# The decomposition of inequality reconsidered: Weakly decomposable measures

Listed:
• Ebert, Udo

## Abstract

The paper characterizes the class of weakly decomposable (aggregable) inequality measures which satisfy a new (weak) decomposition (and aggregation) property. These measures can be decomposed into the sum of the usual within-group and a between-group term which is based on the inequality between all pairs of individuals belonging to the groups involved. The measures therefore depend on the inequality index for two-person distributions and are proportional to the total sum of the inequality values between all pairs of individuals. Extending Gini's mean difference, the Gini coefficient, and the variance of logarithms we characterize three families of measures. By choosing other basic measures further (families of) weakly decomposable measures can be defined.

## Suggested Citation

• Ebert, Udo, 2010. "The decomposition of inequality reconsidered: Weakly decomposable measures," Mathematical Social Sciences, Elsevier, vol. 60(2), pages 94-103, September.
• Handle: RePEc:eee:matsoc:v:60:y:2010:i:2:p:94-103
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File URL: http://www.sciencedirect.com/science/article/pii/S0165-4896(10)00038-7

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## References listed on IDEAS

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## Citations

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Cited by:

1. Mornet, Pauline & Zoli, Claudio & Mussard, Stéphane & Sadefo-Kamdem, Jules & Seyte, Françoise & Terraza, Michel, 2013. "The (α, β)-multi-level α-Gini decomposition with an illustration to income inequality in France in 2005," Economic Modelling, Elsevier, vol. 35(C), pages 944-963.
2. Pauline Mornet, 2013. "A program for weakly decomposable inequality measures by population subgroups," Economics Bulletin, AccessEcon, vol. 33(3), pages 1738-1750.
3. Chameni Nembua, C. & Miamo Wendji, C., 2016. "Ordinal equivalence of values, Pigou–Dalton transfers and inequality in TU-games," Games and Economic Behavior, Elsevier, vol. 99(C), pages 117-133.
4. Chameni Nembua, Célestin & Demsou, Themoi, 2013. "Ordinal equivalence of values and Pigou-Dalton transfers in TU-games," MPRA Paper 44895, University Library of Munich, Germany, revised 09 Mar 2013.
5. Modalsli, Jørgen, 2011. "Inequality and growth in the very long run: inferring inequality from data on social groups," Memorandum 11/2011, Oslo University, Department of Economics.
6. Jørgen Modalsli, 2017. "Decomposing Global Inequality," Review of Income and Wealth, International Association for Research in Income and Wealth, vol. 63(3), pages 445-463, September.
7. Satya R. Chakravarty & Nachiketa Chattopadhyay & Conchita D'Ambrosio, 2016. "On a Family of Achievement and Shortfall Inequality Indices," Health Economics, John Wiley & Sons, Ltd., vol. 25(12), pages 1503-1513, December.
8. Kristof Bosmans & Z. Emel Öztürk, 2018. "An axiomatic approach to the measurement of envy," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 50(2), pages 247-264, February.
9. Mauro Mussini, 2010. "On the Link between Silber and Dagum Decomposition of the Gini Index," Statistics in Transition new series, Główny Urząd Statystyczny (Polska), vol. 11(3), pages 597-614, December.
10. Jørgen Modalsli, 2015. "Inequality in the very long run: inferring inequality from data on social groups," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 13(2), pages 225-247, June.
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12. Francesco Andreoli & Claudio Zoli, 2015. "Measuring the interaction dimension of segregation: the Gini-Exposure index," Working Papers 30/2015, University of Verona, Department of Economics.
13. Rolf Aaberge & Andrea Brandolini, 2014. "Multidimensional poverty and inequality," Discussion Papers 792, Statistics Norway, Research Department.
14. Arthur Charpentier & Stéphane Mussard, 2011. "Income inequality games," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 9(4), pages 529-554, December.
15. Mornet, Pauline, 2016. "On the axiomatization of the weakly decomposable inequality indices," Mathematical Social Sciences, Elsevier, vol. 83(C), pages 71-78.
16. Chiang, Yen-Sheng, 2015. "Inequality measures perform differently in global and local assessments: An exploratory computational experiment," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 437(C), pages 1-11.
17. M. Costa, 2019. "The evaluation of gender income inequality by means of the Gini index decomposition," Working Papers wp1130, Dipartimento Scienze Economiche, Universita' di Bologna.
18. Maria Giovanna Monti & Simone Pellegrino & Achille Vernizzi, 2015. "On Measuring Inequity in Taxation Among Groups of Income Units," Review of Income and Wealth, International Association for Research in Income and Wealth, vol. 61(1), pages 43-58, March.
19. Francesco Porro & Michele Zenga, 2020. "Decomposition by subpopulations of the Zenga-84 inequality curve and the related index $$\zeta$$ζ: an application to 2014 Bank of Italy survey," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 29(1), pages 187-207, March.
20. Michele Zenga, 2016. "On the decomposition by subpopulations of the point and synthetic Zenga (2007) inequality indexes," METRON, Springer;Sapienza Università di Roma, vol. 74(3), pages 375-405, December.

### Keywords

Inequality measures Decomposition Aggregation Gini's mean difference The Gini coefficient Variance of logarithms;

### JEL classification:

• D63 - Microeconomics - - Welfare Economics - - - Equity, Justice, Inequality, and Other Normative Criteria and Measurement
• D31 - Microeconomics - - Distribution - - - Personal Income and Wealth Distribution
• C43 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Index Numbers and Aggregation

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