Attribute decomposition of multidimensional inequality indices

Economics Letters 117 (2012) 189–191

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Attribute decomposition of multidimensional inequality indices Martyna Kobus Faculty of Economic Sciences, University of Warsaw, 00-241 Warsaw, Duga 44/50, Poland

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Article history: Received 30 December 2011 Received in revised form 12 March 2012 Accepted 23 March 2012 Available online 11 April 2012

abstract Inequality indices decomposable by attributes are characterized in the case where association can be ignored. This technique permits checking for decomposition of well-known indices. © 2012 Elsevier B.V. All rights reserved.

JEL classification: D31 D63 Keywords: Multidimensional inequality Attribute decomposability Association

0. Introduction In acknowledgement of well-being as a multidimensional concept, multidimensional social indicators have emerged together with measures of inequality. A vast literature is already devoted to axiomatically derived new measures (Tsui, 1995, 1999; Maasoumi, 1986; Bourguignon, 1999; Gajdos and Weymark, 2005). Abul Naga and Geoffard (2006) introduced the notion of attribute decomposability. A multidimensional index is attribute decomposable if it can be represented as some function of inequality in each attribute and potentially some measure of association. In this paper, the strongest indices, which are decomposable ignoring association, are characterized. The result provides a simple tool to confirm the proof for the validity of decomposition for a given index. Although, in some cases it may be easy to assess a given index as potentially strongly decomposable, there is no simple published method to prove it. Among the indices proposed in the literature only the Tsui index (Tsui, 1995) and the Gajdos and Weymark index (Gajdos and Weymark, 2005) are decomposable. By no means should the strong form of decomposability be treated as an initial case. In a broader paper (Kobus, 2011), attribute decomposable indices with association are characterized in cases where association is measured using a copula.

E-mail address: [email protected]. 0165-1765/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2012.03.024

1. Notation and definitions The range of function f is denoted by R(f ). Bivariate distributions, which can be naturally extended to the case of k attributes, are dealt with throughout this paper. Let us introduce space X = [X1 , X2 ] of real-valued 2 × n matrices. For any x ∈ X by x1 , x2 , we understand the first and second columns (attribute/dimension) and x1i and x2i denote their values for the ith person, respectively; µx1 , µx2 are their means. Conversely, whenever we introduce x1 , x2 (or any other symbol with subscripts 1 , 2 ) we assume that x := (x1 , x2 ). Consider a multidimensional inequality index I : X → R. Below we define axioms that we will use in the characterization of I. Axioms are not always desirable properties. Scale Invariance (SI). The index does not change when attributes are rescaled. For m ∈ R+ define a new distribution of attributes x′ by setting x′ji := mxji , i ∈ {1, . . . , n}, j ∈ {1, 2}. Then, scale invariance implies I (x) = I (x′ ). Pigou–Dalton Transfer (PDT).1 A Pigou–Dalton transfer, that is, a transfer from the richer to the poorer that does not alter income ranking and occurs within any attribute, decreases the index value. By means of a Pigou–Dalton transfer, let x′1 , x′2 obtained from x1 , x2 , leads to I (x′ ) ≤ I (x).

1 Please notice that it differs from multidimensional generalizations of the Pigou–Dalton Transfer used in Tsui (1999), namely, Uniform PD Majorization and Uniform Majorization.

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M. Kobus / Economics Letters 117 (2012) 189–191

Strong Symmetry (SS). The index does not change for any permutation of any attribute (column). Let P1 , P2 be permutation matrices; we define a new distribution x′ by x′1 := P1 x1 and x′2 := P2 x2 . Then, Strong Symmetry implies I (x) = I (x′ ). It should be noted that this differs substantially from standard Symmetry (S) in which P1 = P2 . Consistency (C). The orderings imposed on any dimension when the other is fixed do not depend on the other dimension. Let x1 , x′1 ∈ X1 and µx1 = µx′ we require 1

I ((x1 , x2 )) ≥ I ((x′1 , x2 )) I ((x1 , x′2 )) ≥ I ((x′1 , x′2 )),

iff

∀x2 ,x′2 ∈X2

and analogously let x2 , x′2 ∈ X2 and µx2 = µx′ we require

Fig. 1. The graph of the difference f (0.75, 0.25, 0.5, 1.5) − f (0.25, 0.75, 0.5, 1.5) and the graph of g := 0 for r1 , r2 ∈ (0, 1).

2

I ((x1 , x2 )) ≥ I ((x1 , x2 )) ′

I ((x′1 , x2 )) ≥ I ((x′1 , x′2 )),

iff

∀x1 ,x′1 ∈X1 .

Definition 1. We call a multivariate index strongly decomposable by attributes if there exist univariate indices I1 , I2 fulfilling SI, S and PDT and a function h : R(I1 ) × R(I2 ) × R+ → R increasing with respect to first two arguments such that

µx I (x) = h I1 (x1 ), I2 (x2 ), 1 µx2 



.

(1)

f (x, y, 0.5, 1.5) = f (y, x, 0.5, 1.5) be satisfied for all x, y. In particular, (C) yields f (0.75, 0.25, 0.5, 1.5) − f (0.25, 0.75, 0.5, 1.5) = 0. This, as Fig. 1 shows, is not true for any r1 , r2 ∈ (0, 1). Another Tsui (1995) index is the following

 r /(r +r )  r /(r +r ) 1/n n  x2i 2 1 2 x1i 1 1 2 , IT ((x1 , x2 )) := 1 − µx 1 µx 2 i=1 where r1 , r2 > 0 are parameters. This index is decomposable. Let us recall (1). We set

  r1  1n n  x1i r1 +r2 I1 (x1 ) := 1 − µx1 i=1

2. A characterization theorem Theorem 1. A multivariate index I : X → R is strongly decomposable by attributes if and only if it fulfils scale invariance, Pigou–Dalton Transfer, Strong Symmetry and Consistency. Consistency is not a standard inequality measurement axiom. Consistency is in principle similar to separability axioms in utility theory. If x1 is ranked as more unequal than x′1 , then changes on the second dimension do not affect this ranking. As shown by Theorem 1, strong decomposition implies a highly undesirable property such as Strong Symmetry. The attributes can be permuted in an arbitrary manner and the index is unchanged. Indeed, any association existing between variables is ignored. Thus multidimensional inequality is reduced to aggregation of inequalities in margins.

  r1  1n n  x2i r1 +r2 I2 (x2 ) := 1 − µx2 i=1 h(i1 , i2 ) := 1 − (1 − i1 )(1 − i2 ). The decomposition of the Gajdos and Weymark index is not as straightforward as the decomposition of Tsui (1995) index and the method of proof of Theorem 1 (see Appendix) makes it much easier to construct the decomposition. Gajdos and Weymark (2005) index is the following3

   

IM ((x1 , x2 )) = 1 − 

3. Decomposability of specific indices Following Theorem 1, we show non-decomposability of the Tsui index (Tsui, 1999). Similar techniques can be used to show nondecomposability of the Bourguignon index2 and the Maasoumi index. Let us consider the Tsui index

 r  r 1/(r1 +r2 ) n   x1i 1 x2i 2 ¯IT ((x1 , x2 )) := 1 − 1 , n i=1 µx1 µx 2 where r1 , r2 ∈ (0, 1) are parameters. We will prove that ¯IT does not satisfy (C). For the sake of simplicity, we adhere to the case of two individuals. It is enough to study the following function f (x11 , x12 , x21 , x22 ) =



x11

r 1 

x11 + x12

 +

x12 x11 + x12

x21

r2

x21 + x22

r 1 

x22 x21 + x22

r2

r a1i x1i

i =1

 +

n 

r  1r a2i x˜ 2i

i=1

(µx1 )r + (µx2 )r

   . 

Let us fix some x˜ 1 ∈ X1 , x˜ 2 ∈ X2 and denote µ ˜ 1 = µx˜ 1 , µ ˜ 2 = µx˜ 2 , A1 = (

r

˜ 2i . We set a1i x˜ 1i )r , A2 = i=1 a2i x  r  1r   r n n   r ˜ a x a x + µ 1i 1i 2i 2i x   1  i=1  i=1 I1 (x1 ) = 1 −     (µx1 )r + (µx1 )r (µ2 )r n

i=1

n

r  n r  1r n   r ˜ µ a x + a x 1i 1i 2i 2i  x2    i=1 i=1 I2 (x2 ) = 1 −   r r r   (µ1 ) (µx2 ) + (µx2 ) 

f (i1 , i2 , µ1 , µ2 )

.

Let x21 = x22 = 1. The function (x11 , x12 ) → f (x11 , x12 , 1, 1) is clearly symmetric. That is f (x, y, 1, 1) = f (y, x, 1, 1) for all x, y. For the Consistency condition to apply we must have that

2 Except for the case of α = β in the Bourguignon index.

n 

  =

µ1 µ2

r      1r (1 − i1 )r (1 + µ ˜ r2 ) − A2 + (1 − i2 )r (µ ˜ r1 + 1) − A1   r  . µ1 + 1 µ 2

3 Strictly speaking, we consider the case when dimensions are weighted in the same way. Different weights do not change our argument.

M. Kobus / Economics Letters 117 (2012) 189–191

µx

and x2 , x′2 such that I2 (x2 ) = I2 (x′2 ) and µ 1 x2

Acknowledgements This research was funded by the grant of the National Centre of Science in Poland. I am indebted to Ramses H. Abul Naga, Piotr Milos and Oded Stark for constructive advice. Appendix Proof. Assume that I is strongly decomposable. Let us start with axiom SI. By the scale invariance of the univariate indices we obtain

 I (x ) = h I1 (x1 ), I2 (x2 ), ′

′

′



= h I1 (x1 ), I2 (x2 ),

µx′1



µ2 x′

µx 1 µx 2



  mµx1 = h I1 (mx1 ), I2 (mx2 ), mµx2 = I (x).

I1 (x1 ) := I ((x1 , x˜ 2 µx1 )), I2 (x2 ) := I (˜x1 µx2 , x2 ),

x1 ∈ X 1

1

I1 (x′1 ) = I ((x′1 , x˜ 2 µx′ )) ≤ I ((x1 , x˜ 2 µx1 )) = I1 (x1 ). 1

We define function h by the equation



1

µ x′

, we have

2

I ((x1 , x2 )) = I ((x1 , x2 )). There exists m > 0 and x1 := mx′1 , x′′2 := mx′2 such that µx1 = µx′′ and µx2 = µx′′ , moreover I1 (x′′1 ) = ′

′

′′

2

1

I1 (x′1 ), I2 (x′′2 ) = I2 (x′2 ) and by SI I ((x′′1 , x′′2 )) = I ((x′1 , x′2 )). If we show that I ((x1 , x2 )) = I ((x′′1 , x2 )) and that I ((x′′1 , x2 )) = I ((x′′1 , x′′2 )), then I ((x1 , x2 )) = I ((x′′1 , x′′2 )), which by SI equals I ((x′1 , x′2 )). We start with I ((x1 , x2 )) = I ((x′′1 , x2 )). From C we clearly see that the hypothesis is equivalent to I ((x1 , x˜ 2 µx1 )) = I ((x′′1 , x˜ 2 µx′′ )). 1 The truth of the last statement follows via definition of I1 , namely, ′′ ′′ I1 (x1 ) = I ((x1 , x˜ 2 µx1 )) = I ((x1 , x˜ 2 µx′′ )) = I1 (x1 ), which is true as 1

we established earlier I1 (x′′1 ) = I1 (x′1 ) which further equals I1 (x1 ). To conclude the proof, we must show that h increases with respect to the first two co-ordinates. Indeed, let us pick x2 ∈ X2 , x1 , x′1 ∈ X1 such that I1 (x1 ) > I1 (x′1 ) and µx1 = µx′ . By

definition of I1 , the inequality is equivalent to I ((x1 , x˜ 2 µx1 )) > I ((x′1 , x˜ 2 µx′ )) and C implies I ((x1 , x2 )) > I ((x′1 , x2 )). This gives us 1 the required monotonicity with respect to the first co-ordinate. The second co-ordinate is treated analogously.  References

x2 ∈ X2 .

It can be easily verified that I1 , I2 are inequality indices. Let for example x′1 be a vector obtained from x1 by means of a Pigou–Dalton transfer. Note that µx1 = µx′ . By axiom PDT we have

µx h I1 (x1 ), I2 (x2 ), 1 µx 2

=

1

The remaining axioms can be proven similarly. Now we assume that a function I : X → R satisfies axioms SI, PDT, SS, C. Pick any x˜ 1 ∈ X1 , x˜ 2 ∈ X2 and define ‘‘univariate projections’’



191

µ x′

= I ((x1 , x2 )).

It is necessary to confirm that this is indeed a valid definition of a function h : R(I1 ) × R(I2 ) × R+ → R. To this end, it is sufficient to establish that for any x1 , x′1 such that I1 (x1 ) = I1 (x′1 )

Abul Naga, R.H., Geoffard, P.Y., 2006. Decomposition of bivariate inequality indices by attributes. Economics Letters 90 (3), 362–367. Elsevier. Bourguignon, F., 1999. Comment to multidimensioned approaches to welfare analysis by Maasoumi, E. In: Silber, J. (Ed.), Handbook of Income Inequality Measurement. Kluwer Academic, Boston, Dordrecht, London, pp. 477–484. Gajdos, T., Weymark, J.A., 2005. Multidimensional generalized gini indices. Economic Theory 26 (3), 471–496. Kobus, M., 2011. Attribute decomposability of inequality indices via copula. Available at: http://coin.wne.uw.edu.pl/mkobus/. Maasoumi, E., 1986. The measurement and decomposition of multi-dimensional inequality. Econometrica 54 (4), 991–997. Tsui, K.Y., 1995. Multidimensional generalizations of the relative and absolute inequality indices: the Atkinson–Kolm–Sen approach. Journal of Economic Theory 67 (1), 251–265. Tsui, K.Y., 1999. Multidimensional inequality and multidimensional generalized entropy measures: an axiomatic derivation. Social Choice and Welfare 16 (1), 145–157. Springer.