CPS-UIC Math Forum Gini Index

Integrals and Equity

http://www.rethinkingschools.org/archive/19_03/inte193.shtml

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Measuring inequality of distribution of income by the [Corrado] Gini Index

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Lorenz Curve: Graph of Cumulative Percentage vs. Cumulative Percentage

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Gini Index =¹ 100 × (Gini Coefficient)

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Simple Example: If the population is divided into two groups - the rich 20% and the poor 100 − 20 = 80% and the rich control 90% of the income, the Gini Index is g = 90 −20 = 70. See Fig.2 in http://www.cr1.dircon.co.uk/pdffiles/Lorenz.pdf Thus the Gini Index is the incremental advantage of the rich population.

Gini Coefficient

http://en.wikipedia.org/wiki/Gini_coefficient

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Calculation: The Gini coefficient is calculated 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 underneath the Lorenz curve is B, then the Gini coefficient is A/(A+B).

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Advantages and Disadvantages of the Gini coefficient as a measure of inequality

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Advantages: Anonymity, Scale independence, Population independence

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Disadvantages: Units of Measurement, Households vs. Individuals, Definition of Income.

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Policy Considerations: FRBSF: Economic Letter - Inequality in the United States

http://www.frbsf.org/econrsrch/wklyltr/el97-03.html

Simple Example

Simple Example: If the population is divided into two groups - the rich proportion r and the poor proportion 1− r and the rich control an r + g proportion of the the total income, the Gini Coefficient is g = (r + g) − r. If there was complete equity , the rich would control r and g represents the incremental advantage of the rich.

In the figure, A = (1,1), q = GH = CD, and

Area ∆OCA = Area ∆OCD + Area ∆DCA = 1 2 CD ·OB + 1 2 BF ·OB = 1 2 CD ·OF = 1 2 q,

or

Area ∆OCA Area ∆OFA = q.

Footnotes: