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How Should Financial Intermediation Services be Taxed Ben Lockwood First version

When 100 taxation of pro t is available taxes on services supplied to rms should be deductible from pro t implying the optimality of a VATtype tax As for the rate of tax in the steady state an optimal arrange ment is to set it equal to the rate of t

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How Should Financial Intermediation Services be Taxed Ben Lockwood First version






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HowShouldFinancialIntermediationServicesbe Taxed? ¤ BenLockwood y Firstversion:May312010 Thisversion:30September2013 Abstract Thispaperconsiderstheoptimaltaxationofsavingsintermediationservicesin adynamicgeneralequilibriumsetting,whenthegovernmentcanalsouseconsump- tion,incomeandpro … ttaxes.When100%taxationofpro … tisavailable,taxeson servicessuppliedto … rmsshouldbedeductiblefrompro … t,implyingtheoptimality ofaVAT-typetax.Asforthe rate oftax,inthesteadystate,anoptimalarrange- mentistosetitequaltotherateoftaxoncapitalincome, not consumption.In turn,thecapitalincometaxiszerowhenthewhenanunrestrictedpro … ttaxis available,butinthemorerealisticcasewhensuchataxisnotavailable,thisrate canbepositiveornegative,butgenerallydi¤erenttotheoptimalrateoftaxon consumption. JELClassi … cation :G21,H21,H25 Keywords : … nancialintermediationservices,taxdesign,banks,paymentservices ¤ Thispaperisasubstantiallyrevisedversionofpartofanearlierpaperofthesametitle,Lock- wood(2010).IwouldliketothankSteveBond,MichaelDevereux,ClemensFuest,MiltosMakris,Michael McMahon,RuuddeMooij,DavidRonayneandseminarparticipantsattheUniversityofSouthampton, GREQAM,the2010CBTSummerSymposium,andthe2011IIPFConferenceforhelpfulcommentson anearlierdraft.IalsogratefullyackowledgesupportfromtheESRCgrantRES-060-25-0033,"Business, TaxandWelfare". y CBT,CEPRandDepartmentofEconomics,UniversityofWarwick,CoventryCV47AL,England; Email:B.Lockwood@warwick.ac.uk 1.Introduction Financialintermediationservicesincludesuchimportantservicesasintermediationbe- tweenborrowersandlenders,insurance,andpaymentservices(e.g.creditanddebitcard services).Theseservicescompriseasigni … cantandgrowingpartofthenationaleconomy. Forexample, … nancialintermediationservicesasconventionallyde … nedinthenational accountsincludesactivitiessuchasthetakingofdepositsandthegrantingofcredit, … nancialleasing,investmentinsecuritiesandpropertiesby … nancialintermediaries,in- suranceandpensionfunding,andservicesancillaryto … nancialintermediation 1 .TheEU KLEMSdatabaseshowsthatthissectorcomprised6.5%ofnationaloutputintheUKin 1997,increasingto7.9%by2007.The … guresfortheUS,usingthesamede … nitionof … nancialintermediationservices,are7.3%in1997,risingto8.6%andfortheEurozone, 4.8%,risingto5.3% 2 .Evenexcludinginsurance-whichisbeyondthescopeofthispaper - … nancialintermediationisquantitativelyimportantinOECDcountries 3 . Thequestionofwhether,andhow, … nancialintermediationservicesshouldbetaxed isacontentiousone 4 .Forexample,withinEuropeanUnioncountries,most … nancial servicesarecurrentlyexemptfromVAT,andthereisconsiderabledebateaboutthe possiblebene … tsfrombringingthemintotheVATsystem(delaFeriaandLockwood (2010)).Also,therecentIMFproposalsfora"banktax"tocoverthecostofgovernment interventionsinthebankingsystemincludeaFinancialActivitiesTaxleviedonbank pro … tsandremuneration,oneversionofwhich-FAT1-whichwouldworkverymuchlike aVAT,leviedusingtheadditionmethod(IMF(2010)). Inthepolicyliteratureonthistopic,itislargelyassumedthatwithinaconsump- tiontaxsystem,suchasaVAT,itisdesirabletotax … nancialservicessuppliedatthe standardrateofVAT,andallowprovidersofintermediationservicestoclaimbackVAT theypayoninputs:seee.g.Ebril,Keen,Bodin,andSummers(2001)).However,this policyprescriptionisatvariancewithasmallacademicliteratureonthistopic(Grubert andMackie(1999),Jack(1999),andBoadwayandKeen(2003)),whichsuggeststhatwhile 1 Financialintermediationcomprisesactivities65,66,67intheISIC/NACEsystemofnationalaccounts. Thede … nitionoftheseactivitiescanbefoundforexample,inthehandbookNACE:REV.1,published byEurostat. 2 Authors ’ calculations: … nancialintermediationcompriseslinesJ65-67intheEUKLEMSGrowth andProductivityAccounts(http://www.euklems.net/index.html). 3 Datashowingthesizeofintermediationservicesrelatingtothetakingofdepositsandgrantingof credit only arepresentedfortheUKinSection1.2below. 4 Therearetechnicaldi¢cultiesintaxing … nancialintermediationservices;however,thesedi¢culties arenotinsurmountable-seeSection1.1below. 2 paymentservicesshouldbetaxedatthesamerateasconsumption,intermediationbe- tweenborrowerandlendershouldnotbetaxedatall.However,thisliteratureisbasedon … rst-bestargumentsi.e. … ndingthetaxarrangementthatdoesnotdriveawedgebetween thehouseholdmarginalrateofsubstitutionandthemarginalrateoftransformationin production. Theobjectiveofthispaperistotakeafreshlookatthisquestion,fromataxdesign pointofview.Ourfocusonthemostimportantintermediationservice-intermediation betweenborrowersandlenders 5 .Wesetupandsolvethetaxdesignprobleminadynamic generalequilibriummodeloftheChamley(1986)type,wherethegovernmentchoosesa taxonsavingsintermediation,aswellastheusualtaxesonconsumption(orequivalently, wageincome)andincomefromcapital,to … nanceapublicgood,andwhere … nancial intermediaries,intheformofbanks,areexplicitlymodelled.Realistically,weassume thatsavingsintermediationisnotexplicitlypriced,butchargedforviaaspreadbetween borrowingandlendingratessetbycompetitivebanks.Thisspreadcanbetaxed,atarate thatmaybedi¤erentfromthetaxonconsumption,andthetaxsystemisparameterized sothatsomefraction ofthetaxpaidby … rmson … nancialintermediationinputscanbe creditedagainsttheconsumptiontaxchangedby … rms.InthecaseofaVAT, =1 Themainresultsareasfollows.First,thetaxpaidby … rmson … nancialintermediation inputsshouldbefullycreditedi.e.thetaxshouldbeaVAT,when100%taxationofpro … t ispossible 6 .ThisisanexampleofthegeneralDiamond-Mirrleesproductione¢ciency resultthatintermediateinputsshouldnotbetaxedundertheseconditions.But,atwhat rateshouldtheVATbeset?Here,therearetwomain … ndings.First,theoptimaltax structureisindeterminate,becausethegovernmenthastwoinstruments,acapitalincome taxanda … nancialintermediationtax,tocontrolthemarginalrateofsubstitutionbetween consumptioninsuccessiveperiods.However,itturnsoutthatfromaninformationalpoint ofview,thesimplestoptimaltaxstructureiswherecapitalincometaxanda … nancial intermediationtaxareequal. Inparticular,when100%taxationofpurepro … tispossible,thesimplestoptimaltax structureistosetboththetaxrateoncapitalincome,andthetaxrateon … nancial intermediationservicesequaltozero.Inthemorerealisticcasewhenthereisanupper boundontherateofpro … toflessthan100%,weshowthatasimpleoptimaltaxstructure isagaintosetthetwotaxesatthesamerate.Thesignofthiscommonratethendepends 5 Istudytaxationofpaymentservicesinacompanionpaper,LockwoodandYerushalmi(2013). 6 Firmsmustmakepurepro … tinequilibrium,becausetheymusthavedecreasingreturns,inturn becausetheypossiblyfacedi¤erentborrowingcosts. 3 onthepropertiesoftheproductionfunction;itcanbepositiveornegative.Moreover, thiscommonrateisgenerallydi¤erentfromtheoptimaltaxrateonconsumption. Theseresultsarequitedi¤erentfromtheexistingliterature(seeSection1.2)which generally … ndsthat … nancialintermediationservicesshouldbeuntaxed.However,these are … rst-bestmodels,wherethereis(implicitly)norevenuerequirement.There,the optimalityofleaving … nancialintermediationservicesuntaxedisderivedjustfromthe conditionthatthemarginalrateofsubstitutioninconsumptionisequaltothemarginal rateoftransformation.Ourresultsalsodi¤erfromAuerbachandGordon(2002),which … nds,usingaratherdi¤erentargument,that … nancialintermediationservicesshouldbe taxedatthesamerateasconsumption. Theremainderofthepaperisorganizedasfollows.Section1.1discussessomebasic factsaboutthetaxationof … nancialintermediation,andSection1.2discussesrelated literature.Section2outlinesthemodelandSection3presentsthemainresults.Section 4considersthecasewithout100%pro … ttaxation,Section5considersotherextensions, andSection6concludes. 1.1.TheSizeandTaxTreatmentofFinancialIntermediationServices The … guresquotedatthebeginningofthepapermeasuretheoverallsizeofthe … nancial intermediationsector.Togetanideaofthevalueof … nancialintermediationassociated withthetakingofdepositsandthegrantofcreditonly,wecanlookatFISIM 7 .FISIM iscomputedfromthetransactionsbetweenthebankingsectorandothersectorsofthe economy(forexample,non- … nancial … rmsandhouseholds).Foreachofthesesectors, theloansfrom,anddepositswith,thebankingsector,aremeasuredandthemargins madebythebankingsectorontheseactivitiesarecalculated.Speci … cally,themarginper currencyunitdepositedisareferencerateminustheaveragerateofinterestondeposits, andthemarginpercurrencyunitlentistheaveragerateofinterestonloans,minusa referencerate(Akritidis(2007)).So,FISIMcanbecalculatedbysector,andalsoonloans anddepositsseparately.Asourfocusisprimarilyontaxationofthehouseholdsector,we showconsumptionofFISIMbyhouseholdsandnon-pro … tinstitutionsservinghouseholds fortheUKovertheperiod1997to2012. 7 Thisisanacronymfor" … nancialintermediationservicesindirectlymeasured". 4 Figure1:FISIMConsumedbyHouseholdsasa PercentageofHouseholdConsumptionExpenditure,UK .0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 1997199819992000200120022003200420052006200720082009201020112012 Source:O¢ceofNationalStatistics,UK.ThechartshowsFISIMonloans(seriesIV8X) anddeposits(seriesIV8W)consumedbyhouseholdsandnon-pro … tinstitutions servinghouseholdsasapercentageofaggregateconsumption(seriesRPQM) ThisshowsthatconsumptionofFISIMbyhouseholdsisbetween3%and4%oftotal householdexpenditureintheUKoverthisperiod;notlarge,butnotanegligiblefraction, either.Alsothe … nancialcrisishashadaspositiveimpactonthis … gure,asbankshave increasedtheirspreadsonloanstorepairtheirbalancesheets.So,overall,itcanbeseen that … nancialintermediationservicesareasigni … cantandgrowingpartoftheeconomy intheUK.ThepictureisthesameforotherOECDcountrieswithadeveloped … nancial sector. Asregardsthetaxationof … nancialintermediationservices,inourtheoreticalanalysis below,weassumethatintermediationservicescanbetaxed;speci … cally,thatbankscan chargetaxestohouseholdsand … rmsseparatelyontheirconsumptionofintermediation services.Itisrecognizedthatinpracticetherearetechnicaldi¢cultieswhenthoseservices arenotexplicitlypriced(so-calledmargin-basedservices),becauseitisnotstraightforward todividethe"value-added"betweenborrowerandlender.Inparticular,thisraisesa problemfortheuseofaVATviatheusualinvoice-creditmethod(Ebril,Keen,Bodin andSummers(2001)).Asaresult,thestatusquoinmostcountriesisthatawiderange 5 of … nancialintermediationservicesarenottaxed.Forexample,intheEU,manysuch servicesareexemptasaresultofthe6thVATdirective 8 . However,conceptually,theproblemscanbesolvedinseveraldi¤erentways.One administrativelystraightforwardsystemwouldbetozero-ratesalestoVAT-registered entitiesandtaxsalestonon-registeredentitiese.g.householdsonanaggregatebasis (Huizinga(2002)).Alternatively,PoddarandEnglish(1997)haveproposedacash- ‡ ow VATwithtaxcalculationaccounts;thisisadministrativelymorecomplex,buttheincreas- ingsophisticationofbanks ’ ITsystemsmeansthatthissolutionisbecomingpractical.A recentstudybytheEuropeanCommissioncalculatesthatEU-27taxrevenuemightrise byaround£15billionEuroifintermediationserviceswerebroughtintotheVATsystem, andtaxedatthestandardrate(EuropeanCommission(2011)). 1.2.RelatedLiterature Thereisasmallliteraturedirectlyaddressingtheoptimaltaxationofborrower-lender intermediationandpaymentservices,GrubertandMackie(1999),Jack(1999),andBoad- wayandKeen(2003).Usingasimpletwo-periodconsumption-savingsmodel,thesepapers agreeonapolicyprescription 9 .Givenaconsumptiontaxthatisuniformovertime,pay- mentservicesshouldbetaxedatthisuniformrate,butsavingsintermediationshouldbe leftuntaxed.Theargumentusedtoestablishthisissimple;inatwo-periodconsumption- savingsmodelwiththesame,exogenously … xed,taxonconsumptioninbothperiods, thisarrangementleavesthemarginalrateofsubstitutionbetweencurrentandfuturecon- sumptionundistortedi.e.equaltothemarginalrateoftransformationinproduction. Usingadi¤erentapproach,AuerbachandGordon(2002)donotmakeasharpdistinc- tionbetweenpaymentservicesandsavingsintermediation,andarguethatbothactivities shouldbetaxedatthesamerateasconsumption 10 .Moreprecisely,theyshowthata wagetaxisequivalenttoauniformtaxonconsumptionandintermediationservices. 8 TheSixthVATDirectiveandsubsequentlegislationexemptsawiderangeof … nancialservicesfrom VAT,includinginsuranceandreinsurancetransactions,thegrantingandthenegotiationofcredit,trans- actionsconcerningdepositandcurrentaccounts,payments,transfers,debts,cheques,currency,bank notesandcoinsusedaslegaltenderetc.(CouncilDirective2006/112/ECof28November2006,Article 135). 9 ChiaandWhalley(1999),usingacomputationalapproach,reachtheratherdi¤erentconclusionthat paymentservicesshouldbeuntaxed,butbuttheirmodelisnotdirectlycomparabletotheseothers,as theintermediationcostsareassemedtobeproportionaltothe price ofthegoodsbeingtransacted. 10 AuerbachandGordon(2002)state:"transactionscostscanincludetherealresources...lostwhen investingthesefundssothattheywillbeavailableinalaterperiod"(AuerbachandGordon(2002), p412). 6 However,onecanmakethreecriticismsofthisliterature.First,taxesaretakenas given.Inparticular,consumptiontaxesareassumedequalinbothperiods,andcapital incometaxesaresettozero.Combinedwiththe(implicit)assumptionof … xedlabour supplyinthosemodels 11 ,thetaxsystem(otherthantaxeson … nancialservices)amounts toanon-distortionarytaxonlabourincome.Inthissetting,itisofcourse,optimal formarginalrateofsubstitutioninconsumptiontobeequaltothemarginalrateof transformationinproduction.Itisthennotverysurprisingthatthetaxonborrower- lenderintermediationshouldbezero.Second,astaxesarenotdistortionary,thereisno second-besttaxdesignproblem,sothequestionoftradingo¤distortionsgeneratedby ataxon … nancialservicesagainstotherdistortionarytaxesdoesnotarise.Third,the productionsideoftheeconomyisnotexplicitlymodelled,sothatquestionsofdistortions ininputpricescausedbytaxeson … nancialservicescannotbeaddressed. Second,thereisalsoalesscloselyrelatedliteratureontheuseoftaxationtocontrol "badbanks".Theideahereisthatwhilebanksmayengageinsociallyundesirableactiv- itiesonbothlendinganddeposit-takingmargins,theseshouldbecorrectedbyPigouvian taxes(orregulations)thatapplydirectlytothesedecisionmargins.Therehasrecently beensurgeofliteratureonsuchPigouviantaxes;seee.g.Acharyaet.al.(2010),Bianchi andMendoza(2010),Keen(2010),PerrottiandSuarez(2011),Coulter,Mayer,andVick- ers(2012).Inourmodel,banksaremerelyproducersthatpriceintermediationservices atmarginalcost,sothereisnoroleforPigouviantaxes. 2.TheModel 2.1.Households ThemodelisaversionofAtkeson,ChariandKehoe(1999)withsavingsintermediation bybanksaddedtothebasicstructure.Thereisasinglein … nitelylivedhouseholdwith preferencesoverlevelsofasingleconsumption-capitalgood,leisure,andapublicgoodin eachperiod =0 1 oftheform 1 X =0 ( ( )+ ( )) (2.1) where isthelevelofconsumptioninperiod 2 [0 1] isthesupplyoflabourhours, and ispublicgoodprovision.Utility ( ) isstrictlyincreasingin strictlydecreasing 11 TheexceptionhereisAuerbachandGordon(2002),wherelaboursupplyisvariable.However,in theirmodel,theconsumptiontaxisjustassumedtobeuniform,notoptimised. 7 in andstrictlyconcave,and ( ) isstrictlyincreasingandstrictlyconcavein Finally, 0 1 isadiscountfactor. Inanyperiod thehouseholdisassumedtopayanadvaloremtax on andalso paysproportionaltaxesonlabourandcapitalincome.Usingthewell-knownfactthat aconsumptiontaxisequivalenttoawagetax,weassumew.l.o.g.thatthewagetaxis zero Finally,forthemoment,wesupposethatthehouseholdhasnopro … tincomeinany period: … rmsgeneratepurepro … ts(forreasonsexplainedinSection3.2below),butthese aretaxedat100%.So,inanyperiod,thebudgetconstraintis (1+ )+ +1 = +(1+ ) where isthepost-taxreturntothehouseholdonsavings,and isthewage,and +1 issavings.Finally, =(1 ¡ ) where isthepre-taxreturnonsavingsforthe household,determinedbelow,and isthecapitalincometax. So,followingAtkeson,ChariandKehoe(1999),thepresentvaluebudgetconstraintof thehouseholdcanbeobtainedbyaggregatingoverper-periodbudgetconstraints: 1 X =0 ( (1+ )+ +1 )= 1 X =0 ( +(1+ ) ) (2.2) where isthepriceofoutputinperiod Wenormalizebysetting 0 =1 andassume forconveniencethat 0 =0 i.e.initialcapitaliszero 12 The … rst-orderconditionsfora maximumof(2.1)subjectto(2.2)withrespectto +1 respectivelyare: = (1+ ) (2.3) ¡ = (2.4) =(1+ +1 ) +1 (2.5) where isthemultiplieron(2.2),andweuse(hereandbelow)thenotationthatfor anyanyfunction ( ) ,thepartialderivativeof withrespectto is the cross-derivativeis etc. 2.2.Banks Forsimplicity,weassume100%depreciationofthecapitalgood.So,inthestandard versionofthismodel,without … nancialintermediation,thehouseholdprovides unitsof 12 Thisimpliesthatthegovernmentcannotseta … rst-periodcapitallevyon … xedcapital 0 andthus simpli … estheanalysis(seeAtkeson,ChariandKehoe(1999)). 8 theconsumption-capitalgoodtothe … rmsinperiod ¡ 1 ,and … rms(inaggregate)repay the unitsofconsumption-capitalgoodtothehouseholdinperiod plusinterest In ourversionofthemodel,the unitsoftheconsumption-capitalgoodaredepositedwith abankat ¡ 1 whocanthenprovidethisstock to … rmsasaninputtoproduction. The … rmsthenrepay tothebankat ,plusinterest,and … nally,thebankrepays to thehouseholds,plusinterest. Thecostofintermediatingoneunitofsavingsbetweenthehouseholdand … rm is ~ unitsoflabour.Notethatwetake ~ as … xed,butpossiblyvaryingbetween … rms.This isrealistic;lendingisacomplexprocessinvolvinginitialassessmentoftheborrowervia e.g.creditscoring,structuringandpricingtheloan,andmonitoringcompliancewithloan covenants(GupandKolari(2005,chapter9). Wealsosupposethatthetotalintermediationcost ~ canbedividedbythebank betweenthecostofservicesprovidedtothehousehold, (e.g.safekeepingofdeposits, liquidity)andcostofservicesprovidedto … rm (e.g.monitoring), i.e. ~ = + .Thisisarealisticassumption,becauseinpractice,anyscheme(suchasthetax calculationaccountsofPoddarandEnglish(1997))whichimplementsaVATonmargin- based … nancialintermediationservicesmustsplitthe"value-added"betweenborrower andlender.Weassumethatbankscanborrowandlendata"pure"rateofinterest whichwilleventuallybedeterminedinequilibriuminthecapitalmarket.Finally,we assumethatbanksareperfectlycompetitive,whichcombinedwiththeconstantreturns intermediationtechnology,impliesthattheymakezeropro … tinequilibrium. Finally,weassumethatataxonintermediationservices,orspreadtax,isinoper- ationatrate whichcanbedi¤erentfromtherate on … nalconsumption.Banks mustbreakevenoneachofthetwoactivitiesofprovidingservicestohouseholds,and to … rmsseparately,otherwiseacompetitorbankcouldpro … tablyundercutthem.So, inequilibrium,thecostsofintermediation plusthetaxpaid,areequaltothe spreads ¡ ¡ respectively,where istherateatwhich … rm borrows,and istherateofreturnonsavingsforthehousehold.Thatis: ¡ = (1+ ) ¡ = (1+ ) (2.6) Thisisintuitive:thespreadonbothborrowingandlendingisequaltotherealresource costplusbythetax. 9 2.3.Firms Thereare … rms =1 whichproducethehomogenousconsumptiongoodineachpe- riod 13 Firm producesoutputfromlabourandcapitalviathestrictlyconcaveproduction function ( ) where arecapitalandlabourinputs 14 .Because … rmsmaydi¤er inintermediationcosts … rmsfacedi¤erencesinthecostofcapitali.e. … rm must repay 1+ perunitofcapitalborrowedfromthebank.So,thepro … tof … rm is ( ) ¡ ¡ (1+ ) + where isthetotaltaxpaidonintermediationservicesbythe … rm,and isthe fractionofthetaxpaidonintermediationservicesthatthe … rmcanclaimagainstthe taxpaidonoutput.Ofcourse,intheusualVATsystem, =1 So,substituting from (2.6)pro … tcanbewritten ( ) ¡ ¡ (1+ + (1+(1 ¡ ) )) (2.7) Maximizing(2.7)withrespectto impliesthe … rst-orderconditions: ( )= ( )=1+~ (2.8) where ~ = + (1+(1 ¡ ) ) isthecostofcapitalfor … rm Finally,thecapital andlabourmarketclearingconditionsare: X =1 = X =1 + + X =1 = (2.9) Theseconditions(2.8),(2.9)jointlydetermine and ,givenhouseholdsavingsand laboursupplydecisions. 2.4.Discussion Theabovemodelprovidesageneralframeworkwhichencompassessomeaspectsofthe speci … cmodelsoftaxationof … nancialservices(AuerbachandGordon(2002),Boadway andKeen(2003)),Jack(1999),GrubertandMackie(1999))thathavebeendevelopedso 13 Wealsoassumeforconveniencethatoneunitoftheconsumptiongoodcanbetransformedintoone unitofthepublicgood.This … xestherelativepre-taxpriceof and atunity. 14 Weassumethat … rmsfacedecreasingreturns,becausewithdi¤erentcostsofcapital,andthesame wage,withconstantreturns,onlytheone … rmwiththelowestunitcostwouldoperate,andthiscaseis oflimitedinterest. 10 far.Speci … cally,ignoringpaymentservices,whicharenotdealtwithinthispaper,Boad- wayandKeen(2003)),Jack(1999),GrubertandMackie(1999)aretwo-periodversionsof theabovemodel 15 ,with … xedtaxesand(implicitly) … xedlaboursupply.Auerbachand Gordon(2002)isa … nite-horizonversionofthemodel,withtheadditionalfeature 16 that thereare consumptiongoodsineachperiod.Paymentservicesaredealtwithina companionpaper,LockwoodandYerushalmi(2013). 3.TaxDesign Wetakeaprimalapproachtothetaxdesignproblem.Inthisapproach,anoptimal policyforthegovernmentisachoiceofalltheprimalvariablesinthemodel,inthiscase © +1 ( ) =1 ª 1 =0 tomaximizeutility(2.1)subjecttothecapitalandlabour marketclearingconditions(2.9),aggregateresource,andimplementabilityconstraints. Wearethusassuming,followingChamley(1986),thatthegovernmentcanpre-committo apolicyat =0 Theaggregateresourceconstraintsaysthattotalproductionmustequal tothesumoftheusestowhichthatproductionisput: + +1 + = X =1 ( ) =0 1 (3.1) Theimplementabilityconstraintensuresthatthegovernment ’ schoicesalsosolvethe householdoptimizationproblem,andisinourmodel,quitestandard.Weobtainit bysubstitutingthehousehold ’ s … rst-orderconditions(2.3)-(2.5)in(2.2).Aftersome rearrangement,thisgivesthecondition: 1 X =0 ( + )=0 (3.2) Asisstandardintheprimalapproachtotaxdesign,wecanincorporatetheimple- mentabilityconstraint(3.2)intothegovernment ’ smaximandbywriting = ( )+ ( )+ ( + ) (3.3) where istheLagrangemultiplieron(3.2).If · 0 i.e.consumptionandleisureare complements,itispossibletoshowthat ¸ 0 atthesolutiontothistaxdesignproblem 15 Aminorquali … cationhereisthatBoadwayandKeenallowfora … xedcostofsavingsintermediation e.g. … xedcostsofopeningasavingsaccount.Theseintroduceanon-convexityintohouseholddecision- making,whichgreatlycomplicatestheoptimaltaxproblem,andsoweabstractfromtheseinthispaper. 16 Italsohaslaboursupplyinonlyoneperiod. 11 (seeAppendix).If =0 therevenuefrompro … ttaxationissu¢cienttofundthepublic good, Wewillruleoutthisuninterestingcase,andsowillassumethat 0 atthe optimuminwhatfollows Thegovernment ’ schoiceofprimalvariablesmustmaximize P 1 =0 subjectto(3.1) and(2.9).The … rst-orderconditionswithrespectto are,respectively ; = (3.4) ¡ = (3.5) = ¡ 1 + (3.6) = (3.7) = + =1 (3.8) = =1 (3.9) where arethemultipliersontheresourceconstraintandthecapitalandlabour marketclearingconditionsattime respectively. Moreover,from(3.3), = (1+ (1+ )) = + (3.10) and = (1+ (1+ )) = + (3.11) Here, aregivenbystandardformulaefound,forexample,intheprimalapproach tothestatictaxdesignproblem(AtkinsonandStiglitz(1980)).Inparticular, ¡ 0 underourassumption · 0 anditsmagnitudemeasuresthedegreeofcomplementarity betweenconsumptionandleisure. Webeginbycharacterizingtherateoftaxontheconsumptiongoodviathefollowing result,whichisprovedintheAppendix: Proposition1. Atanydate =0 1 2 theoptimaltaxon … nalconsumptioninad valoremformis 1+ = µ ¡ ¶µ ¡ 1+ ¶ = ¡ (3.12) Notethat(3.12)isaformulaforanoptimalconsumptiontaxthatalsooccursinthe staticoptimaltaxproblem,whentheprimalapproachisused(AtkinsonandStiglitz(1980, p377).Inparticular, isthemarginalbene … tof$1tothegovernment,and isa measureofthemarginalutilityof$1tothehousehold,so ¡ isameasureofthesocial gainfromadditionaltaxationatthemargin.Itsaysthatotherthingsequal,thehigher 12 thedegreeofcomplementaritybetweenconsumptionandleisure, ¡ thehigheris Notealsothatbyourassumptionthat · 0 0 Nowweturntoconsiderthequestionofwhethertaxpaidon … nancialintermediation shouldbedeductibleby … rmsi.e.thechoiceof From(3.8),(3.9),weseethatthe marginalrateofsubstitutionbetweenlabourandcapitalis = + (3.13) whichimpliesthat ¡ = ¡ (3.14) However,fromthe … rst-orderconditionsforthe … rm,(2.8),weseethat = 1+ + (1+(1 ¡ ) ) (3.15) whichimpliesthat ¡ =( ¡ )(1+(1 ¡ ) ) (3.16) If 6 = and 6 =0 equations(3.14),(3.16)canonlyholdsimultaneouslyif =1 So, wehaveshown: Proposition2. Ifthereisheterogeneityinintermediationcosts, ( 6 = some ) and therateoftaxon … nancialservices 6 =0 thenanydate e¢ciencyrequires =1 i.e. fulldeductibilityof by … rms. Theintuitionforthisresultisclear.Equation(3.14)saysthatthemarginalproduct ofcapitalnetoftrueintermediationcostsshouldbeequalacross … rms,whichofcourseis justtheconditionforcapitaltobeallocatede¢cientlyacross … rms.But,condition(3.14) isgenerallynotconsistentwithanon-zero when … rmsareheterogenous.Thisisjust aninstanceoftheDiamond-Mirrleesproductione¢ciencytheorem.Ataxonthebank marginisanintermediatetaxontheallocationofcapital,andgivenourassumptions(a fullsetoftaxinstruments,andnopurepro … ts),thistaxshouldbesettozero.Note alsothatwhenthereisonlyone … rm,thisargumenthasnobite,andthus isleft indeterminate. Wenowturntothequestionofhowthetaxesoncapitalincome, andoninter- mediationservices, shouldbeset.Generally,itcanbeshown,bystraightforward 13 manipulationofthe … rst-orderconditionstotheoptimaltaxproblemandthehousehold and … rmproblems,that: Proposition3. Atalldates =1 2 solve ¡ 1+ ¡ ¢ =1+(1 ¡ )( ¡ (1+ )) (3.17) where = 1+ (1+ ) 1+ (1+ ¡ 1 ) 1+ 1+ ¡ 1 TheproofofthisisintheAppendix.Clearly, arenotuniquelydetermined fromthissinglecondition.Ultimately,thisisbecausetheplannerhastwoinstruments, forcontrollingthemarginalrateofsubstitutionbetweenconsumptionat and +1 However,wecanusethefollowingcriteriontochoosebetweensolutions.Saythata solution to(3.17)is simple ifitsolves(3.17)independentlyoftheprecisevaluesof theeconomicdata Asthetaxauthorityisunlikelytoknowthesevalues,or atleasttosettaxesconditionalonthem,asimplesolutionisadministrativelyconvenient. Nowconsiderthesteadystate.Then, =1 andtherefore(3.17)becomes ¡ 1+ ¡ ¢ =1+(1 ¡ )( ¡ (1+ )) (3.18) Byinspection,theonlysimplesolutionto(3.18)is = =0 Theresultthattheinterest incometaxiszeroisofcourse,theclassicresultofChamley(1986)andJudd(1985);our newresultisthattherateoftaxon … nancialservicesshouldbezero.So,wehaveproved: Proposition4. Atthesteadystate,theonlysimpletaxsystemiswherethetaxon interestincome, andthetaxon … nancialintermediationservices, arebothzero. Twocommentscanbemadehere.First,awayfromthesteadystate, = =0 isgenerallynotoptimal.So,Proposition4-alongwithProposition6below-makes precisetheconditionsunderwhichtheresultoftheexistingliteraturethatsavingsinter- mediationshouldnotbetaxedgeneralizestoasecond-bestenvironment;zerotaxationof intermediationservicesrequires(a)unrestrictedtaxationofpro … t,and(b)asteadystate. Second,thecelebratedresultofChamley(1986)andJudd(1985)thatinthesteadystate, thetaxoncapitalincomeiszerodoesnotholdpreciselyinourmodel,astheplanner hasanadditionaltaxinstrument, However,zerotaxationofcapitalemergesifwealso imposeadministrativesimplicity. 4.LessthanFullTaxationofPurePro … t Recallthatwehavetoassumethat … rmshavedecreasingreturns,becausetheyface di¤erentcostsofcapital.Therefore,theygeneratepurepro … t.Sofar,wehavemade 14 thestrongassumptionthat100%taxationofthispro … tispossibleforthegovernment. However,iswell-knownthatthisisakeyassumptionbehindtheclassicalDiamond- Mirrleesresultthatinputsarenottaxedatthesecond-bestoptimum.Here,weinvestigate towhatextentourresultsgeneralizetothemorerealisticcasewherepurepro … tcannot betaxedat100%.Ideally,weshouldmodelthisviasomekindofincentiveconstraint formanagersorentrepreneursthatconstrainsapro … ttax.However,thatisbeyondthe scopeofthispaper,andfollowingalargeliteratureintaxdesign,wejustassumethatthe pro … ttaxis is … xedatsome 1 . Themainchangetothetaxdesignproblemisthatnowpost-taxpro … tappearsin thebudgetconstraintofthehousehold.Thispost-taxpro … tcanbewrittenas (1 ¡ ) where isaggregatepre-taxpro … t: = P =1 ( ¡ ¡ ) (4.1) andsoincomeinperiod isnow +(1+ ) +(1 ¡ ) Itcanthenbecheckedthat alsoappearsintheimplementabilityconstraintasfollows: 1 X =0 ( + ( +(1 ¡ ) ))=0 (4.2) From(4.1),andthefactthat = weseethat onlydependson = 1 So,theonly … rst-orderconditionstothetaxdesignproblemthatchangeare(3.8), (3.9).Theychangeto: +(1 ¡ ) ( ) = + =1 (4.3) +(1 ¡ ) ( ) = =1 (4.4) The … rstquestioniswhetheraggregateproductione¢ciencyholdsi.e.whether =1 Itiseasytocheck 17 that ( ) ( ) cannotbewritten respectivelyfor somecommonconstant Thisimpliesthatwecannotconcludethatitisoptimaltohave (3.13)holdingatthetaxoptimum.Thisimpliesinturn,thatwecannolongershowthat shouldbefullydeductiblei.e. =1 Thisisnotsurprising:generally,without100% taxationofpro … t,andheterogenous … rms,itiswell-knownthataggregateproduction e¢ciencydoesnothold. 17 Thiscanbeseenfromthefactthat dependson viatheterm ( ¡ )) ¡ ; di¤erentiationofthelatterwithrespectto generateexpressionsthatarenotcommonlyproportional to evenforspecialcasessuchastheCobb-Douglas. 15 But,withone … rm,thequestionofwhetherthee¢cientallocationofcapitalacross … rmsdoesnotarise.So,tolookatthekeyquestionoftheoptimalratefor weassume justone … rm,sothatwecanabstractfromthequestionofwhether canbedeductible from … rmcosts.Infact,withoutlossofgeneralitywecanassumefulldeductibility,sothe costofcapitalforthesingle … rmis ~ = + wherewereplace by toemphasize thatthereisnowasingle … rm.Inthiscase,wecanprovethefollowinganalogueof Proposition3.TheprooffollowstheproofofProposition3closely,exceptthatconditions (4.3),(4.4)replace(3.8),(3.9),andisthusomitted 18 . Proposition5. Atalldates =1 2 solve ¡ 1+ ¡ + ¢ =1+(1 ¡ )( ¡ (1+ )) (4.5) where isde … nedinProposition3,and = (1 ¡ ) 1 1+ µ ¡ ¶µ ( + ) ( ) ¡ ( ) ¶ Notethatif100%pro … ttaxationisavailablei.e. =1 then(4.5)reducesto(3.17). Asbefore, arenotuniquelydeterminedfromthissinglecondition.However,wecan proceedasabove,byfocussingonasolutionwheretheinterestincometaxandthespread taxareequal.Assumeasteadystate,sothat =1 .Also,assumethatthetwotaxes areequal 19 i.e. = (1+ ) Substitutingthisinto(4.5)gives 1+ = =(1 ¡ ) 1 1+ µ ¡ ¶ µ ( ) ¡ ( + ) ( ) ¶ (4.6) Thisisquiteanintuitiveformula.First,when100%pro … ttaxationisavailablei.e. =1 , itreducesto = =0 consistentlywithProposition4.Second, isnon-zeroonly whenitissociallydesirabletotaxmorei.e. ¡ 0 Third,withouttransactionscosts, thesignof isthesameasthesignof ( ) Theintuitionisasfollows.Iftaxationisdistortionaryatthemarginthegovernment wouldliketotaxpro … t(more),asitisanon-distortionarysourceoftaxrevenue;more precisely,itwouldliketoreduce asthisrelaxestheimplementabilityconstraint. Itcannotdothisdirectly.However,ifareductionin reduces thiscanbedone indirectlyviataxingcapitalincome.ThisisinlinewithresultsinStiglitzandDas- gupta(1971)forthecaseofastatic(one-period)economy.ItisalsosimilartoCorreia 18 Itisavailableonrequest. 19 Thespreadtaxisexpressedasapercentageofthecostofintermediationservices,grossofthetax, andthusmustbedividedthroughby 1+ tomakeitcomparableto 16 (1996),whoshows,inthecontextoftheChamleymodel,thatifanuntaxed(orincom- pletelytaxed)thirdfactorofproductioniscomplementarytocapitalintheproduction function,thencapitalincomeshouldbetaxedpositively 20 .Withtransactionscosts,an additionalterm ¡ ( + ) ( ) isadded.Fourth,takingtheratioof 1+ in(4.6),and 1+ in(3.12),andcancellingthecommonfactor 1 1+ ³ ¡ ´ wecanseethat and willdi¤er,althoughwecannotsaygenerallywhichwillbelarger. Togetafeelforwhen willbepositive,considertwocases.First,supposethatthe productionfunctionislinearinlabour,i.e. ( )= ( )+ Then,inthiscase,itis easilycheckedthat = ( ) ¡ 0 ( ) so ( ) ¡ ( + ) ( ) = ¡ 00 ( ) 0 andsotheinterestincometaxandspreadtaxarebothpositive.Ontheotherhand,if theproductionfunctionisCobb-Douglasi.e. ( )= + 1 then = (1 ¡ ¡ ) so ( ) ¡ ( + ) ( ) = ¡ ( + )(1 ¡ ¡ ) 0 andsotheinterestincometaxandspreadtaxarebothnegative.Tosummarize: Proposition6. Assumejustone … rm,andthattheeconomyisinthesteadystate.Then, anoptimaltaxschemeistotaxinterestincomeand … nancialintermediationservicesat thesamerategivenby(4.6).Thiscommontaxcanbepositiveornegative,dependingon thepropertiesoftheproductionfunction. 5.OtherExtensions 5.1.UnitaryTaxationofWageandCapitalIncome Anotherinterestingspecialcaseiswherethereisunitarytaxationofwageandcapital income.Thisisrelevantbecauseinpractice,manycountriestaxwageandnon-wage incomeinaunitaryway,accordingtoasingleprogressiveschedule 21 .Inthissimple 20 Theexactconditionsfor 0 aresomewhatdi¤erentheretoCorreia(1996),asweassumethatthe thirdfactorofproduction,whichgivesrisetopurepro … t,itis … xedsupply,whereasinCorreia(1996), itisinelasticsupply.Thelatterassumptionimposesanadditionalimplementabilitycontrsaintonthe optimaltaxproblem. 21 Awell-knownexceptionhereisthethedualincometaxsystemwhichleviesaproportionaltaxrate onallnetincome(capital,wageandpensionincomelessdeductions)combinedwithprogressivetaxrates ongrosslabourandpensionincome.Thedualincometaxwas … rstimplementedinthefourNordic countries(Denmark,Finland,NorwayandSweden)throughanumberoftaxreformsfrom1987to1993. 17 model,withlineartaxes,unitarytaxationofincomesimplymeanstaxingbothwageand capitalincomeatthesamerate.Here,thereisnoexplicitwageincometax;itisimplicitly de … nedviathebudgetconstraintvia = 1+ Thatis,ifthegovernmentreplaceda consumptiontaxatrate byawageincometaxatrate 1+ therealequilibriuminthe modelwouldbeunchanged.So,withunitarytaxation( = ) ,wecanreplace 1 ¡ by 1 1+ in(3.17)toget: (1+ ) ¡ 1+ ¡ ¢ =1+ +( ¡ (1+ )) (5.1) Thisismosteasilyanalyzedinthesteadystatewhen =1 Then,(5.1)canbeeasily solvedfor togivethefollowingresult: Proposition7. Supposethatthereisunitarytaxationofwageandnon-wageincome. Then,atthesteadystate,optimaltaxes satisfy = h 1 ¡ i So,weseethatwithunitaryincometaxation,thetaxrateonintermediationservices isproportionaltothetaxonconsumption,butisatalowerrate,andcouldbenegative. 5.2.EndogenizingSavingsIntermediationServices Wehave,sofar,treatedtheserviceofsavingsintermediationbybanksinrather ” black box ” fashion.Inparticular,wehavetreated theamountofintermediationservicesper unitofcapitalsuppliedto … rm ,asexogenous.However,itisclearthatbankssupply severaldi¤erentkindsofintermediationservices,notablyliquidityservices(Diamondand Dybvig(1983)),andmonitoringservices(Diamond(1991),BesankoandKanatas(1993), HolmstromandTirole(1997)). Inthisversionofthepaper,wedonotattemptprovideafullymicrofoundedversionof thesekindsofintermediationservices,forseveralreasons.First,itistechnicallydi¢cult toembedsomeexplicitmodelsofintermediationservicesintothedynamicoptimaltax framework.Second,thepayo¤fromdoingsointermsofincreasedinsightsisnotreally proportionatetotheincreasedcomplexity.Intheend,bankintermediationactivity,when explicitlymodelled,may(ormaynot)havespilloversontherestoftheeconomy.Ifthere arespillovers,thentheoptimaltaxisaPiguovianonetointernalisethesespillovers. Ultimately,thisisbecausethegovernmentcanusetheinterestincometaxtocontrolthe household ’ smarginalrateofsubstitutionbetweenpresentandfutureconsumption,and soanytaxonintermediationservicesisafreeinstrumentwhichcanbeusedtointernalize externalitiesarisingfrombankactivity. 18 Thesegeneralpointsareillustratedinapreviousversionofthepaper,Lockwood(2010), where isinterpretedasthelevelofbankmonitoring,alongthelinesofHolmstromand Tirole(1997).Intheirframework,withoutmonitoring,banklendingto … rmsisimpossi- ble,becausetheinformationalrenttheydemandissohighthattheresidualreturntothe bankdoesnotcoverthecostofcapital.So,asmonitoringiscostly,thesociallye¢cient levelofmonitoringisthatlevelwhichjustinducestobanktolend.Inthecasewhere thebankiscompetitive,i.e.where … rmchoosesthetermsoftheloancontractsubject toabreak-evenconstraintforthebank,anassumptioncommonlymadeinthe … nance literature,thisisalsotheequilibriumlevelofmonitoring.Inthiscase,savingsinterme- diationshould not betaxed,becausedoingtowillviolateproductione¢ciency,asinthe casewithheterogenous … rmsanda … xedamountofintermediationservicesperunitof savings.But,inthecasewherethebankisamonopolisti.e.itchoosesthecontract,it willgenerallychoosea higher levelofmonitoringthanthis,inordertoreducethe … rm ’ s informationalrent.So,inthiscase,theoptimaltaxisapositivePigouviantax,setto internalizethisnegativeexternality. 6.Conclusions Thispaperhasconsideredtheoptimaltaxationof … nancialintermediationservicesin adynamiceconomy,whenthegovernmentcanalsousewageandcapitalincometaxes. Theobjectiveofthispaperhasbeentotakeafreshlookatthisquestion,fromatax designpointofview.Wesetupandsolvethetaxdesignprobleminadynamicgeneral equilibriummodeloftheChamley(1986)type,wherethegovernmentchoosestaxessavings intermediation,aswellastheusualtaxesonconsumption(orequivalently,wageincome) andincomefromcapital,to … nanceapublicgood,andwhere … nancialintermediaries,in theformofbanks,areexplicitlymodelled. Ourmain … ndingisthatatthesteadystate,theonlyadministrativelysimpleoptimal taxstructureistosetthetaxrateon … nancialintermediationservicesequaltothetax rateoncapitalincome.When100%pro … ttaxationisavailable,thiscommonrateiszero; inthemorerealisticcaseoflessthan100%pro … ttaxation,thiscommonratecanbe positiveornegative,andisgenerallydi¤erentfromtheoptimaltaxonconsumption. Thereareseveralobviouslimitationsoftheanalysis.The … rstandmostfundamental, isthattheroleofbanksisnotmicrofounded.Thisisclearlyatopicforfuturework: somepreliminaryresultsinthisdirectionaredescribedinSection5.2.Thesecondis therestrictiontolinearincometaxation.TheclassicresultofAtkinsonandStiglitztells usthatwithnon-linearincometaxation,commoditytaxationisredundant,andmore 19 recently,Golosovet.al.(2003)hasrecentlyshownthatthisresultgeneralizestoa dynamiceconomy.Theirresultwouldapply,forexample,inaversionofourmodelwhere householdsdi¤erinskilllevels,andwithoutany … nancialintermediation.Itisatopicfor futureworktointroduce … nancialintermediationinthisenvironment. 7.References Acharya,V.V.,L.H.Pedersen,T.Philippon,andM.Richardson,(2010)"Measuringsys- temicrisk,"WorkingPaper1002,FederalReserveBankofCleveland. 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From(3.4),(3.5),(3.9),(2.8),wehave ¡ = ¡ 1+ (1+ ) 1+ (1+ ) = = 1 = 1 (A.4) And,from(2.3),(2.4): ¡ = 1+ (A.5) So,combining(A.4),(A.5),weget,aftersomerearrangement: 1+ = ( ¡ ) 1+ (1+ ) (A.6) Also,from(A.2),wehave: ¡ = ¡ (1+ (1+ ))= whichimplies = 1 1+ ¡ = ¡ (A.7) Combining(A.6),(A.7)toeliminate andrearranging,weget(3.12)asrequired. ¤ ProofofProposition3. From(3.4),(3.11),weget ¡ 1 ¡ 1 = 1 ~ ¡ 1 = ¡ 1 (A.8) 22 where ~ = (1+ (1+ )) (1+ (1+ ¡ 1 )) .Next,using(4.3),(4.4)toeliminate in(3.6),andalso using(3.7),weget ¡ = ¡ 1 + (A.9) So,thenfrom(A.9), ¡ 1 =( ¡ ( + )) (A.10) = ( ¡ ( + )) Butnow,from(2.8)and(3.15)with =1 wealsohave = 1+ + (A.11) Combining(A.10)and(A.11)gives ¡ 1 = ( 1+ ¡ ) (A.12) Next,combining(A.12)and(A.8),weget: ¡ 1 = ~ ( 1+ ¡ ) (A.13) Finally,from(2.3),(2.5),(2.8),weget: ¡ 1 = (1+ ) 1+ ¡ 1 1+ (A.14) = ¡ 1+(1 ¡ )( ¡ (1+ ¢ ) 1+ ¡ 1 1+ Combining(A.13),(A.14),andeliminating ¡ 1 weget ( 1+ ¡ )=1+(1 ¡ )( ¡ (1+ )) (A.15) where = ~ 1+ 1+ ¡ 1 = (1+ (1+ )) (1+ (1+ ¡ 1 )) 1+ 1+ ¡ 1 asrequired. ¤ 23