## Gini coefficient

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# Gini Coefficient

BIBLIOGRAPHY

The Gini coefficient is the most popular measure of inequality in use today. The measurement is named for its developer, Corrado Gini (18841965), and is based on the Lorenz curve (Sen 1997; Xu 2004). Although the Gini was traditionally used to measure income as a measure of welfare, it is now often used to measure other variables such as expenditures, wealth, and even health.

The clearest way to portray the Gini coefficient is diagrammatically. Take a population of ten individuals, as shown in Table 1. If you sort the population from poorest to richest and graph the cumulative share of income against population, what you get is a Lorenz curve, as depicted by the curve in Figure 1. If every individual in the society had equal income, the graph would be the diagonal. But in any society, the poorest will have a share less than their proportion in the population, so the Lorenz curve will always be below the diagonal. The Gini coefficient is measured as the ratio of the area between the diagonal and the Lorenz curve (X ) and the area under the

Table 1
IndividualShare of PopulationIndividual IncomeCumulative Share of Income
1000
21022
32035
43049
540615
650823
760932
8701143
9801558
10901775
10025100
Total Income 100

diagonal (X + Y ). The formula for the coefficient can be stated as

where n is the population, µ is the mean of incomes, y is income, and i and j are individuals in the population. This is effectively a sum of all pair-wise income inequalities in the population. The Gini coefficient is, thus, a measure of the overall inequality in the population (Sen 1997).

In Table 2 and Figure 2, in addition to the original distribution used above (labeled as B ), we add two other distributions (A and C ). As income inequality increases, the Lorenz curve moves farther away from the diagonal (see distribution C ), and the opposite occurs as inequality decreases (see distribution A ). In a society where income is equally shared, the area X = 0 because the Lorenz curve would coincide with the diagonal and the Gini coefficient would equal 0. In contrast, if all income is held by the richest individual in society, then Y = 0 and the ratio would equal 1. The Gini coefficient thus always lies between 0 and 1. The most common convention nowadays is to multiply the ratio by 100 and then report the Gini coefficient as a number between 0 and 100. One of the great strengths of the Gini is that any redistribution of income from a poorer person to a richer person results in an increase of the coefficient, and thus it captures distribution across the entire population rather than just at the mean.

Despite its popularity, the Gini suffers from the fact that there is no intuitive meaning to any particular magnitude of the coeffecient. Nor does any specific magnitude represent a unique distribution. A Gini of 30, for example, can represent two different distributions, and there is no objective way to differentiate the distributions. This occurs because Lorenz curves may cross. It follows from this that determining distributions that are highly unequal versus those that are not is more an empirical question.

Table 2
Individual Income Cumulative Share of Income
IndividualShare of PopulationA B C A B C
10000000
210124124
320335359
4304465915
540667101522
6508810152332
7609911213243
870111112284355
980151530385868
1090171715537583
100472517100100100
Total Income 100 100 100

In order to give the reader some sense of what levels of inequality different sizes of Gini represent, Table 3 illustrates the average Gini coeffecients and the average level of income accruing to each quintile of the population for thirty-eight countries, from a dataset compiled by Klaus Deininger and Lyn Squire (1996). The thirty-eight countries were picked because they had a measurement of Gini that Deininger and Squire considered reliable and comparable across countries between 1990 and 1995, as well as information on the quintile distributions of income, which gives us more information on the underlying distribution. If we divide the countries evenly into three groups we find that the group with lowest inequality has Ginis that range from 22 to 30, the middle group from 31 to 34, and the most unequal group from 35 to 62. Because this last group has such high variation we divide the group into two at 45.

