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Playing with the Hadamard product in decomposing Gini, concentration, redistribution and re-ranking indexes

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  • Achille Vernizzi

    (Department of Economics, Business and Statistics, University of Milan)

Abstract

Gini and concentration indexes are well known useful tools in analysing redistribution and re-ranking effects of taxes with respect to a population of income earners. Aronson, Johnson and Lambert (1994), Urban and Lambert (2008) decompose Gini and re-ranking indices to analyse potential redistribution effects and the unfairness of a tax systems. Urban and Lambert (2008) consider contiguous income groups which are created by dividing the pre-tax income parade according to the same bandwidth. However, earners may be very often split into groups characterized by social and demographic aspects or by other characteristics: in these circumstances groups can easily overlap. In this paper we consider a more general situation that takes into account overlapping among groups; we obtain matrix compact forms for Gini and concentration indexes, and consequently, for redistribution and re-ranking indexes. In deriving formulae the so called matrix Hadamard product is extensively used. Matrix algebra allows to write indexes aligning incomes in a non decreasing order either with respect to post-tax income or to pre-tax incomes. Moreover, matrix compact formulae allow an original discussion for the signs of the within group, across group, between and transvariation components into which the Atkinson-Plotnick-Kakwany (Plotnick, 1981) re-ranking index can split.

Suggested Citation

  • Achille Vernizzi, 2009. "Playing with the Hadamard product in decomposing Gini, concentration, redistribution and re-ranking indexes," UNIMI - Research Papers in Economics, Business, and Statistics unimi-1085, Universitá degli Studi di Milano.
  • Handle: RePEc:bep:unimip:unimi-1085
    Note: oai:cdlib1:unimi-1085
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    References listed on IDEAS

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    1. Maria Giovanna MONTI, 2007. "A note on the Dagum decomposition of the Gini inequality index," Departmental Working Papers 2007-16, Department of Economics, Management and Quantitative Methods at Università degli Studi di Milano.
    2. Achille Vernizzi, 2007. "Una precisazione sulla scomposizione dell'indice di redistribuzione RE di Aronson-Johnson-Lambert e una proposta di estensione dell'indice di Plotnick," ECONOMIA PUBBLICA, FrancoAngeli Editore, vol. 2007(1-2), pages 145-154.
    3. Dagum, Camilo, 1997. "A New Approach to the Decomposition of the Gini Income Inequality Ratio," Empirical Economics, Springer, vol. 22(4), pages 515-531.
    4. Aronson, J Richard & Johnson, Paul & Lambert, Peter J, 1994. "Redistributive Effects and Unequal Income Tax Treatment," Economic Journal, Royal Economic Society, vol. 104(423), pages 262-270, March.
    5. Lambert, Peter J & Aronson, J Richard, 1993. "Inequality Decomposition Analysis and the Gini Coefficient Revisited," Economic Journal, Royal Economic Society, vol. 103(420), pages 1221-1227, September.
    6. Ivica Urban & Peter J. Lambert, 2008. "Redistribution, Horizontal Inequity, and Reranking: How to Measure Them Properly," Public Finance Review, , vol. 36(5), pages 563-587, September.
    7. Plotnick, Robert, 1981. "A Measure of Horizontal Inequity," The Review of Economics and Statistics, MIT Press, vol. 63(2), pages 283-288, May.
    8. Aronson, J. Richard & Lambert, Peter J., 1994. "Decomposing the Gini Coefficient to Reveal the Vertical, Horizontal, and Reranking Effects of Income Taxation," National Tax Journal, National Tax Association, vol. 47(2), pages 273-294, June.
    9. Maria Giovanna MONTI & Alessandro SANTORO, 2009. "The Gini decomposition: an alternative formulation with an application to tax reform," Departmental Working Papers 2009-30, Department of Economics, Management and Quantitative Methods at Università degli Studi di Milano.
    10. Aronson, J. Richard & Lambert, Peter J., 1994. "Decomposing the Gini Coefficient to Reveal the Vertical, Horizontal, and Reranking Effects of Income Taxation," National Tax Journal, National Tax Association, vol. 47(2), pages 273-94, June.
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    Cited by:

    1. Achille VERNIZZI & Maria Giovanna MONTI & Mauro MUSSINI, 2010. "A Gini and concentration index decomposition with an application to the APK reranking measure," Departmental Working Papers 2010-10, Department of Economics, Management and Quantitative Methods at Università degli Studi di Milano, revised 13 Dec 2010.
    2. Mussini, Mauro, 2013. "On decomposing inequality and poverty changes over time: A multi-dimensional decomposition," Economic Modelling, Elsevier, vol. 33(C), pages 8-18.
    3. 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.
    4. repec:csb:stintr:v:12:y:2010:i:3:p:597-614 is not listed on IDEAS
    5. Simone Pellegrino & Achille Vernizzi, 2013. "On measuring violations of the progressive principle in income tax systems," Empirical Economics, Springer, vol. 45(1), pages 239-245, August.

    More about this item

    Keywords

    Gini and concentration indexes decompositions; Tax redistributive effects; Tax re-ranking effects; Hadamard product.;

    JEL classification:

    • D31 - Microeconomics - - Distribution - - - Personal Income and Wealth Distribution
    • D63 - Microeconomics - - Welfare Economics - - - Equity, Justice, Inequality, and Other Normative Criteria and Measurement
    • H23 - Public Economics - - Taxation, Subsidies, and Revenue - - - Externalities; Redistributive Effects; Environmental Taxes and Subsidies
    • H24 - Public Economics - - Taxation, Subsidies, and Revenue - - - Personal Income and Other Nonbusiness Taxes and Subsidies

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