4. In the town of M, there are 3 types of people: fat, thin (and there are equal numbers of both), and well-fed, who make up a quarter of the town's population. One day, two old friends met at the train station - one thin and one fat. Starting to discuss old times, they, as often happens, moved on to discussing acute social issues and almost got into an argument: the thin one claimed that wages in M were very low, as he, like all 42 thin people, earned only 17 bitcoins a month and could barely afford to buy jeans with cuffs. To which the fat one replied that wages were quite good, as all the fat people in town, according to his data, earned 85 bitcoins a month, which allowed them to even visit barbershops. The salary of the well-fed people, meanwhile, is equal to the average between the salaries of the fat and thin.

Help the friends assess the level of social inequality in M using any known methods, based on the data in the problem. Provide a quantitative assessment and interpret it.
# Solution:

The following are possible evaluation criteria.

1) Coefficient $\mathrm{R} / \mathrm{P} 10 \%$ or decile coefficient: $K_{D}=\frac{85}{17}=5$ - the income of the top $10 \%$ richest is five times that of the bottom $10 \%$ poorest
2) Lorenz coefficient: $L=\frac{1}{2} \sum\left|w_{i}-d_{i}\right|$, where $w_{i}$ is the share of the population in the $i-$th group, and $d_{i}$ is the share of the income of the $i-$th group in the total income.

The total population $N=43 / 0.1=430$, of which the well-fed population is $430-2 \cdot 43=344$, and their wage is $\frac{17+85}{2}=51$

| Indicator | Thin | Well-fed | Fat |
| :--- | :---: | :---: | :---: |
| Number of people in the group | 43 | 344 | 43 |
| Share of the group | 0.1 | 0.8 | 0.1 |
| Income of the group | 731 | 17544 | 3655 |
| Share of the group's income | $0.03(3)$ | 0.8 | $0.1(6)$ |

Then $L=0.5 \cdot[(0.1-0.03(3))+(0.8-0.8)+(0.1(6)-0.1)]=0.06(6) \quad-\quad$ indicates how much the income distribution differs from the population distribution by groups. The closer to zero, the closer the distributions.

3) Gini coefficient: $G=\sum p_{i} q_{i+1}-\sum p_{i+1} q_{i}$, where $p_{i}$ is the cumulative share of the population, and $q_{i}$ is the cumulative share of income.

| Indicator | Thin | Well-fed | Fat |
| :--- | :---: | :---: | :---: |
| Cumulative share of population | 0.1 | 0.9 | 1 |
| Cumulative share of income | $0.03(3)$ | $0.8(3)$ | 1 |

$G=0.1 \cdot 0.8(3)+0.9 \cdot 1-0.03(3) \cdot 0.9-0.8(3) \cdot 1=0.12-\quad$ describes the uniformity of income distribution among groups. The closer to zero, the more uniform.

## Evaluation Criteria:

2 points - attempt to compare using average values.

From 1 to 5 points (depending on the completeness of explanations and accuracy of calculations) for explanations and calculations of each of the three coefficients (maximum 15 points).