Intersecting Lorenz curves and aversion to inverse downside inequality
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Intersecting Lorenz curves and aversion to inverse downside inequality W. Henry Chiu1 Received: 22 January 2019 / Accepted: 16 September 2020 © The Author(s) 2020
Abstract This paper defines and characterizes the concept of an increase in inverse downside inequality and show that, when the Lorenz curves of two income distributions intersect, how the change from one distribution to the other is judged by an inequality index exhibiting inverse downside inequality aversion often depends on the relative strengths of its aversion to inverse downside inequality and inequality aversion. For the class of linear inequality indices, of which the Gini coefficient is a member, a measure characterizing the strength of an index’s aversion to inverse downside inequality against its own inequality aversion is shown to determine the ranking by the index of two distributions whose Lorenz curves cross once. The precise condition under which the same result generalizes to the case of multiple-crossing Lorenz curves is also identified.
1 Introduction The Lorenz curve as an analytical tool has played a central role in the studies of income inequality as the Lorenz criterion (i.e., whether the Lorenz curve of one distribution lies above that of another) coincides precisely with the Pigou–Dalton “principle of transfers”, which says that an income transfer from a poorer to a richer person worsens inequality and captures our usual concept of inequality. It is however well-known that in empirical studies of real-world income distributions e.g., Atkinson (1973) and Davies and Hoy (1985) the Lorenz criterion can typically provide a ranking for only a small minority of all possible pairwise comparisons. This
* W. Henry Chiu [email protected] 1
Economics, School of Social Sciences, University of Manchester, Manchester, UK
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led many authors to propose to strengthen it with the additional principle of “transfer sensitivity” (Shorrocks and Foster 1987) or equivalently “aversion to downside inequality” (Davies and Hoy 1995).1 These authors argue that, for a fixed income gap, the same amount of income transfer from a poorer to a richer person should be considered more disequalizing the lower it occurs in the distribution. The concept parallels that of downside risk aversion proposed by Menezes et al. (1980) since an inequality index exhibiting such transfer sensitivity always assigns a higher value to a distribution that has “more downside inequality” than another in the sense that the former can be obtained from the latter by a sequence of what Menezes et al. (1980) term “mean-variance preserving transformations”, each of which combines a “mean-preserving spread” (equivalently a regressive transfer) with a “mean-preserving contraction” (equivalently a progressive transfer) occurring at higher income levels in a way that the variance is preserved. Using empirical data, Shorrocks and Foster (1987) show that such a strengthening does significantly increase the ranking success rate. These authors ha
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