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 – 4Q. Also, each firm has constant marginal cost equal to 28. There are no fixed costs.The marginal revenue of the two firms are given by:MR1 = 100 – 8q1 – 4q2MR2 = 100 – 4q1 – 8q2.A) (8 points) How much output will each firm produce in the Cournot equilibrium?B) (8 points) What will be the market price of the good?C) (8 points) What is the deadweight loss that results from this duopoly?D) (8 points) How much profit does each firm make?E) (8 points) Suppose Firm 2 produced 10 units of output. How much output should Firm 1 produce in order to maximize profit? (Hint: Use Firm 1’s Reaction Function)

Answer 1

A):- How much output will each firm produce in the Cournot equilibrium?

Each firm’s marginal cost function is MC = 28 and the market demand function is P = 100 – 4(q1 + q2)

Find the best response functions for both firms:

Revenue for firm 1:-

R1 = P*q1 = (100 – 4(q1 + q2))*q1 = 100q1 – 4q12 – 4q1q2.

Firm 1 has the following marginal revenue and marginal cost functions:

MR1 = 100 – 8q1 – 4q2

MC1 = 28

Profit maximization implies:

MR1 = MC1

100 – 8q1 – 4q2 = 28

which gives the best response function:

q1 = 9 - 0.5q2.

By symmetry, Firm 2’s best response function is:

q2 = 9 - 0.5q1.

Cournot equilibrium is determined at the intersection of these two best response functions:

q1 = 9 - 0.5(9 - 0.5q1)

q1 = 4.5 + 0.25q1

This gives q1 = q2 = 6 units This the Cournot solution.

B):- What will be the market price of the good?

Price is (100 – 4*12) = $52.

C):- What is the deadweight loss that results from this duopoly?

Deadweight loss = 0.5*(current price - competitive price)*(competitive quantity - combinded cournot quantity)

Competition has P = MC

100 - 4Q = 28

72 = 4Q

Q = 18 and so P = 28

DWL = 0.5*(52 - 28)*(18 - 12) = $72.

D):- How much profit does each firm make?

Profit to each firm = (52 – 28)*6 = $144

E):- Suppose Firm 2 produced 10 units of output. How much output should Firm 1 produce in order to maximize profit?

Suppose Firm 2 produced 10 units of output. Firm 1 produce q1 = 9 - 0.5q2 or q1 = 9 - 0.5*10 = 4 units in order to maximize profit.