Some axiomatics of inequality measurement, with specific reference to intermediate indices

Working Paper 2017-445

Abstract

In this paper we consider properties for inequality measurement. A property that should be demanded for every inequality measure we call “axiom” otherwise we call it only “desideratum”. The most important new axiom (A6) and the desideratum “Cowell’s Feature (CF)” are motivated carefully. (A6) is more restrictive than Zheng’s (2007a) “unit consistency axiom” for partial inequality orderings, but it is not as restrictive as the overwhelmingly favoured “scale invariance” property. We will show that the combination of these two properties characterizes a type of differentiable inequality measure that the author had already introduced and characterized in 1994, but then with a stronger requirement. Since then, this measure has been widely employed in applied work because it has been perceived to possess some attractive properties. However, the aim of this paper is not only a better characterization of a single type of inequality measure, but also a numerical comparison of different “good” inequality measures that qualify under (A6). Our focus lies on the so-called intermediate measures, being a compromise between the scale invariant “relative inequality measures” and the translation invariant “absolute inequality measures”, where equal absolute changes in all incomes do not affect the inequality value. We present three methods to construct strictly intermediate or centrist inequality measures, which are explained with the help of three examples. Then we undertake a comparison of how these illustrative measures satisfy our axioms. Finally we give a complete summary table showing all the properties of these inequality measures. A last relevant example has been taken from true life.

Authors: Manfred Krtscha.