Hang K. Ryu

Two flexible functional form approaches for approximating the Lorenz curve

Abstract This paper introduces two flexible form approaches to approximate Lorenz curves. The first approach expands the inverse function of an income distribution in an exponential polynomial series and derives the Lorenz curve from it. The required convexity condition can be imposed using a Bayesian method. The second approach approximates the Lorenz curve with a sequence of Berstein polynomial functions. The required convexity condition is automatically established in this approach. We compare these approaches with other well-known fixed functional form approaches. We evaluate the performance of these functional forms by comparing approximation errors, maximum error, and the estimates of the Gini coefficient produced by various approaches.

Estimating the density of unemployment duration based on contaminated samples or small samples

In estimating a density function for the duration of unemployment, we consider two departures from what would be ideal conditions. If the so-called digit preference effect produces local distortion in observed samples, we can apply a maximum entropy density estimation method. To establish the functional form of the density, we maximize entropy subject to moment restrictions. The global shape of the density is determined by the lower ordered sample moments which are not affected much by the digit preference effect. As a by-product of this method, we can establish the local transition structure of the digit preference effect. As a second case of departure from an ideal condition, we consider coarse sample observations where unemployment duration was observed only for 4, 10, 14, 26, and 52 weeks. Once the unemployment duration density is derived, quintile behavior over time, the Lorenz curve, and the Gini coefficient for the distribution of unemployment duration can be obtained. Finally, we discuss the ramifications of only focusing on the headcount ratio of unemployment when other information is available.

Parametric Apppoximations of the Lorenz Curve

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

Some New Functional Forms For Approximating Lorenz Curves

This chapter introduces two flexible functional form approaches to approximate Lorenz curves. The first approach expands the inverse function of an income distribution in an exponential polynomial series and derives the Lorenz curve from it. The required convexity condition can be imposed using a Bayesian method. The second approach approximates the Lorenz curve with a sequence of Bernstein polynomial functions. The required convexity condition is automatically established in this approach. We compare these approaches with another well known fixed functional form approaches. We evaluate the performance of these functional forms by comparing approximation errors, maximum error, and the estimates of the Gini coefficient produced by various approaches.

Comparing Income Distributions Using Index Space Representations

The purpose of this chapter is to introduce a new method to compare income distributions. The methodology used here allows us to examine the relationship between the observed income graduation in 1974 and the observed income graduation in 1990 in order to obtain a different perspective on how (and perhaps why) income inequality in the U.S. increased over that 16 years period. The main feature of this chapter is that we introduce an index space representation to compare two income distributions. This concept will be explained below.

A New Method for Estimating Limited Dependent Variables: An Analysis of Hunger

The issue of hunger in America was considered resolved in the 1970s after attracting so much attention in the 1960’s. President Johnson’s War on Poverty was thought to have solved the problem once and for all. The poverty rate in the United States began to fall drastically with the onslaught of the Great Society Programs in the 1960’s and continued to do so until the early 1970s when the poverty rate started creeping up again. In the early 1980’s the poverty rate again began to level off, only to rise again during the Reagan and Bush eras, cf. Jorgenson (1990).

Capabilities and Earnings Inequality

Economists have been intrigued by the relationship between unobserved heterogeneity among individuals and the shape of the observed size distribution of earnings for over two hundred years. In this chapter, capability is a description of how an individual combines various attributes in the labor market to produce market outcomes such as observed earnings.16 Ability and capability may differ because a given individual may have ability which when combined with other attributes does not result in performing in the labor market in a positive way. Earlier researchers thought that measurable proxies for capability should be normally distributed, while clearly empirical earnings distributions have non-zero skewness and kurtosis, cf. Sahota (1978) and Creedy (1985) for discussion of this early work. The causes of the departure from normality in earnings graduations have been debated for almost a century. So-called “stochastic theories” were predicated on the assumption that stochastic or random occurrences (like luck) played a major role in changing the shapes of the observed distributions.

Coordinate Space Versus Index Space Representations as Estimation Methods: An Application to How Macro Activity Affects the U.S. Income Distribution

This article examines the impact of macroeconomic variables on the level of income inequality in the United States. The innovation is that we introduce what is known as an index space representation to do so. In an index space representation, income shares are considered as a linear combination of what are known as index functions. An index function is a polynomial function whose order is specified by the given index. The macroeconomic variables affect the function parameters corresponding to each index function. In early years, we have only quintile data, but we can still derive a well-behaved Lorenz curve, Gini coefficient, Theils entropy measure, standard deviation of the logarithm of the share function, and Atkinsons inequality measure.