M. Dolores Robles-Fernandez, Ph.D1, Hamid Hamoudi, Ph.D.2, and Marcos Sanz Martín-Bustamante, Ph.D.2. (1) Fundamentos del Análisis Económico, Universidad Complutense de Madrid, Facultad de Cc. Economicas, Campus de Somosaguas s.n., Pozuelo de Alarcón - Madrid, 28223, Spain, (2) Fundamentos del Análisis Económico, Universidad Rey Juan Carlos, Facultad de Cc. Económicas, Pº Artilleros s.n., Madrid, 28032, Spain
Equilibrium Existence in the Linear-city Model of Spatial Competition RevisitedIn this paper we have studied the existence of the sequential equilibrium in the context of the traditional Hotelling model of spatial competition. In our analyis, the transport cost structure is the key feature of the model. We assume a transport cost function which generalizes at the same time the convex and the concave case. We find that the demand function is always connected in the convex case whereas in the concave case this function can be non-connected After that, in order to study perfect Nash price equilibrium, we specify a particular transport cost function, the class of linear-quadratic and convex transport cost function. We propose a decisive change of variable that permit that the profit functions of both firms being symmetric to respect to the mean point between the locations of the firms. This allows us to simplify the analysis, and characterize the exact regions of location pairs for which a price equilibrium exist in a general framework. Our results confirm the general property that the sequential equilibrium fails to exist under linear-quadratic transport costs, in line with Gabszewicz and Thisse (1986), Anderson (1988), Hamoudi and Moral (2005) and Arguedas and Hamoudi (2008). The reason is that no price equilibrium exists for all the possible locations of the firms. As these authors, we confirm the existence of price equilibrium when firms locate sufficiently far in the case where the indifference consumer is located between the firms. Furthermore, we generalize the particular analysis of Anderson (1988), which only considers that one firm is located in the extreme of the city, by studying the equilibrium existence at any firm’s location. A price equilibrium exists if firms are located sufficiently close one from each other. To the point, our main contribution is to characterize the exact regions of location pairs for which a price equilibrium exist. In this sense, our general result closes the analysis on the existence of equilibrium for the class of linear-quadratic and convex transport cost function.