6. Maximum 15 points. Mr. N, maximizing his profit, owns a patent for the production of a unique good and can sell his product only in countries A and B. The demand for his product in these countries is represented by the functions $q_{A}=40-2P$ and $q_{B}=26-P$, where $q_{A}$ and $q_{B}$ are the quantities of the product in countries A and B, and $P$ is its price. Reselling the product produced by Mr. N between countries is associated with significant costs for its buyers, so Mr. N can set different prices for his product in different countries. Regardless of where his product is produced, the delivery costs to another country will be fixed and amount to 1. Mr. N found out that the production costs of his product will not depend on the country in which it is produced and are described by the function $\mathrm{TC}(\mathrm{Q})=8\mathrm{Q}+1$. Mr. N can register his business and produce the product in any of the countries, but different tax systems are in place in each. In country A, a proportional tax system with a fixed rate of $15\%$ is in place. In country B, a progressive multi-tier tax system is in place, with the rates specified in the table:

| Profit | Up to 30 | From 30 to 100 | From 100 to 150 | From 150 and more |
| :--- | :--- | :--- | :--- | :--- |
| Tax, $\%$ | Not taxed | 10 | 20 | 30 |

What will be Mr. N's profit after tax? How much of the product and at what price will it be sold in each of the countries?
# Solution:

Since resale between countries is impossible, the monopolist, maximizing its profit, will increase sales of its product in these countries until the marginal costs of its production become equal to the marginal revenue in each of the countries. That is, until $M R_{A}\left(q_{A}\right)=M R_{B}\left(q_{B}\right)=M C\left(q_{A}+q_{B}\right)$. If this relationship does not hold at a positive volume in each of the countries, then the monopolist has the opportunity to increase its profit by redistributing sales between the countries. Note also that this condition indeed defines the maximum profit, since the demand functions are decreasing, and the marginal costs do not decrease.

Considering the specified relationship, we find:

$M R_{A}\left(q_{A}\right)=20-q_{A}$

$M R_{B}\left(q_{B}\right)=26-2 q_{B}$

$M C\left(q_{A}+q_{B}\right)=8$

From which we find

$q_{A}=12, p_{A}=14$

$q_{B}=9, p_{B}=17$

$q_{A}+q_{B}=21$.

Since production costs in any of the countries are the same, the monopolist's profit before taxation will be

$\Pi=14 * 12+17 * 9-8 * 21-1-1=151$

Taking into account taxation, the monopolist's profit in country A will be

$\Pi^{A}=151 * 0.87=128.35$

Taking into account the progressive multi-tier tax system, the monopolist's profit in country B will be

$\Pi^{B}=30+0.9 * 70+0.8 * 50+0.7 * 1=133.7$

Comparing the profits, we note that production is more profitable to organize in country B. The profit in this case will be 133.7.

## Grading Scheme:

A total of 3 points for the correct justification of the distribution of production between countries. A total of 5 points for the correct determination of sales volumes and prices.

A total of 1 point for the correct determination of profit before taxation.

A total of 1 point for the correct determination of profit after tax when production is organized in country A.

A total of 4 points for the correct determination of profit after tax when production is organized in country B. Of these, 3 points for the correct approach to accounting for the tax system.

A total of 1 point for the correct answer about the location of production.

For each arithmetic error in the calculations, the score was reduced by 1 point.