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From unidimensional to multidimensional inequality: a review

Abstract

We review results concerning the representation of partial orders of univariate distributions via stochastic orders and investigate their applications to some classes of stochastic dominance conditions applied in inequality and welfare measurement. The results obtained in an unidimensional framework are extended to multidimensional analysis. We discuss difficulties arising from aggregation of multidimensional distributions into synthetic indicators that value both inequality in the distribution of each attribute and the association between the attributes. We explore the potential for multidimensional evaluations that are based on the partial orders induced by different criteria of majorization and organize related and equivalent inequality and welfare representations.

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Notes

  1. 1.

    For alternative reviews of multivariate inequality see Savaglio [82], Marshall et al. [64] and Aaberge and Brandolini [3].

  2. 2.

    Both in Dahl [23] and Andreoli and Zoli [5] the criterion of matrix majorization is formalized through transformations obtained by post multiplication of a matrix by a row stochatic matrix. However, for consistency of notation in this work all matrices are transposed with respect to those considered in the mentioned papers and therefore the notion of matrix majorization should be based on transformations generated by pre-multiplication via column stochastic matrices.

  3. 3.

    In order to obtain this result it is sufficient to set the integer n sufficiently large to guarantee that the product \(nz_{i}\) gives an integer number \(s_{i}\) for any unit i in the populations of both \(X^{\prime }\) and \(Y^{\prime }.\) Then, each unit i should be split in \(s_{i}\) units of equal weight 1/n.

  4. 4.

    For general definitions of lift zonoid and Lorenz zonoid see Koshevoy and Mosler [57] and Mosler [70].

  5. 5.

    These concepts are known in the literature under different denominations. They are known either as linear-combinations majorization and positive linear-combinations majorization or as directional majorization and positive directional majorization, see Bhandari [11], Joe and Verducci [51], and Tsui [94], for an overview see Marshall et al. [64] Ch.15. Koshevoy and Mosler [57] suggest the possibility of interpreting the directional majorization in terms of prices (both positive and negative), while Mosler [70] uses the term price majorization for the case where prices are non-negative. In our exposition we retain the explanation in terms of prices making explicit in the terminology when non-negative prices are considered, in this case we use the term positive price majorization.

  6. 6.

    As an extreme case if \(p_{1}=-1\) and \(p_{2}=+1,\) then the potential budget distibution for Z is (0, 0);  and for Y is \((1,-1)\) where clearly the former is less unequal than the latter also in term of the PD principle of transfers. This result goes in the opposite direction of what required by Aversion to CIT.

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Andreoli F. research has also been undertaken at the Luxembourg Institute of Socio-Economic Research, LISER, his research acknowledges financial support from the French National Agency for Research under the project The Measurement of Ordinal and Multidimensional Inequalities (grant ANR-16-CE41-0005-01) and the Luxembourg Fonds National de la Recherche (IMCHILD grant INTER/NORFACE/16/11333934). This research is also part of the project MOBILIFE (grant RBVR-17KFHX) supported by the University of Verona. This paper is partially built on a revised and updated version of C. Zoli lecture notes on “multidimensional inequality” presented at Canazei winter school on Inequality and Social Welfare Theory. We are grateful to the editor and the anonymous referees for helpful suggestions on earlier versions of the paper. The usual disclaimer applies.

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Andreoli, F., Zoli, C. From unidimensional to multidimensional inequality: a review. METRON 78, 5–42 (2020). https://doi.org/10.1007/s40300-020-00168-4

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Keywords

  • Multidimensional inequality
  • Well-being
  • Majorization
  • Stochastic orders
  • Lorenz zonoid
  • Price dominance