Bertrand and the long run

Abstract

We propose a new model of simultaneous price competition, where …rms o¤er personalized prices to consumers, who then independently decide which o¤er 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 …rms behave as if they were price takers, leading to the competitive outcome (but positive pro…ts). Given the unique result for the short-run competition, we are able to investigate the …rms'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.

Personalized prices and uncertainty in monopsony

“…Once more, when the number of suppliers is large (and their capacity small, with respect to the size of the market), max j v j is small and (given regularity) the su¢ cient condition in Proposition 3 is satis…ed. Consequently, the expected values in the left hand side of (5) and (6) for similar values of p 1 and p 2 approach, and then we recover the predictions of the classical monopsony model. We now undertake to prove formally this and previous convergence results.…”

“…The literature on optimal trading mechanisms 6 is not directly relevant, as our interest here is in a second best. Another strand of the literature makes pairwise comparisons between bargaining, auctions and posted prices.…”

“…Nonetheless, it is of note that our personalized pricing scheme is the best mechanism the buyer can devise subject to bilateral commitment in the static context. Insisting on bilateral commitment is the alternative approach to designing credible mechanisms where the principal does not cheat because the incentives are set right, as in Li (2017) and Akbarpour and Li (2018).Finally, it is important to point out that the personalized price setting mechanism that we analyze in this paper can be usefully adapted to the context of competing price setters Burguet and Sákovics (2017a, 2017b, 2019. are witnesses to this.…”

“…When both cost distributions are super-regular relative to the buyer's marginal valuation of each unit, the optimal personalized pricing strategy is a posted price in each market, satisfying (5)-(6).…”

Competitive foreclosure

“…Proposition —as well as our subsequent results—generalizes to asymmetric firms, but in that case, the unique (supplier‐symmetric) equilibrium requires the suppliers to use a different mixed strategy. See Proposition in Burguet and Sákovics ().…”