Question

Suppose that BMW can produce any quantity of cars at a constant marginal cost equal to $20,000 and a fixed cost of $10 billion. You are asked to advise the CEO as to what prices and quantities BMW should set for sales in Europe and in the United States. The demand for BMWs in each market is given by: QE = 4,000,000 ? 100PE and QU = 1,000,000 ? 20PU where the subscript E denotes Europe and the subscript U denotes the United States. Assume that BMW can restrict U.S. sales to authorized BMW dealers only. a. What quantity of BMWs should the firm sell in each market, and what should the price be in each market? b. What should the total profit be? c. If BMW were forced to charge the same price in each market, what would be the quantity sold in each market, the equilibrium price, and the company’s profit?

Answer 1

(a)

In Europe,

QE = 4,000,000 - 100PE

100PE = 4,000,000 - QE

PE = 40,000 - 0.01QE

Total revenue (TRE) = PE x QE = 40,000QE - 0.01QE²

Marginal revenue (MRE) = dTRE/dQE = 40,000 - 0.02QE

Equating MRE & MC,

40,000 - 0.02QE = 20,000

0.02QE = 20,000

QE = 1,000,000

PE = 40,000 - (0.01 x 1,000,000) = 40,000 - 10,000 = $30,000

In US,

QU = 1,000,000 - 20PU

20PU = 1,000,000 - QU

PU = 50,000 - 0.05QU

Total revenue (TRU) = PU x QU = 50,000QU - 0.05QU²

Marginal revenue (MRU) = dTRU/dQU = 50,000 - 0.1QU

Equating MRU & MC,

50,000 - 0.1QU = 20,000

0.1QU = 30,000

QU = 300,000

PU = 50,000 - (0.05 x 300,000) = 50,000 - 15,000 = $35,000

(b)

Aggregate revenue (TR) ($ Million) = TRE + TRU = (PE x QE) + (PU x QU) = (30,000 x 1) + (35,000 x 0.3)

= 30,000 + 10,500 = 40,500

Total cost ($ Million) = MC x (QE + QU) - Fixed costs = 20,000 x (1 + 0.3) - 10,000 = 20,000 x 1.3 - 10,000

= 26,000 - 10,000 = 16,000

Profit ($ Million) = TR - TC = 40,500 - 16,000 = 24,500

(c) In absence of price discrimination, PE = PU = P and market demand (Q) = QE + QU

Q = 4,000,000 - 100P + 1,000,000 - 20P

Q = 5,000,000 - 120P

120P = 5,000,000 - Q

P = (5,000,000 - Q) / 120

TR = P x Q = (5,000,000Q - Q²) / 120

MR = dTR/dQ =(5,000,000 - 2Q) / 120

Equating MR & MC,

(5,000,000 - 2Q) / 120 = 20,000

5,000,000 - 2Q = 2,400,000

2Q = 2,600,000

Q = 1,300,000

P = (5,000,000 - 1,300,000) / 120 = 3,700,000 / 120 = $30,833

QE = 4,000,000 - (100 x 30,833) = 4,000,000 - 3,083,333 = 916,667

QU = 1,000,000 - (20 x 30,833) = 1,000,000 - 616,660 = 383,340

Profit ($ Million) = Q x (P - MC) - FC = 1.3 x (30,833 - 20,000) - 10,000 = 1.3 x 10,833 - 10,000 = 14,082.9 - 10,000

= 4,082.9