Duopolistic price competition with captives

We extend the Bertrand duopolistic competition to include captives. These are consumers that have no choice between the suppliers. Usual population of shoppers are modeled performing a sequential search in order to decide where to buy a homogenous good. These two simple departures from the original setup have sharp consequences. First, we find that duopolistic price competition is not robust to inclusion of captives. The equilibrium results starkly differ and the only possible equilibrium now includes duopolists charging monopolistic prices. Second, addition of sequential search introduces multiplicity of pure strategy Nash equilibria. In this setup, we observe perverse optimal response to competitor's price changes. Notably, we find that the firm might want to reduce the price in response to the competitor's price increase, which is at odds with the usual undercutting principle. Third, we investigate the behavior of equilibrium prices depending on the heterogeneity in consumer risk attitudes. We find that the higher consumer heterogeneity with respect to acceptance of risky gambles leads to higher prices in equilibrium.