Kakwani index

The Kakwani index is a disparity measure that was named after the statistician , economist and economist Nanak Chand Kakwani .

Derivation

From the following term (compare: Atkinson measure !):

${\displaystyle \mu ^{*}=U^{-1}\left[{\frac {1}{N}}\sum \limits _{i=1}^{N}U\left(x_{i}\right)\right]}$

- where represents the equally distributed equivalent income (vector) , i.e. indicates the amount of the equally distributed equivalent income - the following can be seen: ${\displaystyle \mu ^{*}}$

${\displaystyle M^{*}=\mu (1-A)\,.}$

For the underlying welfare function it follows: ${\displaystyle i}$

${\displaystyle W-NU\left(\mu ^{*}\right)+NU\{\mu (1-A)\}\,,}$

which is a monotonically increasing function of and a monotonically decreasing function of . Another function with these properties, but it is as follows: ${\displaystyle u}$ ${\displaystyle A}$

${\displaystyle W-NU\left({\frac {\mu }{1+K}}\right)\,.}$

Here is a measure of inequality (thanks to Kakwani). Adding the (social) welfare function gives: ${\displaystyle K}$

${\displaystyle \mu ^{*}={\frac {\mu }{1+K}}}$

and consequently:

${\displaystyle K={\frac {\mu }{\mu ^{*}}}-1\,,}$

which is in full agreement with in all relationships except for its sensitivity (sensitivity) to income differences compared to inequality . It should be noted that is too reciprocal . ${\displaystyle A}$ ${\displaystyle (1+K)}$ ${\displaystyle (1-A)}$

Web links

• Pramod Kumar Chaubey (eGyanKosh, IGNOU / Indira Gandhi National Open University ): Unit 11: Measures of Inequality - PD / public domain (PDF file; 3.34 MB).