In this chapter, I consider the Hotelling model by assuming that each consumer is characterized by a continuously decreasing demand function (elastic demand function) rather than by a unit demand function. First, I show that when the two firms cannot price discriminate, the Hotelling model with elastic demand functions is not easy to manage, and the equilibrium outcome cannot be characterized explicitly. Then, I assume that firms can set different quantities (spatial Cournot competition) or different prices (spatial Bertrand competition) at different locations. I find the equilibrium in each of the two cases, and I compare them.