Atkinson index

The Atkinson index (also known as the Atkinson measure) is a measure of economic income inequality developed by Anthony Barnes Atkinson. The distinguishing feature of the Atkinson index is its ability to gauge movements in different segments of the income distribution.

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=

\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) \\ 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.

An entropy measure from Atkinson can be computed from the Theil index, T, (example without using $$\varepsilon$$)


 * $$A = 1 - e^{-T}.\,$$