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Multidimensional Poverty Measurement with the Weak Focus Axiom

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Multidimensional Poverty Measurement with the Weak Focus Axiom FLORENT BRESSON? Cemafi, Université de Nice Sophia Antipolis version 0.16?† 23rd February 2009 Abstract The present paper defines an axiomatic framework for multidimensional pov- erty measurement that fits a weak version of the focus axiom and is consistent with Duclos, Sahn, and Younger's (2006) “well-being” approach of poverty identi- fication. This slackening of the tradtional axiomatic framework is appealing for two reasons. First, regarding the issue of poverty identification, the approach is less restrictive than the traditional “union” and “intersection” views. Secondly, concerning the issue of aggregation among attributes for each individual, it al- lows for substitution effects between meagre and non-meagre attributes as well as the existence of varying needs considerations. As an illustration of these de- velopments, we introduce two extensions of family of multidimensional poverty indices defined by Bourguignon and Chakravarty (2003). JEL classification: I32, C00. Key words: Multidimensional poverty measure, focus axiom, poverty identification. 1 INTRODUCTION Since many years, the increasing emphasis on the multidimensional essence of pov- erty has naturally entailled the search of adequate measures for that phenomenon. First attempts have been realized using multidimensional extensions of the head- count index, but, as in the case of monetary poverty (Sen, 1976), such measures exhibit some features that make them inappropriate for the evaluation of poverty changes.

  • since many

  • into account

  • view since

  • individual i?

  • concerning multidimensional

  • any difference between

  • pov- erty measurements

  • multidimensional poverty


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MultidimensionalPovertyMeasurement
withtheWeakFocusAxiom
∗FLORENT BRESSON
Cemafi, Université deNice Sophia Antipolis
†version0.16α
23rdFebruary2009
Abstract
The presentpaper defines an axiomaticframeworkfor multidimensionalpov-
erty measurement that fits a weak version of the focus axiom and is consistent
with Duclos,Sahn,andYounger’s(2006) “well-being”approachofpovertyidenti-
fication. This slackening of the tradtional axiomatic frameworkis appealingfor
two reasons. First, regardingtheissue ofpovertyidentification, the approachis
less restrictive than the traditional “union” and “intersection” views. Secondly,
concerning the issue of aggregation among attributes for each individual, it al-
lows for substitution effectsbetweenmeagreand non-meagreattributes as well
as the existence of varying needs considerations. As an illustration of these de-
velopments, we introduce two extensions of family of multidimensional poverty
indicesdefinedbyBourguignonandChakravarty(2003).
JELclassification: I32,C00.
Keywords: Multidimensionalpovertymeasure,focusaxiom,poverty
identification.
1 INTRODUCTION
Since manyyears, the increasingemphasison themultidimensionalessence of pov-
erty has naturally entailled the search of adequate measures for that phenomenon.
First attempts have been realized using multidimensional extensions of the head-
count index, but, as in the case of monetary poverty (Sen, 1976), such measures
exhibit some features that make them inappropriate for the evaluation of poverty
changes. In order to deal with the non-binary nature of poverty and the necessity
∗Contact: florent.bresson@unice.fr. IwouldliketothankJean-YvesDuclosforhishelpfulcomments
onanearlierversionofthisdocument.
†Theαmeansthatthisversionisaverypreliminaryversionthatislikelytoincludemanymistakes
andtobesubstanciallyrewritten. Asaconsequence,itshouldnotbequotedor cited.
11 INTRODUCTION
toweighdifferentlyindividualswithunequaldegrees ofdeprivations,manyapproa-
ches have been suggested to define more appealing indices. For instance, some in-
dices like the ones defined by Cerioli and Zani (1990) and Cheli and Lemmi (1995)
have been proposed on the basis of the theory of fuzzy sets, and are now widely
1used in empirical studies. Starting with the studies of Chakravarty, Mukherjee,
and Ranade (1998), some other authors have also tried to build poverty measures
2using concepts and tools developed for unidimensional poverty measurement. Of
particular importance is the role the axiomatic approach in this literature since it
underlines the ethical aspects of poverty measurements. Indeed, the mathematical
properties of any indices aimed at the evaluation of some aspect of social welfare
are generally not trivial because they partly reflects the ethical preferences of the
social evaluator with respect to social justice issues (see for instance Dalton, 1920,
3Kolm, 1969, Blackorby and Donaldson, 1980). As poverty measures are generally
designed for the evaluation of economic and social policies, it is necessary for policy
makers to have at their disposal some measures that trully fit the society’s norms
in order to avoid undesirable outcomes. The most refined compass is not useful if it
doesnotreallyindicatethenorthern direction. Andsoisitforsocialindicators.
Uptonow,themeasuressuggestedinthelitteraturehaveshowna greatvariety
of feelingswithrespect to thedesirablecriteria thatshouldbefulfilledby anappro-
priate measure of multidimensional poverty. However, our opinion is that all these
measures may be based on a somehow narrow axiomatic framework with respect
to the identification of the poor as well as for the aggregation of indivual informa-
tions into a single index. In both cases, the proposed measures do not fully take
into account the potential relationships in tems of well-being between the different
attributes which level of deprivations haveto be gathered. For instance, the indices
definedbyTsui(2002)andBourguignonandChakravarty(2003)allowfortrade-offs
between two attributes for which individualsare simultaneouslydeprived, but only
for a limited portion of the poverty domain. Moreover, substitution is not allowed
for the definition of the population of the poor. Indeed, the issue of identification
generallyreliesonthenumberofdeprivationsexperienced byeachsingleagentand
independantly defined with respect to a fixed set of poverty lines. However, argu-
ments can be set against that way of thinking poverty identification. Notably, if we
believe that poverty should be thought in terms of well-being shortages, it may be
relevant to consider that deprivations in each dimension are not independantly de-
fined, and that larger gaps in one dimension may entail greater needs in any other
1Foracomprehensiveviewofthefuzzysetapproachtopovertymeasurement,seeBettiandLemmi
(2006).
2Aside from Bourguignon and Chakravarty (2003) on which is based the measure defined in the
present paper, we can cite the indices proposed by Tsui (2002),Kocklaeuner (2006),Alkire andFoster
(2007)andChakravartyandSilber(2008).
3With respect to the conception of social justice, it is worth noting that the choice of the relevant
informationspaceanddimensions(Sen, 1979)are undoubtedlyasimportant asthe functionalform of
thechosenpoverty index.
22 THEDEFINITION OFTHEPOVERTYDOMAIN
dimensionofpoverty. Forinstance,let’ssupposethatwealthandhealtharerelevant
dimensionsof poverty. Wouldbibliccharacters Croesus andJobhavebeen suffering
from the same disability, it seems reasonable to assume that the former’s wealth
wouldhavemadeitsdisabilitylessunbearablethaninthelatter case. With respect
tothatconcern,itisthennecessarytolookforpoverty measuresthatmakepossible
substitutionsbetweenlevelsofmeagreandnon-meagreattributes.
In the present paper some axioms traditionally used for multidimensional pov-
erty measurements are slackened so as to take that criticism into account. More
precisely, we investigate the effects of the use of weak versions of the focus axiom
and builda complete axiomaticframework that fits this approach. To illustrate our
findings, we define multidimensional extensions of Bourguignon and Chakravarty’s
(2003) family of poverty measures that complies with Duclos, Sahn, and Younger’s
(2006) general view of poverty identification as well as the intuitions behind the
studiesdedicatedtowell-beingcomparisonsofnon-homogeneouspopulations(Bour-
guignon,1989,Atkinson,1992,Ebert,1997,GravelandMoyes,2008).
Thepaperisstructuredasfollows. Thefirstsectionintroducesournotationsand
thedifferent approachesusedintheliteraturetoidentifythesetofpoor individuals
inagivenpopulation. Insection3,webrieflyreviewthemainaxiomsusedtoassess
thevalidityofmultidimensionalpoverty measures,withparticularemphasisonthe
focusaxiom. Section4presentsanewmeasureofmultidimensionalpoverty. Finally,
section 5concludeswithadditionalcommentsonfurtherdevelopments.
2 THE DEFINITION OF THE POVERTY DOMAIN
LetX be an n×m matrix of the m ∈ N\{1} attributes of a population with n ∈ N
members. Each element x ∈R fromX is the jth attribute of the ith individual.ij +
For expositional simplicity, we assume that the welfare of any individual is an in-
creasing function of the quantity of each attribute. X is itself an element of the set
n nW := ∪ ∪ W withW being the set of n× m matrices with non-negativen∈N m∈N m m
elements. Asusual,anyindividualiisconsideredassufferingfromdeprivationwith
respect to attribute j if x < z where z ∈ R is the corresponding exogenousij j j ++
poverty linethatwouldbeusedinaunidimensionalpoverty analysis. Thesetofad-
mmissible vectors of poverty lines is denotedL := ∪ R . Letz andx denote them∈N i++
m-vectors that correspond respectively to the vector of poverty lines and individual
′i’s vector of attributes. For anyx andz,x ∧x is the m-vector obtained from thei i i
minimalvaluesofx andx ′ foreachattribute,i.e. min(x ,x ′ ),...,min(x ,x ′ ) .i i i1 i 1 im im

′ ′ ′In the same spirit, we definex ∨x := max(x ,x ),...,max(x ,x ) . Finally,i i i1 i 1 im im
letx bethe n-vector ofthevaluesofthejth attributeobservedinthepopulation..j
D denotes the deprivation space related to the attributes{t,v...} and corre-t,v...
msponds to the set of pointsx ∈R such that x < z ∀j ∈{t,v...}. A k-deprivationi ij j+
k mspaceD isthesubsetofR corresponding totheset ofvectorsx suchthat x < zi ij j+
32 THEDEFINITION OFTHEPOVERTYDOMAIN
D2
BxB2
AxA2
zz2
D1
λ (x )= c2 iD1,2
λ (x )= c1 i
x x z attribute1A1 B1 1
Figure1: Thedefinitionofthepovertydomainunderdifferentrival
approaches.
2for at least k attributes, i.e. in formal termsD := ∪ D ∀{t,u}⊆ {1,...,m}. Byt=u t,u
m m−1 1definition, we observe D ⊂ D ··· ⊂ D . The poverty domain, that is the set of
malladmissiblepointsinR forwhichindividualsisdeemedpoor,isnotedP. Finally,+
letP ⊆ ...,n}bethesetofpoor individualsandQitscomplement.
Since Sen’s (1976) seminal work, the study of poverty is based on the strict dis-
tinctionbetweeni)theidentificationofthepopulationofthepoor,andii)theestima-
tionofindividualdeprivationdegreesandtheiraggregationforthewholepopulation
into a synthetic index. The issue of the identification of the poor is slightly more
complex inthemultidimensionalcontextthanintheundimensionalonesincemany
rival approaches can be adopted to defineP, even if we confine our attention to the
4,5absoluteviewofpoverty.
Concerning multidimensional poverty, the issue of identification has first been
dominated by two rival approaches, that is the “union” and “intersection” approa-
ches. (Bourguignon and Chakravarty, 2002). The “union” approach states that an
individual i is poor if he is deprived in at least one of the m dimensions, in other
1words ifx ∈ D . On the contrary, the second view consists in tagging someone asi
mpoor if he suffers from deprivation in all dimensions, i.e. ifx ∈ D . The differencei
betweenthetwoapproachesisillustratedonfigure1inthetwo-dimensioncase. For
4An absolute view of poverty consists in using poverty lines that are common to the whole popu-
lation and exogeneously defined. In particular, the poverty lines are not determined by the observed
distributionoftheattributesunderconsideration.
5For the sake of simplicity, we do not review here the so-called fuzzy approach of poverty identi-
fication that is widely used for the analysis of multidimensional poverty. With the fuzzy approach,
inclusioninthesetP isnot realizedina dichotomous waybutusinga continousvariablecorrespond-
ing to the probabilityof belonging toP. As shown in Chakravarty (2006),there are close connections
between the axiomaticapproach usedfor the present paper andthe fuzzyapproach. Indeed, withany
measure Θ satisfying MON and MON , the lower weight of any individual i that is unambiguouslym R
′ ′richerthananypoor individuali canbeinterpretedasi beingmore likelyto belongtoP thani.
4
6bb
attribute22 THEDEFINITION OFTHEPOVERTYDOMAIN
instance, given the thresholds z and z , an individual whose vector of attributes is1 2
(x ,x ) (point A)belongsto thesetP withthe “union”view,butto the setQwithA1 A2
the“intersection” viewsinceheisnotdeprived withrespect tothesecond attribute.
For the measurement of poverty, it may be useful to formalize these different
m mapproaches using a dichotomous identification function ϕ :R ×R → {0,1} that+ ++
takestherespective followingformsfor the“union”and“intersection” approaches:

1 if∃j ∈{1,...,m}suchthat x < z ,ij jUϕ (x ,z) := (2.1)i 0 otherwise,

1 if x < z ∀j ∈{1,...,m},ij jIϕ (x ,z) := (2.2)i
0 otherwise.
U 1The corresponding definitions of the poverty domain are respectively P := D
I mand P := D . In both cases, the poverty frontier is shaped by the hyperplanes
6definedbyeach poverty line. However, thisapproachcanbeseen asrestrictive, not
to say inappropriate in some specific cases. For instance, in the (almost) trivialcase
of the m attributes being different income sources of a person, both the “union”and
the“intersection”approacheswouldundoubtedlyyieldaninconsistantclassification
ofthemembersofthepopulationintothesetsP andQasitcanbeseenfromfigure1.
If the different sources of income are perfectly fungible into the individual’soverall
budget,thepoverty frontiershouldbethestraightlinedefinedbytheequationx +i1
x = z + z (noted λ (x ) = c in figure 1), and the poverty domain the grey areai2 1 2 1 i
below this poverty frontier. While individual A is clearly poor, it is not the case of
B. However, both the “union” and the “intersection” approaches do not make any
difference between AandB forthedefinitionofP.
Thissimpleexemplecanbeextendedtosituationsinwhichthedifferentsources
of income are not perfect substitutes. That would the case when considering assets
that are not perfectly liquid or social benefits that are partially fungible into one’s
budget. Forinstance, ifweare interested inthestudy of householdsfor whichsome
membershavetoliveabroadsoastosendremittancestotheremainingmembers,it
necessary to take into account the imperfect substituabilityof domestic and abroad
incomes due to exchange rate variability and transfer costs. Moreover, as the dif-
ferent members do not live together, it may be preferable for each group to have
at its disposal an income — here, we assume that each group’s income is a distinct
dimension — that is sufficient to meet subsistance needs. In such a case, the pov-
ertyfrontierisundoubtedlyneitherthestraightlineofperfect substituationnorthe
frontiers corresponding tothe“union”and“intersection” approaches.
Ofcourse,themovefrommonetarypovertytomultidimensionalpovertyhasbeen
6An hyperplane is the generalization of the concept of the plane in a vector space with more than
two dimensions. For the generalizations of surfaces, the corresponding concept is the hypersurface.
Here,thehyperplanerelatedtothepoverty linez issimplytheonedefinedbytheequationx = z .j j j
53 THEAXIOMATICFRAMEWORK
motivated to a great extent by the inappropriateness of income as the sole determi-
nantofwell-being(seeforinstanceSen,1987),butthereasoningholdsifweconsider
attributes like health and education to estimate poverty, since the effect of any one
of these attributes is unlikely to be orthogonal to the level of the other one in terms
7ofwell-being.
Asconsequence,itmayberelevanttoadoptamoreflexibleapproachinwhichthe
poverty frontier is implicitly defined by some continuous and increasing well-being
m 8functionλ:R →R. Thisisthe“well-being”approachdefinedbyDuclos,Sahn,and+
Younger(2006)thatcanbesummarizedusingthefollowingidentificationfunction:

1 if λ(x ) < λ(z),iWϕ (x ,z,λ) := (2.3)i
0 if λ(x )> λ(z).i
where λ(z) can be interpreted as a “poverty level” of well being. This approach is
illustrated on figure 1 with the two poverty frontiers defined by λ (x ) = λ (z) and1 i 1
λ (x ) =λ (z). Itdeservestobenotedthateachoneofthesehypersurfacesdefinesa2 i 2
λ msetofpointsF :={x ∈R |λ(x ) =λ(z)}that,insomecases(seesection3.1),mighti iz +
be used during the aggregation step as an alternative toz. In other words, for any
multidimensionalpoverty index Θ :W×L→R andany distributionX, we maym +
′ ′′ ′ ′′ λobserve Θ (X,z ) = Θ (X,z ) ∀{z ,z } ⊂ F when considering the “well-being”m m z
approachofpovertyidentification. Finally,thecorresponding povertydomaincanbe
W m W U Idefined as P := {x ∈ R |ϕ (x ,z,λ) = 1}, with P and P being some specifici i+
realisations.
3 THE AXIOMATIC FRAMEWORK
Unsurprisingly,theaxiomaticframeworkofmultidimensionalpovertymeasurement
9islargelyinspiredbytheoneusedforincomepoverty. However,theextensiontothe
multidimensional context is sometimes not straightforward and many rival axioms
7UsingthecapabilityapproachofSen(1979,1987,1992),itwouldsurelybemorerelevanttoadopt
the “union”approach ifeachattributecorresponds to a uniquefunctionningor capability. However, in
practice,wegenerallyhavetorelyonproxiesthatdonotperfectlyfittheconceptstheyaresupposedto
be the measures of. As a consequence, we argue that more flexible views of poverty identification are
necesseryevenfor non-welfaristapproachesofwell-beinglikethecapabilityapproach.
8It is worth mentioning the intermediate approach of Alkire and Foster (2007) that consists in
tagging someone as poor if the number of meagre attributes is more than k ∈ {1,...,m}, that is if
kx ∈ D . For k = 1 (k = m), the approach corresponds to the traditional “union” (“intersection”) viewi
of poverty identification. Finally, as the social evaluator may not give the same importance to the
mdifferents dimensions, it is then necessary to weigh the different attributes using the m-vector w
Pmsuchthat w = m. Thegeneral expressionoftheidentificationfunctionisthen:ji=1
( Pm I1 if w ϕ (x ,z )> c,AF j ij jj=1ϕ (x ,z,w,c) := Pi m I0 if w ϕ (x ,z ) < c.j ij jj=1
ThisintermediateapproachcanbeconsideredasaspecialcaseofDuclos,Sahn,andYounger’s(2006)
approach.
9See(Zheng,1997)foradetailledreview.
63 THEAXIOMATICFRAMEWORK
can be suggested. In the present paper, we follow the lead of most authors and
put forward that any multidimensional poverty measure Θ should respect somem
properties that directly stem from the issues of poverty identification presented in
thelastsectionaswellasfromethicalconsiderations. Whennecessary,theseaxioms
willbediscussedintheirweakandstrong versions.
3.1 THE FOCUS AND MONOTONICITY AXIOMS
Thepropertiesconsideredinthissectiondealwiththerespectiveconsequencesofat-
tributeslevelimprovementsfor thenon-poorandthepoor. Wepresenttheseaxioms
intheirformalforms before discussingtheirrelevance.
Axiom FOC (Weak focus): Θ (Y,z) = Θ (X,z) if Y is obtained from X byW m m
addingany amountε ∈R to x ∀i∈Q, j ∈{1,...,m}.j ++ ij
Axiom FOC (Strong focus): Θ (Y,z) = Θ (X,z) if Y is obtained from X byS m m
addingany amountε ∈R to any x > z .j ++ ij j
The axiom FOC simply means that the measure is indifferent to the value ofW
the attributes of the non-poor. Thus, it implicitly states than poverty is an absolute
phenomenon. The stronger version FOC differs in that it introduces a distinctionS
between meagre (x < z ) and non-meagre (x > z ) attributes, and may also con-ij j ij j
cern poor individuals. In other words, FOC implies that any increase of the levelS
of a non-meagre attribute does not change the level of poverty. As a result we ob-
serve the following equivalence Θ (x ,z) = Θ (x ∧z,z) under FOC . It is worthm i m i S
noting that both versions ot the focus axiom are perfect substitutes considering an
“intersection” approach ofpoverty identification,butdiffer inother cases.
Asdiscussedinthenextparagraphs,thechoiceof FOC isvindicatedintwodif-W
ferentcasesthat,roughlyspeaking,aretechnicallybutnotethicallyequivalent,that
is i) when any increase in the level of the level of a non-meagre attribute for a poor
individualmay lower poverty due to substitution effects (thereafter called the “sub-
stitution” view), and ii) when varying poverty lines are used (thereafter called the
“varyingneeds” approach). Atfirst glance, FOC seems themore naturalapproachW
whenusingthe“well-being”approachofpovertyidentification,butthisconclusionis
misleading. Indeed, this view relies on a well-being function that yields a complete
mordering ofthedifferent “baskets”x overthedomainR ,andso, maybecharacter-i +
ized by some substitution degree between the different attributes whateverx ∈ P.i
Consequently, any increase of the level of a non-meagre attribute of a poor person
should increase its well-being, hence the corresponding poverty level. However, in
Duclos, Sahn, and Younger (2006), the function λ is used only for the definition of
the space P, not for the assessment of the degree of poverty. This means that it is
possible with this general framework to evaluate poverty on a different basis than
well-being. For instance, using observed functionnings as the relevant information
73 THEAXIOMATICFRAMEWORK
B zz2
λFzAxA2
z (1+δ )1 1
x z z attribute1A1 1 A1
Figure2: Extendedsubstitutionspaceandvaryingpovertylines.
basis for poverty measurement would entail thinking poverty in terms of function-
ingsdeprivations for those who do not achieve the level of well-being corresponding
toz. This slight distinction deserves to be stressed but does not necessarily argue
in favor of FOC , since subsitution effects may also have to be taken into accountS
witha different information basis. So, the choice ofthe relevantversion of thefocus
axiomcruciallydependsonthenatureoftheattributesusedfortheanalysisaswell
asonthewaypoverty isthought.
In fact, the axioms FOC and FOC differ with respect to the definition of theW S
ksets of points S in which substitution effects between two attributes may occur,
whatever their respective position with respect to the corresponding poverty lines.
Letx denote the (m−1)-vector obtained fromx after dropping its jth elementi,−j i
and δ := δ (x ,z) be the excess of the level of the jth attribute, expressed as aj j i,−j
percentage of the corresponding poverty line, up to which substituability between
this attribute and meagre attributes is allowed. For consistency with FOC , weW
have to impose δ = 0 if x > z where z is the counterpart of x withj i,−j i,−j−j −j
10 krespect toz. Using the vectorδ of functions δ , the space S , k ∈ {2,...,m}, canj
be defined as the set of points belonging to the poverty domain for which there is at

kleast k attributes such that x < z 1+δ (x ) . In other words, S is the set inij j j i,−j
which the poverty effect of any decrease in the level of an attribute can be partially
compensated by some increase of the quantity of another attribute. As in the case
k m m−1 2of the deprivation spaces D , we observe S ⊆ S ··· ⊆ S . Under FOC , it canS
k keasily be shown that these substitution spaces are S := {x ∈ D |λ(x ) < λ(z)}iis
whatever poverty identification approach is used. As a consequence, FOC entailsS
kthat substitution effects withinS are possible between the sole meagre attributes.s
10In order to save space, δ (x ,z) will simply be denoted by δ (x ). In section 4.2, we willj i,−j j i,−j
′assume that the functions δ and δ are non-decreasing fonctions of the elements of z in order toj j
complywiththe NDZ axiom(cf. section3.2).
8
bb
attribute23 THEAXIOMATICFRAMEWORK
kOn figure 2, this set is identified by the dashed area. With FOC , any space SW
k δcan be extended to any set included inP but includingS . LetZ denote the outers z
m m kfrontierofS inR . Finally,wenoteS ⊆P thesetofpoorindividualswhichvector+
kx is located in the substitution space S . Figure 2 illustrates this idea in the casei
oftwoattributeswithsubstitutionallowedwithrespecttothefirstattribute. There,
2the setS corresponds to the grey area whileP is obtainedby adding the lightgrey
2 2areatoS . Asstressedabove,thesubsitutionsurfaceS isnecessaryincludedinthe
poverty domainP,butdoes notnecessarilyfitthatpoverty domain.
Anotherwayofvindicatingtheuseoftheaxiom FOC istoassumethatthevec-W
torz isnotappropriateforevaluatingdeprivationsineachdimensionsforeachpoor
individual. For instance, let’s consider income and health as the relevant dimen-
sions for poverty analysis, and two poor individuals A and B with the same income
x < z , but different health levels, a situation depicted in figure 2. More precisely,1 1
suppose that individual B’s health level is exactly equal to the corresponding pov-
erty line z while individual A is handicapped, i.e. x < z . Indeed, this health2 A2 2
shortage implies that A is poorer than B since he suffers from deprivation in that
dimension, but we may go a step further and aknowledge that person A’s handicap
hasalsoconsequencesintheincomedimension. Thehandicapgeneratesspecificex-
penditures (long-term medical treatments, protheses...) and increases the cost of
otherexpendituresliketransportthatsurelymakethepovertylinez inappropriate1
for person Aasitsincomelevelcannotyieldthesameconsumption level asforindi-
vidual B. Thus, while A and B share the same income level, we may also consider
that Asuffersfrom alargerdegree ofdeprivation that B intheincomedimension.
In this exemple, this larger deprivation in the income dimension for the handi-
capped person will be taken into account if the related deprivation level is assessed

