Generalized entropy index

A general formula for measuring income inequality. The Theil index and the mean log deviation are special cases of the formula. The formula is

$GE(\alpha \neq 0,1)={\frac {1}{N\alpha (\alpha -1)}}\sum _{i=1}^{N}\left[\left({\frac {y_{i}}{\overline {y}}}\right)^{\alpha }-1\right]$

$GE(\alpha =1)={\frac {1}{N}}\sum _{i=1}^{N}\left[{\frac {y_{i}}{\overline {y}}}ln\left({\frac {y_{i}}{\overline {y}}}\right)\right]$

$GE(\alpha =0)={\frac {1}{N}}\sum _{i=1}^{N}ln\left({\frac {y_{i}}{\overline {y}}}\right)$

where $y_{i}$ is the income for individual or group i that is part of N and $\alpha$ is the weight given to distances between incomes at different parts of the income distribution and can take any real value. For lower values of $\alpha$, GE is more sensitive to changes in the lower tail of the distribution and vice versa for higher values. Theil's $T_{T}$ is $\alpha =1$ and Theil's $T_{L}$ is $\alpha =0$. $\alpha =2$ is also commonly used.

When the $y_{i}$ of $\alpha =1$ is relpaced with $1/y_{i}$ (for example, income/person becomes persons/income) then $\alpha =1$ is equivalent to $\alpha =0$.

See also

• Theil index
• Atkinson index
• Lorenz curve
• Gini coefficent
• Hoover index
• Robin Hood index
• Suits index
• Income inequality metrics