Models of spatial competition have proven to be very useful in describing differentiated products markets. A serious problem is that nonexistence of Nash equilibria seems endemic. This problem is resolved by modelling the price formation process using the core. The equilibrium is the outcome of a two-stage process. In the first stage, two firms Choose locations simultaneously, looking ahead to the second stage. The second stage has prices determined by an allocation in the core of a cooperative subgame allowing for coalitions of buyers and sellers. The price selection is the joint profit maximum for the duopolists. This selection exists for all location pairs and coincides with the pure strategy Nash equilibrium of duopoly competition when the latter exists. Furthermore, these prices approach the competitive level as the distance between the firms goes to zero, thus capturing the essence of duopoly rivalry. For this price selection, in the location game, the two firms establish themselves at the efficient locations--the first and third quartiles.