宏观经济学 Lecture 3-Ramsey Model

Lecture3:TheRamseyModelLearningObjective{Definetheagent’s(includingfirmandhousehold)optimizationproblem{Solvetheoptimalcontrolproblem–theHamiltonianApproach{DefinetheSteadyStateequilibriumSolowvs.Ramsey{Solowmodel:agentsintheeconomyfollowasimplisticlinearruleforconsumptionandinvestment.{Ramseymodel:agentschooseconsumptionandinvestmentoptimallysoastomaximizetheirindividualutility(orsocialwelfare).ÎEstablishingthemaincharacteristicsofmoderndynamicmacroeconomics(micro-foundation).Ramsey(1928)–optimalgrowthmodelHowmuchanationshouldsave?{Aframeworkforstudyingtheoptimalintertemporalallocationofresources.{Micro-foundation⇒Theoptimizingbehaviorofagentsisexplicit⇒ThesavingrateisendogenousBasicAssumptionsaboutFirms{Profit-maximizingfirmsandutility-maximizinghouseholds{Firm:{AlargenumberofidenticalfirmswithtechnologyspecifiedbyY=F(K,AL),whereAfollowsanexogenouspath,growingataconstantrateg.{Firmsareprice-takersinthemarketsfortheinputstoproduction,i.e.theyadjusttogivenpaths{r(t),w(t)}.{Firmshaveperfectforesight,i.e.theycandeductwhatthefuturewilllooklike.BasicAssumptionsaboutFirms{Profitsfallsontheowners=households.{Inequilibrium:employalllaborandcapital,remuneratedattheirmarginalproduct.{Therealinterestrate:r(t)=∂F(K,AL)/∂K=f’(k(t))(why?)(Notehere,k(t)denotescapitalperunitofefficiencylabor,likely-wisefortherestsmalllettervariables){Therealwage,w(t),willbeexpressedasthewagefortheefficiencyunitoflabor(AL):w(t)=∂F(K,AL)/∂(AL)=f(k(t))-k(t)f’(k(t)){ZeroprofitinequilibriumBasicAssumptionsaboutHouseholds{Hidenticalhouseholds,eachofwhichisofsizeL(t)/H,andhencethesizeofhouseholdsgrowsataconstantraten.{Infinitelife.{Perfectforesight{Households(actingasoneentitysincetheyareidentical)seekstomaximize:{C(t):theconsumptionlevelofeachhouseholdmember.Households’utilityfunction{CRRA(ConstantRelativeRiskAversion)utilityfunction:{Properties:(1)Thecoefficientofrelativeriskaversion–CU’’(C)/U’(C)isθ,aconstantindependentofC.Thelargerisθ,thelesswillingthehouseholdallowsitsCtovaryovertime.(2)Theelasticityofsubstitutionbetweenconsumptionatanytwopointsintimeis1/θ.(3)Whenθ=1,theutilityfunctiontakesthelogfunctionalform.(verify,hint:Hopitalrule)Households’UtilityFunction{Transformc(t)=C(t)/A(t):{WhereB=A(0)1-θL(0)/Handβ=ρ-n-(1-θ)g0Households’BudgetConstraint(ResourceConstraint){Allhouseholdshavethesamechoicesinequilibrium,i.e.eachhouseholdwillholdK(t)/Hunitsofcapital.{Rentingoutcapitalthengivesincome:r(t)K(t)/H{Incomefromsellinglaborw(t)L(t)A(t)/H{Inequilibrium,householdsdonotreceiveprofitsduetotheconstantreturnstoscale.Inadditionweignoredepreciation.Hencethehouseholds’dynamicbudgetconstraintlookslikethefollowing:Households’DynamicBudgetConstraint{Definek=K/AL.then{dynamicbudgetconstraint:holdscontinuouslyforallt.{Note:Wewillbeusingasolutiontechniquethatemploysthisdynamicformulation.Romer,however,workswithasingleinter-temporalbudgetconstraint.Households’LifetimeBudgetConstraint(Romer){Thehouseholds’lifetimebudgetconstraint:{IntroducethenotationR(t)=∫0tr(τ)dτ.eR(t)isthetotalcumulativeinterestpaidfromtime0tot:oneunitofthegoodinvestedattimezeroyieldseR(t)unitsattimetinthefuture.{Thediscountfactorattime0ise-(R(t)+(g+n)t),weusethistoexpressthebudgetconstraintatpresentvalueattime0:Households’budgetconstraint:Anidentity{Thelifetimebudgetistheintegrationsolutiontothedifferentialequation—thedynamicbudget.Toseethis,Multiplyeachterminthedynamicbudgetbye-[R(t)-(n+g)t]:No-Ponzi-gamecondition{IfweletTÆ∞,weget{Wemustruleoutthathouseholdscanaccumulateborrowingforever,otherwiseitwillobviouslybeoptimaltopayforbothconsumptionandinterestonpresentcapitalbyborrowingmore.Wethereforerequirethat:{Thisisoftenreferredtono-Ponzi-gamecondition.APonzigameisaschemeinwhichsomeoneissuesdebtandrollsitoverforever.Suchaschemeallowstheissuertohaveapresentvalueoflifetimeconsumptionthatexceedsthepresentvalueofthelifetimeresources.Inthemodel,ifallowed,allhouseholdswouldbewillingtofollowthisstrategy,whichcannotbeequilibriumsinceeveryoneisidenticalanddoesthesameandthennobodywouldbereadytolendinthecreditmarket.Solvingthemodel—OptimalControlProblem{Romersolvesthemodelbymaximizingtheutilityfunctionunderthelifetimebudgetconstraint.However,findingsuchintegratedconstraintscanbecomeverydifficultinmorecomplexsettings.Wethereforeinsteaduseadifferentapproachbasedonthedynamicbudgetconstraint.{Namely,weareusingtheoptimalcontroltheorytooptimizethehouseholds’utilitysubjecttothedynamicbudgetconstraint.Itisquitestraightforwardtolearntoapplytheprocedureforsolvingsuchproblems.{WestartbyconstructingtheHamiltonianfunctionfortheproblem.Household’sProblem{Subjectto:ThePresentValuedHamiltonianFunction{Thepresentedvaluedfunction:{Thesolutionischaracterizedbythefollowingthreeconditions:ThePresentValuedHamiltonianFunction{Here,ctisreferredtoasacontrolvariablesinceitisdirectlyoperatedbyhouseholds.Anothercontrolvariableisinvestment.ktisthestatevariable,whichcannotbeadjustedinstantlyattimet;istheco-statevariablewhichisalsotheshadowpriceofthestatevariable—capital;alternatively,itisthemarginalutilityvalueofthebudget.{The