The Atkinson index 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.
Definition
The index can be turned into a normative measure by imposing a coefficient to weight incomes. Greater weight can be placed on changes in a given portion of the income distribution by choosing, the level of "inequality aversion", appropriately. The Atkinson index becomes more sensitive to changes at the lower end of the income distribution as increases. Conversely, as the level of inequality aversion falls the Atkinson becomes less sensitive to changes in the lower end of the distribution. The Atkinson index is for no value of highly sensitive to top incomes because of the common restriction that is nonnegative. The Atkinson 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 incomemultiplied 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. For,, the marginal social welfare from income is invariant to income, i.e. marginal increases in income produce as much social welfare whether they go to a poor or rich individual. In this case, the welfare equivalent equally distributed income is equal tomean income, and the Atkinson index is zero. For 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. In this case, the Atkinson index is equal to mean income minus the smallest income, divided by mean income. 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 parameter. Under the utilitarian ethical standard and some restrictive assumptions, is equal to the income elasticity of marginal utility of income. The Atkinson index is defined as: where is individual income and 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. Atkinson index relies on the following axioms:
The index is symmetric in its arguments: for any permutation.
The index is non-negative, and is equal to zero only if all incomes are the same: iff for all.
The index satisfies the principle of transfers: if a transfer is made from an individual with income to another one with income such that, then the inequality index cannot increase.
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:
The index satisfies mean independence, or income homogeneity, axiom: if all incomes are multiplied by a positive constant, the inequality remains the same: for any.
The index is subgroup decomposable. 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:
