How the gamma law of income distribution appears invariant under aggregation*

Some preliminary findings from the project on which this paper is based were presented to the 1993 annual meetings of the American Statistical Association in a paper entitled “An Apparent Invariance of the Size Distribution of Personal Income Conditioned on Education,” which appears in the Proceedings of those meetings. Thanks to Prof. Francois Nielsen for comments.

The Gamma Law of Income Distribution appears to be a scientific law because the gamma pdf 1) fits the range of shapes seen in income distributions, 2) is parsimonious, 3) appears to be scale invariant, i.e., to show invariance under population aggregation, and 4) the gamma pdf's shape parameter provides a convenient descriptor of the range of shapes seen in income distributions, allowing the apparent invariance between education and the shape of the income distribution to be simply described. The Gamma Law of Income Distribution cannot, however, be a scientific law because it is not scale invariant. An unconditional distribution of income is a mixture, i.e., the weighted sum, of variously shaped income distributions. People at different education levels have differently shaped income distributions. These distributions are well fitted by gamma pdfs making the corresponding unconditional distribution a gamma shape mixture. A gamma shape mixture is not in general a gamma pdf. Aggregating the income distributions of population segments together can give rise to gamma shape mixtures. Thus the Gamma Law is not scale invariant. However, under certain conditions a gamma shape mixture can be hard to distinguish from GAM(α*,λ), the gamma pdf whose shape parameter is α*, the weighted average of the α i ’s, the shape parameters of the component gamma pdfs of the mixture. GAM(α*, λ) has the same mean as the shape mixture. These conditions allow the Gamma Law of Income Distribution to appear to be scale invariant These conditions occur in geographically defined populations in the contemporary U.S. They are 1) the distribution of income conditioned on education is itself gamma distributed, 2) is invariant under aggregation, 3) most of the population has attained an education whose corresponding income distribution is fitted by GAM(α i ,λ) where α i > 1, 4) there is a close relationship between the shape of the income distribution and education, and 5) the distribution of people over education is approximately symmetric, unimodal, and peaked at its mode. The Gamma Law of (unconditional) Income Distribution appears to work because a Gamma Law of Income Conditioned on Education exists.

Key words:

• Income distribution
• gamma PDF
• mixture
• gamma law of income distribution

Notes

Some preliminary findings from the project on which this paper is based were presented to the 1993 annual meetings of the American Statistical Association in a paper entitled “An Apparent Invariance of the Size Distribution of Personal Income Conditioned on Education,” which appears in the Proceedings of those meetings. Thanks to Prof. Francois Nielsen for comments.