英文原版经济学系列——动态经济学 277页

DynamicEconomicsQuantitativeMethodsandApplicationsJe´roˆmeAddaandRussellCooperTheMITPressCambridge,MassachusettsLondon,England\nContents1Overview1ITheory2TheoryofDynamicProgramming72.1Overview72.2IndirectUtility72.2.1Consumers72.2.2Firms82.3DynamicOptimization:ACake-EatingExample92.3.1DirectAttack102.3.2DynamicProgrammingApproach122.4SomeExtensionsoftheCake-EatingProblem162.4.1InfiniteHorizon162.4.2TasteShocks202.4.3DiscreteChoice222.5GeneralFormulation242.5.1NonstochasticCase242.5.2StochasticDynamicProgramming292.6Conclusion313NumericalAnalysis333.1Overview333.2StochasticCake-EatingProblem343.2.1ValueFunctionIterations343.2.2PolicyFunctionIterations403.2.3ProjectionMethods413.3StochasticDiscreteCake-EatingProblem463.3.1ValueFunctionIterations47\nviiiContents3.4ExtensionsandConclusion503.4.1LargerStateSpaces503.5Appendix:AdditionalNumericalTools523.5.1InterpolationMethods523.5.2NumericalIntegration553.5.3HowtoSimulatetheModel594Econometrics614.1Overview614.2SomeIllustrativeExamples614.2.1CoinFlipping614.2.2SupplyandDemandRevisited744.3EstimationMethodsandAsymptoticProperties794.3.1GeneralizedMethodofMoments804.3.2MaximumLikelihood834.3.3Simulation-BasedMethods854.4Conclusion97IIApplications5StochasticGrowth1035.1Overview1035.2NonstochasticGrowthModel1035.2.1AnExample1055.2.2NumericalAnalysis1075.3StochasticGrowthModel1115.3.1Environment1125.3.2Bellman’sEquation1135.3.3SolutionMethods1155.3.4Decentralization1205.4AStochasticGrowthModelwithEndogenousLaborSupply1225.4.1Planner’sDynamicProgrammingProblem1225.4.2NumericalAnalysis1245.5ConfrontingtheData1255.5.1Moments1265.5.2GMM1285.5.3IndirectInference1305.5.4MaximumLikelihoodEstimation131\nContentsix5.6SomeExtensions1325.6.1TechnologicalComplementarities1335.6.2MultipleSectors1345.6.3TasteShocks1365.6.4Taxes1365.7Conclusion1386Consumption1396.1OverviewandMotivation1396.2Two-PeriodProblem1396.2.1BasicProblem1406.2.2StochasticIncome1436.2.3PortfolioChoice1456.2.4BorrowingRestrictions1466.3InfiniteHorizonFormulation:TheoryandEmpiricalEvidence1476.3.1Bellman’sEquationfortheInfiniteHorizonProblem1476.3.2StochasticIncome1486.3.3StochasticReturns:PortfolioChoice1506.3.4EndogenousLaborSupply1536.3.5BorrowingConstraints1566.3.6ConsumptionovertheLifeCycle1606.4Conclusion1647DurableConsumption1657.1Motivation1657.2PermanentIncomeHypothesisModelofDurableExpenditures1667.2.1Theory1667.2.2EstimationofaQuadraticUtilitySpecification1687.2.3QuadraticAdjustmentCosts1697.3NonconvexAdjustmentCosts1717.3.1GeneralSetting1727.3.2IrreversibilityandDurablePurchases1737.3.3ADynamicDiscreteChoiceModel1758Investment1878.1OverviewandMotivation1878.2GeneralProblem188\nxContents8.3NoAdjustmentCosts1898.4ConvexAdjustmentCosts1918.4.1QTheory:Models1928.4.2QTheory:Evidence1938.4.3EulerEquationEstimation1988.4.4BorrowingRestrictions2018.5NonconvexAdjustment:Theory2028.5.1NonconvexAdjustmentCosts2038.5.2Irreversibility2088.6EstimationofaRichModelofAdjustmentCosts2098.6.1GeneralModel2098.6.2MaximumLikelihoodEstimation2128.7Conclusion2139DynamicsofEmploymentAdjustment2159.1Motivation2159.2GeneralModelofDynamicLaborDemand2169.3QuadraticAdjustmentCosts2179.4RicherModelsofAdjustment2249.4.1PiecewiseLinearAdjustmentCosts2249.4.2NonconvexAdjustmentCosts2269.4.3Asymmetries2289.5TheGapApproach2299.5.1PartialAdjustmentModel2309.5.2MeasuringtheTargetandtheGap2319.6EstimationofaRichModelofAdjustmentCosts2359.7Conclusion23810FutureDevelopments24110.1OverviewandMotivation24110.2PriceSetting24110.2.1OptimizationProblem24210.2.2EvidenceonMagazinePrices24410.2.3AggregateImplications24510.3OptimalInventoryPolicy24810.3.1InventoriesandtheProduction-SmoothingModel24810.3.2PricesandInventoryAdjustment25210.4CapitalandLabor254\nContentsxi10.5TechnologicalComplementarities:EquilibriumAnalysis25510.6SearchModels25710.6.1ASimpleLaborSearchModel25710.6.2EstimationoftheLaborSearchModel25910.6.3Extensions26010.7Conclusion263Bibliography265Index275\n1OverviewInthisbookwestudyarichsetofappliedproblemsineconomicsthatemphasizethedynamicaspectsofeconomicdecisions.Althoughourultimategoalsaretheapplications,weprovidesomebasictech-niquesbeforetacklingthedetailsofspecificdynamicoptimizationproblems.Thiswayweareabletopresentandintegratekeytoolssuchasdynamicprogramming,numericaltechniques,andsimula-tionbasedeconometricmethods.Weutilizethesetoolsinavarietyofapplicationsinbothmacroeconomicsandmicroeconomics.Over-all,thisapproachallowsustoestimatestructuralparametersandtoanalyzetheeffectsofeconomicpolicy.Theapproachwepursuetostudyingeconomicdynamicsisstruc-tural.Asresearcherswehavefrequentlyfoundourselvesinferringunderlyingparametersthatrepresenttastes,technology,andotherprimitivesfromobservationsofindividualhouseholdsandfirmsaswellasfromeconomicaggregates.Whensuchinferencesaresuc-cessful,wecanthentestcompetinghypothesesabouteconomicbehaviorandevaluatetheeffectsofpolicyexperiments.Toappreciatethebenefitsofthisapproach,considerthefollowingpolicyexperiment.InrecentyearsanumberofEuropeangovern-mentshaveinstitutedpoliciesofsubsidizingthescrappingofoldcarsandthepurchaseofnewcars.Whataretheexpectedeffectsofthesepoliciesonthecarindustryandongovernmentrevenues?Atsomelevelthisquestionseemseasyifaresearcher‘‘knows’’thedemandfunctionforcars.Butofcoursethatdemandfunctionis,atbest,elusive.Furtherthedemandfunctionestimatedinonepolicyregimeisunlikelytobeveryinformativeforanovelpolicyexperi-ment,suchasthisexampleofcarscrappingsubsidies.Analternativeapproachistobuildandestimateamodelofhouseholddynamicchoiceofcarownership.Oncetheparametersof\n2Chapter1thismodelareestimated,thenvariouspolicyexperimentscanbeevaluated.1Thisapproachseemsconsiderablymoredifficultthanjustestimatingademandfunction,andofcoursethisisthecase.Itrequiresthespecificationandsolutionofadynamicoptimizationproblemandthentheestimationoftheparameters.But,asweargueinthisbook,thismethodologyisfeasibleandyieldsexcitingresults.Theintegrationofdynamicoptimizationwithparameterestima-tionisattheheartofourapproach.Wedevelopthisideabyorga-nizingthebookintwoparts.PartIprovidesareviewoftheformaltheoryofdynamicopti-mization.Thisisatoolusedinmanyareasofeconomics,includ-ingmacroeconomics,industrialorganization,laboreconomics,andinternationaleconomics.Asinpreviouscontributionstothestudyofdynamicoptimization,suchasbySargent(1987)andbyStokeyandLucas(1989),ourpresentationstartswiththeformaltheoryofdynamicprogramming.Becauseofthelargenumberofothercon-tributionsinthisarea,ourpresentationinchapter2reliesonexistingtheoremsontheexistenceofsolutionstoavarietyofdynamicpro-grammingproblems.Inchapter3wepresentthenumericaltoolsnecessarytoconductastructuralestimationofthetheoreticaldynamicmodels.Thesenumericaltoolsservebothtocomplementthetheoryinteachingstudentsaboutdynamicprogrammingandtoenablearesearchertoevaluatethequantitativeimplicationsofthetheory.Inourexperi-encetheprocessofwritingcomputercodetosolvedynamicpro-grammingproblemshasprovedtobeausefuldeviceforteachingbasicconceptsofthisapproach.Theeconometrictechniquesofchapter4providethelinkbetweenthedynamicprogrammingproblemanddata.Theemphasisisonthemappingfromparametersofthedynamicprogrammingprob-lemtoobservations.Forexample,avectorofparametersisusedtonumericallysolveadynamicprogrammingproblemthatisusedtosimulatemoments.Anoptimizationroutinethenselectsavectorofparameterstobringthesesimulatedmomentsclosetothemomentsobservedinthedata.PartIIisdevotedtotheapplicationofdynamicprogrammingtospecificareasofeconomicssuchasthestudyofbusinesscycles,consumption,andinvestmentbehavior.Thepresentationofeach1.Thisexerciseisdescribedinsomedetailinthechapteronconsumerdurablesinthisbook.\nOverview3applicationinchapters5through10containsfourelements:pre-sentationoftheoptimizationproblemasadynamicprogrammingproblem,characterizationoftheoptimalpolicyfunctions,estimationoftheparameters,andpolicyevaluationusingthismodel.Whiletheapplicationsmightbecharacterizedasmacroeconomics,themethodologyisvaluableinotherareasofeconomicresearch,intermsofboththetopicsandthetechniques.Theseapplicationsuti-lizematerialfrommanyotherpartsofeconomics.Forexample,theanalysisofthestochasticgrowthmodelincludestaxationandourdiscussionoffactoradjustmentattheplantleveliscertainlyrelevanttoresearchersinlaborandindustrialorganization.Moreoverweenvisionthesetechniquestobeusefulinanyproblemwheretheresearcherisapplyingdynamicoptimizationtothedata.Thechap-terscontainreferencestootherapplicationsofthesetechniques.Whatisnewaboutourpresentationistheuseofanintegratedapproachtotheempiricalimplementationofdynamicoptimiza-tionmodels.Previoustextshaveprovidedamathematicalbasisfordynamicprogramming,butthosepresentationsgenerallydonotcontainanyquantitativeapplications.Othertextspresenttheunder-lyingeconometrictheorybutgenerallywithoutspecificapplications.Ourapproachdoesboth,andthusaimstolinktheoryandapplica-tionasillustratedinthechaptersofpartII.Ourmotivationforwritingthisbookshouldbeclear.Fromtheperspectiveofunderstandingdynamicprogramming,explicitempiricalapplicationscomplementtheunderlyingtheoryofopti-mization.Fromtheperspectiveofappliedmacroeconomics,explicitdynamicoptimizationproblems,posedasdynamicprogrammingproblems,provideneededstructureforestimationandpolicyevaluation.Sincethebookisintendedtoteachempiricalapplicationsofdynamicprogrammingproblems,wehavecreatedaWebsiteforthepresentationofcomputercodes(MATLABandGAUSS)aswellasdatasetsusefulfortheapplications.ThismaterialshouldappealtoreaderswishingtosupplementthepresentationinpartII,andwehopethatWebsitewillbecomeaforumforfurtherdevelopmentofcodes.OurwritingthisbookhasbenefitedfromjointworkwithJoaoEjarque,JohnHaltiwanger,AlokJohri,andJonathanWillis.Wethanktheseco-authorsfortheirgeneroussharingofideasandcom-putercodeaswellastheircommentsonthefinaldraft.Thanksalso\n4Chapter1gotoVictorAguirregabiria,YanBai,JoyceCooper,DeanCorbae,ZviEckstein,SimonGilchrist,HangKang,PeterKlenow,SamKor-tum,Vale´rieLechene,NicolaPavoni,AldoRustichini,andMarcosVeraforcommentsonvariouspartsofthebook.WealsoappreciatethecommentsofoutsidereviewersandtheeditorialstaffatTheMITPress.Finally,wearegratefultoourmanymastersanddoctoralstudentsatTelAvivUniversity,UniversityofTexasatAustin,theIDEIattheUniversite´deToulouse,theNAKEPhDprograminHolland,theUniversityofHaifa,theUniversityofMinnesota,andUniversityCollegeLondonfortheirnumerouscommentsandsug-gestionsduringthepreparationofthisbook.\nITheory\n2TheoryofDynamicProgramming2.1OverviewThemathematicaltheoryofdynamicprogrammingasameansofsolvingdynamicoptimizationproblemsdatestotheearlycontribu-tionsofBellman(1957)andBertsekas(1976).Foreconomists,thecontributionsofSargent(1987)andStokeyandLucas(1989)provideavaluablebridgetothisliterature.2.2IndirectUtilityIntuitively,theapproachofdynamicprogrammingcanbeunder-stoodbyrecallingthethemeofindirectutilityfrombasicstaticcon-sumertheoryorareducedformprofitfunctiongeneratedbytheoptimizationofafirm.Thesereducedformrepresentationsofpay-offssummarizeinformationabouttheoptimizedvalueofthechoiceproblemsfacedbyhouseholdsandfirms.Aswewillsee,thetheoryofdynamicprogrammingtakesthisinsighttoadynamiccontext.2.2.1ConsumersConsumerchoicetheoryfocusesonhouseholdsthatsolveVðI;pÞ¼maxuðcÞcsubjecttopc¼I;wherecisavectorofconsumptiongoods,pisavectorofpricesandIisincome.1Thefirst-orderconditionisgivenby1.AssumethatthereareJcommoditiesinthiseconomy.Thispresentationassumesthatyouunderstandtheconditionsunderwhichthisoptimizationproblemhasasolutionandwhenthatsolutioncanbecharacterizedbyfirst-orderconditions.\n8Chapter2ujðcÞ¼lforj¼1;2;...;J;pjwherelisthemultiplieronthebudgetconstraintandujðcÞisthemarginalutilityfromgoodj.HereVðI;pÞisanindirectutilityfunction.ItisthemaximizedlevelofutilityfromthecurrentstateðI;pÞ.Someoneinthisstatecanbepredictedtoattainthislevelofutility.Onedoesnotneedtoknowwhatthatpersonwilldowithhisincome;itisenoughtoknowthathewillactoptimally.Thisisverypowerfullogicandunderliestheideabehindthedynamicprogrammingmodelsstudiedbelow.Toillustrate,whathappensifwegivetheconsumerabitmoreincome?WelfaregoesupbyVIðI;pÞ>0.Cantheresearcherpredictwhatwillhappenwithalittlemoreincome?Notreallysincetheoptimizingconsumerisindifferentwithrespecttohowthisisspent:ujðcÞ¼VIðI;pÞforallj:pjItisinthissensethattheindirectutilityfunctionsummarizesthevalueofthehouseholdsoptimizationproblemandallowsustodeterminethemarginalvalueofincomewithoutknowingmoreaboutconsumptionfunctions.Isthisallweneedtoknowabouthouseholdbehavior?No,thistheoryisstatic.Itignoressavings,spendingondurablegoods,anduncertaintyoverthefuture.Theseareallimportantcomponentsinthehouseholdoptimizationproblem.Wewillreturntotheseinlaterchaptersonthedynamicbehaviorofhouseholds.Thepointherewassimplytorecallakeyobjectfromoptimizationtheory:theindirectutilityfunction.2.2.2FirmsSupposethatafirmmustchoosehowmanyworkerstohireatawageofwgivenitsstockofcapitalkandproductpricep.ThusthefirmmustsolvePðw;p;kÞ¼maxpfðl;kÞ
wl:lAlabordemandfunctionresultsthatdependsonðw;p;kÞ.AswithVðI;pÞ,Pðw;p;kÞsummarizesthevalueofthefirmgivenfactor\nTheoryofDynamicProgramming9prices,theproductpricep,andthestockofcapitalk.Boththeflexi-bleandfixedfactorscanbevectors.ThinkofPðw;p;kÞasanindirectprofitfunction.Itcompletelysummarizesthevalueoftheoptimizationproblemofthefirmgivenðw;p;kÞ.Aswiththehouseholdsproblem,givenPðw;p;kÞ,wecandirectlycomputethemarginalvalueofallowingthefirmsomeadditionalcapitalasPkðw;p;kÞ¼pfkðl;kÞwithoutknowinghowthefirmwilladjustitslaborinputinresponsetotheadditionalcapital.But,isthisallthereistoknowaboutthefirm’sbehavior?Surelynot,forwehavenotspecifiedwherekcomesfrom.Sothefirm’sproblemisessentiallydynamic,thoughthedemandforsomeofitsinputscanbetakenasastaticoptimizationproblem.Theseareimportantthemesinthetheoryoffactordemand,andwewillreturntotheminourfirmapplications.2.3DynamicOptimization:ACake-EatingExampleHerewewilllookataverysimpledynamicoptimizationproblem.Webeginwithafinitehorizonandthendiscussextensionstotheinfinitehorizon.2SupposethatyouarepresentedwithacakeofsizeW1.Ateachpointoftime,t¼1;2;3;...;T,youcaneatsomeofthecakebutmustsavetherest.Letctbeyourconsumptioninperiodt,andletuðctÞrepresenttheflowofutilityfromthisconsumption.Theutilityfunc-tionisnotindexedbytime:preferencesarestationary.Wecanassumethatuð
Þisrealvalued,differentiable,strictlyincreasing,andstrictlyconcave.Furtherwecanassumelimu0ðcÞ!y.Wecouldc!0representyourlifetimeutilitybyXTðt
1ÞbuðctÞ;t¼1where0aba1andbiscalledthediscountfactor.Fornow,weassumethatthecakedoesnotdepreciate(spoil)orgrow.HencetheevolutionofthecakeovertimeisgovernedbyWtþ1¼Wt
ctð2:1Þ2.Foraverycompletetreatmentofthefinitehorizonproblemwithuncertainty,seeBertsekas(1976).\n10Chapter2fort¼1;2;...;T.Howwouldyoufindtheoptimalpathofcon-sumption,fcgT?3t12.3.1DirectAttackOneapproachistosolvetheconstrainedoptimizationproblemdirectly.ThisiscalledthesequenceproblembyStokeyandLucas(1989).ConsidertheproblemofXTðt
1ÞmaxbuðctÞð2:2ÞTTþ1fctg1;fWtg2t¼1subjecttothetransitionequation(2.1),whichholdsfort¼1;2;3;...;T.Alsotherearenonnegativityconstraintsonconsumingthecakegivenbyctb0andWtb0.Forthisproblem,W1isgiven.Alternatively,theflowconstraintsimposedby(2.1)foreachtcouldbecombined,yieldingXTctþWTþ1¼W1:ð2:3Þt¼1Thenonnegativityconstraintsaresimpler:ctb0fort¼1;2;...;TandWTþ1b0.Fornow,wewillworkwiththesingleresourcecon-straint.Thisisawell-behavedproblemastheobjectiveisconcaveandcontinuousandtheconstraintsetiscompact.Sothereisasolu-tiontothisproblem.4Lettinglbethemultiplieron(2.3),thefirst-orderconditionsaregivenbybt
1u0ðcÞ¼lfort¼1;2;...;Ttandl¼f;wherefisthemultiplieronthenonnegativityconstraintonWTþ1.Thenonnegativityconstraintsonctb0areignored,aswecanassumethatthemarginalutilityofconsumptionbecomesinfiniteasconsumptionapproacheszerowithinanyperiod.T3.Throughout,thenotationfxtg1isusedtodefinethesequenceðx1;x2;...;xTÞforsomevariablex.4.ThiscomesfromtheWeierstrasstheorem.SeeBertsekas(1976,app.B)orStokeyandLucas(1989,ch.3)foradiscussion.\nTheoryofDynamicProgramming11Combiningtheequations,weobtainanexpressionthatlinkscon-sumptionacrossanytwoperiods:u0ðcÞ¼bu0ðcÞ:ð2:4Þttþ1Thisisanecessaryconditionofoptimalityforanyt:ifitisviolated,theagentcandobetterbyadjustingctandctþ1.Frequently(2.4)isreferredtoasaEulerequation.Tounderstandthiscondition,supposethatyouhaveaproposed(candidate)solutionforthisproblemgivenbyfcgT,fWgTþ1.t1t2EssentiallytheEulerequationsaysthatthemarginalutilitycostofreducingconsumptionbyeinperiodtequalsthemarginalutilitygainfromconsumingtheextraeofcakeinthenextperiod,whichisdiscountedbyb.IftheEulerequationholds,thenitisimpossibletoincreaseutilitybymovingconsumptionacrossadjacentperiodsgivenacandidatesolution.Itshouldbeclearthoughthatthisconditionmaynotbesufficient:itdoesnotcoverdeviationsthatlastmorethanoneperiod.Forexample,couldutilitybeincreasedbyreducingconsumptionbyeinperiodtsavingthe‘‘cake’’fortwoperiodsandthenincreasingconsumptioninperiodtþ2?Clearly,thisisnotcoveredbyasingleEulerequation.However,bycombiningtheEulerequationthatholdacrossperiodtandtþ1withthatwhichholdsforperiodstþ1andtþ2,wecanseethatsuchadeviationwillnotincreaseutility.ThisissimplybecausethecombinationofEulerequationsimpliesthatu0ðcÞ¼b2u0ðcÞttþ2sothatthetwo-perioddeviationfromthecandidatesolutionwillnotincreaseutility.Aslongastheproblemisfinite,thefactthattheEulerequationholdsacrossalladjacentperiodsimpliesthatanyfinitedeviationsfromacandidatesolutionthatsatisfiestheEulerequationswillnotincreaseutility.Isthisenough?Notquite.Imagineacandidatesolutionthatsat-isfiesalloftheEulerequationsbuthasthepropertythatWT>cTsothatthereiscakeleftover.Thisisclearlyaninefficientplan:satisfy-ingtheEulerequationsisnecessarybutnotsufficient.TheoptimalsolutionwillsatisfytheEulerequationforeachperioduntiltheagentconsumestheentirecake.Formally,thisinvolvesshowingthatthenonnegativityconstraintonWTþ1mustbind.Infact,thisconstraintisbindinginthesolution\n12Chapter2above:l¼f>0.Thisnonnegativityconstraintservestwoimportantpurposes.First,intheabsenceofaconstraintthatWTþ1b0,theagentwouldclearlywanttosetWTþ1¼
y.Thisisclearlynotfea-sible.Second,thefactthattheconstraintisbindingintheoptimalsolutionguaranteesthatcakedoesnotremainafterperiodT.Ineffecttheproblemispinneddownbyaninitialcondition(W1isgiven)andbyaterminalcondition(WTþ1¼0).Thesetof(T
1)Eulerequationsand(2.3)thendeterminethetimepathofconsumption.LetthesolutiontothisproblembedenotedbyVTðW1Þ,whereTisthehorizonoftheproblemandW1istheinitialsizeofthecake.VTðW1ÞrepresentsthemaximalutilityflowfromaT-periodproblemgivenasizeW1cake.Fromnowon,wecallthisavaluefunction.Thisiscompletelyanalogoustotheindirectutilityfunctionsex-pressedforthehouseholdandthefirm.Asinthoseproblems,aslightincreaseinthesizeofthecakeleadstoanincreaseinlifetimeutilityequaltothemarginalutilityinanyperiod.Thatis,V0ðWÞ¼l¼bt
1u0ðcÞ;t¼1;2;...;T:T1tItdoesn’tmatterwhentheextracakeiseatengiventhatthecon-sumerisactingoptimally.Thisisanalogoustothepointraisedaboveabouttheeffectonutilityofanincreaseinincomeinthecon-sumerchoiceproblemwithmultiplegoods.2.3.2DynamicProgrammingApproachSupposethatwechangetheproblemslightly:weaddaperiod0andgiveaninitialcakeofsizeW0.Oneapproachtodeterminingtheoptimalsolutionofthisaugmentedproblemistogobacktothesequenceproblemandresolveitusingthislongerhorizonandnewconstraint.But,havingdoneallofthehardworkwiththeTperiodproblem,itwouldbenicenottohavetodoitagain.FiniteHorizonProblemThedynamicprogrammingapproachprovidesameansofdoingso.Itessentiallyconvertsa(arbitrary)T-periodproblemintoatwo-periodproblemwiththeappropriaterewritingoftheobjectivefunc-tion.Thiswayitusesthevaluefunctionobtainedfromsolvingashorterhorizonproblem.\nTheoryofDynamicProgramming13Byaddingaperiod0toouroriginalproblem,wecantakeadvan-tageoftheinformationprovidedinVTðW1Þ,thesolutionoftheT-periodproblemgivenW1from(2.2).GivenW0,considertheproblemofmaxuðc0ÞþbVTðW1Þ;ð2:5Þc0whereW1¼W0
c0;W0given:Inthisformulationthechoiceofconsumptioninperiod0determinesthesizeofthecakethatwillbeavailablestartinginperiod1,W1.Now,insteadofchoosingasequenceofconsumptionlevels,wejustfindc0.Oncec0andthusW1aredetermined,thevalueoftheprob-lemfromthenonisgivenbyVTðW1Þ.Thisfunctioncompletelysummarizesoptimalbehaviorfromperiod1onward.Forthepur-posesofthedynamicprogrammingproblem,itdoesnotmatterhowthecakewillbeconsumedaftertheinitialperiod.Allthatisimpor-tantisthattheagentwillbeactingoptimallyandthusgeneratingutilitygivenbyVTðW1Þ.Thisistheprincipleofoptimality,duetoRichardBellman,atwork.Withthisknowledge,anoptimaldecisioncanbemaderegardingconsumptioninperiod0.Notethatthefirst-ordercondition(assumingthatVTðW1Þisdif-ferentiable)isgivenbyu0ðcÞ¼bV0ðWÞ0T1sothatthemarginalgainfromreducingconsumptionalittleinperiod0issummarizedbythederivativeofthevaluefunction.AsnotedintheearlierdiscussionoftheT-periodsequenceproblem,V0ðWÞ¼u0ðcÞ¼btu0ðcÞT11tþ1fort¼1;2;...;T
1.Usingthesetwoconditionstogetheryieldsu0ðcÞ¼bu0ðcÞttþ1fort¼0;1;2;...;T
1,afamiliarnecessaryconditionforanoptimalsolution.SincetheEulerconditionsfortheotherperiodsunderliethecre-ationofthevaluefunction,onemightsuspectthatthesolutiontotheTþ1problemusingthisdynamicprogrammingapproachisidenti-\n14Chapter2caltothatofthesequenceapproach.5Thisisclearlytrueforthisproblem:thesetoffirst-orderconditionsforthetwoproblemsareidentical,andthus,giventhestrictconcavityoftheuðcÞfunctions,thesolutionswillbeidenticalaswell.Theapparenteaseofthisapproach,however,maybemisleading.WewereabletomaketheproblemlooksimplebypretendingthatweactuallyknowVTðW1Þ.Ofcourse,thewaywecouldsolveforthisisbyeithertacklingthesequenceproblemdirectlyorbuildingitrecursively,startingfromaninitialsingle-periodproblem.Onthisrecursiveapproach,wecouldstartwiththesingle-periodproblemimplyingV1ðW1Þ.Wewouldthensolve(2.5)tobuildV2ðW1Þ.Giventhisfunction,wecouldmovetoasolutionoftheT¼3problemandproceediteratively,using(2.5)tobuildVTðW1ÞforanyT.ExampleWeillustratetheconstructionofthevaluefunctioninaspecificexample.AssumeuðcÞ¼lnðcÞ.SupposethatT¼1.ThenV1ðW1Þ¼lnðW1Þ.ForT¼2,thefirst-orderconditionfrom(2.2)is1b¼;c1c2andtheresourceconstraintisW1¼c1þc2:Workingwiththesetwoconditions,wehaveW1bW1c1¼andc2¼:1þb1þbFromthis,wecansolveforthevalueofthetwo-periodproblem:V2ðW1Þ¼lnðc1Þþblnðc2Þ¼A2þB2lnðW1Þ;ð2:6ÞwhereA2andB2areconstantsassociatedwiththetwo-periodprob-lem.Theseconstantsaregivenby1bA2¼lnþbln;B2¼1þb:1þb1þb5.Bythesequenceapproach,wemeansolvingtheproblemusingthedirectapproachoutlinedintheprevioussection.\nTheoryofDynamicProgramming15Importantly,(2.6)doesnotincludethemaxoperatoraswearesub-stitutingtheoptimaldecisionsintheconstructionofthevaluefunc-tion,V2ðW1Þ.Usingthisfunction,theT¼3problemcanthenbewrittenasV3ðW1Þ¼maxlnðW1
W2ÞþbV2ðW2Þ;W2wherethechoicevariableisthestateinthesubsequentperiod.Thefirst-orderconditionis10¼bV2ðW2Þ:c1Using(2.6)evaluatedatacakeofsizeW,wecansolveforV0ðWÞ222implying:1B2b¼b¼:c1W2c2Herec2theconsumptionlevelinthesecondperiodofthethree-periodproblemandthusisthesameasthelevelofconsumptioninthefirstperiodofthetwo-periodproblem.Furtherweknowfromthetwo-periodproblemthat1b¼:c2c3Thisplustheresourceconstraintallowsustoconstructthesolutionofthethree-periodproblem:2W1bW1bW1c1¼2;c2¼2;c3¼2:1þbþb1þbþb1þbþbSubstitutingintoV3ðW1ÞyieldsV3ðW1Þ¼A3þB3lnðW1Þ;where21b2bA3¼lnþblnþbln;2221þbþb1þbþb1þbþb2B3¼1þbþb:Thissolutioncanbeverifiedfromadirectattackonthethree-periodproblemusing(2.2)and(2.3).\n16Chapter22.4SomeExtensionsoftheCake-EatingProblemHerewegobeyondtheT-periodproblemtoillustratesomewaystousethedynamicprogrammingframework.Thisisintendedasanoverview,andthedetailsoftheassertions,andsoforth,willbepro-videdbelow.2.4.1InfiniteHorizonBasicStructureSupposethatforthecake-eatingproblem,weallowthehorizontogotoinfinity.Asbefore,onecanconsidersolvingtheinfinitehori-zonsequenceproblemgivenbyXytmaxbuðctÞyyfctg1;fWtg2t¼1alongwiththetransitionequationofWtþ1¼Wt
ctfort¼1;2;...:Inspecifyingthisasadynamicprogrammingproblem,wewriteVðWÞ¼maxuðcÞþbVðW
cÞforallW:cA½0;WHereuðcÞisagaintheutilityfromconsumingcunitsinthecurrentperiod.VðWÞisthevalueoftheinfinitehorizonproblemstartingwithacakeofsizeW.Sointhegivenperiod,theagentchoosescur-rentconsumptionandthusreducesthesizeofthecaketoW0¼W
c,asinthetransitionequation.Weusevariableswithprimestodenotefuturevalues.ThevalueofstartingthenextperiodwithacakeofthatsizeisthengivenbyVðW
cÞ,whichisdiscountedatrateb<1.Forthisproblem,thestatevariableisthesizeofthecake(W)givenatthestartofanyperiod.Thestatecompletelysummarizesallinformationfromthepastthatisneededfortheforward-lookingoptimizationproblem.Thecontrolvariableisthevariablethatisbeingchosen.Inthiscaseitisthelevelofconsumptioninthecurrentperiodc.Notethatcliesinacompactset.Thedependenceofthestatetomorrowonthestatetodayandthecontroltoday,givenbyW0¼W
c;iscalledthetransitionequation.\nTheoryofDynamicProgramming17Alternatively,wecanspecifytheproblemsothatinsteadofchoosingtoday’sconsumptionwechoosetomorrow’sstate:VðWÞ¼maxuðW
W0ÞþbVðW0ÞforallW:ð2:7ÞW0A½0;WEitherspecificationyieldsthesameresult.Butchoosingtomorrow’sstateoftenmakesthealgebraabiteasier,sowewillworkwith(2.7).Thisexpressionisknownasafunctionalequation,anditisoftencalledaBellmanequationafterRichardBellman,oneoftheorigi-natorsofdynamicprogramming.NotethattheunknownintheBellmanequationisthevaluefunctionitself:theideaistofindafunctionVðWÞthatsatisfiesthisconditionforallW.Unlikethefinitehorizonproblem,thereisnoterminalperiodtousetoderivethevaluefunction.Ineffect,thefixedpointrestrictionofhavingVðWÞonbothsidesof(2.7)willprovideuswithameansofsolvingthefunctionalequation.NotetoothattimeitselfdoesnotenterintoBellman’sequation:wecanexpressallrelationswithoutanindicationoftime.Thisistheessenceofstationarity.6Infactwewillultimatelyusethestation-arityoftheproblemtomakeargumentsabouttheexistenceofavaluefunctionsatisfyingthefunctionalequation.Afinalveryimportantpropertyofthisproblemisthatallinfor-mationaboutthepastthatbearsoncurrentandfuturedecisionsissummarizedbyW,thesizeofthecakeatthestartoftheperiod.Whetherthecakeisofthissizebecauseweinitiallyhavealargecakeandcaneatalotofitorasmallcakeandarefrugaleatersisnotrel-evant.Allthatmattersisthatwehaveacakeofagivensize.Thispropertypartlyreflectsthefactthatthepreferencesoftheagentdonotdependonpastconsumption.Ifthiswerethecase,wecouldamendtheproblemtoallowthispossibility.Thenextpartofthischapteraddressesthequestionofwhetherthereexistsavaluefunctionthatsatisfies(2.7).Fornowweassumethatasolutionexistssothatwecanexploreitsproperties.Thefirst-orderconditionfortheoptimizationproblemin(2.7)canbewrittenasu0ðcÞ¼bV0ðW0Þ:6.Asyoumayalreadyknow,stationarityisvitalineconometricsaswell.Thusmakingassumptionsofstationarityineconomictheoryhaveanaturalcounterpartinempiricalstudies.Insomecaseswewillhavetomodifyoptimizationproblemstoensurestationarity.\n18Chapter2Thismaylooksimple,butwhatisthederivativeofthevaluefunc-tion?Itisparticularlyhardtoanswerthis,sincewedonotknowVðWÞ.However,wecanusethefactthatVðWÞsatisfies(2.7)forallWtocalculateV0.Assumingthatthisvaluefunctionisdifferentiable,wehaveV0ðWÞ¼u0ðcÞ;aresultwehaveseenbefore.SincethisholdsforallW,itwillholdinthefollowingperiod,yieldingV0ðW0Þ¼u0ðc0Þ:SubstitutionleadstothefamilarEulerequation:u0ðcÞ¼bu0ðc0Þ:Thesolutiontothecake-eatingproblemwillsatisfythisnecessaryconditionforallW.Thelinkfromthelevelofconsumptionandnextperiod’scake(thecontrolsfromthedifferentformulations)tothesizeofthecake(thestate)isgivenbythepolicyfunction:c¼fðWÞ;W0¼jðWÞ1W
fðWÞ:SubstitutingthesevaluesintotheEulerequationreducestheprob-lemtothesepolicyfunctionsalone:u0ðfðWÞÞ¼bu0ðfðW
fðWÞÞÞforallW:Thepolicyfunctionsaboveareimportantinappliedresearch,fortheyprovidethemappingfromthestatetoactions.Whenele-mentsofthestateaswellastheactionareobservable,thesepolicyfunctionswillprovidethemeansforestimatingtheunderlyingparameters.AnExampleIngeneral,itisnotactuallypossibletofindclosedformsolutionsforthevaluefunctionandtheresultingpolicyfunctions.Sowetrytocharacterizecertainpropertiesofthesolution,andforsomecases,wesolvetheseproblemsnumerically.Nevertheless,asindicatedbyouranalysisoffinitehorizonprob-lems,therearesomespecificationsoftheutilityfunctionthatallowustofindaclosedformsolutiontothevaluefunction.Suppose,as\nTheoryofDynamicProgramming19above,thatuðcÞ¼lnðcÞ.FromtheresultsoftheT-periodproblem,wemightconjecturethatthesolutiontothefunctionalequationtakestheformofVðWÞ¼AþBlnðWÞforallW:Bythisexpressionwehavereducedthedimensionalityoftheun-knownfunctionVðWÞtotwoparameters,AandB.ButcanwefindvaluesforAandBsuchthatVðWÞwillsatisfythefunctionalequation?Letussupposethatwecan.ForthesetwovaluesthefunctionalequationbecomesAþBlnðWÞ¼maxlnðW
W0ÞþbðAþBlnðW0ÞÞforallW:W0ð2:8ÞAftersomealgebra,thefirst-orderconditionbecomes0bBW¼jðWÞ¼W:1þbBUsingthisin(2.8)resultsinWbBWAþBlnðWÞ¼lnþbAþBlnforallW:1þbB1þbBCollectingthetermsthatmultiplylnðWÞandusingtherequirementthatthefunctionalequationholdsforallW,wefindthat1B¼1
bisrequiredforasolution.Afterthis,theexpressioncanalsobeusedtosolveforA.Thuswehaveverifiedthatourguessisasolutiontothefunctionalequation.WeknowthatbecausewecansolveforðA;BÞsuchthatthefunctionalequationholdsforallWusingtheoptimalconsumptionandsavingsdecisionrules.Withthissolution,weknowthatc¼Wð1
bÞ;W0¼bW:Thistellsusthattheoptimalpolicyistosaveaconstantfractionofthecakeandeattheremainingfraction.ThesolutiontoBcanbeestimatedfromthesolutiontotheT-periodhorizonproblemswhere\n20Chapter2XTt
1BT¼b:t¼1Clearly,B¼limT!yBT.Wewillbeexploitingtheideaofusingthevaluefunctiontosolvetheinfinitehorizonproblemasitisrelatedtothelimitofthefinitesolutionsinmuchofournumericalanalysis.Belowaresomeexercisesthatprovidesomefurtherelementstothisbasicstructure.Bothbeginwithfinitehorizonformulationsandthenprogresstotheinfinitehorizonproblems.exercise2.1Utilityinperiodtisgivenbyuðct;ct
1Þ.SolveaT-periodproblemusingthesepreferences.Interpretthefirst-orderconditions.HowwouldyouformulatetheBellmanequationfortheinfinitehorizonversionofthisproblem?exercise2.2ThetransitionequationismodifiedsothatWtþ1¼rWt
ct;wherer>0representsareturnfromholdingcakeinventories.SolvetheT-periodproblemwiththisstoragetechnology.Interpretthefirst-orderconditions.HowwouldyouformulatetheBellmanequationfortheinfinitehorizonversionofthisproblem?Doesthesizeofrmatterinthisdiscussion?Explain.2.4.2TasteShocksAconvenientfeatureofthedynamicprogrammingproblemistheeasewithwhichuncertaintycanbeintroduced.7Forthecake-eatingproblem,thenaturalsourceofuncertaintyhastodowiththeagent’sappetite.Inothersettingswewillfocusonothersourcesofuncer-taintyhavingtodowiththeproductivityoflaborortheendow-mentsofhouseholds.Toallowforvariationsofappetite,supposethatutilityovercon-sumptionisgivenbyeuðcÞ;whereeisarandomvariablewhosepropertieswewilldescribebelow.ThefunctionuðcÞisagainassumedtobestrictlyincreasing7.Tobecareful,hereweareaddingshocksthattakevaluesinafiniteandthuscountableset.SeethediscussioninBertsekas(1976,sec.2.1)foranintroductiontothecomplexitiesoftheproblemwithmoregeneralstatementsofuncertainty.\nTheoryofDynamicProgramming21andstrictlyconcave.Otherwise,theproblemistheoriginalcake-eatingproblemwithaninitialcakeofsizeW.Inproblemswithstochasticelements,itiscriticaltobepreciseaboutthetimingofevents.Doestheoptimizingagentknowthecurrentshockswhenmakingadecision?Forthisanalysis,assumethattheagentknowsthevalueofthetasteshockwhenmakingcur-rentdecisionsbutdoesnotknowthefuturevaluesofthisshock.Thustheagentmustuseexpectationsoffuturevaluesofewhendecidinghowmuchcaketoeattoday:itmaybeoptimaltoconsumelesstoday(savemore)inanticipationofahighrealizationofeinthefuture.Forsimplicity,assumethatthetasteshocktakesononlytwovalues:eAfeh;elgwitheh>el>0.Furtherwecanassumethatthetasteshockfollowsafirst-orderMarkovprocess,8whichmeansthattheprobabilitythataparticularrealizationofeoccursinthecur-rentperioddependsonlythevalueofeattainedinthepreviousperiod.9Fornotation,letpijdenotetheprobabilitythatthevalueofegoesfromstateiinthecurrentperiodtostatejinthenextperiod.Forexample,plhisdefinedfromp1Probðe0¼eje¼eÞ;lhhlwheree0referstothefuturevalueofe.Clearly,pþp¼1forihili¼h;l.LetPbea22matrixwithatypicalelementpijthatsummarizestheinformationabouttheprobabilityofmovingacrossstates.Thismatrixislogicallycalledatransitionmatrix.Withthisnotationandstructure,wecanturnagaintothecake-eatingproblem.Weneedtocarefullydefinethestateofthesystemfortheoptimizingagent.Inthenonstochasticproblem,thestatewassimplythesizeofthecake.Thisprovidedalltheinformationtheagentneededtomakeachoice.Whentasteshocksareintroduced,theagentneedstotakethisfactorintoaccountaswell.Weknowthatthetasteshocksprovideinformationaboutcurrentpayoffsand,throughthePmatrix,areinformativeaboutthefuturevalueofthetasteshockaswell.108.FormoredetailsonMarkovchains,wereferthereadertoLjungqvistandSargent(2000).9.Theevolutioncanalsodependonthecontrolofthepreviousperiod.Notetoothatbyappropriaterewritingofthestatespace,richerspecificationsofuncertaintycanbeencompassed.10.Thisisapointthatwereturntobelowinourdiscussionofthecapitalaccumula-tionproblem.\n22Chapter2FormallytheBellmanequationiswritten000VðW;eÞ¼maxeuðW
WÞþbEe0jeVðW;eÞforallðW;eÞ;W0whereW0¼W
casbefore.Notethattheconditionalexpectationis00denotedherebyEe0jeVðW;eÞwhich,givenP,issomethingwecancompute.11Thefirst-orderconditionforthisproblemisgivenby0000euðW
WÞ¼bEe0jeV1ðW;eÞforallðW;eÞ:Usingthefunctionalequationtosolveforthemarginalvalueofcake,wefindthat0000000euðW
WÞ¼bEe0je½euðW
WÞ:ð2:9ÞThis,ofcourse,isthestochasticEulerequationforthisproblem.TheoptimalpolicyfunctionisgivenbyW0¼jðW;eÞ:TheEulerequationcanberewritteninthesetermsas0000euðW
jðW;eÞÞ¼bEe0je½euðjðW;eÞ
jðjðW;eÞ;eÞÞÞ:Thepropertiesofthepolicyfunctioncanthenbededucedfromthiscondition.Clearly,bothe0andc0dependontherealizedvalueofesothattheexpectationontherightsideof(2.9)cannotbesplitintotwoseparatepieces.2.4.3DiscreteChoiceToillustratetheflexibilityofthedynamicprogrammingapproach,webuildonthisstochasticproblem.Supposethatthecakemustbeeateninoneperiod.Perhapsweshouldthinkofthisasthewine-drinkingproblem,recognizingthatonceagoodbottleofwineisopened,itmustbeconsumed.Furtherwecanmodifythetransitionequationtoallowthecaketogrow(depreciate)atrater.Thecakeconsumptionexamplebecomesthenadynamic,sto-chasticdiscretechoiceproblem.Thisispartofafamilyofproblemscalledoptimalstoppingproblems.12Thecommonelementinallof011.ThroughoutwedenotetheconditionalexpectationofegiveneasEe0je.12.EcksteinandWolpin(1989)provideanextensivediscussionsoftheformulationandestimationoftheseproblemsinthecontextoflaborapplications.\nTheoryofDynamicProgramming23theseproblemsistheemphasisonthetimingofasingleevent:whentoeatthecake,whentotakeajob,whentostopschool,whentostoprevisingachapter,andsoon.Infact,formanyoftheseproblems,thesechoicesarenotonceinalifetimeevents,sowewillbelookingatproblemsevenricherthanthoseoftheoptimalstoppingvariety.LetVEðW;eÞandVNðW;eÞbethevaluesofeatingsizeWcakenow(E)andwaiting(N),respectively,giventhecurrenttasteshockeAfeh;elg.ThenVEðW;eÞ¼euðWÞandN0VðWÞ¼bEe0jeVðrW;eÞ;whereVðW;eÞ¼maxðVEðW;eÞ;VNðW;eÞÞforallðW;eÞ:Tounderstandthisbetter,thetermeuðWÞisthedirectutilityflowfromeatingthecake.Oncethecakeiseaten,theproblemhasended.SoVEðW;eÞisjustaone-periodreturn.Iftheagentwaits,thenthereisnocakeconsumptioninthecurrentperiod,andinthenextperiodthecakeisofsizeðrWÞ.Astastesarestochastic,theagentchoosingtowaitmusttakeexpectationsofthefuturetasteshock,e0.Theagenthasanoptioninthenextperiodofeatingthecakeorwaitingsomemore.HencethevalueofhavingthecakeinanystateisgivenbyVðW;eÞ,whichisthevalueattainedbymaximizingoverthetwooptionsofeatingorwaiting.Thecostofdelayingthechoiceisdeterminedbythediscountfactorbwhilethegainstodelayareassociatedwiththegrowthofthecake,parameterizedbyr.Furthertherealizedvalueofewillsurelyinfluencetherelativevalueofcon-sumingthecakeimmediately.Ifra1,thenthecakedoesn’tgrow.Inthiscasethereisnogainfromdelaywhene¼eh.Iftheagentdelays,thenutilityinthenextperiodwillhavetobelowerduetodiscounting,andwithprobabil-ityphl,thetasteshockwillswitchfromlowtohigh.Sowaitingtoeatthecakeinthefuturewillnotbedesirable.HenceVðW;eÞ¼VEðW;eÞ¼euðWÞforallW:hhhInthelowestate,mattersaremorecomplex.Ifbandraresuffi-cientlycloseto1,thenthereisnotalargecosttodelay.Further,ifplhis\n24Chapter2sufficientlycloseto1,thenitislikelythattasteswillswitchfromlowtohigh.ThusitwillbeoptimalnottoeatthecakeinstateðW;elÞ.13Herearesomeadditionalexercises.exercise2.3Supposethatr¼1.Foragivenb,showthatthereexistsacriticallevelofplh,denotedbyplhsuchthatifplh>plh,thentheoptimalsolutionisfortheagenttowaitwhene¼elandtoeatthecakewhenehisrealized.exercise2.4Whenr>1,theproblemismoredifficult.Supposethattherearenovariationsintastes:eh¼el¼1.Inthiscasethereisatrade-offbetweenthevalueofwaiting(asthecakegrows)andthecostofdelayfromdiscounting.Supposethatr>1anduðcÞ¼c1
g=ð1
gÞ.Whatisthesolutiontotheoptimalstoppingproblemwhenbr1
g<1?Whathappensifbr1
g>1?Whathappenswhenuncertaintyisadded?2.5GeneralFormulationBuildingontheintuitiongainedfromthecake-eatingproblem,wenowconsideramoreformalabstracttreatmentofthedynamicpro-grammingapproach.14Webeginwithapresentationofthenon-stochasticproblemandthenadduncertaintytotheformulation.2.5.1NonstochasticCaseConsidertheinfinitehorizonoptimizationproblemofanagentwithapayofffunctionforperiodtgivenbyss~ðst;ctÞ.ThefirstargumentofthepayofffunctionistermedthestatevectorðstÞ.Asnotedabove,thisrepresentsasetofvariablesthatinfluencestheagent’sreturnwithintheperiod,butbyassumption,thesevariablesareoutsideoftheagent’scontrolwithinperiodt.ThestatevariablesevolveovertimeinamannerthatmaybeinfluencedbythecontrolvectorðctÞ,thesecondargumentofthepayofffunction.Theconnectionbetweenthestatevariablesovertimeisgivenbythetransitionequation:13.Inthefollowingchapteronthenumericalapproachtodynamicprogramming,westudythiscaseinconsiderabledetail.14.Thissectionisintendedtobeself-containedandthusrepeatssomeofthematerialfromtheearlierexamples.OurpresentationisbydesignnotasformalassaythatprovidedinBertsekas(1976)orStokeyandLucas(1989).Thereaderinterestedinmoremathematicalrigorisurgedtoreviewthosetextsandtheirmanyreferences.\nTheoryofDynamicProgramming25stþ1¼tðst;ctÞ:So,giventhecurrentstateandthecurrentcontrol,thestatevectorforthesubsequentperiodisdetermined.Notethatthestatevectorhasaveryimportantproperty:itcompletelysummarizesalloftheinformationfromthepastthatisneededtomakeaforward-lookingdecision.Whilepreferencesandthetransitionequationarecertainlydependentonthepast,thisdependenceisrepresentedbyst:othervariablesfromthepastdonotaffectcurrentpayoffsorconstraintsandthuscannotinfluencecurrentdecisions.Thismayseemrestrictivebutitisnot:thevectorstmayincludemanyvariablessothatthedependenceofcurrentchoicesonthepastcanbequiterich.Whilethestatevectoriseffectivelydeterminedbypreferencesandthetransitionequation,theresearcherhassomelatitudeinchoosingthecontrolvector.Thatis,theremaybemultiplewaysofrepresent-ingthesameproblemwithalternativespecificationsofthecontrolvariables.WeassumethatcACandsAS.InsomecasesthecontrolisrestrictedtobeinasubsetofCthatdependsonthestatevector:cACðsÞ.Furtherweassumethatss~ðs;cÞisboundedforðs;cÞASC.15Forthecake-eatingproblemdescribedabove,thestateofthesys-temwasthesizeofthecurrentcakeðWtÞandthecontrolvariablewasthelevelofconsumptioninperiodt,ðctÞ.ThetransitionequationdescribingtheevolutionofthecakewasgivenbyWtþ1¼Wt
ct:Clearly,theevolutionofthecakeisgovernedbytheamountofcurrentconsumption.Anequivalentrepresentation,asexpressedin(2.7),istoconsiderthefuturesizeofthecakeasthecontrolvariableandthentosimplywritecurrentconsumptionasWtþ1
Wt.Therearetwofinalpropertiesoftheagent’sdynamicoptimizationproblemworthspecifying:stationarityanddiscounting.Notethatneitherthepayoffnorthetransitionequationsdependexplicitlyontime.Truetheproblemisdynamic,buttimeperseisnotoftheessence.Inagivenstatetheoptimalchoiceoftheagentwillbethesameregardlessof‘‘when’’heoptimizes.Stationarityisimportant15.Ensuringthattheproblemisboundedisanissueinsomeeconomicapplications,suchasthegrowthmodel.OftentheseproblemsaredealtwithbyboundingthesetsCandS.\n26Chapter2bothfortheanalysisoftheoptimizationproblemandforempiricalimplementationofinfinitehorizonproblems.Infact,becauseofsta-tionarity,wecandispensewithtimesubscriptsastheproblemiscompletelysummarizedbythecurrentvaluesofthestatevariables.Theagent’spreferencesarealsodependentontherateatwhichthefutureisdiscounted.Letbdenotethediscountfactorandassumethat0n_s*LoopoverallsizesofthetotalamountofcakeX*c_L=X_L*Minvalueforconsumption*c_H=X[i_s]*Maxvalueforconsumption*i_c=1dountili_c>n_c*Loopoverallconsumptionlevels*c=c_L+(c_H-c_L)/n_c*(i_c-1)i_y=1EnextV=0*Initializethenextvaluetozero*dountili_y>n_y*Loopoverallpossiblerealizationsofthefutureendowment*nextX=R*(X[i_s]-c)+Y[i_y]*Nextperiodamountofcake*nextV=V(nextX)*Hereweuseinterpolationtofindthenextvaluefunction*EnextV=EnextV+nextV*Pi[i_y]*Storetheexpectedfuturevalueusingthetransitionmatrix*i_y=i_y+1endo*Endofloopoverendowment*aux[i_c]=u(c)+beta*EnextV*Storesthevalueofagivenconsumptionlevel*i_c=i_c+1endo*Endofloopoverconsumption*newV[i_s,i_y]=max(aux)*Takethemaxoverallconsumptionlevels*i_s=i_s+1endo*Endofloopoversizeofcake*V=newV*Updatethenewvaluefunction*Figure3.1Stochasticcake-eatingproblem\nNumericalAnalysis39Figure3.2Valuefunction,stochasticcake-eatingproblemOncethevaluefunctioniterationpieceoftheprogramiscom-pleted,thevaluefunctioncanbeusedtofindthepolicyfunction,c¼cðXÞ.ThisisdonebycollectingalltheoptimalconsumptionvaluescicforeveryvalueofXis.Hereagain,weonlyknowthefunctioncðXÞatthepointsofthegrid.Wecanuseinterpolatingmethodstoevaluatethepolicyfunctionatotherpoints.Thevaluefunctionandthepolicyfunctionaredisplayedinfigures3.2and3.3forparticularvaluesoftheparameters.Asdiscussedabove,approximatingthevaluefunctionandthepolicyrulesbyafinitestatespacerequiresalargenumberofpointsonthisspace(nshastobebig).Thesenumericalcalculationsareoftenextremelytime-consuming.Sowecanreducethenumberofpointsonthegrid,whilekeepingasatisfactoryaccuracy,byusinginterpolationsonthisgrid.WhenwehaveevaluatedthefunctioniivðRðXs
ccÞþyÞ,i¼L;H,weusethenearestvalueonthegridtojiiiapproximateRðXs
ccÞþy.Withasmallnumberofpointsontheigrid,thiscanbeaverycrudeapproximation.Theaccuracyofthecomputationcanbeincreasedbyinterpolatingthefunctionvjð:Þ(seetheappendixformoredetails).TheinterpolationisbasedonthevaluesinV.\n40Chapter3Figure3.3Policyfunction,stochasticcake-eatingproblem3.2.2PolicyFunctionIterationsThevaluefunctioniterationmethodcanberatherslow,asitcon-vergesatarateb.ResearchershavedevisedothermethodsthatcanbefastertocomputethesolutiontotheBellmanequationinaninfi-nitehorizon.Thepolicyfunctioniteration,alsoknownasHoward’simprovementalgorithm,isoneofthese.WereferthereadertoJudd(1998)orLjungqvistandSargent(2000)formoredetails.Thismethodstartswithaguessofthepolicyfunction,inourcasec0ðXÞ.Thispolicyfunctionisthenusedtoevaluatethevalueofusingthisruleforever:XV0ðXÞ¼uðc0ðXÞÞþbpiV0ðRðX
c0ðXÞÞþyiÞforallX:i¼L;HThis‘‘policyevaluationstep’’requiressolvingasystemoflinearequations,giventhatwehaveapproximatedRðX
c0ðXÞÞþybyaniXonourgrid.Nextwedoa‘‘policyimprovementstep’’tocomputec1ðXÞ:\nNumericalAnalysis41"#Xc1ðXÞ¼argmaxuðcÞþbpiV0ðRðX
cÞþyiÞforallX:ci¼L;HGiventhisnewrule,theiterationsarecontinuedtofindV1ðÞ;c2ðÞ;...;cjþ1ðÞuntiljcjþ1ðXÞ
cjðXÞjissmallenough.Theconver-gencerateismuchfasterthanthevaluefunctioniterationmethod.However,solvingthe‘‘policyevaluationstep’’cansometimesbequitetime-consuming,especiallywhenthestatespaceislarge.Onceagain,thecomputationtimecanbemuchreducediftheinitialguessc0ðXÞisclosetothetruepolicyrulecðXÞ.3.2.3ProjectionMethodsThesemethodscomputedirectlythepolicyfunctionwithoutcalcu-latingthevaluefunctions.Theyusethefirst-orderconditions(Eulerequation)tobackoutthepolicyrules.Thecontinuouscakeproblemsatisfiesthefirst-orderEulerequationu0ðcÞ¼bREu0ðcÞtttþ1ifthedesiredconsumptionlevelislessthanthetotalresourcesX¼Wþy.Ifthereisacornersolution,thentheoptimalconsump-tionleveliscðXÞ¼X.Takingintoaccountthecornersolution,wecanrewritetheEulerequationasu0ðcÞ¼max½u0ðXÞ;bREu0ðcÞ:ttttþ1Weknowthatbytheiidassumption,theproblemhasonlyonestatevariableX,sotheconsumptionfunctioncanbewrittenc¼cðXÞ.Asweconsiderthestationarysolution,wedropthesub-scripttinthenextequation.TheEulerequationcanthenberefor-mulatedas0000uðcðXÞÞ
max½uðXÞ;bREy0uðcðRðX
cðXÞÞþyÞÞ¼0ð3:4ÞorFðcðXÞÞ¼0:ð3:5ÞThegoalistofindanapproximationcc^ðXÞofcðXÞ,forwhich(3.5)isapproximatelysatisfied.TheproblemisthusreducedtofindthezeroofF,whereFisanoperatoroverfunctionspaces.Thiscanbedonewithaminimizingalgorithm.Therearetwoissuestoresolve.First,\n42Chapter3weneedtofindagoodapproximationofcðXÞ.Second,wehavetodefineametrictoevaluatethefitoftheapproximation.SolvingforthePolicyRuleLetfpiðXÞgbeabaseofthespaceofcontinuousfunctions,andletC¼fcigbeasetofparameters.WecanapproximatecðXÞbyXncc^ðX;CÞ¼cipiðXÞ:i¼1Thereisaninfinitenumberofbasestochosefrom.AsimpleoneistoconsiderpolynomialsinXsothatcc^ðX;CÞ¼cþcXþcX2þ


:012Althoughthischoiceisintuitive,itisnotusuallythebestchoice.Inthefunctionspacethisbaseisnotanorthogonalbase,whichmeansthatsomeelementstendtobecollinear.Orthogonalbaseswillyieldmoreefficientandpreciseresults.3Thechosenbaseshouldbecomputationallysimple.Itselementsshould‘‘looklike’’thefunctiontoapproximate,sothatthefunctioncðXÞcanbeapproximatedwithasmallnumberofbasefunctions.Anyknowledgeoftheshapeofthepolicyfunctionwillbetoagreathelp.If,forinstance,thispolicyfunctionhasakink,amethodbasedonlyonaseriesofpolynomialswillhaveahardtimefittingit.Itwouldrequirealargenumberofpowersofthestatevariabletocomesomewhereclosetothesolution.Havingchosenamethodtoapproximatethepolicyrule,wenowhavetobemorepreciseaboutwhat‘‘bringingFðcc^ðX;CÞÞclosetozero’’means.Tobemorespecific,weneedtodefinesomeoperatorsonthespaceofcontinuousfunctions.ForanyweightingfunctiongðxÞ,theinnerproductoftwointegrablefunctionsf1andf2onaspaceAisdefinedasðhf1;f2i¼f1ðxÞf2ðxÞgðxÞdx:ð3:6ÞATwofunctionsf1andf2aresaidtobeorthogonal,conditionalonaweightingfunctiongðxÞ,ifhf1;f2i¼0.Theweightingfunctionindi-cateswheretheresearcherwantstheapproximationtobegood.Weareusingtheoperatorh:;:iandtheweightingfunctiontoconstructametrictoevaluatehowcloseFðcc^ðX;CÞÞistozero.Thiswillbedone3.PopularorthogonalbasesareChebyshev,Legendre,orHermitepolynomials.\nNumericalAnalysis43bysolvingforCsuchthathFðcc^ðX;CÞÞ;fðXÞi¼0;wherefðXÞissomeknownfunction.WenextreviewthreemethodsthatdifferintheirchoiceforthisfunctionfðXÞ.First,asimplechoiceforfðXÞisFðcc^ðX;CÞÞitself.ThisdefinestheleastsquaremetricasminhFðcc^ðX;CÞÞ;Fðcc^ðX;CÞÞi:CBythecollocationmethod,detailedlaterinthissection,wecanchoosetofindCasminhFðcc^ðX;CÞÞ;dðX
XiÞi;i¼1;...;n;CwheredðX
XiÞisthemasspointfunctionatpointXi,meaningthatdðXÞ¼1ifX¼XianddðXÞ¼0elsewhere.AnotherpossibilityistodefineminhFðcc^ðX;CÞÞ;piðXÞi;i¼1;...;n;CwherepiðXÞisabaseofthefunctionspace.ThisiscalledtheGaler-kinmethod.Anapplicationofthismethodcanbeseenbelow,wherethebaseistakentobe‘‘tent’’functions.Figure3.4displaysasegmentofthecomputercodethatcalcu-latestheresidualfunctionFðcc^ðX;CÞÞwhentheconsumptionruleisapproximatedbyasecond-orderpolynomial.Thiscanthenbeusedinoneoftheproposedmethods.CollocationMethodsJudd(1992)presentsinsomedetailthismethodappliedtothegrowthmodel.ThefunctioncðXÞisapproximatedusingChebyshevpolynomials.Thesepolynomialsaredefinedontheinterval½0;1andtaketheformpiðXÞ¼cosðiarccosðXÞÞ;XA½0;1;i¼0;1;2;...:Fori¼0,thispolynomialisaconstant.Fori¼1,thepolynomialisequaltoX.Asthesepolynomialsareonlydefinedonthe½0;1interval,onecanusuallyscalethestatevariablesappropriately.4The4.ThepolynomialsarealsodefinedrecursivelybypiðXÞ¼2Xpi
1ðXÞ
pi
2ðXÞ,ib2,withp0ð0Þ¼1andpðX;1Þ¼X.\n44Chapter3procedurec(x)*Herewedefineancc=psi_0+psi_1*x+psi_2*x*xapproximationforthereturn(cc)consumptionfunctionbasedendprocedureonasecond-orderpolynomial*i_s=1dountili_s>n_s*Loopoverallsizesofthetotalamountofcake*utoday=U0(c(X[i_s]))*Marginalutilityofconsuming*ucorner=U0(X[i_s])*Marginalutilityifcornersolution*EnextU=0*Initializeexpectedfuturei_y=1marginalutility*dountili_y>n_y*Loopoverallpossiblerealizationsofthefutureendowment*nextX=R(X[i_s]-*Nextamountofcake*c(X[i_s]))+Y[i_y]nextU=U0(c(nextX))*Nextmarginalutilityofconsumption*EnextU=EnextU+nextU*Pi[i_y]*HerewecomputetheexpectedfuturemarginalutilityofconsumptionusingthetransitionmatrixPi*i_y=i_y+1endo*Endofloopoverendowment*F[i_s]=utoday-max(ucorner,beta*EnextU)i_s=i_s+1endo*Endofloopoversizeofcake*Figure3.4Stochasticcake-eatingproblem,projectionmethodpolicyfunctioncanthenbeexpressedasXncc^ðX;CÞ¼cipiðXÞ:i¼1NextthemethodfindsC,whichminimizeshFðcc^ðX;CÞÞ;dðX
XiÞi;i¼1;...;n;wheredðÞisthemasspointfunction.HencethemethodrequiresthatFðcc^ðX;CÞÞiszeroatsomeparticularpointsXiandnotoverthewholerange½XL;XH.ThemethodismoreefficientifthesepointsarechosentobethezerosofthebaseelementspiðXÞ,hereXi¼cosðp=2iÞ.Thismethodisreferredtoasanorthogonalcollocationmethod.Cis\nNumericalAnalysis45Figure3.5Basisfunctions,finiteelementmethodthesolutiontoasystemofnonlinearequations:Fðcc^ðXi;CÞÞ¼0;i¼1;...;n:Thismethodisgoodatapproximatingpolicyfunctionsthatarerel-ativelysmooth.AdrawbackisthattheChebyshevpolynomialstendtodisplayoscillationsathigherorders.TheresultingpolicyfunctioncðXÞwillalsotendtofluctuate.Thereisnoparticularruleforchoos-ingn,thehighestorderoftheChebyshevpolynomial.Obviouslythehighernis,thebetterwillbetheapproximation,butthiscomesatthecostofincreasedcomputation.FiniteElementMethodsMcGrattan(1996)illustratesthefiniteelementmethodwiththesto-chasticgrowthmodel(seealsoReddy1993foranin-depthdiscus-sionoffiniteelements).Tostart,thestatevariableXisdiscretizedoveragridfXisgns.is¼1Thefiniteelementmethodisbasedonthefollowingfunctions:8>>X
Xis
1>>ifXA½Xis
1;Xis;>>Xis
Xis
1>ifXA½Xis;Xisþ1;>>Xisþ1
Xis>>:0elsewhere:\n46Chapter3ThefunctionpisðXÞisasimplefunctionin½0;1,asillustratedinfigure3.5.Itisinfactasimplelinearinterpolation(andanordertwospline;seetheappendixformoreonthesetechniques).Ontheinterval½Xis;Xisþ1,thefunctioncc^ðXÞisequaltotheweightedsumofpisðXÞandpisþ1ðXÞ.HeretheresidualfunctionsatisfieshFðcc^ðX;CÞÞ;piðXÞi¼0;i¼1;...;n:Equivalently,wecouldchooseaconstantweightingfunction:ðXpisðXÞFðcc^ðXÞÞdX¼0;is¼1;...;ns:0Thisgivesasystemwithnequationsandnunknowns,fcgns.ssisis¼1Thisnonlinearsystemcanbesolvedtofindtheweightsfcig.Tossolvethesystem,theintegralcanbecomputednumericallyusingnumericaltechniques;seetheappendix.Asinthecollocationmethod,thechoiceofnsistheresultofatrade-offbetweenincreasedprecisionandahighercomputationalburden.3.3StochasticDiscreteCake-EatingProblemWepresenthereanotherexampleofadynamicprogrammingmodel.Itdiffersfromtheonepresentedinsection3.2intwoways.First,thedecisionoftheagentisnotcontinuous(howmuchtoeat)butdis-crete(eatorwait).Second,theproblemhastwostatevariablesastheexogenousshockisseriallycorrelated.TheagentisendowedwithacakeofsizeW.Ineachperiodtheagenthastodecidewhetherornottoeattheentirecake.Evenifnoteaten,thecakeshrinksbyafactorreachperiod.Theagentalsoexperiencestasteshocks,possiblyseriallycorrelated,andwhichfol-lowanautoregressiveprocessoforderone.Theagentobservesthecurrenttasteshockatthebeginningoftheperiod,beforethedecisiontoeatthecakeistaken.However,thefutureshocksareunobservedbytheagent,introducingastochasticelementintotheproblem.Althoughthecakeisshrinking,theagentmightdecidetopostponetheconsumptiondecisionuntilaperiodwithabetterrealizationofthetasteshock.Theprogramoftheagentcanbewrittenintheform0VðW;eÞ¼max½euðWÞ;bEe0jeVðrW;eÞ;ð3:7ÞwhereVðW;eÞistheintertemporalvalueofacakeofsizeWcondi-tionaloftherealizationeofthetasteshock.HereEe0denotesthe\nNumericalAnalysis47expectationwithrespecttothefutureshocke,conditionalonthevalueofe.ThepolicyfunctionisafunctiondðW;eÞthattakesavalueofzeroiftheagentdecidestowaitoroneifthecakeiseaten.WecanalsodefineathresholdeðWÞsuchthatdðW;eÞ¼1ife>eðWÞ;dðW;eÞ¼0otherwise.Asinsection3.2theproblemcanbesolvedbyvaluefunctioniterations.However,theproblemisdiscrete,sowecannotusetheprojectiontechniqueasthedecisionruleisnotasmoothfunctionbutastepfunction.3.3.1ValueFunctionIterationsAsbefore,wehavetodefine,first,thefunctionalformfortheutilityfunction,andweneedtodiscretizethestatespace.Wewillconsiderr<1,sothecakeshrinkswithtimeandWisnaturallyboundedbetweenW,theinitialsizeand0.InthiscasethesizeofthecaketakesonlyvaluesequaltortW,tb0.HenceC¼friWgisajudi-Sciouschoiceforthestatespace.Contrarytoanequallyspacedgrid,thischoiceensuresthatwedonotneedtointerpolatethevaluefunctionoutsideofthegridpoints.Next,weneedtodiscretizethesecondstatevariable,e.Theshockissupposedtocomefromacontinuousdistribution,anditfollowsanautoregressiveprocessoforderone.WediscretizeeinIpointsIfeigi¼1followingatechniquepresentedbyTauchen(1986)andsum-marizedintheappendix.InfactweapproximateanautoregressiveprocessbyaMarkovchain.Themethoddeterminestheoptimaldis-Icretepointsfeigi¼1andthetransitionmatrixpij¼Probðet¼eijet
1¼ejÞsuchthattheMarkovchainmimicstheAR(1)process.Ofcourse,theapproximationisonlygoodifIisbigenough.InthecasewhereI¼2,wehavetodeterminetwogridpointseLandeH.TheprobabilitythatashockeLisfollowedbyashockeHisdenotedbypLH.Theprobabilityoftransitionscanbestackedinatransitionmatrix:pLLpLHp¼pHLpHHwiththeconstraintsthattheprobabilityofreachingeitheraloworahighstatenextperiodisequaltoone:pLLþpLH¼1andpHLþpHH¼\n48Chapter3i_s=2dountili_s>n_s*Loopoverallsizesofthecake*i_e=1dountili_e>2*Loopoverallpossiblerealizationsofthetasteshock*ueat=u(W[i_s],e[i_e])*Utilityofdoingtheeatingnow*nextV1=V[i_s-1,1]*Nextperiodvalueiflowtasteshock*nextV2=V[i_s-1,2]*Nextperiodvalueifhightasteshock*EnextV=nextV1*p[i_e,1]+nextV2*p[i_e,2]newV[i_s,i_e]=max(ueat,beta*EnextV)*Takethemaxbetweeneatingnoworwaiting*i_e=i_e+1endo*Endofloopovertasteshock*i_s=i_s+1endo*Endofloopoversizeofcake*V=newV*Updatethenewvaluefunction*Figure3.6Discretecake-eatingproblem1.ForagivensizeofthecakeWis¼risWandagivenshocke,j¼LjorH,itiseasytocomputethefirsttermeuðrisWÞ.Tocomputethejsecondtermweneedtocalculatetheexpectedvalueoftomorrow’scake.Givenaguessforthevaluefunctionofnextperiod,vð:;:Þ,theexpectedvalueisisþ1isþ1isþ1Ee0jejvðrWÞ¼pjLvðrW;eLÞþpjHvðrW;eHÞ:TherecursionisstartedbackwardwithaninitialguessforVð:;:Þ.ForagivenstateofthecakeWisandagivenshockej,thenewvaluefunctioniscalculatedfromequation(3.7).Theiterationsarestoppedwhentwosuccessivevaluefunctionsarecloseenough.InnumericalcomputingthevaluefunctionisstoredasamatrixVofsizenWne,wherenWandnearethenumberofpointsonthegridforWande.Ateachiterationthematrixisupdatedwiththenewguessforthevaluefunction.Figure3.6givesanexampleofacomputercodethatobtainsthevaluefunctionvjþ1ðW;eÞgiventhevaluevjðW;eÞ.Thewaywehavecomputedthegrid,thenextperiodvalueissimpletocomputeasitisgivenbyV½is
1;:.Thisruleisvalidif\nNumericalAnalysis49Figure3.7Valuefunction,discretecake-eatingproblemis>1.ComputingV½1;:ismoreofaproblem.Onewayistouseanextrapolationmethodtoapproximatethevalues,giventheknowl-edgeofV½is;:,is>1.Figure3.7showsthevaluefunctionforparticularparameters.Theutilityfunctionistakentobeuðc;eÞ¼lnðecÞ,andlnðeÞissupposedtofollowanAR(1)processwithmeanzero,autocorrelationre¼0:5andwithanunconditionalvarianceof0.2.Wehavediscretizedeintofourgridpoints.Figure3.8showsthedecisionrule,andthefunctioneðWÞ.Thisthresholdwascomputedasthesolutionof:0uðW;eðWÞÞ¼bEe0jeVðrW;eÞ;whichisthevalueofthetasteshockthatmakestheagentindifferentbetweenwaitingandeating,giventhesizeofthecakeW.Wereturnlaterinthisbooktoexamplesofdiscretechoicemodels.Inparticular,wereferthereaderstothemodelspresentedinsections8.5and7.3.3.\n50Chapter3Figure3.8Decisionrule,discretecake-eatingproblem3.4ExtensionsandConclusionInthischapterwereviewedthecommontechniquesusedtosolvethedynamicprogrammingproblemsofchapter2.Weappliedthesetechniquestobothdeterministicandstochasticproblems,tocontin-uousanddiscretechoicemodels.Thesemethodscanbeappliedaswelltomorecomplicatedproblems.3.4.1LargerStateSpacesThetwoexampleswehavestudiedinsections3.2and3.3havesmallstatespaces.Inempiricalapplicationsthestatespaceoftenneedstobemuchlargerifthemodelistoconfrontrealdata.Forinstance,theendowmentshocksmightbeseriallycorrelatedortheinterestrateRmightalsobeastochasticandpersistentprocess.Forthevaluefunctioniterationmethod,thismeansthatthesuc-cessivevaluefunctionshavetobestackedinamultidimensionalmatrix.Alsothevaluefunctionhastobeinterpolatedinseveraldimensions.Thetechniquesintheappendixcanbeextendedtodeal\nNumericalAnalysis51withthisproblem.However,thevaluefunctioniterationmethodquicklyencountersthe‘‘curseofdimensionality.’’Ifeverystatevari-ableisdiscretizedintonsgridpoints,thevaluefunctionhastobeevaluatedbyNnspoints,whereNisthenumberofstatevariables.Thisdemandsanincreasingamountofcomputermemoryandsoslowsdownthecomputation.Asolutiontothisproblemistoeval-uatethevaluefunctionforasubsetofthepointsinthestatespaceandthentointerpolatethevaluefunctionelsewhere.ThissolutionwasimplementedbyKeaneandWolpin(1994).Projectionmethodsarebetterathandlinglargerstatespaces.SupposethattheproblemischaracterizedbyNstatevariablesfX1;...;XNg.TheapproximatedpolicyfunctioncanbewrittenasXNXnjjcc^ðX1;...;XNÞ¼cijpijðXjÞ:j¼1ij¼1jTheproblemisthencharacterizedbyauxiliaryparametersfcg.iexercise3.1SupposethatuðcÞ¼c1
g=ð1
gÞ.Constructthecodetosolveforthestochasticcake-eatingproblemusingthevaluefunctioniterationmethod.Plotthepolicyfunctionasafunctionofthesizeofthecakeandthestochasticendowmentforg¼f0:5;1;2g.Comparethelevelandslopeofthepolicyfunctionsfordifferentvaluesofg.Howdoyouinterprettheresults?exercise3.2Consider,again,thediscretecake-eatingproblemofsection3.3.Constructthecodetosolveforthisproblem,withiidtasteshocks,usinguðcÞ¼lnðcÞ,eL¼0:8,eH¼1:2,pL¼0:3,andpH¼0:7.Mapthedecisionruleasafunctionofthesizeofthecake.exercise3.3Consideranextensionofthediscretecake-eatingproblemofsection3.3.Theagentcannowchooseamongthreeactions:eatthecake,storeitinfridge1orinfridge2.Infridge1,thecakeshrinksbyafactorr:W0¼rW.Infridge2,thecakediminishbyafixedamount:W0¼W
k.Theprogramoftheagentischarac-terizedasVðW;eÞ¼max½VEatðW;eÞ;VFridge1ðW;eÞ;VFridge2ðW;eÞ8>:Fridge20VðW;eÞ¼bEe0VðW
k;eÞ:\n52Chapter3Constructthecodetosolveforthisproblem,usinguðcÞ¼lnðcÞ,eL¼0:8,eH¼1:2,pL¼0:5,andpH¼0:5.Whenwilltheagentswitchfromonefridgetotheother?exercise3.4Considerthestochasticcake-eatingproblem.Supposethatthediscountratebisafunctionoftheamountofcakecon-sumed:b¼Fðb1þb2cÞ,whereb1andb2areknownparametersandFðÞisthenormalcumulativedistributionfunction.Constructthecodetosolveforthisnewproblemusingvaluefunctioniterations.Supposeg¼2,b1¼1:65,pL¼pH¼0:5,yL¼0:8,yH¼1:2,andb2¼
1.Plotthepolicyrulec¼cðXÞ.Comparetheresultwiththatofthecasewherethediscountrateisindependentofthequantityconsumed.Howwouldyouinterpretthefactthatthediscountratedependsontheamountofcakeconsumed?3.5Appendix:AdditionalNumericalToolsInthisappendixweprovidesomeusefulnumericaltoolsthatareoftenusedinsolvingdynamicproblems.Wepresentinterpolationmethods,numericalintegrationmethods,aswellasamethodtoapproximateseriallycorrelatedprocessesbyaMarkovprocess.Thelastsubsectionisdevotedtosimulations.3.5.1InterpolationMethodsWebrieflyreviewthreesimpleinterpolationmethods.Forfurtherreadings,see,forinstance,Pressetal.(1986)orJudd(1996).Whensolvingthevaluefunctionorthepolicyfunction,weoftenhavetocalculatethevalueofthesefunctionsoutsideofthepointsofthegrid.Thisrequiresonetobeabletointerpolatethefunction.Usingagoodinterpolationmethodcanalsosavecomputertimeandspacesincefewergridpointsareneededtoapproximatethefunc-tions.LetusdenotefðxÞthefunctiontoapproximate.Weassumethatweknowthisfunctionatanumberofgridpointsxi,i¼1;...;I.Wedenotebyfi¼fðxiÞthevaluesofthefunctionatthesegridpoints.Weareinterestedinfindinganapproximatefunctionff^ðxÞsuchthatff^ðxÞFfðxÞ,basedontheobservationsfxi;fig.WepresentthreedifferentmethodsanduseasanexamplethefunctionfðxÞ¼xsinðxÞ.Figure3.9displaystheresultsforallthemethods.\nNumericalAnalysis53Figure3.9ApproximationmethodsLeastSquaresInterpolationAnaturalwaytoapproximatefðÞistouseaneconometrictech-nique,suchasOLS,to‘‘estimate’’thefunctionff^ð:Þ.Thefirststepistoassumeafunctionalformforff^.Forinstance,wecanapproximatefwithapolynomialinxsuchasff^ðxÞ¼aþaxþ


þaxN;NxIorx>fi
fi
12>>ai¼
biðxi
xi
1Þ
ciðxi
xi
1Þ;i¼1;...;I;>>xi
xi
1>>>>>>>>>>>>bI>>cI¼
;>>3ðxI
xI
1Þ>>>:2aiþ2biðxi
xi
1Þþ3ciðxi
xi
1Þ¼aiþ1:Solvingthissystemofequationleadstoexpressionsforthecoef-ficientsfai;bi;cig.Figure3.9showsthatthecubicsplineisaverygoodapproximationtothefunctionf.3.5.2NumericalIntegrationNumericalintegrationisoftenrequiredindynamicprogrammingproblemstosolvefortheexpectedvaluefunctionorto‘‘integrateout’’anunobservedstatevariable.Forinstance,solvingtheBell-Ðmanequation(3.3)requiresonetocalculateEvðX0Þ¼vðX0ÞdFðX0Þ,whereFð:Þisthecumulativedensityofthenextperiodcash-on-handX.Ineconometricapplicationssomeimportantstatevariablesmightnotbeobserved.Forthisreasononemayneedtocomputethedeci-sionruleunconditionalofthisstatevariable.Forinstance,inthestochasticcake-eatingproblemofsection3.2,ifXisnotobserved,Ðonecouldcomputec¼cðXÞdFðXÞ,whichistheunconditionalmeanofconsumption,andmatchitwithobservedconsumption.Wepresentthreemethodsthatcanbeusedwhennumericalintegrationisneeded.QuadratureMethodsThereareanumberofquadraturemethods.WebrieflydetailtheGauss-Legendremethod(moredetailedinformationcanbefoundinPressetal.1986).Theintegralofafunctionfisapproximatedasð1fðxÞdxFw1fðx1Þþ


þwnfðxnÞ;ð3:8Þ
1wherewiandxiarenweightsandnodestobedetermined.Integra-tionoveradifferentdomaincanbeeasilyhandledbyoperatinga\n56Chapter3changeoftheintegrationvariable.Theweightsandthenodesarecomputedsuchthat(3.8)isexactlysatisfiedforpolynomialsofdegree2n
1orless.Forinstance,ifn¼2,denotefðxÞ¼xi
1,i¼i1;...;4.Theweightsandnodessatisfyð1w1f1ðx1Þþw2f1ðx2Þ¼f1ðxÞdx;
1ð1w1f2ðx1Þþw2f2ðx2Þ¼f2ðxÞdx;
1ð1w1f3ðx1Þþw2f3ðx2Þ¼f3ðxÞdx;
1ð1w1f4ðx1Þþw2f4ðx2Þ¼f4ðxÞdx:
1Thisisasystemoffourequationswithfourunknowns.Thesolu-tionsarew1¼w2¼1andx2¼
x1¼0:578.Forlargervaluesofn,thecomputationissimilar.Byincreasingthenumberofnodesn,theprecisionincreases.Noticethatthenodesarenotnecessarilyequallyspaced.Theweightsandthevalueofthenodesarepublishedintheliteratureforcommonlyusedvaluesofn.ApproximatinganAutoregressiveProcesswithaMarkovChainInthisdiscussionwefollowTauchen(1986)andTauchenandHus-sey(1991)andshowhowtoapproximateanautoregressiveprocessoforderonebyafirst-orderMarkovprocess.Thiswaywecansim-plifythecomputationofexpectedvaluesinthevaluefunctionitera-tionframework.Toreturntothevaluefunctioninthecake-eatingproblem,weneedtocalculatetheexpectedvaluegivene:0VðW;eÞ¼max½euðWÞ;Ee0jeVðrW;eÞ:Thecalculationofanintegralateachiterationiscumbersome.Sowediscretizetheprocesse,intoNpointsei,i¼1;...;N.Nowwecantreplacetheexpectedvalueby23XNVðW;eiÞ¼max4euðWÞ;pVðrW;ejÞ5;i¼1;...;N:i;jj¼1\nNumericalAnalysis57Figure3.10Exampleofdiscretization,N¼3Asinthequadraturemethod,themethodinvolvesfindingnodesejandweightsp.Aswewillseebelow,theeiandthepcanbei;ji;jcomputedpriortotheiterations.SupposethatetfollowsanAR(1)process,withanunconditionalmeanmandanautocorrelationr:et¼mð1
rÞþret
1þut;ð3:9Þwhereuisanormallydistributedshockwithvariances2.Todis-tcretizethisprocess,weneedtodeterminethreedifferentobjects.First,weneedtodiscretizetheprocessetintoNintervals.Second,weneedtocomputetheconditionalmeanofetwithineachintervals,whichwedenotebyzi,i;...;N.Third,weneedtocomputetheprobabilityoftransitionbetweenanyoftheseintervals,pi;j.Figure3.10showstheplotofthedistributionofeandthecut-offpointseiaswellastheconditionalmeanszi.WestartbydiscretizingthereallineintoNintervals,definedbythelimitse1;...;eNþ1.Astheprocesseisunbounded,e1¼
ytandeNþ1¼þy.Theintervalsareconstructedsuchthatehasantequalprobabilityof1=Noffallingintothem.Giventhenormalityassumption,thecut-offpointsfeigNþ1aredefinedasi¼1eiþ1
mei
m1F
F¼;i¼1;...;N;ð3:10ÞseseNwhereFðÞisthecumulativeofthenormaldensityandseisthepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffistandarddeviationofeequaltos=ð1
rÞ.Workingrecursively,weget\n58Chapter3i
1i
1e¼seFþm:NNowthatwehavedefinedtheintervals,wewanttofindtheaveragevalueofewithinagiveninterval.Wedenotethisvaluebyzi,whichiscomputedasthemeanofeconditionaloneA½ei;eiþ1:ttfððei
mÞ=sÞ
fððeiþ1
mÞ=sÞzi¼EðejeA½ei;eiþ1Þ¼seeþm:tteFððeiþ1
mÞ=sÞ
Fððei
mÞ=sÞeeFrom(3.10),weknowthattheexpressionsimplifiestoei
meiþ1
mzi¼Nsf
fþm:eseseNextwedefinethetransitionprobabilityasp¼PðeA½ej;ejþ1jeA½ei;eiþ1Þi;jtt
1ðeiþ1jþ1N
ðu
mÞ2=ð2s2Þe
mð1
rÞ
rupi;j¼pffiffiffiffiffiffiffiffiffiffieeF2pseeisej
mð1
rÞ
ru
Fdu:sThecomputationofpi;jrequiresthecomputationofanontrivialintegral.Thiscanbedonenumerically.Notethatifr¼0,meaningeisaniidprocess,theexpressionaboveissimply1pi;j¼:NWecannowdefineaMarkovprocessztthatwillmimicanauto-regressiveprocessoforderone,asdefinedin(3.9).zttakesitsvaluesinfzigNandthetransitionbetweenperiodtandtþ1isdefinedasi¼1Pðz¼zjjz¼ziÞ¼p:tt
1i;jByincreasingN,thediscretizationbecomesfinerandtheMarkovprocessgetsclosertotherealautoregressiveprocess.ExampleForN=3,r¼0:5,m¼0,ands¼1,wehavez1¼
1:26;z2¼0;z3¼1:26;and\nNumericalAnalysis59230:550:310:1467p¼40:310:380:315:0:140:310:553.5.3HowtoSimulatetheModelOncethevaluefunctioniscomputed,theestimationortheevalua-tionofthemodeloftenrequiresthesimulationofthebehavioroftheagentthroughtime.Ifthemodelisstochastic,thefirststepistogenerateaseriesfortheshocks,fort¼1;...;T.Thenwegofromperiodtoperiodandusethepolicyfunctiontofindouttheoptimalchoiceforthisperiod.Wealsoupdatethestatevariableandproceedtonextperiod.HowtoProgramaMarkovProcessTheMarkovprocessischaracterizedbygridpoints,fzigandbyatransitionmatrixp,withelementsp¼Probðy¼zj=y¼ziÞ.Weijtt
