilibrium

Abstract

We present a new approach to the theory of imperfect competition and apply it to study price competition among differentiated products. The central result provides general conditions under which there exists a pure-strategy price equilibrium for any number of firms producing any set of products. This includes products with multi-dimen- sional attributes. In addition to the proof of existence, we provide conditions for uniqueness. Our analysis covers location models, the characteristics approach, and probabilistic choice together in a unified framework. To prove existence, we employ aggregation theorems due to Prekopa (1971) and Borell (1975). Our companion paper (Caplin and Nalebuff (1991)) introduces these theorems and develops the application to super-majority voting rules. WE PRESENT A NEW APPROACH to the theory of imperfect competition and apply it to study price competition among differentiated products. The central result is that there exists a pure-strategy price equilibrium for any number of firms producing any set of products. In addition to the proof of existence, we provide conditions for uniqueness. Our model both unites diverse strands of the earlier literature and opens up uncharted areas for future analysis. In particular, we expand the traditional one-dimensional framework to allow for multi-dimen- sional product differentiation. Our approach involves twin restrictions on consumer preferences: one on individuals' preferences, the other on the distribution of preferences across society. These are generalizations of the restrictions supporting 64%-majority rule presented in Caplin and Nalebuff (1988). To prove existence, we apply a new technique of aggregation. This technique is valuable in a variety of other problems. In the companion paper, we use the aggregation result to generalize our earlier work on 64%-majority rule and to characterize the relationship between the distribution of human capital and the distribution of income (Caplin and Nalebuff (1991)). There are additional applications in statistics and in search theory. We begin with a brief review of the early literature on imperfect competition, describing in more detail the existence problem and previous solutions. Section 3 presents our twin assumptions, and shows that they cover many standard cases. In Section 4, we introduce the aggregation theorem and use it in the analysis of demand functions. The proof of existence of equilibrium is in Section