Kinked-Demand Equilibria and Weak Duopoly in the Hotelling Model of Horizontal Differentiation

Abstract

The Hotelling model with finite consumer reservation price is, in its various forms, perhaps the canonical model of horizontal product differentiation. Yet the following key aspects of this model are little understood: (i) the existence of asymmetric price equilibria when consumers have unit demands and (ii) for a very broad set of model specifications, the non-monotonicity of price as a function of consumers' transportation cost, i.e., the degree of product differentiation in the market. We provide a complete characterization of the asymmetric equilibria, show that they exist for a comparatively "wide" range of markets, and argue that their existence is robust to various extensions of the prototype model. Introducing elasticity into consumer demands suppresses the kink in the firms' demand functions and ensures the uniqueness (and symmetry) of the price equilibrium. However, the key perverse comparative static of the symmetric kinked equilibrium, decreasing price as a function of transport cost, survives relaxation of the unit-demand assumption. Indeed the symmetric kinked equilibrium of the unit-demand case is a special case of a general but largely unexplored set of equilibria in the Hotelling model we call weak duopoly. Weak-duopoly equilibria exist for intermediate values of transportation cost for a very broad class of consumer demands. They occupy a comparatively wide range of the model parameter space, and firm behavior in weak duopoly differs fundamentally from that which occurs for lower values of the transportation cost, or when consumers have no reservation price.