Forschungsinstitut zur Zukunft der ArbeitInstitute for the Study of Labor

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Person Equivalent Headcount Measures of Poverty

IZA DP No. 9402

October 2015

Tony CastlemanJames FosterStephen C. Smith

Person Equivalent Headcount

Measures of Poverty

Tony Castleman Catholic Relief Services and IIEP, George Washington University

James Foster

George Washington University

Stephen C. Smith

George Washington University and IZA

Discussion Paper No. 9402 October 2015

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IZA Discussion Paper No. 9402 October 2015

ABSTRACT

Person Equivalent Headcount Measures of Poverty*

Headcount measures of poverty are by far the most common tools for evaluating poverty and gauging progress in global development goals. The headcount ratio, or the prevalence of poverty, and the headcount, or the number of the poor, both convey tangible information about poverty. But both ignore the depth of poverty, so they arguably present distorted views of the spatial distribution of poverty as well as the extent of progress against poverty over time. Additionally, headcount measures can provide incentives for policymakers and NGOs to focus their efforts on the least poor, an observation well understood among policymakers themselves. While other poverty measures mitigate these problems by capturing the intensity as well as the prevalence of poverty, they are often not central to policy discourse because they are perceived to be too “unintuitive” to have traction. There is a need for poverty measures that go beyond traditional headcount measures, but retain their direct interpretation. This paper presents person equivalent (p. e.) headcount measures, which do just that. Our approach draws on the logic of full‐time equivalent jobs, adult equivalent incomes, and other constructs in economics. An initial period is used to calibrate the average depth of poverty among the poor, which then becomes the “person equivalent” underlying the p. e. headcount and the p. e. headcount ratio. We illustrate our methods using $1.25 a day poverty data from 78 countries as provided by the World Bank, and show how the new measures map out different pictures of poverty and progress than traditional headcount measures. Overall, the picture is one of a more rapid decline in global poverty, but with significant redistributions of its burden across regions and countries. For example, p. e. headcounts are much higher than traditional headcounts in Latin America and the Caribbean and Sub Saharan Africa; in South Asia and East Asia and the Pacific the reverse is true. In Kenya the traditional headcount rose by 8 million and the p. e. headcount rose by 11 million; in South Africa the p. e. headcount fell by more than the traditional headcount. We discuss properties of the new measures, outline some generalizations and conclude with recommendations for using this approach in development goals to track progress and direct policy. JEL Classification: I32, O15, D63 Keywords: poverty measurement, headcount, poverty gap, FGT indices,

development goals, inclusive growth, multidimensional poverty Corresponding author: Stephen C. Smith Department of Economics Monroe Hall 306 (2115 G St. NW) George Washington University Washington, D.C. 20052 USA E-mail: ssmith@gwu.edu

* The authors are grateful to the William and Flora Hewlett Foundation for research support through Grant #2013‐8857 to the George Washington University; and to the Institute for International Economic Policy for additional research support. We thank Steve Radelet for discussions and encouragement, and Misato Sato and Sam Gosney for excellent research assistance. We also thank participants of the GWU Symposium on the Economics of Ultrapoverty, the IEA‐World Bank Round Table in Jordan, and seminars at GWU and Brookings.

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1.Introduction

Themostcommontoolsformonitoringpovertyareheadcountmeasures,whichevaluateacountry’spovertylevelusingthenumberorprevalenceofpoorpersonsinthecountry.YetasemphasizedbySen(1976),headcountmeasureshaveseriouslimitationsstemmingfromtheirinabilitytodifferentiateamongthepoor.1Largechangesinpoorincomesareignoredwhentheincomesstaybelowthepovertyline,whilesmallchangesnearthelinecandisproportionatelyaffectmeasuredpoverty.Alternativepovertymeasureshavebeendevelopedthataddressthisproblembyaccountingfortheintensityofpoverty;butthesemeasuresaretypicallyabsentfrompolicydiscussionsastheycanbeviewedaschallengingforpolicymakerstoexplainintuitively,orforthepublictounderstand.

Theexclusiveuseofheadcountmeasurestoevaluatepovertycanhavesignificantimplicationsforpoliciesusedtoaddresspoverty.BourguignonandFields(1990)demonstratedhowusingheadcountmeasuresencouragespolicymakerstoignorethepoorestofthepoorandfocusonthosewithincomesjustbelowthepovertyline.Sen(1992,p.105)contendsthatanygovernmentfocusingsolelyonheadcountmeasures“facesastrongtemptationtoconcentrateontherichestamongthepoor,sincethisisthewaythatthenumberofthepoor...canbemosteasilyreduced.”AsimilarstatementcouldbemadefordevelopmentNGOs,internationalorganizations,orotheraidpartnerswhoseeffortsarejudgedusingpovertyheadcounts.2

Thereisclearlyaneedforpovertymeasuresthathaveastraightforwardinterpretationanalogoustotheheadcountmeasuresandyetappropriatelyreflecttheintensityofpovertyamongthepoor.Thispaperpresentsanewvarietyofpovertymeasures‐calledpersonequivalentheadcountmeasures‐toaddressthisneed.Theaveragedepthofpovertyamongthepooriscalculatedfromaninitialpopulation;thisbenchmark“personequivalent”isusedtotranslatebetweenincomeandpersons.Povertyismeasuredin“peoplespace”bycountingthenumberofpersonequivalents.Theideahasanalogieswiththenotionofafulltime

1Sen(1976,p.219)critiquebeginswiththeobservationthatheadcountmeasuresignorethepovertydepth:“Anunchangednumberofpeoplebelowthe‘povertyline’maygowithasharpriseintheextentoftheshort‐fallofincomefromthepovertyline.”Healsoattacksheadcountmeasuresforignoringthedistributionofincomeamongthepoor.2Ina2012presentation,SteveRadelet,formerChiefEconomistwithUSAID,urgedthedevelopmentcommunitytolookbeyondheadcountmeasures,whichignoreprogressthattakesplacebelowthepovertyline(Risley,2012).Ourpaperwaswritteninresponsetohismessageandtherealworldexampleshedescribedinsubsequentconversations.

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equivalentemployee,whichmeasuresemploymentusingabenchmarkworkweektoaccountforvariationsinthehoursworkedbydifferentemployees.3

Wepresenttwomeasures.Thefirst,calledthepersonequivalentheadcount,isanalogoustotheheadcount,orthenumberofpoor,butinsteadofcountingpersons,itcountspersonequivalents.Thesecond,calledthepersonequivalentheadcountratio,dividesthroughbytheoverallpopulationsize;itisanalogoustothetraditionalheadcountratio,ortheshareofthepopulationthatispoor.Bothareshowntobelinkedtotraditionalgapmeasuresofpoverty,andexhibitanumberofusefulproperties,includingtwothattraditionalheadcountmeasureslack:monotonicity(whichrequirespovertytoriseifapoorperson’sincomefalls)andcontinuity(whichrequiresthemeasurenottochangeabruptlywithasmallchangeinincome).Wenotethatraisingtheincomeofaminimallypoorpersonabovethepovertylinewilllowerapersonequivalentheadcountbylessthanone;raisingtheincomeofapersonfromfarbelowthepovertylinetojustbelowthepovertylinewilllowerthepersonequivalentheadcountbymorethanone.Inaddition,bothmeasuresaredecomposablebysubgroupandhencearesubgroupconsistent.

Thenewmeasuresarerelatedtogapmeasures,butdifferinonekeyrespect‐theirnumericalvalueshavemeaningsthatarevividandintuitive,asheadcountsthatcontrolfortheconditionofthepoor.Traditionalheadcountmeasurescanbemisleadingwhentheconditionsofthepoorchangedramatically.Personequivalentheadcountmeasuresbenchmarktheinitialconditionsofthepoor,andthenemploythisstandardasameasuringrodtocountthenumberofstandardizedpoor,orpersonequivalents.Thepictureofpovertyisalteredinappropriateways:itraisesthelevelofmeasuredpovertywhentheconditionsofthepoorbecomeworse;itlowersitwhentheaverageconditionsarebetter.Theextentofthisalterationinpracticecanbecapturedwiththeelasticityofthepersonequivalentheadcountratiowithrespecttothetraditionalheadcountratio(or“depthelasticity”).

Weillustrateourmethodsusing$1.25adaydatafromPovcalNetattheWorldBank,whicharebasedonhouseholdsurveydata.Weselecttwodatapointsforeachof78countries,onefromthe1990sandasecondfromthe2000s,andshowhowthepictureofpovertyisalteredoverspaceandtimewhenp.e.headcountmeasuresareused.Overall,thepictureisoneofamorerapiddeclineinglobalpoverty,butwithsignificantredistributionsofitsburdenacrossregionsandcountries.Thedepthelasticitiesofindividualcountriesmeasuredaretypicallygreaterthan1.0,butwithwidevariation.4Wealsoillustratehowpovertylevelschangewhenthebenchmark

3Seealsotherelatednotionsofadultequivalentincomescommonlyusedindistributionanalysis,theequallydistributedequivalentincomeofAtkinson(1970),oradultequivalentlaborasinBasuandPham’s(1998)modelofchildlabor.4Acaveatisthatsomeofthecountriesthatwecouldnotincludeduetolackofdatahavebeenestimatedotherwisetohavehighaveragepovertydepth;theseincludeAfghanistan,Congo,Guinea‐Bissau,Eritrea,Haiti,Liberia,SierraLeone,and

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populationisaltered,butotherconclusionsarenotaffected,includingcountrycomparisons,povertygrowthrates,anddepthelasticities.Weconcludewithadiscussionofsomepotentialtopicsforfutureresearch,suchasapplyingtheapproachtothesquaredpovertygapandtotheincreasinglyinfluentialmultidimensionalpovertymeasuresandusingpersonequivalentheadcountmeasuresindevelopmentgoals.

Section2beginswiththebasicdefinitionsandnotationusedinthepaper,whilesection3constructsthenewmeasuresanddiscussestheircharacteristics.Theempiricalexamplesarepresentedinsection4.Section5providesconcludingremarksandfutureextensions.

2.DefinitionsandNotation

Thepopulationsizeisdenotedbyapositiveinteger ,withpersonsrepresentedas1, … , .Thevector , . . . , denotesadistributionofincomeamongthe

population,whileapovertyline 0isusedtoidentifywhenapersonispoor,namely,when .Let , … , bethevectorofnormalizedgaps,where

– / foranyperson whoispoorwhile 0fornonpoor .Thenormalizedgapofapoorpersonexpressestheshortfall – fromthepovertylineasashareofthepovertyline .Apovertymeasure aggregatestheinformationin given toobtainanoveralllevel ; ofpoverty.Asimpleexampleisgivenbythepovertyheadcount ; orthenumberofpoorpeoplein given ,whiletheheadcountratio / istheshareofthepopulationthatispoor.

Headcountmeasuresdonotdistinguishamongthepoor;theyignoretheprogressapoorpersonmakesonthewaytoescapingpoverty.Incontrast,thepovertygapratio , … , ,whichisthemeannormalizedgapinapopulation,clearlydifferentiatesamongthepooraccordingtothedepthoftheirpoverty,andregistersadecreasewheneverapoorperson’sincomerises.Noticethatitcanbewrittenas

/ where istheheadcountratio, istheincomegapratiooraveragenormalizedgapamongthepoor,and istheaverageshortfallamongthepoor.5Ingeneral,theFGTclassofpovertymeasurescanbedefinedfor 0as

, … , ,orthemeanofthenormalizedgapsraisedtothepower .Clearlyistheheadcountratioand isthepovertygapmeasure,while istheFGT

Zimbabwe.Severaloftheomittedcountrieshaveahistoryofviolentconflict–perhapsareasonformissingdata,butalsoalikelycauseofpoverty.5Notethat (or )isanindicatoroftheaverageintensityofpovertyamongthepoor,butisnotagoodoverallmeasureofpoverty.Inparticular,itcanincreasewhenapoorpersonescapespovertyandtheremainingincomesareunchanged,thusviolatingastandardmonotonicityrequirement.

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squaredgappovertymeasurethatisparticularlysensitivetothepoorestpoorandaccountsforinequalityamongthepoor.

Eachofthesepovertymeasuresisrelativeinthatitevaluatesthemagnitudeofpovertyrelativetothepopulationsize,andsatisfiesreplicationinvariance,whichrequiresagivendistributiontohavethesamepovertylevelasoneinwhicheachincomeisreplicated times.Othermeasuresareabsoluteinthattheysatisfylinearreplication,whichrequiresa ‐replicationofadistributiontohave timesthepovertyoftheoriginaldistribution.Theheadcount isoneexample,andthetotalgap ,orthetotalincomenecessarytoraiseallpoorpersonstothepovertyline,isanother.Bothrelativeandabsolutemeasuresarehelpfulinevaluatingincomepovertyacrosspopulationsandtheirsubgroups.

3.PovertyGapsandPersonEquivalents

FollowingSen(1976),therehasbeenashiftinthefocusofpovertymeasurementfromtheidentificationstep,bywhichthesetofthepoorareidentified,totheaggregationstep,bywhichthedataareaggregatedintoanoverallmeasureofpoverty.Variousimprovedaggregatemeasureshavebeenproposedasreplacementsfortheheadcountq(thenumberofthepoor)ortheheadcountratioH(theshareofthepopulationthatispoor).However,thesimplicityoftheheadcountmeasureshascontinuallyledpolicymakersandappliedresearchersbacktothesecrudemeasures.6EventhepovertygapmeasureP1=HIwhichwascritiquedbySen,andpopularizedaspartofthedecomposableclassofFoster,Greer,andThorbecke(1984),isoftendismissedasbeingtoodifficultforpolicymakerstograspanduse.Consequently,manydiscussionsofpovertyignoresignificantvariationsintheintensityofpovertyacrossspaceandtime.Withheadcountmeasures,eachpoorpersoncountsthesame;withthepovertygapandrelatedmeasures,thecontributionofeachpoorpersondependsontheintensityofthepovertytheyexperience.

Inincomepovertymeasurement,thesimplestgaugeofapoorperson’sintensityofpovertyistheshortfall fromthepovertyline.Theaveragedepthorintensityofpovertyamongthepoorcanthenbemeasuredas ortheaverageshortfallamongthepoor.Denotetheaverageshortfallandthepovertyheadcountinaninitialorbenchmarkdistribution, ,by and respectively.Wewanttomeasureprogresswhenchangesoccurandleadtoanewdistribution anditsassociatedaverageintensityandheadcount, and .Forsimplicity,letusinitiallyassumethatthereisnopopulationgrowth,sothat .Atraditionalwayofassessingprogressisbyusingthechangeinheadcounts,withΔ – 0

6FosterandSen(1997)discussedthetradeoffbetweenthedesirablepropertiesofthenewaggregatemeasuresandthesimplicityoftheheadcountmeasuresandother“partialindices”ofpovertythatconveytangibleinformationononeaspectofpoverty.

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indicatinganimprovementinpoverty,Δ – 0indicatingaworsening,andΔ – 0suggestingthatpovertyisunchanged.Thisassessmentmightwellbejustifiediftheaverageintensityofpovertywereheldconstant.However,if isalsochanging,andthedepthofpovertyisseentobeofrelevanceinassessingprogress,thenheadcountsbythemselvescanprovideamisleadingviewofpoverty.Forexample,ifincomesofallextremelypoorpersonsinsocietyrosetocloseto,butjustbelowthepovertyline,thiswouldbeviewedusing asnoprogressatall.Likewise,ifonepersonwhoismarginallypoor(i.e.,havinganincomejustbelowthepovertyline)becamenonpoorandintheprocessallotherpoorpersonsbecomeextremelypoor,thiswouldbeseenasanunambiguousimprovementby .However,suchaconclusionwouldbechallengedif weretakenintoaccount.Weconsideranalternativewayofmeasuringheadcountsthatcontrolsforthechangesintheaverageintensityofpoverty.

Tothisend,weobservethattheinitialintensity canbeusedasameasuringrodininterpretingpovertycomparisonsandevaluatingprogress.Forexamplesupposethatthepovertyline isthetraditional$1.25aday,andthatinitiallythereare1000poorpersonswithanaverageshortfallof 50¢,sothatthetotalshortfallinthepopulationis $500.Inthefollowingperiodsupposethat isunchangedat1000whiletheaverageshortfalldeclinesto 45¢,leadingtoanewtotalshortfallof $450.Althoughthetotalnumberofpoorpersonsisunchanged,progresshasclearlybeenmadetowardsreducingpoverty.Wecanmeasurethisimprovementbydividingthenewtotalshortfall $450bytheoriginalaverageshortfall 50¢toobtainthepersonequivalent(p.e.)headcount

900.Inwords,thepersonequivalentheadcount isthenumberofpoorpersonswithabenchmarkaverageshortfall thatitwouldtaketoaggregateuptothetotalshortfallof .Itmeasuresthepovertygapin“peoplespace”byusingtheaverageshortfallofpoorpersonsastheunitofmeasurement.

Thepersonequivalentheadcountisthenumber thatsolves and,hence,thepersonequivalentheadcountisdefinedas

. (1)

Progresscanbegaugedusingthep.e.headcount,withΔ – 0indicatinganimprovementinpoverty,Δ – 0indicatingaworseningofpoverty,andΔ – 0indicatingthatpovertyisunchanged.TheaboveexamplehasΔ 900– 1000 100,whichsuggeststhattherehasbeenprogresstowardsreducingpovertyontheorderof100personequivalents.Incontrast,thetraditionalheadcountwouldindicatenoprogressatall.Notethattheratio / isatransformationfactorthatconvertstheconventionalheadcount intothepersonequivalentheadcount .Itreinterpretstheaveragegapusingthemeasuringrodofthebenchmarkaveragegap.Inourexample, / 45¢/50¢ 9/10,sothatthenewdistributionwith1000poorpersonsandanaveragegapof45¢isviewedashaving 900personequivalents.

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Thisapproachcanbeextendedtothevariablepopulationsizecasetoobtainameasureofpovertyqethatisabsoluteorindependentofthenumberofthenonpoor.Whenpopulationsizevaries,though,itismoretraditionaltoreporttheheadcountratio,ortheprevalenceofthepoorasashareoftheoverallpopulation.Let/ denotetheinitialheadcountratioand / denotethesubsequent

headcountratio.Thepersonequivalentheadcountratioisdefinedas / .Inourpreviousexample,supposethatthepopulationsizewasinitially 5000anddroppedto 4500.Thenwith 1000,wewouldhave 1/5and2/9sothatpovertyasmeasuredbytheheadcountratiohasrisen,orΔ 0.However,since 900/4500 1/5,thismeansthatthepersonequivalentheadcountratioisunchanged;thenumberofpersonequivalentsinpovertydeclinedatthesamerateastheoverallpopulation.Notethat isthevaluethatsolves

andhencethepersonequivalentheadcountratiomaybedefinedas

. (2)

Asbefore, / isthetransformationfactorconvertingtheheadcountratio intothep.e.headcountratio .Inthisexample, 2/9and / 9/10,andso1/5.

Theabovepresentationhasemployedtheaveragegap asameasureofintensitytogaugetheconditionsofthepoor.Analternativetotheaveragegapistheincomegapratio / ,whichexpressestheaveragegapasapercentageofthepovertyline,ratherthaninmonetaryunits,andistheintensitymeasurebehindthepovertygapratio .Whatwouldchangeiftheincomegapratio ratherthantheaverage(monetary)gap wereusedintheconstructionofthep.e.headcountandheadcountratio?Itiseasytoseethatsince / / ,thetransformationfactorwouldremainthesame,andhencetheresultingpersonequivalentmeasures

and . (3)

areidenticaltothosedefinedin(1)and(2).Intuitivelyspeaking,thepersonequivalentcomparestheintensityinthelaterperiodtothebenchmarkintensity,andtheratioisthesamewhethertheintensityismeasuredinmonetaryunitsorinpovertylineunits.

Thepovertygapratio , … , combines and toobtainameasurethatreflectsboththeprevalenceandintensityofpoverty.ItisasecondindicatorusedtomeasureprogresstowardthepovertygoaloftheMDGsandisreadilyavailableontheWorldBank’sPovcalNetwebsite.Whenpublisheddataexiston , ,and ,thepersonequivalentmeasures and canbeeasilyderivedasfollows.Firstfindthebenchmarkintensitylevel / .Thencalculatethepersonequivalentmeasuresas

/ / (4)

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Inotherwordsthepersonequivalentheadcountratioisfoundbydividingthepovertygapmeasure bythebaseintensitylevel ,whiletheheadcountfurthernetsoutthepopulationsize .

Theexpressionfor inequation(4)makesitclearthatgivenafixedinitialaverageintensitylevel ,theequivalentheadcountratio isproportionatetothepovertygapmeasure .Thus isapovertymeasurethatevaluatesdistributionsinthesamewayas ,buthasanalternativeinterpretationasthenumberofpersonequivalentspercapita.Itsharesthesamepropertiesas ,includingsymmetry,populationreplication,thefocusaxiom,monotonicity,continuityanddecomposability.7Incontrast,theoriginalheadcountratio doesnotsatisfymonotonicityandcontinuity;itignoresallimprovementsintheconditionsofthepoorthatdonotresultinacrossingofthepovertyline,butregistersadiscretechangewhenapoorpersondoescross.Forinstance,ifaprogramtargetingtheultra‐poorsuccessfullyloweredtheirincomeshortfallsby90%,itwouldberegardedby withindifference–theprevalenceofpovertyhassimplynotchanged.Theprogram’sprogresswouldberevealedifthep.e.headcountratio wereused.Inananalogousfashion,equation(1)revealsthatthepersonequivalentheadcountisproportionatetothetotalincomegap andhenceevaluates

distributionsinasimilarway.Thepropertiessatisfiedby (and )includesymmetry,linearreplication,thefocusaxiom,monotonicity,continuityandadditivity,allofwhicharesatisfiedbytheheadcountmeasure apartfrommonotonicityandcontinuity.8Themonotonicityaxiomensuresthat reflectsthechangesintheintensityofpovertyevenwhen isunchanged.

Wehavedescribedtwonewmeasuresofpoverty–thepersonequivalentheadcountmeasure andthepersonequivalentheadcountratio –thatevaluatepovertyin“peoplespace”withthehelpofatransformationfactorbasedontheaveragedepthorintensityofpovertyinabenchmarkperiod.Ifaveragedepthfallsbelowbenchmark,thepersonequivalentheadcountmeasurewillbelowerthanitsrespectivetraditionalheadcountmeasure;ifaveragedepthrisesabovebenchmark,thepersonequivalentheadcountmeasurewillbehigher.Torecap,whenapersonwhowaspoorinaninitialperiodcrossesthepovertyline,theimpactonapersonequivalentheadcountmeasuredependsonthedepthoftheperson’spovertyinthepriorperiod:Iftheinitialincomewasslightlybelowthepovertyline,itwouldhaveasmalleffectonapersonequivalentheadcountmeasure,whileiftheincomewaswellbelowthepovertyline,itwouldhavealargeeffect.

Incertaincontexts,wemightbemainlyinterestedinevaluatingthepercentagechangeinpovertythroughtime.Toevaluatetheinclusiveness(orpro‐poorness)of

7PrecisedefinitionscanbefoundinFosterandSen(1997),Foster(2005)andFosteretal(2013).8Bylinearreplicationitismeantthatareplicationofthedistributionthatresultsinak‐foldincreaseinpopulationleadstoak‐foldincreaseinmeasuredpoverty.Additivityiscapturedin(5)or(6)below.

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growth,forexample,therateofreductioninpovertycanbedividedbythegrowthrateinpercapitaincometoderivethegrowthelasticityofpoverty–ameasureofhowwelltheeconomyisconvertingincomegrowthintopovertyalleviation.Thetraditionalgrowthelasticityusestheheadcountratio ;thepersonequivalentheadcountratio couldalsobeused.Arelatedstatisticcomparesthegrowthelasticitiesof and (orequivalently,thegrowthratesof and )toobtainthedepthelasticityofpoverty %Δ /%Δ ,whichindicateshowprogressintheheadcountmeasureistranslatingintoprogressinthepersonequivalentheadcountmeasure.9

Theabovediscussionappliestoevaluationsofprogressovertimeforagivenpopulation.Theanalysiscanbereadilyextendedtocomparisonsbetweensubgroupsdefinedaccordingtogeographicallocation,demographiccharacteristic,orsomeotherparameter.Forexample,personequivalentheadcountmeasuresfordifferentcountriescanbeconstructedandcomparedusingtheglobal$1.25adaypovertystandardusingabenchmarkintensityleveldrawnfromworlddata.Alternatively,interestmightberegionalinscope,inwhichcasep.e.headcountcomparisonscouldbemadeacrosscountrieswithintheregionusingaregionalbenchmark.Bothexamplesdivideanoveralldistribution intosubgroupdistributionsandevaluatesubgrouppovertyusingapersonequivalentheadcountmeasurebenchmarkedfortheoverallpopulation.Forsimplicityofnotation,letusfocusonthetwo‐subgroupcasewherethedistributioncanbewrittenas , forsubgroupdistributions and ;thesamelogicwouldapplytothemanysubgroupcase.Thebenchmarkintensity isobtainedfromaninitialdistributiondrawnfromthesamegeneralpopulationas ,butpotentiallyatanearliertime

period.Thecase correspondstoananalysisofpovertyoverspace(i.e.,acrosssubgroups);thecasewhere isdrawnfromanearliertimeperiodleadstocomparisonsoverspaceandtime.

Let and betheheadcountandaverageintensitylevelsfordistribution ,andlet and betheassociatedvaluesfordistribution .Bytheadditivityoftheheadcountmeasure weknowthat

(5)

ortheoverallheadcountin isthesumoftherespectiveheadcountsin and .Giventhebenchmarklevelofintensity ,definethepersonequivalentheadcountfordistributions and by / and / .Since

,itimmediatelyfollowsfromdividingthroughby that

(6)

9Italsoindicateshowwellchangesin predictchangesin .Notethatthegrowthelasticityof (orequivalently,of )istheproductofthedepthelasticity andthetraditionalgrowthelasticityofH.

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sothattheoverallpersonequivalentheadcountisthesumoftherespectivepersonequivalentheadcountsin and .Ascomparedtothecrudeheadcount ,thep.e.headcount / ishigherorlowerdependingonwhethertheintensityin is,respectively,higherorlowerthanthebenchmarklevel.Equation(6)thenprovidesthebreakdownofthepersonequivalentheadcountacrossthesubgroups.

If ,theoverallpersonequivalentheadcountin reducesto ,

thetraditionalheadcount,sothat fromequation(6)isequalto fromequation(5).Thedecompositionin(6)isanalternativepovertybreakdownto(5)thataccountsforthedifferentialdepthofpovertyexperiencedbypeopleinthetworegions.Usingpersonequivalentheadcountscanbeinterpretedbyimaginingthatthereisaredistributionofthepopulationofpoorpeopleacrossthesubgroups,withthehigherintensitysubgroupgainingpersonequivalentsandthelowerintensitysubgrouplosing.Iftherearetwoperiods,thenequation(6)canalsobeappliedtothesecondperiodusingthebaseintensitylevel andcomparingacrosstimetoobtainthechangesineachsubgroupandoverall.ItiseasytoshowthatΔ ΔΔ ,sothatthechangeintheoverallp.e.headcountisthesumoftherespectivechangesinp.e.headcountsforthetwosubgroups.

Fromequation(5),orbythedecomposabilitypropertyoftheheadcountratio,itfollowsthat

/ / (7)

andsotheoverallheadcountratioisapopulation‐shareweightedaverageofthesubgroupheadcountratios.Ananalogousargumentusingequation(6)ordecomposabilityfor yields

/ / (8)

whichisthedecompositionformulaforthepersonequivalentheadcountratio.Ifappliedtotheoriginaldistribution, becomes sothat(8)providesanalternativebreakdownoftheheadcountratioaccountingfortheintensitiesofpovertyinthetwogroups.Equation(8)canalsobeusedovertimetolinkprogressinthepersonequivalentheadcountratiotoprogressatthesubgrouplevel.

Theinterpretationsof and dependcentrallyonthebenchmarklevelofintensity,andhencethetimeandregionfromwhichitisdrawn.Thebenchmarkisanaveragevalueinaregion(whichcouldbeaparticularcountry,acollectionofcountries,ortheworld)atagivenpointintime.Apoorpersonwithsmallerthanaverageincomegapwillaccountforlessthanonepersonequivalent;apoorpersonwithalargerthanaveragegapwilladdmorethanone.Ifasubgroupcontainsmanypoorpeoplewhoaredeeplypoor,andtheregionalbenchmarkissufficientlysmall,thenumberofpersonequivalentsinthesubgroupcouldwellexceedthesubgroup

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population,leadingtoap.e.headcountratiobeyondtheusualbounds.10Likewise,aregionwithalargenumberofpoorpersonsjustbelowthepovertylinecouldrecordamuchlowerp.e.headcountthanitstraditionalheadcount,particularlyiftheregionalbenchmarkishigh.Inanycase,thesubgrouplevelsstayinproportionwithoneanotherevenasthebenchmarkchanges.

Thetechnologyofpersonequivalentheadcountsiswellsuitedforformulatingdevelopmentgoalsandtargetsthatgobeyondcrudeheadcountmeasuresandhaveanaturalstartingtimefromwhichtobenchmark.Thedecompositionformulawouldsupportmultilevelanalysesatdifferentgranularities,fromthegloballeveldowntoanindividualhousehold.Themethodscouldbeadaptedtodifferentpurposesbyalteringtheregionusedinbenchmarking.Forexample,tomonitorglobalprogress,theaverageintensityacrosstheglobeinthestartingperiodcouldbeusedasthebenchmark.Aregionaldevelopmentbankmightbeinterestedonlyinreportingprogressinitstargetareaandcoulduseabenchmarkfromageographicregionsuchassub‐SaharanAfrica.Moreover,acountrycouldmonitoritsownprogressusingtheinitialcountrywideaverageintensityasabenchmark.And,asnotedabove,evenwhenthebenchmarkisvariedtoreflectthedifferentpurposesandscopeofanalyses,theresultsarequiteconsistent.Thepersonequivalentheadcountmeasuresobtainedusingonebenchmarkareproportionaltothoseobtainedwithanother,preservingtheirrelativemagnitudesacrossspace,andtheirgrowthratesacrosstime.Thereisnopossibilityofmisalignedincentivesforthepartiesconductingtheanalysesatdifferentlevels.

Forexample,supposethattheinitialaverageintensityis 0.50inacountry,0.75intheregion,and 0.25intheworld.Ifthecountry’spovertygap

ratiowere0.12intheinitialperiodand0.06inasubsequentperiod,thepersonequivalentheadcountratiosasreckonedusingthecountrybenchmarkwouldinitiallybe0.24andsubsequently0.12.Ifinsteadtheregionalbenchmarkwereused,thenthepersonequivalentheadcountratioswouldbe0.16and0.08,respectively,or / 2/3timesthesevalues,whileattheworldbenchmark,thevalueswouldbe0.48and0.24,respectively,or / 2timesaslargeastheinitialvalues.Notethatthetrendsinpovertyforagivencountryareconsistentirrespectiveofwhichbenchmarkisemployed.Moreover,ifthegoalweretolowerthepersonequivalentheadcountratioby50%oftheinitialvalue,allthreelevelswouldtrackprogressconsistentlyandwouldmeetthegoalatthesametime.Inthissense,thegoalwouldberobusttothechoiceofbenchmark.11

10Inasimilarfashion,thenumberoffulltimeequivalentemployeesatacompanycanexceedthenumberofpersonsemployed.11Thiswouldnotbetrueifthegoalweretolowerpersonequivalentpovertytoaparticularabsolutelevel(orindeedbyanabsoluteamount),sinceabsolutelevelsandchangesdependonthebenchmark.Inthiscaseitwouldcrucialtospecifythebenchmarkaheadoftime.

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4.AnIllustration:GlobalPoverty

ThepersonequivalentapproachtoevaluatingpovertyisillustratedusingpovertydatapublishedbytheWorldBankonPovcalNet,whichinturnisbasedonunderlyinghouseholdsurveydata.12Inordertogaugeprogressovertime,werestrictconsiderationtocountriesforwhichdataexistforatleastoneyearinaninitialrangeof1992‐2000andoneyearinthelaterrangeof2005‐2010.Atotalof78countriesfromsixregionssatisfythiscriterion.Weconstructa“developingworld”madeupofthesecountries,andextractthe$1.25adayfigurestocreatecountrypovertydatafortwotimeperiods.Decompositionformulasallowheadcountratiosandthepovertygapratiostobecalculatedforthedevelopingworldandforregions.Weapplytheformula / (orequivalently/ )todatafromtheinitialperiodtoderivetheappropriatebenchmarklevel,

whichinturnisusedtoproducepersonequivalentheadcountsandpersonequivalentheadcountratios.Table1reportsthepovertystatisticsforourfull78‐countrysampleoverthetwoperiodsusingtheglobalbenchmarklevelof 39.5¢perday(orequivalently

0.316).Theglobalheadcount droppedby512millionpersonsduringthisperiod,andsincetheaverageincomeshortfallamongthepooralsodeclinedbymorethan4¢,to =35.2¢perday,thedropinpersonequivalentswas625million,113millionmorethanthedropinheadcount.Theconventionalheadcountratiofell44%from =0.36to =0.20betweenthetwoperiods,whilethepersonequivalentheadcountratiodecreasedby50%from =0.36to =0.18,onceagainreflectingthedecreaseinaveragedepth.Theglobaldepthelasticitywasabout1.1,indicatingthatforeveryonepercentdropintheheadcountratiotherewasa1.1%declineinthepersonequivalentheadcountratio.Table2exploresregionalpovertylevelsandtrendsusingthesameglobalbenchmark.13Figures1and2depictthelevelsgraphicallyusingmapsoftheregions.ThemovetopersonequivalentheadcountsfromtraditionalheadcountsresultsinincreasesinpovertyratesinSub‐SaharanAfrica(SSA)andLatinAmericaandtheCaribbean(LAC)anddecreasesintheotherregions;thisistrueforboth

12PovCalNetwasaccessedJuly2015athttp://iresearch.worldbank.org/PovcalNet/index.htm13NotethatdatafromtheLatinAmericaandCaribbeancountriesinPovCalNetuseincomedata,whereasallothercountries(exceptLatvia)useconsumptiondata.AfewcountriesinLAChavebothconsumptionandincomedataavailablefromthesameyear.Forthesecountriespovertyratesaresubstantiallyhigherusingincomedata,andthedifferencesareevengreaterforthepovertygapthantheyarefortheconventionalheadcount.Giventhesedifferencesbetweenconsumptionandincomedata–whichthepersonequivalentheadcountmeasurehighlights–comparisonsbetweenLACandotherregionsshouldbeinterpretedwithcaution.

14

periods.AcomparisonofSouthAsia(SA)andSSAisparticularlyinformative.ThepovertyheadcountinSAwasinitially608million,farhigherthaninSSAat260million,andSAcontinuedtodominateSSAinthenumbersofpoorpeoplebyawidemarginofover180millioninthesecondperiod,evenastheheadcountfellinSAandroseinSSA.However,whenviewedthroughthelensofpersonequivalentheadcounts,theinitiallevelsofthetworegionsareseentobemuchcloserand,inthesecondperiod,SSAactuallyovertakesSAbymorethan35millionpersonequivalents.Incorporatingthedepthofpovertypaintsaratherdifferentpictureofpovertyandprogressinthetworegionsthanheadcountalone.Headcountratiostakeintoaccountthedifferentialpopulationsizesacrosscountriesandthroughtime.Thedatafor showstrongdeclinesforallregions,withthesuccessstoryofEastAsiaandthePacific(EAP)beingrepresentedbyasharplyfallingvalueof (from0.38to0.11).NoticethatthedeclineintheheadcountratioinLAC(from0.09to0.05)isalsoimpressiveinpercentageterms.Indeed,thedataonpersonequivalentheadcountratiosalsoshowprogressin forbothregions;butdifferencesinaverageintensityshiftthevaluesforEAPdownandLACup,withtheresultthatbothregionsreachthesamelevelof (namely,0.07)inthesecondperiod.ReturningtothecaseofSSAandSA,theinitialvaluesfor arenotdissimilar(at0.59and0.50,respectively).However,regionaldifferencesinintensitygenerateawidedivergenceintheinitialperiod’svaluesof forSSAandSA(0.81and0.44,respectively),whiledifferencesinprogressaccentuatethisfurthersothatthefinalperiod valuesare,respectively,0.62and0.23.Depthelasticitiesweregreaterthan1.0infourregions(SSA,SA,ECA,andEAP),indicatingfasterreductionin than .LAChadanelasticityof0.97,indicatingsimilarratesofreductioninthetwoheadcountmeasures.TheMiddleEastandNorthAfrica(MENA)hadadepthelasticityof0.7.Apparently,thereductionin wasaccompaniedbyariseinthedepthofpoverty,leadingtoasmallerimprovementinthep.e.headcountratio thaninH.Table3containspovertystatisticsforelevenofthe78countriesusingthesameglobalbenchmarkasabove.Weexaminehowthelevelsof and inacountrydifferfromoneanotherandhowthisalterstherateatwhichpovertychanges.Fourofthecountries(China,India,Egypt,andSouthAfrica)followtheleadoftheglobalfigures,andtheEAPandSAregions,byhavingsmallervaluesfor than inbothperiods;fourothercountries(Bolivia,Brazil,Kenya,andMozambique)followtheLACandSSAregionsbyhavinghigher valuesinbothperiods,indicatingthattheaveragedepthisgreaterthanthebenchmarklevelsforbothperiods.Inthethreeremainingcountries(Vietnam,Nepal,andNiger),thelevelof ishigherthanHintheinitialperiodandlowerthanHinthesecond,indicatingfasterpovertyreductioninthesecountrieswhendepthofpovertyistakenintoaccountthanwhenonlytheprevalenceofpovertyismeasured.Indeed,allcountriesexceptforBrazilandEgyptexhibitahigherrateofchangein thanin ,andthushavedepthelasticitiesthatexceed1.0.Forexample,Nepal’sconventionalheadcountratiodeclined4.4%peryearanditsp.e.headcountratiodeclinedby5.4%peryear,leadingtoanelasticityof1.2.Twocountries,KenyaandBolivia,sawpovertyrisebetweenthetwoperiods;

15

largeincreasesintheirheadcountratiosweremagnifiedevenfurtherbytheirelasticitiesof1.5and2.3,respectively.Adepthelasticitygreaterthanonemeansthattheaveragedepthofpovertyworsenedincountrieswheretheprevalenceofpovertyworsenedandimprovedincountrieswheretheprevalenceofpovertyimproved.ForBrazilandEgyptthepictureisdifferent.Brazil’sconventionalheadcountratiodeclinedmorequickly(3.7%peryear)thanitsp.e.headcountratio(3.0%peryear)resultinginadepthelasticityof0.8;Egypt’sp.e.headcountratioincreasedslightly(0.7%peryear)whileitsconventionalheadcountratiodeclined(2.5%peryear),yieldingacasewherethedepthelasticitytakesonanegativevalueof‐0.3.Figure3graphstheconventionalheadcountandpersonequivalentheadcountratiosforthesecountriesinthe2005‐2010period,andFigure4graphstheannualratesofchangeforthetworationsbetweenthetwoperiods.Table4presentsthesecondperioddataforcountriesinSub‐SaharanAfricausingacontemporaneousregionalbenchmarkof =50.0¢perday(theaverageincomeshortfallinthe2005‐2010periodintheSSAcountries).Notethewiderangeinthepersonequivalentheadcountratios(from0.06to1.2)ascomparedtotheconventionalheadcountratios(rangingfrom0.14to0.88),suggestingthattheintensityishigherthantheregionalaverageforsomecountrieswithhighprevalenceofpoverty,andlowerthanaverageforsomecountrieswithlowprevalence.Cameroon,Niger,andSouthAfrica,forexamplehavemuchlowerp.e.headcountsthantraditionalheadcounts,whileMadagascarandZambiahavemuchhigherp.e.headcounts.NotethattherelativepictureacrossSSAcountrieswouldbethesameifadifferentbenchmarkwereused,suchastheregionalbenchmarkcomputedfromperiodonedataortheglobalbenchmarkusedabove: and wouldsimplybeshiftedproportionallytoreflectthenewstandard.

Table5showshowanindividualcountry,Niger,canuseitsownbaseyearaverageincomeshortfall(61.7¢perdayin1994)tobenchmarkitsprogressincombatingpoverty.Between1994and2007,thenumberofindividualsinNigerearninglessthan$1.25perdaydroppedfrom6.9millionto6.0million,amodestdecline.Thepersonequivalentheadcountdroppedfrom6.9millionto3.4millionduringthesameperiod,averylargedecrease.Thepersonequivalentheadcountratiodecreasedatanannualizedrateof‐5.3%peryearand,asdiscussedabove,thispercentagechangewouldbethesameregardlessofwhetherthebenchmarkusedisthecountry’saverageincomeshortfall,theregion’s,ortheworld’s.However,foritsowninternalassessment,acountrymayfocusonthelocallybenchmarkedfigures,whileknowingthefindingswillbeconsistentwithglobally,regionally,orevenarbitrarilybenchmarkedfigures.

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5.Conclusions

Inthispaperwehavepresentedpersonequivalentheadcountmeasures,whichliketraditionalheadcountmeasuresareevaluatedin“peoplespace”;butinsteadofcountingpoorpersons,thesemeasurescountpersonequivalents,asbenchmarkedbytheaveragedepthofpovertyinagivenplaceandtime.Theresultingmeasuresareintuitivetoexplainandeasytocalculate,butatthesametimesatisfymonotonicityandaresensitivetothedepthofpovertylikethepovertygapmeasurestowhichtheyarerelated.Byexplicitlyaccountingfortheconditionsofthepoor,theyremovetheincentivetofocusontheleastdeprivedsegmentsofthepoor.Whenanextremelypoorpersonescapespoverty,thishasagreaterimpactonthemeasuresthanwhenamarginallypoorpersoncrossesthepovertyline.Andifapoorpersonmakesgoodprogresstowardsescapingpoverty,buthasnotyetcrossedthepovertyline,thisisregardedbypersonequivalentheadcountmeasuresasapositiveachievementratherthansomethingtobeignored.

Wethenappliedournewmeasuresto$1.25‐a‐dayglobalpovertydatatoshowhowtheypaintadifferentpictureofpovertyandprogressthanconventionalheadcountmeasures.Forexample,personequivalentheadcountsaremuchhigherinSub‐SaharanAfricaandinLatinAmericaandtheCaribbeanthantraditionalheadcounts,andlowerinSouthAsiaandEastAsiaandthePacific.Interestingly,SSAjoinstheSAandEAPregionsinregisteringfasterpercentagedeclinesinpersonequivalentheadcountratiosthanintraditionalheadcountratios,whileinLACtheratesofdeclineinthetwomeasuresarequitesimilar.Ananalysisbycountrylikewiseprovidesnewinsightsintocountryexperiences,withlargercountrieslikeChina,India,andBrazilcloselymatchingtheirregionalresultsandotherslikeSouthAfricadepartingwidelyfromtheirregionalpicture.

