Analytical solution of a multi-dimensional Hotelling model with quadratic transportation costs

We present an analytical solution to the two-dimensional two-stage Hotelling model with quadratic transportation costs. We assume that consumers' choice is tempered by a logit function, which characterizes consumers' heterogeneity. As in the one-dimensional case, stores aggregate spatially when consumers' heterogeneity is strong enough. When it decreases, we show that stores differentiate in only one dimension. The analytical solution allows us to give a precise interpretation of this effect through the comparison of consumers' elasticity under differentiation along one or two characteristics. Finally, we extend our results to a hypercube of any dimension.