IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2203.09000.html
   My bibliography  Save this paper

Lorenz map, inequality ordering and curves based on multidimensional rearrangements

Author

Listed:
  • Yanqin Fan
  • Marc Henry
  • Brendan Pass
  • Jorge A. Rivero

Abstract

We propose a multivariate extension of the Lorenz curve based on multivariate rearrangements of optimal transport theory. We define a vector Lorenz map as the integral of the vector quantile map associated with a multivariate resource allocation. Each component of the Lorenz map is the cumulative share of each resource, as in the traditional univariate case. The pointwise ordering of such Lorenz maps defines a new multivariate majorization order, which is equivalent to preference by any social planner with inequality averse multivariate rank dependent social evaluation functional. We define a family of multi-attribute Gini index and complete ordering based on the Lorenz map. We propose the level sets of an Inverse Lorenz Function as a practical tool to visualize and compare inequality in two dimensions, and apply it to income-wealth inequality in the United States between 1989 and 2022.

Suggested Citation

  • Yanqin Fan & Marc Henry & Brendan Pass & Jorge A. Rivero, 2022. "Lorenz map, inequality ordering and curves based on multidimensional rearrangements," Papers 2203.09000, arXiv.org, revised Apr 2024.
  • Handle: RePEc:arx:papers:2203.09000
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2203.09000
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Puccetti, Giovanni & Scarsini, Marco, 2010. "Multivariate comonotonicity," Journal of Multivariate Analysis, Elsevier, vol. 101(1), pages 291-304, January.
    2. Alfred Galichon & Marc Henry, 2012. "Dual theory of choice under multivariate risks," SciencePo Working papers Main hal-01024582, HAL.
    3. José María Sarabia & Vanesa Jorda, 2020. "Lorenz Surfaces Based on the Sarmanov–Lee Distribution with Applications to Multidimensional Inequality in Well-Being," Mathematics, MDPI, vol. 8(11), pages 1-17, November.
    4. Stéphane Bonhomme & Jean-Marc Robin, 2009. "Assessing the Equalizing Force of Mobility Using Short Panels: France, 1990-2000," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 76(1), pages 63-92.
    5. repec:hal:spmain:info:hdl:2441/5rkqqmvrn4tl22s9mc0p30p95 is not listed on IDEAS
    6. Victor Chernozhukov & Alfred Galichon & Marc Hallin & Marc Henry, 2014. "Monge-Kantorovich Depth, Quantiles, Ranks, and Signs," Papers 1412.8434, arXiv.org, revised Sep 2015.
    7. Arthur B. Kennickell & R. Louise Woodburn, 1999. "CONSISTENT WEIGHT DESIGN FOR THE 1989, 1992 AND 1995 SCFs, AND THE DISTRIBUTION OF WEALTH," Review of Income and Wealth, International Association for Research in Income and Wealth, vol. 45(2), pages 193-215, June.
    8. Thibault Gajdos & John Weymark, 2005. "Multidimensional generalized Gini indices," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 26(3), pages 471-496, October.
    9. Koshevoy, G. A. & Mosler, K., 1997. "Multivariate Gini Indices," Journal of Multivariate Analysis, Elsevier, vol. 60(2), pages 252-276, February.
    10. A. B. Atkinson & F. Bourguignon, 1982. "The Comparison of Multi-Dimensioned Distributions of Economic Status," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 49(2), pages 183-201.
    11. Rüschendorf, L. & Rachev, S. T., 1990. "A characterization of random variables with minimum L2-distance," Journal of Multivariate Analysis, Elsevier, vol. 32(1), pages 48-54, January.
    12. Rolf Aaberge & Andrea Brandolini, 2014. "Multidimensional poverty and inequality," Discussion Papers 792, Statistics Norway, Research Department.
    13. Gleb A. Koshevoy & Karl Mosler, 2007. "Multivariate Lorenz dominance based on zonoids," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 91(1), pages 57-76, March.
    14. Franklin M. Fisher, 1956. "Income Distribution, Value Judgments, and Welfare," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 70(3), pages 380-424.
    15. Jesse Bricker & Lisa J. Dettling & Alice Henriques Volz & Joanne W. Hsu & Lindsay Jacobs & Kevin B. Moore & Sarah Pack & John Edward Sabelhaus & Jeffrey P. Thompson & Richard Windle, 2017. "Changes in U.S. Family Finances from 2013 to 2016: Evidence from the Survey of Consumer Finances," Federal Reserve Bulletin, Board of Governors of the Federal Reserve System (U.S.), vol. 103(3), September.
    16. Edward N. Wolff, 2021. "Household Wealth Trends in the United States, 1962 to 2019: Median Wealth Rebounds... But Not Enough," NBER Working Papers 28383, National Bureau of Economic Research, Inc.
    17. repec:hal:spmain:info:hdl:2441/64itsev5509q8aa5mrbhi0g0b6 is not listed on IDEAS
    18. Francesco Andreoli & Claudio Zoli, 2020. "From unidimensional to multidimensional inequality: a review," METRON, Springer;Sapienza Università di Roma, vol. 78(1), pages 5-42, April.
    19. Banerjee, Asis Kumar, 2010. "A multidimensional Gini index," Mathematical Social Sciences, Elsevier, vol. 60(2), pages 87-93, September.
    20. Serge-Christophe Kolm, 1977. "Multidimensional Egalitarianisms," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 91(1), pages 1-13.
    21. repec:hal:wpspec:info:hdl:2441/5rkqqmvrn4tl22s9mc0p30p95 is not listed on IDEAS
    22. Gastwirth, Joseph L, 1971. "A General Definition of the Lorenz Curve," Econometrica, Econometric Society, vol. 39(6), pages 1037-1039, November.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Mohamad Khaled & Paul Makdissi & Prasada Rao & Myra Yazbeck, 2023. "A Unidimensional Representation of Multidimensional Inequality: An Econometric Analysis of Inequalities in the Arab Region," Working Papers 2304E Classification- D63, University of Ottawa, Department of Economics.
    2. Mohamad A. Khaled & Paul Makdissi & D.S. Prasada Rao & Myra Yazbeck, 2023. "A unidimensional representation of multidimensional inequality, with an application to the Arab region," Discussion Papers Series 659, School of Economics, University of Queensland, Australia.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Yanqin Fan & Marc Henry, 2020. "Vector copulas," Papers 2009.06558, arXiv.org, revised Apr 2021.
    2. Asis Kumar Banerjee, 2019. "Economic Properties of Statistical Indices: The Case of a Multidimensional Gini Index," Journal of Quantitative Economics, Springer;The Indian Econometric Society (TIES), vol. 17(1), pages 41-56, March.
    3. Asis Banerjee, 2014. "A multidimensional Lorenz dominance relation," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 42(1), pages 171-191, January.
    4. Thi Kim Thanh Bui & Guido Erreygers, 2020. "Multidimensional Inequality in Vietnam, 2002–2012," Economies, MDPI, vol. 8(2), pages 1-31, April.
    5. Henar Diez & Mª Casilda Lasso de la Vega & Ana Marta Urrutia, 2007. "Unit-Consistent Aggregative Multidimensional Inequality Measures: A Characterization," Working Papers 66, ECINEQ, Society for the Study of Economic Inequality.
    6. Fan, Yanqin & Henry, Marc, 2023. "Vector copulas," Journal of Econometrics, Elsevier, vol. 234(1), pages 128-150.
    7. Thibault Gajdos & John Weymark, 2005. "Multidimensional generalized Gini indices," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 26(3), pages 471-496, October.
    8. Oliver Grothe & Fabian Kächele & Friedrich Schmid, 2022. "A multivariate extension of the Lorenz curve based on copulas and a related multivariate Gini coefficient," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 20(3), pages 727-748, September.
    9. Francesco Andreoli & Claudio Zoli, 2020. "From unidimensional to multidimensional inequality: a review," METRON, Springer;Sapienza Università di Roma, vol. 78(1), pages 5-42, April.
    10. Francesco Andreoli & Claudio Zoli, 2023. "Robust dissimilarity comparisons with categorical outcomes," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 60(3), pages 397-437, April.
    11. Galichon, Alfred & Henry, Marc, 2012. "Dual theory of choice with multivariate risks," Journal of Economic Theory, Elsevier, vol. 147(4), pages 1501-1516.
    12. José María Sarabia & Vanesa Jorda, 2020. "Lorenz Surfaces Based on the Sarmanov–Lee Distribution with Applications to Multidimensional Inequality in Well-Being," Mathematics, MDPI, vol. 8(11), pages 1-17, November.
    13. John A. Weymark, 2003. "The Normative Approach to the Measurement of Multidimensional Inequality," Vanderbilt University Department of Economics Working Papers 0314, Vanderbilt University Department of Economics, revised Jan 2004.
    14. Karl Mosler, 2023. "Representative endowments and uniform Gini orderings of multi-attribute welfare," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 21(1), pages 233-250, March.
    15. Koen Decancq, 2020. "Measuring cumulative deprivation and affluence based on the diagonal dependence diagram," METRON, Springer;Sapienza Università di Roma, vol. 78(2), pages 103-117, August.
    16. Chiara Gigliarano & Karl Mosler, 2009. "Constructing indices of multivariate polarization," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 7(4), pages 435-460, December.
    17. Maria Ana Lugo & Koen Decancq, 2009. "Measuring Inequality of Well-Being with a Correlation-Sensitive Multidimensional Gini Index," Economics Series Working Papers 459, University of Oxford, Department of Economics.
    18. Oscar de J. Gálvez-Soriano & Paulina Benitez-Blacio, 2018. "How to Measure the Multidimensional Inequality with Household Surveys: The Mexican Case," Remef - Revista Mexicana de Economía y Finanzas Nueva Época REMEF (The Mexican Journal of Economics and Finance), Instituto Mexicano de Ejecutivos de Finanzas, IMEF, vol. 13(2), pages 175-193, Abril-Jun.
    19. Masato Okamoto, 2009. "Decomposition of gini and multivariate gini indices," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 7(2), pages 153-177, June.
    20. Marcello Basili & Paulo Casaca & Alain Chateauneuf & Maurizio Franzini, 2017. "Multidimensional Pigou–Dalton transfers and social evaluation functions," Theory and Decision, Springer, vol. 83(4), pages 573-590, December.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:2203.09000. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.