THUSEM macro chapter02_Optimal Growth

THUSEMmacrochapter02_OptimalGrowthChapter2:OptimalGrowth1TheRamsey-Cass-KoopmansModel1.1Assumptions1).TheenvironmentTheeconomyiscomposedofmanyidentical¯rmsandidenticalhouseholds.Thenumbersof¯rmsandhouseholdsarebothsu±cientlylargesothatnonehassigni¯cantin°uenceonmarketprices.2).FirmsEach¯rmhasaccesstotheproductiontechnologyY=F(K;AL)whichsatis¯esthesameassumptionsasinchapter1(theSolowmodel).Firmshireworkersandrentcapitalincompetitivefactormarkets,andselltheiroutputinacompetitivegoodsmarket.Firmsmaximizepro¯tsandareownedbythehouseholds,soanypro¯tstheyearngotothehouseholds.Sinceall¯rmsarealike,wecannormalizethenumberof¯rmstooneandviewtheproductionfunctionasanaggregateproductionfunctionoperatedbyarepresentative¯rm.Thepro¯tfunctionoftherepresentative¯rmineachperiodisgivenby¦t=Yt¡(rt+±)Kt¡wtLtwhereristherealinterestrate,±istherateofcapitaldepreciation,andwistherealwage.Note1:Theoutputpriceisnormalizedtooneanditisthesameasthepriceofcapital,sincethereisonlyonetypeofgoodsinthiseconomy.Note2:Thecompetitiverentforoneunitofcapitalequalstheusercostofcapital,whichincludestheopportunitycost(interestrater)plusdepreciationcost(±).3).HouseholdsThereareHidenticalhouseholds,whereHisa¯xednumber.EachhouseholdiscomposedofMtidenticalmembers.Eachmembersuppliesoneunitoflaborateverypointintime.Thesizeofeachhouseholdgrowsatraten:_Mt=nMtwhichimpliesthatthetotalpopulation(laborsupply)growsattheratensinceLt=HMt.Eachhouseholddecidesateachpointintimeitstotalincomebetweenconsumptionandsavingsoastomaximizeitslifetimeutilityforallmembers.Thehousehold'sutilityfunctionisgivenbyU=Z1t=0e¡½tu(Ct)Ltdt;whereCistheconsumptionofamember,u(C)isthecorrespondingutilitylevel,LtHisthenumberofmembersofthehousehold,and½isthediscountrateforthefuture.SincehouseholdsareidenticalandHis¯xed,wecanexpresstheaggregateeconomy'sutilityfunctionasHU=Z1t=0e¡½tu(Ct)Ltdt:Clearly,maximizingeachhousehold'sutilityisthesameasmaximizingtheaggregateutilityHU.HencewecanalsonormalizeHtoonewithoutlossofgenerality:H=1.Thus,theeconomycanbeviewedascomposedofansinglerepresentativehouseholdwhosuppliesallthelaborintheeconomyandownsallthecapitalstock.Whytomaximizelife-timeutilityinsteadofinstantaneousutilityatapointoftime,u(Ct)?Aren'ttheythesamething?Thereasonisthatsavingtodaya®ectsconsumptiontomorrow.Sincecurrentdecisiona®ectsthefuture,maximizinginstantaneousutilitymaynotimplymaximizinglife-timeutility.Thisisthusdi®erentfromthe¯rm'sproblem(Note:Ifthe¯rm'sproblemistodeterminethelevelofinvestment,insteadofcapital,thenwehavetoconsiderlife-timepro¯tsratherthaninstantaneouspro¯tatonepointintime.Whyarewenotdoingthatthen?Well,inthissimpleeconomyweassumethatsavingequalsinvestmentandhouseholdsowncapital,hencethe¯rm'sinvestmentproblemissolvedbythehousehold'ssavingproblem.Hencewhatisleftfor¯rmstosolveissimplyhowmuchexistingcapitaltobeutilized(rented),ratherthanhowmuchnewcapitaltobeinstalled).Theinstantaneousutilityfunctionisassumedtotaketheformu(C)=C1¡µ:Note1:Theutilityfunctionisconcaveonlyifµ¸0.Note2:Thecoe±cientofrelativeriskaversionisgivenby¡Cu00=u0=µ,whichmeasuresindividual'sattitudetowardsrisk.Note3:Theelasticityofintertemporalsubstitutionbetweenconsumptionatanytwopointsintimeis1=µ:u0(Ct)u(Ct+¢)=PtPt+¢!µCtt+¢¶¡µ=Ptt+¢!¡@³CtCt+¢´@³PtPt+¢´Ptt+¢CtCt+¢=1µ:Inourcontext,thepriceratiobetweenconsumptionatanytwopointsintimeisafunctionoftheinterestrateandthediscountrate.Hence,byestimatingtheresponsivenessofconsumptiontochangesininterestratecangiveusanempiricalmeasureofµ.Note4:C1¡µ¡11¡µ=lnCwhenµ=1.Hence,u0=C¡µ=1=Cwhenµ=1.1.2OptimalBehaviorofFirmsandHouseholds1).Firm'sbehavior@¦@K=F0K¡(r+±)=0!F0K=r+±@¦=F0L¡w=0!F0L=w:Note1:Constantreturnstoscale!F0KK+F0LL=F(K;AL):(0)Proof:CRS!F(¸K;¸AL)=¸F(K;AL)(1)Sincethisholdsforall¸;di®erentiatingbothsidesof(1)withrespectto¸gives@FK+@F(AL)=F(K;AL):Setting¸=1gives@FK+@F(AL)=F(K;AL):(2)[Alternatively,di®erentiatingbothsidesof(1)withrespecttoKandLrespectivelygives@F@(¸K)¸=¸@F@K!@F@(¸K)=@F@Kand@F@(¸AL)¸A=¸@F@(AL)A!@F@(¸AL)=@F@(AL)].Inaddition,since@F(K;AL)@L=@F(K;AL)@(AL)A;(2)thenimplies@FK+@FL=F(K;AL):¥Hence,CRSandperfectcompetitionimplythat¯rmsearnzeropro¯t;namely,revenue=totalfactorcosts:F(K;AL)=F0KK+F0LL=(r+±)K+wL:DividingbothsidesbyALgivesf(k)=F0Kk+F0L1A=(r+±)k+w1A:(3)Note2:F0K=f0(k)=(r+±):Proof:F(K;AL)=ALf(k)!F0K=ALf0k(k)@k@K=ALf0k(k)1AL=f0(k):¥Note3:F0L=A[f(k)¡f0(k)k]=A[f(k)¡(r+±)k]=w:Proof:F(K;AL)=ALf(k)!F0L=Af(k)+ALf0(k)¡¡KAL2¢=A[f(k)¡f0k(k)k]=A[f(k)¡(r+±)k].Alternatively,byequation(3),wealsohavew(=F0L)=A[f(k)¡(r+±)k]:¥Implications:Inasteadystate(oralongabalancedgrowthpath)wheref0k(¹k)=const,technologyprogressinlabore±ciencyhasnoe®ectontherealinterestrate(r),butwillraisetherealwagecontinuously:_wtt=_Att:Theintuitionforthisisasfollows.ImagingthatLisconstantovertime,thenasAtgrowsovertime,YandKwillalsogrowatthesamerateonaBGP.Since(rt+±)=f0(¹k)isconstant,implyingthatthetotalearningsofcapital,(r+±)Kt,alsogrowsatthesamerate.SinceLisconstant,undercompetitiontherealwagehastogrowatthesamerateaswellinordertoensurethatthetotalearningsoflabor,wtLt,growsovertime.Thus,wgrowswithlaborimprovingtechnology.Thispredictionisquiteconsistentwiththedata.ThisistheotherreasonforchoosingtheproductionfunctionF(K;AL)insteadofF(AK;L)orAF(K;L):2).Household'sbehaviorNote1:Interestratecompounding:Indiscretetime,1unitofgoodsinvestedattime0yields¦t¿=0(1+r¿)unitsattheendoft.Toconvertthisintocontinuoustimeanalogue,we¯rstreplace(1+r¿)byer¿.Thenwenotethat¦t¿=0er¿=ePt¿=0r¿;whosecontinuoustimeanalogueiseRt¿=0r¿d¿´eRt.Similarly,togetpresent-valuediscountingfromtto0,wereplace1t¿=0¿by1Rt=e¡Rt.a).BudgetConstraintSupposetherepresentativehousehold'sinitialcapitalholdingsisK0anditisKsattimet=s.Thentherepresentativehousehold'slife-timebudgetconstraintuptotimesise¡RsKs+Zst=0e¡RtCtLtdt=K0+Zst=0e¡RtwtLtdt;(4)wheretheLHSisthepresentvalueofcapitalholdingsasoftimespluslife-timetotalconsumption,theRHSistheinitialcapitalholdingsplusthepresentvalueoflife-timelaborincome.Toseefurtherwhatthisimplies,di®erentiatingbothsidesw.r.t.s(thisislegitimatesincetheinequalityholdsforanys)gives(note:_Rs=dRs¿=0r¿d¿ds=rs):¡rse¡RsKs+e¡Rs_Ks+e¡RsCsLs=e¡RswsLsorCtLt+_Kt=rtKt+wtLt(5)!CtLt+St=(rt+±)Kt+wtLt(50)whereSdenotessaving,whichequalsinvestmentsincecapitalistheonlyassetthehouseholdholds:St=It=_Kt+±Kt:TheLHSoftheinstantaneousbudgetconstraint(50)istotalhouseholdspendingattimet,andtheRHSistotalhouseholdincomeattimet,whichequalscapitalincomereceivedfromrentingcapitaltotherepresentative¯rmandlaborincomereceivedfromworking.Note:(5)canberewritteninper-e®ective-workerform:ct+_kt=(rt¡g¡n)kt+!t;where(r¡g¡n)istherateofreturntoper-e®ective-workercapital.b).Theno-Ponzi-gameconditionTakinglimitofthelife-timebudgetconstraint(4)ass!1,wehavelims!1e¡RsKs+Z1t=0e¡RtCtLtdt=K0+Z1t=0e¡RtwtLtdt;whereweassumelims!1e¡RsKs¸0;whichsaysthatthepresentvalueofassetholdingscannotbenegativeinthelimit.Thisisonewaytoruleout¯nancingcapitalbyrepeatedborrowing,hencetheeconomycannothavedebtinthelimit(note:wedoallowSt<0forsomet).Hence,thelife-timebudgetconstraintisZ1t=0e¡RtCtLtdt·K0+Z1t=0e¡RtwtLtdt:c).Transformationtoper-e®ectivelaboreconomy(x´X)Utilityfunction:SinceCisconsumptionpermember,wecande¯neCAasconsumptionpere®ectivemember.Thelife-timeutilityoftherepresentativehouseholdisthusU=Z1t=0e¡½tA1¡µtc1¡µtLtdt=A1¡µ0L0Z1t=0e¡¯tc1¡µtdt;where¯´[½¡(1¡µ)g¡n].toensurethatUis¯nite,weneedtoassume¯>0[Note:Morerigorously,weneed¯>(1¡µ)_c.SinceinS-S,_c=0,hence¯>0].Withoutlossofgenerality,wecanassumeA0=L0=1.Budgetconstraint(de¯ne!t´wtAtandk´KAL):Z1t=0e¡RtCtLtdt·K0+Z1t=0e¡RtwtLtdtcanbewrittenasA0L0Z1t=0e¡Rte(g+n)tctdt·K0+A0L0Z1t=0e¡Rte(g+n)t!tdtorZ1t=0e¡Rte(g+n)tctdt·k0+Z1t=0e¡Rte(g+n)t!tdt:No-Ponzi-gamecondition:lims!1e¡RsAsLsKsss=A0L0lims!1e¡Rse(¹g+n)sks=0:d).MaximizationTheconstrainedmaximizationproblemcanbesolvedusingtheLagrangemethod:maxct(Z1t=0e¡¯tc1¡µt+¸·k0+Z1t=0e¡Rte(g+n)t!tdt¡Z1t=0e¡Rte(g+n)tctdt¸)where¸>0istheLagrangianmultiplier.TheFOCw.r.t.ctis,1e¡¯tc¡µt=¸e¡Rte(g+n)t;whichsaysthatontheoptimalconsumptionpaththemarginalutilityshouldevolveaccordingto:c¡µt=¸e(¯+g+n)te¡Rt:Toseewhatthisimpliesforconsumptiongrowth,takinglogonbothsidesgives,¡µlnct=ln¸+(¯+g+n)t¡Rt;di®erentiatingw.r.t.timegives,¡µ_ctt=(¯+g+n)¡rt:(1)or_ctt=1(rt¡¯¡g¡n):(6)Thisequationsaysthatontheoptimalpath,consumptionshouldbegrowingatrater¡¯¡g¡nµ.Inparticular,consumptionperworkerisrisingiftherateofreturntocapital(r)exceedstherateatwhichthehouseholddiscountsfutureper-e®ective-workerconsumption,andisfallingifthe1IfwedonotwanttousetheEuler'sprinciple,thanonewaytoderivethis¯rst-orderconditionistousea¯nitelifemodel,lettingtheterminaldate(s)tobearbitrary,¯rstchoosing(ct)accordingtotheLeibniz'sruletoget:Zst=0e¡¯tu0(ct)dt=¸Zst=0e¡Rt+(g+n)tdtSincethishastoholdforanys,di®erentiatingtheabove¯rst-orderconditionagainwithrespecttotheterminaldata(s)gives:e¡¯su0(cs)=¸e¡Rs+(g+n)s:reverseholds.Thesmallerisµ{thelessmarginalutilitychangesasconsumptionchanges{themoreresponsiveisconsumptiontowardsdi®erencesbetweenrand(¯+g+n).Anyconsumptionpathdi®erentfromthisoneisnotutilitymaximizing.Equation(6)isknownastheEulerequationforthismaximizationproblem.Aintuitivewaytoderive(6)isasfollows.Supposec¤tmaximizesU.Thenconsiderasmallreductioninconsumptionatthispointoftimet,dc¤t;andinvestthisamountassavingforashorttime,dt,andthenconsumingthereturnsattimet+dt:Thisshouldleavelife-timeutilityunchangedifc¤tisoptimal.Namely,marginalcost=marginalbene¯tatoptimum.Sincethemarginalutilityattimetisc¡µt;thereductioninutility(marginalcost)is(c¤t)¡µdc¤t.Sincetheinstantaneousrateofreturnforper-e®ectiveunitofsavingisrt¡g¡n,thenetre-turnafterperioddtise[rt¡g¡n]dtdc¤t.Notethattheutilityvalueofthisreturndependsonthemarginalutilityofconsumptiondtperiodslater,asweneedtotakeintoaccountofdiminishingmarginalutility.Afterdt,thelevelofconsumptionbecomesc¤t+dt.Supposethegrowthrateofconsumptionis_ctct,thenwealsohavec¤t+dt=c¤te_ctctdt.Theutilityvalueofthenetreturnisthenhc¤te_ctctdti¡µ£e(rt¡g¡n)dtdc¤t¤:Sinceweneedtocomparethepresentvaluesofcostsandbene¯ts,wethereforehavee¡¯t(c¤t)¡µdc¤t=e¡¯(t+dt)hc¤te_ctctdti¡µhe(rt¡g¡n)dtdc¤ti:Dividingbye¡¯t(c¤t)¡µdc¤tandtakinglogsyields0=¡¯dt¡µ_cttdt+(rt¡g¡n)dt;rearranginggives(6):_ctct=1µ(rt¡¯¡g¡n);(6)whichsaysthatthegrowthrateofper-e®ective-workerconsumptionmustbe1µ(rt¡¯¡g¡n)inorderfortheentireconsumptionpath,fc¤tg1t=0,tobeoptimalateachpointintime(!nofurtherroomforimprovement).1.3TheDynamicsoftheEconomyUptothispoint,wehavenotsolvedfortheoptimalconsumptionpolicyyet.Anoptimalcon-sumptionpolicyisamappingbetweenthestateoftheeconomy(e.g.,thecurrentassetholdings,kt)andtheoptimalconsumptionlevel,sinceingeneralequilibrium,frt;!tgareallfunctionsofkt.Wehavetwodi®erentialequationsthatneedtobesolvejointlytodeterminetheconsumptionpolicy:theEulerequation(6)andtheinstantaneousbudgetconstraint,CtLt+_Kt=rtKt+wtLt;whichcanbetransformedto(asintheSolowmodel):ct+(g+n)kt+_kt=f(kt)¡±kt:1).ThePhaseDiagramThedynamicsofcisdescribedby_ctt=1(f0(k)¡±¡¯¡g¡n)!theconstantconsumptionlocusisdescribedby_ct=0orbyf0(k)=¯+±+g+n;whichisaverticallineinthec¡kspacewithinterceptwiththehorizontalaxisatk¤.Ifkt¯+±+g+n;!_ct>0orct";andconversely,ifkt>k¤;thenf0(k)<¯+±+g+n;!_ct<0orct#.Thedynamicsofcapitalisdescribedbytheinstantaneousbudgetconstraint,_k=f(kt)¡(±+g+n)kt¡ct;t!theconstantcapitallocusisdescribedby_kt=0orbyct=f(k)¡(±+g+n)k;whichisahump-shapedcurveinthec¡kspace.Abovethiscurve,consumptionistoohigh,hence_kt<0orkt#;belowthiscurve,consumptionistoolow,hence_kt>0orkt#.Note1:Thecrosspointofthetwoconstantlocuscurvesistheuniquesteadystateatwhichboth_ctand_ktequal0.Note2:TheoptimalconsumptionpathisaSaddlePath(Figure1),whichindicatesuniqueequilibrium.Note3:Thesolutiontothistwo-variabledi®erentialequationsystemisessentiallythepolicyct=c(kt)forarbitraryt:Hence,givenk0,weneedonlytoknowc0=c(k0),thentheEulerequationandtheinstantaneousbudgetconstraintwillguidetheeconomytowardsthesteadystatealongthesaddlepath.However,thetwodi®erentialequations(theEulerequationandtheinstantaneousbudgetconstraint)arenotsu±cienttogivetheoptimalpolicyruleunlesssomeextraconditionsaregiven[di®erentialequationsgiveonlytheoptimalrateofchange,notthelevels.Forexample,a¯rst-orderdi®erentialequation,_x=f(x;t);hasageneralsolution,x=g(x0;t);withanarbitraryconstant,x0.x0needstobedeterminedinordertopindownaparticularpath].Inthismodel,theextraconditionsaretheno-Ponzi-gameconditionandthelife-timebudgetconstraintinstrictequality.Ofteninpractice,thesaddlepathisfoundbyusingstationaritycriterionorthetransversalitycondition(moreonthispointlater).2).TheGoldenruleandbalancedgrowthpathTheS-Ssatis¯esthemodi¯edgoldenrule(=Goldenruleif¯=0):f0(k)=¯+±+g+n:Themodi¯ed-golden-rulesavingratsis1¡cy=1¡f(k)¡(g+n+±)kf(k)=(g+n+±)kf(k)=(g+n+±)®=®µg+n+±¶<®:Thebalancedgrowthrateforhouseholdvariables,fYt;Kt;CtLtg;are(g+n).Hence,optimalsavingdoesnotcontributetolong-termgrowth!Note1:IntheSolowmodel,wecan¯ndtheGolden-Rulesavingrateonlyinthesteadystate,notonatransitionalpath.Thecurrentmodelforoptimalgrowthdoespreciselythat{to¯ndGoldenrulesavingrateontransitionalpath.Note2:IntheSolowmodel,thereexistauniquesteadystateforanygivenarbitrarysavingrate.Inthecurrentmodel,however,asteadystateexistsforonlyonepossiblesavingrate,whichisthemodi¯edgoldenrulesavingrate.Therearenosteadystatesforanyotherarbitrarysavingratesbecausetheotherarbitrarysavingratesarenotoptimal.Note3:TheS-SintheSolowmodelisstable,soistheS-Sinthecurrentmodel.TheoptimalpathbeingasaddlepathdoesnotreallyimplythattheS-Sisunstable.Itsimplyshowsthattheoptimaltransitionalpathisunique.Thehouseholdcertainlyhasfreedomtodeviatefromandreturntothesaddle-pathanywhereandanytimeitwants,butthatwon'tbeanoptimalthingtodo!1.4TransitionalDynamics{GraphicAnalysis1).OptimalResponsestoUnexpected&PermanentEnvironmentalChanges(½)AlthoughwecannotanalyzetransitionaldynamicsbychangingthesavingrateaswedidintheSolowmodel,butwecanchangethemodel'sexogenousparameters,suchasthediscountrate,astheya®ecttheoptimalrateofsaving.Note1:Ahigher½impliesalowersavingrate,becausethehouseholddevaluefutureconsump-tionfurtherdownrelativetocurrentconsumptionwhen½ishigher[whatwouldbeagooddailylifeexampleforanincreasein½?].Note2:Since¯=½¡(1¡µ)g¡n;!¯"if½".2).OptimalResponsestoUnexpected&TransitoryEnvironmentalChanges(½)3).OptimalResponsestoExpected&PermanentEnvironmentalChanges(½)4).OptimalResponsestoExpected&TransitoryEnvironmentalChanges(½)1.5TransitionalDynamics{AlgebraicAnalysisLinearizationaroundtheS-S(again:-).Theconsumptionequationgives(denote¿=¡¹cµf00(¹k)):_ct=ct1(f0(kt)¡¯¡±¡g¡n)¼@_c@c[ct¡¹c]+@_c@k£kt¡¹k¤=¹cµf00(¹k)£kt¡¹k¤´¡¿£kt¡¹k¤;Thecapitalequationgives_kt=f(kt)¡(g+n+±)kt¡ct¼@_k[ct¡¹c]+@_k£kt¡¹k¤=¡[ct¡¹c]+¡f0(¹k)¡(g+n+±)¢£kt¡¹k¤=¡[ct¡¹c]+¯£kt¡¹k¤:Hencewehave:_ct=¡¿£kt¡¹k¤_kt=¡[ct¡¹c]+¯£kt¡¹k¤Note0:Theabovetwolinearequationsgivealinearphasediagram,wheretheslopeofthelinefor_kis¯>0.Di®erentiatingthecapitalequationagainandsubstitutingout_cgives:¢¢kt=¿£kt¡¹k¤+¯_kt;whichhasthecharacteristicroots:f¹1;¹2g=1³¯§p´[Note:¹1+¹2=¯;¹1¹2=¡¿].Clearly,oneofthemispositiveandtheothernegative.Hencethesolutionis(let¹1<0and¹2>0)kt¡¹k=a1e¹1t+a2e¹2t;!_kt=a1¹1e¹1t+a2¹2e¹2t!ct¡¹c=¯£kt¡¹k¤¡_kt=(¯¡¹1)a1e¹1t+(¯¡¹2)a2e¹2t=¹2a1e¹1t+¹1a2e¹2tTodeterminethetwoarbitraryconstantsfa1;a2g,wehavek0=¹k+a1+a2c0=¹c+¹2a1+¹1a2whichimpliesµk0¡¹kc0¡¹c¶=µ11¹2¹1¶µa1a2¶!