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(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.

ENGR.ECONOMIC ANALYSIS
ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN:9780190931919
Author:NEWNAN
Publisher:NEWNAN
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Kindly solve part (b)

1. Consider a Cournot duopoly. The inverse demand function of the market is given by p = 1 – Q, where p is the market price, and Q = q1+ q2 is the aggregate output. The marginal costs of the two firms are ci = 0.1 and c2 = 0.4. (a) Solve for the Nash equilibrium of the game including firm outputs, market price, aggregate output, and firm profits. (b) Now suppose these two firms play a 2-stage game. In stage 1, they produce capacities qi and ī2, 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- pi – īj if pi > Pj if pi = Pj · if pi < Pi (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; = mand for firm i + j, is D; (pi, P;) = 1-Pi 2 1- Pi 1- ğı – 2, for i = 1, 2. (c) This part continues from part (b). Prove that each firm charges the market clearing price p; = 1– ı – 72, for i = 1,2, is the unique pure strategy Nash equilibrium.
Transcribed Image Text:1. Consider a Cournot duopoly. The inverse demand function of the market is given by p = 1 – Q, where p is the market price, and Q = q1+ q2 is the aggregate output. The marginal costs of the two firms are ci = 0.1 and c2 = 0.4. (a) Solve for the Nash equilibrium of the game including firm outputs, market price, aggregate output, and firm profits. (b) Now suppose these two firms play a 2-stage game. In stage 1, they produce capacities qi and ī2, 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- pi – īj if pi > Pj if pi = Pj · if pi < Pi (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; = mand for firm i + j, is D; (pi, P;) = 1-Pi 2 1- Pi 1- ğı – 2, for i = 1, 2. (c) This part continues from part (b). Prove that each firm charges the market clearing price p; = 1– ı – 72, for i = 1,2, is the unique pure strategy Nash equilibrium.
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