Does less inequality among households mean less inequality among individuals?

Consider an income distribution among households of the same size in which individuals, equally needy from the point of view of an ethical observer, are treated unfairly. Individuals are split into two types, those who receive more than one half of the family budget and those who receive less than one half. We look for conditions under which welfare and inequality quasi-orders established at the household level still hold at the individual one. A necessary and sufficient condition for the Generalized Lorenz test is that the income of dominated individuals is a concave function of the household income: individuals of poor households have to stand more together than individuals of rich households. This property also proves to be crucial for the preservation of the Relative and Absolute Lorenz criteria, when the more egalitarian distribution is the poorest. Extensions to individuals heterogeneous in needs and more than two types are also provided.     

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.     

Fuzzy measurement of income inequality: Some possibility results on the fuzzification of the lorenz ordering

Summary This paper starts from the premise that the concept of income inequality is ill-defined, and hence, it studies the measurement of income inequality from a fuzzy set theoretical point of view. It is argued that the standard (fuzzy) transitivity concepts are not compatible with fuzzy inequality orderings which respect Lorenz ordering. For instance, we show that there does not exist a max-min transitive fuzzy relation on a given income distribution space which ranks distributions unambiguously according to the Lorenz criterion whenever they can actually be ranked by it. Weakening the imposed transitivity concept, it is possible to escape from the noted impossibility theorems. We introduce some alternative transitivity concepts for fuzzy relations, and subsequently, construct a class of fuzzy orderings which preserve Lorenz ordering and satisfy these alternative transitivities. It is also shown that fuzzy measurement can be used to construct confidence intervals for the crisp conclusions of inequality indices.I wish to thank Ashish Banerjee, Kaushik Basu, Larry Blume, Gary Fields, Semih Koray, Tapan Mitra, Antony Shorrocks, Sinan Unur and two anonymous referees of this journal for insighthul comments and suggestions. I am also grateful to the participants of the 1993 Midwest Mathematical Economics Conference held in University of Wisconsin at Madison and the 2nd International Meeting of the Society for Social Choice and Welfare held in University of Rochester.     

A Sufficient Condition for Generalized Lorenz Order

In this paper, a sufficient condition for non-negative random variables to be ordered in the Generalized Lorenz sense is presented. This condition does not involve inverse distribution functions. Applications of this result to several income distribution models are given. Journal of Economic Literature Classification Numbers: D36, D69.     

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.     

Second‐Order Discrimination and Generalized Lorenz Dominance

We propose a definition of second‐order discrimination that does not require the reference distribution to first‐order dominate the comparison one, and allows rankings of discrimination patterns when both the reference and the comparison distributions differ. It involves comparing the probabilities that randomly selected individuals in the reference and comparison distributions belong to subgroups having the same cumulative mean income, yields orderings of distributions equivalent to those from generalized Lorenz dominance, and allows orderings of discrimination patterns, partial or complete, across pairs of distributions. We compare discrimination against U.S. seniors (inter‐distributional inequality between seniors and non‐seniors) by ethnicity.     

A stochastic dominance approach to the measurement of discrimination

This note suggests a bridge between stochastic dominance (Rothschild and Stiglitz, 1970 [17], 1973 [18]), inequality measurement (Atkinson, 1970 [1]) and discrimination measurement (Gastwirth, 1975 [10]). Discrimination orderings are defined and illustrated through discrimination curves, in the same spirit as stochastic dominance analysis. The main result, which links the second order discrimination curve and the Gastwirth discrimination index, also generalizes the equivalence between Generalized Lorenz dominance and second order stochastic dominance.     

Is the Pareto–Lévy law a good representation of income distributions?

Mandelbrot (Int Econ Rev 1:79–106, 1960) proposed using the so-called Pareto–Lévy class of distributions as a framework for representing income distributions. We argue in this article that the Pareto–Lévy distribution is an interesting candidate for representing income distributions because its parameters are easy to interpret and it satisfies a specific invariance-under-aggregation property. We also demonstrate that the Gini coefficient can be expressed as a simple formula of the parameters of the Pareto–Lévy distribution. We subsequently use income data for Norway and seven other OECD countries to fit the Pareto–Lévy distribution as well as the Generalized Beta type II (GB2) distribution. The results show that the Pareto–Lévy distribution fits the data better than the GB2 distribution for most countries, despite the fact that GB2 distribution has four parameters whereas the Pareto–Lévy distribution has only three.     

STABILITY OF THAI BAHT: TALES FROM THE TAILS

We demonstrate how the EVT‐based signalling approach for currency crises can be applied to an individual country with a small sample size. Using Thai historical data, first, we study the tail characteristics of the distributions of two Thai baht instability measures and 21 economic fundamentals. Then, we test asymptotic dependence between the currency instability measures and lagged economic fundamentals. Empirically, we find that the distributions of both currency instability measures and economic variables are heavy tailed. Assuming a normal distribution for the variables tends to underestimate the probability of extreme events. Furthermore, most of the economic variables which are usually used as signalling indicators for currency crises are asymptotically independent of the currency instability measures. Signals issued by these variables are thus not reliable. Nevertheless, the non‐parametric EVT approach facilitates the selection of economic indicators with credible signals and high crisis prediction success.     

