Question

The market demand curve is P = 260 – Q, where Q is the output of Firm 1 and Firm 2, q1 + q2. The products of the two firms are identical.

a. Firm 1 and Firm 2 have the same cost structure: AC = MC = $20. If the firms are in a competitive duopoly, how much profit does each firm earn?

b. Now suppose that Firm 2's production costs increase to AC = MC = $80. If the firms continue their competition, how much profit does each firm earn?

Answer 1

Part A

Each firm’s marginal cost function is MC = 20 and the market demand function is P = 260 – (q1 + q2)

Find the best response functions for both firms:

Revenue for firm 1

R₁ = P*q₁ = (260 – (q₁ + q₂))*q₁ = 260q₁ – q₁² – q₁q₂.

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

MR₁ = 260 – 2q₁ – q₂

MC₁ = 20

Profit maximization implies:

MR₁ = MC₁

260 – 2q₁ – q₂ = 20

which gives the best response function:

q₁ = 120 - 0.5q₂.

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

q₂ = 120 - 0.5q₁.

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

q₁ = 120 - 0.5(120 - 0.5q₁)

q₁ = 60 + 0.25q₁

This gives q₁ = q₂ = 80 units This the Cournot solution. Price is (260 – 160) = $100 . Profit to each firm = (P – AC)*Q = (100 – 20)*80 = $6400

Part B

The marginal cost of firm 1 is 20 and that of firm 2 is 80. The market demand function is P = 260 – Q, Where Q is the sum of each firm’s output Q1 and Q2_.

Find the best response functions for firm 1:

Revenue for firm 1

R₁ = P*Q1 = (260 – (Q1 + Q2))*Q1 = 260Q1 – Q1² – Q1Q2.

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

MR₁= 260 – 2Q1 – 2Q2

MC₁ = 20

Profit maximization implies:

MR₁ = MC₁

260 – 2Q1 – Q2 = 20

which gives the best response function:

Q1 = 120 - 0.5Q2.

Find the best response functions for firm 2:

Revenue for firm 2

R₂ = P*Q2 = (260 – (Q1 + Q2))*Q2 =260Q2 – Q2² – Q1Q2.

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

MR₂= 260 – 2Q2 – Q1

MC₂ = 80

Profit maximization implies:

MR₂ = MC₂

260 – 2Q2 – Q1= 80

which gives the best response function:

Q2 = 90 - 0.5Q1.

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

Q2 = 90 – 0.5(120 – 0.5Q2)

0.75Q2 = 60

Q2 = 40 units

Q1 = 120 – 0.5*40 = 100 units

Price = 260 – (100 + 40) = 120

Profit to firm 1 = (120 – 20)*100 = $10000

Profit to firm 2 = (120 – 80)*40 = $1600.