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Atkinson index

The index can be turned into a normative measure by imposing a coefficient ${\displaystyle \varepsilon }$ to weight incomes. Greater weight can be placed on changes in a given portion of the income distribution by choosing ${\displaystyle \varepsilon }$, the level of "inequality aversion", appropriately. The Atkinson index becomes more sensitive to changes at the lower end of the income distribution as ${\displaystyle \varepsilon }$ increases. Conversely, as the level of inequality aversion falls (that is, as ${\displaystyle \varepsilon }$ approaches 0) the Atkinson becomes less sensitive to changes in the lower end of the distribution. The Atkinson index is for no value of ${\displaystyle \varepsilon }$ highly sensitive to top incomes because of the common restriction that ${\displaystyle \varepsilon }$ is nonnegative.^[2]

The Atkinson ${\displaystyle \varepsilon }$ 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. Thus the Atkinson index gives the share of current income which could be sacrificed, without reducing social welfare, if perfect inequality were instated. For ${\displaystyle \varepsilon =0}$, (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.

For ${\displaystyle \varepsilon =+\infty }$ (infinite aversion to inequality) the marginal social welfare of income of the poorest individual is infinitely larger than any even slightly richer individual, and the Atkinson social welfare function is equal to the smallest income in the sample. In this case, the Atkinson index is equal to mean income minus the smallest income, divided by mean income. As in large typical income distributions incomes of zero or near zero are common, the Atkinson index will tend to be one or very close to one for very large ${\displaystyle \varepsilon }$.

The Atkinson index then varies between 0 and 1 and is a measure of the amount of social utility to be gained by complete redistribution of a given income distribution, for a given ${\displaystyle \varepsilon }$ parameter. Under the utilitarian ethical standard and some restrictive assumptions (a homogeneous population and constant elasticity of substitution utility), ${\displaystyle \varepsilon }$ is equal to the income elasticity of marginal utility of income.

The Atkinson index is defined as:

${\displaystyle A_{\varepsilon }(y_{1},\ldots ,y_{N})={\begin{cases}1-{\frac {1}{\mu }}\left({\frac {1}{N}}\sum _{i=1}^{N}y_{i}^{1-\varepsilon }\right)^{1/(1-\varepsilon )}&{\mbox{for}}\ 0\leq \varepsilon \neq 1\\1-{\frac {1}{\mu }}\left(\prod _{i=1}^{N}y_{i}\right)^{1/N}&{\mbox{for}}\ \varepsilon =1\\1-{\frac {1}{\mu }}\min \left(y_{1},...,y_{N}\right)&{\mbox{for}}\ \varepsilon =+\infty \end{cases}}}$

where ${\displaystyle y_{i}}$ is individual income (i = 1, 2, ..., N) and ${\displaystyle \mu }$ 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).

Atkinson index satisfies the following properties:

1. The index is symmetric in its arguments: ${\displaystyle A_{\varepsilon }(y_{1},\ldots ,y_{N})=A_{\varepsilon }(y_{\sigma ($1)},\ldots ,y_{\sigma (N)})} for any permutation ${\displaystyle \sigma }$.
2. The index is non-negative, and is equal to zero only if all incomes are the same: ${\displaystyle A_{\varepsilon }(y_{1},\ldots ,y_{N})=0}$ iff ${\displaystyle y_{i}=\mu }$ for all ${\displaystyle i}$.
3. The index satisfies the principle of transfers: if a transfer ${\displaystyle \Delta >0}$ is made from an individual with income ${\displaystyle y_{i}}$ to another one with income ${\displaystyle y_{j}}$ such that ${\displaystyle y_{i}-\Delta >y_{j}+\Delta }$, then the inequality index cannot increase.
4. The index satisfies population replication axiom: if a new population is formed by replicating the existing population an arbitrary number of times, the inequality remains the same: ${\displaystyle A_{\varepsilon }(\{y_{1},\ldots ,y_{N}\},\ldots ,\{y_{1},\ldots ,y_{N}\})=A_{\varepsilon }(y_{1},\ldots ,y_{N})}$
5. The index satisfies mean independence, or income homogeneity, axiom: if all incomes are multiplied by a positive constant, the inequality remains the same: ${\displaystyle A_{\varepsilon }(y_{1},\ldots ,y_{N})=A_{\varepsilon }(ky_{1},\ldots ,ky_{N})}$ for any ${\displaystyle k>0}$.
6. The index is subgroup decomposable.^[3] This means that overall inequality in the population can be computed as the sum of the corresponding Atkinson indices within each group, and the Atkinson index of the group mean incomes:

${\displaystyle A_{\varepsilon }(y_{gi}:g=1,\ldots ,G,i=1,\ldots ,N_{g})=\sum _{g=1}^{G}w_{g}A_{\varepsilon }(y_{g1},\ldots ,y_{gN_{g}})+A_{\varepsilon }(\mu _{1},\ldots ,\mu _{G})}$

where ${\displaystyle g}$ indexes groups, ${\displaystyle i}$, individuals within groups, ${\displaystyle \mu _{g}}$ is the mean income in group ${\displaystyle g}$, and the weights ${\displaystyle w_{g}}$ depend on ${\displaystyle \mu _{g},\mu ,N}$ and ${\displaystyle N_{g}}$. The class of the subgroup-decomposable inequality indices is very restrictive. Many popular indices, including Gini index, do not satisfy this property.

• Income inequality metrics
• Generalized entropy index
• Gini index

^ The Atkinson index is related to the generalized entropy (GE) class of inequality indexes by ${\displaystyle \epsilon =1-\alpha }$ - i.e an Atkinson index with high inequality aversion is derived from a GE index with small ${\displaystyle \alpha }$. GE indexes with large ${\displaystyle \alpha }$ are sensitive to the existence of large top incomes but the corresponding Atkinson index would have negative ${\displaystyle \varepsilon }$. For a hypothetical Atkinson index with ${\displaystyle \epsilon }$ being negative, the implied social utility function would be convex in income, and the Atkinson index would be nonpositive.

• Atkinson, AB (1970) On the measurement of inequality. Journal of Economic Theory, 2 (3), pp. 244–263, doi:10.1016/0022-0531(70)90039-6. 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, ISBN 978-0-7190-5732-8.
• Sen A, Foster JE (1997) On Economic Inequality, Oxford University Press, ISBN 978-0-19-828193-1. (Python script for a selection of formulas in the book)
• World Income Inequality Database Archived 2011-03-13 at the Wayback Machine, from World Institute for Development Economics Research
• Income Inequality, 1947–1998, from United States Census Bureau.

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.
• AMATLAB Inequality Package Archived 2008-10-04 at the Wayback Machine, 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.