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 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 $$\scriptstyle L(x)$$ is $$\scriptstyle \max (x - L(x))$$.

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 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. $$E_\text{total}$$ then would be the sum of incomes of all $$N$$ quantiles and $$A_\text{total}$$ would be the sum of the income earners in all $$N$$ quantiles.

Computation of the Robin Hood index $$H$$:



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

For comparison, here also the computation of the symmetrized Theil index $$T_s$$ is given:



T_s = {\frac{1}{2}} \sum_{i=1}^N \color{OliveGreen} \ln{\frac{{E}_i}{{A}_i}} \color{Black} \left( {\frac{{E}_i}{{E}_\text{total}}} - {\frac{{A}_i}{{A}_\text{total}}} \right). $$

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.)
 * Robert SAPOLSKY: Sick of Poverty, Scientific American, December 2005