Gini coefficient

The extreme cases are represented by the most equal possible society in which every person receives the same income (G = 0), and the most unequal society (with N individuals) where a single person receives 100% of the total income and the remaining N − 1 people receive none (G = 1 − 1/N).

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 f − u. A more graded distribution with these same values u and f will always have a higher Gini coefficient than f − u.

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,[26] if u = 1% of the world's population owns f = 50% of all wealth, the wealth Gini coefficient is at least 49%.

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 ${\displaystyle y_{1}\leq y_{2}\leq \cdots \leq y_{n}}$,[27]

${\displaystyle G={\frac {1}{n}}\left(n+1-2\left({\frac {\sum _{i=1}^{n}(n+1-i)y_{i}}{\sum _{i=1}^{n}y_{i}}}\right)\right).}$

This may be simplified to:

${\displaystyle G={\frac {2\sum _{i=1}^{n}iy_{i}}{n\sum _{i=1}^{n}y_{i}}}-{\frac {n+1}{n}}.}$

The Gini coefficient can also be considered as half the relative mean absolute difference. For a random sample S with values ${\displaystyle y_{1}\leq y_{2}\leq \cdots \leq y_{n}}$, the sample Gini coefficient

${\displaystyle G(S)={\frac {1}{n-1}}\left(n+1-2\left({\frac {\sum _{i=1}^{n}(n+1-i)y_{i}}{\sum _{i=1}^{n}y_{i}}}\right)\right)}$

is a consistent estimator of the population Gini coefficient, but is not in general unbiased. In simplified form:

${\displaystyle G(S)=1-{\frac {2}{n-1}}\left(n-{\frac {\sum _{i=1}^{n}iy_{i}}{\sum _{i=1}^{n}y_{i}}}\right).}$

There does not exist a sample statistic that is always an unbiased estimator of the population Gini coefficient.

For a discrete probability distribution with probability mass function ${\displaystyle f(y_{i}),}$ , where ${\displaystyle f(y_{i})}$ is the fraction of the population with income or wealth ${\displaystyle y_{i}>0}$, the Gini coefficient is:

${\displaystyle G={\frac {1}{2\mu }}\sum \limits _{i=1}^{n}\sum \limits _{j=1}^{n}\,f(y_{i})f(y_{j})|y_{i}-y_{j}|}$

where

${\displaystyle \mu =\sum \limits _{i=1}^{n}y_{i}f(y_{i}).}$

If the points with non-zero probabilities are indexed in increasing order ${\displaystyle (y_{i}<y_{i+1})}$, then:

${\displaystyle G=1-{\frac {\sum _{i=1}^{n}f(y_{i})(S_{i-1}+S_{i})}{S_{n}}}}$

where

${\displaystyle S_{i}=\sum _{j=1}^{i}f(y_{j})\,y_{j}\,}$ and ${\displaystyle S_{0}=0.}$ These formulas are also applicable in the limit, as ${\displaystyle n\rightarrow \infty .}$

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 for f(x):

${\displaystyle F(x)=\int _{0}^{x}f(x)\,dx}$

and L(x) is the Lorenz function:

${\displaystyle L(x)={\frac {\int _{0}^{x}x\,f(x)\,dx}{\int _{0}^{\infty }x\,f(x)\,dx}}}$

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:

${\displaystyle B=\int _{0}^{1}L(F)\,dF.}$

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, and specifying that F(y) is zero for all negative values, the Gini coefficient is given by:

${\displaystyle G=1-{\frac {1}{\mu }}\int _{0}^{\infty }(1-F(y))^{2}\,dy={\frac {1}{\mu }}\int _{0}^{\infty }F(y)(1-F(y))\,dy}$

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)

${\displaystyle G={\frac {1}{2\mu }}\int _{0}^{1}\int _{0}^{1}|Q(F_{1})-Q(F_{2})|\,dF_{1}\,dF_{2}.}$

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....[28] 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 ${\displaystyle \sigma }$, then ${\displaystyle G=\operatorname {erf} \left({\frac {\sigma }{2}}\right)}$ where ${\displaystyle \operatorname {erf} }$ is the error function ( since ${\displaystyle G=2\Phi \left({\frac {\sigma }{\sqrt {2}}}\right)-1}$, where ${\displaystyle \Phi }$ is the cumulative distribution function of a standard normal distribution).[29] In the table below, some examples for probability density functions with support on ${\displaystyle [0,\infty )}$ are shown. The Dirac delta distribution represents the case where everyone has the same wealth (or income); it implies no variations between incomes.[30]

Income Distribution function	PDF(x)	Gini Coefficient
Dirac delta function	${\displaystyle \delta (x-x_{0}),\,x_{0}>0}$	0
Uniform distribution[31]	${\displaystyle {\begin{cases}{\frac {1}{b-a}}&a\leq x\leq b\\0&\mathrm {otherwise} \end{cases}}}$	${\displaystyle {\frac {(b-a)}{3(b+a)}}}$
Exponential distribution[32]	${\displaystyle \lambda e^{-x\lambda },\,\,x>0}$	${\displaystyle 1/2}$
Log-normal distribution[29][33]	${\displaystyle {\frac {1}{x\sigma {\sqrt {2\pi }}}}e^{-{\frac {1}{2}}\left({\frac {\ln \,(x)-\mu }{\sigma }}\right)^{2}}}$	${\displaystyle {\textrm {erf}}(\sigma /2)=2\Phi \left({\frac {\sigma }{\sqrt {2}}}\right)-1}$
Pareto distribution[34]	${\displaystyle {\begin{cases}{\frac {\alpha k^{\alpha }}{x^{\alpha +1}}}&x\geq k\\0&x<k\end{cases}}}$	${\displaystyle {\begin{cases}1&0<\alpha <1\\{\frac {1}{2\alpha -1}}&\alpha \geq 1\end{cases}}}$
Chi distribution[35]	${\displaystyle f(x;k)={\begin{cases}{\dfrac {x^{k-1}e^{-x^{2}/2}}{2^{k/2-1}\Gamma \left({\frac {k}{2}}\right)}},&x\geq 0\\0,&x<0\end{cases}}}$	${\displaystyle (-1)^{k}\left|I_{-1}(k,{\tfrac {1}{2}})\right|}$
Chi-squared distribution[36]	${\displaystyle {\frac {2^{-k/2}e^{-x/2}x^{k/2-1}}{\Gamma (k/2)}}}$	${\displaystyle {\frac {2\,\Gamma \left({\frac {1+k}{2}}\right)}{k\,\Gamma (k/2){\sqrt {\pi }}}}}$
Gamma distribution[28]	${\displaystyle {\frac {e^{-x/\theta }x^{k-1}\theta ^{-k}}{\Gamma (k)}}}$	${\displaystyle {\frac {\Gamma \left({\frac {2k+1}{2}}\right)}{k\,\Gamma (k){\sqrt {\pi }}}}}$
Weibull distribution[37]	${\displaystyle {\frac {k}{\lambda }}\,\left({\frac {x}{\lambda }}\right)^{k-1}e^{-(x/\lambda )^{k}}}$	${\displaystyle 1-2^{-1/k}}$
Beta distribution[38]	${\displaystyle {\frac {x^{\alpha -1}(1-x)^{\beta -1}}{B(\alpha ,\beta )}}}$	${\displaystyle \left({\frac {2}{\alpha }}\right){\frac {B(\alpha +\beta ,\alpha +\beta )}{B(\alpha ,\alpha )B(\beta ,\beta )}}}$
Log-logistic distribution[39]	${\displaystyle {\frac {(\beta /\alpha )(x/\alpha )^{\beta -1}}{\left(1+(x/\alpha )^{\beta }\right)^{2}}}}$	${\displaystyle 1/\beta }$

• ${\displaystyle \Gamma (\,)}$ is the Gamma function
• ${\displaystyle B(\,)}$ is the Beta function
• ${\displaystyle I_{k}(\,)}$ is the Regularized incomplete beta function

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 (X_k, Y_k) are the known points on the Lorenz curve, with the X_k indexed in increasing order (X_{k – 1} < X_k), so that:

• X_k is the cumulated proportion of the population variable, for k = 0,...,n, with X₀ = 0, X_n = 1.
• Y_k is the cumulated proportion of the income variable, for k = 0,...,n, with Y₀ = 0, Y_n = 1.
• Y_k should be indexed in non-decreasing order (Y_k > Y_{k – 1})

If the Lorenz curve is approximated on each interval as a line between consecutive points, then the area B can be approximated with trapezoids and:

${\displaystyle G_{1}=1-\sum _{k=1}^{n}(X_{k}-X_{k-1})(Y_{k}+Y_{k-1})}$

is the resulting approximation for G. More accurate results can be obtained using other methods to approximate the area B, such as approximating the Lorenz curve with a quadratic function across pairs of intervals or building an appropriately smooth approximation to the underlying distribution function that matches the known data. If the population mean and boundary values for each interval are also known, these can also often be used to improve the accuracy of the approximation.

The Gini coefficient calculated from a sample is a statistic, and its standard error, or confidence intervals for the population Gini coefficient, should be reported. These can be calculated using bootstrap techniques, mathematically complicated and computationally demanding even in an era of fast computers.[40] Economist Tomson Ogwang made the process more efficient by setting up a "trick regression model" in which respective income variables in the sample are ranked, with the lowest income being allocated rank 1. The model then expresses the rank (dependent variable) as the sum of a constant A and a normal error term whose variance is inversely proportional to y_k:

${\displaystyle k=A+\ N(0,s^{2}/y_{k})}$

Thus, G can be expressed as a function of the weighted least squares estimate of the constant A and that this can be used to speed up the calculation of the jackknife estimate for the standard error. Economist David Giles argued that the standard error of the estimate of A can be used to derive the estimate of G directly without using a jackknife. This method only requires using ordinary least squares regression after ordering the sample data. The results compare favorably with the estimates from the jackknife with agreement improving with increasing sample size.[41]

However, it has been argued that this depends on the model's assumptions about the error distributions and the independence of error terms. These assumptions are often not valid for real data sets. There is still ongoing debate surrounding this topic.

