Generalized Lorenz Curves Research Articles

A Dominance Criterion for Measuring Income Inequality from a Centrist View: The Case of Australia

This paper introduces a new Lorenz dominance criterion that allows ranking income distributions according to centrist measures a la Seidl and Pfingsten (1997). In doing so, it defines α-Lorenz curves by adapting the generalized Lorenz curves to this case. In addition, it provides an empirical illustration of these tools using Australian income data for the period 2001–2008. The results suggest that despite the reduction of relative inequality, inequality increased for most centrist value judgments.

Dominance testing for ‘pro-poor’ growth with an application to European growth

This article introduces statistical testing procedures to evaluate ‘pro-poor’ growth. Our measure of ‘pro-poorness’ follows Kakwani (J Quant Econ 16(1):67–80, 2000), Kakwani and Pernia (Asian Dev Rev 18(1):1–16, 2000), and Son (Econ Lett 82:307–314, 2004), who decompose the generalized Lorenz ordinates into a growth effect and an inequality effect. We derive an asymptotic distribution-free covariance matrix for the decomposed generalized Lorenz curves. Using this decomposition (and our standard errors), we test for pro-poor dominance in the growth process. We illustrate our test for the pro-poor dominance by evaluating the degree of pro-poor growth in five European countries.

Lorenz Curves Determine Partial Orders for Comparing Network Structures

Networks can be studied from different points of view. In this paper, it is shown that the traditional Lorenz curve and some of its generalisations can be used for characterising inequality in network properties. Each type of Lorenz curve determines a partial order in a set of networks and a Gini-type index can be associated to each of these curves. The following types of Lorenz curves, each related to a different type of inequality in network properties and resulting in a different partial order, are briefly discussed: (a) Classical Lorenz curves and evenness measures; (b) intrinsic diversity profiles (or k-dominance curves) and associated measures of diversity; (c) generalised Lorenz curves as introduced by Shorrocks, that are not scale invariant but take absolute numbers into account; (d) weighted Lorenz curves for comparisons with an internal or external standard; and (e) Lorenz-type curve introduced to perform source per source comparisons of items. This paper claims that the Lorenz curves and the Gini index are universal tools for studying inequality, including inequality in network properties.

On the Concept of Equity in Opportunity

Measuring the equity of opportunity in a given society is an essential ingredient in the formulation of policies and programs that promote inclusive growth. In this paper, equity of opportunity is defined and measured through the theoretical framework of the social opportunity function, a concept similar to the social welfare function. The functional and graphical distribution of opportunity is discussed through the generalized Lorenz curve and the Bonferroni curve, while complete ranking of distributions is achieved through their related numerical indices: the concentration index and the Bonferroni index of opportunity, respectively. The concepts of relative and absolute measures of equity of opportunity are then introduced and a social opportunity index that considers both the amount and distribution of opportunity is developed. These measures are used to analyze changes in the opportunities for health care and education in the Philippines from 1998 to 2007.

Reranking and Pro-Poor Growth: Decompositions for China and Vietnam

Reranking in the move from one income distribution to another makes it impossible to infer from changes in Lorenz and generalised Lorenz curves how income growth among those toward the bottom of the initial income distribution compares to that among those toward the top, and whether there has been income growth among those who were initially poor. Decompositions allowing for reranking indicate that economic growth in China and Vietnam has been better for households who were initially poor than changes in the Lorenz and generalised Lorenz curve and poverty growth curve would suggest.

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.

A Discrepancy Index for the Study of Participation with an Application to the Case of Higher Education in Italy

This article proposes and applies a simple method to measure the distance from a situation of uniform participation. First, a discrepancy index based on the use of generalized Lorenz curves is presented. This index can be expressed in terms of means and Gini indices of relevant characteristics in the populations of participants and that of a control group. A multivariate extension is also illustrated. The discrepancy index is then used to analyse access into higher education in Italy during the nineties. Discrepancy between participants and non-participants is especially high in terms of parents’ wealth and education, less in terms of income. Also, discrepancy is higher during recessions. Multivariate analysis reveals that “cultural constrains” are more important than financial constraints. Hence, efforts to equalize opportunity for access into higher education should be as much focussed on motivation and social background as they are on financial constraints.

Privation et mesure du bien-être social

Nous introduisons un nouveau critere pour effectuer des comparaisons de bien-etre, qui repose sur la notion d’absence de privation. Ce quasi-ordre constitue une alternative naturelle a l’approche standard en economie normative, qui consiste a comparer les courbes de Lorenz generalise des distributions de revenu. Le quasi-ordre de Lorenz generalise satisfait aux principes des transferts selon lequel le bien-etre social augmente suite a un transfert progressif. Le critere que nous proposons verifie une condition moins exigeante que le principe des transferts : le bien-etre social doit augmenter uniquement pour des combinaisons de transferts progressifs particulieres, ou les positions des individus impliques dans ces transformations jouent un role crucial.

Convex rearrangements, generalized Lorenz curves, and correlated Gaussian data

We propose a statistical index for measuring the fluctuations of a stochastic process ξ . This index is based on the generalized Lorenz curves and (modified) Gini indices of econometric theory. When ξ is a fractional Brownian motion with Hurst index α ∈ ( 0 , 1 ) , we develop a complete picture of the asymptotic theory of our index. In particular, we show that the asymptotic behavior of our proposed index depends critically on whether α ∈ ( 0 , 3 4 ) , α = 3 4 , or α ∈ ( 3 4 , 1 ) . Furthermore, in the first two cases, there is a Gaussian limit law, while the third case has an explicit limit law that is in the second Wiener chaos.

Testing for Generalized Lorenz Dominance

The Generalized Lorenz dominance can be used to take account of differences in mean income as well as income inequality in case of two income distributions possessing unequal means. Asymptotically distribution-free and consistent tests have been proposed for comparing two generalized Lorenz curves in the whole interval [p1, p2] where 0 < p1 < p2 < 1. Size and power of the test has been derived.

Strong laws for generalized absolute Lorenz curves when data are stationary and ergodic sequences

We consider generalized absolute Lorenz curves that include, as special cases, classical and generalized L L -statistics as well as absolute or, in other words, generalized Lorenz curves. The curves are based on strictly stationary and ergodic sequences of random variables. Most of the previous results were obtained under the additional assumption that the sequences are weakly Bernoullian or, in other words, absolutely regular. We also argue that the latter assumption can be undesirable from the applications point of view.

Education Vouchers, Growth and Income Inequality

This paper uses a growth model with public and private education alternatives to investigate the implications of education vouchers for economic growth and the evolution of income inequality. The results indicate that introducing education vouchers can increase economic growth. Families that switch from public to private education due to vouchers experience higher incomes, leading to growth in the tax base which in turn raises public education expenditures and increases the growth of the whole economy. Vouchers have an ambiguous effect on income inequality. Gini coefficients indicate increased inequality while other measures indicate decreased inequality. Comparing income distributions using generalised Lorenz curves, the distribution under a voucher scheme dominates the distribution without vouchers. This implies that in the long run, vouchers offer a welfare improvement. In short run welfare comparisons however, more than half the population is worse off with vouchers, which is consistent with repeated failures of voucher referenda in the US. The results add a new dimension on which vouchers can be evaluated in the continuing policy debate.

Density inference for ranking European research systems in the field of economics

The econometrics of inequality measurement is adapted in order to investigate the publishing habits of the economists of seven European countries: Belgium, France, Germany, Italy, Netherlands, Spain and UK. California is added for comparison. Data come from the CD-ROM of the Journal of Economic Literature. A Bayesian model selection procedure helps to select a Weibull distribution as an admissible reduction of the generalised Beta-II. Analytical upper partial moment curves and modified generalised Lorenz curves are derived for the Weibull model and serve to detect a group of three dominant european countries: The Netherlands, the UK and Belgium.

