Estate taxes, consumption externalities, and altruism Journal of Public Economics

14 Pages·2008·0.29 MB·English

JournalofPublicEconomics92(2008)1751–1764
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Journal of Public Economics
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☆
Estate taxes, consumption externalities, and altruism
Jaime Alonso-Carreraa, Jordi Caballéb,⁎, Xavier Raurichc
aDepartamentodeFundamentosdelAnálisisEconómicoandRGEA,UniversidadedeVigo,Fac,CC,EconómicaseEmpresariais,CampusAsLagoas-Marcosende,36310Vigo,Spain
bUnitatdeFonamentsdel'AnàlisiEconomicaandCODE,UniversitatAutònomadeBarcelona,EdificiB,08193Barcelona,Spain
cDepartamentdeTeoriaEconòmicaandCREB,UniversitatdeBarcelona,FacultatCC,EconòmiquesiEmpresarials,Av,Diagonal690,08034Barcelona,Spain
a r t i c l e i n f o a b s t r a c t
Articlehistory: We study how the introduction of consumption externalities affects the optimality of the
Received9November2005 dynamicequilibriuminaneconomydisplayingdynasticaltruism.Whenthebequestmotiveis
Receivedinrevisedform10January2008 inoperativeconsumptionexternalitiesaffecttheintertemporalmarginbetweenyoungandold
Accepted1February2008
consumptionandthusmodifytheintertemporalpathofaggregateconsumptionandcapital.
Availableonline20February2008
Theoptimaltaxpolicythatsolvesthisintertemporalsuboptimalityconsistsofataxoncapital
income and a pay-as-you-go social security system. The latter solves the excess of capital
JELclassification:
accumulation due tothe inoperativeness of the bequest motive and the former solves the
E21
suboptimalallocationofconsumptionduetoconsumptionexternalities.Whenthebequest
E13
motiveisoperativeconsumptionexternalitiesonlycausean intratemporalmisallocation of
E62
consumption but do not affect the optimality of the capital stock level. This suboptimal
Keywords: allocation of consumption implies in turn that the path of bequest deviates also from
Consumptionexternalities optimality.Theoptimaltaxpolicyinthiscaseconsistsofanestatetaxandacapitalincometax.
Bequests
©2008ElsevierB.V.Allrightsreserved.
Optimaltaxrates
1.Introduction
In this paper we aim at analyzing the suboptimal allocation arising from consumption externalities in an overlapping
generation model (OLG) with dynastic altruism. We use this model to show how the interaction between consumption
externalitiesandaltruismaffectstheoptimalityoftheequilibriumpath.Inourmodeltheconsumptionexternalitywilltakethe
formofareferencelevelofconsumptionthatisusedtocomparetheutilityderivedfromownconsumption.Wewillassumethat
thisreferenceconsumptionisaweightedaverageoftheconsumptionofalltheagentslivinginthesameperiod.
Severalauthorshaveanalyzedtheimplicationsofaconsumptionreferencearisingfromownpastconsumption(orinternal
habits)inOLGmodels.ExamplesofthisstrandoftheliteraturearethepapersofLahiriandPuhakka(1998)andWendner(2002),
whostudytheeffectofhabitsonsavinginapureexchangeeconomy;andAlonso-Carreraetal.(2007),delaCroix(1996)anddela
Croix and Michel (1999, 2001), who analyze several related stability issues. Obviously, when the reference is the others'
consumption,thensuboptimalityoftheequilibriumpathislikelytoarise.Inaframeworkwithinfinitely-livedagents,Alonso-
Carreraetal.(2004and2005),FisherandHof(2000),Guo(2005),LiuandTurnovsky(2005),LjungqvistandUhlig(2000),and
TurnovskyandMonteiro(2007),amongmanyothers,havecharacterizedtheoptimaltaxratesthatsolvethiskindofsuboptimality.
IntheframeworkofOLGmodels,Abel(2005)showsthatacapitalincometaxandapay-as-you-gosocialsecuritysystemconstitute
theoptimaltaxpolicy.Weextendthelatterpaperbyintroducingconsumptionexternalitiesinthemodelwithdynasticaltruismof
☆ ThispaperhasbenefitedfromthevaluablecommentsoftworefereesandthecoeditorofthisjournalandparticipantsintheSimposiodeAnalisisEconomico
(Oviedo).FinancialsupportfromtheSpanishMinistryofEducationandScienceandFEDERthroughgrantsSEJ2005-03753,SEJ2006-03879,andSEJ2006-05441;
theGeneralitatofCataloniathroughtheBarcelonaEconomicsprogram(XREA)andgrantsSGR2005-00447andSGR2005-00984;andtheXuntadeGaliciathrough
grantPGIDIT06PXIC300011PNaregratefullyacknowledged.
⁎ Correspondingauthor.UniversitatAutònomadeBarcelona,Departamentd'Economiaid'HistòriaEconòmica,EdificiB,08193Bellaterra(Barcelona),Spain.Tel.:
[phone];fax:[phone].
E-mailaddresses:[email protected](J.Alonso-Carrera),[email protected](J.Caballé),[email protected](X.Raurich).
0047-2727/$–seefrontmatter©2008ElsevierB.V.Allrightsreserved.
doi:10.1016/j.jpubeco.2008.02.004
1752 J.Alonso-Carreraetal./JournalofPublicEconomics92(2008)1751–1764
Barro(1974).FollowingAbel'sanalysis,weassumethattheseexternalitiestaketheformofaweightedaverageoftheconsumption
ofthetwotypesofagentslivinginthesameperiod.ByintroducingaltruismandthepossibilityofbequestsinAbel'smodelwecan
analyze new phenomena, like the potential suboptimality of the level of bequests and the interaction between consumption
externalitiesandthelackofoptimalityofthebequestconstrainedequilibrium.
TheanalysisofoptimalityinAbel(2005)isbasedonthecomparisonbetweenthemodelofDiamond(1965)withconsumption
externalitiesandthecorrespondingplannerproblemwhentheobjectivefunctionoftheplannerisjustthediscountedsumofthe
utilities of all the agents in the economy. This analysis does not allow to distinguish the deviation from optimality due to
consumptionspilloversfromtheoneduetotheinoperativenessofthebequestmotive.Incontrast,theanalysisinourpaperis
based on the comparison between the model of Barro (1974) with consumption externalities and the corresponding planner
problem.AsshownbyAbel(2005),whenthebequestmotiveisinoperative,suboptimalityisduetobothconsumptionexternalities
and the typical capital overaccumulationproblem (Cass,1972). However, when the bequest motive is operative consumption
externalitiesaretheonlysourceofsuboptimality.Therefore,ouranalysisshowshowthelackofoptimalityduetoconsumption
externalitiesdependsontheoperativenessofthebequestmotive.
