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

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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

, Andrea Mannberg

aDepartmentofEconomics,UmeåSchoolofBusinessandEconomics,UmeåUniversity,SE-90187,Sweden

bTromsøUniversityBusinessSchool,9037Tromsø,Norway

a rt 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 ra c t

Thispaperanalyzeswhethermarginaltaxationoflaborandcapitalincomeareusefulsec- ondbestinstrumentsforinternalizingtheexternalitiescausedbyconspicuoushousing consumption,whenthegovernmentisunabletoimplementafirstbestcorrectivetaxon housingwealth.Therationaleforstudyingincometaxationinthisparticularcontextisthat firstbesttaxesonhousingwealthmaybeinfeasible(atleastinashortertimeperspec- tive),whileincometaxesindirectlyaffectboththelevelandcompositionofaccumulated wealth.Weshowthatasuboptimallylowtaxonhousingwealthprovidesanincentivefor thegovernmenttosubsidizefinancialsavingandtaxlaborincomeatthemargin.

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

1. Introduction

Thispaperexamineswhethermarginallaborandcapitalincometaxation/subsidizationmightbeusefulmeansofexter- 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).¹Ifconcernsforrelativeconsumptionare

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).

1AcademicworkonconspicuousconsumptiondatesbackatleasttoVeblen(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 (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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.²Therefore, 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.³Inaddition,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.⁵Therefore,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).⁶Inother

2 SeealsoZahirovic-HerbertandChatterjee(2011),whofindthatpeoplearewillingtopaymoreforhomeswithanameattachedtoit,andPatacchini andVenanzoni(2014),whofindsignificantpeer-effectsonthedemandforhousingqualityinUSA.

3 InSweden,forinstance,thetaxonhousingpropertyis0.75percentofthevalueofthepropertyuptoamaximumlimit(wherethevalueattachedtoeach propertybythetaxauthorityistypicallylowerthanthemarketvalue).Thecorrespondingrateisbetween0.2and0.7percentinNorway,wherethemunic- 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 (2008,2010,2014),WendnerandGoulder(2008),Knell(2010),andBilanciniandD’Antoni(2012).

words,mostpreviousworkdoesnotdistinguishbetweenpositionalandnon-positionalgoods,andthereissurprisingly littleresearchonhowtaxorotherpolicyinstrumentsoughttorespondtorelativeconcernsfordurableconsumption.One importantexceptionfromcategory(b)isEckerstorferandWendner(2013)dealingwithoptimallinearcommoditytaxation inatwo-goodeconomy,whereoneofthegoodsispositionalandtheothernon-positional.Theirmaincontributionisto examinetaxpolicyimplicationsofnon-atmosphericandasymmetricconsumptionexternalities,andbystudyingthejoint taxpolicyimplicationsofrelativeconsumptionconcernsandinequality-aversion.⁷Yet,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

7Amongotherthings,theyshowthatiftheexternalityisnon-atmospheric(meaning,intheirstudy,thatindividualsdifferintheircontributiontothe externality),theprincipleoftargetingisnolongerfullyapplicablesuchthatbothcommoditytaxestypicallyareusedtocorrectfortheexternalitycaused bytheconsumptionofthepositionalgood.Eckerstorfer(2014)extendstheanalysistoaproblemofoptimalmixedtaxationunderasymmetricinformation basedonastaticmodelwithtwonon-durableconsumptiongoods(inadditiontothelabor–leisurechoice),wherethegovernmentdecidesonanoptimal nonlinearincometaxandasetoflinearcommoditytaxes.

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

9Ourabstractionfromthepriceformationprocessandfromlocationissuesdoesnotmeanthatwethinktheseissuestobeunimportant.However, inthispaper,weareprimarilyinterestedinanalyzingtaxpolicytocounteractpositionalexternalities.Aslongastheconsumershavepreferencesfor theirrelativeconsumptionofhousing(whichisthebasicassumptiononwhichourpaperrests),thepolicyincentivescharacterizedbelowwouldalsobe presentinamoregeneralframework(althoughpossiblycombinedwithotherpolicyincentivesdependingonthefunctioningofthehousingmarket).The modelpresentedbelowisanalyticallyconvenientandallowsustoidentifybasicmechanismsbehindasecondbestoptimalpolicydesignedtocorrectfor positionalconsumptionexternalities.

consumption),whilethenon-durablegoodiscompletelynon-positional.¹⁰Individualsalsoderivedisutilityfromworkeffort.

Tosimplifythenotation(sincewearefocusingonefficiencyaspectsofpublicpolicy),wealsoassumethatindividualsofthe samegenerationareidenticalandnormalizethenumberofindividualspergenerationtoone.

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

Thelifetimeutilityfunctionfacinganindividualofgenerationtisgivenby

Ut=U(c0,t,lt,ht,h¯t,c1,t+1)=u0(c0,t)+(lt)+(ht)+˚(ht−h¯t)+u1(c1,t+1) (1) Eq.(1)meansthattheindividualderivesutilitybothfromhis/herabsoluteconsumption,ht,andrelativeconsumption, ht−h¯t,ofhousing,aswellasfromhis/herabsoluteconsumptionofthenon-durablegoodanduseofleisure.Althoughthe housingchoiceismadeduringthefirstperiod,theutilityderivedfromhousingcanbeinterpretedasreferringtoboth periodsoflife.Thereareseveralpossiblewaysofconceptualizingaunitofhousing,andonecan(atleast)thinkofheither asapurelyquantitativemeasure(e.g.,thenumberofsquaremeters)orintermsofunitsofqualityofhousing(e.g.,finishes andfixtures).¹¹Evenifbothinterpretationsmayberelevantfromtheperspectiveofrelativeconcerns,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,r_t+1,Pt,P_t+1,and ¯ht asexogenous. By substitutingthebudgetconstraintintotheobjectivefunctiontoreplacetheconsumptionofnon-durablegoods,weobtain adecision-problemins_t,h_t,andl_twiththefirstorderconditions¹³

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,U_t,h=∂Ut/∂ht,Ut,c1 =∂Ut/∂c1,t+1,andU_t,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).¹⁴Theoptimalmarginal taxratescanthenbederivedthroughacomparisonofsocialandprivatefirstorderconditions.Theresourceconstraintfor theeconomyasawholeinperiodtcanbewrittenas¹⁵

yt+kt(1+rt)−Pt(ht−ht−1(1−ı))−c0,t−c1,t−kt+1=0 (6) wherektisthephysicalcapitalstockdefinedsuchthats_t−1=ktforallt.Theresourceallocationpreferredbythegovernment canbederivedbychoosingc_0,t,lt,c_1,t,htandkt forallttomaximizethesocialwelfarefunctionsubjecttotheresource constraintand ¯ht=ht forallt.Thelatterfollowsbecausetheindividualsareidentical,andthe(externalitycorrecting) governmentrecognizeshowthereferenceconsumption isdetermined.TheLagrangeancorrespondingtothisdecision- problemisgivenby

L=W+

t

t

wheretistheLagrangemultiplieroftheresourceconstraintinperiodt.Define MRS^t_h,c

0= Ut,h

Ut,c0

>0, MRS^t_l,c

0= Ut,l

Ut,c0

<0, MRS^t_c0,c1=Ut,c0

Ut,c1

>0

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

MRS^t_c0,c1=(1+rt+1) (8a)

MRS^t_h,c

0+ U_t,h¯

Ut,c0

−Pt+ 1

MRS^t_c0,c1Pt+1(1−ı)=0 (8b)

−MRS^t_l,c

0=wt (8c)

Now, let t=ht−h¯t, and notice from Eq. (1) that U_t,h¯=−∂˚(t)/∂t=−˚t,, and U_t,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)=

˛tMRS^t_h,c

0forallt.

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

14See,e.g.,Stiglitz(1982).

15Notethattheanalysispresupposesthatht≥h_t−1(1−ı);otherwise,oldconsumerswouldnotbeabletoconsumealltheirhousingwealth.

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.¹⁷Ifwefurtherassumethatanindividualownsahousefor30years,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.¹⁸AscanbeseenfromObservation1,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)

whereb_0,tandb_1,t+1aretreatedasfixedincomesinthefirstandsecondperiodoflifeforgenerationt.Thisgivesthefirst ordercondition

−Ut,c0Pt(1+ t)+Ut,h+Ut,c1Pt+1(1−ı)=0 (11) whichimplicitlydefinestheindividual’sdemandforhousingasafunctionofPt(1+ t),P_t+1(1−ı),b_0,t,b_1,t+1and ¯ht,i.e.,

ht=h

Pt(1+ t),Pt+1(1−ı),b0,t,b1,t+1,h¯t

NoticethatEq.(12)isinterpretableasaconditionaldemandfunctioninthesenseofmeasuringthisdemandconditionalon theindividual’sincomeandsaving(whicharepartofb_0,tandb_1,t+1).BysubstitutingEqs.(10a),(10b)and(12)intoEq.(1), thecorrespondingconditionalindirectutilityfunctioncanbewrittenas

Vt=V

Pt(1+ t),P_t+1(1−ı),b0,t,b_1,t+1,lt,h¯t

b0,t−Pt(1+ t)ht,lt,ht,h¯t,b_1,t+1+P_t+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 secondperiod,thiscommoditytaxwouldstillbedesignedsuchthatthemarginalexternalityistaxedaway.

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)−T1(strt+1)rt+1]=0 (15a)

Vt,b0wt[1−T₀(yt)]+Vt,l=0 (15b)

inwhichV_t,b0=∂Vt/∂b0,t,V_t,b1 =∂Vt/∂b_1,t+1andV_t,l=∂V_t/∂l_t. 3.1. Secondbestoptimalincometaxation

Asexplainedabove,thesavingsandlaborsupplychoicesbygenerationtaffecttheconditionaldemandforhousingvia b0,tandb1,t+1,Itis,therefore,instructivetobeginbyderivingcomparativestaticsofthehousingdemandwithrespecttob0,t

andb_1,t+1.Sincethegovernmentaimsatinternalizingthepositionalexternalityandincorporatesintoitsdecision-problem that ¯ht=ht,wedifferentiateEq.(11)withrespecttoht,b_0,tandb_1,t+1,whileusing ¯ht=ht.Thisgives

dht

db0,t = Ut,c0c0Pt(1+ t)

>0 (16a)

dht

db1,t+1 =−Ut,c1c1P_t+1(1−ı)

<0 (16b)

whereUt,c0c0<0,Ut,c1c1<0and

=Ut,c0c0[Pt(1+ t)]²+Ut,hh+Ut,c1c1[P_t+1(1−ı)]²+U_t,hh¯<0. Doublesub-script(e.g.,c₀c₀)denotessecondorderpartialderivative.

Withafixedtaxonhousing,thesocialdecision-problemisasecondbestproblem,whichcanbewrittenas max

b0,tb1,t,k_t,l_t∀tW(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).¹⁹ByusingttodenotetheLagrangemultiplieroftheresourceconstraintinperiodt(asbefore), thesocialfirstorderconditionsforb_0,t,b_1,t+1,k_t+1,andlt(theconditionscharacterizinganoptimalallocationforgeneration t)become

b0,t:∂W

∂Vt

Vt,b0+V_t,h¯

dht

db0,t

+t

−1+ tPt dht

db0,t

=0 (18a)

b1,t+1:∂W

∂Vt

Vt,b1+V_t,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:

19Withgeneral,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.

(1994)intheirstudyonoptimalmixedtaxation.Itdoesnotmatterfortheresultsderivedbelowwhichoptionischosen.

Proposition1. Ifthemarginaltaxonhousingwealthisfixedat tforallt,theoptimalmarginalincometaxrateswilltakethe form

T₁(strt+1)rt+1=(1+rt+1) ˘t

1+rt+1˛tMRS^t_h,c

0− tPt MRS^t_c0,c1 dht

db_1,t+1 − dht

db0,t

wtT₀(yt)=−MRS^t_l,c

0

˘t

1+rt+1˛tMRS^t_h,c

0− tPt

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=˛tMRS^t_h,c

0 and˘t=1+r_t+1,weobtainT₁(str_t+1)=T₀(yt)=0.Thesetwospecialcasesalsosuggest thefollowingmoregeneralcorollarytoProposition1²⁰:

Corollary1. IfPt t<˛tMRS^t_h,c

0,Proposition1impliesT₁(str_t+1)<0andT₀(yt)>0.

Proof. SeeAppendix.

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

ThemultiplierMRS^t_c0,c1(dht/db1,t+1)−dht/db0,t<0inthemarginalcapitalincometaxformulainProposition1appears becausethecapitalincomesubsidyconstitutesanindirectinstrumenttoreducethehousingwealth,andworksthroughthe effectofsavingsonthehousingdemand.Assuch,themoreanincreaseinsavingsreducesthedemandforhousing(through adecreaseinb_0,tandcorrespondingincreaseinb_1,t+1),themoreeffectivewillbethesavingssubsidyasaninstrumentto reducethepositionalexternality.Ontheotherhand,ifdht/db_0,tanddht/db_1,t+1areclosetozero,thesavingssubsidymaynot beausefulinstrumenttoinfluencetheaccumulationofhousingwealth,inwhichcasethesubsidywouldbeclosetozero ornotusedatall.Themultiplierinthemarginallaborincometaxformula,dht/db_0,t>0,hasananalogousinterpretation.

Finally,letusbrieflyreturntothespecialcasediscussedinSection2whereı=1,whichisinterpretabletomeanthat individualsrenttheirhomesinsteadofowningthem.Notethatthisspecialcasegivesthesamequalitativeresultsasthose derivedinProposition1andCorollary1.Weshouldstillsubsidizethesavingandtaxthelaborincomeatthemarginunderthe samegeneralconditionstocounteractthepositionalconsumptionexternality.However,whilethepolicyruleformarginal laborincometaxationwouldbeidenticaltothatderivedintheproposition,sincethisformuladoesnotdependdirectlyonı, themarginalsavingsubsidyislikelytobesmallerifindividualsrenttheirhomesbecausedh_t/db_1,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 IfPt t>˛tMRS^t_h,c

0,Proposition1insteadimpliesthatT₁(strt+1)>0andT₀(yt)<0.However,thiswouldimplyascenariowherethegovernmentsets thehousingtaxhigherthanthevalueofthemarginalexternality.Thisseemsunrealisticsincethegovernmentinthiscasecouldchoosethefirstbestoptimal taxinsteadofthesecondbestsolution.Wethereforedonotconsiderthisscenarioinouranalysis.