Atkinson measure

The Atkinson measure (after Anthony Atkinson [1944–2017]) describes a set of inequality measures with which, for example, income or wealth inequality in a society can be calculated.

Origin / history

The Dalton Index introduced by Hugh Dalton is not invariant to positive linear transformations in personal income welfare functions . ${\displaystyle D}$

In 1970 Atkinson tried to redefine the index in such a way that it had the corresponding invariance.

definition

Every Atkinson measure is based on a concave utility function . How strongly the Atkinson measure reacts to inequalities is determined by this underlying utility function.

Usually, an Arrow welfare function is used, which determines, by means of a parameter indicating the inequality aversion , how great the welfare difference of an additional euro is between a person with a high and a low income. The larger the epsilon, the more the Atkinson measure reacts to inequality. Is this means that the distribution of income is socially insignificant. ${\displaystyle \varepsilon }$ ${\displaystyle \varepsilon =0}$

This Atkinson index is defined as follows:

${\displaystyle A=A_{\varepsilon }=A_{\varepsilon }(y_{1},\ldots ,y_{n})={\begin{cases}1&{\mbox{für}}\ \varepsilon =0\\1-{\frac {1}{\mu }}\left({\frac {1}{n}}\sum _{i=1}^{n}y_{i}^{1-\varepsilon }\right)^{1/(1-\varepsilon )}&{\mbox{für}}\ \varepsilon >0\land \varepsilon \neq 1\\1-{\frac {1}{\mu }}\left(\prod _{i=1}^{n}y_{i}\right)^{1/n}&{\mbox{für}}\ \varepsilon =1,\end{cases}}}$

where is the individual income ( i = 1, 2, ..., N ) and the average income. ${\displaystyle y_{i}}$ ${\displaystyle \mu }$

properties

The Atkinson index has the following properties:

1. Symmetry in the arguments: for all permutations . ${\displaystyle A_{\varepsilon }(y_{1},\ldots ,y_{N})=A_{\varepsilon }(y_{\sigma (1)},\ldots ,y_{\sigma (N)})}$ ${\displaystyle \sigma }$
2. The index is between zero and one. and for everyone ${\displaystyle 0\leq A_{1}\leq 1}$ ${\displaystyle 0\leq A_{\varepsilon }\leq 1-n^{-\epsilon }<1}$ ${\displaystyle \varepsilon \neq 1}$
3. The index is only zero if income is equal : iff. for everyone . ${\displaystyle A_{\varepsilon }(y_{1},\ldots ,y_{N})=0}$ ${\displaystyle y_{i}=\mu }$ ${\displaystyle i}$
4. Invariance versus multiplication: If the population is replicated identically (multiple times), the index remains the same: ${\displaystyle A_{\varepsilon }(\{y_{1},\ldots ,y_{N}\},\ldots ,\{y_{1},\ldots ,y_{N}\})=A_{\varepsilon }(y_{1},\ldots ,y_{N})}$
5. Invariance to inflation: If all incomes are multiplied by a positive constant, the index remains the same: for all ${\displaystyle A_{\varepsilon }(y_{1},\ldots ,y_{N})=A_{\varepsilon }(ky_{1},\ldots ,ky_{N})}$ ${\displaystyle k>0}$
6. The index can be broken down into subgroups. It applies

${\displaystyle A_{\varepsilon }(y_{g,i_{g}}:i_{g}=1,\ldots ,n_{g};g=1,\ldots ,G)=\sum _{g=1}^{G}w_{g}A_{\varepsilon }(y_{g,1},\ldots ,y_{g,n_{g}})+A_{\varepsilon }(\mu _{1},\ldots ,\mu _{G})}$, where the number of subgroups indicates the average income of the subgroup , and the weights for a function independent of the concrete situation f. ${\displaystyle G}$ ${\displaystyle \mu _{g}}$ ${\displaystyle g}$ ${\displaystyle w_{g}=f(\mu _{g},\mu ,N,N_{g})}$

application

For the Theil index resulting from the “generalized entropy class” it applies that it can be converted into an entropy measure developed by Atkinson, which also appeared in the literature as the “normalized Theil index ”. The measure is calculated from the function . ${\displaystyle \varepsilon =1}$ ${\displaystyle I_{1}}$ ${\displaystyle 1-e^{-T}}$

See also

• Champernowne index

literature

Original article:

• Anthony B. Atkinson: On the Measurement of Inequality. In: Journal of Economic Theory . Vol. 2 (3), 1970. pp. 244-263.

To deepen:

• Yoram Amiel: Thinking about inequality. Cambridge 1999.
• Frank Alan Cowell: Measurement of Inequality. In: Anthony B. Atkinson, François Bourguignon (eds.): Handbook of Income Distribution. Vol. 1, Amsterdam et al. 2000. pp. 87-166.
• Amartya Sen, James Eric Foster: On Economic Inequality. Oxford University Press, Oxford 1996. ISBN 0-19-828193-5 . ( Python script with important formulas from the book, including formulas for calculating the Atkinson index)

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).

Individual evidence

1. ↑ Shorrocks, AF (1980). The class of additively decomposable inequality indices. Econometrica , 48 (3), 613-625, doi : 10.2307 / 1913126 .
2. ↑ "Generalized Entropy Class", Janes E. Foster in Annex A.4.1 (p. 142) by Amartya Sen: On Economic Inequality. 1973/1997.
3. ↑ Anthony B. Atkinson developed various measures. The measure related to the Theil index can be found in Lionnel Maugis in: Inequality Measures in Mathematical Programming for the Air Traffic Flow Management Problem with En-Route Capacities (publication for IFORS 96). 1996.
4. ↑ Juana Domínguez-Domínguez, José Javier Núñez-Velázquez: The Evolution of Economic Inequality in the EU Countries During the Nineties ( Memento of the original from March 25, 2009 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. (PDF; 330 kB). 2005.