Decomposability of multidimensional inequality indices by attributes is considered a highly desired property. Naga and Geoffard (2006) provided for it in case of three bivariate indices. To this end, they introduced the notion of a copula function into inequality measurement theory which, as a measure of association, is a natural concept for the study of decomposability. We show that the decomposition obtained is unrelated to copulas, and prove that two indices do not admit decomposition if association is indeed measured via copula. Most notably, the proof reveals a necessary property of indices decomposable via copulas which is similar to well-known separability property.multidimensional inequality; decomposition by attributes; copula function

Martyna Kobus

English

Research Papers in Economics

Warsaw 2010
Working Papers
No. 7/2010 (30)
Martyna Kobus
Decomposition of Bivariate
Inequality Indices by Attributes
Revisited
Working Papers 
 
 
Working Paper No. 7/2010 (30) 
 
Decomposition of Bivariate Inequality Indices by Attributes Revisited 
 
Martyna Kobus 
Faculty of Economic Sciences 
University of Warsaw 
 
[address] 
[e-mail] 
Abstract 
Decomposability of multidimensional inequality indices by attributes is considered a highly desired 
property. Naga and Geoffard (2006) provided for it in case of three bivariate indices. To this end, 
they introduced the notion of a copula function into inequality measurement theory which, as a 
measure of association, is a natural concept for the study of decomposability. We show that the 
decomposition obtained is unrelated to copulas, and prove that two indices do not admit 
decomposition if association is indeed measured via copula. Most notably, the proof reveals a 
necessary property of indices decomposable via copulas which is similar to well-known 
separability property. 
Keywords: 
multidimensional inequality; decomposition by attributes; copula function 
 
JEL: 
D31, D63 
 
Working Papers contain preliminary research results. 
Please consider this when citing the paper. 
Please contact the authors to give comments or to obtain revised version. 
Any mistakes and the views expressed herein are solely those of the authors.1 Deﬁnitions and notation
Decomposition of multidimensional inequality indices by attributes allows us
to judge how much of the overall inequality comes from inequality in diﬀerent
attributes (income, years of schooling etc.). Although two observation units
can have the same level of joint inequality, its sources can be very diﬀerent
and require thus diﬀerent treatment, therefore this type of knowledge is useful
in policy making and, in general, in comparisons of inequality patterns across
regions, nations and points in time.
Following Naga and Geoﬀard (2006) (henceforth NG), a population consists
of n individuals and each individual i has resources xi = (xi1;xi2) 2 R2. The
joint distribution is represented by matrix X :=
2
6
4
x1
. . .
xn
3
7
5 2 Mn, the set of all
n  2 matrices with strictly positive elements. We also write X =: [X1;X2],
where Xj gives the distribution for the j-th attribute (an n  1 vector).
A bidimensional inequality index is a real valued function I : Mn ! R+.
Underlying the index is a social welfare function W : Mn ! R of the form:
W(X) = 1
n
Pn
i=1 u(xi), where u is a real-valued utility function with standard
properties. Given the distribution X = [X1;X2] we deﬁne 1 := (X1);2 :=
(X2) to be means of X1 and X2 respectively. Let now  := (X) be such
that W(x) = u(1;2), that is, for distribution X, (X) is a fraction of
the sum-total of each attribute that would give the same level of welfare
as X, provided each attribute were distributed equally. The corresponding
inequality index is given by I(X) := 1   (X). NG consider three diﬀerent
forms of utility function and hence, three diﬀerent indices, which will be
presented later.
Denote by F12 the cumulative distribution function (cdf) of X and by
Fj the cdf of the marginal distribution of the j-th attribute Xj. By Sklar’s
theorem1 there exists a copula function c : [0;1]  [0;1] ! [0;1] such that
for any (x1;x2) 2 R2;F12(x1;x2) = c[F1(x1);F2(x2)]. Copula and marginal
distributions characterize joint distribution fully and copula captures the
association between the two variables, therefore it seems to be indeed a very
useful mathematical concept to be used for decomposition.
We say that an index  is decomposable by attributes if it can be decom-
posed into indices 1;2 and a measure of association , that is, if there exist
1It can be found for instance in Klement and Mesiar, 2005: theorem 14.2.3.
1four R ! R functions h;g1;g2;, where h;g1;g2 are monotonically increas-
ing, such that:2
h[(F12)] = g
1[1(F1)] + g
2[2(F2)] + [(c)] (1)
2 Decomposition by attributes and discrete
copulas
It is known that for copulas deﬁned on n-atomic simple distributions (such
as the one we have here), there exists a bijection between the set of copulas
and the set of nn permutation matrices (Proposition 7.3.24, Klement and
Mesiar, 2005; see also Mesiar, 2005). There are n! copulas on n-atomic
distributions. Furthermore, copulas change when the relative order between
attributes changes. Proposition 1 in NG gives explicit expression for , which
is:
 =
n
P
i x
i1x

i2 P
i x
i1
P
i x

i2
(2)
Let us now make the maximum element in attribute X1 higher,  changes
then but the relative order of two attributes does not change so copula does
not change either. It is impossible then that  measures the association of
attributes via copula function, because it necessarily uses information about
the joint distribution other than that which is encoded in copula.
If we correct (1), that is, we make  indeed dependent on the joint distri-
bution only through copula c, then two out of three indices presented in NG
2A careful reader will note the diﬀerence between the above deﬁnition and the one used
in NG, which is:
h[(F12)] = g1[1(F1)] + g2[2(F2)] + [(c(F1;F2))]
Clearly, deﬁnition in NG is ﬂawed. Since there is a unique copula (to be precise, it is
unique on the image of Fj) associated with F12 the only meaningful interpretation of
c(F1;F2) is that it is in fact cF12(F1;F2). This however equals F12 (as denoted above),
so we can always perform such decomposition by simply subtracting the inequality in
marginal distributions (1;2 or their functions) from an overall inequality ((F12) or
its function); what is left potentially depends on the entire information about a given
distribution. However, the reason for us to decompose indices at all is exactly because we
want an overall inequality to be decomposed into components which do not depend on the
entire information. Otherwise, every index is trivially decomposable.
2are not decomposable. As we now present, one index is decomposed trivially.
Let utility function be of the form: u(a;b) = lna +  lnb; ; > 0. The
deﬁnition of (X) implies
(ln(X) + ln1) + (ln(X) + ln2) = W(X)
and hence
ln(X) = ( + )
 1 (W(X)   ln1    ln2)
Finally the corresponding index is
I(X) = 1   exp

( + )
 1 (W(X)   ln(X1)    ln(X2))
	
We can easily observe that
W(X) = 
1
n
n X
i=1
lnxi1
| {z }
=:V (X1)
+
1
n
n X
i=2
lnxi2
| {z }
=:V (X2)
The index can be written as
I(x) = 1   exp

(V (X1)   ln(X1)) +  (V (X2)   ln(X2))
 + 

Let J be the univariate entropy index 3
J(x) =
1
n
n X
i=1
ln

(x)
xi

;
where (x) is the mean value of vector x. Then our decomposition takes the
form
I(x) = 1   exp

 
J(X1) + J(X2)
 + 

:
As we can see, due to the additively separable utility function underlying
this index there is not any association between attributes.
3Also known as the Theil second measure or Mean Logarithmic Deviation, see Theil
(1967).
33 Non-decomposability of indices using copulas
We now prove that two indices presented in Naga and Geoﬀard (2006) are not
decomposable if association between attributes is to be measured by copulas
as postulated by the authors. We present the proof for one of the indices
as the other one is similar in reasoning.4 Let utility function be of the form
u(a;b) = ab; ; > 0; +  < 1:. The deﬁnition of  implies
((X1)(X))
 ((X2)(X))
 = W(X)
Easily we get
(X) =

W(X)
(X1)(X2)
 1
+
and the corresponding index is
I(X) = 1  

W(X)
(X1)(X2)
 1
+
We will now show that this index does not admit any decomposition of the
form
I(X) = k(I1(X1);I2(X2);c(X)); (3)
where I1;I2 are indices for one attribute, c(X) is the unique discrete cop-
ula corresponding to the distribution of X and k is a strictly coordinate-wise
increasing function. To this end we will ﬁnd X1;X0
1;X2;X0
2 such that the cop-
ulas corresponding to distributions [X1;X2];[X1;X0
2] and to [X0
1;X2];[X0
1;X0
2]
are the same and
I(([X1;X2]) = I([X
0
1;X2]); I([X1;X
0
2]) 6= I([X
0
1;X
0
2]): (4)
This yields the contradiction with the existence of a decomposition (3). In
fact, the ﬁrst condition implies that I1(X1) = I1(X0
1) and for the same reason
the second one implies that I1(X1) 6= I1(X0
1).
Now we will show that we can ﬁnd such X1;X0
1;X2;X0
2. We will work with
f(X) := (+)ln(1   I(X)) which is a monotone transform of I. We check
that
f(X) = lnW(X)   ln(X1)    ln(X2):
4The functional form of the other index is similar u(a;b) =  ab; ; < 0
4The conditions (4) write as
lnW(X1;X2)   lnW(X
0
1;X2)   (ln(X1)   ln(X
0
1)) = 0;
lnW(X1;X
0
2)   lnW(X
0
1;X
0
2)   (ln(X1)   ln(X
0
1)) 6= 0:
Further we restrict to the case of two persons5
x
11x

12 + x
21x

22
x0
11x

12 + x0
21x

22

(x0
11 + x0
21)

(x11 + x21)
 = 1;
x
11x
0
12 + x
21x
0
22
x0
11x
0
12 + x0
21x
0
22

(x0
11 + x0
21)

(x11 + x21)
 6= 1
For simplicity of exposition we ﬁx x12 = 1;x22 = 21=;x0
12 = 1;x0
22 = 1001=
and x11 = x0
11 = 1 then
1 + 2x
21
1 + 2x0
21

(1 + x0
21)

(1 + x21)
 = 1;
1 + 100x
21
1 + 100x0
21

(1 + x0
21)

(1 + x21)
 6= 1:
We also deﬁne functions
h1(a;) :=
(1 + a)
1 + 2a ; h2(a;) :=
(1 + a)
1 + 100a
Finally, the conditions write as
h1(x0
21;)
h1(x21;)
= 1;
h2(x0
21;)
h2(x21;)
6= 1:
Now we ﬁx x21 = 1:01. We have veriﬁed numerically that for  2 (0:02;0:5)6
the equation
h1(x0
21;)
h1(1:01;) = 1 has a non-trivial root (i.e. diﬀerent from 1) r()
and checked that
h2(r();)
h2(1:01;) 6= 1.7
5This can be extended to the case of n persons by giving the rest e.g. zero income.
6For other  the same procedure can be applied with diﬀerent choice of x12;x22 etc.
7For the two matrices (rows individuals; columns attributes):
  1 1
1:01 2
1


,
  1 1
r() 2
1


the
corresponding copula c : f0; 1
2;1g2 7! f0; 1
2;1g is the following: c(1
2; 1
2) = 1
2;c(1
2;1) =
1
2;c(1; 1
2) = 1
2;c(1;1) = 1 and c(0;x) = c(x;0) = 0.
50.1 0.2 0.3 0.4 0.5
10
20
30
40
50
(a) Non-trivial root r() of h1(x;) =
h1(1:01;)
0.1 0.2 0.3 0.4 0.5
0.80
0.85
0.90
0.95
1.00
(b) Value of
h2(r();)
h2(1:01;)
4 Axiom and the usefulness of copulas in in-
equality theory
The above proof rests on step (4), which suggests the following axiom has to
be fulﬁlled by indices decomposable into attributes via copula:
Multivariate Separability Axiom (MSA) For any X2;X0
2 2 Rn we require
I((X1;X2))  I((X
0
1;X2)) ^ c(X1;X2) = c(X
0
1;X2) iﬀ
I(X1;X
0
2)  I(X
0
1;X
0
2) ^ c(X1;X
0
2) = c(X
0
1;X
0
2) 8X1;X0
12Rn;
and analogously for any X1;X0
1 2 Rn we require
I((X1;X2))  I((X1;X
0
2)) ^ c(X1;X2) = c(X1;X
0
2) iﬀ
I((X
0
1;X2))  I((X
0
1;X
0
2)) ^ c(X
0
1;X2) = c(X
0
1;X
0
2) 8X2;X0
22Rn:
Axiom MSA means that the orderings imposed on any dimension when
the other attribute and copula do not change is independent on the other
variable or, in other words, X1(orX2) is separable from X2(orX1) given the
copula.
The main result of NG provides for a decomposition of three bivariate indices.
Clearly (2) is similar to correlation and can be interpreted as a measure of
association, but it does not measure association via copula. On a deeper
level, however, we wonder about the usefulness of decomposing inequality
indices via copula on n-atomic distributions of variables that possess natural
scale e.g. incomes. To see why, let us assume we decomposed the index in
this way and we have two people with e.g. almost the same incomes and
6we change incomes by a small amount that is enough yet to reverse the
ranking. Then, relative order has changed so the value of the inequality
index will change greatly, which clearly does not make much sense. This
will happen except for the case when an index is deﬁned by the functions
which are deﬁned over domains that correspond to diﬀerent copulas and
are ”glued” in a continuous manner.8 On the other hand, copula which is
inherently scale-free, seems to be a perfect notion to measure association
between ordinal variables that appear a lot in social measurement. These
are usually assigned some arbitrary measure units, whereas order is in fact
the only reliable information we have about them.
References
[1] Klement E.P, Mesiar R., Logic, algebraic, analytic and probabilistic as-
pects of triangular norms, Amsterdam, Elsevier, 2005.
[2] Mesiar R., Discrete copulas - what they are, Joint EUSFLAT-LFA, 2005.
[3] Naga A., Geoﬀard P.Y, Decomposition of bivariate inequality indices by
attributes, Economics Letters, Elsevier, vol. 90(3), pages 362-367, 2006.
8It is not clear if this can be done in a non-trivial way for more than two individuals.
7

Decomposition of Bivariate Inequality Indices by Attributes Revisited

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics