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.     

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.     

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.     

INFERENCE FOR DOMINANCE RELATIONS

The use of partial orders has been popularized as a way to conduct social evaluations using only minimal normative assumptions. Generically, this process involves comparing continuously indexed curves that are uniquely determined by the cumulative distributions of the individual attributes under study. In the literature on income poverty and inequality, for example, pairwise comparisons of entire income distributions and their respective Lorenz curves are routinely performed in order to characterize rankings of poverty, inequality, and welfare. In this article, we focus on the inferential problem that arises whenever such comparisons are made in the absence of census data. Statistical inference in these situations is particularly complex due to the fact that comparing curves invariably gives rise to four possibilities: the true population curves are equal, the first curve lies below the second, the second lies below the first, or the curves cross. To address this four‐decision problem, we introduce a two‐stage test that has good power and fine control over misclassification error rates.     

Pro‐Poorness Orderings

An indicator of pro‐poorness of a growth profile associated with a distribution of income is a measure of the extent to which growth is biased towards the poor. This paper proposes a general approach to pro‐poorness, called the progressive sequential averaging principle (PSA), relaxing the requirement of rank preservation due to growth. An endogenous benchmark for evaluating the growth of poor comes out naturally from this principle. A dominance relation on the basis of the above approach for a class of growth profiles is introduced through a simple device, called the PSA curve and its properties are examined in relation to the standard dominances in terms of the generalized Lorenz curve and the inverse generalized Lorenz curve. The paper concludes with an application to evaluate growth profiles experienced by the United States between 2001–07 and 2007–13.     

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...     

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     

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.     

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.     

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.     

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.     

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.     

EFFECT OF INCOME DISTRIBUTION ON THE ENVIRONMENTAL KUZNETS CURVE

This paper augments the model of Andreoni and Levinson by analyzing the effect of income distribution on the inverted U‐shaped relationship between some forms of pollution and income, the so‐called ‘environmental Kuznets curve’. In a context in which pollution abatement technology shows increasing returns to scale and an inverse U‐shaped pollution–income path is present, this study demonstrates the existence of a majority voting equilibrium, and concludes that the inverted U‐shaped relationship is between median income and environmental degradation rather than between per capita income and environmental degradation. Our results suggest that an increase in equality in income distribution improves environmental quality and social efficiency. The implication of the model for the empirical estimation of environmental Kuznets curves is examined using a panel data set of 36 countries over a 20‐year period. Estimation results using different models show that income distribution might be an important factor in the empirical estimation of these curves.     

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.     

PROGRESSIVITY AND REDISTRIBUTION IN NON‐REVENUE NEUTRAL TAX REFORMS: THE LEVEL AND DISTANCE EFFECTS

Kakwani and Reynolds–Smolensky indices are used in the literature to measure the progressivity and redistributive capacity of taxes. These indices may, however, show some limits when used to make normative assessments about non‐revenue neutral tax reforms. Two approaches have traditionally been taken to overcome this problem. The first of these consists of comparing after‐tax income distributions through generalized Lorenz (concentration) curves. The second approach is based on the decomposition of changes in the Reynolds–Smolensky index into changes in the average tax rate and variations in progressivity. Nonetheless, this decomposition between the average tax rate and progressivity may be further exploited to obtain some information that can be relevant to assess tax reforms. The main aim of this study is to draw up some indicators that can be useful to quantify the effects of non‐revenue neutral tax reforms. These indicators are used to investigate the last personal income tax reforms that have taken place in Spain.     

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.     

Income Inequality and Political Polarization: Time Series Evidence Over Nine Decades

Rising income inequality and political polarization have led some to hypothesize that the two are causally linked. Properly interpreting such correlations is complicated by multiple factors driving these phenomena, potential feedback between inequality and polarization, measurement issues, and the statistical challenges of modeling non‐stationary variables. We find that a more precise measure of inequality (the inverted Pareto–Lorenz coefficient) is more consistently and statistically related to polarization in the short and long runs than the less precise top 1 percent share of income. We find bi‐directional causality between polarization and inequality, consistent with theoretical conjecture and less formal evidence in previous studies.     

An alternative functional form for estimating the Lorenz curve

We propose a simple single parameter functional form for the Lorenz curve. The new specification is fitted to existing data sets and is shown to provide a better fit than existing single parameter Lorenz curves for the given data.     

Redistribution in the age of austerity: evidence from Europe 2006–2013

We examine the relationship between changes in a country’s public sector fiscal position on inequality at the top and bottom of the income distribution during the age of austerity from 2006 to 2013. We use a parametric Lorenz curve model and Gini-like indices of inequality as our measures to assess distributional changes. Based on Statistics of Income and Living Conditions (EU-SILC) and IMF data for 12 European countries, we find that more severe adjustments to the cyclically adjusted primary balance (i.e., more austerity) are associated with a more unequal distribution of income driven by rising inequality at the top. The data also weakly suggests a decrease in inequality at the bottom. The distributional impact of austerity measures reflects the reliance on regressive policies and likely produces increased incentives for rent-seeking while reducing incentives for workers to increase productivity.     

Soaring dragons,roaring tigers,growling bears

We perform a comparative analysis of regional growth and convergence in China, Russia and India over the period 1993–2003 by means of non‐parametric methods and kernel density estimates. Our results indicate that wealthy regions were largely responsible for the rapid growth in all three countries. For China and India, capital dissipation was identified as the major determinant of regional growth. In Russia, capital deepening impeded positive changes in labour productivity, leaving technological change as the only source of regional growth. Furthermore, we find that the increasing regional income inequality in all three countries was driven by technological change which more than offset the convergence resulting from capital deepening in China and India.