The Atkinson index (also known as the Atkinson measure or Atkinson inequality measure) is a measure of income inequality developed by British economist Anthony Barnes Atkinson. The measure is useful in determining which end of the distribution contributed most to the observed inequality.

Definition

The Atkinson index is defined as:

is individual income (i = 1, 2, ..., N is the mean income.

In other words, the Atkinson index is the complement to 1 of the ratio of the Hölder generalized mean of exponent 1−ε to the arithmetic mean of the incomes (where as usual the generalized mean of exponent 0 is interpreted as the geometric mean).

Interpretation

The index can be turned into a normative measure by imposing a coefficient is nonnegative.

parameter is often called the "inequality aversion parameter", since it regulates the sensitivity of the implied social welfare losses from inequality to income inequality as measured by some corresponding generalised entropy index. The Atkinson index is defined in reference to a corresponding social welfare function, where mean income multiplied by one minus the Atkinson index gives the welfare equivalent equally distributed income, (no aversion to inequality), the marginal social welfare from income is invariant to income, i.e. marginal increases in income produce as much social welfare whether they go to a poor or rich individual. In this case, the welfare equivalent equally distributed income is equal to mean income, and the Atkinson index is zero.

.

parameter. Under the utilitarian ethical standard and some restrictive assumptions (a homogeneous population and constant elasticity of substitution is equal to the income elasticity of marginal utility of income.

Relationship to generalized entropy index

is equivalent (under a monotonic rescaling) to a generalized entropy index

Properties

The Atkinson index satisfies the following properties:

1. .

2. .

3. The index satisfies the principle of transfers, then the inequality index cannot increase.

4. .

5. The index is (non-additively) subgroup decomposable and the corresponding generalized entropy index is additively subgroup decomposable. This means that overall inequality in the population can be computed as the sum of the corresponding GE indices within each group, and the GE index of the group mean incomes:

   1. . The class of the additively-decomposable inequality indices is very restrictive; in fact, only the generalized entropy indices are additively decomposable. Many popular indices, including Gini index, do not satisfy this property.

See also

• Income inequality metrics

• Generalized entropy index

• Gini index

Footnotes

References

• Atkinson, AB (1970) On the measurement of inequality. Journal of Economic Theory, 2 (3), pp.244–263, . The original paper proposing this inequality index.

• Allison PD (1978) Measures of Inequality, American Sociological Review, 43, pp.865–880. Presents a technical discussion of the Atkinson measure's properties. There is an error in the formula for the Atkinson index, which is corrected in Allison (1979).

• Allison, PD (1979) Reply to Jasso. American Sociological Review 44(5):870–72.

• Biewen M, Jenkins SP (2003). Estimation of Generalized Entropy and Atkinson Inequality Indices from Complex Survey Data. IZA Discussion Paper #763. Provides statistical inference for Atkinson indices.

• Lambert, P. (2002). Distribution and redistribution of income. 3rd edition, Manchester Univ Press, .

• Sen A, Foster JE (1997) On Economic Inequality, Oxford University Press, . (Python script for a selection of formulas in the book)

• World Income Inequality Database , from World Institute for Development Economics Research

• Income Inequality, 1947–1998, from United States Census Bureau.

External links

Software:

• 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.

• Stata inequality packages: ineqdeco to decompose inequality by groups; svygei and svyatk to compute design-consistent variances for the generalized entropy and Atkinson indices; glcurve to obtain generalized Lorenz curve. You can type ssc install ineqdeco etc. in Stata prompt to install these packages.