Multidimensional Poverty Measurement with the Weak Focus Axiom

Jean-Yves Duclos

Description

Niveau: Supérieur, Doctorat, Bac+8
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

Sujets

• Duclos
• Bresson

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 21
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

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: fl[e-mail]. 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 individualsandQitscomp