A.3.1. Partial Rankings and Intersection Quasi‐Orderings
A.3.1. Partial Rankings and Intersection Quasi‐Orderings
Any specific statistical measure of income inequality (such as the Gini coefficient, the coefficient of variation, or the Theil measure) generates a ‘complete’ ranking that orders every pair of income distributions. So does any fully articulated complete welfare function defined over the space of income vectors. In contrast, a relation devised as a partial ordering, like the Lorenz relation, can be silent on many pairs and only record unambiguous comparisons. The very basis of its comparison, to wit, one Lorenz curve being higher everywhere (or at least, higher somewhere and lower nowhere), makes the ranking relation potentially incomplete (depending on whether the Lorenz curves to be compared cross or not).
In contrast to a designed partial order, it is also possible to arrive at a partial order on the basis of the rule of going by the congruence of different complete orderings; for example, the shared rankings of distinct complete orders generated by different statistical measures of income inequality. OEI‐1973 was much concerned with derived incomplete relations based on ‘intersections’ of complete orders—what were called ‘intersection quasi‐orderings’ (pp. 72–4). Intersection quasi‐orderings are based on unanimity according to a given set of criteria, or—equivalently—on the intersection of the orderings generated by these criteria. If the multiple criteria are welfare functions (or, alternatively, inequality measures), the intersection quasi‐ordering offers verdicts that are independent of the choice of a specific welfare function among the admitted ones (or of a particular inequality measure in the acceptable class of inequality indicators). In this section the ‘intersection (p.133) approach’ will be discussed in the context of welfare functions, and later on—in section A.4.2—the approach will be applied to the class of relative measures of inequality.
The ‘Atkinson Theorem’ regarding Lorenz dominance, for fixed mean comparisons, can itself be seen as linking up two intersection quasi‐orderings. Indeed, Lorenz dominance, which reflects the intersection of a class of inequality comparisons, coincides with the intersection quasi‐ordering generated by permissible classes of welfare functions (such as the sum total of individual utilities with a strictly concave individual utility function shared by all—the class considered by Atkinson himself).30 Subsequent work has explored important aspects of these quasi‐orderings, particularly how each treats cross‐mean comparisons and how each can be strengthened or made ‘more complete’. We now know, for example, that the Lorenz ranking is the intersection quasi‐ordering generated by all ‘relative’ inequality measures, and that the welfare intersection quasi‐ordering has its own ‘generalized’ Lorenz curve which indicates when that intersection ranking holds.
The additive welfare functions of Atkinson's Theorem have the form:
The theorems presented in Chapter 3 of OEI‐1973 effectively show that to get the former result, additivity is not needed, and that strict concavity can be relaxed to strict S‐concavity.32 This generalizes this part of the Atkinson result to a much broader class of welfare functions for which the Lorenz ranking is decisive.33 While this sufficiency result about what Lorenz ranking entails is clearly a generalization of Atkinson's theorem, the converse—that is, the necessity result which tells us what entails the Lorenz ranking—is subsumed by Atkinson's Theorem. It is redundant to check that all strictly S‐concave welfare functions give the same ranking before pronouncing that there is a Lorenz dominance here, since unanimity over the strictly smaller—additive and strictly concave—class ensures the same conclusion, viz. that is a case of Lorenz dominance (see OEI‐1973, pp. 54–5). One of the implications of this relationship is that unanimity over the smaller, additive, and strictly concave class of welfare functions ensures unanimity over the larger, general class of welfare functions—without additivity and with only strict S‐concavity. Consequently, for the special case of unanimous welfare judgements across different welfare functions, additivity and strict concavity represent no additional restriction at all.
(30) The additive case based on the sum total of individual utilities is a special application of summing individual u(y i) for all individuals i, where the u function is strictly concave, whether or not it is interpreted to be the individual utility of person i. This broader case was the one with which Atkinson himself was concerned. The result can be shown to be obtainable from other classes of not necessarily additive social welfare functions (on which see OEI‐1973, Ch. 3).
(31) Strict concavity, monotonicity, and additivity are well‐understood general properties of such real‐valued and vector‐argument functions. Replication invariance was discussed earlier, and requires that if x is obtained from y by a replication of any length (so that x = (y, . . . , y)), we have W(x) = W(y). This effectively ensures that W reflects welfare in per capita terms. Finally, symmetry requires that W(x) = W(y) whenever x is obtained from y by a permutation.
(32) Strict S‐concavity is a weaker requirement, given symmetry, than not only strict concavity but also strict quasi‐concavity. In fact, strict S‐concavity is as far as we can go; it is equivalent to the Pigou–Dalton transfer condition, with symmetry. The Pigou–Dalton transfer condition is satisfied if any transfer from a poorer to a richer person reduces social welfare W (see OEI‐1973, pp. 56, 64).