# Robin Hood index

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Robin Hood index

The Robin Hood index, also known as the Hoover index, is a measure of income inequality. It is equal to the portion of the total community income that would have to be redistributed (taken by force from the richer half of the population and given to the poorer half) for there to be perfect equality.

It can be graphically represented as the longest vertical distance between the Lorenz curve, or the cumulative portion of the total income held below a certain income percentile, and the 45 degree line representing perfect equality.

Mathematically, the Robin Hood index for Lorenz curve $scriptstyle L\left(x\right)$ is $scriptstyle max \left(x - L\left(x\right)\right)$.

The Robin Hood index is typically used in applications related to socio-economic class (SES) and health. It is conceptually the simplest inequality index used in econometrics. A better known inequality measure is the Gini coefficient which is based on the Lorenz curve.

Computation

For the formula, a notation [The notation using E and A follows the notation of a small calculus published by Lionnel Maugis: "Inequality Measures in Mathematical Programming for the Air Traffic Flow Management Problem with En-Route Capacities" (für IFORS 96), 1996] is used, where the amount $N$ of quantiles only appears as upper border of summations. Thus, inequities can be computed for quantiles with different widths $A$. For example, $E_i$ could be the income in the quantile #i and $A_i$ could be the amount (absolute or relative) of earners in the quantile #i. $E_ ext\left\{total\right\}$ then would be the sum of incomes of all $N$ quantiles and $A_ ext\left\{total\right\}$ would be the sum of the income earners in all $N$ quantiles.

Computation of the Robin Hood index $H$:

: $H = \left\{frac\left\{1\right\}\left\{2 sum_\left\{i=1\right\}^N color\left\{Blue\right\} left| color\left\{Black\right\} \left\{frac$E}_i}E}_ ext{total} - {fracA}_i}A}_ ext{total} color{Blue} ight| color{Black}.

For comparison [For an explanation of the comparison with Henri Theil's index see: Theil index] , here also the computation of the symmetrized Theil index $T_s$ is given:

: $T_s = \left\{frac\left\{1\right\}\left\{2$ sum_{i=1}^N color{Blue} ln{fracE}_i}A}_i left( color{Black} {fracE}_i}E}_ ext{total} - {fracA}_i}A}_ ext{total} color{Blue} ight) color{Black}.

References

Literature

* Edgar Malone HOOVER jr.: "The Measurement of Industrial Localization", Review of Economics and Statistics, 1936, Vol. 18, No. 162-171
* Edgar Malone HOOVER jr.: "An Introduction to Regional Economics", 1984, ISBN 0075544407
* Philip B. COULTER: "Measuring Inequality", 1989, ISBN 0-8133-7726-9 (This book describes about 50 different inequality measures - it's a good guide, but it contains some mistakes, so watch out.)
* Robert SAPOLSKY: "Sick of Poverty", Scientific American, December 2005

* Free Inequality Calculator: [http://www.poorcity.richcity.org/calculator Online] and [http://luaforge.net/project/showfiles.php?group_id=49 downloadable scripts] (Python and Lua) for Atkinson, Gini, and Hoover inequalities

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