Parametric Lorenz Curves: Models and Applications

The Lorenz curve (LC) is an important instrument for analyzing the size of distribution of income or wealth and inequality. Finding an appropriate functional form is an important practical and theoretical problem. In this chapter we study parametric models for the LC and some important applications. The basic properties that a function should satisfy in order to be a genuine LC are discussed. Next, we study the different ways for generating parametric families of LCs, as well as some of their basic properties, including their relationship with the underlying income distribution function. The basic parametric models proposed in the literature are studied, including the Pareto, lognormal and other important families of LCs. Some general strategies to obtain extensions and generalizations of the basic parametric models are presented. One of the main applications of LCs is the study of inequality. We begin studying different measures of inequality together with their expressions in terms of the LC. These measures include the Gini index and some of their generalizations proposed by Kakwani (1980) and Yitzhaki (1983). Their corresponding expressions for the proposed parametric families of LCs will be obtained. The Lorenz ordering is also studied. The Lorenz ordering is a partial order that allows the comparison of two distributions when its corresponding LCs do not intersect. Some basic properties of this order are studied, including the effect of transformations, its relations with other partial orderings and their application to important parametric income distributions. The recent proposal of multivariate versions of the LC are studied. Finally, some applications of the Lorenz curve are presented.