ROBUST MULTIDIMENSIONAL POVERTY COMPARISONS*€¦ · Such comparisons thus relieve the analyst or...

ROBUST MULTIDIMENSIONAL POVERTY COMPARISONS*

Jean-Yves Duclos, David E. Sahn and Stephen D. Younger

We demonstrate how to make poverty comparisons using multidimensional indicators of well-being,showing in particular how to check whether the comparisons are robust to aggregation proceduresand to the choice of multidimensional poverty lines. In contrast to earlier work, our methodologyapplies equally well to what can be defined as �union�, �intersection� or �intermediate� approaches todealing with multidimensional indicators of well-being. To make this procedure of some practicalusefulness, the article also derives the sampling distribution of various multidimensional povertyestimators, including estimators of the �critical� poverty frontiers outside which multidimensionalpoverty comparisons can no longer be deemed ethically robust. The results are illustrated using datafrom a number of developing countries.

Since the seminal work of Amartya Sen (1979, 1985, 1987), it is common to assertthat poverty is a multi-dimensional phenomenon, a proposition that most econo-mists accept in theory. Yet in practice, the vast majority of empirical work on povertyuses a one-dimensional yardstick to judge a person’s well-being, usually expendituresor income per capita or per adult equivalent. Our purpose in this article is to showthat it is possible to bring the empirical literature closer to the theoretical rhetoricby making quite general poverty comparisons when deprivation is measured inmultiple dimensions.

We note at the outset that there is a branch of the poverty measurement litera-ture that considers multiple dimensions of well-being, but these papers invariablyaggregate the multilple measures of well-being into a one-dimensional index,essentially returning to a univariate analysis. The best-known example is the HumanDevelopment Index (HDI) of the UNDP (1990), which uses a weighted average oflife expectancy, literacy and GDP per capita for a population, though several morehave been proposed recently.1 Any such index requires a specific aggregation ruleto sum up the components of the index, and any such rule is necessarily arbitrary.This leaves open the possibility that two equally valid rules for aggregating acrossseveral dimensions of well-being could lead to contradictory conclusions aboutwhich groups have higher poverty.2

To avoid that, we develop poverty comparisons that are valid for a broad class ofaggregations rules. This is in the spirit of the dominance approach to poverty com-parisons, as initially developed by Atkinson (1987) and Foster and Shorrocks

* Duclos� research was supported by SSHRC, FCAR and IDRC’s MIMAP programme. Sahn’s and Younger’sparticipation was supported by the USAID and AERC. An early version of this work was written as a supportingpaper for the World Development Report (2000), regarding which we are grateful to Christian Grootaert forhis comments and support. We also thank Rob Strawderman and Steve Westin for technical advice, NicolasBeaulieu and Wilson Perez for their excellent research assistance, and Peter Lambert, Paul Makdissi and LarsOsberg for very useful comments.

1 See, for example the papers presented at a recent WIDER conference on Inequality, Poverty, and HumanWell-Being, http://www.wider.unu.edu/conference/conference-2003–2/conference2003–2.htm.

2 Such rules have been the focus of some of the recent literature: see for instance Tsui (2002), Dutta et al.(2003), and Bourguignon and Chakravarty (2003). Bourguignon and Chakravarty (2003) also give severalinteresting examples in which poverty orderings vary with the choice of aggregation rules.

The Economic Journal, 116 (October), 943–968. � The Author(s). Journal compilation � Royal Economic Society 2006. Published by

Blackwell Publishing, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA.

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(1988a,b,c) in a unidimensional context.3 One well-appreciated advantage of this ap-proach is that it is capable of generating poverty orderings that are robust to the choiceof a poverty index over broad classes of indices – the orderings are �poverty-measurerobust�. Such comparisons thus relieve the analyst or policy maker from the task ofchoosing one particular (and ethically arbitrary) poverty measure. To understand ourgeneralisation of dominance comparisons to multiple dimensions of well-being, it ishelpful to think of a one-dimensional poverty index as an aggregation rule within thatdimension. That is, the index aggregates all of the information in an income distri-bution into a single scalar poverty measure. Univariate dominance comparisons arevalid for broad classes of such rules, i.e., broad classes of poverty measures. Ourmultidimensional comparisons will be similarly valid for broad classes of aggregationrules used in any one dimension, and also for broad classes of aggregation rules usedacross dimensions of well-being.

In contrast to earlier work on multidimensional comparisons, our orderings are also�poverty-line robust�, in the sense of being valid for the choice of any poverty frontierover broad ranges. Given the well-known sensitivity of many poverty comparisons to thechoice of poverty lines, and the difficulty of choosing the �right� poverty line, especiallyfor many non-income dimensions of well-being, this would appear to be an importantcontribution.

For most of the article, we limit ourselves to the case of two measures of well-being, though we do provide an example of a three-dimensional comparison. Intheory, extending our results to more than two dimensions is straightforward. Inpractice, though, most existing datasets in developing countries are probably notlarge enough to support tests on more than a few dimensions of well-being. This isbecause the curse of dimensionality (Bellman, 1961) affects our non-parametricestimators.

We begin in Section 1 with the theory of making multidimensional poverty com-parisons. One of the first conceptual challenges of poverty analysis in multipledimensions is deciding who is �poor�. We consider this question in Section 1.1. Theliterature distinguishes between intersection and union definitions of poverty. If wemeasure well-being in the dimensions of income and height, say, then a person couldbe considered poor if her income falls below an income poverty line or if her heightfalls below a height poverty line. We may define this as a union definition of multidi-mensional poverty. An intersection definition, however, would consider a person to bepoor only if she falls below both poverty lines.4 In contrast to earlier work, the tests thatwe develop are valid for both definitions – and also valid, in fact, for any choice ofintermediate definitions for which the poverty line in one dimension is a function ofwell-being measured in the other. We also show how the concept of a poverty line inone dimension extends naturally to a �poverty frontier� in multiple dimensions, with theshape of that frontier determined by whether we are interested in union, intersection,or intermediate poverty measures.

3 Atkinson and Bourguignon (1982, 1987) and Bourguignon (1989) first used this approach in the contextof multidimensional social welfare. See also Crawford (1999) for a recent application.

4 For further recent discussion of this, see inter alia Bourguignon and Chakravarty (2002, 2003), Atkinson(2003) and Tsui(2002).

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Section 1.2 presents our main theorem for multidimensional poverty comparisonsthat are first-order in each dimension.5 Section 1.3 extends these results to higher-order poverty comparisons. Section 1.4 compares our method to two other methodsthat researchers might use to consider poverty in multiple dimensions.

Section 1.5 presents a different approach to multidimensional poverty comparisons.Rather than asking, �Is poverty lower for A than for B over all reasonable povertyfrontiers?� we ask, �What is the area of poverty frontiers over which we can be sure thatpoverty is lower for A than for B?� This approach provides one useful way to get aroundthe need to make an arbitrary choice of �reasonable� limits for the range of admissiblepoverty frontiers. The procedure also makes the dominance conditions �locate thedisagreements that are crucial� (Atkinson, 2003).

While much of the article is a contribution to the theory of multidimensional povertymeasurement, the results are also important for applied poverty analysts. The Mille-nium Development Goals, for example, focus attention on deprivation in multipledimensions. Many policy makers take these goals seriously, and they have clearly helpedto broaden the development policy debate beyond a narrow focus on reducing incomepoverty. The methods that we propose here should provide applied researchers with anattractive tool for analysing these broader definitions of poverty. To demonstrate that,Section 3 gives a series of examples that highlight some of the subtleties of the article’stheoretical results, using data from several developing countries.

One of the key points in Section 3 is that it is possible for a set of univariate analysesdone independently for each dimension of well-being to conclude that poverty in A islower than poverty in B while a multivariate analysis concludes the opposite, and viceversa.6 The key to these possibilities is the interaction of the various dimensions of well-being in the poverty measure and their correlation in the sampled populations. Weargue that a reasonable poverty measure should allow the level of deprivation in onedimension to affect our assessment of how much poverty declines if there is animprovement in another dimension. For example, we might think that an increase inincome for a poor person should cause a larger decline in a poverty measure if therecipient is also relatively deprived in the dimensions of health, education etc. �One-at-a-time� comparisons of poverty in the dimensions of income, education, and healthcannot capture these interdependencies, while our multidimensional measures can. Inpractice, populations exhibiting higher correlations between measures of well-beingwill be poorer than those that do not, relative to what one would expect from makingunivariate comparisons alone.

