Social Welfare Comparisons

One of the most fruitful extensions of Atkinson (1970) has been the development of methods for comparisons of income distributions in terms of their social welfare, as well as in terms of their inequality. As in Tony’s original article, the contributions refer to both dominance (partial orderings) and scalar indices (complete orderings) for social welfare comparisons.

Social welfare comparisons are sensitive to differences in real income levels between two distributions as well as inequality differences and allow for some trade-off between the two (Tony’s inequality comparisons ignore differences in mean income by construction). For example, it may be the case that increases in real income levels are judged sufficient to compensate for some increase in income inequality. Shorrocks (1983) showed that unanimous social welfare orderings according to all measures consistent with essentially the same social welfare function as Tony’s were equivalent to orderings by non-intersecting generalised Lorenz curves. A generalised Lorenz curve is simply a standard Lorenz curve scaled up vertically at each point by the mean income of the distribution. (See also Kakwani 1984.) A dominance result was also proved by Saposnik (1981), who showed that unanimous orderings of all symmetric increasing (but not necessarily concave) social welfare functions were equivalent to orderings by non-intersecting cumulative distribution functions of income. These second- and first-order social welfare dominance results have proved particularly useful because they also have applications in poverty comparisons (see below) and to examinations of trends in social welfare over time and across countries. Scalar indices of social welfare have also been used for these purposes, and they are often simply Tony’s 1970 equally distributed equivalent income or monotonic transformations of it: see, for example, Jenkins (1997) or Gruen and Klasen (2008). A feature of these ‘real income’ indices, as Sen (1976a) labelled them, is that choosing different values for the inequality aversion parameter allows checking of the sensitivity of results to differences in social judgements.

When Lorenz or generalised Lorenz curves cross, we do not have unanimous inequality and social welfare orderings. Several authors proved how orderings can be derived, nonetheless, in such cases if more assumptions are made about the social welfare function: in particular if it is assumed that not only does a rank-preserving transfer from a richer person to a poorer person reduce inequality, but also that the inequality reducing impact of this is larger the poorer the recipient of the transfer (transfer sensitivity). Dardanoni and Lambert (1988) and Shorrocks and Foster (1987) show that where generalised Lorenz curves cross just once, unanimous orderings of distributions consistent with this revised class of evaluation functions depend on the variances of the distributions. Davies and Hoy (1995) develop results for the case where Lorenz curves intersect potentially several times. As it happens, the key result for the case of a single crossing of Lorenz curves was already in an unpublished paper circulated by Tony in 1973, subsequently published as a ‘Rediscovered Classic’ in 2008 by the Journal of Economic Inequality (Atkinson 2008a).