Question

B Problem 4 Suppose the inverse demand for a product produced by a single firm is given by. P = 200 – 5 Q and this firm has a marginal cost of production of MC = 20 + 2 Q Answer the following questions: If the firm cannot price-discriminate, what is the profit-maximizing price and level of output? If the firm cannot price-discriminate, what are the levels of producer and consumer surplus in the market? What is the deadweight loss? If the firm can practice perfect price discrimination, what output level will it choose? What are the levels of producer and consumer surplus and deadweight loss under perfect price discrimination?

Answer 1

(1) If firm cannot price-discriminate, it will equate Marginal revenue (MR) with MC.

P = 200 - 5Q

Total revenue (TR) = P x Q = 200Q - 5Q²

MR = dTR/dQ = 200 - 10Q

Equating MR & MC,

200 - 10Q = 20 + 2Q

12Q = 180

Q = 15

P = 200 - (5 x 15) = 200 - 75 = 125

From demand function, When Q = 0, P = 200 (Reservation price)

Consumer surplus (CS) = Area between demand curve & market price = (1/2) x (200 - 125) x 15 = (1/2) x 75 x 15

= 562.5

From MR function, when Q = 15, MR = 200 - (10 x 15) = 200 - 150 = 50

From MC function, When Q = 0, MC = 20 (Minimum acceptable price)

Producer surplus (PS) = Area between MC curve and market price = (1/2) x 15 x [(125 - 20) + (125 - 50)]**

= (1/2) x 15 x (105 + 75) = (1/2) x 15 x 180 = 1,350

When P = MC, 200 - 5Q = 20 + 2Q

7Q = 180, or Q = 25.71

P = 200 - (5 x 25.71) = 200 - 128.55 = 71.45

Deadweight loss = (1/2) x Change in price x Change in quantity = (1/2) x (125 - 71.45) x (25.71 - 15)

= (1/2) x 53.55 x 10.71 = 286.76

(2) When firm can perfectly price discriminate, it will equate P with MC.

As obtained above, P = 71.45 and Q = 25.71

CS = (1/2) x (200 - 71.45) x 25.71 = (1/2) x 128.55 x 25.71 = 1,652.51

Since P = MC, Producer surplus = 0 and deadweight loss = 0.