Bertrand and the long run

We propose a new model of simultaneous price competition, where firms offer personalized prices to consumers, who then independently decide which offer to accept, if any. Even with decreasing returns to scale, this decentralized market mechanism has a unique equilibrium, which is independent of any exogenously imposed rule for rationing or demand sharing. In equilibrium, the firms behave as if they were price takers, leading to the competitive outcome (but positive profits). Given the unique result for the short-run competition, we are able to investigate the firms’ ex ante capital investment decisions. While there is underinvestment in the long-run equilibrium, the overall outcome is more competitive than one-shot Cournot competition.