Parametric Apppoximations of the Lorenz Curve

Abstract

The purpose of this chapter is to discuss parametric approximations to the Lorenz curve. There is a relatively large literature on distribution-free statistical Lorenz curves which have been used to analyze the welfare implications of Atkinson’s (1970) notion of Lorenz dominance. This work is discussed in other places in this book. The reader may then question the value of imposing a functional form on the Lorenz curve when an empirical Lorenz curve can be constructed “distribution-free.”2 Another problem with that body of work, however, is that it has focused on Lorenz ordinates for quintiles, deciles, etc., in a quest for “crossings” which yield information about social welfare implications. As such, those methods do not consider the shape of the entire Lorenz curve, but rather, only provide information about discrete piece-wise segments. Another related question is why should a researcher be interested in approximating the Lorenz curve at all when the Lorenz curve can be calculated directly from the empirical data? There are several reasons to do so as we argue in (1996). Suppose you have 65,000 or more observations of income receiving units (as is the case for the Current Population Survey for 1990 and beyond). It is now possible to graphically represent this information quite well with all these thousands of data points with modern personal computers with immense computing power. The problem still remains, however, as to how to describe these empirical Lorenz curves mathematically and statistically and how to summarize the inequality inherent in these empirical Lorenz curves. Using parametric approximations to the Lorenz curve, we can summarize thousands of observations by estimating just a few parameters and find that we can approximate the empirical Lorenz curve very well. Certainly if this is the case (and we argue below that it is), then a parametric representation is certainly parsimonious and worthwhile. Another reason to estimate the Lorenz curve directly is that one can then estimate the density function at any point, cf. Kakwani (1993). Arnold (1983, 1986, 1987) and Arnold et al. (1987) suggested methods to examine nested families of the Lorenz curve. The parametric approximation of the Lorenz Curve is also useful because it makes the construction of inequality measures possible. The most popular measure of inequality is of course the Gini coefficient. The Gini has a natural interpretation from the Lorenz curve and the two are frequently discussed in tandem. Finally, the use of a functional form for the Lorenz curve allows us to detect possible “laws” from our data that it would not otherwise be possible to detect.3