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← Older revision		Revision as of 10:38, 15 March 2013
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	The ~~person who wrote~~ the ~~post~~ is ~~called Jayson Hirano~~ and ~~he completely digs~~ that ~~title~~. ~~He functions~~ as a ~~bookkeeper~~. ~~Her family members life~~ in ~~Ohio but her husband wants them~~ to ~~transfer~~. ~~One~~ of the ~~issues she loves most~~ is ~~canoeing~~ and ~~she's been doing it~~ for ~~quite~~ a ~~while~~.<br><br>~~My page~~ - [http://~~kjhkkb~~.net/xe/~~notice~~/~~374835 tarot card readings~~]		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]]'' [http://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 index is defined as:

			:<math>A_\varepsilon(y_1,\ldots,y_N)=
			\begin{cases}
			1-\frac{1}{\mu}\left(\frac{1}{N}\sum_{i=1}^{N}y_{i}^{1-\varepsilon}\right)^{1/(1-\varepsilon)}
			& \mbox{for}\ \varepsilon \in \left[0,1\right)\cup\left(1,+\infty\right) \\
			1-\frac{1}{\mu}\left(\prod_{i=1}^{N}y_{i}\right)^{1/N}
			& \mbox{for}\ \varepsilon=1,
			\end{cases}
			</math>

			where <math>y_{i}</math> is individual income (''i'' = 1, 2, ..., ''N'') and <math>\mu</math> is the [[mean]] income.

			The entropy measure developed by Atkinson<ref>Anthony Barnes Atkinson developed various metrics. The Atkinson index which is related to the Theil index has been described by Lionnel Maugis in ''Inequality Measures in Mathematical Programming for the Air Traffic Flow Management Problem with En-Route Capacities'' (published on occasion of IFORS 96), 1996</ref> can be computed from a "normalized [[Theil index]]".<ref>Juana Domínguez-Domínguez, José Javier Núñez-Velázquez: ''[http://www.uib.es/congres/ecopub/ecineq/papers/100Dominguez-Nunez.pdf The Evolution of Economic Inequality in the EU Countries During the Nineties].'' 2005</ref> This, however, only applies to the Theil index <math>{I_1}</math>, which is derived from the "generalized entropy class"<ref>James E. Foster in annexe A.4.1 (p.142) of: Amartya Sen, ''On Economic Inequality'', 1973/1997</ref> with <math>{\epsilon} = 1</math>. The Atkinson index is computed using the function <math>1-e^{- T}</math>.

			Atkinson index relies on the following axioms:
			# 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>.
			# The index satisfies the [[Income_inequality_metrics#Transfer_principle\|principle of transfers]]: if a transfer <math>\Delta>0</math> is made from an individual with income <math>y_i</math> to another one with income <math>y_j</math> such that <math>y_i - \Delta > y_j + \Delta</math>, then the inequality index cannot increase.
			# 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 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 Atkinson indices within each group, and the Atkinson index of the group mean incomes:
			:<math>
			A_\varepsilon(y_{gi}: g=1,\ldots,G, i=1,\ldots,N_g) = \sum_{g=1}^G w_g A_\varepsilon( y_{g1}, \ldots, y_{g,N_g}) + A_\varepsilon(\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 subgroup-decomposable inequality indices is very restrictive. Many popular indices, including [[Gini index]], do not satisfy this property.

			== See also ==
			* [[Income inequality metrics]]
			* [[Generalized entropy index]]
			* [[Gini index]]

			==Footnotes==
			<references />

			==References==
			* Atkinson, AB (1970) On the measurement of economic inequality. ''Journal of Economic Theory'', 2 (3), pp. 244–263, {{doi\|10.1016/0022-0531(70)90039-6}}. The original paper proposing this inequality index.
			* Allison PD (1978) Measures of Inequality, ''American Sociological Review'', 43, pp. 865–880. Presents a technical discussion of the Atkinson measure's properties.
			* 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], from [[World Institute for Development Economics Research]]
			* [http://www.census.gov/hhes/www/income/incineq/p60204/p60204txt.html Income Inequality, 1947-1998], from [[United States Census Bureau]].

			== External links ==
			'''Software:'''
			* [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 [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], 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 <tt>ssc install ineqdeco</tt> etc. in Stata prompt to install these packages.

			{{DEFAULTSORT:Atkinson Index}}
			[[Category:Index numbers]]
			[[Category:Welfare economics]]
			[[Category:Econometrics]]

en>SmackBot: remove Erik9bot category,outdated, tag and general fixes

2009-12-16T12:37:56Z

remove Erik9bot category,outdated, tag and general fixes

New page

The person who wrote the post is called Jayson Hirano and he completely digs that title. He functions as a bookkeeper. Her family members life in Ohio but her husband wants them to transfer. One of the issues she loves most is canoeing and she's been doing it for quite a while.<br><br>My page - [http://kjhkkb.net/xe/notice/374835 tarot card readings]

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en>SmackBot: remove Erik9bot category,outdated, tag and general fixes