Generalized entropy index

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.

Properties

The GE index satisfies the following properties:

1. The index is symmetric in its arguments: ${\displaystyle GE(\alpha ;y_{1},\ldots ,y_{N})=GE(\alpha ;y_{\sigma (1)},\ldots ,y_{\sigma (N)})}$ for any permutation ${\displaystyle \sigma }$.
2. The index is non-negative, and is equal to zero only if all incomes are the same: ${\displaystyle GE(\alpha ;y_{1},\ldots ,y_{N})=0}$ iff ${\displaystyle y_{i}=\mu }$ for all ${\displaystyle i}$.
3. The index satisfies the principle of transfers: if a transfer ${\displaystyle \Delta >0}$ is made from an individual with income ${\displaystyle y_{i}}$ to another one with income ${\displaystyle y_{j}}$ such that ${\displaystyle y_{i}-\Delta >y_{j}+\Delta }$, then the inequality index cannot increase.
4. 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: ${\displaystyle GE(\alpha ;\{y_{1},\ldots ,y_{N}\},\ldots ,\{y_{1},\ldots ,y_{N}\})=GE(\alpha ;y_{1},\ldots ,y_{N})}$
5. The index satisfies mean independence, or income homogeneity, axiom: if all incomes are multiplied by a positive constant, the inequality remains the same: ${\displaystyle GE(\alpha ;y_{1},\ldots ,y_{N})=GE(\alpha ;ky_{1},\ldots ,ky_{N})}$ for any ${\displaystyle k>0}$.
6. The GE indices are the only additively decomposable inequality indices.^[1] This means that overall inequality in the population can be computed as the sum of the corresponding GE indices within each group, and the GE index of the group mean incomes:

${\displaystyle GE(\alpha ;y_{gi}:g=1,\ldots ,G,i=1,\ldots ,N_{g})=\sum _{g=1}^{G}w_{g}GE(\alpha ;y_{g1},\ldots ,y_{gN_{g}})+GE(\alpha ;\mu _{1},\ldots ,\mu _{G})}$

where ${\displaystyle g}$ indexes groups, ${\displaystyle i}$, individuals within groups, ${\displaystyle \mu _{g}}$ is the mean income in group ${\displaystyle g}$, and the weights ${\displaystyle w_{g}}$ depend on ${\displaystyle \mu _{g},\mu ,N}$ and ${\displaystyle N_{g}}$. The class of the additively-decomposable inequality indices is very restrictive. Many popular indices, including Gini index, do not satisfy this property.^[1][3]

Relationship to other indices

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 ${\displaystyle \alpha }$.

The formula for deriving an Atkinson index with inequality aversion parameter ${\displaystyle \epsilon }$ under the restriction ${\displaystyle \epsilon =1-\alpha }$ is given by: $${\displaystyle A=1-[\epsilon (\epsilon -1)GE(\alpha )+1]^{(1/(1-\epsilon ))}\qquad \epsilon \neq 1}$$ $${\displaystyle A=1-e^{-GE(\alpha )}\qquad \epsilon =1}$$

Note that the generalized entropy index has several income inequality metrics as special cases. For example, GE(0) is the mean log deviation a.k.a. Theil L index, GE(1) is the Theil T 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

1. Shorrocks, A. F. (1980). "The Class of Additively Decomposable Inequality Measures". Econometrica. 48 (3): 613–625. doi:10.2307/1913126. JSTOR 1913126.
2. Pielou, E.C. (December 1966). "The measurement of diversity in different types of biological collections". Journal of Theoretical Biology. 13: 131–144. Bibcode:1966JThBi..13..131P. doi:10.1016/0022-5193(66)90013-0.
3. STEPHEN, JENKINS. "CALCULATING INCOME DISTRIBUTION INDICES FROM MICRO-DATA" (PDF). National Tax Journal. University of Oregon.