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04-07-2021 | Original Paper

Revisiting comparisons of income inequality when Lorenz curves intersect

Authors: James Davies, Michael Hoy, Lin Zhao

Published in: Social Choice and Welfare | Issue 1/2022

Abstract

The main contribution of Davies and Hoy (Am Econ Rev 85:980–986, 1995) was a “necessary and sufficient” condition for comparing inequality between income distributions according to the principle of transfer sensitivity (PTS). Chiu (Soc Choice Welf 28:375–399, 2007) showed that although the condition is sufficient, it is not necessary. In this paper, we provide the correct necessary and sufficient condition, and demonstrate with a simple example how the corrected condition allows for more pairs of distributions to be ranked by PTS. The correction clarifies the connection between Lorenz curve comparisons and inequality rankings when the curves intersect.

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In the favourable composite transfer defined by Shorrocks and Foster (1987), the progressive transfer occurs in a lower range of incomes than does the regressive transfer, although the two transfers may occur over intersecting income ranges. See Davies and Hoy (1995) for a discussion of the normative appeal of placing greater emphasis on inequality that occurs for relatively lower incomes within the distribution.

In the literature, Proposition 2 of Davies and Hoy (1995) is usually cited as an equivalence between PTS and the comparison of variances for appropriately truncated subpopulations of a pair of income distributions. See, for example, Davies and Shorrocks (2000) and Cowell (2011) for citations in monographs, and Aaberge (2009), Gajdos and Weymark (2012) and Ibragimov et al. (2018) for citations in other publications.

This is clear upon comparing his Lemma 2 and Footnote 11 with our Proposition 1 here.

Theorem 5.2.3 in Athreya and Lahiri (2006, p.155) states integration by parts under mild conditions. Since cumulative distribution functions are always nondecreasing right-continuous and the quadratic function in our derivations never has any point of discontinuity, it follows as a corollary of Theorem 5.2.3 that in our situation, integration by parts is workable for general distributions including discontinuities.

It can be shown that \(z_i^*\) is also the unique zero root of S(y) on \([G^{-1}(P_i),F^{-1}(P_i)]\). Indeed, applying integration by parts to (2) yields \(P_iG^{-1}(P_i)+\int _{(G^{-1}(P_i),z_i^*]}G(y)dy=P_iF^{-1}(P_i)-\int _{(z_i^*,F^{-1}(P_i)]}F(y)dy\), and to \(\int _{[\underline{y},G^{-1}(P_i)]}ydG(y)=\int _{[\underline{y},F^{-1}(P_i)]}ydF(y)\) yields \(\int _{[\underline{y},G^{-1}(P_i)]}G(y)dy-P_iG^{-1}(P_i)=\int _{[\underline{y},F^{-1}(P_i)]}F(y)dy-P_iF^{-1}(P_i)\). Summing up the two equations results in \(S(z_i^*)=0\).

Aaberge R (2009) Ranking intersecting Lorenz curves. Soc Choice Welf 33:235–259 CrossRef

Athreya KB, Lahiri SN (2006) Measure theory and probability theory. Springer Texts in Statistics, 1st edn. Springer, New York, NY. https://doi.org/10.1007/978-0-387-35434-7 CrossRef

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Chiu WH (2007) Intersecting Lorenz curves, the degree of downside inequality aversion, and tax reforms. Soc Choice Welf 28:375–399 CrossRef

Cowell F (2011) Measuring inequality. Oxford University Press, Oxford CrossRef

Davies J, Hoy M (1995) Making inequality comparisons when Lorenz curves intersect. Am Econ Rev 85:980–986

Davies J, Shorrocks AF (2000) The distribution of wealth. In: Atkinson A, Bourguignon F (eds) Handbook of income distribution, vol 1. North-Holland, Amsterdam

Gajdos T, Weymark JA (2012) Introduction to inequality and risk. J Econ Theory 147:1313–1330 CrossRef

Ibragimov M, Ibragimov R, Kattuman P, Ma J (2018) Income inequality and price elasticity of market demand: the case of crossing Lorenz curves. Econ Theor 65:729–750 CrossRef

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Shorrocks AF, Foster JE (1987) Transfer sensitive inequality measures. Rev Econ Stud 54:485–497 CrossRef

Title: Revisiting comparisons of income inequality when Lorenz curves intersect
Authors: James Davies
Michael Hoy
Lin Zhao
Publication date: 04-07-2021
Publisher: Springer Berlin Heidelberg
Published in: Social Choice and Welfare / Issue 1/2022
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-021-01343-w

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