The Generalized Beta Distribution as a Model for the Distribution of Income: Estimation of Related Measures of Inequality

  • James B. McDonald
  • Michael Ransom
Part of the Economic Studies in Equality, Social Exclusion and Well-Being book series (EIAP, volume 5)


The generalized beta (GB) is considered as a model for the distribution of income. It is well known that its special cases include Dagum’s distribution along with the Singh-Maddala distribution. Related measures of inequality such as the Gini Coefficient, Pietra Index, or Theil Index are expressed in terms of the parameters of the generalized beta. This paper also explores the use of numerical integration techniques for calculating inequality indexes. Numerical integration may be useful since in some cases it may be computationally very difficult to evaluate the equations that have been derived or the equations are not available. We provide examples from the distribution of family income in the United States for the year 2000.


Income Inequality Income Distribution Gini Index Lorenz Curve Inequality Measure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Aitchison, J. and J. A. C. Brown (1969) The Lognormal Distribution with Special References to Its Uses in Economics, Cambridge University Press, Cambridge.Google Scholar
  2. Ammon, O. (1895) Die Gesellschaftsordnung und ihre Natu¨rlichen Grundlagen, Jena.Google Scholar
  3. Amoroso, L. (1924-1925) Ricerche Intorno alla Curva dei Redditi Annali di Matematica, Pura ed Applicata, Series 4-21, 2, 123-157.Google Scholar
  4. Bandourian, R., J. B. McDonald and R. S. Turley (2003) A Comparison of Paramet-ric Models of Income Distribution across Countries and over Time, Estadística, 55, 135-152.Google Scholar
  5. Bartels, C. P. A. and H. van Metelen (1975) Alternative Probability Density Functions of Income, Vrije University Amsterdam: Research Memorandum 29.Google Scholar
  6. Bordley, R. F., J. B. McDonald and A. Mantrala (1996) Something New, Something Old: Parametric Models for the Size Distribution of Income, Journal of Income Distribution, 6, 91-103.Google Scholar
  7. Cox, D. R. and D. V. Hinkley (1974) Theoretical Statistics, Chapman and Hall, London.Google Scholar
  8. Dagum, C. (1977) A New Model for Personal Income Distribution: Specification and Estimation, Economie Appliqu ée, 30, 413-437.Google Scholar
  9. Dagum, C. (1980) The Generation and Distribution of Income, the Lorenz Curve and the Gini Ratio, Economie Appliqu ée, 33, 327-367.Google Scholar
  10. Dastrup, S. R., R. Hartshorn and J. B. McDonald (2007) The Impact of Taxes and Transfer Payments on the Distribution of Income: A Parametric Comparison, Journal of Economic Inequality, 5, 353-369.CrossRefGoogle Scholar
  11. Dorfman, R. (1979) A Formula for the Gini Coefficient, Review of Economics and Statistics, 61, 146-149.CrossRefGoogle Scholar
  12. Gastwirth, J. L. (1971) A General Definition of the Lorenz Curve, Econometrica, 39, 1037-1039.CrossRefGoogle Scholar
  13. Gastwirth, J. L. (1972) The Estimation of the Lorenz Curve and Gini Index, Review of Economics and Statistics, 54, 306-316.CrossRefGoogle Scholar
  14. Gibrat, R. (1931) Les In égalit és E´conomiques, Librairie du Recueil Sirey, Paris.Google Scholar
  15. Gini, C. (1912) Variabilita’ e Mutabilita, Studio Economicogiuridici, Universita di Cagliari. 3:2a. Reprinted in C. Gini (1955), 211-382.Google Scholar
  16. Israelsen, D. and J. B. McDonald (2003) Measurement Error and the Distribution of Income, Journal of Income Distribution, 12, 21-31.Google Scholar
  17. March, L. (1898) Quelques Exemples de Distribution des Salaires, Journal de la Soci ét é Statistique de Paris, pp. 193-206 and 241-248.Google Scholar
  18. McDonald, J. B. (1981) Some Issues Associated with the Measurement of Income Inequality, in C. Taillie, G. P. Patil and B. Balderssari (eds.) Statistical Distributions in Scientific Work, vol. 6, Reidel, Boston.Google Scholar
  19. McDonald, J. B. (1984) Some Generalized Functions for the Size Distribution of Income, Econometrica, 52, 647-663.CrossRefGoogle Scholar
  20. McDonald, J. B. and R. J. Butler (1990) Regression Models for Positive Random Variables, Journal of Econometrics, 43, 227-251. CrossRefGoogle Scholar
  21. McDonald, J. B. and M. Ransom (1979a) Alternative Parameter Estimators Based on Grouped Data, Communications in Statistics: Theory and Methods, A8, 899-917.Google Scholar
  22. McDonald, J. B. and M. Ransom (1979b) Functional Forms, Estimation Techniques and the Distribution of Income, Econometrica, 47, 1513-1525.Google Scholar
  23. McDonald, J. B. and Y. J. Xu (1995) A Generalization of the Beta Distribution with Applications, Journal of Econometrics, 66, 133-152, Erratum: Journal of Econometrics, 69: 427-428.Google Scholar
  24. Pareto, V. (1895) La Legge della Domanda, Giornale degli Economisti, 10, 59-68. English Translation in Rivista di Politica Economica, 87 (1997), 691-700.Google Scholar
  25. Pareto, V. (1897) Cours d’Economie Politique, Rouge, Lausanne.Google Scholar
  26. Parker, S. C. (1999) The Generalized Beta as a Model for the Distribution of Earnings, Economics Letters, 62, 197-200.CrossRefGoogle Scholar
  27. Rainville, E. D. (1960) Special Functions, MacMillan, New York.Google Scholar
  28. Salem, A. B. Z. and T. D. Mount (1974) A Convenient Descriptive Model of Income Distribution: The Gamma Density, Econometrica, 42, 1115-1127.CrossRefGoogle Scholar
  29. Sarabia, J. M., E. Castillo and D. J. Slottje (2002) Lorenz Ordering between Mcdon-alds Generalized Functions of the Income Size Distribution, Economics Letters, 75, 265-270.CrossRefGoogle Scholar
  30. Singh, S. K. and G. S. Maddala (1976) A Function for the Size Distribution of Incomes, Econometrica, 44, 963-970.CrossRefGoogle Scholar
  31. Taillie, C. (1981) Lorenz ordering within the Generalized Gamma Family of Income Distributions, in C. Taillie, G. P. Patil and B. Balderssari (eds.) Statistical Distributions in Scientific Work, vol. 6, pp. 181-192, Reidel, Boston.Google Scholar
  32. Thurow, L. C. (1970) Analyzing the American Income Distribution, American Economic Review, 48, 261-269.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • James B. McDonald
    • 1
  • Michael Ransom
    • 1
  1. 1.Department of EconomicsBrigham Young UniversityProvoUSA

Personalised recommendations