Bertrand equilibrium in a differentiated duopoly

This paper studies the stability of price competition in a horizontally differentiated duopoly. The firms' demand is derived from a distribution of consumer preferences. This description of the consumer sector is applicable to a large class of differentiated commodity markets, including spatial competition models. We show that there is a (pure) price setting equilibrium when consumer tastes are sufficiently dispersed. Further conditions on the dispersedness of preferences guarantee uniqueness of the equilibrium. In addition, we examine the relation between consumer preferences and the competitiveness and efficiency of the equilibrium outcome. This paper investigates the stability of price competition in a horizontally differentiated duopoly. The duopolists' demand is derived from a distribution of preference characteristics over the population of consumers. We show that competition between the firms results in a (pure) price equilibrium when consumer tastes are sufficiently dispersed. The competitiveness of the equilibrium is closely related to the diversity of consumer types. When the support of the preference distribution shrinks to a single point, the equilibrium approaches the Bertrand outcome of a homogeneous good market. We further show that a sufficient degree of preference dispersion guarantees uniqueness of the equilibrium. Finally, we discuss the firms' incentives for product differentiation from the viewpoint of social efficiency. The attractiveness of Bertrand's (1883) approach to the theory of oligopoly lies in the fact that in his model prices are chosen by economic agents rather than by a fictitious auctioneer. Yet, modelling price competition leads to a number of problems, especially with homogeneous goods. As already Bertrand (1883) observed, in the case of equally efficient firms and constant marginal costs the price setting equilibrium coincides with the competitive outcome. This extreme prediction, that holds even with only two firms, appears paradoxical and economically uninteresting for oligopolistic competition. In the absence of the constant returns to scale assumption, the Bertrand model faces up to a further drawback, namely the problem of nonexistence of equilibrium. This problem was first pointed out by Edgeworth (1925) in his analysis of a capacity constrained oligopoly. In his famous article on the stability of competition, Hotelling (1929) took the view that these problems originate in the abstraction of homogeneous goods. Under this assump