Soc Choice Welfare (2002) 19: 265–280

2002 9 9 9 9

On measuring deprivation adjusted for group disparities S. Subramanian, Manabi Majumdar Madras Institute of Development Studies, 79, Second Main Road, Gandhinagar, Adyar, Chennai – 600 020, Tamilnadu, India (e-mail: [email protected]) Received: 1 February 1999/Accepted: 9 October 2000

Abstract. This paper is concerned to present a procedure for ‘adjusting’ a realvalued index of deprivation in such a way that the resulting measure is a summary statistic of both the average level of deprivation and the extent of inequality which obtains in its distribution.

1 Introduction In recent times – and most notably in succesive volumes of the UNDP’s Human Development Report (HDR) – a considerable amount of concern has been in evidence in the matter of deriving summary indicators of human development, capability failure, and related measures. Increasing attention has also been devoted to reckoning disparities in the distribution of achievements (or deprivations) across subgroups or individuals within a population. To this end, e¤orts have been made in the literature, correlatively, to adjust aggregate indices such that they reflect inter-group or inter-individual disparities in wellbeing achievements or failures. In terms of this approach, a commonly shared objective has been to employ the formal tool of ‘adjustment’, wherein the idea is to derive generalized indices of achievement or impoverishment which are ‘adjusted’ in such a way as to buttress information on ‘average’ performance with information on the inequality of its distribution. For examples of e¤orts in this direction, the reader is referred to – among others – Anand and Sen (1995); Jayaraj and Subramanian (1999); Majumdar (1999); and Hicks (1997). Particular thanks are due to Professor Prasanta Pattanaik who, in a related context, suggested the procedure for moving from the ‘two-groups’ case to the ‘many-groups’ case discussed in Sect. 2.5 of the paper. The paper has also benefited from the comments of two anonymous referees of the journal. The usual caveat applies.

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In this connection, particular mention must be made of the UNDP’s work (see the Human Development Reports from 1995 onwards) in compiling countrywide estimates of a Human Development Index (HDI) which is adjusted for gender-based disparities in the attainment of achievement: the resulting ‘adjusted’ index is called the Gender-adjusted Development Index (GDI). The UNDP’s GDI is based on a measure advanced by Anand and Sen (1995), of which a mirror-image (relating to deprivation rather than achievement) has been independently derived in Jayaraj and Subramanian (1999). In this paper, we advance an alternative methodology for deriving an ‘adjusted’ index of deprivation. Our approach to deriving an index of deprivation which is sensitized to inter-group di¤erentials in its distribution relies on standard results in welfare theory and inequality measurement; and we attempt to provide a simple axiomatic rationalization for our measure. Section 2 of the paper deals with methodological issues in deriving an ‘adjusted’ index of deprivation, in moreor-less general terms. Section 3 concludes. The paper also carries two Appendices, to which we have relegated some formalisms in order to avoid clutter in the main text.

2 An ‘adjusted’ measure of deprivation 2.1 Motivation In this section, we derive a procedure for obtaining an index of deprivation which reckons, in addition to the average level of deprivation, the extent of inequality in the distribution of group-specific deprivation levels in order to arrive at an overall estimate of (‘inequality-adjusted’) deprivation in the society. To this end, we first derive a measure of inequality in the distribution of deprivation across subgroups. We call this measure G
. It should be emphasized that in the context of the present paper, G
plays a wholly instrumental role, despite whatever independent interest it may have as an index of group-related dispersion: its place in the general scheme of things is only as a component of the generalised ‘inequality-adjusted’ deprivation index that is subsequently derived. Next, we construct – using a simple axiomatic framework – an ‘adjusted’ index of deprivation for the special case in which the population is partitioned into just two subgroups. By way of comparison, we then briefly review the AnandSen (1995) measure on which the UNDP’s Gender Adjusted Development Index is based. Thereafter, we extend our results for the ‘two-groups’ case to the ‘many-groups’ case: this presents us with the final ‘adjusted’ index we are after. Finally, by way of an interesting curiosum, we take note of the relationship between our ‘adjusted’ index of deprivation and certain well-known real-valued measures of income-poverty. 2.2 Measuring inequality in the distribution of deprivation across sub-groups Suppose we have K ðb 2Þ mutually exclusive and completely exhaustive groups indexed by i ¼ 1; . . . ; K. Let Di , which is a real number, be the deprivation level

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267

(or, in general, ‘illfare’ indicator) of the ith group, and let the groups be indexed in non-increasing order, viz. D1 b D2 b    b DK . We shall take it that the overall level of deprivation for the society as a whole, D, is decomposable, namely that D can be written as a population-share P K weighted average of the group-specific deprivation levels, so that D ¼ i¼1 ti Di , where ti is the population share of group i. It is convenient, at this stage, to define, for every i, Ti to be the cumulative proportion of the population who belong to groups whose deprivation levels are lower than or equal to the deprivation level of the ith group. Formally, for every i ¼ 1; . . . ; K: Ti :¼

K X

tj :

j¼i

Under one particular approach to inequality measurement, a measure of the extent of dispersion of group-specific deprivation levels around the mean level of deprivation can be written, in general form, as: E¼

K X

vi Di =D  1;

ð1Þ

i¼1

where the dispersion index E is rendered ‘equity-conscious’ by the requirement that, other things equal, a group with higher deprivation receives a larger weight, so that vi declines as i increases (recall that the groups have been indexed in non-increasing order of deprivation); further, the group-specific weights Pand, K are required to add up to unity, viz. i¼1 vi ¼ 1 (this is a simple normalization requirement). We shall now specify a particular set of weights fvi
g satisfying the desired properties discussed above. Let v^i :¼ ðK þ 1  iÞti þ Ti , i ¼ 1; . . . ; K; and let the normalized weight vi
for the ith group be given by vi
¼ v^i =

K X

v^j ;

i ¼ 1; . . . ; K:

ð2Þ

j¼1

It is easy to verify that K X j¼1

v^j

¼

K X

! ½ðK þ 1  jÞtj þ Tj 

¼ K þ 1;

ð3Þ

j¼1

so that, from (2) and (3), we now obtain: vi
¼ v^i =ðK þ 1Þ;

i ¼ 1; . . . ; K:

ð4Þ

Notice now that when ti ¼ t ð¼ 1=KÞ for all i ¼ 1; . . . ; K, as i increases, v^i (and therefore vi
) declines: for, under the assumed condition, ðK þ 1  iÞti ¼ ðK þ 1  iÞ=K and Ti ¼ ðK þ 1  iÞ=K for all i ¼ 1; . . . ; K, whence v^i ð¼ ðK þ 1  iÞti þ Ti Þ ¼ ð2=KÞðK þ 1  iÞ, which declines as i increases. The weight vi
on the ith group’s deprivation level is obtained additively from two components: the first ½ðK þ 1  iÞti  is the population-share weighted

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rank-order of the ith group; and the second ðTi Þ is simply the cumulative proportion of the population who belong to groups whose deprivation levels are not higher than the ith group’s deprivation level. We shall designate by G
the dispersion index obtained by employing the set of weights fvi
g on the right hand side of (1): G
¼ ½1=DðK þ 1Þ

K X

½ðK þ 1  iÞti þ Ti Di  1:

ð5Þ

i¼1

Because of the rank-order weighting formula employed in (5), one can easily discern that G
bears a close family resemblance to the familiar Gini coe‰cient of inequality1, so widely invoked in the literature on the measurement of income-inequality. Indeed, if the grouping we resorted to were an ‘atomistic’ one, that is, if each person in a community of n individuals were to be regarded as constituting a group by herself or himself, then it would be the case that ti ¼ 1=n and Ti ¼ 1  ði  1Þ=n for all i ¼ 1; . . . ; n; so that, for such an ‘atomistic’ grouping, the P value of G
– in view of (5) – would be given by the expression n ½2=nðn þ 1ÞD i¼1 ðn þ 1  iÞDi  1, which readers will be quick to recognize as corresponding to one ‘standard’, popular way of writing the Gini coe‰cient of interpersonal inequality (see, for example, Sen (1973, Eq. 2.8.3), or Anand (1983, p. 314)). The place of G
in the general scheme of things will become clearer as we proceed. For the moment, we consider how we may derive an ‘adjusted’ index of deprivation for the special case in which K ¼ 2. The following sub-section draws on Majumdar (1999). 2.3 An ‘adjusted’ deprivation index for the two-groups case Consider the special case in which the number of groups, K, is 2. Let the groups be indexed 1 and 2 respectively; and let D1 and D2 be the deprivation levels of groups 1 and 2 respectively, with the groups indexed in such a way that D1 b D2 . We shall take it that the deprivation index for the society as a whole, D1; 2 , is decomposable, so that the latter can be written as D1; 2 ¼ t1 D1 þ t2 D2 where ti ði ¼ 1; 2Þ is the population share of group i. Let D1; 2 be an ‘adjusted’ index of deprivation for the society as a whole, where the ‘adjustment’ assumes the form of supplementing information on the average level of deprivation, D1; 2 , with information on the extent of inequality in the deprivation levels of the two groups. We may wish the index D1; 2 to be governed by certain ‘reasonable’ properties, which we discuss and state below in the form of a set of axioms. To begin with, we could take ðD1  D2 Þ to be a simple and plausible measure of the extent of inequality in group-specific deprivation levels. It is reasonable to require that the comprehensive index of deprivation, D1; 2 , be a func1 Given the a‰nity between G
and the Gini inequality measure, it might be a matter of independent interest to investigate the conditions under which alternative distributions of group deprivation levels can be compared in terms of the criterion of Lorenzdominance.

Deprivation and group disparities

269

tion of both the mean level of deprivation, D1; 2 , and the inequality measure ðD1  D2 Þ. We could specialize the functional dependence of D1; 2 on D1; 2 and ðD1  D2 Þ to a linear form: this is essentially arbitrary, but it does have the merit of simplicity. Stated formally, we have: Axiom 1 (Linearity in mean and dispersion). The ‘adjusted’ index of deprivation D1; 2 is a linear function of D1; 2 and ðD1  D2 Þ: D1; 2 ¼ aD1; 2 þ bðD1  D2 Þ;

ð6Þ

where a and b are real numbers. Second, it seems reasonable to require that when there is no inequality in the distribution of deprivation across the two subgroups, the ‘adjusted’ index should coincide with the mean deprivation level. This is a simple normalization condition, and can be stated formally as follows: Axiom 2 (Normalization). When D1 ¼ D2 , D1; 2 ¼ D1; 2 . Third, we borrow a property of ‘subgroup consistency’ from the poverty measurement literature (see Foster and Shorrocks, 1991): this property demands that, other things equal, overall deprivation should respond positively to an increase in the deprivation of any subgroup. Stated in a diluted form, we have: Axiom 3 (Weak subgroup consistency). Ceteris paribus, if deprivation in any subgroup increases, the value of the ‘adjusted’ deprivation index should not decline, viz. qD1; 2 =qDi b 0, i ¼ 1; 2. Fourth, we could bring into the picture a measure of ‘level sensitivity’ by requiring that when the subgroup population shares are the same, a given increase in subgroup deprivation should cause a greater increase in ‘adjusted’ deprivation the more deprived the subgroup is. Stated in a weak form, this property is captured in Axiom 4 (Non-decreasing non-negative responsiveness to subgroup deprivation). When t1 ¼ t2 ð¼ 1=2Þ, qD1; 2 =qD1 b qD1; 2 =qD2 b 0. Consider now a specific ‘adjusted’ index of deprivation, parametrized by the quantity d which ranges between zero and unity, and is given by D1; 2 ðdÞ ¼ dD1 þ ð1  dÞD1; 2 ;

d A ½0; 1:

ð7Þ

The following proposition is now true. Proposition 1. The only class of ‘adjusted’ deprivation indices satisfying Axioms 1–4 is that given by Eq. (7). Proof. Appendix 1. Notice that the parameter d in (7) is in the nature of an index of ‘betweengroup inequality aversion’: as d increases, aversion to inter-group disparity increases. From (7) it is clear that D1; 2 ð0Þ ¼ D1; 2 ;

ð8Þ

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S. Subramanian, M. Majumdar

and D1; 2 ð1Þ ¼ D1 :

ð9Þ

d ¼ 0 corresponds to sort of Benthamite (‘average’) utilitarian rule (or rather, in the present context, a rule of average ‘illfare’ or deprivation): there is no concern here at all with how the total deprivation is distributed across the subgroups. By contrast, when d ¼ 1, we have a sort of Rawlsian (‘maximax’) rule: ‘adjusted’ deprivation for the society as a whole is identified simply with the deprivation level of the worse-o¤ of the two groups. These are the polar cases. The Benthamite rule has been often criticized for its complete insensitivity to distributional considerations; and the Rawlsian rule has also attracted censure for its ‘extreme’ insistence on upholding a sort of ‘dictatorship of the weakest’. We note in passing that in the special, two-groups case under review, a ‘safe’ choice of d might be to peg it at a value of one-half: figuratively speaking, one’s aversion to between-group inequality is then located on the mid-point of a straight line drawn from Bentham to Rawls. 2.4 The Anand-Sen/UNDP approach It is useful to consider the ‘adjusted’ index for the two-groups case we have derived in the previous sub-section in the light of the ‘Gender Adjusted Development Index’ employed by the UNDP’s Human Development Reports from 1995 onward, and which is based on an ‘adjusted’ measure proposed by Anand and Sen (1995). While Anand and Sen conduct their analysis in the space of achievements, our own approach here is related to the space of deprivations. For purposes of consistent treatment, we shall, in what follows, adapt the AnandSen formulation to reflect measurement in the space of deprivations; and in the process, we shall also take some (harmless) notational liberties with that formulation. As before, we shall consider a situation in which the number of groups, K, is 2. (Typically, where the concern is with gender di¤erences, as in the case of the UNDP’s GDI index, the two groups would refer to the populations of females and males respectively.) Let D1 and D2 be the values of the deprivation indices for groups 1 and 2 respectively, with D1 b D2 . The deprivation index for the society as a whole, D1; 2 , is simply the population-share weighted average of the group-specific deprivation levels: D1; 2 ¼ t1 D1 þ t2 D2 , where ti ði ¼ 1; 2Þ is the population share of group i. As before Ti will refer to the proportion of the population with deprivation levels not exceeding that of the ith group. Anand and Sen begin with a ‘social valuation function’, which is some transform of any given level of deprivation to its ‘social value’. The specific type of valuation function employed by Anand and Sen is the ‘constant elasticity marginal valuation’ type, which we shall denote by VA , where the subscript A is to remind us of its use by Atkinson in his (1970) work on the welfare basis of inequality measurement. Concretely, for each i ¼ 1; 2, we have: VAi ¼ ð1=lÞDil ;

l b 1:

ð10Þ

The parameter l can be interpreted as an indicator of ‘between group inequal-

Deprivation and group disparities

271

ity aversion’, with such aversion being an increasing function of l. Only values of l b 1 are considered, so as to ensure that the social valuation function is ‘equity-conscious’. Let WA be an ‘Atkinson-type’ ‘social illfare function’ (SIF), which is simply a measure of society-wide illfare arising from aggregating the (population share weighted) social valuations of all group deprivations: WA ¼

K X

ti VAi ¼ ð1=lÞðt1 D1l þ t2 D2l Þ:

ð11Þ

i¼1

Next, let DA be a level of deprivation such that if both groups 1 and 2 share this degree of deprivation, then the resulting social illfare is the same as that which obtains under the given distribution ðD1 ; D2 Þ. DA is the ‘equally distributed equivalent deprivation’ (or ede deprivation, for short); and, in view of (10) and (11), DA will be determined through the equation ð1=lÞDAl ¼ ð1=lÞðt1 D1l þ t2 D2l Þ;

ð12Þ

from which one obtains – for the ‘Atkinson-type’ SIF – the following expression for the ede deprivation: DA ¼ ðt1 D1l þ t2 D2l Þ 1=l :

ð13Þ

DA is, precisely, the Anand-Sen measure of ‘adjusted deprivation’. Now, if we relate inequality to the increment in social illfare attributable to its presence, then a natural measure of such inequality – call it IA – would be given by: IA ¼ ðDA  D1; 2 Þ=D1; 2 ¼ ½ðt1 D1l þ t2 D2l Þ=D1; 2   1:

ð14Þ

In (14), the extent of between-group disparity is expressed as the proportionate di¤erence between the ede deprivation and the mean deprivation. Notice that it is always the case that DA b D1; 2 : the underlying ‘equity conscious’ SIF exhibits the property of accepting a higher level of average deprivation in exchange for a more equitable distribution of that deprivation across sub-groups. From (14), it is immediately clear that D1; 2 ð1 þ IA Þ :¼ DA : the adjusted measure of deprivation is just the average level of deprivation enhanced by a factor representing the extent of intergroup inequality in the distribution of deprivations. Are there any correspondences between the Anand-Sen measure and our own ‘adjusted’ measure D1; 2 , captured in Eq. (7)? The answer, straightforwardly, is ‘yes’. To see the nature of the correspondence, consider an alternative form of the social valuation function, VB , such that, for each i ¼ 1; 2: VBi ¼ ½ðK  1  diÞ þ dðTi =ti ÞDi ;

d A ½0; 1:

ð15Þ

Notice from (15) that VBi transforms any given level of deprivation Di into its ‘social value’ through a linear operator: ‘social value’ is proportional to the level of deprivation, and the proportionality factor incorporates a Borda rankorder weight – hence the subscript B in VB . Like l in (10), d in (15) is a parameter which can be interpreted as an indicator of between-group disparity aversion, with aversion being an increasing function of d.

272

S. Subramanian, M. Majumdar

Next, given (15), recalling the definition of the ‘equally distributed equivalent deprivation’, and noting (i) that K ¼ 2; (ii) that t1 þ t2 ¼ 1; and (iii) that t1 D1 þ t2 D2 ¼ D1; 2 , it is a routine matter to verify that the ede deprivation DB corresponding to the SIF WB is given by: DB ¼ dD1 þ ð1  dÞD1; 2 :

ð16Þ

DB is, of course, our own ‘adjusted’ measure of deprivation. Precisely analogously to the way in which we have derived the inequality index IA in (14), we can now define the between-group disparity measure IB corresponding to the SIF WB : IB ¼ ðDB  D1; 2 Þ=D1; 2 ¼ ½fdD1 þ ð1  dÞD1; 2 g=D1; 2   1:

ð17Þ

Once more, we can see that D1; 2 ð1 þ IB Þ :¼ DB : as in the case of the Anand-Sen measure, our own ‘adjusted’ measure of deprivation is just the mean enhanced by a between-group inequality coe‰cient. The di¤erence resides in the precise inequality coe‰cients employed. Briefly, both the Anand-Sen and our own adjusted measures can be seen to be essentially the equally distributed equivalent deprivation levels corresponding to alternative social illfare functions founded on alternative social valuation functions of deprivation: the AnandSen measure exploits an ‘Atkinson-type’ approach, while our measure exploits a ‘Borda-type’ approach. Both the correspondence and the di¤erence between the two measures should now be apparent. It may be added that there has also been some deviation in method: while we could have derived our index by beginning with the postulation of an arbitrary social illfare function, we have chosen instead to provide an axiomatic characterisation of the index. What remains to note is that in (13), for l ¼ 1, DA becomes just D1; 2 while for l ! y, DA ! D1 2; precisely analogously, in (16), for d ¼ 0, DB ¼ D1; 2 while for d ¼ 1, DB ¼ D1 . These are, respectively, the ‘Benthamite’ and ‘Rawlsian’ outcomes discussed earlier. 2.5 From two groups to many groups: The general ‘adjusted’ index derived We have just seen in the two preceding sub-sections that in the special case where we have only two groups, j and k, with Dj b Dk , it is possible to rationalize an ‘adjusted’ deprivation index – Dj; k – which satisfies certain seemingly acceptable properties and is given by Dj; k ðdÞ ¼ dDj þ ð1  dÞDj; k ;

d A ½0; 1;

where Dj; k is the population-share weighted average of Dj and Dk ½Dj; k ¼ ðtj Dj þ tk Dk Þ=ðtj þ tk Þ and d is in the nature of an index of ‘between-group 2 The parameter l has been employed in social cost benefit analyses of projects, as a means of according di¤erential distributional weights to consumption benefits from a project accruing to the poor and the non-poor: for example, the parameter ‘n’ employed by Squire and van der Tak (1975) is essentially the parameter ‘l’ discussed above, with suitable contextual adaptation.

Deprivation and group disparities

273

disparity aversion’, with aversion being an increasing function of d. The question arises: given the ‘adjusted’ deprivation index Dj; k for the ‘two-groups’ case, is it possible to seek an extension for the ‘many-groups’ (i.e. groups exceeding two in number) case? To this end, consider the following. Suppose, in general, that we have K ðb 2Þ groups. Let Di be the deprivation level of the ith group, with the groups indexed in non-increasing order of deprivation ðDi b Diþ1 ; i ¼ 1; . . . ; K  1Þ. We shall assume that the overall (‘unadjusted’) deprivation measure for the society as a whole – call it D – is PK decomposable, so that D ¼ i¼1 ti Di , where ti is the population-share of group i ði ¼ 1; . . . ; KÞ. Now, for every pair of groups f j; kg, where j ¼ 1; . . . ; K  1 and k > j, we have the ‘adjusted’ deprivation index Dj; k ðdÞ ¼ dDj þ ð1  dÞDj; k ; d A ½0; 1. An overall society-wide ‘adjusted’ index – call it D
ðdÞ – can now be obtained as a weighted sum of all the pairwise adjusted indices Dj; k ðdÞ: D
ðdÞ ¼

K1 X K X

wj; k Dj; k ðdÞ;

ð18Þ

j¼1 k¼jþ1

where wj; k is the weight attached to the adjusted index for the pair of groups f j; kg. With normalization in mind, we could require the pair-wise weights to add up to unity: K 1 X K X

wj; k ¼ 1:

ð19Þ

j¼1 k¼jþ1

A simple and natural weighting scheme which suggests itself is weighting according to population-shares. Specifically, define ^ j; k :¼ tj þ tk ; w

j ¼ 1; . . . ; K  1; k > j: ð20Þ P K1 P K ^ a; b ¼ K  1; and we could It is a simple matter to verify that a¼1 b¼aþ1 w specify the following set of normalized weights fwj;
k g, given by ^ j; k = wj;
k ¼ w

K 1 X K X

^ a; b ¼ ðtj þ tk Þ=ðK  1Þ; w

j ¼ 1; . . . ; K; k > j:

ð21Þ

a¼1 b¼aþ1

Employing the set of weights fwj;
k g presented in (21) on the right hand side of (18), we now obtain an aggregate, society-wide index of ‘adjusted’ deprivation: D
ðdÞ ¼ ½1=ðK  1Þ

K 1 X K X

ðtj þ tk ÞDj; k ðdÞ:

ð22Þ

j¼1 k¼jþ1

Are there more transparent, readily interpretable and ‘usable’ expressions available for D
ðdÞ? This is the content of our next proposition: Proposition 2. The aggregate society-wide index of ‘adjusted’ deprivation P K1 P K D
ðdÞ ¼ ½1=ðK  1Þ j¼1 k¼jþ1 ðtj þ tk ÞDj; k ðdÞ can be written, equivalently, in the following two forms: D
ðdÞ ¼ D½1 þ dfðK þ 1Þ=ðK  1ÞgG
;

ð23Þ

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S. Subramanian, M. Majumdar

PK where G
:¼ ½1=DðK þ 1Þ i¼1 ½ðK þ 1  iÞti þ Ti Di  1 is a measure of inequality in the distribution of deprivation across subgroups; and K X D
ðdÞ ¼ ½1=ðK  1Þ ½ðK  1  diÞti þ dTi Di : ð24Þ i¼1

Proof. Appendix 2. (23) indicates that the ‘adjusted’ deprivation index D
is simply the mean deprivation D enhanced by a factor incorporating the extent of inequality, G
, in the distribution of deprivation levels across the groups into which the population has been classified. This expression for D
(with suitable contextual adaptation) is strongly reminiscent of Sen’s (1976b) index of ‘real national income’, given by mð1  GÞ, where m is average income and G is the Gini coefficient of inequality in the interpersonal distribution of income. In standard welfare theory – see, for example, Brent (1986) – the social welfare function is a ‘two objective’ function, increasing in ‘e‰ciency’ (captured by the size of mean income) and declining in inequality. Precisely analogously, underlying the ‘adjusted’ deprivation index D
is a two objective ‘illfare’ function, increasing in both the average level of deprivation and the intergroup disparity of its distribution. Notice now from (23) that D
ð0Þ ¼ D:

ð25Þ

This is the ‘average Benthamite’ case we have discussed earlier. Further, D
ð1Þ ¼ D½1 þ fðK þ 1Þ=ðK  1ÞgG
:


ð26Þ

For ‘large’ values of K, D ð1Þ can be approximated to Dð1 þ G Þ, where G
, as we have seen in Sect. 2.2, is the analogue of the Gini coe‰cient of interpersonal inequality. In connection with the Gini coe‰cient, it is worth recalling Sen’s (1973, p. 33) observation: ‘‘Suppose the welfare level of any pair of persons is equated to the welfare level of the worse-o¤ person of the two. Then if the total welfare of the group is identified with the sum of the welfare levels of all pairs, we get the welfare function underlying the Gini coe‰cient’’. He goes on to point out that ‘‘. . . there is an analogy here with Rawls’ (1971) ‘maximin’ criterion of justice but applied pairwise’’. We have already noted in Sect. 2.3 that the Rawlsian criterion comes into play in determining the value of the pair-wise ‘adjusted deprivation index’ Dj; k ðdÞ when d ¼ 1. The obvious implications of these observations for an interpretation of D
ð1Þ, in terms of the ‘Rawlsian connection’, need no further labouring. (24) funishes an alternative, convenient way in which the index D
ðdÞ can be written: namely, as a weighted sum of the group-specific deprivation levels. What is of interest is the precise weighting scheme one chooses to employ. At a general level, one could write an overall deprivation index – call it D – as a weighted sum of the deprivation levels of all groups: D¼

K X i¼1

ai Di ;




Deprivation and group disparities

275

where the ai are the relevant weights. The central problem revolves around the determination of the weights fai g. If the weights one chooses are the sub-group specific population shares (viz., ai ¼ ti for all i), then D collapses to D – just the average level of deprivation for the society. However, if one wishes D to reflect a concern for equity, one must choose the ai in such a way that a greater weight is attached to a group with higher deprivation. One way of doing this is to employ the Borda rule: if the ai are chosen as an appropriate mix of group-specific population shares and group-specific rank-orders, then D becomes the index D
ðdÞ, as presented in (24). Apart from being a particularly convenient expression to employ in computational exercises, (24) also makes it immediately apparent that qD
ðdÞ=qDi ¼ ½1=ðK  1Þ½ðK  1  diÞti þ dTi  b 0 Ei ¼ 1; . . . ; K when d b 0: that is, for d b 0, D
ðdÞ satisfies weak subgroup consistency. Further, consider the case in which ti ¼ t :¼ 1=K Ei ¼ 1; . . . ; K. Then, given (24), it can be verified that D
ðdÞ ¼ ½1=KðK  1Þ

K X ½K  1 þ dðK þ 1Þ  2diDi :

ð27Þ

i¼1

Consider now any two groups a and b such that Da > Db (so that a < b). In view of (27), it is easy to check that ðqD
ðdÞ=qDa Þ  ðqD
ðdÞ=qDb Þ ¼ ½2d=KðK  1Þðb  aÞ > 0

for d > 0; ð28Þ

since b > a. (28) indicates that when d is positive, D
ðdÞ satisfies a strengthened version of axiom 4 which we may call increasing non-negative responsiveness to subgroup deprivation: this is the property that when all groups are of the same size, a given increase in subgroup deprivation causes overall deprivation to rise by more for subgroup a than for subgroup b when a is more deprived than b. 2.6 A link with the income-poverty measurement literature: An aside By way of a minor, but possibly interesting, digression, we consider the relationship between the ‘adjusted’ deprivation index D
ðdÞ and certain well-known income-poverty measures. Let z be the poverty line (so that any person with income not exceeding z is poor), and let Q (respectively, Q) be the set of poor (respectively, nonpoor) persons in a population of n persons. Let x ¼ ðx1 ; . . . ; xi ; . . . ; xn Þ be an ordered n-vector of incomes, with xi a xiþ1 , i ¼ 1; . . . ; n  1. It is conventional in the income-poverty literature to measure the ith person’s deprivation level Di by i’s income-shortfall ratio ðz  xi Þ=z if i is poor; and his deprivation level is taken to be zero if i is nonpoor: Di ¼ ðz  xi Þ=z Ei A Q; ¼0

Ei A Q:

ð29Þ

Now it can be shown that for an atomistic grouping of the population in which each individual is seen as constituting a group by herself or himself, the value of the index D
ðdÞ when d is zero is given – employing (24) and (29) – by:

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S. Subramanian, M. Majumdar

D
ð0Þ ¼ ð1=nÞ

X ½ðz  xi Þ=z:

ð30Þ

iAQ

The right hand side of (30) is just the very well-known poverty index called the per capita income-gap ratio, R (it is, in fact, a member of the Foster et al. (1984) Pa family of poverty measures, and is realized for a ¼ 1). Further, if d ¼ 1, and if the population size is ‘large’ (viz. as n ! y), then under the atomistic grouping it can be shown – given (24) and (29) – that the ‘adjusted’ deprivation index D
ðdÞ is given by X ðn þ 1  iÞðz  xi Þ: ð31Þ D
ð1Þ ¼ ½2=nðn  1Þz iAQ

The right hand side of (23) is, as it happens, Thon’s (1979) poverty index, T, which is a variant of Sen’s (1976a) poverty index (and which – unlike the Sen measure – satisfies a strong version of the so-called ‘transfer’ axiom widely invoked in the poverty-measurement literature). In general, under the ‘atomistic’ grouping, for ‘large’ values of n, it can be shown that D
ðdÞ ¼ dT þ ð1  dÞR:

ð32Þ


That is, for the deprivation function presented in (24), D ðdÞ turns out – under the relevant specification of the individual deprivation indices Di , and under the appropriate restrictions on grouping and the size of the total population – to be a convex combination of two well-known income-poverty measures, T and R.

3 Concluding observations In this paper we have presented a fairly straightforward exercise aimed at sensitizing overall indices of deprivation or ‘illfare’ to the phenomenon of intergroup disparities in levels of deprivation. This could be seen as a contribution to a renewed academic interest in the study of well-being achievements and failures, with specific reference to the di¤erential intensity with which such achievements and failures are borne by di¤erent identifiable subgroups of the population when the latter is classified into socio-economically meaningful categories3 such as by caste, gender, ethnicity and sector of residence. A particularly fruitful domain of application of the formal results obtained in this paper is that constituted by the class of decomposable4 ‘illfare’ indices, an example of 3 It is important to stress that the partitioning of the population into subgroups must be e¤ected with some attention to relevance and coherence. Specifically, a division of the population according to gender has a certain ‘natural’ normative significance arising, among other things, from the fact that people usually have no volition in choosing their sex. But partitioning on the basis of place of residence or occupation, for example, could carry with it some biases arising from the phenomenon of self-selection. 4 Decomposability is a crucial requirement. For a properly exhaustive treatment of this property in measures of deprivation and disparity, the reader is referred to Anand (1983) and Shorrocks (1988).

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277

which is the UNDP’s (1997) ‘Human Poverty Index’. Indeed, in a companion piece (Majumdar and Subramanian 2001), we have sought an application of this paper’s measurement concerns to Indian data on capability failure in the dimensions of literacy, infant survival prospects and the ability to have a decent standard of living, after classifying the population (wherever possible) by gender, caste and sector of residence. Studies such as these often confirm that while measures of central tendency tell a sad enough story of human su¤ering, the picture becomes a good deal worse when one takes account also of groupspecific dispersions around the central measure. Appendix 1: Statement and Proof of Proposition 1 Proposition 1. The only class of ‘adjusted’ deprivation indices satisfying Axioms 1–4 is given by D1; 2 ðdÞ ¼ dD1 þ ð1  dÞD1; 2 ;

d A ½0; 1:

ðA1:1Þ

Proof. (a) Su‰ciency. In proving Proposition 1, we shall repeatedly make use of the following relations: t1 þ t2 ¼ 1;

ðA1:2Þ

and

t1 D1 þ t2 D2 ¼ D1; 2 :

ðA1:3Þ

From (A1.1), we have: D1; 2 ðdÞ ¼ D1; 2 þ dðD1  D1; 2 Þ ¼ D1; 2 þ d½D1  ðt1 D1 þ t2 D2 Þ

ðusing ðA1:3ÞÞ

¼ D1; 2 þ d½D1  ft1 D1 þ ð1  t1 ÞD2 g ðusing ðA1:2ÞÞ; or D1; 2 ðdÞ ¼ D1; 2 þ dð1  tÞðD1  D2 Þ;

ðA1:4Þ

viz. D1; 2 ðdÞ is a linear function of D1; 2 and ðD1  D2 Þ: so D1; 2 satisfies Axiom 1. Next, it is immediately apparent from (A1.4) that when D1 ¼ D2 , D1; 2 ðdÞ ¼ D1; 2 , so D1; 2 ðdÞ satisfies Axiom 2. Further, from (A1.1) and (A1.3), we have: D1; 2 ðdÞ ¼ dD1 þ ð1  dÞðt1 D1 þ t2 D2 Þ;

or

D1; 2 ðdÞ ¼ ½d þ ð1  dÞt1 D1 þ ð1  dÞt2 D2 ; whence qD1; 2 ðdÞ=qD1 ¼ d þ ð1  dÞt1 b0, since d; t1 A ½0; 1; and qD1; 2 ðdÞ=qD2 ¼ ð1  dÞt2 b 0, since, again, d; t2 A ½0; 1: that is D1; 2 ðdÞ satisfies Axiom 3. Finally, note from (A1.5) that when t1 ¼ t2 ¼ 1=2, qD1; 2 ðdÞ=qD1 ¼ ð1 þ dÞ=2 b qD1; 2 ðdÞ=qD2 ¼ ð1  dÞ=2 b 0: so D1; 2 ðdÞ also satisfies Axiom 4. (b) Necessity. We begin by noting that, in view of Axioms 1 and 2, we have: a ¼ 1;

ðA1:6Þ

whence, by virtue of (6) (in the text), D1; 2 ¼ D1; 2 þ bðD1  D2 Þ:

ðA1:7Þ

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S. Subramanian, M. Majumdar

Substituting for D1; 2 from (A1.3) into (A1.7), and collecting terms involving D1 and D2 yields: D1; 2 ¼ ðt1 þ bÞD1 þ ðt2  bÞD2 :

ðA1:8Þ

By Axiom 3 (Weak Subgroup Consistency), we would require that t1 þ b b 0 and t2  b b 0; the latter condition can be stated equivalently as ðA1:9Þ

b a t2 :

Axiom 4 (Non-decreasing Non-negative Responsiveness to Subgroup Deprivation) would now require that, when t1 ¼ t2 ¼ 1=2, ðqD1; 2 =qD1 ¼Þ 1=2 þ b b 1=2  b ð¼ qD1; 2 =qD2 Þ, that is, ðA1:10Þ

b b 0:

From (A1.9) and (A1.10) we have: b A ½0; t2 , viz. b can be written as some nonnegative fraction d of t2 : b ¼ dt2 ;

d A ½0; 1:

ðA1:11Þ

Substituting for b from (A1.11) into (A1.7) yields: D1; 2 ðdÞ ¼ D1; 2 þ dt2 ðD1  D2 Þ ¼ ðt1 D1 þ t2 D2 Þ þ ðdt2 D1  dt2 D2 Þ ðusing A1:3Þ ¼ ðt1 þ dt2 ÞD1 þ ð1  dÞt2 D2

ðcollecting termsÞ

¼ ½t1 þ dð1  t1 ÞD1 þ ð1  dÞð1  t1 ÞD2

ðusing A1:2Þ

¼ ½t1 ð1  dÞ þ dD1 þ ð1  dÞð1  t1 ÞD2 ¼ dD1 þ ð1  dÞ½t1 D1 þ ð1  t1 ÞD2  ¼ dD1 þ ð1  dÞD1; 2

ðusing A1:2 and A1:3Þ; as desired:

This completes the proof of Proposition 1. (Q.E.D.) Appendix 2: Statement and Proof of Proposition 2 Proposition 2. The aggregate, society-wide index of ‘adjusted’ deprivation D
ðdÞ ¼ ½1=ðK  1Þ

K 1 X K X

ðtj þ tk ÞDj; k ðdÞ

ðA2:1Þ

j¼1 k¼jþ1

can be written, equivalently, in the following two forms: ðA2:2Þ D
ðdÞ ¼ D½1 þ dfðK þ 1Þ=ðK  1ÞgG
; PK
where G :¼ ½1=DðK þ 1Þ i¼1 ½ðK þ 1  iÞti þ Ti Di  1 is a measure of inequality in the distribution of deprivation across subgroups; and D
ðdÞ ¼ ½1=ðK  1Þ

K X i¼1

½ðK  1  diÞti þ dTi Di :

ðA2:3Þ

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279

Proof. First, substituting for Dj; k ðdÞ from (18) (in the text) into (A2.1) yields: D
ðdÞ ¼ ½1=ðK  1Þ

K 1 X K X

ðtj þ tk Þ½dDj þ ð1  dÞDj; k :

ðA2:4Þ

j¼1 k¼jþ1

Let us now perform two separate summations: first, a summation of all the terms which have d as a multiplicative constant; and second, a summation of all the terms which have ð1  dÞ as a multiplicative constant. Let us designate the two summations SðdÞ and Sð1  dÞ respectively. Then, through a sequence of simple if tedious steps (which have been omitted here but are available on request from the authors), it can be verified that " # K K X X SðdÞ ¼ ½d=ðK  1Þ ðK þ 1  iÞti Di þ Ti Di  2D ; ðA2:5Þ i¼1

i¼1

and Sð1  dÞ ¼ ð1  dÞD:

ðA2:6Þ


Harking back now to the expression for G presented in the statement of Proposition 2, some routine manipulation will confirm that K X

½ðK þ 1  iÞti þ Ti Di ¼ Dð1 þ G
ÞðK þ 1Þ:

i¼1

Substituting for

PK

i¼1 ½ðK

ðA2:7Þ

þ 1  iÞti þ Ti Di frpm (A2.7) into (A2.5) yields:

SðdÞ ¼ ½d=ðK  1Þ½Dð1 þ G
ÞðK þ 1Þ  2D:

ðA2:8Þ




Recalling that D ðdÞ ¼ SðdÞ þ Sð1  dÞ, from (A2.6) and (A2.8), it requires some very elementary manipulation to verify that D
ðdÞ ¼ D½1 þ dfðK þ 1Þ= ðK  1ÞgG
, which, precisely, is (A2.2), as desired. Further, given (A2.4), (A2.5) and (A2.6), we have: " # K X
fðK þ 1  iÞti þ Ti gDi  2D þ ð1  dÞD; D ðdÞ ¼ ½d=ðK  1Þ i¼1

which, again through a series of steps that are available on request, can be shown to be amenable to being written, equivalently, as: D
ðdÞ ¼ ½1=ðK  1Þ  P K i¼1 ½ðK  1  diÞti þ dTi Di : this, precisely, is (A2.3), as desired. This completes the proof of the proposition. (Q.E.D.)

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