The Lorenz curve in economics and econometrics

This paper surveys selected applications of the Lorenz curve and related stochastic orders in economics and econometrics, with a bias towards problems in statistical distribution theory. These include characterizations of income distributions in terms of families of inequality measures, Lorenz ordering of multiparameter distributions in terms of their parameters, probability inequalities for distributions of quadratic forms, and Condorcet jury theorems.Lorenz curve, Lorenz order, majorization, income distribution, income inequality, statistical distributions, characterizations, Condorcet jury theorem.

Ranking Intersecting Lorenz Curves

This paper is concerned with the problem of ranking Lorenz curves in situations where the Lorenz curves intersect and no unambiguous ranking can be attained without introducing weaker ranking criteria than first-degree Lorenz dominance. To deal with such situations two alternative sequences of nested dominance criteria between Lorenz curves are introduced. At the limit the systems of dominance criteria appear to depend solely on the income share of either the worst-off or the best-off income recipient. This result suggests two alternative strategies for increasing the number of Lorenz curves that can be strictly ordered; one that places more emphasis on changes that occur in the lower part of the income distribution and the other that places more emphasis on changes that occur in the upper part of the income distribution. Both strategies turn out to depart from the Gini coefficient; one requires higher degree of downside and the other higher degree of upside inequality aversion than what is exhibited by the Gini coefficient. Furthermore, it is demonstrated that the sequences of dominance criteria characterize two separate systems of nested subfamilies of inequality measures and thus provide a method for identifying the least restrictive social preferences required to reach an unambiguous ranking of a given set of Lorenz curves. Moreover, it is demonstrated that the introduction of successively more general transfer principles than the Pigou-Dalton principle of transfers forms a helpful basis for judging the normative significance of higher degrees of Lorenz dominance. The dominance results for Lorenz curves do also apply to generalized Lorenz curves and thus provide convenient characterizations of the corresponding social welfare orderings.generalized Gini families of inequality measures, rank-dependent measures of inequality, Gini coefficient, partial orderings, Lorenz dominance, Lorenz curve, general principles of transfers

Modeling inequality and spread in multiple regression

We consider concepts and models for measuring inequality in the distribution of resources with a focus on how inequality varies as a function of covariates. Lorenz introduced a device for measuring inequality in the distribution of income that indicates how much the incomes below the u$^{th}$ quantile fall short of the egalitarian situation where everyone has the same income. Gini introduced a summary measure of inequality that is the average over u of the difference between the Lorenz curve and its values in the egalitarian case. More generally, measures of inequality are useful for other response variables in addition to income, e.g. wealth, sales, dividends, taxes, market share and test scores. In this paper we show that a generalized van Zwet type dispersion ordering for distributions of positive random variables induces an ordering on the Lorenz curve, the Gini coefficient and other measures of inequality. We use this result and distributional orderings based on transformations of distributions to motivate parametric and semiparametric models whose regression coefficients measure effects of covariates on inequality. In particular, we extend a parametric Pareto regression model to a flexible semiparametric regression model and give partial likelihood estimates of the regression coefficients and a baseline distribution that can be used to construct estimates of the various conditional measures of inequality.Comment: Published at http://dx.doi.org/10.1214/074921706000000428 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

The Transmuted Weibull-Pareto Distribution

A new generalization of the Weibull-Pareto distribution called the transmuted Weibull-Pareto distribution is proposed and studied. Various mathematical properties of this distribution including ordinary and incomplete moments, quantile and generating functions, Bonferroni and Lorenz curves and order statistics are derived. The method of maximum likelihood is used for estimating the model parameters. The flexibility of the new lifetime model is illustrated by means of an application to a real data set

Single Crossing Lorenz Curves and Inequality Comparisons

Since the order generated by the Lorenz criterion is partial, it is a natural question to wonder how to extend this order. Most of the literature that is concerned with that question focuses on local changes in the income distribution. We follow a different approach, and define uniform $\alpha$-spreads, which are global changes in the income distribution. We give necessary and sufficient conditions for an Expected Utility or Rank-Dependent Expected Utility maximizer to respect the principle of transfers and to be favorable to uniform $\alpha$-spreads. Finally, we apply these results to inequality indices.Inequality measures, Intersecting Lorenz Curves, Spreads

Mobility and Long term Equality of Opportunity

The aim of this paper is to propose a methodology for evaluating long-term income distributions according to the equality of opportunity principle; we propose partial and complete rankings of long term income distributions and show the relationship between the inequality of oppor- tunity in the single periods of time and inequality of opportunity in the long run. We show that this relationship can be interpreted in terms of intragenerational mobility. In general, it is possible to state that mobility can act as an equalizer of opportunities when the accounting period is extended.Equality of opportunity, income mobility, inequality, social welfare

Measuring Inequality Change in an Economy with Income Growth

This paper analyzes how to measure changes in inequality in an economy with income growth. The discussion distinguishes three stylized kinds of economic growth: high income sector enrichment, low income sector enrichment, high income sector enlargement, in which the high income sector expands and absorbs persons from the low income sector. The two enrichment types pose no problem for assessing inequality change in the course of economic growth: for high income sector enrichment growth, inequality might reasonably be said to increase, whereas for low income sector enrichment, inequality might be said to decrease. These adjustments are non-controversial and non-problematical. Where problems arise is in the case of high income sector enlargement growth. In that case, the two alternative approaches have been shown in this paper to yield markedly results: The traditional inequality indices generate an inverted-U pattern of inequality. That is, inequality rises in the early stages of high income sector enlargement growth and falls thereafter. The new approach suggested here, based on axioms of gap inequality and numerical inequality, generates a U pattern of inequality. That is, inequality falls in the early stages of high income sector enlargement growth and rises thereafter. The discrepancy between the familiar indices and the alternative approach based on axioms of gap inequality and numerical inequality bears further scrutiny. Two courses of action are possible. One might try to axiomatize inequality in ways that generate an inverted-U pattern in high income sector enlargement growth, thereby rationalizing the continued use of the usual inequality indices with the inverted-U property. Alternatively, one might retain the axioms proposed here, embed them into a more formal structure, and construct a family of inequality indices consistent with them. Others might wish to pursue the first course; I am at work on the second

Asymptotic Distribution Theory of Empirical Rank-dependent measures of Inequity

A major aim of most income distribution studies is to make comparisons of income inequality across time for a given country and/or compare and rank different countries according to the level of income inequality. However, most of these studies lack information on sampling errors, which makes it difficult to judge the significance of the attained rankings. The purpose of this paper is to derive the asymptotic properties of the empirical rank-dependent family of inequality measures. A favourable feature of this family of inequality measures is that it includes the Gini coefficients, and that any member of this family can be given an explicit and simple expression in terms of the Lorenz curve. By relying on a result of Doksum [14] it is easily demonstrated that the empirical Lorenz curve, regarded as a stochastic process, converges to a Gaussian process. Moreover, this result forms the basis of the derivation of the asymptotic properties of the empirical rank-dependent measures of inequality.

Third-Degree Stochastic Dominance and the von-Neumann-Morgenstern Independence Property

This paper is an investigation of the third-degree stochastic dominance order which has been introduced in the context of risk analysis and is now receiving an increased attention in the area of inequality measurement. After observing that this partial order fails to satisfy the von Neumann-Morgenstern property in the space of random variables, we introduce strong and local third-degree stochastic dominance. We motivate these two new binary relations and o\ufb00er a complete and simple characterizations in the spirit of the Lorenz characterization of the second-degree stochastic order. The paper compares our results with the closest literature. JEL Classification Numbers: D31, D63