The generalized entropy class of inequality indices is derived for Generalized Beta of the Second Kind (GB2) income distributions, thereby providing a full range of topsensitive and bottom-sensitive measures. An examination of British income inequality in 1994/95 and 2004/05 illustrates the analysis.inequality, generalized entropy indices, generalized Beta of the second kind distribution, GB2 distribution, Singh-Maddala distribution, Dagum distribution

Stephen P. Jenkins

English

Research Papers in Economics

   
 
Working Paper Series 
 
 
 
 
 
 
 
 
 
 
 
 
Inequality and the GB2 income 
distribution 
 
Stephen P. Jenkins   
 
 
 
 
 
 
 
ECINEQ WP 2007 – 73    
 
ECINEC 2007-73 
July 2007  
 
www.ecineq.org  
 
 
 
 
Inequality and the GB2 income distribution 
 
 
Stephen P. Jenkins* 
University of Essex 
 
27 May 2007  
 
Abstract  
 
The generalized entropy class of inequality indices is derived for Generalized Beta of 
the Second Kind (GB2) income distributions, thereby providing a full range of top-
sensitive and bottom-sensitive measures. An examination of British income inequality 
in 1994/95 and 2004/05 illustrates the analysis. 
 
JEL Classification: C16, C46, D31. 
Key words: inequality, generalized entropy indices, generalized Beta of the second kind 
distribution, GB2 distribution, Singh-Maddala distribution, Dagum distribution 
                                                 
