The shapes of distributions of personal incomes in USA have been investigated based on the data for 1993 to 2008. Comparisons between four models utilizing various number of parameters have been performed. The studies showed that the empirical data is described the best by the three-parameter Dagum model. Values of the models parameters indicate that the distribution of personal incomes can be regarded as zero-modal one. However, one-parameter exponential model shows a good agreement with data and can be treated as a good approximation of empirical distribution with the exception of the region with very high incomes. The high-income region is characterized by the relatively great number of events and is described much better by the Dagum distribution

A Orłowski

K Karpio

P Łukasiewicz

CiteSeerX

Vol. 121 (2012) ACTA PHYSICA POLONICA A No. 2-B
Proceedings of the 5th Symposium on Physics in Economics and Social Sciences, Warszawa, Poland, November 25–27, 2010
The Models of Personal Incomes in USA
P. Łukasiewicza, K. Karpioa and A. Orłowskia,b
aKatedra Informatyki SGGW, Nowoursynowska 159, 02-776 Warszawa, Poland
bInstytut Fizyki PAN, al. Lotników 32/46, 02-668 Warszawa, Poland
The shapes of distributions of personal incomes in USA have been investigated based on the data for 1993
to 2008. Comparisons between four models utilizing various number of parameters have been performed. The
studies showed that the empirical data is described the best by the three-parameter Dagum model. Values of the
models parameters indicate that the distribution of personal incomes can be regarded as zero-modal one. However,
one-parameter exponential model shows a good agreement with data and can be treated as a good approximation
of empirical distribution with the exception of the region with very high incomes. The high-income region is
characterized by the relatively great number of events and is described much better by the Dagum distribution.
PACS: 89.65.-s, 89.65.Cd
1. Introduction
For a majority of countries we observe similar, charac-
teristic shape of the income distribution. In the majority
of societies we deal with the predominant number of en-
tities (persons, families, households) of similar incomes
as well as with a relatively small number of entities of
high or very high incomes. That’s why income distribu-
tions are single-modal with large right-sided asymmetry,
additionally characterized by the “fat tail” in the range
of very high incomes. Rarely we deal with zero-modal
distributions. That is a case in poor countries, where a
majority of entities gain incomes concentrated in a range
of small incomes. However, in developed countries we
may also obtain distributions of incomes similar to zero-
modal ones. That takes place in the case of personal
incomes in USA. In [1] authors approximated personal
income distribution in USA with one-parametric expo-
nential function given by the equation:
fE(x) =
1
a
exp(−x
a
), (1)
where x indicates individual income, whereas a parame-
ter is equal to average income.
Model (1) is of course zero-modal. In this paper we
investigate in more details a shape of the distribution of
personal incomes in USA. We compare the results ob-
tained in [1] with the results for three other models. The
functions proposed have greater number of parameters,
so they shall better compliance with empirical distribu-
tions. Moreover, depending on the shape of empirical
distribution those models can became zero or one-modal.
We are interested in answers to the following questions:
(i) do the distribution of the personal incomes in USA
can be regarded as zero-modal, or is it rather an one-
modal distribution significantly “moved” towards small
incomes (ii) do the models with greater number of pa-
rameters significantly improve agreement with empirical
data (iii) can the exponential model be regarded as a
good approximation of the income distribution in USA
(iv) do the models proposed well describe distributions
of very high incomes?
2. Models of incomes distributions
One of the directions of studies regarding incomes is
search for mathematical functions which approximate
empirical distributions. In literature there are propos-
als for various types of such functions. Some of them,
as for example the Pareto model or log-normal and
gamma distributions are currently rarely used (they have
rather historical meaning). In some cases these func-
tions can well approximate specific ranges of a distribu-
tion, e.g. incomes greater that a certain threshold. Very
high accuracy with empirical distribution is characteris-
tic for Dagum and Singh-Maddala models. These models
are the three-parameter density functions with relatively
simple analytical forms. There are also studies explor-
ing the usefulness of some non-elementary functions, like
beta distribution, generalized beta distribution, normal-
Laplace distribution, generalized normal-Laplace distri-
butions and others [2–5]. In this paper we use four mod-
els: exponential, Weibull, Dagum, and Singh-Maddala.
Density function of the two-parameter Weibull distri-
bution has a form:
fW (x) =
b
a
xb−1 exp(−x
b
a
), (2)
where x > 0. Also a and b parameters are positive. For
b = 1 this function reduces to (1). When 0 < b ≤ 1 the
Weibull distribution is zero-modal. For b > 1 it is single-
modal distribution. This model has been used, among
others, during studies of incomes in [6]. Models (1) and
(2) are characterized by the so-called “thin tails”.
Density function of the Dagum distribution [7] is de-
scribed by the equation
fD(x) =
abc
xb+1(1 + ax−b)c+1
, (3)
where x > 0, and the parameters fulfill the following
conditions: a > 1, b > 0, and c > 0. This distribution is
zero-modal when 0 < bc < 1 and single-modal for bc > 1.
The Singh-Maddala distribution [8] can be expressed
in the form of
fSM (x) =
abcxb−1
(1 + axb)c+1
, (4)
where x > 0, and a > 0, b > 0, c > 0, bc > 1. This
(B-82)
The Models of Personal Incomes in USA B-83
distribution is zero-modal when 0 < b < 1 and single-
modal for b > 1.
Studies performed in various countries show that mod-
els (3) and (4) exhibit high conformance to empirical dis-
tributions of incomes. Very often the Dagum model is
utilized [9–12]. The other advantage of these functions
is a small number of finite moments. Curves (3) and
(4) have “fat tails” what is their advantage because em-
pirical distributions are usually extended in the range of
incomes exceeding average. Models (2), (3), and (4) are
universal, they may describe zero-modal distributions as
well as single-modal ones.
3. Income data and models estimation
Data analyzed in this paper contain information,
among others, about personal incomes in USA in 1993
to 2008. Files with data have been downloaded from
http://www.census.gov [14]. Data for 1993, 1996, 2004,
and 2008 have been collected within the project “Survey
of Income and Program Participation” (SIPP), whereas
data for 1998, 2000 and 2002 within the “Annual Demo-
graphic Survey” (March CPS Supplement). The variable
“total person’s income” has been studied. Data under-
gone preliminary selection: zero values (lack of data)
have been eliminated, monthly income has been recal-
culated into annual income. Incomes are expressed in k$
(thousand of dollars).
The a parameter’s estimator of the model (1) is, of
course, equal to arithmetic average from the sample. The
parameters of the models (2), (3), and (4) have been
evaluated with the means of the maximum likelihood
method. Data have been grouped in 1800 bins with the
width 0.6 k$ each. Vector θ̂ of the values of a model’s
parameters have been calculated by maximizing the func-
tion:
ln(L(θ)) = ln
(
n!
n1!n2! . . . n1800!
1800∏
i=1
(pi(θ))ni
)
=
= const +
1800∑
i=1
ni ln(F (ci; θ)− F (ci−1; θ)), (5)
where F is a cumulative distribution function (CDF) of
the theoretical distribution, θ is the vector of parameters
being evaluated, ci−1, ci are the lower and upper edges
of the i-th bin, ni is a number of incomes in an i-th bin,
and n = n1 + n2 + . . . + n1800.
For graphical presentation as well as to determine mea-
sures of compliance between theoretical and empirical
distributions data have been grouped in 300 bins with
the width 10/3 k$ each (the same way as in [1]). For each
evaluated model the sum of squared residuals (the sum
of squared errors or SSE ) and the sum of absolute values
of the residuals (the sum of absolute errors or SAE ) are
calculated, according to the equations
SSE =
300∑
i=1
[
ni
n
− pi(θ̂)]2,
SAE =
300∑
i=1
|ni
n
− pi(θ̂)|. (6)
In addition, two other (but related) measures are com-
puted:
Wr = (1− 12SAE)× 100%,
Wp =
300∑
i=1
min(
ni
n
, pi(θ̂))× 100%. (7)
In the case of correct choice of a theoretical distribution
we shall observe good compatibility between values of
empirical ci and theoretical qi quantiles. As the measure
of such a compatibility a correlation coefficient squared
ρ2 between quantiles ci and qi has been used [13]. Cor-
relation coefficients ρ have been calculated based on the
199 quantiles of i/200 rank (i = 1, 2, ..., 199), evaluated
on the basis of individual data.
4. Shapes of income distributions
The results of the models estimation are presented in
the Table. For each year the best compatibility is ob-
tained for Dagum model. It yields the lowest values of
SSE and SAE as well as the highest values of Wr, Wp,
and ρ2, thus being the best choice with respect to all
quality measures. For years from 1996 to 2000 the second
model, judged by the level of compatibility, is the Singh-
Maddala model, while the third one is the exponential.
Starting from 2002 we observe the inverse order between
the two latter models but the differences are small. For
all years except 1993, the worst agreement between the
model and the empirical distributions is observed for the
Weibull distribution. For each investigated year the esti-
mated Dagum models are always the zero-modal distri-
butions (b̂ · ĉ < 1), the estimated Singh-Maddala models
are always the single-modal distributions (b̂ > 1), and
the estimated Weibull models are always zero-modal dis-
tributions (b̂ ≤ 1). Plots of the density functions for 1996
and 2008 are presented in Fig. 1.
Fig. 1. Models of income distributions for 1996 and
2008. There are logarithmic scales on the bottom plots.
Models: (1) Exponential (2) Weilbull (3) Dagum (4)
Singh-Maddala.
B-84 P. Łukasiewicz, K. Karpio, A. Orłowski
TABLE
Results of the fits of income models.
Year Distribution Parameters
Count â b̂ ĉ SSE SAE Wr Wp ρ2
2008 Exponential 30.163 − − 0.00079 0.12463 93.77 93.74 0.9075
Weibull 22.896 0.9284 − 0.00176 0.17019 91.49 91.27 0.9159
288,454 Dagum 32,897.3 2.7086 0.2964 0.00071 0.09969 95.02 94.50 0.9479
Singh-Maddala 0.0048 1.0378 6.9577 0.00108 0.12613 93.69 93.74 0.9323
2004 Exponential 26.440 − − 0.00048 0.11134 94.43 94.40 0.9729
Weibull 19.056 0.9123 − 0.00185 0.17334 91.33 91.01 0.9785
203,231 Dagum 33,598.4 2.7982 0.2819 0.00040 0.06787 96.61 95.97 0.9924
Singh-Maddala 0.0062 1.0453 6.1825 0.00091 0.12492 93.75 93.81 0.9871
2002 Exponential 31.879 − − 0.00061 0.12733 93.63 93.61 0.9075
Weibull 24.178 0.9299 − 0.00133 0.17110 91.45 91.24 0.9157
145,209 Dagum 29,473.9 2.6752 0.3148 0.00024 0.08369 95.82 95.46 0.9495
Singh-Maddala 0.0055 1.0847 5.1106 0.00084 0.13071 93.46 93.58 0.9371
2000 Exponential 28.591 − − 0.00075 0.12677 93.66 93.63 0.9843
Weibull 26.535 0.9806 − 0.00101 0.13917 93.04 92.97 0.9856
93,970 Dagum 29,473.9 2.7434 0.3183 0.00027 0.07094 96.45 96.17 0.9987
Singh-Maddala 0.0053 1.1228 5.1245 0.00046 0.08999 95.50 95.67 0.9956
1998 Exponential 26.712 − − 0.00076 0.12799 93.60 93.57 0.9827
Weibull 21.953 0.9480 − 0.00159 0.16335 91.83 91.65 0.9859
92,080 Dagum 29,384.0 2.7973 0.3089 0.00036 0.07029 96.49 96.16 0.9976
Singh-Maddala 0.0068 1.1333 4.3675 0.00049 0.09885 95.06 95.26 0.9957
1996 Exponential 23.611 − − 0.00087 0.11869 94.07 94.02 0.9901
Weibull 18.726 0.9363 − 0.00214 0.16545 91.73 91.47 0.9919
256,408 Dagum 22,020.6 2.7926 0.2977 0.00055 0.08041 95.98 95.49 0.9995
Singh-Maddala 0.0070 1.0971 5.2941 0.00083 0.11057 94.47 94.64 0.9964
1993 Exponential 17.330 − − 0.0021 0.1285 93.57 93.50 0.9885
Weibull 18.624 1.0215 − 0.0016 0.1148 94.26 94.26 0.9874
142,162 Dagum 26,359.9 3.0252 0.2719 0.0011 0.1047 94.76 94.13 0.9966
Singh-Maddala 0.0022 1.0499 23.4414 0.0014 0.1091 94.54 94.62 0.9902
For all empirical distributions the existence of rela-
tively large number of very high incomes is observed. In
Fig. 2 we present the best fits of all four models (ex-
ponential, Weibull, Dagum, and Singh-Maddala) to the
empirical distribution of incomes for 2004. Part b) of
Fig. 2 is an amplification of the high incomes region. In
Fig. 3 the same models are fitted to the empirical dis-
tributions of incomes for 1998 and 2004. This time the
logarithmic scale is used to highlight differences in the
models quality as applied to the high incomes regions. It
is clear that the Dagum model is the best approximation
to the empirical distributions. The exponential model
does not explain large number of events observed in the
