Construction of concentration measures for General Lorenz curves using Riemann-Stieltjes integrals

Abstract

Lorenz curves were invented to model situations of inequality in real life and applied in econometrics (distribution of wealth or poverty), biometrics (distribution of species richness), and informetrics (distribution of literature over their producers). Different types of Lorenz curves are hereby found in the literature, and in each case a theory of good concentration measures is presented. The present paper unifies these approaches by presenting one general model of concentration measure that applies to all these cases. Riemann-Stieltjes integrals are hereby needed where the integrand is a convex function and the integrator a function that generalizes the inverse of the derivative of the Lorenz function, in case this function is not everywhere differentiable. Calling this general measure C we prove that, if we have two Lorenz functions f, g such that f < g, then C(f) < C(g). This general proof contains the many partial results that are proved before in the literature in the respective special cases.