6. Maximum 15 points. A school economics teacher believes it is extremely important to know the basics of both microeconomics and macroeconomics. Therefore, for his subject, he has introduced the following final grading system. First, the student's study of the basics of microeconomics ( $O_{\text {micro }}$ ) is assessed out of 5 points, and then the basics of macroeconomics ( $O_{\text {macro }}$ ) are assessed out of 5 points. The final grade in economics is formed as the minimum value of two quantities: $0.25 * O_{\text {micro }}+0.75 * O_{\text {macro }}$ and $0.75 * O_{\text {micro }}+0.25 * O_{\text {macro }}$ (the final value is rounded to the nearest integer according to the rules of mathematical rounding). Student Ivanov seriously approached the planning of time for studying each school subject and decided that he would spend no more than 4.6 time units on studying economics. However, studying the basics of one area of economics is more difficult for Ivanov than the other. Thus, each unit of time spent on studying macroeconomics increases his final score (in points) in this area by exactly 1.5 points, while the same unit of time invested in preparing for microeconomics increases the final score in this area by 2.5 points.

(a) What is the maximum grade that student Ivanov can get in economics?

(b) Explain why the final grade in economics reflects the teacher's opinion that it is extremely important to know both microeconomics and macroeconomics. Will this always be the case with such a grading system?
# Solution:

(a) If the minimum value of the two is determined by the expression $0.75 * O_{\text {мИкро }} + 0.25 * O_{\text {мАкро }}$, i.e., $O_{\text {мИкро }}O_{\text {мАкро }}$, then it is more advantageous for Ivanov to spend an additional unit of time studying macroeconomics.

Thus, to achieve the highest grade, Ivanov should not spend time studying only one area of economics. How should he correctly allocate his time? As long as the expressions $0.25 * O_{\text {мИкро }} + 0.75 * O_{\text {мАкро }}$ and $0.75 * O_{\text {мИкро }} + 0.25 * O_{\text {мАкро }}$ are not equal, as shown above, it will always be more advantageous to switch to studying one of the areas, increasing the resulting grade. Therefore, the maximum final grade can only be achieved if the time is allocated such that $0.25 * O_{\text {мИкро }} + 0.75 * O_{\text {мАкро }} = 0.75 * O_{\text {мИкро }} + 0.25 * O_{\text {мАкро }}$. From this, $O_{\text {мИкро }} = O_{\text {мАкро }}$. To achieve this, with the full use of the time resource, it is necessary to spend $5 / 8$ of the total time on studying macroeconomics, and the rest on studying microeconomics. Then the calculated final grade in economics will be $O_{\text {мИкро }} = O_{\text {мАкро }} = 3 * 2.5 * 4.6 / 8 = 4.3125$, and after mathematical rounding, it will be 4 points.

It should be noted that since the grade was rounded down, Ivanov can achieve a grade of 4 without using the entire planned time resource of 4.6 time units, and can spend the remaining time on another activity.

(b) Using this formula, it turns out that the student will know both microeconomics and macroeconomics. However, if the coefficients for the grades for knowledge of microeconomics and/or macroeconomics, and the coefficients of the labor intensity of studying both areas of economics were different, it could result in a boundary solution where the student would spend time preparing only for one area of economics (for example, if the labor intensity ratio was $1: 5$ with the given formula for the final grade).

## Grading Scheme:

(a) Total 10 points:

- 5 points for a complete justification that the solution will be internal (that $O_{\text {мИкро }} = O_{\text {мАкро }}$);
- 5 points for finding the maximum grade.

(b) Total 5 points:

- 2 points for explaining the internal solution;
- 3 points for explaining the possibility of a boundary solution.