Income inequality measures and the middle class

Abstract

We study how the presence of the middle class in the sense of Gevorgyan-Malykhin affects the value of income inequality measures including the Gini coefficient J and the Hoover index H. It is proved that in the presence of the middle class (1) $J \leqslant \frac{1}{2}\frac{{L'\left( 0 \right)}}{2}$J≤12L′(0)2 (where L is the Lorenz function), (2) $H \leqslant \frac{1}{2}$H≤12, (3) the longest vertical distance between the diagonal and the Lorenz curve (which is equal to H) is attained at ${z_0} 0. Tight upper and lower bounds for the differential deviation in terms of the Gini coefficient are found as well.