Champernowne index

The Champernowne index or Champernowne chain index ( English : Champernowne's [chain] index ) - after the British mathematician and economist David Gawen Champernowne (1912–2000) - is a disparity measure derived from the Atkinson measure .

definition

The following applies to the Atkinson index : ${\displaystyle A}$

${\displaystyle A=1-\left[{\frac {1}{n}}\sum \limits _{i=1}^{n}\left({\frac {I_{i}}{\mu }}\right)^{1-\varepsilon }\right]^{\frac {1}{1-\varepsilon }}\,;\quad \varepsilon \neq 1\,,}$

where stands for the income of the -th economic sector (or household), for the average income ( arithmetic mean ) and for the number of sectors. ${\displaystyle I_{i}}$ ${\displaystyle i}$ ${\displaystyle \mu }$ ${\displaystyle n}$

Once is, the index collapses to the Champernowne index , which is defined as follows: ${\displaystyle \varepsilon =1}$ ${\displaystyle A}$ ${\displaystyle C}$

${\displaystyle C={\textit {CII}}=1-{\frac {\hat {\mu }}{\mu }}=1-{\frac {M}{\mu }}}$with as well ${\displaystyle \mu ={\bar {I}}_{\text{arithm}}={\frac {1}{n}}\sum \limits _{i=1}^{n}I_{i}}$ ${\displaystyle {\hat {\mu }}=M={\bar {I}}_{\text{geom}}=\left(\prod \limits _{i=1}^{n}I_{i}\right)^{\frac {1}{n}}={\sqrt[{n}]{\prod \limits _{i=1}^{n}I_{i}}}\,,}$

${\displaystyle C=1-{\frac {\sqrt[{n}]{\prod \limits _{i=1}^{n}I_{i}}}{{\frac {1}{n}}\sum \limits _{i=1}^{n}I_{i}}}\,.}$

Derivation

This chain index makes use of the concept of the geometric mean . It is very well known that if the distribution is not uniform, the geometric mean is smaller than the arithmetic mean. The opposite number of the quotient of the geometric mean to the arithmetic mean can truly be viewed as an index of the inequality of distribution ( inequality / disparity ), a so-called inequality measure . Formally, the index can be noted as follows:

${\displaystyle C={\textit {CII}}=1-{\frac {\hat {\mu }}{\mu }}\,,}$

where and indicate the previous arithmetic and geometric means of income distribution. It is easy to see that this index values varies between and . ${\displaystyle \!\ \mu }$ ${\displaystyle {\hat {\mu }}}$ ${\displaystyle 0}$ ${\displaystyle 1}$

Obviously one can think of a further measure in which the geometric mean is substituted by harmonic means .
These measures are sensitive to the transfer of income and are even greater if the transfer takes place at the “lower end” of the distribution.

You are sensitive to the transfer of income between two people. This can be checked by well as corresponding by and exchanged, and one can determine the direction of the (supply) change. It is also possible to use differential calculus . ${\displaystyle x_{j}}$ ${\displaystyle x_{k}}$ ${\displaystyle (x_{j}-d)}$ ${\displaystyle (x_{k}+d)}$

The big problem with these indices is that they cannot be defined if there is an income of . ${\displaystyle 0}$

literature

• Rao Bhanoji, MK Ramakrishnan: Income Inequality in Singapore: Impact of Economic Growth and Structural Change, 1966–1975. NUS Press, Singapore 1981 p. 26.
• Amartya Sen : On Some Debates in Capital Theory . In: Economica. Vol. 41, No. 163. Blackwell Publishing, 1974, pp. 328-335.

Web links

• Pramod Kumar Chaubey (eGyanKosh, IGNOU / Indira Gandhi National Open University ): Unit 11: Measures of Inequality - PD / public domain (PDF; 3.34 MB).
• Athanasios Kampas, Spyridon Mamalis ( Aristoteles University of Thessaloniki ): Assessing the Distributional Impacts of Transferable Pollution Permits: The Case of Phosphorus Pollution Management at a River Basin Scale (PDF; 76 kB).