Question

a.) Two identical firms compete as a Cournot duopoly. The market demand is P=100-2Q, where Q stands for the combined output of the two firms, Q=q1 +q2. The marginal cost for each firm is 4. Derive the best-response functions for these firms expressing what q1 and q2 should be.

b.) Continuing from the previous question, identify the price and quantity that will prevail in the Cournot duopoly market

c.) Now suppose two identical firms compete as a Bertrand duopoly. The market demand is P=100-2Q, where Q stands for the combined output of the two firms, Q=q1+q2. The marginal cost for each firm is 4. Identify the price and quantity in this market.

Answer 1

a) Each firm’s marginal cost function is MC = 4 and the market demand function is P = 100 – 2(q1 + q2) where Q is the sum of each firm’s output q₁ and q_2.

Find the best response 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 = 4

Profit maximization implies:

MR₁ = MC₁

100 – 4q₁ – 2q₂ = 4

which gives the best response function:

q₁ = 24 - 0.5q₂.

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

q₂ = 24 - 0.5q₁.

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

q₁ = 24 - 0.5(24 - 0.5q₁)

q₁ = 12 + 0.25q₁

This gives q₁ = q₂ = 16 units This the Cournot solution. Price is (100 – 2*32) = $36

c) Under Bertrand competition, P = MC for identical goods. Hence price is 4. Market quantity is Q = (100 - 4)/2 = 48 units, 24 units by each.