Gini coefficient (Gini coefficient), Lorenz coefficient

In early 20th century, Italian economist Guinea, in 1922, proposed quantitative measurement of income distribution differences in the level of indicators. It is based on the Lorenz curve to find out the criteria for determining the degree of equality (such as).

The area between the actual income distribution curve and the absolute equality curve of income distribution is a, and the area at the bottom right of the actual income distribution curve is B. And the quotient of a divided by a+b represents the degree of inequality. This value is called the Gini coefficient or the Lorenz coefficient. If a is zero, the Gini coefficient is zero, which means that the income distribution is completely equal; if B is zero, the coefficient is 1, and the income distribution is absolutely unequal. The coefficient can take any value between 0 and 1. The more the income distribution tends to be equal, the smaller the radian of Lorenz curve, the smaller the Gini coefficient, the more the income distribution tends to be unequal, the larger the radian of the Lorenz curve, the greater the Gini coefficient. If personal income tax can equalize income, the Gini coefficient will become smaller.

The formula for calculating Gini coefficients is:

　　

where x represents the proportion of the population of each group, Y represents the proportion of the income of each group, and V represents the cumulative proportion of the income of each group, and I=1,2,3,...,n,n represents the number of groups grouped.

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The Gini coefficient, as stipulated by the relevant United Nations organizations:

• If below 0.2 indicates an absolute average of income;

• 0.2-0.3 means the comparative average;

• 0.3-0.4 is relatively reasonable;

• 0.4-0.5 that the income gap is larger;

• Above 0.5 indicates a disparity in income disparity.

Economists usually use the Gini index to show the distribution of wealth in a country or region. This index is between 0 and one, the lower the value, the more evenly the distribution of wealth among members of society is, and vice versa.

0.4 is usually used as a "cordon" of income distribution gaps. The Gini index in the general developed countries is between 0.24 and 0.36, and the United States is high at 0.4. Both mainland China and Hong Kong have a Gini coefficient of more than 0.4.

In addition, the Lorenz curve is about the relationship between the percentage of the market's total shipping value and the cumulative percentage of the market from small to large vendors. The smaller the radian of the Lorenz curve, the lower the Gini coefficient.

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Gini coefficient is the statistical index of the United Nations, which is used to measure the inequality of income distribution in various countries, regions, races and industries, and it is also the most important basis for judging the distribution of income by many countries and even most of our scholars. The advantage is that simple packet data can be used to concisely summarize the distribution of residents ' income as described in Lorentz curve, and reflect the overall income disparity with a numerical value. But it is also because of this generalization that it has lost some of the information represented by the Lorentz curve, and there are some shortcomings in measuring and explaining the income distribution situation.

Gini coefficient can not accurately reflect the shape of Lorentz curve, so it can not accurately reflect the unequal degree of income distribution. The formula of the Gini coefficient shows that the value of the Gini coefficient depends entirely on the size of the area between the Lorentz curve and the absolute average line, the income structure of each income class can not be known from the numerical value, and the corresponding Lorentz curve, the area may be the same, that is, the same Gini coefficient.

For illustrative purposes, see the table below. In this example, the Lorentz curves of Case 1 and Condition 2 are respectively shown in the L1 and L2 of the curve. As shown, the income distribution structure expressed by L1 and L2 is obviously different, the Lorentz curve of the situation 2 is more inclined than the Lorentz curve of the situation 1, and the relative poverty degree of the low-income group in L1 is much higher than that of L2, if from the angle of enlarging domestic demand and guaranteeing social stability, It is more urgent to adopt policy measures to adjust the income structure for the situation described by L1, thus increasing the income level of the lower income groups. However, according to the definition of Gini coefficient, the Gini coefficients calculated by these two curves are equal, and if only based on Gini coefficient, the policy-makers cannot make the most reasonable decision. Wilson (1987) has done more in-depth research on this issue. After a study of American data, he found that the Gini coefficient could underestimate the extent of the disparity between the rich and poor in the United States, as the income gap between blacks and whites has widened, while the Gini coefficient has shrunk in real terms. He further pointed out that this result is precisely due to the Gini coefficient corresponding to the Lorentz curve of the non-uniqueness, so the Gini coefficient to reflect the gap between the rich and poor is not accurate, at least not comprehensive. The degree of inequality in social distribution is caused by the gap between the two income distributions, the gap formed by income below the average income of society as a whole, and the gap created by income higher than the average income of society as a whole. The Gini coefficient calculates the sum of the two gaps, but does not reflect the size of the two gaps.

Gini coefficients in two different cases Gini

Group number		1	2	3	coefficient
Scenario 1	Income I	1/6	3/6	2/6	13/36
	Population p	3/6	2/6	1/6
Scenario 2	Income I	1/6	2/6	3/6	13/36
	Population p	2/6	3/6	1/6

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The roulunds curve is the curve of income distribution presented by economist Max Laurenz in 1905, and the Italian economist Corrado Kini defined the Gini coefficient on this basis.

In economics, the Lorenz curve is the curve of the cumulative distribution function established on the previous wealth distribution data, which reflects the proportion of each allocation by the value of the variable y%. It is often used to describe the distribution of income, that is, the percentage of households that represent part (income-like) of a family as a proportion of the total social household, with the y% representing the portion of the household's income as a percentage of the overall social income. The curve can also be used to describe the distribution of social capital. In these applications, economists often use it to measure whether society (mainly social income) is fair. probability density function (f(x)) or cumulative distribution function (f(x)):

L (f) =∫−∞x (f) x f (x) d x∫−∞∞x f (x) d x =∫0 f x (f′) d f′∫0 1 x (f′) d F′{\displayst Yle L (f) ={\frac {\int _{-\infty}^{x (f)}xf (x) \,dx}{\int _{-\infty}^{\infty}xf (x) \,dx}}={\frac {\int _{0}^{F}x (f ') \,dF '}{\int _{0}^{1}x (F ') \,df '

}}

This curve in the development of economics, in addition to the common Gini coefficient of income distribution, as well as the distribution of land, the extent of education degree distribution.

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Draw a rectangle, the height of the rectangle to measure the percentage of social wealth, divides it into five equal parts, each divided into 20 of the total wealth of society. On the long side of the rectangle, 100 of families from the poorest to the richest have left-to-right, divided into 5 equal parts, the first of which represents 20 of the households with the lowest income. In this rectangle, the percentage of all possessions owned by each 100-minute family is accumulated, and the corresponding points are drawn in the graph, and a curve is the Lorentz curve.

　　

Obviously, the degree of curvature of Lorentz curve is of great significance. In general, it reflects the unequal level of income distribution. The greater the degree of bending, the more unequal the income distribution, and vice versa. In particular, if all income is concentrated in the hands of one person, while the rest of the population has nothing, and the distribution of income is completely unequal, the Lorentz curve becomes a polyline OHL; On the other hand, if any percentage of the population equals its percentage of income, so that the cumulative percentage of population equals the cumulative percentage of income, Then the income distribution is completely equal, the Lorentz curve becomes through the origin of the 45-degree line ol.

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Gini coefficient (Gini coefficient), Lorenz coefficient