Question

Two firms, firm 1 & firm 2, in a Cournot duopoly are facing the market demand given by P = 140 – 0.4Q, where P is the market price and Q is the market quantity demanded. Firm 1 uses old technology and has (total) cost of production given by C(q1) = 200 + 15q1, where q1 is the quantity produced by firm 1. Firm 2 has managed to introduce a new technology to lower the per unit cost, and its (total) cost of production is now given by C(q2) = 200 + 10q2, where q2 is the quantity produced by firm 2. Referring to SCENARIO 2, find the deadweight loss in the Cournot duopoly equilibrium

Answer 1

Market demand is P = 140 – 0.4(Q1 + Q2). Marginal revenue functions for firm 1 and firm 2 are:

TR1 = 140Q1 - 0.4Q1^2 - 0.4Q1Q2

MR1 = 140 - 0.8Q1 - 0.4Q2

Similarly, MR2 = 140 - 0.8Q2 - 0.4Q1

MC1 is 15 and MC2 = 10. Hence we find the cournot duopoly output

140 - 0.8Q1 - 0.4Q2 = 15 140 - 0.8Q2 - 0.4Q1 = 10

125 = 0.8Q1 + 0.4Q2 130 = 0.4Q1 + 0.8Q2

Q1 = 156.25 - 0.5Q2

Use this in BRF for firm 2

130 = 0.4*(156.25 - 0.5Q2) + 0.8Q2

130 = 62.5 - 0.2Q2 + 0.8Q2

Q2 = 67.5/0.6 = 112.5

Q1 = 156.25 - 0.5*112.5 = 100

Hence the market price is 140 - 0.4*(100 + 112.5) = $55. Market quantity = 112.5 + 100 = 212.5.

In perfect competition, price = marginal cost. Here one firm has a lower marginal cost so it will serve the entire market. Competitive price P = 10 and quantity = 140 - 0.4Q = 10 or Q = 325 units.

Deadweight loss = 0.5*(duopoly price - competitive price)*(competitve quantity - duopoly quantity)

= 0.5*(55 - 10)*(325 - 212.5)

= $2531.25