Skip to main content
Log in

κ-generalized models of income and wealth distributions: A survey

  • Review
  • Network Economics
  • Published:
The European Physical Journal Special Topics Aims and scope Submit manuscript

Abstract

The paper provides a survey of results related to the “κ-generalized distribution”, a statistical model for the size distribution of income and wealth. Topics include, among others, discussion of basic analytical properties, interrelations with other statistical distributions as well as aspects that are of special interest in the income distribution field, such as the Gini index and the Lorenz curve. An extension of the basic model that is most able to accommodate the special features of wealth data is also reviewed. The survey of empirical applications given in this paper shows the κ-generalized models of income and wealth to be in excellent agreement with the observed data in many cases.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Chatterjee, S. Yarlagadda, B.K. Chakrabarti, Econophysics of Wealth Distributions (Springer-Verlag Italia, Milan, 2005)

  2. V.M. Yakovenko, in Encyclopedia of Complexity and System Science, edited by R.A. Meyers (Springer-Verlag, Berlin, 2009), p. 2800

  3. V.M. Yakovenko, J. Barkley Rosser, Jr., Rev. Mod. Phys. 81, 1703 (2009)

    Article  ADS  Google Scholar 

  4. B.K. Chakrabarti, A. Chakraborti, S.R. Chakravarty, A. Chatterjee, Econophysics of Income and Wealth Distributions (Cambridge University Press, New York, 2013)

  5. J.E. Stiglitz, The Price of Inequality: How Today’s Divided Society Endangers Our Future (W. W. Norton & Company, New York, 2012)

  6. T. Piketty, Capital in the Twenty-First Century (The Belknap Press of Harvard University Press, Cambridge MA, 2014)

  7. A.B. Atkinson, Inequality: What Can Be Done? (Harvard University Press, Cambridge MA, 2015)

  8. J.E. Stiglitz, The Great Divide: Unequal Societies and What We Can Do About Them (W. W. Norton & Company, New York, 2015)

  9. V. Pareto, Giorn. Econ. 10, 59 (1895)

    Google Scholar 

  10. V. Pareto, 1896, reprinted in Œeuvres complètes de Vilfredo Pareto, Tome 3: Écrits sur la courbe de la répartition de la richesse, edited by G. Busino (Librairie Droz, Geneva, 1965), p. 1

  11. V. Pareto, Cours d’économie politique (Macmillan, London, 1897)

  12. V. Pareto, Giorn. Econ. 14, 15 (1897)

    Google Scholar 

  13. C. Kleiber, S. Kotz, Statistical Size Distributions in Economics and Actuarial Sciences (John Wiley & Sons, New York, 2003)

  14. F. Clementi, M. Gallegati, G. Kaniadakis, Eur. Phys. J. B 57, 187 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  15. F. Clementi, T. Di Matteo, M. Gallegati, G. Kaniadakis, Physica A 387, 3201 (2008)

    Article  ADS  Google Scholar 

  16. F. Clementi, M. Gallegati, G. Kaniadakis, J. Stat. Mech. 2009, P02037 (2009)

    Article  Google Scholar 

  17. F. Clementi, M. Gallegati, G. Kaniadakis, Empir. Econ. 39, 559 (2010)

    Article  Google Scholar 

  18. F. Clementi, M. Gallegati, G. Kaniadakis, J. Econ. 105, 63 (2012)

    Article  Google Scholar 

  19. F. Clementi, M. Gallegati, G. Kaniadakis, J. Stat. Mech. 2012, P12006 (2012)

    Article  MathSciNet  Google Scholar 

  20. F. Clementi, M. Gallegati, The Distribution of Income and Wealth: Parametric Modeling with the κ-Generalized Family (Springer International Publishing, Switzerland, 2016)

  21. G. Kaniadakis, Physica A 296, 405 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  22. G. Kaniadakis, Phys. Rev. E 66, 056125 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  23. G. Kaniadakis, Phys. Rev. E 72, 036108 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  24. G. Kaniadakis, Eur. Phys. J. B 70, 3 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  25. G. Kaniadakis, Eur. Phys. J. A 40, 275 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  26. B. Trivellato, Math. Method. Oper. Res. 69, 1 (2009)

    Article  MathSciNet  Google Scholar 

  27. D. Imparato, B. Trivellato, in Algebraic and Geometric Methods in Statistics, edited by P. Gibilisco, E. Riccomagno, M.P. Rogantin, H.P. Wynn (Cambridge University Press, New York, 2010), p. 307

  28. B. Trivellato, Int. J. Theor. App. Finance 15, 1250038 (2012)

    Article  MathSciNet  Google Scholar 

  29. G. Kaniadakis, Entropy 15, 3983 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  30. M. Cravero, G. Iabichino, G. Kaniadakis, E. Miraldi, A.M. Scarfone, Physica A 340, 410 (2004)

    Article  ADS  Google Scholar 

  31. B. Trivellato, Entropy 15, 3471 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  32. E. Moretto, S. Pasquali, B. Trivellato, Physica A 446, 246 (2016)

    Article  MathSciNet  Google Scholar 

  33. D. Rajaoarison, D. Bolduc, H. Jayet, Econ. Lett. 86, 13 (2006)

    Article  Google Scholar 

  34. D. Rajaoarison, Econ. Lett. 100, 396 (2008)

    Article  Google Scholar 

  35. S. Landini, in The Distribution of Income and Wealth: Parametric Modeling with the κ-Generalized Family, by F. Clementi, M. Gallegati (Springer International Publishing, Switzerland, 2016), p. 93

  36. R. D’Addario, Giorn. Econ. Ann. Econ. 33, 205 (1974)

    Google Scholar 

  37. C.P.A. Bartels, H. van Metelen, Alternative probability density functions of income: A comparison of the lognormal-, Gamma- and Weibull-distribution with Dutch data. Research Memorandum No. 29 (1975)

  38. C.P.A. Bartels, Economic aspects of regional welfare, income distribution and unemployment (Martinus Nijhoff, Leiden, 1977)

  39. P. Espinguet, M. Terraza, Econ. Appl. 36, 535 (1983)

    Google Scholar 

  40. J.B. McDonald, Econometrica 52, 647 (1984)

    Article  Google Scholar 

  41. N. Atoda, T. Suruga, T. Tachibanaki, Econ. Stud. Quart. 39, 14 (1988)

    Google Scholar 

  42. R.F. Bordley, J.B. McDonald, A. Mantrala, J. Income Distrib. 6, 91 (1996)

    Google Scholar 

  43. K. Brachmann, A. Stich, Trede, All. Stat. Arch. 80, 285 (1996)

    Google Scholar 

  44. T. Tachibanaki, T. Suruga, N. Atoda, J. Japan. Statist. Soc. 27, 191 (1997)

    Article  Google Scholar 

  45. B. Mandelbrot, Int. Econ. Rev. 1, 79 (1960)

    Article  Google Scholar 

  46. M.O. Lorenz, Publ. Am. Stat. Assoc. 9, 209 (1905)

    Google Scholar 

  47. C. Gini, Atti Reale Istit. Veneto Sci. Lett. Art. 73, 1203 (1914)

    Google Scholar 

  48. M. Okamoto, Extension of the κ-generalized distribution: New four-parameter models for the size distribution of income and consumption. LIS Working Paper No. 600 (2013), available at: http://www.lisdatacenter.org/wps/liswps/600.pdf

  49. A.A. Drăgulescu, V.M. Yakovenko, Eur. Phys. J. B 20, 585 (2001)

    Article  ADS  Google Scholar 

  50. P.D. Allison, Am. Sociol. Rev. 43, 865 (1978)

    Article  Google Scholar 

  51. F.A. Cowell, Eur. Econ. Rev. 13, 147 (1980a)

    Article  Google Scholar 

  52. F.A. Cowell, Rev. Econ. Stud. 47, 521 (1980b)

    Article  Google Scholar 

  53. A.F. Shorrocks, Econometrica 48, 613 (1980)

    Article  MathSciNet  Google Scholar 

  54. F.A. Cowell, K. Kuga, J. Econ. Theory 25, 131 (1981a)

    Article  MathSciNet  Google Scholar 

  55. F.A. Cowell, K. Kuga, Eur. Econ. Rev. 15, 287 (1981b)

    Article  Google Scholar 

  56. H. Theil, Economics and Information Theory (North-Holland, Amsterdam, 1967)

  57. C. Kleiber, Econ. Lett. 57, 39 (1997)

    Article  Google Scholar 

  58. C. Rao, Linear Statistical Inference and its Applications (John Wiley & Sons, New York, 1973)

  59. J.K. Ghosh, Higher Order Asymptotics (Institute of Mathematical Statistics and American Statistical Association, Hayward CA, 1994)

  60. R.J. Schoenberg, Comput. Econ. 10, 251 (1997)

    Article  Google Scholar 

  61. C. Dagum, in Encyclopedia of Statistical Sciences, Second Edition, Volume 5, edited by N. Balakrishnan, C.B. Read, B. Vidakovic (John Wiley & Sons, New York, 2006), p. 3363

  62. C. Dagum, Statistica 66, 235 (2006)

    MathSciNet  Google Scholar 

  63. J.B. Hagerbaumer, Rev. Econ. Stat. 59, 377 (1977)

    Article  Google Scholar 

  64. G. Pyatt, C.-N. Chen, J. Fei, Q. J. Econ. 95, 451 (1980)

    Article  Google Scholar 

  65. Y. Amiel, F.A. Cowell, A. Polovin, Economica 63, S63 (1996)

    Article  Google Scholar 

  66. S.P. Jenkins, M. Jäntti. Methods for Summarizing and Comparing Wealth Distributions. ISER Working Paper No. 2005-05 (2005), available at: https://www.iser.essex.ac.uk/publications/working-papers/iser/2005-05.pdf

  67. F.A. Cowell, Measuring Inequality (Oxford University Press, New York, 2011)

  68. G. Kaniadakis, M. Lissia, A.M. Scarfone, Physica A 340, 41 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  69. G. Kaniadakis, M. Lissia, A.M. Scarfone, Phys. Rev. E 71, 046128 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  70. E. Parzen, J. Am. Stat. Assoc. 74, 105 (1979)

    Article  MathSciNet  Google Scholar 

  71. W.J. Reed, Physica A 319, 469 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  72. W.J. Reed, J. Income Distrib. 13, 7 (2004)

    Google Scholar 

  73. D.G. Champernowne, Econ. J. 63, 318 (1953)

    Article  Google Scholar 

  74. S.K. Singh, G.S. Maddala, Econometrica 44, 963 (1976)

    Article  Google Scholar 

  75. C. Dagum, Econ. Appl. 30, 413 (1977)

    Google Scholar 

  76. M. Okamoto, Econ. Bull. 32, 2969 (2012)

    Google Scholar 

  77. W.J. Reed, M. Jorgensen, Commun. Stat.-Theor. M. 33, 1733 (2004)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Fabio Clementi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Clementi, F., Gallegati, M., Kaniadakis, G. et al. κ-generalized models of income and wealth distributions: A survey. Eur. Phys. J. Spec. Top. 225, 1959–1984 (2016). https://doi.org/10.1140/epjst/e2016-60014-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1140/epjst/e2016-60014-2

Navigation