metadata: authors: 0: author: Datt,Gaurav
count: World
docna: 0: docna: Distribution-sensitive multidimensional
            poverty measures
docty: Policy Research Working Paper
abstracts: cdata!: This paper presents axiomatic arguments
            to make the case for distribution-sensitive multidimensional
            poverty measures. The commonly-used counting measures
            violate the strong transfer axiom which requires regressive
            transfers to be unambiguously poverty-increasing and they
            are also invariant to changes in the distribution of a given
            set of deprivations amongst the poor. The paper appeals to
            strong transfer as well as an additional cross-dimensional
            convexity property to offer axiomatic justification for
            distribution-sensitive multidimensional poverty measures.
            Given the nonlinear structure of these measures, it is al
            also shown how the problem of an exact dimensional
            decomposition can be solved using Shapley decomposition
            methods to assess dimensional contributions to poverty. An
            empirical illustration for India highlights distinctive
            features of the distribution-sensitive measures.
repnme: repnme: Distribution-sensitive multidimensional
            poverty measures
display_title: Distribution-sensitive
            multidimensional poverty measures
listing_relative_url: /research/2018/02/29610604/distribution-sensitive-multidimensional-poverty-measures
topicv3: Poverty m1326687 695,Nutrition m1326779 644,Poverty Measurement and Analysis m1326941 707,Education m1327048 470,Social Assistance m1327201 1466
publishtoextweb_dt: 2018-02-20T00:00:00Z
no_of_pages: 55
author_keys: 0: author_key: 428678
model_extractions: WPS8346 Policy Research Working Paper 8346 Distribution-Sensitive Multidimensional Poverty Measures _Gaurav Datt_ Development Economics Vice Presidency Strategy and Operations Team February 2018
Policy Research Working Paper 8346

# **Abstract**

This paper presents axiomatic arguments to make the case for distribution-sensitive multidimensional poverty measures. The commonly-used counting measures violate the strong transfer axiom which requires regressive transfers to be unambiguously poverty-increasing and they are also invariant to changes in the distribution of a given set of deprivations amongst the poor. The paper appeals to strong transfer as well as an additional cross-dimensional convexity property to offer axiomatic justification for distribution-sensitive multidimensional poverty measures. Given the nonlinear structure of these measures, it is al also shown how the problem of an exact dimensional decomposition can be solved using Shapley decomposition methods to assess dimensional contributions to poverty. An empirical illustration for India highlights distinctive features of the distribution-sensitive measures.

This paper is a product of the Strategy and Operations Team, Development Economics Vice Presidency. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The author may be contacted at gaurav.datt@monash.edu.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
# **Distribution‐sensitive multidimensional poverty measures**

## **Gaurav Datt**<sup></sup>

_Keywords_ : multidimensional poverty; poverty measurement; transfer axiom; crossdimensional convexity; Shapley decomposition; India _JEL_ classification: I3, I32, D63, O1  Gaurav Datt (corresponding author) is Associate Professor at the Department of Economics, and Deputy Director of the Centre for Development Economics and Sustainability, Monash Business School, at Monash University, Clayton VIC 3800, Australia; his email address is: gaurav.datt@monash.edu. The research undertaken for the paper was not supported by any grant from funding agencies in public, commercial, or not‐ for‐profit sectors. The author owes a special debt to Birendra Rai for many rounds of productive discussions. For useful comments and feedback, the author also thanks Aaron Nicholas, Ranjan Ray, Arijit Sen, Suman Seth, and participants at workshops in Melbourne, Australia (May 2013) and Prato, Italy (June 2016). The paper also benefited greatly from the anonymous referee and editorial comments received from the World Bank Economic Review. The author alone is responsible for the contents of the paper.
Recent estimates suggest that globally about 1.5 billion people live in multidimensional poverty (UNDP, 2015). Over the last decade or so, there has been a veritable explosion of literature on the measurement of multidimensional poverty. Following the initial Atkinson and Bourguignon (1982) paper and important later contributions of Tsui (2002), Bourguignon and Chakravarty (2003) and Alkire and Foster (2007, 2011a), the literature on both the theory and applications of multidimensional poverty measurement has grown rapidly.<sup>1</sup> With regards to applied work, measures within the “counting” approach (which focus on counting the number of dimensions in which people are deprived) have been influential, in particular the class of measures proposed by Alkire and Foster (2011a). One indicator of this influence is the publication of the Multidimensional Poverty Index (MPI) – which is a special case of the Alkire-Foster (hereafter AF) class of measures – for more than a 100 countries in the United Nations Development Program’s annual Human Development Report since 2010 (UNDP, 2010). This influence is only likely to grow with the recent adoption by the UN of the new set of Sustainable Development Goals (SDGs) which encompass a wide range of welfare dimensions many of which are also central to multidimensional poverty measurement.<sup>2</sup> This paper revisits the measurement of multidimensional poverty from an axiomatic perspective, taking the AF measures as a point of departure. At its core, it presents axiomatic arguments for a class of “distribution-sensitive” measures. The arguments centre on the specific formulation of the dominance axioms<sup>3</sup> , in particular, the transfer and the rearrangement axioms. The paper proposes stronger versions of these dominance axioms – a stronger version of the transfer axiom requiring regressive transfers (defined later) to be poverty-increasing, while also introducing a cross-dimensional convexity axiom requiring multiple deprivations to matter more than the sum of their parts, which is shown to be equivalent to a stronger version of the rearrangement axiom.

2
Two central ideas underlie this paper and the distribution-sensitive measures: (i) regressive transfers in any single dimension reduce social welfare, and (ii) multiple deprivations have compounding negative effects on individual and social welfare. The first has a long pedigree in the uni-dimensional poverty literature (see Sen, 1976, for instance) and there are no compelling reasons to eschew it in a multidimensional setting. The second is specific to the multidimensional context, and refers to the cumulative effects of multiple disadvantage on the severity of poverty, whereby the effect of disadvantage in one dimension is exacerbated by disadvantage in another. Multiple disadvantage can thus make it harder to graduate out of poverty. Recent evidence from Banerjee et al. (2015) on the success of multi-faceted interventions in tackling ultra-poverty lends support to the notion of multiple disadvantage as poverty traps. Hence, from this perspective, measures that disregard the extra burden of multiple deprivations could miss out on an important feature of the hardship experienced by those thus deprived.

The paper shows that poverty measures that are inclusive of _all_ deprivations (union-based) and allow for the extra burden of multiple deprivations are distribution-sensitive in two respects: first, they satisfy the strong transfer axiom, and second, they are sensitive to inequality in the distribution of deprivations. The positive contribution of the paper is in bringing these two central ideas together to make a case for this class of distributionsensitive multidimensional poverty measures which also satisfy a range of other desirable axioms proposed in the literature, without imposing any additional data requirements for practical applications. There are precursors to this class of measures in the literature in Bourguignon and Charavarty (2003), Chakravarty and D’Ambrosio (2006) and Alkire and Foster (2011a) – in that the particular mathematical functional form for the distributionsensitive measures has been anticipated in these papers, though mainly in passing without exploring the special value of such measures in incorporating distributional considerations as discussed in this paper.

3
A property of multidimensional poverty measures that has been found to be attractive from a practical and policy perspective is that of dimensional decomposition (sometimes also referred to as factor decomposition), i.e. the ability to decompose the poverty measure into mutually exclusive and exhaustive contributions of each of the included dimensions. The AF measures, for instance, allow for a simple linear decomposition by dimension once the poor have been identified. However, this facility of linear decomposition is no longer available for distribution-sensitive poverty measures when they allow for compounding effects of deprivations across multiple dimensions. Another contribution of the paper is to show how an exact dimensional decomposition can be implemented in this case drawing upon the Shapley decomposition methods proposed by Shorrocks (2013). The Shapley decompositions are quite general and can be readily applied to any multidimensional poverty measure, including the AF measures. They also have the advantage of fully accounting for the contributions of dimensions both through the identification of the poor and the aggregation of their individual poverty levels.

The paper presents an application for India to illustrate key properties of distributionsensitive poverty measures and how they compare with the AF class of measures. Besides presenting results on the implementation of Shapley dimensional decompositions, the application highlights several ways in which the use of distribution-sensitive measures could matter for measurement and analysis. First, it illustrates how much of multidimensional poverty could be potentially missed by measures that are not union-based and rely on a minimum number (or proportion) of deprivations to identify the poor. Second, it shows how the use of distribution-sensitive poverty measures can influence the profile of the poor and hence subgroup contributions to poverty. Third, it offers an example of how a maximal concentration of a given set of deprivations, while reflected in higher values of distributionsensitive measures, can be consistent with a _fall_ in counting measures. Fourth, using a “mean-inequality” type of decomposition, it shows how the contribution of inequality in the distribution of deprivations to distribution-sensitive measures could be assessed.

4
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M(a,k;y) M(a,k; y) M(0,k;y)
<!-- Start of picture text --> az=0 a>0 a2>1<br>M(a,k;y)<br>Yirj’ < Vint j! y' € {1,2,...d} i’ i”<br>yi yj! yy Dy;<br>y y J i<br>UT yyy < aj Virjr = Virjt — Ai;! Vpngr = irr ayy O<AS1 Vij =<br>Vij GD) #IWH), C93<br>y y<br>M(a,k;y ) > M(a,k;y)<br>yee yi Yirj' < Vir"j'<br>j'<br>Vip > Vi"'j j#i' i’<br><!-- End of picture text -->
<!-- Start of picture text --> i’ yur<br>Jil<br>M(a,k;y)<br>=| 67 0 | > y -| 11020 |<br>z=([10 ] k=2<br>9°) = con (0.2)% (0.3)(0.4)% | > g)= (o.n» (0.2)* (0.8)%0 |<br>M(a,k;y ) < M(a,k;y) a<br>a>1<br>k<br>k<br><!-- End of picture text -->
Conversely, it is easy to check that for ��1 the violation will not occur if the recipient remains poor after the transfer. Fundamentally, the violation arises because the “dual cutoff” assigns a zero weight to any deprivations of non-poor persons.

The statement of the problem also suggests an immediate solution: make every deprivation count by setting ��1 (or, more generally for unequally weighted dimensions, setting the poverty cut-off at no more than the minimum of dimensional weights). This is the so-called _union_ approach to identification whereby a person is considered poor if deprived in at least one dimension. It yields the following multidimensional poverty measure: This is the same as the additive multidimensional extension of the FGT (Foster, Greer, Thorbecke, 1984) measure in Bourguignon and Chakravarty (2003). While Alkire and Foster (2011a) present their dual cut-off measures as a generalization of the BourguignonChakravarty measure, the generalization comes at a price (most directly, the violation of the strong transfer axiom as above) that we may not want to pay.

Alkire and Foster (2011a) briefly consider the union method, but conclude that it is not appropriate in all circumstances. They offer two arguments. First, “deprivation in certain single dimensions may be reflective of something other than poverty”. Second, “when the number of dimensions is large, the union approach will often identify most of the population as being poor... Consequently, a union-based poverty methodology may not be helpful in distinguishing and targeting the most extensively deprived”. For convenience, let’s term these as the mismeasurement argument (what is measured is not “true” deprivation or poverty) and the targeting argument (what is measured is not helpful for targeting). On closer scrutiny, however, neither argument is particularly compelling.

In relation to the mismeasurement issue, it is arguable that the censoring implied by the dual cut-off method presents a rather amputative approach to dealing with measurement error, 9
which has several problems. First, it assumes that _all_ deprivations of an individual must be “fortuitous” or measured with error if the total number of deprivations for the individual falls short of the poverty cut-off. Second, measurement errors are unlikely to be discontinuous at deprivation thresholds such that it is also possible (on account of measurement error) for someone to be deemed non-deprived in a particular dimension when in fact she is.<sup>8</sup> With dual cut-off censoring, it is then possible for this individual to be deemed non-poor, and thus exacerbating the problem by not only ignoring her mismeasured deprivation, but also all her other deprivations too. Third, measurement errors are also likely to differ across dimensions: some dimensions are harder to measure accurately than others. Yet, dual cutoff censoring as a response to measurement error is completely non-specific with respect to dimensions. Fourth, dual cut-off as a response to measurement error also carries the problematic implication that measurement error matters for those with fewer deprivations than the cross-dimensional cut-off, but may be ignored for those with more deprivations. For all these reasons, the dual cut-off approach does not appear to be a methodologically appropriate response to measurement error.

In relation to the targeting argument, it is useful to note that given subgroup decomposability, the union-based poverty measure can be written as population share ( �� )-weighted sum of poverty measures of � mutually exclusive groups comprising respectively of those who are deprived in exactly � dimensions for ��1, 2, ... � .

Thus, if budgetary resources or other considerations do not allow policymakers to target the full set of those facing _any_ deprivation, it is always possible to drill down a union-based poverty measure and focus on subsets of those deprived in multiple dimensions. For instance, one could start with those deprived all � dimensions, then move on to those deprived in ��1 dimensions, and so on. Alternatively, it is possible to define a target group as bottom �% of the population when individuals are ranked from the poorest to the least 10
<!-- Start of picture text --> m;(a;y)<br>y y<br>Vij = Vij Gp)#@j) M(a;y ) > M(a;y)<br>k a>0<br>k=1<br>y y<br>j' i’<br>@7) Vrjt 2 Zr > Very" Vij = Vij<br>Gp)N#C SI) M(a;y )<M(a;y)<br>k<br>k<br>k=1<br><!-- End of picture text -->
<!-- Start of picture text --> k<br><!-- End of picture text --> <!-- Start of picture text --> Y= | (€1)* (Ez) (§3)* eo > PY )= leo (€2)* (€3)% co<br>y y<br>y y<br>M(a,k;y) = M(a,k;y ) a<br><!-- End of picture text -->
The above example illustrates a case where (in moving from � to �<sup>∗</sup> ) there is a change in the joint distribution of achievements but no change in the censored marginal distributions. Note however that the invariance of AF measures to such changes in joint distribution relates to ���, �; �� measures, and not to partial indices, such as � and � , which indeed do not satisfy many of the (axiomatic) properties proposed for the AF measures. For instance, it is true that in the example above, in moving from distribution � to �<sup>∗</sup> , while poverty incidence, � , is halved, the intensity of poverty amongst the poor, � , doubles, which is why the product �����0, �; �� remains unchanged. But that only highlights the key issue. As the halving of � is a desirable change and the doubling of � is the opposite, how do we form a judgement on the overall change? The judgement implicit in the AF measures ���, �; �� , that these two effects always exactly cancel each other out, is contentious. As argued below, the distribution-sensitive ���, �; �� measures for ��1 incorporate the alternative judgement that the same set of deprivations if shared more (less) equally imposes a lesser (greater) burden on society.

Such distributional invariance of AF measure, ironically, renders them somewhat similar _in that respect_ to indices such as the Human Development Index (HDI) or the Human Poverty Index (HPI) which depend only on the marginal distributions of indicators they aggregate.<sup>11</sup> Strictly, the joint density across dimensions matters for the AF measures at the identification stage (in determining whether someone is poor or not), but not at the aggregation stage. Put differently, the AF measures depend on the censored marginal distributions, and that renders them insensitive to greater or lesser concentration of a given set of deprivations amongst the poor so long as the set of poor remains the same.

It is notable that this issue persists even in the union case when ��1 and there is no censoring. In general, ���; �� will be unchanged for any rearrangement of a _given_ set of deprivations for all values of α . In this respect, the AF class of measures (including the special union case with ��1 ) are rather utilitarian in their reliance on a “sum-ranking”<sup>12</sup> , i.e., the aggregation of individual deprivations by summing them up.

13
It is arguable however that a given deprivation is more burdensome for an individual if also accompanied by deprivation in other dimensions. A similar view is echoed by Stiglitz, Sen and Fitoussi (2009, p. 15) in their discussion of measures of the quality of life: “...the consequences for quality of life of having multiple disadvantages far exceed the sum of their individual effects”. The notion of compounding negative effects of multiple disadvantages is also deeply rooted in the sociological concept of the _underclass_ (e.g. Wilson, 1987, 2006), in policy discussions of social exclusion (e.g. Levitas et al, 2007), as well as in some formulations of “poverty traps” (e.g. Durlauf, 2006; Banerjee and Duflo, 2011). Common to these diverse perspectives is the idea that through various mechanisms multiple disadvantages can reinforce each other in the reproduction of poverty for certain groups, in undermining the long-term “life chances” or future prospects of the most disadvantaged in society.<sup>13</sup> It is thus eminently arguable that the concentration or dispersion of deprivations across individuals matters, and a case can be made for the requirement that multidimensional poverty measures be sensitive to the distribution of a given set of deprivations. This can be achieved by requiring the _individual_ multidimensional poverty measure to be convex over individual deprivations in different dimensions, and can be axiomatized as follows.

**Cross-dimensional convexity** :  An increment in deprivation �<sup>�</sup> for individual � induces a greater increase in measured poverty for individual � , the greater individual � ’s deprivation in any other dimension ���′ .<sup>14</sup>

There is no real mystery to the idea that if you are hurting on many fronts, an additional blow could be completely crushing. The cross-dimensional convexity axiom appeals to this idea of the compounding (negative) effect of multiple deprivations on the overall well-being of the individual. It asserts that the severity of the effects of an increase in deprivation in any one dimension increases not only with the level of deprivation in that dimension but also with the level of deprivation in other dimensions. Put differently, it is a way of capturing the added 14
n B M(a,B;y) = ~~-)Ie{=~~ Y g fora>Oandfp 21 M(a,B;y) M(a,B;y ~~)==~~ ?-<sup>iz (@</sup> Bi ¥) m;(a, Bi y;)D=(= ( ~~G2~~ jagd ° M(a,B;y) M(a;y) B=1 Bot M(a,B;y) B
M(a,B;y) M(a@,B;y) az0,p21 a>0 B2=1 az>0,p>1 a>1p21 az1,p>1 M(a,B;y) M(a, B;y) n B M(a,B;y) = ~~=~~ ) ~~(4)~~ for B>=1 i=1 M(a,B;y) M(a, B;y)
n _ w ~~aV~~ v ~~(S)~~ k ~~e£~~ i=1 n ~~—~~ ) \(a.g%, + e298)Aso i=1 M(a@,B;y) d=a A= ap A>1 6=1 Asa<1 Asa<1 A=0>1 A>O A<@ WL) aA<0 BK<l MaGQiiZ) [Ma (vis 2)” y>o y<i1 y>i y=1
y>1 M(a,B;y) n d M(a,k;y) = ~~—~~ n ) (/a" > gf for nonnegative a,K i=1 j=1 M(a,k;y) aK>t1 a>1 J K>1)
<!-- Start of picture text --> M(a,B;y)<br>1B>1 a>1p>1<br>M(a,B;y)<br><!-- End of picture text --> <!-- Start of picture text --> M(a,B;y)<br>M(a,B;y)<br>Y yyDyi<br>yi! yin vi<br>Cin 2 Cyr > O y<br>j' i Ye = ery! Yin = Vij" Vij = Vij<br><!-- End of picture text -->
<!-- Start of picture text --> Gp)e#{@),WI93 yy ~Dyjn Yi" yin<br>M(a,B;y ) < M(a,B;y) j' wy<br>M(a,k;y ) < M(a,k;y)<br>M(a,k;y ) = M(a,k;y)<br>M(a,k;y)<br>M(a,B; y) p>1<br>i’ i"<br>yi! yi"<br>yy~Dyjyn yy Dyin<br>M(a,B;y ) > M(a,B;y)<br>M(a,k;y)<br>B>1<br>B>1<br>M(a,B;y)<br><!-- End of picture text -->
**(iv)** The satisfaction of cross-dimensional convexity for ��1 can be interpreted as asserting _complementarity_ of multiple deprivations in different dimensions.  There has been some confusion in terminology regarding this in the literature.  The axiom of cross-

dimensional convexity implies a _complementarity between different dimensional shortfalls_ for an individual in the sense that a person’s multidimensional poverty �� increases by a larger amount in response to increased deprivation in one dimension ( ��� ), the greater this person’s deprivation in another dimension ( ���� ). That is, (for continuous functions) the cross-partial derivative of individual poverty to dimensional deficits is positive. But since it follows that a positive cross-partial derivative with respect to dimensional deficits also implies a positive cross-partial derivative with respect to dimensional achievements. However, the latter is what Bourguignon and Chakravarty (2003) refer to as the dimensions being substitutes. Whether we call them complements or substitutes, what matters for �<sup>�</sup> ����,�;�� cross-dimensional convexity is being positive. The key point is that however one ��������� may want to think of complementarity or substitutability of dimensions on the production side (how one attribute or dimension is transformed into another), _when there is a shortfall in multiple dimensions_ , cross-dimensional convexity asserts that it hurts more than the sum of single dimensional shortfalls. And it is in this precise sense that under cross-dimensional convexity the shortfalls in different dimensions are deemed complementary.<sup>21</sup> The issue of whether production complementarities between attributes pose a challenge to the normative appeal of a property like strong rearrangement or cross dimensional convexity has featured frequently in the literature in various forms; for instance, the argument is made that considerations of complementarity between attributes should make room for departures from cross-dimensional convexity.<sup>22</sup> One can discuss the key issue at two levels: (i) in the stark case of perfect complementarity between two attributes in the specific sense that the 21
possession of one attribute (e.g. refrigerators) is of no use without the possession of the other (e.g. electricity); (ii) in the context of the more general and larger question of whether or how complementarity should influence multidimensional poverty measurement.<sup>23</sup> In the first case, cross-dimensional convexity (CDC) would seem to favour a more even distribution of electricity connections and refrigerators, even though refrigerators without electricity of no use (abstracting from the possibility that they can be sold for something else). However, a reasonable response here could be to redefine the two underlying deprivations to take into account this particular feature of the dimensions being considered. Thus, in the specific example, the two deprivations could be redefined as: one, not having access to electricity, and two, having access to electricity and not having a refrigerator. With this redefinition, handing out refrigerators to those without an electricity connection will not serve to reduce the second deprivation, opening the door for a consistent application of CDC.

Turning to the more general issue of complementarity between attributes, one can argue with some justification that how production complementarities between attributes (how one attribute is transformed into another) should inform policy choices is better handled independently of how a social welfare function is specified (viewing multidimensional poverty measures as social welfare functions). Trying to incorporate such production complementarities into the specification of social welfare functions on the other hand can be confounding. One can illustrate this point with the following example. Consider two attributes, nutrition (N) and education (E) and two persons, A and B, with the following deprivation matrix, �′ : Person A has an education deficit of 80% of the education threshold and nutrition deficit of 20% of the nutrition threshold, while person B has a 60% deficit of both attributes. Thus, A 22
g" E N g" =_ A B g" g' CDC => M(g") > M(qg') g' g” E N gg” _— A B g gg” ~~__~~ M(g') M(g"") < M(g’)
n d B d M(a,B;y) = ~~-~~ ) > 9% for a=0,f =1 and > =1 Mit \fa j=1 M(a,k;y) k-1
M(a;y) <u>agij9Gij</u> #0 B=1 M(q,B;y) j M(a,B;y) j d!

J J d-1 J d d d= 10 S axd ‘ d d J = 20-1 J s SP sy?

, A t sW 6(t)
t Sy? ot’) J !

M C;(M) GM) = >. w(t)M(t) — >. w(t')M(t’) tes? res) w(t) = ~~COR NEO~~ and w(t’) = ~~(SO E—1- sey~~ M(t) M(t’) : i ¢_,Cj(M) = M C;(M)/M M(a,B;y) M(a,k;y)
# **VI. An application to India**

This section illustrates the ideas introduced in the preceding sections of the paper with an application to India using data from the Third National Family Health Survey (NFHS3) for 2005-6. In part, the motivation for using these data is that they have recently been used by Alkire and Seth (2009, 2013) for an analysis of multidimensional poverty in India following the AF methodology, and thus offer a useful point of comparison. The empirical implementation thus closely follows Alkire-Seth definitions of dimensional indicators, cut-offs and weights (as set out in Table 4.1 in Alkire and Seth, 2013).<sup>27</sup> In brief, Alkire-Seth use a slightly modified version of the UNDP MPI framework for India, under which there are 10 unequally-weighted indicators or dimensions, each represented by a binary variable indicating the presence or absence of the particular deprivation for a person. Amongst the 10 indicators, there are two for education with weights of 1/6<sup>th</sup> each, two for health also with weights of 1/6<sup>th</sup> each, and six indicators related to the standard of living each with a weight of 1/18<sup>th</sup> .<sup>28</sup> The cross-dimensional cut-off is set at 1/3; thus, a person is deemed poor if deprived in at By comparis-third of weighted dimensions. With this setup, multidimensional poverty measures ���, �; �� and ���, �; �� are constructed below for ��1/3 and different values of � .<sup>29</sup> The illustration aims to highlight five points as noted in the introduction.

_(i) A significant fraction of multidimensional poverty can be potentially missed by not counting deprivations of those below the cross-dimensional cut-off._ Table 1 reports multidimensional poverty measures both in the aggregate as well as disaggregated by the breadth of dimensional deprivation. The estimates show that about 47% of the Indian population are deprived in one-third or more of the 10 weighed dimensions and are thus deemed to be poor under the dual cut-off approach. However, there is another 41% of the population who also experience deprivations, albeit in less than one-third of weighted dimensions. If these deprivations were to be counted in, multidimensional poverty would be 29% higher, rising from 0.243 (the value of ���, ��1⁄3� ) to 0.314 (the value of ���, �� 1� ). The deprivations of those deemed non-poor by the dual cut-off account for 22 and 10 27
per cent respectively of ���, ��1� and ���, ��2� . If the dimensional cut-offs are considered reasonable, it seems hard to contend that more than one-fifth of all deprivations, which are assigned a weight of zero under the AF measures, represent mismeasured, accidental or chosen deprivations.

_(ii) Distribution-sensitive measures can matter for the poverty profile and subgroup contributions._ The implications of using the distribution-sensitive measures are further illustrated by considering the profile of multidimensional poverty across India’s 29 states using measures with ��1, 2, 3 incorporating increasing complementarity across multiple deprivations. The southern state of Kerala – well-known for its superior social indicators – has the lowest level of multidimensional poverty by all measures. Figure 1 expresses poverty measures for all states relative to Kerala, and shows that inter-state variation in multidimensional poverty can be seriously understated by measures that attach little or no importance to the extra burden of multiple deprivations. For instance, multidimensional poverty in Bihar is 3.5 times that in Kerala for measures with ��1 , but it is 17 times higher for measures with ��3 . Measures with no or lower orders of distribution sensitivity will tend to compress the poverty profile.