*  Address for correspondence: Jenkins: ISER, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, 
stephenj@essex.ac.uk..  1 Introduction
Parametric functional forms have received considerable attention in the lit-
erature on earnings and income distribution. Although a large number of
functional forms have been proposed, the four-parameter Generalized Beta
of the Second Kind (GB2) model is now widely acknowledged to give an
excellent description of income distributions, providing ￿ne goodness-of-￿t
with relative parsimony, while also including many other models as spe-
cial or limiting cases. See, inter alia, Bordley et al. (1999), Brachmann et
al. (1999), Butler and McDonald (1989), McDonald (1984), and McDonald
and Xu (1995). Feng et al. (2006) address issues of time-inconsistency in
top-coded US Current Population Survey earnings data by ￿tting GB2 dis-
tributions that account for top-coding, and derive a consistent time series
of Gini coe¢ cients from the estimates. Parker￿ s (1999) model of optimising
￿rm behaviour predicts that the earnings distribution has the GB2 shape.
Despite widespread use of the GB2 distribution, it is remarkable that in-
equality in the ￿tted distribution has been summarized in terms of the Gini
coe¢ cient alone. Although commonly used, the Gini is but one of many mea-
sures available, and it incorporates particular assumptions about the way in
which income di⁄erences in di⁄erent parts of the distribution are summa-
rized. (It is relatively sensitive to income di⁄erences around the mode.) In
other forms of income distribution research, generalized entropy (GE) and
Atkinson indices are widely used to assess inequality trends and di⁄erences ￿
these one-parameter families have the advantage that variations in inequality
aversion are straightforwardly incorporated. This paper provides formulae for
generalized entropy indices in the GB2 model, and hence also for the impor-
tant special cases of the three-parameter Singh-Maddala and Dagum models,
thereby making a full range of top-sensitive and bottom-sensitive measures
available to analysts.
The only GE index mentioned in Kleiber and Kotz￿ s (2003) encyclopaedic
survey of the GB2 and related distributions is the Theil index for the Singh-
Maddala model. Cowell and Flachaire (2007) provide GE index formulae for
the Singh-Maddala model, but using a di⁄erent parameterization from the
standard one that is employed by McDonald (1984) and Kleiber and Kotz
(2003). There appear to be no extant GE index formulae for the Dagum
distribution, which is surprising given Kleiber￿ s (1996) argument that the
Dagum distribution is likely to provide a better ￿t to income data than the
Singh-Maddala distribution. This paper￿ s derivations for the GB2 model are
illustrated with an examination of the change in income inequality in Britain
between 1994/95 and 2004/05.
12 Generalized entropy indices
Consider the distribution of a random variable y (￿ income￿ ), which takes
strictly positive values. The generalized entropy (GE) class of inequality
measures, I(￿), is de￿ned as1
I(￿) =
￿￿￿￿￿ ￿ 1
￿(￿ ￿ 1)
; ￿ 6= 0;1 (1)
where
￿￿ =
Z
y
￿dF(y): (2)
and F(y) is the cumulative distribution function (cdf) for y. The mean
logarithmic deviation (MLD) index is
I(0) = lim
￿!0I(￿) = log￿ ￿ v0 (3)
where ￿0 =
R
logydF(y) and ￿ ￿ E(y) is the mean of y. The Theil index is
I(1) = lim
￿!1I(￿) =
￿
￿
v1
￿
￿ log￿ (4)
where ￿1 =
R
y logydF(y). I(2) is half the squared coe¢ cient of variation.
Parameter ￿ 2 (￿1;1) characterizes the sensitivity of I(￿) to income
di⁄erences in di⁄erent parts of the income distribution. The more positive
that ￿ is, the more sensitive is I(￿) to income di⁄erences at the top of the
distribution; the more negative that ￿ is, the more sensitive is I(￿) to income
di⁄erences at the bottom of the distribution. In empirical work, the range
of values for ￿ is typically restricted to [￿1;2] because, otherwise, estimates
may be unduly in￿ uenced by a small number of very small incomes or very
high incomes.
For each member of the Atkinson (1970) class of inequality indices, A(￿),
there is an ordinally equivalent member of the GE class (but not vice versa).
Speci￿cally, for inequality aversion parameter ￿ = 1 ￿ ￿,
A(￿) = 1 ￿ [￿(￿ ￿ 1)I(￿) + 1]
1
￿ ; ￿ < 1;￿ 6= 0
= 1 ￿ exp[￿I(0)]; ￿ = 0. (5)
Since A(￿) can be computed from I(￿), this paper focuses on the derivation
of I(￿) in the GB2 distribution case.
1On the characterization of the GE class of inequality indices, see Bourguignon (1980),
Cowell (1980), and Shorrocks (1980, 1984).
23 The GB2 distribution
The GB2 distribution has pdf
f(y) =
ayap￿1
bapB(p;q)[1 + (y=b)a]
p+q;y > 0 (6)
where parameters a;b;p;q, are each positive, B(u;v) = ￿(u)￿(v)=￿(u + v)
is the Beta function, and ￿(:) is the Gamma function (McDonald 1984).
Parameter b is a scale parameter, and a;p; and q are each shape parameters.
The kth moment of the GB2 distribution is
E(y
k) =
bk￿(p + k
a)￿(q ￿ k
a)
￿(p)￿(q)
(7)
and exists only if ￿ap < k < aq.
The Singh-Maddala distribution is the special case of the GB2 distribu-
tion when p = 1; the Dagum distribution is the special case when q = 1.
For a discussion of other special cases, see McDonald (1984) and Kleiber and
Kotz (2003).
4 GE inequality indices and the GB2 distri-
bution
Expressions for each GE index, I(￿), other than for the cases ￿ = 0;1, can
be derived by straightforward substitution, using the expressions for ￿￿ and
￿ given by (2) and (7). In particular, the bottom-sensitive index I(￿1) is
given by
I(￿1) = ￿
1
2
+
￿(p ￿ 1
a)￿(q + 1
a)￿(p + 1
a)￿(q ￿ 1
a)
2￿2(p)￿2(q)
: (8)
The top-sensitive index I(2) is given by
I(2) = ￿
1
2
+
￿(p)￿(q)￿(p + 2
a)￿(q ￿ 2
a)
2￿2(p + 1
a)￿2(q ￿ 1
a)
: (9)
Expressions for the MLD and Theil indices can be derived noting that the
expression for I(￿) can be written as
I(￿) =
g(￿)
h(￿)
(10)
3where g(￿) = ￿￿￿￿￿ ￿ 1, with ￿￿ = b￿￿(p + ￿
a)￿(q ￿ ￿
a)=￿(p)￿(q) from (7),
and h(￿) = ￿(￿ ￿ 1). Hence, using L￿ Hb opital￿ s rule, I(0) = ￿g0(0) and
I(1) = g0(1) , where g0(￿) = (￿￿￿)0￿￿ + ￿￿￿(￿￿)0. We therefore require
expressions for (￿￿￿)0 and (￿￿)0 evaluated at the limits ￿ ! 0, and ￿ ! 1.
It can be shown that
(￿
￿￿)
0 = ￿￿
￿￿ log￿ (11)
and
(￿￿)
0 = ￿￿
￿
 (p + ￿
a)
a
￿
 (q ￿ ￿
a)
a
+ logb
￿
(12)
given digamma function  (z) = ￿0(z)=￿(z). Hence, applying the appropriate
limits,
I(0) = ￿(p +
1
a
) + ￿(q ￿
1
a
) ￿ ￿(p) ￿ ￿(q) ￿
 (p)
a
+
 (q)
a
(13)
and
I(1) =
 (p + 1
a)
a
￿
 (q ￿ 1
a)
a
￿ ￿(p +
1
a
) ￿ ￿(q ￿
1
a
) + ￿(p) + ￿(q): (14)
To derive the expression for I(￿) in the special case of the Singh-Maddala
model, set p = 1 and note that ￿(1) = 1. For the Dagum model, set q = 1
instead.
5 Empirical illustration: income inequality in
Britain, 1994/95 and 2004/05
The derivations are illustrated with analysis of income inequality in Britain in
￿scal years 1994/95 and 2004/05. Estimation is based on the unit record data
used to calculate the o¢ cial income statistics, derived from the Family Re-
sources Surveys of 1994/95 and 2004/05. ￿ Income￿is the distribution among
individuals of needs-adjusted post-tax post-transfer household income, with
each individual assumed to receive the income of the household to which
s/he belongs. For further details of the construction of the distributions, see
Department for Work and Pensions (2006). Observations with income equal
to zero were excluded from the calculations (182 observations in the 1994/95
￿le and 302 observations in the 2004/05 ￿le).
4Estimates of the GB2 parameters for each year are shown in Table 1,
together with inequality index estimates implied by them.2 According to
probability plots and quantile plots (not shown), the GB2 distribution ￿ts
the data well.
The estimated GB2 shape parameters changed markedly over the decade,
with a notable rise in a combined with a sharp fall in both p and q. Put
another way, the distribution was well-characterized by a Fisk distribution
in 1994/95 (the GB2 case when p = q = 1), but could not be described thus
a decade later. These changes contrast with the trend in GB2 parameters
for 1984￿ 1993 reported by Brachmann et al. (1996) for household income in
Germany, and for 1948￿ 1980 for US white family income reported by Butler
and McDonald (1989). In both cases, there was a secular decline in a and a
rise in p and q.
<Table 1 near here>
The rise in a combined with a fall in p and q implies that neither distri-
bution Lorenz-dominates the other one (Kleiber 1999), so conclusions about
whether inequality increased or decreased depend on the inequality index
used. As it happens, the GB2 estimates of the Gini coe¢ cient and each of
four GE indices increased between 1994/95 and 2004/05, and the increase
for the GE indices is greater the more positive that ￿ is. However, of the
￿ve indices, it is only for I(2) ￿for which the estimated increase is some 28
per cent ￿that the increase is statistically signi￿cant. (In this case the test
statistic for the relevant t-test is 2.5, but it is markedly less than 2 for the
other four indices.)
The signi￿cant rise in top-sensitive index I(2) suggests that the principal
changes over the decade in the British income distribution occured at the very
