Location, location, location

We analyze the canonical location-then-price duopoly game with general log- concave consumer densities. A unique pure-strategy equilibrium to the two-stage game exists if the density is not "too asymmetric" and not "too concave." These criteria are satisfied by many commonly used densities. Equilibrium locations are closer, and prices lower, the tighter the density. Our results apply also to a vertical differentiation specification. Symmetric densities that are "too concave" have no symmetric equilibrium, although asymmetric ones may exist. Finally, product differentiation is always excessive. Under symmetry the equilibrium dispersion lies between 3/2 and 3 times the optimum dispersion.