1. (10 points) Consider a duopoly market in which the two firms have no costs of
ID: 1216325 • Letter: 1
Question
1. (10 points) Consider a duopoly market in which the two firms have no costs of production. They produce homogeneous goods and inverse market demand is given by P d (Q) = 1 ? Q, where Q denotes the total output of the two firms.
(a) (5 points) Calculate the profit, ?^N , at the static Cournot-Nash equilibrium.
(b) (5 points) Suppose each firm each produces output q0 ? (0, 1/2) in a period. Calculate each firm’s associated profit, ? ^0 . Calculate a firm’s maximum profit, ?^d , if it “cheats” (“defects”) optimally against a rival that produces q0.
(c) (5 points) Now suppose the firms interact over an infinite horizion, repeatedly playing the Cournot stage game an infinite number of times. Future payoffs are discounted by the common per-period discount factor ? = 1/3. Determine the most profitable common ouput per firm, q0, that the firms can sustain in each period as a (subgame perfect) equilibrium supported by a Grim Trigger Strategy that threatens reversion to the static Nash equilibrium.
Explanation / Answer
1.a.
P = 1 – Q
Let q1 and q2 are the quantities of 1st and 2nd firm respectively.
P = 1 – (q1 + q2)
TR = P×q1 = 1q1 – q1^2 – q1q2
MR = Derivative of TR with respect to q1
= 1 – 2q1 – q2
The equilibrium condition is MR = MC
1 – 2q1 – q2 = 0 ………………………. (i)
Again,
TR = P×x2 = 1q2 – q1q2 – q2^2
MR = Derivative of TR with respect to q2
= 1 – q1 – 2q2
The equilibrium condition is MR = MC
1 – q1 – 2q2 = 0………………………. (ii)
Solving (i) and (ii), q1 = 1/3 quantities and q2 = 1/3 quantities
Q = q1 + q2
= 1/3 + 1/3
= 2/3
Therefore price, P = 1 – Q
= 1 – 2/3
= 1 – 2/3
= 1/3
TR = P × Q = 1/3 × 2/3 = 2/9
TC = 0
Profit = TR – TC
= 2/9 – 0
= 2/9 (Answer)
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