4. Suppose that BMW can produce any quantity of cars at a constant marginal cost

Question

4. 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 – 100 PE and QU = 1,000,000 – 20PU
where the subscript E denotes Europe, the subscript U denotes the United States. Assume that BMW can restrict U.S. sales to authorized BMW dealers only. Prices and costs are in dollars (not thousands of dollars as your book may indicate).
a. What quantity of BMWs should the firm sell in each market, and what should the price be in each market? What should the total profit be?
b. 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?

Explanation / Answer

a. What quantity of BMWs should the firm sell in each market, and what will
the price be in each market? What will the total profit be?

With separate markets, BMW chooses the appropriate levels of QE and
QU to maximize profits, where profits are:

= TR TC = (QEPE + QUPU) {(QE +QU)*20,000+10,000,000,000}

Solve for PE and PU using the demand equations, and substitute the
expressions into the profit equation:

= QE(40,000(QE/100)) + QU(50,000 (QU/20)) - {(QE +QU)*20,000 + 10,000,000,000}

Differentiating and setting each derivative to zero to determine the
profit-maximizing quantities, we get:

(/QE) = 40,000 (QE/50) 20,000 = 0, or QE = 1,000,000 cars

and

(/QU) = 50,000 (QU/10) 20,000 = 0, or QU = 300,000 cars.

Substituting QE and QU into their respective demand equations, we may
determine the price of cars in each market:

1,000,000 = 4,000,000 - 100PE, or PE = $30,000

and

300,000 = 1,000,000 - 20PU, or PU = $35,000.

Substituting the values for QE, QU, PE, and PU into the profit equation,
we have:

= {(1,000,000)*($30,000) + (300,000)*($35,000)} - {(1,300,000)*(20,000)) +
10,000,000,000}, or

= $4.5 billion. Which is your answer to part a.

b. 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?

If BMW charged the same price in both markets, we substitute Q = QE + QU into the demand equation and write the new demand curve as follows:

Q = 5,000,000 - 120P, or in inverse for as P = (5,000,000/120) (Q/120).

Since the marginal revenue curve has twice the slope of the demand curve:

MR = (5,000,000/120) (Q/60)

To find the profit-maximizing quantity, set marginal revenue equal to
marginal cost:

(5,000,000/120) (Q/60) = 20,000, or

Q* = 1,300,000.

Substituting Q* into the demand equation to determine price:

P = (5,000,000/120) (1,300,000/120) = $30,833.33. Your equilibrium price!

Substituting into the demand equations for the European and American
markets to find the quantity sold :

QE = 4,000,000 - (100)($30,833.3), or QE = 916,667 This is the quantity for the Euro market

and

QU = 1,000,000 - (20)($30,833.3), or QU = 383,333 This is the quantity for U.S. market

Substituting the values for QE, QU, and P into the profit equation, we
find:

= {1,300,000*$30,833.33} - {(1,300,000)(20,000)) + 10,000,000,000}, or

= $4,083,333,330. This is the company's total profit!

I hope this has answered your questions, I have tried to bear in mind each significant portion and highlight where we have information to be used later. Please let me know if anything is unclear! <(^_^)b

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