A particular scale-invariant index of poverty is subjected to careful analysis. This leads to a new perspective, not seen before, on the family of subgroup-consistent and scale-invariant poverty indices. Parametric families of new poverty indices are presented which offer the analyst a degree of flexibility in the choice of transfer sensitivity and distribution sensitivity which has not been available before now.Poverty measurement, Distributional properties.

Kristof Bosmans

Lucio Esposito

Peter Lambert

English

Research Papers in Economics

 
 
 
Working Paper Series 
 
 
 
 
 
 
 
 
 
New perspectives on a more-or-less familiar 
poverty index, with extensions  
 
Kristof Bosmans 
Lucio Esposito 
Peter Lambert 
 
 
 
 
 
ECINEQ WP 2009 – 116   
 
 
ECINEQ 2009-116 
June 2009 
 
www.ecineq.org  
 
New perspectives on a more-or-less familiar 
poverty index, with extensions 
 
Kristof Bosmans 
Maastricht University 
 
Lucio Esposito 
Università degli Studi di Pavia  
 
Peter Lambert
* 
University of Oregon 
 
 
Abstract 
A particular scale-invariant index of poverty is subjected to careful analysis. This leads 
to a new perspective, not seen before, on the family of subgroup-consistent and scale-
invariant poverty indices. Parametric families of new poverty indices are presented 
which offer the analyst a degree of flexibility in the choice of transfer sensitivity and 
distribution sensitivity which has not been available before now.  
 
Keywords: Poverty measurement, Distributional properties. 
JEL Classification: D31, D63. 
 
 
 
                                                 
* Addresses of correspondence: Department of Economics, 1285 University of Oregon, Eugene, OR 97403-1285, 
USA. Email: plambert@uoregon.edu. Tel: +1-541-346-4670, Fax: +1-541-346-1243.   1 
 
 
 
1. Introduction 
In this paper we examine some properties and extensions of one particular poverty index 
which is already passingly familiar, if not well-known, to poverty analysts. Our 
investigation leads to a new perspective on the entire family of subgroup-consistent and 
scale-invariant poverty indices, and to a clarification of the different roles of transfer 
sensitivity and distribution sensitivity in poverty measurement. New poverty indices are 
presented which offer the analyst a greater degree of flexibility in these regards than has 
heretofore been available.  
   2 
 
2. The more-or-less familiar poverty index 
In a society of n individuals, let      y = (y1,y2,...,yn)   ++
n  be the vector of incomes 
arranged in non-decreasing order, where yi is the income of the i
th individual, and let 
   z   ++ be an exogenously given poverty line. The individuals i for whom  yi < z  are the 
poor ones. Let yq be the largest poor income, so that the headcount ratio is 
 
H =
q
n
. The 
poverty index we are interested in, which we denote  P
  , takes the form:  
(1)   
   
P
 (y;z) =
1
n
z
yi i=1
q
  . 
The reason for our notation will become apparent later. 
   P
   may not appear to be very familiar, but it has familiar antecedents, involving 
the income gap ratio, the Atkinson (1970) inequality index, and what Chakravarty (1983, 
p. 81) describes as "a fairly natural translation of a relative inequality index of a censored 
income distribution into a relative poverty index". Let the generic Atkinson (1970) utility 
of income function be 
   
Ue(x) =
x
1 e
1  e
 for    0 < e   1 and    Ue(x) = lnx  for    e =1. First, 
inequality among the poor is 
   
Ie
P =1 
 
P
µ
P , where   
P is the equally distributed equivalent 
poor income defined by 
   
Ue  
P ( ) =
1
q
Ue(yi)
i=1
q
  , which, for e = 2, becomes 
   
1
 
P =
1
q
1
yi i=1
q
  =
P
 
zH
; and second, the income gap ratio is 
   
IGR =1 
µ
P
z
 
where
   
µ
P =
1
q
yi
i=1
q
  . Putting these two together, we see that  P
   can be expressed in terms 
of some well-known ingredients: 
(2)   
   
P
  =
H
(1  IGR)(1  Ie=2
P )
.   3 
   P
   is also strongly related to the second family of poverty indices suggested by 
Clark, Hemming and Ulph (1981). Defining welfare over basic incomes 
   bi = min{yi,z}, 1  i   n, and using the generic utility function  Ue  to do this, the authors 
define their poverty index as 
   
P e
CHU 2(y;z) =1 
 b
z
 where   b is the equally distributed 
equivalent basic income.
1 This yields 
   
P e
CHU 2(y;z) =1 
1
n
n  q +
yi
z
 
   
 
   
1 e
i=1
q
 
 
 
 
 
 
 
 
 
 
 
 
   
 
 
 
   
1
1 e
 
when    0 < e   1, where   b is the equally distributed equivalent basic income defined by 
   
Ue  b ( ) =
1
n
Ue(bi)
i=1
n
  . When e = 2, we arrive at another way to view our chosen poverty 
index  P
  : 
 (3)   
   
P
  =
1
1  P e=2
CHU 2(y;z)
  (1  H) . 
  Finally, we point out that the poverty contribution function inherent in  P
   actually 
forms a building-block for the entire class of subgroup-consistent and scale-invariant 
indices of poverty. These are the poverty functions      P(y;z): ++
n    ++    +  which 
evaluate aggregate poverty as a normalized sum of individual poverty contributions 
   p(yi;z): 
(4)   
   
P(y;z) =
1
n
p
i=1
n
  (yi;z), 
where    p(yi;z) = 0 if  yi   z  and    p(yi;z) > 0  otherwise,    p(yi;z) = p( yi; z) for all 
   > 0, and    p(yi;z)  is continuous and non-increasing in  yi  for    yi  (0,z). Our  P
   is in 
this form, as is the Watts (1968) index, for which 
   
p(yi,z) = ln
z
yi
 
   
 
    , and the ubiquitous 
                                                 
1 The poverty index of Chakravarty (1983) is also defined in terms of the equally distributed equivalent 
basic income, and reduces to    P
e
CHU 2 when the generic utility function U
e  is invoked.  
   4 
‘FGT index’ of Foster, Greer and Thorbecke (1984), call it  P   , for which 
   
p(yi;z) =  i ( )
 
, where 
 
 i =
z   yi
z
 is person i’s normalized poverty gap and 
        0.
2 
  The individual poverty contributions of all such indices are, in fact, transformations 
of the poverty contribution of our  P
  . For as Foster and Shorrocks (1991) show, for a 
poverty index satisfying (4) and the accompanying restrictions, 
(5)   
   
p(yi;z) =  
z
yi
 
   
 
     
for some continuous and non-decreasing function   . The ‘building block’ for this class 
of poverty indices is thus the individual poverty function 
z
yi
 of our  P
  . In the graphical 
illustration of Figure 1, 
   
 (
z
yi
) is plotted as a continuous line when    is the identity 
function, namely, for our index  P
  , and dotted lines show the pattern of values 
   
 (
z
yi
) 
takes for the Watts index and for  P   ,   = 0,1,2,3. The function    in (5) can be seen as a 
transformation apt to choose which property or properties inherent in the hyperbolic 
functional form 
 
z
yi
 are desired, and which not, in an index in the class defined by (4). 
 
 
z
y
 
   
 
    =
z
y
 enjoys upper unboundedness and discontinuity at any finite poverty line. The 
property of upper unboundedness is rejected for all members of the  P    class but is 
accommodated by the Watts index; poverty line discontinuity is retained by  P    for    = 0 
but not by the other members of the  P    class, nor by the Watts index.  
 [FIGURE 1 ABOUT HERE] 
                                                 
2 Although Foster, Greer and Thorbecke (1984) do not specify the nature of the parameter   , thus 
allowing it to be taken as any non-negative real number, values for the parameter are usually drawn from 
the set of nonnegative integers, as we shall assume in this paper.   5 
 
3. A new result, linking P  with the class of FGT indices 
The central insight of this paper comes by a very simple application of the theory of the 
convergent geometric series, and it links  P
   firmly with the  P    class. Recalling that 
 
 i =
z   yi
z
 is person i’s normalized poverty gap, i < q, and that 
 
 i ( )
 
 is that person’s 
contribution to the index  P   , observe that: 
(6)    
   
z
yi
= (1 
z   yi
z
)
 1 =
1
1   i
=  i
0 +  i
1 +  i
2 +  i
3 +   = lim
M  
 i ( )
 
 =0
M
  . 
The poverty contribution function inherent in  P
   is thus the infinite sum of those 
inherent in the FGT indices  P    across integer values of   . In turn, one can write  P
   
itself as an infinite sum of FGT indices: 
(7)   
   
P
  =
1
n
z
yi
=
i=1
q
  lim
M  
1
n
 i ( )
 
i=1
q
 
 =0
M
  = lim
M  
P  
 =0
M
  . 
There are several implications. 
  First, in view of what has gone before, one can now think of the FGT class as 
actually providing the building block class for all scale-invariant and subgroup-consistent 
poverty indices as in (4). Second, a discussion of transfer sensitivity is now in order. The 
poverty index in (4) does not necessarily even satisfy the transfer principle as written. For 
that,  P would have to be such that if      x = (x1,x2,...,xn)   ++
n  is obtained from y by an 
income transfer    from poor individual j to poor individual h, where 
 
yj < yh  and where 
   0 <  < z   yh , then    P(y;z) < P(x;z).
3 If    p(yi;z)  is (infinitely) differentiable in  yi  
for   yi  (0,z), this requires that 
   
 p
 yi
< 0 and 
   
 
2p
 yi
2 > 0; the subsequent degree of transfer 
sensitivity in poverty measurement theory (represented by the so-called Transfer 
Sensitivity Axiom) requires that 
   
 
3p
 yi
3 < 0 . The requirements for further degrees of transfer 
                                                 
3 This version of the transfer principle accords with the Minimal Transfer Axiom of Zheng (1997, p. 132).   6 
transitivity follow the scheme 
   
 
  p
 yi
  < 0 for    = (1,3,5,7,...) and 
   
 
  p
 yi
  > 0 for 
   = (2,4,6,8,...). Despite the lack of expressed transfer properties for the general scale-
invariant poverty index in (4), clearly our building block poverty index  P
   satisfies 
transfer sensitivity of all degrees – or, infinite transfer sensitivity as we might say, hence 
our notation – and in the FGT class,  P    possesses  (   1)
th  degree transfer sensitivity but 
no higher (hence for   = 0,1 there is no transfer sensitivity, for   = 2 the Pigou-Dalton 
condition is satisfied, for   = 3 the Transfer Sensitivity Axiom is met, etc.). If   1 > 2 , 
then 
   
 P  1  accommodates a higher level of transfer sensitivity than 
   
P  2 : there is no upper 
limit to the assignation of transfer sensitivity in the FGT class.
4 Here is a link, then, 
between scale invariance and transfer sensitivity for subgroup-consistent poverty indices, 
unremarked upon in previous literature. Clearly the transformation function    in (5) 
conditions the transfer sensitivity, or not, of an index in the class defined by (4), just as it 
conditions the upper (un)boundedness and poverty line (dis)continuity properties of the 
index. 
  Finally, it is instructive to examine the distribution sensitivity of  P
   and of  P    in 
the sense of Zheng (2000), who sets this notion on a sound footing à la Pratt-Arrow in the 
case of inequality aversion. Zheng’s measure, which for the poverty index in (4) takes the 
form 
   
sp(x;z) =  
 
2p
 y
2
 p
 y
  (x < z) and quantifies the sensitivity of    P(y;z) to income 
transfers in (relative) marginal terms, is 
   
s
 (y;z) =
2
y
 for  P
   and 
   
s (y;z) =
   1
z   y
 for P   . 
                                                 
4 Note, however, that in the limit, as       , no transfer property is met because the poverty index 
“approaches a Rawlsian measure which considers only the position of the poorest poor” (Foster et al. 1984, 
p. 763). The interest in higher degrees of transfer sensitivity was introduced into the poverty literature by 
Kakwani (1980). Elegant theory for strengthened versions of the Pigou-Dalton principle is fully expounded 
by Fishburn and Willig (1984), and essentially postulates that, “any combination of a socially desirable 
transfer [or series of transfers] with its inverse at uniformly higher levels of income will have positive 
social benefit” (p. 323). Zheng (1999) observes, in regard to the required continuity of higher and higher 
derivatives of the poverty contribution function, that “even if one is persuaded [in respect of the first 
derivative] ... it becomes harder to argue ... all the way up to the (k-2)
th degree” (p. 367).   7 
Thus  P
  has declining distribution sensitivity and  P    has increasing distribution 
sensitivity.
5  P
   is more distribution-sensitive at low (poor) income levels whilst  P    is 
more so at high (poor) income levels. The crossover is at 
   
y  =
2z
  +1
 which, of course, 
approaches zero as      . For the general poverty index in (4), with poverty 
contribution function 
   
p(y;z) =  
z
y
 
   
 
     as in (5), 
   
sp(y;z) = s
 (y;z)+
z
y
2 .
     z
y
 
   
 
   
    z
y
 
   
 
   
, 
whence its distribution sensitivity relative to that of the building block index  P
   is 
conditioned by the concavity/convexity property of   .  
 
4. Parametric extensions of P  
Here we introduce two extended families of poverty indices, we shall call them 
 
P  
   and 
   
P   , 
   where     and   >    are non-negative integers. Just as  P
  has been revealed to be 
the sum ad infinitum of the FGT indices  P    for integer values of   , these new indices 
are truncated sums of FGT indices, the one an infinite sum and the other a finite sum. 
These introduce enhanced flexibility relative to the FGT family, as will be seen. 
  The following is our formulation for the poverty contribution function 
 
p 
  of a 
new poverty index 
 
P  
  : 
(8) 
   
p 
 (yi;z) =  i ( )
  z
yi
=
 i ( )
 
1   i
=  i
  +  i
  +1 +  i
  +2 +  i
  +3 +   = lim
M  
 i ( )
 
 = 
M
  . 
This results from the product of an FGT poverty contribution function, that inherent in 
 
P   , and the poverty contribution function of our  P
  and bears comparison with Sen’s 
                                                 
5 As Zheng (2000, p. 123) points out, satisfaction of the transfer sensitivity axiom is a necessary but not a 
sufficient condition for a poverty index to exhibit diminishing distribution sensitivity. In Esposito and 
Lambert (2008), it is argued that transfer and distribution sensitivity in poverty measurement stem not from 
an egalitarian view, valuing equality per se, but from a prioritarian attitude.  
   8 
(1976) Axiom N, which states that when transfer properties are not of interest – i.e. in the 
case of a perfectly egalitarian distribution of incomes below the poverty line – aggregate 
poverty should be given by a multiplicative functional form between two intuitively 
appealing distribution insensitive indices such as the headcount index and the income gap 
ratio. The multiplicative form we have in (8), albeit between poverty contribution 
functions and not between full-blown poverty indices, may be thought of as playing an 
analogous role when poor incomes differ.
6 
  The aggregate poverty measure corresponding to (8) is an infinite sum of FGT 
indices, beginning with 
 
P   : 
(9)   
     
P  
 (y;z) = lim
M  
P  
 = 
M
  (y;z) = P   (y;z)+ P   +1(y;z)+ . 
The integer    can be interpreted as the degree of transfer sensitivity of the least transfer-
sensitive summand included in the measure.  
Finally, we introduce the further extended class of poverty indices  
(10)   
   
P   , 
  (y;z) =
1
n
p  , 
  (yi;z)
i=1
n
  =
1
n
p 
 (yi;z)  p 
 (yi;z) { }
i=1
n
   
in which     and   >    are both non-negative integers. This index is a finite sum of FGT 
indices:  
(11)   
     
P   , 
  (y;z) = P   (y;z)+ P   +1(y;z)+ + P   1(y;z) 
and it embraces both of the following two properties: 
i)   for    =   +1 the members of the  P    class are generated; 
ii)   for       we get the members of the 
 
P  
   class. 
                                                 
6 Sen’s own acknowledgment that “the multiplicative form chosen in Axiom N though simple, is arbitrary” 
(p. 227) is appeased by the work of Basu (1985), in which Axiom N is shown equivalent to three more 
elementary properties whose desirability is more readily appreciable. It is interesting to check the 
performance of  P
 
   with respect to Basu’s three axioms. While Axioms 2 and 3, which concern 
monotonicity in first differences of an assumed function of the headcount and income gap ratio, are met in 
the case of our product function, Axiom 1, concerning extreme values, is not because 
   
p
  (y
i;z)  [0,1]  
whilst 
z
y
 [1, ) .   9 
These properties of 
   
P   , 
   are illustrated in Table 1. 
[TABLE 1 ABOUT HERE]  
  The new classes of indices permit the important distinction between transfer 
sensitivity and distribution sensitivity to be made much clearer. The measure 
   
P   , 
   takes 
the transfer sensitivity of the final FGT measure in sum (11), while its distribution 
sensitivity is an average of the transfer sensitivities of all the FGTs in sum (11). By 
consequence, the measure allows to some extent to combine relatively high degrees of 
transfer sensitivity with relatively low degrees of distribution sensitivity or vice versa. 
The simple FGT indices do not allow this much flexibility, nor do any other poverty 
indices we know of.  
 
References    
Atkinson, A.B., 1970, On the measurement of inequality, Journal of Economic Theory 2, 
244-263. 
Basu, K., 1985, Poverty measurement: a decomposition of the normalization axiom, 
Econometrica 53, 1439-1444. 
Chakravarty, S.R., 1983, Ethically flexible measures of poverty, Canadian Journal of 
Economics 16, 74-85. 
Clark, S., R. Hemming and D. Ulph, 1981, On indices for the measurement of poverty, 
Economic Journal 91, 515-526. 
Esposito, L. and P.J. Lambert, 2008, Poverty measurement: prioritarianism and the fourth 
‘I’ of poverty, Mimeo, University of Oregon. 
Fishburn, P. C. and R.D. Willig, 1984, Transfer principles in income redistribution, 
Journal of Public Economics 25, 323-328. 
Foster, J. E., J. Greer and E. Thorbecke, 1984, A class of decomposable poverty 
measures, Econometrica 52, 761-766. 
Foster, J. E. and A. Shorrocks, 1991, Subgroup consistent poverty indices, Econometrica 
59, 687-709. 
Kakwani, N.C., 1980, On a class of poverty measures, Econometria 48, 437-446. 
Watts, H.W., 1968, An economic definition of poverty, in: D.P. Moynihan, ed., On 
Understanding Poverty (Basic Books, New York) 316-329. 
Zheng, B., 1997, Aggregate poverty measures, Journal of Economic Surveys 11, 123-
162. 
Zheng, B., 1999, On the power of poverty orderings, Social Choice and Welfare 16, 349-
371. 
Zheng, B., 2000, Minimum distribution-sensitivity, poverty aversion and poverty 
orderings, Journal of Economic Theory 95, 116-137.   10 
Figure 1: Poverty contribution functions of P
 , the Watts index  
and P    for    = 0,1,2,3 
 
  
 
 
 
yi                                               z 
p(yi;z) 
P
∞ 
Watts 
Index 
Pα=0 
Pα =1 
Pα =2 
Pα =3   11 
Table 1: Different combinations of parameters    and   ,    >  . 
 
 
    
   
 
0 
 
1 
 
2 
 
3 
 
4 
 
  
   
  0 
 
X 
 
   P  =0 
 
   P  =0+   P  =1 
 
   P  =0+   P  =1+   P  =2 
 
   P  =0+   P  =1+   P  =2+   P  =3 
 
   
P   =0
   
 
1 
 
X 
 
X 
 
   P  =1 
 
   P  =1+   P  =2 
 
   P  =1+   P  =2+   P  =3 
 
   
P   =1
   
   
X 
 
X 
 
X 
 
   P  =2 
 
   P  =2+   P  =3 
 
   
P   =2
   
 
3 
 
X 
 
X 
 
X 
 
X 
 
   P  =3 
 
   
P   =3
   
 
4 
 
X 
 
X 
 
X 
 
X 
 
X 
 
   
P   =4
   
 
  
 
 
X 
 
X 
 
X 
 
X 
 
X 
 
X 
 

New perspectives on a more-or-less familiar poverty index, with extensions

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