1startinperiod1.Theprocesszisinitializedat,say,zi.Next,wethavetoassignavalueforz2.Tothisend,usingtherandomgenera-torofthecomputer,wedrawauniformvariableuin½0;1.Thestateinperiod2,j,isdefinedast=1oldind=1*Variabletokeeptrackofstateinperiodt-1*y[t]=z[oldind]*Initializefirstperiod*dountilt>T*Loopoveralltimeperiods*u=uniform(0,1)*Generateauniformrandomvariable*sum=0*Willcontainthecumulativesumofpi*ind=1*Indexoverallpossiblevaluesforprocess*dountilu<=sum*Looptofindoutthestateinperiodt*sum=sum+pi[oldind,ind]*Cumulativesumofpi*ind=ind+1endoy[t]=z[ind]*Stateinperiodt*oldind=ind*Keeptrackoflaggedstate*t=t+1endoFigure3.11SimulationofaMarkovprocess\n60Chapter3XjXjþ1pi;lm,themodelisunderidentified.Inthelattercaseestimationcannotbeachievedastherearetoomanyunknownparameters.So,ifkam,theestimatorofycomesfromminððmðyÞ
mÞ0W
1ðmðyÞ
mÞ:yInthisquadraticform,Wisaweightingmatrix.Asexplainedbelow,thechoiceofWisimportantforobtaininganefficientestimatorofywhenthemodelisoveridentified.UsingSimulationsInmanyapplicationstheproceduresoutlinedabovearedifficulttoimplement,eitherbecausethelikelihoodofobservingthedataorthemomentsisdifficulttocomputeanalyticallyorbecauseitinvolvessolvingtoomanyintegrals.Putdifferently,theresearcherdoesnothaveananalyticrepresentationofMðyÞ.Ifthisisthecase,thenesti-mationcanstillbecarriedoutnumericallyusingsimulations.Consideragaintheiidcase,whereI¼2.Thesimulationapproachproceedsinthefollowingway:First,wefixy,theparameterofMðyÞ.Second,usingtherandomnumbergeneratorofacomputer,wegenerateSdrawsfusgfromauniformdistributionover½0;1.Weclassifyeachdrawasheads(denotedi¼1)ifusI
1Þ,wehaveasituationwherethemodelisagainunderidentified.GiventhemaximumlikelihoodestimateofP,therearemultiplecombinationsoftheparametersthat,throughthemodel,cangenerateP.Again,theresearcherneedstobringaddi-tionalinformationtotheproblemtoovercometheindeterminacyof\n74Chapter4theparameters.Sointhecoin-flippingexampleaphysicaltheorythatinvolvesmorethanasingleparametercannotbeusedtoesti-matefromdatathatyieldasingleprobabilityofheads.Alternatively,ifkeðW;yÞ,thentheindividualwillconsumethecake.eðW;yÞhasnoanalyticalexpressionbutcanbesolvednumericallywiththetoolsdevelopedinchapter3.Theprobabilityofnotcon-sumingacakeofsizeWinagivenperiodisthenPðeeðrtWÞÞ:t1121t1Insection4.3.2theshockswereiid,andthisprobabilitycouldeasilybedecomposedintoaproductoftterms.Ifeisseriallycorrelated,thenthisprobabilityisextremelydifficulttowriteasetiscorrelatedwithallthepreviousshocks.6Fortperiodswehavetosolveamul-tipleintegralofordert,whichconventionalnumericalmethodsofintegrationcannothandle.Inthissectionwewillshowhowsimu-latedmethodscanovercomethisproblemtoprovideanestimateofy.Thedifferentsimulationmethodscanbeclassifiedintotwogroups.Thefirstgroupofmethodscomparesafunctionoftheobserveddatatoafunctionofthesimulateddata.Heretheaverageistakenbothonthesimulateddrawsandonallobservationintheoriginaldatasetatonce.Thisapproachiscalledmomentcalibration.Itincludesthesimulatedmethodofmomentsandindirectinference.6.Forinstance,ife¼reþuwithu@Nð0;s2Þ,theprobabilitythatthecakeistt
1tteateninperiod2isp¼PðeeðWÞÞ21122¼PðeeðWÞje0.\nConsumption141Thisformulationoftheconsumers’constraintsaresimilartotheonesusedthroughoutthisbookinourstatementofdynamicpro-grammingproblems.Theseconstraintsareoftentermedflowcon-straints,sincetheyemphasizetheintertemporalevolutionofthestockofassetsbeinginfluencedbyconsumption.Aswewillsee,itisnaturaltothinkofthestockofassetsasastatevariablesandcon-sumptionasacontrolvariable.Thereisanalternativewaytoexpresstheconsumer’sconstraintsthatcombinesthesetwoflowconditionsbysubstitutingthefirstintothesecond.Aftersomerearranging,ityieldsa2c1y1þþc0¼ða0þy0Þþ:ð6:2Þr1r0r0r0Theleftsideofthisexpressionrepresentstheexpendituresoftheconsumerongoodsinbothperiodsoflifeandonthestockofassetsheldatthestartofperiod2.Therightsidemeasuresthetotalamountofresourcesavailabletothehouseholdforspendingoveritslifetime.Thisisatypeofsourcesasopposedtousesformulationofthelifetimebudgetconstraint.Thenume´raireforthisexpressionofthebudgetconstraintisperiod0consumptiongoods.Maximizationof(6.1)withrespecttoðc0;c1Þsubjectto(6.2)yieldsu0ðcÞ¼l¼bru0ðcÞð6:3Þ001asanecessaryconditionforoptimality,wherelisthemultiplieron(6.2).Thisisanintertemporalfirst-ordercondition(oftentermedtheconsumer’sEulerequation)thatrelatesthemarginalutilityofcon-sumptionacrosstwoperiods.Itisbesttothinkofthisconditionasadeviationfromaproposedsolutiontotheconsumer’soptimizationproblem.So,givenacandi-datesolution,supposethattheconsumerreducesconsumptionbyasmallamountinperiod0andincreasessavingsbythesameamount.Thecostofthedeviationisobtainedbyu0ðcÞin(6.3).Thehousehold0willearnr0betweenthetwoperiodsandwillconsumethoseextraunitsofconsumptioninperiod1.Thisleadstoadiscountedgaininutilitygivenbytherightsideof(6.3).Whenthisconditionholds,lifetimeutilitycannotbeincreasedthroughsuchaperturbationfromtheoptimalpath.Asinourdiscussionofthecake-eatingprobleminchapter2,thisisjustanecessarycondition,since(6.3)capturesaveryspecialtype\n142Chapter6ofdeviationfromaproposedpath:reduceconsumptiontodayandincreaseittomorrow.Formoregeneralproblems(morethantwoperiods)therewillbeotherdeviationstoconsider.But,eveninthetwo-periodproblem,theconsumercouldhavetakenthereducedconsumptioninperiod0andusedittoincreasea2.Ofcourse,thereisanotherfirst-orderconditionassociatedwith(6.1):thechoiceofa2.Thederivativewithrespecttoa2isgivenbyl¼f;wherefisthemultiplieronthenonnegativityconstraintfora2.Clearly,thenonnegativityconstraintbindsðf>0Þifandonlyifthemarginalutilityofconsumptionispositiveðl>0Þ.Thatis,itissuboptimaltoleavemoneyinthebankwhenmoreconsumptionisdesirable.This(somewhatobviousbutveryimportant)pointhastwoimpli-cationstokeepinmind.First,inthinkingaboutperturbationsfromacandidatesolution,wearerighttoignorethepossibilityofusingthereductioninc0toincreasea2asthisisdefinitelynotdesirable.Sec-ond,andperhapsmoreimportant,knowingthata2¼0isalsocriticaltosolvingthisproblem.LookingattheEulerequation(6.3)aloneguaranteesthatconsumptionisoptimallyallocatedacrossperiods,butthisconditioncanholdforanyvalueofa2.So,applying(6.3)isbutonenecessaryconditionforoptimality;a2¼0mustbeincludedaswell.Witha2¼0,theconsumer’sconstraintsimplifiestoc1y1þc0¼a0þy0þ1w0;ð6:4Þr0r0wherew0islifetimewealthfortheagentintermsofperiod0goods.Theoptimalconsumptionchoicesdependonthemeasureoflifetimewealthðw0Þandtheintertemporaltermsoftradeðr0Þ.Intheabsenceofanycapitalmarketrestrictions,thetimingofincomeacrossthehouseholdslifetimeisirrelevantfortheirconsumptiondecisions.Instead,variationsinthetimingofincome,givenw0,arereflectedinthelevelofsavingsbetweenthetwoperiods.11.Thishasawell-understoodimplicationforthetimingoftaxes.Essentiallyagovernmentwithafixedlevelofspendingmustdecideonthetimingofitstaxes.Ifweinterpretthein-comeflowsinourexampleasnetoftaxes,thenintertemporalvariationintaxes(holdingfixedtheirpresentvalue)willonlychangethetimingofhouseholdincomeandnotitspresentvalue.Thustaxpolicywillinfluencesavingsbutnotconsumptiondecisions.\nConsumption143Asanexample,supposeutilityisquadraticinconsumption:d2uðcÞ¼aþbcc;2wherewerequirethatu0ðcÞ¼bdc>0.InthiscasetheEulercon-ditionsimplifiestobdc0¼br0ðbdc1Þ:Withthefurthersimplificationthatbr0¼1,wehaveconstantcon-sumption:c0¼c1.Notethatthispredictionisindependentofthetimingofincomeoverperiods0and1.Thisisanexampleofamoregeneralphenomenontermedconsumptionsmoothing.Thesmooth-ingeffectwillguideourdiscussionofconsumptionpolicyfunctions.6.2.2StochasticIncomeWenowadduncertaintytotheproblembysupposingthatincomeinperiod1ðy1Þisnotknowntotheconsumerinperiod0.FurtherweusetheresultofA2¼0andrewritetheoptimizationproblemmorecompactlyasmaxEy1jy0½uðc0ÞþbuðR0ðA0þy0c0Þþy1Þ;c0wherewehavesubstitutedforc1usingthebudgetconstraint.Notethattheexpectationistakenherewithrespecttotheonlyunknownvariableðy1Þconditionalonknowingy0,period0income.Infactweassumethaty1¼ry0þe1;wherejrjA½0;1.Heree1isashocktoincomethatisnotforecastableusingperiod0information.Intheoptimizationproblemthecon-sumerisassumedtotaketheinformationaboutfutureincomecon-veyedbyobservedcurrentincomeintoaccount.TheEulerequationforthisproblemisgivenbyu0ðcÞ¼EbRu0ðRðAþycÞþyÞ:0y1jy0000001Noteherethatthemarginalutilityoffutureconsumptionisstochas-tic.Thusthetrade-offintheEulerequationreflectsthelossofutilitytodayfromreducingconsumptionrelativetotheexpectedgain,whichdependsontherealizationofincomeinperiod1.\n144Chapter6ThespecialcaseofquadraticutilityandbR0¼1highlightsthedependenceoftheconsumptiondecisiononthepersistenceofincomefluctuations.ForthiscasetheEulerequationsimplifiestoc0¼Ey1jy0c1¼R0ðA0þy0c0ÞþEy1jy0y1:Solvingforc0andcalculatingEy1jy0y1yieldsR0ðA0þy0Þry0R0A0ðR0þrÞc0¼þ¼þy0:ð6:5Þð1þR0Þð1þR0Þð1þR0Þð1þR0ÞThisexpressionrelatesperiod0consumptiontoperiod0incomethroughtwoseparatechannels.First,variationsiny0directlyaffecttheresourcescurrentlyavailabletothehousehold.Second,varia-tionsiny0provideinformationaboutfutureincome(unlessr¼0).From(6.5)wehaveqc0ðR0þrÞ¼:qy0ð1þR0ÞIntheextremecaseofiidincomeshocksðr¼0Þ,consumerswillsaveafractionofanincomeincreaseandconsumetheremainder.Intheoppositeextremeofpermanentshocksðr¼1Þ,currentcon-sumptionmovesoneforonewithcurrentincome.Thensavingsdoesnotrespondtoincomeatall.Thesensitivityofconsumptiontoincomevariationsdependsonthepermanenceofthoseshocks.2Bothextremesreflectafundamentalpropertyoftheoptimalproblemofconsumptionsmoothing.Bythisproperty,variationsincurrentincomearespreadovertimeperiodsinordertosatisfytheEulerequationconditionthatmarginalutilitytodayisequaltothediscountedmarginalutilityofconsumptiontomorrow,giventhereturnR0.Ineffect,consumptionsmoothingistheintertemporalexpressionofthenormalityofgoodspropertyfoundinstaticdemandtheory.Butourexamplehelpshighlightaninterestingaspectofcon-sumptionsmoothing:asthepersistenceofshocksincreases,sodoestheresponsivenessofconsumptiontoincomevariations.Thisactu-allymakesgoodsense:ifincomeincreasestodayarelikelytopersist,thereisnoneedtosaveanyofthecurrentincomegainasitwillreappearinthenextperiod.Thesethemesofconsumptionsmooth-2.Ifr>1,thenqc0=qy0willexceed1.\nConsumption145ingandtheimportanceofthepersistenceofshockswillreappearthroughoutourdiscussionoftheinfinitehorizonconsumeropti-mizationproblem.6.2.3PortfolioChoiceAsecondextensionofthetwo-periodproblemistheadditionofmultipleassets.Historicallytherehasbeenacloselinkbetweentheoptimizationproblemofaconsumerandassetpricingmodels.Wewillexplaintheselinksasweproceed.Webeginherewithasavingsprobleminwhichtherearetwoassets.Assumethatthehouseholdhasnoinitialwealthandcansavecurrentincomethroughthesetwoassets.OneisnonstochasticandhasaoneperiodgrossreturnofRs.ThesecondassetisriskywithareturndenotedbyRR~randameanreturnofRr.Letarandasdenotetheconsumer’sholdingsofassettypej¼r;s.Assets’pricesarenor-malizedat1inperiod0.Theconsumer’schoiceproblemcanthenbewrittenasmaxuðyarasÞþERR~rrss0RR~rbuðaþRaþy1Þ:ar;asHerewemakethesimplifyingassumptionthaty1isknownwithcertainty.Thefirst-orderconditionsareu0ðyarasÞ¼bRsE0rrss0RR~ruðRR~aþRaþy1Þand0ðyarasÞ¼bERR~r0rrssu0RR~ruðRR~aþRaþy1Þ:Notewehavenotimposedanyconditionsregardingtheholdingoftheseassets.Inparticular,wehaveallowedtheagenttobuyorsellthetwoassets.SupposethatuðcÞisstrictlyconcavesothattheagentisriskaverse.Furthersupposethatwesearchforconditionssuchthatthehouseholdiswillingtoholdpositiveamountsofbothassets.Thenwewouldexpectthattheagentwouldhavetobecompensatedfortheriskassociatedwithholdingtheriskyasset.Thiscanbeseenbyequatingthesetwofirst-orderconditions(whichholdwithequality)andthenusingthefactthattheexpectationoftheproductoftworandomvariablesistheproductoftheexpectationsplustheco-variance.Thismanipulationyields\n146Chapter6cov½RR~r;u0ðRR~rarþRsasþyÞRs¼Rrþ1:ð6:6ÞE0rrssRR~ruðRR~aþRaþy1ÞThesignofthenumeratoroftheratioontherightdependsonthesignofar.Iftheagentholdsboththerisklessandtheriskyasset(ar>0andas>0),thenthestrictconcavityofuðcÞimpliesthatthecovariancemustbenegative.Inthiscase,RrmustexceedRs:theagentmustbecompensatedforholdingtheriskyasset.Iftheaveragereturnsareequal,thentheagentwillnotholdtheriskyassetða¼0Þand(6.6)willhold.Finally,ifRrislessthanRs,rtheagentwillselltheriskyassetandbuyadditionalunitsoftherisklessasset.6.2.4BorrowingRestrictionsAfinalextensionofthetwo-periodmodelistoimposearestrictionontheborrowingofagents.Toillustrate,consideraveryextremeconstraintwheretheconsumerisabletosavebutnottoborrow:c0ay0.Thustheoptimizationproblemoftheagentismax½uðc0ÞþbuðR0ðA0y0c0Þþy1Þ:c0ay0Denotethemultiplierontheborrowingconstraintbym,thefirst-orderconditionisgivenbyu0ðcÞ¼bRu0ðRðAþycÞþyÞþm:0000001Iftheconstraintdoesnotbind,thentheconsumerhasnonnegativesavingsandthefamiliarEulerequationforthetwo-periodproblemholds.However,ifm>0,thenc0¼y0andu0ðyÞ>bRu0ðyÞ:001TheborrowingconstraintislesslikelytobindifbR0isnotverylargeandify0islargerelativetoy1.Animportantimplicationofthemodelwithborrowingcon-straintsisthatconsumptionwilldependonthetimingofincomereceiptsandnotjustW0.Thatis,imaginearestructuringofincomethatincreasedy0anddecreasedy1,leavingW0unchanged.Intheabsenceofaborrowingrestriction,consumptionpatternswouldnotchange.But,iftheborrowingconstraintbinds,thenthisrestructur-ingofincomewillleadtoanincreaseinc0andareductioninc1as\nConsumption147consumption‘‘follows’’income.Totheextentthatthischangeinthetimingofincomeflowscouldreflectgovernmenttaxpolicy(ytisthenviewedasafter-taxincome),thepresenceofborrowingrestric-tionsimpliesthatthetimingoftaxescanmatterforconsumptionflowsandthusforwelfare.Theweaknessofthisandmoregeneralmodelsisthatthebasisfortheborrowingrestrictionsisnotprovided.Soitisnotsurprisingthatresearchershavebeeninterestedinunderstandingthesourceofborrowingrestrictions.Wereturntothispointinalatersection.6.3InfiniteHorizonFormulation:TheoryandEmpiricalEvidenceWenowconsidertheinfinitehorizonversionoftheoptimalconsumptionproblem.Weareinterestedinseeinghowthebasicintuitionofconsumptionsmoothingandotheraspectsofoptimalconsumptionallocationscarryovertotheinfinitehorizonsetting.Inadditionweintroduceempiricalevidenceintoourpresentation.6.3.1Bellman’sEquationfortheInfiniteHorizonProblemConsiderahouseholdwithastockofwealthdenotedbyA,acurrentflowofincomey,andareturnonitsinvestmentsoverthepastperiodgivenbyR1.Thestatevectoroftheconsumer’sproblemisðA;y;R1Þ,andtheassociatedBellmanequationis00vðA;y;R1Þ¼maxuðcÞþbEy0;RjR1;yvðA;y;RÞforallðA;y;R1Þ;cwherethetransitionequationforwealthisgivenbyA0¼RðAþycÞ:Weassumethattheproblemisstationarysothatnotimesubscriptsarenecessary.3Thisrequires,amongotherthings,thatincomeandreturnsbestationaryrandomvariablesandthatthejointdistributionofðy0;RÞdependonlyonðy;RÞ.1Thetransitionequationhasthesametimingasweassumedinthetwo-periodproblem:interestisearnedonwealthplusincomeless3.Weassumethatthereexistsasolutiontothisfunctionequation.Thisrequires,asalways,thatthechoicebebounded,perhapsbyaconstraintonthetotaldebtthatahouseholdcanaccumulate.\n148Chapter6consumptionovertheperiod.Furthertheinterestratethatappliesisnotnecessarilyknownatthetimeoftheconsumptiondecision.ThustheexpectationinBellman’sequationisoverthetwounknownsðy0;R0Þwherethegivenstatevariablesprovideinformationonfore-castingthesevariables.46.3.2StochasticIncomeToanalyzethisproblem,wefirstconsiderthespecialcasewherethereturnonsavingsisknownandtheindividualfacesuncer-taintyonlywithrespecttoincome.Wethenbuildonthismodelbyaddinginaportfoliochoice,endogenouslaborsupply,andborrow-ingrestrictions.TheoryThecasewestudyis00vðA;yÞ¼maxuðcÞþbEy0jyvðA;yÞ;ð6:7ÞcwhereA0¼RðAþycÞforallðA;yÞ.Thesolutiontothisproblemisapolicyfunctionthatrelatesconsumptiontothestatevector:c¼fðA;yÞ.Thefirst-orderconditionis000uðcÞ¼bREy0jyvAðA;yÞ;ð6:8ÞwhichholdsforallðA;yÞ,wherevðA0;y0ÞdenotesqvðA0;y0Þ=qA0.A00Using(6.7)tosolveforEy0jyvAðA;yÞyieldstheEulerequation000uðcÞ¼bREy0jyuðcÞ:ð6:9ÞTheinterpretationofthisequationisthatthemarginallossofreducingconsumptionisbalancedbythediscountedexpectedmar-ginalutilityfromconsumingtheproceedsinthefollowingperiod.Asusual,thisEulerequationimpliesthataone-perioddeviationfromaproposedsolutionthatsatisfiesthisrelationshipwillnotincreaseutility.TheEulerequation,(6.9),holdswhenconsumptiontodayandtomorrowisevaluatedusingthispolicyfunction.InthespecialcaseofbR¼1,thetheorypredictsthatthemarginalutilityofconsumptionfollowsarandomwalk.Ingeneral,onecannotgenerateaclosed-formsolutionofthepolicyfunctionfromtheseconditionsforoptimality.Still,some4.Ifthereareothervariablesknowntothedecisionmakerthatprovideinformationonðy0;RÞ,thenthesevariableswouldbeincludedinthestatevectoraswell.\nConsumption149propertiesofthepolicyfunctionscanbededuced.GiventhatuðcÞisstrictlyconcave,onecanshowthatvðA;yÞisstrictlyconcaveinA.Asarguedinchapter2,thevaluefunctionwillinheritsomeofthecur-vaturepropertiesofthereturnfunction.Usingthisand(6.8),thepolicyfunction,fðA;yÞ,mustbeincreasinginA.Else,anincreaseinAwouldreduceconsumptionandthusincreaseA0.Thiswouldcon-tradict(6.8).Asaleadingexample,considerthespecificationofutilityc1g1uðcÞ¼;1gwhereg¼1isthespecialcaseofuðcÞ¼lnðcÞ.Thisiscalledthecon-stantrelativeriskaversioncase(CRRA),sincecu00ðcÞ=u0ðcÞ¼g.Usingthisutilityfunction,werewrite(6.9)as0gc1¼bRE;cwheretheexpectationistakenwithrespecttofutureconsumptionwhich,throughthepolicyfunction,dependsonðA0;y0Þ.Asdis-cussedinsomedetailbelow,thisequationisthenusedtoestimatetheparametersoftheutilityfunction,ðb;gÞ.EvidenceHall(1978)studiesthecasewhereuðcÞisquadraticsothatthemar-ginalutilityofconsumptionislinear.Inthiscaseconsumptionitselfispredictedtofollowarandomwalk.Hallusesthisrestrictiontotestthepredictionsofthismodelofconsumption.Inparticular,ifcon-sumptionfollowsarandomwalk,thenctþ1¼ctþetþ1:Thetheorypredictsthatthegrowthinconsumptionðetþ1Þshouldbeorthogonaltoanyvariablesknowninperiodt:Etetþ1¼0.Hallusesaggregatequarterlydatafornondurableconsumption.Heshowsthatlaggedstockmarketpricessignificantlypredictconsumptiongrowth,whichviolatesthepermanentincomehypothesis.5Flavin(1981)extendsHall’sanalysis,allowingforageneralARMAprocessfortheincome.Incomeiscommonlyfoundasapre-dictorofconsumptiongrowth.Flavinpointsoutthatthisfindingis5.Sargent(1978)alsoprovidesatestforthepermanentincomehypothesisandrejectsthemodel.\n150Chapter6notnecessarilyinoppositionwiththepredictionofthemodel.Cur-rentincomemightbecorrelatedwithconsumptiongrowth,notbecauseofafailureofthepermanentincomehypothesisbutbecausecurrentincomesignalschangesinthepermanentincome.However,shealsorejectsthemodel.Theimportanceofcurrentincometoexplainconsumptiongrowthhasbeenseenasevidenceofliquidityconstraints(seesection6.3.5).Anumberofauthorshaveinvestigatedthisissue.6However,mostofthepapersuseaggregatedatatotestthemodel.Blundelletal.(1994)testthemodelonmicrodataandfindthatwhenonecontrolsfordemographicsandhouseholdcharacteristics,currentincomedoesnotappeartopredictconsumptiongrowth.MeghirandWeber(1996)explicitlytestforthepresenceofliquidityconstraintsusingaU.S.paneldataanddonotfindanyevidence.6.3.3StochasticReturns:PortfolioChoiceWehavealreadyconsideredasimpleportfoliochoiceproblemforthetwo-periodproblem,sothisdiscussionwillbeintentionallybrief.Weelucidatetheempiricalevidencebasedonthismodel.TheoryAssumethatthereareNassetsavailable.LetR1denotetheN-vectorofgrossreturnsbetweenthecurrentandpreviousperiodandletAbethecurrentstockofwealth.Letsidenotetheshareofasseti¼1;2;...;Nheldbytheagent.Normalizingthepriceofeachassettobeunity,thecurrentconsumptionoftheagentisthenXc¼Asi:iSubstitutingthisintotheBellmanequation,wehave!!XX0vðA;y;R1Þ¼maxuAsiþbER;y0jR1;yvRisi;y;R;siiið6:10ÞwhereRiisthestochasticreturnonasseti.NotethatR1isonlyinthestatevectorbecauseoftheinformationalvalueitprovidesonthereturnoverthenextperiod,R.6.See,forinstance,Zeldes(1989b)andCampbellandMankiw(1989).\nConsumption151Thefirst-orderconditionfortheoptimizationproblemholdsfori¼1;2;...;N,anditis!X00uðcÞ¼bER;y0jR1;yRivARisi;y;R;iwhereagainvAðÞisdefinedasqvðÞ=qA.Using(6.10)tosolveforthederivativeofthevaluefunction,weobtain000uðcÞ¼bER;y0jR1;yRiuðcÞfori¼1;2;...;N;where,ofcourse,theleveloffutureconsumptionwilldependonthevectorofreturns,R,andtherealizationoffutureincome,y0.ThissystemofEulerequationsformsthebasisforfinancialmodelsthatlinkassetpricestoconsumptionflows.Thissystemisalsothebasisfortheargumentthatconventionalmodelsareunabletoexplaintheobserveddifferentialbetweenthereturnonequityandrelativelysafebonds.Finally,theseconditionsarealsousedtoesti-matetheparametersoftheutilityfunction,suchasthecurvatureparameterinthetraditionalCRRAspecification.ThisapproachisbestseenthroughareviewofHansenandSingleton(1982).Tounderstandthisapproach,recallthatHallusestheorthogonalityconditionstotestamodelofoptimalconsumption.NotethatHall’sexercisedoesnotestimateanyparametersastheutilityfunctionisassumedtobequadraticandtherealinterestrateisfixed.Instead,Hallessentiallytestsarestrictionimposedbyhismodelattheassumedparametervalues.ThelogicpursuedbyHansen-Singeltongoesastepfurther.Insteadofusingtheorthogonalityconstraintstoevaluatethepre-dictionsofaparameterizedmodel,theyusetheseconditionstoesti-mateamodel.Evidently,ifoneimposesmoreconditionsthanthereareparameters(i.e.,iftheexerciseisoveridentified),theresearchercanbothestimatetheparametersandtestthevalidityofthemodel.EmpiricalImplementationThestartingpointfortheanalysisistheEulerequationforthehousehold’sproblemwithNassets.Werewritethatfirst-ordercon-ditionhereusingtimesubscriptstoshowthetimingofdecisionsandrealizationsofrandomvariables:u0ðcÞ¼bERu0ðcÞfori¼1;2;...;N;ð6:11Þttitþ1tþ1whereRitþ1isdefinedastherealreturnonassetibetweenperiods\n152Chapter6tandtþ1.Theexpectationhereisconditionalonallvariablesobservedinperiodt.Unknowntþ1variablesincludethereturnontheassetsaswellasperiodtþ1income.ThepoweroftheGMMapproachderivesfromthisfirst-ordercondition.Whatthetheorytellsusisthatwhileexpostthisfirst-orderconditionneednothold,anydeviationsfromitwillbeunpredictablegivenperiodtinformation.Thatis,theperiodtþ1realizationsay,ofincome,mayleadtheconsumertoincreasecon-sumptionisperiodtþ1,thusimplyingexpostthat(6.11)doesnothold.Thisdeviationisconsistentwiththetheoryaslongasitisnotpredictablegivenperiodtinformation.Formally,defineeiðyÞastþ1bRu0ðcÞiitþ1tþ1etþ1ðyÞ1u0ðcÞ1fori¼1;2;...;N:ð6:12ÞtThuseiðyÞisameasureofthedeviationforanasseti.Wehavetþ1addedyasanargumentinthiserrortohighlightitsdependenceontheparametersdescribingthehousehold’spreferences.HouseholdoptimizationimpliesthatEðeiðyÞÞ¼0fori¼1;2;...;N:ttþ1Letztbeaq-vectorofvariablesthatareintheperiodtinformationset.7ThisrestrictiononconditionalexpectationsimpliesthatEðeiðyÞnzÞ¼0fori¼1;2;...;N;ð6:13Þtþ1twherenistheKroneckerproduct.SothetheoryimpliestheEulerequationerrorsfromanyoftheNfirst-orderconditionsoughttobeorthogonaltoanyoftheztvariablesintheinformationset.ThereareNqrestrictionscreated.TheideaofGMMestimationisthentofindthevectorofstructuralparametersðyÞsuchthat(6.13)holds.Ofcourse,appliedeconomistsonlyhaveaccesstoasample,sayoflengthT.LetmTðyÞbeanNqvectorwherethecomponentrelatingassetitooneofthevariablesjinzt,zt,isdefinedby1XTijðetþ1ðyÞztÞ:Tt¼17.Thetheorydoesnotimplywhichofthemanypossiblevariablesshouldbeusedwhenemployingtheserestrictionsinanestimationexercise.Thatis,thequestionofwhichmo-mentstomatchisnotansweredbythetheory.\nConsumption153TheGMMestimatorisdefinedasthevalueofythatminimizes0JTðyÞ¼mTðyÞWTmTðyÞ:HereWTisanNqNqmatrixthatisusedtoweightthevariousmomentrestrictions.HansenandSingleton(1982)usemonthlyseasonallyadjustedaggregatedataonU.S.nondurableconsumptionornondurablesandservicesbetween1959and1978.Theyuseasameasureofstockreturns,theequallyweightedaveragereturnonallstockslistedontheNewYorkStockExchange.TheychooseaconstantrelativeriskaversionutilityfunctionuðcÞ¼c1g=ð1gÞ.Withthisspecificationtherearetwoparameterstoestimate,thecurvatureoftheutilityfunctiongandthediscountfactorb.Thusy¼ðb;gÞ.Theauthorsuseasinstrumentslaggedvaluesofðct;Ritþ1Þandestimatethemodelwith1,2,4,or6lags.Dependingonthenumberoflagsandtheseriesused,theyfindvaluesforgwhichvarybetween0.67and0.97andvaluesforthediscountfactorbetween0.942and0.998.Asthemodelisoveridentified,thereisscopeforanoveridentificationtest.Dependingonthenumberoflagsandtheseriesused,thetestgivesmixedresultsastherestrictionsaresometimessatisfiedandsome-timesrejected.Notethattheauthorsdonotadjustforpossibletrendsintheesti-mation.Supposethatlogconsumptionischaracterizedbyalineartrend:ct¼expðatÞcc~t;wherecc~tisthedetrendedconsumption.Inthatcaseequation(6.12)isrewrittenasbeagRcc~giitþ1tþ1etþ1ðyÞ1g1fori¼1;2;...;N:cc~tHencetheestimateddiscountfactorisaproductbetweenthetruediscountfactorandatrendeffect.Ignoringthetrendwouldresultinabiasforthediscountrate.6.3.4EndogenousLaborSupplyOfcourse,itisnaturaltoaddalaborsupplydecisiontothismodel.Wecanthinkofthestochasticincome,takenasgivenabove,as\n154Chapter6comingfromastochasticwageðwÞandalaborsupplydecisionðnÞ.Inthiscaseconsiderthefollowingfunctionalequation:A000vðA;wÞ¼maxUAþwn;nþbEw0jwvðA;wÞforallðA;wÞ:A0;nRHerewehavesubstitutedinforcurrentconsumptionsothattheagentischoosinglaborsupplyandfuturewealth.Notethatthelaborsupplychoice,givenðA;A0Þ,ispurelystatic.Thatis,thelevelofemploymentandthuslaborearningshasnody-namicaspectotherthansupplementingtheresourcesavailabletofinancecurrentconsumptionandfuturewealth.Correspondinglythefirst-orderconditionwithrespecttothelevelofemploymentdoesnotdirectlyinvolvethevaluefunction,anditisgivenbywUcðc;nÞ¼Unðc;nÞ:ð6:14ÞUsingc¼AþwnðA0=RÞ,thisfirst-orderconditionrelatesntoðA;w;A0Þ.Denotethisrelationshipasn¼jðA;w;A0Þ.Itcanthenbesubstitutedbackintothedynamicprogrammingproblemyieldingasimplerfunctionalequation:000vðA;wÞ¼maxZðA;A;wÞþbEw0jwvðA;wÞ;A0whereZðA;A0;wÞ1UðAþwjðA;w;A0ÞðA0=RÞ;jðA;w;A0ÞÞ:ThissimplifiedBellmanequationcanbeanalyzedbystandardmethods,thusignoringthestaticlaborsupplydecision.8Onceasolutionisfound,thelevelofemploymentcanthenbedeterminedfromtheconditionn¼jðA;w;A0Þ.ByasimilarmodelMaCurdy(1981)studiesthelaborsupplyofyoungmenusingthePanelStudyonIncomeDynamics(PSID).Theestimationofthemodelisdoneinseveralsteps.First,theintraperiodallocation(6.14)isestimated.Thecoefficientsarethenusedtogetattheintertemporalpartofthemodel.Toestimatetheparametersoftheutilityfunction,onehastoobservehoursofworkandconsumption,butinthePSID,totalcon-sumptionisnotreported.Toidentifythemodel,MaCurdyusesa8.Thisissimilartothetrickweusedinthestochasticgrowthmodelwithendogenousemployment.\nConsumption155utilityfunctionthatisseparablebetweenconsumptionandlaborsupply.Theutilityfunctionisspecifiedasuðc;nÞ¼gco1gno2,tt1tt2ttwhereg1tandg2taretwodeterministicfunctionsofobservedchar-acteristicsthatmightaffectpreferencessuchasage,education,andnumberofchildren.Withthisspecificationthemarginalutilityofleisure,Unðc;nÞ,isindependentoftheconsumptiondecision.Using(6.14),hoursofworkcanbeexpressedaslnwt1lnðntÞ¼þðlnUcðct;ntÞlng2tlno2Þ:o21o21Whilethefirsttermintheright-hand-sideisobserved,thesecondtermcontainstheunobservedmarginalutilityofconsumption.Ucðct;ntÞcanbeexpressedasafunctionoftheLagrangemultiplierassociatedwiththewealthconstraintinperiod0:l0Ucðct;ntÞ¼t:bð1þr1Þ...ð1þrtÞTheauthortreatstheunobservedmultiplierl0,asafixedeffectandusespaneldatatoestimateasubsetoftheparametersoftheutilityfunctionfromfirstdifferences.Inanextstepthefixedeffectisbackedout.Atthispointsomeadditionalidentificationassumptionsareneeded.AspecificfunctionalformisassumedfortheLagrangemultiplier,writtenasafunctionofwagesoverthelifecycleandini-tialwealth,allofthembeingunobservedinthedataset.TheauthorusesthenfixedcharacteristicssuchthateducationoragetoproxyfortheLagrangemultiplier.Theauthorfindsthata10percentin-creaseintherealwageinducesaoneto5percentincreaseinhoursworked.Eichenbaumetal.(1988)analyzethetimeseriespropertiesofahouseholdmodelwithbothasavingsandalaborsupplydecision.Theypayparticularattentiontospecificationsinwhichpreferencesarenonseparable,bothacrosstimeandbetweenconsumptionandleisurecontemporaneously.TheyestimatetheirmodelusingGMMontimeseriesevidenceonrealconsumption(excludingdurables)andhoursworked.Theyfindsupportfornontimeseparabilityinpreferences,thoughinsomecasestheyfoundlittleevidenceagainstthehypothesisthatpreferenceswereseparablewithinaperiod.\n156Chapter66.3.5BorrowingConstraintsTheModelandPolicyFunctionTheextensionofthetwo-periodmodelwithborrowingconstraintstotheinfinitehorizoncaseisdiscussedbyDeaton(1991).9Oneofthekeyadditionalinsightsfromextendingthehorizonistonotethateveniftheborrowingconstraintdoesnotbindinaperiod,thisdoesnotimplythatconsumptionandsavingstakethesamevaluesastheywouldintheproblemwithoutborrowingconstraints.Simplyput,consumersanticipatethatborrowingrestrictionsmaybindinthefuture(i.e.,inotherstates),andthisinfluencestheirchoicesinthecurrentstate.FollowingDeaton(1991),letx¼Aþyrepresentcashonhand.ThenthetransitionequationforwealthimpliesthatA0¼RðxcÞ;wherecisconsumption.Intheeventthatincomevariationsareiid,wecanwritetheBellmanequationforthehouseholdas0vðxÞ¼maxuðcÞþbEy0vðRðxcÞþyÞð6:15Þ0acaxsothatthereturnRisearnedontheavailableresourceslesscon-sumption,xc.Notethatincomeisnotastatevariablehereasitisassumedtobeiid.Hencecashonhandcompletelysummarizestheresourcesavailabletotheconsumer.Theborrowingrestrictiontakesthesimpleformofcaxsothattheconsumerisunabletoborrow.Ofcourse,thisisextremeandentirelyadhoc,butitdoesallowustoexploretheconsequencesofthisrestriction.AsarguedbyDeaton,theEulerequationforthisproblemmustsatisfyu0ðcÞ¼maxfu0ðxÞ;bREu0ðc0Þg:ð6:16ÞThus,eithertheborrowingrestrictionbindssothatc¼xoritdoesn’tsothatthemorefamiliarEulerequationholds.Onlyforlowvaluesofxwillu0ðxÞ>bREu0ðc0Þ,andonlyinthesestates,asarguedforthetwo-periodproblem,willtheconstraintbind.Toemphasizeanimportantpoint:eveniftheu0ðxÞ0,arenormallydistributedwithmean0andvariances2ands2respectively.TheconsumernufacesabudgetconstraintWtþ1¼ð1þrÞðWtþYtCtÞ:Theconsumercanborrowandsavefreely.However,undertheassumptionthatthereisaprobabilitythatincomewillbezeroandthatthemarginalutilityofconsumptionisinfiniteatzero,thecon-sumerwillchoosenevertoborrowagainstfutureincome.HencetheoutcomeofthemodelisclosetotheoneproposedbyDeaton(1991)10.SeealsoCarroll(1992).\nConsumption161andpresentedinsection6.3.5.Notethatinthemodeltheagentcanonlyconsumenondurables.Theauthorsignorethedurabledecision,orequivalentlyassumethatthisdecisionisexogenous.Thismightbeastrongassumption.Ferna´ndez-VillaverdeandKrueger(2001)arguethatthejointdynamicsofdurablesandnondurablesareimportanttounderstandthesavingsandconsumptiondecisionsoverthelifecycle.Definethecashonhandasthetotalofassetsandincome:Xt¼WtþYt;Xtþ1¼RðXtCtÞþYtþ1:DefineVtðXt;PtÞasthevaluefunctionatageTt.Thevaluefunc-tionisindexedbyageasitisassumedthattheconsumerhasafinitelifehorizon.Thevaluefunctiondependsontwostatevariables:thecashonhand,whichindicatesthemaximallimitthatcanbecon-sumed,andtherealizationofthepermanentcomponent,whichpro-videsinformationonfuturevaluesofincome.TheprogramoftheagentisdefinedasVtðXt;PtÞ¼max½uðCtÞþbEtVtþ1ðXtþ1;Ptþ1Þ:CtTheoptimalbehaviorisgivenbytheEulerequation:u0ðCÞ¼bREu0ðCÞ:tttþ1Asincomeisassumedtobegrowingovertime,cashonhandandconsumptionarealsononstationary.Thisproblemcanbesolvedbynormalizingthevariablesbythepermanentcomponent.Denotext¼Xt=Ptandct¼Ct=Pt.ThenormalizedcashonhandevolvesasRxtþ1¼ðxtctÞþUtþ1:Gtþ1Ntþ1UndertheassumptionthattheutilityfunctionisuðcÞ¼cð1gÞ=ð1gÞ,theEulerequationcanberewrittenwithonlystationaryvariables:u0ðcÞ¼bREu0ðcGNÞ:tttþ1tþ1tþ1Asthehorizonoftheagentisfinite,onehastopostulatesometer-minalconditionfortheconsumptionrule.Itistakentobelinearinthenormalizedcashonhand:cTþ1¼g0þg1xTþ1.GourinchasandParker(2002)usethisEulerequationtocomputenumericallytheoptimalconsumptionrule.Normalizedconsumptionisonlyafunctionofthenormalizedcashonhand.Bydiscretizing\n162Chapter6Figure6.3Optimalconsumptionrulethecashonhandoveragrid,theproblemissolvedrecursivelybyevaluatingctðxÞateachpointofthegridusingu0ðcðxÞÞtðð0R¼bRð1pÞuctþ1ðxctÞþUGtþ1NdFðNÞdFðUÞGtþ1Nð0RþbRpuctþ1ðxctÞGtþ1NdFðNÞ:Gtþ1NThefirsttermontheright-hand-sidecalculatestheexpectedvalueofthefuturemarginalutilityconditionalonazeroincome,whilethesecondtermistheexpectationconditionalonastrictlypositiveincome.Theintegralsaresolvedbyaquadraturemethod(seechap-ter3).Theoptimalconsumptionrulesareobtainedbyminimizingthedistancebetweentheleft-handsideandtheright-handside.Figure6.3displaystheconsumptionruleatdifferentages.1111.Thefigurewascomputedusingthefollowingparameterization:b¼0:96,g¼0:5,s2¼0:0212,s2¼0:044,p¼0:03.g¼0:0196,andg¼0:0533.WearegratefultoGour-un01inchasandParkerforprovidinguswiththeircodesanddata.\nConsumption163Figure6.4ObservedandpredictedconsumptionprofilesOncetheconsumptionrulesaredetermined,themodelcanbesimulatedtogenerateaveragelifecycleprofilesofconsumption.Thisisdoneusingtheapproximatedconsumptionrulesandbyaveragingthesimulatedbehaviorofalargenumberofhouseholds.ThesimulatedprofilesarethencomparedtoactualprofilesfromU.S.data.Figure6.4displaysthepredictedconsumptionprofilefortwovaluesoftheintertemporalelasticityofsubstitutionaswellastheobservedconsumptionprofilesconstructedfromtheU.S.ConsumerExpenditureSurvey.12Moreformally,theestimationmethodisthesimulatedmethodofmoments(seechapter4).Theauthorsminimizethedistancebetweenobservedandpredictedconsumptionatdifferentages.Asneitherthecashonhandnorthepermanentcomponentofincomearedirectlyobserved,theauthorsintegrateoutthestatevariablestocalculatetheunconditionalmeanof(log)consumptionatagivenage:ðlnCtðyÞ¼lnCtðx;P;yÞdFtðx;P;yÞ;12.Seefootnote11fortheparameterization.\n164Chapter6whereyisthevectorofparameterscharacterizingthemodelandwhereFtðÞisthedensityofthestatevariablesforindividualsofaget.Characterizingthisdensityisdifficultasithasnoclosedformsolution.HencetheauthorsusesimulationstoapproximatelnCtðyÞ.Denote1XIt1XSgðyÞ¼lnClnCðXs;Ps;yÞ:IitStttti¼1s¼1Thefirstpartistheaveragelogconsumptionforhouseholdsofaget,andItisthenumberofobservedhouseholdinthedataset.ThesecondpartistheaveragepredictedconsumptionoverSsimulatedpaths.yisestimatedbyminimizing0gðyÞWgðyÞ;whereWisaweightingmatrix.Theestimatedmodelisthenusedtoanalyzethedeterminantofsavings.Therearetworeasonstoaccumulatesavingsinthismodel.First,itcushionstheagentfromuninsurableincomeshocks,toavoidfacingalowmarginalutility.Second,savingsareusedtofinanceretirementconsumption.GourinchasandParker(2002)showthattheprecautionarymotivedominatesatleastuntilage40,whereasolderagentssavemostlyforretirement.6.4ConclusionInthischapterwedemonstratedhowtousetheapproachofdynamicprogrammingtocharacterizethesolutionofthehouse-holdsoptimalconsumptionproblemandtolinkitwithobservations.Thediscussionwentbeyondthesavingsdecisiontointegrateitwiththelaborsupplyandportfoliodecisions.Asinotherchapters,wegavenumerousextensionsthatareopen-endedfortheresearchertoconsider.(Inthenextchapterweturntooneofthese,durablegoods.)Further,therearenumerouspolicyexercisesthatcanbeevaluatedusinganestimatedmodelofthehouseholdconsumptionchoice,includedavarietyofpoliciesintendedtoinfluencesavingsdecisions.1313.RustandPhelan(1997)provideagoodexampleinexploringtheeffectsofsocialsecuritypoliciesonlaborsupplyandretirementdecisionsinadynamicprogrammingframework.\n7DurableConsumption7.1MotivationSofartheconsumptiongoodswelookedatareclassifiedaseithernondurablesorservices.Thisshouldbeclearsinceconsumptionexpendituresaffectedutilitydirectlyintheperiodofthepurchaseandthendisappeared.1However,durablegoodsplayaprominentroleinbusinesscyclesasdurableexpendituresarequitevolatile.2Inthischapterwestudytwoapproachestounderstandingdurableconsumption.Thefirstisanextensionofthemodelsstudiedinthepreviouschapterinwhicharepresentativeagentaccumulatesdura-blestoprovideaflowofservices.HerewepresenttheresultsofMankiw(1982)whicheffectivelyrejectstherepresentativeagentmodel.3Thesecondmodelintroducesanonconvexityintothehousehold’soptimizationproblem.Themotivationfordoingsoisevidencethathouseholdsdonotcontinuouslyadjusttheirstockofdurables.Weexplorethisoptimizationproblemthroughthespecificationandestimationofadynamicdiscretechoicemodel.1.Inamodelofhabitformation,pastconsumptioncaninfluencecurrentutilityeveniftheconsumptionisofanondurableoraservice.Thenthestatevectorissupple-mentedtokeeptrackofthatexperience.Fordurablegoodswewillsupplementthestatevectortotakethestockofdurablesintoaccount.2.AccordingtoBaxter(1996),thevolatilityofdurableconsumptionisaboutfivetimesthatofnondurableconsumption.3.Tobecomplete,asweexplain,therearealsomaintainedassumptionsaboutpref-erences,shocks,andthelackofadjustmentcosts.\n166Chapter77.2PermanentIncomeHypothesisModelofDurableExpendituresWebeginwithamodelthatbuildsonthepermanentincomehypothesisstructurethatweusedinthepreviouschaptertostudynondurableexpenditures.Wefirstprovidethetheoreticalpropertiesofthemodelandthendiscussitsempiricalimplementation.7.2.1TheoryTomodelexpendituresonbothdurableandnondurablegoods,weconsiderhouseholdbehaviorinwhichtheconsumerhasastockofwealthA,astockofdurablegoodsD,andcurrentincomey.Theconsumeruseswealthpluscurrentincometofinanceexpendituresoncurrentnondurableconsumptioncandtofinancethepurchaseofdurablegoodseatarelativepricep.Therearetwotransitionequationsforthisproblem.OneistheaccumulationequationforwealthgivenbyA0¼RðAþy
c
peÞ:Theaccumulationequationfordurablesissimilartothatusedforcapitalheldbythebusinesssector:D0¼Dð1
dÞþe;ð7:1ÞwheredAð0;1Þisthedepreciationrateforthestockofdurables.Utilitydependsontheflowofservicesfromthestockofdurablesandthepurchasesofnondurables.Intermsoftiming,assumethatdurablesboughtinthecurrentperiodyieldservicesstartinginthenextperiod.So,aswithcapital,thereisatimelagbetweentheorderandtheuseofthedurablegood.4Withthesedetailsinmind,theBellmanequationforthehouse-holdisgivenby0000VðA;D;y;pÞ¼maxuðc;DÞþbEy0;p0jy;pVðA;D;y;pÞD0;A0forallðA;D;y;pÞð7:2Þwith4.Ofcourse,otherpossibleassumptionsontimingareimplementableinthisframe-workasweshowlaterinthechapter.\nDurableConsumption167A0c¼Aþy

pðD0
ð1
dÞDÞð7:3ÞRandthetransitionforthestockofdurablesgivenby(7.1).Themaxi-mizationgivesrisetotwofirst-orderconditions:000ucðc;DÞ¼bREy0;p0jy;pVAðA;D;yÞð7:4Þand000ucðc;DÞp¼bEy0;p0jy;pVDðA;D;yÞ:Inbothcasestheseconditionscanbeinterpretedasequatingthemarginalcostsofreducingeithernondurableordurableconsump-tioninthecurrentperiodwiththemarginalbenefitsofincreasingthe(respective)statevariablesinthenextperiod.Usingthefunctionalequation(7.2),wecansolveforthederiva-tivesofthevaluefunctionandthenupdatethesetwofirst-orderconditions.Thisimpliesthat00ucðc;DÞ¼bREy0;p0jy;pucðc;DÞð7:5Þand00000pucðc;DÞ¼bEy0;p0jy;p½uDðc;DÞþpð1
dÞucðc;DÞ:ð7:6ÞThefirstconditionshouldbefamiliarfromtheoptimalconsump-tionproblemwithoutdurables.Themarginalgainofincreasingconsumptionisoffsetbythereductioninwealthandthusconsump-tioninthefollowingperiod.Inthisspecificationthemarginalutilityofnondurableconsumptionmaydependonthelevelofdurables.So,totheextentthereisaninteractionwithintheutilityfunctionbetweennondurableanddurablegoods,empiricalworkthatlookssolelyatnondurableconsumptionmaybeinappropriate.5Thesecondfirst-orderconditioncomparesthebenefitsofbuyingdurableswiththemarginalcosts.Thebenefitsofadurableexpendi-turecomesfromtwosources.First,increasingthestockofdurableshasdirectutilitybenefitsinthesubsequentperiod.Second,astheEulerequationcharacterizesaone-perioddeviationfromaproposedsolution,theundepreciatedpartoftheadditionalstockissoldandconsumed.Thisisreflectedbythesecondtermontheright-hand5.Thatis,movementinthemarginalutilityofconsumptionofnondurablesmaybetheconsequenceofvariationsinthestockofdurables.Wereturntothispointinthediscussionofempiricalevidence.\n168Chapter7side.Themarginalcostofthedurablepurchaseisthereductioninexpendituresonnondurablesthattheagentmustincur.Aslightvariationintheproblemassumesthatdurablespurchasedinthecurrentperiodprovideservicesstartinginthatperiod.Sincethisformulationisalsofoundintheliterature,wepresentithereaswell.Inthiscasethedynamicprogrammingproblemis00000VðA;D;y;pