An Anatomy of the Sen and

Sen-Shorrocks-Thon Indices:

Multiplicative Decomposability

and Its Subgroup Decompositions

Kuan Xu and Lars Osberg�

First Version: January 1999Current Version: June 1999

Abstract

This paper proposes a uni�ed framework for the Sen and Sen-

Shorrocks-Thon indices of poverty intensity, which shows an explicit

connection between the two indices and the underlying social evalu-

ation function. This paper also identi�es the common multiplicative

decomposition of the two indices which allows intuitive geometric in-

terpretation and easy numerical computation. It also allows further

subgroup decompositions, which are useful to policy analysts but not

possible prior to the multiplicative decomposition.

JEL Class�cation: C000, H000, O150

Keywords: poverty intensity, poverty rate, poverty gap, subgroup de-

composition, equally-distributed-equivalent-income, social evaluation

function, Gini index

�We would like to thank Satya R. Chakravarty, Gordon Anderson and Michael Hoy andparticipants at the Canadian Economics Association 33rd Annual Meeting for their helpfulcomments. Kuan Xu would like to thank SSHRC and Dalhousie University and LarsOsberg would like to thank SSHRC for providing �nancial assistance. Mailing Address:Department of Economics, Dalhousie University, Halifax, NS, Canada B3H 3J5. E-MailAddress: Kuan.Xu@Dal.Ca and Lars.Osberg@Dal.Ca. This is a draft. All comments arewelcome.

1

1 Introduction

In 1976, Sen proposed an axiomatic approach to poverty research and anindex of poverty intensity (the Sen index or the S index). Noting that the Sindex does not satisfy the strong upward transfer axiom,1 is not replicationinvariant, and is not continuous in individual incomes, Shorrocks (1995) pro-posed a modi�ed Sen index for measuring poverty intensity. This modi�edSen index is identical to the limit of Thon's modi�ed Sen index [Thon (1979,1983)]. Thus, we call it the Sen-Shorrocks-Thon index (the SST index).

Both the S and SST indices were proposed based on a set of well-justi�edand commonly agreead axioms [see Sen (1976) and Shorrocks (1995)], it isalso useful to know: (i) what underlying social evaluation function the Sand SST indices are related to; and (ii) which they are not. Sen (1976)addressed the �rst question in the context of the S index. Shorrocks (1995)was a signi�cant extension of the S index. Blackorby and Donaldson (1978,1980) and, later, Chakravarty (1983, 1997) provided a linking bridge thatthis paper uses to show that the S and SST indices are the special casesof two more general poverty indices when the underlying social evaluationfunction is chosen to be the Gini social evaluation function.

Proposing these indices is only the �rst step in measuring poverty inten-sity. From a policy point of view, it is also useful to understand the causalrelationship between poverty intensity and its contributing components [see,for example, and Birdsall and Londono (1997), Myles and Picot (1999), andPhipps (1999) among others]. In this paper, we therefore examine the multi-plicative decomposition of these indices and geometric interpretation, as wellas further subgroup decompositions of the decomposed components of theindices.

In the literature on poverty and inequality, the research focus has beenon the development of poverty indices that exhibit additive decomposabil-

1The strong upward transfer axiom states that an acceptable measure of poverty shouldalways register an increase in poverty whenever a pure transfer of income is made fromsomeone below the poverty line to someone who has more income. This axiom will bereviewed later in the paper. This property is not possessed by the poverty rate, theaverage poverty gap ratio, and the originally formulated Sen index.

2

ity. This type of decomposability requires that for any partitioning of thepopulation|subgroup decomposition, an aggregate index of poverty inten-sity can be computed as a weighted average of its subgroup indices of povertyintensity, thereby allowing the researcher to identify policy impacts on sub-groups of the population. Unfortunately, additive decomposition only suits aclass of additively decomposable poverty indices related to Theil (1967) anda class of additively decomposable indices proposed by Foster, Greer andThorbecke (1984).2

According to Foster, Greer and Thorbecke (1984), the subgroup mono-tonicity axiom is required for any additive decomposition.3 Many well-knownindices of poverty intensity such as the S index [Sen (1976)], Thon index(1979, 1983), SST index [Shorrocks (1995)],4 BD index [Blackorby and Don-aldson (1980)] and C index [Chakravarty (1983)] do not satisfy this axiomin general.5 As Foster, Greer and Thorbecke (1984) pointed out, the abovewell-known indices satisfy the Sen criteria but are not decomposable.

The S and SST indices allow a multiplicative decomposition into a setof familiar and commonly used poverty measures|the poverty rate (oftencalled the headcount ratio), the average poverty gap ratio (often called theincome gap ratio), and the Gini index of poverty gap ratios. This multiplica-tive decomposition allows intuitive geometric interpretation, easy numeri-cal computation, and further subgroup decompositions of the decomposedcomponents. As well, multiplicative decomposition enables the percentagechange over time, or the percentage di�erence between two distributions ofincome, to be expressed as the sum of the percentage changes of their com-ponents.

Multiplicative decomposition of this type has not been a major focus ofthe literature because the shortcomings of the poverty rate and the averagepoverty gap ratio led to much research e�ort being devoted to developingnew indices that would not su�er the same shortcomings. Multiplicative

2See Chapter 7 of Chakravarty (1990) for a detailed survey of the additive decomposi-tion of the poverty intensity indices.

3This axiom states that the aggregate poverty intensity index should be a strictlyincreasing function of individual subgroup poverty intensity indices.

4It is also called the modi�ed-Sen index.5The Chakravarty index with the symmetric means of order r (r < 1)is an exception.

The Kakwani (1977) index (the K index) also shares the similar decomposition di�cultybut it is not the focus of the paper.

3

decomposition was brie y mentioned by Clark, Hemming, and Ulph (1981)for the S index. Osberg and Xu (1997, 1999), based on the work of Shorrocks(1995), proposed and used the multiplicative decomposition of the SST indexin two empirical studies. Graphical analysis that focuses on the poverty rate,the average poverty gap ratio and the inequality of poverty gap ratios ofthe population has been proposed by Jenkins and Lambert (1997) and Xuand Osberg (1998). However, a detailed, uni�ed analysis of multiplicativedecomposition of the S and SST indices and their relationship is not yetavailable and further subgroup decompositions of the index components havenot yet been considered in the literature.6

The main results of this paper are as follows:(i) The S index is a BD index with Gini social evaluation function (see

Lemma 3). This provides an explicit link between the S index and its under-lying social evaluation function.

(ii) The SST index is a C index with Gini social evaluation function (seeLemma 4). This provides an explicit link between the SST index and itsunderlying evaluation function.

(iii) The S index has a multiplicative decomposition which can be ex-pressed as the product of the poverty rate, the average poverty gap ratioof the poor and one plus the Gini index of the poverty gap ratios of thepoor|this term is called the index of equality|(see Proposition 1).

(iv) The SST index can be expressed as the product of the poverty rate,the average poverty gap ratio of the poor and one plus the Gini index of thepoverty gap ratios of the population (see Proposition 2).

(v) Both the S and SST indices share the same multiplicative decompo-sition, which o�er intuitive geometric interpretation, easy numerical compu-tation, and further subgroup decompositions.

The geometric interpretation of the SST index in this paper is similar tothe three `I's poverty curve summarized by Jenkins and Lambert (1997) and

6However, it is known that the poverty rate, the average poverty gap ratio and theGini index permit subgroup decomposition [see, for example, Foster, Greer and Thorbecke(1984) and Lambert and Aronson (1993)]. The Gini index also allows factor decomposition[see, for example, Fei, Ranis, and Kuo (1978) and Shorrocks (1982, 1983)]. Blackorby,Donaldson and Auersperg (1981) also considered subgroup decomposition of the indicesof income inequality. Their approach requires that the social evaluation function satisfya separability condition. Since the Gini social evaluation function used in the S and SSTindices fails to satisfy the condition, this paper does not adopt their approach.

4

the deprivation pro�le used by Shorrocks (1955) and Osberg and Xu (1997).Multiplicative decomposition also relates to the Foster, Greer and Thorbecke(1984) decomposable poverty measure with a parameter � [the FGT(�)] inthe following sense: (a) The poverty rate is FGT(0); and (b) the averagepoverty gap ratio of the population, which is the product of the poverty rateand the average poverty gap ratio of the poor, is FGT(1). Since the FGT(�)is additively decomposable, the multiplicative structure of the S and SSTindices permits further subgroup decomposition.

(vi) The SST index is a linear transformation of the S index and vice versa(Proposition 3). This result is due to the fact that the poor and non-poorsubpopulations are not overlapping. This addition to Shorrocks (1995) givesa simple link between the SST and S indices. The SST index is the upperbound of the S index (Proposition 4). This is an extension of Chakravarty(1990, Theorem 6.9).

(vii) The multiplicative decomposition of the S and SST indices can belinearized by taking logarithmic transformation so that the logarithm of theS and SST indices is additively decomposable (Corollary 1)|A useful resultfor empirical comparisons.

(viii) The two decomposed components of the S and SST indices, thepoverty rate and the average poverty gap ratio, allow further additive sub-group decomposition (Proposition 5).

(ix) The Gini index of poverty gap ratios also allows the decompositioninto between-group Gini index, within-group Gini indices and a residual term(Proposition 6).7

Although there has been some concerns about the complexity of the Giniindex subgroup decomposition, this result is still useful to policy analystsgiven that the Gini index is the most-used inequality measure in social policyresearch and it is an essential part of the two widely-used poverty indices.

(x) Because the Gini index of poverty gap ratios allows subgroup decom-position, all three multiplicative components of the S and SST indices allowfurther subgroup decomposition (Corollaries 2 and 3).

7In Osberg and Xu (1997, 1999), we demonstrate that in empirical work [LuxemburgIncome Study countries and Canadian provinces] the Gini index of poverty gap ratios isnearly constant over time in these datasets while Myles and Picot (1999) show that theGini index of poverty gap ratios does not vary much for Canadian children over time. Thisimplies that for many purposes, decomposition of the poverty rate and average povertygap ratio will su�ce.

5

Appendix A presents a numerical example which can be used to illustrateand verify the our theoretical results. But this example cannot be used toexplain: (i) how each of the S and SST indices can be properly interpretedin the context of social welfare evaluation; and (ii) how each index is relatedto its decomposed components. The aim of the rest of the paper is to answerthese questions.

The remainder of the paper is as follows. In section 2, the notation and ba-sic axioms used in this paper are introduced. Section 3 provides some prelim-inary discussion of social evaluation function, equally-distributed-equivalent-income, and the Gini index and its subgroup decomposition. In section 4,two general poverty indices|the BD index and the C index|and two specialcases|the S index and SST index|are discussed. The �rst three sectionsprepare the foundation for our analysis of multiplicative decomposition. Insection 5, we discuss the multiplicative decomposition of the S and SST in-dices. After that, we will establish the geometric interpretation of the S andSST indices in Section 6. Section 7 explores further subgroup decomposi-tions of the decomposed components of the S and SST indices. Section 8concludes.

2 Notation and Basic Axioms

2.1 Notation

Before discussing the poverty indices, let us introduce the notation. Let Dn

be the positive-orthant of the Euclidean n-space, Rn. Let y 2 Dn or

y = [y1; y2; : : : ; yn]> (1)

be a column vector with (individual or family) incomes sorted in non-decreasingorder such that

y1 � y2 � � � � � yn:

Let ey = [ey1; ey2; : : : ; eyn]> (2)

be y with incomes sorted in non-increasing order such that

ey1 � ey2 � � � � � eyn:6

The notation \ e " will be used for sorting a vector y in opposite order.For example, if a vector x has elements in non-increasing order, then thevector ex contains the identical elements ranked in non-decreasing order andvice versa. As we can see later, this notation will be applicable to any lineartransformation of the income vector.

Let the poverty line be z > 0. A censored income vector is obtained by

setting�yi= yi if yi < z and

�yi= z otherwise,8 that is

�y = [

�y1;

�y2; : : : ;

�yn]

> = [y1; y2; : : : ; yq; z; : : : ; z]>: (3)

The income vectors of the poor, yp 2 Dq and eyp 2 Dq, are the truncated

income vectors generated from�y by deleting z's:

yp = [y1; y2; : : : ; yq]>; (4)

where y1 � y2 � � � � � yq, and

eyp = [ey1; ey2; : : : ; eyq]>; (5)

where ey1 � ey2 � � � � � eyq. The q poor individual incomes are all less thanthe poverty line z. The poverty rate therefore is given by q

n.

De�ne 1 as a column vector of ones with an appropriate dimension, whichwill be obvious in each context. The poverty gap ratio vector of the pop-

ulation is de�ned as z1��

y

zwhere the poor have poverty gap ratios z�yi

z,

i = 1; 2; : : : ; q, and the non-poor have zero poverty gap ratios. The ele-

ments in z1��

y

zare ranked in non-increasing order because the elements in

�y

are ranked in non-decreasing order. Similarly, the poverty gap ratio vectorof the poor is given by z1�yp

zwhere the poor have poverty gap ratios z�yi

z,

i = 1; 2; : : : ; q, and the non-poor's zero poverty gap ratios are excluded. Theelements in z1�yp

zare ranked in non-increasing order because the elements

in yp are ranked in non-decreasing order.When considering subgroup decompositions later, we also need to intro-

duce some additional notation. Let yi(k) represent the income of individuali (i = 1; 2; : : : ; n) in subgroup k (k = 1; 2; : : : ; l). The income vector forsubgroup k is

y(k) = [y1(k); y2(k); : : : ; ynk(k)]>; (6)

8We use the weak de�nition of the poor here|a poor person's income is less than thepoverty line|as it is generally treated in the literature.

7

where y1(k) � y2(k) � � � � � ynk(k). The number of the elements in y(k), nk,depends on the number of individuals belonging to subgroup k. Note thatn =

Plk=1 nk. The vector of income of the poor in subgroup k, yp(k), is given

byyp(k) = [y1(k); y2(k); : : : ; yqk(k)]

>; (7)

where y1(k) � y2(k) � : : : � yqk(k). The number of the elements in yp(k), qk,depends on the number of poor individuals belonging to subgroup k. Notethat q =

Plk=1 qk.

To discuss the subgroup decomposition of the Gini index of income later(Lemma 2), the subgroup income vectors, y(1);y(2); : : : ;y(l), need to be stackedinto a vector y(�) as

y(�) = [y>(1);y>(2); : : : ;y

>(l)]

>

so that the subgroup income vectors, y(1);y(2); : : : ;y(l), are sorted by thesubgroup average incomes in non-decreasing order. It should be noted thaty and y(�) may di�er if the incomes of di�erent subgroups overlap in y sincethe incomes in y are sorted by the size of the individual incomes and thosein y(�) are sorted according to the size of the individual incomes within each

subgroup and the size of the subgroup average incomes across subgroups.�y

and�y(�) can be de�ned similarly as y and y(�), respectively.The poverty gap ratio vector may be also de�ned for subgroup k as

z1�yp(k)

zwhere

z�yi(k)

z, k = 1; 2; : : : ; l and i = 1; 2; : : : ; qk. To discuss the

subgroup decomposition of the Gini index of poverty gap ratios later (Propo-sition 5), the vectors of subgroup poverty gap ratios need to be stacked intoa vector as

z1� yp(�)

z=

"�z1� yp(1)

z

�>;

�z1� yp(2)

z

�>; : : : ;

�z1� yp(l)

z

�>#>;

wherez1�yp(1)

z;z1�yp(2)

z; : : : ;

z1�yp(l)

zare sorted by the subgroup average poverty

gap ratios in non-decreasing order. It should be noted that z1�yp

zand

z1�yp(�)

z

may di�er if the poverty gap ratios of di�erent subgroups overlap in z1�yp

z,

since the poverty gap ratios in z1�yp

zare sorted by the size of the individual

poverty gap ratios and the poverty gap ratios inz1�yp(�)

zare sorted by the

size of the individual poverty gap ratios within each subgroup and the sizeof subgroup average poverty gap ratios across subgroups.

8

De�nition 1 The average of an income vector y is given by

�(y) =1

n

nXi=1

yi: (8)

Because the order of the elements in y is not relevant in computing themean,

�(y) =�(ey): (9)

The average income of subgroup k is denoted as �(y(k)). The average poverty

gap ratio of the population (the poor) is denoted �( z1��

y

z) [�( z1�yp

z)]. In

general, �(�) is used to denote the average of a particular vector.

2.2 Axioms

Chakravarty (1990) has a detailed summary of the development of povertymeasurement axioms and their sources.

Axiom 1 (Focus axiom) The poverty index should be independent of thenon-poor population.

Axiom 2 (Weak monotonicity axiom) A reduction in a poor person's in-come, holding other incomes constant, must increase the poverty index.

Axiom 3 (Impartiality axiom) The poverty index may be de�ned over or-dered income pro�les without loss of generality.

Axiom 4 (Weak transfer axiom) An increase in the poverty index occursonly if the poorer of the two individuals involved in an upward transfer ofincome is poor and if the set of the poor people does not change.

Axiom 5 (Strong upward transfer axiom) An increase in the poverty indexoccurs if the poorer of the two individuals involved in an upward transfer ofincome is poor.

[Note that the strong upward transfer axiom implies the weak transferaxiom since Axiom 5 allows the poor subpopulation to be either the same orto change while Axiom 4 does not allow the poor subpopulation to change.]

9

Axiom 6 (Continuity axiom) The poverty index varies continuously withincomes.

Axiom 7 (Replication invariance axiom) The poverty index does not changeif it is computed based on an income distribution that is generated by the k-fold replication of an original income distribution.

When the weak de�nition of the poor is adopted (i.e., the poor all have anincome less than the poverty line), the �rst four axioms are equivalent to re-quiring that for a �xed number of poor persons, the poverty index should bea decreasing, strictly S-convex function of income of the poor or should corre-spond to an increasing strictly S-concave social evaluation function in a neg-ative monotonic way [see Donaldson and Weymark (1986) and Chakravarty(1990)].

The poverty rate ( qn) as a poverty measure gives the proportion of the

population who have income under the poverty line. This measure clearlysatis�es the focus axiom but it violates both weak monotonicity and weaktransfer axioms. The average poverty gap ratio (1

q

Pqi=1

z�yiz) as a poverty

measure provides the normalized deprivation of the poor. But it violatesthe weak transfer axiom. These dissatisfactions led Sen (1976) to work onthe S index based on the focus, monotonicity, and weak transfer axioms.Although the S index satis�es the �rst four axioms, it violates the last threeaxioms|the strong upward transfer, continuity, and replication invarianceaxioms. Shorrocks (1995) proposed the modi�ed Sen index (the SST index).The SST index satis�es the strong upward transfer axiom|which impliesthe weak transfer axiom, the continuity axiom and the replication invarianceaxiom.

3 SEF, EDEI, and the Gini Index

As is well known, the S and SST indices are not additively decomposable. Toshow that they are multiplicatively decomposable, we need to introduce theconcept of the equally-distributed-equivalent-income (EDEI), or the repre-sentative income proposed by Atkinson (1970), Kolm (1969), and Sen (1973),and used by Blackorby and Donaldson (1978). For a particular social evalu-ation function (SEF), an EDEI given to every individual could be viewed asidentical in terms of social welfare to an actual income distribution.

10

Assumption 1 W is a homothetic (ordinal) SEF:

W (y) =�(W (y)) (10)

where � is increasing and W is positively linearly homogeneous.

De�nition 2 Let � be the EDEI and 1 be a column vector of ones with anappropriate dimension. Then,

W (� � 1) = W (y) (11)

orW (� � 1) = W (y): (12)

Given W is positively linearly homogeneous, we can recover the EDEI � =�(y) as

� =W (y)

W (1)= �(y): (13)

The SEF, W , and the EDEI, �, have an one-to-one corresponding relation-ship.

The EDEI is an important concept for us to relate the S index to the BDindex and the SST index to the C index.

One of the most used SEF's is the Gini SEF. The S index di�ers fromits general form|the BD index|because it adopts the Gini SEF and hencethe Gini EDEI. Similarly, the SST index di�ers from its general form|the Cindex|because it adopts the Gini SEF and hence the Gini EDEI. Becauseof these, we need to introduce the Gini SEF and Gini EDEI.

De�nition 3 The Gini SEF is9

WG(y) =1

n2

nXi=1

(2n� 2i+ 1)yi:

9This is because

W (1) =1

n2

nXi=1

(2n� 2i+ 1) = 1:

11

Its corresponding EDEI function is

�G(y) =1

n2

nXi=1

(2n � 2i + 1)yi (14)

or

�eG(ey) = 1

n2

nXi=1

(2i� 1)eyi (15)

with �G(y) = �eG(ey).10It is easy to see that the Gini SEF attaches a higher weight to a lower

level of income and vice versa. The weight is determined by the rank of anincome rather than the size of the income.

De�nition 4 The Gini index can be de�ned in terms of the Gini EDEI andthe mean income as

G(y) =1 ��G(y)

�(y)= 1 �

1

n2�(y)

nXi=1

(2n � 2i+ 1)yi (16)

or eG(ey) =1 � �eG(ey)�(ey) = 1�

1

n2�(ey)nXi=1

(2i � 1)eyi; (17)

where y (ey) has elements in non-decreasing (non-increasing) order.11

It is useful to note that G(y) = eG(ey) in De�nition 4 but G(�) and eG(�)have di�erent functional forms and the elements in y and ey are sorted dif-ferently.

Lemma 1 The Gini index has the following property:

G(y) = 1 ��G(y)

�(y)= �G(ey) = �

1�

�G(ey)�(ey)

!: (18)

10This is because yi = eyn�i+1 and eyi = yn�i+1.11The two equations are identical because �(y) =�(ey), yi = eyn�i+1, and eyi = yn�i+1.

12

Proof: See Appendix B.1. 2From Lemma 1 and De�nition 4,

1 +G(y) = 1 �G(ey) = �G(ey)�(ey) : (19)

These de�nitions and lemmas will be used in our analysis of the S and SSTindices later in this paper.

The following two de�nitions are crucial to the subgroup decompositionof the Gini index.

De�nition 5 The Lorenz curve for the population income vector y is de�nedas

L(r

n) =

1

n�(y)

rXi=1

yi; (20)

for r = 1; 2; : : : ; n. The between-group Lorenz curve is de�ned as

LB(r

n) =

1

n�(y)

0@ Xs�k�1

ns�(y(s)) + nk�(y(k))i

nk

1A ; (21)

where r =P

s�k�1 ns+i = 1; 2; : : : ; n when k = 1; 2; : : : ; l and i = 1; 2; : : : ; nk.The within-group Lorenz curve for subgroup k is given by

Lk(r

nk) =

1

nk�(y(k))

rXi=1

yi(k); (22)

for r = 1; 2; : : : ; nk.The concentration curve is de�ned as

C(r

n) =

1

n�(y)

0@ Xs�k�1

ns�(y(s)) + nk�(y(k))Lk(i

nk)

1A ; (23)

where r =P

s�k�1 ns+i = 1; 2; : : : ; n when k = 1; 2; : : : ; l and i = 1; 2; : : : ; nk.

13

De�nition 6 The Gini index can be de�ned in terms of the Lorenz curve ofthe population income vector y as12

G(y) =2

n

nXi=1

�i

n� L(

i

n)�: (24)

The between-group Gini index can be de�ned in terms of the between-groupLorenz curve as

GB(y) =2

n

nXi=1

�i

n� LB(

i

n)�: (25)

The within-group Gini index can be de�ned in terms of the within-groupLorenz curve as

Gk(y(k)) =2

nk

nkXi=1

�i

nk� Lk(

i

nk)�: (26)

It should be noted that the between-group Gini index, GB(y), is de�nedas the Gini index as G(y) but with every income in every subgroup k beingreplaced by the relevant subgroup mean income while the Gini index of thepopulation and the Gini index of subgroup k are de�ned as usual.

Lemma 2 The Gini index of the population income vector can be decom-posed into three components|the between-group, within-group, and overlap-ping components as

G(y) =GB(y)+lX

k=1

akGk(y(k)) +R(y;y(�)); (27)

where ak =n2k

n2�(y(k))

�(y) is the income share going to subgroup k and R(y;y(�))is a residual which is zero if the subgroup income ranges do not overlap or ify = y(�).

12This is consistent with De�nition 4 because

G(y) = 2n

Pn

i=1

�in�

1n�(y)

Pi

j=1 yj

�= 2

n2

Pn

i=1 i�2

n2�(y)

Pn

i=1

Pi

j=1 yj

= 1� 1n�(y)

Pn

i=1(2n� 2i+ 1)yi:

14

Proof: SeeYitzhaki and Lerman (1991), Lambert and Aronson (1993).2

The above results will be used in our analysis of the S and SST indiceslater in this paper.

4 Two General Poverty Indices and Their

Special Cases with Gini SEF

The link between the S and SST indices can be better understood based ontwo general poverty indices introduced by Blackorby and Donaldson (1980)(the BD index) and Chakravarty (1983) (the C index). Since these twoindices have a direct link to the Gini SEF, the analysis in this section permitsan explicit interpretation of the S and SST indices in term of social welfare.The S index [Sen (1976)] can be viewed as a special case of the BD indexwhile the SST index [Shorrocks (1995)] can be considered as a special case ofthe C index. Alternatively, the BD index can be viewed as a generalizationof the S index while the C index can be taken as a generalization of the SSTindex.

Consistent with the S index, the BD index focuses on the incomes ofthe poor yp or the truncated income distribution by excluding the non-poorpopulation. It is de�ned as follows:

De�nition 7 The BD index is de�ned as

IBD(yp) =q

n

"z � �(yp)

z

#(28)

where � is the EDEI function of yp for some increasing and strict S-concaveSEF.

Please note that the EDEI function is generic and may be related tosome SEF as shown in De�nition 2. For the ease of comparison, the S indexis de�ned as follows:

De�nition 8 The S index is de�ned as

IS(yp) =q

n

" 1

q

qXi=1

z � yi

z

!+

1�

1

q

qXi=1

z � yi

z

!!G(yp)

#: (29)

15

Lemma 3 The BD index IBD(yp) with the Gini EDEI, �G(yp), is the Senindex, that is

IS(yp) = IGBD(yp) =q

n

"z � �G(yp)

z

#: (30)

Proof: See Blackorby and Donaldson (1980, 1054{1055). 2As can be shown later, the S index can be decomposed into the poverty

rate, the average poverty gap ratio and the Gini index of poverty gap ratiosof the poor.

Equation (30) provides a mathematical structure based on which one cansee why the S index does not satisfy the strong upward transfer, continu-ity and replication invariance axioms as found by Shorrocks (1995). It isalso clear from Equation (30) that the S index is explicitly related to theunderlying Gini SEF.13

To propose the C index, Chakravarty (1983) adopted the idea of Thon

(1979) and Takayama (1979) and focused on the censored income vector�y.

The C index is de�ned as follows:

De�nition 9 The C index is de�ned as

IC(�y) =

z � �(�y)

z; (31)

where � is the EDEI function for some increasing and strict S-concave SEF.

