Relative consumption of housing: marginal saving subsidies and income taxes as a second-best policy?

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This is the published version of a paper published in Journal of Economic Behavior and Organization.

Citation for the original published paper (version of record):

Aronsson, T., Mannberg, A. (2015)

Relative consumption of housing: marginal saving subsidies and income taxes as a second-best

policy?.

Journal of Economic Behavior and Organization, 116: 439-450

http://dx.doi.org/10.1016/j.jebo.2015.05.011

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Journal

of

Economic

Behavior

&

Organization

jo u r n al ho me p ag e :ww w . e l s e v i e r . c o m / l o c a t e / j e b o

Relative

consumption

of

housing:

Marginal

saving

subsidies

and

income

taxes

as

a

second-best

policy?

夽

Thomas

Aronsson

a,∗

_,

_Andrea

_Mannberg

a,b

a_{DepartmentofEconomics,UmeåSchoolofBusinessandEconomics,UmeåUniversity,SE-90187,Sweden} b_{TromsøUniversityBusinessSchool,9037Tromsø,Norway}

a

r

t

i

c

l

e

i

n

f

o

Articlehistory: Received8August2014

Receivedinrevisedform13May2015 Accepted19May2015

Availableonline29May2015

JELclassification: D62 H21 H23 Keywords: Relativeconsumption Housing Taxation Behavioraleconomics

a

b

s

t

r

a

c

t

Thispaperanalyzeswhethermarginaltaxationoflaborandcapitalincomeareuseful sec-ondbestinstrumentsforinternalizingtheexternalitiescausedbyconspicuoushousing consumption,whenthegovernmentisunabletoimplementafirstbestcorrectivetaxon housingwealth.Therationaleforstudyingincometaxationinthisparticularcontextisthat firstbesttaxesonhousingwealthmaybeinfeasible(atleastinashortertime perspec-tive),whileincometaxesindirectlyaffectboththelevelandcompositionofaccumulated wealth.Weshowthatasuboptimallylowtaxonhousingwealthprovidesanincentivefor thegovernmenttosubsidizefinancialsavingandtaxlaborincomeatthemargin.

©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCC BY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Thispaperexamineswhethermarginallaborandcapitalincometaxation/subsidizationmightbeusefulmeansof exter-nalitycorrectionwhenconsumershavepositionalpreferencesforadurablegood,inwhichcasetheindividual’sconsumption ofthisgoodimposesexternalitiesonotherpeople.Housingconstitutesanobviousexamplebygivingtheowneradirect consumptionbenefitandatthesametimebeinganassetthroughwhichtofundfutureconsumption.Althoughouranalysis isapplicabletoanydurablegoodhavingtheseproperties,wewillrefertoitashousinginwhatfollows.Thiswillbefurther explainedandmotivatedbelow.

Thereisagrowingbodyofevidenceshowingthatpeopleareconcernedwiththeirrelativeconsumption.Atypicalfinding inthisliteratureisthatindividualwell-beingincreasesiftheindividual’sownconsumptionorincomeincreasesrelativetothe consumptionorincomeofreferentothers(e.g.,Easterlin,2001;Johansson-Stenmanetal.,2002;BlanchflowerandOswald, 2004;Luttmer,2005;SolnickandHemenway,1998,2005;ClarkandSenik,2010).1_{Ifconcernsforrelativeconsumptionare}

夽 Theauthorswouldliketothankanassociateeditor,twoanonymousreferees,DavidGranlund,andMagnusWikströmforhelpfulcommentsand suggestions.ResearchgrantsfromtheSwedishResearchCouncil(ref421-2010-1420)andfromtheWallander-HedeliusResearchFoundation (W2010-0368:1)arealsogratefullyacknowledged.

∗ Correspondingauthor.Tel.:+46907865017.

E-mailaddresses:thomas.aronsson@umu.se(T.Aronsson),andrea.mannberg@umu.se(A.Mannberg).

1_{AcademicworkonconspicuousconsumptiondatesbackatleasttoVeblen(1899),althoughtheassociatedpolicyimplicationswerebrieflytouched}

uponalreadybyMill(1848).AneconomictheoryofrelativeconsumptionwasfirstpresentedinDuesenberry(1949).

http://dx.doi.org/10.1016/j.jebo.2015.05.011

0167-2681/©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense

drivenbythedesiretosignalstatusorwealth,onewouldexpectthatclearlyvisiblegoodsaremorepositionalthanlessvisible goods,i.e.,thattheutilityofconsumingvisiblegoodstoalargerextentisdrivenbypreferencesforrelativeconsumption. Evidencefromsurvey–experimentalstudiessuggeststhatthisisalsothecase,sincevisiblegoodssuchashousesandcars havebeenfoundtobemorepositionalthanother,lessvisiblegoods(Alpizaretal.,2005;SolnickandHemenway,2005; Carlssonetal.,2007).CalculationspresentedinAlpizaretal.(2005)showthatthedegreeofpositionality(theextentto whichrelativeconsumptionmatterscomparedtoabsoluteconsumption)forhousingissubstantial:onaverage,about50 percentoftheutilitygainofadditionalexpendituresonhousingmaybeduetoincreasedrelativeconsumption.2_Therefore, individuals’choicesofhousingseemtoimposesubstantialexternalitiesonotherpeople.

Ifhousing,atleastinpart,representsconspicuousconsumption,afirstbestpolicywouldbetotaxhousingwealth suchthattheexternalitythateachindividualimposesonotherpeoplebecomesfullyinternalized.Yet,althoughtaxeson housingwealthareusedinmanycountries,thetaxratesareoftenquitelow;atleastbycomparisonwiththemagnitudeofthe positionalexternalitymentionedabove.Thisargumentwillbesubstantiatedbelow,whereweshowthatourmodelcombined withempiricalevidenceofrelativeconsumptionconcernswouldimplyanannualtaxonhousingwealthofbetween2and 3percentof themarketvalueunderreasonableassumptions. However,in manycountries(Denmarkbeinga notable exception)propertytaxesaresubstantiallybelowthisrate.3_{Inaddition,taxesonowner-occupiedhousingareinpractice} oftenlowerthanofficialtaxratesimply.Thisissosincemortgageinterestisoftendeductiblefromtaxableincomeandsince imputedrentsgountaxed.Thisimpliesthatcapitalintermsofhousingispreferentiallytreatedincomparisontoothertypesof capital(e.g.,Gervais,2002).Onereasonforsettingsuchlowtaxratesmightbethatpropertytaxesarepoliticallycontroversial, andhomeownersconstituteaninfluentialgroupinsociety.InSweden,forinstance,theHomeowners’Associationwasformed partlyforthepurposeofcollectiveactionagainsttaxationofhousingwealthandmostlikelycontributedtothesignificant reductionineffectivetaxratesduringthelatestdecade.Also,sincetaxesonhousingwealth(orpropertyingeneral)are oftenlocalorregional,thepolicyincentivesimplicitinsuchtaxesmaynotcorrectlyreflectpositionalexternalities;atleast notiftheconsumptioncomparisonsgobeyondthelocalorregionaljurisdiction.4

Therefore,ifanoptimalcorrectivetaxonhousingwealthisnotfeasible,itisimportanttoconsiderotherinstrumentsto correctfortheexternalitiescausedbyconspicuousconsumptionofhousing.Inthispaper,wefocusonanoptimalmixof laborandcapitalincometaxation,whichisdefinedconditionalontheexistingtaxonhousingwealth.Akeyroleofrealestate istofundfutureconsumption,suggestingthatamarginalsavingsubsidy(negativemarginalcapitalincometax)provides anincentiveforindividualstosavemoreinfinancialassetsandspendlessonhousing.Inturn,thispolicycounteractsthe positionalconsumptionexternalitythathousinggivesriseto.Onemaythusconjecturethatmarginalsavingsubsidies/taxes arekeysecondbestinstrumentsforinternalizingexternalitiesassociatedwithdurablegoodsconsumption.Yet,sincesuch subsidies/taxesarestillonlyindirectinstrumentsforcorrection,thereisroomforotherinstrumentsaswell,andweconsider anoptimallaborincometaxalongsidethesavings-orientedpolicydescribedabove.Theideahereisthatpositionalconcerns mayleadindividualstoincreasetheirlaborsupply,andamarginallaborincometaxpolicycanbedesignedtocounteract thisincentive.5_{Therefore,themixtureoflaborandcapitalincometaxesconstitutesaninterestingcombination,sinceitwill} affectboththelevelandcompositionofaccumulatedwealth.

Ourstudycontributestoalargeliteratureontaxandotherpolicyresponsestoconsumptionpositionalityandtosome extenttotheampleliteratureontaxationofowner-occupiedhousing.Thelatterhasshownthatpreferentialtaxtreatmentof owner-occupiedhousingcausesoverconsumptionofhousingservices(see,e.g.,Skinner,1996;Gervais,2002).However,to thebestofourknowledge,thisliteraturedisregardsexternalitiescausedbypositionalpreferences.Theliteraturedealingwith optimalpolicyresponsestopositionalexternalitiesfocusesalmostexclusivelyon(a)positionalconcernsfornon-durable goods,and(b)typicallyalsoonmodeleconomieswithonesingleprivateconsumptiongood(inadditiontoleisure).6_Inother

2 _{SeealsoZahirovic-HerbertandChatterjee(2011),whofindthatpeoplearewillingtopaymoreforhomeswithanameattachedtoit,andPatacchini}

andVenanzoni(2014),whofindsignificantpeer-effectsonthedemandforhousingqualityinUSA.

3 _{InSweden,forinstance,thetaxonhousingpropertyis0.75percentofthevalueofthepropertyuptoamaximumlimit(wherethevalueattachedtoeach}

propertybythetaxauthorityistypicallylowerthanthemarketvalue).Thecorrespondingrateisbetween0.2and0.7percentinNorway,wherethe munic-ipalitiesfreelydecideontheimplementation(about30percentofthemunicipalitiesdidnotimplementsuchataxin2009).Denmarkappliesasystemwith tworates:1percentifthemarketvalueofthepropertyislessthan3millionDKKand3percentotherwise.ThecorrespondingtaxinFinlandis0.32–0.75per centofthemarketvalue.InGermany,thetaxratevariesbetween0.26and0.35percent.GreatBritainalsoappliesazerorateexceptforhomeswithveryhigh marketvalues.PropertytaxesintheU.S.arebasedonthemarketvalueandtheratesvarybetweenstates(althoughthetaxisformallycollectedatthelocal level);inCalifornia,themaximumrateis1percent(TheSwedishHomeowners’Association,http://www.villaagarna.se/;Wikipedia,https://wikipedia.org; GermanyTradeandInvest,http://www.gtai.de/GTAI/Navigation/EN/Invest/Investment-guide/The-tax-system/taxation-of-property.html;International living,http://internationalliving.com/real-estate/countries/france/taxes/;http://internationalliving.com/real-estate/countries/spain/taxes/;Propertiesin Europe,http://www.properties-in-europe.com/infoitalytax.htm;http://soumi.fi).

4 _{Recentevidencesuggeststhatsocialreferencegroupsarenotformedsolelybasedonthelocalenvironment;possiblyduetotechnologicaldevelopments}

ofsocialmediaandtheInternet.Forinstance,Becchettietal.(2010)foundthattheimportanceofsocialcomparisonsbetweencountrieshasincreasedover time,andClarkandSenik(2010)foundthatthatInternetaccessispositivelycorrelatedwithrelativeconsumptionconcerns.

5 _{GoerkeandHellesheim(2013)showinatheoreticalmodelthatindividualsundercertainconditionssupplymorelaboriftheyareconcernedwiththeir}

relativeconsumptionthantheywouldinanundistortedeconomywithoutrelativeconcerns.Empiricalevidencepointinginthisdirectionispresentedby

BowlesandPark(2005).Theyconsideramodelwhereindividualsderiveutilityfromtheirownconsumptionrelativetotheconsumptionofareference

groupwithhigherincome,andshowempiricallythatincreasedinequalityisassociatedwithalargernumberofworkhours.

6 _{See,e.g.,BoskinandSheshinski(1978),Ng(1987),Tuomala(1990),LjungqvistandUhlig(2000),DuporandLiu(2003),AronssonandJohansson-Stenman}

words,mostpreviousworkdoesnotdistinguishbetweenpositionalandnon-positionalgoods,andthereissurprisingly littleresearchonhowtaxorotherpolicyinstrumentsoughttorespondtorelativeconcernsfordurableconsumption.One importantexceptionfromcategory(b)isEckerstorferandWendner(2013)dealingwithoptimallinearcommoditytaxation inatwo-goodeconomy,whereoneofthegoodsispositionalandtheothernon-positional.Theirmaincontributionisto examinetaxpolicyimplicationsofnon-atmosphericandasymmetricconsumptionexternalities,andbystudyingthejoint taxpolicyimplicationsofrelativeconsumptionconcernsandinequality-aversion.7_{Yet,theirstudyneitheraddressesthe} implicationsofrelativeconsumptionofdurablegoods,northeoptimalmixoflaborandcapitalincometaxationwhena flexibledirectinstrumenttotargettheexternalityisunavailable,whicharethekey-ingredientsofthepresentpaper.Toour knowledge,theonlyearlierstudydealingwithpolicyimplicationsofrelativeconcernsfordurablegoodsisAronssonand Mannberg(2013).Theyconsideranoverlappinggenerationsmodelwhereeachconsumerlivesforthreeperiodsandanalyze thejointtaxpolicyimplicationsofpositionalconcernsforhousingandaself-controlproblemcausedbyquasi-hyperbolic discounting.Theircontributionistoshowhowtheoptimalmixoftaxesonhousingwealthandcapitalincomevariesover theindividuallife-cycle,aswellashowitdependsonwhetherconsumershavenaïveorsophisticatedattitudestotheir self-controlproblems.

Thepresentpapercontributestotheliteratureinprimarilytwoways.First,weanalyzeindirectinstrumentstocorrect fortheexternalitiescausedbyconspicuousconsumptionwhenadirectinstrumentisnotavailable;ascenarioofclear practicalrelevanceforreasonsdiscussedabove.Oursettingalsomeansthatweextendthestudyofoptimalincometaxation indynamiceconomieswheretheconsumershavepositionalpreferencesbeyondthestandardframeworkwithonlyone private,non-durableconsumptiongood(inwhich caseincometaxesconstitutedirect instrumentsfor correction).8 _As such,ourstudytakesadifferentdirectionthanthepaperbyEckerstorferandWendner(2013),whichfocusesonoptimal commoditytaxationinamodel-economywithpositionalandnon-positionalgoods.Second,thereisnotmuchresearch onthetaxpolicyimplicationsofpositionaldurablegoods;theonlyearlierstudythatweareawareofisAronssonand Mannberg(2013)referredtoabove.Wesupplementtheirstudybyconsidering(i)adifferentpolicyproblem,(ii)another mixoftaxinstruments,and(iii)byexaminingasecondbestoptimaltaxpolicy.Assuch,thepresentpaper’sfocusondurable consumptionalsocontributestotheliteratureonamoregenerallevel.

