\Leftist",\Rightist"andIntermediateDecompositionsofPovertyVariationswi - PDF document

“Leftist”,“Rightist”andIntermediateDecompositionsofPovertyVariationswithanApplicationtoChinafrom1990to2003.FlorentBresson¤KellyLabar†CERDI-UNIVERSITÉd'AUVERGNEVERSION1‡September13,2007AbstractThispaperinvestigatestheinuenceofinvarianceaxiomsinthedecompositionofobservedpovertyvariationsintogrowthandinequalityeffects.Afteracompleteandcriticalreviewoftheinvarianceaxiomssuggestedintheliterature,weshowthatfewinformationisneededfortheorderingoftheeffectsrespectivelyobtainedthroughscale,translationandintermediatein-variance.UsingChinesedatafortheperiod1990-2003,wendthatsomecommonlyobservedresultsofthedecompositionarecontingenttotheinvarianceaxiomchoiceswhilstotherarerobusttochangesinethicalpreferences.JELclassication:I32,D63,D31.Keywords:Poverty,inequalityeffect,growtheffect,decomposition,scaleinvariance,transla-tioninvariance,intermediateinvariance,China.IntroductionDoesmultiplyingtheincomesofeachmembersofapopulationbythesamescalarincreases,decreasesorleavesincomeinequalityunchanged?Doesaddingthesameabsoluteamountofincometoeachmemberofapopulationincreases,decreasesorleavesinequalityunchanged?Itisveryinterestingtonotethatpeoplemaygiveverydifferentanswerstoquestionsrelatedtoaxiomaticchoices,andthusexpresssoheterogeneousfeelingsabouthowinequalityshouldbedenedandmeasured.Usingquestionnaireswithlargesamplesofstudents(generallyunder-graduatestudentsineconomics),AmielandCowell(1992,1997,1999,2001)noticedthatveryfewrespondantswerelikelytosupportmostofthecoretraditionalaxiomsusedintheinequal-ityandpovertymeasurementliterature.Inparticular,themajorityoftherespondantswasnotinagreementwiththeclassicalopinionthatdoublingeachincomeinadistributiondoesnotchangethedegreeofinequality.Suchareactionagainstthisscaleinvarianceaxiomisnotreallysurpris-ingsincethereisnounanimousapprovalofthisaxiomamongeconomists.Forinstance,many ¤orent.bresson@u-clermont1.fr†kelly.labar@cerdi.u-clermont1.fr‡TheauthorswouldliketothanksJean-LouisCombesandRolandKpodarfortheirhelpfulcomments.1 famousscholarslikeDalton(1920)orKolm(1976a)alsoexpressedheterodoxviewsabouthowadditionalincomesshould(orcould)bedividedamongindividualssoastopreservethedegreeofinequality.1Questioningthedesirabilityofthepropertiesofanyinequalityorpovertymeasureisnotatrivialexercisesinceitmayhaveadirectimpactonpolicydecisions.Internationalincomein-equalities,thatisinequalityamongcountries,areagoodillustrationoftheimportanceoftheheterogeneityoffeelingsaboutinequalityanditsconsequences.Manyindividualswillfocusontheincreasingabsolutedifferencesbetweenmeanincomeswhileotherswilljustconsiderthede-creasingrelativedifferencesbetweennations.Therstonesmaycertainlyconcludethatinequal-itieshaverisenduringthelastdecades,whilethesecondwouldsupporttheoppositepointofview.Theconsequenceisthatverydifferentpolicies,inparticularaidanddevelopmentpolicies,couldberecommendedonthebasisofsuchheterogeneousinterpretationsofobservedtrends.However,thesubjectofthepresentpaperisnotinternationalincomeinequalitiesbutpresentssimilarinterpretationissues.Herewewouldliketoemphasizetheimportanceofaxiomaticchoicesontheanalysisofpovertyvariations.SincethepioneeringdevelopmentsofJainandTendulkar(1990);KakwaniandSubbarao(1990)andDattandRavallion(1992),thedecompositionsofpovertyvariationsintogrowthandinequalityeffectshavebecomeverypopularinempiricalstudiessinceitisaveryelegantwayofestimatingtherelativecontributionoftheincreaseinmeanincomeandofthechangesintherelativedistributionofincomes.Inthepresentstudieswestressthecrucialroleofethicalpreferencesinvolvedinthegeneralconceptionofinequalitysinceitdenesthefrontierbetweenwhatcanbeconsideredas“pure”growth,thatisgrowthwithoutinequalitychange,and“pure”redistribution,thatischangeintherelativedistributionwithaconstantmeanincome.Thisremarkisparticularlyrelevantforsomepovertymeasuresliketheheadcountindexthatarecompatiblewithmanyrivalaxiomsandthusthatleaveroomforpersonaljudgments.Consequently,thesamevariationofpovertymaybemostlyattributedto“pure”growthor“pure”redistributiondependingofindividualtastes,aresultthatmayleadtogreatmisunderstandingsandinefcientpolicyrecommendationsifresearchersdonotexplicitlyexplainstheaxiomaticbasisinvolvedintheirdecompositionofobservedpovertytrends.Inthepresentpaper,werstreviewthedifferenttechniquesusedforthedecompositionofpovertyspells(section1)andthenthedifferentinequalityviewswhichhavebeenpresentedandformalizedintheinequalityandpovertymeasurementliterature(section2).Moreprecisely,wefocusoninequalityviewsthatareattributedto“rightist”and“leftist”politicalopinionsaccord-ingtoKolm(1976a).A“rightist”isbasedontheopinionthatinequalitydoesnotchangewhenincomesgrowatthesamerateasmeanincomethroughthecurseofeconomicdevelopmentwhereas“leftist”individualsfeelthatthedegreeofinequalityisconstantwheneconomicagents'incomesincreasebythesameamountasmeanincomedoes.2Ourreviewalsoincludesinterme- 1Concerningtheoppositionbetweenscaleinvarianceandtranslationinvariancethatwillbetreatedinthenextsections,Kolm(1976a,p.419)arguethat“itisnolesslegitimatetoattachtheinequalitybetweentwoincomestotheirdifferencethantotheirratio.”2The“leftist”and“rightist”labelsarelinkedtothefrenchpoliticalcontextandtheideologicaldifferencesbetweenleft-wingandright-wingopinions.Kolm(1976a)introducestheseexpressionswithareferencetodebatesthatoccuredfortheGrenelleagreementsin1968whichdecreedthesameproportionalincreaseinwagesforallemployees.Kolmreports(p.419)that“theRadicalsfeltbitterandcheated;intheirview,thiswidelyincreasedincomesinequality.”In2 1THEDECOMPOSITIONOFPOVERTYSPELLS diateviewsthatconstitutescompromisesbetweenthesetwopolarcasesandforwhichweprovideanewdenition.Amongtheformalizedintermediateviewsthatarepresentedinthepaper,wearguethatthenon-linearviewsuggestedbyKrtscha(1994)andYoshida(2005)isthesolerelevantwayofexpressinganintermediateinequalityviewforpovertydecompositions.Consideringthetheoreticalimplicationsofinvariancerelatedaxiomchangesonthegrowth-inequalitydecompositionsofpovertyspells(section3),weshowthatfewinformationisneededsoastoorderthedifferentgrowthandinequalityeffectsbasedonthese“rightist”,“leftist”andintermediateviewsinthecaseoftheheadcountindex.Consequently,wecaneasilypredictwhichinterpretationdifferencesshouldbeobservedwithinthissetofinequalityconceptionsforthesameobservedpovertyvariations.AnapplicationonChinesedataduringtheperiod1990-2003isthenprovidedinsection4.Theconclusionsthatregularlyappearinempiricalstudies(Fanetal.,2002;WanandZhang,2006;ChenandRavallion,2007)arethatgrowthisthemaincontributortopovertyreductionandthatrelativedistributionchangeshampersthebenecialeffectsofgrowthonpoverty.Ouraimisthustotestiftheseconclusionarerobusttochangesininequalityviews.Ourresultsshowthatup-holdersofthe“leftist”viewwillconsiderthatinequalitychangeshavecontributedtotheincreaseofthenumberofpoorpeopleinChinaduringthewholeperiodwhilstthosewhobelieveina“rightist”viewwillsupporttheoppositeconclusion.Anotherimportantconsequenceisthatconsideringrivalinequalityviewsmayreverseconclusionswhencomparingthevalueofthees-timatedeffectsbetweenmanysubperiods.InthecaseofChina,thesemethodologicalconsid-erationsshouldbeseenascrucialsinceitmodiesconclusionsthatconcernsaboutonefthoftheworldpopulation.Ontheotherhand,sometraditionalconclusionsarestrengthenedliketheneedtoimproveredistributionsoastoghtextremepoverty.Finally,section5concludeswithsomeremarksabouteconomists'practices.Inparticular,wearguethateconomistsshould,atleast,beawareofthenormativeimplicationsofthetoolsthattheyuseforpurelypositiveanalysisofobservedeconomicphenomena.1ThedecompositionofpovertyspellsInthepresentpaper,ourattentionisconnedtoabsolutepovertymeasures£,whichcanbefullycharacterizedbyapovertylinez,themeanincome¹andavectorofinequalitymeasures¼thataccountforallinequalityfeaturesoftheobserveddistribution.3Thus,povertyattimetisgiven section2,weshowthatthe“leftist”viewimpliesamoreegalitarianwayofsharingadditionalincomesamongindivid-ualsthatthe“rightist”view..Astheexpressionsof“leftist”and“rightist”viewsarecommonintheinvariance-relatedliteratureanddonotyieldconfusionslikethetermsabsoluteandrelative(seenote3),wewillmakeanintensiveuseofthemthroughoutthepaper.However,weareconsciousthattheseexpressionsmaynotcorrespondtotherealityofpoliticaldoctrinesandmovements,especiallyoutsidethefrenchcontext.Forajusticationofthedifferencesbetween“leftist”and“rightist”inequalityviewsintermsofutilityfunctions,seenote15.3Inthepresentstudy,absolutepovertyreferstotheuseofanabsolutepovertylinewhichisonlydenedbytheamountneededtosatisfysome“basic”needs(seeSen,1983,1985,forfurtherdetails).Soitcontrastswithrelativepovertyinwhichthepovertylineissetwithrespecttotheobserveddistributionofincome.Sometime,absolute(rel-ative)povertycorrespondstopovertyviewswhichcomplywithtranslation(scale)invarianceaxioms(cf.section2).Theadjectivesabsoluteandrelativearealsousedinthecontextofinequalitymeasurementandreferstoindicesthatarerespectivelydenedasdifferencesandratiosofmeanincomewiththecorrespondingequallydistributedequiva-lentincome(Atkinson,1970;Kolm,1976a,b),i.e.thepercapitaincomewhichifequallysharedamongthepopulation3 1THEDECOMPOSITIONOFPOVERTYSPELLS by:£tÆ£(zt,¹t,¼t).(1.1)Inordertocomparevaluesof£atdifferentpointsoftime,ztisheldconstant(weassumeincomearemeasuredinrealterms).SoztÆz.Consistentwiththisassumptionandequation(1.1)istheintuitionthatvariationsof£canbedecomposedintodifferentcomponentsthatcanbeattributedtogrowthandvariationsininequality.Inmathematicalterms,ourintentionistoget:£tÅk¡£tÆGt,tÅkÅDt,tÅk(1.2)whereGandDarerespectivelythegrowthandinequalitycomponentsofpoverty.4Thegrowthcomponentisthevariationofthepovertymeasurethatisonlyduetochangeinmeanincome,thatiswheninequalityisheldconstant.Similarly,theinequalitycomponentisthevariationofthepovertymeasurethatcanbeattributedtovariationsoftheelementsof¼.ThistechniquewasinitiatedbyJainandTendulkar(1990);KakwaniandSubbarao(1990)andDattandRavallion(1992)andisnowstandardinthepovertyliterature.Itshouldbeacknowledgedthatthisdecom-positionisapurelystatisticaldecompositionanddiffersfromtheeconometricanalysis,likeChenandRavallion(2007),inthesensethatitdoesnotaccountforthecorrelationsbetweengrowthandvariationsofthedegreeofinequality(whateverthedirectionofthecausality,ifitdoesexist).Thedecompositionofpovertyspellscanbecarriedindifferentways,dependingonwhetherinitialornalvaluesareusedforthexedelementofeachcomponent.Inthepresentpaper,wechoosetofocusonthetwomostwidelyuseddecompositiontechniquesnamelytheonesuggestedbyDattandRavallion(1992)andtheShapleydecompositiondevelopedbyShorrocks(1999)andKakwani(2000).TheDattandRavallion(1992)procedureischaracterizedbytheuseoftheinitialvaluesasreferencesforthecomputationofeacheffectsandthusbythepresenceofaresidualterm.Thegrowthandinequalityeffectsarethendenedbythefollowingequations:Gt,tÅkÆ£(z,¹tÅk,¼t)¡£(z,¹t,¼t),(1.3)Dt,tÅkÆ£(z,¹t,¼tÅk)¡£(z,¹t,¼t).(1.4)Inthecontextofamulti-periodanalysis,thistechniqueprovestobetime-transitivewhenthesamedistributionisusedasreferenceforthecomputationoftheeffectsforeachperiod.5Forrecentapplicationofthistechnique,seeFanetal.(2002)forurbanChinafrom1992to1998,Contreras(2003)forChileduringtheperiod1990-1996,Kappeletal.(2005)forUgandafrom1992to2002andFerreiraetal.(2006)forBrazilduringtheperiod1981-2004.However,thismethodhasbeenheavilycriticizedsinceitgenerallydoesnotprovideaperfectdecompositionof£tÅk¡£t.Theevidenceshowsthattheresidualcomponentofthisdecomposi-tionisgenerallyimportantandcannotbeeasilyinterpreted.Toavoidthisshortcoming,Shorrocks wouldyieldthesametotalwelfareastheobservedincomedistribution(forashortreviewofthelinksbetweenabsoluteindices,relativeindices,scaleinvarianceandtranslationinvariance,seeFleurbaey,1996).Inordertoavoidconfusion,wewillnotmakeuseoftheseexpressionsthroughouttherestofthepaper.4InthecaseoftheDattandRavallion(1992)approach,thedecompositionisnotexactandaresidualtermshouldbeadded.5Thispropertyiscalledsub-periodadditivityinDattandRavallion(1992).4 2INVARIANCEANDTHEDECOMPOSITIONOFPOVERTYVARIATIONS (1999),usingtheShapley-valuefromthecooperativegametheory,andKakwani(2000),usinganaxiomaticapproach,provideadecompositionframework,sothati)thedecompositionisexact(thereisnoresidualcomponents),ii)thevariationofthepovertymeasureispositive(negative)whenboththegrowthandtheinequalitycomponentsarepositive(negative),andiii)thevalueofthegrowth(inequality)componentbetweentandtÅkistheoppositeofitsvaluebetweentÅkandt.AccordingtothisShapleydecomposition,wegetthefollowingvaluesforeachcomponent:G0t,tÅkÆ¡£(z,¹tÅk,¼t)¡£(z,¹t,¼t)¢Å¡£(z,¹tÅk,¼tÅk)¡£(z,¹t,¼tÅk)¢ 2,(1.5)D0t,tÅkÆ¡£(z,¹t,¼tÅk)¡£(z,¹t,¼t)¢Å¡£(z,¹tÅk,¼tÅk)¡£(z,¹tÅk,¼t)¢ 2.(1.6)RecentillustrationsofthisdecompositiontechniqueincludeKolenikovandShorrocks(2005)forRussiainthemid90s,Baye(2006)forCameroonduringtheperiod1984-1996andWanandZhang(2006)forruralChinaduringtheperiod1988-2000.6Despiteitsattractiveness,theShapleydecompositionisnotthepanaceasinceitcanbeprovedthattheestimatedeffectsarenottimetransitive.7However,sinceourobjectiveistoquestioneconomistscurrentpractices,itdoesnotmatterwhichparticulardecompositiontechniqueistherightone.8Thisexplainswhythispaperfocusesonthedecompositionscorrespondingtoequations(1.3)to(1.6).2InvarianceandthedecompositionofpovertyvariationsFromatechnicalpointofview,theestimationofthegrowthandinequalitycomponentsofthepovertyspellsimpliesthecomputationofintermediate,orcounterfactual,valuesforthechosenpovertymeasure,thatisthevaluesthatwouldbereachedbythepovertymeasureifonly¹or¼changedbetweenthedatestandtÅk.Thedesignoftheseintermediatevaluesrequiresanex-plicitformulationofwhatinequalitymeans,inparticularwhichethicalvaluesareinvolvedintheconceptofinequalityusedfortheanalysis.Ofparticularinterestforthedecompositionexerciseistheconceptofinvariancethatwillbeextensivelydiscussedthroughthenextparagraphs.ConsideranincomedistributionXofsizenï¿¿2withn2N¤.IncomesaredenedonthesetD®:[®,Å1).EachdistributionXisthendrawnfromthesetD®ÆSn2N¤Dn®.SometimeD®is 6KolenikovandShorrocks(2005)primeinterestisnotdynamicbutregionaldecompositionofthevariationsofpoverty.7Inordertogetgrowthandinequalitycomponentsthatrespectthisproperty,Kakwani(2000)suggestsusingthefollowingformulaforGt,tÅkandIt,tÅk:˜Gt,tÅkÆ1 ssXjÆ1³G0t,jÅG0j,tÅk´,˜Dt,tÅkÆ1 ssXjÆ1³I0t,jÅD0j,tÅk´.whenthetotalobservedperiodis1toswith16t6tÅk6s.However,eveniftheseeffectsaretimetransitiveandyieldaperfectdecomposition,theypresenttheundesirablefeatureofbeingpath-dependentsincetheyalsodependoftheincomedistributionsduringtheperiods1...t¡1,tÅ1...tÅk¡1andtÅkÅ1...s.SotwoeconomieswiththesameincomedistributionsintandtÅkmaypresentdifferentvaluesof˜Gt,tÅkand˜Dt,tÅkiftheydonotsharethesameevolutionduringtheperiodofanalysis.Toourknowledge,Kakwani(2000)isthesoleapplicationoftheseformula.8Formorecriticsoftheaforementioneddecompositiontechniques,seeMuller(2006).5 2INVARIANCEANDTHEDECOMPOSITIONOFPOVERTYVARIATIONS restrictedtothenonnegativeorstrictlypositiveorthantofthen-dimensionalEuclideanspaceRnwiththeorigindeleted.SuchsetswillberespectivelynotedDÅandDÅÅ.EachvectorXisorderedsothatx16x2...6xn.AninequalityindexªisamappingofD®intoRÅsuchthatª(X1)Ǫ(X2)impliesthatX1isconsideredaslessunequalthanX2.Forthesakeofsimplicity,atraditionalassumptionisª(¹I)Æ08¹2RÅÅwithIbeingan-vectorof1.9Wealsoimposeasminimumrequirementstherespectofthecoreanonymity,conti-nuityandpopulationaxioms.10Inthefollowingparagraphs,wewillmakeuseofthePigou-Daltonprincipleoftransferssuchthatprogressive(regressive)transferslower(increase)inequality.1112TherespectoftheanonymityaxiomandofthePigou-DaltonprincipleoftransfersimplythatªisS-convex(Dasguptaetal.,1973).13However,thisprincipleoftransferscanbedebated(seeforinstanceAmielandCowell1992orChateauneufandMoyes2005)andwillsometimeconictwithotheraxioms.So,thoughitwillbeconsideredasadesiredproperty,ªmaysometimenotrespectthePigou-Daltonprincipleoftransfers.Invarianceisthepropertyofanyinequalitymeasureªsuchthat:ª¡©(X)¢Æª(X),(2.1)where©isacontinuousincreasingfunction©:D!D.Suchanaxiomisnecessaryforthecom-parisonofincomedistributionswithdifferentmeans.14Soinvariancecanbeseenasthewayofsharinganadditionalincomeinordertoleavethejudgmentoninequalityunchanged.15Re- 9InthecaseoftheinequalitymeasuredenedbyAlonso-VillaranddelRio(2007a),thisconditionmaynotberespectedsinceitsdomaingenerallydoesnotincludedistributionswereincomesareequallyshared.Theunderlyinginvarianceaxiomispresentedinsection2.3.1.10Chakravarty(1999)isafairlycomprehensivereviewofthemostcommonaxiomsusedintheinequalitymeasure-mentrelatedliterature.Anonymity,alsocalledsymmetry,horizontalequityorequaltreatmentofequals,meansthatª(PX)ƪ(X)withPbeinganypermutationmatrixofsizen£n.ContinuityimpliesthatmarginalvariationsofanyelementofXdonotcauselargevariationsofthemeasureP.Finally,ameasurerespectsthepopulationaxiom,alsocalledreplicationinvarianceaxiom,ifam-replicationofXexhibitthesamedegreeofinequalityasX,whateverX2D.11Atransferisprogressive(regressive)ifitincreases(decreases)theincomeofanindividualattheexpenseof(infavourof)arichestindividualwithoutchangingtheirrelativepositioninthedistribution.Aweakerversionoftheprincipleversionwouldrequireregressive(progressive)transfersnottoincrease(lower)thevalueofª.12Thispropertyiscalled“rectiance”inKolm's(1976a)seminalpaper.13Foranybistochasticn£n-matrixB,thatisasquarematrixwhichcontainsonlypositiveelementsandwhichcolumnsandrowssumtoone,afunctionisS-convexifª(BX)6ª(X).IfstrictS-convexityisrequired,thenweshouldobserveª(BX)Ǫ(X)forallbistochasticmatricesexceptpermutationsmatrices.14Ebert(2004)stressedthatinvarianceonlydenesrelationsbetweendistributionsforwhichwefeelindifferentwithrespecttoinequality.Soitisofnohelpforrankingdistributionsthatarenotinthesameiso-inequalityset.15Anotherjusticationforthevariousinvarianceaxiomspresentedherecanbefoundinthenormativeapproachofinequalitymeasurement.SinceKolm(1969);Atkinson(1970)andSen(1973),inequalitymeasuresareoftenderivedfromsocialevaluationfunctionsW:D®!Rwhichprovideaquasi-orderingofincomedistributionsfromthesetD®.Inotherwords,Wreectstheopinionsofthesocialevaluator(theobserver)intermsofdistributivejustice.Letaandbbesomenegativeconstantparameters.Kolm(1969,theorems13and14)showsthatsocialevaluationfunctionthatcomplieswithanonymityandIndependence—asocialevaluationfunctionfulllsIndependenceifitcanbeexpressedasPif(yi)withf:R!Rbeinganincreasingfunction—,areoftheform:WÆnXiÆ1aybiorWÆnYiÆ1yai,ifthesocialevaluatorbelievesina“rightist”viewand:WÆnXiÆ1aebyi,6 2INVARIANCEANDTHEDECOMPOSITIONOFPOVERTYVARIATIONS cently,Zheng(2004)hasshedlightonanaxiomwhichiscloselylinkedtoinvariance,namelytheunit-consistencyaxiom.16Unit-consistencyrequirestheinequalityorderingtobeinvariantwithrespecttochangesinthecommonunitofmeasureadoptedtoevaluatethedistributions.Sowehavetoobserveª(¸X)Ƥ¡ª(X)¢8X2D®where¸isapositivescalarand¤isacontinuousmonotonefunctionfromRÅtoRÅ.Inotherwords,twoincomedistributionsshouldberankedinthesamemanneraccordingtoªwhenincomesaremeasuredineurosorindollars.Aswewillseeinthenextsections,unit-consistencyisnecessarywhenconsideringinequalityviewsthatdonotrelyonscaleinvariance.Mostoftheseaxioms,evenslightlymodied,aresharedbypovertymeasures.Forinstance,continuityisgenerallyreplacedbyrestrictedcontinuitysuchthat£isaleftcontinuousfunctionofxforallxÇz.Themainadditionisthefocusaxiomwhichstatesthattheonlyrelevantin-formationrelatedtothenon-poormembersofthepopulationistheirnumber.17Soapovertymeasure£isnotaffectedbyanyincrementoftheincomeofnon-poorperson.ThisexplainswhypovertymeasuresareoftenconsideredasarestrictionofinequalitymeasuresonthesubsetXpoftheincomedistributionsuchthateachelementofXpisnotgreaterthanthepovertylinez.Asaconsequence,thefollowingexpressionsareperfectsubstitutes£(z,¹,¼)Æ£(z,X)Æ£(z,Xp,n).Anadditionalrequirementistheweakmonotonicityaxiomwhichimposesonapovertymeasurenottodecreaseifapoorperson'sincomedecreases.Finally,£shouldbenondecreasinginz.18Inthefollowingsections,wenowdetailssomeparticularversionsoftheinvarianceaxiomandpresenttheirimplementationforthecalculationofgrowthandinequalityeffectsofpovertyvariations.2.1ScaleinvarianceThemostwidelyusedinvarianceaxiomisthescaleinvarianceaxiom,suchthat:ª(¸X)ƪ(X)8¸È0.(2.2)Thescaleinvarianceaxiommeansthatdoublingeachincomeoftheobserveddistribution ifthesocialevaluator'spreferencesareinaccordancewiththe“leftist”view.ApplyingthefamousresultsofArrowandPratt,Atkinson(1970)emphasizesthatthersttwofunctionalformsreectsaconstantrelativeinequality(orrisk)aversionandthethirdoneaconstantabsoluteinequality(orrisk)aversion.ThefunctionWisusedtodenetheequallydistributedequivalentincome˜x,thatisthelevelofpercapitaincomewhich,ifequallydistributed,wouldprovidethesamelevelofsocialwelfareastheobserveddistribution.Thenthenaturalformofscaleandtranslationinequalityindicesisrespectively:ªrÆ1¡˜x ¹,ªaƹ¡˜x.AccordingtoKolm(1976a),ªrshouldbeconsideredasameasureofinequality“perpound”andªaasameasureofinequality“perperson”.16ThispropertywasalreadydetailedinAczélandMoszner(1994).Kolm(1995)andZoli(2003)alsoconsideredthisdesiredpropertyandcalleditrespectively“unitinvariance”and“weakcurrency-independence”.17Formathematicalconvenience,aweakdenitionofpoverty—anindividualispoorifhisincomeisstrictlyinferiortothepovertyline—isgenerallypreferred(seeforinstanceDonaldsonandWeymark,1986).18Ifweputforwardthat£isstrictlyincreasinginz,monotonicityisimplicitlyassumed.Formoredetailsaboutthedifferentpovertyaxiomsandtheirinterrelations,seeZheng(1997).7 2INVARIANCEANDTHEDECOMPOSITIONOFPOVERTYVARIATIONS doesnotaffectinequalityasmeasuredbyª.Inmathematicalwords,ªcomplieswithscalein-varianceifitishomogeneousofdegreezero.19Withthemeasurementofpoverty,amarkedlymodiedversionofthescaleinvarianceaxiomhastobeinvoked,thatis:20£¡¸z,¸Xp,n¢Æ£¡z,Xp,n¢8¸È0(2.3)whichislessrestrictivethantherstversionsincethesoleconditionimposedontheincomeofthenon-pooristoremaingreaterthanz.Mostinequality(e.g.Ginicoefcient,Atkinsonindex,generalizedentropyindexes)andpoverty(e.g.Wattsindex,Senindex,Foster,GreerandThor-beckeindexes)measuresusedinempiricalanalysisrelyonthesescaleinvarianceaxioms.ThesamepropertyholdsforthetraditionalLorenzcurve.Inthecontextofthedecompositionofpovertyspells,scaleinvarianceisoftenusedinanimplicitmannerforthecomputationoftheintermediatevaluesofthepovertymeasuresincetheseonearegenerallydenedwithrespecttotheLorenzcurve.21Inthetwoperiodcase,therespectofthescaleinvarianceimpliesequations(1.3)and(1.4)tobecomputedasfollow:GSt,tÅ1Æ£¡z,¸t,tÅ1Xpt¢¡£¡z,Xpt¢,(2.4)DSt,tÅ1Æ£¡z,¸tÅ1,tXptÅ1¢¡£¡z,Xpt¢.(2.5)where¸t,tÅkƹtk ¹t.22Extensiontoequations(1.5)and(1.6)isstraightforward.23Scaleinvarianceisfrequentlyseenasadesirablefeatureforaninequalitymeasuresoasitsvaluedoesnotdependontheunitusedforthemeasurementofincomes.ManyauthorslikeZheng(2004)arguethatthisisaratherstrongrequirementforaninequalityorapovertymeasureandthatoneonlyneedtherankingofdifferentdistributionstobepreservedwhenincomeareexpressedinadifferentmeasuringunit.2425Thisunit-consistencyaxiomisweakerthanscaleinvariancesinceitallowsfordifferentwayofthinkinginequalitywhilekeepingthesoledesirablecharacteristicofscaleinvariance.So,ifinequalityisconsideredfromanordinalpointofview, 19InKolm(1969),inequalitymeasuresthatfulllsscaleinvariancearecalled“intensive”.20MitraandOk(1995)aredubiousabouttheusefulnessofinvarianceaxiominthecontextofpovertysinceitwouldbeanon-sensefromapracticalpointofviewtocompare£¡©(Xp),©(z)¢with£(Xp,z).“Sincethevalueofapovertyindexisexplicitlyafunctionofthepovertyline,itdoesnotmakesensetocomparethepovertylevelsoftwoincomedistributionswithetwodifferentpovertylines.”Weadvocatethattheknowledgeofthepropertiesofpovertymeasuresareessentialwhenlinkingpovertytoinequality.Thenonsensewouldbetomakeuseofinequalityandpovertystudiesthatarenotbasedonthesameethicalground.Inthecontextofpovertyspellsdecompositionsbetweengrowthandinequalitycomponents,ourfeelingisthattheknowledgeoftheinvarianceaxiomunderlyingthechosenpovertymeasureisessentialsinceitpredeterminestherelativecontributionofgrowthandredistributiontopovertyreduction.Moreover,invariancepropertiesofthepovertymeasuresarecrucialforrelativepovertymeasures,thatiswhenthepovertylinedependsontheobservedincomedistribution.21Mostofthetime,povertymeasuresrespectasoleinvarianceaxiom.Soanexplicitformulationofwhichinvari-anceaxiomisusedmaybeconsideredassecondary.22Inthepresentsection,wewillassumentobeconstant.Thusitcaneasilybedroppedinordertosavespace.However,resultsdonotchangewhennvariesthroughtime.23Inordertosavespace,wedonotreportthecorrespondingformulafortheShapleydecompositionbutusethemintheapplicationdevelopedinsection4.24Theunit-consistencyaxiomisfurtheranalyzedforinequalityandpovertymeasurementinZheng(2007a,2005,2007b,c).25Theargumentthatchangesintheunitofmeasurementdonotnecessarilyhavethesameeffectsthatchangesofsize,wasearlierpresentedbyAczélandMoszner(1994)inthemoregeneralcontextofeconomicindices.8 2INVARIANCEANDTHEDECOMPOSITIONOFPOVERTYVARIATIONS scaleinvarianceisnottheuniquewayofthinkinginequalityanymore.However,oneshouldnotethatthescaleinvarianceaxiommayndlittlesupportinpresenceofnegativeincomessinceitinducesafailureofthePigou-Daltonprincipleoftransfers.26More-over,ethicalvaluesassociatedwiththemeasurementofinequalityandpovertyarenotunani-mouslyshared,evenwithinwelfareeconomists.Inparticular,thereisnounanimousagreementontheinvarianceaxiomthatshouldbeused.Forinstance,Dalton(1920)arguedthatapplyingthesamepositiverateofgrowthtoallincomedecreasesthedegreeofinequalityoftheincomedis-tribution.Usingquestionnairesonsamplesofundergraduatestudents,AmielandCowell(1992;1999;2001)andHarrisonandSeidl(1994b)observedthatscaleinvariancewasgenerallynotsup-portedforinequalityanalysisbyamajorityoftherespondentsandthatmanyrivalinvarianceaxiomswerepreferredbysomerespondents.27Amajorimplicationofthesesstudiesisthatin-equalitymeasurementtoolsshouldreecttheheterogeneityoffeelingsandmoraljudgmentsaboutinequalitysinceonecannotdiscriminatebetweenvalueswithoutethical,yetsubjective,arguments.Asthedesignofpoverty-reducingpoliciesrequirestheuseoftoolsthatareconsistentwithpolicymakers'ethicalvalues,oneshouldbecautiousofasystematicuseofindexesbasedonthescaleinvarianceaxiom.Thus,wehavetoexaminerivalversionsoftheinvarianceaxiomandtheirimplicationsforthedecompositionofpovertyspellsintogrowthandinequalitycompo-nents.2.2TranslationinvarianceTherstrivalinvarianceaxiomthatiscommonlytreatedintheliteratureisthetranslationin-varianceaxiomwhich,accordingtoKolm(1976a)isassociatedtoa“leftist”viewofinequality(inKolm'swordsscaleinvariancecorrespondstoa“rightist”view).Aninequalitymeasureissaidtorespectthetranslationinvarianceaxiomif:ª(XűI)ƪ(X)8±2R(2.6)whichimpliesthatanyequalincrementordecrementofeachincomeofthedistributionleavestheinequalityindexunchanged.2829Thelessrestrictiveversionofthetranslationinvarianceax- 26Forinstance,itmaybedifculttoarguethatthedistributionsX1Æ{¡2,20}andX2Æ{¡4,40}exhibitthesamedegreeofinequality.Anacceptationofthestatementª(X1)ƪ(X2)wouldimplyafailureofthePigou-Daltontrans-fersprinciple.Forinstanceaprogressivetransferof2unitswouldleadtothedistributionX3Æ{¡2,38}suchthatª(X3)6ª(X2).Therespectofbothscaleinvarianceandtransferprinciplewouldleadtothehardlyjustiableconclu-sionthatª(X3)6ª(X1).Zoli(2003)showsthatthesoleinvarianceaxiomwhichiscompatiblewithbothS-convexityandincomesdenedonRisthetranslationinvarianceaxiomfornï¿¿3,aresultthatwasalreadyobservedbyKolm(1976a)inthecontextofthe“centrist”inequalityview.27PrimaryinterestofAmielandCowell(2001)isdifferenceofinequalityandriskperceptions,buttheauthorschoosetofocusoninvarianceperceptions.AmielandCowell(1997)alsoperformedanempiricalinvestigationofstudents'agreementaboutaxiomscommonlyusedinthepovertymeasurementrelatedliteraturebutdidnotexaminecompliancewithinvarianceaxioms.28Foradetailedexaminationofinequalityindicesbasedontranslationinvariance,seeBlackorbyandDonaldson(1980).However,wecanmentionthevarianceasawidelyusedtranslation-invariantinequalitymeasure.Otherabso-lutepovertyindicesaresuggestedinMitraandOk(1995)andZheng(2007c).29HarrisonandSeidl(1994a)arguethatonecaneasilyndsomeinequalitymeasurethatisbasedonacombined9 2INVARIANCEANDTHEDECOMPOSITIONOFPOVERTYVARIATIONS iomthatissuitableforpovertyanalysisis:£¡zűI,XpűI,n¢Æ£¡z,Xp,n¢8±2R.(2.7)Inthiscase,£shouldnotbedenedanymorewithrespecttotheLorenzcurve.Thecounter-partofthetraditionalLorenzcurvefortranslationinvariantinequalitymeasuresistheabsoluteLorenzcurveLa(Moyes,1987).Thusatranslationinvariantpovertymeasurecanbewrittenas£(¹,La).Underthisaxiom,theestimationofthegrowthandinequalitycomponentsofpovertyspellswilldifferfromtheonecorrespondingtoscaleinvariantpovertymeasures.30Inthetwoperiodcase,Gt,tÅkandDt,tÅknowbecome:GTt,tÅkÆ£¡z,Xptűt,tÅkI¢¡£¡z,Xpt¢,(2.8)DTt,tÅkÆ£¡z,XptÅkűtÅk,tI¢¡£¡z,Xpt¢,(2.9)where±t,tÅkƹtÅk¡¹t.2.3IntermediateinvarianceThesetofinvarianceaxiomsisnotrestrictedtoscaleandtranslationinvariances,andmanyri-valaxioms,theso-calledintermediateinvarianceaxioms,havebeendevelopedduringthelastdecade.Intermediateviewsarebasedontheintuitionthatanequiproportionaladditiontoallincomesshouldincreaseinequalitywhileanequal-incrementtoallincomesshouldreducein-equality.Therstreasonofconsideringanintermediateviewisofcoursethatitmaybethewaysomepeoplefeelinequalityshouldbedened.Inthecontextofpovertyanalysis,itwillalsobeusefultoconsiderfamiliesofintermediateinequalityviewswhenthedecompositionsbasedonscaleandtranslationinvariancedonotyieldthesameconclusions.Asinthecontextofinequal-ityorderings,itmaybewisetouseintermediateinequalityviewssoastondcut-offvaluesoftheethicalparametersinvolvedinthedenitionofeachintermediateviewsuchthatconclusionschangewhenthisparticularvalueiscrossed.Thusitcanbeseenasawayofassessingtherobust-nessofaconclusionobtainedthroughscaleortranslationinvariance. viewwhichsatisesbothscaleandtranslationinvariancelike:ª(X)ƪµX¡min{xijiÆ1....n}I n¡¹¡min{xijiÆ1...n}¢¶ThisviewdiffersfromKolm;Kolm's(1969;1976b)“synthetic”solutionwhichsuggestsusinginequalitymeasuresthatarescaleinvariantintheirrelativeformandtranslationinvariantintheirabsoluteform(seenote15forthedenitionoftherelativeandabsoluteforms).Forthesakeofsimplicity,wewillsupposethatthedifferentinvarianceviewsarerivalandthusshouldnotbere-spectedsimultaneously.Suchanhypothesisisstandardintherelatedliteraturebutwouldmeritafurtherexaminationsinceempiricalevidenceshowsthatsomeindividualsmayfeelinaccordancewithbothscaleandtranslationinvari-ance,andingthatmostauthorsseeastheresultsofmistakes(seeforinstanceAmielandCowell1992).30DuclosandWodon(2004)alsoconsideredtranslationinvarianceinthecontextofthesocialevaluationof“pro-poor”characterofgrowth,anissuethatiscloselylinkedtothedecompositionofpovertyspellsintogrowthandin-equalityeffects.However,theauthorsdidnotinvestigatetheimplicationsofachangeinthechoseninvarianceaxiom,nordotheyillustratetheirapproachwithanempiricalapplication.10 2INVARIANCEANDTHEDECOMPOSITIONOFPOVERTYVARIATIONS Kolm(1976a,b)wasthersttogiveaformaltreatmenttoinequalityindicesbasedonanin-termediateaxiom.31His“centrist”viewisdenedbytherelation:ª¡¯(X¡I")ÅI"¢Æ¯ª(X)8¯È0,"2]¡1,0].(2.10)Inthesecase,notethatinvarianceisimplicitlydenedsincethetransformedincomedistri-butionofequation(2.10)is¯timesasunequalastheoriginaldistribution.Thusthisinequalityviewispoorlyoperationalforthedecompositionofpovertyspells.Inordertoavoidsuchanundesirablefeature,manyauthorsformulatedintermediateinequal-ityviewswhichexplicitlydenestheiso-inequalitysetofdistributionsthatcorrespondstoanydistributionX.Beforereviewingthevariousintermediateinequalityviewssuggestedinthelit-erature,itmaybeusefultostatepreciselywhatismeantbyintermediateinequalityviews.Thequestionisnottrivialsincetheconceptisgivendifferentmeaningsbyauthorsoftheeld.Gen-erally,aninequalityviewisintermediateifanequiproportionalincreaseinallincomesraisesthedegreeofincomeinequality,whereasanequalincrementdecreasesit.32Inthepresentpaperwewillconsiderclassesofintermediateviewswiththehelpofageneralparametrizeddenitionwhichstatesthattheequallyunequalincomedistributionsoughttobeexpressedasweightedmeansofthecorrespondingtransformeddistributionsunderscaleandtranslationinvariance.33Hence,wesuggestusingadenitionbasedonthefollowinglemma:Lemma1.Aninequalityviewissaidintermediateifthetransformeddistribution©I(X,¹Y)whichisconsideredasexhibitingthesamedegreeofinequalityasXandwithmeanincome¹Y,respectsthefollowingcondition:ª¡©I(X,¹Y)¢Æªµu(¹X,¹Y)X¹Y ¹XÅ¡1¡u(¹X,¹Y)¢¡XÅ(¹Y¡¹X)I¢¶Æª(X)(2.11)withu(¹X,¹Y)2[0,1]8¹Y2RÅÅ.Proof.Weknowthatwhateverthechoseninequalityview,everytransformedincomedistribution©(X)islocatedonthetwo-dimensionsub-spaceSXdenedbythevectorsIandX.Ontheother 31Intherelatedliterature,itiscommontofocusoninvarianceaxiomsthatareboundedbytranslationandscalein-variance.AmielandCowell(1992)andHarrisonandSeidl(1994b)studiesshowthatthereis(little)supportforextreme“rightist”—multiplyingincomesbythesameconstant¸È1decreasesinequality—andextreme“leftist”—addingthesameamount±È0toallincomesincreasesinequality—views.However,thereiscurrentlylittleformaltreatmentofsuchviews.Anexceptionisthe“ultrarightist”reference-pointinequalityviewdenedbyEbert(2004).However,itcanbeprovedthatthisgeneralizationofBossertandPngsten(1990)isnotunit-consistent.32InBossertandPngsten's(1990)andZoli's(2003)words,thispropertyisrespectivelycalled“compromisecondi-tion”and“compromiseinequalityequivalence”.33Anotherdenition,whichisderivedfromBossertandPngsten(1990)andZheng(2007c,a),requirestheformu-lationofanintermediatenessaxiomwhichimpliesthattheintermediateinequality(poverty)orderingoftwodistribu-tionsshouldbethesameastheorderingcorrespondingto“leftist”and“rightist”viewswhentheirinequality(poverty)orderingofthetwodistributionisidentical.Accordingtothisdenition,aninequality(poverty)indexisintermediateifitrespectstheintermediatenessaxiomandincludessomevectorofparameterssuchthatscaleinvariantpovertymea-suresandtranslationinvariantinequality(poverty)measuresareobtainedforsomelimitingspeciccombinationsoftheseparameters.Infact,thisparametrizeddenitiondoesnotdeneanintermediateview,butaclassofinterme-diateviews.Moreover,itisgenerallypossibletoextendthedomainofthevaluesoftheseparameterssoastoincludeextremeviews.11 2INVARIANCEANDTHEDECOMPOSITIONOFPOVERTYVARIATIONS hand,thesetofallincomedistributionswithmeanincomeequalto¹YisonthehyperplanedenedbytheequationPniÆ1yiÆn¹Y.AsthishyperplaneisdenedbythenormalvectorIwhichisbydenitionincludedinthesubspaceSX,itsintersectionwithSXisnon-emptyanddenesauniquerayL.Bydeni-tion,LpassesthroughthedistributionsX¹Y ¹X,XÅ(¹Y¡¹X)IandI¹Y,and,moregenerally,includesalldistributions©(X,¹Y)withmeanincome¹Y.Moreover,sinceitisastraightline,everydistribution©(X,¹Y)canbeexpressedasalinearcombinationofthedistributionsX¹Y ¹XandXÅ(¹Y¡¹X)I.Italsodeservestobestressedthatthecloseradistribution©(X,¹Y)istothedistributionI¹Y,themoreequalwillitbeconsidered.Themostgeneraldenitionofanintermediateinequalityviewisthataninequalityviewisin-termediatewhenanequiproportionaladdition(subtraction)toallincomesincreases(decreases)inequalitywhileanequal-increment(decrement)toallincomesreduces(increases)inequality.Consequently,anydistribution©I(X,¹Y)derivedfromanintermediateinequalityviewisneces-sarilylocatedbetweenthepointsX¹Y ¹XandXÅ(¹Y¡¹X)IonL.Thus,wecanusethefollowingexpressionof©I(X,¹Y):©I(X,¹Y)Æu(¹X,¹Y)X¹Y ¹XÅ¡1¡u(¹X,¹Y)¢¡XÅ(¹Y¡¹X)I¢(2.12)withu(¹X,¹Y)2[0,1]8¹Y2RÅÅ.Pluggingequation(2.12)intoequation2.1giveageneralde-nitionofintermediateinvarianceaxiomsthroughequation(2.11). Inthecaseoftheintermediateinvarianceaxiomsthatwillbereviewedinthenextparagraphs,theweighingtermucanbeexpressedasu(¹X,¹Y,½),½beingsomesetofparameters.Forsomecombinationsoftheseparameters,u(¹X,¹Y,½)Æ1(=0)and©I(X,½)becomestheequallyun-equaldistributionwithmeanincome¹Ycorrespondingtoscale(translation)invariance.Onecanalsonotethat,foragiveninitialdistributionX,umaydependonthevalueofthemeanofthenaldistribution.Inthiscase,uisnotconstantandtheintermediateviewmaytendto“leftist”or“rightist”viewsinequalityasmeanincomeincreases.Figures1illustratesthispropertyofintermediateviewsinthecaseofathree-persondistri-butionXÆ{x1,x2,x3}.PerfectequalityisrepresentedbythestraightlinethroughthepointsOandM.Alldistributionswithmeanequalto¹YareontheplanedenedbythepointsA,BandC.Ifincomesarenon-negative,thesetofdistributionswithmeanequalto¹YisrestrictedtothesurfaceABC.AllequallyunequaldistributionsissuedfromdistributionXareonthesubspacedenedbythevectors¡¡!OXand¡¡!OM.TheraythroughthepointsXandOistheiso-inequalitylinecorrespondingtoscaleinvariance.ItintersectsthesurfaceABCatXS.Thetranslationinvari-anceiso-inequalityrayisthestraightlinethroughXandsupportedby¡¡!OM.TheprojectionofXaccordingtothis“leftist”viewonthesurfaceABCisthepointXT.ItcanbeeasilyseenthatanytransformationofXwithmean¹YisonthesegmentLM,thatistheintersectionofsurfaceABCandthesubspacedenedby¡¡!OXand¡¡!OM.34ThepointMistheonecorrespondingtoanequaldistributionwhileLrepresentsthemostunequaldistributionwithmean¹Ythatcanbe 34WedonotconsiderthewholepartofthisintersectionsincepointsalongthelinethroughthepointsLandMbutclosertoAthanMarejustpermutationsoftheincomedistributionsobservedonthesegmentLM.12 2INVARIANCEANDTHEDECOMPOSITIONOFPOVERTYVARIATIONS directlyobtainedfromX,thatisfromalinearcombinationof¡¡!OXand¡¡!OM.SincebydenitionanyincomedistributionXIwhichisobtainedthroughanintermediatetransformationofXisconsideredasmoreequalthanXSandmoreunequalthanXTwhentotalincomeincreases,XIisnecessarilyonthesegmentXSXTandcanbeexpressedasalinearcombinationofthevectors¡¡¡!OXSand¡¡¡!OXT. Figure1:“Leftist”,“rightistandintermediateequallyunequalincomedistributions.2.3.1LinearintermediateinvarianceAsanalternativetothe“centrist”viewofKolm,BossertandPngsten(1990)suggestaninterme-diateinvarianceaxiomsuchthat:ª³XÅ'¡´XÅ(1¡´)I¢´Æª(X)8'2Rs.t.XÅ'¡´XÅ(1¡´)I¢2D®(2.13)where´2[0,1]reectsethicalpreferences.35Ebert(1997)noticesthatincomeshouldbedenedonD®sothat®Æ¡1¡´ ´.HealsodemonstratesthattherespectofthePigou-Daltonprincipleoftransfersimposesthecondition'È¡1/´.Soastoeasetheinterpretationoftheparameter',we 35Ebert(2004)suggeststhattheparameter´ofequation(2.13)canbedenedontherangeRÅinordertoextendintermediateinequalityto“ultra-rightist”viewsofinequality.For´È1,incomeshavetobegreaterthan®È0whichrepresentsthelevelofincomeneededforthesatisfactionofbasicneeds.This“ultra-rightist”viewofEbert(2004),alsocalled“referencepoint”inequality,impliesthateachadditionalincomehastobedistributedinproportionofeachindividualdisposableincome,thatisthedifferencebetweentheactualincomeand®,inordertopreservethedegreeofinequality.ThisviewwasalsoexpressedbutnotformalizedinDalton(1920).13 2INVARIANCEANDTHEDECOMPOSITIONOFPOVERTYVARIATIONS mayfollowZheng(2007a)andrewrite(2.13)as:ªµX¡¹XI ´(¹X¡1)Å1Å´¶ÆªµY¡¹YI ´(¹Y¡1)Å1Å´¶(2.14)whereYisthetransformeddistribution.IfweintendtotransformthedistributionXinordertogetanincomedistributionwithmean¹Ywithoutchanginginequality,equation(2.14)implies'ƹY¡¹X ´(¹X¡1)Å1.Usingthegeneralformofequation(2.11),itcanbeshownthat:u(¹X,¹Y,½)Æu(¹X,´)Æ´¹X ´(¹X¡1)Å1.(2.15)ManyauthorslikeZheng(2004)havestressedthatsuchanintermediatetransformationtendstobehavelikea“rightist”transformationasmeanincomeincreaseswhen´È0.Thiscanbeseenfromequation(2.15)sincelim¹X!Å1u(¹X,´)Æ18´È0.Usingtheintermediateinvarianceaxiomdenedbyequation(2.13)andagivenvalueof´yieldsthefollowingrelationforthemeasurementofpoverty:£³zÅ'¡´zÅ1¡´¢,XpÅ'¡´XpÅ(1¡´)I¢,n´Æ£¡z,Xp,n¢8'È¡1/´(2.16)andthustherespectiveexpressionsofthegrowthandinequalitycomponentsofpoverty:GIt,tÅkÆ£³z,XptÅ't,tÅk¡´XptÅ(1¡´)I¢´¡£¡z,Xpt¢,(2.17)DIt,tÅkÆ£³z,XptÅkÅ'tÅk,t¡´XptÅkÅ(1¡´)I¢´¡£¡z,Xpt¢,(2.18)where't,tÅkƹtk¡¹t ´(¹t¡1)Å1.However,Zheng(2004)demonstratedthatanyinequalitymeasurebasedontheviewdevelopedbyBossertandPngsten(1990)violatesunit-consistency.Thus,itshouldnotbeusedforthemeasurementofinequalityandpovertysinceitmayleadtonon-robustcon-clusions.Inordertogetalinearfamilyoftransformationsthatdonottendtobehavelikethe“rightist”view,PngstenandSeidl(1997)haveproposedtheso-calledray-invarianceaxiom.Aninequal-ityviewrespectsaray-invarianceaxiomifequallyunequaldistributionsarealongaraywhichincludestheobservedincomedistribution.Thisrayisdenedbyavectordrawnfromthen-simplexandwhichhastorespectthefollowingconditions:i)thevectorLorenz-dominatestheoriginaldistributionX;ii)thevectorreectsanunequaldistribution¡6Æn¡1I¢.36Thisviewdiffersfromtheonedescribedthroughequation(2.13)inasmuchasthepartoftheincrementalincomethatisnotequallysharedbetweeneachincomereceivers,isnotnecessarilydistributedinpro-portionofeachincome'sshareintheinitialdistribution.Ethicalpreferencesarethendescribedthroughan-vectorthatunfortunatelycannotbeeasilyinterpreted.37AparticularcaseofPng- 36Ageneralexpressionofray-invariancewasalsoprovidedbyKrtscha(1994)underthenameof“weakrelativeinequality”.Theauthorarguedthatthisisnotasuitablewayofthinkinginequalitysincethereisnoreasonthatalargeadditionalamountshouldbesharedinthesamemannerthatasmallerone.37Moreover,Zoli(2003)observesthatindividualswiththesamelevelofincomemayreceiveadifferentamount.Adifferenttreatmentofidenticalincome-receiverscanbeseenasaviolationoftheanonymityaxiom.Thesameremarkseemstoholdfortheinequalityviewdenedthroughequation(2.19).However,conditionsimposedonVinAlonso-14 2INVARIANCEANDTHEDECOMPOSITIONOFPOVERTYVARIATIONS stenandSeidl(1997)ray-invarianceisthe(V,À)-invariancedescribedbydelRioandRuiz-Castillo(2000)andgeneralizedbyAlonso-VillaranddelRio(2007a),whichimposestheuseofareferencedistributionVofsizenVandthefollowingrelationforinequalitymeasurement:ªµXÅ¿µÀVX ¹VÅ(1¡À)I¶¶Æª(X)8¿2Rs.t.XÅ¿µÀVX ¹VÅ(1¡À)I¶2D®.(2.19)whereÀ2[0,¶]reectsethicalpreferencesandVXistheprojectionofthereferencedistributionVintothesubspaceSXdenedbythevectorsXandI.ForÀÆ0,the“leftist”inequalityviewisobtainedwhereasÀƶcorrespondstothe“rightist”view.38InappendixA,wedemonstratethatavalidequationforthecomputationofVXis:VX ¹VÆ1 ¶X ¹Xŵ1¡1 ¶¶I.(2.20)with:¶Ævuuuut n¡1PniÆ1³yi ¹Y¡1´2 n¡1VPnViÆ1³vi ¹V¡1´2.(2.21)InordertosimplifywecanchoseXasthereferencedistribution.Inthiscase,thelinkwithequation(2.11)isstraightforwardsinceu(¹X,¹Y,½)ÆÀ.39WithanyotherregulardistributionV,itcaneasilybeprovedthatlemma1stillholdsundercertainconditions(cf.appendixB).40ContrarytoBossertandPngsten's(1990)intermediateview,oneshouldnotethatthevalueoftheparameterÀiscontingenttothechoiceofareferencedistribution.Ifweconsidertwodis-tributionsXandYwithdifferentmeans,onecaneasilydemonstratethatthetransformationofYintoXusingYasthereferencewillrequireavalueÀ0thatisdifferentfromtheonecorrespondingtoatransformationofXintoYusingXasthereferenceexceptforÀÆ1andÀÆ0.Otherwise,weobserve(delRioandRuiz-Castillo,2000,proposition1)À0ÆÀ¹Y (1¡À)¹XÅÀ¹Y.41 VillaranddelRio(2007a)ensureanonymitytoberespected(seenote40).38IndelRioandRuiz-Castillo(2000)andAlonso-VillaranddelRio(2007a),theparameterÀisdenedontheunitinterval.However,foranydistributionX,itcanbeshownthatÀÆ1correspondstoan“ultra-rightist”viewuntilVXisequaltoXuptoascalefactor.39Zheng(2004,proposition2.4)arguedthatinequalitymeasuresbasedonPngstenandSeidl(1997)tendtobehaveliketranslationinvariantmeasuresasmeanincomeincreaseswhenÀÇ1.Ourresultthatu(¹X,¹Y,½)isconstantinvalidatestheseproposition.40InAlonso-VillaranddelRio(2007a),ray-invariantinequalityviewsarecharacterizedbysubstitutingthereferencedistributionV(orX)bytheEuclideandistanceÂ2"0,r nPi³xi n¹X¡1 n´2#betweenthechosenvectorofincrementsÀV ¹VÅ(1¡À)IandtheonecorrespondingtoequalincrementsI n.Thevector(Â,À)denesauniqueintermediateviewsinceauniquevectorÀV ¹VÅ(1¡À)Iisassociatedwitheachtwo-dimensionsubspaceforgivenvaluesofÂandÀ.Thisviewimplicitlyintroducesanew(andmaybecontroversial)axiomforthemeasurementofinequalitythatsuggeststhattwodistributionswiththesamemeanareequallyunequaliftheirsize-normalizedEuclideandistancefromthevectorofperfectequalityisthesame.Inpractice,suchageneralizationofdelRioandRuiz-Castillo's(2000)viewwillbehelpfultocomparethedifferentvaluesofÀwhenthedecompositionofpovertyspellsisrealizedformanysubperiods(cf.appendixA).41IfthesizeofdistributionsXandYisrespectivelynandmwithn6Æm,therelationbetweenÀ0andÀbecomes:À0ÆÀm¹Y (1¡À)n¹XÅÀm¹Y.15 2INVARIANCEANDTHEDECOMPOSITIONOFPOVERTYVARIATIONS Therelatedinvarianceaxiomforpovertymeasurementofthisinequalityviewis:£ÃzÅ¿µÀVX,b ¹VÅ1¡À¶,XpÅ¿ÃÀVpX ¹VÅ(1¡À)I!,n!Æ£¡z,Xp,n¢.(2.22)whereVpXisthebottompartofVXsothatXpandVpXareofthesamesize,andVX,bistheb-thelementofVXsothatVX,b¡16zÇVX,b.Theparameter¿isrestrictedinthesamewayasinequa-tion(2.19).Withsuchkindofinequalityview,wegetthefollowingformulaforthecomputationofGt,tÅkandDt,tÅk:GRt,tÅkÆ£Ãz,XptÅ¿t,tÅkÃÀVpt ¹VÅ(1¡À)I!!¡£¡z,Xpt¢,(2.23)DRt,tÅkÆ£Ãz,XptÅkÅ¿tÅk,tÃÀVptÅk ¹VÅ(1¡À)I!!¡£¡z,Xpt¢,(2.24)with¿t,tÅkƹtÅk¡¹tandVtandVtÅkbeingtherespectiveprojectionsofVinthetwo-dimensionsubspacesincludingXtandXtÅkanddenedthroughequation2.20.42ItcaneasilybeproventhatdelRioandRuiz-Castillo's(2000)inequalityviewcomplieswithunit-consistency(cf.appendixC)andsoissuitableforpovertyanalysis.4344However,amajorissuewithequations(2.23)and(2.24)isthatthedecompositionofpovertyvariationsmaynotprovidetheresultscorrespondingtoscaleinvarianceinamultiperiodanalysis.Asthevalueoftheparameter¶varywithfromadistributiontoanother,wehavetoadoptitsmin-imalvalueforallcomparisonsinordertoavoid“ultra-rightist”views.Whenusingthisparticularvalue¶¤,wewillthenobtainintermediatedecompositionsforsomeperiodsand“rightist”decom-positionsfortheperiodswhichinitialornaldistributionistheonethatdenes¶¤.Thisresultispuzzlingsinceitwouldbeanon-sensetocompareonthebasisofthesame(V,¶¤)-intermediateviewintermediateeffectsforaperiodwith“rightist”effectsforotherperiods.Consequently,wearguethattheviewdevelopedbydelRioandRuiz-Castillo(2000)isnotsuitableforpovertyanal-ysis.Moreover,linearintermediateinvarianceaxiomsmaynotbeanappropriatewayofmodellingindividual'tastesandfeelings.AmielandCowell(2001)resultssuggestthatmanypeoplemay ItcanbeeasilyshownthatÀ0ÈÀif¹YȹX.42Alternatively,ifXtisalwayschosenasthereferencedistribution,analternativeformulationforequation(2.24)is:Dt,tÅkÆ£Ãz,XptÅkÅ¿tÅk,tÃÀXpt ¹tÅ(1¡À)I!!¡£³z,Xpt´.Fromapracticalpointofview,itmaybeeasiertouseequation(2.24),sinceobserveddistributionsXtandXtÅkarenotnecessarilyofthesamesize.43Analternativewayofprovingunit-consistency(Alonso-VillaranddelRio,2007a)istodeneaninequalitymea-surebasedontheexpressedinequalityviewandwhichrespectsbasicinequalityaxiomsandthentodemonstratethatthemeasureisunit-consistent.44Zheng(2007a)recentlydemonstratesthatthesoleunit-consistentintermediateinequalityviewwhichLorenz-criterioncanbeexpressedasaquasilinearweightedmeanoftherelativeandabsoluteLorenzcurvesisthenon-linearinequalityviewproposedbyKrtscha(1994)andYoshida(2005).ThisresultisconsistentwithourndingssinceonecanprovethattheintermediateLorenzcurvecorrespondingtodelRioandRuiz-Castillo's(2000)viewisnotintermediateinthesenseofZheng(2007a)andthuscangenerallynotbeexpressedasaquasilinearweightedmeanoftherelativeandabsoluteLorenzcurves.16 2INVARIANCEANDTHEDECOMPOSITIONOFPOVERTYVARIATIONS thinkinequalityinawaythatinvolvesnon-linearinvarianceaxioms.Suchviewsarepresentedinthenextsection.2.3.2Non-linearintermediateinvarianceIntheprecedingsection,weconsideredinequalityviewssuchthatdistributionsthatareconsid-eredasequivalenttodistributionXfromaninequalitypointofview,arealignedonauniqueraythroughX.Inthepresentsectionwefocusonnon-linearintermediateviews.ThedifferencewithlinearintermediateviewsisthatthecompletesequenceofequallyunequaldistributionsofsizendenesacurvethroughX.45Recently,Zoli(2003)andYoshida(2005)developedsomenon-linearintermediateviewofinequalitythatdonotbreakwithbasicdesirablepropertieslikethelineartransformationspre-sentedabove.Zoli(2003)rstsuggesteda“exibleinequalityequivalence”transformationsuchthat:ªµµ!¹Å· ¹Å·¶¾(X¡¹I)Å!¹I¶Æª(X)8!2RÅs.t.µ!¹Å· ¹Å·¶¾(X¡¹I)Å!¹I2DÅ(2.25)where¾and·areethicalpreferenceparametersrespectivelydenedontheunitintervalandonRÅ.For¾Æ1,wegetequation(2.13)with·Æ1¡´ ´and!Æ'¡´(¹¡1)Å1¢ ¹Å1.BossertandPngsten(1990)intermediateinequalityisaparticularcaseofZoli(2003)non-linearinequalityview.InthespiritofKrtscha's(1994)faircompromiseinequalityview,asingle-parameterversionofthisgen-eralinvarianceaxiom,the¾-invarianceaxiom,issuggestedbyYoshida(2005)with·Æ0.Zheng(2007a)demonstratesthatZoli(2003)'s“exibleinequalityequivalence”canbeusedtodenein-equalitymeasuresthatrespecttheunit-consistencyaxiomonlyif·Æ0.Thus,weonlyfocusinthepresentstudyonthe¾-invariancedenedthroughequation(2.26):46ª¡!¾XÅ¡!¡!¾¢¹I¢Æª(X)8!2RÅ.(2.26)Asfortheprecedingintermediateinequalityviews,itcanbeusefultoexpresstheequallyunequalincomevectorwithmean¹YandcorrespondingtodistributionXusingequation(2.11). 45Hagenaars(1987)wasapparentlythersttodeneapovertymeasurethatdoesnotcomplywithscaleortrans-lationinvariance.Herfamousmeasure,£H(x,z)Æ1 nPqiÆ11¡logxi logzwhereqisthelengthofthevectorXp,implicitlyreliesonthefollowingnon-linearintermediateinvarianceaxiom:ª(Xº)ƪ(X)8º2RÅÅ,X2D1.whichcorrespondstoanultra-rightistviewforanypositiverateofgrowth(xjÈ18jandºÈ1).ThisinequalityviewisconsideredbyEbert(2004)asnon-coherentsinceasequenceofaprogressivetransferandanincreaseinmeanincomethatdoesnotchangethedegreeofinequality,doesnotyieldthesamedistributionastheconversesequence.However,wecanquestionifthis“transfer-consistency”axiomisreallydesirable.Ontheotherhand,itshouldbestressedthatthisinequalityviewisnotsuitableforpovertyandinequalitymeasure-mentsinceunit-consistencyisnotrespectedforall¸2RÅÅ.Forinstanceif£H(X,z)È£H(Y,z),H(¸X,¸z)willbegreaterthatH(¸Y,¸z)ifandonlyif¸È1 z.Thusunit-consistencyisviolatedfor¸2³0,1 z´.Moreover,itseemsthatnoultra-rightistinequalityorpovertymeasurecancomplywithunit-invariance.Infact,ultra-rightistviewsrequireincomestobedenedonthesetD®½DÅÅ.Sincetherealwaysexistsomestrictlypositivescalar¸suchthat¸XÝD®,unit-consistencycannotberespected.46Zoli(2003)alsoconsideredthisspecialcaseandcalledit“proportionalinequalityequivalence”.17 3INVARIANCE,THEMEASUREMENTOFPOVERTYANDTHEDECOMPOSITIONOFITSVARIATIONS UsingYoshida(2005)originalversionofequation(2.26),wederivethefollowingexpressionoftheweighingfunction:u(¹X,¹Y,½)Æu(¹X,¹Y,¾)Ƴ¹Y ¹X´¾¡1 ¹Y ¹X¡1.(2.27)This¾-invarianceaxiomimpliesthat,inordertokeepinequalityunchanged,anyincrementalincomeshouldbedividedintoinnitesimalamountsthataresequentiallysharedsuchthat100¾percentaredistributedinproportionoftheincomerelativesharesand100(1¡¾)percentequallyamongincomereceivers.Alonso-VillaranddelRio(2007b)notethatfor¾È0,thisinequalityviewtendstobehavelikea“leftist”viewastheinitialmeanincomeincreases.Here,wewouldliketostressthatthisstatementdependsontheassumptionmadeabouttherelationbetween¹Xand¹Y.Ifweconsideraconstantdifferencebetweentheinitialandnalmeanincomes,wehavetorecognizethatlim¹X!Å1u(¹X,¹Y,¾)Æ08¾Ç1.Ontheotherhand,foragivenpositivegrowthrategƹY¡¹X ¹X,itcanbeseenfromequation(2.27)thatlim¹X!Å1u(¹X,¹Y,¾)Æ(1Åg)¾¡1 g8¾.47Inotherwords,the¾-invarianceaxiomkeepsbeingintermediateifweconsiderconstantgrowthrates.Itcanalsobeseenfromequation2.27thatthevalueof¾suchthattheintermediatecounterfactualdistributionisthearithmeticmeanorthecounterfactual“leftist”and“rightist”incomedistributions,is˜¾Ælog¡1³¹Y ¹X´log³1 2³¹Y ¹XÅ1´´'0.5inmostcases.Forinstanceifmeanincomeincreasesby10%overtheperiodofinterest,thevalueof˜¾isapproximatelyequalto0.51.Thus,evenifwecomparetheresultsofintermediatedecompositionsovermanyperiodswithdifferentgrowthrates,itisreasonabletoacceptthesamevalueof¾foreachperiodasstandingforthesameintermediateinequalityview.Theweakercounterpartofequation(2.26)inpovertyanalysisis:£¡!¾zÅ¡!¡!¾¢¹,!¾XpÅ¡!¡!¾¢¹I,n¢Æ£¡z,Xp,n¢8!2RÅ.(2.28)andthecorrespondingvalueofGt,tÅkandDt,tÅkare:GKt,tÅkÆ£³z,!¾t,tÅkXptų!t,tÅk¡!¾t,tÅk´¹tI´¡£¡z,Xpt¢,(2.29)DKt,tÅkÆ£³z,!¾tÅk,tXptÅkų!tÅk,t¡!¾tÅk,t´¹tÅkI´¡£¡z,Xpt¢,(2.30)with!t,tÅkƹtk ¹t.3Invariance,themeasurementofpovertyandthedecompositionofitsvariations3.1TheheadcountindexMostpovertyindexesrespectauniqueinvarianceaxiom.Asaconsequencethereisnouncer-taintyabouttheinvarianceaxiomthatshouldbeadoptedforthedecompositionofvariations 47Aninterestingfeatureisthatthevalueofu(¹X,¹Y,¾)doesnotdependofthemonetaryunitchosenforthemeasurementofincomes.Consequently,foragivenvalueof¾,wewillgetthesamegrowthandinequalityeffectsifincomesaremeasuredindollarsorinthousanddollars.18 3INVARIANCE,THEMEASUREMENTOFPOVERTYANDTHEDECOMPOSITIONOFITSVARIATIONS ofsuchmeasures.Howeveritcanbeeasilyshownthatthemostwidelyusedpovertyindex,theheadcountindex,isthesolemeasurethatisconsistentwithalltheinvarianceaxiomspresentedthroughtheprecedinglines.ForagivenincomedistributionX,weknowthattheheadcountindexhissimply:h(z,X)Æ¥(XjxiÇz) ¥(X)(3.1)where¥isafunctionreturningthelengthofthespeciedvector.Aparticularfeatureofthehead-countindexamongthetraditionalpovertymeasuresispresentedinthefollowingproposition:Proposition1.Continuousincreasingfunctionsoftheheadcountindexarethesolepovertymea-suresthatrespectbothscale,translationandintermediateinvarianceaxioms.Proof.Intherstparagraphsofsection2,aninvarianceaxiomisgivenageneraldenitionwiththehelpofacontinuousincreasingfunction©onD®.Thisprecludestheuseofextreme“left-ist”views,sincetheclassoffunctions©oughttoberestrictedtorankpreservingfunctions.Sowhateverthespecicformof©,weshouldobserve©(xp)6©(z)6©(xpÅ1)forxp6z6xpÅ1,©(xi)Ç©(xp)8iÇpand©(xi)È©(xp)8iÈp.Consequently¥¡©(X)j©(xi)Ç©(z)¢Æ¥(XjxiÇz)andh¡©(z),©(X)¢Æh(z,X).Wecanconcludethattheheadcountindexcomplieswithallinvari-anceaxiomswhichimplytransformationsofincomesthatareincludedbetweenthoseinducedbyscaleandtranslationinvarianceaxioms.Toprovethattheheadcountisthesoletraditionalpovertymeasurethatiscompatiblewiththevariousinvarianceaxiomspresentedearlier,wecanmakeuseoftheresultsofZheng(1994,proposition2)whichstatesthatthesolepovertymeasuresthatrespectbothscaleandtranslationinvarianceaxiomsaretheheadcount-relatedpovertyindexes,i.e.povertyindexesthataredenedascontinuousincreasingfunctionsofthesizeofthedistributionandthenumberofpoor.48 Proposition1meansthatthedecompositionofvariationsoftheheadcountindexintogrowthandinequalitycomponentscanbehandledinmanyways.So,thecoupleofequations(2.4,2.5),(2.8,2.9),(2.17,2.18),(2.23,2.24)and(2.29,2.30)areallconsistentwiththeaxiomaticofthehead-countindex.Thechoiceofaparticulardecompositionframeworkreliesentirelyonindividualperceptionsandtastesaboutinequality.However,forreasonsthathavebeenalreadydetailledintheprecedinglines,wearguethatresearchersshouldmakeuseofthesole“rightist”,“leftist”andnon-linearintermediatedecompositions,andthenusethesolecoupleofequations(2.4,2.5),(2.8,2.9)and(2.29,2.30).3.2ImplicationsforthedecompositionofpovertyvariationsInthefollowingparagraphs,wetrytosketchtheconsequencesontheestimatedgrowthandin-equalityeffectsofamovefroma“rightist”toa“leftist”view.Itcaneasilybeshownthatwedonot 48AccordingtoZheng(1994)alessrestrictivedenitionofthefamilyofheadcount-relatedpovertyindexescanbeadoptedifwedonotimposetherespectofthepopulation,weakmonotonicityandsubgroupconsistencyaxioms.19 3INVARIANCE,THEMEASUREMENTOFPOVERTYANDTHEDECOMPOSITIONOFITSVARIATIONS needtoconsiderexplicitlyintermediateinvariancetodeneanorderingof“leftist”,intermedi-ateand“rightist”effectssincetherstandthelastdenestherangeofthesecond.Thisresultissummarizedinproposition2.Proposition2.Thevalueofanyintermediategrowth(inequality)effectofobservedvariationsoftheheadcountindexisalwayscomprisedbetweenthevaluesofthe“leftist”and“rightist”growth(inequality)effects.Proof.Thedemonstrationisadirectimplicationoflemma1.Astheintermediateequallyunequaldistributions©I(X,¹Y)areweightedmeanofthe“leftist”an“rightist”counterfactualdistribu-tionsX¹Y ¹XandXÅ(¹Y¡¹X)I,itscumulativedistributionfunction(CDF)isboundedbetweentheCDFsofX¹Y ¹XandXÅ(¹Y¡¹X)I.Asaresult,theeffectsobtainedthroughtheequations(1.3)to(1.6)underscaleandtranslationinvarianceareboundsfortheeffectsobtainedthroughinterme-diateinvarianceaxioms. Figure2illustratesthispropositioninthecaseofathree-person(orthree-group)distribution.ThetriangleABCisthesameasingure1butcanbereducedtoasimplexforthesakeofsimplic-ity.Inthiscase,thedistanceofpointwithrespecttothepointsA,BandCrespectivelyindicatestheshareofeachindividualintotalincome.ThepointszB1,zC1,zA2,zC2,zA3andzB3aretheprojec-tionsofthepovertylinezforeachindividualalongtheaxisAB,BCandAC(seegure2).Then,thepovertystatusofeachmemberofthepopulationdependsofthepositionofthedistributionwithrespecttothelineszB1zC1,zA2zC2andzA3zB3.Inthisexample,weconsiderthat¹Yislargerthanz.ThentherstindividualfallintopovertyifthepointdistributionisontherightofthelinezB1zC1andnon-poorotherwise.IfXTandXSarethecorrespondingequallyunequaldistributionstoXwithmean¹Y,weknowthateveryintermediatedistribution©I(X,¹Y)willbelocatedonthesegmentXTXS.PointsXI,XI0andXI00arepotentialcounterfactualdistributionscorrespondingtointermediatetransformationsofX.Whateverthelocationof©I(X,¹Y),wecanseethatthevalueoftheheadcountindexisalwayscomprisedbetweenh³z,X¹Y ¹X´andh¡z,XÅ(¹Y¡¹X)I¢. Figure2:Invarianceandpovertyvariationsinthethree-personsimplex.20 3INVARIANCE,THEMEASUREMENTOFPOVERTYANDTHEDECOMPOSITIONOFITSVARIATIONS 3.2.1“Leftist”vs“Rightist”growtheffectsInthefollowingparagraphs,weshowthatthecomputationofthegrowthandinequalityeffectsofvariationsoftheheadcountindexundereachinvarianceaxiomaremostofthetimenotnecessaryifonejustintendtocomparethemagnitudeoftheeffects.Proposition3statesthatamovefroma“rightist”toamore“leftist”inequalityviewisnotlikelytochangethesignoftheestimatedgrowtheffect.Proposition3.Whateverinvarianceaxiomisconsidered,thesignofthegrowtheffectisthesameastheobservedgrowthrate.Proof.Thepropositionisjusttheresultoftheapplicationoftheweakmonotonicityaxiomwheneveryincomexiisincreased. Itdeservestobeemphasizedthatwhencontinuousdistributionsofincomeareconsideredandtheprobabilityofobservinganincomeequaltozisnonzero,thegrowtheffectisalwaysdifferentfromzero.49Proposition4.InthecontextoftheDattandRavallion(1992)decompositionoftheheadcountindex,the“leftist”growtheffectislowerthanthe“rightist”growtheffectifandonlyiftheobservedgrowthrateispositive(negative)andthenalmeanincomeisabove(below)thepovertyline.Proof.UsingtheDattandRavallion(1992)approach,therelationbetweenthedifferentgrowtheffectsuniquelydependsonthesignoftheobservedrateofgrowthandtherelativepositionofthenalmeanincomeandthepovertyline.Thus,wehavetoconsiderthefollowingdifferentfourcasessituations:i)Let'sconsiderrstthemostcommoncaseofapositiverateofgrowth(¹YȹX)and¹YÈz,withXandYbeingrespectivelytheinitialandnalincomedistributions.Ifmeanincomesarehigherthanthepovertyline,poorindividualsgainlessfrom“pure”growthunderscaleinvariancethanundertranslationinvariance,i.e.:xjÅ(¹Y¡¹X)Èxj¹Y ¹X8j2{1,...p}.(3.2)If¹XÇz,then:(xjÅ(¹Y¡¹X)ï¿¿xj¹Y ¹X8j2{1,...sjxs¡1ǹX6xs},xjÅ(¹Y¡¹X)Çxj¹Y ¹X8j2{sÅ1,...pjxs¡1ǹX6xs}.(3.3)Inthislastcase,weareonlyinterestedinindividualswhichrankareintheset{1,...sjxs¡1ǹX6xs}sincealltheotherpoorindividualsbecomenon-poorwhentheirincomeareincreasedbyproportionalorequalincrements.Asaconsequence,whatevertherespectivepositionof¹Xandz,h¡z,XÅ(¹Y¡¹X)I¢6h³z,X¹Y ¹X´.HenceGT6GSasGisanincreasingfunctionofh¡©(X),z¢. 49AnanalyticaldemonstrationformarginalchangesofmeanincomeusingscaleinvariancecanbefoundinKak-wani(1993).21 3INVARIANCE,THEMEASUREMENTOFPOVERTYANDTHEDECOMPOSITIONOFITSVARIATIONS ii)Ontheotherhandifthegrowthrateisnegative(¹YǹX),wehavetoconsidertheevolu-tionofnon-poorincome,ormorepreciselytheincomeofthenon-poorthatwouldbecomepoorafterthe“pure”growtheffect.Thecomparisonofthesecounterfactualincomeswithscaleandtranslationinvarianceyields:(xjÅ(¹Y¡¹X)6xj¹Y ¹X8j2{p,...sjxs¡1ǹX6xs},xjÅ(¹Y¡¹X)Èxj¹Y ¹X8j2{sÅ1,...njxs¡1ǹX6xs}.(3.4)sincezǹYǹX.Asindividualsfromtheset{sÅ1,...njxs¡1ǹX6xs}donotcrossthepovertyline,wecanfocusontherstlineofequation(3.4).Thusweobserveh¡z,XÅ(¹Y¡¹X)I¢ï¿¿h³z,X¹Y ¹X´andconcludeGTï¿¿GS.iii)Forapositiverateofgrowthbut¹YÇz,theincomeofthepoorindividualsincreaseac-cordingtoequation(3.3).Thistime,onlyindividualswhichrankisintheset{sÅ1,...pjxs¡1ǹX6xs}cancrossthepovertyline.Consequently,h¡z,XÅ(¹Y¡¹X)I¢ï¿¿h³z,X¹Y ¹X´.HenceGTï¿¿GS.iv)Consideringthelastsituationofanegativegrowthratewith¹YÇz,weknowthatincomeofthenon-poorbecome:xjÅ(¹Y¡¹X)Èxj¹Y ¹X8j2{pÅ1,...n}.(3.5)if¹XÇzand:(xjÅ(¹Y¡¹X)6xj¹Y ¹X8j2{p,...sjxs¡1ǹX6xs},xjÅ(¹Y¡¹X)Èxj¹Y ¹X8j2{sÅ1,...njxs¡1ǹX6xs}.(3.6)otherwise.Inthesecondcase,allmembersfromtheset{p,...sjxs¡1ǹX6xs}becomepoorwhateverinvarianceaxiomisconsidered.Thusresultsdifferonlywithrespecttotheevolutionoftherichestpartofthenon-poorpopulation.Consequently,weshouldobserveGT6GSsinceh¡z,XÅ(¹Y¡¹X)I¢6h³z,X¹Y ¹X´. Figure3givessomeinsightabouttherationaleunderlyingproposition4foratwo-person(ortwo-group)distributionandapositiverateofgrowth.50StartingfromthepointXwithcoordi-nates(x1,x2),theequallyunequaldistributionscorrespondingtoscaleandtranslationinvari-ancearerespectivelyrepresentedbythelinesXXSandXXT.Foranaldistributionwithmeanincome¹Y,thecorrespondingcounterfactualincomescanbefoundatthepointswhereeachofthesecurvescrossthelineXSXTwhichrepresentsthesetofincomedistributionswithmean¹Y.Thus,weobtainthetwodistributions(xS1,xS2)and(xT1,xT2).IfthepovertylineissettozǹY,onlyoneindividualisconsideredaspoorintheinitialdistribution.Whereasthescaleinvariancetransformationofincomesdoesnotchangethevalueoftheheadcountindex(therstindividualisstillpoor),thesharingoutoftheadditionalincomeundertranslationinvariancelowerspoverty 50Thecaseofnegativegrowthratecanbeeasilyderivedfromgure4ifdistributionY(pointE)ischosenastheinitialdistribution.22 3INVARIANCE,THEMEASUREMENTOFPOVERTYANDTHEDECOMPOSITIONOFITSVARIATIONS sincenobodyisconsideredaspooranymore.Consequently,weobservesGTÇGSÆ0.Ontheotherhand,ifthepovertylineissettoz0ȹY,theresultsarereversed.Theinitialdistributionpresentsà100%povertyratewhichdoesnotchangewithatranslationinvariancetransformationofincomesbyishalvedusingthescaleinvarianceaxiom.Thus,weobserveGSÇGTÆ0. Figure3:“Leftist”vs“rightist”growtheffectswithapositiverateofgrowth.Mostofthetime,theShapleydecompositionprovidesthesameorderingofthe“leftist”and“rightist”growtheffectsastheDattandRavallion(1992)decomposition.Thisresultanditsex-ceptionsaresummarizedinthefollowingproposition:Proposition5.TheShapleydecompositiontechniqueyieldsthesameorderingofthe“leftist”and“rightist”growtheffectsthantheDattandRavallion(1992)decompositionexceptifthepovertylineliesbetweentheinitialandnalmeanincomes,and:hµz,X¹Y ¹X¶¡hµz,Y¹X ¹Y¶Èh¡z,XÅ(¹Y¡¹X)I¢¡h¡z,YÅ(¹X¡¹Y)I¢.(3.7)Proof.Theextensionofproposition4totheShapleydecompositionisstraightforwardandyieldsthesameresultexceptwhenthepovertylineliesbetweenthemeanincomeoftheinitialandnaldistributions.Thesetwoparticularcasesare:i)If¹YÈzȹX,thedifferencebetween“rightist”and“leftist”counterfactualincomesofthepoorindistributionXaredescribedbyequation(3.3)andthenh¡z,XÅ(¹Y¡¹X)I¢6h³z,X¹Y ¹X´.Ontheotherhand,thecounterfactualincomesofthenon-poorindistributionYarerankedasfollows:(yjÅ(¹X¡¹Y)6yj¹X ¹Y8j2{q,...rjyr¡1ǹY6yr},yjÅ(¹X¡¹Y)Èyj¹X ¹Y8j2{rÅ1,...njyr¡1ǹY6yr}.(3.8)withqÆ¥(YjyiÇz).Sincetheincomeofallmembersoftheset{q,...rjyr¡1ǹY6yr}fallbelowthepovertyline,onlythesecondlineofequation(3.8)canbeconsidered.Asaconsequence,h¡z,YÅ(¹X¡¹Y)I¢6h³z,Y¹X ¹Y´.23 3INVARIANCE,THEMEASUREMENTOFPOVERTYANDTHEDECOMPOSITIONOFITSVARIATIONS ii)Consideringthesituationwith¹XÈzȹY,theincomeofthenon-poorindividualindis-tributionXchangeaccordingtoequation(3.6)andh¡z,XÅ(¹Y¡¹X)I¢6h³z,X¹Y ¹X´.Asgrowthisnegative,wehavetofocusontheevolutionoftheincomeofthepoorindistributionY.Thesecounterfactualincomesexhibitthefollowingrelation:(yjÅ(¹X¡¹Y)ï¿¿yj¹X ¹Y8j2{1,...rjyr¡1ǹY6yr},yjÅ(¹X¡¹Y)Çyj¹X ¹Y8j2{rÅ1,...qjyr¡1ǹY6yr}.(3.9)Aseveryindividualj2{rÅ1,...qjyr¡1ǹY6yr}isnotpooranymorewhateverinvarianceax-iomisconsidered,onlymembersfromtheset{1,...rjyr¡1ǹY6yr}matter.Weconcludethath¡z,YÅ(¹X¡¹Y)I¢6h³z,Y¹X ¹Y´.Whatevertheorderingofthemeanincomeoftheinitialandnaldistributions,developmentsofcasesi)andii)yieldsthesameexpressionofthedifferencebetweenthe“rightist”and“leftist”growtheffects:G0S¡G0TÆ1 20BBB@hµz,X¹Y ¹X¶¡h¡z,XÅ(¹Y¡¹X)I¢| {z }È0Åh¡z,YÅ(¹X¡¹Y)I¢¡hµz,Y¹X ¹Y¶| {z }Ç01CCCA.(3.10)Thus,accordingtoequations(3.10),therankingofG0SandG0Tcannotbeknownuntilthein-termediatevaluesh¡z,©(X)¢andh¡z,©(Y)¢arecomputed.ForG0TÈG0S,rearrangingthesecondtermofequation(3.10)yieldsequation(3.7). Forconvenience,thecombinedresultsofpropositions3,4and5aresummarizedintable1.Table1:Comparisonofthegrowtheffectsunderscaleandtranslationinvariance. ConditionDecompositiontechnique GrowthPovertylineDattandRavallion(1992)Shapley ¹YȹX¹YÈzGT6GS60G0T6G0S60oraG0S6G0T60¹YȹX¹YÇzGS6GT60G0S6G0T60¹XȹY¹YÈzGTï¿¿GSï¿¿0G0Tï¿¿G0Sï¿¿0¹XȹY¹YÇzGSï¿¿GTï¿¿0G0Sï¿¿G0Tï¿¿0orbG0Tï¿¿G0Sï¿¿0 a:ifzȹXandh³z,X¹Y ¹X´¡h³z,Y¹X ¹Y´ï¿¿h¡z,XÅ(¹Y¡¹X)I¢¡h¡z,YÅ(¹X¡¹Y)I¢.b:if¹XÈzandh³z,X¹Y ¹X´¡h³z,Y¹X ¹Y´ï¿¿h¡z,XÅ(¹Y¡¹X)I¢¡h¡z,YÅ(¹X¡¹Y)I¢.Inmostcases,thevalueofthepovertylineisbelowthoseoftheinitialandnalmeanin-come.Asaconsequence,weshouldexpectthe“leftist”growtheffecttobeinferior(superior)tothe“rightist”growtheffectforpositive(negative)observedgrowthrates.24 3INVARIANCE,THEMEASUREMENTOFPOVERTYANDTHEDECOMPOSITIONOFITSVARIATIONS 3.2.2“Leftist”vs“Rightist”inequalityeffectsNow,weturntothe“leftist”and“rightist”inequalityeffectsofheadcountindexvariations.Con-trarytogrowtheffects,thesignoftheseinequalityeffectscannotbederivedneitherfromthesolecomparisonofthepovertylineandtheinitialandnalmeanincomes,norfromtheuseofinequalitymeasures.Forinstance,adecreaseininequalityaccordingtoanyscaleinvariantinequalitymeasure,donotnecessarilyimpliesthatthecorrespondinginequalityeffectisnega-tive.51Moreover,noLorenzdominancecriterioncanbeusedforthepurposeofheadcountindexcomparisons.InthecontextoftheDattandRavallion(1992)decompositionframework,thesoleorderingcriterionthatmaybehelpfulistherst-degreestochasticdominancecondition(Atkin-son,1987)betweentheinitialdistributionandthecounterfactualdistributionderivedfromthenaldistribution.Inotherwords,theonlywayofgettingthesignofD,whateverinvarianceaxiomischosen,istocomputeitsvalue.Proposition6.UsingtheDattandRavallion(1992)decompositionoftheheadcountindex,the“leftist”inequalityeffectislargerthanthe“rightist”inequalityeffectifandonlyiftheobservedgrowthrateispositive(negative)andtheinitialmeanincomeisabove(below)thepovertyline.Proof.AsforthecomparisonofthedifferentgrowtheffectsusingtheDattandRavallion(1992)decompositionframework,theorderingofthe“leftist”and“rightist”growtheffectsonlydependsonthesignofthegrowthrateandtherelativepositionof¹Xandz.Consequently,thefourfol-lowingcasesmustbeseparatelytreated:i)Supposerstthat¹YȹXand¹XÈz.ForthecomputationoftheinequalityeffectusingtheDattandRavallion(1992)decompositiontechnique,thefocusoughttobeputonthetrans-formationofnon-poorindividualsincomefromdistributionYinthecontextofapositiverateofgrowth.Thecomparisonofthetransformedincomesisgivenbyequation(3.8).Sinceonlynon-poorindividualswhoseincomesarelowerthan¹Yaresusceptibletocrossthepovertyline,weareonlyinterestedintherstlineofequation(3.8).Asaresult,h¡z,YÅ(¹X¡¹Y)I¢ï¿¿h³z,Y¹X ¹Y´,andthenDTï¿¿DS.ii)Foranegativegrowthratebut¹Xstilllargerthatz,wehavetoconsidertwodifferentcases,dependingontherelativepositionof¹Yandz.If¹YÈz,thecomparisonofthecounterfactualincomesisgivenby:yjÅ(¹X¡¹Y)Èxj¹X ¹Y8j2{1,...q}.(3.11)Ontheotherhand,if¹YÇz,therankingisgivenbyequation(3.9).Sinceallmembersfromtheset{rÅ1,...qjyr¡1ǹY6yr}crossthepovertyline,onlytheverypoorestwillmakethediffer-enceforthecomparisonofthe“leftist”and“rightist”effects.Inbothsituations,wethusndh¡z,YÅ(¹X¡¹Y)I¢6h³z,Y¹X ¹Y´andconcludeDT6DS.iii)Consideringthesituationofapositivegrowthrateand¹XÇz,thevalueoftheinequalityeffectsdependsonthewaynon-poorincomeindistributionYchange.Twodifferentcasescanbemet.With¹YÈz,thesituationisdescribedbyequation(3.8).Sincetheincomeofallmembers 51Bresson(2007)showsinthecontextoftheanalyticalderivationofaclassofinequalityelasticitiesofpovertythatanincreaseininequalitymayjustaswellresultinanincreaseoradecreaseofthelevelofpovertywhatevertherespectivepositionofmeanincomeandthepovertyline.25 3INVARIANCE,THEMEASUREMENTOFPOVERTYANDTHEDECOMPOSITIONOFITSVARIATIONS oftheset{q,...rjyr¡1ǹY6yr}fallbelowthepovertyline,onlythesecondlineofequation(3.8)canbeconsidered.Assuming¹YÇzgives:yjÅ(¹X¡¹Y)Èxj¹X ¹Y8j2{q,...n}.(3.12)Inbothsituations,h¡z,YÅ(¹X¡¹Y)I¢6h³z,Y¹X ¹Y´andwendDt6DS.iv)Finally,foranegativegrowthrateand¹XÇz,theorderingofpoorindividualincomesisgivenbyequation(3.9).Asonlythosefromtheset{rÅ1,...qjyr¡1ǹY6yr}maycrossthepovertyline,wendh¡z,YÅ(¹X¡¹Y)I¢ï¿¿h³z,Y¹X ¹Y´andconcludeDtï¿¿DS. Figure4isthecounterpartofgure3forthecomputationoftheinequalityeffectsusingtheDattandRavallion(1992)decompositionframework.Inthiscase,thenaldistributionYisre-portedasitisneededtondthecorrespondingequallyunequaldistributionswithmeanincome¹X.Inordertoimprovethereadabilityofthegure,thecoordinates(y1,y2)arepermuted,butthismodicationisofnoconsequenceforourpurpose.Ifthepovertylineissettoz,bothindi-vidualsareconsideredaspoorintheoriginaldistribution(pointX).Withscaleinvariance,thecounterfactualdistributionYSwhichexhibitthesamedegreeofinequalityasYdoesnotchangethevalueofthepovertyindexsinceeveryincomeremainsbelowthepovertyline.Onthecontrary,thetranslationinvariancetransformationYTofdistributionYyieldsacounterfactualdistribu-tionwithonlyhalfofthepopulationbeingpoorsinceyT2ÈzÈyT1.Thus,wendDTÇDSÆ0.Ontheotherhand,withapovertylinez0thatislargerthantheinitialmeanincome,theinitialvalueoftheheadcountindexiszeroanddoesnotchangeifweadopta“rightist”view.Withthe“leftist”inequalityview,thenumberofpoorincreasesastherstindividualincomefallsbelowthepovertyline.Asaconsequence,weconcludeDTÈDSÆ0.Proposition7.WiththeShapleydecompositionoftheheadcountindex,the“leftist”inequalityeffectishigherthanthe“rightist”inequalityeffectifandonlyiftheobservedgrowthrateispositive(negative)andthenalmeanincomeisabove(below)thepovertyline,exceptifthepovertylineliesbetweentheinitialandnalmeanincomes,and:hµz,X¹Y ¹X¶¡hµz,Y¹X ¹Y¶Çh¡z,XÅ(¹Y¡¹X)I¢¡h¡z,YÅ(¹X¡¹Y)I¢.(3.13)Proof.InthecaseoftheShapleydecomposition,thedemonstrationistrivialsinceweknowthatD0Æ¢h¡G0.As¢hremainsthesamewhateverinvarianceaxiomhasbeenadopted,thedifferencebetweenD0SandD0TissimplytheoppositeofthedifferencebetweenG0SandG0T. Anoticeablefeatureofpropositions6and7isthattheorderingoftheinequalityeffectsunderscaleandtranslationinvariancedoesnotdependofthesignoftheseeffects.Generallythevalueofthepovertylineisbelowtheobservedmeanvaluesoftheinitialandnalincomedistributions.Inthissituation,weshouldexpectthe“leftist”inequalityeffecttobesuperior(inferior)tothe“rightist”inequalityeffectforpositive(negative)observedgrowthrates.Consideringtherelativecontributionofgrowthandredistributiontovariationsoftheheadcount,movingfroma“rightist”26 3INVARIANCE,THEMEASUREMENTOFPOVERTYANDTHEDECOMPOSITIONOFITSVARIATIONS Note:inordertoimprovethereadabilityofthegure,in-comesofthetwoindividualsarepermutedwithrespecttodistributionX.Figure4:“Leftist”vs“rightist”inequalityeffectswithapositiverateofgrowth.Table2:Comparisonoftheinequalityeffectsunderscaleandtranslationinvariance. ConditionDecompositiontechnique GrowthPovertylineDattandRavallion(1992)Shapley ¹YȹX¹XÈzDTï¿¿DSD0Tï¿¿D0S¹YȹX¹XÇzDT6DSD0T6D0SoraD0Tï¿¿D0S¹XȹY¹XÈzDT6DSD0T6D0SorbD0Tï¿¿D0S¹XȹY¹XÇzDTï¿¿DSD0Tï¿¿D0S a:if¹YÈzandh³z,X¹Y ¹X´¡h³z,Y¹X ¹Y´ï¿¿h¡z,XÅ(¹Y¡¹X)I¢¡h¡z,YÅ(¹X¡¹Y)I¢.b:ifzȹYandh³z,X¹Y ¹X´¡h³z,Y¹X ¹Y´ï¿¿h¡z,XÅ(¹Y¡¹X)I¢¡h¡z,YÅ(¹X¡¹Y)I¢.27 4ANAPPLICATIONTOPOVERTYINCHINA,1990-2004 toa“leftist”inequalityviewincreasesthecontributionofgrowthtothereductionofpovertywithrespecttoredistributionforapositiverateofgrowth.Thisresulthasmajorimplicationsfortheevaluationof“pro-poor”growth.Ifgrowthissaid“pro-poor”whenobservedpovertyreductionishigherthatthereductionthatwouldoccurunderdistributionneutrality(KakwaniandPernia,2000),leavingscaleinvarianceforintermediateand“leftist”inequalityviewsmakesgenerallytheoccurrenceof“pro-poor”growthmorescarcewhenmeanincomeincreases.Attheotherhand,negativegrowthismorelikelytobedeemed“pro-poor”underthetranslationinvarianceaxiomthanunderthescaleinvarianceaxiom.4AnapplicationtopovertyinChina,1990-2004Mostofthetime,empiricalstudiesrelatedtoincomeinequalityandpovertyare(implicitly)basedonthepriorthatinequalityandpovertyshouldbeanalyzedthroughscaleinvarianttools.Thismayreectthemainstreamviewineconomicsbutnotnecessarilythedominantviewofpolicy-makersandcitizens.Asstatedearlier,theheterogeneityofinequalityperceptionsisarelevantjusticationforanalyzingthesensibilityofresultstoethicalpreferencesinpovertystudies.52Em-piricalstudiesthatdonotrelyonscaleinvariancearescarce.SuchstudiesincludedelRioandRuiz-Castillo(2001)ontheevolutionofinequalityinSpainfrom1980to1991andAtkinsonandBrandolini(2004)oninternationalandglobalincomeinequalitiesinthelastcentury.Inthepresentsection,wewanttoillustratetheimportanceofachoiceofaparticularaxiomforthedecompositionofpovertyvariationsintogrowthandinequalitycomponentsusingChi-nesedata.ConsideringChinaisofprimeimportance:recentpublications(Bhalla,2004;Sala-iMartin,2004,2006)relatedtotheevolutionoftheworldincomedistributionhavestressedhowtheirresultsweresensitivetochangesintheChinesedistribution.Moreovermanyauthors(BesleyandBurgess,2003;ChenandRavallion,2004)haveemphasizedthecrucialroleofChinaintheachievementoftheglobalobjectiveofhalvingextremepovertyduringtheperiod1990-2015.SuchanimportantcontributiontoglobalpovertyreductionisgenerallyattributedtotheimpressiveeconomicperformancesofChinaduringthelastdecade(ChenandRavallion,2007).Inthefol-lowingparagraphs,weemphasizethatthisconclusioniscontingenttoaxiomaticchoicesandmaynotholdwhenmovingfroma“rightist”toa“leftist”view.4.1DataThedatausedinthispaperstemfromthe1990,1996,1999and2003roundsoftheChinaHealthandNutritionSurvey(CHNS).TheCHNSisanongoinglongitudinalsurveythatcoversnineprovinces(Guangxi,Guizhou,Henan,Hubei,Hunan,Jiangsu,Liaoning,HeilongjiangandShangdong).Al-thoughthesurveyisnotnationallyrepresentative,theseprovinceswereselectedtoprovidesig-nicantvariabilityingeography,economicdevelopmentandhealthindicators,sothattheymaybeconsideredtoberoughlyrepresentativeofthewholepopulationofthecountry. 52OneshouldhaveinmindthattheuseofdifferentmeasuresliketheTheilandtheGinicoefcientinagivenempiricalstudyalsoimpliesaxiomaticchangesandsoinvolvesamixofdifferentethicalpreferences.28 4ANAPPLICATIONTOPOVERTYINCHINA,1990-2004 Amultistagerandom-clustersamplingprocedurewasusedtodrawthesamplefromeachoftheprovinces.Countiesintheeightprovinceswerestratiedbyincome(low-,middle-andhigh-incomegroups)withpercapitaincomeguresfromtheStateStatisticalOfce,andaweightedsamplingschemewasusedtoselectfourcountiesrandomlyineachprovince(onelowincome,twomiddleincome,andonehighincome).Aprobability-proportional-to-sizesamplingwasthenchosentoselectthesamplefromtheseunits.Inaddition,urbanareasthatwereinitiallynotwithinthecounty-stratawerelaterincorporatedbyincludingtheprovincialcapitalandalow-incomecityfromeachprovince.Withineachcounty,thetownshipcapitalwasaddedandthreevillageswerechosenrandomly.Withineachcity,urbanandsub-urbanneighborhoodswereran-domlypickedout.Thesamerandomselectionprocedurewasusedtochoosetheneighborhoodsfortownshipsandvillages.Incomedataaredividedbetweenincomesissuedfromagriculture,business,paidactivities,subventionsandremittances.Theagriculturalincomescomefromshing,farming,cropsgrow-ing,gardeningandrearing.Businessincomesarerelatedtohandicraftandsmallbusinesses.Paidactivitiesrepresentallthejobsforwhichindividualsarewageearners(includingworkinagricul-turalandbusinessactivities)andincludebonusesreceivedallalongtheyear.Subventionsaredis-tributedbyenterprisesortheStateforhousing,food,energy,childbearing,childcare,health...53Finally,remittancesrepresentmoneysentbackbychildrentotheirparentsornancialhelpfromfriendsorrelatives.Forself-employmentinagricultureorbusiness,weconstructnetincomede-nedastheincomegeneratedbytheproductssoldplusthemonetaryvalueofproductskeptbythehousehold,minusthecostsengagedfortheproduction.Wedonotconsiderobservationsforwhichinformationsrelatedtocostsorincomesweremissing.Theaggregationofallkindsofrevenueconstraintsustoconsideryearlyincomeforthewholehouseholdandconsequentlytoassumethattotalincomeisequallysharedbetweeneachmem-bers.However,wedonotusethedirectnumberofhouseholdmemberstoobtaintheindividualincome,butassumesomepossibleeconomiesofscaleinthehousehold.Consequently,weusethemethodologysuggestedby(Deaton,1997)andnormalizetheincomesbydividingthembynawherenisthenumberofthehouseholdmembersandaisanequivalencefactor.Inourempiri-calapplication,wewilluseavalueofaÆ0.8,avaluethanwaschosenbyWanandZhang(2006)fortheirestimationsonthesameCHNSdata.Togetrealincomes,weusetheconsumerpricein-dexes(CPIs)providedtheChineseNationalBureauofStatistics.WeconsiderprovincialCPIs,withadistinctionbetweenurbanandruralareas,foralltheyearsconsidered,withthe1990yearasthereference.Inordertoaccountforthespacialpricedifferencesinthereferenceyear,incomesareadjustedusingtheprovincial(ruralandurban)deatorsconstructedbyBrandtandHolz(2006).Intable3arepresentedthevaluesoftheheadcountindexforthedifferentperiodsofobser-vation.Inthispaper,weconsiderthetraditionalUS$1.08andUS$2.16(lattermentionedasUS$1andUS$2forconvenience)perdaypovertylinesin1996PPP.InthecontextofChina,thedeni-tionofarelevantpovertylineistheobjectofgreatdebates(Fanetal.,2002;Hanmeretal.,2004;Gregoryetal.,2005;ChenandRavallion,2007).Someauthorsarguethatitisimportanttodistin- 53Forthe1991survey,foodcouponsreceivedbyhouseholdsareisolatedfromthesubventions.Weconsequentlyincludetheminthesubventiontomakedatabasecomparable.29 4ANAPPLICATIONTOPOVERTYINCHINA,1990-2004 guishtheruralandurbanareas(ChenandRavallion,2007),andevenconsiderspecicpovertylinescorrespondingtoadequateconsumptionbasketsforeacharea(Gregoryetal.,2005).Nev-ertheless,wechoosethecommonlyusedUS$1andUS$2linesaswerealizeageneralanalysisofpovertyinChinaandasthesetwomeasuresaretheonesusedinthecontextoftheMillenniumDevelopmentGoals.Thebottompartofthecumulativedistributionfunctionsforeachsurveyarereportedingure5.Table3:ValuesoftheheadcountindexinChinaduringtheperiod1990-2003. Year1990199619992003 US$116.26.811.213.3(0.004)(0.003)(0.005)(0.004)US$236.514.317.918.0(0.004)(0.004)(0.005)(0.004) Note:standarderrorsinparenthesesusingabootstrapprocedurewith200replications. 0200040006000800010000 0.00.20.40.60.81.0 income (yuan base 1990)probabily 1990 Note:ThesolidanddashedverticallinesrepresentstheUS$1andUS$2perdaypovertylines.Figure5:ThedistributionofincomesinChinafrom1990to2003.Consideringthe3,wenotethatafteradropinpovertybetween1990and1996,aslightin-creaseoccursinChinasincetheendofthe1990's.ThisseemssurprisingasweknowthatChinahasexperiencedahugegrowthsincethebeginningofthe1980's.Ontheotherhand,ourguresarenottotallysupportedbyotherstudiesrelatedtopovertyinChinaandwhichtendtodemon-strateaconstantdecreaseofpovertysincethemovementofreformsinitiatedbytheendofthe1970's.Nevertheless,someotherauthorsunderlineshortepisodesofincreaseintheheadcountindex.Forinstance,ChenandRavallion(2007)forthewholeChinaandWanandZhang(2006)30 4ANAPPLICATIONTOPOVERTYINCHINA,1990-2004 fortheruralareasnoteaslightincreasein2000.Thedifferenceswecanstressbetweenourresultsandotherstudiesonesareessentiallyduetothedatastructureandourdenitionofincome.TheCHNSdatabaseishighlydetailedandmanyrivalhypothesiscanbedoneconcerningwhichin-comeandcostsmustornotbeconsidered.Thiscanhaveimportantimpactontheresults.Withthiscaveatinmind,ourconclusionsconcerningChina'spovertyneedtobetakenwithcaution.However,asthisisnotthecentralgoalofourpaper,problemsconcerningdatawillnotinuencethemajorresultsconcerningthedifferencesbetweendecompositionsdonewithscaleinvarianceandtheonesbasedontranslationinvariance.Wecanhaveacloserlookatthelevelofpovertylookingattheincomesdistributiongivenongure5.Thankstothisgure,weclearlyseethattherehasbeenahugedecreaseinpovertybetween1990and1996usingtheUS$2povertylinebutthatafterthisdate,andforthepoorestindividuals,nosignicantevolutioncanbedrawn.Thesedistributionsemphasizeadecreaseofinequalitiesforthehighestquartilebutnotforthelowestones.ThisisincoherencewiththeevolutionofthedistributionofwealthinChinaasweseesincefewyearsthedevelopmentofanewmiddleclasswhichbeginstobalancewiththeenrichmentofanarrowshareoftheChinesepopulation.4.2“Leftist”vs“rightist”effectsInthisparagraph,wefocusonthecomparisonsofthedifferencesoftheestimatedeffectsob-tainedthroughthetwolimitingviewspresentedintheprecedingsections,thatisthosebasedonscaleandtranslationinvariance.Tables4and5respectivelypresenttheestimationsusingtheDattandRavallion(1992)andtheShapleydecompositiontechniquesfortheperiod1990-2003andthesub-periods1990-1996,1996-1999and1999-2003.54Theguresincludedinthesetablesgivethetotalvariationsoftheheadcountindex,thegrowthandinequalityeffectsinper-centagepointsaswellastheirrelativecontribution(trade-off)topovertyreduction.Forinstance,lookingattheresultsbasedonscaleinvarianceintable4fortheperiod1990-2003andfortheUS$1povertyline,wecanobservethatpovertyhasdecreasedbyabout2.9percentagepoints.Decomposingthisevolutionintogrowthandinequalityeffects,wendthatpovertywouldhavedecreasedby8.9percentagepointsthankstogrowthifinequalitieshadremainedstableduringtheperiod.Inparallel,ifthegrowthratehadbeennull,theevolutionofinequalitieswouldhaveincreasedpovertyby0.7percentagepoints.ThesameinterpretationsholdfortheShapleyde-compositionresultsgivenintable5.Asnotedearlier,ourobjectiveisnottopromoteanyofthetwotechniques,buttoemphasizethejudgmentdifferencesinvolvedbyachangeintheconceptionofinequality.Byandlarge,thetwodecompositionstechniquesyieldthesameconclusions,theonlysalientdifferencebeingwiththe“leftist”inequalityeffectfortheperiod1990-1996usingtheUS$1povertyline(2.55percentagepointsfortheShapleydecompositionversus¡1.63withtheDattandRavallion(1992)approach).Atrst,itisimportanttostressthatthetheoreticalresultssummedupintables1and2areconrmedbytheempiricalresultspresentedintables4and5.Weareinthecaseofapositive 54EstimationsandguresareobtainedwithR2.5.1(RDevelopmentCoreTeam,2007).Scriptsareavailableuponrequest.31 4ANAPPLICATIONTOPOVERTYINCHINA,1990-2004 Table4:“Rightist”and“leftist”decompositionsofpovertyspellsinChina1990-2003usingtheDattandRavallion(1992)technique. AxiomScaleinvarianceTranslationinvariance 1990-19961996-19991999-20031990-20031990-19961996-19991999-20031990-2003 US$1povertylineTotal-9.444.442.04-2.96-9.444.442.04-2.96(%point)[¡10.2,¡8.67][3.65,5.16][1.18,2.96][¡3.93,¡1.99][¡10.2,¡8.67][3.65,5.16][1.18,2.96][¡3.93,¡1.99](0.004)(0.004)(0.005)(0.005)(0.004)(0.004)(0.005)(0.005)Growth-2.7-1.36-1.39-8.98-16.2-6.85-11.3-16.3(%point)[¡3.54,¡1.52][¡1.59,¡1.17][¡1.65,¡0.97][¡9.74,¡8.07][¡16.9,¡11.4][¡7.32,¡6.4][¡11.9,¡10.7][¡17,¡15.7](0.005)(0.001)(0.002)(0.004)(0.015)(0.002)(0.003)(0.003)Inequality-8.135.912.980.76-1.6319.113.524.2(%point)[¡9.21,¡7.29][5.1,6.64][2.11,3.82][¡0.31,1.73][¡5.64,1.88][17.3,20.7][11.3,15][21.8,26.3](0.005)(0.004)(0.005)(0.005)(0.019)(0.009)(0.01)(0.012)Residual1.39-0.110.465.268.35-7.84-0.19-10.9(%point)[0.85,2.02][¡0.40,0.38][0.10,0.69][4.58,5.93][4.85,8.73][¡9.57,¡5.83][¡1.73,2.06][¡13.1,¡8.38](0.003)(0.002)(0.002)(0.003)(0.01)(0.009)(0.01)(0.012)Trade-off0.33-0.23-0.46-11.89.91-0.35-0.83-0.67(G/D)[0.17,0.45][¡0.29,¡0.18][¡0.68,¡0.28][¡128,117][¡108,97.7][¡0.40,¡0.32][¡1.01,¡0.73][¡0.75,¡0.60](0.073)(0.027)(0.103)(5335)(213)(0.022)(0.071)(0.038)US$2povertylineTotal-22.23.560.072-18.6-22.23.560.072-18.6(%point)[¡23.2,¡21.2][2.58,4.46][¡0.95,1.14][¡19.7,¡17.5][¡23.2,¡21.2][2.58,4.46][¡0.95,1.14][¡19.7,¡17.5](0.005)(0.005)(0.005)(0.006)(0.005)(0.005)(0.005)(0.006)Growth-4.93-3.83-2.39-17.9-21.1-12.2-17.9-36.6(%point)[¡6.91,¡3.34][¡4.28,¡3.12][¡2.78,¡1.65][¡19.1,¡16.3][¡29.7,¡12.7][¡14.8,¡11.1][¡18.6,¡12.3][¡37.4,¡35.7](0.009)(0.003)(0.003)(0.007)(0.044)(0.012)(0.017)(0.004)Inequality-19.77.822.01-10.4-13.418.313.19.04(%point)[¡21.3,¡18.1][6.79,8.74][1.11,3.15][¡11.8,¡8.95][¡17.2,¡9.76][16.6,19.9][10.3,14.6][6.7,10.9](0.008)(0.005)(0.005)(0.007)(0.019)(0.009)(0.01)(0.011)Residual2.47-0.430.459.7312.3-2.474.928.99(%point)[1.93,3.3][¡1.05,0.10][¡0.26,0.66][8.73,10.5][7.51,17.4][¡3.05,¡0.41][1.74,6.12][7.1,11.5](0.004)(0.003)(0.003)(0.005)(0.025)(0.007)(0.011)(0.012)Trade-off0.25-0.48-1.191.721.58-0.67-1.37-4.05(G/D)[0.16,0.37][¡0.58,¡0.39][¡2.14,¡0.59][1.45,2.08][0.75,3.04][¡0.80,¡0.62][¡1.54,¡1.1][¡5.48,¡3.33](0.056)(0.048)(0.386)(0.161)(0.576)(0.051)(0.107)(0.586) Note:95%condenceintervalsinbracketsandstandarderrorsinparenthesesusingabootstrapprocedurewith500replications.32 4ANAPPLICATIONTOPOVERTYINCHINA,1990-2004 Table5:“Rightist”and“leftist”decompositionsofpovertyspellsinChina1990-2003usingtheShapleytechnique. AxiomScaleinvarianceTranslationinvariance 1990-19961996-19991999-20031990-20031990-19961996-19991999-20031990-2003 US$1povertylineTotal-9.444.442.04-2.96-9.444.442.04-2.96(%point)[¡10.2,¡8.67][3.65,5.16][1.18,2.96][¡3.93,¡1.99][¡10.2,¡8.67][3.65,5.16][1.18,2.96][¡3.93,¡1.99](0.004)(0.004)(0.005)(0.005)(0.004)(0.004)(0.005)(0.005)Growth-2-1.42-1.16-6.35-12-10.8-11.4-21.7(%point)[¡2.6,¡1.08][¡1.6,¡1.16][¡1.35,¡0.84][¡6.9,¡5.58][¡13.8,¡7.75][¡11.7,¡9.77][¡12.3,¡10.2][¡22.9,¡20.3](0.004)(0.001)(0.001)(0.003)(0.016)(0.005)(0.005)(0.006)Inequality-7.445.863.213.392.5515.213.418.8(%point)[¡8.59,¡6.56][5.09,6.54][2.32,4][2.38,4.24][¡1.81,4.4][14.2,16.1][12.2,14.3][17.5,19.9](0.005)(0.004)(0.004)(0.005)(0.016)(0.005)(0.005)(0.006)Trade-off0.26-0.24-0.36-1.87-4.71-0.70-0.84-1.16(G/D)[0.12,0.37][¡0.29,¡0.19][¡0.50,¡0.23][¡2.6,¡1.48][¡29.9,33.3][¡0.75,¡0.66][¡0.90,¡0.78][¡1.22,¡1.1](0.066)(0.025)(0.067)(0.295)(259.8)(0.023)(0.032)(0.028)US$2povertylineTotal-22.23.560.072-18.6-22.23.560.072-18.6(%point)[¡23.2,¡21.2][2.58,4.46][¡0.95,1.14][¡19.7,¡17.5][¡23.2,¡21.2][2.58,4.46][¡0.95,1.14][¡19.7,¡17.5](0.005)(0.005)(0.005)(0.006)(0.005)(0.005)(0.005)(0.006)Growth-3.7-4.04-2.17-13-15-13.5-15.5-32.1(%point)[¡5.39,¡2.3][¡4.5,¡3.39][¡2.6,¡1.67][¡14.2,¡11.7][¡21.1,¡8.87][¡15.5,¡12.2][¡16.4,¡11.2][¡33.2,¡30.6](0.008)(0.003)(0.003)(0.006)(0.031)(0.01)(0.013)(0.006)Inequality-18.57.612.24-5.54-7.231715.513.5(%point)[¡20.2,¡16.7][6.64,8.47][1.3,3.23][¡6.85,¡4.2][¡13.2,¡1.16][15.8,19][11.5,16.4][12.3,14.5](0.009)(0.005)(0.005)(0.007)(0.031)(0.009)(0.012)(0.006)Trade-off0.2-0.53-0.962.352.07-0.79-0.99-2.37(G/D)[0.11,0.31][¡0.62,¡0.44][¡1.7,¡0.63][1.76,3.34][0.66,17.1][¡0.85,¡0.74][¡1.06,¡0.92][¡2.56,¡2.23](0.052)(0.045)(0.276)(0.394)(8.37)(0.029)(0.036)(0.087) Note:95%condenceintervalsinbracketsandstandarderrorsinparenthesesusingabootstrapprocedurewith500replications.33 4ANAPPLICATIONTOPOVERTYINCHINA,1990-2004 growthrate(¹YȹX)withinitialandnalmeanincomesthatarebothabovethepovertyline(¹YÈzand¹XÈz).Consequently,wendinallcasesthatthenegativegrowtheffectismoreimportantinthetranslationinvariancecasethaninthescaleinvarianceone.Moreoverthetrans-lationinequalityeffectislargerthantheeffectbasedonscaleinvariance.Whenthevaluesarenegative,a“leftist”observerwouldthenattributealowercontributionofinequalitychangestovariationsoftheheadcountindexthana“rightist”observer.Ontheotherhand,withapositiveinequalityeffect,hewouldthinkthatinequalitychangeshampersmorepovertyalleviationthanthe“rightist”one.Ofcourse,theseresultsholdforboththeDattandRavallion(1992)andtheShapleydecompositiontechniques.Whatisalsointerestingisthedifferencesrelativetothetrade-offsbetweenthegrowthandtheinequalityeffects.Theycanbefoundinthetables1and2andaremeasuredbytheratioG/D.Weclearlyseethattheweightsgiventogrowthrelativetoinequalitiesintheexplanationofob-servedpovertytrendsarehighlydifferentbetweenthe“leftist”and“rightist”views.Thesamephe-nomenonhappenswhateverpovertylineanddecompositionapproacharechosen.Forexample,fortheUS$1povertyline,theShapleygrowtheffectmeasuredwiththescaleinvarianceisapprox-imatelyfourtimeslessimportantthantheinequalityeffectwhenweconsidertheevolutionofpovertybetween1990and1996,whereasitis4.71timesmoreimportantthantheinequalityef-fectoncewemovetothetranslationinvariancecase.Forthesameperiodandthesamepovertyline,theDattandRavalliontrade-offsare0.33forthescaleinvarianceand9.91forthetranslationone.Noclearrelationshipappearsbetweenthetypeofinvariancechosenandthemoreorlesshightrade-offsthatareobserved.Thisunderlinesevenmoretheneedforasensitivityanalysisofthepovertydecompositiontoinvariancepreferences.Toillustratetheimportanceoftheaxiomaticchoice,let'shavealookattheresultsoftheDattandRavalliondecompositionfortheperiod1990-2003andfortheUS$1povertyline.Wenotethatthegrowtheffectforthetranslationinvariance(GTÆ¡16.3)isnearlytwotimeshigherthanthatforthescaleinvariance(GSÆ¡8.98).Consideringtheinequalityeffect,thedifferenceisevenlarger,the“leftist”effect(DSÆ24.2)beingmorethan30timeshigherthanthe“rightist”one(DTÆ0.76).Totestthestatisticalsignicanceoftheseresults,wecomputed95%condenceintervalsforeacheffectusingabootstrapprocedurewith500replicationsandresamplingatthehouseholdlevel.Inmostcases,wendthatthesedifferencesaresignicantasintervalcrossingsarerarelynoted.Themostimpressiveconsequenceofinvarianceaxiomchangesisthatamodicationofethi-calpreferencesmayinduceachangeinthesignoftheinequalityeffect.Comparingtheresultsofthescaleandtranslationinvariance,weobserveopposite(andsignicantlydifferent)signsfortheinequalityeffectsfortheperiod1990-2003withtheUS$2povertylinewhateverdecompositiontechniqueischosen.Thesamephenomenonisobservedintable5forthesub-period1990-1996andtheUS$1povertylinebutonlywiththeShapleydecompositionapproach.Butascanbeseenthankstothecondenceinterval,thepositive“leftist”effectisnotstatisticallydifferentfromzero.Consequentlyforthissub-periodandthispovertyline,movingfroma"rightist"toa"leftist"pointofviewimpliesthattheeffectofthechangesintherelativedistributionofincomesonpovertyisnotsignicantlydifferentfromzeroanymore.34 4ANAPPLICATIONTOPOVERTYINCHINA,1990-2004 Concomitantlytothisresult,anothermajorobservationcanbemadeaboutthedependencyoftheorderingofeffectstoinvariancechoicesformanysub-periods.Forinstance,theShapleygrowtheffectfortheUS$1povertylineishigherduringthe1996-1999periodwhenusingthescaleinvariance(G0S96¡99Æ¡1.42Ç¡1.16ÆG0T99¡03)andduringthe1999-2003whenthetranslationin-varianceischosen(G0S96¡99Æ¡10.8È¡11.4ÆG0T99¡03).Thesamekindofobservationscanbemadeonthesamesubperiods,forboththeShapleygrowtheffectsattheUS$2povertylineandtheDattandRavalliongrowtheffectsattheUS$1povertyline.Consequently,itseemsthatmovingfroma“rightist”toa“leftist”pointofviewimpliesadifferentperceptionoftherangeoftheimpactofgrowthorinequalityonpoverty.However,itisimportanttostressthatdifferencesbetweentheseestimatedgrowtheffectsarenotstatisticallysignicantexceptforthescaleinvarianceinequalityeffectsobtainedwiththeShapleydecompositiontechniqueattheUS$2povertyline.Anotherimportantfactisthatalltheseresultscruciallydependonthelevelofthepovertyline.Figures6and7presentthevalueofthedifferentestimatedeffectsaswellasobservedpovertyvari-ationsasafunctionofthepovertyline.Atrst,theseguresconrmthemeaningfuldifferencesthatwendbetweenthescaleandthetranslationinvariancedecompositions.Astherangeofobservedvaluesforthetranslationinvarianceiswider,thecurvesrelativetoGTandDTarere-spectivelybelowandabovetheonesforGSandDS.Mostofthetime,theevolutionsoftheeffectsforthetwotypesofinvarianceareparallel.Nevertheless,weclearlyseeonthegure6aforthe1990-2003periodandonthegure6cfortheperiod1996-1999adivergenceofthegrowtheffectsbetweenthescaleandthetranslationinvariancesasthepovertylineincreases.Therefore,itisim-portant,asstressedinthepovertyorderingliterature(Atkinson,1987),toanalysethesensibilityofresultstothelevelofthepovertyline.554.3IntermediateeffectsIntheprecedingsection,wehaveshownthatchoosingthescaleinvarianceasthesolerelevantinequalityviewinthecontextofthedecompositionofpovertyvariationsprovideconclusionsthatmaynotbesharedbyindividualswhichpreferencesareclosertoviewsbasedontranslationinvariance.Inthefollowingparagraphs,weintroducesintermediateinvarianceintheempiricalanalysissoastogetamoresubtleanddeeperanalysisoftheeffectsofchangesinethicpreferencesrelatedtoinvariance.First,wehavetoremindthattheintermediateinvarianceaxiomweareusingistheoneofYoshida(2005)sinceitisthesoledescribedinsection2.3thatissuitableforpovertydecomposi-tions.Consequently,theparameterwhichisdeterminanttothisanalysisis¾,asitdescribesposi-tionofindividualsbetweenthe“rightist”and“leftist”views.Theestimatedeffectscorrespondingtointermediatepositionsarereportedingure8fortheDattandRavallion(1992)decompositiontechniqueandingure9whenusingtheShapleydecompositionapproach.Itcanbeseenfromeachoftheseguresthatproposition2isrespectedsincecurvesaremonotonically(weakly)in-creasingordecreasing.However,itisparticularlyinterestingtonotethattheestimatedeffectsaresometimesstableonsomesignicantportionsofthedenitionintervaloftheparameter¾.This 55Foracomprehensivesurveyofpovertyorderings,seeZheng(2000).35 4ANAPPLICATIONTOPOVERTYINCHINA,1990-2004 (a)1990-2003 400600800100012001400 -0.6-0.4-0.20.00.2 poverty line (yuan, base 1990)variation DhGSDSGTDT (b)1990-1996 400600800100012001400 -0.6-0.4-0.20.00.2 poverty line (yuan, base 1990)variation DhGSDSGTDT (c)1996-1999 400600800100012001400 -0.6-0.4-0.20.00.2 poverty line (yuan, base 1990)variation DhGSDSGTDT (d)1999-2003 400600800100012001400 -0.6-0.4-0.20.00.2 poverty line (yuan, base 1990) DhGSDSGTDT Figure6:“Rightist”and“leftist”decompositionsofpovertyspellsinChina1990-2003usingtheDattandRavallion(1992)technique.36 4ANAPPLICATIONTOPOVERTYINCHINA,1990-2004 (a)1990-2003 400600800100012001400 -0.6-0.4-0.20.00.2 poverty line (yuan, base 1990)variation DhGSDSGTDT (b)1990-1996 400600800100012001400 -0.6-0.4-0.20.00.2 poverty line (yuan, base 1990)variation DhGSDSGTDT (c)1996-1999 400600800100012001400 -0.6-0.4-0.20.00.2 poverty line (yuan, base 1990)variation DhGSDSGTDT (d)1999-2003 400600800100012001400 -0.6-0.4-0.20.00.2 poverty line (yuan, base 1990) DhGSDSGTDT Figure7:“Rightist”and“leftist”decompositionsofpovertyspellsinChina1990-2003usingtheShapleytechnique.37 4ANAPPLICATIONTOPOVERTYINCHINA,1990-2004 isinparticularthecaseingure8aforthegrowtheffectsobtainedthroughtheDattandRavallion(1992)decompositiontechniqueontheperiods1990-2003,1996-1999and1999-2003.(a)Growtheffect,zÆUS$1 0.00.20.40.60.81.0 -0.15-0.10-0.05 svariation 1990-1996 (b)Inequalityeffect,zÆUS$1 0.00.20.40.60.81.0 -0.050.050.150.25 s 1990-1996 (c)Growtheffect,zÆUS$2 0.00.20.40.60.81.0 -0.35-0.25-0.15-0.05 svariation 1990-1996 (d)Inequalityeffect,zÆUS$2 0.00.20.40.60.81.0 -0.2-0.10.00.1 s 1990-1996 Figure8:Intermediate(Yoshida,2005)decompositionofpovertyspellsinChina1990-2003usingtheDattandRavallion(1992)technique.Wehavenotedintheprecedingparagraphtwomajordifferencesonceweadopttranslationinvariance:achangeinthesignoftheinequalityeffectfortheperiod1990-2003and1990-1997,andaninversionoftheorderingofthegrowtheffectsonpovertyvariationsbetweentheperi-ods1996-1999and1999-2003.Theseresultstranslatesdifferencesbetweentwooppositeethicpreferences.Thankstotheintermediatemethodology,weareabletouseacontinuumofethicpreferencesandconsequentlydeterminethelevelsof¾whichcorrespondtoareversaloftheconclusions.Concerningthechangeofsignunderlinedfortheinequalityeffectrelatedtothewholeperiod,thesensibilityoftheresultscanbeappreciatedfromgures8dand9d.Theydescribetheevolu-tionofthedifferenteffectswiththeparameter¾.Weclearlyseethatthechangesofsignoccurforavalueof¾Æ0.5fortheDattandRavalliondecompositionandaround0.9fortheShapleydecom-position.Whenlookingatthegure10a,whichaddsthecondenceintervalstoinequalityeffects38 4ANAPPLICATIONTOPOVERTYINCHINA,1990-2004 (a)Growtheffect,zÆUS$1 0.00.20.40.60.81.0 -0.20-0.15-0.10-0.05 svariation 1990-1996 (b)Inequalityeffect,zÆUS$1 0.00.20.40.60.81.0 -0.050.000.050.100.15 s 1990-1996 (c)Growtheffect,zÆUS$2 0.00.20.40.60.81.0 -0.30-0.20-0.10 svariation 1990-1996 (d)Inequalityeffect,zÆUS$2 0.00.20.40.60.81.0 -0.15-0.050.050.15 s 1990-1996 Figure9:Intermediate(Yoshida,2005)decompositionofpovertyspellsinChina1990-2003usingtheShapleytechnique.39 4ANAPPLICATIONTOPOVERTYINCHINA,1990-2004 forthewholeperiod,interestingdifferencesappearbetweentwoconsidereddecompositionap-proaches.Forbothofthem,westressvaluesof¾forwhichtheinequalityeffectissometimespositive,neutralornegative.UsingtheDattandRavallion(1992)decompositiontechnique,thepositiveimpactisfoundfor¾inferiorto0.42,theneutralonefor¾comprisedbetween0.42and0.62,andthenegativeonefor¾superiorto0.62.Withthesecondtechnique,thecut-offsarere-spectively0.82and0.91.Consequently,theobservationofapositiveShapleyinequalityeffects(D0È0)whichtranslatetheperceptionofanharmfulimpactofinequalityvariationsonpovertyduringtheperiod1990-2003,occursuntilhighervaluesof¾thanfortheDattandRavallionones.Ifweconsiderthatinvariancepreferencesareuniformlydistributedalongthevaluesof¾,thisimpliesthatwiththeShapleymethodology,morepeoplewilltendtohavea“leftist”viewoftheinequalityimpactthanthosewhowillhaveamore“rightist”position.Moreover,wenotethatthecondenceintervalsdonotoverlap,suggestingthatthedifferencesbetweenthetwomethodolo-giesarealsosignicant.56Fortheperiod1990-1996withtheShapleydecompositiontechnique,wealsopreviouslynotedachangeofsign.Thisisconrmedbythegure9basthecorrespondingcurvefortheinequalityeffectcrossesthex-axesfor¾Æ0.3.However,wehavenotedintable5thatthe“rightist”inequalityeffectobtainedthroughtheShapleydecompositiontechniqueisnotsignicantlydifferentfromzero.Figure10bfocusesonthecurveweareinterestedin,andgivesthecondenceintervals.Weemphasizenowclearlythatindividualswhoseethicalpreferencesarebelow¾Æ0.53considerthatinequalitieschangeshadanonsignicantimpactontheevolutionofpovertyduringtheperiod1990-1996andthatindividualswhosepreferencesareabovethisvaluemayfeelthatitsig-nicantlycontributestopovertyalleviation.ItisalsointerestingtonotethattheinequalityeffectsDareneversignicantlydifferentfromthosecorrespondingtotheShapleydecomposition.Asnotedearlier,ethicalpreferenceschangesmayreversetheorderingoftheeffectsbetweenmanyperiods.IfwetakealookatthegrowtheffectsissuedfrombothDattandRavallion(1992)andShapleydecompositionstechniques,wendthatwiththeUS$2povertyline,the1996-1999growtheffectisroughlyequaltothe1999-2003oneforrespectivevaluesof¾thatareapproxi-matelycomprisedbetween0.4and0.72.Forvaluesthatarelowerthan0.4,individualsconsiderthatthegrowthreducingeffecthasbeenlargerbetween1996and1999thanbetween1999and2003.Theconverseconclusionholdsfor¾È0.72.4.4SomemorewordsaboutgrowthandredistributioninChinaComingbacktotheChinesecontext,wecandrawimportantconclusionsontheevolutionofpovertyandtheroleplayedbygrowthandrelativedistributionchanges.First,ifwelookatthewholeperiod1990-2003,weunderlineadecreaseinpovertythatismostlyduetothehighgrowthratesthatwereobservedduringthisperiod.Thisresultispartic-ularlyrobustsinceitisconsistentwithallinequalityviewsandpovertylinesconsideredinthe 56Theseareonlyobservationsasourgoalisnottopromoteoneortheothermethodology.Howeveritisimportanttostressthesedifferencesastheygiverisetooppositeconclusionsinsomecases.Forinstance,whenweconsiderthevalueof¾comprisedbetween0.62and0.82,weclearlyseethattheinequalityeffectispositiveconsideringtheShapleydecompositionandnegativeoncewemovetotheDattandRavallion'sone.Consequently,thisunderlinesthepotentialneedforaclearchoicebetweenthetwomethodologies,orthedenitionathirdprocedure.40 4ANAPPLICATIONTOPOVERTYINCHINA,1990-2004 (a)1990-2003,zÆUS$2 0.00.20.40.60.81.0 -0.100.000.050.100.15 svariation Datt & Ravallion (b)1990-1996,zÆUS$1 0.00.20.40.60.81.0 -0.08-0.040.000.04 s Datt & Ravallion Note:Theareabetweenthedashedcurvesrepresentsthe95%condenceintervalobtainedthroughaboot-strapprocedurewith500replications.Figure10:Sensibilityoftheinequalityeffectstoethicalpreferencesparameter¾.analysis.Ontheotherhand,theeffectsofinequalitychangesonthevariationofpovertyareal-wayspositivewhenweconsidertheUS$1povertylinebutdependsontheethicpreferencesintermsofinvariancewhenwelookattheresultsfortheUS$2povertyline.Inthelatercase,in-equalitiestendtodecreasepovertyconsideringthe“rightist”viewbutincreaseitonceweadoptthe“leftist”pointofview.ItiswellknownthatChinahasexperiencedahugegrowthsincethere-formsmovementinitiatedattheendofthe1970's.Butinparalleltothiseconomicdevelopment,inequalitiesraiseddramatically.Theharmfulimpactofinequalitiesonpovertyfortheperiod1990-2003,stressedintheUS$1caseandintheUS$2caseonlyforthetranslationinvariancede-composition,arethusnotsurprising.TheseresultsareconsistentwiththosefoundinstudiesrelatedtoChinesepoverty.Forinstance,thearticlesofChenandWang(2001);Fanetal.(2002);Hanmeretal.(2004);Gregoryetal.(2005);WanandZhang(2006)andChenandRavallion(2007)alldemonstratethepositiveroleofgrowthindecreasingpovertyinChinabutthenegativeroleofincreasinginequalities.Explanationsofthisphenomenoncanbefoundintherurallatedevelop-ment(Hanmeretal.,2004;Gregoryetal.,2005),orintheevolutionofthelabormarket(Fanetal.,2002),butouranalysiscannotgiveanysupporttothesehypotheses.Withacloserlookattheevolutionofpovertythroughtime,wenoteaveryimportantdecreasebetween1990and1996butthenaslightincreasebetween1996and2003.ThisunstableevolutionofpovertyhasbeenpreviouslyemphasizedinsomestudiesonpovertyinChina(ChenandRaval-lion,2007;Gregoryetal.,2005).Consideringtherecentincreaseofpoverty,weseethatitsmaincauseisthepositiveimpactofinequalities,aresultthatdoesnotdependonthechoseninequal-ityview.Thismeansthatwhateverdecompositionprocedure,invariancepreferencesandpovertylinearechosen,growinginequalitiesinChinahaveworsenedthesituationofthepoorestpopula-tion.Ina“pro-poor”growthanalysisàlaKakwaniandPernia(2000),both“leftist”,intermediateand“rightist”observerswouldthenconsiderthatgrowthcanbedeemed“anti-poor”ontheperi-ods1996-1999and1999-2003,andsurelymorewhenmovingfroma“rightist”toa“leftist”point41 5CONCLUDINGREMARKS ofview.Thepoliticalrecommendationissuedfromtheseconclusionsjointheonespreviouslydonebyotherresearchers:thereisaimportantneedtoassociategrowthwithamore“pro-poor”redistributivepolicyiftheChinesegovernmentwantstosucceedinalleviatingextremepoverty.5ConcludingremarksInthesefewlines,wewillnotdrawpolicyrecommendationsbutmethodologicalones.Fromourpointofview,theissuesillustratedinthepresentpapercouldleadtothethreefollowingattitudes:i)standardizationofthepractices,ii)consistencywithpersonalethicalpreferences,iii)sensibilityanalysistoethicalpreferences.Attitudei)consistsinthedenitionofastandardviewthatshouldbeusedbyeveryeconomist.Themainargumentinfavorofthisstrategyisthatitdenesacommonanalyzingframeworkandhelpstomakedifferentstudiescomparable.Moreover,weshouldrecognizethatitcorrespondstotheattitudethatpresentlyseemstoprevailsincemosteconomistsimplicitlyfeelinaccordancewithinequalityviewsbasedonscaleinvariance.However,wewouldliketostressthateconomistsshouldbeawareofthenormativeimplicationsofthisparticularaxiomaticchoiceandofthepo-tentialdiscrepancybetweentheethicalpreferencesreectedbyscaleinvarianceandtheirownpersonalpreferences.Atleastshouldtheyclearlyexpressonwhichaxiomsarebasedtheiranaly-seswhenchosenmeasuresarecompatiblewithmanyrivalaxioms.Thesecondstrategyreectstheoppositestrategy.Itimplieseconomiststomakeuseonthesolemeasuresthatareconsistentwiththeirownpersonalethicalpreferences.Amajorproblemisthatknowingoneself,oratleasthisownfeelings,andexpressingthesepreferencesthroughrigorousmathematicalpropertiesisadifculttask.57Kolm(1995,p.301)observesthat“theviewconcerningthecomparativejusticeofcovariationsinincomesdependsonthesettingoftheques-tion,and,ofcourse,onthepoliticalreadingofthissetting.Itdependsonthelevelsoftherealincomes,andinparticularontheaveragelevelandonthelevelsofthelowestandofthehighest;ontheconceivedsolidarityordutyofsolidarity;ofcourseontheoriginofthesetransformations;onpastandexpectedhistory;onthefactthattheconsideredvariationisanincreaseoradecrease;andsoon.”Ofcoursethisattituderaisestheproblemofthecomparabilityofresearchers'workssincemostindividualsarenotlikelytospeakthesame“language”.Finally,attitudeiii)consistsinnotchoosingforthereaderwhichinequalityviewheshouldadopt,andpresentingasensibilityanalysesoftheresultstoaxiomaticchangessoasthereadercanndwhichresultsthisownconceptionofinequality.58Asillustratedbyourapplicationon 57Itisparticularlyinterestingtonotethatafterhoursandendeavoursdevotedtothereviewofthedifferentinvari-anceaxiomssuggestedintheliteratureandtheirimplications,theauthorsofthepresentpaperarestillnotabletoexpresspreciselytheirownfeelingsonthisprecisesubject.Moreover,economists'tastesmaybetosomeextentendogenousasnotedbyAmielandCowell(1992,p.22):“Widerangingpolicydecisionscanbeinuencedbyideasaboutinequality;theseideasare,inturn,inuencedbythewayindividualsaretrainedtothinkabouttheissues.”58Kolm(1969,p.148)advocatesthattheeconomist“isanobserverofcitizens'valuejudgementsandopinions,asheisanobserveroftheirtastesconcerningconsumers'goods.[...]Usefulnormativeeconomicsisthereforeapositivesciencesinceitsbasisistheobjectiveobservationofsubjectiveopinions.”Thuscitizens'preferencesshouldbegiventhepre-eminenceovertheeconomist'stastes.42 ATHE(V,À)-INVARIANCEAXIOMINPRACTICE Chinesedata,consideringdifferentinequalityviewscanreverseconclusionsbutmayalsoim-provetherobustnessofsomeresults,andthusgivemoreconvincingargumentsforpolicyrecom-mendations.Thiscorrespondstoatraditionalattitudeinwelfareeconomics,inparticularforthedescriptionofpovertyandinequalitytrends.Forinstance,whencomparingdifferentdistribu-tions,itiscommontomakeuseofmanydifferentmeasuresliketheAtkinson's(1970)inequalitymeasures,whicharebasedonclassicalvonNeumannandMorgensternutilitarianism,andGiniindiceswhicharederivedfromrank-dependantexpectedutilitymodels(seeGajdos,2001,forareview).Inthesamespirit,itiscommontondstudiesthatusebothFosteretal.(1984)andSen's(1976)povertymeasures.59Concerningthespecicsubjectofthedecompositionofobservedpovertyvariations,atti-tudesii)andiii)callforthedevelopmentofappropriateinequalityandpovertymeasures.Inthepresentpaper,wefocusontheheadcountindexsinceitisthesoleknownpovertymeasure(cf.proposition1)thatsimultaneouslycomplieswithalltheaforementionedinvarianceaxioms,andthusleavesroomtoindividualpreferencesfortheinterpretationofitsvariations.Indeed,theheadcountindexisconsideredbymostauthorsasapoormeasureofpovertysinceitdoesnotaccountfortheintensityandinequalitydimensionsofpoverty.Asaconsequence,manydistribution-sensitivepovertymeasureshavebeenproposedby(Watts,1968;Sen,1976;Kakwani,1980;Clarketal.,1981;Fosteretal.,1984;Hagenaars,1987),buteachoneisconsistentwithauniqueinvarianceaxiomsothatcomparisonsoftheir“leftist”,“rightist”andintermediatedecom-positionsarenotpossible.Asaresult,theevaluationoftherelativecontributionofgrowthandredistributiontopovertyalleviationusingdistribution-sensitivemeasuresforvariousinequalityviewscanonlybeperformedwiththehelpofclassesofinvariance-sensitivepovertymeasures,thatispovertymeasureswhichfeaturessomeparametersthatreectinvariancepreferences.Re-centpropositionsbyZheng(1997)aretoourknowledgethesoletentativetoprovidesuchtoolsandshouldinspirefurtherresearch.60AppendicesAThe(V,À)-invarianceaxiominpracticeThemajorconcernwiththeimplementationofthe(V,À)-invarianceaxiomisthepresenceofthereferencedistributionVthatgenerallycannotbedirectlyusedtondthecounterfactualincomesdistributionsneededforthecomputationofthegrowthandinequalityeffects.Forconvenience,supposethatVisonthetwo-dimensionsubspaceSXdenedbythevectorsX ¹XandI.Wealso 59Inthesetwoexamples,oneshouldnotethatorderingcriterionshavebeendevelopedsoastodetermineinwhichcasesdifferentinequalityandpovertyindiceswouldrespectivelyyieldthesameconclusions.Consequentlywesuggestthatfurtherresearchcouldbedevotedtothendindoforderingconditionsformanyinvarianceaxioms.60TheindicessuggestedbyZheng(2007c)arenotincludedinthepresentpaperbecausetheyarenotyetwelldoc-umented.Moreprecisely,itcanbedemonstratedthattheproposedKrtscha-typepovertyindexdoesnotcomplywiththenon-linearinvarianceaxiompresentedinsection2.3.2.MoreovertheDalton-Hagenaarsindexdevelopedbytheauthorreliesonanunknowninvarianceaxiom.Withoutexplicitformulationofthetransformationsthatpreservesinequality,thecomputationofthecounterfactualincomescannotbedone.Consequently,thisprecludesadecompo-sitionofthevariationsofthismeasure.43 ATHE(V,À)-INVARIANCEAXIOMINPRACTICE considerthatYisofsizenYthatmaybedifferentfromn.Intheoriginalversionoftheaxiom(delRioandRuiz-Castillo,2000),adistributionYcanbedirectlycomparedwithadistributionXonlyifitbelongstoSX.Forthepurposeofdecomposingpovertyspells,VcangenerallynotbeusedforthecomputationoftheinequalityeffectDsincethetransformeddistributionwouldnotbeonthesamesubspaceSYasY.InordertomakethecomparisonfeasiblewitheachdistributionY2D®,oneneedstondthedistributionV0 ¹V0thatcorrespondstotheprojectionoftherefer-encedistributionV ¹VintothesubspaceSY.Inthespiritof(Alonso-VillaranddelRio,2007a),wecandeneV0 ¹V0asthedistributioninSYwhichexhibitsthesameEuclideandistancefromperfectequalityasV ¹V.Thus,V0mustbechosensoastorespect:vuut n¡1nXiÆ1µvi ¹V¡1¶2Ævuut n¡1YnYXiÆ1Ãv0i ¹V0¡1!2.(A.1)TogetauniquedistributionV0,wehavetoaddsomeconstraintstoequation(A.1).Ifwenor-malizethedistributionsV0andYbytheirmeanvalue,weknowthatV0hastomeetthefollowingcondition:V0 ¹V0ƳY ¹YÅ(1¡³)I³2RÅÅ.(A.2)Asaconsequence:vuut n¡1YnYXiÆ1Ãv0i ¹V0¡1!2Ævuut n¡1YnYXiÆ1µ³yi ¹YÅ(1¡³)¡1¶2,(A.3)Ƴvuut n¡1YnYXiÆ1µyi ¹Y¡1¶2.(A.4)Rearrangingequation(A.4)andusingequation(A.1),weobtain:³Ævuuuut n¡1PniÆ1³vi ¹V¡1´2 n¡1YPnYiÆ1³yi ¹Y¡1´2.(A.5)Concerningthevalueof¶,scaleinvarianceisobtainedwhen:¶V0 ¹V0Å(1¡¶)IÆY ¹Y.(A.6)Consequently:vuut n¡1YnYXiÆ1öv0i ¹V0Å(1¡¶)¡1!2Ævuut n¡1YnYXiÆ1µyi ¹Y¡1¶2,(A.7)whichyields,usingequation(A.1):¶Æ1 ³.(A.8)44 CTHE(V,À)-INVARIANCEAXIOMANDUNIT-CONSISTENCY BThe(V,À)-invarianceaxiomandlemma1IftheprojectionVXofchosenreferencedistributionisnotequaltoXuptoascalefactor,wehave:VX ¹VƳ1X ¹Xų2I.(B.1)Inordertogetatransformeddistribution©(X,¹Y)whichmeanvalueisequalto¹Y,thetwoparametersmustrespectthefollowingcondition³2Æ1¡³1.Moreover,sinceVXshouldLorenzdominatethedistributionX,weobserve³1È1.Consequently,weget:©(X,¹Y)ÆXÅ(¹Y¡¹X)µÀVX ¹VÅ(1¡À)I¶,(B.2)ÆXÅ(¹Y¡¹X)µÀµ³1X ¹XÅ(1¡³1)I¶Å(1¡À)I¶,(B.3)ÆÀ³1X¹Y ¹XÅ(1¡À³1)¡XÅ(¹Y¡¹X)I¢.(B.4)Comparingwithequation(2.12),wecanconcludethatlemma1willbefullledifandonlyifVischosensothat³16À¡1.Anotherimportantconclusionisthatthetransformeddistribu-tioncorrespondingtoscaleinvariancecanbeobtainedfromXonlyifVXisequaltoXuptoascalefactor.Ontheotherhand,noconditionisimposedforthevalueof³1soastoobtainthetransformeddistributionthatwouldcorrespondtotranslationinvariance.CThe(V,À)-invarianceaxiomandunit-consistencyInarecentpaper,Zheng(2004)arguedthatanyinequalitymeasurebasedonthe(V,À)-inequality(delRioandRuiz-Castillo,2000)viewviolatesunitconsistency.HisdemonstrationisbasedonthecorrespondingintermediateLorenzcriterionwhichisdenedby:61L(X,j,À):Æ1 njXiÆ0Àxi ¹XÅ(1¡À)(xi¡¹XÅ1).(C.1)andisweightedmeanoftherelativeandabsoluteLorenzcurves.62TocomparethedistributionsXandY,wehavetodrawthecorrespondingintermediateLorenzcurveforY.Followingtherationaleofequation(C.1),Zheng(2004)gets:L(Y,j,À):Æ1 njXiÆ0Àyi ¹YÅ(1¡À)(yi¡¹YÅ1).(C.2) 61ThedenitionoftheadequateLorenzcriterionisquiteeasysinceoneonlyneedtheusetheequationoftheinvarianceaxiomonthegeneralizedLorenzcurvedenedbyShorrocks(1983)soastonormalizemeanincometounity.62ThisisaslightlymodiedversionoftheabsoluteLorenzcurvesinceoriginalversionbyMoyes(1987)is:L(X,j):Æ1 njXiÆ0xi¡¹X.45 CTHE(V,À)-INVARIANCEAXIOMANDUNIT-CONSISTENCY Inordertoprovethatunit-consistencyisnotrespected,XandYmustbechosensoasthereexistsavalueÀ¤2[0,¶]suchthatthetwodistributionscanbeconsideredasexhibitingthesamedegreeofinequality.Inotherwords,onedistributionshouldberelative-Lorenzdominatedbytheotherwhichabsolute-Lorenzdominatestheformer.LetXrelative-LorenzdominatesY.SoonecanndÀ¤suchthat:1 njXiÆ0À¤xi ¹XÅ(1¡À¤)(xi¡¹XÅ1)Æ1 njXiÆ0À¤yi ¹YÅ(1¡À¤)(yi¡¹YÅ1)(C.3)Ifasmallerunitofincomeisthenused(eachincomeisscaledupbythesameconstant,therespectofunit-consistencyimpliesthatweshouldstillfeelthatthetwodistributionsareequallyunequalforÀÆÀ¤.However,multiplyingthevectorXandYby¸È1yields:1 njXiÆ0À¤xi ¹XÅ(1¡À¤)(¸xi¡¸¹XÅ1)61 njXiÆ0À¤yi ¹YÅ(1¡À¤)(¸yi¡¸¹YÅ1)(C.4)sincewehavesupposedthatXisabsolute-LorenzdominatedbyY,thatistosay:1 njXiÆ0xi¡¹XÅ161 njXiÆ0yi¡¹yÅ1(C.5)Zheng(2004)thenconcludesonthebasisof(C.3)thatunit-consistencyisviolated.How-ever,thisresultisduetoamisunderstandingofdelRioandRuiz-Castillo's(2000)approach.Theauthorassumesthatthepartoftheincrementalincomethatisnotequallysharedbetweenin-comereceiversmustbedistributedinproportionoftheirrespectiverelativecontributiontototalincome.63Infact,thispartmustbedistributedwithrespecttoincomesharesofareferencedistri-butionwhichshouldbethesamewhencomparingXandYsoasauniquevalueofÀcanbeused.IfXischosenasthereference-distribution,therealintermediateLorenzcurvecorrespondingtoYis:L(Y,j,À):Æ1 njXiÆ0yiÅ(1¡¹Y)µÀxi ¹XÅ1¡À¶.(C.6)Unit-consistencyisnotviolatedifthedifferencesL(¸Y,j,À)¡L(Y,j,À)andL(¸X,j,À)¡L(X,j,À)areequalforÀÆÀ¤.Weobserve:L(¸Y,j,À¤)¡L(Y,j,À¤)Æ(¸¡1)1 njXiÆ0yi¡¹YµÀ¤xi ¹XÅ1¡À¤¶(C.7)L(¸X,j,À¤)¡L(X,j,À¤)Æ(¸¡1)1 njXiÆ0xi¡¹XµÀ¤xi ¹XÅ1¡À¤¶(C.8)Adding(¸¡1)1 nPjiÆ0À¤xi ¹xÅ1¡À¤toeachmemberyields:L(¸Y,j,À¤)¡L(Y,j,À¤)Å(¸¡1)1 njXiÆ0À¤xi ¹XÅ1¡À¤Æ(¸¡1)L(Y,j,À¤)(C.9) 63ThesamemistakecanbeobservedinZoli(2003).46 REFERENCES L(¸X,j,À¤)¡L(X,j,À¤)Å(¸¡1)1 njXiÆ0À¤xi 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