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

Q_E = 4,000,000 - 100P_E

and

Q_U = 1,000,000 - 20P_U

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.

1. 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?
2. 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?

Short answer

1. BMW should sell 1,000,000 cars in Europe at $30,000, and 300,000 cars in U.S. at $35,000. The total profit will be $4.5 billion.
2. BMW should sell 916,667 cars in Europe and 383,333 cars in the U.S. at $30,833.33. The total profit will be $4.083 billion.

Step-by-step solution

Explanation for part (a)

The optimum production level for BMW takes place where the marginal revenue is equal to the marginal cost.

The price and quantity in the European market are calculated below:

$$\begin{gathered} {\text{Q}}_{\text{E}}{\text{= 4,000,000 - 100P}}_{\text{E}} \\ {\text{P}}_{\text{E}}{\text{= 40,000 - 0.01Q}}_{\text{E}} \\ {\text{TR}}_{\text{E}}{\text{= 40,000Q}}_{\text{E}}{\text{- 0.01Q}}_{\text{E}}^{\text{2}} \\ {\text{MR}}_{\text{E}}{\text{= 40,000 - 0.02Q}}_{\text{E}} \\ \text{MC = 20,000} \\ {\text{MR}}_{\text{E}}\text{= MC} \\ {\text{40,000 - 0.02Q}}_{\text{E}}\text{= 20,000} \\ {\text{0.02Q}}_{\text{E}}\text{= 20,000} \\ {\text{Q}}_{\text{E}}\text{= 1,000,000} \\ {\text{P}}_{\text{E}}\text{= 40,000 - 0.01}\left(\text{1,000,000}\right) \\ \text{= 40,000 - 10,000} \\ \text{= \$ 30,000} \end{gathered}$$

In Europe, 1,000,000 cars will be produced at $30,000.

The price and quantity in the U.S. market are calculated below:

$$\begin{gathered} {\text{Q}}_{\text{U}}{\text{= 1,000,000 - 20P}}_{\text{U}} \\ {\text{P}}_{\text{U}}{\text{= 50,000 - 0.05Q}}_{\text{U}} \\ {\text{TR}}_{\text{U}}{\text{= 50,000Q}}_{\text{U}}{\text{- 0.05Q}}_{\text{U}}^{\text{2}} \\ {\text{MR}}_{\text{U}}{\text{= 50,000 - 0.01Q}}_{\text{U}} \\ \text{MC = 20,000} \\ {\text{MR}}_{\text{U}}\text{= MC} \\ {\text{50,000 - 0.1Q}}_{\text{U}}\text{= 20,000} \\ {\text{0.1Q}}_{\text{U}}\text{= 30,000} \\ {\text{Q}}_{\text{U}}\text{= 300,000} \\ {\text{P}}_{\text{U}}\text{= 50,000 - 0.05}\left(\text{300,000}\right) \\ \text{= 50,000 - 15,000} \\ \text{= \$ 35,000} \end{gathered}$$

In the U.S., 300,000 cars will be produced at $35,000.

The total profit is calculated below:

$$\begin{gathered} \mathrm{\pi }=\mathrm{TR}-\mathrm{TC} \\ =(30,000\times 1,000,000)+\left(35,000\times 300,000\right)-\left(10,000,000+20,000\left(1,30,000\right)\right) \\ =30,000,000,000+10,500,000,000-10,000,000,000+26,000,000,000 \\ =\$4.5\mathrm{billion} \end{gathered}$$

The total profit will be $4.5 billion.

Explanation for part (b)

The total demand, when the same price is charged, will be:

$$\begin{gathered} Q=Q_E+Q_U \\ Q=4,000,000-100P_E+1,000,000-20P_U \\ Q=5,000,000-120P \\ P=\frac{5,000,000}{120}-\frac{Q}{120} \end{gathered}$$

The optimum price and quantity in each market are calculated below:

$$\begin{gathered} \text{TR =}\frac{\text{5,000,000}}{\text{120}}\text{Q -}\frac{{\text{Q}}^{\text{2}}}{\text{120}} \\ \text{MR =}\frac{\text{5,000,000}}{\text{120}}\text{-}\frac{\text{Q}}{\text{60}} \\ \text{MC = 20,000} \\ \text{MR = MC} \\ \frac{\text{5,000,000}}{\text{120}}\text{-}\frac{\text{Q}}{\text{60}}\text{= 20,000} \\ \text{5,000,000 - 2Q = 2,400,000} \\ \text{2Q = 2,600,000} \\ \text{Q = 1,300,000} \\ \text{P =}\frac{\text{5,000,000}}{\text{120}}\text{-}\frac{\text{1,300,000}}{\text{120}} \\ \text{= 41,666.67 - 10,833.33} \\ \text{= \$ 30,833.33} \end{gathered}$$

The quantity of each market at $30,833.33 is calculated below:

$$\begin{gathered} Q_E=4,000,000-100\left(30,833.33\right) \\ =4,000,000-3,083,333 \\ =916,667 \\ {Q}_U=1,000,000-20\left(30,833.33\right) \\ =1,000,000-616666.6 \\ =383333 \end{gathered}$$

BMW should sell 916,667 cars in Europe and 383,333 cars in the U.S. at $30,833.33.

Total profit is calculated below:

$$\begin{gathered} \mathrm{\pi }=\mathrm{TR}-\mathrm{TC} \\ =(30,833.33\times 1,300,000)-\left(10,000,000+20,000\left(1,30,000\right)\right) \\ =40,083,329,000-10,000,000,000+26,000,000,000 \\ =\$4.083\mathrm{billion} \end{gathered}$$

The total profit will be $4.083 billion.