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How Should Financial Intermediation Services be Taxed Ben Lockwood First version How Should Financial Intermediation Services be Taxed Ben Lockwood First version

How Should Financial Intermediation Services be Taxed Ben Lockwood First version - PDF document

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How Should Financial Intermediation Services be Taxed Ben Lockwood First version - PPT Presentation

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 ID: 3918

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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