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 = 160 - 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 = 160 – 4q₁ – 2q₂

• MR₂ = 160 – 2q₁ – 4q₂.

A) How much output will each firm produce in the Cournot equilibrium and what will be the market price of the good?

B) What is the deadweight loss that results from this duopoly?

C) How much profit does each firm make?

D) Suppose Firm 2 produced 30 units of output. How much output should Firm 1 produce in order to maximize profit?

Answer 1

Each firm’s marginal cost function is MC = 10 and
the market demand function is
=> P = 160 – 2Q
=> P = 160 - 2(q₁₊q₂)

Finding best response functions for both firms:

Revenue for firm 1
R₁ = P*q₁ = (160 - 2(q₁₊q₂))*q₁ = 160q₁ – 2q₁² – 2q₁q₂.

Firm 1 has the following marginal revenue and marginal cost functions:
MR₁ = 160 – 4q₁ – 2q₂
MC₁ = 10

Profit maximization implies:
MR₁ = MC₁
160 – 4q₁ – 2q₂ = 10
which gives the best response function:
q₁ = 37.5 - 0.5q₂.

By symmetry, Firm 2’s best response function is:
q₂ = 37.5 - 0.5q₁.

Cournot equilibrium is determined at the intersection of these two best response functions:
q₁ = 37.5 - 0.5(37.5 - 0.5q₁)
q₁ = 18.75 + 0.25q₁
q₁ = 25

A) This gives q₁ = q₂ = 25 units This the Cournot solution.
     Price is (160 – 2*50) = $60

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

Competition has P = MC
160 - 2Q = 10
150 = 2Q
Q = 75 and so P = 10

dead weight loss = 0.5*(60 - 10)*(75 - 50) = $625

C) Profit to each firm = (60 – 10)*25 = $1250

D) Suppose Firm 2 produced 30 units of output.
     Firm 1 produce q₁ = 37.5 - 0.5q₂ or q₁ = 37.5 - 0.5*30 = 22.5 units (rounding to 23 units) in order to maximize profit