Generalized entropy index

The formula for general entropy for real values of $\backslash alpha$ is:

$$GE(\backslash alpha)\; =\; \backslash begin\{cases\}$$

\frac{1}{N \alpha (\alpha-1)}\sum_{i=1}^N\left[\left(\frac{y_i}{\overline{y}}\right)^\alpha - 1\right],& \alpha \ne 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), $y\_i$ is the income for case i and $\backslash alpha$ is a parameter which regulates the weight given to distances between incomes at different parts of the income distribution. For large $\backslash alpha$ the index is especially sensitive to the existence of large incomes, whereas for small $\backslash alpha$ the index is especially sensitive to the existence of small incomes.

The GE index satisfies the following properties:

1. The index is symmetric in its arguments: $GE(\backslash alpha;\; y\_1,\backslash ldots,y\_N)=GE(\backslash alpha;\; y\_\{\backslash sigma(1)\},\backslash ldots,y\_\{\backslash sigma(N)\})$ for any permutation $\backslash sigma$.
2. The index is non-negative, and is equal to zero only if all incomes are the same: $GE(\backslash alpha;\; y\_1,\backslash ldots,y\_N)\; =\; 0$ iff $y\_i\; =\; \backslash mu$ for all $i$.
3. The index satisfies the principle of transfers: if a transfer $\backslash Delta>0$ is made from an individual with income $y\_i$ to another one with income $y\_j$ such that $y\_i\; -\; \backslash Delta\; >\; y\_j\; +\; \backslash Delta$, 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: $GE(\backslash alpha;\; \backslash \{y\_1,\backslash ldots,y\_N\backslash \},\backslash ldots,\backslash \{y\_1,\backslash ldots,y\_N\backslash \})=GE(\backslash alpha;\; y\_1,\backslash ldots,y\_N)$
5. The index satisfies mean independence, or income homogeneity, axiom: if all incomes are multiplied by a positive constant, the inequality remains the same: $GE(\backslash alpha;\; y\_1,\backslash ldots,y\_N)\; =\; GE(\backslash alpha;\; ky\_1,\backslash ldots,ky\_N)$ for any $k>0$.
6. The GE indices are the only additively decomposable inequality indices. 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:

::: $$

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 $g$ indexes groups, $i$, individuals within groups, $\backslash mu\_g$ is the mean income in group $g$, and the weights $w\_g$ depend on $\backslash mu\_g,\; \backslash mu,\; N$ and $N\_g$. The class of the additively-decomposable inequality indices is very restrictive. Many popular indices, including Gini index, do not satisfy this property.{{Cite journal |last=STEPHEN |first=JENKINS |title=CALCULATING INCOME DISTRIBUTION INDICES FROM MICRO-DATA |url=https://pages.uoregon.edu/plambert/900/28stephenNTJ.pdf |journal=National Tax Journal |publisher=University of Oregon |volume=}}

An Atkinson index for any inequality aversion parameter can be derived from a generalized entropy index under the restriction that $\backslash epsilon=1-\backslash alpha$ - i.e. an Atkinson index with high inequality aversion is derived from a GE index with small $\backslash alpha$.

The formula for deriving an Atkinson index with inequality aversion parameter $\backslash epsilon$ under the restriction $\backslash epsilon\; =\; 1-\backslash alpha$ is given by:

$$A=1-[\backslash epsilon(\backslash epsilon-1)GE(\backslash alpha)\; +\; 1]^\{(1/(1-\backslash epsilon))\}\; \backslash qquad\; \backslash epsilon\backslash ne1$$

$$A=\; 1-e^\{-GE(\backslash alpha)\}\; \backslash qquad\; \backslash epsilon=1$$

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.

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

{{Reflist}}

{{DEFAULTSORT:Generalized Entropy Index}}

Category:Income inequality metrics

Category:Information theory