We investigate a dynamic duopoly game with horizontal product differentiation, to show that the standard approach to spatial competition fails to produce a pure strategy equilibrium in prices when treated in a differential game framework. This holds independently of the shape of the transportation cost function. Then, we introduce an endogenous costs associated with the choice of location and characterise the open-loop and closed-loop equilibria of the model, showing that in the closed-loop case firms invest more in product differentiation and less in advertising, than they do in the open-loop setting. This happens because the gains from product differentiation can be more easily internalised than those associated with advertising.