On Inequality Comparisons

Abstract

answer. There are two points of contention. One is the issue of cardinality vs. ordinality. Practitioners of the cardinal approach compare distributions by means of summary measures such as a Gini coefficient, variance of logarithms, and the like. For purposes of ranking the relative inequality of two distributions, the cardinality of the usual measures is not only a source of controversy, but it is also redundant.3 Accordingly, some researchers prefer an ordinal approach, adopting Lorenz domination as their criterion. The difficulty with the Lorenz criterion is its incompleteness, affording rankings of only some pairs of distributions but not others. Current practice in choosing between a cardinal or an ordinal approach is now roughly as follows: Check for Lorenz domination in the hope of making an unambiguous comparison; if Lorenz domination fails, calculate one or more cardinal measures. This raises the second contentious issue: which of the many cardinal measures in existence should one adopt? The properties of existing measures have been discussed extensively in several recent papers.4 Typically, these studies have started with the measures and then examined their properties. In this paper, we reverse the direction of inquiry. Our approach is to start by specifying as axioms a relatively small number of properties which we believe a good index of inequality should have and then examining whether the Lorenz criterion and the various cardinal measures now in use satisfy those properties. The key issue is the reasonableness of the postulated properties. Work to date has shown the barrenness of the Pareto criterion.5 Only recently have researchers begun to develop an alternative axiomatic structure.6 The purpose of this paper is to contribute to such a development.