Question

Suppose demand in a market is given by x = 250 - Px. Marginal cost of any of two firms operating is MC = $4.

What is the Cournot duopoly equlibrium price and quantity?

Answer 1

Each firm’s marginal cost function is MC= 4 and the market demand function is Px = 250 – x

Where x is the sum of each firm’s output x₁ and x_2.

Find the best response functions for both firms:

Revenue for firm 1

R₁ = P*x₁ = (250 – (x₁ + x₂))*x₁ = 250x₁ – x₁² – x₁x₂.

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

MR₁ = 250 – 2x₁ – x₂

MC₁ = 4

Profit maximization implies:

MR₁ = MC₁

250 – 2x₁ – x₂ = 4

which gives the best response function:

x₁ = 123 - 0.5x₂.

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

x₂ = 123 - 0.5(123 - 0.5x₂)

= 123 - 61.5 + 0.25x₂

x₂ = 82 and so x₁ is also 82. Total quantity is 82 + 82 = 164 and price = 250 - 164 = $86

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

x₂ = 123 - 0.5x₁.

x₁ = x₂ = 173.33