Guillermina Jasso[42] and Angus Deaton[43] independently proposed the following formula for the Gini coefficient:

${\displaystyle G={\frac {N+1}{N-1}}-{\frac {2}{N(N-1)\mu }}(\sum _{i=1}^{n}P_{i}X_{i})}$

where ${\displaystyle \mu }$ is mean income of the population, P_i is the income rank P of person i, with income X, such that the richest person receives a rank of 1 and the poorest a rank of N. This effectively gives higher weight to poorer people in the income distribution, which allows the Gini to meet the Transfer Principle. Note that the Jasso-Deaton formula rescales the coefficient so that its value is one if all the ${\displaystyle X_{i}}$ are zero except one. Note however Allison's reply on the need to divide by N² instead.[44]

FAO explains another version of the formula.[45]

According to UNICEF, Latin America and the Caribbean region had the highest net income Gini index in the world at 48.3, on an unweighted average basis in 2008. The remaining regional averages were: sub-Saharan Africa (44.2), Asia (40.4), Middle East and North Africa (39.2), Eastern Europe and Central Asia (35.4), and High-income Countries (30.9). Using the same method, the United States is claimed to have a Gini index of 36, while South Africa had the highest income Gini index score of 67.8.[52]

Taking income distribution of all human beings, worldwide income inequality has been constantly increasing since the early 19th century (and will keep on increasing over the years) . There was a steady increase in the global income inequality Gini score from 1820 to 2002, with a significant increase between 1980 and 2002. This trend appears to have peaked and begun a reversal with rapid economic growth in emerging economies, particularly in the large populations of BRIC countries.[53]

The table below presents the estimated world income Gini coefficients over the last 200 years, as calculated by Milanovic.[54]

More detailed data from similar sources plots a continuous decline since 1988. This is attributed to globalization increasing incomes for billions of poor people, mostly in countries like China and India. Developing countries like Brazil have also improved basic services like health care, education, and sanitation; others like Chile and Mexico have enacted more progressive tax policies.[56]

Education Gini index estimates the inequality in education for a given population.[59] It is used to discern trends in social development through educational attainment over time. A study across 85 countries by three World Bank economists, Vinod Thomas, Yan Wang, and Xibo Fan, estimated Mali had the highest education Gini index of 0.92 in 1990 (implying very high inequality in educational attainment across the population), while the United States had the lowest education inequality Gini index of 0.14. Between 1960 and 1990, China, India and South Korea had the fastest drop in education inequality Gini Index. They also claim education Gini index for the United States slightly increased over the 1980–1990 period.

Though India's education Gini Index has been falling from 1960 through 1990, most of the population still has not received any education, while 10 percent of the population received more than 40% of the total educational hours in the nation. This means that a large portion of capable children in the country are not receiving the support necessary to allow them to become positive contributors to society. This will lead to a deadweight loss to the national society because there are many people who are underdeveloped and underutilized.[60]

Similar in concept to the Gini income coefficient, the Gini opportunity coefficient measures inequality in opportunities.[61][62][63] The concept builds on Amartya Sen's suggestion[64] that inequality coefficients of social development should be premised on the process of enlarging people's choices and enhancing their capabilities, rather than on the process of reducing income inequality. Kovacevic, in a review of the Gini opportunity coefficient, explained that the coefficient estimates how well a society enables its citizens to achieve success in life where the success is based on a person's choices, efforts and talents, not their background defined by a set of predetermined circumstances at birth, such as gender, race, place of birth, parent's income and circumstances beyond the control of that individual.

In 2003, Roemer[61][65] reported Italy and Spain exhibited the largest opportunity inequality Gini index amongst advanced economies.

In 1978, Anthony Shorrocks introduced a measure based on income Gini coefficients to estimate income mobility.[66] This measure, generalized by Maasoumi and Zandvakili,[67] is now generally referred to as Shorrocks index, sometimes as Shorrocks mobility index or Shorrocks rigidity index. It attempts to estimate whether the income inequality Gini coefficient is permanent or temporary and to what extent a country or region enables economic mobility to its people so that they can move from one (e.g., bottom 20%) income quantile to another (e.g., middle 20%) over time. In other words, the Shorrocks index compares inequality of short-term earnings, such as the annual income of households, to inequality of long-term earnings, such as 5-year or 10-year total income for the same households.

Shorrocks index is calculated in several different ways, a common approach being from the ratio of income Gini coefficients between short-term and long-term for the same region or country.[68]

A 2010 study using social security income data for the United States since 1937 and Gini-based Shorrock's indices concludes that income mobility in the United States has had a complicated history, primarily due to the mass influx of women into the American labor force after World War II. Income inequality and income mobility trends have been different for men and women workers between 1937 and the 2000s. When men and women are considered together, the Gini coefficient-based Shorrocks index trends imply long-term income inequality has been substantially reduced among all workers, in recent decades for the United States.[68] Other scholars, using just 1990s data or other short periods have come to different conclusions.[69] For example, Sastre and Ayala conclude from their study of income Gini coefficient data between 1993 and 1998 for six developed economies that France had the least income mobility, Italy the highest, and the United States and Germany intermediate levels of income mobility over those five years.[70]