Abstract
Since Pareto (1897) discovered the functional form of the distribution of high incomes efforts have been made to find a form that covers the entire income range. As was true of Pareto’s own work, most of these efforts were descriptive in the sense of looking for a distribution that would fit a set of data believed to be representative. Important steps in the search for appropriate distributions were the introduction of the generalized gamma for this specific purpose (Amoroso, 1925) and the application of the log normal (Gibrat, 1931) and stable distributions (Mandelbrot, 1960; Zolotarev, 1986.). An empirical comparison of the large family of distributions by McDonald (1984) served to clarify their respective merits; thus, he reported poor performance of the log normal.
I am indebted to Joshua Angrist, Richard Barakat, Gary Chamberlain, Herman Chernoff, Richard Freeman, Lawrence Katz, Benoit Mandelbrot, James Medoff, Donald Rubin, and Hal Stern for helpful conversations, and Ivor Frischknecht, Robert Plunkett, and Tycho Stahl for excellent research assistance. They are not responsible for any defects in the paper. The Harvard Institute for Economic Research provided financial support.
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Notes
- 1.
As I hope to show elsewhere, these distributions are also useful in the analysis of daily price changes in financial markets.
- 2.
This property recalls Mandelbrot’s concept of “self-similarity,” which plays a central role in his work. Self-similarity, however, requires that the micro and macro distributions agree in their parameters; this is not necessary for self-mixability.
- 3.
In addition, Mandelbrot suggests that the plot of U against V has a point of inflection for some large value of V. Inflection is apparently required for the distribution to be stable, but in the data there is not much evidence that such a point exists. Since the possibility of inflection would complicate the analysis without improving the fit, it is not considered here. The conic distributions are in effect approximations to the (asymmetric) stable distribution that ignore the inflection.
- 4.
If this condition is not satisfied U is a rational function of V, but the resulting distribution function does not appear to be capable of describing actual incomes or earnings and will not be considered further.
- 5.
It is possible, however, that the minimizing value of V given by equation (22.11) is less than V 0. In that case, it can be shown that equation (22.14) should be replaced by
$C^* = \left| {\begin{array}{*{20}c} 1 & {c_1 } & {c_3 } \\ {c_1 } & {c_2 } & {c_4 } \\ 0 & 1 & { - V_0 } \\\end{array}} \right|,$((22.15))which again should be nonnegative.
- 6.
If the density did vanish at an interior point, the distribution would break up into two or more non-overlapping segments (the “rich” and the “poor,” if there were two segments). Conceivably, this pattern may have existed in feudal societies, but it is not of contemporary interest.
- 7.
Note that U Q = 0 from Condition I.
- 8.
See the discussion following expression (22.19).
- 9.
The two squares collapse into a single one if and only if equation (22.39) is false, in which case the conic-quadratic distribution degenerates to the Pareto.
- 10.
- 11.
Divide the denominator into the numerator and then take the limit as V gets large, using l’Hospital’s rule.
- 12.
The cases where either condition does not hold are not of interest in the applications addressed in this paper, and will not be pursued. They could be relevant to other applications.
- 13.
LDT note: At this point in the paper, HSH intended to include a figure showing the effect on the shape of the CQ density of different values of c 1. (The values used for the other parameters were approximately those estimated for all men over 18 who reported earnings in 1979.) Although he had obviously constructed the figure at some point, it (as with the figure noted in Footnote 4 above) was missing from the copy from which this chapter is taken. His discussion of the figure is as follows: For the middle and large values of c 1 , most of the distribution has the expected shape, but in the lower tail there is a second mode. Since CQ is defined as having a density of zero at the lower limit of v, this second mode is internal [i.e., an artifact of the functional form]. For the smaller value of c 1 , the only mode is at the lower end. The question of modality, which the Achilles heel of CQ (and also of CL), is discussed further in Section 22.5.3; it does not appear to affect the goodness of fit.
- 14.
Confirmation that the slope of the upper asymptote is −α is obtained by dividing expression (22.73) by (U + βW) and taking limits of the right-hand side as W gets large. That the slope of the lower asymptote (as W goes to zero) is −β is obtained similarly (though through the use of l’Hospital’s rule).
- 15.
This is one of the three indices of inequality proposed by Atkinson (1970); the other two depend upon parameters that are difficult—perhaps impossible—to estimate. The index used here equals the fraction by which aggregate income could be reduced, while keeping aggregate utility constant, if that income were distributed equally among all recipients. From the viewpoint of economic theory, this is a more meaningful measure of income equality than the Gini coefficient. The validity of Atkinson’s index, however, depends upon three questionable assumptions: (1) that individual utilities can be aggregated; (2) that the utility function is Bernoullian; and (3) that output remains unchanged under an egalitarian redistribution of income.
- 16.
LDT note: The programs were written by HSH in True Basic.
- 17.
As in published census tables, the sample medians are calculated by linear interpolation within the interval in which the median is located.
- 18.
There is also “bottom-coding,” which is especially relevant to earnings from self-employment, since they can be negative. In PUMS, the largest recorded loss is $9995. The relatively few individuals with negative earnings are always included in the lowest earnings interval. It should also be mentioned that the interview forms do not have room for incomes exceeding $999,995, which is a limitation (although presumably minor) on the accuracy of the published tabulations.
- 19.
- 20.
CG is also more difficult to estimate because some of the four conditions derived in Section 22.2.1 are not automatically satisfied.
- 21.
In this context, one anomaly has to be mentioned. In the early part of the period analyzed in Section 22.7, including the census year 1980, there were so few women with high earnings that the maximum-likelihood estimation effectively ignores them. As a result, the parameter estimates for women in those early years are based upon women with low and medium earnings only, and those with high earnings sometimes appear outside the “Pareto” asymptote defined in Section 22.3.1. This anomaly is not encountered in recent years.
- 22.
LDT note: As with the figures mentioned in Footnote 14, figures 20.2 and 20.3 are unfortunately not available.
- 23.
Another measure of inequality, the logarithmic variance defined in equation (22.104), also declined markedly for all persons from 1.4346 to 1.2367 in 1990. Although not entirely clear-cut, these findings cast serious doubt on the widespread belief that during the period under review “the rich got richer and the poor got poorer.” The conflict between the three inequality measures is attributable to the fact that the Lorenz curves (cf. Section 22.4.4) for 1976 and 1990 have interior points in common. As it happens, these curves are so close together as to be visually indistinguishable.
- 24.
In the present context they are natural numbers, but without this restriction, they can also be used to represent fractional moments.
- 25.
Admittedly, expression (22.116) is only a sufficient condition, but if necessary can be safely conjectured.
- 26.
If the complete function is represented by an integral extending from 0 to infinity, then the incomplete function is the same integral extending from 0 to its second argument. The relevant integral representing the (complete) Macdonald function is given by Erdélyi [1953, Vol. II, p. 82, Equation (21)]. Agrest and Maksimov (1971) actually have two versions of the incomplete Macdonald function, of which only the “Bessel” form is need here.
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Taylor, L.D. (2010). Conic Distributions of Earned Incomes. In: Consumer Demand in the United States. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-0510-9_22
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