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.
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.
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 \(S(z_i^*)=0\).