µa1a2¶=1¹1¡¹2µ¹1¡1¡¹21¶µk0¡¹kc0¡¹c¶ora2=112¡c0¡¹c¡¹2(k0¡¹k)¢:Note1:c0isachoicevariable,notastatevariable.Note2:Unlessc0ischosensuchthata2=0,thesystemwillneverconvergetotheS-Sbecause¹2>0.Thisimpliesc0¡¹c=¹2(k0¡¹k)a1=k0¡¹k:Hencethesolutionsare:kt¡¹k=(k0¡¹k)e¹1t(7)ct¡¹c=¹2(kt¡¹k)=(c0¡¹c)e¹1t:(8)Thesaddlepathisthusgivenbytheoptimalconsumptionpolicyinequation(8).Notethattheslopeofthissaddlepathissteeperthanthe_k=0line:¹2(=¯¡¹1)>¯:Note3:Theoptimalrateofsavingonthetransitionalpathisnotaconstant(notice:yt¼¹y+f0(¹k)£kt¡¹k¤):st=yt¡ctt=¹y¡¹c+(f0(¹k)¡¹2)£kt¡¹k¤¹y+f(¹k)£kt¡¹k¤:Nowconsiderachangein¯duetoachangein½[note:d¯d½=1].(7)impliesdktd¯=(1¡e¹1t)d¹kd¯¡¹ke¹1td¹1d¯t(7)alsoimplies_kt=¹1(kt¡¹k);henced_ktd¯=d¹1d¯(kt¡¹k)+¹1µdktd¯¡d¹kd¯¶(8)impliesdct=d¹c+d¹2(kt¡¹k)+¹2µdkt¡d¹k¶:ThesecomparativestaticsaremuchhardertoanalyzethanthoseintheSolowmodelbecauseherenotonlydoestheS-Schange,butalsodoesthespeedofconvergence,¹.Hence,itisbettertousecomputertoconductsimulations(wewilldothislaterafterthemid-termexamintheRBCsection).Q:Givenananticipatedpermanentchangein¯thatwilltakeplacelaterinperiodt=¿¸0;what'stheoptimalresponseofconsumptionnowinperiodt=0?Solution:Thesystemdynamicsarecharacterizedbykt¡¹k=a1e¹1t+a2e¹2t(A)ct¡¹c=¹2a1e¹1t+¹1a2e¹2twherea1=112¡¹1(k0¡¹k)¡(c0¡¹c)¢a2=112¡c0¡¹c¡¹2(k0¡¹k)¢:andthesaddlepathischaracterizedbykt¡¹k=(k0¡¹k)e¹1t(B)ct¡¹c=¹2(kt¡¹k)=(c0¡¹c)e¹1tStartingfromthesteadystatefk0;c0g=©¹k;¹cª.Supposetheoptimalconsumptionchangeatt=0is¢c:Thenc0=¹c+¢c:Thenwehavea1=¡112(¢c)a2=112(¢c):¿periodslater,thevaluesoffkt;ctgatt=¿arethencharacterizedbyk¿¡¹k=¢c12(¡e¹1¿+e¹2¿)(A0)c¿¡¹c=¢c12(¡¹2e¹1¿+¹1e¹2¿)Att=¿,apermanentchangein¯takesplace.Hencethesteadystatechangesfrom©¹k;¹cªtofk¤;c¤g.Fortheconsumptionpathattime¿tobeoptimal,itmustberightonthenewsaddlepathsatisfyingthesaddle-pathproperty(B):c¿¡c¤=¹2(k¿¡k¤)Substitutingoutfk¿;c¿gusing(A0)yields¹c¡c¤+¢c12(¡¹2e¹1¿+¹1e¹2¿)=¹2µ¹k¡k¤+¢c12(¡e¹1¿+e¹2¿)¶whichimplies¢c¹1¡¹2(¹1¡¹2)e¹2¿=¹2(¹k¡k¤)¡(¹c¡c¤)or¢c=£¹2(¹k¡k¤)¡(¹c¡c¤)¤e¡¹2¿:(B0)Sinceinthelinearmodel,the_k=0locushasaslopeof¯<¹2.Sincethe_k=0locushasnotshifted,itpassesthroughboththenewandtheoldsteadystates,fk¤;c¤gand©¹k;¹cª,implyingthat¹c¡c¤=¯(¹k¡k¤):Substitutingthisinto(B0)yields¢c=(¹2¡¯)(¹k¡k¤)e¡¹2¿:Hence,¢c>0ifk¤<¹k;and¢c<0ifk¤>¹k.Furthermore,dj¢cjd¿<0;namely,theabsolutevalueof¢cdecreaseswiththewaitingperiod¿.Forexample,if¿=0(i.e.,thechangeisunanticipated),then¢c=(¹2¡¯)(¹k¡k¤);whichimpliesthatc0jumpsinstantaneouslytoalevelthatisonthenewsaddlepath:c0¡c¤=¹2(¹k¡k¤).Ontheotherhand,if¿!1(i.e.,theanticipatedchangeneverarrives),then¢c!0.Homework:Analyzethee®ectofachangeingovernmentspending.