Although these observations do not encompass the whole range of measured distributions, they do provide us with some more information. On average, developing countries, particularly in Africa and Latin America, dominate the group of highly unequal countries, whereas the group of low inequality is dominated by European countries, particularly those that formerly belonged to the socialist bloc. For countries with low inequality, the richest 20 percent of the population earned less than 40 percent of national income, whereas for those with high inequality the richest earned over 47 percent, averaging close to 60 percent. In contrast, for the poorest 20 percent in the low-inequality countries earned between 7 and 11 percent of national income, whereas in the highly unequal countries they only earned between 2 and 4 percent of national income.

Since the late 1980s a number of strides have been made in estimating the Gini from functional forms including the diagrammatic methods discussed above, as well as in decomposing it so that one can estimate the impact of different components of income on inequality. For a good summary of this literature, see Xu (2004).

Table 3
GiniQuintile 1Quintile 1Quintile 1Quintile 1Quintile 1Number of Observations
Low Inequality270.090.140.180.230.3613
Range for Variable21-300.07-0.110.13-0.150.17-0.200.21-0.250.33-0.39

Average Inequaliy330.070.120.170.240.4013
Range for variable31-340.05-0.080.11-0.130.16-0.190.21-0.250.37-0.42

Moderately High Inequality390.050.110.160.240.447
Range for Variable35-450.05-0.070.10-0.120.14-0.170.21-0.260.40-0.49

High Inequality540.030.070.120.200.576
Range for variable45-520.02-0.040.05-0.090.09-0.180.18-0.240.47-0.65

SEE ALSO Development Economics; Inequality, Income; Inequality, Political; Kuznets Hypothesis; Poverty, Indices of

## BIBLIOGRAPHY

Anand, Sudhir. 1983. Inequality and Poverty in Malaysia: Measurement and Decomposition. New York: Oxford University Press.

Deaton, Angus. 1997. The Analysis of Household Surveys. Baltimore, MD: Johns Hopkins University Press.

Deininger, Klaus, and Lyn Squire. 1996. A New Data Set Measuring Income Inequality. World Bank Economic Review 10: 565591.

Sen, Amartya K. 1973. On Economic Inequality. Oxford: Clarendon.

Sen, Amartya K. 1997. On Economic Inequality. Expanded ed. with a substantial annexe by James E. Foster and Amartya Sen. Oxford: Clarendon.

Xu, Kuan. 2004. How Has the Literature on Ginis Index Evolved in the Past 80 Years? Department of Economics Working Papers, Dalhousie University. http://economics.dal.ca/RePEc/dal/wparch/howgini.pdf.

Mwangi wa Gĩthĩnji

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Gini coefficient A statistic conventionally used in order to describe in summary form the degree of inequality in distributions (usually of income and wealth). It is calculated by determining the area between a Lorenz curve of the distribution (that is a graph of the cumulative income share against the cumulative share of recipients) and the line of perfect equality. This area is then expressed as a percentage of the whole triangular area under the diagonal. The resulting statistic reduces the position of the curve to a single figure. A distribution of perfectly equal incomes has a Gini coefficient of zero. As inequality increases, and the actual Lorenz curve bellies out, the Gini coefficient moves towards its maximum value of 1 (only one recipient receives all income).

The Gini coefficient is one of several measures of inequality (others include the coefficient of variation and inter-decile ratio) which purport to rest exclusively on the mathematical properties of a distribution. Such indices tend to give conflicting answers when they are asked to determine whether any particular distribution is more unequal than another. For example, one statistic may be more sensitive than another to a scatter of low incomes, and each indicator tends to attach different weights to different forms of inequality within the distribution. More sophisticated measures—such as the Theil index and Atkinson index—have therefore been devised, which attempt to express in a single number the degree of inequality in a distribution as a whole, but in ways which explicitly weight inequalities in different parts of the distribution—in line (for example) with the observer's judgements about the relative value of each additional unit of currency accruing to a rich as against a poor person. ( Atkinson's formula for his index contains a coefficient which expresses the researcher's aversion to inequality.)

A good account of the various properties of the different measures will be found in Henry Phelps Brown , Egalitarianism and the Generation of Inequality (1988)