Question

Suppose there are two firms in a market who each simultaneously choose a quantity. Firm 1’s quantity is q1, and firm 2’s quantity is q2. Therefore the market quantity is Q = q1 + q2. The market demand curve is given by P = 100 – 2Q. Also, each firm has constant marginal cost equal to 10. There are no fixed costs. The marginal revenue of the two firms are given by: MR1 = 100 – 4q1 – 2q2 MR2 = 100 – 2q1 – 4q2. A) How much output will each firm produce in the Cournot equilibrium? B) What will be the market price of the good? C) What is the deadweight loss that results from this duopoly? D) How much profit does each firm make? E) Suppose Firm 2 produced 25 units of output. How much output should Firm 1 produce in order to maximize profit?

Answer 1

P = 100 - 2q1 - 2q2

(A)

For firm 1, setting MR1 = MC,

100 - 4q1 - 2q2 = 10

4q1 + 2q2 = 90...........(1) [Best response, firm 1]

For firm 2, setting MR2 = MC,

100 - 2q1 - 4q2 = 10

2q1 + 4q2 = 90...........(2) [Best response, firm 1]

(2) x 2 yields

4q1 + 8q2 = 180.........(3)

4q1 + 2q2 = 90...........(1)

(3) - (1) yields:

6q2 = 90

q2 = 15

q1 = (90 - 4q2)/2 [from (2)] = [90 - (4 x 15)]/2 = (90 - 60)/2 = 30/2 = 15

(B)

Q = 15 + 15 = 30

P = 100 - 2 x 30 = 100 - 60 = 40

(C)

When P = MC for efficient outcome,

100 - 2Q = 10

2Q = 90

Q = 45

P = MC = 10

When Q = 30 & P = 40 for duopoly,

Deadweight loss = (1/2) x Change in P x Change in Q = (1/2) x (40 - 10) x (45 - 30) = (1/2) x 30 x 15 = 225

(D)

Profit, firm 1 = q1 x (P - MC) = 15 x (40 - 10) = 15 x 30 = 450

Profit, firm 2 = q2 x (P - MC) = 15 x (40 - 10) = 15 x 30 = 450

(E)

From Best response function of Firm 1,

4q1 + 2q2 = 90

4q1 + 2 x 25 = 90

4q1 + 50 = 90

4q1 = 40

q1 = 10