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 = 225 - 3Q. Also, each firm has constant marginal cost equal to 9. There are no fixed costs.

The marginal revenue of the two firms are given by:

• MR₁ = 225 – 6q₁ – 3q₂
• MR₂ = 225 – 3q₁ – 6q₂.

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 40 units of output. How much output should Firm 1 produce in order to maximize profit?

Answer 1

P = 225 - 3Q = 225 - 3q1 - 3q2

(A)

For firm 1, equating MR1 and MC,

225 - 6q1 - 3q2 = 9

6q1 + 3q2 = 216

2q1 + q2 = 72..........(1) (Best response, firm 1)

For firm 2, equating MR2 and MC,

225 - 3q1 - 6q2 = 9

3q1 + 6q2 = 216

q1 + 2q2 = 72..........(2) (Best response, firm 2)

Cournot equilibrium is obtained by solving (1) and (2). Multiplying (2) by 2,

2q1 + 4q2 = 144........(3)

2q1 + q2 = 72..........(1)

(3) - (1) yields:

3q2 = 72

q2 = 24

q1 = 72 - 2q2 [From (2)] = 72 - (2 x 24) = 72 - 48 = 24

(B)

Q = q1 + q2 = 24 + 24 = 48

P = 225 - (3 x 48) = 225 - 144 = 81

(C)

In social optimal output, P = MC.

225 - 3Q = 9

3Q = 216

Q = 72

P = MC = 9

Deadweight loss = (1/2) x Change in P x Change in Q = (1/2) x (81 - 9) x (72 - 48) = (1/2) x 72 x 24 = 864

(D)

Profit, Firm 1 = Q1 x (P - MC) = 24 x (81 - 9) = 24 x 72 = 1728

Profit, Firm 2 = Q2 x (P - MC) = 24 x (81 - 9) = 24 x 72 = 1728

NOTE: As per Answering Policy, 1st 4 parts are answered.