Question

24. Cournot duopolists face a market demand curve given by P = 90 - Q where Q is total market demand. Each firm can produce output at a constant marginal cost of 30 per unit. There are no fixed costs. Determine the (1) equilibrium price, (2) quantity, and (3) economic profits for the total market, (4) the consumer surplus, and (5) dead weight loss.

25. If the duopolists in question 24 behave according to the Stackelberg Leader-Follower model, determine the (1) equilibrium price, (2) quantity, and (3) economic profits for the total market and (4) the consumer surplus, and (5) dead weight loss.

26. If the duopolists in question 24 behave, instead, according to the Bertrand model, determine the (1) equilibrium price, (2) quantity, and (3) economic profits for the total market and (4) the consumer surplus, and (5) dead weight loss.

27. If the duopolists in question 24 behave as a shared monopoly, determine the (1) equilibrium price, (2) quantity, and (3) economic profits for the total market and (4) the consumer surplus, and (5) dead weight loss.

Answer 1

24. Each firm’s marginal cost function is MC = 30 and the market demand function is P = 90 – (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 = (90 – (q₁ + q₂))*q₁ = 90q₁ – q₁² – q₁q₂.

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

MR₁ = 90 – 2q₁ – q₂

MC₁ = 30

Profit maximization implies:

MR₁ = MC₁

90 – 2q₁ – q₂ = 30

which gives the best response function:

q₁ = 30 - 0.5q₂.

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

q₂ = 30 - 0.5q₁.

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

q₁ = 30 - 0.5(30 - 0.5q₁)

q₁ = 15 + 0.25q₁

This gives q₁ = q₂ = 20 units This the Cournot solution. Price is (90 – 40) = $50 . Profit to each firm = (50 – 30)*20 = $400 or 800 in total. CS = 0.5*(90 - 50)*40 = $800. DWL = 0.5*(50 - 30)*(60 - 40) = $200.