91Ó°ÊÓ

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:

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

The total profit will be $4.5 billion.

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

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:

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

The total profit will be $4.083 billion.