# Atkinson index: Wikis

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# Encyclopedia

The Atkinson index (also known as the Atkinson measure) is a measure of economic income inequality developed by Anthony Barnes Atkinson. The distinguishing feature of the Atkinson index is its ability to gauge movements in different segments of the income distribution.

The index can be turned into a normative measure by imposing a coefficient $\varepsilon$ to weight incomes. Greater weight can be placed on changes in a given portion of the income distribution by choosing $\varepsilon$, the level of "inequality aversion", appropriately. The Atkinson index becomes more sensitive to changes at the lower end of the income distribution as $\varepsilon$ approaches 1. Conversely, as the level of inequality aversion falls (that is, as $\varepsilon$ approaches 0) the Atkinson becomes more sensitive to changes in the upper end of the income distribution.

The Atkinson index is defined as:

$A= \begin{cases} 1-\frac{1}{\mu}\left(\frac{1}{N}\sum_{i=1}^{N}y_{i}^{1-\varepsilon}\right)^{1/(1-\varepsilon)} & \mbox{for}\ \varepsilon \in \left[0,1\right) \ 1-\frac{1}{\mu}\left(\prod_{i=1}^{N}y_{i}\right)^{1/N} & \mbox{for}\ \varepsilon=1, \end{cases}$

where yi is individual income (i = 1, 2, ..., N) and μ is the mean income.

The entropy measure developed by Atkinson[1] can be computed from a "normalized Theil index"[2]. This, however, only applies to the Theil index I1, which is derived from the "generalized entropy class"[3] with ε = 1. The Atkinson index is computed using the function 1 − e T.

## Footnotes

1. ^ Anthony Barnes Atkinson developed various metrics. The Atkinson index which is related to the Theil index has been described by Lionnel Maugis in Inequality Measures in Mathematical Programming for the Air Traffic Flow Management Problem with En-Route Capacities (published on occasion of IFORS 96), 1996
2. ^ Juana Domínguez-Domínguez, José Javier Núñez-Velázquez: The Evolution of Economic Inequality in the EU Countries During the Nineties. 2005
3. ^ James E. Foster in annexe A.4.1 (p.142) of: Amartya Sen, On Economic Inequality, 1973/1997

## References

• Paul D. Allison, Measures of Inequality, American Sociological Review, 43 (December 1978), pp. 865-880, presents a technical discussion of the Atkinson measure's properties.
• Amartya Sen, James E. Foster: On Economic Inequality, Oxford University Press, 1996 (Python script for a selection of formulas in the book)
• Income Inequality, 1947-1998, from the United States Census Office.

• Software:
• Free Online Calculator computes the Gini Coefficient, plots the Lorenz curve, and computes many other measures of concentration for any dataset
• Free Calculator: Online and downloadable scripts (Python and Lua) for Atkinson, Gini, and Hoover inequalities
• Users of the R data analysis software can install the "ineq" package which allows for computation of a variety of inequality indices including Gini, Atkinson, Theil.
• A MATLAB Inequality Package, including code for computing Gini, Atkinson, Theil indexes and for plotting the Lorenz Curve. Many examples are available.