美聯(lián)儲(chǔ)-具有不確定私營(yíng)部門前景的最優(yōu)貨幣政策（英）

FinanceandEconomicsDiscussionSeriesFederalReserveBoard,Washington,OptimalMonetaryPolicywithUncertainPrivateSectorForesightChristopherGustandDavidLopez-Salido2024-059Gust,Christopher,andDavidLopez-Salido(2024).“OptimalMonetaryPolicywithUncertainPrivateSectorForesight,”FinanceandEconomicsDiscussionSe-/10.17016/FEDS.2024.059.NOTE:StafworkingpapersintheFinanceandEconomicsDiscussionSeries(FEDS)arepreliminarymaterialscirculatedtostimulatediscussionandcriticalcomment.TheanalysisandconclusionssetfortharethoseoftheauthorsanddonotindicateconcurrencebyothermembersoftheresearchstafortheBoardofGovernors.ReferencesinpublicationstotheFinanceandEconomicsDiscussionSeries(otherthanacknowledgement)shouldbeclearedwiththeauthor(s)toprotectthetentativecharacterofthesepapers.OptimalMonetaryPolicywithUncertainPrivateSectorForesightAbstractCentralbanksoperateinaworldinwhichthereissubstantialuncertaintyregardingtheprivate-sectorexpectations.Wemodelprivatesectorexpectationsusiandpricingdecisions.Inthissetting,contrarytostandardNewKeynesian(NK)models,wein且ationexpectationsdeviatepersistentlyfcharacterizeoptimaltime-consistentmonetarypolicywhenthereismentconsiderationsmodifytheoptimalleaning-against-the-windprincipleintheNKliteratureJELClassification:C11,E52,E70Keywords:Finitehorizonplanning,optimaltime-consistentpolicyunderuncertainty,leaningagainstthewind,attenuationprinciple.*FederalReserveBoard.Gust:christopher.gust@.L/opez-Salido:david.lopez-salido@.This11Introductionuncertaintyafectourunderstandingoftheprinciplesunderlyingthedesignofoptimalpolicy.1Inthispaper,wecontributetothisliteraturebystudyingoptimaltime-consistentpolicywhenpolicymakersareuncertainaboutthin?nite-horizonplanningproblemsislimited.Inparticular,weusetheNe?rmsareboundedlyrationalbecausetheyhavela?nitehorizon.AnappealingfeatureoftheNK-FHPmodelthatwedemontarget.ThisfeaturemakestheNK-FHPmodelsuitedtostudythedesignofoptimalpolicywhenThereareadditionalb?rmsallowsusto且exiblyapproximatestarklydiferentwaysandthetransmissionofmonetarypolicyoccursrelativelyquickly.Incontrast,whenagentshshortplanninghorizons,agentsarenotf f 2Ontheempiricalside,theNK-FHPframeworkhasbeenshowntobeafruitfulwaytompredictabilityofforecasterrors.Inpasthe“hybrid”NKmodelthatfeaturesrationalexpectations,habitpersistenceinconsumption,importantlimitedanduncertainforesightisfortheconductofmonetaOuranalysisalsobuildsontheprominentworkofClarida,Gali,andGertler(1999),whostudytheNK-FHPmodelisequivalenttothe“l(fā)eaningagainstthewind”strategydiscussedinClarida,Gali,andGertler(1999).But,whenagentshcymakeractingunderdiscretionfollowsamoWhenpolicymakersareuncertainabouttheshareofagentswithdiferenoptimal,time-consistent,policyundeformation.ThisresultsstandsincontrasttotheclassicresultinBrainard(1967),whoshowedthatuncertaintyabouttheefectofpolicthatuncertaintyaboutexpectationsformationincreasesthel3arelativelymoreaggressivepolicypaththanundisknownwithcertainty.Inparticular,withuncertainexpectationsformation,thedistributionforurebyshowingthattheinftheFOMC’sSurveyofEconomicProjections(SEP)inMarchof2023.Weusethisinformationstrategies,includingonethatwoLiteraturereview.BuildingonClarida,Gali,andGertler(1999),thispaperbridgestwostrandsoftheliteratureonoptimalmonetarypolicy.The?rstanalyzesoptimalmonetarypolicywhenninghorizonare?xed.Relativetotheirwork,westudyoptimalmonetarypolicyunderdiscretionoutsidetheirplanninghorizons.Inisuncertaintyregardingtheformation 4futuremonetarypolicystillafecteconomicoutcomes.Ourpaperisalsorelatequivalence.Ourmodeldifersfromtheseearlierpapers,asweexpltigatethemechanismqintheaftermathoftheforfuturework.2OptimalPolicyinaFinite-HorizonPlanningModeltaintyusingdiferent“modes”orregimesthatfollowaMarkovprocess.Forthediferentmodes,horizonswhoseexpectationstakeintostructuralrelationshipsfarintothefuture,a acentralbankobservesthecurrentdistributi 3ThedynamictargetcriteriaimpliedbyoptimalpolicyintheFHPmodelisdistinctfromthoseimpliedbytheadaptivelearningmodelsofG/ati(2023)andMoln/arandSantoro(2014).Inthosemodels,thecentralbankadjuststhestatictargetingrulebyrespondingtotermsthatre丑ecttheexpecteddiscountedvalueofthefuturepathofoutputgaps.Inourmodel,thecentralbankadjuststhestatictargetingrulebyrespondingtoaterminvolvingtheexpecteddiscountedvalueofthefuturepathofin丑ationgaps,asdeviationsofin丑ationfromacentralbank’stargetcanleadtoanundesirabledriftinagent’slonger-runin丑ationbeliefs.4Similartoourpaper,KimuraandKurozumi(2007)alsoprovidemicrofoundationsforinertialin丑ationdynamics.Intheircase,itarisesbecauseasubsetof?rmsdonotoptimallychoosetheirpricesbutsettheirpricesbasedonlaggedin丑ation.Incontrast,the?rmsinourmodelchoosetheirpricesoptimallybutareboundedlyrationalbecauseoftheir?niteplanninghorizonsandthelearningtheydoabouteventsoutsideoftheirplanninghorizons.5thefuture.foftheoutputgap.Themonetaryauthoritthusareboundedlyrational,thecentraboutexpectationsformationtoace2.1HeterogeneousPlanningInthissubsectionweprov(2018).decisionsattimetbasedonformulatingstπ=βEτπ+κy+uτ(1)y=Eτy-σ(i-Eτπ-r),(2)forτ=t+k-jandj>0wherejindexesthenumberofperiodsleftinanagents’planninghorizonTheexpectationsoperator,Et,inequations(1)and(subjectiveexpectationsoperator,EsuchthatforvariableZt+k-jwithk>j≥0,thefollowing6EZt+k-j=EtZ+k-j.(3)rationalexpectationsintwoways.First,theyonlyformulatefully-statecontingentplansforkperiodsratherthantheirin?nitelifetimes.Sut=ρuut-1+eut(4)r=ρrr-1+ert,(5)deviations,σuandσr,respectively.fAsk→∞,itcanbeshownthatthe‘canonical’NKmodelusedtostudyoptimalpolicyinCGali,andGertler(1999)isaspecialcaseofequations(6)and(7).Inthatcase,households’andrelationshipsoverthei2.2Longer-RunLearning7π+k=κy+k+βvpt+ut+k(8)y+k=-σ(i+k-rt+k)+vht,(9)outsideoftheirplanninghorizons.Whiletheymaketheirtimetdecisionstakingvpandupdatevptandvhtaccordingto:vpt+1=(1-γp)vpt+γpπt(10)vht+1=(1-γh)vht+γh(yt+σπt),(11)Whilelonger-runlearningintroducestwoextraparameters,γpandγh,ithasboththeoreticalandempiricalbene?ts.Onthetheoreticalside,itgivestheFHPmodelattractivepropertiesinInationScares.Firms’beliefsabouteventsoutsideitsplanninghorizonplayaparticularlyimportantroleinouranalysis.Inparticular,vptcanbeinterpretedasa?rm’slonger-runbeliefs6Inmakingtheirtimetdecisionsforpricesandspending,householdsand?rmsignorethattheirvaluefunctionschangeovertimeandthusthelearningframeworkweadoptusesan‘a(chǎn)nticipatedutility’approach(e.g.,CogleyandSargent(2008)).7Forconvenience,wehaverescaleda?rm’svaluefunctionbytheprobabilitythatthe?rmcanre-optimizetheirprice.Thus,relativetoWoodford(2018),vpt=(1-θp)t,where1-θpistheprobabilitythataFHP?rmhastheopportunitytore-optimizeitspriceandtisthecontinuationvalueatdatetofsucha?rm.8vpt=√piπt-1-i,(12)(2005)andOrphanidesandWilliams(2022).Accordingly,theFHPmodelwithlonger-runlearningliterature.Later,wediscusshowoptimal(time-consistent)2.3ModelUncertaintyofhouseholdsand?rmswithhorizonk0atdatet.Thevariablemtisrandomwithmt∈{0,1}.constanttransitionprobabilities:Pmn=Pr{mt+1=njmt=m},m,n=0,1(13)withthematrixPdenotingthe2×2matrixπt=ωtπ0+(1ωt)π1(14)yt=ωty0+(1—ωt)y1,tractablesinceconditionalonmt,themodelequationsare93OptimalMonetaryPolicyandtheoutputfromtheirβτ-t(16)3.1PrivateSector(Policy)Beliefsanin?nitehorizon,theirperceptionsaboutfuturemonetarypolicydiferfromthepolicythatthecentralbankcanimplement.Inparticular,theyperceivethatinformulatingpolicythecentraltsoi=itfork=k0,k1.Fortheirdatetforecastsofpolicyovertheremainderoftheirk-periodsubjecttoequations(1)and(2).Inequation(17),st=(ut,r)denotesthevectorofshocks.Atthelastperiodoftheirplanninghorizon(j=0),theyperceivethatthecentralbankminimizes:centralbank,liketheydo,takesthecontinuationvalu willbecorrect,andtherewillb+k-j=—+k-j.(19)Agents’PerceivedLAW.Theagents’perceivedtargetingruleissimilartothetargetingruleunderoptimaldiscretioninthecanonicalNKmodel(e.g.,Clarida,Gali,andGertler(1999)),Inparticular,thisconditionimpliesthatagentsbelievethatthecentralbankwouldpursuealeanagainstthewind(LAW)policyateachfuturedateoftheirpisabovetarget,theyexpectinterestrate);andvice-versawhenitisbeloStickyExpectationsofInation.Todescribetheactualpolicythatacentralbankim-aboutin且ation.9Intuitively,optimalpolicyisperceivedas?rms’continuationvaluefunction,vpt..takeintoaccounttheefectsofthesupplyshockoverthat,ifthein且ation-outputtradeofgetsincreasinglysmall(i.e.,hdecreasestoward0),ap(k)3.2OptimalTargetingRuleunderdiscretion.Whilethecentralboptimalpolicythatisactuallyimplementedandpras:dst+1(21)wherethefunctionf(st+1jst)denotestheconditionaldensityfortheshockstoaggregatesupplyanddemand.Inaddition,Pr(mt+1=njmt)denotestheconditionalprobabilitiesinthetransitionmatrix,PandΠt三π(vpt,st,mt;it)andYt三y(vpt,st,mt;it)denotefunctionsthatdetermineΠt=βEtΠt+1+κYt+ut(22)Yt=EtYt+1—σ(it—EtΠt+1—r)(23)vpt+1=(1—γp)vpt+γpΠttheoutputgapinthecanonicalNKmodelexceptthatthefunctions,EtΠt+1,andEtYt+1re且ectEtΠt+1=ωtEtΠ1(vpt,ut)+(1—ωt)EtΠ1(vpt,ut)EtYt+1=ωtEtY1(vpt,ut)+(1—ωt)EtY1(vpt,ut)Tosettheoptimaltime-consistentpolicy,th{k0,k1}.However,afewadditionalcommentsareinorder.First,thecentralbanktakesintoaccountthatitsinterest-ratAccordingly,insettingthecurrentpolicyrate,it,thecentralbankstakesintoaccountthatvpt+1fWilliams(2005).Asshownintheappendix,theoptimaltargetingπt+γpβEtWpt+1=-(24)Wpt=βaptπt+β[1-γp(1-βapt)]EtWpt+1(25)t)andtheoutputgap(yt)aregirespectively.Thetermapt=ωtap(k0)+(1-ωt)ap(k1)re且ectsthemarginalefectofachangeinWp(vpt,st,mt)ismarginalefectofvptonthecentralbank’slossfunctionandEtWpt+1satis?es:Expression(24)extendsthethecelebratedLAWprincipleofClaridaetal.sideofthetargetingrulere且ectsthestaticLAWprinciplederivedbyClaridaetal.(1999):Ift>0,asaresultofthecospushestheoutputgapintonegativeterritory(yt<0).Thenewtargetingcriterion,however,isnowincorporatesanadditionalterm,EtWpt+1.Tounderstandthisterm,notethatthevariableisre且ectedinthetermEtWpt+1>0.ThattermimpliesthatitisoptimalforacentralbanktoactpreemptivelybypuAcentralbank’suncertaintyaboutevariationintheshareofagentswithdiferentplanninghorizons.Iftheshareofagentswithshortplanninghorizonsisgreater,thisdiscountfactor,(β[1-γp(1-βapt)]),ishigher,intensifyingaγp=0).Ifalloftheprivate-sectoragentshadin?niteplanninghorizonsstaticLAWprincipleofClarida,Gali,andGertler(1999):πt=-λyt.Similarly,if?rmsdonotrunin且ationarybeliefs.Asaresult,equation(24)alsosimpli?esandsatis?estofClarida,Gali,andGertler(1999).Certainty-EquivalentEconomy.Toassesstheroleofuncertainexpectationsformation,horizon:anouncertaintybenchmark.Becausecertainty-equivalence(CE)issatis?edinthiwithuncertainexpectationsformation.IntheCEeconomy,satis?es:E-γpβEtW(27)WE=βap(kCE)π+β[(1-γp)+βγpap(kCE)]EtW1(28)(24)and(25)withexpressions(27)and(28).Themarginalefectofvptonexpectedin且ationnextperiod,apt,doesnotvaryinthecertainty-equivalenteconomy.Instead,itis?xedat,ap(kCE),themultiplicativeshock,mt.Asdiscussedabove,thismultiplicativeshockimpliesthatcertainty-kCE→∞,in且ationscaresdonotoccurandtheoptimaltarprinciple:π=-λy.4ResultsparametersoftheNK-FHPmodelfromGust,Herbst,andL/opez-Salido(2022)andGust,Herbst,andL/opez-Salido(2024).Thosepaapolicymaker’sobjective.Todoso,weuseinformationfromtProjections(SEP)inMarchof2023toconsaboutexpectationsf4.1ParameterValues=0.5.Forthepersisfvariationsintheseparameters.whichallagentshaverationalexpectations.Accordingly,ωtsatis?es:ωt=1ifmt=0andωt=0ifmt=1.Thevalueofk0=4ishigherthanthemeanestimatereportedinGust,Herbst,andexpectationsfromthesurveyofprof0.9andP11=0.9675.Theseprobabilitiesimplythoftherationalexpectations,canonicalNKmodelisregimeoftheNK-FHPmodel.TheyalsoimplythattheunconditionalorergodicprobabilitiesofthemodeswithrationalexpectationsandFHPexpectationsare75%and25%,respectively.Fortheremainingparameters,Foracentralbank’spreferenceparameterinthelossfunctioofoutputfrompotentialinitslossfunction(Debortolietal.(2019)).4.2SEP-ConsistentBaselineMarch2023SummaryofEconomicProjections(SEP).Toconstructtheseshocks,weassumethatthefederalfundsrateit=ρiit-1+(1-ρi)(φππt+φyyt)+emt(29)outputgapimpliedbytheSEP.ToconstructtheimpliedpathforthesevariablesfromtheSEP,welinearlyinterpolatetheannualmedianprojectionsshownintheSEPandconstructtheoutputoftheunemploymentratefromthemedianSEPparticipant’sestimateofitsloTheleft-handsideofFigure1showstheSEP-consistentpathofthesevariables.Athere,SEPparticipaofoutputimpliedbySEPparticipantsprojectionsoftheunemployment-rateisabovepotentSEPparticipantalsoprojectsthatunderappropriatemonetarypolicythefederalfundsratewillpeaklaterthisyeaWeusethecanonicalNKmodelwithREtoinferthepathofshocks.Thatmodel,likethe modelpathsofoutput,in且ation,and 10TheviewthatFederalReservepolicyprocedureshavegenerallyinvolvedinterestratesmoothingwasintroducedbyMankiwandMiron(1986).Onthisissue,seealsothediscussioninGoodfriend(1987)andthereferencestherein.TherestoftheparametersofthisinterestraterulearefromTaylor(1999).11SeeGust,Herbst,andL/opez-Salido(2022)foracomparisonofthecostofapre-announced,permanentdisin丑atintheNK-FHPmodeltothecanonicalNKmodel.TheyshowthatcanonicalNKmodelimpliesarelativelycostlessdisin丑ationandthatthemodelwith?nitehorizonplanningcanhelpaccountforhistoricalestimatesoftheoutputFigure1:TheSEP-ConsistentBaseline0.50.0 0.5 1.0OutputGap1.51.00.50.0 0.52022Q42023Q42024Q42025Q42026Q42027Q4ADShock2022Q42023Q42024Q42025Q42026Q42027Q44.54.03.53.02.52.0Inflation1.00.20.02022Q42023Q42024Q42025Q42026Q42027Q4Cost-PushShock2022Q42023Q42024Q42025Q42026Q42027Q45.04.54.03.53.02.5PolicyRate2022Q42023Q42024Q42025Q42026Q42027Q4Interest-RateRuleShock0.40.20.0 0.2 0.42022Q42023Q42024Q42025Q42026Q42027Q4Note:TheSEP-consistentbaselineisconstructedbyinterpolatingquarterlydatausingtheMarch2023SEP.ThecanonicalNKmodelisusedtoinfertheshockstoaggregatedemand,aggregatesupply(cost-push),andtheinterest-rateruleoverthe2023Q1-2027Q4period.fftheprojectionperiod,suggestingthataninertialversionofaTaylorrule?tstheSEP-consistentForsimulationsoftheNK-FHPmodel,wealcontinuationvaluefunctions,upin2022Q4.Inthemodelwithuncertaintyaboutexpectationsformation,thisvalueimpliestrend in且ationaround2.4%in2022Q4.Wefollowasi 12Interestingly,thecanonicalNKmodelgeneratesacombinationofdemandandsupplyshocksthatresonateswiththeevidencepresentedinBlanchardandBernanke(2023).Theseauthors?ndanimportantroleforpandemic-inducedsupplyconstraintsaswellaspersistentlyhigheraggregatedemandassettingofthein丑ationin2022.impliesuht=3.1in2024.3TheTransmissionofShocksintheNK-FHPModelFigure2:TheEfectofanAggregate-SupplyShockintheNKModelwithRationalandFHPExpectationsInflationOutputGapInflation0.0 0.1 0.2 0.31.00.80.60.40.20.00.0 0.1 0.2 0.3051015200051015200.10.0 0.10.0 0.1 0.2 0.30510150.0ExpectedInflationTrend05101520Note:The?gureshowstheefectofashocktoaggregatesupplythatincreasesannualizedin丑ationby1percentagepointinthecanonicalNKmodelwithrationalexpectations(darkline)andtheNKmodelwithFHPexpectations(redline).MonetarypolicyisassumedtofollowaTaylorrule.inthepolicyruleimpliesaninitialfallintherealpolicyrateinthecanonicalNKmodel,therealpolicyrisesabovesteadystatewithinthe?rstyearan?rsttwoyearsaftertheshock.ThistighteninginpolicyalongwiththetemporarynatureofthefIntheNK-FHPmodel,theefectsoftheshocksonin且ation,output,andthepolicyratearepersistentlyintheFHPmodel.13Thisriseinthetrendcomponentleadstoanin且ationraelevatedwellforlongeraswell.Asaresult,outputfaarealsonotablymorepersistentintheNK-FHPmodelthaninthecanonicalNKmodel.144.4OptimalPolicyunderUncertaintyrealinterestrate,andtrendin且ationinthecanonicalNKmodel(labelledRE)andFHPmperfectcertainty.Forthecatargetingruleapplies,whichinthecaseofthecanonicalπt=-λyt.ThistargetingruleandtheSEP-consistentpolicyresultinbroadlysimilaroutcomes.thanundertheSEP-consistentbaseline.Outputremainsclosertopotentialunderoptimalpolicy, 13Thetrendcomponentisde?nedbyignoringtheefectsoftheshocksonEtΠt+1sothatthetrendcomponentonlyre丑ectstheefectsofagents’valuefunctionsonexpectedin丑ation.14Gust,Herbst,andL/opez-Salido(2022)showthatthetransmissionofmonetaryshocksoccursmoregraduallyintheNK-FHPmodelthaninthecanonicalNKmodelwiththepeakefectonin丑ationoccurringconsiderablylater.Figure3:OptimalPolicyunderDiferentExpectationalAssumptions0.5Output0.55.00.00.0 0.5 1.03.0 1.52.02022Q42023Q42024Q42025Q4PolicyRate2026Q42027Q48877663.5553.0442.5332.02022Q42023Q42024Q42025Q42026Q42027Q4Inflation2022Q42023Q42024Q42025Q42026Q42027Q4ExpectedInflationTrendUncertain2023Q42024Q42025Q42026Q42027Q4Note:The?gureshowsthemeanresponsesunderoptimaldiscretionarypoliciesusing50,000drawsofshockscenteredaroundtheaggregatedemandandsupplyshocksassociatedwiththeSEP-consistentbaseline.TheblacklineshowstheoptimalpolicypathsundercanonicalNKmodelmodewithrationalexpectations;thereddashedlineshowstheoptimalpolicypathsusingtheNK-FHPmodelmodewithinertialexpectations;and,thedashedblueshowstheoptimalpolicypathswhenthereisuncertaintyaboutexpectationsformation.modelwithREbutisnecesconditionaldistributionsothatateachdatdeclineineconomicactivity,whichthecentralbankMeanSquaredLossesfromInflationGap2.52.01.51.00.50.0Figure4:ThePolicyTradeofFrontierMeanSquaredLossesfromInflationGap2.52.01.51.00.50.0CertaintyEquivalentUncertaintyCertaintyEquivalentUncertainty01234MeanSquaredLossesfromOutputGapFigure3indicatesthatpolicyrespondsmoreaggressivelyunderuncertaintythanintheCEplanninghorizonintheeconomyinwhichtheplanninghorizonisuncertainisthesameasforthatmode.15Thus,thecanonicalNKmodelwithrationalexpectationshasthesameaverageplanninghorizonastheeconomywithanuncertainplanninghorizonandweuseitasourcertainty-equivalentbenchmark.Figure3showsthat,relativetothatCEbenchmarkeconomy,thecentralbanktightenspolicynotablymorethanundercertaintyequivalencebecauseintheeconomywithandk1.objective.PolicyTradeofFrontier.ThemoreaggressiveresponseofmonetarypolicyunderuncertaintyFigure4highlightshowuncerlence.IntheCEeconomyofthecanonicalNKmodel,ifacentrofin且ationfromtargetisrelativelylow.InunderuncertaintythaDistributionofMacroeconomicOutcomes.Figure3comparedthemeanoutcomesofWhenthecentralbankfacesuncertaintyaboutrighttails.GainsfromaRisk-ManagementApproach.Inan