The generalized entropy index has been proposed as a measure of
income inequality in a population.[1] It is derived from
information theory as a measure of
redundancy in data. In
information theory a measure of
redundancy can be interpreted as non-randomness or
data compression; thus this interpretation also applies to this index. In addition, interpretation of
biodiversity as entropy has also been proposed leading to uses of generalized entropy to quantify biodiversity.[2]
Formula
The formula for general entropy for real values of is:
where N is the number of cases (e.g., households or families), is the income for case i and is a parameter which regulates the weight given to distances between incomes at different parts of the
income distribution. For large the index is especially sensitive to the existence of large incomes, whereas for small the index is especially sensitive to the existence of small incomes.
An
Atkinson index for any inequality aversion parameter can be derived from a generalized entropy index under the restriction that - i.e. an Atkinson index with high inequality aversion is derived from a GE index with small . Moreover, it is the unique class of inequality measures that is a monotone transformation of the
Atkinson index and which is additive decomposable. Many popular indices, including
Gini index, do not satisfy additive decomposability.[1][3]
The formula for deriving an Atkinson index with inequality aversion parameter under the restriction is given by: