The measure is useful in determining which end of the distribution contributed most to the observed inequality.
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).
The index can be turned into a normative measure by imposing a coefficient
Greater weight can be placed on changes in a given portion of the income distribution by choosing
The Atkinson index becomes more sensitive to changes at the lower end of the income distribution as
Conversely, as the level of inequality aversion falls (that is, as
approaches 0) the Atkinson becomes less sensitive to changes in the lower end of the distribution.
highly sensitive to top incomes because of the common restriction that
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.
(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.
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
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
Under the utilitarian ethical standard and some restrictive assumptions (a homogeneous population and constant elasticity of substitution utility),
is equivalent (under a monotonic rescaling) to a generalized entropy index with parameter
The formula for deriving an Atkinson index with inequality aversion parameter
The Atkinson index satisfies the following properties: Software: