In economics, the Gini coefficient ( ), also known as the Gini index or Gini ratio, is a measure of statistical dispersion intended to represent the income inequality, the wealth inequality, or the consumption inequality within a nation or a social group. It was developed by Italian statistician and sociologist Corrado Gini.
The Gini coefficient measures the inequality among the values of a frequency distribution, such as income levels. A Gini coefficient of 0 reflects perfect equality, where all income or wealth values are the same. In contrast, a Gini coefficient of 1 (or 100%) reflects maximal inequality among values, where a single individual has all of the income while all others have none.Note: The Gini coefficient could be near one only in a large population where a few persons have all the income. In the special case of just two people, where one has no income, and the other has all the income, the Gini coefficient is 0.5. For five people, where four have no income, and the fifth has all the income, the Gini coefficient is 0.8. See: Bellù, L.G. and Liberati, P. 2006. Inequality Analysis: The Gini Index. EASYPol: Resources for policy making. Rome, FAO.
Corrado Gini proposed the Gini coefficient as a measure of inequality of income or wealth.Gini, Corrado (1936). "On the Measure of Concentration with Special Reference to Income and Statistics", Colorado College Publication, General Series No. 208, 73–79. For OECD countries in the late 20th century, considering the effect of taxes and transfer payments, the income Gini coefficient ranged between 0.24 and 0.49, with Slovakia being the lowest and Mexico the highest. African countries had the highest pre-tax Gini coefficients in 2008–2009, with South Africa having the world's highest, estimated to be 0.63 to 0.7. However, this figure drops to 0.52 after social assistance is taken into account and drops again to 0.47 after taxation. Slovakia has the lowest Gini coefficient, with a Gini coefficient of 0.238. Various sources have estimated the Gini coefficient of the global income in 2005 to be between 0.61 and 0.68.
There are multiple issues in interpreting a Gini coefficient, as the same value may result from many different distribution curves. The demographic structure should be taken into account to mitigate this. Countries with an aging population or those with an increased birth rate experience an increasing pre-tax Gini coefficient even if real income distribution for working adults remains constant. Over a dozen variants of the Gini coefficient exist, devised by various scholars.
History The Italian statistician Corrado Gini developed the Gini coefficient and published it in his 1912 paper Variabilità e mutabilità ().Gini, C. (1909). "Concentration and dependency ratios" (in Italian). English translation in Rivista di Politica Economica, 87 (1997), 769–789. Building on the work of economist Max Lorenz, Gini proposed using the difference between the hypothetical straight line depicting perfect equality and the actual line depicting people's incomes as a measure of inequality. In this paper, he introduced the concept of simple mean difference as a measure of variability.
He then applied the simple mean difference of observed variables to income and wealth inequality in his work On the measurement of concentration and variability of characters in 1914. Here, he presented the concentration ratio, which further developed into today's Gini coefficient. Secondly, Gini observed that improving methods introduced by Lorenz, Chatelain, or Séailles could also achieve his proposed ratio.
In 1915, Gaetano Pietra introduced a geometrical interpretation between Gini's proposed ratio and between the observed area of concentration and maximum concentration. This altered version of the Gini coefficient became the most commonly used inequality index in upcoming years.
According to data from the OECD, the Gini coefficient was first officially used country-wide in Canada in the 1970s. Canadian index of income inequality ranged from 0.303 to 0.284 from 1976 to the end of the 1980s. The OECD has published more data on countries since the start of the 21st century. The Central European countries of Slovenia, Czechia, and Slovakia have had the lowest inequality index of all OECD countries ever since the 2000s. Scandinavian countries also frequently appeared at the top of the equality list in recent decades.
Definition
The Gini coefficient is an index for the degree of inequality in the distribution of income/wealth, used to estimate how far a country's wealth or income distribution deviates from an equal distribution.
The Gini coefficient is usually defined mathematically based on the Lorenz curve, which plots the proportion of the total income of the population (y-axis) that is cumulatively earned by the bottom x of the population (see diagram). The line at 45 degrees thus represents perfect equality of incomes. The Gini coefficient can then be thought of as the ratio of the area that lies between the line of equality and the Lorenz curve (marked A in the diagram) over the total area under the line of equality (marked A and B in the diagram); i.e., . If there are no negative incomes, it is also equal to 2A and due to the fact that .
Assuming non-negative income or wealth for all, the Gini coefficient's theoretical range is from 0 (total equality) to 1 (absolute inequality). This measure is often rendered as a percentage, spanning 0 to 100. However, if negative values are factored in, as in cases of debt, the Gini index could exceed 1. Typically, we presuppose a positive mean or total, precluding a Gini coefficient below zero.
An alternative approach is to define the Gini coefficient as half of the relative mean absolute difference, which is equivalent to the definition based on the Lorenz curve. The mean absolute difference is the average absolute difference of all pairs of items of the population, and the relative mean absolute difference is the mean absolute difference divided by the average, , to normalize for scale. If xi is the wealth or income of person i, and there are n persons, then the Gini coefficient G is given by:
When the income (or wealth) distribution is given as a continuous probability density function p(x), the Gini coefficient is again half of the relative mean absolute difference:
where is the mean of the distribution, and the lower limits of integration may be replaced by zero when all incomes are positive.Dorfman, Robert. "A Formula for the Gini Coefficient." The Review of Economics and Statistics, vol. 61, no. 1, 1979, pp. 146–49. JSTOR, . Accessed 2 January 2023.
Calculation
While the income distribution of any particular country will not correspond perfectly to the theoretical models, these models can provide a qualitative explanation of the income distribution in a nation given the Gini coefficient.
Example: Two levels of income The extreme cases are represented by the most equal possible society in which every person receives the same income (), and the most unequal society (with N individuals) where a single person receives 100% of the total income and the remaining people receive none ().
A simple case assumes just two levels of income, low and high. If the high income group is a proportion u of the population and earns a proportion f of all income, then the Gini coefficient is . A more graded distribution with these same values u and f will always have a higher Gini coefficient than .
For example, if the wealthiest u = 20% of the population has f = 80% of all income (see Pareto principle), the income Gini coefficient is at least 60%. In another example, if u = 1% of the world's population owns f = 50% of all wealth, the wealth Gini coefficient is at least 49%.
Alternative expressions
In some cases, this equation can be applied to calculate the Gini coefficient without direct reference to the Lorenz curve. For example, (taking y to indicate the income or wealth of a person or household): For a population of n individuals with values ,
This may be simplified to:
The Gini coefficient can also be considered as half the relative mean absolute difference. For a random sample S with values , the sample Gini coefficient
is a consistent estimator of the population Gini coefficient, but is not in general unbiased. In simplified form:
There does not exist a sample statistic that is always an unbiased estimator of the population Gini coefficient.
Discrete probability distribution
For a discrete probability distribution with probability mass function , where is the fraction of the population with income or wealth , the Gini coefficient is:
where
If the points with non-zero probabilities are indexed in increasing order , then:
where and These formulas are also applicable in the limit, as
Continuous probability distribution
When the population is large, the income distribution may be represented by a continuous probability density function f(x) where f(x) dx is the fraction of the population with wealth or income in the interval dx about x. If F(x) is the cumulative distribution function (CDF) for f(x):
and L(x) is the Lorenz function:
then the Lorenz curve L(F) may then be represented as a function parametric in L(x) and F(x) and the value of B can be found by integration:
The Gini coefficient can also be calculated directly from the cumulative distribution function of the distribution F(y). Defining μ as the mean of the distribution, then specifying that F(y) is zero for all negative values, the Gini coefficient is given by:
The latter result comes from integration by parts. (Note that this formula can be applied when there are negative values if the integration is taken from minus infinity to plus infinity.)
The Gini coefficient may be expressed in terms of the quantile function Q(F) (inverse of the cumulative distribution function: Q(F(x)) = x)
Since the Gini coefficient is independent of scale, if the distribution function can be expressed in the form f(x,φ,a,b,c...) where φ is a scale factor and a, b, c... are dimensionless parameters, then the Gini coefficient will be a function only of a, b, c.... For example, for the exponential distribution, which is a function of only x and a scale parameter, the Gini coefficient is a constant, equal to 1/2.
For some functional forms, the Gini index can be calculated explicitly. For example, if y follows a log-normal distribution with the standard deviation of logs equal to , then where is the error function ( since , where is the cumulative distribution function of a standard normal distribution).Crow, E. L., & Shimizu, K. (Eds.). (1988). Lognormal distributions: Theory and applications (Vol. 88). New York: M. Dekker, page 11. In the table below, some examples for probability density functions with support on are shown. The Dirac delta distribution represents the case where everyone has the same wealth (or income); it implies no variations between incomes.
{| class="wikitable" style="float: left; margin-left: 1em;" |- ! Income Distribution function !! PDF(x) !! Gini Coefficient |- | Dirac delta function || || 0 |- | Uniform distribution
| Exponential distribution
| Log-normal distributionFor the log-normal with = 0, = 0; = 0.
| Pareto distribution
| Chi distribution
| Chi-squared distribution
| Gamma distribution
| Weibull distribution
| Beta distribution
|Log-logistic distribution
is the Gamma function is the Beta function is the Regularized incomplete beta function
Other approaches
Sometimes the entire Lorenz curve is not known, and only values at certain intervals are given. In that case, the Gini coefficient can be approximated using various techniques for interpolating the missing values of the Lorenz curve. If (Xk, Yk) are the known points on the Lorenz curve, with the Xk indexed in increasing order (Xk – 1 < Xk), so that: