Atkinson index: Difference between revisions

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{{Short description|Inequality measure}}

The '''Atkinson index''' (also known as the '''Atkinson measure''' or '''Atkinson inequality measure''') is a measure of [[income inequality]] developed by British economist [[Anthony Barnes Atkinson]]. The measure is useful in determining which end of the distribution contributed most to the observed inequality.<ref>''[[inter alia]]'' [httphttps://www.census.gov/prod/2011pubs/p60-239.pdf "Income, Poverty, and Health Insurance Coverage in the United States: 2010"], [[U.S. Census Bureau]], 2011, p.10</ref>

==Definition==

The index can be turned into a [[Normative economics|normative]] measure by imposing a [[coefficient]] <math>\varepsilon</math> to weight incomes. Greater weight can be placed on changes in a given portion of the income distribution by choosing <math>\varepsilon</math>, the level of "inequality aversion", appropriately. The Atkinson index becomes more sensitive to changes at the lower end of the income distribution as <math>\varepsilon</math> approaches 1. Conversely, as the level of inequality aversion falls (that is, as <math>\varepsilon</math> approaches 0) the Atkinson becomes more sensitive to changes in the upper end of the income distribution.

The Atkinson <math>\varepsilon</math> parameter is often called the "inequality aversion parameter", since it quantifies the amount of social utility that is assumed to be gained from complete redistribution of resources. For <math>\varepsilon=0</math>, (no aversion to inequality) it is assumed that no social utility is gained by complete redistribution and the Atkinson index (<math>A_\varepsilon</math>) is zero. For <math>\varepsilon=\infty</math> (infinite aversion to inequality), it is assumed that infinite social utility is gained by complete redistribution in which case <math>A_\varepsilon=1</math>. The Atkinson index (<math>A_\varepsilon</math>) then varies between 0 and 1 and is a measure of the amount of social utility to be gained by complete redistribution of a given income distribution. Based on one's value judgement concerning the social utility of complete redistribution, as embodied in the <math>\varepsilon</math> parameter, different income distributions may be compared by calculating the Atkinson index at that <math>\varepsilon</math> value, with lower values of <math>A_\varepsilon</math> indicating lower social utility to be gained, higher values indicating more. Lower values of <math>A_\varepsilon</math> thus indicate a more equal distribution than higher values, given a particular degree of inequality aversion.

The Atkinson index is defined as:

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\begin{cases}

1-\frac{1}{\mu}\left(\frac{1}{N}\sum_{i=1}^{N}y_{i}^{1-\varepsilon}\right)^{1/(1-\varepsilon)}

& \mbox{for}\ 0 \leq \epsilonvarepsilon \neq 1 \\

1-\frac{1}{\mu}\left(\prod_{i=1}^{N}y_{i}\right)^{1/N}

& \mbox{for}\ \varepsilon=1, \\

1-\frac{1}{\mu}\min \left(y_1,...,y_N\right)

& \mbox{for}\ \varepsilon=+\infty

\end{cases}

</math>

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In other words, the Atkinson index is the complement to 1 of the ratio of the [[Generalized mean|Hölder generalized mean]] of exponent 1−ε to the arithmetic mean of the incomes (where as usual the generalized mean of exponent 0 is interpreted as the [[geometric mean]]).

==Interpretation==

Atkinson index relies on the following axioms:

The index can be turned into a [[Normative economics|normative]] measure by imposing a [[coefficient]] <math>\varepsilon</math> to weight incomes. Greater weight can be placed on changes in a given portion of the income distribution by choosing <math>\varepsilon</math>, the level of "inequality aversion", appropriately. The Atkinson index becomes more sensitive to changes at the lower end of the income distribution as <math>\varepsilon</math> approaches 1increases. Conversely, as the level of inequality aversion falls (that is, as <math>\varepsilon</math> approaches 0) the Atkinson becomes moreless sensitive to changes in the upperlower end of the distribution. The Atkinson index is for no value of <math>\varepsilon</math> highly sensitive to top incomes because of the common restriction that <math>\varepsilon</math> is nonnegative.<ref> The Atkinson index is related to the generalized entropy (GE) class of inequality indexes by <math>\epsilon=1-\alpha</math> - i.e an Atkinson index with high inequality aversion is derived from a GE index with small <math>\alpha</math>. GE indexes with large <math>\alpha</math> are sensitive to the existence of large top incomes but the corresponding Atkinson index would have negative <math>\varepsilon</math>. For a hypothetical Atkinson index with <math>\epsilon</math> being negative, the implied social utility function would be convex in income, distributionand the Atkinson index would be nonpositive.</ref>

The Atkinson <math>\varepsilon</math> parameter is often called the "inequality aversion parameter", since it regulates the sensitivity of the implied social welfare losses from inequality to income inequality as measured by some corresponding generalised entropy index. The Atkinson index is defined in reference to a corresponding social welfare function, where mean income multiplied by one minus the Atkinson index gives the ''welfare equivalent equally distributed income''. Thus the Atkinson index gives the share of current income which could be sacrificed, without reducing social welfare, if perfect inequality were instated. For <math>\varepsilon=0</math>, (no aversion to inequality), the marginal social welfare from income is invariant to income, i.e. marginal increases in income produce as much social welfare whether they go to a poor or rich individual. In this case, the welfare equivalent equally distributed income is equal to mean income, and the Atkinson index is zero.

For <math>\varepsilon=+\infty</math> (infinite aversion to inequality) the marginal social welfare of income of the poorest individual is infinitely larger than any even slightly richer individual, and the Atkinson social welfare function is equal to the smallest income in the sample. In this case, the Atkinson index is equal to mean income minus the smallest income, divided by mean income. As in large typical income distributions incomes of zero or near zero are common, the Atkinson index will tend to be one or very close to one for very large <math>\varepsilon</math>.

The Atkinson index then varies between 0 and 1 and is a measure of the amount of social utility to be gained by complete redistribution of a given income distribution, for a given <math>\varepsilon</math> parameter. Under the utilitarian ethical standard and some restrictive assumptions (a homogeneous population and [[constant elasticity of substitution]] utility), <math>\varepsilon</math> is equal to the income elasticity of [[marginal utility]] of income.

==Relationship to generalized entropy index==

The Atkinson index with inequality aversion <math>\varepsilon</math> is equivalent (under a monotonic rescaling) to a [[generalized entropy index]] with parameter <math>\alpha = 1 - \varepsilon</math>

The formula for deriving an Atkinson index with inequality aversion parameter <math>\epsilon</math> from the corresponding GE index under the restriction <math>\varepsilon = 1-\alpha</math> is given by:

<math display="block">A=1-[\varepsilon(\varepsilon-1)GE(\alpha) + 1]^{(1/(1-\varepsilon))} \qquad \varepsilon\ne1</math>

<math display="block">A= 1-e^{-GE(\alpha)} \qquad \varepsilon=1</math>

==Properties==

The Atkinson index relies onsatisfies the following axiomsproperties:

# The index is symmetric in its arguments: <math>A_\varepsilon(y_1,\ldots,y_N)=A_\varepsilon(y_{\sigma(1)},\ldots,y_{\sigma(N)})</math> for any permutation <math>\sigma</math>.

# The index is non-negative, and is equal to zero only if all incomes are the same: <math>A_\varepsilon(y_1,\ldots,y_N) = 0</math> iff <math>y_i = \mu</math> for all <math>i</math>.

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# The index satisfies population replication axiom: if a new population is formed by replicating the existing population an arbitrary number of times, the inequality remains the same: <math>A_\varepsilon(\{y_1,\ldots,y_N\},\ldots,\{y_1,\ldots,y_N\})=A_\varepsilon(y_1,\ldots,y_N)</math>

# The index satisfies mean independence, or income homogeneity, axiom: if all incomes are multiplied by a positive constant, the inequality remains the same: <math>A_\varepsilon(y_1,\ldots,y_N) = A_\varepsilon( ky_1,\ldots,ky_N)</math> for any <math>k>0</math>.

# The index is (non-additively) subgroup decomposable and the corresponding generalized entropy index is ''additively'' subgroup decomposable.<ref>Shorrocks, AF (1980). The class of additively decomposable inequality indices. ''Econometrica'', 48 (3), 613–625, {{doi|10.2307/1913126}}</ref> This means that overall inequality in the population can be computed as the sum of the corresponding AtkinsonGE indices within each group, and the AtkinsonGE index of the group mean incomes:

::: <math>

A_\varepsilonGE(\alpha; y_{gi}: g=1,\ldots,G, i=1,\ldots,N_g) = \sum_{g=1}^G w_g A_\varepsilonGE(\alpha; y_{g1}, \ldots, y_{g,N_ggN_g}) + A_GE(\varepsilon(alpha; \mu_1, \ldots, \mu_G)

</math>

::where <math>g</math> indexes groups, <math>i</math>, individuals within groups, <math>\mu_g</math> is the mean income in group <math>g</math>, and the weights <math>w_g</math> depend on <math>\mu_g, \mu, N</math> and <math>N_g</math>. The class of the subgroupadditively-decomposable inequality indices is very restrictive; in fact, only the generalized entropy indices are additively decomposable. Many popular indices, including [[Gini index]], do not satisfy this property.

== See also ==

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* Allison, PD (1979) Reply to Jasso. ''American Sociological Review'' 44(5):870–72.

* Biewen M, Jenkins SP (2003). Estimation of Generalized Entropy and Atkinson Inequality Indices from Complex Survey Data. [http://ideas.repec.org/p/iza/izadps/dp763.html IZA Discussion Paper #763]. Provides statistical inference for Atkinson indices.

* Lambert, P. (2002). ''Distribution and redistribution of income''. 3rd edition, Manchester Univ Press, {{ISBN |978-0-7190-5732-8}}.

* Sen A, Foster JE (1997) ''On Economic Inequality'', Oxford University Press, {{ISBN |978-0-19-828193-1}}. ([http://www.poorcity.richcity.org/oei/#Atkinson Python script] for a selection of formulas in the book)

* [http://www.wider.unu.edu/research/Database/en_GB/database/ World Income Inequality Database] {{Webarchive|url=https://web.archive.org/web/20110313002049/http://www.wider.unu.edu/research/Database/en_GB/database/ |date=2011-03-13 }}, from [[World Institute for Development Economics Research]]

* [httphttps://www.census.gov/hhes/www/income/incineq/p60204/p60204txt.html Income Inequality, 1947–1998], from [[United States Census Bureau]].

== External links ==

'''Software:'''

* [https://archive.today/20121204174230/http://www.wessa.net/co.wasp Free Online Calculator] computes the Gini Coefficient, plots the Lorenz curve, and computes many other measures of concentration for any dataset

* Free Calculator: [http://www.poorcity.richcity.org/calculator.htm Online] and [https://web.archive.org/web/20041012085224/http://luaforge.net/project/showfiles.php?group_id=49 downloadable scripts] ([[Python (programming language)|Python]] and [[Lua programming language|Lua]]) for Atkinson, Gini, and Hoover inequalities

* Users of the [http://www.r-project.org/ R] data analysis software can install the "ineq" package which allows for computation of a variety of inequality indices including Gini, Atkinson, Theil.

* A [http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=19968 MATLAB Inequality Package] {{Webarchive|url=https://web.archive.org/web/20081004090028/http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=19968 |date=2008-10-04 }}, including code for computing Gini, Atkinson, Theil indexes and for plotting the Lorenz Curve. Many examples are available.

* [[Stata]] inequality packages: [http://ideas.repec.org/c/boc/bocode/s366007.html ineqdeco] to decompose inequality by groups; [http://ideas.repec.org/c/boc/bocode/s453601.html svygei and svyatk] to compute design-consistent variances for the generalized entropy and Atkinson indices; [http://ideas.repec.org/c/boc/bocode/s366302.html glcurve] to obtain generalized Lorenz curve. You can type <ttkbd>ssc install ineqdeco</ttkbd> etc. in Stata prompt to install these packages.

{{DEFAULTSORT:Atkinson Index}}

[[Category:IndexIncome numbersinequality metrics]]

[[Category:Welfare economics]]

[[Category:Econometrics]]