11withrespecttoz =z 1+δ (x ) insteadofz . Inotherwordsz 1+δ (x ) canA1 1 1 A2 1 1 1 A2
be thought as the specific income poverty line for those whose health level is equal
δto x , andZ the set of poverty lines issued from the vectorsz andδ. The idea ofA2 z
poverty lines that varies with the value of other attributes has already be explored
in the unidimensional poverty and inequality literatures, but its extension to mul-
tidimensional evaluations of welfare has not yet been the subject of comprehensive
12investigations. Moreover,authorsgenerallyconsiderattributes,likethesizeofthe
household,thatarenotrelevantasadimensionofpoverty. Varyingpovertylinesare
alsoimplicitlyconsidered in Duclos, Sahn,andYounger(2006),butour work differs
δinthatthesetofpovertylinevectorsZ doesnotnecessarilymergewiththepovertyz
11As a consequence, the expression “meagre attribute” refers to those attributes j which level is` ´
below their corresponding specific poverty line z 1 + δ (x ,z) when using this “varying needs”j j i,−j
viewofpoverty aggregation.
12Ontheuseofdifferentpovertylinesforheterogenouspopulationsinaunidimensionalcontext,see
notably Bourguignon (1989),Atkinson (1992), Chambaz and Maurin (1998) and Duclos and Makdissi
(2005). To our knowledge, Gravel and Moyes (2006, 2008) are the sole studies that explicitly investi-
gatesthatissueinamultidimensionalsetting.
93 THEAXIOMATICFRAMEWORK
λfrontierF asinfigure2.z
Thecriticisms inducedbythe “substitution”andthe“varyingneeds” approaches
presentedinthepreviousparagraphscanbesummarizedwiththefollowingversion
ofthefocusaxiom:
Axiom FOC (Extended strong focus): Θ (Y,z) = Θ (X,z) if Y is obtainedE m m
from X by adding any amount ε ∈ R to any person i such that x > z 1 +j ++ ij j

δ (x ) .j i,−j
The axiom FOC is weaker than FOC since it does not impose any conditionE S
on Θ for values z 6 x < z 1+δ (x ) , but stonger than FOC as it adds am j ij j j i,−j W

restriction on Θ for z 1 + δ (x ) 6 x < z˜ where z˜ is the jth coordinate ofm j j i,−j ij j j
λtheprojection ofx alongthecorresponding axisonthehypersurfaceF . Theaxiomi z
mFOC isequivalentto FOC ifandonlyifS =P.E W
Now,weconsiderincrementsthatmayexplicitlyleadtopoverty variations:
Axiom MON (Weakmotonicity): Θ (Y,z)6 Θ (X,z) ifY is obtained fromXW m m
byaddingany amount ε ∈R tox ∀i∈P, j ∈{1,...,m}.j ++ ij
Axiom MON (Monotonicity): Θ (Y,z) < Θ (X,z) if Y is obtained from X bym m
addingany m-vectorεwhose elements are strictly positive tox ∀i∈P.i
Axiom MON (Restricted strong monotonicity): Θ (Y,z) < Θ (X,z) ifY isR m m
mobtained fromX byaddingany amount ε ∈R to x ∀i∈S , j ∈{1,...,m}.j ++ ij
The monotonicity axioms are the complement of the focus axioms. The axiom
MON significates that any increase of the quantity of an attribute does not in-W
crease the level of poverty, while MON states that any increase of the quantity of
all the attributes of a poor person lowers the level of poverty. Finally, MON im-R
poses Θ to decrease for any improvement concerning individuals in the substitu-m
tion space. These stronger versions are defined so as to comply with either FOC ,W
FOC or FOC . While MON and MON both implies the respect of MON , there isE S R W
no clear relationship between MON and MON . To conclude with the monotonicityR
axioms, we can observe that, contrary to the situation expressed with the “substi-
tution” view, MON significates that the set of admissible functions δ have to beW j
restricted to those that are non-increasing with respect to elements ofx with the−j
“varyingneeds” view.
3.2 OTHER TRADITIONAL BASIC AXIOMS
To define a suitable family of poverty measures, further restrictions have to be im-
posed. The next set of axioms is traditional and is generally widely recognized as
ethically robust. Axioms are presented so as to comply with either FOC , FOCW E
or FOC , and thus may slightly differ from the versions usually presented in theS
literature.
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