Playing with the Hadamard product in decomposing Gini, concentration, redistribution and re-ranking indexes
AbstractGini 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.
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Bibliographic InfoPaper provided by Department of Economics, Management and Quantitative Methods at Università degli Studi di Milano in its series Departmental Working Papers with number 2009-50.
Date of creation: 31 Jul 2009
Date of revision:
Gini and concentration indexes decompositions; Tax redistributive effects; Tax re-ranking effects; Hadamard product;
Other versions of this item:
- 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.
- 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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- Mussini, Mauro, 2013. "On decomposing inequality and poverty changes over time: A multi-dimensional decomposition," Economic Modelling, Elsevier, vol. 33(C), pages 8-18.
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