An Exponential Family of Lorenz Curves.

• Author: Sarabia, Jose-Maria

Jose-Maria Sarabia [*]

Enrique Castillo [+]

Daniel J. Slottje [++]

A new method for building parametric-functional families of Lorenz curves, generated from an initial Lorenz curve (which satisfies some regularity conditions), is presented. The method is applied to the exponential family since they use the exponential Lorenz curves as their generating curves. Several properties of these families are analyzed, including the population function, inequality measures, and Lorenz orderings. Finally, an application is presented for data from various countries. The family is shown to perform well in fitting the data across countries. The results are very robust across data sources.

1. Introduction

   The purpose of this paper is to introduce a parametric family of Lorenz curves that are obtained by a general method. In a recent paper, Sarabia, Castillo, and Slottje (1999) (SCS) introduced a method that allowed for the building of hierachies of Lorenz curves when some regularity conditions are satisfied. They introduced the Pareto family, which was found to be a flexible form and which fits actual income distribution data well. This paper introduces another family, the exponential family, which also has interesting characteristics. The exponential family involves more complex estimation with a form that is somewhat less flexible but in return gives a robust performance in fitting actual data across countries, as we will show here. The researcher or policy maker is provided another effective tool in the ongoing effort to quantify, analyze, and understand economic inequality.

   The strategy used here is to apply a Lorenz curve hierarchy that contains (as special cases) Lorenz curves derived from this general method. In section 2 we introduce the notation and some necessary background information. The general method is presented in section 3, which starts from an initial Lorenz curve [L.sub.0](p) (which is called the generating curve) and builds a family with an increasing number of parameters. These in turn can be interpreted in terms of elasticities of [L.sub.0](p). Also in section 3 we introduce the exponential family of Lorenz curves and discuss some of its properties as population functions and inequality measures and for undertaking Lorenz orderings. In section 4 we present a method for estimating Lorenz curves and apply it to the two families specified previously. Since the goodness of fit is one important criterion in the evaluation of these (and any) models, we use a method due to Gastwirth (1972) and actually incorporate his procedure into the estimation process, as will b e clear in section 4. An example of an application of our new methodology is presented in section 5. Finally, in section 6 we conclude the paper.

2. Notation and Previous Results

   In this section we use the Lorenz curve as defined by Gastwirth (1971). That is,

   DEFINITION 1. Given a distribution function F(x) with support in the subset of the positive real numbers and with finite expectation [micro], we define a Lorenz curve as

   [L.sub.F](p) = [[micro].sup.-1] [[[integral].sup.p].sub.0] [F.sup.-1](x) dx, 0 [less than or equal to] p [less than or equal to] 1, (1)

   where

   [F.sup.-1](x) = sup{y: F(y) [less than or equal to] x}.

   A characterization of the Lorenz curve that is attributed to Gaffney and Anstis by Pakes (1981) is given by the following theorem:

   THEOREM 1. Assume that L(p) is defined and continuous in the interval [0,1] with second derivative L"(p). The function L(p) is a Lorenz curve i.f.f.

   L(0) = 0, L(1) = 1, L'([0.sup.+]) [greater than or equal to] 0 for p [epsilon] (0, 1) [L.sup.n](p) [greater than or equal to] 0. (2)

   Lorenz curves allow establishing a ranking in a set of distributions functions. If two distribution functions have associated Lorenz curves that do not intersect, then they can be ordered without ambiguity in terms of welfare functions that are symmetric, increasing, and quasiconcave (Atkinson 1970; Dasgupta, Sen, and Sarret 1973; Shorrocks 1983). A distribution function [F.sub.x](x) is said to have less inequality in the Lorenz sense than a distribution function [G.sub.Y](y) if their Lorenz curves [L.sub.F](p) and [L.sub.G](p) satisfy the condition [L.sub.F](p) [greater than or equal to] [L.sub.G](p) for all p, where the sign [greater than] applies for at least one p [epsilon] (0, 1)'. In this case we write X [[less than or equal to].sub.L] Y. From the definition of the Lorenz curve (Eqn. 1), it is evident that the Lorenz partial order is invariant with respect to scale transformations, that is, X [[less than or equal to].sub.L] Y i.f.f. [lambda]X [[less than or equal to].sub.L] vY for all [lambda], v [grea ter than] 0.

   THEOREM 2. Let L(p) be a Lorenz curve and consider the transformation

   [L.sub.[alpha]](p) = [P.sup.[alpha]]L(p), [alpha] [greater than or equal to] 0. (3)

   Then, if [alpha] [greater than or equal to] 1, [L.sub.[alpha]](p) is a Lorenz curve, too. In addition, if 0 [less than or equal to] [alpha] [less than] 1 and [L.sup.m](p) [greater than or equal to] 0, [L.sub.[alpha]](p) is also a Lorenz curve.

   THEOREM 3. If L(p) is a Lorenz curve,

   [L.sub.[gamma]](p) = L[(p).sup.[gamma]], [gamma] [greater than or equal to] 1 (4)

   is a Lorenz curve. Since [L.sub.[gamma]](p) is an increasing convex transform of L(p) and [L.sub.[gamma]](0) = 0 and [L.sub.[gamma]](1) = 1, [L.sub.[gamma]](p) is a Lorenz curve as well. We now present several examples to demonstrate the usefulness of these theorems.

   One well-known form of the Lorenz curve is that attributable to Rasche et al. (1980). Other forms are due to Kakwani and Podder (1973) and Kakwani (1980). Rasche et al. (1980) showed that Kakwani's Lorenz curve does not satisfy all the requirements for a Lorenz curve. Using our Theorem 1, we find a modified Lorenz curve:

   L(p; a, [beta]) = p - ap[(1 - p).sup.[beta]], 0 [less than or equal to] a [less than or equal to] 1; 0 [less than] [beta] [less than or...