# 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 ${\displaystyle \varepsilon }$ 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 ${\displaystyle \varepsilon }$ approaches 1. Conversely, as the level of inequality aversion falls (that is, as ${\displaystyle \varepsilon }$ approaches 0) the Atkinson becomes more sensitive to changes in the upper end of the income distribution.

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

The Atkinson index is defined as:

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

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 relies on the following axioms:

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