Question

1. Total market demand for pillows in San Francisco is given by P = 125 - 0.5Q. There are 2 suppliers of pillows in the market, who each have a constant marginal cost of $5 per pillow. Assuming no fixed costs, if the 2 firms compete against each other in a Cournot duopoly, how much lower will the 2 firms' combined profits be compared to if they colluded and acted as a monopoly? (Write answer without a negative sign and without the dollar sign.)

Answer 1

In monopoly, MR = MC.

TR = PQ = 125Q - 0.5Q²

MR = dTR/dQ = 125 - Q

125 - Q = 5

Q = 120

P = 125 - 0.5 x 120 = 125 - 60 = 65

Profit = Q x (P - MC) = 120 x (65 - 5) = 120 x 60 = 7200

In Cournot model,

P = 125 - 0.5Q1 - 0.5Q2, where Q = Q1 + Q2

For firm 1,

TR1 = P x Q1 = 125Q1 - 0.5Q1² - 0.5Q1Q2

MR1 = TR1/Q1 = 125 - Q1 - 0.5Q2

Setting MR1 = MC,

125 - Q1 - 0.5Q2 = 5

Q1 + 0.5Q2 = 120............(1) [Best response, firm 1]

For firm 2,

TR2 = P x Q2 = 125Q2 - 0.5Q1Q2 - 0.5Q2²

MR2 = TR2/Q2 = 125 - 0.5Q1 - Q2

Setting MR2 = MC,

125 - 0.5Q1 - Q2 = 5

0.5Q1 + Q2 = 120............(2) [Best response, firm 2]

Multiplying (2) by 2,

Q1 + 2Q2 = 240...............(3)

Q1 + 0.5Q2 = 120...............(1)

(3) - (1) yields:

1.5Q2 = 120

Q2 = 80

Q1 = 240 - 2Q2 [from (3)] = 240 - 2 x 80 = 240 - 160 = 80

Q = 80 + 80 = 160

P = 125 - 0.5 x 160 = 125 - 80 = 45

Profit, Firm 1 = Q1 x (P - MC) = 80 x (45 - 5) = 80 x 40 = 3200

Profit, Firm 2 = Q2 x (P - MC) = 80 x (45 - 5) = 80 x 40 = 3200

Total profit = 3200 + 3200 = 6400

Decrease in profit = 7200 - 6400 = 800