Constructing indices of multivariate polarization

  • Chiara Gigliarano
  • Karl Mosler


Multivariate indices of polarization are constructed to measure effects of non-income attributes like wealth and education. Polarization is considered as the presence of groups which are internally homogeneous, externally heterogeneous, and of similar size. We propose a class of polarization indices which is built from measures of relative groups size and from decomposable indices of socio-economic inequality. For the latter, we employ the special inequality indices of Maasoumi (Econometrica 54:991–997, 1986), Tsui (J Econ Theory 67:251–265, 1995; Soc Choice Welf 16:145–157, 1999) and Koshevoy and Mosler (J Multivar Anal 60:252–276, 1997). Then, postulates for multidimensional polarization measurement are stated and discussed. The approach is illustrated by an empirical application to the population of the East and West Germany with polarization defined on income and education.


Polarization index Decomposable inequality indices Multidimensional inequality Multivariate social evaluation 

JEL Classifications

C43 D63 


  1. 1.
    Atkinson, A., Bourguignon, F.: The comparison of multidimensional distributions of economic status. Rev. Econ. Stud. 49, 183–201 (1982)CrossRefGoogle Scholar
  2. 2.
    Bhattacharya, N., Mahalanobis, B: Regional disparities in household consumption in India. J. Am. Stat. Assoc. 62, 143–161 (1967)CrossRefGoogle Scholar
  3. 3.
    Blackorby, C., Donaldson, D., Auersperg, M.: A new procedure for the measurement of inequality within and among population subgroups. Can. J. Econ. 14, 665–685 (1981)CrossRefGoogle Scholar
  4. 4.
    Bourguignon, F., Chakravarty, S.R.: The measurement of multidimensional poverty. J. Econ. Inequality 1, 25–49 (2003)CrossRefGoogle Scholar
  5. 5.
    Chakravarty, S.R., Majumder, A.: Inequality, polarization and welfare: theory and applications. Aust. Econ. Pap. 40, 1–13 (2001)CrossRefGoogle Scholar
  6. 6.
    Chakravarty, S.R., Weymark, J.A.: Axiomatizations of the entropy numbers equivalent index of industrial concentration. In: Eichhorn, W. (ed.) Measurement in economics, pp. 437–484. Kluwer, London (1988)Google Scholar
  7. 7.
    D´Ambrosio, C.: Household characteristics and the distribution of income in Italy: an application of a social distance measures. Rev. Income Wealth 47, 43–64 (2001)CrossRefGoogle Scholar
  8. 8.
    Davis, J., Huston, J.: The shrinking middle-income class: a multivariate analysis. East. Econ. J. 18, 277–285 (1992)Google Scholar
  9. 9.
    Donaldson, D., Weymark, J.: A single-parameter generalization of the Gini indices of inequality. J. Econ. Theory 22, 67–86 (1980)CrossRefGoogle Scholar
  10. 10.
    Duclos, J., Esteban, J., Ray, D.: Polarization: concepts, measurement, estimation. Econometrica 72, 1737–1772 (2004)CrossRefGoogle Scholar
  11. 11.
    Esteban, J., Gradín, C., Ray, D.: An extension of a measure of polarization, with an application to the income distribution of five OECD countries. J. Econ. Inequality 5, 1–19 (2007)CrossRefGoogle Scholar
  12. 12.
    Esteban, J., Ray, D.: On the measurement of polarization. Econometrica 62, 819–851 (1994)CrossRefGoogle Scholar
  13. 13.
    Gradín, C.: Polarization by sub-populations in Spain, 1973–91. Rev. Income Wealth 46, 457–474 (2000)CrossRefGoogle Scholar
  14. 14.
    Kolm, S.: Multidimensional egalitarianisms. Q. J. Econ. 91, 1–13 (1977)CrossRefGoogle Scholar
  15. 15.
    Koshevoy, G., Mosler, K.: Multivariate Gini indices. J. Multivar. Anal. 60, 252–276 (1997)CrossRefGoogle Scholar
  16. 16.
    Lasso de la Vega, M.C., Urrutia, A.M.: A new factorial decomposition for the Atkinson measure. Econ. Bull. 4, 1–12 (2003)Google Scholar
  17. 17.
    Maasoumi, E.: The measurement and decomposition of multidimensional inequality. Econometrica 54, 991–997 (1986)CrossRefGoogle Scholar
  18. 18.
    Maasoumi, E., Nickelsburg, G.: Multivariate measures of well-being and an analysis of inequality in the Michigan data. J. Bus. Econ. Stat. 6, 327–334 (1988)CrossRefGoogle Scholar
  19. 19.
    Marshall, A., Olkin, I.: Inequalities: Theory of Majorization and its Applications. Academic, New York (1979)Google Scholar
  20. 20.
    Milanovic, B.: A New Polarization Measure and Some Applications. Mimeo, Development Research Group, World Bank (2000)Google Scholar
  21. 21.
    Mosler, K.: Majorization in economic disparity measures. Linear Algebra Appl. 199, 91–114 (1994)CrossRefGoogle Scholar
  22. 22.
    Rodríguez, J.G., Salas, R.: Extended bi-polarization and inequality measures. Res. Econ. Inequal. 9, 69–83 (2003)CrossRefGoogle Scholar
  23. 23.
    Tsui, K.: Multidimensional generalizations of the relative and absolute inequality indices: the Atkinson-Kolm-Sen approach. J. Econ. Theory 67, 251–265 (1995)CrossRefGoogle Scholar
  24. 24.
    Tsui, K.: Multidimensional inequality and multidimensional generalized entropy measures: an axiomatic derivation. Soc. Choice Welf. 16, 145–157 (1999)CrossRefGoogle Scholar
  25. 25.
    Wang, Y., Tsui, K.: Polarization orderings and new classes of polarization indices. J. Public Econ. Theory 2, 349–363 (2000)CrossRefGoogle Scholar
  26. 26.
    Wolfson, M.C.: When inequalities diverge. Am. Econ. Rev. 48, 353–358 (1994)Google Scholar
  27. 27.
    Wolfson, M.C.: Divergent inequalities: theory and empirical results. Rev. Income Wealth 43, 401–421 (1997)CrossRefGoogle Scholar
  28. 28.
    Zhang, X., Kanbur, R.: What difference do polarisation measures make? An application to China. J. Dev. Stud. 37, 85–98 (2001)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2008

Authors and Affiliations

  1. 1.Department of EconomicsUniversità Politecnica delle MarcheAnconaItaly
  2. 2.Universität zu KölnKölnGermany

Personalised recommendations