IDEAS home Printed from https://ideas.repec.org/p/inq/inqwps/ecineq2012-253.html
   My bibliography  Save this paper

Classical inequality indices, welfare functions, and the dual decomposition

Author

Listed:
  • Oihana Aristondo

    (BRIDGE Research Group, Universidad del País Vasco)

  • José Luis García-Lapres

    () (PRESAD Research Group, IMUVA, Universidad Valladolid)

  • Casilda Lasso de la Vega

    () (BRIDGE Research Group, Universidad del País Vasco)

  • Ricardo Alberto Marques Pereira

    () (Dipartimento di Informatica e Studi Aziendali, Universitμa degli Studi di Trento)

Abstract

We consider the classical inequality measures due to Gini, Bonferroni, and De Vergottini and we present a brief review of the three inequality indices and the associated welfare functions, in the correspondence scheme introduced by Blackorby and Donaldson, and Weymark. The three classical inequality indices incorporate different value judgments in the measurement of inequality, leading to different behavior under income transfers between individuals in the population. The welfare functions associated with the Gini, Bonferroni, and (normalized) De Vergottini indices are Schur-concave OWA functions, with larger weights for lower incomes. We examine the dual decomposition and the orness degree of the three welfare functions in the standard framework of aggregation functions on the [0; 1]n domain, and show that it offers interesting insight on the distinct and complementary nature of the classical inequality indices.

Suggested Citation

  • Oihana Aristondo & José Luis García-Lapres & Casilda Lasso de la Vega & Ricardo Alberto Marques Pereira, 2012. "Classical inequality indices, welfare functions, and the dual decomposition," Working Papers 253, ECINEQ, Society for the Study of Economic Inequality.
  • Handle: RePEc:inq:inqwps:ecineq2012-253
    as

    Download full text from publisher

    File URL: http://www.ecineq.org/milano/WP/ECINEQ2012-253.pdf
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Garcia-Lapresta, Jose Luis & Llamazares, Bonifacio, 2001. "Majority decisions based on difference of votes," Journal of Mathematical Economics, Elsevier, vol. 35(3), pages 463-481, June.
    2. Dorfman, Robert, 1979. "A Formula for the Gini Coefficient," The Review of Economics and Statistics, MIT Press, vol. 61(1), pages 146-149, February.
    3. Donaldson, David & Weymark, John A., 1980. "A single-parameter generalization of the Gini indices of inequality," Journal of Economic Theory, Elsevier, vol. 22(1), pages 67-86, February.
    4. Yaari, Menahem E, 1987. "The Dual Theory of Choice under Risk," Econometrica, Econometric Society, vol. 55(1), pages 95-115, January.
    5. Oihana Aristondo & José Luis García-Lapresta & Casilda Lasso de la Vega & Ricardo Alberto Marques Pereira, 2011. "The Gini index,the dual decomposition of aggregation functions, and the consistent measurement of inequality," Working Papers 203, ECINEQ, Society for the Study of Economic Inequality.
    6. Sen, Amartya, 1973. "On Economic Inequality," OUP Catalogue, Oxford University Press, number 9780198281931.
    7. Porath Elchanan Ben & Gilboa Itzhak, 1994. "Linear Measures, the Gini Index, and The Income-Equality Trade-off," Journal of Economic Theory, Elsevier, vol. 64(2), pages 443-467, December.
    8. Kolm, Serge-Christophe, 1976. "Unequal inequalities. II," Journal of Economic Theory, Elsevier, vol. 13(1), pages 82-111, August.
    9. Claudio Zoli, 1999. "Intersecting generalized Lorenz curves and the Gini index," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 16(2), pages 183-196.
    10. Donaldson, David & Weymark, John A., 1983. "Ethically flexible gini indices for income distributions in the continuum," Journal of Economic Theory, Elsevier, vol. 29(2), pages 353-358, April.
    11. Encarnación M. Parrado-Gallardo & Elena Bárcena-Martín & Luis J. Imedio-Olmedo, 2014. "Inequality, Welfare, and Order Statistics," Research on Economic Inequality, in: John A. Bishop & Juan Gabriel Rodríguez (ed.), Economic Well-Being and Inequality: Papers from the Fifth ECINEQ Meeting, volume 22, pages 383-399, Emerald Publishing Ltd.
    12. Bossert, Walter, 1990. "An axiomatization of the single-series Ginis," Journal of Economic Theory, Elsevier, vol. 50(1), pages 82-92, February.
    13. Giovanni Maria Giorgi, 2005. "A methodological survey of recent studies for the measurement of inequality of economic welfare carried out by some Italian statisticians," Econometrics 0509007, University Library of Munich, Germany.
    14. Blackorby, Charles & Donaldson, David, 1980. "A Theoretical Treatment of Indices of Absolute Inequality," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 21(1), pages 107-136, February.
    15. Satya Chakravarty, 2007. "A deprivation-based axiomatic characterization of the absolute Bonferroni index of inequality," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 5(3), pages 339-351, December.
    16. Weymark, John A., 1981. "Generalized gini inequality indices," Mathematical Social Sciences, Elsevier, vol. 1(4), pages 409-430, August.
    17. Mehran, Farhad, 1976. "Linear Measures of Income Inequality," Econometrica, Econometric Society, vol. 44(4), pages 805-809, July.
    18. Kolm, Serge-Christophe, 1976. "Unequal inequalities. I," Journal of Economic Theory, Elsevier, vol. 12(3), pages 416-442, June.
    19. Satya R. Chakravarty & Pietro Muliere, 2003. "Welfare indicators: A review and new perspectives. 1. Measurement of inequality," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(3), pages 457-497.
    20. Atkinson, Anthony B., 1970. "On the measurement of inequality," Journal of Economic Theory, Elsevier, vol. 2(3), pages 244-263, September.
    21. Blackorby, Charles & Donaldson, David, 1978. "Measures of relative equality and their meaning in terms of social welfare," Journal of Economic Theory, Elsevier, vol. 18(1), pages 59-80, June.
    22. Giovanni Maria Giorgi & Michele Crescenzi, 2005. "A look at the Bonferroni inequality measure in a reliability framework," Econometrics 0507004, University Library of Munich, Germany.
    23. Maes, Koen C. & Saminger, Susanne & De Baets, Bernard, 2007. "Representation and construction of self-dual aggregation operators," European Journal of Operational Research, Elsevier, vol. 177(1), pages 472-487, February.
    24. Gastwirth, Joseph L, 1972. "The Estimation of the Lorenz Curve and Gini Index," The Review of Economics and Statistics, MIT Press, vol. 54(3), pages 306-316, August.
    25. Giovanni Maria Giorgi & Riccardo Mondani, 2005. "Sampling distribution of the Bonferroni inequality index from exponential population," Econometrics 0507008, University Library of Munich, Germany.
    26. Yaari, Menahem E., 1988. "A controversial proposal concerning inequality measurement," Journal of Economic Theory, Elsevier, vol. 44(2), pages 381-397, April.
    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. Mariateresa Ciommi & Chiara Gigliarano & Giovanni Maria Giorgi, 2019. "Bonferroni And De Vergottini Are Back: New Subgroup Decompositions And Bipolarization Measures," Working Papers 439, Universita' Politecnica delle Marche (I), Dipartimento di Scienze Economiche e Sociali.
    2. Silvia Bortot & Ricardo Alberto Marques Pereira, 2013. "The binomial Gini inequality indices and the binomial decomposition of welfare functions," Working Papers 305, ECINEQ, Society for the Study of Economic Inequality.
    3. Elena Bárcena-Martin & Jacques Silber, 2017. "The Bonferroni index and the measurement of distributional change," METRON, Springer;Sapienza Università di Roma, vol. 75(1), pages 1-16, April.

    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. Satya Chakravarty, 2007. "A deprivation-based axiomatic characterization of the absolute Bonferroni index of inequality," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 5(3), pages 339-351, December.
    2. Rolf Aaberge, 2007. "Gini’s nuclear family," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 5(3), pages 305-322, December.
    3. Silvia Bortot & Ricardo Alberto Marques Pereira & Thuy H. Nguyen, 2015. "Welfare functions and inequality indices in the binomial decomposition of OWA functions," DEM Discussion Papers 2015/08, Department of Economics and Management.
    4. Satya R. Chakravarty, 2009. "Equity and efficiency as components of a social welfare function," International Journal of Economic Theory, The International Society for Economic Theory, vol. 5(2), pages 181-199, June.
    5. Rolf Aaberge, 2009. "Ranking intersecting Lorenz curves," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 33(2), pages 235-259, August.
    6. Rolf Aaberge & Magne Mogstad, 2009. "On the Measurement of Long-Term Income Inequality and Income Mobility," ICER Working Papers 09-2009, ICER - International Centre for Economic Research.
    7. Luis José Imedio Olmedo & Elena Bárcena Martín, 2007. "Dos familias numerables de medidas de desigualdad," Investigaciones Economicas, Fundación SEPI, vol. 31(1), pages 191-217, January.
    8. Aaberge, Rolf, 2001. "Axiomatic Characterization of the Gini Coefficient and Lorenz Curve Orderings," Journal of Economic Theory, Elsevier, vol. 101(1), pages 115-132, November.
    9. Claudio Zoli, 2002. "Inverse stochastic dominance, inequality measurement and Gini indices," Journal of Economics, Springer, vol. 9(1), pages 119-161, December.
    10. Rolf Aaberge & Magne Mogstad, 2011. "Robust inequality comparisons," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 9(3), pages 353-371, September.
    11. Chateauneuf, Alain & Gajdos, Thibault & Wilthien, Pierre-Henry, 2002. "The Principle of Strong Diminishing Transfer," Journal of Economic Theory, Elsevier, vol. 103(2), pages 311-333, April.
    12. 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.
    13. Yoram Amiel & Frank A Cowell, 1997. "Inequality, Welfare and Monotonicity," STICERD - Distributional Analysis Research Programme Papers 29, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    14. Duclos, Jean-Yves & Jalbert, Vincent & Araar, Abdelkrim, 2000. "Classical Horizontal Inequity and Reranking: an Integrated Approach," Cahiers de recherche 0002, Université Laval - Département d'économique.
    15. Rolf Aaberge & Ugo Colombino, 2005. "Designing Optimal Taxes With a Microeconometric Model of Household Labour Supply," Public Economics 0510013, University Library of Munich, Germany.
    16. repec:ebl:ecbull:v:3:y:2003:i:19:p:1-16 is not listed on IDEAS
    17. Jean-Yves Duclos & Abdelkrim Araar, 2003. "An Atkinson-Gini family of social evaluation functions," Economics Bulletin, AccessEcon, vol. 3(19), pages 1-16.
    18. Fabio Maccheroni & Pietro Muliere & Claudio Zoli, 2005. "Inverse stochastic orders and generalized Gini functionals," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(3), pages 529-559.
    19. Ida Petrillo, 2017. "Ranking income distributions: a rank-dependent and needs-based approach," SERIES 03-2017, Dipartimento di Economia e Finanza - Università degli Studi di Bari "Aldo Moro", revised Jul 2017.
    20. Satya R. Chakravarty & Nachiketa Chattopadhyay & Conchita D'Ambrosio, 2016. "On a Family of Achievement and Shortfall Inequality Indices," Health Economics, John Wiley & Sons, Ltd., vol. 25(12), pages 1503-1513, December.
    21. Elisa Pagani, 2015. "Certainty Equivalent: Many Meanings of a Mean," Working Papers 24/2015, University of Verona, Department of Economics.

    More about this item

    Keywords

    income inequality and social welfare; classical Gini; Bonferroni; and De Vergottini inequality indices; welfare functions; aggregation functions; WA and OWA functions; dual decomposition; ornessClassification-JEL: D63; I32;
    All these keywords.

    JEL classification:

    • D63 - Microeconomics - - Welfare Economics - - - Equity, Justice, Inequality, and Other Normative Criteria and Measurement
    • I32 - Health, Education, and Welfare - - Welfare, Well-Being, and Poverty - - - Measurement and Analysis of Poverty

    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:inq:inqwps:ecineq2012-253. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Maria Ana Lugo). General contact details of provider: https://edirc.repec.org/data/ecineea.html .

    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 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.

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

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.