_(iii) Greater concentration of deprivations amongst the poor can_ reduce _dual cut-off poverty measures._ Another respect in which the distribution-sensitive measures and the AF class of measures may evaluate certain joint distributions of deprivations differently is shown in Table 2. Starting with the actual distribution of deprivations, Table 2 considers a rearrangement of the deprivations of those deemed poor under the dual cut-off approach such that their concentration amongst the poor is maximized. Under this rearrangement, the marginal distributions for all dimensions remain unchanged. Such maximal concentration implies that many of the erstwhile poor are no longer so because post-rearrangement they are deprived in less than one-third of weighted dimensions. As a result, ����1/3� , the incidence of AF poverty, declines from 47.4% to 33.1%. ���, ��1⁄3� also declines from 0.243 to 0.223. By contrast, there is only a small decline in ����0� , the incidence of poverty under the 28
M(a,B=1) B=1 M(a,B =2) M(a,B;y) g* M(a,B;g9") ce) . a g* " " ~~in~~ 97- ~~5~~ i= Gij M(a,B;y) B p= 1 M(a,1;9%) = M(a,1; 9”) B=2 B=3 M(a,k;y) M(a,B= 1)
_(v) Shapley dimensional decompositions of poverty._ Table 4 presents the Shapley dimensional decompositions of multidimensional poverty for the distribution-sensitive ���, �; �� measures as well as the ���, �; �� measures following the methods discussed in section 5. The decompositions offer a measure of the contribution of each dimension to overall poverty. The results indicate that the four dimensions of nutrition, sanitation, fuel and assets are the prime contributors to multidimensional poverty in the country accounting for about half of overall multidimensional poverty, well-above their combined dimensional weight of one-third. For these data, it turns out that the Shapley dimensional contributions for the distribution-sensitive measure ���, ��2� are almost identical to those for ���, ��1⁄3� , though there are some differences with respect to those for ���, ��1� . It also turns out that for the AF measure ���, ��1⁄3� itself, the post-identification dimensional contributions are not very different to the Shapley contributions which also build in contributions through identification. The similarity of dimensional decompositions across distribution-sensitive and other measures, while true of these data, is however unlikely to be a general result.

# **VII. Conclusion**

This paper contributes to the methodological literature on multidimensional poverty measurement. All poverty measures incorporate a set of social judgements. This is true of uni-dimensional as well as multidimensional measures of poverty, and amongst the latter, of counting measures as well as social welfare measures (invoking the distinction introduced by Atkinson, 2003). While there will always be room for some disagreement on the most appropriate set of social judgements, this paper has argued the case for distributionsensitive multidimensional poverty measures, denoted as ���, �; �� in the paper, which straddle the space between counting and social welfare measures, and could also be thought of as generalized counting measures. These measures, are distribution-sensitive in two ways: they satisfy the strong transfer axiom requiring regressive transfers to be poverty30
increasing, and they satisfy cross-dimensional convexity (also shown to be equivalent to strong rearrangement) whereby the cumulative effect of multiple deprivations is deemed to be greater than the sum of their parts. The parameters � and � calibrate these two elements of distribution-sensitivity. Indeed, by allowing � to vary by dimension, the distribution-sensitive measures also open up the further possibility of varying degree of interdependence across any pair of dimensions.

An empirically useful property of the counting measures of multidimensional poverty is that they are additively decomposable across subgroups of population as well as across dimensions. Dimensional decomposability has been deemed particularly attractive from a policy perspective as it offers a means of assessing how deprivations in different dimensions contribute to overall poverty, thus offering guidance on dimensional priorities for poverty reduction. But simple additive dimensional decomposition is lost when cross-dimensional convexity is introduced. The paper however demonstrates how an exact Shapley dimensional decomposition can be implemented to assess the contributions of individual dimensions. The decomposition methods described in the paper are very general in nature and can be readily applied to virtually any poverty measure, not limited to distributionsensitive measures alone.

The paper has also sought to illustrate key properties of the distribution-sensitive measures with Indian data. Using the counting measures as a point of comparison, the application offers examples of how the use of distribution-sensitive measures can matter for assessing the magnitude of poverty, for the profile of the poor and subgroup contributions, and for the contribution of inequality in distribution of deprivations to multidimensional poverty. In many instances, such points of difference will also carry different implications for a variety of policy issues, including those related to dimensional priorities or the targeting of particular subgroups. The results for the Indian application are intended to be mainly illustrative. As in the case of uni-dimensional FGT measures relative to the widely-used headcount index, the use-value of distribution-sensitive multidimensional poverty measures will only be 31
established through empirical practice. Additional applications will help clarify further the empirical and policy significance of distribution-sensitive measures. The fundamental appeal of these measures however ultimately rests on the properties they satisfy and how reasonable these properties are deemed to be.

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