When the bequest motive is operative consumption externalities only cause an intratemporal misallocation of resources,
whereastheycauseanintertemporalmisallocationwhenthebequestmotiveisinoperative.Intheformercase,theyonlyaffectthe
intratemporalmarginbetweenconsumptionofthetwogenerationslivinginthesameperiod.Thisimpliesthattheymodifythe
allocationofconsumptionbetweenthetwolivinggenerations,butdonotaffecttheoptimalityoftheintertemporalpathsofsaving
andoutput.Infact,thissuboptimalallocationofconsumptionisassociatedwithasuboptimallevelofbequest.Incontrast,when
thebequestmotiveisinoperative,consumptionexternalitiesaffecttheintertemporalmarginofconsumptionalongthelifecycleof
agents.Giventhatconsumptionspilloversaffecttheintertemporalpathofaggregateconsumption,theyalsomodifythepathsof
savingandproduction.Theexternalityassociatedwithyoungconsumptionreducesthestockofcapitaland,hence,reducesthe
overaccumulation of capital due to the inoperativeness of the bequest motive. Obviously, the externality associated with old
consumptionhastheoppositeeffectonthestockofcapitalandthusincreasesthegapbetweentheoptimalstockofcapitalandthe
oneobtainedinacompetitiveequilibrium.
Notethatthenatureofsuboptimalitycruciallydependsontheoperativenessofthebequestmotiveandthisoperativeness
dependsinturnontheintensityofconsumptionexternalities.Inthisrespectweshowthatinthelongrunastrongeryoung
consumptionreferencereducesthecriticallevelofaltruismabovewhichthebequestmotiveisoperative,whereasalargerold
consumptionreferencerisesthiscriticallevelofaltruism.
Theoptimalvaluesoftaxratesalsodependontheoperativenessofthebequestmotive.Whenthebequestmotiveisinoperative,the
optimaltaxpolicyconsistsofataxoncapitalincomeandapay-as-you-gosocialsecuritysystem.Thelattersolvestheexcessofcapital
accumulationandtheformersolvesthesuboptimalallocationofconsumptionduetoconsumptionexternalities.Notethattheoptimal
capital income tax rate modifies the intertemporal path of aggregate consumption and, thus, solves the lack of optimality due to
consumptionspillovers.Whenthebequestmotiveisoperative,theoptimaltaxpolicyconsistsofanestatetaxandacapitalincometax.
Moreover,inthiscasethepay-as-you-gosocialsecuritysystemisnotrequiredasthereisnooveraccumulationofcapital.
Thepaperisorganizedasfollows.Section2presentsthemodel.Section3characterizesthecompetitiveequilibriumwhenthe
bequestmotiveisoperativeandwhenitisnot.Section4characterizesthesocialplannersolution.InSection5weconductthe
analysisofoptimalitybycomparingthesolutionachievedbytheplannerwiththecompetitivesolution.Section6characterizesthe
optimaltaxrates.Section7concludesthepaper.AlltheproofsappearintheAppendixA.
2.Themodel
LetusconsideranOLGmodelwhereN identicalindividualsareborninperiodt.Theseindividualslivefortwoperiods.Each
t
individualhasanoffspringattheendofthefirstperiodofhislifeandthenumberofchildrenperparentisn≥1.AsinDiamond
(1965),eachagentsuppliesinelasticallyoneunitoflaborinthefirstperiodofhislifeandisretiredinthelastperiodofhislife.We
indexeachgenerationbytheperiodinwhichitsmemberswork.
Individualsareassumedtobealtruistictowardstheirchildren.Letb betheamountofbequestthatanoldindividualleavesto
t
eachoftheirchildreninperiodt.Weimposetheconstraintthatparentscannotforcetheirdescendentstogivethemgifts,
btz0: ð1Þ
Eachyoungindividualdistributeshislaborincomeandhisinheritancebetweenconsumptionandsaving.Therefore,thebudget
constraintfacedbyanindividualduringhisfirstperiodoflifeis
wtþbt¼ctþst; ð2Þ
wherec istheamountofconsumptionofayoungagent,w isthelaborincomeands istheamountsaved.Inthesecondperiodof
t t t
lifeindividualsreceiveareturnontheamountoftheirsaving,whichisdistributedbetweenconsumptionandbequestsfortheir
children.Therefore,thebudgetconstraintofanoldindividualis
Rtþ1st¼xtþ1þnbtþ1; ð3Þ
whereR isthegrossrateofreturnonsavingandx istheamountofconsumptionofanoldindividual.
t+1 t+1
J.Alonso-Carreraetal./JournalofPublicEconomics92(2008)1751–1764 1753
Theutilityfunctionofanindividualbelongingtogenerationtis
Vt¼Uðcˆt;xˆtþ1ÞþbVtþ1; for t¼0;1;2; N ð4Þ
where V represents the indirect utility of each of his descendants, the parameter β∈[0,1) is the altruism factor,1 and the
t+1
variablescˆ andxˆ representtheeffectiveconsumptioninthefirstandsecondperiodsoflife,respectively,ofarepresentative
t t+1
individualbelongingtogenerationt.Weassumethatindividualsdonotderiveutilityfromtheabsolutelevelofconsumptionbut
from the comparison between their consumption and some consumption reference. In particular, we assume the following
functionalformsforeffectiveconsumption:
cˆt¼ct(cid:2)gvyt; ð5Þ
and
xˆtþ1¼xtþ1(cid:2)dvotþ1; ð6Þ
whereγ∈[0,1)andδ∈[0,1)provideameasureoftheintensityoftheconsumptionreference.2Theseconsumptionreferencesare
assumedtobeaweightedarithmeticaverageofthepercapitaconsumptionofthetwolivinggenerations.Ontheonehand,we
assumethat
(cid:1) (cid:3) (cid:1) (cid:3)
vyt ¼NtNcttþþhhyyNNtt(cid:2)(cid:2)11xt¼ nþnhy ctþ nþhyhy xt; ð7Þ
whereθy∈[0,1]istheweightofconsumptionofarepresentativeoldconsumerinthespecificationofthereferenceforyoung
consumers.Ontheotherhand,weassumethat
(cid:1) (cid:3) (cid:1) (cid:3)
votþ1¼hoNthþo1Ncttþþ11þþNNttxtþ1¼ hohnoþn1 ctþ1þ hon1þ1 xtþ1; ð8Þ
whereθo∈[0,1]istheweightofconsumptionofarepresentativeyoungconsumerinthespecificationofthereferenceforold
consumers.Notethattherestrictionsimposedonthevaluesoftheparametersθyandθoimplythatwearegivingalargerweightto
theaverageconsumptionoftheagentsbelongingtothesamegeneration.Letusdefineey¼ n andeo¼ hon :Then,Eqs.(7)and
nþhy honþ1
(8)canberewrittenasfollows:
vyt ¼eyctþð1(cid:2)eyÞxt; ð9Þ
and
votþ1¼eoctþ1þð1(cid:2)eoÞxtþ1: ð10Þ
AsinAbel(1986)orLaitner(1988),weassumethatthefunctionU(∙,∙)istwicecontinuouslydifferentiableandadditiveinitstwo
arguments.Therefore,wewillusethefollowingfunctionalform:
Uðcˆt;xˆtþ1Þ¼uðcˆtÞþquðxˆtþ1Þ; ð11Þ
whereρN0isthetemporaldiscountfactor.Weassumethatu′(z)N0,u″(z)b0forzN0andtheInadaconditionslimz→0u′(z)=∞and
limz→∞u′(z)=0.
Inperiod0thereisanoldgenerationofN−1individualsborninperiod−1.Whentheyareold,theindividualsofgeneration−1
havepreferencesrepresentedbytheutilityfunction:
quðxˆ ÞþbV : ð12Þ
0 0
Thereisasinglecommodityinthiseconomy,whichcanbedevotedtoeitherconsumptionorinvestment.Letusassumethat
ˆ
thiscommodityisproducedbymeansofaneoclassicalproductionfunctionF(K,L),whereK isthecapitalstockandL isthe
t t t t
amountoflaborusedinperiodt.Capitaldepreciateseveryperiodattherateν∈[0,1].Theproductionfunctionpercapitaisˆf(k),
t
wherektisthecapitalstockpercapita.Weassumethatˆf′(k)N0,ˆf″(k)b0forallkN0andtheInadaconditionslimk→0ˆf′(k)=∞and
limk→∞ˆf′(k)=0.Toeasethenotation,weintroducethefunctionf(kt)=ˆf(kt)+(1−ν)ktand,hence,limk→0f′(k)=∞andlimk→∞f′(k)=1−ν.
Asfirmsbehavecompetitively,therentalpricesofthetwoinputsequaltheirmarginalproductivities
Rt¼fVðktÞuRðktÞ; ð13Þ
wt¼fðktÞ(cid:2)fVðktÞkt¼fˆðktÞ(cid:2)fˆVðktÞktuwðktÞ: ð14Þ
1 Eachparentcaresequallyaboutthefelicityoftheirnchildren.Thus,theintercohortutilitydiscountβcouldberewrittenasβ=nρβ′,whereρwouldbethe
temporaldiscountfactorandβ′isthepureinterpersonal(fromparentstochildren)discountfactor.
2 WeassumeanadditivespecificationforeffectiveconsumptioninsteadofthemultiplicativeformulationofAbel(2005)inordertoguaranteeconcavityofthe
socialplanner'sutilityfunction(seeAlonso-Carreraetal.,2005).
1754 J.Alonso-Carreraetal./JournalofPublicEconomics92(2008)1751–1764
Inequilibriumthecapitalstockinstalledinperiodt+1isequaltotheaggregatesavinginperiodtand,thus,wehave
nktþ1¼st; for t¼0;1;2; N : ð15Þ
3.Competitiveequilibrium
Ontheonehand,theproblemfacedbyeachindividualbelongingtogenerationt,t=0,1,2,…istomaximize(4)withrespectto
{c, x , b } subject to (1), (2), (3), (5) and (6), which is equivalent to solving the following dynamic programming
t t+1 t+1
problem:
(cid:4) (cid:5) (cid:4) (cid:5)
VtðbtÞ¼fsmt;batþx1gfu wtþbt(cid:2)st(cid:2)gvyt þqu Rtþ1st(cid:2)nbtþ1(cid:2)dvotþ1 þbVtþ1ðbtþ1Þg; ð16Þ
with b ≥0, for vy, vo , w and R given for all t. On the other hand, the problem faced by the individuals belonging to
t+1 t t+1 t t+1
generation−1whentheyareold(i.e.,inperiod0)isequivalenttothefollowing:
(cid:6) (cid:4) (cid:5) (cid:7)
max qu R0k0(cid:2)nb0(cid:2)dvo0 þbV0ðb0Þ ; ð17Þ
fb0g
withb ≥0,fork ,vo,andR given.Notethatk istheendowmentofcapitalofanoldindividualinperiod0.
0 0 0 0 0
UsingtheenvelopetheoreminEq.(16)weobtain,
AV
Abt¼uVðcˆtÞ; for t¼0;1;2; N : ð18Þ
t
Using(18),weobtainthefirstorderconditionsforproblem(16)correspondingtothederivativewithrespecttos,
t
(cid:4) (cid:5) (cid:4) (cid:5)
uVct(cid:2)gvyt ¼qRtþ1uVxtþ1(cid:2)dvotþ1 ; for t¼0;1;2; N : ð19Þ
Moreover,thefirstorderconditionforproblems(17)and(16)withrespecttobequestsb is
t
(cid:4) (cid:5) (cid:4) (cid:5)
nquVxt(cid:2)dvot zbuVct(cid:2)gvyt ; for t¼0;1;2; N ; ð20Þ
wheretheconditionholdswithequalityifb N0.Eq.(19)characterizestheoptimalallocationofconsumptionalongthelifetimeof
t
anindividual.Ifthebequestmotiveisoperative,thenEq.(20)characterizestheoptimalallocationofconsumptionbetweenthe
twoconsecutivegenerationst−1andt.Thisequationtellsusthat,whenthebequestmotiveisoperative(b N0),theutilitylossof
t
parentsarisingfromalargeramountofbequestmustbeequaltothediscountedutilitygainoftheirdirectdescendants.
Thecompetitiveequilibriumofthiseconomyisapath{k ,c,x,b}∞ that,foragiveninitialvalueofk ,solvesthesystemof
t+1 t t t t=0 0
difference equations composed of (19) and (20), together with (1), (2), (3), (9), (10), (13), (14), (15) and the transversality
condition
(cid:4) (cid:5)
tlYimlbtuVct bt¼0: ð21Þ
Theprevioustransversalityconditionstatesthatthepresentvalueofthelongrunamountofbequeststendstozero.
Wewillrestrictouranalysistosteadystateequilibria,thatis,competitiveequilibriawherethevariablesk,c,x andb areall
t t t t
constant.3Tothisend,wecombine(19)with(9)and(10)toobtain
uV½ð1(cid:2)geyÞc(cid:2)gð1(cid:2)eyÞx(cid:3)(cid:2)qRuV½ð1(cid:2)dð1(cid:2)eoÞÞx(cid:2)deoc(cid:3)¼0;
whichafterusing(2),(3),(13),(14)and(15)becomes
(cid:8) (cid:9) (cid:8) (cid:9)
hðk;bÞuuVcˆðk;bÞ (cid:2)qRðkÞuVxˆðk;bÞ ¼0 ð22Þ
with
cˆðk;bÞ¼ð1(cid:2)geyÞðwðkÞþb(cid:2)nkÞ(cid:2)gð1(cid:2)eyÞnðfVðkÞk(cid:2)bÞ
and
xˆðk;bÞ¼ð1(cid:2)dð1(cid:2)eoÞÞnðfVðkÞk(cid:2)bÞ(cid:2)deoðwðkÞþb(cid:2)nkÞ:
Thefunctionh(k,b)isonlydefinedforthevalueskN0andb≥0forwhichcˆ(k,b)N0andˆx(k,b)N0.
Thefollowingtwolemmasprovidetwousefulpropertiesofthefunctionh(k,b).Theproofsofthetwolemmasfollowdirectly
fromadirectcomputationofthecorrespondingpartialderivativesofh(k,b)andthecompetitiverentalprices(13)and(14).The
nextlemmareferstothebehaviorofthefunctionhwhenthestationaryamountofbequestsvaries:
3 Wewillsuppressthetimesubindexwhenwerefertothesteadystateequilibriumvalueofavariable.
J.Alonso-Carreraetal./JournalofPublicEconomics92(2008)1751–1764 1755
Lemma3.1. h uAhb0forallbN0:
b Ab
Thenextlemmaprovidesasufficientconditionunderwhichthefunctionh(k,b)isincreasinginthecapitalstockpercapitafor
allb≥0:
Lemma3.2. Assumethatthefollowingconditionholds:
jfWðkÞjbminfn=k;fVðkÞ=kg: ð23Þ
Then,h uAhN0forallb≥0.
k Ak
Notethatcondition(23)canbewrittenas
fWðkÞkþnN0; ð24Þ
and
fWðkÞkþfVðkÞN0: ð25Þ
Clearly,theinequality(24)impliesthatthestationaryfirstperiodconsumptioncisdecreasinginthestationarycapitalstock,
whereastheinequality(25)impliesthatthestationarysecondperiodconsumptionxisincreasinginthestationarycapitalstockat
asteadystateequilibrium.
AsinAbel(1987),wearegoingtointroducetheassumptionthatthesteadystateequilibriumvaluek¯ofcapitalinaneconomy
withnobequestsisuniqueandcompatiblewithpositiveeffectiveconsumptionsandthath(k,0)≷0wheneverk≷k¯.Moreover,we
willmakethegenericassumptionthattheslopeofh(k,0)atk¯,h (k¯,0),isdifferentfromzero.Obviously,Lemma3.2providesa
k
sufficientconditionunderwhichthisassumptionholds.Moreover,noticethattheseassumptionsareimmediatelysatisfiedwhenu
islogarithmic,FˆisCobb–Douglasandthevaluesoftheparametersδandγaresufficientlysmall.
AssumptionA. Thereexistsauniquestrictlypositivevaluek¯ satisfyingh(k,0)=0withˆc(k¯,0)N0andˆx(k¯,0)N0.Moreover,h
k
(k¯,0)N0.
Letuscombineconditions(19),(20)whenitisjustbinding,and(13),allofthemevaluatedatthesteadystatewithnobequests,
toobtainthethresholdvalueofthealtruismfactorβabovewhichbequestsarepositive,
bP¼ nP : ð26Þ
fVðkÞ
Notethatthisthresholdvalueβ¯ coincideswiththeoneobtainedinWeil(1987).Aswewillsee,consumptionexternalitieswill
modifythevalueofβ¯ throughtheireffectonthevaluek¯ ofthestationarycapitalstock.
Proposition3.3. a)IfAssumptionAholdsandβ≤β¯ thentheuniquesteadystateequilibriumsatisfiesthefollowingequations:
P
b¼0;
(cid:10) (cid:11)
P
h k;0 ¼0;
(cid:10) (cid:11) (cid:10) (cid:11)
Pc¼f Pk (cid:2)PkfV Pk (cid:2)nPk;
and
(cid:10) (cid:11)
Px¼nfV Pk Pk;
wherek¯,b¯,c¯andx¯arethesteadystatevaluesofcapital,bequestsandconsumptionwhenyoungandwhenold,respectively.
b)IfβNβ¯ thenthesteadystateisunique,exhibitsastrictlypositiveamountofbequestsandsatisfiesthefollowingequations:
(cid:10) (cid:11)
fV k⁎ ¼n; ð27Þ
b
(cid:10) (cid:11)
h k⁎;b⁎ ¼0;
(cid:10) (cid:11) (cid:10) (cid:11)
c⁎¼f k⁎ (cid:2)k⁎fV k⁎ (cid:2)nk⁎þb⁎;
and
(cid:10) (cid:11)
x⁎¼nfV k⁎ k⁎(cid:2)nb⁎;
wherek*,b*,c*andx*arethesteadystatevaluesofcapital,bequestandconsumptionwhenyoungandwhenold,respectively.
1756 J.Alonso-Carreraetal./JournalofPublicEconomics92(2008)1751–1764
ObservethatthecapitalstockwhenthebequestmotiveisoperativeisimplicitlygiveninEq.(27)andisthustheoneassociated
with the modified golden rule. However, the capital stock when the bequest motive is inoperative is smaller than the one
correspondingtothatrule.
Proposition3.4. a)LetAssumptionAandcondition(25)hold.Then,thesteadystateequilibriumwithinoperativebequestmotiveis
locallysaddlepathstableforvaluesofδandγsufficientlyclosetozero.b)Thesteadystateequilibriumwithoperativebequestmotiveis
locallysaddlepathstable.
Whereasthesteadystatewithpositivebequestsisalwayslocallysaddlepathstable,thelocalsaddlepathstabilityofthesteady
statewithzerobequestscanbeobtainedunderAssumptionA,condition(25),whichmeansthatsecondperiodconsumptionis
locallyincreasinginthestationarycapitalstock,andwhentheintensityofconsumptionexternalities,whichissummarizedbythe
valueoftheparametersδandγ,issmall.However,asisshownintheAppendixA,whentheintensityofconsumptionexternalities
issufficientlylarge,thesteadystatewithzerobequestsmaynotbesaddlepathstableundertheseconditions.4Forthesakeof
completeness,weshowinTable1ofAppendixBhowthestabilitypropertiesofasteadystatewithinoperativebequestmotivevary
withthevalueoftheparametersδandγ.
Thenextpropositionprovidessomecomparativestaticsresultswhentheintensityofconsumptionexternalitiesvaries:
Proposition3.5. a)IfβNβ¯,thenAk⁎¼0; Ak⁎¼0; Ab⁎N0; Ab⁎b0.b)IfAssumptionAholdsandβbβ¯,thenAkPN0,andAkPb0.
Ag Ad Ag Ad Ad Ag
Thepreviouspropositiontellsusinitspart(a)that,iftheamountofbequestispositive,thenconsumptionexternalitiesmodify
theintergenerationaldistributionofconsumptionbutdonotaffectthelongrunvalueofcapital.Inotherwords,consumption
externalitiesneithermodifytheamountofsavingnortheoutputlevel,buttheymodifytheallocationofconsumptionbetween
youngand oldgenerations.Thischange intheallocationofconsumption isachieved byadjustingtheamountofbequest.An
increaseinthevalueoftheparameterγraisesthemarginalvaluationofyoungconsumptionand,asfollowsfrom(20),thisresults
inautilitygainfromalargeramountofinheritances.Incontrast,asδincreasesagentsarewillingtoincreaseoldconsumptionand
thisrequiresareductionintheamountofbequest.
An increase in γ induces agents to increase young consumption, whereas an increase in δ induces agents to increase old
consumption.Accordingly,part(b)ofProposition3.5tellsusthat,whentheequilibriumamountofbequestsiszero,agentsrise
young(old)consumptionbydecreasing(increasing)theamountofsaving.Thisexplainstheeffectofδandγonthecapitalstock
sincesavingscoincidewiththestockofcapitalinequilibrium.
Wenextshowhowtheoperativenessofthebequestmotiveisaffectedbyconsumptionspillovers:
Proposition3.6. AbPN0andAbPb0.
Ad Ag
Accordingtothepreviousproposition,ifyoungindividualsbecomemoresensitivetotheothers'consumption(γincreases),
positive bequests are more likely to appear in a stationaryequilibrium. The converse occurs when the effect of consumption
spillovers on old consumers becomesstronger (δincreases). Clearly, theseresults showthat the introductionof consumption
externalitiesaffectstheoperativenessofthebequestmotiveand,thus,affectsthedynamicoptimalityoftheequilibriumpath
(Cass,1972).
4.Thesocialplannersolution
Weassumethatthesocialplannergivesthesameweighttoalltheindividualsbelongingtothesamegenerationinhisobjective
function and the intergenerational discount factor coincides with private altruistic factor β. Therefore, the social planner
maximizes5
Xl
Uˆt¼b(cid:2)1½quðð1(cid:2)dð1(cid:2)eoÞÞx0(cid:2)deoc0Þ(cid:3)þ bt½uðð1(cid:2)geyÞct(cid:2)gð1(cid:2)eyÞxtÞþquðð1(cid:2)dð1(cid:2)eoÞÞxtþ1(cid:2)deoctþ1Þ(cid:3); ð28Þ
t¼0
subjecttotheresourceconstraint
x
fðktÞ¼ctþntþnktþ1; for t¼0;1;2; N ð29Þ
withk given.NotethatthefirsttermintheexpressionforUˆ inEq.(28)referstotheoldgenerationinperiod0.Thus,theLagrangeanof
0 t
theplanner'smaximizationproblemcanbewrittenas
Xl
o¼b(cid:2)1½quðð1(cid:2)dð1(cid:2)eoÞÞx0(cid:2)deoc0Þ(cid:3)þ bt½uðð1(cid:2)geyÞct(cid:2)gð1(cid:2)eyÞxtÞþquðð1(cid:2)dð1(cid:2)eoÞÞxtþ1(cid:2)deoctþ1Þ(cid:3)
Xl (cid:10) x (cid:11) t¼0
þ kt fðktÞ(cid:2)ct(cid:2)nt(cid:2)nktþ1 :
t¼0
4 InrelatedOLGmodelswhereindividualpreferencesarenotsubjecttoconsumptionexternalities,delaCroixandMichel(2001)andAlonso-Carreraetal.
(2007)showthatthesteadystatewithinoperativebequestissaddlepathstablewhenhkN0.However,thisconditionisnotsufficienttoguaranteesaddlepath
stabilitywhenconsumptionexternalitiesarepresent.
5 Notethatthisobjectivefunctioncoincideswiththeplanner'sobjectivefunctioninAbel(2005).
J.Alonso-Carreraetal./JournalofPublicEconomics92(2008)1751–1764 1757
Thecorrespondingfirstorderconditionswithrespecttoc,x,andk are
t t t+1
(cid:10) (cid:11) (cid:10) (cid:11)
ð1(cid:2)geyÞbtu0 cˆpt (cid:2)deobt(cid:2)1qu0 xpt (cid:2)kt¼0; ð30Þ
(cid:10) (cid:11) (cid:10) (cid:11)
k
(cid:2)gð1(cid:2)eyÞbtu0 cˆp þ½1(cid:2)dð1(cid:2)eoÞ(cid:3)bt(cid:2)1qu0 xˆp (cid:2) t¼0; ð31Þ
t t n
and
(cid:10) (cid:11)
ktþ1f0 kptþ1 (cid:2)nkt¼0; ð32Þ
for t=0,1, 2,…, where the superscript p denotes an optimal pathfromthe planner's viewpoint. Combining(30) and (31), we
obtain
0 (cid:10) (cid:11)1
uVxˆp (cid:10)q(cid:11) b
@ (cid:10) t(cid:11)A ¼ ; ð33Þ
uVcˆpt I n
where
1(cid:2)gðð1þnÞey(cid:2)nÞ
Iu (cid:4) (cid:4) (cid:5) (cid:5):
1(cid:2)d 1(cid:2) 1þ1 eo
n
Using (30), (31) and (32), we get
0 (cid:10) (cid:11)1
@uV(cid:10)cˆptþ1(cid:11) A(cid:1)b(cid:3)¼ (cid:10)1 (cid:11): ð34Þ
uVcˆp n fV kp
t tþ1
n o
l
Thesocialplannersolutionisapath kp ;cp;xp that,foragiveninitialvalueofkp,solvesthesystemofdifferenceEqs.(29),
tþ1 t t t¼0 0
(33)and(34),togetherwiththetransversalitycondition
h (cid:10) (cid:11) (cid:10) (cid:11) i
tlYiml btuVcˆpt ð1(cid:2)geyÞ(cid:2)bt(cid:2)1quVxˆpt deo ktþ1¼0:
Proposition4.1. Thereexistsauniquesteadystateinthesocialplannerproblem.Thissteadystateisgivenby
n
fVðkpÞ¼ ; ð35Þ
b
(cid:10) (cid:11)
uVxˆp Ib
(cid:10) (cid:11)¼ ; ð36Þ
uVcˆp nq
and
xp
fðkpÞ¼cpþ þnkp; ð37Þ
n
wherekp,cpandxparethesteadystatevaluesofcapital,consumptionwhenyoungandwhenold,respectively.
Thefollowingpropositioncharacterizesthedynamicsaroundthesteadystateoftheplanner'ssolutionanditsproofisomitted
sinceissimilartothatofpart(b)ofProposition3.4:
Proposition4.2. Thesteadystateofthesocialplannersolutionislocallysaddlepathstable.
5.Planneroptimality
Therearetwodifferentsourcesofdeviationsfromtheplanner'soptimalsolutioninoureconomy:thecontemporaneousconsumption
externalitiesandtheinoperativenessofthebequestmotive.Inwhatfollows,wewillanalyzetheeffectsofthesedifferentsourcesof
suboptimality,whichinteractinanon-obviousway.Tothisend,wefirstcomparethesocialplannersolutionwiththecompetitive
equilibriumsolutionwhenbequestsarepositive,andthenweperformthesamecomparisonwhenbequestsarezero.
1758 J.Alonso-Carreraetal./JournalofPublicEconomics92(2008)1751–1764
5.1.Suboptimalityoftheequilibriumpathwhenthebequestmotiveisoperative
Whenthebequestmotiveisoperative,consumptionexternalitiesaretheonlysourceofdeviationfromplanneroptimality.Note
thatthedemographicstructureofourOLGmodelmakesthiscasedifferentfromthemodelwithaninfinitely-livedrepresentative
agent,whichweconsideredinAlonso-Carreraetal.(2004).Whentheindividualsofthesamefamilyareeffectivelylinkedthrough
positive bequests, the path of aggregate consumption in a given period is optimal in spite of contemporaneous externalities.
However,theexistenceofcontemporaneousconsumptionexternalitieswhenthereareagentswithdifferentageslivinginthe
sameperiodmakessuboptimaltheintratemporalallocationofconsumptionbetweenthetwogenerations.6
Wecancomparethesocialplannersolution,whichischaracterizedbyEqs.(29),(33)and(34),withthecompetitiveequilibrium
solutionwhenthebequestmotiveisoperative,whichischaracterizedbyEqs.(19),(20)and(29).Wefirstrewritetheequilibrium
solutionbyrearranging(20)asfollows:
(cid:1) (cid:3)
uVðxˆtÞ q¼b; ð38Þ
uVðcˆtÞ n
wherethelefthandside(LHS)istheprivatemarginalrateofsubstitution(MRS)betweenyoungandoldconsumptionatperiodt
andtherighthandside(RHS)istheprivatemarginalrateoftransformation(MRT).Thus,Eq.(38)determinestheintratemporal
allocationofconsumptionbetweenagentsbelongingtodifferentgenerationsalongtheequilibriumpath.Combining(19)with
(20),weobtain
(cid:1) (cid:3)(cid:1) (cid:3)
uVðcˆtþ1Þ b ¼ 1 ; ð39Þ
uVðcˆtÞ n fVðktþ1Þ
wheretheLHSistheprivateMRSbetweenyoungandoldconsumptionofindividualsbelongingtogenerationtandtheRHSisthe
corresponding private MRT. It then follows that Eq. (39) determines the intertemporal allocation of consumption along the
equilibriumpath.
Wefollowasimilarreasoningwiththesocialplannersolution.Ontheonehand,(33)determinestheoptimalintratemporal
allocationofconsumptionbetweenagentsbelongingtodifferentgenerations.Ontheotherhand,(34)determinestheoptimal
intertemporalallocationofconsumption.ThisequationistheKeynes–Ramseyequationthat,givenaninitialconditiononthestock
ofcapital,characterizestheoptimalpathofsavingandthustheoptimalpathofcapitalstockandoutput.
From the comparison of (33) and (34) with (38) and (39) we see that consumption externalities affect the intratemporal
allocationofconsumptionbetweengenerations,astheprivateandsocialMRSdonotcoincide,butdonotaffecttheintertemporal
allocationofconsumption.Therefore,consumptionexternalitiesdonotaffecttheoptimalityofthecapitalpathwhenthebequest
motiveisoperative.Giventhatproductionandsavingsareatitsoptimallevel,thissuboptimalallocationofconsumptionbetween
youngandoldagentsisassociatedwithasuboptimalpathofbequests.Thesizeofthedifferencebetweentheprivateandsocial
MRSisgivenbythevalueofthevariableI,whichprovidesameasureoftheinefficiencyduetoconsumptionexternalities.Thus,the
amountofbequestissuboptimallysmall(large)whentheexternalitiesmakeagentsvalueold(young)consumptioninexcessto
young(old)consumption.Infact,itcanbeshownthatbequestsaresuboptimallysmall(large)wheneverIN(b)1.Notealsothatthe
twoconsumptionexternalitiesresultinoppositeeffectsand,infact,whenI=1thecompetitiveequilibriumisefficienteventhough
consumptionexternalitiesarepresent.
5.2.Suboptimalityoftheequilibriumpathwhenthebequestmotiveisinoperative
Whentheamountofbequestiszero,theequilibriumischaracterizedbyEqs.(3),(19)and(29).Weproceedtocomparethese
equationswiththosecharacterizingthesocialplannersolution.First,wecombine(33)with(34)toobtain
0 (cid:10) (cid:11)1
@uV(cid:10)xˆptþ1(cid:11) A(cid:10)q(cid:11)¼ (cid:10)1 (cid:11); ð40Þ
uVcˆp I fV kp
t tþ1
wheretheLHSisthesocialMRSbetweenconsumptionwhenyoungandwhenoldandtheRHSisthecorrespondingsocialMRT.We
cancomparethisequationwiththeequilibriumEq.(19),whichcanberewrittenas
(cid:4) (cid:5)!
uVxˆ(cid:4)tþ(cid:5)1 q¼ 1 ;
uVcˆt fVðktþ1Þ
6 InAlonso-Carreraetal.(2004)thereisnoheterogeneityamongagentssothattheoptimalityoftheaggregateconsumptionpathimpliesanoptimal
intratemporalallocationofconsumption.Wealsoproveinthatpaperthatcontemporaneousconsumptionexternalitiesgiverisetosuboptimalconsumption
pathsonlyiftheyinteractwiththeconsumptionofpreviousperiods.
J.Alonso-Carreraetal./JournalofPublicEconomics92(2008)1751–1764 1759
wheretheLHSistheprivateMRSandtheRHSisthecorrespondingprivateMRT.Thisequationdrivestheintertemporalmargin
between consumptionwhenyoung and when old along an equilibriumpath with inoperativebequestmotive. Notethat this
equilibriummargindiffersfromtheoptimalmargininboththeMRSandintheMRT.Thedifferencebetweenthesocialandthe
privateMRSisduetoconsumptionexternalitiesandthedifferencebetweenthesocialandtheprivateMRTarisesbecausethe
competitivestockofcapitaldiffersfromitsplannedcounterpartwhenthebequestmotiveisnotoperative.Toseethis,notethat
(34) does not hold along an equilibrium path when bequests are zero. Moreover, as follows from (3) and (19), consumption
spilloversmodifytheequilibriumstockofcapitaland,hence,theprivateMRTwhenbequestsarezero.Thus,whenthebequest
motiveisnotoperative,consumptionexternalitiesaffectboththeMRSandtheMRTandthusgiverisetobothasuboptimallevelof
productionandasuboptimalallocationofconsumption.
Thelongrunvalueofthestockofcapitalisnotoptimalinthiscase.Theoptimallongrunvalueofthestockofcapitalkpis
obtainedfrom(34),whereasthelongrunvaluek¯ofthestockofcapitalinthecompetitiveequilibriumcomesfromtheequationh
(k¯,0)=0.Asfollowsfromthecondition(26)fortheoperativenessofthebequestmotive,k¯Nkp.Thus,theinoperativenessofthe
bequestmotiveimpliesasuboptimallylargecapitalstock.Thestockofcapitalinasteadystateequilibriumcanbeexpressedasa
functionofγandδ,k¯(γ,δ)whereastheoptimalstockofcapitalcanbewrittenasafunctionofthealtruismfactor,kp(β).Theoverall
inefficiencycould be measured bythe difference k¯(γ, δ)−kp(β). This inefficiencycan be divided into two components: (i) the
inefficiency due to the inoperativeness of the bequest motive; and (ii) the inefficiency due to consumption externalities. The
formerinefficiencyisgivenbyk¯(0,0)−kp(β).Sincethestrengthofaltruismrisestheoptimalstockofcapital(see(35))butdoesnot
affecttheequilibriumstockofcapitalwhenthebequestmotiveisinoperative,thisinefficiencydecreasesasβincreases.Onthe
otherhand,theinefficiencyduetoconsumptionexternalitiesisgivenbyk¯(γ,δ)−k¯(0,0).FromProposition3.5,wecanconclude
thatthesizeofthisinefficiencydecreaseswithγandincreaseswithδ.
6.Optimaltaxes
Inthissectionweproceedtocharacterizetheoptimaltaxratestoimplementanoptimalallocationwhenconsumptionexternalities
interactwithaltruism.Weconsiderataxpolicyconsistingofanestatetax,acapitalincometaxandasystemoflump-sumtaxes.
Concerningthelump-sumtaxes,weassumethatyoungagentspayalump-sumtaxτyandtherevenuesaredevotedtofinancealump-
t
sumsubsidytotheold−τo.Therefore,thesetwotaxratesarerelatedbythefollowingbalancedbudgetconstraint:
t
so¼(cid:2)nsy: ð41Þ
t t
Wealsoassumethatyoungagentspayanestatetaxontheinheritancetheyreceiveandthattherevenuesaccruingfromthesetaxesare
returnedtothesameyoungagentsbymeansofalump-sumsubsidy.Thus,thereisasecondgovernmentbudgetconstraint,which
is
/yt ¼sbtbt; ð42Þ
where/yisalump-sumsubsidyandτbistheestatetaxrate.Finally,weassumethatoldagentspayacapitalincometaxonthereturnsof
t t
savingsandthattheserevenuesarereturnedtotheseagentsbymeansofalump-sumsubsidy.Therefore,anothergovernmentbudget
constraintis
/ot ¼sktstfVðktþ1Þ; ð43Þ
whereϕoisalump-sumsubsidyandτkisthecapitalincometaxrate.
t t
Consideranindividualbelongingtogenerationt=0,1,2,…whomaximizes(4)withrespectto{c,x ,b }subjectto(1)and
t t+1 t+1
thefollowingconstraints:
(cid:10) (cid:11)
ct¼wt(cid:2)syt þ/yt þ 1(cid:2)sbt bt(cid:2)st; ð44Þ
(cid:10) (cid:11)
xtþ1¼ 1(cid:2)sktþ1 Rtþ1stþ/otþ1(cid:2)nbtþ1(cid:2)sotþ1; ð45Þ
whichamountstosolvethefollowingdynamicprogrammingproblem:
(cid:10) (cid:10) (cid:11) (cid:11) (cid:10)(cid:10) (cid:11) (cid:11)
VtðbtÞ¼fsmt;batþx1gfu wt(cid:2)syt þ/yt þ 1(cid:2)sbt bt(cid:2)st(cid:2)gvyt þqu 1(cid:2)sktþ1 Rtþ1stþ/otþ1(cid:2)nbtþ1(cid:2)sotþ1(cid:2)dvotþ1 þbVtþ1ðbtþ1Þg;
withb ≥0,forvy,vo ,w andR givenfort=0,1,2,….
t+1 t t+1 t t+1
Theproblemfacedbytheindividualsbelongingtogeneration−1inperiod0isthefollowing:
n (cid:10)(cid:10) (cid:11) o
mfba0gx qu 1(cid:2)sk0 R0k0þ/o0(cid:2)nb0(cid:2)so0(cid:2)dvo0ÞþbV0ðb0Þ ;
withb ≥0,fork ,vo,andR given.
0 0 0 0
1760 J.Alonso-Carreraetal./JournalofPublicEconomics92(2008)1751–1764
Thefirstorderconditionsoftheprevioustwoproblemswithrespecttos andb are
t t
(cid:4) (cid:5) (cid:10) (cid:11) (cid:4) (cid:5)
uVct(cid:2)gvyt ¼q 1(cid:2)sktþ1 Rtþ1uVxtþ1(cid:2)dvotþ1 ð46Þ
and
(cid:4) (cid:5) (cid:10) (cid:11) (cid:4) (cid:5)
nquVxt(cid:2)dvot z 1(cid:2)sbtþ1 buVct(cid:2)gvyt ; ð47Þ
fort=0,1,2,…,wherethelastconditionholdswithequalityifb N0.
t
Thecompetitiveequilibriumofthiseconomywithtaxesisapath{k,c,x,b,ϕy,ϕo}∞ that,foragiveninitialvalueofk ,solves
t t t t t t t=0 0
the system of difference equations composed of (46) and (47), together with (13), (14), (15), (41), (42), (43), (44), (45) and the
transversalitycondition(21)foragivenpathof{τtyn,τot,τtb,τtk}t∞=0. o
Letusdenotethepathofoptimaltaxratesas ∽sy;∽so;∽sb;∽sk l .Theseoptimaltaxratesaresuchthatmaketheequilibrium
t t t t
t¼0
pathof{k,c,x}coincidewiththeoptimalpath{kp,cp,xp},characterizedby(29),(33)and(34).Byusing(3),weobtainthatthe
t t t t t t
optimalpathofbequestimpliedbytheplanners'solutionisgivenby
(cid:4) (cid:5) xp
bp¼fVkp kp(cid:2) t : ð48Þ
t t t n
Apositiveoptimalamountofbequestimpliesthatthebequestmotiveisoperativealonganequilibriumpaththatattainsthefirst
bestsolutionandanegativeoptimalamountofbequestimpliesthatthebequestmotiveisinoperativealongthisequilibriumpath.
Tocharacterizetheoptimaltaxrates,wemustdistinguishbetweenthesetwocases.
Whentheoptimalamountofbequestispositive,theequilibriumpathassociatedtotheoptimaltaxratesischaracterizedby
(29),(45)andthefollowingtwoequations,whichareobtainedfromcombining(46)and(47):
0 1
(cid:1) (cid:3) (cid:1) (cid:3)
Rðktþ1Þ uuVðcVðˆctˆþtÞ1Þ ¼ nb @(cid:10)1(cid:2)∽sb (cid:11)1(cid:10)1(cid:2)∽sk (cid:11)A; ð49Þ
tþ1 tþ1
and
(cid:1) (cid:3)(cid:1) (cid:3)
nq uVðxˆtÞ ¼1(cid:2)∽sb: ð50Þ
b uVðcˆtÞ t
FromcomparingthepairofEqs.(49)and(50)with(33)and(34),weobtainthefollowingresult:
Proposition6.1. Whentheoptimalamountofbequestispositive,theoptimaltaxratesareτ~b=1−Iandτ~k=1−1.
t t I
Iftheoptimalamountofbequestispositiveandconsumptionexternalitiesaretheonlysourceofsuboptimality,thenlump-sumtaxes
areirrelevantduetoRicardianequivalence.Inthiscase,becausethespecificationoftheconsumptionexternalityallowsadifferentlevelof
distortioninyouthandoldage,anappropriatetaxonbequests(estatetax)couldoutweightheeffectofexternalities(see(50)).However,
theestatetaxaffectsthedegreeofcapitalaccumulationaccordingtoEq.(49).Therefore,iftheoptimaltaxrateonbequestsispositive,we
shouldsubsidizecapitalaccumulationsoasnottocreateadistortionwherenoneoriginallyexisted.Conversely,ifasubsidyonbequests
counterbalancestheeffectofconsumptionexternalities,weshouldintroducepositivetaxationoncapitalincometobringtheamountof
capitaltoitsoptimallevel.Inthisrespectnotethatthesignofτ~kistheoppositeofthatofτ~b.
t t
Whentheoptimalamountofbequestisnegative,theequilibriumpathassociatedwiththeoptimaltaxratesischaracterizedby
b =0andEqs.(29),(45)and(46).Weproceedtocharacterizetheoptimaltaxrates.Fromthecomparisonbetween(40)and(46),it
fotllowsthattheoptimalcapitalincometaxrateis∽sk ¼1(cid:2)1.Thistaxsolvesthesuboptimalityduetoconsumptionexternalities,
t I
whereasthesuboptimalityduetotheinoperativenessofthebequestmotiveisthussolvedthroughlump-sumtaxes.Toobtainthe
latteroptimaltaxes,weuse(13),(15),(43)and(45)toobtain
xt¼nfVðktÞkt(cid:2)sot: ð51Þ
Fromthecomparisonbetween(48)and(51),weobtainthatτ~o=nbpand,using(41),wegetτ~y=−bp,wherebpisdefinedinEq.(48).
t t t t t
Giventhatbpb0,weobtainthatτ~ob0andτ~yN0.Thus,theoptimalsystemoflump-sumtaxescorrectstheexcessofcapitalby
t t t
meansofreducingtheincomeoftheyoungagentsandincreasingtheincomeoftheoldagents.Inthissense,theoptimallump-
sumtaxesareapay-as-you-gosocialsecuritysystem.Theseresultsaresummarizedinthefollowingproposition:
Proposition6.2. Whentheoptimalamountofbequestisnegative,theoptimaltaxratesare∽sk¼1(cid:2)1;∽so¼nbpand∽sy¼(cid:2)bp.
t I t t t t
Wehavethusshownthatifagentsarealtruistictheoptimaltaxpolicydependsontheoperativenessofthebequestmotivealongan
optimalpath.Ontheonehand,ifbequestsarezerothen,asinAbel(2005),theoptimaltaxpolicyconsistsofataxoncapitalincomeanda
pay-as-yougosocialsecuritysystem.Thelattersolvestheexcessofcapitalaccumulationduetotheinoperativenessofthebequestmotive
andtheformersolvesthesuboptimalallocationofconsumptionduetoconsumptionexternalities.Notethattheoptimalcapitalincome
taxratewillbepositiveornegative,dependingontherelativeintensityofthetwoexternalities.Thus,thisoptimaltaxisconstructedina
waythatstimulatesordeterssavingdependingontheoveralleffectofexternalitiesontheallocationofconsumption.