A note on pro-poor growth

This paper proposes a ‘poverty growth curve’ that measures whether economic growth is pro-poor or not pro-poor. Our methodology is developed based on Atkinson's theorem linking the generalized Lorenz curve and changes in poverty. This methodology is applied to both Thailand's unit record household surveys and international cross-country data. Empirical study illustrates that our approach to measuring pro-poor growth provides conclusive results in the majority of cases.

A local hierarchy theory for acyclic digraphs

Local hierarchy theory focuses on direct links in acyclic digraphs. In- and out-degrees are used to determine the local hierarchical number for each vertex in the graph. Together, these local hierarchical numbers form a vector through which hierarchical properties are studied. The main tool, leading to a partial order of acyclic digraphs is a form of generalized Lorenz curve. Gini-like measures respecting this partial order can be derived. Local hierarchy theory is then the theory related to this particular partial order. Results have possible applications in administration and business organizational charts and in citation analysis. In the latter, a direct link represents a reference or a citation of a document. Finally, we study rooted trees as a concrete example of local hierarchy theory.

Generalized Lorenz curves and convexifications of stochastic processes

We investigate convex rearrangements, called convexifications for brevity, of stochastic processes over fixed time intervals and develop the corresponding asymptotic theory when the time intervals indefinitely expand. In particular, we obtain strong and weak limit theorems for these convexifications when the processes are Gaussian with stationary increments and then illustrate the results using fractional Brownian motion. As a theoretical basis for these investigations, we extend some known, and also obtain new, results concerning the large sample asymptotic theory for the empirical generalized Lorenz curves and the Vervaat process when observations are stationary and either short-range or long-range dependent.

Distribution-free statistical inference for generalized Lorenz dominance based on grouped data

One income distribution is preferable to another under any increasing and Schur-concave (S-concave) social welfare function (SWF) if and only if the generalized Lorenz (GL) curve of the first distribution lies above that of the second. Thus, testing for GL dominance of one distribution over another is of interest. The paper focuses on inference based on grouped data and makes two contributions: (i) it gives a new formula for the asymptotic variance–covariance matrix of a vector of sample GL curve ordinates, interpreting it as a method-of-moments (MM) estimator, and (ii) it proposes a new test for multivariate inequality restrictions, of which GL dominance is a special case. For the Japanese household income data grouped into deciles, the test accepts the null hypothesis that income distribution in Japan improved from 1979 to 1994.

Generalized Lorenz curves and convexifications of stochastic processes

We investigate convex rearrangements, called convexifications for brevity, of stochastic processes over fixed time intervals and develop the corresponding asymptotic theory when the time intervals indefinitely expand. In particular, we obtain strong and weak limit theorems for these convexifications when the processes are Gaussian with stationary increments and then illustrate the results using fractional Brownian motion. As a theoretical basis for these investigations, we extend some known, and also obtain new, results concerning the large sample asymptotic theory for the empirical generalized Lorenz curves and the Vervaat process when observations are stationary and either short-range or long-range dependent.

The Effect of Social Transfers in Western Europe: An Empirical Analysis of the Countries Using Generalized Lorenz Curves Katja Holsch

This paper aims at examining the impact of different transfers on income distribution in five Western European countries by carrying out an empirical analysis using generalized Lorenz curve comparisons. The results obtained are then investigated by relating them to a classification of European social transfer systems. We conclude that clear differences can be found with regards to transfer arrangements and underlying strategies. This does not necessarily mean that those differences result in clearly distinct transfer rankings for every country: some transfers like unemployment and disability benefits have an obvious impact in countries with quite different underlying strategies and transfer arrangements.

Changes in income inequality and welfare under economic transition: evidence from urban China

The paper examines the changes in income inequality and welfare in urban China from 1981 to 1999. While the Lorenz curves estimated using Kakwani's interpolation method indicate that the level of income inequality in urban China has increased significantly since 1981, welfare comparisons based on generalized Lorenz curves suggest that the rise in real average income has more than compensated for the increase in inequality and has therefore brought about continuous improvement in welfare since 1981, except in 1988 and 1989 due to high inflation rates. Nevertheless, it is becoming very critical for China to develop welfare programmes and a social security system to provide a guaranteed living standard for low-income households.