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1.
Gleb Koshevoy 《Social Choice and Welfare》1997,15(1):1-14
The distribution of d commodities among n individuals is described by an n×d row stochastic matrix. We present a geometric approach to order such matrices. For a row stochastic matrix the Lorenz zonotope
is investigated, which is a higher dimensional generalization of the Lorenz curve. The Lorenz zonotope is a convex polytope.
The inclusion of Lorenz zonotopes defines an ordering between row stochastic matrices, which is a multivariate majorization.
For a cone in nonnegative d-space, a cone extension of the Lorenz zonotope and the respective inclusion ordering are introduced. We study this class
of orderings and establish equivalence with known majorizations. It is provided a finite set of inequalities to which the
ordering is equivalent.
Received: 16 February 1994/Accepted: 22 May 1996 相似文献
2.
The Journal of Economic Inequality - We propose an extension of the univariate Lorenz curve and of the Gini coefficient to the multivariate case, i.e., to simultaneously measure inequality in more... 相似文献
3.
Rolf Aaberge 《Social Choice and Welfare》2009,33(2):235-259
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. 相似文献
4.
Rolf Aaberge 《Social Choice and Welfare》2000,17(4):639-653
The purpose of this paper is to propose and justify the use of a few measures of inequality for summarizing the basic information
provided by the Lorenz curve. By exploiting the fact that the Lorenz curve can be considered analogous to a cumulative distribution
function it is demonstrated that the moments of the Lorenz curve generate a convenient family of inequality measures, called
the Lorenz family of inequality measures. In particular, the first few moments, which often capture the essential features
of a distribution function, are proposed as the primary quantities for summarizing the information content of the Lorenz curve.
Employed together these measures, which include the Gini coefficient, also provide essential information on the shape of the
income distribution. Relying on the principle of diminishing transfers it is shown that the Lorenz measures, as opposed to
the Atkinson measures, have transfer-sensitivity properties that depend on the shape of the income distribution.
Received: 20 July 1998/Accepted: 10 September 1999 相似文献
5.
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. 相似文献
6.
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. 相似文献
7.
Patrick Moyes 《Journal of Economic Inequality》2007,5(3):279-303
It is well-known that, when the Lorenz curves do not cross, the ranking of distributions provided by the Gini index is identical
to the one implied by the Lorenz criterion. This does not preclude inequality as measured by the Gini index to increase while
the Lorenz curves cross. A suitable modification of the Gini coefficient allows the Lorenz quasi-ordering to coincide with
the ranking generated by the application of unanimity over the class of extended Gini indices. Recently the Lorenz quasi-ordering
and the underlying principle of transfers have come under attack, while new criteria – the differentials, deprivation and
satisfaction quasi-orderings – have been proposed for providing unambiguous rankings of distributions. We suggest to weaken
the principle of transfers by imposing additional restrictions on the progressive transfers, which take into account the positions
on the income scale of the donors and beneficiaries. We identify the subclasses of extended Gini indices that satisfy these
weaker versions of the principle of transfers and we show that the application of unanimity among these classes generate rankings
of distributions that coincide with those implied by the differentials, deprivation and satisfaction quasi-orderings.
相似文献
8.
Karl Mosler 《Journal of Economic Inequality》2004,2(2):89-103
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. 相似文献
9.
Karl Mosler 《Journal of Economic Inequality》2005,2(2):89-103
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. 相似文献
10.
Gini’s nuclear family 总被引:1,自引:0,他引:1
Rolf Aaberge 《Journal of Economic Inequality》2007,5(3):305-322
The purpose of this paper is to justify the use of the Gini coefficient and two close relatives for summarizing the basic
information of inequality in distributions of income. To this end we employ a specific transformation of the Lorenz curve,
the scaled conditional mean curve, rather than the Lorenz curve as the basic formal representation of inequality in distributions
of income. The scaled conditional mean curve is shown to possess several attractive properties as an alternative interpretation
of the information content of the Lorenz curve and furthermore proves to yield essential information on polarization in the
population. The paper also provides asymptotic distribution results for the empirical scaled conditional mean curve and the
related family of empirical measures of inequality.
相似文献
11.
Lorenz curves and second-order dominance criteria, the fundamental tools for stochastic dominance, are known to be sensitive
to data contamination in the tails of the distribution. We propose two ways of dealing with the problem: (1) Estimate Lorenz
curves using parametric models and (2) combine empirical estimation with a parametric (robust) estimation of the upper tail
of the distribution using the Pareto model. Approach (2) is preferred because of its flexibility. Using simulations we show
the dramatic effect of a few contaminated data on the Lorenz ranking and the performance of the robust semi-parametric approach
(2). Since estimation is only a first step for statistical inference and since semi-parametric models are not straightforward
to handle, we also derive asymptotic covariance matrices for our semi-parametric estimators. 相似文献
12.
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. 相似文献
13.
Two preimputations of a given TU game can be compared via the Lorenz order applied to the vectors of satisfactions. One preimputation
is `socially more desirable' than the other, if its corresponding vector of satisfactions Lorenz dominates the satisfaction
vector with respect to the second preimputation. It is shown that the prenucleolus, the anti-prenucleolus, and the modified
nucleolus are maximal in this Lorenz order. Here the modified nucleolus is the unique preimputation which lexicographically
minimizes the envies between the coalitions, i.e. the differences of excesses. Recently Sudh?lter developed this solution
concept. Properties of the set of all undominated preimputations, the maximal satisfaction solution, are discussed. A function
on the set of preimputations is called collective satisfaction function if it respects the Lorenz order. We prove that both
classical nucleoli are unique minimizers of certain `weighted Gini inequality indices', which are derived from some collective
satisfaction functions. For the (pre)nucleolus the function proposed by Kohlberg, who characterized the nucleolus as a solution
of a single minimization problem, can be chosen. Finally, a collective satisfaction function is defined such that the modified
nucleolus is its unique maximizer.
Received: 18 October 1996 / Accepted: 31 January 1997 相似文献
14.
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. 相似文献
15.
It is possible to partially order cities according to the informativeness of neighborhoods about their ethnic groups. It is also possible to partially order cities with two ethnic groups according to the Lorenz criterion. We show that a segregation order satisfies four well-established segregation principles if and only if it is consistent with the informativeness criterion. We then use this result to show that for the two-group case, the Lorenz and the informativeness criteria are equivalent. 相似文献
16.
Inequality orderings and unit consistency 总被引:1,自引:0,他引:1
Buhong Zheng 《Social Choice and Welfare》2007,29(3):515-538
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. 相似文献
17.
W. Henry Chiu 《Social Choice and Welfare》2007,28(3):375-399
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 相似文献
18.
Typical welfare and inequality measures are required to be Lorenz consistent which guarantees that inequality decreases and welfare increases as a result of a progressive transfer. We explore the implications for welfare and inequality measurement of substituting the weaker absolute differentials and deprivation quasi-orderings for the Lorenz quasi-ordering. Restricting attention to distributions of equal means, we show that the utilitarian model – the so-called expected utility model in the theory of risk – does not permit one to make a distinction between the views embedded in the differentials, deprivation and Lorenz quasi-orderings. In contrast it is possible within the dual model of M. Yaari (Econometrica 55 (1987), 99–115) to derive the restrictions to be placed on the weighting function which guarantee that the corresponding welfare orderings are consistent with the differentials and deprivation quasi-orderings respectively. Finally we drop the equal mean condition and indicate the implications of our approach for the absolute ethical inequality indices. 相似文献
19.
Typical welfare and inequality measures are required to be Lorenz consistent which guarantees that inequality decreases and welfare increases as a result of a progressive transfer. We explore the implications for welfare and inequality measurement of substituting the weaker absolute differentials and deprivation quasi-orderings for the Lorenz quasi-ordering. Restricting attention to distributions of equal means, we show that the utilitarian model - the so-called expected utility model in the theory of risk - does not permit one to make a distinction between the views embedded in the differentials, deprivation and Lorenz quasi-orderings. In contrast it is possible within the dual model of M. Yaari (Econometrica
55 (1987), 99–115) to derive the restrictions to be placed on the weighting function which guarantee that the corresponding welfare orderings are consistent with the differentials and deprivation quasi-orderings respectively. Finally we drop the equal mean condition and indicate the implications of our approach for the absolute ethical inequality indices. 相似文献
20.
Asis Kumar Banerjee 《Social Choice and Welfare》2014,42(1):171-191
This paper seeks to extend the unidimensional notion of Lorenz dominance to the multidimensional context. It formulates a definition of a multidimensional Lorenz dominance relation (MLDR) on the set of alternative distributions of well-being in an economy by incorporating a generalization of the well-known transfer principle of unidimensional theory suggested in recent literature. It also proposes two conditions which an MLDR may reasonably be required to satisfy. The paper notes that the existing literature does not seem to contain an example of an MLDR satisfying these two conditions and suggests one that does. The suggested MLDR does not seem to have appeared in the literature before. 相似文献