The Gini coefficient and discontinuity

Abstract

This article reveals a discontinuity in the mapping from a Lorenz curve to the associated cumulative distribution function. The problem is of a mathematical nature—based on an analysis of the transformation between the distribution function of a bound random variable and its Lorenz curve. It will be proven that the transformation from a normalized income distribution to its Lorenz curve is a continuous bijection with respect to the (Formula presented.) ([0,1])-metric—for every q ≥ 1. The inverse transformation, however, is not continuous for any q ≥ 1. This implies a more careful attitude when interpreting the value of a Gini coefficient. A further problem is that if you have estimated a Lorenz curve from empirical data,then you cannot trust that the associated distribution is a good estimate of the true income distribution.