# Standing on Atkinson's Shoulders: Subsequent Research

Tony’s 1970 article stimulated much research by others, keen to build on his key insights and to extend them. Tony continued to develop his work as well of course. In this section, I classify developments under four main headings: the nature of the social welfare function, social welfare comparisons, poverty comparisons, and multidimensional comparisons.

## Variations in the Social Welfare Function

Other works have generalised Tony’s 1970 results or taken his ideas in a different direction. Early works addressed issues relating to the social welfare function. For example, Dasgupta et al. (1973) generalised Tony’s Lorenz dominance result to all Schur-concave evaluation functions, not only concave ones. Blackorby and Donaldson (1978) showed how to uncover the social judgement hidden in any equality index, while Newbery (1970) suggested rejecting the Gini coefficient because of its incompatibility with an additively separable function, causing Sheshinski (1972) to respond that additive separability has no particular appeal. Others have explored different concepts of inequality—for instance, when inequality refers to absolute differences in income rather than relative differences as in the Atkinson (1970) approach. It turns out that there is a family of absolute inequality indices (with a parameter encapsulating the degree of absolute rather than relative inequality aversion) that is analogous to Tony’s class of relative inequality indices (Kolm 1976a, b). Moyes (1987) introduced the concept of an absolute Lorenz curve and showed that unanimous orderings by all standard absolute inequality indices were equivalent to orderings by non-intersecting absolute Lorenz curves—a dominance result analogous to Tony’s 1970 Lorenz dominance result for relative income differences.

In the Atkinson tradition, the properties of inequality indices are not derived directly but are related to the properties which are placed on the social welfare function. An important sub-branch of the inequality literature has taken a different approach, characterising general classes of inequality measures by specifying properties on the measures directly ab initio. The four basic axioms placed on inequality indices in this tradition are labelled symmetry, scale invariance, replication invariance, and the principle of transfers. Fields and Fei (1978) and Foster (1985) show that orderings according to all inequality indices satisfying these axioms are equivalent to orderings by non-intersecting Lorenz curves, which is of course the Atkinson 1970 result. Moreover, if one adds one more axiom, that of decomposability by population subgroup, then one gets the so-called single-parameter generalised entropy class of inequality indices (see, for example, Bourguignon 1979; Cowell 1980; Shorrocks 1980, 1984). But it turns out that this class includes the entire class of Atkinson inequality indices (monotonically transformed). Thus, Atkinson indices not only link directly to fundamental welfare economic foundations in terms of social welfare functions, but they are also decomposable—a property which has turned out to be remarkably useful in empirical applications (Cowell and Jenkins 1995). (See Atkinson (1993) and Jenkins (1995) for examples that exploit this property to help explain why UK inequality rose so much during the 1980s.)