InSection2,wepresent thebenchmarkmodelandbrieflydescribehowa governmentmayimplementa firstbest optimumthroughmarginaltaxationofhousingwealth.Section3assumesthatthepossibilitytotaxhousingwealthis restrictedandexaminestheoptimalsecondbesttaxationoflaborandcapitalincome.Theresultsshowthatifthemarginal taxonhousingwealthfallsshortofthevalueofthemarginalpositionalexternality,thesecondbestoptimalpolicyfeatures amarginalsavingsubsidyandamarginallaborincometax.InSection4,weextendtheanalysisbyconsideringtwoissues thatthebenchmarkmodeldoesnotaddress:non-separabilitybetweenleisureandhousing,andbequestmotives.Section5 summarizesandconcludesthepaper.

2. Abenchmarkmodel

Themodelpresentedhereaimsatcapturingtwoaspectsofhousing:(1)thepossibilitythatindividualsderiveutilityfrom theirrelativeconsumptioncomparedtoreferentothers,and(2)thattheindividualinvestsinhousingwealthwhenyoung andconsumesthishousingwealthalongwithotheraccumulatedwealthwhenold.Toaccomplishthistaskinthesimplest possibleway,weabstractfromthepriceformationprocessinassumingthatthesupplyofhousingunitsisinfinitelyelastic ineachperiod,andthattheindividualcansellanyremainingunitsofthehousewhenold.Assuch,andbyanalogytoalarge literatureonoptimaltaxation,weassumethattheproductiontechnology(inourcaseinboththedurableandnon-durable goodssectors)issuchthattheproducerandfactorpricesarefixedineachperiod,althoughnotnecessarilyconstantover time.9

Theeconomyisrepresentedbyanoverlappinggenerationsmodel,inwhichindividualslivefortwoperiods;theywork inthefirstandareretiredinthesecond.Eachindividualderivesutilityfromtheconsumptionoftwogoods:adurablegood referredtoashousingandanon-durablegood.Housingisassumedtobeapositionalgoodinthesensethatindividuals deriveutilityfromtheirrelativeconsumptioncomparedtoreferentothers(inadditiontotheutilitygainoftheabsolute

7_{Amongotherthings,theyshowthatiftheexternalityisnon-atmospheric(meaning,intheirstudy,thatindividualsdifferintheircontributiontothe}

externality),theprincipleoftargetingisnolongerfullyapplicablesuchthatbothcommoditytaxestypicallyareusedtocorrectfortheexternalitycaused bytheconsumptionofthepositionalgood.Eckerstorfer(2014)extendstheanalysistoaproblemofoptimalmixedtaxationunderasymmetricinformation basedonastaticmodelwithtwonon-durableconsumptiongoods(inadditiontothelabor–leisurechoice),wherethegovernmentdecidesonanoptimal nonlinearincometaxandasetoflinearcommoditytaxes.

8_{SeeLjungqvistandUhlig(2000)andAronssonandJohansson-Stenman(2010,2014).}

9_{Ourabstractionfromthepriceformationprocessandfromlocationissuesdoesnotmeanthatwethinktheseissuestobeunimportant.However,}

inthispaper,weareprimarilyinterestedinanalyzingtaxpolicytocounteractpositionalexternalities.Aslongastheconsumershavepreferencesfor theirrelativeconsumptionofhousing(whichisthebasicassumptiononwhichourpaperrests),thepolicyincentivescharacterizedbelowwouldalsobe presentinamoregeneralframework(althoughpossiblycombinedwithotherpolicyincentivesdependingonthefunctioningofthehousingmarket).The modelpresentedbelowisanalyticallyconvenientandallowsustoidentifybasicmechanismsbehindasecondbestoptimalpolicydesignedtocorrectfor positionalconsumptionexternalities.

consumption),whilethenon-durablegoodiscompletelynon-positional.10_{Individualsalsoderivedisutilityfromworkeffort.} Tosimplifythenotation(sincewearefocusingonefficiencyaspectsofpublicpolicy),wealsoassumethatindividualsofthe samegenerationareidenticalandnormalizethenumberofindividualspergenerationtoone.

Anindividualenteringtheeconomyatthebeginningofperiodt,tobereferredtoasgenerationt,isyoung(inthesense ofbeingactiveinthelabormarket)inperiodtandoldinperiodt+1.Letc0,tandc1,t+1denotetheconsumptionofthe non-durablegoodbytheyoungandoldgenerationt,respectively,ltdenoteslaborsupply,whilehtdenoteshousingconsumption. Thelifetimeutilityfunctionfacinganindividualofgenerationtisgivenby

Ut=U(c0,t,lt,ht, ¯ht,c1,t+1)=u0(c0,t)+(lt)+(ht)+˚(ht− ¯ht)+u1(c1,t+1) (1) Eq.(1)meansthattheindividualderivesutilitybothfromhis/herabsoluteconsumption,ht,andrelativeconsumption, ht− ¯ht,ofhousing,aswellasfromhis/herabsoluteconsumptionofthenon-durablegoodanduseofleisure.Althoughthe housingchoiceismadeduringthefirstperiod,theutilityderivedfromhousingcanbeinterpretedasreferringtoboth periodsoflife.Thereareseveralpossiblewaysofconceptualizingaunitofhousing,andonecan(atleast)thinkofheither asapurelyquantitativemeasure(e.g.,thenumberofsquaremeters)orintermsofunitsofqualityofhousing(e.g.,finishes andfixtures).11_{Evenifbothinterpretationsmayberelevantfromtheperspectiveofrelativeconcerns,whichisthekeyissue} inourpaper,wepreferthelatter(qualityoriented)interpretationprimarilybecauseourmodellacksaspatialdimension. Thevariable ¯htdenotesthereferenceconsumptioninperiodt.Weassumethatthefunctionsu0(·),(·),˚(·),andu1(·)are increasingandstrictlyconcaveintheirrespectivearguments,while(·)isdecreasingandstrictlyconcave.Allgoodsare normal.Also,theindividualisassumedtoactasanatomisticagentandtreat ¯htasexogenous.Theseparablestructureof Eq.(1)ismaintainedthroughouttheanalysisinSections2and3andwillallowustosignkeycomparativestatics(seeEqs. (16a)and(16b)).However,tosimplifythepresentation(andavoidunnecessarynotationintheformulas),thefirstorder conditionswill,nevertheless,bewrittenintermsofthegeneralfunctionU(·).

Letyt=wtltdenotelaborincome,wtthewagerate,stsaving,rttheinterestrate,andPtthepriceperunitofhousingin periodt.Beforeimposinganyrestrictiononthepossibilitytotaxhousingwealth,theindividual’sbudgetconstraintcanbe writtenasfollows:12

c0,t=yt−st−Ptht−0(Ptht)−T0(yt) (2)

c1,t+1=st(1+rt+1)−T1(strt+1)+Pt+1ht(1−ı) (3) whereıdenotestherateofdepreciationofahouse,0(·)isataxonhousingwealthpaidwhenyoung,T0(·)alaborincome taxpaidwhenyoung,andT1(·)acapitalincometaxpaidwhenold.Whenyoung,theindividualconsumes(andinvestsin) housing,consumesthenon-durablegood,andsavesonthecapitalmarket;whenold,he/sheusesthefinancialwealthand housingwealthfornon-durableconsumption.Thelaborandcapitalincometaxfunctionsareassumedtobenonlinearsuch thatthegovernmentmaycarryoutanyredistributionbetweengenerationsonalump-sumbasis(ifitwishestodoso).As willbedescribedingreaterdetailbelow,thisenablesustoconcentrateoncorrectiveaspectsofmarginaltaxation.

Anindividualofgenerationtchooseshousing,non-durableconsumption,andworkhourstomaximizetheutilitygiven byEq.(1)subjecttothebudgetconstraintinEqs.(2)and (3),whiletreatingwt,rt+1,Pt,Pt+1,and ¯ht asexogenous. By substitutingthebudgetconstraintintotheobjectivefunctiontoreplacetheconsumptionofnon-durablegoods,weobtain adecision-probleminst,ht,andltwiththefirstorderconditions13

st:−Ut,c0+Ut,c1[(1+rt+1)−T1(s0,trt+1)rt+1]=0 (4a) ht:−Ut,c0[Pt+Pt0(Ptht)]+Ut,h+Ut,c1Pt+1(1−ı)=0 (4b)

lt:Ut,c0wt(1−T0(yt))+Ut,l=0 (4c)

whereUt,c0=∂Ut/∂c0,t,Ut,h=∂Ut/∂ht,Ut,c1 =∂Ut/∂c1,t+1,andUt,l=∂Ut/∂lt,denotepartialderivativesoftheutilityfunction,

while0(Ptht), T0(yt)and T1(s0,trt+1)arethemarginaltaxratesonhousingwealth,laborincome,andcapital income, respectively.Notealsothattheoldconsumermakesnoactivedecisionotherthansellingthehouse:he/shethenspendsthe

10 _{Theassumptionthatthenon-durablegoodiscompletelynon-positionalisasimplification,yetoflittlepracticalimportanceaslongasnon-durables}

arelesspositionalthandurablegoods.

11 _{Wearegratefultoarefereeforpointingoutdifferentpossibleinterpretations.}

12 _{Sinceeachindividualonlymakesonehousinginvestmentinourmodel,andgiventhattheindividualpaysataxonhousingwealthinthefirstperiod,}

asecondperiodtaxonhousingwealth(tobepaidbeforethehouseissold)wouldbecomeredundant.

13 _{Notethatthetwoperiodstructuretosomeextenthidesthepossibilitythatindividualsfinancethehousingpurchasebyborrowingonthecapital}

market,sincefinancialsaving,st,mustbepositivetogenerateacapitalstocktobeusedinproductioninperiodt+1(seebelow).However,suchborrowing

isperfectlyconsistentwiththemodelsetoutaboveandjustimpliesthatstdecreases.Thefirstperiod(whentheindividualis“young”)inourtwoperiod

settingisinterpretabletomeantheindividual’swholeworkinglifeuptotheretirementage(aroundage65–67inSweden).Itisclearlyreasonableto expectthatarepresentativeindividualhasaccumulatedfinancialwealthattheageofretirement,despitethathe/sheborrowedresourcestofinancethe housingpurchasesome30yearsearlier.Thisiswhatourmodelrequires,sincetheworkinglifeoftheindividualismergedintoasingleperiod.Datafrom StatisticsSwedensupportsthisviewbyshowingthatpeopleaged65–74onaveragehavepositiverealwealth(inwhichhousingwealthisincluded)and positivefinancialwealth.ThisdataispubliclyavailableatthehomepageofStatisticsSweden(www.scb.se).

remainingwealthforconsumptionofthenon-durablegood.Forpurposesofcomparison,inSection4weconsideracase wheregenerationtinsteadleavesthehouseasabequesttothenextgeneration.

2.1. Optimaltaxpolicyinthebenchmarkmodel

Thegovernmentisfirstmovervis-à-vistheprivatesectorandaimsatcorrectingthepositionalexternalities.Weassume ageneralsocialwelfarefunctionthatisincreasinginthelifetimeutilityfacedbyeachgeneration

W=W(U0,U1,...). (5)

Notethatnonlineartaxationoflaborincomeandcapitalincomeaccompaniedbyaflexibletaxonhousingwealthwillallow thegovernmenttoperfectlycontrolthehoursofwork,theconsumptionofdurableandnon-durablegoods,andthecapital stockintheeconomy.Therefore,wefollowconventionintheliteratureonoptimalnonlinearandmixedtaxationinwriting thesocialdecision-problemasadirectdecision-problemintermsofworkhours,privateconsumption,andthecapitalstock, insteadofintermsofparametersofthetaxfunctions(suchasinterceptsandmarginaltaxrates).14_{Theoptimalmarginal} taxratescanthenbederivedthroughacomparisonofsocialandprivatefirstorderconditions.Theresourceconstraintfor theeconomyasawholeinperiodtcanbewrittenas15

yt+kt(1+rt)−Pt(ht−ht−1(1−ı))−c0,t−c1,t−kt+1=0 (6) wherektisthephysicalcapitalstockdefinedsuchthatst−1=ktforallt.Theresourceallocationpreferredbythegovernment canbederivedbychoosingc0,t,lt,c1,t,htandkt forallttomaximizethesocialwelfarefunctionsubjecttotheresource constraintand ¯ht=ht forallt.Thelatterfollowsbecausetheindividualsareidentical,andthe(externalitycorrecting) governmentrecognizeshowthereferenceconsumption isdetermined.TheLagrangeancorrespondingtothis decision-problemisgivenby L=W+



t t



yt+kt(1+rt)−Pt(ht−ht−1(1−ı))−c0,t−c1,t−kt+1



(7) wheretistheLagrangemultiplieroftheresourceconstraintinperiodt.Define

MRSt_h,c 0= Ut,h Ut,c0 >0, MRSt_l,c 0= Ut,l Ut,c0 <0, MRSt_c0,c1=Ut,c0 Ut,c1 >0

tobethemarginalratesofsubstitutionbetweenhousingandfirstperiodconsumption;betweenworkeffortandfirstperiod consumption;andbetweenthefirstandsecondperiodconsumption,respectively,foranindividualofgenerationt.The socialfirstorderconditionscanthenbewrittenasfollows:

MRSt_c0,c1=(1+rt+1) (8a) MRSt_h,c 0+ U_{t, ¯h} Ut,c0 −Pt+ 1 MRSt_c0,c1Pt+1(1−ı)=0 (8b) −MRSt l,c0=wt (8c)

Now, let t=ht− ¯ht, and notice from Eq. (1) that U_{t, ¯h}=−∂˚(t)/∂t=−˚t,, and Ut,h=t,h+˚t, where t,h=∂(ht)/∂ht.Wecanthendefinethedegreeofpositionalityforhousinginperiodtasthefractionoftheoverallutility gainfromanadditionaldollarspentonhousingthatisduetoincreasedrelativeconsumption,i.e.,

˛t=− U_{t, ¯h} Ut,h = ˚t, t,h+˚t, ∈(0,1). (9) Thedegreeofpositionalityisalsointerpretableasthemarginalexternalityperunitofhousingconsumption.Assuch,it willplayakeyroleintheanalysisofmarginaltaxpolicybelow.Thefirstbestoptimaltaxpolicycanthenbesummarizedas followsbasedonEqs.(4),(8)and(9):

Observation 1. The optimal tax policyin the benchmark model satisfies T0(wtlt)=0, T1(s0,trt+1)=0 and Pt0(Ptht)= ˛tMRSth,c0forallt.

Clearly,sincethegovernmentofthebenchmarkmodelhasaccesstoaflexibletaxonhousingwealth,itmayusethistax tofullyinternalizetheexternalitycausedbyconspicuoushousingconsumption.Thisexplainswhythemarginallaborand capitalincometaxratesareequaltozero:sincetherearenootherdistortionsthantheexternalitycausedbyeachconsumer’s desiretohavemoreunitsofhousingthanotherpeople,andsincethereisnoneedtodistortbehaviorjusttoraiserevenue,

14_{See,e.g.,Stiglitz(1982).}

15_{Notethattheanalysispresupposesthath}

theonlyreasonfortaxinghousingwealthinthismodelistointernalizethepositionalexternality.Consequently,thepolicy ruleforthistaxdependsonhoweachindividual’shousingchoiceaffectsthewell-beingofotherpeople(asreflectedinthe degreeofpositionality).Itdoesnotexplicitlydependonotheraspectsofthehousingmarketsuchasthechangeintheprice overtheindividuallife-cycle.16

Thelefthandsideofthetaxformulaforhousingmeasuresthemarginaltaxonhousingwealthtimesthepricepaid perunitofhousinginperiodt,i.e.,thetaxpaymentforanadditionalunitofhousing,whichissetequaltothedegreeof positionalitytimesthemarginalwillingnesstopayforhousinginperiodt.Therefore,if˛tisintheneighborhoodof0.5,as suggestedbyempiricalevidencepresentedintheintroduction,about50percentoftheconsumer’smarginalwillingnessto payforhousingrepresentssocialwasteandshouldbetaxedaway.Toexemplifyandgivesomesubstancetotheargument madeintheintroductionthatexistingtaxesonhousingwealthoftenfallshortofthemarginalpositionalexternality,we assumethatPt=Pt+1/(1+rt+1)(inwhichcasethereisno“bubble-component”intheprice),andthatıisbetween2and4per cent.17_{Ifwefurtherassumethatanindividualownsahousefor30years,ourmodelimpliesthattheoptimalyearlytaxon} housingwealthisbetween1.3and2.7percentif˛t=0.4,andbetween2and4percentif˛t=0.5.ExceptforDenmark,the countriessurveyedabove(seefootnote3)applymuchlowerrates.

Aninterestingspecialcaseofourmodelariseswhenı=1,inwhichthehomeisnolongerinterpretableasadurableasset totheindividual.Thisspecialcasecanbeinterpretedsuchthattheindividualsrenttheirhomesinsteadofowningthem,also meaningthattheoptimaltaxonhousingwealthoughttobere-interpretedasanordinarycommoditytaxattachedtotherent payment.18_{AscanbeseenfromObservation1,sincetheoptimalmarginaltaxonhousingwealthdoesnotdependdirectly} onı,andprovidedthattherelativeconcernsareonlyassociatedwiththequalityofthehome(asweassumeabove),the optimalpolicyruletakesexactlythesameformirrespectiveofwhetherindividualsownorrenttheirhomes.Theintuition is,ofcourse,thattheexternalityeachindividualimposesonotherpeopleisindependentofownership.Weshowinthenext sectionthatthisqualitativeresultalsocarriesovertothecasewhereaflexibletaxonhousingwealth(orcommoditytaxon rentedhomes)isnotavailabletothegovernment;yetwithaslightlymodifiedinterpretationsinceoneofthepolicyrules dependsexplicitlyonıinthatcase.

3. Suboptimalmarginaltaxationofhousingwealth

Supposethat thegovernmentisunabletoimplementtheoptimaltaxonhousingwealthdescribed intheprevious section.Thetaxisnowgivenby0(Ptht)= tPtht,inwhich tisanonnegativetaxratesuchthat t≤ ¯ t,where ¯ tdenotes anupperlimit.Wecaninterprettheupperlimitasreflectingtheresistanceagainsttaxesonhousingpropertydescribed intheintroduction.Thisformulationisalsoconvenientasitencompassesaneconomywithouttaxationofhousingwealth (inwhich ¯ t=0forallt)andthefirstbestoptimaltaxpolicycharacterizedabove(wheretheconstraintdoesnotbind)as specialcases.Inthissection,weassumethattheconstraintisbinding,suchthatthetaxrateonhousingwealthisfixed. Thequestionisthenwhethermarginaltaxation/subsidizationoflaborandcapitalincomemaybeusefulassupplementary instrumentstocorrectforpositionalexternalities.

Toaddressthisquestion,itisconvenienttomodeltheindividual’sdecision-problemintwostages.Inthefirststage,we choosehttomaximizelifetimeutilityinEq.(1)subjecttothefollowingbudgetconstraint:

b0,t=c0,t+Ptht(1+ t) (10a)

b1,t+1=c1,t+1−Pt+1ht(1−ı) (10b)

whereb0,tandb1,t+1aretreatedasfixedincomesinthefirstandsecondperiodoflifeforgenerationt.Thisgivesthefirst ordercondition

−Ut,c0Pt(1+ t)+Ut,h+Ut,c1Pt+1(1−ı)=0 (11)

whichimplicitlydefinestheindividual’sdemandforhousingasafunctionofPt(1+ t),Pt+1(1−ı),b0,t,b1,t+1and ¯ht,i.e., ht=h



Pt(1+ t),Pt+1(1−ı),b0,t,b1,t+1, ¯ht



. (12)

NoticethatEq.(12)isinterpretableasaconditionaldemandfunctioninthesenseofmeasuringthisdemandconditionalon theindividual’sincomeandsaving(whicharepartofb0,tandb1,t+1).BysubstitutingEqs.(10a),(10b)and(12)intoEq.(1), thecorrespondingconditionalindirectutilityfunctioncanbewrittenas

Vt=V



Pt(1+ t),Pt+1(1−ı),b0,t,b1,t+1,lt, ¯ht



=U



b0,t−Pt(1+ t)ht,lt,ht, ¯ht,b1,t+1+Pt+1ht(1−ı)



(13)

16 _{Althoughpricechangesovertheindividuallife-cycledonotleadtoachangeinthepolicyruleforthecorrectivetax,theymay,nevertheless,affectits}

levelviathedemandforhousing.Forinstance,achangeinthesellingpricemayinfluencethecorrectivetaxeitherthroughthedegreeofpositionalityor themarginalwillingnesstopayforhousing(orboth).

17 _{Hardingetal.(2007)estimatethegrossannualrateofdepreciationto3percentforthetypicalhomeintheU.S.andthedepreciationnetofmaintenance}

to2percent.Aslightlyhigheropportunitycostmightbemotivatedfromcommutingcostsandothertime-costsassociatedwithatypicalhome.

18 _{Thisinterpretationdoesnotdependontheassumptionthattheindividualonlypaysrentonce.Evenifweweretointroducearentpaymentinthe}

wherehtisgivenbyEq.(12).Inthesecondstage,wederivethesavingsandlaborsupplybehaviorbychoosingstandltto maximizetheindirectutilityfunctiongivenbyEq.(13)subjectto

b0,t=yt−st−T0(yt) (14a)

b1,t+1=st(1+rt+1)−T1(strt+1). (14b)

Thefirstorderconditionsforthisproblemcanbewrittenas −Vt,b0+Vt,b1[(1+rt+1)−T  1(strt+1)rt+1]=0 (15a) Vt,b0wt[1−T  0(yt)]+Vt,l=0 (15b) inwhichVt,b0=∂Vt/∂b0,t,Vt,b1 =∂Vt/∂b1,t+1andVt,l=∂Vt/∂lt.

3.1. Secondbestoptimalincometaxation

Asexplainedabove,thesavingsandlaborsupplychoicesbygenerationtaffecttheconditionaldemandforhousingvia b0,tandb1,t+1,Itis,therefore,instructivetobeginbyderivingcomparativestaticsofthehousingdemandwithrespecttob0,t andb1,t+1.Sincethegovernmentaimsatinternalizingthepositionalexternalityandincorporatesintoitsdecision-problem that ¯ht=ht,wedifferentiateEq.(11)withrespecttoht,b0,tandb1,t+1,whileusing ¯ht=ht.Thisgives

dht db0,t = Ut,c0c0Pt(1+ t) >0 (16a) dht db1,t+1 = −Ut,c1c1Pt+1(1−ı) <0 (16b) whereUt,c0c0<0,Ut,c1c1<0and =Ut,c0c0[Pt(1+ t)] 2₊ Ut,hh+Ut,c1c1[Pt+1(1−ı)] 2₊ U_{t,h ¯h}<0. Doublesub-script(e.g.,c0c0)denotessecondorderpartialderivative.

Withafixedtaxonhousing,thesocialdecision-problemisasecondbestproblem,whichcanbewrittenas max

b0,tb1,t,kt,lt∀t

W(V0,V1,...,)

s.t.: yt−b0,t+kt(1+rt)+ tPtht−b1,t−kt+1=0 forall t

(17) wheretheresourceconstraintishereexpressedintermsofthe“savings-adjusted”measuresofdisposableincomegiven inEqs.(14a)and(14b).Theresourceconstraintisderivedbyusing tPtht+T0(yt)+T1(st−1rt)=0andst−1=ktincombination withEqs.(14a)and(14b).19_Byusing

ttodenotetheLagrangemultiplieroftheresourceconstraintinperiodt(asbefore), thesocialfirstorderconditionsforb0,t,b1,t+1,kt+1,andlt(theconditionscharacterizinganoptimalallocationforgeneration t)become b0,t:∂W ∂Vt



Vt,b0+Vt, ¯h dht db0,t



+t



−1+ tPt dht db0,t



=0 (18a) b1,t+1:∂W ∂Vt



Vt,b1+Vt, ¯h dht db1,t+1



+t tPt dht db1,t+1 −t+1= 0 (18b) kt+1:t+1(1+rt+1)−t=0 (18c) lt: ∂W ∂VtVt,l+twt=0. (18d) ByusingEqs.(18a)–(18d)togetherwith(1+rt+1)−Vt,b0/Vt,b1=T1(strt+1)rt+1andwt+Vt,l/Vt,b0=wtT0(yt)fromtheprivate firstorderconditionsgiveninEqs.(15a)and(15b),respectively,wecanderivethefollowingresult:

19_{Withgeneral,nonlineartaxesonlaborandcapitalincome,thegovernmentisabletocontrolthehoursofwork,disposableincome,andthecapital}

stock,althoughitnolongerhasadirectinstrumenttocontrolthehousingchoice.Instead,since tisfixed,thegovernmentisonlyabletoinfluencethe

revenuefromthehousingwealthtaxindirectlyviatheeffectsthatthelaborandcapitalincometaxeshaveonh(throughb0,andb1),whichexplainsthe

resourceconstraintinEq.(17).InawaysimilartotheanalysisinSection2,therefore,wehavestillwrittentheresourceconstraintintermsofthedirect decision-variables.AnalternativewouldbetowritetheresourceconstraintasinEq.(6)above,whichcanbederivedbysubstitutingEqs.(10a)and(10b)

intoEq.(17),andthenusethatc0andc1arealsofunctionsofb0andb1.Thelatterapproach(yetmodifiedtosuittheirmodel)wastakenbyEdwardsetal.

Proposition1. Ifthemarginaltaxonhousingwealthisfixedat tforallt,theoptimalmarginalincometaxrateswilltakethe form T1(strt+1)rt+1=(1+rt+1)

_˘ t 1+rt+1˛t MRSt_h,c 0− tPt



MRSt_c0,c1 dht db1,t+1− dht db0,t



wtT0(yt)=−MRStl,c0

_˘ t 1+rt+1˛t MRSt_h,c 0− tPt

_dh t db0,t forallt,where˘t=(∂W/∂Vt)(Vt,b0/t+1)>0.

Proof. SeeAppendix.

IneachtaxformulainProposition1,thesecondtermwithinsquarebracketsontherighthandsideisproportionaltothe actual(andpossiblysuboptimal)taxonhousingwealth,whilethefirsttermreflectsthevalueofthemarginalexternality. Beforeweinterprettheresultsingreaterdetail,notethatProposition1containstwousefulspecialcases.First,if t=0,

Proposition1andEqs.(16a)and(16b)togetherimplyT1(strt+1)<0andT0(yt)>0.Second,thefirstbestpolicysummarized byObservation1alsofollowsasaspecialcaseofthemoregeneralresultderivedintheproposition:inafirstbestresource allocationwherePt t=˛tMRSth,c0 and˘t=1+rt+1,weobtainT



1(strt+1)=T0(yt)=0.Thesetwospecialcasesalsosuggest thefollowingmoregeneralcorollarytoProposition120_:

Corollary1. IfPt t<˛tMRSth,c0,Proposition1impliesT



1(strt+1)<0andT0(yt)>0. Proof. SeeAppendix.

Corollary1meansthatifthehousingwealthtaxperunitofhousingfallsshortofthemarginalexternality,inwhichcasethis taxistoolowtoachievefullexternalitycorrection,thereisanincentiveforthegovernmenttocounteracttheexternality furtherthroughamarginalsavingsubsidyandmarginallaborincometaxpolicy.Theintuitionisthatincreasedsaving infinancialwealthleadstolessinvestmentinhousing,i.e.,financialsavingbecomesamoreattractivemeansoffunding futureconsumptionattheindividuallevel.Asaconsequence,thepositionalconsumptionexternalityattachedtohousing decreases.Similarly,reducedlaborincomereducestheresourcesavailableforalltypesofwealthaccumulation.

ThemultiplierMRSt_c0,c1(dht/db1,t+1)−dht/db0,t<0inthemarginalcapitalincometaxformulainProposition1appears becausethecapitalincomesubsidyconstitutesanindirectinstrumenttoreducethehousingwealth,andworksthroughthe effectofsavingsonthehousingdemand.Assuch,themoreanincreaseinsavingsreducesthedemandforhousing(through adecreaseinb0,tandcorrespondingincreaseinb1,t+1),themoreeffectivewillbethesavingssubsidyasaninstrumentto reducethepositionalexternality.Ontheotherhand,ifdht/db0,tanddht/db1,t+1areclosetozero,thesavingssubsidymaynot beausefulinstrumenttoinfluencetheaccumulationofhousingwealth,inwhichcasethesubsidywouldbeclosetozero ornotusedatall.Themultiplierinthemarginallaborincometaxformula,dht/db0,t>0,hasananalogousinterpretation.

Finally,letusbrieflyreturntothespecialcasediscussedinSection2whereı=1,whichisinterpretabletomeanthat individualsrenttheirhomesinsteadofowningthem.Notethatthisspecialcasegivesthesamequalitativeresultsasthose derivedinProposition1andCorollary1.Weshouldstillsubsidizethesavingandtaxthelaborincomeatthemarginunderthe samegeneralconditionstocounteractthepositionalconsumptionexternality.However,whilethepolicyruleformarginal laborincometaxationwouldbeidenticaltothatderivedintheproposition,sincethisformuladoesnotdependdirectlyonı, themarginalsavingsubsidyislikelytobesmallerifindividualsrenttheirhomesbecausedht/db1,t+1=0ifı=1.Theintuition isthatthehomeisnolongeradurablegoodthroughwhichindividualsmayfundfutureconsumptionand,therefore,not analternativetofinancialwealthfortheindividual.Inthatcase,themarginalsavingsubsidyworksmuchlikethemarginal laborincometaxinthesenseofjustreducingtheamountofresourcesavailableforconsumptioninthefirstperiod.This suggeststousthattheroleofmarginalsavingsubsidiesisweakerifindividualsrenttheirhomescomparedtowhenthey ownthem.

4. Extensions

Resultsalwaysdependontheassumptionsmade,andthereferencemodelexaminedaboveisnoexception.Inthissection, webrieflyconsidertworelevantextensionsoftheanalysissetoutinSections2and3byexaminingtheimplicationsof(i) non-separabilitybetweenhousingandleisureand(ii)abequestmotivesuchthatgenerationtleavesthehomeasabequest togenerationt+1.Theseextensionsserveasasensitivityanalysistocheckwhetherthequalitativeresultsofthebenchmark modelholdinslightlydifferentsettings.

20 _IfP

t t>˛tMRSh,ct 0,Proposition1insteadimpliesthatT 

1(strt+1)>0andT0(yt)<0.However,thiswouldimplyascenariowherethegovernmentsets

thehousingtaxhigherthanthevalueofthemarginalexternality.Thisseemsunrealisticsincethegovernmentinthiscasecouldchoosethefirstbestoptimal taxinsteadofthesecondbestsolution.Wethereforedonotconsiderthisscenarioinouranalysis.

4.1. Non-separabilitybetweenleisureandhousing

Bygeneralizingtheutilityfunctionofthebenchmarkmodeltoincludenon-separabilitybetweenleisureandhousing, Eq.(1)canberewrittenas

Ut=U(c0,t,lt,ht, ¯ht,c1,t+1)=u0(c0,t)+ (lt,ht)+˚(ht− ¯ht)+u1(c1,t+1) (19) wherethesub-utilityfunction (·)replaces(·)and(·)inEq.(1);theotherpartsarethesameasbefore.Assuch,weassume that (·)isdecreasinginlandincreasinginh.

ThischangeoffunctionalformhasnoimplicationforthefirstbestpolicyrulepresentedinSection2.Ifthedegreeof positionalityisproperlyredefinedtoreflectEq.(19),i.e.,˛t=−U_{t, ¯h}/Ut,h=˚t,/( t,h+˚t,)where t,h=∂ (lt,ht)/∂ht,the firstbestpolicycanbewritteninexactlythesameformatasinObservation1.Therefore,ifthegovernmentisequippedwith aflexibletaxonhousingwealth,thebestitcandoisstilltofullyinternalizetheexternalitythroughthisdirectinstrument, inwhichcasethereisnoneedtodistortthesavingbehaviororlabor–leisurechoiceoftheconsumers.

ThesecondbestpolicypresentedinSection3ismodified.WhentheutilityfunctionisgivenbyEq.(19)insteadofEq.(1), generationt’sconditionaldemandforhousingwillalsodependonthehoursofwork,lt,i.e.,Eq.(12)extendsto

ht=h(Pt(1+ t),Pt+1(1−ı),b0,t,b1,t+1,lt, ¯ht). (20) Itisstraightforwardtoderivedht/dlt<0(>0)if t,lh=∂2 (lt,ht)/∂lt∂ht<0(>0),whichisinterpretabletomeanthathousingis complementarywith(substitutablefor)leisure.Thecomparativestaticswithrespecttob0,tandb1,t+1arestillgivenbyEqs.

(16a)and(16b),respectively.Assuch,itisstilldesirabletosubsidizesavingatthemarginundertheconditiondescribedin Corollary1,andtheoptimalmarginalsavingsubsidytakesthesameformasinProposition1.However,theoptimalmarginal laborincometaxratechangessuchthattheformulainProposition1isnowreplacedby

wtT0(yt)=

_˘ t 1+rt+1˛t MRSt_h,c 0− tPt



_dh t dlt − MRSt_l,c 0 dht db0,t



. (21)

Evenif˛tMRSth,c0> tPt,therighthandsideofEq.(21)canbeeitherpositiveornegative.Ifleisureandhousingaresubstitutes,

oriftheyareweakenoughcomplementssuchthatdht/dlt−MRS_l,ct ₀dht/db0,t>0,itisstilloptimaltotaxlaborearnings atthemargin,whereasthepolicyturnsintoamarginalsubsidyifleisureandhousingarestrongenoughcomplementsto implydht/dlt−MRSt_l,c0dht/db0,t<0.

4.2. Bequestmotives

InthebenchmarkmodelexaminedinSections2and3,weassumedthatindividualsselltheirhouseswhenbecomingold andthenspendallremainingwealthonnon-durableconsumption.Yet,adurablegoodmaynotonlybeusedtofundfuture consumptionofthecurrentowner,itmayalsoprovideasourceofwealthforfuturegenerations.Analternativeformulation ofthemodelwouldthusbetoassumethatoldindividualsgivetheirhomesasabequesttothenextgeneration.Allowingfor altruismofthistypemakestheanalysissubstantiallymorecomplex,andwethereforeonlybrieflydiscusstheimplications ofbequestsinahighlystylizedcase.21

Theimportantimplicationofaltruismfromourperspectiveisthatrelativeconsumptionconcernsforhousingwilllead tointergenerationalexternalities.Thisaspectcanbecapturedinasimplifiedmodelwhereonlyonearbitrarygeneration tleavesthehomeasabequesttogenerationt+1,whileallothergenerationsareassumedtobehaveinthesamewayas inSections2and3.Suchaframeworkmakesitpossibletoexaminethepolicyimplicationsofaltruismwithouthavingto alterthetime-separablestructureinotherpartsofthemodel.22_{Inthissetting,generationt+1mayincreaseitshousing} consumptionthroughaninvestmentintheinheritedhome.Ourmodelthenimpliesthatgenerationtdirectlyaffectsthe positionalexternalityinperiodt+1, ¯ht+1,viaitschoiceofhousing.Also,toavoidtheextracomplicationsthatwouldfollow ifgenerationtandthegovernmentpreferdifferentintergenerationaltradeoffs,wehaveassumedthatthesocialwelfare functionisgivenbyadiscountedsumofgenerationalutilities,andthatgenerationtdiscountstheutilityofgenerationt+1 usingthesamediscountrateasthegovernment.

Ifgenerationttreatsthedecision-variablesofgenerationt+1asexogenous,itisstraightforwardtoshowthatthefirstbest policypresentedinObservation1continuestoapply.Theintuitionisthatgenerationteffectivelyfacesthemarginaltaxon housingwealthbothinperiodtandt+1,andtheexternalityitimposesongenerationt+1isproportionaltotheexternality thateachindividualofgenerationt+1imposesonotherpeopleofthesamegeneration.Therefore,theintergenerational externalitycausedbygenerationtisinternalizedthroughthemarginaltaxonhousingwealthinperiodt+1,whilethe

21_{Calculationsandproofsforthestylizedbequestcaseareavailablefromtheauthorsuponrequest.}

22_{Animplicationofaltruismisthatthebequest-givinggeneration’sfirstorderconditionforhousingbecomesdependentonthenextgeneration’s}

decision-variables,whichcomplicatestheanalysisconsiderably.Toavoidmostofthesecomplications,whilestillbeingabletoaddresstheintergenerational externalitymentionedabove,weassumethatallgenerationsotherthantsellthehouseduringthesecondperiodoflife.

remainingwithin-generationexternalityisinternalizedthroughthemarginaltaxonhousingwealthinperiodt.Thismeans, inturn,thattheoptimalmarginallaborandcapitalincometaxesareequaltozero.

Incasethemarginaltaxonhousingwealthispredetermined(thescenarioexaminedinSection3),thesecondbestoptimal taxpolicyimplementedforgenerationtgenerallydiffersfromthatofthebenchmarkmodel.Morespecifically,themarginal savingsandlaborincometaxesimplementedforgenerationtcontaintwoanalogouscomponents:onereferringtothe discrepancybetweenthemarginalpositionalexternalityandthemarginaltaxonhousingwealthinperiodtandtheotherto thecorrespondingdiscrepancyinperiodt+1.Onlythefirstcomponentwaspresentinthebenchmarkmodel,whichisseen fromProposition1.Theintuitionis,ofcourse,thatthetaxpolicyisnowdesignedtocorrectbothforwithin-generationand between-generationexternalities.Asaconsequence,theresultsaremoredifficulttointerprethere.However,ifthemarginal taxonhousingwealthissufficientlysmallcomparedtothemarginalpositionalexternalityinperiodt+1,thesecondbest optimalpolicyincludesamarginalsavingsubsidyandmarginallaborincometaxforgenerationtunderthesameconditions asinCorollary1,inwhichcasethequalitativeresultfromthebenchmarkmodelremainsvalid.

5. Conclusions

Thispaperdealswiththepolicyimplicationsofrelativeconsumptionconcernsfordurablegoods,exemplifiedbyhousing, whereeachindividual’sconsumptionchoiceimposesnegativepositionalexternalitiesonotherpeople.Withoutanyother distortion,afirstbestpolicywouldinthiscasebetotaxhousingwealthsuchthatthemarginaltaxexactlycorresponds tothevalueofthemarginalexternality.However,taxesonhousingwealtharepoliticallycontroversial,andthetaxrates observedinmanycountriesaremuchlowerthanaratethatwouldinternalizesuchapositionalityexternalityifcalculated basedonempiricalevidence.Wethereforeconsiderasecond-bestsolutionwheremarginaltaxes(orsubsidies)onsavings andlaborincomeareusedtosupplementthetaxonhousingwealth.Ouranalysisshowsthataslongastheexistingmarginal taxonhousingwealthfallsshortofthemarginalexternalitythatrelativehousingconsumptiongivesriseto,asecond-best optimalsolutionisachievedthroughamixofmarginalsavingsubsidiesandmarginallaborincometaxes.

Thereareseveralissuesworthfurtherexamination.First,althoughouranalysisbrieflyaddressesaltruismand intergen-erationaltransferofhousing,additionalquestionsremaintobeaddressed.Forinstance,iftaxesonhousingwealthcannot beusedtotheirfullpotential(ascenarioaddressedbothinSections3and4),taxesoninheritanceorgiftsalsoconstitute potentialsecondbestinstrumentsthroughwhichtocorrectforpositionalexternalities.Thisisclearlyaninterestingtopic forfutureresearch.Second,ifthetaxonhousingwealthisrestricted,aconsumptiontaxorsubsidyonthenon-positional goodmayserveasacomplementtotheincometaxpolicyexaminedabove.Suchanextensionisinterestinginitsownright, andwouldalsoallowustointegratetheanalysiscarriedoutherewithearlierworkreferredtointheintroduction.23_Third, althoughourmodeldistinguishesbetweenhomeownershipandrenting(whererentingisequivalenttothespecialcase wheretherateofdepreciationisequaltoone),itdoesnotallowforachoicebetweenthesetwopossibilities.Anobvious extensionwouldbetoconsideramodelwhereeachindividualcaneitherbuyorrentahome,andallowsuchchoicestobe associatedwithdifferentrelativeconcerns.Thiskindofanalysiswould(mostlikely)necessitateadiscretechoicemodeland isclearlycomplexenoughtomotivateitsownpaper.

Finally,wehaveassumedawaythatconspicuousconsumptionmayinfluenceprices;anissueofpotentialimportance forunderstandingthehousingmarket.Thissimplificationisofnomajorconcernforthequalitativeresultsderivedinthe presentpaper,sincetheefficiencycostsofrelativeconsumptionwouldbedrivenbythemechanismslaidoutabovealso inamoregeneralmodel.Priceformationis,nevertheless,importantforourunderstandingofwealthaccumulationand distributionmoregenerallyand,therefore,arelevanttopictoaddressinfutureresearch.Italsoopensupforstudyingthe interestingcasewhereindividualsarepositionalintermsofhousingwealth(orpossiblybothintermsofhousingattributes andwealth).Inourmodel,wherethepriceisfixed,itdoesnotmatterforthequalitativeresultswhetherindividualsare positionalintermsoftheattributes(orquantity),h,orintermsofhousingwealth,Ph.Inamoregeneralmodelwithan endogenoushousingprice,ontheotherhand,thisdistinctionwillmostlikelymatter,andmaypossiblystrengthentherole ofthemarginalsavingsubsidyandmarginallaborincometaxexaminedabove,asthereduceddemandforhousingislikely tocontributetoalowerpriceaswell.

Appendix.

ProofofProposition1. ByusingEq.(9),Eqs.(18a)and(18b)canberewrittenasfollows: ∂W ∂VtVt,b0= ∂W ∂VtVt,b0˛t MRSt_h,c 0 dht db0,t +t



1− tPt dht db0,t



(A1)

23 _{Wehavedonesomepreliminaryanalysishere.Moreprecisely,wehaveanalyzedwhethertheintroductionofasmalltaxorsubsidyonthenon-positional}

goodconsumedinthefirstperiodmayleadtohigherwelfare,giventhatthemarginalsavingsubsidyandmarginallaborincometaxareoptimallychosen (asdescribedinProposition1).Ifthemarginaltaxonhousingwealthfallsshortofthemarginalpositionalexternality,wefoundthatintroducingasmalltax onthenon-positionalgoodleadstohigherwelfareifitdecreasesthecompensateddemandforhousing.Thesecalculationsareavailablefromtheauthors uponrequest.

∂W ∂VtVt,b1= ∂W ∂VtVt,b0˛t MRSt_h,c 0 dht db1,t+1 −t tPt dht db1,t+1 +t+1. (A2) Eqs.(A1),(A2)and(18c)canbeusedtoderive

t+1



(1+rt+1)−Vt,b0 Vt,b1



= ∂W ∂VtVt,b0˛t MRSt_h,c 0



Vt,b0 Vt,b1 dht db1,t+1 − dht db0,t



−t tPt



Vt,b0 Vt,b1 dht db1,t+1− dht db0,t



(A3) whileEqs.(A1)and(18d)canbecombinedsuchthat

t



wt+ Vt,l Vt,b0



=− Vt,l Vt,b0



∂W ∂VtVt,b0˛t MRSt_h,c 0−t tPt



dht db0,t. (A4) Substituting(1+rt+1)−Vt,b0/Vt,b1=T1(strt+1)rt+1 intoEq.(A3),andwt+Vt,l/Vt,b0 =wtT0(yt)intoEq.(A4),whileusing MRSt_c0c1=Ut,c0/Ut,c1=Vt,b0/Vt,b1,MRS

t

l,c0=Ut,l/Ut,c0=Vt,l/Vt,b0,and˘t=(∂W/∂Vt)(Vt,b0/t+1),weobtainthemarginal

incometaxratesinProposition1.

Proof of Corollary 1. Ascan be seen from the two marginal tax formulas in Proposition 1, a sufficient condition forCorollary 1toholdisthat˛tMRS_h,ct ₀ > tPt implies˘t≥1+rt+1.Byusingt+1(1+rt+1)=tfromEq.(18c)and ˘t= (∂W/∂Vt)(Vt,b0/t+1)>0,thefirstorderconditionforb0,tasgivenbyEq.(18a)canberewrittenas

˘t



1−˛tMRSth,c0 dht db0,t



=(1+rt+1)



1− tPt dht db0,t



. (A5)

ByusingEq.(10a),wecanderive 1− tPt



dht db0,t



= dc0,t db0,t +Pt



dht db0,t



>0

sincebothgoodsarenormal,meaningthat1−˛tMRSt_h,c0(dht/db0,t)>0fromEq.(A5).Therefore,byreorganizingEq.(A5) suchthat ˘t 1+rt+1 = 1− tPt(dht/db0,t) 1−˛tMRSt_h,c0(dht/db0,t) (A6) wecanseethat˛tMRSt_h,c0 > tPtimplies˘t>1+rt+1.

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