A Theory of Out­of­Wedlock Childbearing: Lewis Team Theory of Out of Wedlock Childbearing:...

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  • ATheoryofOutofWedlockChildbearing:LewisTeamNotes

    1

    PresentedbyMarriumKhanLaurenRussell

    March30,2015

    1Thispictureistakenfromhttp://www.slate.com/articles/news_and_politics/the_best_policy/2008/03/_and_baby_makes_two.html

  • TableofContents

    1. Introduction

    2. SingleMother,UnidentifiedFather

    3. EstablishmentofPaternityandtheGainstoMarriage

    4. OutofWedlockFatherhood

    5. MarriageMarketsandChildbearingwithinMarriage

    6. MarriageMarketsandOutofWedlockChildbearing

    7. DiscussionandConclusions

    8. Extensions

    9. EmpiricalEvidence

    11. Bibliography

    12. MathAppendixA

    13. MathAppendixB

    14. MathAppendixC

  • 1. IntroductionInhispaperATheoryofOutofWedlockChildbearing,RobertWillispresentsatheoreticalframeworktoexplainthesteadygrowthinthepercentageofchildbirthstounmarriedcouplesthatoccurredbetweenthe1960sandtoday.Inhismodel,heconsidersmenandwomenasseparatedecisionmakerswhoarebothseekingtooptimizetheirindividualutility,andthechildisconsideredacollectivegood.Intheir1985papertitledChildrenasCollectiveGoodsandDivorceSettlements,WeissandWillisarguethat,whenparentsliveseparately,theyreachaninefficient,noncooperativeequilibriumwherebothparentsgivetoofewresourcestotheirchildrenbecausetheyareunabletomonitortheothersspendingactivity.Willisthentakesthatmodelandbuildsuponittoexplainwhysomemenmightfindsinglefatherhoodpreferabledespitethisinefficiency.Somemenmightprefertohavechildrenoutofwedlockataloworzerocosttotheminsteadofhavingchildrenwithinmarriagewheretheymustsharethecostwiththeirwife.Thiswilloccurifamanisabletofindasufficientnumberofwomenwillingtohavehischildrenwithlittleornosupportfromhim.Thishappensifwomensincomesarehighabsolutelyandrelativelytotheincomesofmenandifwomenaremorenumerousthanmen.Therefore,therewillbeafractionoflowincomemenwhochoosetofathermanychildrenoutofwedlockbymorethanonewomanfromthelowerendoftheincomedistributionifthewomenwillassumeallfinancialresponsibilityforthechildren.Inthiscase,expendituresonchildrenendupbeinglowerthantheywouldbeiftheparentsweremarried.Aswewillshowlaterinournotes,thishassubstantialimpactonminoritycommunities(mainlyblacksandhispanics)becauseofthedistinctlyhigheroccurrenceofsinglemotherhood.Thispaperhas269citationsandhasbeenusedasabasisforunderstandingtheimpactofoutofwedlockchildbearingonoverallchildwelfare.Variable Description

    cf Mothersconsumption

    n Numberofchildrenmotherbears

    q Qualityperchild

    yf Mothersfullincome

    Marginalcostofchildqualityrelativetoadultconsumption

    kf Mothersnonlaborincome

    wf Motherswage

  • Motherstotalsupplyoftime

    us Utilityofsinglemother

    qs Singlemothersdemandfunctionforchildquality

    If Differencebetweenmothersutilityofhavingonechildandnothavingachild

    cm Fathersconsumption

    ym Fathersfullincome

    u Arbitraryutilityassignedtosinglemother

    vs Utilityofsinglefather

    f Mothersaltruism

    m Fathersaltruism

    T Incometransferbyfathertomother

    T * Optimalincometransferbyfather

    P Numberofpartnersfatherhas

    dij d=1iffatherliveswithithchildofjthhousehold.

    Elasticityof (n)

    Elasticityoftransferswithrespecttopartners

    ElasticityofB(P)withrespecttoP

    P Reservationnumberofpartners

    Sm Supplyofmentooutofwedlockfatherhood

    R Populationratiofemaletomale

    OWF Demographicavailabilityfunction

  • 2. SingleMother,UnidentifiedFatherAwomanconsideringmotherhoodhasautilityfunction

    (c , n, q)u f andabudgetconstraint

    .nqyf = cf +

    Variable Explanation

    cf mothersconsumption

    n numberofchildrenmotherbears

    q qualityperchild

    yf mothersfullincome

    marginalcostofchildqualityrelativetoadultconsumption

    Fromtheabovebudgetconstraint,wenoticethat,givenanincome asthenumberofchildren,,yf n,orthedesiredqualityperchild,q,increase,themothersnonchildrelatedexpenditure, ,cf mustdecrease.Thisisbecausealargerportionofthemothersincomemustnowbeallocatedtochildrearing.Iflaborincomeisapartof andchildrenarerelativelytimeinsensitive,thenmothersincomeyf andmarginalcostofchildservices( )arepositivefunctionsofherwage.Awomansincome functioncanthusbewrittenas:

    ,yf = kf +wf

    Variable Explanation

    kf mothersnonlaborincome

    wf motherswage

    motherstotalsupplyoftime

    where isawomansnonlaborincomesuchasunearnedincomeorincomefromwelfarekf programsforsinglemothers.Since ,thenifawomanworksmorehours,shewillspend(w ) = f lesstimewithherchildandthiswillincreasethemarginalcostofchildservices.

  • Oneoftheassumptionsofthismodelisthatawomancaneitherchoosetoremainchildlessorhaveachild.Usingthisassumption,herutilityfunctionwhenshedecidestoremainchildlesswillbe:

    (c , 0, 0) (y , 0, 0) (y )u f = u f f Whenawomandecidesnottohaveachildthen ,whichmeans implying .n = 0 nq = 0 yf = cf Somothersutilityinachildlessstateisdependentonherincomeassheisspendingitallonprivateconsumption.Ontheotherhand,herutilityfunctionwithonechildis:

    (c , 1, q) (y q (y , ), 1, q) (y , )u f = u f s f f Here representswomansdemandfunctionforchildqualitythatisdependentonher( qs , )yf totalincomeandmarginalcostofchildservices.Givenshehasonechildandallocatesherincomeoptimallybetweenherownconsumptionandchildexpenditures,theMRSbetweenqandwillbeequalto .cf

    RSM f = ucuq =

    ItisimportanttorememberherethatMRSfdenotestherateatwhichthemotheriswillingtodecreaseherownprivateconsumptiontoreceiveonemoreunitofchildqualitywhileremainingatthesamelevelofoverallwelfare.Awomansdecisiontobecomeamothercanbeunderstoodby:

    .(y , ) (y )I f = f f > 0 Iftheutilityshegetsfromhavingachildisgreaterthantheutilityofremainingchildless,shewillchoosetobecomeamother.Here,thefertilitydemandisassumedtobenormalsothereisapositiverelationshipbetweenincomeandchildbearing.Thismeansthatwithhigherincomethequalityofchildwillalsoincrease.Hence,normalityholdswhen .AI /y f f = y y > 0 womansmarginalutilityofconsumptionishigherwhenshehasachildascomparedtoachildlessstate.Therefore,awomanwouldvalueconsumptionmoreasamotherbecauseyouhavetoconsumelessingeneralinordertoprovideforchildcare.Analternativeexplanationofthisisthatthewomansmarginalutilityofprivateconsumptionisdiminishingandindependentofparenthood(becausepositiveexpendituresonthechildimplieshigherconsumptionlevelinachildlessstate).If,however, thenthewomanwilldecidetoremainchildlessunlessherincomeincreasesI f < 0 pastacriticalvalue( ),holding constant.Anincreasein willreduceutilityinastatewith childrenbutitwillhavenoeffectonutilityfromchildlessstate.Soawomanwillchoosetobecomeamotherif andcriticalvalueincreasesas increases .()yf > / ) ( > 0

  • 3. EstablishmentofPaternityandGainstoMarriageTheutilityfunctionformalesisrepresentedbyv=v(cm,n,q).

    Variable Explanation

    cm mansconsumptiononallgoods(excludinganychildrelatedexpenses)

    n numberofchildren

    q qualityperchild

    Thisutilityfunctionassumesthatamansutilityfromchildrendoesnotdependoninteractingwiththemorcohabitingwiththemotherofhischildren.(Willismakesthisassumptioninordertosimplifytheanalysis.)Therefore,amalewithoutanyidentifiedchildrenhastheutilityfunctionv=v(ym,0,0).Here,ymreplacescmbecauseallofhisincomeisspentonconsumptionnotrelatedtochildcare.Thiscanbeseeninthefollowingequationrepresentinghisbudgetconstraint:

    ym=cm+ nq Foramanwithnoidentifiedchildren, isequaltozero.Thus,ym=cm.Therefore,whenanq singlewomandecidestohaveachild(implyingthatherincomeisgreaterthan( )andthe) fatherdoesnotrecognizeorplantofinanciallysupportthechild,sheincursallofthecostsandspendsq=qs(yf, )onthechild.

    Willisthenmovestoconsiderthecasewhereafatherrecognizesthathehasachildandobservesthelevelofchildquality.Itisimportanttonotethatthefathersutilityisincreasedbyestablishingpaternity(i.e.recognizingthathesthefather)ifhe:

    1) hasapositivepreferenceforchildren 2

    2) and,ifthemothersexpenditureonthechildissufficientlyhigh.Thiswouldbethecaseofasinglefatherwhosechildisbeingfinanciallysupportedsolelybyitsmother.Theutilityforthistypeoffatherisdenotedvsandisrepresentedby:

    vs=v(ym,1,qs(yf, ))>v(ym,0,0) Here,itisimportanttonotethattheutilityofasinglefatherwhereonlythemotherfinanciallysupportsthechildisgreaterthantheutilityofamanwithnoidentifiedchildrenbecausethesinglefatherhasapositivepreferenceforchildren.

    2Fathermighthaveaminimumstandardforchildqualitybelowwhichhewouldprefernothavingachild.If,however,mothersstandardofchildqualityisatleastashighasthefathers,thenshecanvoluntarilyhavethechild.Thisconstraintwillnotbebindingforthefather.

  • However,becausethefatherhasapositivepreferenceforchildren,thissituationisnotparetooptimal.Thatis,apairingotherthan(us,vs)canbechosensothatboththemotherandfatherarebetteroff.Thisnewpairingcanbereachedbyincreasingthelevelofchildexpendituretoalevelgreaterthanqs(yf, ),assumingthatbothparentssharetheexpense.In ordertoprovethis,itisfirstimportanttonotethathavingapositivepreferenceforchildrenimpliesthatthefatherwillalwaysvalueincreasesinq.Therefore,MRSm=vq/vc>0,meaningthatthefatheriswillingtodecreasehisownconsumptioninordertoincreaseqforhischild.SinceMRSf= ,itfollowsthat

    MRSf+MRSm> ifthethefatherisidentified,butonlythemotherfinanciallysupportsthechild.ThisinequalityviolatestheSamuelsoncondition forefficiencyintheprovisionofacollectivegood.This3

    conditionimpliesthat ,whereMRTisthemarginalrateoftransformationofRS RTn

    i=1M i = M

    parentsincomeintochildquality. However,ifthemothersutilityismaximizedsubjecttothejointbudgetconstraintof

    bothparties,y=yf+ym+ nq

    andtheconstraint vs,whichensuresthatfatherisnotmadeworseoffbysharingtheexpense,v themothersutilitywillexceed ,whichistheresultoftheutilitymaximizationforasingleu motherwithanoncontributingfatherThatis,whenbothparentsshareinthecostofchildrearing,themotherisbetteroffthanw