Hotelling's competition with general customer distributions

Hotelling and his critics assumed uniformly distributed demand. In this note the solution is characterized for any distribution F and some propositions follow. For any F an equilibrium does not exist if the duopolists are located too closely together, but another distribution can be always found for which an equilibrium does exist. When we observe duopolies in equilibrium with closer and closer proximity we can expect the density in the area of contention to go up and prices and profits to go down. Hotelling's known suggestion that any duopolist has an advantage to move his location towards the other, and thus creating a tendency for the differentiation to diminish, holds whenever an equilibrium exists.