(b) Now suppose these two firms play a 2-stage game. In stage 1, they produce capacities ₁ and 2, which are equal to the Nash equilibrium quantities of the Cournot game characterised by part (a). In stage 2, they simultaneously decide on their prices pi and p2. The marginal cost for each firm to sell up to capacity is 0. It is impossible to sell more than capacity. The residual de- 1-Piāj if pi > Pj if Pi = Pj. 1-Pi mand for firm i #j, is Di (pi, Pj) = 1- Pi if pi < Pj (Note, here we assume that the efficient/parallel rationing ap- plies). Prove that it is a Nash equilibrium of the second stage subgame that each firm charges the market clearing price p; = 1-91-92, for i = 1, 2.
(b) Now suppose these two firms play a 2-stage game. In stage 1, they produce capacities ₁ and 2, which are equal to the Nash equilibrium quantities of the Cournot game characterised by part (a). In stage 2, they simultaneously decide on their prices pi and p2. The marginal cost for each firm to sell up to capacity is 0. It is impossible to sell more than capacity. The residual de- 1-Piāj if pi > Pj if Pi = Pj. 1-Pi mand for firm i #j, is Di (pi, Pj) = 1- Pi if pi < Pj (Note, here we assume that the efficient/parallel rationing ap- plies). Prove that it is a Nash equilibrium of the second stage subgame that each firm charges the market clearing price p; = 1-91-92, for i = 1, 2.
Chapter15: Imperfect Competition
Section: Chapter Questions
Problem 15.5P
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