Dynamic price competition with capacity constraints and strategic ...

p Bp45°p −δV 3p + δV 3: (2,0): (1,1): (0,2)pδV 3δV 3p p + δV3p δV3− p pAFigure 3: Period 1 acceptances as a function of prices.seller’s price will never be more than δV 3 above his price. He will sell 1 unit if the prices are withinδV 3 , and 0 units otherwise. Since we show that p − p ≥ 2δV 3 , we establish the following importantresult.Proposition 3 In the monopsony model, splitting of orders by the buyer between the two sellersoccurs in equilibrium with positive probability: when the difference of the two prices is smaller thanδV 3 ,thebuyerbuysoneunitfromeachseller.In Appendix A4, we also prove that the lowest price, p, offered by the sellers in a mixed strategyequilibrium of the monopsony model is greater than δV 3 . It immediately then follows that:Proposition 4 In the monopsony model, the expected profit of each seller is greater than δV 3 .Thus, in equilibrium, the sellers receive rents above satisfying the residual demand after thebuyer bought the other seller’s capacity (or the static Bertrand competition), δV 3 . Why is this thecase? By Lemma 4, a seller knows that, if he makes no sales in period 1, his expected profit isδV 3 .This gives a seller the incentive to raise his price above δV 3 to take a chance of not making a sale inperiod 1, since by Lemma 5 a seller knows that even if he hasthehighestpricehewillmakeasellas long as the price difference is less than δV 3 . Since there is no cost of increasing his price aboveδV 3 while there is a potential benefit, the seller can improve his payoff.14