# Assignment 4. 20 points

## Variant 1

In country Alpha, only two goods, X and Y, are produced from a single production factor - factor Z, of which there are exactly 100 units in Alpha. The amount of good X produced from factor Z can be described by the function $X=\frac{\sqrt{Z_{X}}}{2}$, and one unit of good Y is produced from each unit of factor Z. Country Alpha can buy and sell both goods on the world market at prices $P_{X}=8$ and $P_{Y}=1$. The residents of the country always consume goods in sets: for every 3 units of good X, 184 units of good Y are consumed. The government of the country is concerned only with ensuring that the residents can consume as many such sets of goods as possible.

(a) Find the quantities of goods X and Y that will be produced and consumed by the residents of this country. Graphically represent the found values and the production possibilities frontier of the country on one diagram.

(b) On the world market, an increase in the price of good X by a factor of $\alpha$ is expected, while it is known that the price of good Y will not change. Do you think that country Alpha can benefit from the increase in the price of good X? If it can, find all such values of $\alpha$ for which the country will benefit, or explain why Alpha cannot benefit from the increase in the price of good X.
# Solution:

(a) Since the country can sell and buy goods on the world market, to achieve the consumption of the largest quantity of good sets, it needs to achieve the highest income from trade. For this, the country needs to use its resources as efficiently as possible. First, let's find the equation of the production possibilities frontier (PPF) of the country.

$Z_{X}+Z_{Y}=100$ according to the problem. At the same time, $Z_{X}=4 X^{2}, Z_{Y}=Y$.

Then the PPF equation is: $4 X^{2}+Y=100$.

The country's income, which uses its resources most efficiently in this case, is $T R=8 X+Y=8 X+100-4 X^{2}$. The maximum of this function, whose graph is a parabola opening downwards, is achieved at a production volume of $X_{\text {prod }}=1, Y_{\text {prod }}=100-4=96$.

The maximum income of the country is then $8 * 1+1 * 96=104$.

Considering the problem's condition, if the country consumes $\mathrm{X}$ units of good X, then it will consume $184 \mathrm{X} / 3$ units of good Y. Since the country will spend 104 monetary units on a set of goods in such a proportion, $8 X_{\text {cons }}+Y_{\text {cons }}=104$, where $Y_{\text {cons }}=184 X_{\text {cons }} / 3$ and the residents of the country will be able to consume $X_{\text {cons }}=1.5$ and $Y_{\text {cons }}=92$.

Let's graphically represent the obtained production and consumption volumes.

![](https://cdn.mathpix.com/cropped/2024_05_06_30b7fccfc6f6ad7fdcc0g-07.jpg?height=654&width=717&top_left_y=204&top_left_x=310)

(b) Now let's check whether the country can benefit from an increase in the price of good X. If an increase in the price of this good will allow the country to consume a larger quantity of each good than in part (a), then it will benefit from such a price change.

First, let's find out how much of the good the country will produce if the price of good X increases by a factor of $\alpha$. Then the country's income, which uses its resources most efficiently in this case, is $T R=8 \alpha X+Y=8 \alpha X+100-4 X^{2}$. The maximum of this function, whose graph is a parabola opening downwards, is achieved at a production volume of $X_{\text {prod }}=\alpha, Y_{\text {prod }}=100-4 \alpha^{2}$.

The maximum income of the country is then

$8 \alpha * \alpha+1 *\left(100-4 \alpha^{2}\right)=4 \alpha^{2}+100$.

If the country's income with the change in the price of good X is greater than the expenses on the set of goods $X_{\text {cons }}=1.5, \quad Y_{\text {cons }}=92$, i.e., $4 \alpha^{2}+100>8 \alpha * 1.5+1 * 92$, then the country will benefit from an increase in the price of good X. The inequality will be satisfied for $\alpha>1$ only if $\alpha>2$, that is, if the price of good X increases by more than two times.

## Grading Criteria:

(a) Evaluation of solution steps (maximum 10 points):

correctly found the PPF equation - 1 point;

correctly set up the problem of maximizing the country's income - 1 point;

correctly justified the maximum of the problem of maximizing the country's income (at least once in one of the points) - 1 point;

correctly found the production volumes - 1 point;

correctly found the ratio of goods X and Y - 1 point;

correctly set up the income equation to find the consumption volumes - 1 point;

correctly found the consumption volumes of goods - 1 point;

correctly graphically represented the PPF (schematically) - 1 point;

correctly graphically represented the line of equal income (maximum) - 1 point;

correctly graphically represented the country's consumption volumes on the line of maximum income - 1 point.
(b) Evaluation of solution steps (maximum 10 points):

correctly set up the problem of maximizing the country's income - 1 point;

correctly found the production volumes depending on $\alpha-2$ points;

correctly justified the idea of comparing incomes and wrote the corresponding inequalities - 4 points;

correctly found the possible values of $\alpha-2$ points;

made the correct conclusion about the possible values of $\alpha$, at which the country will benefit from the price change - 1 point.

## Penalties:

Any arithmetic error that did not lead to a distortion of the essence of the result is penalized by 1 point.

Any arithmetic error that led to a distortion of the result in essence is penalized by 5 points.

The absence of explanations for the calculations in part (b) is penalized by 2 points.

If in part (b) only a correct example of the value of $\alpha$, at which the country benefits from the price change, is provided, then the maximum score for this part with correct other calculations is 5 points.

## Variant 2

In country Beta, only two goods X and Y are produced from one production factor - factor Z, of which there are exactly 100 units in country Beta. The quantity of produced good X from factor Z can be described by the function $X=\frac{\sqrt{Z_{X}}}{4}$, and one unit of good Y is produced from each unit of factor Z. Both goods can be bought and sold by country Beta on the world market at prices $P_{X}=64$ and $P_{Y}=1$. The residents of the country always consume goods only in sets: for every 3 units of good X, 136 units of good Y are consumed. The government of the country is concerned only with ensuring that the residents can consume as many such sets of goods as possible.

(a) Find the quantities of goods X and Y that will be produced and consumed by the residents of this country. Graphically represent the found values and the production possibilities frontier of the country on one graph.

(b) On the world market, a decrease in the price of good X by a factor of $\alpha$ is expected, while it is known that the price of good Y will not change. Do you think country Beta can benefit from a decrease in the price of good X? If it can, find all such values of $\alpha$ at which the country will benefit, or explain why Beta cannot benefit from a decrease in the price of good X.

## Solution:

(a) Since the country can sell and buy goods on the world market, to achieve the consumption of the largest quantity of good sets, it needs to achieve the highest income from trade. For this