Econometrica

In this paper, we have developed an approach to inequality comparisons which differs from the conventional one. Beginning by postulating three axioms, we showed that the axiomatic system so constructed is sufficient to justify the Lorenz criterion for inequality comparisons. However, like the Lorenz criterion, the axiomatic system is incomplete. Past researchers have achieved completeness by the use of cardinal inequality measures. We showed that many but by no means all of the commonly used indices satisfy our three axioms. The ones which do satisfy the axioms agree on the ranking of distributions whose Lorenz curves do not intersect. However, when Lorenz curves do intersect, the various measures partition the income distribution space differently. Since the three axioms are insufficient to determine the specific partition to use, the use of any of the conventional measures implicitly accepts the additional welfare judgments associated with that measure. The key issue for inequality comparisons is the reasonableness of the ordering criterion, which in the case of cardinal measures is the index itself. An axiomatic approach is probably the ideal method for confronting this issue, because the reasonable properties (i.e., the axioms) are postulated explicitly. At minimum, this approach facilitates communication by enabling (and indeed requiring) one to set forth clearly his own viewpoints and value judgments for scrutiny by others. But in addition, to the extent that one person's judgments (such as those in our three axioms) are acceptable to others, controversies over inequality comparisons may be resolved. We have seen that our three axioms are incomplete insofar as they cannot determine the ordinal ranking uniquely. A feasible and desirable direction for future research is to investigate what further axioms could be introduced to complete the axiomatic system or at least to reduce further the zones of ambiguity. It is conceivable that beyond some point the search for new axioms may turn out to be unrewarding. In that case, inequality comparisons will always be subject to arbitary specifications of welfare weights. The selection of suitable weights by whatever reasonable criterion one cares to exercise is a less desirable but possibly more practical alternative than a strictly axiomatic approach. Our research has hopefully made clear that inequality comparisons cannot be made without adopting value judgments, explicit or otherwise, about the desirability of incomes accruing to persons at different positions in the income distribution. Even the Lorenz criterion, which permits us to rank the relative inequality of different distributions in only a fraction of the cases, embodies such judgments. The traditional inequality indices such as those considered in Section 3, to the extent they complete the ordering, embody some value judgments