high incomes region.
It is intersting to note that the shape of the investi-
gated empirical distributions of incomes exhibit a rather
peculiar and irregular behavior in the region of high in-
comes. For each year there is a range of (high) incomes
that is more populated than its neighborhood from both
sides. For example, in 1998 there were about 440 ob-
servations within range of 330–480 k$, in 2004 about 780
observations were registered within range of 380–500 k$),
while in 2008 about 1800 observations fell into 300–420
k$ interval. This effect seems not to be properly un-
derstood and suggests that more detailed analysis of the
high incomes region is desirable. We plan to investigated
this and related issues in a forthcoming paper.
Fig. 2. Fits to incomes distribution for 2004. High in-
comes region is magnified in part b). Models: (1) Ex-
ponential (2) Weilbull (3) Dagum (4) Singh-Maddala.
The Models of Personal Incomes in USA B-85
Fig. 3. Fits to incomes distributions for 1998 and 2004.
Logarithmic scale is used to better resolve the fit quality
for the high incomes regions. Models: (1) Exponential
(2) Weilbull (3) Dagum (4) Singh-Maddala.
5. Final conclusions
1. The best agreement with empirical distributions is
observed for the Dagum model. For all the inves-
tigated years the zero-modal distributions are ob-
tained.
2. The Singh-Maddala model is characterized by the
lower level of compatibility with empirical data
than the Dagum model. For all the investigated
years the single-modal distributions are obtained.
3. The exponential model can be considered as a good
approximation of personal income distributions but
only up to a certain threshold value of income.
4. The exponential model does not explain the in-
comes behavior in the high incomes range. In that
region the best approximation is provided by the
Dagum model.
5. The quality of the exponential model (being the
special case of the two-parametric Weibull model)
cannot be improved by incorporating a second pa-
rameter.
References
[1] A. Drăgulescu, V. M. Yakovenko, Evidence for the
exponential distribution of income in the USA, The
European Physical Journal B 20, 585-589 (2000).
[2] J.B. McDonald, Y.J. Xu, A generalization of the beta
distribution with applications, Journal of Economet-
rics 66, 133-152 (1995).
[3] J.B. McDonald, Some Generalized Functions for the
Size Distribution of Income, Econometrica 52 (3),
647-663 (1984).
[4] Fan Wu, Applications of The Normal Laplace and
Generalized Normal Laplace Distributions, A Thesis
Submitted in Partial Fulfillment of the Requirements
for the Degree of Master of Science in the Department
of Mathematics and Statistics, University of Victoria,
2008.
[5] M. Jagielski, R. Kutner, Study of Households’ Income
in Poland by Using the Statistical Physics Approach,
Acta Physica Polonica A 117 (4), 615-618 (2010).
[6] C.P.A. Bartels, H. van Metelen, Alternative Proba-
bility Density Functions of Income, Vrije University
Amerstadam: Research memorandum 29, 1975.
[7] C. Dagum, A New Model of Personal Income Distri-
bution: Specification and Estimation, Economic Ap-
pliquee, XXX (3), 413-437 (1977).
[8] S.K. Singh, G.S. Maddala, A Function for Size Dis-
tribution of Incomes, Econometrica 44 (5), 963-970
(1976).
[9] R. Bandourian, J.B. McDonald, R.S. Turley, A Com-
parison of Parametric Models of Income Distribution
Across Countries and Over Time, Luxembourg In-
come Study Working Paper 305, 2002.
[10] C. Dagum, A. Lemmi, A Contribution to the Analy-
sis of Income Distribution and Income Inequality and
a Case Study: Italy, Econometric Society Meeting,
Chicago, 1987.
[11] P. Łukasiewicz, A.J. Orłowski, Probabilistic Models
of Income Distributions of Polish Households, Quan-
titative Methods in Economic Research III, 122-130
(2003) [in Polish].
[12] P. Łukasiewicz, A.J. Orłowski, Probabilistic Models of
Income Distributions, Physica A 344 (1-2), 146-151
(2004).
[13] R.B. D’Agostino, M.A. Stephens, Goodness-of-fit
Techniques, Marcel Dekker, Inc., New York, 1986.
[14] U.S. Census Bureau, http://www.census.gov .


The Models of Personal Incomes in USA

http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=A52F2FDD6A1F61C028C374B15BCF18B8?doi=10.1.1.1054.6317&rep=rep1&type=pdf

The Models of Personal Incomes in USA

Abstract

Similar works

Full text

Available Versions

CiteSeerX