Mostofthecountriesweexaminedhaveadepthelasticity‐orthepercentagechangeinp.e.headcountratiooverthepercentagechangeinthetraditionalheadcountratio‐thatisgreaterthanone.SouthAfricalowereditsheadcountratioagreatdeal,butwithadepthelasticityof 1.6,theperformanceintermsofpersonequivalentswasevenmoreimpressive.Kenyahadasimilardepthelasticityof1.5,butsincetheheadcountratiorose,thisindicatedanevenmoredramaticincreaseinp.e.headcountratio.OthercountrieslikeBrazilwith 1.0hadtheirimprovementsinp.e.headcountmeasuresmutedascomparedtoheadcountratios.Twocountries–EgyptandMauritania–exhibitednegativedepthelasticities,buttheyalsohadverysmallchangesinheadcount(andp.e.headcount)ratios.Weillustratedhowthebenchmarkunderlyingpersonequivalentmeasuresisalteredbyusingdifferentgeographicareas,fromaglobaltoregionalorevencountrylevel.Differentbenchmarksresultindifferentvaluesforacountry’spersonequivalentheadcountmeasures,butsincecountriesareaffectedproportionally,rankingsareconsistentandgrowthratesareunchanged.Thisconsistencymakestheperson

17

equivalentheadcountratioespeciallyappropriateforuseinmultileveldevelopmentgoals.14

Thepersonequivalentheadcountmeasuresprovideanintuitivewayofincorporatinginformationonthedepthofpoverty,butsomemightcontendthattheymovetoofarafieldfromtraditionalheadcountmeasures.Indeed,iftheconditionofapersonchangeddiscontinuouslyasthepovertylineiscrossed,itcouldmakesensetoretainthisfeatureinameasureofpoverty.Oneapproachcouldbetoconstruct“hybrid”measuressuchas 1 or1 where 0,1 representstheextentonebelievesthatthediscontinuity(andhence or )isimportant.15Nowpovertyisevaluatednotonlybycountingthepoororcountingpersonequivalents,butthroughacompromisebetweenthetwoperspectives.Thispracticalapproach,however,canreintroduceanincentivetofocusontheminimallypoor‐atleastinthesimplifiedperfectinformationscenarioofBourguignonandFields(1990).Itwouldbeinterestingtoseewhetherthehybridmeasurescanprovidebenefitsinothermorerealisticenvironments,suchaswheninformationasymmetries(say,betweenpolicymakersandaidworkers)playasignificantrole.

Onepossiblecritiqueofthepresentationuptonowisitsexclusivefocusonmonetarypoverty.Asemphasizedinthe2000WorldDevelopmentReportoftheWorldBank,povertygoesbeyondmonetaryresources:itdependscentrallyonotherkeydimensionsthatshouldalsobeincludedwhenidentifyingthepoorandmeasuringpoverty.Ofcoursealloftheabovemeasurementtechnologywillapplydirectlytoanyother(cardinal)singledimensionalvariable(e.g.schoolingornutrition),thusidentifyingpersonswhoaredeprivedinthatvariableandmeasuringtheirlevelsofdeprivation.16

However,itisnowgenerallyrecognizedthatthemultipledimensionsmustbesimultaneouslyobservedinordertoidentifywhoispoorandtoevaluatehowmuchpovertytheyhave.Andthistypicallyrequiresexpandeddataandanewmeasurementtechnology.Distributionsarenowmatrices,thesinglepovertylinebecomesavectorof“deprivationcutoffs,”andanoverallmeasureisconstructedby

14Amultileveldevelopmentgoalsetstargetsandevaluatesoutcomesforseverallevelsofpopulationaggregation.15FosterandShorrocks(1991,p.699)derivestheclass 1 axiomaticallywhere isacontinuous,decomposablemeasureand 0,1 .Asimilarformarisesinthemeasurementofultra‐poverty.SeeFosterandSmith(2015).16Forexample,theothermajortargetofthefirstMDGishalvinghunger.Hungerisgenerallyexpressedintermsofthefractionofthepopulationprojectedtosufferfromabelow‐minimumcaloricintake.Yetsurelypeoplemovingtowardthatminimum,evenifnotyetcrossingit,alsorepresentsprogressagainsthunger.Thepersonequivalentapproachcouldbeavaluablecomplementtoexistingheadcountmetricsforinternationalpovertygoals.

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aggregatingacrossdimensionsforthepersonsidentifiedaspoor.Unliketheunidimensionalcase,identificationisnotasimplematterinthemultidimensionalcontext;indeed,mosttheoreticalpresentationsdonotprovideapracticalmethodforidentifyingthemultidimensionallypoor,butinsteadfallbackona“union”approach,equatingpovertywithbeingdeprivedinanydimension.Moreover,theindicatorsavailableformultidimensionalpovertyanalysisareoftenordinal,renderingmanytheoreticalsolutionstotheidentificationandaggregationstepsinapplicable.Themultidimensionalheadcountratio isoneindexthatworkswellwithordinaldata;however,bothitandthemultidimensionalheadcount = sufferfromtheflawsoftheirunidimensionalcousinsashighlightedinthispaper.

Thechallenge,then,ofmultidimensionalpovertymeasurementhasbeentosolvetheidentificationandaggregationstepsinawaythatisconsistentwithordinaldata,butgoesbeyondcrudeheadcountmeasures.OnemethodologythatdoesthisisfoundinAlkireandFoster(2011).17Apersonisdeprivedinagivendimensioniftheachievementlevelisbelowadeprivationcutoffforthedimension.Eachdeprivationhasa“value”andapersonispoorornotdependingontheextentoftheperson’smultiplicityofdeprivations,asmeasuredbythedeprivationcountorsumofthesevalues(wherethemaximumsumofallvaluesisfixedat ,thenumberofdimensions).Forexample,ifeachdeprivationhasthesamevalue,thenthedeprivationcountisthenumberofdeprivationsthepersonisexperiencingatthesametime.Apersonispoorifthedeprivationcountmeetsorexceedsapovertycutoffsetbetween0and .Apoorperson’sintensityofpovertyismeasuredasthedeprivationcountdividedbyitsmaximum .Theaverageintensity,denoted ,isthesumoftheintensitiesofthepoordividedbytheirnumber .Theadjustedheadcountratioisthengivenby .

Notethattheformofthismeasureisentirelyanalogoustothatofthepovertygapratio ,whichunderliesthepersonequivalentheadcountmeasuresformonetarypoverty.Couldourtechnologybeappliedinthemultidimensionalcasetotransformtheadjustedheadcountratiointoamultidimensionalpersonequivalentheadcountratio?Ifso,thenitcouldofferhelpfulinterpretationsforthemanyapplicationsof incommonuse,includingofficialmeasuresinseveralcountriesandtheMultidimensionalPovertyIndex(MPI)publishedintheannualHumanDevelopmentReportbytheUnitedNations.18Thiswouldbeausefuldirectiontopursueinfuturework.

Inaddition,wecouldalsoconsidermonetarypovertymeasuresthatstresstheconditionsofthepoorestpoorandtakeintoaccountinequalityamongthepoor.ThedistributionsensitivemeasuresofSen(1976),Watts(1968)andFoster,GreerandThorbecke(1984)allhavethischaracteristicand,asnotedbyBourguignonandFields(1990),theyprovideapositiveincentiveforfocusingonthepoorestpoor

17SeealsoAlkire,etal(2015).18SeeAlkireandSantos(2014).

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first.Woulditbepossibletoconstructpersonequivalentheadcountmeasuresforeachthatwouldappropriatelyreflectinequality?

Considerthecaseofthesquaredpovertygap ofFoster,GreerandThorbecke.Asnotedabove, placesgreaterweightonpersonswhoarefurtherbelowthepovertylinebysquaringthenormalizedgaps beforeaveraging.Clearly, where istheheadcountratioand istheaverageof amongthepoor(analternativeintensitymeasurethataccountsforinequalityamongthepoor).Letting beitsbenchmarklevel,wecandefine / and asthepersonequivalentmeasuresassociatedwith .Anassociated“severityelasticity”couldevaluatetheelasticityof withrespectto ,or% /% .

PreliminaryresultsusingPovcalNetdatasuggesthowaccountingfordistributionsensitivitybyusingP2influencesthepictureofglobalpoverty.Globaltotalsarevirtuallyunchanged(seeTable6).However,theregionalpicturebecomesevenmorepronounced,with and movingfurtherinthedirectionstakenby and(seeTable7).NowtheEAPregionbeginswith524Mpersonequivalentsandends

upwithonly94million.Incontrast,SSArosefromaround455millioninthe1990sto471millioninthe2000s‐almosttwicethenumberofpersonequivalentsinSAandfivetimesthenumberinEAP.Theseverityp.e.headcountratio(He’)19forLACof0.11issubstantiallyhigherthanthatofEAP(0.07).Applyingthepersonequivalentapproachtodistributionsensitivemeasuresofpovertyisaninterestingtopicforfuturework.

Wehaveemphasizedthesuitabilityofthepersonequivalenttechnologyfordefiningandtrackingmultileveldevelopmentgoals.WenowconcludewithabriefdiscussionofitsrelevancetotheMillenniumDevelopmentGoals(MDGs)andthepost2015agenda.RecallthatthemainindicatorforthepovertyportionofGoal1oftheMDGshasbeenthe$1.25adayheadcountratioH.ThepovertygapratioP1isalsolistedasacomplementaryindicator,butforreasonsofsimplicityhaslargelybeenabsentfromallbutthemosttechnicaldiscussions.Forexample,initsassessmentofprogressintheMDGs,theUnitedNations(2014)presentsonlyheadcountsorheadcountratiosasindicatorsofsuccessorfailureinreachingpovertygoals.TheWorldBank’s(2010)assessmentusesP1,butonlytohelpexplainthewhypoorercountriesmighthavelowgrowthelasticitiesandslowerprogressinreducingH.Wewouldarguethatdepthshouldbeincludedintothemix,bothwhenevaluatingtheinitialdistributionofpovertyandinmonitoringtheprogressofcountries.Thepersonequivalenttechnologyprovidesasimpleandintuitivewayofdoingjustthis.

Endingextrememonetarypoverty‐interpretedbytheUNandWorldBankasreducingthe$1.25adayheadcountratiotonomorethan3%ofthepopulation‐hasemergedfromthepost‐2015discussionasapossiblepovertygoal.Assumingthattheglobalpopulationwillbe9billion,thiscouldtranslatetoaheadcountof270millionleftbehind.Ifthisgoalwereachievedandonly270millionpeopleremained19WhencomparingHewithHe’,forclaritywerefertotheformerasthedepthp.e.headcountratio,andthelatterastheseverityp.e.headcountratio.

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belowthe$1.25perdaypovertyline,itstandstoreasonthatthisgroupcouldcontainsomeofthemostdeeplydeprived,difficult‐to‐reachpersonsonearth.Thereisnothinginthegoalthatwouldpreventthemfromhavinganaveragedepthofpovertythatistwicetheaveragedepthofpovertyamongthepoorin2015.Usingthe2015averagedepthasthebenchmark,thiswouldmeanapersonequivalentheadcountofoverahalfabillion.Shouldthisreallybeseenasanendtoextremepoverty?Restatingthegoalintermsofpersonequivalentheadcountratiosremovestheambiguityabouttheconditionsofthoseleftbehind.Monitoringprogresswiththismeasureensuresthatthedepthofpovertyisalsobeingevaluatedthroughtime.Assessinginitialconditionsusingpersonequivalentheadcountmeasurespresentsamorecompleteguidetothechallengesthatlieahead.

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References

Alkire,S.,Foster,J.,Santos,M.,Seth,S.,Ballon,P.,andRoche,J.,(2015)MultidimensionalPovertyMeasurementandAnalysis.Forthcoming,OxfordUniversityPress.

Alkire,S.,andFoster,J.(2011).‘CountingandMultidimensionalPovertyMeasurement.’JournalofPublicEconomics,95(7‐8):476‐487.

Alkire,S.,andSantos,M.(2014).‘MeasuringAcutePovertyintheDevelopingWorld:RobustnessandScopeoftheMultidimensionalPovertyIndex.’WorldDevelopment,59:251‐274.

Atkinson,A.(1970).‘OntheMeasurementofInequality.’JournalofEconomicTheory,2:244‐263.

Basu,K.,andPhamH.V.(1998)."TheEconomicsofChildLabor."AmericanEconomicReview88(2):412‐27.

Bourguignon,F.,andFields,G.(1990).‘PovertyMeasuresandAnti‐PovertyPolicy.’RecherchesEconomiquedeLouvain,56(3‐4),409‐427.

Foster,J.,Greer,J.,andThorbecke,E.(1984)."AClassofDecomposablePovertyMeasures".Econometrica.352:761–766.

Foster,J.(2005).‘PovertyIndices.’InPoverty,InequalityandDevelopment:EssaysinHonorofErikThorbecke(A.deJanvryandR.Kanbur,eds.).KluwerAcademicPublishers.

Foster,J.,Lokshin,M.,Sajaia,Z.,andSeth,S.(2013)AUnifiedApproachtoMeasuringPovertyandInequality:TheoryandPractice.WorldBankPress.

Foster,J.E.,Sen,A.K.,1997.“OnEconomicInequality.AfteraQuarterCentury.”AnnextotheenlargededitionofOnEconomicInequalitybySen,A.K.Oxford:ClarendonPress.

Foster,J.,andSmith,S.(2014)‘MeasuringUltrapoverty:ANewClassofIndicesandanApplicationtoRuralEthiopiaPanelData.’Unpublished.

Foster,J.,andShorrocks,A.(1991).‘SubgroupConsistentPovertyIndices.’Econometrica,59(3):687‐709.

Risley,H.(2012).“TheEconomicPerspective:CausesandConstraintstoReachingtheUltraPoor”USAIDMicrolinksBlog,https://www.microlinks.org/blogs/economic‐perspective‐causes‐and‐constraints‐reaching‐ultra‐poor

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Sen,A.(1976).‘Poverty:AnOrdinalApproachtoMeasurement.’Econometrica,44(2):219‐231.

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Appendix:CountriesandYearsCambodia 1994 2009China 1996 2010Indonesia 1996 2010LaoPDR 1997 2007Philippines 1994 2009Thailand 1994 2010Vietnam 1992 2008Albania 1996 2008Armenia 1998 2010Azerbaijan 1995 2008Belarus 2000 2010Croatia 1998 2008Georgia 1996 2010Hungary 1998 2007Kazakhstan 1996 2010KyrgyzRep. 1993 2010Latvia 1996 2009Lithuania 1998 2008Macedonia 2000 2008MoldovaRep 1997 2010Poland 1996 2010Romania 1998 2010RussianFed 1996 2006Tajikistan 1999 2009Turkey 1994 2010Ukraine 1995 2010Argentina* 1995 2010Bolivia 1993 2008Brazil 1995 2009Chile 1994 2009Colombia 1996 2010CostaRica 1995 2009DominicanR 1996 2010Ecuador 1999 2010ElSalvador 1995 2009Guatemala 1998 2006Honduras 1996 2008Mexico 1994 2010Nicaragua 1993 2005Panama 1995 2010

Paraguay 1995 2010Peru 1997 2010Uruguay* 1995 2005Venezuela 1995 2006EgyptArabR 1995 2008IranIslamicR1994 2005Jordan 1997 2010Morocco 1998 2007Tunisia 1995 2010YemenRep 1998 2005Bangladesh 1995 2010India 1993 2009Nepal 1995 2010Pakistan 1996 2007SriLanka 1995 2009BurkinaFaso 1994 2009Burundi 1992 2006Cameroon 1996 2007CentralAfrR 1992 2008Côted'Ivoire 1995 2008Ethiopia 1995 2010Ghana 1998 2005Guinea 1994 2007Kenya 1994 2005Madagascar 1993 2010Malawi 1997 2010Mali 1994 2010Mauritania 1995 2008Mozambique 1996 2007Niger 1994 2007Nigeria 1996 2009Rwanda 2000 2010Senegal 1994 2005SouthAfrica 1995 2008Swaziland 1994 2009Tanzania 1991 2007Uganda 1996 2009Zambia 1996 2010

*ArgentinaandUruguaydataareurbanonly.

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Table 1: Person Equivalent Headcount Measures: Full Sample (1990s Global Benchmark)

Range of

Years Population n (millions)

Headcount q (millions)

Headcount ratio H

Person-equivalent headcount qe (mill.)

Person-equivalent headcount ratio He

% change in H

% change in He

Depth Elasticity

World (78 countries)

1992-2000 4,316 1,547 .36 1,547 .36 -44.3% -50.4% 1.1

2005-2010 5,189 1,035 .20 922 .18 Benchmark is global average income shortfall in 1992-2000 period: 39.5¢ per day

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Table 2: Person Equivalent Headcount Measures by Region (1990s Global Benchmark)

Region Range of

Years Population n (millions)

Headcount q (millions)

Headcount ratio H

Person-equivalent headcount qe (mill.)

Person-equivalent headcount ratio He

% change in H

% change in He

Depth Elasticity

East Asia & Pacific

1992-97 1,635 615 .38 575 .35 -71% -79% 1.1

2007-10 1,842 201 .11 134 .07

Europe & Central Asia

1993-2000 399 15 .04 12 .03 -74% -83% 1.1

2007-10 402 4 .01 2 .01

Latin America & Caribbean

1993-99 457 43 .09 60 .13 -49% -48% 0.97

2005-10 535 26 .05 38 .07

Middle East & North Africa

1994-98 179 7 .04 4 .02 -35% -25% 0.73

2005-10 213 5 .02 3 .02

South Asia 1993-96 1,210 608 .50 537 .44

-37% -49% 1.3 2007-10 1,553 489 .32 354 .23

Sub-Saharan Africa

1992-2000 441 260 .59 359 .81 -16% -23% 1.4

2005-10 625 308 .49 390 .62 Benchmark is global average income shortfall in 1992-2000 period: 39.5¢ per day

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Table 3: Person Equivalent Headcount Measures for Selected Countries (1990s Global Benchmark)

Country Year Population n (millions)

Headcount q (millions)

Headcount ratio H

Person-equivalent headcount qe (mill.)

Person-equivalent headcount ratio He

Annual % change in H

Annual % change in He

Depth Elasticity

Bolivia 1993 7 0.6 .09 0.8 .11

1.5% 3.5% 2.3 2008 10 1.0 .16 1.6 .17

Brazil 1995 162 16 .110 21 .13

-3.7% -3.0% 0.8 2009 193 9 .05 15 .08

China 1996 1,218 455 .37 427 .35

-5.4% -5.8% 1.1 2010 1,338 123 .09 86 .06

Vietnam 1993 68 44 .64 51 .74

-4.8% -5.5% 1.1 2008 85 14 .17 10 .12

India 1993 921 455 .49 395 .43

-2.1% -2.8% 1.3 2009 1,190 388 .33 282 .24

Nepal 1995 21 14 .68 17 .81

-4.4% -5.4% 1.2 2010 27 6 .24 4 .16

Egypt 1995 61 1.5 .025 .66 .011

-2.5% 0.7% -0.3 2008 76 1.3 .017 .88 .012

Kenya 1994 27 8 .29 8 .30

4.6% 7.1% 1.5 2005 36 16 .43 19 .54

Mozambique 1996 16 13 .81 21 1.3

-2.2% -3.3% 1.5 2007 23 14 .61 19 .82

Niger 1994 9 7 .78 11 1.2

-3.5% -5.3% 1.5 2007 14 6 .42 5 .37

South Africa 1995 39 8 .21 6 .17

-2.6% -4.1% 1.6 2008 50 7 .14 4 .07

Benchmark is global average income shortfall in 1992-2000 period: 39.5¢ per day

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Table 4: Countries in Sub-Saharan Africa (2000s Regional Benchmark)

Country Year Population n (millions)

Headcount q (millions)

Headcount ratio H

Person-equivalent headcount qe (mill.)

Person-equivalent headcount ratio He

Burkina 2009 15 6.7 .44 5.5 .37 Burundi 2006 8 6.5 .81 7.3 .91

Cameroon 2007 19 5.2 .28 3.5 .18 Central African Republic 2008 4 2.6 .63 3.3 .78

Cote d’Ivoire 2008 18 6.4 .35 5.8 .32 Ethiopia 2010 87 34 .39 23 .26 Ghana 2005 21 6.1 .29 5.3 .25 Guinea 2007 10 4.0 .39 3.3 .33 Kenya 2005 36 16 .43 15 .42

Madagascar 2010 21 18 .88 26 1.2 Malawi 2010 15 10.8 .72 12.9 .86

Mali 2010 14 7.1 .51 5.8 .41 Mauritania 2008 3.4 .80 .23 .58 .17

Mozambique 2007 23 14 .61 15 .65 Niger 2007 14 6.0 .42 4.2 .29

Nigeria 2009 155 96 .62 107 .67 Rwanda 2010 11 6.8 .63 7.2 .66 Senegal 2005 11 3.8 .34 3.0 .27

South Africa 2008 50 6.8 .14 2.8 .06 Swaziland 2009 1.2 .46 .40 .45 .38 Tanzania 2007 41 28 .68 29 .70 Uganda 2009 33 12 .38 10 .30 Zambia 2010 13 9.8 .74 14 1.0

Benchmark is Sub-Saharan Africa region’s average income shortfall in 2005-2010: 50.0¢ per day

28

Table 5: Person Equivalent Headcount Measures in Niger (1994 Country Benchmark)

Year Population

n (millions) Headcount q (millions)

Headcount ratio H

Person-equivalent headcount qe (mill.)

Person-equivalent headcount ratio He

Annual % change in H

Annual % change in He

Depth Elasticity

1994 8.9 6.9 .78 6.9 .78 -3.5% -5.3% 1.5

2007 14.2 6.0 .42 3.4 .24

Benchmark is Niger’s average income shortfall in 1994: 61.7¢ per day

Table 6: Global Person-Equivalent Poverty including Squared Gaps (1990s Global Benchmark)

Years n (mill.) q (mill.) H qe (mill.) He qe2 (mill.) He2 % H % He % He2 Depth Elasticity

Severity Elasticity

World (78 countries)

1992-2000 4,321 1,547 .36 1,547 .36 1,547 .36 -44.3% -50.4% -51.8% 1.1 1.2

2005-2010 5,189 1,035 .20 922 .18 897 .17 Benchmark for qe and He is global average income shortfall in 1992-2000 period: 39.5¢ per day.

Benchmark for qe2 and He2 is global average squared income shortfall in 1992-2000: 22¢2 per day, equivalent to a gap of 47¢ per day.

29

Table 7: Person Equivalent Poverty among Regions including Squared Gaps (1990s Global Benchmark)

Region Years n (mill.) q (mill.) H qe (mill.) He qe2 (mill.)

He2 % H % He % He2 Depth Elasticity

Severity Elasticity

East Asia & Pacific

1992-97 1,635 615 .38 575 .35 524 .32 -71% -79% -84% 1.1 1.2

2007-10 1,842 201 .11 134 .07 94.4 .05

Europe & Central Asia

1993-2000 399 15 .04 12 .03 11 .03 -74% -83% -80% 1.1 1.1

2007-10 402 4 .01 2 .01 2.3 .01

Latin America & Caribbean

1993-99 457 43 .09 60 .13 90 .20 -49% -48% -46% 0.97 0.93 2005-10 535 26 .05 38 .07 59 .11

Middle East & North Africa

1994-98 179 7 .04 4 .022

2.8 .016

-35% -.25% -5% 0.73 .14 2005-10 213 5 .02 3 .01

6

3.2 .015

South Asia 1993-96 1,210 608 .50 537 .44 464 .38

-37% -49% -55% 1.3 1.5 2007-10 1,553 489 .32 354 .23 268 .17

Sub-Saharan Africa

1992-2000 441 260 .59 359 .81 456 1.0 -16% -23% -27% 1.4 1.7

2005-10 625 308 .49 390 .62 471 .75

Benchmark for qe and He is global average income shortfall in 1992-2000 period: 39.5¢ per day. Benchmark for qe2 and He2 is global average squared income shortfall in 1992-2000: 22¢2 per day, equivalent to a gap of 47¢ per day.

30

Figure 1: Regional Headcounts 1992-2000

61515

43

7

608

260

Headcountq(millions)1992‐2000

57512

60

4

537

359

Person‐equivalentHeadcountqe(millions)1992‐2000

31

Figure 2: Regional Headcounts 2005-2010

2014

26

5

489

308

Headcountq(millions)2005‐2010

1342

38

3

354

390

Person‐equivalentHeadcountqe(millions)2005‐2010

32

H

HH

H

H

H

H

H

H

H

HHe

HeHe

He

He

He

He

He

He

He

He

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

Bolivia2008

Brazil2009

China

2010

Vietn

am2008

India

2009

Nepal

2010

Egypt

2008

Kenya

2005

Mozam

b.

2007

Niger2007

South

Africa2008

CountryandDate

Figure3:ComparisonofHeadcountRatioandPerson‐EquivalentHeadCountRatioforSelectedCountries

33

H

H

HH

H

H

H

H

H

H

H

He

He

He He

He

He

He

He

He

He

He

‐10,0%

‐5,0%

0,0%

5,0%

10,0%

Bolivia

1993‐2008

Brazil

1995‐2009

China

1996‐2010

Vietn

am1993‐2008

India

1993‐2009

Nepal

1995‐2010

Egypt

1995‐2008

Kenya

1994‐2005

Mozam

bique

1996‐2007

Niger

1994‐2007

SouthAfrica

1995‐2008

Annual%Change

CountryandDate

Figure4:ComparisonofAnnualPercentageChangeinHeadcountRatioandPerson‐EquivalentHeadCountRatiofor

SelectedCountries