Question

Consider the Cournot competition example in the lecture notes. Inverse demand function is P(Q) = 31 ? 2Q. However, make the change that firm B’s cost function is CB(Q) = 2Q. Firm A’s cost function remains the same at CA(Q) = Q.
a) Determine firm A’s best response function Q*A (QB).
b) Determine firm B’s best response function Q*B (QA).
c) How much quantity is each firm producing in the Cournot
d) What is the price at which output goods are sold in the Cournot Nash equilibrium? Carefully argue your answer.
e) How much demand is firm A facing in the Cournot Nash equilibrium?

Answer 1

Two firms are A and B with costs CA(q1) = q1 and CB(q2) = 2q2. Firm A has a marginal cost of 1 while B has a marginal cost of 2. The two firms compete in quantities

The market demand function is P = 31 – 2Q

Where Q is the sum of each firm’s output q₁ and q_2.

Find the best response functions for both firms:

a) Revenue for firm A

R₁ = P*q₁ = (31 – 2(q₁ + q₂))*q₁ =31q₁ – 2q₁² – 2q₁q₂.

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

MR₁ = 31 – 4q₁ – 2q₂

MC₁ = 1

Profit maximization implies:

MR₁ = MC₁

31 – 4q₁ – 2q₂ = 1

which gives the best response function:

q₁ = 7.5 - 0.5q₂.

b) Find the best response functions for firm B

Revenue for firm B

R2 = P*q₂ = (31 – 2(q₁ + q₂))*q₂ =31q₂ – 2q₂² – 2q₁q₂.

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

MR₂ = 31 – 4q₂ – 2q₁

MC₂ = 2

Profit maximization implies:

MR₂ = MC₂

31 – 4q₁ – 2q₂ = 2

which gives the best response function:

q₂ = 7.25 - 0.5q₁.

c) Solve the two equations

q₂ = 7.25 - 0.5*(7.5 - 0.5q₂)

q2 = 3.5 + 0.25q2

This gives q2 = 4.667 and q1 =  7.5 - 0.5*4.6667 = 5.167

d) Price = 31 - 2*( 4.667 + 5.167) = 11.33

e) Demand is P = 31 - 2q1 - 2*4.667

P = 21.67 - 2q1

This is the required demand.