# Task 2. Maximum 30 points

In a certain country, there are two regions - northern and southern. The northern region has 24 inhabitants, while the southern region has four times fewer. Both regions are capable of producing goods X in quantity $x$ and Y in quantity $y$. Due to climatic peculiarities, the production capabilities of the regions differ. The production possibilities curve of one representative of the northern region is given by the equation $y=13.5-9x$, and for the southern region by $-y=24-1.5 \cdot x^{2}$. Both goods are produced only for export, and sales are possible only in sets. Each set consists of 1 unit of good X and 9 units of good Y, and the number of sets can be non-integer. The price of a set is 2000 monetary units. Each region maximizes the total revenue from selling sets. The entire income earned by a region is evenly distributed among the inhabitants of that region.

(a) Find the Gini coefficient in the country if both regions operate independently of each other.

(b) The southern region proposed to the northern region to combine their production efforts in creating sets and to share the earned money, distributing it evenly among all the inhabitants of the country. The administration of the northern region initially refused this proposal, believing it could reduce the total income of the region. However, after carefully calculating everything, they stated that the northern region is willing to transfer its production resources if the southern region pays an amount equal to the previous total income of the northern region, plus a symbolic fee of 661 monetary units. Determine how the Gini coefficient will change if the southern region agrees to the terms of the northern region.
# Solution:

(a) In the northern region, there are 24 residents, and in the southern region, there are $\frac{24}{4}=6$. Both regions operate independently of each other. Let's find the production capabilities of each region. Given that the individual production possibility frontier (PPF) of each resident in the northern region is linear and identical to that of other residents, the total PPF of the northern region is $y=13.5 \cdot 24 - 9x = 324 - 9x$. Let $k_{n}$ and $k_{s}$ be the number of sets produced by the northern and southern regions, respectively. It is known that for the northern region, the equation $k_{n}=x=\frac{y}{9}$ holds. Let's find the maximum number of sets produced by the northern region: $9k_{n}=324-9k_{n}, k_{n}=18$. By selling the sets, the northern region will earn $18 \cdot 2000 = 36000$ monetary units, and the income per resident will be $\frac{36000}{24}=1500$ monetary units. The individual PPFs of residents in the southern region are quadratic and identical to each other, with strictly increasing opportunity costs. To achieve the maximum total income, any production volume should be evenly distributed among all residents. The total PPF of the southern region is $y=24 \cdot 6 - 1.5 \cdot 6 \cdot \left(\frac{x}{6}\right)^2 = 144 - 0.25x^2$. For the southern region to produce the maximum number of sets, the equation $k_{s}=x=\frac{y}{9}$ must hold. Clearly, $k_{s}>0$. The suitable root of the equation is $k_{s}=12$. By selling the sets, the southern region will earn $12 \cdot 2000 = \mathbf{24000}$ monetary units, and the income per resident will be 4000. Let's determine the Gini coefficient. The Gini coefficient can be determined using the well-known formula for two population groups $G=x-y$, where $x$ is the share of the less affluent group in the total population, and $y$ is the share of the income of the less affluent group in the total income of the entire population. This formula can be applied if the incomes within the group are uniformly distributed. A resident of the northern region is less affluent compared to a resident of the southern region, as $\mathbf{1500}<\mathbf{4000}: x=\frac{24}{24+6}=0.8; y=\frac{36000}{36000+24000}=0.6. G=0.8-0.6=\mathbf{0.2}$.

## Answer: the Gini coefficient is 0.2.

(b) If the southern region agrees to receive the production capabilities of the northern region, then the southern region will have PPFs $y=144-0.25x^2$ and $y=324-9x$. Aiming to maximize income from the sale of products, the southern region will choose a point on the combined PPF. Let's construct the combined PPF. AI $1=0.5x_1$, AI $2=9$. The first 18 units of $x$ are more profitable for the southern region to produce on its own PPF, the next 36 units of $x$ on the production capabilities of the northern region, and then continue producing $x$ on the PPF of the southern region. We get that the combined PPF has the form:

$$
\begin{aligned}
& y=468-0.25x^2, x \in [0; 18] \\
& y=549-9x, x \in (18; 54] \\
& y=18x-0.25x^2-180, x \in (54; 60]
\end{aligned}
$$

The maximum income from selling sets is achieved when $x=\frac{y}{9}$. This equality is achieved on the second segment of the combined PPF at $x=30.5$. Using the resources of both regions, the maximum number of sets that can be produced is $k_{s}=\mathbf{30.5}$. The income from selling the sets will be 61000 monetary units. According to the terms of the deal, the southern region must pay the northern region $36000+661=36661$ monetary units. The southern region will have 24339 monetary units left. A resident of the northern region remains less affluent than a resident of the southern region, as $\frac{36661}{24}<\frac{24339}{6}$. Let's find the Gini coefficient: $G=\frac{24}{30}-\frac{36661}{61000}=\mathbf{0.199}$. We get that the Gini coefficient has decreased by 0.001.

Answer: the Gini coefficient has decreased by $0.001$.

## Grading Criteria:

(a):

- The number of residents in the southern region is determined (1 point)
- The optimal number of sets for the northern region is determined (2 points)
- The optimal number of sets for the southern region is determined (4 points)
- The maximum income of the northern region is determined (1 point)
- The maximum income of the southern region is determined (2 points)
- The population is divided by income (1 point)
- The Gini coefficient is determined (4 points)

## (b):

- The production capabilities of the southern region are determined (4 points)
- The optimal number of sets is determined (2 points)
- The total income of the southern region is determined (1 point)
- The total income of the northern region is determined (2 points)
- The population is divided by income (1 point)
- The Gini coefficient is determined (3 points)
- The change in the Gini coefficient is determined (2 points)

## Alternative methods of solution, if fully and correctly justified, are scored in full.