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Session Number: Parallel Session 6D Time: T hursday , August 2 6 , P M Paper Prepared for the 31st General Conference of The International Association for Research in Income and Wealth St. Gallen, Switzerland, August 22 - 28, 2010 On the “ Pro - Poorness ” of Growth in a Multidimensional Context V alerie B er enger and F lorent B resson For additional information please contact: Name: Valerie Berenger Affiliation : University of Nice Email Address : valerie.berenger@unice.fr. This paper is posted on the following website: http://www.iariw.org Onthe“Pro-Poorness”ofGrowthinaMultidimensionalContextVALÉRIEBÉRENGERFLORENTBRESSONyversion0.7.1z12thMarch2010AbstractThispaperrepresentsarstattempttogathertheissuesofgrowth“pro-poorness”andmultidimensionalpovertyassessments.Moreprecisely,wesuggesttheuseofse-quentialdominanceprocedures(Bourguignon,1989,Atkinson,1992,JenkinsandLambert,1993)totestthe“pro-poorness”ofobservedgrowthspellswhenpovertyismeasuredonthebasisofincomeandsomeotherdiscretewell-beingattribute.SequentialproceduresarealsousedtogetgraphicaltoolsthatareconsistentwiththespiritofChenandRavallion's(2003)growthincidencecurveandSon's(2004)povertygrowthcurve.Contrarytotraditionalunidimensionaltests,ourmethod-ologyallowstotakeintoaccounttheimportanceofdeprivationcorrelationsattheindividuallevelandthusmayreverseresultsobservedwiththetraditionaltoolsofgrowth“pro-poorness”check.AnillustrationofourapproachisnallygivenusingTurkishdatafortheperiod2003-2005.JELclassication:I32,C00.Keywords:“Pro-poor”growth,growthincidencecurve,sequentialsto-chasticdominance.1INTRODUCTIONThedenitionoftheMillenniumDevelopmentGoalsin2000bytheinternationalcom-munitywasamajorbreakdownwiththepreviousparadigmoftheWashingtoncon-sensusanditsimplicitreferenceto“trickledown”theories.OneremarkablefeaturewastherehabilitationofChenery,Ahluwalia,Bell,Duloy,andJolly's(1974)advocacyinfavorofintroducingredistributiveconcernsintogrowthpoliciesinthedeveloping Cema,UniversitédeNiceSophiaAntipolis.Contact:valerie.berenger@unice.fr.yLÉO,CNRS-Universitéd'Orléans.Contact:orent.bresson@unice.fr.zThemeansthatthisversionisapreliminaryversionthatislikelytoincludesomemistakes.Asaconsequence,itshouldnotbequotedorcitedwithoutexplicitpermissionoftheauthor.1 1INTRODUCTION world.1Indeed,sincelate1990s,manysocialresearchershaveforcefullyarguedinfavorofassigningtogrowthonlyaninstrumentalrolewithrespecttopovertyissues.Inotherwords,povertyalleviationshouldnotberegardedasadesirablesideeffectofgrowthbuttheultimategoaltobereachedinthespiritofformerWorldBank'spresidentRobertMcNamaradesiretoshiftthefocustowardstargetedpovertyreduc-tion.However,howbestthisgoalcanbemetwasanopenandcomplexquestion,andresearchershavespentmanyeffortsinyieldinglessonsfromtheempiricsofgrowthandpoverty.Inparticularhasbeendevelopedaliteraturerelatedtotheidenticationof“pro-poor”growthspells,i.e.growthspellsthatcorrespondstoamarkedimprove-mentregardingthestateofpoverty.Thenatureofthisbiasinfavorofthepoorhasentailedmanydebates,notablyconcerningthedesirabiltyofobservingapovertyalle-viationeffectofinequalityreductiontotagagrowthpatternas“pro-poor”(KakwaniandPernia,2000,Ravallion,2004,Zepeda,2004,Osmani,2005),butthetheoreticalframeworktobeusedforempiricalassessmentisnowwelldened(Duclos,2009).ThecommitmentoftheinternationalcommunitytotheachievementoftheheightMillenniumDevelopmentGoalswasalsoanofcialrecognitionofthemultidimen-sionalnatureofpoverty.Itiswellknown(seeforinstanceSen,1987,1992,Streeten,1994)thatthelinkagesbetweenincome(orexpenditure)andwell-beingarenotstraight-forwardandhingeonmanydeterminantslikeidiosyncraticcharacteristicsormarketfactors.Asaresult,theefciencyofpovertyreducingpoliciesshouldalsobeassessedonthebasisofthesatisfactionofnon-incomeneedslikehealth,educationorpartic-ipationtosociallife.Ifpovertyhastobethoughtandmeasuredtakingamultidi-mensionalapproach,itisthennecessarytohavealookatthe“pro-poor”natureofgrowthbeyondthesolemonetaryaspectsofpoverty.Therecentabundantliteratureonthesetwoconceptshasevolvedinaparallelway.Surprisingly,veryfewattempshavebeencarriedoutinordertoincludetheadditionalinformationaasociatedwithotherdimensionsofwell-beingalongsidethemonetaryonewithintheassessmentofthe“pro-poor"natureofgrowth.Atourknowledge,theonlystudiesthatdealwiththatissueareKlasen(2008)andGrosse,Harttgen,andKlasen(2008)thatsuggestmakinguseofthetoolsdevelopedfor“pro-poor”growthteststoinvestigatethedistributionofchangeswithrespecttonon-incomeattributes.Thenon-incomegrowthincidencecurveproposedinthesestudiesallowsforwiden-ingthescopeof“pro-poor”growthanalysesandmayhighlightpotentialdiscrepenciesbetweenprogressesinthemonetaryandnon-monetarydimensions.However,thesegraphicaltoolsonlyfocusonthemarginaldistributionsofwell-beingattributesandthus,donottakeintoaccounttheadditionalinformationprovidedbythejointdis-tributionofthatattributes.Manyauthors(AtkinsonandBourguignon,1982,Tsui,2002,BourguignonandChakravarty,2002)havestressedtheimportanceofcorrela-tionsbetweenthedistributionsofthedifferentattributesinmultidimensionalpoverty 1Today,suchpoliciesaregenerallycalledinclusivegrowthpolicies.2 2“PRO-POORNESS”WITHUNIDIMENSIONALPOVERTY 2“PRO-POORNESS”WITHUNIDIMENSIONALPOVERTYLetyi2Rbethelevelofsomemonetaryvariable,likeincomeorexpenditure,fortheithpersonofagivenpopulationofsizen2N.2Thedistributionofincomeamongthepopulationcanthenbedescribedbythen-vectory:=fy1;:::;yng.Inordertoeasethecomparisonsbetweendistributionsofdifferentsizes,itisoftenpreferabletousetheunivariatecumulativedistributionfunction(cdf)F(z;y).Thatcdfreturnstheproba-bilityp2[0;1]ofpickingoutofyanincomewhichvalueislessthanthethresholdz.Itisworthnotingthat,inthecontextofpovertyanalysis,thecdfcorrespondstothewidelyusedpovertymeasureknownastheheadcountindex0.Inthepresentsection,monetarypovertyisrstassessedusingthefollowingclass1ofadditivepovertymeasures:(y;z):=Zz0(y;z)dF(y;y)(1)with(z;z)=0,@=@z�0,@=@y60ifyz,@=@y=0ify&#x-531;&#x.230;zsothatthemeasurecomplieswiththetraditionalaxiomsoffocus,weakmonotonicity,continu-ity,anonymity,population,non-decreasingnesswithrespecttothepovertylineandsubgroupadditivity.3Thatclassofsubgroupadditivepovertymeasures(FosterandShorrocks,1991)isverygeneralandincludesthemostwidelyusedpovertymeasuresliketheonesuggestedbyWatts(1968)andFoster,Greer,andThorbecke(1984).4Herewewouldliketostresstheparticularimportanceoftheanonymityaxiomthatstatesthatincomeisthesolerelevantvariabletobeusedtodiscriminatepeopleforpovertyanalysis.Inequation(1),therespectoftheanonymityaxiomthenentailsthattheindividualpovertyfunctionisthesameforeachindividual.Thiscrucialassumptionwillbepartiallyslackenedinsection3whenindividualswithdifferentneedswillbeconsidered.Asstressedintheliterature(KakwaniandPernia,2000,ChenandRavallion,2003,Kraay,2006),whetheranobservedgrowthpatternis“pro-poor”ornotcruciallydependsonthesocialevaluator'sdenitionofwhatmaybea“pro-poor”growth.Inparticular,itreliesonthewayanyadditionalincomeshouldbesharedbetweenthedifferentmembersofthepopulationsoastogetagrowthpatternthatisneither“pro-poor”nor“anti-poor,”butethically“neutral.”Asourworkisorthogonalwithrespecttothisspecicpoint,wereferheretothegeneraldenitionof“pro-poor”growthproposedbyDuclos(2009).ForDuclos,theassessmentofthe“pro-poorness”ofgrowthbetweenyearstandt+1alwaysimpliesthecomparisonofthepovertylevelint+1withthelevelthatwouldhavebeenobservedforsomeconterfactualdistributiondened 2Forthesakeofsimplicity,wewillconsiderthatydenotestheincomelevel,butthischoicedoesnotprecludeusinganyotherconceptsthatwouldberelevanttoassessmonetarypoverty.3SeeZheng(1997)foracomprehensivereviewoftheaxiomaticframeworkusedforunidimensionalpovertyanalysis.4Intherstcase,theindividualfunctionisdenedby(x;z):=logxlogz.WiththeFoster,Greer,andThorbecke's(1984)classofpovertymeasures,thatfunctionbecomes(x;z):=(1x=z),�0.4 2“PRO-POORNESS”WITHUNIDIMENSIONALPOVERTY measuresdenedinequation(1)isrestrictedtoindicesthatrespect@2=@y2�0,wegettheclassofpovertymeasures2thatbelongsto1andcomplieswiththeweaktransferaxiom(Sen,1976).Accordingtotheweaktransferaxiom,anincomelossforapoorindividualdoesnotraisepovertyifitisatleastcompensatedbyanincreaseofthesameamountforapoorerperson.Robustnesstestsbasedonthisaxiomarethenmorepowerfulthanrstorderstochasticconditionsastheydonotrequireincomeimprovementateachquantileofthepopulationduringtheperiodofinterest.Moreprecisely,secondorderdominancetestsrequiretheuseofthepovertygapfunctionGsuchthat:G(z;y):=Zz0(zy)dF(y;y):(9)Thatfunctionsimplyreturnstheaverageshortfallwithrespecttothepovertylinezgiventheincomedistributiony.Therelationshipbetweengrowth“pro-poorness”andtheclassofpovertymeasures2isthensummarizedbythefollowingproposition:Proposition2.Foragivencounterfactualscenario\randagivenmaximumvaluez+forthepovertyline,thestatementthatthegrowthpatternobservedbetweenyearstandt+1is“pro-poor”isweaklyrobustwithrespecttothechoiceofthepovertymeasureamongthefamily2andthevalueofthepovertylinezifandonlyif:G(z;yt+1)Gz;\r(yt)
608z6z+;(10)withaleastonevaluez2[0;z+]suchthat:G(z;yt+1)Gz;\r(yt)
0:(11)A“pro-poorness”testproposedbySon(2004)andrelatedtotheclassofpovertymeasures2isbasedonthepovertygrowthcurve(PGC)thatplotsthegrowthrateofthemeanincomeofthebottomppercentofthepopulationwhenindividualsarerankedbyincreasingorderofincome.Moreformally,thePGCisdenedby:g2(p;yt;yt+1):=Zp0F1(u;yt+1) F1(u;yt)1du:(12)ItcaneasilybecheckedthatthecomparisonoftheobservedPGCwiththeonecorrespondingtothecounterfactualdistributionyieldsacriterionthatisequivalenttotheonepresentedinproposition2.8Corollary2.Forsomegivencriterion\randagivenmaximumvaluez+forthepovertyline,thestatementthatthegrowthpatternobservedbetweenyearstandt+1is“pro-poor”isweaklyrobustwithrespecttothechoicesofthepovertymeasureamongthe 8OnthepowerofthePGCfor“pro-poorness”tests,seeDavis(2007).7 3“PRO-POORNESS”WITHMULTIDIMENSIONALPOVERTY andsubgroupadditivityaxioms.12Moreover,itisalsoassumedthat:@ @yi(yi;xk;zk)6@ @yi(yi;xk+1;zk+1);8k2f1;:::;K1g:(16)Itcaneasilybeseenthat11.Thatclassofunidimensionalpovertymeasuresisobtainedforzk=z8k2f1;:::;Kgandwhenthevalueof@=@ydoesnotvarywiththelevelofxi.Finally,theconditionexpressedinequation(16)isstandardintheliteratureonmultidimensionalinequalitiesandpoverty(AtkinsonandBourguignon,1982,Tsui,2002)andisrelatedtotheaxiomknownasthenon-decreasingnessundercorrelationswitches.Thataxiomstipulatesthat,giventwoindividualswithendow-ments(yA;xA)and(yB;xB),apermutationofthevaluesofthesetwovectorssothatAcanbesaidunambiguouslypoorerthanB,shouldnotlowerthepovertylevel,otherthingsbeingequal.Wenowturntotheissueof“pro-poorness”evaluation.Intheprevioussection,theassessmentofthe“pro-poorness”ofgrowthwasperformedonthebasisofacounterfac-tualincomedistribution\r(yt).Thatdenitionof“pro-poorness”isconsistentwiththeincome-basedapproachofpovertybutmaynotbeappropriatewhenotherattributesaretakenintoaccount.Indeed,inthecontextofoursetting,weareconcernedwiththeevolutionofthewholematrixX,sothatgrowthbetweentheyearstandt+1willbedeemed“pro-poor”foragivencounterfactualbenchmark,agivenpovertymeasureandagivensetofpovertylineszifandonlyif:(Xt+1;z)(Xt);z
60:(17)Themaindifferencewiththedenitioncorrespondingtoequation(2)consistsinthedenitionofthecounterfactualscenariothatgivesmorelatitudeforthesocialevaluator.Indeed,aswemayobservesimultaneousvariationsofthevectorsyandx,itisthennecessarytoaskwhethertheevaluationshouldbeperformedonthebasisofacounterfactualdistributionforthedistributionoftheindexx.Wethenhavetodistinguishthesituationsinwhichthecounterfactualmatrix(Xt)isobtainedfromXtbysimplychangingitsincomevectoryt,andcasesinwhichthenon-incomevectorxtisnotnecessarilyleavedunchanged.Lettherstsituationbecalledthe“incomepro-poorness”ofgrowthandthesecondone“well-beingpro-poorness”ofgrowth.Toavoidconfusion,let\ryand\rxrespectivelydenotethefunctionsusedtodenethecounterfactualdistributionsoftheincomeandnon-incomevariables.Thecaseof“well-beingpro-poorness”deservessomeinterestbecausethecounter-factualdistribution\rx(xt)ofthenon-incomeitemmaybeslightlymorecomplexthattheonecorrespondingtoindividualincomes.Themostimportantquestioniswhether\rx(xt)shouldbeexogeneouslyorendogeneouslydenedwithrespecttotheobserved 12Foracomprehensivereviewoftheaxiomsusedformultidimensionalpovertymeasurement,seeBresson(2009).11 3“PRO-POORNESS”WITHMULTIDIMENSIONALPOVERTY growthpattern.Intheformercase,wemaychoosetodene\rx(xt)usingtheinitialandnaldistri-butionsxtandxt+1,anddissociateitfromobservedchangesintheincomedimension.Itisworthemphasizingthat,duetotheparticularnatureofthevariablex,thechoiceof\rxisobviouslyknottierthanwithincome.Hencetherelativeandabsolutecoun-terfactualfunctions\rrand\racannotbeusedwithourgeneralsettingasthevariablexisordinal—itwouldnotmakesenseforinstancetoapplyagivengrowthrateonqualitativedata.Infact,thiscriticismprevailsforallconceptionsofthecoun-terfactualscenariothatrelyonadistributionalneutralapproachofgrowthandusesmean-baseddenitionsofinequality.Withrespecttothatissue,apromisingsolu-tionistheuseofmedian-basedapproacheswithinequalitybeingthoughtintermsof“distance”tothemedianvalue(AllisonandFoster,2004).13Finally,withthecounter-factualfunction\rp,thenon-incomevectorissimplytheinitialdistributionxtsothat“incomepro-poorness”isaparticularcaseof“well-beingpro-poorness.”Onthecontrary,wemayfeelthat\rx(xt)shouldbecomputedonthebasisofsomestatisticalortheoreticalrelationshipbetweenxandy,andthecounterfactualdistri-butionofincome\ry(yt).Morecomplexdesignscanalsobechosen,usingCGEmodelswithmicro-simulationsexercicessoastofullytaketheeffectsofeconomicgrowthintoaccount.Finally,whateverthechosenprocedure,thenatureofthevariableusedfortheindexxhasalsotobetakenintoaccountasitmaybebounded(Klasen,2008).Inordertosavespace,wenowintroducethefollowingnotation:
\rt;t+1Fk(z;y):=qk(xt+1)Fk(z;yt+1)qk\rx(xt)
Fkz;\ry(yt)
;(18)withqk(x)beingtheshareofobservationsfromxwhichvaluesareequaltoxk.Thepropertiesoftheclassofpovertymeasures1thenleadtothefollowingresult:Proposition3.Foragivencounterfactualscenarioandagivenvectorz+ofmax-imumvaluesforthespecicpovertylines,thestatementthatthegrowthpatternob-servedbetweenyearstandt+1is“pro-poor”isweaklyrobustwithrespecttothechoiceofapovertymeasurefromtheclass1andthevalueofthepovertylineszifandonlyif:jXk=1
\rt;t+1Fk(z;y)608z6z+j;j2f1;:::;Kg;(19)withaleastoneintegerj2f1;:::;Kgandonevaluez2[0;z+j]suchthat:jXk=1
\rt;t+1Fk(z;y)0:(20)Thecriterionsuggestedinproposition3referstotheonerstsuggestedinBour- 13Here,theword“distance”doesnotrefertothetraditionaleuclideandistancebuttothenumberofcategoriesseparatingtwovaluesoftheindexx.12 3“PRO-POORNESS”WITHMULTIDIMENSIONALPOVERTY guignon(1989)anddevelopedbyAtkinson(1992),JenkinsandLambert(1993),andChambazandMaurin(1998),butappliedtothequestionoftheassessmentof“pro-poor”growth.Theseconddifferencewithrespecttothesestudiesisthatthehet-erogeneityofthepopulationisnotgraspedbythehouseholdsize,butbyanysetofindividualcharacteristicsthatcanbeconsideredasrelevantdimensionsofpoverty.Intheprevioussection,wehaveseenthattheconditionstobemettoconcludeinarobustmannerwhethergrowthhasbeen“pro-poor”couldalsobeexpressedwiththehelpoftheGIC(cfcorollary1).Inmostcases,thatequivalencecannotbeobserved,exceptwhenthecounterfactualdistribution\rx(xt)isthesameasthedistributionofxobservedinyeart+1.Forthatparticularcase,itisnecessarytodenethekpartialquantilefunctionas:F1k(p;X):=minnyit2yktjFyit;ykt�po;(21)withyktbeingthesubsetofvaluesfromytcorrespondingtoindividualswhichvalueoftheindexxisnotgreaterthanxk.ThefunctionF1k(p;X)returnsthevalueofin-comeycorrespondingtothep-thcentileofthesubpopulationoftype1tokrankedbyincreasingvalueofincome.Fork=K,thatfunctionsimplybecomesthetraditionalquantilefunctionpresentedinequation(5).Usingthatinstrument,wecanthenpro-posetheuseofthefollowingk-GIC:g1;k(p;Xt;Xt+1):=F1k(p;Xt+1) F1k(p;Xt)1;(22)thatcorrespondstotheincomegrowthrateofthep-thpercentileofthesubpopulationoftype1tokconsideringthenon-incomeattribute.Dominancecanthenassessedbycomparingthevaluesofthatfunctionwiththecorrespondingk-GICforthecounter-factualdistribution(Xt)forthebottompartofthepopulation.Ourresultsarethensummarizedbythefollowingcorollary:Corollary3.Foragivencounterfactualscenario\randagivenvectorz+ofmaximumvaluesforthespecicpovertylines,thestatementthatthegrowthpatternobservedbetweenyearstandt+1is“pro-poor”isweaklyrobustwithrespecttothechoiceofpovertymeasureamongthefamily1ifandonlyif:g1;k(p;Xt;Xt+1)g1;kp;(Xt);Xt+1
�08p6F(z+k;ykt);k2f1;:::;Kg;(23)withaleastoneintegerj2f1;:::;Kgandonevaluep2h0;F(z+j;yjt)isuchthat:g1;j(p;Xt;Xt+1)g1;jp;(Xt);Xt+1
�0:(24)Theconditionsuggestedwithcorollary3suitsbestsituationssuchthatthedis-tributionofthenon-monetaryvariablexistime-invariant.However,thatresultcan13 3“PRO-POORNESS”WITHMULTIDIMENSIONALPOVERTY easilybeextendedtothecaseofvariabledistributionsofthatindexx.Indeeditcanbeshownthatasufcient,butnotnecessary,conditionforgrowthtobedeemed“pro-poor”betweentandt+1giventhecourterfactualscenarioandthesetofpovertylinesz+istocomplysimultaneouslywiththeconditionsexpressedincorollary3and:F(xj;xt+1)6Fxj;\rx(xt)
8j2f1;:::;Kgsuchthatzj�0:(25)3.2SECOND-ORDERSTOCHASTICSEQUENTIALDOMINANCEANDk-PGCAsinthecaseofhomogenouspopulations,thetestsuggestedintheprevioussectionmaybeunconclusive.Inordertheincreasethepowerofthetest,itisthennecessarytoturntoareducedsetofpovertymeasures2.Startingwiththeconditionsusedtodenetheclass1,weimposethefollowingadditionalrestriction:@2 @y2(yi;xk;zk)�@ @y2(yi;xk+1;zk+1)�0;8k2f1;:::;K1g:(26)Thecondition(26)canbedecomposedintwoparts.Therstonerelatestothenon-negativityofthesecond-orderderivativeofthefunctionwithrespecttoincome.Thisnon-concavityassumptioniswell-knowninthepovertyandinequalityliterature,andsignicatesthatprogressivetransfersofincome—atransferissaidprogressiveifitreducesinequalities—withinthesetofindividualswiththesamevalueoftheindexxdonotraisethepovertylevel.Thesecondpartofcondition(26)isthenon-increasingnessof@2=@y2withrespecttothevalueofx.Thatassumptionindicatesthattherearediminisingreturnsofprogressivetransfersaswemovetolessneedyindividualsforgivenlevelsofincome.14,15LetGk(z;y)denotethevalueofG(z;y)whenF(z;y)isreplacedbyFk(z;y)inequation(9).Thatfunctionindicatesthevalueoftheaverageincomegapamongindividualsofthek-thtypeforagivenincomepovertylinez.Usingthefollowingnotation:
\rt;t+1Gk(z;y):=qk(xt+1)Gk(z;yt+1)qk\rx(xt)
Gkz;\ry(yt)
;(27)wegetthe“pro-poorness”conditionexpressedinproposition4whenpovertymeasuresoftheclass2areconsidered.Proposition4.Foragivencounterfactualscenarioandagivenvectorz+ofmax-imumvaluesforthespecicpovertylines,thestatementthatthegrowthpatternob-servedbetweenyearstandt+1is“pro-poor”isweaklyrobustwithrespecttothechoice 14AsemphasizedinLambertandRamos(2002),itisworthmentionningthataclassofpovertymea-suresthatbelongsto1andincludes2canalsobeusedifonlythenon-concavityofisassumed.ItisthennecessarytoturntothesequentialdominancecriterionproposedbyBourguignon(1989)togetarobustevaluationofgrowth“pro-poorness”usingthatintermediateclassofpovertymeasures.15ForadiscussiononthatgeneralizationofthePigou-Daltontransferprinciple,seeinparticularEbert(2000).14 4ILLUSTRATION:TESTINGTHEPRO-POORNESSOFGROWTHINTURKEY... ofpovertymeasureamongthefamily2ifandonlyif:jXk=1
\rt;t+1Gk(z;y)608z6z+j;j2f1;:::;Kg;(28)withaleastoneintegerj2f1;:::;Kgandonevaluez2[0;z+j]suchthat:jXk=1
\rt;t+1Gk(z;y)0:(29)Asintheprevioussection,itmaybeinterestingtolookforanalternativewayofexpressingproposition4whenthemarginaldistributionsxt+1and\rx(xt)donotdiffer.Letg2;kbethek-PGC,thatisthefunctionthatreturnsthemeangrowthrateofthebottomppercentsofthesubpopulationoftype1tokconsideringthenon-incomeattribute,i.e.:g2;k(p;Xt;Xt+1):=Zp0F1k(u;Xt+1) F1k(u;Xt)1du:(30)Corollary4.Foragivencounterfactualscenario\randagivenvectorz+ofmaximumvaluesforthespecicpovertylines,thestatementthatthegrowthpatternobservedbetweenyearstandt+1is“pro-poor”isweaklyrobustwithrespecttothechoiceofpovertymeasureamongthefamily2ifandonlyif:g2;k(p;Xt;Xt+1)g2;kp;(Xt);Xt+1
�08p6F(z+k;ykt);k2f1;:::;Kg;(31)withaleastoneintegerj2f1;:::;Kgandonevaluep2h0;F(z+j;yjt)isuchthat:g2;j(p;Xt;Xt+1)g2;jp;(Xt);Xt+1
�0:(32)4ILLUSTRATION:TESTINGTHEPRO-POORNESSOFGROWTHINTURKEY2003-2005Theproposedmethodologyisnowappliedusingdatafromthe2003,2004and2005Turkishhouseholdconsumptionandexpendituresurveys(HICES)providedbytheTurkishStatisticsInstitute(Turkstat).Turkeyisaninterestingcaseoverwhichtotestthepro-poornessofgrowth.Afterthe2001crisis,Turkeyenteredaperiodofhighgrowthandstructuraltransformations.Followingareboundin2001,annualgrowthratesaveragednearly7%overtheyears2003-2007.Accordingtotheinternationalstandards,povertyislowincomparisonwithotherMENAcountries,butinequalitiesremainhighandaretoalargeextentdrivenbyhighdifferentialsacrossregions.Moreover,despiteimprovementsinsocialindicators,educationrecordsprettyandweaklevelsincomparisonswithcountrieswithequivalentlevelsofGDPpercapita(Akkoyunlu-Wigley,2008).Thecountryalsofaceswideincomeandeducationgaps15 4ILLUSTRATION:TESTINGTHEPRO-POORNESSOFGROWTHINTURKEY... betweenurbanandruralareas(WorldBank,2005and2008)andeducationseemstoholdanimportantroleinunderstandingdiscrepanciesofdevelopmentwithinthecountry(Duman,2008).Inordertoillustratetheusefulnessofourmethodology,povertyisheredenedusingeducationalongsidethemoretraditionalincomecomponent.Theincomecom-ponentcorrespondstothedisposableequivalentindividualincomeadjustedbytheOECDequivalencescale.Inordertotakeinationintoaccount,allincomesareex-pressedinreferencetothe2003consumerpriceindexprovidedbyTurkstat.Edu-cationdeprivationsaremeasuredonthebasisofeducationlevelattainments.Ourdatasetsallowthedistinctionbetweenthefollowingsixcategories:illiterate,literatebutwithoutcompletingschool,primaryschool,primaryeducation,secondaryeduca-tionandoccupationaleducationequaltosecondaryschool,highschoolandhigherstudies.Sincechildrenhavenotachievedtheirnaleducationallevel,theanalysisfocusesontheadultpopulation(olderthan20years).Inthespiritofourframework,weputthereasonableassumptionsthatwell-beingisanincreasingfunctionofedu-cationattainmentsandthatincomeimproveswell-beingthemoreatloweducationallevels.Consequently,eachsamplehasbeensplitintosixgroupsofeducationallev-elsrankedbydecreasingneedswithrespecttoincome:illiteratepersonsarethusassociatedwithhighestneedsandhighschoolandhigherstudieswiththelowestones.Regardingthepro-poornessofgrowth,theillustrationsrelyonaverytradi-tionalcounterfactualscenario,thatisarelativeapproachofpro-poorgrowthfortheincomedimension(\ry(yt)=\rr(yt)=ytt+1 t)whileusingjusttheobservedchangesforeducation(\rx(xt)=xt).Insection3,wementionnedthattheclassesofbidimensionnalpovertymeasuresusedfor“pro-poorness"checksimpliesthedenitionofdifferentmonetarypovertylinesforeachvalueofthenon-monetaryattributes.Forthesakeofsimplicity,wede-neageneralincomepovertylineexpressedassomepercentageofthemedianincomeasusuallydonefortheanalysisofpovertyinOECDcountries,andconsiderthatthispovertylineisappropriatefortheleastdeprivedgroupregardingeducation.Moreprecisely,asstochasticdominancetestsaredesignedtoassesstherobustnessofpov-ertycomparisonstothelevelofthepovertyline,wehaveoptedforaveryconservativemaximumvalueofthatincomepovertyline,thatis90%ofthemedianincomeforthewholepopulation.Withthechoiceofastrictlypositivevalueofthepovertylineforpersonswithhigheducationendowments,wehaveassumedthatimprovingtheeducationlevelofanyindividualshouldraiseitslevelofwell-beinginasignicantmannerbutneverresultsinamoveoutofpoverty.Fortheremainingeducationalgroups,insteadofchoosingsomeparticularvalues,wepreferedleavingthatissueunansweredapriorisinceonecanhardlyconcludehowimportantshouldbetheincomelevelofapoorlyeducatedindividualsoastoescapepovertyregardingthestandardforawell-educatedperson.Nethertheless,westillre-quiredthesespecicincomepovertylinestobeneverinferiortotheonescorrespond-16 4ILLUSTRATION:TESTINGTHEPRO-POORNESSOFGROWTHINTURKEY... (a)Urbanpopulation. -0.04-0.03-0.02-0.010.000.01 income 505005000 education level (b)Ruralpopulation. -0.04-0.020.000.02 income 505005000 education level Figure2:Pro-poorgrowthcheck:rstordersequentialdominancechecksforTurkey,2003-2004,urbanandruralpopulations.However,thepicturebecomesslightlydifferentonceweturntomultidimensionalpovertywiththeinclusionoftheeducationdimension.Consideringtheurbanarea,eachcurveisbelowzeroforthebottompartoftheincomerangeandtheestimatedval-ues^z+karealwaysgreaterthanthevaluechosenforthehighlyeducatedindividuals,sothattheclassical“pro-poor"resultisconrmedbythemultidimensionalanalysis.Fortheruralarea,relyingsolelyonthestandardmonetaryanalysiswithoutputtingsomeemphasisonthepoorlyeducatedhouseholdswouldleadtothewrongconclusionthatthegrowthpatternwasbiasedinfavoroftheneediestbetween2003and2004.Indeed,focusingonthersttwogroupsofeducationattainment(thecontinuousanddashedlightgraycurvesingure2b)showthattheshareoflowincomeandlowedu-cationindividualshasnotdeacreasedasmuchasitwouldhavebeenthecasewitha“neutral"growthpatternduringtheperiod.Thesecondillustrationisrelatedtotheusefulnessofasecond-ordersequentialstochasticdominancecheck.Lookingatgure3a,itcanbeseenthatwecannotcon-cludewhetergrowthwas“pro-poor"inarobustmannerconsideringthewholeTurkishpopulationforthe2004-2005growthspellsincedominancecurvesaresometimeaboveandsometimebelowzeroforincomelevelsbelowtheincomepovertylinecorrespond-ingtothehighlyeducatedhouseholds.However,sincethecurvesareabovethatlevelfortheverybottompartoftheincomerange,itmaybeinterestingtofocusonthemorelimitedset2ofdistribution-sensitivemultidimensionalpovertymeasuresandconsequentlytoturntothesecond-ordersequentialstochasticdominanceprocedure.Contrarytotherst-orderprocedure,thesecond-orderonereliesontheuseofthe18 5CONCLUDINGREMARKS (a)Firstorderdominance. -0.010.000.010.02 income 505005000 education level (b)Secondorderdominance. -6-4-20246 income 50500 education level Figure3:Pro-poorgrowthcheck:rstandsecondordersequentialdominanceforTurkey,2004-2005,wholepopulation.incomegaps,thatistheextentofincomeshortfallswithrespecttothepovertylinetimesthevalueofthecorrespondingmultidimensionnalheadcountindex.Theresults,plottedingure3bshowthatthejointdistributionofeducationandincomein2005isdominatedbythecorrespondingcounterfactualbasedonthe2004distributionuptosomeadmissiblepovertyfrontier.Inotherwords,wecanconcludeinarobustmannerthatgrowthcanbedeemed“anti-poor",intherelativesense,inTurkeyduringtheperiod2004-2005consideringindiceswithinthesetofdistribution-sensitivemultidimensionalpovertymeasures.Itisworthnotingthatthemainresultsaresimilartotheoneobtainedwiththetraditionnalrst-andsecond-orderdominancechecksinthatcase(c.f.thethickblackcurvesingure3aand3b),butourapproachyieldsmoreinformationsonthedistributionoftheeconomicgrowth“cake"asitshowsunambiguouswelfareimprovementatthertorderforthegroupsofilliterateandliteratebutwithoutcompletingschoolindividuals.5CONCLUDINGREMARKSInthispaper,wehaveproposedtoextendtheuseofsequentialstochasticdominancetechniquesinordertoassessrobustjudgmentsofthe“pro-poorness”ofgrowthwithintheframeworkofmultidimensionalapproachtopovertymeasurement.Indeedthetraditionaltoolsusedtocheckfor“pro-poor”growthfocusonthesolemonetaryaspectofpoverty.Asiswell-known,theinclusionofotherdimensionsofpovertyinducesachangeinthepovertydenition.Inparticular,theanonymityaxiomthatmonetaryindexofpovertyshouldsatisfyisnotalwaystrueandisnotethicallyanymoreac-19 APROOFOFPROPOSITIONSANDCOROLLARIES3TO4 ceptable.Here,weproposetousethesequentialdominanceproceduressuggestedbyBourguignon(1989)anddevelopedbymanyauthorslikeAtkinson(1992)andJenkinsandLambert(1993)inordertodenerst-orderandsecond-orderdominancecriteriathatmakeitpossibletoassessweakrobustnessof“pro-poor”growthinincomeaswellasinotherwell-beingattributesforaclassofpovertymeasuresandawiderangeofpovertylines.Unliketothetraditionalstudiesusingsequentialstochasticdominance,thehet-erogeneityofthepopulationisnotdenedonthebasisofthehouseholdssizesandcompositions.Onthecontrary,individuals'needsdifferaccordingtonon-incomeat-tributesasinthestudyofDuclosandÉchevin(2009)forpovertymeasurement.Un-liketheattemptmadebyGrosse,Harttgen,andKlasen(2008)whoextendChenandRavallion's(2003)growthincidencecurve(GIC)tonon-monetarydimensionsofpov-erty,ourmethodologytakesintoaccountthechangesinthejointdistributionofthewell-beingattributes.Forthispurpose,itonlyaddstwoweakconditionstothetra-ditionalmathematicalconditionsusedforunidimensionalpovertymeasurement.Therstoneisthattheincomepovertylinedoesnotincreasewiththelevelofthenon-monetaryindicator.Thesecondoneimposesthemarginalcontributionofincometowell-beingtodecreasewiththelevelofnon-incomeattributes.Asaspecialcaseofourapproach,itispossibletodenetheequivalenceofGICandPGCcurves,namedk-GICandk-PGCcurves,thatarebasedonpartialquantilefunctionsandmaybeusedtogetrobustconclusionswhenthemarginaldistributionofthenon-monetaryattributeisleftunchanged.Itisworthnotingthattheuseofthesecurvescanbeextendedsoastotakechangesinthedistributionofthenon-incomeattributesintoaccount.Finally,thoughthesocialevaluatorhasmorelatitudetodenethecounterfactualsituationinordertomakejudgmentsofthe“pro-poorness”ofgrowth,thedenitionofthatcoun-terfactualischallengingfromanempiricalpointofviewasitentailsconsideringtherelationshipsbetweenincomeandthenon-incomeattributes.Ourfeelingisthatthisissueshouldbeamatterofscrutinyforfurtherempiricalstudies.APROOFOFPROPOSITIONSANDCOROLLARIES3TO4A.1FIRST-ORDERSTOCHASTICSEQUENTIALDOMINANCETheproofproposition3issimilartoJenkinsandLambert(1993,proposition1).Werstintroducethefollowingnotation:
\rt;t+1fk(z;y):=qk(xt+1)fk(z;yt+1)qk\rx(xt)
fkz;\ry(yt)
;(33)withfbeingthedensityfunction,thatistherst-orderderivativeofF.Usingequa-20 APROOFOFPROPOSITIONSANDCOROLLARIES3TO4 ifqk(xt+1)=qk\r(xt)
holds8k2f1;:::;jg.Multiplyingeachtermin(39)byPjk=1qk(xt+1),thesecondtermsimplybecomesG(z;yjt+1)Gz;\ry(yjt)
.Condition(46),canthenberewrittenas:G(z;yjt+1)6Gz;\ry(yjt)
8y6zj;j2f1;:::;Kg:(48)Moreover,integratingG(z;yj)bypartsyields:G(z;yj)=Zz0(zy)dF(y;yj);(49)=Zz0F(y;yj)dy:(50)Pluging(50)inrelation(48)andinvertingF,weget:Zp0F1(u;yjt+1)du�Zp0F1u;\ry(yjt)
du8p6F(zj;yt);j2f1;:::;Kg:(51)Dividingeachtermofequation(51)byRp0F1j(u;yjt)duandsubstractingoneyieldsthecomparisonofthek-PGCusedforcorollary4.23 REFERENCES DUCLOS,J.-Y.(2009):“Whatis“pro-poor”?,”SocialChoiceandWelfare,32(1),37–58.DUCLOS,J.-Y.,ANDD.ÉCHEVIN(2009):“HealthandIncome:aRobustComparisonofCanadaandtheUS,”Cahierderecherche/WorkingPaper09-09,CIRPÉE.DUCLOS,J.-Y.,ANDP.MAKDISSI(2005):“SequentialStochasticDominanceandtheRobustnessofPovertyOrderings,”ReviewofIncomeandWealth,51(1),63–87.DUCLOS,J.-Y.,D.SAHN,ANDS.YOUNGER(2006):“RobustMultidimensionalPovertyComparisons,”TheEconomicJournal,116,943–968.EBERT,U.(2000):“SequentialGeneralizedLorenzDominanceandTransferPrinci-ples,”BulletinofEconomicResearch,52(2),113–122.FOSTER,J.,J.GREER,ANDE.THORBECKE(1984):“AClassofDecomposablePovertyMeasures,”Econometrica,52(3),761–766.FOSTER,J.,ANDA.SHORROCKS(1988):“PovertyOrderingsandWelfareDominance,”SocialChoiceandWelfare,5,179–198. (1991):“SubgroupConsistentPovertyIndices,”Econometrica,59(3),687–709.GASTWIRTH,J.(1971):“AGeneralDenitionoftheLorenzCurve,”Econometrica-NotesandComments,39(6),1037–1039.GRAVEL,N.,ANDP.MOYES(2008):“BidimensionalInequalitieswithanOrdinalVari-able,”CahiersduGREThA2008-14,GREThA,UniversitéMontesquieuBordeauxIV.GROSSE,M.,K.HARTTGEN,ANDS.KLASEN(2008):“MeasurinPro-PoorGrowthinNon-IncomeDimensions,”WorldDevelopment,36(6),1021–1047.JENKINS,S.,ANDP.LAMBERT(1993):“RankingIncomeDistributionsWhenNeedsDiffer,”ReviewofIncomeandWealth,39(4),337–356.KAKWANI,N.,ANDE.PERNIA(2000):“WhatisPro-PoorGrowth?,”AsianDevelop-mentReview,18(1),1–16.KAKWANI,N.,ANDH.SON(2008):“PovertyEquivalentGrowthRate,”ReviewofIn-comeandWealth,54(4),643–655.KLASEN,S.(2008):“EconomicGrowthandPovertyReduction:MeasurementIssuesUsingIncomeandNon-IncomeIndicators,”WorldDevelopment,36(3),420–445.KOLM,S.-C.(1977):“MultidimensionalEgalitarianisms,”QuarterlyJournalofEco-nomics,91(1),1–13.KRAAY,A.(2006):“WhenisGrowthPro-Poor?EvidencefromaPanelofCountries,”JournalofDevelopmentEconomics,80,198–227.25