A further output of the article (Section 2) is to provide the sampling distribution ofmany of the estimators that are useful for multidimensional poverty analysis, in such away that one may make inferences about the true population value of poverty measures.Previous work on multidimensional poverty comparisons has ignored sampling vari-ability, yet this is fundamental if the study of multidimensional poverty comparisons isto have any practical application.

5 We will define this precisely in Section 1.2, but it is analogous to first-order poverty comparisons in onedimension.

6 To our knowledge, the literature does not include poverty analyses that look at several dimensions of well-being �one-at-a-time�, but the rise of the MDGs could well bring that about as analysts first study incomepoverty, then health poverty, then education poverty etc.

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1. Multiple Indicators of Well-being

1.1. Poverty Frontiers and Poverty Aggregation in Two Dimensions

Let x and y be two indicators of individual well-being.7 These could be, for instance,income, expenditures, caloric consumption, life expectancy, height, cognitive ability,the extent of personal safety and freedom etc. Denote by

kðx; yÞ : <2 ! < @kðx; yÞ@x

� 0;@kðx; yÞ@y

� 0

�� ð1Þ

a summary indicator of individual well-being, analogous to but not necessarily the same asa utility function. Note that the derivative conditions in (1) simply mean that differentindicators can each contribute to overall well-being. We make the differentiabilityassumptions for expositional simplicity but they are not strictly necessary, so long ask(x, y) is non-decreasing over x and y. It should be clear that the weak inequalities onthese derivatives impose few constraints on the precise value of these contributions.

We then assume that an unknown poverty frontier separates the poor from the rich.We can think of this frontier as a series of points at which overall well-being of anindividual is precisely equal to a �poverty level� of well-being, and below which indi-viduals are in poverty. This frontier is assumed to be defined implicitly by a locus of theform k(x, y) ¼ 0, and is analogous to the usual downward-sloping indifference curveson the (x, y) space. The set of the poor is then obtained as:

KðkÞ ¼ ðx; yÞ ðkðx; yÞ � 0jf g: ð2Þ

Consider Figure 1 with poverty lines zx and zy in dimensions of well-being x and y.k1(x, y) gives an �intersection� poverty index: it considers someone to be in poverty only

y

x

zy

zx

0

λ2(x,y)=0

λ3(x,y)=0

λ1(x,y)=0

Fig. 1. Union and Intersection Poverty Indices

7 For expositional simplicity, we focus on the case of two dimensions of individual well-being. We willillustrate the extension to more than two dimensions in Section 3.



if she is poor in both of the two dimensions of x and y, and therefore if she lies within thedashed rectangle of Figure 1. k2(x, y) (the L-shaped, dotted line) gives a union povertyindex: it considers someone to be in poverty if she is poor in either of the two dimen-sions, and therefore if she lies below or to the left of the dotted line. Finally, k3(x, y)provides an intermediate approach. Someone can be poor even if y > zy, if her x valueis sufficiently low to lie to the left of k3(x, y) ¼ 0. Alternatively, someone can be non-poor even if y < zy if her x value is sufficiently high to lie to the right of k3(x, y) ¼ 0.8

To define multidimensional poverty indices precisely, let the joint distribution of xand y be denoted by F(x, y). For analytical simplicity, we focus in this article on classesof additive multidimensional poverty indices.9 An additive poverty index that combinesthe two dimensions of well-being can be defined generally as P(k),

PðkÞ ¼Z Z

KðkÞpðx; y; kÞdF ðx; yÞ; ð3Þ

where p(x, y; k) is the contribution to poverty of an individual with well-being indicatorsx and y. By the well-known �poverty focus axiom� (Foster, 1984), this is such that

pðx; y; kÞ � 0 if kðx; yÞ � 0¼ 0 otherwise.

�ð4Þ

The multidimensional headcount is obtained when p(x, y; k) ¼ 1 wheneverk(x, y) � 0. The p function in (4) is thus the weight that the poverty measure attachesto someone who is �poor,� i.e. inside the poverty frontier. That weight could be 1 (for aheadcount) but it could take on many other values as well, depending on the povertymeasure of interest. By the focus axiom, it has to be zero for those outside the povertyfrontier.

1.2. Poverty Dominance in Two Dimensions

With F(x) being the distribution function for x, recall first that the usual unidimen-sional stochastic dominance curve is given (for x) by

P aðzÞ ¼Z z

0ðz � xÞadF ðxÞ ð5Þ

for an integer a � 0. Equation (5) is thus an average of poverty gaps (z � x) to thepower a, and it is also the popular Foster, Greer and Thorbecke (1984) (FGT) povertyindex. A bi-dimensional stochastic dominance surface can then be defined as:

P ax ;ay ðzx ; zyÞ ¼Z zy

0

Z zx

0ðzx � xÞax ðzy � yÞay dF ðx; yÞ ð6Þ

for integers ax � 0 and ay � 0. We generate the dominance surface by varying thepoverty lines zx and zy over an appropriately chosen domain, with the height of the

8 As noted in (1), k(x, y) is a very general function. Specific examples of bivariate poverty measures thatlead to a variety of such functions, including the three in Figure 1, can be found in Bourguignon andChakravarty (2003).

9 This is a usual assumption, often obtained as a product of a subgroup decomposability axiom – see forinstance Foster et al. (1984) and Bourguignon and Chakravarty (2002).



surface determined by (6). This function looks like a two-dimensional generalisation ofthe FGT index, and it could be interpreted as such. It is important to highlight,however, that the poverty comparisons that we make will be valid for broad classes ofpoverty functions, not this one alone.10 A second important feature of the dominancesurface is that it is influenced by the covariance between x and y, the two measures ofwell-being, because the integrand is multiplicative. Rewriting (6), we find indeed that

P ax ;ayðzx ; zyÞ ¼Z zy

0ðzx � xÞax dF ðxÞ

Z zx

0ðzy � yÞay dF ðyÞ

þ cov ðzx � xÞax ; ðzy � yÞay� �

:

ð7Þ

The height of the dominance surface is therefore the product of the two unidimen-sional curves plus the covariance in the poverty gaps in the two dimensions. Thus, thehigher the correlation between x and y, the higher the dominance surfaces, otherthings equal.

Our poverty comparisons make use of orders of dominance, sx in the x and sy in the ydimensions, which will correspond respectively to sx ¼ ax þ 1 and sy ¼ ay þ 1. Theparameters ax and ay also capture the aversion to inequality in poverty in the x and inthe y dimensions, respectively. P 0,0(zx, zy) generates a cumulative density function,analogous to a poverty incidence curve in a univariate analysis. P1,0(zx, zy) sums thepoverty gaps in x (given by max(zx � x, 0)) for those that are poor in y, and thennormalises this sum by the size of the total population. P1,1(zx, zy) sums the product ofthe poverty gaps in x and in y, again normalised by the size of the total population, andcan therefore be thought of as a bidimensional average poverty gap index, with theweights on the poverty gaps in one dimension being the poverty gaps in the otherdimension. Analogous interpretations exist for other combinations of ax and ay values.

To describe the class of poverty measures for which the dominance surfaces definedin (6) are sufficient to establish poverty orderings, assume first that the general povertycontribution in (3) is left differentiable11 with respect to x and y over the set K(k), up tothe relevant orders of dominance, sx for derivatives with respect to x and sy for deriv-atives with respect to y. Denote by px the first derivative12 of p(x, y;k) with respect to x;by py the first derivative of p(x, y; k) with respect to y ; by pxy the derivative of p(x, y; k)with respect to x and to y; and treat similar expressions accordingly.

We then define the following class €P1;1ðk�Þ of bidimensional poverty indices:

€P1;1ðk�Þ ¼ PðkÞ

KðkÞ � Kðk�Þpðx; y; kÞ ¼ 0; whenever kðx; yÞ ¼ 0

px � 0 and py � 0 8x; ypxy � 0; 8x; y:

��

8>>><>>>:

9>>>=>>>;

ð8Þ

The first line on the right of (8) defines the largest poverty set to which the poormust belong: the poverty set covered by the P(k) indices should lie within the maximalset K(k�). The second line assumes that the poverty indices are continuous along the

10 One of the more remarkable features of the results that follow is that they are also useful for unionpoverty definitions, even though the function in (6) is itself an intersection definition.

11 This differentiability assumption is made for expositional simplicity. It could be relaxed.12 The derivatives include the implicit effects of x and y on k(x, y).



poverty frontier. This excludes the multidimensional poverty headcount, which is dis-continuous at the poverty frontier. It implies among other things that small measure-ment errors should not have un-commensurate impacts on measured poverty. Thethird line of assumptions says that indices that are members of €P1;1 are weaklydecreasing in x and in y. This implies that an increase in either x or y cannot be bad forpoverty reduction. Given the interpretation of these variables, this would seem a nat-ural assumption. For the indices to be non-degenerate, we must also have that px < 0,py < 0 and pxy > 0 over some ranges of x and y. Note also that the inequalities in (8) areweak, which is different from the strong inequalities that are often found in the lit-erature. This is consistent, however, with the way in which we will proceed to testdominance – we will test for strict ordering of the dominance surfaces, instead of theweak orderings often tested in the empirical literature.

The last line assumes that the marginal poverty benefit of an increase in either x or ydecreases with the value of the other variable. Apart from our exclusion of the head-count, which we address below, this is the only assumption in (8) that implies debatablerestrictions on the class €P1;1ðk�Þ, so we discuss it in some detail. Atkinson and Bour-guignon (1982) and Bourguignon and Chakravarty (2002) refer to it as a property ofnon-decreasing poverty under a �correlation-increasing switch�. To see what thisimplies, consider Figure 2. A first individual, initially located at a with an associated lowx, sees his value of y also brought low by a movement from a to b. A second individual,who has more of x than the first individual, moves from c to d with a correspondingincrease in his value of y by the same amount that the first individual sees his y fall. Thisis a �correlation-increasing switch�. The marginal distributions of both x and y are

b

a d

c

x

y

λ(x,y)=0

Fig. 2. A Correlation-increasing Switch



unaffected by this switch. The correlation of deprivation and the incidence of multipledeprivation is, however, higher after than before the switch.13

The pxy � 0 assumption may also be understood as a �substitutability� assumption.The more someone has of x, the less is overall poverty deemed to be reduced if hisvalue of y is increased. This assumption would seem to be ethically justified in manycases.14 Governments, for instance, are often urged to care first for those who sufferfrom multiple deprivation, even though it may sometimes be more costly in budgetresources to reach those individuals. An improvement in access to health services forthose who are poorer in total expenditures would also seem socially more desirablethan for those who are relatively better off in total expenditures.

But one can also think of other cases in which a strong complementarity in the pro-duction of two dimensions of well-being might force us to reverse our assumption aboutpxy. For instance, for a poverty analysis in the dimensions of education and nutritionalstatus of children, there are production complementarities because better-nourishedchildren learn better. If this complementarity is strong enough, it may overcome the usualethical judgement that favours the multiply-deprived, so that overall poverty woulddecline by more if we were to transfer education from the poorly nourished to the betternourished, despite the fact that it increases the correlation of the two measures of well-being. Similarly, one might argue that human capital should be granted to those with ahigher survival probability (because these assets would vanish following their death).Increasing the correlation of deprivations, and increasing the incidence of multipledeprivation, would then be good for poverty reduction. Bourguignon and Chakravarty(2002) derive dominance criteria for this second possibility. We do not pursue this avenuehere. Our main reason is that this second approach would appear to limit drastically thescope for �poverty-frontier robust� orderings, in particular if robustness is sought overunion/intersection/intermediate poverty-frontier definitions.15

Denote by DQ ¼ QA � QB the difference between a function Q for A and for B. Theclass of indices defined in (8) then gives rise to the following €P1;1 bi-dimensionaldominance condition:

Theorem 1. ( €P1;1 poverty dominance)

DPðkÞ > 0; 8PðkÞ 2 €P1;1ðk�Þ; ð9Þ

iff DP 0;0ðx; yÞ > 0; 8ðx; yÞ 2 Kðk�Þ: ð10Þ

Proof. See the Appendix.

13 For further discussion of this concept, with examples, see Tsui (2002).14 See for instance Tsui (2002) for an advocate of this route.15 To see this, note that tests for classes of indices that obey the complementarity condition use dominance

surfaces that are effectively defined by union poverty indices; see e.g. Atkinson and Bourguignon (1982) andBourguignon and Chakravarty (2003). These tests thus include individuals lying beyond the intersection andintermediate poverty sets defined by the dominance surfaces (6) that are central to our proofs. The corres-ponding poverty tests with pxy � 0 also suppose that there can be no welfare interaction between the twoindicators of well-being whenever one’s combination of indicators lies outside the inner intersection area(thus supposing that pxy ¼ 0 whenever x > zx or y > zy in Figure 1).



Condition (10) requires that the bi-dimensional dominance surface be higher for A thanfor B for all intersection poverty frontiers which lie in K(k�). (The notation in (10)uses (x, y) and not (zx, zy) to stress that the comparison of the surfaces is to be doneover an entire area, not only at some arbitrary combination of poverty lines.) If con-dition (10) is met, Theorem 1 says that poverty will be unambiguously higher in A thanin B for all of the poverty indices that are members of the €P1;1ðk�Þ class of povertymeasures defined in (8). Note that this allows for a wide area of poverty frontiers – allthose such that K(k)�K(k�). This is an interesting and surprising feature of Theorem 1:even though the theorem is applicable to intersection, union, and intermediatepoverty measures, one has only to check the intersection-like dominance surface in (6)over an appropriately defined domain to test for poverty dominance. Because of itsgenerality, this result allows for powerful orderings of multidimensional poverty acrossA and B.

To see what Theorem 1 implies more clearly, return to Figure 1 and consider firstan intersection poverty frontier for K(k�). In this case, the relevant domain for thetest would be a rectangle such as the one defined by the axes and the upper bounds(zx, zy). Thus, to establish a robust poverty comparison on this domain, we must checkthat A�s dominance surface is above B�s at every point over this rectangle. Note,however, that once this is established, we are assured of a robust poverty ordering notonly at that precise intersection poverty frontier defined by (zx, zy), but also for anyother poverty frontier which �fits� into this rectangle. These alternative poverty fron-tiers would include all of the intermediate frontiers (of the type of k3(x, y) ¼ 0 inFigure 1) that could fit in the rectangle defined by (0, 0) and (zx, zy). This is despitethe fact that the bi-dimensional dominance surfaces are themselves only intersectionpoverty indices.

For a union poverty frontier k�, the test domain would be an L-shaped regiondefined in Figure 1 by K(k2). Again, condition (10) requires that the dominancesurface be higher for A than for B over all points within that region. If that isestablished, we are assured of a robust poverty ordering for all other union, inter-section, or intermediate poverty frontiers and poverty sets which lie within thattesting area. The extension to more general outer frontiers such as k3(x, y) ¼ 0 inFigure 1 follows naturally.

1.3. Higher Order Dominance Tests

For higher-order dominance, we may increase the order in one dimension or in bothsimultaneously. Either approach adds further assumptions on the effects of changes ineither x or y on aggregate poverty and thus limits the applicable class of povertymeasures. These further assumptions are analogous to those found in the unidimen-sional dominance literature, and require that indices react increasingly favourably toincreases in living standards at the bottom of the distribution of well-being. Theassumptions further require that the reactions of the indices to changes in one indi-cator be greater the lower the level of the other indicator of well-being.

To illustrate this, assume in addition to the above conditions for €P1;1 that the first-order derivative px(x, y; k) is continuous whenever k(x, y) ¼ 0. Further suppose thatequalising transfers in x at a given value of y should, if enactable, weakly reduce



aggregate poverty, and that this effect is decreasing in the value of y.16 We then obtainthe following class of bi-dimensional poverty indices:

€P2;1ðk�Þ ¼ PðkÞ

PðkÞ 2 €P1;1ðk�Þ

pxðx; y; kÞ ¼ 0 whenever kðx; yÞ ¼ 0

pxxðx; y; kÞ � 0 8x;

and pxxyðx; y; kÞ � 0; 8x; y

��

8>>>>><>>>>>:

9>>>>>=>>>>>;: ð11Þ

This leads to the following dominance condition:

Theorem 2. ( €P2;1 poverty dominance)

DPðkÞ > 0; 8PðkÞ 2 €P2;1ðk�Þ

iff DP 1;0ðx; yÞ > 0; 8ðx; yÞ 2 Kðk�Þ:ð12Þ

Proof. See Appendix.

This tests simultaneous dominance of order 2 for x and of order 1 for y by checkingwhether P1,0(x, y), the average poverty gap in x progressively cumulated in thedimension of y, is greater in A than in B, regardless of which intersection povertyfrontier (x, y) is chosen within K(k�). The ordering properties are analogous to thoseof Theorem 1.

Although it may not prove necessary, we can move to higher orders of dominance inthe x dimension. The classes of poverty indices belonging to €Psx ;1ðk�Þ becomeincreasingly restricted as sx increases. For €P3;1ðk�Þ for instance, poverty indices mustobey the principle of transfer sensitivity17 in x, and react more to a favourable com-posite transfer the lower the value of y. Higher values of sx imply compliance withhigher-order principles of transfers.18

In addition, we can simultaneously increase both sx and sy. The procedures, classes ofpoverty indices, and dominance relationships are analogous to those described above.For instance, the conditions for membership in €P2;2ðkÞ require that the poverty indicesbe convex in both x and y, and that they therefore obey the principle of transfers inboth of these dimensions. They also require that this principle be stronger in onedimension of well-being the lower the level of the other dimension of well-being.Finally, they also impose that the second-order derivative in one dimension ofwell-being be convex in the level of the other indicator of well-being. The domi-nance condition then checks whether P1,1(x, y) is greater in A than in B for all(x, y) 2 K(k�).

16 We say �if enactable� because, as a referee has pointed out, some dimensions of well-being such aschildren’s heights are not literally transferable from one person to another.

17 For a definition, see for instance Kakwani (1980) and Shorrocks and Foster (1987).18 For an illustration of this in the unidimensional setting, see Fishburn and Willig (1984).



1.4. Relevance of the Methods

The methods that we propose above are more general than two other methods thatresearchers might use to consider poverty in multiple dimensions. One approach is toaggregate multiple dimensions of poverty into one univariate index, using arbitraryweights on each individual welfare measure. The best-known example is the HumanDevelopment Index (UNDP, 1990), which first constructs a scalar summary measure ofeach welfare variable, and then aggregates those measures into the HDI using arbitraryweights. An alternative approach in the same vein would be to sum each person’swelfare variables, and then create a summary measure of that weighted sum for thepopulation. In either case, the method arbitrarily reduces the problem of povertycomparison from many dimensions to one. Our proposed tests clearly generalise thisrather restrictive approach.19

The second approach is to compare many indicators of well-being independently.Such univariate comparisons are also a special case of our approach. To see this, recallthat the dominance surface Psx�1,sy�1(x, y) is cumulative in both dimensions. Hence,integrating out one dimension leaves the univariate dominance curve for the otherdimension of well-being. In terms of Figure 1, the domain of separate univariate testswould be (for the y variable) a vertical line up to zy at the maximum possible value of x,and (for the x variable) a horizontal line up to zx at the maximum possible value of y.

There are two ways in which our test could differ from this �one-at-a-time� approach.First, it is possible that the univariate dominance curve for A lies above that for B atboth x ¼ 1 and y ¼ 1 for the relevant range of poverty lines but that A is not above Bat one or more interior points in the test domain shown in Figure 2. In this case, theone-at-a-time approach would conclude that poverty is higher in A than B but ourbivariate approach would not. Because the bivariate approach checks the joint distri-bution of all indicators of well-being, it is thus able to show the correlation across suchindicators, which is of ethical importance since it helps to capture the joint incidence ofdeprivation in multiple dimensions. One-at-a-time analysis fails to do this.20

Alternatively,it is possible for the dominance surfaces to cross in the y dimension atx ¼ 1 and/or in the x dimension at y ¼ 1, but for A�s surface to be above B�s for alarge area of interior points in the test domain. In this case, the one-at-a-time approachwould not be able to establish a ranking of poverty but our test would do so forintersection definitions of poverty and some intermediate definitions. Allowing forunion definitions, however, would require including the margins of the surface in itstest domain and would therefore not lead to a robust ordering of poverty.21

To visualise these methodological differences, consider Figure 3, which graphs atypical dominance surface. A larger �hump� in the middle of the surface corresponds to

19 Dutta et al. (2003) show that indices like the Human Development Index are very special and limitedcases of more general multivariate welfare comparisons.

20 We are grateful to a referee for pointing out that the one-at-a-time approach would provide necessaryand sufficient conditions for a (smaller) class of union poverty measures that would allow no interactionbetween the indicators of well-being, namely, those for which pxy ¼ 0. See also the end of the proof ofTheorem 1 in the Appendix.

21 In this case, one could first make the simpler univariate comparisons in each dimension, proceeding tothe more difficult multivariate comparisons only if univariate dominance is found in each dimension.Crawford (1999) develops this approach.



a larger positive correlation between the two well-being variables. Also, the univariatedominance curve for one dimension is found at the upper extreme of the dominancesurface’s other dimension. On Figure 3, the univariate curve for the log of householdexpenditures lies on the extreme right of the surface, while that for the height-for-agez-score (to which we return below) is behind it.

When we make dominance comparisons, we test for the difference between twosurfaces each like the one shown in Figure 3. Figure 4 depicts such a difference for thecase in which one surface has highly correlated welfare variables while the second doesnot (the difference in these surfaces �bulges� in the middle). Although differences inthe univariate dominance curves in both dimensions clearly cross the origin (at theextreme left and right of the Figure), there is a significant interior section where thefirst surface is entirely above the second. Hence, there are intersection and inter-mediate poverty frontiers for which the first distribution has more poverty than thesecond. Conversely, we could think of shifting Figure 4 down such that the univariatedifferences are all negative. The first distribution would then dominate the second inboth dimensions individually, but there would still be a section in the middle where thefirst surface would lie above the second. Thus, there would be no bivariate povertydominance due to the first distribution showing too much incidence of multipledeprivation. We will give further examples of this in Section 3.

1.5. Bounds to Multidimensional Dominance

Implementing the approaches to multidimensional poverty dominance developedabove requires implicitly specifying a maximum poverty set K(k�). Although there may

–3.8

2

–3.3

2

–2.8

2

–2.3

2

–1.8

2

–1.3

2

–0.8

2

–0.3

2

0.18

0.68

1.18

9.50

10.25

11.00

11.75

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Cum

ulat

ive

dist

ribu

tion

Height-for-age z-score

log

HH

expe

nditu

re p

.c.

0.9-10.8-0.90.7-0.80.6-0.70.5-0.60.4-0.50.3-0.40.2-0.30.1-0.20-0.1

Fig. 3. Dominance Surface for Ghanaian Children, 1989



be some intuitive feel that a very large set K(k�) is not sensible, there is rarely reliableempirical evidence about what its precise value should be. Specifying this value a prioriis thus necessarily subject to some degree of arbitrariness.

An alternative approach that gets around such arbitrariness is to estimate directly fromthe samples the maximum K(kþ) for which multidimensional poverty dominance holdsin the sample. This critical set is delimited by a critical poverty frontier, since this will delimitthe area of poverty frontiers which may not be exceeded for a robust multidimensionalordering of poverty to be possible. The researcher can then judge whether these criticalsets and frontiers are sufficiently wide to justify a conclusion of poverty dominance.

To develop this idea further, assume that a critical poverty set exists in the twopopulations of multidimensional well-being being compared. Assume therefore that Binitially dominates A but that their dominance surfaces eventually cross and that theranking of the dominance surfaces is thus eventually reversed. Hence, for a given valueof y, let fþx ðyÞ then be the first crossing point22 of the surfaces in the x dimension, withP

ax ;ay

A ½fþx ðyÞ; y� ¼ Pax ;ay

B ½fþx ðyÞ; y�. Carrying out this exercise for a range of y leads to theestimation of a critical poverty frontier kþ½fþx ðyÞ; y� � 0. By the results derived above,this procedure will provide an estimate of the space K(kþ) in which we can locate all ofthe possible poverty frontiers (union, intersection, or intermediate) for which there isnecessarily more poverty in A than in B for all poverty indices that are members of€Paxþ1;ayþ1ðkþÞ. This procedure can be applied for any desired orders of bi-dimensionaldominance sx ¼ ax þ 1 and sy ¼ ay þ 1, and can be generalised to more dimensions.

–2.5

–1.7

35–0

.97

–0.2

05 0.56 1.32

52.

09 2.85

53.

624.

385

–2.5

–1.9

9–1

.48

–0.9

7–0

.46

0.050.561.071.582.092.

63.

113.624.134.64

–0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Dif

fere

nce

in s

urfa

ces

xy

0.3-0.350.25-0.30.2-0.250.15-0.20.1-0.150.05-0.10-0.05–0.1-0

Fig. 4. Example of Difference in Dominance Surfaces, Intersection Dominance Without MarginalDominance

22 Note that fþx ðyÞ will depend on the orders of dominance (sx, sy) considered and should formally bewritten as fþx ðy; sx ; syÞ. For expositional simplicity, however, we do not make this dependence explicit.



Note that this poverty frontier kþ(x, y) ¼ 0 also locates the (x, y) frontier for which thebi-dimensional intersection FGT poverty indices would be exactly the same in the twodistributions.

2. Estimation and Inference

We now consider the estimation of the tools derived above for multidimensionalpoverty analysis. In this, we generalise to more than one dimension some of the resultsof Davidson and Duclos (2000).

2.1. Dominance Surfaces

Suppose first that we have a random sample of N independently and identically dis-tributed observations drawn from the joint distribution of x and y. We can write theseobservations of xL and yL, drawn from a population L, as ðxL

i ; yLi Þ, i ¼ 1, . . ., N. A

natural estimator of the bidimensional dominance surfaces Pax,ay(zx, zy) is then:

Pax ;ay

L ðzx ; zyÞ

¼Z zy

0

Z zx

0ðzy � yÞayðzx � xÞax dFLðx; yÞ

¼ 1=NXNi¼1

ðzy � yLi Þ

ayðzx � xLi Þ

ax I ðyLi � zyÞI ðxL

i � zxÞ

¼ 1=NXNi¼1

ðzy � yLi Þ

ay

þðzx � xLi Þ

ax

þ

ð13Þ

where F denotes the empirical joint distribution function, I(Æ) is an indicator functionequal to 1 when its argument is true and 0 otherwise, and xþ ¼ max(0, x). For arbitraryax and ay, (13) has the convenient property of being a simple sum of IID variables, evenallowing for the fact that x and y will generally be correlated across observations.

The following theorem allows us to perform statistical inference in the case in whichwe have a sample from each of two populations, A and B, that may or may not havebeen drawn independently from each other.

Theorem 3. Let the joint population moments of order 2 of ðzy � yAÞayþ ðzx � xAÞax

þ andðzy � yBÞay

þ ðzx � xBÞaxþ be finite. Then N1=2 P ax;ay

A ðzx; zyÞ � P ax;ay

A ðzx; zyÞ� �

andN1=2 P ax;ay

B ðzx; zyÞ � P ax;ay

B ðzx; zyÞ� �

are asymptotically normal with mean zero, with asymptoticcovariance structure given by (L, M ¼ A, B):

limN!1

N cov Pax ;ay

L ðzx ; zyÞ; Pax ;ay

M ðzx ; zyÞ� �

¼ E ðzy � yLÞay

þðzx � xLÞax

þ ðzy � yM Þay

þðzx � xM Þax

þ� �� P

ax ;ay

L ðzx ; zyÞPax ;ay

M ðzx ; zyÞ:

ð14Þ

Proof. See the Appendix.



When the samples from the populations A and B are independent, the variance ofeach of P

ax ;ay

A ðzx ; zyÞ and Pax ;ay

B ðzx ; zyÞ can be found by using (14) and by replacing N byNA and NB respectively. The covariance between the two estimators is then zero. Theelements of the asymptotic covariance matrix can be estimated consistently using theirsample equivalents.

2.2. Critical Frontiers

To establish the sampling distribution of estimators of the critical frontier fþx ðyÞ, as-sume that within some bottom area x 2 [0, cx] and at a given value of y, the popu-lation dominance surface for A lies above that for B, but that these surfaces cross(exactly) in the population at some higher critical point fþx ðyÞ. For a fixed value of y, anatural estimator fþx ðyÞ of the location of that point can be defined by the first pointabove y at which the sample ordering of the dominance surface changes. If the sampledominance surface for A were to lie always above that for B above y, then we could setfþx ðyÞ to an arbitrarily large value (denote it by zþx ). Formally, fþx ðyÞ is then defined as:23

fþx ðyÞ ¼ sup xjDP ax ;ay ðx; yÞ � 0 and x � zþx� �

: ð15Þ

Applying this estimator over a range of y leads to an estimator of the critical frontierkþ(x, y) ¼ 0. Under these conditions, the sampling distribution of fþx ðyÞ is given by thefollowing Theorem 4. For this, however, it is expositionally convenient to define anFGT index with a negative ax as:

P�1;ay ðzx ; zyÞ ¼Z zy

0ðzy � yÞay f ðyjx ¼ zxÞdyfxðzxÞ ð16Þ

¼ E ðzy � yÞay

þjx ¼ zx

� �fxðzxÞ ð17Þ

where fx(zx) is the marginal density of x at zx and f(y|x) is the conditional density of y atx. Both are assumed to exist. This leads to the following theorem.

Theorem 4. Let the joint population moments of order 2 of (xA)(ax)(yA)(ay) and(xB)(ax)(yB)(ay�1) exist. If the samples from A and B are independent, assume that the ratio r ¼NA/NB of their respective sample size tends to a constant as NA and NB tend to infinity. When thesamples are dependent, we have N ¼ NA ¼ NB. Under the conditions mentioned above (in par-ticular, that fþx ðyÞ exists in the population), and setting d ¼ ax when ax > 0 and d ¼ 1 whenax ¼ 0, we have for dependent samples that N1=2 fþx ðyÞ � fþx ðyÞ

h iis asymptotically normal with

mean zero and asymptotic variance given by

limN!1

var N 1=2 fþx ðyÞ � fþx ðyÞh in o

¼

d Pax�1;ay

A fþx ðyÞ; y� �

� Pax�1;ay

B fþx ðyÞ; y� �n o� ��2

var ðy � yAÞay

þ fþx ðyÞ � xA� �ax

þ

n o�þ var ðy � yBÞay

þ fþx ðyÞ � xB� �ax

þ

n o

�2cov ðy � yAÞay


þ ; ðy � yBÞay


þ

n o�ð18Þ

23 Recall that DP ¼ PA � PB.



and for independent samples that N1=2A fþx ðyÞ � fþx ðyÞh i

is asymptotically normal with mean zeroand asymptotic variance given by

limNA!1

var N1=2A fþx ðyÞ � fþx ðyÞh in o

¼

d Pax�1;ay

A fþx ðyÞ; y� �

� Pax�1;ay

B fþx ðyÞ; y� �n o� ��2

var ðy � yAÞay


þ

n o�þ r var ðy � yBÞay


þ

n o�: ð19Þ

Proof. See Appendix.

3. Examples

As a first example, consider the question: are rural people poorer than the urban onesin Viet Nam? Many studies, of Viet Nam and elsewhere, find that people living in ruralareas tend to be poorer when judged by expenditures or income alone. However, it ispossible that people are better nourished in rural than urban areas, ceteris paribus,because they have tastes for foods that provide nutrients at a lower cost, or because unitprices of comparable food commodities are lower. In such cases, including an indicatorof nutritional status may change the relative well-being of rural and urban residents.To test this, we measure welfare in two dimensions: per capita household expendi-tures and nutritional status, as measured by a child’s gender and age-standardisedheight, transformed into standard deviations from the median of a healthy population,known as z-scores. Stunted growth in children is widely used as an indicator ofmalnutrition and poor health. The sample comes from the Viet Nam Living StandardsMeasurement Survey carried out in 1993.24 This is a nationally representative house-hold survey that collected detailed expenditure and anthropometric data. The latter,however, are available only for children younger than 60 months, so our sampleis actually these children only, rather than all of the members of the householdsinterviewed.

The test described in (10) requires comparison of the two dominance surfaces ofurban and rural children in Viet Nam: the difference between those two surfaces isshown in Figure 5 for sx ¼ sy ¼ 1. The y axis measures the height-for-age z-score(stunting); the x axis measures the per capita expenditures for the child’s household;and the z axis measures the cumulative proportion of children that fall below the pointsdefined in the (x, y) domain. The poorest children are in the front left-hand corner ofthe graph. If the rural dominance surface is above the urban surface over the relevantarea of poverty frontiers (values of per capita expenditures and stunting), poverty ishigher (more people are below the given well-being levels in each dimension) in ruralareas. This conclusion is then robust to the choice of poverty indices in the class€P1;1ðk�Þ.

24 Information on the LSMS surveys is available in Grosh and Glewwe (1998). Information about thespecific survey that we use is available at the LSMS website: http://www.worldbank.org/html/prdph/lsms/country/vn93/vn93bid.pdf.



In theory, we should test over the entire area defined by K(k�), but it is morepractical to choose a grid of points in the (x, y) domain over which to test.25 Here weuse a grid that is 20 20, spread evenly over the entire domain of the log of per capitaexpenditures and the height-for-age z-score. Following Howes (1996), we test for asignificant difference in the dominance surface at each point of the grid, and reject thenull of non-dominance of A by B only if all of the test statistics (t-statistics) have theright sign and are significantly different from 0.

Figure 5 indicates clearly that, over almost the entire range of expenditures andstunting, rural children are poorer than urban. Table 1 shows whether these statementsare statistically significantly at the 5% level. A negative sign indicates that the urbandominance surface is significantly below the rural one, a positive sign indicates theopposite, and a zero indicates that the difference is not statistically significant. Thenegative differences are statistically significant for any reasonable pair of poverty lines(except at the very bottom right of Table 1. Hence, by Theorem 1, the conclusion thatrural children are poorer than urban ones is valid for almost any intersection, union orintermediate poverty frontier.

Our second example tests for first-order poverty dominance in three dimensions. Weask whether poverty declined in Ghana between 1993 and 1998, using data from theDemographic and Health Surveys. The three welfare variables that we consider are for

–4.1

9

–3.2

9

–2.8

2

–2.4

9

–2.1

6

–1.8

3

–1.5

1

–1.0

9

–0.4

9

0.72

6.08

6.54

6.77

7.03

7.41

–0.5

–0.4

–0.3

–0.2

–0.1

0

0.1

Height-for_age z-score

ln(H

H e

xpen

ditu

re p

er

capi

ta)

Fig. 5. Urban Minus Rural Dominance Surface for Viet Nam

25 The computation of these statistics, as well as of their sampling distribution, can be done using thesoftware DAD (Duclos et al., 2002), freely available at http://www.mimap.ecn.ulaval.ca.



Tab

le1

Tes

tR

esu

lts

for

Dif

fere

nce

betw

een

Dom

inan

ceSu

rfac

esfo

rU

rban

and

Ru

ral

Chi

ldre

nin

Vie

tN

am,

1993

log

of

ho

use

ho

ldex

pen

dit

ure

per

cap

ita/

hei

ght-

for-

age

z-sc

ore

�4.

19�

3.64�

3.29�

3.02�

2.82�

2.66�

2.49�

2.31�

2.16�

2.00�

1.83�

1.67�

1.51�

1.32�

1.09�

0.84�

0.49

0.01

0.72

5.47

6.08

��

��

��

��

��

��

��

��

��

��

6.26

��

��

��

��

��

��

��

��

��

��

6.37

��

��

��

��

��

��

��

��

��

��

6.47

��

��

��

��

��

��

��

��

��

��

6.54

��

��

��

��

��

��

��

��

��

��

6.61

��

��

��

��

��

��

��

��

��

��

6.66

��

��

��

��

��

��

��

��

��

��

6.71

��

��

��

��

��

��

��

��

��

��

6.77

��

��

��

��

��

��

��

��

��

��

6.84

��

��

��

��

��

��

��

��

��

��

6.89

��

��

��

��

��

��

��

��

��

��

6.95

��

��

��

��

��

��

��

��

��

��

7.03

��

��

��

��

��

��

��

��

��

��

7.08

��

��

��

��

��

��

��

��

��

��

7.16

��

��

��

��

��

��

��

��

��

��

7.27

��

��

��

��

��

��

��

��

��

��

7.41

��

��

��

��

��

��

��

��

��

��

7.59

��

��

��

��

��

��

��

��

��

��

7.88

��

��

��

��

��

��

��

��

��

��

9.41

��

��

��

��

��

��

��

��

��

00

Not

es:

Sx¼

1,Sy¼

1.A

neg

ativ

esi

gnin

dic

ates

that

the

urb

and

om

inan

cesu

rfac

eis

sign

ifica

ntl

yb

elo

wth

eru

ral

on

e,a

po

siti

vesi

gnin

dic

ates

the

op

po

site

,an

da

zero

ind

icat

esth

atth

ed

iffe

ren

ceis

no

tst

atis

tica

lly

sign

ifica

nt.



children under five years old: their survival probability, their height-for-age z-score(stunting), and an index of their household’s assets.26 We compare dominance sur-faces for these three measures in 1993 and 1998, the two years for which DHS dataexist. While we cannot graph the resulting four-dimensional surface, Figure 6 sum-marises the results of the statistical test. We use a 20 20 20 grid of test points, andeach horizontal layer in Figure 6 is similar to Table 1 in the previous example.27 A lightgrey plus �þ� sign indicates that the 1998 surface is significantly above the 1993 surface;a circle indicates that the 1998 surface is significantly below the 1993 surface; and asolid black point indicates that they are statistically indistinguishable at the 5% signi-ficance level. It is clear from the Figure that there is no robust poverty dominanceresult. Over some of the domain, poverty does seem to have declined between 1993 and1998. But in significant areas, particularly for low values in the asset dimension, thereverse is true.

In addition to showing that our tests are possible in more than two dimensions, thisexample shows the importance of checking for the robustness of poverty comparisonsusing tests such as those that we employ. For the intersection headcount, shown bypoints on the dominance surfaces, a judicious choice of the poverty lines could leadone to conclude that poverty worsened, improved, or did not change, depending on

0 0.5 1 1.5 −2 0

2

0.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

Asset index Stunting

Pr(S

urvi

val)

Fig. 6. Test Results for Difference between 1993 and 1998 First-order Dominance Surfaces forGhanaian Children

26 Information on the estimation of survival probabilities and the asset index may be found in Sahn et al.(1999) and Sahn and Stifel (2000).

27 We have excluded some of the horizontal layers to make the graph more legible.



the specific choice. None of these results would be robust, but any might seemplausible if it appeared on its own.

The next two examples highlight the difference between using bivariate dominancetests vs. one-at-a-time univariate tests on the same variables. Table 2 gives the results fortests of the differences in the dominance surfaces for stunting and child survivalprobability in Cameroon and Madagascar. The data come from the 1997 DemographicHealth Surveys (DHS) in those countries.28 The �one-at-a-time� dominance curves aregiven in the last row of the Table (for survival probability) and in the last column (forstunting). These univariate comparisons would conclude that poverty is worse inMadagascar than in Cameroon, whether measured by stunting or survival probability.Nevertheless, the bivariate comparison shows several internal points where the surfacesare not significantly different, including two where the point estimate of the differenceis in fact positive. So our method would not come to the same conclusion, findinginstead that there is no statistically significant, first-order poverty ordering of these twopopulations.

Table 3 shows the other possibility for different conclusions. These results are alsofor tests of the differences between first-order dominance surfaces for stunting andchild survival probability, in Colombia and the Dominican Republic, and come fromthe DHS surveys for those countries, carried out in 1995 and 1996 respectively. Inthis case, there is dominance on one margin (for survival probability), but not theother (stunting), so the one-at-a-time approach would not find poverty to benecessarily lower in one population than the other. However, Colombia’s domin-ance surface is significantly below the Dominican Republic’s over a very large rangeof the interior points, suggesting that under an intersection definition of poverty,

Table 2

Test Results for Difference Between Dominance Surfaces for Children in Cameroon andMadagascar, 1997

Height-for-agez-score/Survivalprobability 0.83 0.86 0.88 0.89 0.90 0.91 0.99 1.00

�4.19 � � 0 0 � � . . . � ��3.66 0 0 0 0 � � . . . � ��3.35 0 � � � � � . . . � ��3.13 0 � � � � � . . . � ��2.88 0 � � � � � . . . � ��2.66 � � � � � � . . . � ��2.50 � � � � � � . . . � �

. . . . . . . . . . . . . . . . . . . . . . . . . . .0.46 � � � � � � . . . � �5.39 � � � � � � . . . � .

Notes. Sx ¼ 1, Sy ¼ 1.A negative sign indicates that Madagascar’s dominance surface is significantly above Cameroon’s, a positivesign indicates the opposite, and a zero indicates that the difference is not statistically significant.Ellipsis indicates that all intervening signs are negative.

28 Information on these surveys is available at http://www.measuredhs.com. The child survival probabilityis estimated as in Sahn et al. (1999).



and several intermediate ones as well, poverty was robustly lower in Colombia thanin the Dominican Republic.

Turning our attention now to the analysis of critical poverty frontiers, Figure 7 showstwo such frontiers, for the €P1,1 and €P2,2 classes of poverty measures, respectively, usingchildren’s height-for-age z-score and the log of their households� per capita expendituresas measures of well-being.29 The data are from the 1999 Uganda National HouseholdSurvey, and the comparison is between rural residents in the Eastern region and urbanresidents in the Northern region. Up to these critical frontiers, poverty is lower in ruralEastern Uganda than it is in urban Northern Uganda for all poverty measures in therespective class. Note that the critical frontier is close to the origin for €P1,1, so thatrelatively few poverty frontiers fit within the critical frontier.30 For €P2,2, however, thecritical frontier extends much farther, so that a rather large set of intersection andintermediate poverty measures in this class conclude that poverty is lower in rural Easternthan urban Northern Uganda. Of course, the price of this result is that it is valid only for asmaller class €P2,2 of poverty measures, a class that requires inter alia poverty indices to fallfollowing equalising transfers in dimensions x or y – see Section 1.3.

Table 3

Test Results for Difference Between Dominance Surfaces for Children in Colombia and theDominican Republic, 1995 and 1996

Height-for-agez-score/Survivalprobability 0.906 0.927 0.938 0.947 0.953 0.985 0.987 0.989 0.991 0.995 1.000

�2.85 � � � � � . . . � � � 0 0 0�2.36 � � � � � . . . � � � � 0 0�2.07 � � � � � . . . � � � � 0 0�1.85 � � � � � . . . � � � 0 0 0�1.67 � � � � � . . . � � � 0 0 0�1.47 � � � � � . . . � � � 0 0 þ�1.33 � � � � � . . . � � � 0 0 þ�1.17 � � � � � . . . � � � 0 þ þ�1.04 � � � � � . . . � � 0 0 þ þ�0.92 � � � � � . . . � � � 0 0 þ�0.76 � � � � � . . . � � � 0 þ þ�0.62 � � � � � . . . � � � 0 þ þ�0.49 � � � � � . . . � � � 0 þ þ�0.35 � � � � � . . . � � � � þ þ�0.12 � � � � � . . . � � � � þ þ

0.07 � � � � � . . . � � � � 0 þ0.34 � � � � � . . . � � � � 0 þ0.68 � � � � � . . . � � � � 0 þ1.05 � � � � � . . . � � � � � þ5.92 � � � � � . . . � � � � � 0

Notes: Sx ¼ 1, Sy ¼ 1.A negative sign indicates that the Domincan Republic’s dominance surface is significantly above Colombia’s,a positive sign indicates the opposite, and a zero indicates that the difference is not statistically significant.Ellipsis indicates that all intervening signs are negative.

29 The figure actually shows the critical frontier minus two standard deviations.30 As a point of reference, the median z-score is �1.22 and the median log of per capita expenditures is

8.52.



4. Conclusion

In this article, we have shown that it is possible to make very general poverty com-parisons for multiple dimensions of well-being. These comparisons have severalattractive features:

(i) In the spirit of the stochastic dominance literature, they can be tested forrobustness over broad classes of poverty aggregation procedures, both within agiven dimension of well-being and across the different dimensions of interest.

(ii) A special consideration for the multivariate case is whether poverty is defined asthe intersection or union of poverty in each dimension. The methods that wedescribe are valid for both, as well as for any choice of intermediate cases.

(iii) The poverty comparisons can be tested for robustness over a broad area ofpoverty frontiers. Alternatively, one can estimate a critical poverty frontier up towhich multidimensional poverty dominance necessarily holds.

(iv) The comparisons involve statistical tests that make use of the sampling distri-bution of multidimensional dominance surfaces.

The importance of these methods rests on two considerations. The first is ethical:there is widespread acknowledgement that well-being and poverty are multidimensionalin nature. We take this as given, based either on Sen’s philosophical arguments thatpoverty should be viewed in terms of multidimensional capabilities and functionings, oron the realistic recognition that a person’s well-being has dimensions that cannot bepurchased and that transcend levels of income. The second consideration is practical: to

–4.0

–3.5

–3.0

–2.5

–2.0

–1.5

–1.0

–0.5

0.0

0.5

1.0

7.4 7.6 7.8 8 8.2 8.4 8.6 8.8 9 9.2ln(household expenditure per capita)

Hei

ght-

for-

Age

z-S

core

(Sx,Sy) = (2,2) (Sx,Sy) = (1,1)

Fig. 7. Critical Poverty Frontier, Rural Eastern region vs. Urban Northern region in Uganda (criticalFrontier Minus Two Standard Deviations)



what extent will multidimensional comparisons differ from unidimensional ones? Giventhe relatively weak correlations that are often observed between income and otherwelfare variables, it should not be surprising to find cases where poverty comparisons inmultiple dimensions differ from comparisons in only one of those dimensions, some-thing that we found repeatedly in preparing the examples for this article.

More generally, we have shown that our multidimensional comparisons can alsodiffer from univariate comparisons in each individual dimension in two ways. Onecould find dominance in each dimension separately if the margins of the dominancesurfaces differ in each dimension, but may not find multidimensional dominance if thesurfaces cross in the surfaces� interiors. This draws attention to the importance ofcapturing the incidence of multiple deprivation. Alternatively, the one-dimensionaldominance curves may cross, ruling out univariate dominance, but the interiors of themultidimensional surfaces may be uniformly different, allowing multivariate domin-ance for intersection or intermediate poverty definitions. The empirical importance ofthese two possibilities remains to be firmly established. Our admittedly limitedexperience based on comparisons of the DHS surveys is that the first is rare, while thesecond is fairly common. But a firm sense of the importance of our more generalmethods must await further practice with other samples and other variables.

Appendix

Proof of Theorem 1. Denote the points on the outer poverty frontier k�(x, y) ¼ 0 as zx(y) fora point above y and zy(x) for a point above x. The derivative conditions in (1) imply thatzð1Þx ðyÞ � 0 and zð1Þy ðxÞ � 0, where the superscript (1) indicates the first-order derivative of thefunction with respect to its argument. Note that the inverse of zx(y) is simply zy(x):x � zx[zy(x)].

We then proceed by first integrating (3) by parts with respect to x, over an interval of y rangingfrom 0 to zy. zy can extend to infinity if needed, e.g., for a union poverty frontier. 0 stands for alower bound that could also be set arbitrarily low, thus allowing for negative values of well-beingindicators. This gives:

P zxðyÞ; zy

� �¼Z zy

0pðx; y; k�ÞF ðxjyÞ½ �jzx ðyÞ

0 f ðyÞdy

�Z zy

0

Z zx ðyÞ

0pxðx; y; k�ÞF ðxjyÞf ðyÞdxdy: ð20Þ

The first term on the right-hand-side of (20) is zero since F(x ¼ 0|y) ¼ 0 and since we assumedthat p[zx(y), y;k�] ¼ 0. Hence, it is here that the continuity assumption at the poverty frontiers istechnically important. To integrate by parts with respect to y the second term, define a generalfunction K ðyÞ ¼

R g ðyÞ0 hðx; yÞlðx; yÞdx and note that:

dK ðyÞdy

¼ g ð1ÞðyÞh g ðyÞ; y½ �l g ðyÞ; y½ �

þZ g ðyÞ

0

@hðx; yÞ@y

lðx; yÞdx

þZ g ðyÞ

0hðx; yÞ @lðx; yÞ

@ydx: ð21Þ



Reordering (21) and integrating it from 0 to c, we find:

�Z c

0

Z g ðyÞ

0hðx; yÞ @lðx; yÞ

@ydxdy

¼ �K ðcÞ þ K ð0Þ þZ c

0g ð1ÞðyÞh g ðyÞ; y½ �l g ðyÞ; y½ �dy

þZ c

0

Z g ðyÞ

0

@hðx; yÞ@y

lðx; yÞdxdy: ð22Þ

Now in (22) replace c by zy, g(y) by zx(y), h(x, y) by px(x, y; k�), l(x, y) by F(x, y) and K(y) by itsdefinition K ðyÞ ¼

R g ðyÞ0 hðx; yÞlðx; yÞdx. This gives:

P zxðyÞ; zy

� �¼ �

Z zx ðzyÞ

0pxðx; zy; k

�ÞP 0;0ðx; zyÞdx ð23Þ

þZ zy

0zð1Þx ðyÞpx zxðyÞ; y; k�½ �P 0;0 zxðyÞ; y½ �dy ð24Þ

þZ zy

0

Z zx ðyÞ

0pxyðx; y; k�ÞP 0;0ðx; yÞdx dy: ð25Þ

For the sufficiency of condition (10), recall that zð1Þx ðyÞ � 0, px � 0, and pxy � 0, with strictinequalities for either of these inequalities over at least some inner ranges of x and y. Hence, ifDP0,0(x, y) > 0, for all y 2 [0, zy] and for all x 2 [0, zx(y)] (that is, for all (x, y) 2 K(k�)), then itmust be that DP(k�) > 0 for all of the indices that use the poverty set K(k�) and that obey the firsttwo lines of conditions in (8). But note that for other poverty sets K(k) � K(k�), the relevantsufficient conditions are only a subset of those for K(k�). The sufficiency part of Theorem 1 thusfollows.

For the necessity part, assume that DP0,0(x, y) � 0 for an area defined over x 2 ½c�x ; cþx � andy 2 ½c�y ; cþy �, with cþx � zy and cþx � zxðyÞ. Then any of the poverty indices in €P1;1ðk�Þ for whichpxy < 0 over that area, pxy ¼ 0 outside that area, and for which px(x, zy; k�) ¼ px[zx(y), y; k�] ¼ 0,will indicate that DP < 0. Condition (10) is thus also a necessary condition for the orderingspecified in Theorem 1. (Note, however, that necessary and sufficient conditions for a subclass ofunion poverty indices with pxy ¼ 0 would only involve the marginal or univariate distributions –this can be seen by inspection of (25).)

Proof of Theorem 2. Integrating (25) once more by parts with respect to x, and imposing thecontinuity conditions characterising the indices €P2;1ðk�Þ in (11), we find:

Pðk�Þ ¼Z zx ðzyÞ

0pxxðx; zy; k

�ÞP 1;0ðx; zyÞdx

þZ zy

0pxy zxðyÞ; y; k�½ �P 1;0 zxðyÞ; y½ �dy

�Z zy

0

Z zx ðyÞ

0pxxyðx; y; k�ÞP 1;0ðx; yÞdxdy:

ð26Þ

The rest of the proof is as for Theorem 1.



Proof of Theorem 3. For each distribution, the existence of the appropriate population mo-ments of order 1 lets us apply the law of large numbers to (13), thus showing that P

ax ;ay

K ðzx ; zyÞ is aconsistent estimator of P

ax ;ay

K ðzx ; zyÞ. Given also the existence of the population moments of order2, the central limit theorem shows that the estimator in (13) is root-N consistent and asymp-totically normal with asymptotic covariance matrix given by (14). When the samples aredependent, the covariance between the estimator for A and for B is also provided by (14).

Theorem 3 thus provides the formula needed to estimate the sampling variability of any pointon the dominance surfaces and for any choice of multidimensional poverty lines in the multi-dimensional FGT poverty indices. Extension of the result of Theorem 3 to any additive multi-dimensional poverty indices is straightforward, and simply requires substituting in (14) therelevant functions p(x, y;k) for ðzy � yÞay

þðzx � xÞax

þ .

Proof of Theorem 4. The proof can be established along the lines of the proof of Theorem 3 inDavidson and Duclos (2000). To see this, note that the conditions of Theorem 4 assume that theappropriate joint population moments exist, and that the critical frontier fþx ðyÞ also exists in thepopulation. Furthermore, since this frontier is assumed to be where the dominance surfacesexactly cross, we have that D½@P sx�1;sy�1ðx; yÞ=@x�jx¼fþx ðyÞ < 0. Note that this derivative is given bydfP sx�2;sy�1

A ½fþx ðyÞ; y� � Psx�2;sy�1B ½fþx ðyÞ; y�g, with d ¼ sx � 1 when sx > 1, and d ¼ 1 when sx ¼ 1.

When sx ¼ 1, we also have P�1;sy�1L ½fþx ðyÞ; y� ¼E½ðy � yLÞðsy�1Þ

þ jx ¼ fþx ðyÞ�fx ½fþx ðyÞ�; L ¼ A; B.

Again, the elements of the asymptotic covariance matrix can be estimated consistently by simplyusing their sample estimates. Estimating P sx�2;sy�1½fþx ðyÞ; y� is also easily done when sx > 1. Esti-mating E½ðy � yLÞðsy�1Þ

þ jx ¼ fþx ðyÞ�fx fþx ðyÞ� �

is slightly more complicated, but can be done consis-tently using non-parametric regression procedures. In particular, we use in the illustration aGaussian kernel, K(u) ¼ (2p)�0.5 exp �0.5u2

, and estimate E½ðy � yLÞðsy�1Þþ jx ¼ fþx ðyÞ� fx fþx ðyÞ

� �as:

ðnhÞ�1Xn

i¼1

Kfþx ðyÞ � xL

i

h

� y � yL

i

�ðsy�1Þþ : ð27Þ

Universite Laval and Institut d �An�alisi Econ�omica (CSIC), UAB, BarcelonaCornell UniversityCornell University

Submitted: 1 July 2004Accepted: 17 June 2005

References

Atkinson, A.B. (1987). �On the measurement of poverty�, Econometrica, vol. 55, pp. 749–64.Atkinson, A.B. (2003). �Multidimensional deprivation: constrasting social welfare and counting approaches�,

Journal of Economic Inequality, vol. 1, pp. 51–65.Atkinson, A.B. and Bourguignon, F. (1982). �The comparison of multi-dimensional distributions of economic

status�, chapter 2 in Social Justice and Public Policy, London: Harvester Wheatsheaf.Atkinson, A.B. and Bourguignon, F. (1987). �Income distribution and differences in needs�, in (G.R. Feiwel, ed.),

Arrow and the Foundations of the Theory of Economic Policy, pp. 350–70, New York: New York Press.Bellman, R. (1961). Adaptive Control Processes: A Guided Tour, Princeton: Princeton University Press.Bourguignon, F. (1989). �Family size and social utility: income distribution dominance criteria�, Journal of

Econometrics, vol. 42, pp. 67–80.Bourguignon, F. and Chakravarty, S.R. (2002). �Multi-dimensional poverty orderings�, DELTA, Paris.Bourguignon, F. and Chakravarty, S.R. (2003). �The measurement of multidimensional poverty�, Journal of

Economic Inequality, vol. 1, pp. 25–49.Crawford, I.A. (1999). �Nonparametric tests of stochastic dominance in bivariate distributions, with an

application to UK�, University College London Discussion Papers in Economics 99/07.



Davidson, R. and Duclos, J.-Y. (2000). �Statistical inference for stochastic dominance and the for the meas-urement of poverty and inequality�, Econometrica, vol. 68, pp. 1435–65.

Duclos, J.-Y., Araar, A. and Fortin, C. (2002), �DAD: a software for distributive analysis/analyse distributive�,MIMAP, International Research Centre, Government of Canada and CIRPEE, Universit Laval available at(http://www.mimap.ecn.ulaval.ca).

Dutta, I., Pattanaik, P.K. and Xu, Y. (2003). �On measuring deprivation and the standard of living in amultidimensional framework on the basis of aggregate data�, Economica, vol. 70, pp. 197–221.

Fishburn, P.C. and Willig, R.D. (1984). �Transfer principles in income redistribution�, Journal of PublicEconomics, vol. 25, pp. 323–8.

Foster, J.E. (1984). �On economic poverty: a survey of aggregate measures�, in (R.L. Basmann andG.F. Rhodes, eds.), Advances in Econometrics, vol. 3, pp. 215–51. Connecticut: JAI Press.

Foster, J.E., Greer, J. and Thorbecke, E. (1984). �A class of decomposable poverty measures�, Econometrica,vol. 52 (3), pp. 761–76.

Foster, J.E. and Shorrocks, A.F. (1988a). �Poverty orderings�, Econometrica, vol. 56, pp. 173–7.Foster, J.E. and Shorrocks, A.F. (1988b). �Poverty orderings and welfare dominance�, Social Choice Welfare, vol. 5,

pp. 179–98.Foster, J.E. and Shorrocks, A.F. (1988c). �Inequality and poverty orderings�, European Economic Review, vol. 32,

pp. 654–62.Grosh, B. and Glewwe, P. (1998). �Data watch: the World Bank’s living standards measurement study

household surveys�, Journal of Economic Perspectives, vol. 12, pp. 187–96.Howes, S. (1996). �A new test for inferring dominance from sample data�, Discussion Paper, STICERD,

London School of Economics.Kakwani, N.C. (1980). �On a class of poverty measures�, Econometrica, vol. 48, pp. 437–46.Sahn, D.E. and Stifel, D.C. (2000). �Poverty comparisons over time and across countries in Africa�, World

Development, vol. 28(12), pp. 2123–55.Sahn, D.E., Stifel, D.C. and Younger, S. (1999). �Inter-temporal changes in welfare: preliminary results from

nine African countries�, CFNPP Working Paper no.94.Sen, A.K., (1979). �Personal utilities and public judgment: or what’s wrong with welfare economics?�,

Economic Journal, vol. 89(355), pp. 537–58.Sen, A.K. (1985). Commodities and Capabilities, Amsterdam: North-Holland.Sen, A.K. (1987). �The standard of living: lecture II, lives and capabilities�, in (G. Hawthorn, ed.), The Standard

of Living, pp. 20–38, Cambridge: Cambridge University Press.Shorrocks, A.F., and Foster, J. (1987). �Transfer sensitive inequality measures�, Review of Economic Studies,

vol. 54, pp. 485–97.Tsui, K. (2002). �Multidimensional poverty indices�, Social Choice and Welfare, vol. 19, pp. 69–93.United Nations Development Program (1990). Human Development Report, New York: Oxford University Press.

968 [ O C T O B E R 2006]T H E E C O N O M I C J O U R N A L


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