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# Atkinson index

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### Atkinson index

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.[1]

## Definition

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 \varepsilon parameter is often called the "income aversion parameter", since it quantifies the amount of social utility that is assumed to be gained from complete redistribution of resources. For \varepsilon=0, (no aversion to inequality) it is assumed that no social utility is gained by complete redistribution and the Atkinson index (A_\varepsilon) is zero. For \varepsilon=\infty (infinite aversion to inequality), it is assumed that infinite social utility is gained by complete redistribution in which case A_\varepsilon=1. The Atkinson index (A_\varepsilon) 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. Based on one's value judgement concerning the social utility of complete redistribution, as embodied in the \varepsilon parameter, different income distributions may be compared by calculating the Atkinson index at that \varepsilon value, with lower values of A_\varepsilon indicating lower social utility to be gained, higher values indicating more. Lower values of A_\varepsilon thus indicate a more equal distribution than higher values, given a particular degree of inequality aversion.

The Atkinson index is defined as:

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 \epsilon \neq 1 \\ 1-\frac{1}{\mu}\left(\prod_{i=1}^{N}y_{i}\right)^{1/N} & \mbox{for}\ \varepsilon=1, \end{cases}

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

Atkinson index relies on the following axioms:

1. The index is symmetric in its arguments: A_\varepsilon(y_1,\ldots,y_N)=A_\varepsilon(y_{\sigma(1)},\ldots,y_{\sigma(N)}) for any permutation \sigma.
2. The index is non-negative, and is equal to zero only if all incomes are the same: A_\varepsilon(y_1,\ldots,y_N) = 0 iff y_i = \mu for all i.
3. The index satisfies the principle of transfers: if a transfer \Delta>0 is made from an individual with income y_i to another one with income y_j such that 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: 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: A_\varepsilon(y_1,\ldots,y_N) = A_\varepsilon( ky_1,\ldots,ky_N) for any k>0.
6. The index is subgroup decomposable.[2] 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:
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_{g,N_g}) + A_\varepsilon(\mu_1, \ldots, \mu_G)

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

## Footnotes

1. ^ inter alia "Income, Poverty, and Health Insurance Coverage in the United States: 2010", U.S. Census Bureau, 2011, p.10
2. ^ Shorrocks, AF (1980). The class of additively decomposable inequality indices. Econometrica, 48 (3), 613–625, doi:10.2307/1913126

## References

• 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, from World Institute for Development Economics Research
• Income Inequality, 1947–1998, from United States Census Bureau.