Abstract
The purpose of this chapter is to introduce a new method to compare income distributions. The methodology used here allows us to examine the relationship between the observed income graduation in 1974 and the observed income graduation in 1990 in order to obtain a different perspective on how (and perhaps why) income inequality in the U.S. increased over that 16 years period. The main feature of this chapter is that we introduce an index space representation to compare two income distributions. This concept will be explained below.
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Notes
This chapter follows Ryu and Slottje (1997).
The domain of population coordinate is z = [0, + 1].
Doubling the income of the richest group will affect the entire share function but doubling the income of the poorest group will not affect the other group’s shares.
We have indicated that entropy is not uniquely determined in the continuous case, but the measurement is relative to the coordinate system. However, it can be shown that when we change the coordinate system the ME density functions will not be affected except for multiplication by a Jacobian factor. Before a coordinate transformation each small length element (dx) is given equal weight but after a coordinate transformation the new length element (e.g., dy) is given equal weight.
The share function described by a0 =-log I, a1 = 0.5, and a2-0, a3 = 0, ⋯ is not normalized because the sum of shares of all individuals are not constrained to be one. However, this normalization problem can be resolved simply by multiplying everyone’s share by a fixed constant so we do not have to worry about the normalization problem in our artificial experiment.
Suppose we compare the sample moments of the 1974 income distribution with those of the 1990 income distribution. Using the central limit theorem, we can show that the distribution of the sample moments is normal and its variance approaches zero as the sample size increases to infinity. Therefore, the hypothesis of identical distributions for 1974 and 1990 will be rejected unless the sample moments for the 1990 income distribution exactly coincide with those of the 1974 income distribution.
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© 1998 Springer-Verlag Berlin Heidelberg
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Ryu, H.K., Slottje, D.J. (1998). Comparing Income Distributions Using Index Space Representations. In: Measuring Trends in U.S. Income Inequality. Lecture Notes in Economics and Mathematical Systems, vol 459. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-58896-9_5
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DOI: https://doi.org/10.1007/978-3-642-58896-9_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-64229-9
Online ISBN: 978-3-642-58896-9
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