高级宏观经济学学习笔记英文

MacroeconomicsLectureNotesTOPICS2.TheSolowGrowthmodel2-52.TheRamseyGrowthmodel6-113.RealBusinessCycleTheory12-204.TraditionalKeynesianTheoriesofFluctuations21-235.MicroeconomicFoundationsofIncompleteNominalAdjustments24-396.Consumption40-447.Investment45-508.InflationandMonetarypolicy51-901Growth1.1TheSolowGrowthModelAtanypointintime,theeconomyhassomeamountoflabor(L),capital(K)andknowledge(A)toproduceoutput(Y).TheproductionfunctioncanbewrittenasY(t)=F(K(t),A(t)L(t)),(1)wheretdenotestime.Theproductionfunctionisassumedtohaveconstantreturnstoscaleincapitalandeffectivelabor,(F(ck,cAL)=cF(K,AL)wherec≥0).Hence,wecanwrite1ALF(K,AL)=F(KAL,1).(2)Definingk=K/AL,y=Y/ALandf(k)=F(K,1),wecanrewriteequation(2)as:y=F(k)(3)Here,theoutputpereffectivelabordependsonthecapitalpereffectivelabor.The”intensive”or”percapita”formoftheproductionfunctionf(k)satisfiesf(0)=0,f′(k)>0,f′′(k)<0.Alsotoensurethatthepathoftheeconomydoesnotdiverge,weassumelimk→0f′(k)=∞,limk→∞f′(k)=0.LaborandknowledgegrowatconstantratesL˙(t)=nL(t)(4)A˙(t)=gA(t)(5)wherenandgareexogeneousparametersandX˙(t)isshortfordX(t)/d(t).Then4impliesthatgrowthrateofLisnandthatofAisg.Outputisdividedbetweenconsumption(C)andinvestment(I).Invest-mentisafractionofoutput(s)andisexogenousandconstant.Capitaldepreciatesatarateδ.ThusK˙(t)=sY(t)−δK(t).(6)Notethatthesumofn,gandδisassumedtobepositive.21.1.1ThedynamicsofthemodelSincewewillsolvethemodelforaunitofeffectivelabor,wecanusethechainruletoshowthatcapitalperunitofeffectivelabork=K/ALcanbewrittenas:k˙(t)=K˙(t)A(t)L(t)−K˙(t)[A(t)L(t)]2[A(t)L˙(t)+L(t)A˙(t)]=K˙(t)A(t)L(t)−K(t)A(t)L(t)L˙(t)L(t)−K(t)A(t)L(t)A˙(t)A(t).(7)Droppingthet,thisequationcanfinallybewrittenask˙=sf(k)−(n+g+δ)k(8)Thefirstcomponentistheactualinvestmentpercapitaandthesecondtermisbreakeveninvestment(n+g+δ)k),theamountofinvestmenttopreventkfromfalling(duetodepreciationandgrowthofquantityofeffectivelabor)fromitsexistinglevel.Ifk<k∗then(˙k)>0andkisrising.Ifk>k∗then(˙k)<0andkisfalling.Inthatcase,kconvergestok∗.(seefig1.2inRomer.)1.1.2ThebalancedgrowthpathTheSolowgrowthmodelimpliesthatregardlessofitsstartingpoint,theeconomyconvergestoabalancedgrowthpath—eachvariableofthemodelgrowsataconstantrate.Onthebalancedgrowthpath,thegrowthrateofoutputperworkerissolelydeterminedbytherateoftechnologicalprogress.1.1.3ChangingthesavingratePolicycanimpactthesavingrate,henceactualinvestmentshiftsupwardsshiftingthepositionofk∗.SeeanexperimentinFigure1.4wheresavingincreases.Insum,changeinthesavingratehasaleveleffectastheoutputperworkerincreasesbutitdoesnoteffecttherateofoutputgrowthperworker.Onlytechnologicalprogresshavegrowtheffects.Welfaredependsonconsumption.Hence,ajumpinsavingswillini-tiallyreduceconsumptionwhichwillincreaseafterwards.Thequestionifconsumptionincreasesordecreasesfromtheinitialleveldependsonwhether3f′(k∗)—marginalproductofcapital—ismoreorlessthann+g+δ.Thevalueofk∗whichmaximizesthelevelofconsumptionisknownasthegolden-rulelevel.1.1.4CentralQuestionsreSolowModelTherearetwosourcesofvariationovertimeoracrossnations.Differencesofcapitalperworkerandtheeffectivenessoflabor(A).Butonlytheeffec-tivenessoflaborcancreatepermanentgrowthinoutputperworker.Hence,variationsintheaccumulationofphysicalcapitaldonotaccountforasig-nificantpartofeitherworldwideeconomicgrowthorcrosscountryincomedifferences.Theothersourceofgrowthistheeffectivenessoflabor.Butthemodelissilentaboutit.Apossibilityisthatitrepresentsknowledge.Otherwise,education,skillsoflaborforce,thestrengthofpropertyrights,qualityofinfrastructure...1.2EmpiricalApplicationsSolowmodelledtoseveralpapers.Growthaccounting:Howmuchofgrowthisduetoincreasesinvariousfactorsofproductionandhowmuchofitstemsfromotherfactors.ResearcherslookedattheSolowresidual(R(t))whichisameasureofthecontributionoftechnologicalprogress.Y˙Y−L˙L=αK[K˙K−L˙L]+R(9)GrowthinAsiancountriesareattributedtorapidincreaseintheircapitalstockratherthantechnologicalimprovements.Also,researchersinvestigatedwhypoorcountriesgrowfasterusingtheideashereanddiscussedaboutconvergenceandconditionalconvergence:Convergence:Researchersaskwhetherpoorcountriestendtogrowfasterthanrichcountries?Theabsoluteconvergencehypothesis,positsthefollow-ing:consideragroupofcountries,allofwhichhavehaveaccesstothesametechnology(f(k)),thesamepopulationgrowthrate(n)andthesamesavingspropensity(s),andonlydifferintermsoftheirinitialcapital-laborratio,k.Therearethreereasonstoexpectsuchconvergence.First,theSolowgrowthmodelpredictscountriestoconvergetotheirbalancedgrowthpaths;poorcountriescatchuptorichones.Secondreturnoncapitalishigherinpoor4countries.Hencethereisincentiveforcapitaltoflowintopoorcountriesfromrichcountriescausingconvergence.Third,overtimeaccesstonewtechnol-ogyhelpspoorcountriescatchupwiththerich.EmpiricallyBaumol(1986)estimatesln[(YN)i,1979]−ln[(YN)i,1870]=a+bln[(YN)i,1870]+�i(10)andexpectstofindb=−1toconcludeconvergence.Butthereareproblemsinthisstudy.Theconditionalconvergencehypothesisstatesthatifcountriespossessthesametechnologicalpossibilitiesandpopulationgrowthratesbutdifferinsavingspropensitiesandinitialcapital-laborratio,thenthereshouldstillbeconvergencetothesamegrowthrate,butjustnotnecessarilyatthesamecapital-laborratio.Themodelalsopredictsthat—Thegrowthratewillbehighwhenthecapitalperworkerislowanditwillslowdownasitgetshigher.—AlowvalueofcapitalperworkeralsoimpliesthatMPKwillbehighmeaningthatrealinterestratewillbehigh.—Realinterestratefallsascapital’smarginalproductdeclinesasaneconomydevelops.52TheRamsey–Cass-KoopmansModelTheRamseymodelissimilartotheSolowmodelbutthedynamicsoftheeconomicaggregatesaredeterminedatthemicroeconomiclevel.Hence,sav-ingisnolongerconstantandexogenous;i.e.themodelallowsforendogenousconsumption–savingschoices.Althoughimportant,largenumbersofpost-graduatestudentsandevenmoreundergraduatesremainlargelyunawareofitsinfluenceorevenitsexistence.Thereasonissimple:theRamseymodelrequiresalevelofmathematicaltechniquewhichiswellbeyondthattaughtinmostmainstreamundergraduateeconomicsdegrees.Evenifwecannotconveythefullmathematicalcomplexityofamodeltostudents,itisstilldesirabletogivethemainflavorofitsresultsanditsimplications.2.1AssumptionsFirmsTherearealargenumberofidenticalfirms.EachhasaccesstotheproductionfunctionY=F(K,AL),whichsatisfiesthesameassumptionsasthatintheSolowmodel.FirmstakeAasgiven(itgrowsexogenouslyatrateg).Theproductionfunctioncanthereforebeexpressedinintensiveformasy=f(k),wherekrepresentsKperunitofeffectivelabor.Firmsareassumedtobecompetitiveinbothinputandoutmarkets,sothatcapitalandlaboreachreceivetheirmarginalproducts:rt=f′(kt)andWt=At[f(kt)−ktf′(kt).Thewageperunitofeffectivelaboriswt=f(kt)−ktf′(kt).HouseholdsTherearealargenumberofhouseholdsandtheyownfirmsandcapital;divideitsincome(receivedfromcapitalandlabor)betweenconsumptionandsavingstomaximizeitslifelongutility.Thesizeofeachhouseholdgrowsatraten.Eachmemberifhouseholdsupplies1unitoflaborateverypointintime.Theyrentcapitaltofirms.ThereareHnumberofhouseholdsandthereisnodepreciationforsimplicity.ThehouseholddivideincomebetweenconsumptionandsavingstomaximizeitslifetimeutilitywhichtakestheformU=∫∞t=0e−ρtu(C(t))L(t)Hdt(11)whereC(t)istheconsumptionofeachmemberofthehousehold,u(·)istheutilityfunction,L(t)isthetotalpopulationandL(t)/Histhenumberof6membersinahouseholdandρisthediscountfactor.Theu(C(t))L(t)/Histhehousehold’stotalutilitywheretheinstantaneousutilityfunction,u(C(t))isoftheformu(C(t))=C(t)1−θ1−θ,θ>0,ρ−n−(1−θ)g>0(12)Notethatθdeterminesthehousehold’swillingnesstoshiftconsumptionbe-tweentwoperiods(1/θcanbedefinedastheelasticityofintertemporalsub-stitutionbetweenconsumptionatanytwopointsintime.)Asθgoesto0,marginalutilityfallsmoreslowlyasconsumptionrisessothatthehouseholdismorewillingtoallowconsumptiontovaryovertime.Thelastassumption,ρ−n−(1−θ)g>0,ensuresthatlifetimeutilitydoesnotdiverge.2.1.1ThebehaviorhouseholdsandfirmsHousehold’sbudgetconstraint:Therepresentativehouseholdtakerandwasgiven.SincethehouseholdhasL/Hmembers,itslaborincomeattisW(t)L(t)/HandconsumptionexpendituresareC(t)L(t)/H.Sinceinterestratesovertimecanchange,wewriteR(t)as(=∫tτ=0r(τ)dτ).Oneunitofgoodinvestedatt=0yieldseR(t)unitsofthegoodattimet(showingtheeffectsofcontinuouscompounding).ThehouseholdsconsumptionisC(t)L(t)/Handitsinitialwealthis1/Hofwealthattime0orK(0)/HThehouseholdsbudgetconstraintistherefore:∫∞t=0e−R(t)CtLtHdt≤K0H+∫∞t=0e−R(t)WtLtHdt.(13)Theconstrainimpliesthatthehouseholdcannotspendmorethanitsinitialwealth.Inotherwords,onecannotissuedebtandrollitforever.Theissuermustpayoffthedebt.Household’smaximizationproblem:Therepresentativehouseholdwantstomaximizeitslifetimeutilitysubjecttobudgetconstraint.Definec(t)astheeffectivelaborconsumption,consump-tionperworker,C(t),canbewrittenasA(t)c(t).Hence,thehouseholdsobjectivefunctioncanbewrittenintheintensiveformasU=B∫∞t=0e−βtc(t)1−θ1−θdt(14)7whereB≡A(0)1−θL(0)/Handβ≡ρ−n−(1−θ)g,whereβ>0.Wecanalsowritethehouseholdbudgetconditionintheintensiveform∫∞t=0e−R(t)c(t)e(n+g)tdt≤k(0)+∫∞t=0e−R(t)w(t)e(n+g)tdt(15)HouseholdBehavior:Thehouseholdchoosesthepathofc(t)tomaximizelifetimeutility(14)subjecttobudgetconstraint(15).Tosolvetheproblem,wesetuptheLagrangianL=B∫∞t=0e−βtc(t)1−θ1−θdt(16)+λ[k(0)+∫∞t=0e−R(t)w(t)e(n+g)tdt−∫∞t=0e−R(t)c(t)e(n+g)tdt]Thehouseholdchoosescateachpointintime.HencemaximizingthelagrangianwegetBe−βtc(t)−θ=λe−R(t)e(n+g)t(17)whichcharacterizesthehouseholdbehavior.Takingthelogandthetimederivativeofbothsidesweobtainc˙(t)c(t)=r(t)−ρ−θgθ(18)whichimpliesthatconsumptionperworkerrisesaslongasrealreturnrisgreaterthanthediscountfactorρ.Thisequationisalsoknownastheeulerequationanddescribeshowconsumptionbehavesovertime.Tointerpret(20),recallthatconsumptionperworkerisC(t)=c(t)A(t),hencewecanwriteC˙(t)C(t)=r(t)−ρθ.(19)Equation(19)impliesthatconsumptionperworkerisrisingifrealreturnexceedshousehold’sdiscountrate.82.1.2TheDynamicsoftheEconomyRecallingthatr(t)=f′(k(t)),wecanwrite20asc˙(t)c(t)=f′(k(t))−ρ−θgθ(20)andthedynamicsforcapitalstockcanbewrittenask˙(t)=f(k(t))−c(t)−(n+g)k(t)(21)Combiningthetwoequationsinaphasediagramwecancomputethemodi-fiedgoldenrulek∗whichhappenstobesmallerthanthegolden-rulelevelofkgivenbythepeakofthek˙(t)schedule(seefigure2.3inyourbook).Welfare:Sincemarketsarecompetitiveandtherearenoexternalities,thefirstwelfaretheoremfrommicroeconomicshold.Thedecentralizedequi-libriumproduceshighestpossibleutilityamongallocationsthattreatallhouseholdsinthesameway.2.2PropertiesofthebalancedgrowthpathThepropertiesoftheeconomyatequilibriumisidenticaltothatoftheSoloweconomyonthebalancedgrowthpath.Capital,outputandconsumptionperunitofeffectivelaborareconstant,andatthatpointsavingsrateisalsoconstant!Totalcapitalstockandoutputandconsumptiongrowatraten+g.Hence,centralimplicationsoftheSolowgrowthmodeldoesnothingeonconstantsavingrate.Theonlydifferenceisthattheeconomycannotoperateonacapitallevelabovethegolden-rulecapitalstockk∗.Furthermore,theeconomydoesnotconvergetothebalancegrowthpaththatgivesthemaximumsustainablelevelofconsumption.Thereasonisthathouseholdsvaluepresentvalueofconsumptionmorethanfuturelevelofconsumption,kconvergestoalevelbelowk∗knownasthemodifiedgolden-rulecapitalstock.2.2.1TheeffectsofafallinthediscountrateGiventhesaddlepath(seefigure2.5inyourbook),wecanshowthatapermanentunexpecteddropinthediscountfactorρcausehouseholdstodis-countfutureconsumptionless.Thisisanaloguetoariseinsavingrate.A9theinstanceofchangeinρ,thecurrentconsumptiondropssothattheecon-omyisnowonanewsaddlepath.Thereafter,cincreasestoahigherlevelascapitalstockbuildsup.Eventuallybothcandkarehigherthantheoriginallevels(seefig2.6inyourbook).ThisresultsissimilartothatintheSolowgrowthmodelwhenthesavingrateincreasesexceptthatthefractionofout-putthatissavedduringtheprocessisnotconstant.Weobservetemporaryincreasesinthegrowthratesofcapitalperworkerandoutputperworker.2.2.2TheEffectsofGovernmentPurchasesSupposethegovernmentbuysoutputofG(t)perunitofeffectivelaborattimet.Assumethatthisgovernmentspendingdoesnotaffectutilityfromprivateconsumptionnordoesitaffectfutureoutputastheyareforpublicconsumption.Thepurchasesarefinancedbylumpsumtaxesandthegov-ernmentbalancesitsbudgeteveryperiod.(Ifdebtwasallowed,withinthecontextofthismodeldeficitfinancingwouldnotmakesignificantdifferenceonanyoftheimportantvariables;RicardianEquivalence).Hence,investmentisnowthedifferencebetweenoutputandthesumofprivateconsumptionandgovernmentpurchases.k˙(t)=f(k(t))−c(t)−G(t)−(n+g)k(t)(22)whichmeansthatahighervalueforGshiftsthe(˙k)=0scheduledown(seefigure2.8inyourbook).Thebudgetconstraintofthehousehold(15)willchangeduetoimposedtaxestocoverthegovernmentexpenditures(T=G)∫∞t=0e−R(t)c(t)e(n+g)tdt≤k(0)+∫∞t=0e−R(t)[w(t)−G(t)]e(n+g)tdt(23)SupposethatG(t)isinitiallyzeroandthenincreasespermanentlytoalevelGH.Thenthe(k˙=0)locusshiftsdownbyamountoftheincreaseinG.Butwhatdoesthetransitionpathlooklike?SincethegovernmentpurchasesdonotaffecttehEulerequation,thec˙=0isunaffected.Tosatisfytheintertemporalbudgetconstraint,householdsmustbeonthesaddlepathonceGhasincreased.However,atthetimewhenthepolicyischanged,kispredetermined,sotheonlywaytogettothesaddlepathisifconsumptionfallsimmediatelybythefullamountoftheincreaseingovernmentspending.Inotherwords,thereisnogradualtransitionandprivateconsumptionis10completelycrowdedoutbypublicconsumption.Meaningthepermanentincreaseingovernmentexpensesmeansapermanentincreaseintaxesleadingtoareductionhousehold’slifetimewealth.Thiscontrastswiththetraditionalapproachwhereconsumptiondependsonlytodisposableincomeandrespondslessthanone–for–onewiththatin-come.Hence,ariseingovernmentexpensescrowdsoutinvestmentsothecapitalstockfallandtherealinterestratestartstorise.Butthatassumes,householdsdonotdoanyintertemporaloptimization.Here,privatecon-sumptionfallsduetoapurewealtheffecttheincreaseinGcauseswealthtofallandtheoptimalwayforhouseholdstorespondisbyloweringconsump-tionimmediatelyinproportion.Hereinvestmentisnotcrowdedout.Amorecomplicatedsituationisanunexpectedtemporaryincreaseingovernmentexpenses.Thisleadstoatemporaryincreaseininterestrates.Thisanalysissuggeststhathouseholdsexpectstheirfutureconsumptionwillbehigherinthefuturethanitisinpresent.Annaturalexampleishighgovernmentpurchasesduringwars.Barro(1987)teststhispredictionusingdataonBritishmilitaryspendingandinterestratesandfindsstatisticallysignificantrelationshipbetweenthevariables.2.3IssuesforfutureanalysisCapitalaccumulationcannotexplainlong-rungrowthorcross-countryin-comedifferences.Weshouldunderstandwhateffectivenessoflabormean.Doesitrepresentknowledge?Researchshowsthatknowledgeaccumulationiscentraltoworldwidegrowthbutnottocross-countrydifferences.Incomedifferencesacrosscountriesmaybeattributabletodifferencesinhumancapi-tal.Possiblyinstitutionsareimportant,too.Thisisbasicallycalledthenewgrowththeory.113RealBusiness–CycleTheory3.1OverviewUnderstandingaggregatefluctuationsisacentralgoalofmacroeconomics.Facts:i)theydonotexhibitregularorcyclicalpattern.Outputdeclinesvaryinsizeandspaceconsiderably.Hence,economiststurnedawayfromfromattemptingtoexplainthesefluctuationsascombinationsofdetermin-isticcyclesofdifferentlengths;businesscyclesarerandom.Issueisthepropagationmechanismsandthetypeofshocks.ii)Fluctuationsaredis-tributedunevenlyacrosscomponentsofGDP(seetable4.2inyourbook).Ingeneralinvestmentvariesthemostwhileconsumptionisrelativelystable.iii)outputgrowthisdistributedroughlysymmetricallyaroundthemean.Also,lowgrowthisquicklyfollowedbyextremelyhighgrowth.iv)Aggre-gatefluctuationsbeforeandafterthegreatdepressiondonotappeartobedifferentfromeachother,suggestingthatcharacterofforcesthatdeterminefluctuationsdidnotchange.Thebehaviorofsomeimportantmacrovariablescanbesummarizedasfollows:i)employmentfallsii)lengthofaverageworkweekfalls.iii)Declineinemploymentissmallrelativetodeclineinoutput.Thusproductivity(outputperworkerhour)fallsduringrecessions.TherelationshipbetweenmovementsinoutputandemplymentisknownasOkun’slaw;a3%shortfallofGDPleadsto1%dropinemployment.iv)Inflationshowsnoclearpatternv)realwagefallsslightlyvi)Nominalandrealinterestratesdecline.vii)Realmoneystocksshownoclearpattern.3.2TheoriesofFluctuationsRealbusinesscyclemodelsareWalrasian-competitivemodelwithoutanyexternalitiesasymmetricinformation,missingmarketsorothermarketim-perfections.TheRamseymodelisabaselinemodelwhichweextendtoincorporaterealshocks:technologyshocksorgovernmentspending.Changethemodeltoallowforvariationsinemploymentforbusinesscyclesfeatureprocyclicallabourinputs.Inthathouseholdsutilitywillalsodependontheamounttheyworkaswellasconsumptionwhileemploymentisdeterminedbytheintersectionoflabordemandandsupply.RBCmodelsarederivedfrommicroeconomicfoundationsandtheirsuc-cessisassessedthroughcalibration.InRBCmodels,eachstageofthebusi-12nesscycleisviewedasanequilibrium.Thisdoesnotmeanthatworkerspreferslumpstobooms,justthatslumpsrepresentundesired,undesirableandunavoidableshiftsintheconstraintsthatpeopleface,butthatgiventhoseconstraints,marketsreactefficientlyandpeopleachievethebestout-comesthatcircumstancespermit.However,manybelievethattechnologyshocksandthepropagationmech-anismareoflittlerelevancetoactualfluctuations,andthatfailureofnominalpricesandwagestoadjustfullytothosedisturbancearecentraltofluctu-ations;Keynesiantheoriesoffluctuations.However,Keynesianmodelsdonotaddresstheissuewhythesevariablesbehaveastheydoandwhatcausestheirbehaviortochange.ModernKeynesianmodelsincorporatesthericherspecificationsoftheRBCmodels.Afewmajorpuzzlesare1Whydolaborinputsvary?Specifically,ifthereiscyclicalunemployment,whatshockcauseslabordemandtogodown?Oncelabordemandgoesdown,whyisthisreflectedinunemploymentratherthanwagecuts?2Decreasingreturnstoscaleinlaborimpliescountercyclicalproductivity,butproductivityisprocyclical.Why?3Whyarerecessionssopersistent?TheRBCtheoryexplainsprocyclicalproductivityquitedirectly—boomsaregooddrawsoftechnologygrowthrecessionsarebaddraws.Thesecondpuzzleisemployment.InthebaselineRBCmodel,employ-mentfluctuationsarejustintertemporalsubstitutionofleisure.Insometimeperiods,laborislessproductivethanothers.Theoptimalactionforworkersistoworkmoreinproductiveperiods,lessinunproductiveperiods.Thisexplainsunemployment.Thethirdpuzzleispersistence.InthebaselineRBCmode,capitalaccu-mulationisthe“internalpropagationmechanism”-thethingthatconvertsshockswithoutpersistenceintohighlypersistentshockstooutput.Ifthetechnologylevelisbelowaverage,outputislow,soinvestmentislow,sothenextperiod’scapitalstockisalsobelowaverage.Soevenifthetechnologylevelreturnstonormalnextperiod,outputwillbebelownormal.Afourthobservation(notsomuchapuzzle)iswhyinvestmentspendingismorevariablethanconsumptionspending.ThisisnotsohardtoexplaininthebaselineRBCmodel(oranyothermodel):anagentwiththepreferencetosmoothconsumptionovertimewillinvestinproductiveperiodsandeat13capitalinunproductiveperiods.3.3ABaselineRBCmodelThemodelisadiscreet-timevariationoftheRamseymodel.Theeconomyconsistsofalargenumberofidentical,price–takingfirmsandalargenumberofidentical,price–takinghouseholds,wholiveinfinitelylong.Theinputstoproductionarecapital(K),labor(L)and“technology”(A).TheproductionfunctionisCobb–DouglasYt=Kαt(AtLt)1−α0<α<1.(24)Y=C+I+Gand