Atkinson index: Difference between revisions

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The index can be turned into a [[Normative economics|normative]] measure by imposing a [[coefficient]] <math>\varepsilon</math> to weight incomes. Greater weight can be placed on changes in a given portion of the income distribution by choosing <math>\varepsilon</math>, the level of "inequality aversion", appropriately. The Atkinson index becomes more sensitive to changes at the lower end of the income distribution as <math>\varepsilon</math> increases. Conversely, as the level of inequality aversion falls (that is, as <math>\varepsilon</math> approaches 0) the Atkinson becomes less sensitive to changes in the lower end of the distribution. The Atkinson index is for no value of <math>\varepsilon</math> highly sensitive to top incomes because of the common restriction that <math>\varepsilon</math> is nonnegative.<ref> The Atkinson index is related to the generalized entropy (GE) class of inequality indexes by <math>\epsilon=1-\alpha</math> - i.e an Atkinson index with high inequality aversion is derived from a GE index with small <math>\alpha</math>. GE indexes with large <math>\alpha</math> are sensitive to the existence of large top incomes but the corresponding Atkinson index would have negative <math>\varepsilon</math>. For a hypothetical Atkinson index with <math>\epsilon</math> being negative, the implied social utility function would be convex in income, and the Atkinson index would be nonpositive.</ref>

The Atkinson <math>\varepsilon</math> 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 percentageshare byof which meancurrent income by which could be reducedsacrificed, without reducing social welfare, if perfect inequality were instated. For <math>\varepsilon=0</math>, (no aversion to inequality), 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 to mean income, and the Atkinson index is zero.

For <math>\varepsilon=\infty</math> (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. 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<math>\varepsilon=0</math>.

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 <math>\varepsilon</math> parameter. Under the utilitarian ethical standard and some restrictive assumptions (a homogeneous population and [[constant elasticity of substitution]] utility), <math>\varepsilon</math> is equal to the income elasticity of marginal utility of income.