top of the distribution. This is con￿rmed by the GB2 estimates of the Lorenz
curves (not shown), which indicate imperceptible changes in income shares
at the bottom of the income distribution but increases in income shares at
the top. For example, the GB2 estimate of the income share of the richest
￿ve per cent increased from 16.5 per cent to 17.3 per cent between 1994/95
and 2004/05, and the income share of the richest one per cent from 5.6 per
cent to 6.3 per cent.
2A program for ￿tting a GB2 distribution to unit record data using the statistical
software Statatm (StataCorp 2003), versions 8.2 and later, is provided by Jenkins (2007).
Stata users can install the program directly by typing ssc install gb2fit. The GE
indices, and associated standard errors computed using the delta method, can be derived
after estimation using the nlcom command.
5If British inequality trends over the decade had been assessed using the
GB2-estimated Gini coe¢ cient alone, a number of important dimensions of
the change would not have been picked up. The ability to calculate a range
of indices incorporating di⁄erent assumptions about aggregation of income
di⁄erences in di⁄erent income ranges is a signi￿cant extension to the utility
of the GB2 model for analysis of income and earnings distributions.
6 Acknowledgements
ISER￿ s core-funding from the Economic and Social Research Council and
the University of Essex is gratefully acknowledged. Thanks to Emmanuel
Flachaire for suggesting the method to derive I(0) and I(1), and to Mark
Bryan, Shuaizhang Feng, Lucinda Platt, and Steve Pudney for comments.
7 References
Atkinson, A.B., 1970, On the measurement of inequality, Journal of Economic
Theory 2, 244￿ 263.
Bordley, R.F., J.B. McDonald and A. Mantrala, 1996, Something new,
something old: parametric models for the size distribution of income, Journal
of Income Distribution 6, 91￿ 103.
Bourguignon, F. 1989, Decomposable inequality measures, Econometrica
47, 901￿ 920.
Brachmann, K., A. Stich and M. Trede, 1996, Evaluating parametric
income distribution models, Allegemeine Statistiches Archiv 80, 285￿ 298.
Butler, J.B. and J.B. McDonald, 1989, Using incomplete moments to
measure inequality, Journal of Econometrics 42, 109￿ 119.
Cowell, F.A., 1980, On the structure of additive inequality measures,
Review of Economic Studies 47, 521￿ 531.
Cowell, F.A. and E. Flachaire, 2007, Income distribution and inequal-
ity measurement: the problem of extreme values, Journal of Econometrics,
forthcoming.
Department for Work and Pensions, 2006, Households Below Average In-
come. An Analysis of the Income Distribution 1994/5 ￿2004/5. (Corporate
Document Services, Leeds). http://www.dwp.gov.uk/asd/hbai/hbai2005/contents.asp
Feng, S., R.V. Burkhauser and J.S. Butler, 2006, Levels and long-term
trends in earnings inequality: overcoming Current Population Survey censor-
ing problems using the GB2 distribution, Journal of Business and Economic
Statistics 24, 57￿ 62.
6Jenkins, S.P., 2007, gb2fit: Stata module to ￿t Generalized Beta of the
Second Kind distribution by maximum likelihood, Statistical Software Com-
ponents Archive S456823. http://ideas.repec.org/c/boc/bocode/s456823.html
Kleiber, C., 1996, Dagum versus Singh-Maddala income distributions,
Economics Letters 53, 265￿ 268.
Kleiber, C., 1999, On the Lorenz order within parametric families of
income distributions, Sankhya B 61, 514￿ 517.
Kleiber, C. and S. Kotz, 2003, Statistical Size Distributions in Economics
and Actuarial Sciences. (John Wiley, Hoboken, NJ).
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 distri-
bution with applications, Journal of Econometrics 66, 133￿ 152. Erratum:
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of earnings, Economics Letters 62, 197￿ 200.
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StataCorp, 2003, Stata Statistical Software: Release 8.0. (Stata Corpo-
ration, College Station TX).
71994/95 2004/05
Parameter estimates
a 2.994 4.257
(0.057) (0.179)
b 227.840 341.965
(4.602) (4.602)
p 1.063 0.682
(0.072) (0.042)
q 1.015 0.635
(0.058) (0.037)
Inequality indices
(0.040) (0.039)
Gini 0.327 0.329
(0.003) (0.003)
I(￿1) 0.217 0.220
(0.005) (0.005)
I(0) 0.182 0.186
(0.003) (0.004)
I(1) 0.198 0.211
(0.005) (0.006)
I(2) 0.310 0.397
(0.017) (0.030)
Log-likelihood ￿ 196,960 ￿ 204,850
N (households) 26,033 25,790
N (individuals) 62,055 59,804
Estimated standard errors shown in parentheses.
Table 1: Estimates of GB2 parameters and inequality indices, Britain,
1994/95 and 2004/05.
8

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