Study about Gini Coefficient and Discontinuity: Contribution to the Analysis of a Transformation

Abstract

This chapter reveals a discontinuity in the mapping from a Lorenz curve to the associated cumulative distribution function. The Gini co-efficient is an important tool for analyzing income or wealth distribution within a country or region, but it should not be mistaken for an absolute measurement of income or wealth. A high-income country and a low-income country might have the same Gini co-efficient, even with rather different income distributions. The issue is mathematical in nature and is based on an examination of how a bounded random variable's distribution function gets converted into its Lorenz curve. It will be proven that the transformation from a finite income distribution to its Lorenz curve is a continuous bijection with respect to the L^q ([0,1])-metric – for every q \(\ge\) 1. The inverse transformation, however, is not continuous for any q \(\ge\) 1. This implies a more careful attitude when interpreting the value of a Gini coefficient.  Another issue is that you cannot trust the associated distribution to be an accurate representation of the underlying income distribution if you computed a Lorenz curve using empirical data. Generalisations in several directions are possible when calculating the Gini coefficient using Lorenz curves. One that connects the Lorenz curve to variance is included here.