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Playing with the Hadamard product in decomposing Gini, concentration, redistribution and re-ranking indexes
Achille Vernizzi, Department of Economics, Business and Statistics, University of Milan
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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.
SUBJECT AREA:
D31, D63, H23, H24
SUGGESTED CITATION:
Achille Vernizzi,
"Playing with the Hadamard product in decomposing Gini, concentration, redistribution and re-ranking indexes"
(July 2009).
UNIMI - Research Papers in Economics, Business, and Statistics.
Statistics and Mathematics.
Working Paper 45.
http://services.bepress.com/unimi/statistics/art45
Paper presented by C. Tommasi.
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