This paper examines the notion of intermediate inequality and its measurement. Specifically, we investigate whether the intermediateness of an intermediate measure can be preserved through repeated (affine) inequality-neutral income transformation. For all existent intermediate measures of inequality, we show that the intermediateness cannot be preserved through the transformation; each intermediate measure tends to either a relative measure or an absolute measure. This observation is then generalized to the class of unit-consistent inequality measures. An inequality measure is unit-consistent if inequality rankings by the measure are not affected by the measuring units in which incomes are expressed. We show that the unit-consistent class of intermediate measure of inequality consists of generalizations of an existent intermediate measure and, hence, the intermediateness also cannot be retained in the limit through transformations.
Zheng, B. (2004), "ON INTERMEDIATE MEASURES OF INEQUALITY", Bishop, J.A. and Amiel, Y. (Ed.) Studies on Economic Well-Being: Essays in the Honor of John P. Formby (Research on Economic Inequality, Vol. 12), Emerald Group Publishing Limited, Bingley, pp. 135-157. https://doi.org/10.1016/S1049-2585(04)12005-X
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