Estimating and testing the compensated double-log demand model

In spite of the proliferation of flexible functional forms for consumer demand systems, the double-log demand model continues to be popular, especially in applied work calling for single-equation models. It is usually estimated in uncompensated form. It can also be estimated in compensated form, by deflating the income variable alone using Stone's price index. The compensated form has the same right-hand side as a single-equation version of the popular linear approximation to the Almost Ideal demand model, facilitating the construction of a test for choosing between the two alternatives. This paper demonstrates these results, develops the specification test, and illustrates its application using US meat consumption data. Simulations suggest that the test is well-behaved with good power in typical applications.     

Inverse stochastic dominance,inequality measurement and Gini indices

We investigate the relationship between the third degree inverse stochastic dominance criterion introduced in Muliere and Scarsini (1989) and inequality dominance when Lorenz curves intersect. We propose a new definition of transfer sensitivity aimed at strengthening the Pigou-Dalton Principle of Transfers. Our definition is dual to that suggested by Shorrocks and Foster (1987). It involves a regressive transfer and a progressive transfer both from the same donor, leaving the Gini index unchanged. We prove that finite sequences of these transfers and/or progressive transfers characterize the third degree inverse stochastic dominance criterion. This criterion allows us to make unanimous inequality judgements even when Lorenz curves intersect. The Gini coefficient becomes relevant in these cases in order to conclusively rank the distributions.     

Using BS-PSD-LDA approach to measure operational risk of Chinese commercial banks

The research of operational risk management among Chinese commercial banks is still in its preliminary stage. Operational risk events are rare and data is hard to collect. This leads to very small data samples. Besides, a large number of empirical researches show that the distributions of operational losses are often skewed with fat tails. To address these issues, this paper puts forward a loss distribution approach (LDA) based on bootstrap sampling and piecewise-defined severity distribution (BS-PSD-LDA). The approach divides data samples into two distinct parts (high-frequency low-severity losses and low-frequency high-severity losses), and fits the two parts by lognormal distribution and Generalized Pareto distribution respectively. Using hand-collected samples of 426 operational losses in Chinese commercial banks during 1994–2010, we estimate the magnitude of operational losses using the BS-PSD-LDA method. We show that our method provides a better fit than the traditional parametric methods. Besides, the method using historical simulation of nonparametric method seems to offer a good fit to the sample as well. However, we believe that the BS-PSD-LDA approach offers improvement from the perspective of satisfying risk control requirement of the regulatory authority and ensuring the efficiency of funds' utilization.     

Economic policy and income distribution in China

"Using the Kakwani interpolation method to reconstruct rural and urban size distributions, the authors present Lorenz curve estimates for Chinese income distribution by period, 1952-1986. The level of inequality is one of the lowest in the world. The change in inequality over time conforms to the experience of other less-developed countries. Although rural inequality increased after 1978, national inequality fell, due to a reduction in the rural-urban income gap."     

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.     

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.     

The probability of multidimensional poverty: A new approach and an empirical application to EU-SILC data

This paper proposes a novel method to analyze multidimensional poverty by using a large set of feasible weights to summarize the information about the poor, which enables remaining agnostic about the relative importance given to different poverty dimensions. This method allows for the calculation of the individual probability of being poor in a multidimensional perspective. The distribution of individual probabilities can then be combined with Generalized Lorenz dominance techniques to derive unanimous consent for a wide class of social welfare functions with a minimum load of value judgments. The innovations proposed here allow to move from a dual definition of poverty, where poor and non-poor individuals are classified in a mutually exclusive context, to a continuous measure of deprivation capturing both the extensive and intensive margin of multidimensional poverty. The empirical application of the method consists of measuring multidimensional poverty in ten selected countries using four waves of EU-SILC data (2008–2014).     

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.     

Household Characteristics and the Distribution of Income In Italy: An Application of Social Distance Measures

The Lorenz criterion of preferable distributions fails to distinguish adequately between convergence to the global mean and clustering around local means. This concern has motivated independent work by Wolfson, and Esteban and Ray on the notion of polarization. In this paper I build on this recent work by providing a new method that characterizes changes in the entire distribution, rather than focussing only on dispersion. In particular, the approach proposed offers a new decomposition method of within- and between-group components that differs from the classical method of additively decomposable inequality indices. The new method can monitor which factors modified the entire distribution, where precisely on the distribution these factors had an effect, and what determined the variation in the level of social distance between groups or geographic areas. Summary statistics of the observed movements and of distance between and divergence among the estimated and the counterfactual distributions are provided as well as a new index of social distance. The new method is then applied to Italian data on income distribution between 1987 and 1995.     

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.