Approximating unanimity orderings: An application to Lorenz dominance

Stochastic dominance and Lorenz dominance are examples of orderings which require unanimous agreement among an infinite set of indices. This paper considers various subsets of inequality measures that respect Lorenz dominance, and assesses the extent to which a small number of indices can reproduce the Lorenz ordering. Using income data for 80 countries, our results suggest that Lorenz dominance can be predicted with 99% accuracy using just 3 or 4 inequality measures, as long as two of them focus on the extreme upper and lower tails of the distribution. In contrast, confining attention to the index families and parameter ranges normally considered may fail to detect the majority of occasions when Lorenz curves intersect. These results lead us to question the faith placed in procedures based on a finite set of inequality indices, and to suggest that similar lessons will apply to other types of unanimity orderings.     

Parametric Lorenz Curves and the Modality of the Income Density Function

Similar looking Lorenz curves can imply very different income density functions and potentially lead to wrong policy implications regarding inequality. This paper derives a relation between a Lorenz curve and the modality of its underlying income density: given a parametric Lorenz curve, it is the sign of its third derivative which indicates whether the density is unimodal or zeromodal (i.e., downward‐sloping). The density modality of several important Lorenz curves such as the Pareto, Weibull, Singh–Maddala parametrizations and hierarchical families of Lorenz curves are discussed. A Lorenz curve performance comparison with Monte Carlo simulations and data from the UNU–WIDER World Income Inequality Database underlines the relevance of the theoretical result: curve‐fitting based on criteria such as mean squared error or the Gini difference might lead to a Lorenz curve implying an incorrectly‐shaped density function. It is therefore important to take into account the modality when selecting a parametric Lorenz curve.     

A GENERAL METHOD FOR CREATING LORENZ CURVES

A general method to construct parametric Lorenz models of the weighted‐product form is offered in this paper. Initially, a general result to describe the conditions for the weighted‐product model to be a Lorenz curve, created by using several component parametric Lorenz models, is given. We show that the key property for an ideal component model is that the ratio between its second derivative and its first derivative is increasing. Then, a set of Lorenz models, consisting of a basic group of models, along with their convex combinations, is proposed, and it is shown that any model in the set possesses this key property. We introduce the concept of balanced fit, which provides a means of assigning weights, according to the preferences of the practitioner, to two alternative objectives for developing Lorenz curves in practice. These objectives are generating an acceptable Lorenz curve and improving the accuracy of the density estimation. We apply the balanced fit approach to income survey data from China to illustrate the performance of our models. We first show that our models outperform other popular traditional Lorenz models in the literature. Second, we compare the results generated by the balanced fit approach applied to one of the Lorenz models that we develop with those generated by the kernel method to show that the approach proposed in the paper generates plausible density estimates.     

Interpolating the Lorenz Curve: Methods to Preserve Shape and Remain Consistent with the Concentration Curves for Components

C¹‐class interpolation methods that preserve monotonicity and convexity and are thus suitable for the estimation of the Lorenz curve from grouped data are not widely known. Instead, parametric models are usually applied for such estimation. Parametric models, however, have difficulty in accurately approximating every part of income/expenditure distributions. This paper proposes two types of C¹‐class shape‐preserving interpolation methods. One is a piecewise rational polynomial interpolation (proposed independently by Stineman and Delbourgo) that enables consistent interpolation of the concentration curves for income/expenditure components, attaining approximately the same accuracy as that of the existing methods when applied to decile‐grouped data or to more detailed aggregation. Another is a Hybrid interpolation that employs pieces of curves derived from parametric models on end intervals. Empirical comparisons show that the Hybrid interpolation (with the assistance of parametric models for class‐boundary estimation) outperforms the existing methods even when applied to quintile‐grouped data without class boundaries.     

Note

Functional classes of Lorenz curves are derived from a generalization of a relative poverty notion. All these Lorenz curves compare individual income to the average of all larger or all smaller incomes. The parameters of the Lorenz curves are effectively computed from empirical income data by least square regressions. Best fits are analyzed and resulting functional Gini indices are compared to empirical Gini indices.First version received: September 2002/Final version received: April 2003     

A NEW FUNCTIONAL FORM FOR ESTIMATING LORENZ CURVES

There are several functional forms for estimating Lorenz curves from grouped data. Based on studies of the Spanish distribution of income, we propose a new functional form which provides very good fits. Our specification contains the Pareto Lorenz curve as a particular case, and allows one to compute easily, with the provided formulae, the Gini, Kakwani, and Chakravarty Inequality Indexes.     

Carbon Lorenz curves

The purpose of this paper is twofold. First, it exhibits that standard tools in the measurement of income inequality, such as the Lorenz curve and the Gini-index, can successfully be applied to the issues of inequality measurement of carbon emissions and the equity of abatement policies across countries. These tools allow policy-makers and the general public to grasp at a single glance the impact of conventional distribution rules such as equal caps or grandfathering, or more sophisticated ones, on the distribution of greenhouse gas emissions. Second, using the Samuelson rule for the optimal provision of a public good, the Pareto-optimal distribution of carbon emissions is compared with the distribution that follows if countries follow Nash–Cournot abatement strategies. It is shown that the Pareto-optimal distribution under the Samuelson rule can be approximated by the equal cap division, represented by the diagonal in the Lorenz curve diagram.     

Calculating Poverty Measures from the Generalised Beta Income Distribution

Data for measuring poverty are frequently available in a summary form that describes the proportion of income or expenditure for each of a number of population proportions. While various discrete poverty measures can be applied directly to data in this limited form, they typically require an arbitrary approach to within‐group interpolation. This problem can be overcome by fitting either a parametric income distribution or a Lorenz curve to the grouped data and computing the required quantities from estimated parameters. The Lorenz curve approach is widely used by the World Bank, but can encounter problems. As an alternative, in this article we show how to calculate several poverty measures from parameters of the generalized beta income distribution, and its popular special cases. An analysis of poverty changes in countries from South and Southeast Asia is used to illustrate the methodology.     

Intermediate Inequality and Welfare: The Case of Spain, 1980–81 to 1990–91

The 1990–91 household expenditures distribution in Spain dominates, in the relative ("rightist") Lorenz sense, the 1980–81 distribution, but the latter dominates the former in the absolute ("leftist") Lorenz sense. This situation constitutes a textbook case for intermediate or "centrist" notions of inequality and social welfare. This paper presents the first empirical application of this sort, using the intermediate inequality concept introduced in Del Río and Ruiz-Castillo (2000). The data reveal that there is a decrease in household expenditures inequality for a relatively small set of centrist attitudes.     

The concept and measurement of horizontal inequity

This paper discusses the concept of horizontal equity and presents three formal principles as a basis for making comparisons of the extent of horizontal inequity produced by different redistributions. A condition analogous to the Lorenz criterion used for inequality comparisons is established under which the horizontal inequities of redistributions can be ranked without recourse to a cardinal measure of inequity. Ordinal comparisons are not possible in many cases. The three principles suggest three properties that ‘reasonable’ measures of horizontal inequity should possess. A class of measures which satisfies the properties is proposed. Most other existing measures are found not to satisfy them.     

On the Relative Equitability of a Family of Taxation Rules

A taxation problem consists of collecting a given amount of taxes out of a population whose gross income vector is given. A tax rule R is more equitable than a rule R′ when the after‐tax income vector generated by R Lorenz dominates that generated by R′. In this paper, we present a one‐parameter family of tax rules whose members are fully ranked in terms of relative equitability, for any (finite) income distribution.     

The Lorenz curve as an archetype: A historico-epistemological study

In 1905, Max O. Lorenz suggested a simple method of measuring the concentration of wealth, based on the visual representation of income distribution. The Lorenz curve is now very popular and can be considered as canonical. However, the path leading from Lorenz's original work to contemporary interpretations of his graph has been anything but simple. We thus propose to trace the origin, the evolution and the various subsequent interpretations of the Lorenz graph. We argue that the original Lorenz curve has been shifted in epistemological status as well as inverted in graphic appearance.     

Fitting parametric lorenz curves to grouped income distributions — A critical note

Empirical Economics - The paper surveys various parametric Lorenz curves to be fitted to grouped income data in order to obtain an estimate for the Gini measure of inequality. The curves are fitted...     

Bargaining power and enforcement in credit markets

In a credit market with enforcement constraints, we study the effects of a change in the outside options of a potential defaulter on the terms of the credit contract, as well as on borrower payoffs. The results crucially depend on the allocation of “bargaining power” between the borrower and the lender. We prove that there is a crucial threshold of relative weights such that if the borrower has power that exceeds this threshold, her expected utility must go up whenever her outside options come down. But if the borrower has less power than this threshold, her expected payoff must come down with her outside options. In the former case a deterioration in outside options brought about, say, by better enforcement, must create a Lorenz improvement in state-contingent consumption. In particular, borrower consumption rises in all “bad” states in which loans are taken. In the latter case, in contrast, the borrower's consumption must decline, at least for all the bad states. These disparate findings within a single model permit us to interpret existing literature on credit markets in a unified way.     

Lorenz rankings of rules for the adjudication of conflicting claims

For the adjudication of conflicting claims, we develop three general approaches to obtain Lorenz rankings of rules. Our first approach concerns a parameterized family that contains several important rules (Thomson in Soc Choice Welf 31:667?C692, 2008). We give a condition that the parameters defining two members of the family should satisfy for one of them to Lorenz dominate the other. Our second approach exploits the concept of ??consistency?? (Young in Math Oper Res 12:398?C414, 1987). We derive a criterion to deduce Lorenz domination for arbitrarily many claimants from Lorenz domination in the two-claimant case. Our third approach is based on the notion of an ??operator?? on the space of rules (Thomson and Yeh in J Econ Theory 143:177?C198, 2008). We develop conditions under which operators preserve the Lorenz order, or reverse it. As corollaries of our general theorems, we obtain rankings of most of the rules that have been discussed in the literature.     

THE LORENZ ORDER AND THE EFFECTS OF TAXATION POLICIES

To any finite population of n individuals with associated incomes x ₁…, x _n we can associate a Lorenz curve. By associating this population with a random variable X representing the income of a randomly chosen individual of the population, the concept of a Lorenz curve and the associated partial order (the Lorenz Order) based on nested Lorenz curves is readily extended to be defined in the class of all non-negative integrable random variables. In this context well known results on inequality attenuating and inequality rank preserving taxation policies are found to admit simple more general proofs. Some results on the effects of random taxation are also reviewed. The effects of applying different taxation policies within subpopulations lead one to consider questions regarding inequality attenuation results in mixture settings. It is observed that, more generally, inequality comparisons can be unambiguously made between any non-negative variables even if measured in dissimilar units.     

Growth and Wage Inequality in a Dual Economy

Who benefits from economic growth? This paper analyses the distributional impact of different types of growth within a two‐sector model. The paper first presents necessary and sufficient conditions for unambiguous changes in wage inequality in a dual economy, based on analysis of the entire Lorenz curve. These conditions are then applied to the Harris–Todaro model with an urban non‐agricultural sector and rural agriculture. It is shown that capital accumulation or technical progress in agriculture can shift the Lorenz curve inwards and reduce wage inequality, while the effects of development in non‐agriculture are typically ambiguous.     

RELATIVE INEQUALITY, ABSOLUTE INEQUALITY, AND WELFARE: LARGE SAMPLE TESTS FOR PARTIAL ORDERS

Two general welfare criteria, mean-relative Lorenz and mean-absolute Lorenz dominance, induce partial orders on income distributions. We propose asymptotically distribution-free inference procedures, based on the union-intersection principle, for these two welfare criteria. Unlike classical tests, our procedures allow one to distinguish among dominance, equality, and noncomparability. We show that union-intersection tests must be used to test for partial orders, and that the statistical ordering is acyclic. The tests are applied to compare the UK distribution of real family income to five other countries.     

Carbon dioxide emission and income: A temporal analysis of cross-country distributional patterns

This paper explores the relationship between the inter-country income inequality and CO₂ emission and temporal shifts in such a relationship. It also examines how the mean per capita CO₂ emission and its distributional inequality are related to the corresponding mean and the distributional inequality of income. The analysis is based on a cross-country panel data set at the level of country-group. Here environmental damage is treated as a private good and the technique of Lorenz and specific concentration curve analysis have been used as the basic analytical framework to argue that distributional inequality of income should be an explanatory variable in the Environmental Kuznets Curve relationship, along with the mean income level. In the empirical exercise, Johansen's cointegration analysis technique is used to explore existence of statistically significant cointegrating vector(s) relating mean emission and Specific Concentration Ratio of emission to mean income level and Lorenz Ratio of income, using a set of country-group specific time series data set which covers four country-groups (viz., Africa, America, Asia and Europe) and the World as a whole. The empirical results confirm that the inter-country income inequality has significant effect on the mean emission level and inter-country inequality of emission level for most of the country-groups considered.     

Testing Markov switching models

In this article, we propose a new test for Markov switching models. Unlike the tests in the existing literature (e.g. Hansen, 1992; Garcia, 1998; Cho and White, 2007), we focus on testing the null of two regimes, instead of one single regime, in a switching framework. To implement our test, we propose a Markov switching model with absorbing states and examine whether the absorption probabilities are close to the boundary of the parameter space. We exploit recent advances by Andrews (2001) and conduct inference in the proposed model.