Gini coefficient

The Gini coefficient is a measure of inequality of a distribution. It is defined as a ratio with values between 0 and 1: the numerator is the area between the Lorenz curve of the distribution and the uniform (perfect) distribution line; the denominator is the area under the uniform distribution line. It was developed by the Italian statistician Corrado Gini and published in his 1912 paper "Variabilità e mutabilità" ("Variability and Mutability"). The Gini index is the Gini coefficient expressed as a percentage, and is equal to the Gini coefficient multiplied by 100. (The Gini coefficient is equal to half of the relative mean difference.)

The Gini coefficient is often used as an income inequality metric. (See that article for discussion of the social aspects of income inequality.) Here, 0 corresponds to perfect income equality (i.e. everyone has the same income) and 1 corresponds to perfect income inequality (i.e. one person has all the income, while everyone else has zero income).

The Gini coefficient can also be used to measure wealth inequality. This use requires that no one have a negative net wealth. It is also commonly used for the measurement of the discriminatory power of rating systems in credit risk management.

Calculation

The Gini coefficient is defined as a ratio of the areas on the Lorenz curve diagram. If the area between the line of perfect equality and Lorenz curve is A, and the area under the Lorenz curve is B, then the Gini coefficient is A/(A+B). Since A+B = 0.5, the Gini coefficient, G = A/(.5) = 2A = 1-2B. If the Lorenz curve is represented by the function Y = L(X), the value of B can be found with integration and:

<math>G = 1 - 2\,\int_0^1 L(X) dX </math>

In some cases, this equation can be applied to calculate the Gini coefficient without direct reference to the Lorenz curve. For example:

• For a population with values y_i, i = 1 to n, that are indexed in non-decreasing order ( y_i ≤ y_i+1):

<math>G = \frac{1}{n}(n+1 - 2\,\frac{\Sigma_{i=1}^n \; (n+1-i)y_i}{\Sigma_{i=1}^n y_i})\,</math>

• For a discrete probability function f(y), where y_i, i = 1 to n, are the points with nonzero probabilities and which are indexed in increasing order ( y_i < y_i+1):

<math>G = 1 - \frac{\Sigma_{i=1}^n \; f(y_i)(S_{i-1}+S_i)}{S_n}</math>

where:

<math>S_i = \Sigma_{j=1}^i \; f(y_j)\,y_j\,</math> and <math>S_0 = 0\,</math>

• For a cumulative distribution function F(y) that is piecewise differentiable, has a mean μ, and is zero for all negative values of y:

<math>G = 1 - \frac{1}{\mu}\int_0^\infty (1-F(y))^2dy</math>

Since the Gini coefficient is half the relative mean difference, it can also be calculated using formulas for the relative mean difference.

For a random sample S consisting of values y_i, i = 1 to n, that are indexed in non-decreasing order ( y_i ≤ y_i+1), the statistic:

<math>G(S) = \frac{1}{n-1}(n+1 - 2\,\frac{\Sigma_{i=1}^n \; (n+1-i)y_i}{\Sigma_{i=1}^n y_i})\,</math>

is a consistent estimator of the population Gini coefficient, but is not, in general, unbiased. Like the relative mean difference, there does not exist a sample statistic that is in general an unbiased estimator of the population Gini coefficient. Confidence intervals for the population Gini coefficient can be calculated using bootstrap techniques.

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 by 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.

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:

<math>G_1 = 1 - \sum_{k=1}^{n} (X_{k} - X_{k-1}) (Y_{k} + Y_{k-1})</math>

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.

Income Gini coefficients in the world

A complete listing is in list of countries by income equality; the article economic inequality discusses the social and policy aspects of income and asset inequality.

While most developed European nations tend to have Gini coefficients between 0.24 and 0.36, the United States Gini coefficient is above 0.4, indicating that the United States has greater inequality. Using the Gini can help quantify differences in welfare and compensation policies and philosophies. However it should be borne in mind that the Gini coefficient can be misleading when used to make political comparisons between large and small countries (see criticisms section).

Correlation with per-capita GDP

Poor countries (those with low per-capita GDP) have Gini coefficients that fall over the whole range from low (0.25) to high (0.71), while rich countries have generally low Gini coefficient (under 0.40).

US income gini coefficients over time

Gini coefficients for the United States at various times, according to the US Census Bureau:

• 1970: 0.394
• 1980: 0.403
• 1990: 0.428
• 2000: 0.462
• 2005: 0.469^[1]

Some argue this rise corresponds to the lowering of the highest tax bracket, for example, from 70% in the 1960s to 35% by 2000. However, many other variables that could affect the Gini coefficient have changed during this period as well. For example, much technological progress has occurred, eliminating formerly middle-class factory jobs in favor of the service sector; additionally, the economy has shifted towards professions that require higher education.

Advantages of Gini Coefficient as a measure of inequality

• The Gini coefficient's main advantage is that it is a measure of inequality by means of a ratio analysis, rather than a variable unrepresentative of most of the population, such as per capita income or gross domestic product.

• It can be used to compare income distributions across different population sectors as well as countries, for example the Gini coefficient for urban areas differs from that of rural areas in many countries (though the United States' urban and rural Gini coefficients are nearly identical).

• It is sufficiently simple that it can be compared across countries and be easily interpreted. GDP statistics are often criticised as they do not represent changes for the whole population; the Gini coefficient demonstrates how income has changed for poor and rich. If the Gini coefficient is rising as well as GDP, poverty may not be improving for the majority of the population.

• The Gini coefficient can be used to indicate how the distribution of income has changed within a country over a period of time, thus it is possible to see if inequality is increasing or decreasing.

• The Gini coefficient satisfies four important principles:
  • Anonymity: it does not matter who the high and low earners are.
  • Scale independence: the Gini coefficient does not consider the size of the economy, the way it is measured, or whether it is a rich or poor country on average.
  • Population independence: it does not matter how large the population of the country is.
  • Transfer principle: if income (less than the difference), is transferred from a rich person to a poor person the resulting distribution is more equal.

Disadvantages of Gini Coefficient as a measure of inequality

• The Gini coefficient measured for a large economically diverse country will generally result in a much higher coefficient than each of its regions has individually. For this reason the scores calculated for individual countries within the EU are difficult to compare with the score of the entire US.

• Comparing income distributions among countries may be difficult because benefits systems may differ. For example, some countries give benefits in the form of money while others give food stamps, which may not be counted as income in the Lorenz curve and therefore not taken into account in the Gini coefficient.

• The measure will give different results when applied to individuals instead of households. When different populations are not measured with consistent definitions, comparison is not meaningful.

• The Lorenz curve may understate the actual amount of inequality if richer households are able to use income more efficiently than lower income households. From another point of view, measured inequality may be the result of more or less efficient use of household incomes.

• As for all statistics, there will be systematic and random errors in the data. The meaning of the Gini coefficient decreases as the data become less accurate. Also, countries may collect data differently, making it difficult to compare statistics between countries.

• Economies with similar incomes and Gini coefficients can still have very different income distributions. This is because the Lorenz curves can have different shapes and yet still yield the same Gini coefficient. As an extreme example, an economy where half the households have no income, and the other half share income equally has a Gini coefficient of ½; but an economy with complete income equality, except for one wealthy household that has half the total income, also has a Gini coefficient of ½.

• The Gini coefficient is used to measure income more than net worth, which can be misleading. For example, is Sweden egalitarian? Sweden has a low Gini coefficient for income distribution but a high Gini coefficient for wealth (0.1% of Swedish shareholders control 25% of all shares^{[citation needed]}).

• Too often only the Gini coefficient is quoted without describing the proportions of the quantiles used for measurement. As with other inequality coefficients, the Gini coefficient is influenced by the granularity of the measurements. For example, five 20% quantiles (low granularity) will yield a lower Gini coefficient than twenty 5% quantiles (high granularity) taken from the same distribution.

As one result of this criticism, additionally to or in competition with the Gini coefficient entropy measures are frequently used (e.g. the Atkinson and Theil indices). These measures attempt to compare the distribution of resources by intelligent players in the market with a maximum entropy random distribution, which would occur if these players acted like non-intelligent particles in a closed system following the laws of statistical physics.

Notes

1. ^ Note that the calculation of the index for the United States was changed in 1992, resulting in an upwards shift of about 0.02 in the coefficient. Comparisons before and after that period may be misleading. (Data from the US Census Bureau.)

References

• Anand, Sudhir (1983). Inequality and Poverty in Malaysia. New York: Oxford University Press. 
• Brown, Malcolm (1994). "Using Gini-Style Indices to Evaluate the Spatial Patterns of Health Practitioners: Theoretical Considerations and an Application Based on Alberta Data". Social Science Medicine 38: 1243-1256. 
• Chakravarty, S. R. (1990). Ethical Social Index Numbers. New York: Springer-Verlag. 
• Dixon, PM, Weiner J., Mitchell-Olds T, Woodley R. (1987). "Bootstrapping the Gini coefficient of inequality". Ecology 68: 1548-1551. 
• Dorfman, Robert (1979). "A Formula for the Gini Coefficient". The Review of Economics and Statistics 61: 146-149. 
• Gastwirth, Joseph L. (1972). "The Estimation of the Lorenz Curve and Gini Index". The Review of Economics and Statistics 54: 306-316. 
• Gini, Corrado (1912). "Variabilità e mutabilità" Reprinted in Memorie di metodologica statistica (Ed. Pizetti E, Salvemini, T). Rome: Libreria Eredi Virgilio Veschi (1955).
• Gini, Corrado (1921). "Measurement of Inequality and Incomes". The Economic Journal 31: 124-126. 
• Mills, Jeffrey A.; Zandvakili, Sourushe (1997). "Statistical Inference via Bootstrapping for Measures of Inequality". Journal of Applied Econometrics 12: 133-150. 
• Morgan, James (1962). "The Anatomy of Income Distribution". The Review of Economics and Statistics 44: 270-283. 
• Xu, Kuan (January, 2004). "How Has the Literature on Gini's Index Evolved in the Past 80 Years?". Department of Economics, Dalhousie University. Retrieved on June 1, 2006.

See also

• List of countries by income equality
• List of countries by Human Development Index
• Human Poverty Index
• Welfare economics
• Income inequality metrics
• ROC analysis
• Social welfare (political science)
• Pareto distribution
• Robin Hood index
• Relative mean difference

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