South Africa Inequality: Generalized Entropy Measure

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 ${\displaystyle \alpha }$ is:

$${\displaystyle GE(\alpha )={\begin{cases}{\frac {1}{N\alpha (\alpha -1)}}\sum _{i=1}^{N}\left[\left({\frac {y_{i}}{\overline {y}}}\right)^{\alpha }-1\right],&\alpha \neq 0,1,\\{\frac {1}{N}}\sum _{i=1}^{N}{\frac {y_{i}}{\overline {y}}}\ln {\frac {y_{i}}{\overline {y}}},&\alpha =1,\\-{\frac {1}{N}}\sum _{i=1}^{N}\ln {\frac {y_{i}}{\overline {y}}},&\alpha =0.\end{cases}}}$$

where N is the number of cases (e.g., households or families), ${\displaystyle y_{i}}$ is the income for case i and ${\displaystyle \alpha }$ is a parameter which regulates the weight given to distances between incomes at different parts of the income distribution. For large ${\displaystyle \alpha }$ the index is especially sensitive to the existence of large incomes, whereas for small ${\displaystyle \alpha }$ 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 ${\displaystyle \epsilon =1-\alpha }$ - i.e. an Atkinson index with high inequality aversion is derived from a GE index with small ${\displaystyle \alpha }$. 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 ${\displaystyle \epsilon }$ under the restriction ${\displaystyle \epsilon =1-\alpha }$ is given by:

$${\displaystyle A=1-[\epsilon (\epsilon -1)GE+1]^{(1/(1-\epsilon ))}\qquad \epsilon \neq 1}$$

$${\displaystyle A=1-e^{-GE}\qquad \epsilon =1}$$

Note that the generalized entropy index has several income inequality metrics as special cases. For example, GE(0) is the mean log deviation, GE(1) is the Theil index, and GE(2) is half the squared coefficient of variation.

See also

• Atkinson index
• Gini coefficient
• Hoover index (a.k.a. Robin Hood index)
• Income inequality metrics
• Lorenz curve
• Rényi entropy
• Suits index
• Theil index

References