This paper considers such a situation in a duopoly where demand is composed of overlapping generations of consumers, and firms are infinitely lived. Firms can price differently to their previous customers than to their new customers. The new customers can either have bought the competing product in the previous period or be new in the market. Equilibrium prices are shown to be lower because of the temptation of attracting the competitor' s previous customers. If consumers are more patient about the future, competition is even more intense and firms are less able to defend their clientele. If firms are more patient about the future the equilibrium involves less intense competition. With patient firms and consumers, prices are lower than when firms are not able to recognize their past customers.