Robin Hood Index

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The Robin Hood Index (based on Robin Hood ) - also Ricci Protection Index (after the Italian economist Umberto Ricci [1879–1946] and the American economist Robert R. Schutz [* 1915]), maximum leveling rate ( after Lindahl ) (according to Erik Robert Lindahl [1891–1960]) or Piètra-Index or Pietra-Index ( Gaetano Piètra [1879–1961], Italian statistician) - is a measure of income inequality ( unequal distribution measure ) over geographical areas and is of the Lorenz curve derived.

definition

The blue line shows the difference ${\displaystyle x-L(x)}$

Mathematically of the Lorenz curve related Robin Hood index equal to the value , which is the longest Lorenz curves tendon , the longest vertical, between the Lorenz curve of the perfect equality (z. B. for income) and the measured Lorenz curve can be drawn. Theoretically, the height of the rectangle that surrounds the Lorenz curve corresponds to the largest possible maximum. Consequently, the quotient of the determined vertical and the height of the rectangle is a measure between 0% and 100%. As a rule, however, the vertical direction of the Lorenz curve is already normalized to 1 or 100%, so that the formation of the quotient can be omitted. ${\displaystyle L(x)}$ ${\displaystyle \!\ \max[x-L(x)]}$

The Robin Hood Index is conceptually one of the simplest measures of inequality . It is used in econometrics as well as sociology . The name comes from the fact that it is equal to the amount of total national income that would have to be redistributed in society (taken from the richer half of the population and given to the poorer half) in order to create financial equality. If the existing income is in fact, with the total amount normalized to 1, then the members of the poorer half lack the amount , with the index running from 1 to the position from which more than is earned. The total amount to be redistributed is therefore: ${\displaystyle x_{1}\leq x_{2}\leq \dotsb \leq x_{n}}$ ${\displaystyle {\tfrac {1}{n}}-x_{i}}$ ${\displaystyle i}$ ${\displaystyle j}$ ${\displaystyle {\tfrac {1}{n}}}$

${\displaystyle \sum _{i=1}^{j}\left({\frac {1}{n}}-x_{i}\right)={\frac {j}{n}}-\sum _{i=1}^{j}x_{i}=x-L(x)}$ With ${\displaystyle x={\frac {j}{n}}\,,}$

and the value is at its maximum according to the construction, because if you increase , only negative summands are added to the sum on the left. ${\displaystyle x-L(x)}$ ${\displaystyle j}$

See also

• Gini coefficient
• Theil index
• Hoover inequality
• Atkinson measure
• Suits index
• Generalized entropy
• Diversity index

literature

• Horst Rinne: Pocket book of statistics. 4th edition Harri Deutsch Verlag, Frankfurt a. M. 2008. ISBN 978-3-8171-1827-4 . 1060 p.
• Peter P. Eckstein: Repetition Statistics: Descriptive Statistics - Stochastics - Inductive Statistics. 6th edition Gabler Verlag, Wiesbaden 2006. ISBN 978-3-8349-0464-5 . 388 pp.

Web links

• Bruce P. Kennedy, Ichiro Kawachi, Deborah Prothrow-Stith: Income distribution and mortality: cross sectional ecological study of the Robin Hood index in the United States , BMJ , 1996