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.     

Detecting a change in wealth concentration without the knowledge of the wealth distribution

Consumption is commonly used as a proxy for permanent income. We go a step further by establishing the link between the distribution of consumption and that of permanent income in terms of dominance orderings. We introduce two new dominance orderings, the Generalized Top Lorenz test and the related affluence ordering. If consumption is a concave function of permanent income, we get an indirect and robust method to detect a change in permanent income concentration when the full stream of income receipts along the life cycle is unknown and only consumption data are available. Our application on US data for the period 1980?C2002 points out the difficult start in life of people belonging to the ??Baby loser generation?? (people born in the sixties) with respect to the previous and following cohort.     

Distorted Lorenz curves: models and comparisons

The economic literature contains many parametric models for the Lorenz curve. A number of these models can be obtained by distorting an original Lorenz curve $L$ by a function $h$ , giving rise to a distorted Lorenz curve ${\widetilde{L}}=h\circ L$ . In this paper, we study, in a unified framework, this family of curves. First, we explore the role of these curves in the context of the axiomatic structure of Aaberge (2001) for orderings on the set of Lorenz curves. Then, we describe some particular models and investigate how changes in the parameters in the baseline Lorenz curve $L$ affect the transformed curve ${\widetilde{L}}$ . Our results are stated in terms of preservation of some stochastic orders between two Lorenz curves when both are distorted by a common function.     

Robust inequality comparisons

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, Aaberge (Soc Choice Welf 33:235–259, 2009) introduced two alternative sequences of nested dominance criteria for Lorenz curves, which proved to characterize two separate systems of nested subfamilies of inequality measures. This paper uses the obtained characterization results to arrange the members of two different generalized Gini families of inequality measures into subfamilies according to their relationship to Lorenz dominance of various degrees. Since the various criteria of higher degree Lorenz dominance provide convenient computational methods, these results can be used to identify the largest subfamily of the generalized Gini families, and thus the least restrictive social preferences, required to reach unambiguous ranking of a set of Lorenz curves. We further show that the weight-functions of the members of the generalized Gini families offer intuitive interpretations of higher degree Lorenz dominance, which generally has been viewed as difficult to interpret because they involve assumptions about third and higher derivatives. To demonstrate the usefulness of these methods for empirical applications, we examine the time trend in income and earnings inequality of Norwegian males during the period 1967–2005.     

Averaging Lorenz curves

A large number of functional forms has been suggested in the literature for estimating Lorenz curves that describe the relationship between income and population shares. The traditional way of overcoming functional-form uncertainty when estimating a Lorenz curve is to choose the function that best fits the data in some sense. In this paper we describe an alternative approach for accommodating functional-form uncertainty, namely, how to use Bayesian model averaging to average the alternative functional forms. In this averaging process, the different Lorenz curves are weighted by their posterior probabilities of being correct. Unlike a strategy of picking the best-fitting function, Bayesian model averaging gives posterior standard deviations that reflect the functional-form uncertainty. Building on our earlier work (Chotikapanich and Griffiths, 2002), we construct likelihood functions using the Dirichlet distribution and estimate a number of Lorenz functions for Australian income units. Prior information is formulated in terms of the Gini coefficient and the income shares of the poorest 10% and poorest 90% of the population. Posterior density functions for these quantities are derived for each Lorenz function and are averaged over all the Lorenz functions.     

Kernel density estimation on grouped data: the case of poverty assessment

Grouped data have been widely used to analyze the global income distribution because individual records from nationally representative household surveys are often unavailable. In this paper we evaluate the performance of nonparametric density smoothing techniques, in particular kernel density estimation, in estimating poverty from grouped data. Using Monte Carlo simulations, we show that kernel density estimation gives rise to nontrivial biases in estimated poverty levels that depend on the bandwidth, kernel, poverty indicator, size of the dataset, and data generating process. Furthermore, the empirical bias in the poverty headcount ratio critically depends on the poverty line. We also undertake a sensitivity analysis of global poverty estimates to changes in the bandwidth and show that they vary widely with it. A comparison of kernel density estimation with parametric estimation of the Lorenz curve, also applied to grouped data, suggests that the latter fares better and should be the preferred approach.     

Utility-gap Dominances and Inequality Orderings

The justification for using Lorenz dominance as an inequality ranking condition has been based on the aggregate social welfare comparison and the Pigou–Dalton principle of transfers. Since both the aggregating aspect of the social welfare function and certain implications of the principle of transfers are debatable, ordering conditions stronger than Lorenz dominance are worth exploring. A particularly interesting direction to pursue is to follow the frequently invoked notion that inequality is the “gap” between the rich and the poor. This paper follows this notion to formally propose a unified utility-gap concept and characterizes several utility-gap based conditions as general stronger-than-Lorenz-dominance ranking criteria. Specifically, we propose utility-gap dominance which requires all pair-wise utility-gaps in one distribution to be uniformly smaller than those of the other distribution. We then explore a conceptually weaker dominance concept – quasi dominance – which imposes conditions only on the gap between each person’s utility and some reference utility point of the distribution. I am grateful to two anonymous referees and Peter Lambert for their very constructive comments and suggestions on an earlier version of the paper. The usual caveat applies.     

Estimating trends in US income inequality using the Current Population Survey: the importance of controlling for censoring

We analyze trends in US size-adjusted household income inequality between 1975 and 2004 using the most commonly used data source—the public use version of the March Current Population Survey. But, unlike most researchers, we also give substantial attention to the problems caused by the topcoding of each income source in the CPS data. Exploiting our access to Census Bureau internal CPS data, we examine estimates from data incorporating imputations for topcoded incomes derived from cell means and estimates from data multiply-imputed from parametric distribution models. Our analysis yields robust conclusions about inequality trends. The upward trend in US income inequality that began in the mid-1970s and increased in the 1980s slowed markedly after 1993.     

Inequality orderings and unit consistency

The paper examines the implications of the newly proposed unit consistency axiom for partial inequality orderings. We first show that some intermediate Lorenz dominance conditions violate the axiom. We then characterize a class of intermediate Lorenz orderings and demonstrate that the only unit-consistent member is the one related to Krtscha (Models and measurement of welfare and inequality. Springer, Heidelberg, 1994)’s intermediate notion of inequality which has recently been investigated by Zoli (A surplus sharing approach to the measurement of inequality. Discussion paper no. 98/25, University of York, 1998; Logic, game, theory and social choice. Tilburg University Press, Tilburg, 1999) and Yoshida (Soc Choice Welf 24:557–574, 2005). Finally, we provide a general characterization for unit-consistent Lorenz orderings and the Krtscha-type dominance again turns out to be the only one that is intermediate and unit-consistent.     

Intersecting Lorenz Curves, the Degree of Downside Inequality Aversion, and Tax Reforms

This paper clarifies the conceptual distinction of downside inequality aversion (or transfer sensitivity) as a normative criterion for judging income distributions from the Pigou-Dalton principle of transfers. We show that when the Lorenz curves of two income distributions intersect, how the change from one distribution to the other is judged by an inequality index exhibiting downside inequality aversion often depends on the relative strengths of its downside inequality aversion and inequality aversion. For additive inequality indices or their monotonic transformations, a measure characterizing the strength of an index’s downside inequality aversion against its inequality aversion is shown to determine the ranking by the index of two distributions whose Lorenz curves cross once. The precise condition under which the same result generalizes to the case of multiple-crossing Lorenz curves is also identified. The results are particularly useful in understanding the distributional impact of tax reforms. I received exceptionally helpful comments from Mike Hoy, Peter Lambert, the Editor, Buhong Zheng, and an anonymous referee. The remaining errors and shortcomings are my own – W.H. Chiu     

Characterizations of egalitarian binary relations as transitive closures with a special reference to Lorenz dominance and to single-crossing conditions

This article considers, in the context of the fixed-population constant-sum comparison of income distributions, a number of intransitive binary relations smaller than Lorenz dominance. We determine their transitive closure, and we study how they relate to each other and to other relations that have appeared in the literature. Among other results, we provide alternative characterizations of Lorenz dominance.     

Inequality in the relative differential sense with continuous distributions

Existing literature has studied the relative differential inequality (RDI) concept as an inequality comparison for discrete distributions, where RDI dominance occurs if there is Lorenz dominance between the distributions for all subsets of individuals of corresponding rank. The paper investigates RDI in the context of continuous distributions, providing continuous counterparts to existing discrete results, and others, including discussion of the uniform, Pareto and log-normal families of continuous distributions. The usefulness of RDI for comparative static analysis in theoretical economic models based on continuous distributions is indicated.      

Welfare rankings of income distributions: A rôle for the variance and some insights for tax reform

When generalized Lorenz curves cross, it is not possible to rank the underlying income distributions by the unanimous preference of all additively separable symmetric inequality-averse social welfare functions. But in many cases, unanimous preference results in terms of inequality-aversion are nevertheless available. When generalized Lorenz curves cross once, variance is decisive in determining a robust welfare ranking, and can provide a rationale for choosing equity over efficiency where these two desiderata conflict. Welfare recommendations for certain types of income tax reform are implied, including cases of yield-increasing redistribution.This paper has benefitted considerably from the perceptive comments of an anonymous referee. Valentino Dardanoni gratefully acknowledges financial assistance from the Banco di Sicilia and the Ente Luigi Einaudi of Rome.     

Restricted Lorenz dominance of economic inequality in one and many dimensions

The paper investigates Lorenz dominance and generalized Lorenz dominance to compare distributions of economic status in one and several attributes. Restrictions of these dominance relations are developed that focus on central parts of the distributions and facilitate their comparison.     

Restricted Lorenz dominance of economic inequality in one and many dimensions

The paper investigates Lorenz dominance and generalized Lorenz dominance to compare distributions of economic status in one and several attributes. Restrictions of these dominance relations are developed that focus on central parts of the distributions and facilitate their comparison.     

Dynamic Discrete-time Duration Models: Estimation via Markov Chain Monte Carlo

Discrete-time or grouped duration data, with one or multiple types of terminating events, are often observed in social sciences or economics. In this paper we suggest and discuss dynamic models for flexible Bayesian nonparametric analysis of such data. These models allow simultaneous incorporation and estimation of baseline hazards and time-varying covariate effects, without imposing particular parametric forms. Methods for exploring the possibility of time-varying effects, as for example the impact of nationality or unemployment insurance benefits on the probability of reemployment, have recently gained increasing interest. Our modeling and estimation approach is fully Bayesian and makes use of Markov Chain Monte Carlo (MCMC) simulation techniques. A detailed analysis of unemployment duration data, with full-time job, part-time job and other causes as terminating events, illustrates our methods and shows how they can be used to obtain refined results and interpretations.     

On the Utility of Robust and Resampling Procedures1

Abstract Kassab (1990) makes an important methodological contribution by urging the use of robust regression methods in the study of community economic impacts and by indicating the utility of the bootstrap in assessing standard errors in robust regression. By introducing the notion of a contaminating distribution, we reconcile differences between her claim that ordinary least squares (OLS) regression is biased when outliers are present and standard linear model theory that does not make assumptions about the shape of the residual distribution in proving OLS an unbiased estimator. The contaminating distribution provides a framework for rural sociologists to link their statistical assumptions to a substantive understanding of the phenomena being studied. We suggest an alternative regression estimation strategy that may be more robust than the technique she uses. We also discuss an approach to bootstrapping that is more appropriate for macro-level social indicator data than the one she describes. An appendix discusses the software available for implementing these methods.     

Poverty, income distribution and CGE micro-simulation modeling: Does the functional form of distribution matter?

This paper explores income distribution modeling approaches for poverty analysis in a CGE micro-simulation context. Income distribution functional forms such as the lognormal, Pareto, beta distribution and empirical methods are currently used in CGE models in parallel with the estimation of FGT poverty indices. The particular methods or functional forms used in this context are not always clearly defined and justified. In this paper, we investigate and provide better criteria for selecting a functional distribution for poverty analysis. To achieve this, we apply parametric estimation to seven functional forms and compare the results to a purely “empirical” method. The results showed that no single form is more appropriate in all instances or for all household subgroups. The choice of a modeling approach should be motivated by a search for best fit and should be based on appropriate statistical tests. Selecting inappropriate distributional forms can lead to biased results in terms of poverty analysis. Introducing functional forms in the empirical approach can also provide greater confidence in the results obtained.

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An erratum to this article can be found at     

A new approach to measure socioeconomic inequality in health

This paper introduces a new approach to rank and measure socioeconomic inequality in health. A novel feature of the approach is the use of an income-health matrix that relates socioeconomic class with health status; each row of the matrix corresponds to a socioeconomic class and contains the respective probability distribution of health. By invoking a monotone assumption on the income-health matrix, which is a direct translation of the well-known hypothesis of social gradients in health outcomes, we derive a set of welfare-dominance and inequality-dominance conditions for ranking health distributions. Unlike other existing health inequality measures, our conditions require no cardinal specification of ordinal heath data and, hence, are robust. We then apply the dominance conditions to compare socioeconomic health inequality in the US and Canada using the newly released JCUSH data.     

More on the measurement of inequality

The article, written in 1973, examines what comparisons of income distributions can be made when Lorenz curves cross, employing the concept of third-order stochastic dominance. “More on the measurement of inequality” as reviewed by Prof. P. Lambert will appear in the Rediscovered Classics section, along with the editorial.