Kolm index

The Kolm index is a measure of inequality for describing absolute concentration, which is named after the French economist and econometrician Serge-Christophe Kolm .

definition

Let be the number of incomes such that . This indicates the number of different incomes and each income has recipients (income recipients). The number of equal couples with a given income is: ${\displaystyle N}$ ${\displaystyle N=n\cdot m}$ ${\displaystyle n}$ ${\displaystyle m}$

${\displaystyle {\frac {m\cdot (m-1)}{2}}}$

and the total number of equal incomes would be:

${\displaystyle {\frac {n\cdot m\cdot (m-1)}{2}}\,.}$

Obviously the total number of all pairs would be:

${\displaystyle {\frac {N\cdot (N-1)}{2}}-{\frac {n\cdot m\cdot (n\cdot m-1)}{2}}\,.}$

One can use an 'equality index' in the form of the term:

${\displaystyle {\frac {n\cdot m\cdot (m-1)}{n\cdot m(n\cdot m-1)}}={\frac {m-1}{N-1}}}$

think. Consequently, the inequality measure is calculated by subtracting from : ${\displaystyle 1}$

${\displaystyle {\frac {1-(m-1)}{N-1}}-{\frac {N-m}{N-1}}={\frac {m\cdot (n-1)}{n\cdot m-1}}={\frac {n\cdot m-m}{n\cdot m-1}}\,.}$

If the income has recipients, the index is as follows: ${\displaystyle x_{1}}$ ${\displaystyle f_{i}}$

${\displaystyle K=1-{\frac {\sum f_{i}^{2}-N}{N\cdot (N-1)}}={\frac {N^{2}-\sum f_{i}^{2}}{N\cdot (N-1)}}\,.}$

In 1976 Kolm defined the index as follows:

${\displaystyle K={\frac {1}{\kappa }}\cdot \ln \left[{\frac {1}{n}}\cdot \sum \limits _{i=1}^{n}e^{\kappa \left(\mu -y_{i}\right)}\right]\,;\quad \kappa >0\,,}$

wherein the inequality aversion indicates. ${\displaystyle \kappa }$

motivation

Kolm's intention to develop this curious index is based, in his opinion, on forming an idea of ​​the fact that there are many simple (calculation) ways of approaching a measure of inequality.

literature

• Serge-Christophe Kolm: Unequal inequalities I and II. Journal of Economic Theory 12 u. 13. 1976. pp. 416-442 et al. 82-111.
• Serge-Christophe Kolm: Intermediate measures of inequality. Technical report, CGPC, 1996.

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).