Abstract
Two related problems are dealt with in this article, concerning some popular inequality indices proposed by Gini, Pietra–Ricci and Theil: (1) the calculation of the index when only a frequency distribution is available, thus needing some kind of approximation; and (2) a reasonable decomposition of the index calculated for a mixture, with components related to ‘within’ and ‘between’ inequalities, and possibly to the separate contributions of each group to the overall inequality. Beside the proposals arising from the specific structure of each inequality index, a general approach for identifying the within component is utilized, which is based on the fixation of a given number of fictitious individuals (called aggregate units), common to every group. Regarding the Gini index, a general expression is obtained for the approximation problem, while the within inequality is more easily managed by the recourse to aggregate units. The decomposition of the Pietra–Ricci index displays three components, clearly ascribable to within inequality, to a mixture effect and to a mean effect. Regarding the Theil index, some simple and very accurate approximation formulae are obtained. An application of all the indices and their decompositions has been made for the 2004 income distribution for Italy (Bank of Italy Survey).
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