We provide, for the class of relative bidimensional inequality indices, adecomposition of inequality into two univariate Atkinson-Kolm-Senindices and a third statistic which depends on the joint distribution ofresources.multidimensional inequality, relative indices, correlationincreasing transfers, copulas.

Pierre-Yves Geoffard

Ramses H. Abul Naga

English

Research Papers in Economics

 
 
 
Decomposition of Bivariate Inequality Indices 
by Attributes 
 
Ramses H. Abul Naga
∗  
 
and  
 
Pierre-Yves Geoffard
†, 
 
 
 
 
 
 
 
 
 
 
DARP 83       The Toyota Centre 
February  2006       Suntory  and  Toyota  International   
       Centres  for  Economics  and  Related 
       D i s c i p l i n e s  
       London  School  of  Economics 
       H o u g h t o n   S t r e e t  
       L o n d o n   W C 2 A   2 A  
 
     
       (+44  020)  7955  6674 
                                                 
∗ Institute for Health Economics and Management (Université de Lausanne). 
† PSE (Paris Jourdan), Institute for Health Economics and Management (Université de Lausanne) and 
CEPR. 
 
 Abstract 
 
We provide, for the class of relative bidimensional inequality indices, a 
decomposition of inequality into two univariate Atkinson-Kolm-Sen 
indices and a third statistic which depends on the joint distribution of 
resources. 
 
Keywords: multidimensional inequality, relative indices, correlation 
increasing transfers, copulas. 
 
JEL codes: D31, D63. 
  Distributional Analysis Research Programme 
 
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© Authors: Ramses H. Abul Naga and Pierre-Yves Geoffard 
All rights reserved. Short sections of text, not to exceed two paragraphs, may be 
quoted without explicit permission provided that full credit, including © notice, is 
given to the source. 
 When measuring inequality over two attributes, it may be of practical
interest to express inequality as a function of the underlying level of inequal-
ity in each attribute. This note provides, for the class of relative bidimen-
sional inequality indices, a decomposition of inequality into two univariate
Atkinson-Kolm-Sen [AKS] indices 1 and a third statistic which depends on
the joint distribution of resources.
1 Notation and deﬁnitions
A Population consists of n individuals. Individual i has resources xi
. =
(xi1,x i2), where xi ∈ R2
++. The joint distribution is a matrix X . =
⎡
⎢
⎣
x1
. . .
xn
⎤
⎥
⎦ ∈
Mn, the set of all n×2 matrices with strictly positive elements. Accordingly,
the jth column of X, Xj, gives a distribution (an n × 1 vector) for the jth
attribute in the population, and we also write X =[ X1 X2].
A bidimensional inequality index is a real valued function I(X):Mn →
R+. Underlying I is a social welfare function W(X):Mn → R. We take
W(.) to be the average of individual welfare levels:
ADD : W(X)=1
n
P
i u(xi) for any X (additive separability).
CON : W(.) is continuous in X.
PAR: W(.) is strictly increasing in all its arguments (Pareto principle).
EQUAL : W(.) is a strictly quasi-concave function (preference for equal-
ity).
RSINV : If Λ is an 2×2 diagonal matrix with strictly positive elements,
then W(XA)=W(XB) ⇐⇒ W(XAΛ)=W(XBΛ), (ratio-scale invariance).
The above axioms on W entail that u(.) is continuous, increasing, concave
and ratio-scale invariant. In what follows, we shall refer to the set of axioms
ADD, CON, PAR, EQUAL and RSINV as the basic axioms.
Let µ . =( µ1,µ 2) denote the vector of sample means and let wo . = W(X),
be the level of welfare attained by X. Then, if W(.) satisﬁes the basic ax-
ioms, we may deﬁne a scalar θ(X) in the unit interval, such that u(θµ)=
1
n
P
i u(xi)=wo. Starting from a distribution X, af r a c t i o nθ of the sum-
total of each attribute would lead to the same level of welfare as X, provided
each attribute were equally distributed in the population. Therefore, θ is a
1See Atkinson (1970), Sen (1973), Kolm (1977) and Tsui (1995).
2measure of equality in X, and
I(X)=1− θ(X) (1)
is the corresponding relative inequality index (Kolm, 1977). Likewise, below
we shall deﬁne equality indices θj for each separate attribute. Our aim is
to decompose θ as a function of statistics θ1, θ2, pertaining to the marginal
distributions and a term κ which depends on the correlation between the two
attributes.
Denote by F12 the cumulative distribution function of X,a n db yFj the
marginal distribution of the j-th attribute Xj. By Sklar’s theorem (Sklar,
1959), there exists a copula function c :[ 0 ,1]×[0,1] → [0,1] such that for any
(x1,x 2), F12(x1,x 2)=c[F1(x1),F 2(x2)]. The copula captures the association
between the two variables, and the joint distribution F12 may be identiﬁed
with its two marginal distributions and its copula function.
In general terms, we shall say that θ(X) is decomposable into two indices
θ1, θ2, and a measure of association κ if there exist four R → R functions
h(.), g1(.), g2(.),a n dχ(.),w h e r eh, g1,a n dg2 are monotonically increasing,
such that:
h[θ(F12)] = g
1[θ1(F1)] + g
2[θ2(F2)] + χ[κ(c(F1,F 2))] (2)
where κ(.) measures the association between attributes via the copula func-
tion.
Note that there are diﬀerent ways to deﬁne equality in a given dimension,
in particular with regard to whether the distribution over the other attribute
has been equalized. Two polar cases may thus be deﬁned. First, consider
as i t u a t i o nw h e r eo n l yx2 is redistributed in the population. Then, we may
deﬁne a scalar ρ2(X) such that 1
n
P
i u(xi1,ρ 2µ2)=wo. Hence, ρ2 is a measure
of equality in X2 holding the distribution X1 constant. Next, there exists
a scalar γ1(X), such that u(γ1µ1,ρ 2µ2)=1
n
P
i u(xi1,ρ 2µ2). The scalar γ1
is the fraction of µ1 that, if equally distributed in the population, would
attain the same level of welfare as X, conditional on everyone receiving an
endowment ρ2µ2 of x2.B o t hγ1 and ρ1 provide measures of equality in the
ﬁrst attribute, and we derive below inequalities relating γj and ρj.
When u(.) is additively separable in x1 and x2 we show that γj = ρj,a n dθ
may be decomposed in terms of γ1 and γ2. In the more general context where
u(.) is not separable across attributes, we show that θ may be decomposed
in terms of γ1, γ2, and a third index κ that captures the correlation between
3the two attributes. Finally, we generalize our decomposition in the context
of p>2 attributes.
2 Inequality and the correlation between at-
tributes
It may certainly be advocated that changes in the joint distribution which
increase the correlation between attributes for given marginal distributions,
ought to modify inequality. Formally, let xl and xm denote any two rows
of X, and deﬁne yi
. = xi for all i 6= l,m and ylj
. =m a x {xlj;xmj},y mj
. =
min{xlj;xmj} j =1 ,2; following Tsui (1999), we shall state that Y is a
correlation increasing transformation (CIT) of X. Given that a CIT does
not aﬀect the marginal distributions, it amounts to a transfer which increases
the rank correlation between attributes (i.e. Kendall’s Tau). The next axiom
formalizes the argument W(Y ) ≤ W(X):
CITAV:F o ra n yX and Y, if Y is a CIT of X, then W(Y ) ≤ W(X),
(aversion to correlation increasing transformations).
To provide some intuition for the results to follow, suppose x1 is income
and x2 is health. With u12 < 0, the marginal utility of income is higher
for the unhealthy. Therefore, a redistribution of income from a person in
good health to a less healthy one may increase welfare, even if the latter is
initially wealthier. The following lemma (Atkinson and Bourguignon, 1982)
formalizes this argument.
Lemma Assume W(.) satisﬁes the basic axioms. If in addition W has
an aversion (preference) for CITs, then u12 ≤ 0( u12 ≥ 0).
If W(.) satisﬁes the basic axioms then u(.) is one of three forms (Aczél,
1988; Tsui, 1995):
u(xi1,x i2) . = x
α
i1x
β
i2 α,β > 0,α + β ≤ 1 (3)
u(xi1,x i2) . = −x
α
i1x
β
i2 α,β < 0 (4)
u(xi1,x i2) . = αlnxi1 + β lnxi2 α,β > 0 (5)
The ﬁrst of these forms has u12 > 0, i.e. W(.) is increasing in CITs.F o rt h e
second form u12 < 0, while for the latter u12 =0 , i.e. W(.) is insensitive to
CITs. We begin with this simpler case.
4Under (5), u(x1,x 2) is additively separable in x1 and x2; accordingly,
w
o = αlnθµ1 + β lnθµ2 = αlnγ1µ1 + β lnγ2µ2 (6)
Upon deﬁning ω . = α/(α + β), we obtain the elasticity decomposition
lnθ(X)=ωlnγ1(X1)+( 1− ω)lnγ2(X2) when u12 =0 (7)
so that a 1% increase in equality of the distribution X1 results in a ω %
increase in the overall level of equality θ. Notice also here that ρj = γj =
exp
£
1
n
P
i lnxij − lnµj
¤
.
For the general case u12 6=0 , it is usually not possible to modify the
distribution X1 while holding γ2 constant. The elasticity decomposition is
given in the following proposition.
Proposition 1: Assume W(.) satisﬁes the basic axioms. Then
lnθ(X)=ωlnγ1(X1)+( 1− ω)lnγ2(X2)+
1
(α + β)
lnκ(X) (8)
where γj is an AKS index for Xj and κ is given by
κ . =
n
P
i xα
i1x
β
i2 P
i xα
i1
P
i x
β
i2
(9)
Proof:C o n s i d e r ﬁrst (3). From the deﬁnition of θ, it follows that wo =
(θµ1)α(θµ2)β =( γ1µ1)α(ρ2µ2)β. Therefore, (α+β)lnθ = αlnγ1 +β lnρ2. In
other terms,
lnθ = ωlnγ1 +( 1− ω)lnγ2 +( 1− ω)ln(ρ2/γ2) (10)
Likewise, wo =( ρ1µ1)α(γ2µ2)β = 1
n
P
i(ρ1µ1)αx
β
i2. Hence,
γ2 =
1
µ2
Ã
1
n
X
i
x
β
i2
!1/β
(11)
Since wo = 1
n
P
i xα
i1x
β
i2 = 1
n
P
i xα
i1(ρ2µ2)β, we obtain
ρ2 =
1
µ2
ÃP
i xα
i1x
β
i2 P
i xα
i1
!1/β
(12)
5Together, (11) and (12) give (1 − ω)ln(ρ2/γ2)=( l nκ)/(α + β), which com-
pletes the proof for (3). For (4), it is readily veriﬁed that γ2 and ρ2 are also
given as in (11) and (12) above. ¤
If Y is a CIT of X, it is the case that Fj(Yj)=Fj(Xj). Hence, γj(Y )=
γj(X). On the basis of (8), we therefore obtain
lnθ(Y ) − lnθ(X)=
1
(α + β)
[lnκ(Y ) − lnκ(X)] (13)
Since κ is a measure of association between the two attributes, it is desirable
that this function be rising in correlation increasing transformations. The
next proposition formalizes this argument.
Proposition 2: If Y is a CIT of X, then κ(Y ) ≥ κ(X).
Proof:F o rj =1 ,2 let xj < xj. If Y is a CIT of X,w em a yt a k exl
. =
(x1,x 2),x m
. =( x1,x2),y l
. =( x1,x2), ym
. =( x1,x 2) and yi
. = xi for all i 6= l,m.
Since the marginal distributions are unaﬀected by CITs, it follows from (9)
that κ(Y ) ≥ κ(X) when the diﬀerence ∆ . = xα
1x
β
2 + xα
1x
β
2 − xα
1x
β
2 − xα
1x
β
2 is
positive. Since ∆ . =( xα
1 − xα
1)(x
β
2 − x
β
2), we have that ∆ ≥ 0 when α and β
are of the same sign. ¤
R e t u r n i n gt o( 1 3 )w en o t et h a tw h e nt h es o c i a lw e l f a r ef u n c t i o ni sa v e r s e
to correlation increasing transformations (α,β < 0),aCIT results in a re-
duction in θ, that is, an increase in inequality. In other terms, the change in
inequality that results from a CIT is captured via κ.
3 Some inequalities
Recall that ρ2 is a measure of equality in X2 holding the distribution X1
constant, whereas γ2 measures the level of equality in X2 having equalized
the distribution X1. Consider (4) where W(.) is averse to CITs.T h e n ,i ft h e
correlation between xα
1 and x
β
2 is positive, we expect the level of equality in
X2 to be greater after equalizing the distribution X1 than before, i.e. that
γ2 ≥ ρ2. We have:
Proposition 3: In the context of (3),γ j ≤ ρj iﬀ the correlation between
xα
1 and x
β
2 is positive. For (4),γ j ≥ ρj iﬀ the correlation between xα
1 and
x
β
2 is positive. For (5), γj = ρj.
6Recall that κ is a measure of association between x1 and x2. It is greater
than (less than) unity when xα
1 and x
β
2 are positively (negatively) correlated.
The index κ is a simple transform of the ratio of ρj to γj :
κ =
µ
ρ2
γ2
¶β
=
µ
ρ1
γ1
¶α
(14)
In the context of (3), α,β > 0 while in (4) α,β < 0, from which the results
of the proposition follow.
4 Generalization
A ﬁnal point we turn to is the decomposition of inequality in the general
context of p>2 attributes. Deﬁne xi
. =( xi1,...,x ip). For u(xi) to satisfy
the basic axioms it must take one of the following forms: u(xi) . = x
α1
i1 ·····
x
αp
ip ,α j > 0 for all j, and
P
j αj ≤ 1; u(xi) . = −x
α1
i1 ·····x
αp
ip , αj < 0 for all j
and ﬁnally, u(xi) . =
P
j αj lnxij, where αj > 0 for all j. Propositions 1 and
2 generalize to higher dimensions. The decomposition of inequality in the
context of p attributes is given by
Ã
X
j
αj
!
lnθ =
X
j
αj lnγj +l nκ (15)
κ . =
n
P
i x
α1
i1 ·····x
αp
ip P
i x
α1
i1 ·····
P
i x
αp
ip
(16)
As before, γj denotes an AKS equality index for the distribution Xj, whereas
now κ is a function of the product moment between the variables x
α1
1 ,...,x
αp
p
thus generalizing the earlier bivariate measure of association (9).
7References
Aczél J. (1988): A Short Course on Functional Equations,D o r d r e c h t ,
Reidel.
Atkinson A. (1970): ”On the Measurement of Inequality”, Journal of
Economic Theory 2, 244—263.
Atkinson A. and F. Bourguignon (1982): ”The Comparison of Multi-
Dimensioned Distributions of Economic Status”, Review of Economic Studies
49, 183-201.
Kolm S. (1977): ”Multidimensional Egalitarianisms”, Quarterly Journal
of Economics 91, 1-13.
Sen A. (1973): On Economic Inequality, Oxford, Clarendon Press.
Sklar A. (1959): ”Fonctions de Répartition à n dimensions et leurs marges”,
Publications de l’Institut de Statistique de l’Université de Paris,8 ,2 2 9 - 2 3 1 .
Tsui K-Y (1995): ”Multidimensional Generalizations of the Relative and
Absolute Inequality Indices: The Atkinson-Kolm-Sen Approach”, Journal of
Economic Theory 67, 251—265.
Tsui K-Y (1999): ”Multidimensional Inequality and Multidimensional
Entropy Measures: An Axiomatic Derivation”, Social Choice and Welfare
16, 145-157.
8

Decomposition of Bivariate Inequality Indicesby Attributes

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Decomposition of Bivariate Inequality Indicesby Attributes

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