Bertrand and the long run
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We propose a new model of simultaneous price competition, based on firms offering personalized prices to consumers. In a market for a homogeneous good and decreasing returns, the unique equilibrium leads to a uniform price equal to the marginal cost of each firm, at their share of the market clearing quantity. Using this result for the short-run competition, we then investigate the long-run investment decisions of the firms. While there is underinvestment, the overall outcome is more competitive than the Cournot model competition. Moreover, as the number of firms grows we approach the competitive long-run outcome.