Abstract
Two parametric models for income distributions are introduced. The models fitted to log(income) are the 4-parameter normal-Laplace (NL) and the 5-parameter generalized normal-Laplace (GNL) distributions. The NL model for log(income) is equivalent to the double-Pareto lognormal (dPlN) distribution applied to income itself. Definitions and properties are presented along with methods for maximum likelihood estimation of parameters. Both models along with 4- and 5-parameter beta distributions, are fitted to nine empirical distributions of family income. In all cases the 4-parameter NL distribution fits better than the 5-parameter generalized beta distribution. The 5-parameter GNL distribution provides an even better fit. However fitting of the GNL is numerically slow, since there are no closed-form expressions for its density or cumulative distribution functions. Given that a fairly recent study involving 83 empirical income distributions (including the nine used in this paper) found the 5-parameter beta distribution to be the best fitting, the results would suggest that the NL be seriously considered as a parametric model for income distributions.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Aitchison, J. and J. A. C. Brown (1969) The Lognormal Distribution with Special References to Its Uses in Economics, Cambridge University Press, Cambridge.
Ammon, O. (1895) Die Gesellschaftsordnung und ihre Naturlichen Grundlagen, Jena.
Bandourian, R., J. B. McDonald and R. S. Turley (2002) A Comparison of Parametric Models of Income Distribution across Countries and over Time, Luxembourg Income Study Working Paper, No. 305.
Barndorff-Nielsen, O. (1977) Exponential Decreasing Distributions for the Logarithm of Particle Size, Proc. R. Soc. Lond., 353, 401-419.
Bartels, C. P. A. and H. van Metelen (1975) Alternative Probability Density Func-tions of Income, Vrije University Amsterdam: Research Memorandum 29.
Champernowne, D. G. (1953) A Model of Income Distribution, Economic Journal, 63, 318-351.
Dagum, C. (1977) A New Model for Personal Income Distribution: Specification and Estimation, Economie Appliqu ée, 30, 413-437.
Dagum, C. (1983) Income Distribution Models, in S. Kotz, N. L. Johnson and C. Read (eds.) Encyclopedia of Statistical Sciences, vol. 4, John Wiley, New York.
Dagum, C. (1996) A Systemic Approach to the Generation of Income Distribution Models, Journal of Income Distribution, 6, 105-126.
Gibrat, R. (1931) Les In égalit és E'conomiques, Librairie du Recueil Sirey, Paris.
Johnson, N. L., S. Kotz and N. Balakrishnan (1994) Continuous Univariate Distributions, vol. 1, 2nd ed., Wiley, New York.
Kotz, S., T. J. Kozubowski and K. Podg órski (2001) The Lapace Distribution and Generalizations, Birkhauser, Boston.
Luxembourg Income Study (2004) www.lis.ceps.lu.
McDonald, J. B. (1984) Some Generalized Functions for the Size Distribution of Income, Econometrica, 52, 647-663.
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.
Pareto, V. (1897) Cours d’Economie Politique, Rouge, Lausanne.
Parker, S. C. (1999) The Generalized Beta as a Model for the Distribution of Earnings, Economics Letters, 62, 197-200.
Reed, W. J. (2003) The Pareto Low of Incomes - An Explanation and An Extension, Physica A, 319, 579-597.
Reed, W. J. (2004) On Pareto’s Law and the Determinants of Pareto Exponents, Journal of Income Distribution, 13, 7-17.
Reed, W. J. (2007) Brownian-Laplace Motion and Its Use in Financial Modelling, Communications in Statistics Theory and Methods, 36, 473-484.
Reed, W. J. and M. Jorgensen (2004) The Double Pareto-Lognormal Distribution - A New Parametric Model for Size Distribution, Communications in Statistics Theory and Methods, 33, 1733-1753.
Singh, S. K. and G. S. Maddala (1976) A Function for the Size Distribution of Incomes, Econometrica, 44, 963-970.
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.
Thurow, L. C. (1970) Analyzing the American Income Distribution, American Economic Review, 48, 261-269.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Reed, W.J., Wu, F. (2008). New Four- and Five-Parameter Models for Income Distributions. In: Chotikapanich, D. (eds) Modeling Income Distributions and Lorenz Curves. Economic Studies in Equality, Social Exclusion and Well-Being, vol 5. Springer, New York, NY. https://doi.org/10.1007/978-0-387-72796-7_11
Download citation
DOI: https://doi.org/10.1007/978-0-387-72796-7_11
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-72756-1
Online ISBN: 978-0-387-72796-7
eBook Packages: Business and EconomicsEconomics and Finance (R0)