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.
Definition
The index can be turned into a normative measure by imposing a coefficient
The Atkinson
The Atkinson index is defined as:
where
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).
Atkinson index relies on the following axioms:
- The index is symmetric in its arguments:
A ε ( y 1 , … , y N ) = A ε ( y σ ( 1 ) , … , y σ ( N ) ) for any permutationσ . - The index is non-negative, and is equal to zero only if all incomes are the same:
A ε ( y 1 , … , y N ) = 0 iffy i = μ for alli . - The index satisfies the principle of transfers: if a transfer
Δ > 0 is made from an individual with incomey i y j y i − Δ > y j + Δ , 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:
A ε ( { y 1 , … , y N } , … , { y 1 , … , y N } ) = A ε ( y 1 , … , y N ) - The index satisfies mean independence, or income homogeneity, axiom: if all incomes are multiplied by a positive constant, the inequality remains the same:
A ε ( y 1 , … , y N ) = A ε ( k y 1 , … , k y N ) for anyk > 0 . - 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: