P.KRUSELL,B.KURU - PDF document

P.KRUSELL,B.KURU¸SÇU,ANDA.A.SMITH,JR.consumptionsets;theotherdescribestemptationutility),whichplaysakeyroleindetermininghowactualconsumptionchoicesdepartfromwhatcom-mitmentutilitywoulddictate.Inthisframework,theconsumer’swelfareofagivenset,where,forexample,couldbeanormalbudgetset,isgivenbybyu(x)+v(x),thesumofthecommitmentandtemptationthetemptationutilityevaluatedatthemosttemptingchoice(i.e.,themaximalleveloftemptation).Theconsumer’sactualchoicemaximizesthesumofthecommitmentandtemptationutilities.Inaddition,theconsumerexperiencesa(utility)costofself-control,maxv(x),whichincreasestotheextentthattheconsumer’sactualchoicedeviatesfromsuccumbingcom-pletelytothetemptation.Theconsumer’sactualchoicethenrepresentsacom-promisebetweencommitmentutilityandthecostofself-control.UsingtheGul–Pesendorfermodel,itisstraightforwardtoasknormativequestions.Thepurposeofthispaperistoexamineoptimaltaxpolicywiththeirmodel.Inparticular,welookatRamseytaxation,thatis,weconsiderwhetherandhowlineartax-transferschemescanbeusedtoimproveconsumerwelfare.Linearityhereisarestrictioninouranalysis:ifanynonlineartaxationschemeisallowed,onecould(trivially)circumventtheself-control/temptationprob-lemsbyimplementingacommandpolicyinwhichtheconsumer’schoicesetisreducedtoasingleton.Thereareobviousreasonswhysuchschemesarenotattractivetouseinpractice,butmoreimportantlyweshowthatevenwiththeratherweakinstrumentweofferthegovernment,itispossibletoimproveutil-ityinmostcases,andthatinsomecasesalineartaxcanactuallyachievethefulloptimum.WediscussandmotivatelinearityinmoredetailinSectionOverall,thequestionhereishowdifferent(linear)distortionratesaffectthewelfareofconsumerswhosufferfromtemptationandself-controlproblems.Tobeginouranalysis,welookatatwo-periodmodelwithgeneralpreferences,exceptthatwespecializetemptationutilitytoreectimpatience,sincethisistheobjectofourstudy.Inthetwo-periodmodel,theconsumptionsetfacedbyanagentistheusualtriangle,andtaxesaltertheprecisenatureofthetrian-gle.Werstanalyzeapartialequilibriumeconomy(i.e.,pricesareexogenous)inwhichweletthegovernmentuseatax-transferschemethat—forthecon-sumer’sactualchoice—usesupnonetresourcesandthusisself-nancing.Forexample,thegovernmentcanmakeconsumptioninperiod1moreexpensiverelativetoconsumptioninperiod2bysubsidizingperiod-2consumption,andtotheextenttheconsumerrespondsbybuyingmoreconsumptioninperiod2thanhisendowment,thegovernmentmustusealump-sumtaxinperiod1tobalanceitsbudget.Weshowthat,ingeneral,taxationcanimprovewelfareandthatatemptationtowardimpatiencecallsforsubsidizingconsumptioninperiod2.Wealsoexaminethesizeofthesesubsidies.Subsidizingperiod-2consumptionimproveswelfarebecauseitmakestemp-tationlessattractive.Toseethis,rememberthattheconsumer’sactualchoicemaximizesthesumofthecommitmentandtemptationutilities.Becauseoftheenvelopetheorem,asmallincrease(fromzero)inthesubsidyforperiod-2 P.KRUSELL,B.KURU¸SÇU,ANDA.A.SMITH,JR.thespecialcaseinGulandPesendorfer)inwhichtemptationutilityputszeroweightonthefuture(sothattheconsumeristemptedtoconsumehisentirewealthtoday).FortheLaibsoncase,inparticular,weshowthatthecon-sumerdiscountsutilityusingthe“long-run”discountrate,therebyresolvingtheproblemofwhichselftousewhenevaluatingwelfareinthemultiple-selvesSpecializingfurthertologarithmicperiodutility,wesolvefullyforthelaissez-faireandoptimaloutcomes.Wendthatoptimalinvestmentsubsidyratesincreaseovertimeinanynite-horizonmodel.Moreimportantly,as,theoptimumcallsforaconstantsubsidyrateoninvestment,incon-trasttotheChamley–Juddresult.Finally,intheLaibsoncaseinwhichcon-sumerssuccumbcompletelytotemptation,weshow,foranyperiodutilityfunc-tionfeaturingconstantrelativeriskaversion,thatlineartaxesare,infact,notrestrictive,butinsteaddeliverrst-bestwelfareoutcomes(i.e.,thewelfareout-comeassociatedwiththecommandoutcome)andthat,asinthelogarithmiccase,theoptimumcallsforasubsidytosavingsaslooksatthetwo-periodmodelandSectionlooksatthemodel.Sectionconcludes.ProofsofPropositionsaregatheredin.AllotherproofsareprovidedintheSupplementalmaterialKrusell,Kuru¸sçu,andSmithTHETWO-PERIODMODELForillustratingtemptationandself-controlproblemsinthesavingscontext,atwo-periodmodelcapturesmuchoftheessence,andinthissectionwepro-videsomegeneralresultsforthissetting.InSectionwethenexaminesomeaspectsofthefurtherdynamicsthatappearinmodelswithmoreperiods.Atypicalconsumerintheeconomyvaluesconsumptiontoday()andto-morrow().Specically,theconsumerhasGul–Pesendorferpreferencesrep-resentedbytwofunctionsv(c,whereiscommitmentutilityistemptationutility.Thedecisionproblemofatypicalconsumerthenisv(csubjecttoabudgetconstraintthatwespecifybelow.Theconsumer’sactualchoicemaximizesthesumv(cofthecommitmentandtemp-tationutilities,andforanychoicebundle,thecostofself-controlisv(c.Wemakethreeassumptions.SeeProposition TEMPTATIONANDTAXATIONv(caretwicecontinuouslydifferentiable u2(c1c2)v1(c1c2) forallspecializestothecasewheretemptationutilityistiltedtowardcurrentconsumptionmorethaniscommitmentutility.BudgetConstraintsEachconsumerisendowedwithunitsofcapitalatthebeginningoftherstperiodandwithoneunitoflaborineachperiod.Consumersrentthesefactorsatgivenprices.Let)and)bethegrossreturnonsavingsandwagerateintherst(second)period,respectively,andletbethepricevectordenedasWewillspecifythedeterminationofpricesinthefollowingsubsections.Giventheseprices,theconsumer’sbudgetsetisB(kP)istheconsumer’sassetholdingatthebeginningofperiod2(i.e.,hissavingsinperiod1).Insertingthedenitionsofthefunctionsintotheconsumer’sob-jectivefunctionandcombiningterms,atypicalconsumer’sdecisionproblemB(kP)v(cB(kP)Inthistwo-periodproblem,thetemptationpartoftheproblem(i.e.,thesecondmaximizationproblemintheobjectivefunction)playsnoroleindeterminingtheconsumer’sactionsinperiod1.Thetemptationpartoftheproblemdoes,however,affecttheconsumer’swelfare,aswediscussbelowinSectionLetting,theconsumer’sintertemporalrst-orderconditionis ¯u2(c1c2)=u1(c1c2)+v1(c1c2) Itisstraightforwardtoseethattheintertemporalconsumptionallocation(which,ineffect,maximizes)representsacompromisebetweenmaximiz-andmaximizing.Incontrast,theallocationthatmaximizesthetemp-tationutilitysatises.Assumptionsimplythat TEMPTATIONANDTAXATIONchoiceset—asanaturalbenchmarkagainstwhichtoevaluatetheefcacyoflineartaxation.Thegovernment’sobjective,then,istochoosethetaxrateandtransfersothatanindividual’swelfareismaximized(subjecttothegovernment’sbudgetconstraint).Withachangeintaxes,individualsareinducedtobehavediffer-ently;inaddition,temptationchangesbecausetaxationchangestheshapeofthebudgetsets.ItisthusnotaprioriclearhowtaxesinuenceequilibriumPartialEquilibriumInthissection,weexaminetheeffectsofproportionaltaxesandsubsidiesforaxedpricevectorPropositionstatesthatitisoptimaltosubsidizesavingsinthiscase.Inthepartial-equilibriumtwo-periodmodeltheoptimalin-vestmenttaxisnegativeAsbecomesclearintheproofoftheproposition,theoptimalinvestmentsubsidyispositiveiftherepresentativeconsumer’sactualsavingisgreaterthanwhatwouldhavebeenchosenhadhesuccumbedtotemptation.Thiscanbeexplainedintuitively.ConsiderincreasingtheinvestmentsubsidyfromThemarginaleffectofthisincreaseon,evaluatedattheconsumptionbundleactuallychosenbytheconsumer,,iszerointhetwo-periodmodelsincetheconsumerischoosinghissavingoptimally.However,themar-ginaleffectofthisincreaseonthemaximalleveloftemptation,,isnegative.Thegovernmentsetstaxratessoastobalancethebudgetbasedonequilibriumbehavior:inequilibrium,theconsumerpaysalump-sumtaxequaltotheamountoftheinvestmentsubsidyreceived.Whentheinvestmentsub-sidyispositive,aconsumerwhodeviatestosavelesswouldthusnotreceiveaslargeaninvestmentsubsidy,whilepayingthesametax.Therefore,succumbingtotemptationisnowlessattractive:increasingtheinvestmentsubsidyreducesthemaximalleveloftemptation.Withoptimaltaxation,thus,theconsumerisinducedtosavemore,sothathisintertemporalconsumptionallocationistiltedmoretowardthefuturethanintheabsenceoftaxation.Atthesametime,thechangeintheslopeofthecon-sumer’sbudgetconstraintreduces(otherthingsequal)thetemptationfacedbytheconsumer.Thenetresultistoincreasetheconsumer’swelfare.Whenconsumersaretemptedtoconsume“toomuch,”thecommandpolicyisequivalenttotheimpositionofasavingsoor,withtheoorchosentoreplicatethecommandallocation.Thepartial-equilibriumeconomycouldalsobeviewedasoneinwhichtherearetwolinearproductiontechnologies,oneusingcapital(withxedmarginalreturns)andoneusinglabor(withxedmarginalreturns P.KRUSELL,B.KURU¸SÇU,ANDA.A.SMITH,JR.GeneralEquilibriumInthissection,westudyeconomiesinwhichpricesarenolongerexogenousbutinsteadadjusttoclearperfectlycompetitivemarkets.Thereareimportantdifferencesbetweenthecaseswithandwithoutintertemporalproduction.Inanendowmenteconomy,taxpolicycannotinuenceaggregatesavingsand,consequently,hasnoinuenceonequilibriumwelfare.Inaneconomywithproduction,bycontrast,taxpolicycaninuenceaggregatesavingsandwelfare,thoughtoasmallerextentthaninpartialequilibriumbecauseofequilibriumeffectsonprices.AnEndowmentEconomyThissectionconsiderstheroleoftaxationinanendowmenteconomy.Propositionstatesthata(linear)investmenttaxcannotimprovewelfareinanendowmenteconomy.InanendowmenteconomyaninvestmenttaxhasnoeffectonequilibriumwelfareInanendowmenteconomy,pricesadjustsothatconsumerschoosetoholdtheendowmentatallpointsintime.Theproportionaltax,therefore,can-notinuence(realized)consumption.Furthermore,taxesarenotusefulfordecreasingthedisutilityofself-controleither.Inequilibrium,theslopeofthebudgetlineattheendowmentpointisgivenfrompreferences(wherecom-mitmentutilityandtemptationutilitybothmatter).Becauseequilibriumcon-sumptioncannotchangeinresponsetotaxes,thisslopedoesnotchange:theslopeisdeterminedbythenet-of-taxreturnonsavings,andanychangeintaxratessimplychangesthebefore-taxreturn.Thus,taxesdonotinuencethechoicesetofconsumers:whatevertemptationsconsumersface,theycannotbeinuencedbyaproportionaltax.AnEconomyWithIntertemporalProductionInthissection,weexaminetheeffectsofproportionaltaxesinageneral-equilibriumeconomywithproduction.Letbeastandardneoclassicalaggre-gateproductionfunctionandlettherebestandardgeometricdepreciationat.Themaindifferencebetweentheendowmenteconomyandtheecon-omywithintertemporalproductionisthatprices(wagesandinterestrates)areNonlineartaxes,ofcourse,wouldchangetheconsumer’sequilibriumchoicesetandwould,therefore,affectthedisutilityofself-control.Inaddition,westudyaneconomywitharepresen-tativeconsumer;inaneconomywithheterogeneousconsumerswhodifferin,say,theirshort-rundiscountrates,itisconceivablethatlineartaxationcouldaffectindividualallocations,andhencewelfare,eveniftheaggregateallocationisxed.Weleavetheexaminationofthispossibilitytofutureresearch. P.KRUSELL,B.KURU¸SÇU,ANDA.A.SMITH,JR.sothat1regulatestemptationimpatiencerelativetocommitmentimpa-tience.Inthiscase,theconsumer’sproblemcanbewrittenasP) \b) P) 1Š+\bc1Š2 Intheexpressionabove, 1Š+\bc1Š2 1ŠŠ[c1Š1 1Š+\bc1Š2 isthecostofself-control.AsweshowinSection,whereweextendthemodeltoods,underthesefunctional-formassumptions,theGul–Pesendorfermodeloftemptationandself-controlneststhemultiple-selvesmodel.Inaddition,thetractabilityprovidedbytheseassumptionsallowsustoobtainsomeadditionalanalyticalresults.Weconsiderrstthecaseoflogarithmic(period)utility.Thefollowingpropositiongivesexplicitsolutionsforbothlaissez-faireandoptimalsaving.Inthetwo-periodmodelwithlogarithmicutilitysavingsareafraction\b) \b)ofpresent-valuewealthTheoptimalinvestmenttaxis(\b andtheassociatedsavingsfractionis Theoptimalsubsidydependsonlyonpreferenceparametersandnotonthespecicationoftechnology.Specically,itincreasesastemptationgrowslarger:itdecreasesasincreases(therebyreducingthegapbetweenthe“long-run”discountrateandthe“short-run”discountrate)anditincreasesinthestrength,,ofthetemptation.Toobtainthesavingsrateundercommitment,set0(or1)intheexpressionforthelaissez-fairesavingsrate;toobtainthesavingsratewhensuccumbingtotemptation,let.Thelaissez-fairesavingsratethenliesinbetweenthesetwoextremes.Moreover,theoptimalsavingsrateisidenticaltothelaissez-fairesavingsrateundercommitment.Thisresultholdsbecause,inthespecialcaseoflogarithmicutility,theratiooftemptationconsumptiontoactualconsumptionisaconstantthatdependsonlyonpreferenceparameters.Thisfactimplies,inturn,thatthecostofself-controldependsonlyonprefer-enceparametersand,inparticular,doesnotdependeitheronpricesortaxes.Changesinthesubsidy,consequently,leavethecostofself-controlunchangedwhenutilityislogarithmic,andthegovernmentineffectchoosestheoptimalsubsidyratesimplytomaximizecommitmentutility.Withlogarithmicutility,thecompetitiveequilibriumallocationwithopti-maltaxationcoincideswiththecommand(orcommitment)outcome,thatis,theallocationthatobtainswhenthegovernmentchoosestheconsumerbyrestrictinghisconsumptionsettoasingleton(orwhentheconsumerdoes P.KRUSELL,B.KURU¸SÇU,ANDA.A.SMITH,JR.,thelawofmotionforaggregatecapital,andthesequenceoftransfersandtaxes.Theproblemoftheprice-takingagentinperiod,usingrecursivenotation(aprimesymbolizesnext-periodvalues),isgivenbycku(c)wherethetemptationfunctionisu(c)withabudgetconstraint(whichappliesforbothactualandtemptationchoices)givenbyTheinvestmentsubsidyisallowedtodependontime,andthelump-sumvarieswithsoastoensurethatthegovernment’sbudgetbalances.Theconsumer’sactualsavingsaredeterminedbya“realized”deci-sionrule;similarly,savingswhensuccumbingtotemptationaredeterminedbya“temptation”decisionrule1:Atime-recursivecompetitiveequilibriumforthisecon-omyconsistsofapairofdecisionrules,apairofvaluefunctions,pricingfunctions,andalawofmotionforaggregatecapital,suchthat(i)givensolvesthemaximizationproblemaboveand,(ii)pricesaregivenby,(iii)thelawofmotionforaggregatecapitalisconsistentwiththeindividualdecisionrule,thatis,,and(iv)thegovernmentbudgetbalancesineachperiod:Werequirethegovernmenttorunabalancedbudgetinthisdenition,butthisrequirementisnotrestrictive,becauseaRicardian-equivalencere-sultobtainsstraightforwardlyinthisenvironment(i.e.,giventhesequenceofinvestmentsubsidiesandaccompanyingbalanced-budgetlump-sumtaxes,governmentdecitsand/orsurplusesnancedbyincrementallump-sumtaxes/subsidieswouldhavenoeffectonequilibriumallocations).Withborrowingconstraints,Ricardianequivalencemightfailtoholdinthemodeloftemp-tationandself-controlevenifthetimingoftaxesdoesnotinuenceactualconsumptionchoicesorequilibriuminterestrates.Inparticular,borrowingconstraintscouldstillaffectwelfareiftheyrestrictthetemptationchoicebutnottheactualchoice. P.KRUSELL,B.KURU¸SÇU,ANDA.A.SMITH,JR.convergestotheLaibsonmodelthatisthevaluefunctionsandconsumptionchoicesoftheconsumerintheGPsettingaregivenby (ckargmax \bUThislimitoffersaresolutiontotheproblemofwhichoftheconsumer’sselvestousewhenassessingwelfareinthemultiple-selvesmodel.Specically,inthislimittheconsumersuccumbscompletelytotemptation,butheeval-uateswelfarebydiscountingusingthediscountfactorsincommitmentutil-ity.Forthecaseoflogarithmicutility,weobtainasimilarresultregardlessoftheextenttowhichtheconsumersuccumbstotemptation(i.e.,forGivenalawofmotionforaggregatecapitalasequenceoftaxesandtransfersu(c)(c)thevaluefunctionandconsumptionchoicesoftheagentaregivenby(c)(ckargmax(c)\b)UisaconstantthatdependsonlyonpreferenceparametersThisresultholdsbecause,asinthetwo-periodmodelwithlogarithmicutil-ity,bothactualandtemptationconsumptionareproportional(atanypointintime)tolifetimeincome,withtheconstantofproportionalitydependingonlyonpreferenceparameters(andnotonpricesortaxes).Thus,theratioofac-tualtotemptationconsumptiondependsonlyonpreferenceparametersatanypointintime.Asinthetwo-periodmodel,theself-controlcost(theconstantinProposition)dependsonlyonpreferenceparametersanddoesnotvaryeitherwithpricesorwithpolicy. TEMPTATIONANDTAXATIONForanynitehorizon,theoptimalsubsidyratewillinfactincreaseovertime,andfortheinnite-horizoncase,theoptimalsubsidyrateistime-WeturnnowtothedeterminationofoptimalpolicyunderCRRAutilityforthelimitingcaseinwhich.Propositionshowsthattheobjectiveofthegovernmentisthesameasinthelogarithmiccase,exceptthattheconstanttermintheobjectivefunction(whichcapturesthecostofself-controlinthelogarithmiccase)isequaltozero.Theoptimalpolicy,therefore,istoreplicatethecommitmentallocation,asstatedformallyinPropositionUnderCRRAutilityinthelimitingcasetheoptimalsequenceofinvestmenttaxesimplementsthecommitmentallocationandgener-atestherst-bestwelfareoutcomefortheconsumerWhentheconsumersuccumbscompletelytotemptation,therefore,restrict-ingthesetoftaxinstrumentstoalinearclassdoesnotpreventthegovernmentfromachievingtherst-bestwelfareoutcome.Moreover,wecanagainshowthatas,theoptimumcallsforasubsidytosavings,incontrasttotheChamley–Juddresult.CONCLUSIONANDREMARKSThispapermakesclearthatwhenconsumerssufferfromtemptationandself-controlproblems,lineartaxschedulescanimproveconsumers’welfare,eventhoughsuchschedulesarenotverypowerfultoolsforrestrictingcon-sumers’choicesets.Thedirectionofthechangeistheexpectedone:whentemptationischaracterizedby“excessive”impatience,optimalpolicyistosub-sidizesavings.Moreover,inthespecialcaseinwhichconsumerssuccumbcom-pletelytotemptation(i.e.,themultiple-selvesmodel),lineartaxesdeliverrst-bestwelfareoutcomes.AsdiscussedinSection,itwouldbeveryinterestingtoextendthepresentanalysistononlineartaxation,especiallywhenthereisconsumerheterogene-ityandprivateinformationabouttypesmakesitcostlytousenonlinear(andlinear)schemes.Itwouldalsobeinterestingtoconsiderpolitical-economycon-straintsontaxes;inpractice,weobservearangeoftaxpolicyoutcomesthatdonotappeartolineupwiththeoreticalprescriptions.Forexample,wesuggest)thattheChamley–Juddprescriptionthat“optimaltaxesoncapi-TheusualsettingfortheChamley–Juddresultisaninnite-horizoneconomy.Here,weob-tainresultsfortheinnite-horizoneconomybystudyingthelimitofasequenceofnite-horizoneconomiesasthehorizongrowslong. P.KRUSELL,B.KURU¸SÇU,ANDA.A.SMITH,JR.talincomeshouldbezerointhelongrun”couldbesharpenedto“...shouldbenegative...,”butinrealitythesetaxesarepositive,andinsomeplaceslarge,anditishighlylikelythattheseoutcomeshavepolitical-economyunderpin-Integratingsuchconstraintsisanimportanttopicthatwehopetoad-dressinfuturework.ROOFOF:Letbethelifetimeincomefromperiodwhichisgivenby +···foragivenpricesequence(Forsimplicity,weassumethattaxesandtransfersarezerointheproofoftheproposition,butitisstraightforwardtoadapttheprooftoallowfornonzerotaxes.)Thebudgetconstraintoftheagentattimeintermsofconsump-tioninthatperiodandthelifetimeincomefromnextperiodonisgiven Toprovethisproposition,weshowthattheoptimizationproblemoftheconsumerinperiodtakestheform 1Š\fYtŠYt+1 \b)U 1Š\fYtŠYt+1 \bUisgivenby isaconstantthatdependsonutilityparameters,prices,andtimeperiodInaddition,startingfromthelastperiod,weshowthat,asandthatthevaluefunctionoftheconsumerforeachisgivenby 1Š\fYtŠYt+1 See,forexample,Acemoglu,Golosov,andTsyvinski TEMPTATIONANDTAXATIONTheoptimalallocationmustsatisfytheEulerequation r( ThegovernmentimplementsthisallocationbychoosingtaxratessuchthattheEulerequationfortheconsumerisequivalenttothegovernment’sEulerequa-tionabove.TheproofofPropositionshowsthatthecompetitiveequilibriumallocationsatisestheEulerequation ct=Mt+1r(¯kt+1) it \b) \b) \b)\b) +···+\b)Thus,thegovernmentchoosesitsuchthat it,whichdeliversiT(\b iT(\b \b)it(\b +···+\b)Q.E.D.CEMOGLU,D.,M.GOLOSOVA.T(2008):“PoliticalEconomyofMechanisms,”,76,619–641.619–641.2080]AMADOR,M.,I.WG.-M.A(2006):“CommitmentvsFlexibility,”,74,365–396.365–396.2068]CHAMLEY,C.(1986):“OptimalTaxationofCapitalIncomeinGeneralEquilibriumWithInnite,54,607–622.607–622.2065]GUL,F.,W.P(2001):“TemptationandSelf-Control,”,69, (2004):“Self-ControlandtheTheoryofConsumption,”,72,119–158. 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