# 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 index is defined as:

$A_\varepsilon(y_1,\ldots,y_N)= \begin{cases} 1-\frac{1}{\mu}\left(\frac{1}{N}\sum_{i=1}^{N}y_{i}^{1-\varepsilon}\right)^{1/(1-\varepsilon)} & \mbox{for}\ \varepsilon \in \left[0,1\right)\cup\left(1,+\infty\right) \\ 1-\frac{1}{\mu}\left(\prod_{i=1}^{N}y_{i}\right)^{1/N} & \mbox{for}\ \varepsilon=1, \end{cases}$

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

The entropy measure developed by Atkinson[2] can be computed from a "normalized Theil index".[3] This, however, only applies to the Theil index ${I_1}$, which is derived from the "generalized entropy class"[4] with ${\epsilon} = 1$. The Atkinson index is computed using the function $1-e^{- T}$.

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.[5] 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.