For the ease of comparison, the SST index is de�ned below:

De�nition 10 The SST index of poverty intensity is de�ned as either

ISST (�y) =

1

n2

nXi=1

(2n � 2i+ 1)z�

�yi

z(32)

or

ISST (yp) =1

n2

qXi=1

(2n� 2i+ 1)z � yi

z: (33)

13It should be noted that Sen (1976) started from Axioms R, M, and N which have Ginisocial welfare implications.

16

Lemma 4 The C index, IC(�y), with the Gini EDEI, �G(

�y), is the SST

index, ISST (yp).

Proof: See Chakravarty (1997). 2As can be shown later, the SST index can be decomposed into the poverty

rate, the average poverty gap ratio and the Gini index of poverty gap ratiosof the population.

Lemma 4 provides a mathematical structure based on which one cansee why the SST index satis�es the strong upward transfer, continuity andreplication invariance axioms as found by Shorrocks (1995). As the case forthe S index, Lemma 4 shows that the SST index is explicitly related to theGini SEF.

5 CommonMultiplicative Decomposition of

the S and SST indices

As noted previously, both S index and SST index do not permit additivedecomposition although they possess desirable properties. However, one cananalyze the decomposition issue from the multiplicative point of view, whichallows linearization in that the logarithm of the index is additive decompos-able. We will �nd that both the S index and SST index permit a commonmultiplicative decomposition into the poverty rate, the average poverty gapratio and the Gini index of poverty gap ratios.

The following proposition states that the S index permits multiplicativedecomposition.

Proposition 1 The S index has a multiplicative decomposition which can beexpressed as

IS(yp) =q

n�

�z1� yp

z

��1�G

�z1� yp

z

��(34)

or

IS(yp) =q

n�

�z1� yp

z

� 1 +G

gz1� yp

z

!!; (35)

where z1�yp

z(gz1�yp

z) has elements in non-increasing (non-decreasing) order.

17

Proof: From Lemma 3 and De�nition 3,

IS(yp) =q

n

q2z �

Pqi=1(2q � 2i+ 1)yi

q2z

!: (36)

The above equation can then be rewritten as14

IS(yp) =q

n

2

q

qXi=1

z � yi

z�

1

q2

qXi=1

(2i� 1)z � yi

z

!: (37)

Further manipulation gives

IS(yp) =q

n

0@2��z1� yp

z

��

��z1�yp

z

�q2�

�z1�yp

z

� qXi=1

(2i� 1)z � yi

z

1A : (38)

Based on De�nition 4 the above equation can be rewritten as

IS(yp) =q

n�

�z1� yp

z

��1 + eG�z1� yp

z

��(39)

or

IS(yp) =q

n�

�z1� yp

z

� 1 +G

gz1� yp

z

!!; (40)

where z1�yp

z(gz1�yp

z) has elements in non-increasing (non-decreasing) order.

Based on Lemma 1, the S index can also be expressed as

IS(yp) =q

n�

�z1� yp

z

��1�G

�z1� yp

z

��: (41)

2

14Here we use1

q2

qXi=1

(2i � 1)z

z= 1

and2

q

qXi=1

z

z= 2:

18

As can be seen from the above proposition, we can present the S index asthe product of the poverty rate, the average poverty gap ratio, and the Giniindex of poverty gap ratios of the poor.

It is also interesting to compare the multiplicative decomposition of theoriginal S index with the one presented here. As Sen (1976) pointed out,for a large q, the S index can be written as the one given in equation (29)where the Gini index is for incomes of the poor. In this paper, we show thatthe S index can be written as in equation (35) where the Gini index is forpoverty gap ratios of the poor. Equation (35) appears to be a bit simplerthan equation (29). As we show later, unlike equation (29), equation (35)permits a clear geometric interpretation.

The following proposition states that the SST index permits similar mul-tiplicative decomposition.

Proposition 2 The SST index has a multiplicative decomposition which canbe expressed as

ISST (�y) =

q

n�

�z1� yp

z

�0@1�G

0@z1� �y

z

1A1A (42)

or

ISST (�y) =

q

n�

�z1� yp

z

�0B@1 +G

0B@ gz1�

�y

z

1CA1CA ; (43)

where z1��

y

z(gz1�

�

y

z) has elements in non-increasing (non-decreasing) order.

Proof: Based on Lemma 4, the SST index can be written as

ISST (�y) =

z � �G(�y)

z=

1

n2

nXi=1

(2n � 2i+ 1)z�

�yi

z: (44)

Bymultiplying and dividing the right-hand-side of equation (44) by ��z1�

�

y

z

�=

19

�1n

Pni=1

z��

yiz

�, the SST index becomes

ISST (�y) =

�

�z1�

�

y

z

�0BB@ 2n

n2�

�z1�

�

yz

� �Pni=1

z��

yiz

�� 1

n2�

�z1�

�

yz

� �Pni=1(2i� 1) z�

�

yiz

�1CCA :

(45)According to De�nition 4, this above equation can be simpli�ed further into

ISST (�y) = �

0@z1� �y

z

1A0@1 + eG0@z1� �

y

z

1A1A (46)

or

ISST (�y) = �

0@z1� �y

z

1A0B@1 +G

0B@ gz1�

�y

z

1CA1CA ; (47)

where z1��

y

z(gz1�

�

y

z) has elements in non-increasing (non-decreasing) order.

Because of ��z1�

�

y

z

�= q

n��z1�yp

z

�and Lemma 1, the SST index can be

written as15

ISST (�y) =

q

n�

�z1� yp

z

�0@1�G

0@z1� �y

z

1A1A : (48)

2

As can be seen from the above proposition, we can present the SST indexas the product of the poverty rate, the average poverty gap ratio, and theGini inequality measure of relative deprivations of the population.

From the multiplicative decomposition of the S and SST indices, we notethat the two indices di�er only by the argument of G(�). The S index has

a component G� gz1�yp

z

�while the SST index has a component G

gz1�

�

y

z

!.

It is known that the poverty gap ratios of the non-poor subpopulation arezeros and the poor and the non-poor subgroups do not overlap in the censored

15Note that Osberg and Xu (1997) also decomposed the SST index. We do it here usingthe C index with the Gini EDEI.

20

income vector�y. As shown in the following lemma, the Gini index of poverty

gap ratios of the population can be decomposed into two components: (1)the Gini index of subgroup average poverty gap ratios between the non-poorand poor subpopulations and (2) the product of the poverty ratio and theGini index of poverty gap ratios of the poor.

Lemma 5 The Gini index of poverty gap ratios of the population, G

gz1�

�

y

z

!,

is the sum of the Gini index of the average poverty gap ratios between thenon-poor and the poor subpopulations,

�1� q

n

�, and the poverty-rate-weighted

Gini index of poverty gap ratio of the poor, q

nG

� gz1�yp

z

�,as follows:

G

0B@ gz1�

�y

z

1CA =�1 �

q

n

�+q

nG

gz1� yp

z

!: (49)

Alternatively,

G

0@z1� �y

z

1A =�q

n� 1

�+q

nG

�z1� yp

z

�: (50)

Proof: From De�nition 4, we have

G

0@z1� �y

z

1A = 1�1

n2�

�z1�

�

y

z

� nXi=1

(2n� 2i+ 1)z�

�yi

z; (51)

where the poverty gap ratio vector of the population, z1��

y

zhas elements in

non-increasing order (i.e., the poor subpopulation takes the top partitionof the column vector while the non-poor subpopulation takes the bottompartition). Similarly, from De�nition 4, we have

G

�z1� yp

z

�= 1�

1

q2��z1�yp

z

� qXi=1

(2q � 2i+ 1)z � yi

z; (52)

where the poverty gap ratio vector of the population, z1�yp

z, has elements in

non-increasing order. It is known that

�

0@z1� �y

z

1A =q

n�

�z1� yp

z

�; (53)

21

where the order of z1��

y

zor z1�yp

zis irrelevant. From equation (51) we get

G

0@z1� �y

z

1A = 1�q

n

1

q2��z1�yp

z

� qXi=1

(2q � 2i+ 1)z � yi

z�2

�1�

q

n

�: (54)

It can be further rewritten as

G

0@z1� �y

z

1A =q

n

8<:1� 1

q2��z1�yp

z

� qXi=1

(2q � 2i+ 1)z � yi

z

9=;��1 �

q

n

�:

(55)Thus,

G

0@z1� �y

z

1A =�q

n� 1

�+q

nG

�z1� yp

z

�: (56)

Applying Lemma 1 to G on the left-hand-side, the above relation can beexpressed alternatively as

G

0B@ gz�

�y

z

1CA = ��q

n� 1

��q

nG

�z � yp

z

�: (57)

Applying Lemma 1 to G on the right-hand-side of equation (22), the equationbecomes

G

0B@ gz1�

�y

z

1CA =�1 �

q

n

�+q

nG

gz1� yp

z

!: (58)

2

The above lemma is a special case of the Gini index decomposition givenin Lemma 2.

Proposition 3 The SST index and the S index are related in the followingway:

ISST (�y) =

q

nIS(yp) + 2

q

n

�1�

q

n

��

�z1� yp

z

�: (59)

Proof: Combining the results in Lemma 5 and Proposition 2 gives

ISST (�y) =

q

n�

�z1� yp

z

��2�1 �

q

n

�+q

n

�1 �G

�z1� yp

z

���(60)

22

or

ISST (�y) =

q

n�

�z1� yp

z

� 2�1 �

q

n

�+q

n

1 +G

gz1� yp

z

!!!: (61)

Further manipulation of equations (60) and (61) gives equation (59). 2According to Chakravarty (1990, Theorem 6.9), if the SEF is completely

strictly recursive, then

IBD(yp) < IC(�y): (62)

In other words, the BD index is bounded above by the C index. It is knownthat the Gini SEF, which is the underlying SEF for the S and SST indices,is not completely strict recursive, the following proposition shows that thesimilar relationship holds.

Proposition 4 The S index is bounded above by the SST index, i.e.,

IS(yp) < ISST (�y): (63)

Proof: When q < n, q

n> 0. The above condition implies that �G(

�y) < z

and �G(yp) < z. From the strict S-concavity of �G, that is based on thestrict S-concavity of W , we have

�G(�y) <

q

n�G(yp) +

�n� q

n

�z: (64)

Equation (64) implies

ISST (�y) =

z � �G(�y)

z>

q

n

z � �G(yp)

z

!= IS(yp): (65)

Thus,

ISST (�y) > IS(yp): (66)

2

From the above discussion, it is apparent that the multiplicative decom-position structure allows further analysis of poverty intensity in terms of thepoverty rate, the average poverty gap ratio, and the Gini index of poverty gapratios. The following corollary demonstrates that it is possible to transform

23

the multiplicative decomposition of the S and SST indices into the addi-tive decomposition of the indices via the logarithmic transformation. This isuseful for empirical research of poverty intensity and its decomposition. Toexplain this in the following corollary, we use I for either the S index or theSST index, � for the average poverty gap ratio, and G for the Gini index ofpoverty gap ratios of either the poor or the population.

Corollary 1 Since the S and SST indices of poverty intensity take the formof�

I =q

n�(1 +G); (67)

then, by adding the time subscript t and taking log-di�erence, we get

�I = �q

n+��+�(1 +G); (68)

where �x = lnxt � lnxt�1 �xt�xt�1

xt�1represents the percentage change in x.

Depending on the purpose of research, one may use the same poverty linezt = zt�1 = z for It and It�1 and their components or di�erent poverty lineszt and zt�1, respectively, for It and It�1 and their components.

Osberg and Xu (1997, 1999) used this multiplicative decomposition for theSST index to the international and regional comparative studies. StatisticsCanada has also adopted this methodology to analyze low-income intensityamong Canadian children [Myles and Picot (1999)].

The multiplicative decomposition also allows policy makers to set up threespeci�c anti-poverty policy \targets" (rate, gap, and inequality) in reducingpoverty intensity. These targets may be used to monitor the e�ectiveness ofthe anti-poverty policy.

6 Geometric Interpretation and Further De-

compositions

Multiplicative decomposition allows the S and SST indices to be interpretedgeometrically and allows further subgroup decomposition. We focus on thegeometric interpretation �rst.

24

6.1 Geometric Interpretation

In this subsection, we give the S index a simple geometric interpretationthat is somewhat di�erent from that of Sen (1976) but is quite close to thatof the SST index proposed by Shorrocks (1995). For comparison purpose,we also present and interpret the SST index geometrically. Let the relativedeprivation measure be xi = z�yi

zif z > yi, xi = 0 otherwise, for i =

1; 2; : : : ; n. The xi's are sorted in non-increasing order. Let

H =q

n; (69)

I = �

z1� yp

z

!; (70)

G(exp) = G

0@ gz1� yp

z

1A ; (71)

and

G(e�x) = G

0B@ gz1�

�y

z

1CA : (72)

The �rst q xi's are positive for the poor who are deprived and the rest who arenot deprived have zeros. The deprivation pro�le can be graphed by plotting1n

Pri=1 xi against

r

nfor r = 1; 2; : : : ; n in a unit square box. As shown in

Figure 1, the poverty pro�le starts from the origin, reaches out concavely tothe point a and then becomes horizontal from a to HI. The point H is thepoverty rate, and the point HI represents the average poverty gap ratio ofthe population. Arc 0a represents the inequality of poverty gap ratios of thepoor. Since the deprivation measures fxig are sorted in non-increasing order,the concave arc 0a is in fact an inverted Lorenz curve. The dotted straightline linking 0 and a would be a segment of the poverty pro�le if the poor hadidentical incomes. Since the non-poor have zero deprivation, the horizontalsegment aHI of the deprivation pro�le has no signi�cant information butshows the non-poor account for the 1 �H proportion of the population.

[Insert Figure 1 about here]

25

To show the S index has a nice geometric interpretation that is similarto that of the Gini index, let triangle 0H 0H be area E and triangle 0Ha bearea C and the space between arc 0a and the dotted line linking 0 and a beD. Thus,

Area E =1

2H: (73)

Area C =1

2H2 � I: (74)

Area D can be computed from the fact that the Gini index of poverty gapratios of the poor is given by16

G(exp) = Area D

Area C 0=

Area D

Area C: (75)

Using equations (74) and (75) yields

Area D = Area C �G(exp) = 1

2H2 � I �G(exp)

The S index is simply the ratio of the sum of areas C and D to area E, i.e.,

IS(yp) =12H

2�I+ 12H

2�I �G(exp)12H

= H � I � (1 +G(exp)): (76)

Using equations (69), (70), and (71), equation (76) can be shown to beidentical to equation (35).

For completeness, we also analyze the geometric interpretation of the SSTindex in the similar fashion. Let the lower triangle of the unit box in Figure1 be area A and the rectangule at the lower right-hand corner of the unitbox be area B. Thus

Area A =1

2(77)

andArea B = (1�H) �H � I = H �H2 � I: (78)

From equation (43), the SST index can be expressed as

ISST (y) = H � I � (1 +G(e�x)) (79)

16Note that Area C0, the trangle formed by two dotted straight lines and the verticalaxis, is identical to Area C.

26

using equations (69), (70), and (72). Further, using equations (49), equation(79) becomes

ISST (y) = H � I � (2 �H +H �G(exp)): (80)

Now compute the ratio of the sum of areas B, C, and D to area A, i.e.,

Area B+Area C+Area D

Area A=

H �I �[(1�H)+ 12H+ 1

2H�G(exp)]12

= H � I � (2 �H +H �G(exp)): (81)

Comparing equations (80) and (81), we prove that, indeed, the SST index isthe ratio of the sum of areas B, C, and D to area A.

6.2 Further Decompositions

Because the S and SST indices can be decomposed into three familiar andcommonly used components. Subgroup decompositions can also be appliedto decomposed components, q

n, �, and G.

De�nition 11 The poverty rate and the average poverty gap ratio for sub-

group or region k is de�ned and computed as qknk

and ��z1�yp(k)

z

�, respectively.

Proposition 5 The poverty rate and the average poverty gap ratio are re-lated to their subgroup counterparts, respectively, as follows:

q

n=

lXk=1

wk

qk

nk; (82)

and

�

�z1� yp

z

�=

lXk=1

wk�

�z1� yp(k)

z

�; (83)

where wk =nkn.

Proof: The decomposition based on Foster, Greer and Thorbecke (1984)is quite intuitive. 2

Although it is possible to decompose the Gini index by factor into pseudofactor Gini indices [see Fei, Ranis, and Kuo (1978) and Shorrocks (1982,

27

1983)], this is not our focus. Instead, to be consistent with subgroup decom-position of the poverty rate and average poverty gap ratio given in Proposi-tion 5, our focus is on the subgroup decomposition of the Gini index of thepoverty gap ratios of the poor (for the S index) or the population (for theSST index).

Proposition 6 The Gini index G( z1�yp

z) can be decomposed into the between-

group, within-group, and overlapping components as follows:

G(z1� yp

z) = GB(

z1� yp

z)+

lXk=1

bkGk(z1� yp(k)

z)+R(

z1 � yp

z;z1� yp(�)

z);

(84)

where bk =q2k

q2

�

�z1�yp(k)

z

��

�z1�yp

z

� . Similarly, the Gini index G( z1��

y

z) can be decom-

posed into the between-group, within-group, and overlapping components asfollows:

G(z1�

�y

z) = GB(

z1��y

z)+

lXk=1

ckGk(z1�

�y(k)

z)+R(

z1��y

z;z1�

�y(�)

z); (85)

where ck =n2k

n2

�

z1�

�

y(k)z

!�

�z1�

�

yz

� .

Proof: These are the results from a direct application of Lemma 2. 2Finally, we may present the multiplicative decomposition of the S and

SST indices and their components' subgroup decompositions in the followingcorollary.

Corollary 2 The S index can be expressed as

IS(yp) =�Plk=1wk

qknk

� hPlk=1wk�

�z1�yp(k)

z

�i�n1 �

hGB(

z1�yp

z) +

Plk=1 bkGk(

z1�yp(k)

z) +R( z1�yp

z;z1�yp(�)

z)io;

(86)

28

and the SST index can be expressed as

ISST (yp) =�Plk=1wk

qknk

� hPlk=1wk�

�z1�yp(k)

z

�i�

(1�

"GB(

z1��

y

z) +

Plk=1 ckGk(

z1��

y(k)

z) +R( z1�

�

y

z;z1�

�

y(�)

z)

#):

(87)

Proof: Combining the results in Propositions 5 and 6 gives the resultsin Corollary 2.2

Although both the S and SST indices permit further subgroup decompo-sitions of their decomposed components, the further subgroup decompositionof the S index appears to be more attractive since the R term in the S index,

which depends on z1�yp

zand

z1�y(�)

z, may be substantially less than the R

term in the SST index, which depends on z1��

y

zand

z1��

y(�)

z. This is because

yp contains only the incomes of the poor while�y is the censored income vec-

tor of the poor and the non-poor subpopulations. This observation makesthe SST index and its further subgroup decomposition less appealing. How-ever, according to Proposition 3, the SST index can be expressed as a linearfunction of the S index. Hence, we have the following corollary:

Corollary 3 The SST index can be expressed as

ISST (�y) =

�Plk=1wk

qknk

� hPlk=1 wk�

�z1�yp(k)

z

�i�f2

h1�

�Plk=1wk

qknk

�i+�Pl

k=1 wkqknk

��h1 �

hGB(

z1�yp

z) +

Plk=1 bkGk(

z1�yp(k)

z) +R( z1�yp

z;z1�yp(�)

z)iig:

(88)

Proof: Combining the results in Lemma 5 and Propositions 2 and 5 givesthe result in Corollary 3. 2

Although the mathematical presentation of the further decompositionsappear to be somewhat complex, the S and SST can be computed directlyfrom the most-used poverty and inequality measures (the poverty rate, theaverage poverty gap ratio and the Gini index of poverty gap ratios), whichalso enable subgroup decomposability. As the numerical example in Ap-pendix A shows, both multiplicative and subgroup decompositions of the Sand SST indices are intuitive and easy to compute.

29

7 Concluding Remarks

This paper analyzes, from a theoretical point of view, the underlying socialevaluation function for the Sen (or S) and the Sen-Shorrocks-Thon (or SST)indices and presents a uni�ed multiplicative decomposition framework for theS and SST indices of poverty intensity. The underlying social evaluation func-tion gives the two indices clear social welfare interpretation. The commonmultiplicative decomposition allows the intuitive geometric interpretations,easy numerical computations, and further subgroup decompositions.

The paper �rst shows that the BD index is a generalization of the S indexwhile the S index is a BD index with Gini social evaluation function. The Cindex is a generalization of the SST index while the SST index is a C indexwith Gini social evaluation function. We note that the S and SST indiceshave a common multiplicative decomposition which indicates that the indicescan be viewed as the product of the poverty rate, the average poverty gapratio of the poor and one plus the Gini index of the poverty gap ratios.

The S and SST indices can be computed easily as follows:

The Sindex

!=

povertyrate

! povertygap

!0B@ 1 + Gini indexof poverty gapsof the poor

1CAand

The SSTindex

!=

povertyrate

! povertygap

!0B@ 1 + Gini indexof poverty gapsof the population

1CA :

This multiplicative decomposition gives the two indices much more straight-forward interpretation of so-called poverty intensity, allows the indices to becomputed much more easily via the commonly known poverty measures (thepoverty rate and the average poverty gap ratio) and inequality measures (theGini index of the poverty gap ratios), and also permits the indices to have theGini-index-like geometric interpretation. These are vital to policy analystsand public debate.

The multiplicative decomposition of the poverty indices is similar to thethree `I's of poverty curves summarized by Jenkins and Lambert (1997)|incidence (poverty rate), intensity (average poverty gap), and inequality di-mensions of aggregate poverty (inequality measure). This decomposition also

30

relates to the Foster, Greer and Thorbecke (1984) decomposable poverty mea-sure with a parameter � [the FGT(�)] in the following sense. The povertyrate is the FGT(0) and the average poverty gap ratio of the population, whichis the product of the poverty rate and the average poverty gap ratio of thepoor, is FGT(1).

In addition, this paper also shows that the SST index is a linear trans-formation of the S index and vice versa. This result is due to the fact thatthe poor and non-poor subpopulations are not overlapping. This addition toShorrocks (1995) gives a simple link between the SST and S indices.

The practical implication of the multiplicative decomposition is that theS and SST indices can be linearized by taking logarithmic transformationso that the percentage change in the S and SST indices are additively de-composable. This result is very useful for empirical comparisons across timeacross regions and counties.

The three decomposed components of the S and SST indices can havefurther subgroup decomposition. One of the critical issue is to show that theGini index of poverty gap ratios also allows the decomposition into between-group Gini index, within-group Gini indices and residual term for discreteincome distributions. This paper presents the subgroup decomposition of theGini index for discrete income distributions, which is a minor addition to thecase for continuous income distributions explained in Lambert and Aronson(1993).

31

References

[1] Anand, S. (1983). Inequality and Poverty in Malaysia: Measurementand Decomposition, Oxford University Press, New York.

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Appendix A

A Numerical Example

To make sense of the theoretical work on the poverty index decomposi-tions, we provide a numerical example in this Appendix.

Suppose that he subgroup indicators (F = female; M= male) and incomesof the income distribution is given in Table 1. In this income distribution,there are ten individual incomes from two subgroups. If the poverty line isdrawn at half of the median income, the poverty line is 7 and the poverty gapratios can be computed from the incomes (e.g., the poorest individual has apoverty gap ratio of 0.5714). The poverty rate for this income distribution is40 % and the average poverty gap ratio of the poor is 0.3571. While the Giniindex of poverty gap ratios of the poor (ignoring the zero poverty gap ratios)is 0.2500, the Gini index of poverty gap ratios of the population (taking intoconsideration the zeros) is 0.7000. As this paper shows, the S index (0.1786)is the product of the poverty rate (0.4000), the average poverty gap ratios(0.3571) and one plus the Gini index of the poverty gap ratios of the poor (1+ 0.2500 = 1.2500). The SST index (0.2429) is the product of the povertyrate (0.4000), the average poverty gap ratios (0.3571) and one plus the Giniindex of the poverty gap ratios of the population (1 + 0.7000 = 1.7000).

The subgroup poverty measures contributing to the aggregate povertymeasures are somewhat di�erent. The poverty rate for subgroup F (M) is50% (33.33%). The poverty gap for subgroup F (M) is 0.5000 (0.2143).The poverty rate and average poverty gap ratio are higher in subgroup Fthan in subgroup M. When examining the inequality of poverty gap ratiosof the poor across subgroups, it is found that among the overall Gini indexof poverty gap ratios of the poor (0.2500), the between-group Gini index ishigh (0.2000), the weighted sum of subgroup Gini indices is low (0.050) [theGini index for subgroup F (M) is 0.0714 (0.1667)], and the residual termcapturing the overlapping e�ect is nil (0). When examining the inequalityof poverty gap ratios of the population, it is found that among the overallGini index of poverty gap ratios of the population (0.7000), the between-group Gini index (0.3000) is only slightly higher than the weighted sum ofsubgroup Gini indices (0.2800) [the Gini index for subgroup F (M) is 0.5357(0.7222)], and the residual term capturing the overlapping e�ect is positive

36

Subgroup Income Poverty Gap Ratio

F 3 .5714F 4 .4286M 5 .2857M 6 .1429M 14 .0000F 14 .0000M 16 .0000M 18 .0000F 27 .0000M 55 .0000

Table 1: An Example: Income and Poverty Gap Ratio Distributions, PovertyLine = 7

(0.1200). These changes from the previous case are due to the zero povertygap ratios in the population.

Although the rate and gap are higher in subgroup F than in subgroup M,the inequality of poverty gap ratios is higher in subgroup M than in subgroupF. As shown in Table 1, in subgroup M the poorest individual's poverty gapratio (0.2857) almost doubles the next poorest individual's (0.1429) whilein subgroup F the poorest individual's poverty gap ratio (0.5714) is far lessthan doubling the next poorest individual's (0.4286).

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Appendix B

Appendix B.1 Proof of Lemma 1Based on De�nition 4, G(y) = eG(ey). Thus

G(y) = 1 �1

n2�(ey)nXi=1

(2i� 1)eyi: (89)

The above equation can be further rewritten as

G(y) = 1�1

n2�(ey) 2n

nXi=1

eyi � nXi=1

(2n� 2i+ 1)eyi!: (90)

After some manipulation, it can be written as

G(y) = �

1�

1

n2�(ey)nXi=1

(2n� 2i+ 1)eyi!

(91)

or

G(y) = �

1�

�G(ey)�(ey)

!: (92)

According to De�nition 4, we have17

G(y) =�G(ey): (93)

2

17Fei, Ranis, and Kuo (1978) also proved the result using a somewhat di�erent approach.But this lemma di�ers from Fei, Ranis, and Kuo (1978) in that it gives the Gini index aclear social welfare interpretation.

38

0

1

1

B

A

CC’

E

H

HI

H’ O’

aD

Figure 1: Geometric Interpretation of the Sen Index and the Sen-Shorrocks-Thon Index

39