Question

Assume there is a duopoly. Assume that the market demand is : P=100-2Q       Assume the good can be produced at a constant marginal cost of 0 and that both firms have the same cost. Assume the firms act like Cournot firms.

1. What is the equation for firm 1’s demand curve?

2. What it the equation for firm 2’s demand curve?

3. What is the equation for firm 1’s reaction function?

4. What is the equation for firm 2’s reaction function?

5. What is the equilibrium price?

6. What is Firm 1’s output and profit? What is Firm 2’s output and profit?

Answer 1

Each firm’s marginal cost function is MC = 0 and the market demand function is P = 100 - 2Q Where Q is the sum of each firm’s output q₁ and q_2.

1) Demand function for firm 1 = P = 100 - 2q1 - 2q2

2) Demand function for firm 2 = P = 100 - 2q1 - 2q2

3) Find the reaction functions for both firms:

Revenue for firm 1

R₁ = P*q₁ = (100 – 2(q₁ + q₂))*q₁ = 100q₁ – 2q₁² – 2q₁q₂.

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

MR₁ = 100 – 4q₁ – 2q₂

MC₁ = 0

Profit maximization implies:

MR₁ = MC₁

100 – 4q₁ – 2q₂ = 0

which gives the best response function/reaction function:

q₁ = 25 - (1/2)q₂.

4) By symmetry, Firm 2’s best response function/reaction function is:

q₂ = 25 - (1/2)q₁.

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

q₂ = 25 - (1/2)(25 - (1/2)q₂)

= 25 - 12.5 + 0.25q₂

q₂ = 12.5/0.75 = 16.67

q₁ = q₂ = 16.67

Thus,

Q = q₁ + q₂ = 33.34

5) Price = P = 100 – 2*33.34 = $33.33.

6) Profit for both firms will be equal and given by:

R - C = (33.33) (16.67) - 0 = $555.55