FamilyPolicies:WhatDoesTheStandardEndogenousFertilityModelTellUs?Thoma - PDF document

FamilyPolicies:WhatDoesTheStandardEndogenousFertilityModelTellUs?ThomasBaudinParisSchoolofEconomicsUniversityofParisIPanthéon-SorbonneCentred’EconomiedelaSorbonne106-112Boulevarddel’Hôpital75013ParisAbstractThereisageneralconsensusintheeconomicliteratureandineconomicinstitutionsaboutthelegitimacyofpoliciessubsidizingeducation.Thislegitimacyliesinthefactthateducationisasourceofpositiveexternalities.Inastandardframeworkofendogenousfertility,thepresentpapershowsthatthisresultisstillvalidbutthatsubsidizingeducationalsorequirestaxingbirths.Educationsubsidiesdecreasethenetcostofchildrensuchthatparentscanexhibitatoohighfertilityrate.Furthermore,whenhealthisintroducedasanothersourceofexternalities,themodelshowsthathealthexpenditureshouldnotalwaysbesubsidized.Indeed,thetaxationofbirthsplaystheroleofanindirectsubsidyonhealthexpenditurebecauseitdecreasesthecostofhealthrelativetothecostofchildren’squantity.Whenexternalitiesoneducationareveryhighrelativetopositiveexternalitiesonhealth,theindirectsubsidyonhealthcanexceedthesubsidythatisreallyneeded.Thenhealthexpenditurehastobetaxed.JELCodes:H21,I0,J13,J18KeyWords:Fertility,Education,FamilyPolicy,Mortality,QualityQuantityTrade-o IamgratefultoBertrandWigniolle,PierrePestieau,DaviddelaCroixandVictorHillerfortheirinvaluablehelp.DiscussionswiththeparticipantsoftheEUREQuateam’sworkshopinmacroeconomicshavebeenveryenlightening. 1IntroductionThereisageneralconsensusintheeconomicliteratureandineconomicinstitutionsaboutthelegitimacyofpoliciessubsidizingeducation.Thislegitimacyliesinthefactthateducationisasourceofpositiveexternalities[Hanushek&Welch(2006)].Inthispaper,Iusethestandardframeworkofendogenousfertilitywithatrade-obetweenqualityandquantityshowthatthisresultisstillvalidbutthatsubsidizingeducationalsorequirestaxingbirthsforalargesetofSocialWelfarefunctionsincludingBenthamiteandMillianones.Indeed,educationsubsidiesdecreasethenetcostofchildrensuchthatparentscanexhibitatoohighfertilityrate.Followingthisresult,healthisintroducedasanothersourceofpositiveexternalitiesreducingchildmortality.Asaresult,despiteitsstatusofpositiveexternality,healthexpenditureshouldnotalwaysbesubsidized.Indeed,thetaxationofbirthsplaystheroleofanindirectsubsidyonhealthexpenditurebecauseitdecreasesthecostofhealthrelativetothecostofthequantityofchildren.Inordertoreachthesamenumberofsurvivingchildren,parentstendtohavefewerchildreninbetterhealth.Whenexternalitiesoneducationareveryhigh,thetaxonbirthshasalsotobehigh.Hence,ifthepositiveexternalitiesonhealtharelow,theindirectsubsidycanexceedthesubsidythatisreallyneeded.Thenhealthexpenditurehastobetaxed.The"standardframework"ofendogenousfertilitycomesfromtheseminalworksofBeckeretal.[1973,1976,1988].Itconsistsinamodelwhereparentsvaluethenumberoftheirspring(quantity)aswellastheirfuturehumancapital(quality).Theymaximizetheirexpectedutilitysubjecttoanonlinearbudgetconstraint.Thenatrade-obetweenqualityandquantitytakesplace.ThisfundamentalcontributionofBeckerhasbeenfollowedbythemajorimprovementsofGaloretal.[1999,2002],DelaCroix&Doepke[2003],Kalemli-Ozcan[2003],etc,resultinginauniedframework.Surprisingly,thereareveryfewstudiesexploringtheoptimalitypropertiesofthetrade-obetweenqualityandquantityinthis Noticethat,inthispaper,familypoliciesarelimitedtotax-transferpolicies.Inreality,familiypoliciesincludealargesetofinstrumentlike,forinstance,coercivepoliciesandpreventiveactions.AMilliansocialwelfarefunctionconsistsintheaverageutilityoftherepresentativeagentwhileaBenthamitesocialwelfarefunctionvaluethetotalutilitydistributedintheeconomy.Thisnonlinearityisfundamentalinmodelsoftrade-obetweenqualityandquantity.Becausequalityisprovidedtoeachchild(withorwithoutequity),itscostcruciallydependsonthequantitychoices.Thentheparentalbudgetconstraintisnolongerlinear. edframework.Thequestionofoptimalfertilityhasbeenstudiedinotherframeworks.Samuelson[1975],Deardor[1976]andMichel&Pestieau[1993]explorethequestionoftheoptimalpopulationgrowthrateinanoverlappinggenerationmodelwithexogenousfertility.AmodelwithendogenousfertilityhasbeenproposedbyMichel&Wigniolle[2007]andgeneralizedbyGolosovetal.[2007].TheirinterestfocusesontheParetooptimalityofequilibria.However,theydonotdealeitherwiththequality-quantitytrade-oorwiththequestionofoptimalfamilypolicies.Boulding[1964]proposedimplementingamarketoftradableprocreationrights.ThisideaisexploredbyDelaCroix&Gosseries[2007].Itnallyconsistsinasystemoftaxorsubsidyonthequantityofchildren.However,theydonotexplainthereasonswhygovernmentsarenotsatisedwiththeirnationalfertilityGroezenetal.[2003]proposesamodelofendogenousfertilityanddealswiththequestionofoptimalfamilypolicies.Hearguesthat,inthepresenceofaPayAsYouGo(PAYG)pen-sionsystem,childrenareasourceofpositiveexternalitiesbecausetheirmarginalproductionwillnancethepensionsystem.Itimpliesthatthecompetitivefertilityrateistoolow,andsoachildallowancehastobeimplemented.However,ifthereisnoPAYGpensionsystem,thecompetitivefertilityisoptimal.Groezenetal.donotdealwiththetrade-obetweenqualityandquantitywhichpartlycausesthislastresult.Nerloveetal[1984]studytheoptimalityoffertilitybehaviorsinaframeworkwhereparentsvalueboththequantityandthewellbeingoftheirospringbyconsideringthatparentalutilitypositivelydependsontheconsumptiontheyallocatetotheirchildren.Inthisframework,theyndthattaxingbirthcanbeanoptimalscalpolicywhenthesocialwelfarefunctionisMillian.Conversely,whentheSocialWelfarefunctionisBenthamite,theyndthatanoptimaleconomicpolicyalwaysconsistsinachildallowance.TheseresultssensibelydierfromminebecauseNerloveetaldonotreallydealwiththeintergenerationalcienciesoftheparentalqualityquantitytrade-oMypaperismorecloselyrelatedtothecontributionsofSpiegel[1993]andBalestrino Inthatsense,thepresentpaperhastobeconsideredasacomplementtothisliterature.Loupias&Wigniolle[2004]showthat,inaclosedframework,ageneralizedAllais-Samuelson-Diamondgoldenrulecanbereachedonlyiffertilityissubsidized.Theyalsoproposealternativeformulationswhichwillbementionnedintherestofthepaper. [2000].Theybothdealwithoptimalscalschemeswhenthereisatrade-obetweenqualityandquantity.Theirmainconclusionisthatataxationofbirthscanconstituteanoptimalfamilypolicy.Thisresultcruciallycomesfromtheassumptionthatthesocialplannertriestoreduceinequities.Inthesemodels,ataxonbirthsisanecientinstrumentforreducinginequities.Inthepresentpaper,acomplementaryapproachisproposed.Theexistenceoftax-trans-feronbirthsisnotconditionaleitherontheexistenceofdierencesbetweenthegovernment’sobjectiveandparentalpreferencesoronaproblemofunobservabilityofbehaviors.Indeed,theimplementationofbirthtaxorchildallowancecomesfromboththeexistenceofexter-nalitiesinthehumancapitalaccumulationprocessandfromthenonlinearityoftheparentalbudgetconstraintwithregardtoqualityandquantity.Thesetwofeaturesarecommontothelargestmajorityofpapersdealingwiththequalityquantitytrade-o.DepartingfromtheMillianWelfarefunction(theSocialPlanermaximizestheutilityoftherepresentativeagent(averageutility))oraddingalternativeexternalitieswillenrichtheanalysis.However,itwillneveraectmymainmechanismbecauseitresultsfromthefundamentalassumptionsoftheusualtheoryofendogenousfertility.Asmentionnedprecedently,themodel’smainassumptionistheexistenceofexternali- Balestrinoetal.proposeamodelofoptimaltaxationwhereparentsareheterogenous.ParentalchoicesareallParetoecient.However,thegovernmentischaracterizedbyaBenthamitefunctionofSocialWelfare,soittriestoreducewelfareinequalitiesbetweengroupsofparents.Moreover,thisgovernmentfacesamimickingproblemàlaStiglitz(theworkforceparticipationisnotobservable).Fertilitybeingobservable,taxingbirthscanhelpthegovernmenttodierentiateparentswhoarereallypoorfrommimickers.AnotherenlighteningcontributioncomesfromCignoandPettini[2002]whondasimilarresultwithoutmimickingproblems.Spiegel[1993]proposesamodeloftrade-obetweenqualityandquantitywithaRawlsiansocialplanner.Heshowsthatapolltaxonbirthsenablesthegovernmenttodecentralizethesocialoptimumoftheeconomy.However,thatinstrumentisaperfectsubstituteforataxonthesecondperiodconsumption.InSpiegel’sframework,ifthegovernmentdoesnotvaluewelfareinequalities,notaxonbirthsisrequired.InBalestrinoetal,evenifthesocialplannerdislikeswelfareinequalities,theobservabilityofabilitieswouldmaketheindividualindirectutilitesobservable.Thenlumpsumtransferswouldensureanoptimalredistri-butionofwelfare.Notaxonbirthswouldbenecessary.Forinstance,Iassumethatthereisnoinequalitiesintermofwealthorabilitiesbetweenagents.So,educationalpoliciesdonotndtheirlegitimacyinthegovernementwillingnesstoreduceinequalities.Inreality,thereductionofinequalitiesisamajormotivationtoimplementeducationalpolicies.However,inthepresentframework,Idonotneedtointroduceinequalitiestoobatinmyresults.Doingsowouldonlyenrichthemodel.Behaviorsbeingobservable,thepresenceofinequalitieswouldnecessitatetoimplementadditionaltransfersbetweenfamilies. tiesinhumancapitalaccumulation.Whenparentschoosetheiroptimaltrade-obetweenqualityandquantity,theydonotconsiderthattheireducationinvestmentwillimprovetheoveralleciencyofthehumancapitalaccumulationprocess.Itimpliesthat,atthecompet-itiveequilibrium,theytendtounder-investineducation.So,anoptimaleconomicpolicyistheimplementationofasubsidyoneducationspending.However,thebudgetconstraintinthestandardmodeloftrade-obetweenqualityandquantityisnotlinear.Itimpliesthatreducingthecostsofqualityalsoreducesthenetcostofquantity.Inconsequence,whenitisoptimaltosubsidizeeducation,itisalsooptimaltotaxbirths.Thiscentralresultisrobusttotheintroductionofa"natalistbias"inthesocialplanner’spreferences(theBenthamiteutilityfunctionwillbeaspecialcaseof"natalistbias")andtotheextensiontoendogenouschildmortality.Theintroductionofendogenouschildmortalityisanimportantextensionofthisframe-workbecauseitchangesthenatureofthetrade-obetweenqualityandquantity.Indeed,nowparentsdonotonlyhavetodecidehowtoallocatetheirspendingsbetweenqualityandquantity,theyalsohavetodecidetheiroptimalstrategytoreachtheirdesirednumberofchildren.Inotherwords,theyfaceanalternativetrade-obetweenqualityandquantityofsurvivingchildreninwhichtheirhealthexpenditurewillbeasourceofexternalities.Intheextendedmodel,higherparentalhealthexpenditurereduceschildmortality.Fur-thermore,theaveragelevelofhealthspendinghasanegativeimpactonchildmortality.Theliteratureofdevelopmenteconomicsprovidesstrongevidencethatoverallhealthqualityisoneofthemaindeterminantofindividualhealthquality.Forinstance,Dasgupta[1993]showsthat45percentofalldeathsindevelopingcountriescanbeimputedtoinfectiousandparasiticdisease.Privatehealthexpenditurehelpsreducetheprobabilityofbeinginfectedwhenanagentisincontactwithdiseases.Soahigheraveragelevelofhealthexpenditurere-ducesdeathprobabilitiesinallfamilies.Thispositiveexternalityimpliesthatprivatehealthexpenditureistoolowatthecompetitiveequilibrium.InthisextendedframeworkwhichonlyconsiderstheMilliancaseforsimplicity,reachingoptimalityrequires,onceagain,subsidizingeducationandtaxingbirths.Now,thetaxationofbirthsplaystheroleofanindirectsubsidyonhealthexpenditure.Indeed,itincreasesthecost ThisisinlinewithGaloretAl[1999,2002],DelaCroix&Doepke[2003],Kalemli-Ozcan[2003]etc. ofquantityrelativetothecostofhealth.Parentstendtoincreasetheirhealthexpenditureandtodecreasethenumberofbirthstoreachthesamenumberofsurvivingchildren.Forstrongexternalitiesonhealthexpenditure,theindirecttaxwillnotbesucienttoreachoptimalhealthexpenditureatthecompetitiveequilibrium.Soprivatehealthexpenditurehastobesubsidized.Therecommendationtotaxbirthsincomplementtosubsidiesforeducationandhealth,canbeanalyzedinthelightofsomeempiricalevidencefromChinaandSub-SaharanAfrica.Theybothfaceaproblemofoverpopulationandimplementalternativestrategiestoreducefertility.Mypaper’sresultsareobviouslytheoreticalandcannotreproducetheverycomplexdemographic,economicandpoliticsituationsofthesecountries.However,itprovidessomeincentivestoputintoquestionstheirstrategies.Chinaisexperimentingwithaspeciscalschemeonbirthsthatsubsidizestherstbirthandstronglytaxessubsequentones.However,empiricalstudiessuchasthoseofKanbur&Zhang[2003]andFan&Zhang[2000]showthatinvestmentineducationandhealthisinsucientinChina.Thepresentpaperproposesanalternativescalschemethatwouldreallocatepublicfundsfromtherstbirthsubsidytothepromotionofeducationandhealth,withoutlossofeciencyinbirthcontrol.Sub-SaharanAfricancountrieshaveimplementedseveralfamilyplanningprogramswhichstronglypromoteinvestmentinhealthandeducation.However,arecentreportoftheWorldBank[2007]showsthatthispolicyhasbeeninecientinreducingthenetfertilityrateinalargemajorityofthesecountries.Thepaperarguesthatthesepolicieshavebeeninecientbecausetheydidnotincreasetherelativecostofquantity.Itshowsthatmoreattentionshouldbepaidtotheimplementationofascalschemethatwouldexplicitlysanctionbirths.Therestofthepaperisorganizedasfollows.InSection2,thebenchmarkmodelispresented.Itsrecommendationsintermsoffamilypoliciesarediscussed.InSection3,endogenouschildmortalityandpublichealthexpenditureareintroduced.Section4discussesthepaper’sempiricalimplicationsforChinaandSub-SaharanAfrica.Section5concludes. 2TheBenchmarkmodel2.1TheCompetitiveEquilibriumThemodelconsistsinanoverlappinggenerationeconomywithagentswholivefortwoperiods:childhoodandadulthood.Duringchildhood,anagentreceiveseducationfromhisparentanddoesnotconsume.Whenhebecomesadult,hehastochoosehisconsumptionlevel,thenumberofhischildrenandtheireducation.Forsimplicity,familiesaremonoparental.Parentsexhibitaltruismfortheirchildreninthesensethattheyvaluetheirfuturehumancapital.Theparentalutilityfunctionisdenoted:,ξN(1).,.,.isstrictlyincreasingandconcaveinitsargumentsdenotesthenumberofchildrenborninthefamilyanddthefractionofchildrenwhosurvivetoageve.Iassumethatparentsvaluethenumberofsurvivingchildrenandnotthenumberofchildrenborn.Itimpliesthatchildmortalityisasourceofdisutility.isexogenousinthisSectionbutwillbeendogenizedthereafter.ThereisnouncertaintyaboutthereproductivesuccessofafamilyFinally,representsthehumancapitalinofanadultbornin.FollowingDelaCroix&Doepke[2003],parentsnanceaschoolingtimeandtheaveragehumancapitalofteachersequalstheaveragehumancapitalinthepopulation.Thereisalsoanintrafamilytransmissionofhumancapital:thehumancapitalofparentspositivelyinuencesthefuturehumancapitalofchildren.Becauseparentsdonotdecidetheirownhumancapitallevel,thetransmissionofhumancapitalintothefamilyisanexternality.Moreover,the Idonotconsideraltruismgoingfromchildrentoparentsas,forinstance,inErhlich&Lui[199?]andBelan&Wigniolle[????].NerloveetAl[1984]arguethat,inthiscase,childrenandcapitalmarketsaresubstituebecausechildren’saltruismensurestheirparenttoreceivesupportintheiroldage.Then,apolicyintroducingmoreecientcapitalmarketscanmakefertilitydecreasing.Alternativerepresentationsofutilitycouldhavebeenchosenlike:,ξ,N,ξNAsitwillbecomeclearinthefollowingsections,thesealternativespecicationswouldleadtoexactlythesameresults.So,unlikethemodelsofSah[1991]andKalemli-Ozcan[2003]whichassumeuncertainty,parentswillnotovershoottheirnumberofchildrentoensurethecomplianceoftheiroptimalfertilityrate.Becausechilddeathisassumedtooccurbeforeageve,parentscanrapidlyensurethereplacementofdeadchildren.AsinBecker[1976],Galor&Al[1999,2002],DelaCroix&Doepke[2003]andKalemliOzcan[2003]thepaperassumesthatparentsdirectlyvaluethefuturehumancapitaloftheirchildren.Theydonotvaluetheirfuturewellbeing.Inotherwords,altruismislimitedtoonegeneration. averagelevelofhumancapitalinthepopulationhasapositiveimpactonthechildren’sfuturehumancapital.Thissecondexternalityrepresentstheinuenceoftheeciencyoftheschoolsystem( istheteachers’productivity)andthepresenceofpeereects.Humancapitalisaccumulatedthroughthefollowingprocess (2)Thefunctionisstrictlyincreasingandconcaveregardingeducationalinvestment.Notethat,followingequationcanbeexpressedasafunctionof suchthat: andThemaximizationofutilityissubjecttothefollowingbudgetconstraint: +¸+ (3)representsthenumberofsurvivingchildrenattheendofperiod.Eachchildborntakesapartrtofitsparent’stimeallocationthatisnormalizedtooneMoreovereachsurvivingchildconsumesanextrapartofthistime.Sothequantitycostofasurvivingchildisgreaterthanthecostofanonsurvivingchild.Thetotalcostofquantityisequalto .Itincludestheineectivecostsengagedfornonsurvivingchildren.Consequentlyitnegativelydependsonthechildsurvivalrate.Thecostofoneunitofeducationisnotaectedbythevariationsinthechildmortalityrate.Indeed,noeducationalinvestmentisengageduntilachildreachesageve.Thetotalcostofeducationisconcaveinoneunitofeducationcanbenetmorethanonechild. denotesthecostofgivingunitsofeducationtochildrenwithandIfeducationisapurepublicgoodinthefamily()=1),providingunitsofeducationtoonechildimpliesthesamecostasprovidingunitstochildren.Ifeducationisapureprivategoodinthefamily(),oneunitofeducationbeneonlyonechild.Thenthetotalcostofeducationequalstheunitariancostofeducationtimesthenumberofsurvivingchildren. Noticethatforallfunction,....,,....representsthepartialderivativeofwithregardtoNotethat .θ>isascalarthatallowstherelativeeducationcoststovary. Thepriceofthenalgoodisnormalizedtoone.Itisproducedinquantity,followingalineartechnology:(4)isaproductivityfactorandisthetotalamountofhumancapitalintheworkforce.Atthelabormarket’sequilibrium, (5)Noticethat,ex-post,attheequilibriumofthelabormarket, Byassumption,thereisnoinequalityofhumancapital.Theworkforceparticipationofaparentconsistsinhisremainingtimeafterchildbearing,andteachersdonotparticipateintheproductionofthenalgood.Furthermore,asthelabormarketiscompetitive,thewageequalstheworkers’marginalproductivity:(6)AparentbornindetermineshisoptimaldemandsbymaximizingwithrespecttoandsubjecttoandThisproblemcanbesolvedbymaximizingtheobjectivefunctionwithrespecttoand +¸ ()¡ (7)Toensureglobalconcavityoftheproblem,itsHessianMatrixisassumedtobenegativesemi-deniteThecompetitiveequilibriumisdescribedbytheset satisfyingequations(6)andthefollowingFirstOrderConditions: (8) θAh(9)Theexistenceofexternalitiesonhumancapitalaccumulationimpliesthatthecompetitiveequilibriumcannotbeoptimal.Thenextsectionsderivethesocialoptimumoftheeconomyandcompareittothecompetitiveequilibrium. Notethat,dependsonthefamily’shumancapital,theaveragehumancapitalandtheeducationalchoicesofparents.Asparentsknowthelevelof whentheydetermine;choosingisequivalenttochoosing 2.2TheSocialOptimumThepresenceofexternalitiesmakesprivatechoicesoneducationinecient.Parentsdonotconsiderthepositiveeectoftheireducationalinvestmentontheoveralleciencyofhumancapitalaccumulation.Consequently,theynaturallytendtounderinvestineducation.Intuitively,theimplementationofasubsidyoneducationshouldbesucienttocorrectthisdistortion.Equilibriumwouldbeensuredbytheexistenceofalumpsumtransfer.However,doingsoimplicitlyassumesthateducationisapurepublicgoodwithinthefamilyandthattheobjectiveofthesocialplanneristhesameastheobjectiveoftherepresentativeagent.ningthesocialplanner’sobjectivefunctionisnotstraightforward.Thecrucialpointliesinhispreferenceforthesizeofpopulations.AlargesetofpapersdealingwithoptimalityinendogenousfertilitymodelsattributeaMillianobjectivetotheSocialPlanner.Inthisrepresentation,thesocialplannertriestomaximizetheutilityoftherepresentativeagentC,X,hatthesteadystate.Doingsoimpliesthatheisinterestedinthewell-beingoftherepresentativeagentwithouttakingintoaccountthesizeofthepopulationenjoyingC,X,handinequitiesmatters.ThisrepresentationofsocialpreferencesisoftenopposedtotheBenthamitesocialwelfarefunctions.WithaBenthamiteutility,theSocialPlannertriestomaximizethetotalutilitydistributedintheeconomy.Inthepresentpaper,IproposeageneralsocialwelfarefunctionwhichincludetheMillianandBenthamitecasesaswellasintermediaryones.Todoso,a"natalistbias"denotedisintroducedintheSocialPlanner’spreferences.TheSocialWelfarefunction,atthesteadystate,isthenC,X,hThisformulationisageneralizationoftheMilliancasewhererepresents SeeGroezenetal.[2003],Wigniolle&Loupias[2004],Zhang[2003],Zhang&Zhang[2007],etc.ThisformulationcanalsobeincludedintheA-EciencyproblemsfromGolosovetal..20NoticethatRawlsianobjectivesarenotincludedinthissimpleformulation.Noticethat,asinGroezenetal.[2003],Wigniolle&Loupias[2004],Zhang[2003],Zhang&Zhang[2007],etc.,Ionlyfocusonthesocialwelfarefunctionatthesteadystate.ThedecentralizationofthedynamicaloptimalpathwouldrequiretodenetheSocialPlannerobjectiveasfollows: ¶1Y!(# the"socialplanner’sNatalistBias".FollowingNerloveetAl[1984],when)=1+themaximandofastationaryBenthamitesocialwelfarefunctionreducestotheWelfaresocialfunctioninMoregenerally,foragiven,ahighervalueofmeansthattheSocialplannerexhibitsahighernatalistbias.Inotherwords,ceterisparibus,hepreferslargergenerations.isassumedtobestrictlyincreasingandconcavein2223Then,thesocialplannermaximizessubjecttothefollowingresourceconstraint TheoptimalsteadystateisdescribedbythesetsatisfyingequationandthefollowingFirstOrderConditions: ³b´³b´+µ ++0³b´³b´¶b0 0=µb Obviously,attheoptimalsteadystate,alltheexistingexternalitiesaretakenintoac-count.Inthiseconomy,externalitiesconcerntheaccumulationofhumancapital.Whenparentsinvestineducation,theyimprovethefuturehumancapitaloftheirchildren,suchthat,inturn,theyimproveboththefutureaveragelevelofhumancapitalinthewholeecon-omyandtheirdynasty’slevelofhumancapital.However,parentsdonottakeintoaccounttheirpositiveimpactonthefutureeciencyoftheaccumulationprocess.Itimpliesthattheytendtounderinvestineducation.Furthermore,thepreferencesofparentsdierfromthepreferencesofthesocialplanner.Parentsarenotconcernedwithpro-natalismorequitybetweengenerations.Consequently,theycouldhavetoofewchildren.Howevertheexternalitiesoneducationincreasetheirfertilityratebecausequalityandquantityaresubstitutes.Thecompetitiveequilibriumcan simplymeansthatdistributing tooneagentislessvaluablethandistributing agents.ensurestheexistenceofthetrade-obetweentheutilitydistributedtotherepresentativeagentandthesizeofthegenerationenjoyingit.Noticethatnootherexternalityonfertilityisintroduced.Doingsowouldenrichthemodelofalternativemechanismsbutwillnotchangeitsmainresults.Pestieau(avoiravecpollutionetespacevital)etautrescommeGaloravecimpactpositifdelapopsurleniveautechnologiquedanslesetatsmalthusiens.Toensureglobalconcavityoftheproblem,itsHessianmatrixisassumedtobenegativesemi-de thenbecharacterizedbyoverorunderfertility.Theimplementationofaneconomicpolicyisrequired.2.3TheOptimalTax-TransferPolicyInordertodecentralizethesocialoptimum,thegovernmenthastoimplementapublicpolicywhichmakesthecompetitivesteadystateconvergetotheoptimalone.AnoptimalpolicymakesthesetidenticaltothesetThefollowingsub-Sectionsdiscusstheoptimaltax-transferpoliciesinthegeneralcase()andinthespeciccasewhereeducationisapurepublicgoodinsidethefamily)=1)Iassumethatthegovernmentcanobservetheagent’sbehaviors,itallowstodecentralizetherstorderoptimum.Thisassumptionisstrongbutfundamentalbecauseitshowsthatthestandardproblemoftrade-obetweenqualityandquantity(externalitiesonhumancapitalandnonlinearityoftheparentalbudgetconstraint)structurallygeneratesinoptimallaissezfaireequilibriaregardingbotheducationandfertilitychoices.Addingalternativeexternalitiesonfertilityorstudyingsecondorderoptimawillinterestinglycontrastthisresult.Tosummarize,educationchoicesarenotoptimalbecausethereisanexternalityoneducationinvestment.Asubsidyoneducationspendinghastobeimplementedtocorrectthisexternality.Fertilitychoicesarenotoptimalfortworeasons.First,thesocialplannerdoesnotexhibitthesamepreferencesforquantityasindividuals.Secondly,whenthecoststructureisnotlinear(),theimplementationoftheeducationsubsidydecreasesthetotalquantitycosts.Ataxorasubsidyonbirthshastobeimplemented.Obviously,suchafamilypolicywillnotberequiredinthespeciccasewherethesocialplannerexhibitsnoFertilityBias(educationisapurepublicgoodinthefamily()=12.3.1OptimalTax-TransferPolicyinthegeneralcaseProposition1Whatevertheintensityofthesocialplanner’snatalistbias,apolicyofed-ucationsubsidiesisoptimalwhenitiscombinedwithafamilypolicythatcanbeeithera Atthecompetitivesteadystate, Noticethat,asinNerloveetal[1984],Ionlyfocusonnoncoercivepoliciesandonlineartaxation.Withoutinqualities,focusingonlineartaxationisnotastrongassumptionbecauseredistributionisnotamatterofconcern. taxorasubsidyonbirths.Thegovernmentbudgetconstrainthastobebalancedbytheimplementationofalumpsumtaxoneachfamily.Proof.TheeconomicpolicydescribedinProposition1leadstothefollowingcompetitivesteadystate: ++0())()¶00 θAh)(1 ∙θe )representsasubsidy(respatax)oneachchildbirth.)denotesasubsidy(respatax)oneducationalinvestment.Whenparentsinvestinoneunitofeducation,theyonlypayapartofthisinvestment.isthelumpsumtransfermakingthegovernmentbudgetconstraintbalanced.Equationrepresentsthegovernmentbudgetconstraint;equationsarejusttheexpressionofequationsand(9)whentheeconomicpolicyisimplemented.Observingsystemsandanypolicymakingthesub-systemsandidentical,decentralizesthesocialoptimum.Indeed,implythatissatised.Itfollowsthat +i³b´[b+(b02+b03 ³b´bb01 =0() () 00() ()01µ1b +¸³b+b(b02+b03)i¶=() 0+b b01µ1()b canbeexpressedas: ³b´(b02+b03) ³b´bb01 Noticethat impliesthatisalwayspositive.Theoptimaleducationpolicyisalwaysasubsidy.andbeingdierentfromzero,andarealsodierentfromzero:atax-transferpolicyonbirthsandalumpsumtaxareeectivelyrequiredtoreachtheoptimalsteadystate. Aneducationsubsidyhastobeimplementedbecausethehumancapitalaccumulationprocessisaectedbyexternalities.Parentsdonotinternalizeallthereturnsoftheirinvest-mentineducation.Thentheirlaissezfaireinvestmentinchildren’shumancapitalisslowerthantheoptimalone.Theoptimalscalpolicyonbirthshastwodeterminants.Therstoneisthesocialplanner’snatalistbias.Ifthesocialplannerexhibitsastrongpreferenceforlargepopulations,thelaissezfairefertilityratetendtobesmallerthanattheoptimum.Theseconddeterminantoftheoptimalpolicyonbirthsistheoptimaleducationpolicyitself.Thenonlinearityoftheparentalbudgetconstraintimpliesthatareductionintheeducationcostsdecreasesthetotalnetcostofasurvivingchild.Thenparentstendtohavemorechildrenatthelaissezfaireequilibriumthanattheoptimum.OnemainissueofthatpaperistodeterminetheconditionswherebirthshavetobetaxedProposition2Forlowintensitiesofthesocialplanner’snatalistbiassuchthattotaxbirthsisanoptimalfamilypolicy.Proof.Aftersomecalculuson,thefollowingconditioncanbeobtained: C,X,h Ahe h+ i¶e+ impliesthat Thevalueofisdeterminedbythemodel’skeyvariables.Whentheelasticityofutilitytoconsumption(C,X,h)ishigh,parentsconsumealargepartoftheirincomeandhavefewchildren.Therefore,allotherthingsbeingequal,thecompetitivefertilityrateislowandthetaxlevelhasnottobeveryhighandcouldevenbecomeasubsidy.Whentheprivatereturnsofinvestmentinhumancapitalarehigh(lowvaluesofrelativetoitssocialreturns( ),thetaxwillbelow.Indeed,thisimpliesthat denotetheelasticityofwithregardto.So beingtwiceerentiable. thedistortionsoneducationalchoicesarelow,sotheeducationalsubsidyislow.Becausethetaxonbirthscorrectsthedistortionprovokedbythesubsidyoneducation,itslevelwillbelowtoo.Corollary3WhentheSocialWelfarefunctionisMillian(),theoptimaltaxtransferpolicyforfertilityisnecessarilyataxonbirths.Proof.isalwayssatis Indeed,whenthesocialplannerhasthesamepreferencesasparents,initially,atthecompetitivesteadystate,fertilitybehaviorsareoptimal.However,whenthesocialplannerimplementssubsidiesoneducationalinvestment,thecostofquantityalsodecreases.Thenoverfertilityappearsandataxonbirthshastobeimplemented.Thisresultiscrucialformodelsoftrade-obetweenqualityandquantity.Itimpliesthatimplementinggenerouseducationpoliciescouldrequirerestrictivefamilypolicieswheneducationisnotapurepublicgoodinthefamily.Thefollowingsub-sectionexplorestheprecedingoptimalscalschemeinthespeciccasewhereeducationisapurepublicgoodinsidethefamily.2.3.2OptimalPolicywheneducationisapurepublicgoodinthefamilyInthiscase,thecostofprovidingunitsofeducationtoonechildisthesameasthecostofprovidingunitsofeducationtoaninnitenumberofchildren.Itimpliesthattheprecedingresultsaremodied.Proposition4Wheneducationisapurepublicgoodinthefamily,taxingbirthsisnevernecessarytodecentralizetheoptimalsteadystate.Furthermore,ifthesocialplannerdoesnotexhibitanatalistbias,notaxortransferonfertilityisrequiredtoreachtheoptimalsteadystate. Proof.thescalschemedecentralizingtheoptimalsteadystateistheexpres-sionofsystemwith)=1and +i[b+(b02+b03 bb01 =0() () =() 0+b b01µ1b BytheproofofProposition3,.Educationhastobesubsidized.ItisstraightforwardthatWhenitfollowsfromthatand Thefundamentalresultsofthemodelhavenotreallychanged.Equationisstillsat-ed.However,educationpoliciesnolongerdistortfertilitybehaviors.Indeed,aseducationisapurepublicgoodintothefamily,totalcostsofeducationarenotinuencedbythenum-berofchildrenenjoyingtheeducationalinvestment.Soonlythedistancebetweenthesocialplanner’spreferencesandthehousehold’spreferencescanmakethefertilitybehaviornonoptimal.Withoutthisbias,competitivefertilitychoicesareoptimalandnotaxortransferonfertilityisrequired.Asarstmajorresult,inastandardmodeloftrade-obetweenqualityandquantity,atax-transferpolicyonfertilityisalwaysrequiredtoreachtheoptimalsteadystateifeducationisnotapurepublicgood.Inotherwords,withouttheimplementationofataxorasubsidyonbirths,aneducationpolicyisnotcompletelyeective.InthefollowingSection,themodelisextendedtoincludeprivatehealthexpenditure.Despitethemodicationofthenatureofthetrade-obetweenqualityandquantity,theneedtotaxbirthswillnotbecanceledbytheintroductionofhealthexpenditure.3OptimalTax-Transferpolicywithhealthexpendi-tureThechildsurvivalprobabilityisnowendogenous.Parentscanengageinhealthexpenditureinordertoreducetheirchildren’smortalityrate.InlinewithShakraborty[2004],thechild survivalprobabilityisnow: Parentalexpenditureonhealthhasastrictlypositiveandconcaveinuenceonthechil-dren’ssurvivalprobability,so ) 02( ) Thisexpenditurerepresentsthehealthcareprovidedbyparentstochildren.Parentalhealthcarecoversalargesetofexpendituresuchashygiene,sanitationimprovementsandecientnutrition. denotestheaveragehealthexpenditureintheeconomy.InlinewithDasgupta[1993], ) and ) Theintroductionofanexternalityonhealthexpenditureimpliesthattheparentalchoiceswillnotbeecientatthecompetitiveequilibrium.Intuitively,onecanexpectthatthecompetitivelevelofhealthexpenditurewillbeinferiortoitsoptimallevel.However,theexistenceofeducationalineciencycouldalterthisresultbecause,aspreviouslyshown,itdecreasesthetotalcostofquantity.3.1TheCompetitiveEquilibriumParentsnowhavetodeterminehealthexpenditurefortheirchildren.Inotherwords,theychoose.Theadditionofanexternalityonhealthspendingimpliesthatprivatehealthinvestmentwillnotbeoptimal.Thenthegovernmentintroducesasubsidyhealthexpenditureincomplementtothepreviousscalsystem.Thegovernmentbudgetconstraintisnow ( Whenthescalschemeisimplemented,thefamilialbudgetconstraintis:+(1 ( +(1 Nowthenalgoodcaneitherbeconsumedorinvestedinhealth.Parentshavetomaxi-mizetheobjectivefunctionwithregardtoandandwithrespectAshealthexpendituredoesnotentertheobjectivefunction,parentsdeterminetheir Tosimplifytheresults,)=1Educationisapureprivategood. optimalhealthexpenditurebyminimizing ( Itfollowsthat,atthecompetitivesteadystate: ξ(s∗,s∗)]2X∗wh∗(28)ParentsequalizethemarginalreturnandthemarginalcostofhealthexpenditureThemarginalbenetofhealthexpenditure ξ(s∗,s∗)]2X∗wh∗´consistsinthereductionofthetotalcostofquantity.Inotherwords,equation(28)determinestheoptimalparentalspendingonhealthtohavesurvivingchildren.Thecompetitivesteadystateisnowdescribedbytheset satisfyingequations(6)andthefollowingrstorderconditionswithregardto φ+(1 ()0 =(1Equationsshowthatthetaxationofbirthsincreasesthemarginalcostofquantityandincreasesthemarginalbenetsofhealthexpenditure.3.2TheSocialOptimumForsimplicity,isassumed.ThesocialplannermaximizesaMillianSocialWelfarefunctionC,X,h.Heholdsanewmaximizationinstrumentandhefacesanewresourceconstraint: ( Atthesteadystate Thesocialplannerdeterminestheoptimalhealthexpenditurebyminimizing s,sXAhwithregardtoDoingso,heequalizesthemarginalsocialcostofhealthspending(equaltoone)toitsmarginalsocialreturn.Obviously,themarginalsocialbenetofhealthspendingishigherthanthemarginalprivatebenet(calculatedinequationFormally,theoptimaldecisionrulefor [(bb AsmentionnedintheBenchmarkmodel,ahigherchildsurvivalratedecreasesthecostofquantity. Theoptimalequilibriumnowresultsfromthemaximizationofthefollowingobjectivefunctionwithregardtoand s,X,hThentheSocialOptimumisdescribedbythesetsatisfyingtheequationandthefollowingconditions: 0=³+h+b(b)´ (b)b0 0=µb 3.3TheOptimalTax-TransferPolicyAnoptimalpolicyhastomakeidenticalsystems(34)Inconsequence,theoptimalscalschemeis: ()++b+b[b02+b03]i bbb01 = 01µb (b)++b¡+ ¢¸1¶( ) ( )+( ) (1[ ()++(+ 01¸=( ) Ahε )(1[ ()++(+ OptimalvaluesofarethesameasinthepreviousSection(giventhattheoptimalvaluesofandhavechanged).ItimpliesthatProposition1stillapplies.Inotherwords,whatevertheintensityofthesocialplanner’snatalistbias,apolicyofeducationandhealthsubsidiesisoptimalwhenitiscombinedwithatax-transferpolicyonbirths.Here,becausethesocialplannerexhibitsnonatalistbias,theoptimalfamilypolicyalwaysconsistsinataxonbirths.Thegovernmentbudgetconstraintstillhastobebalancedbytheimplementationofalumpsumtaxoneachfamily.Proposition5Whentheexternalityonhealthexpenditureisstrongsuchthat ) theoptimalhealthpolicyconsistsinasubsidy. Proof.Itisstraightforwardtoshowthatparentalhealthexpenditureisnotoptimalatthecompetitivesteadystate.Atthecompetitivesteadystate(withouttaxation),imply σAhN.Attheoptimalsteadystate,imply )+( ) σAhN.ItfollowsthatHoweverdoesnotensurethathealthexpenditureshouldalwaysbesubsidized.indicatesthattheoptimalvalueofhealthsubsidiesis: ()+() Ã1b Then,ispositiveifthefollowingconditionholds: ()b Whentheexternalityon isstrongsuchthat ,healthexpenditurehastobesubsidizedbecauseparentstendtolargelyunderinvestinhealth.Conversely,externalitiesoneducationarestrongsuchthat ( ) ,healthexpenditurehastobetaxed.Thisresultcomesfromthenonlinearityofthecostsstructure.Indeed,theexistenceofanexternalityonhealthexpenditureimpliesthatparentsdonotinternalizeallthereturnsontheirinvestmentinchildren’shealth.Thecomparisonofwithandindicatesthathealthexpenditureatthecompetitivesteadystateislowerthanattheoptimalsteadystate.However,wheneducationissubsidized,ataxonbirthshastobeimplemented.Doingso,thecostofquantityisincreasedrelativetothecostofhealth,soparentstendtoincreasetheirhealthexpenditure.Thetaxonbirthsplaystheroleofanindirectsubsidyonhealth.Finally,thesignofisdeterminedbythedierencebetweentheintensityoftheexternalityonhealthexpenditureandthesizeoftheindirectsubsidy.Iftheexternalityonhealthisrelativelystrong( ),theindirectsubsidywillnotbesucienttoreachwillbepositive.Conversely,iftheexternalityonhealthisrelativelyweak( theindirectsubsidyexceedsthehealthsubsidythatisreallyneeded.Sowillbenegative:healthexpenditurewillbetaxed. positivelydependsonwhich,inturn,positivelydependson Tosummarize,thepresentpaperprovidestworesults.First,wheneveritisoptimaltosubsidizeeducationhealth,itisoptimaltoimplementatax-transferpolicyonbirths.Thispolicyalwaysconsistsinataxonbirthswhenthesocialplannerhasnonatalistbias.Second,whenthesocialreturnsonhealthexpenditurearenotsucientlyhigh,theoptimalfamilyplanningprogramoftheeconomyconsistsinthepromotionofeducationnancedbythetaxationofhealthandbirthsandalumpsumtax.Conversely,whenthesocialreturnsonhealthexpenditurearehigh,theoptimalfamilyplanningprogramoftheeconomyconsistsinthepromotionofeducationandhealthnancedbythetaxationofbirthsandalumpsumtax.Thisoptimalpolicyhas,infact,twomainobjectives.Therstoneistomodifytheparentaltrade-obetweenqualityandquantity.Moreprecisely,thegovernmenthastoinciteparentstotransferapartoftheirspendingonfertilitytowardeducationinvestment.Thesecondobjectiveistomodifytheparentaltrade-obetweenfertilityandhealth.Inordertoreachthesamenumberofsurvivingchildren,parentsareincitedtomakelesschildreninbetterhealth.4SomeEmpiricalIssuesAtStakeInthissection,Idiscussthemaintheoreticalconclusionofthemodelinthelightofsomeempiricalevidence.Thisdiscussionshowsthatthemodel’stheoreticalconclusionscouldenrichthesetoffamilypoliciesthatareimplementedincountriesfacingoverpopulationproblems.Obviously,thestrongsimplicityofthemodeldoesnotallowtoreproducetheverycomplexdemographicpuzzlesofthesecountries.Itimposestolimitthediscussionongeneralstatements.Countrieswhichfaceover-populationproblemsimplementactivepoliciestoslackentheirpopulationgrowthrate.Twoexamplesareparticularlyilluminating:ChinaandSub-SaharanAfrica.Althoughthesetworegionsbothfaceoverpopulationproblems,theirfamilypolicieshavebeennoticeablydierent.Inthelightofthetheoreticalscalschemeproposedinthispaper,thisSectionprovidesabriefreectionontheimprovementsthatcouldbemadetothecurrentpoliciesimplementedinthesecountries. FollowingDasgupta[1994],thesocialreturnsonhealthexpenditurearehigh.Then,thepromotionofeducationandhealthnancedbythetaxationofbirthsisamorerealisticconclusion. ArecentreportoftheWorldBank[2007]underlinesthat31ofthe35countrieswiththehighestfertilityratescomefromSub-SaharanAfrica.Forthemajorityofthem,fertilityrateshavenotchangedoverthelastdecadesandareallgreaterthansixchildrenperwoman.However,thevastmajorityofthesecountrieshaveimplementedfamilyplanningprogramsincollaborationwithinternationalorganizationssuchastheWorldBank.TheWorldBank’sreport[2007]emphasizesthatthemainfactorinthehighfertilityratesisthepersistenthighlevelofthedesirednumberofchildren.Inotherwords,thetoohighfertilityratesinSub-SaharanAfricadonotcomefromthelackoffamilyplanningprogramsavailable.Itarguesthateortshavetobemadetoreducethedesiredfertility.Todoso,itrecommendsimprovingeducationandhealthprogramsatthelocallevel.However,educationindicatorshaveallbeenincreasingsincethesixties.Morerecently,thenetprimaryschoolenrolmentrateincreasedfrom50to70percentbetween1990and2006.Inthesameperiod,theyouthandadultliteracyratesincreased.Thisnoticeableimprovementineducationrateshasnotbeensucienttoreducefertilityrates.Thepresentpaperdoesnotrecommendincreasingtheamountspentonthefamilyplan-ningprograms.Itproposescomplementingfamilyplanningprogramswithtaxesonbirthshelpingtonanceeducationandhealth.Withouttaxingbirths,theseprogramsreducethenetcostofthechildren’squantity,implyingthepersistenceofahighdesirednumberofchildren.Obviously,theSub-SaharanAfricanpopulationpuzzlecannotbereducedtoasimplemodeloffertility.Morecomplexproblemsofpoliticalinstability,starvationandHIVpan-demythatarewellbeyondthescopeofthispaper,haveadirectandsignicanteectonfertilityandeducationbehavior.Thepossibilityofimplementingtaxesonbirthsinapopu-lationthatislargelyengagedinaninformaleconomyisparticularlyquestionable.However,theincreaseofquantitycostshastobecontemplatedasaninstrumentoffuturefamilyplanningprograms.Chinaalsoimplementsafamilypolicytoreduceitspopulationgrowthrate.However,itsstrategydiersfromthestrategyoffamilyplanningprogramsinSub-SaharanAfrica.Since1980,ChinahasimplementedaOne-Childpolicywhichstronglyconstrainsfamilies’ InSub-SaharanAfrica,theyouthliteracyratewas64%in1990and73%in2006.Theadultliteracyratewas54%in1990and61%in2006.SeeAppendix1foramorecompletedescription. fertility.Itconsistsinasystemwhichprovidesgeneroussubsidiesfortherstbirthandimposesveryhightaxesonthesubsequentbirths.Ifparentsdecidetohaveasecondchildwithoutbeingallowedtodoso,theylosealargepartoftheirretirementpension,theirchildcareallowanceandothersocialadvantages.Furthermore,somephysicalsanctionshavebeenimplementedinruralareas.Thisscalschemeisrelativelydierentfromtheoneproposedinthispaper.TheChinesepolicydoesnottaxallthebirthsatthesamerate.Therstbirthissubsidizedwhereasthesubsequentbirthsareheavilytaxed.Thehighlevelofthetaxonsubsequentbirthsisaveryecientincentivetohaveonlyonechild.Thenthelargemajorityoffamiliesaresubsidizedtoreachthetargetofonechildperfamily.ItimpliesthattheChineseOne-ChildPolicyisaverycostlyfamilypolicy.Itcannotnanceeducationandhealthpolicies.So,nothingensuresthattherelativecostsofeducationandhealthreachtheiroptimalvalue.Indeed,alargeliteraturestressestheinsuciencyofpublicexpenditureonhealthandeducationinChineseruralareaswherethelargemajorityofthepopulationisconcentrated(forexample,seeKanbur&Zhang[2003]andFan&Zhang[2000]).TheresultsofthispaperindicatethatsomemarginalchangesintheOne-ChildpolicycouldimprovetheoveralleciencyoftheChinesefamilyplanningpolicy.Itproposestaxingallbirthssuchthatthefamilypolicydoesnotimplyeectivecosts.TheamountsavedbytheChinesegovernmentcouldbeinvestedinmoreambitiouseducationandhealthpoliciesreducingthelargeinequalitiesexistingbetweenurbanandruralareas.Theoretically,thissystemwouldnotincreasetheoverallcostoftheChinesefamilyplanningprogramandwouldleadtothesamefertilityrates.However,itwouldincreasehealthandeducationinvestment.Furthermore,theChinesefamilypolicyiscoercivewhiletheeconomicpolicyproposedinthispaperisnoncoercive.Then,ifthetwopoliciesareequallyecient,thenoncoecivepolicyisalwayswelfareimproving.5ConclusionThepresentpaperanalysesoptimalfamilypoliciesinthestandardmodeloftrade-obetweenqualityandquantity.Giventhenonlinearityoftheparentalbudgetconstraint,tosubsidizeeducationandhealthwillbeoptimalifatax(orasubsidy)onbirthsisalsoimplemented.Indeed,asubsidyoneducationreducesboththecostofeducationalinvestmentandthe totalcostoffertility.ThisresultappliesforalargesetofsocialwelfarefunctionsliketheMillianandtheBenthamiteones.Obviously,themodelconcludesthattaxingbirthswithoutnancingeducationandhealthisnotoptimaleither.Finally,thescalschemeproposedinthismodelisquitesimple:educationandhealthexpenditurearepromotedbythetaxationofbirthsandlumpsumtransfers.Theimple-mentationofthisschemecouldimprovetheoveralleciencyofthecurrentfamilypoliciesimplementedinChinaandSub-SaharanAfrica.Themainobjectiveofthepresentinvestiga-tionwastoexplorethefamilypolicyrecommendationsofthestandardendogenousfertilitymodel.Asanaturalextensionofthiswork,futureresearchshouldintegratecountries’speci-citiestomakequantitativepropositionsofeconomicpolicyandtoproposeamoreprecisediscussionofempiricalevidences.References[1]BalestrinoA.,CignoA.&PettiniA.[2000]"EndogenousFertilityandtheDesignofFamilyTaxation"InternationalTaxandPublicFinance,Springer,vol.9(2),p175-193[2]BeckerG.S&BarroR.J.,[1988]."AReformulationoftheEconomicTheoryofFertility,"TheQuarterlyJournalofEconomics,MITPress,vol.103(1),p1-25,February[3]BeckerG.S.&LewisH.G.[1973]"OntheInteractionBetweentheQuantityandQual-ityofChildren"TheJournalofPoliticalEconomy,vol.81Part2NewEconomicApproachtoFertility,p279-288[4]BeckerG.S.&TomesN.[1976]"ChildEndowmentsandtheQuantityandQualityofChildren"TheJournalofPoliticalEconomy,vol.84Part2,EssaysinLaborEconomicsinHonorofH.G.Lewis,p143-162[5]Belanwigniolle[6]BouldingK.[1964]."TheMeaningoftheTwentiethCentury".London:GeorgeAllenandUnwinLtd. 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