Symmetric and Asymmetric Equilibria in a Spatial Duopoly

We describe a spatial duopoly in a Hotelling model with quadratic transportation costs where consumers are distributed according to a symmetric density whose degree of concentration is variable. By solving the two-stage game in prices and locations as a function of the concentration index, we analyse the effects on the firms’optimal choices in a unbounded strategy space of an increasing agglomeration of consumers in the middle. Traditional horizontal differentiation-locational models assume that consumers are uniformly distributed over the characteristics space. With a few exceptions, the situations in which the consumers' preferences are concentrated on a subsection of the available varieties have been neglected. This issue was successfully addressed by Tabuchi and Thisse (1995), who explicitly solved the price-location problem for two firms in the presence of a symmetric triangular consumers’ distribution. They showed that in this case any symmetric location cannot be an equilibrium, due to a discontinuity of the reactions functions generated by the non-differentiability of the consumers’ density at its modal value; rather, their model exhibits two subgame perfect asymmetric equilibria characterised by strong product differentiation. In this paper, we assume that consumers are distributed according to a trapezoid distribution. This allows a simple parametrization of the degree of consumers' concentration, which includes the uniform and the triangular distribution as limit cases, and makes possible to solve the price-location problem as a function of the concentration index. Therefore we are able to find a more general explicit solution which covers those previously discussed in the literature. The basic results of the paper are the following. A symmetric equilibrium exists for all values of the concentration parameter, provided that the density is differentiable at the centre of its support. A higher degree of the consumers’ concentration around the middle induces firms to move inwards, in order to locate closer to the growing share of consumers: competition in the highly populated central area of the market reduces differentiation and strengthens price competition. The overall equilibrium shows clearly that the demand effect outweighs the strategic effect. However the symmetric equilibrium may be not unique. When concentration becomes sufficiently high, two asymmetric specular equilibria coexist with the symmetric one. They arise for a degree of concentration lower than that implied by a triangular distribution, with price-location choices collapsing in the limit to those identified by Tabuchi and Thisse. At these equilibria one firm locates in the central area of the market, while the other locates outside the market space. These results are consistent with the idea that a higher concentration of consumers around the centre induces firms to reduce the optimal product differentiation and offer theoretical support to the intuition that homogeneity of consumers might have important implications in terms of reducing the firms' market power. However, our findings suggest that in models of spatial competition realistic representations of the demand side may generate a ‘strange’ interplay between the strategic effect and the demand effect which may cause a failure of the uniqueness property and weakens the economic interpretation of equilibria.