Customer poaching with differentiated products and switching costs

We consider a dynamic two-period model where two firms offer products that are differentiated a la Hotelling. Consumers purchase products in a first period, and in a second period consumers are locked-in to their first-period choice of producer with a switching cost. In the second period firms are able to price discriminate based on consumers purcase history from period 1. We show that i) firms will approach their rival's customers by low prices in the second period (customer poaching) and that inefficient switching will occur, ii) second-period prices are dependent on first-period market shares, a result in contrast to some of the received literature. Finally, iii) with high enough switching costs first-period prices is below the level in a static setting, and more so the higher the switching costs and the more differentiated the products are.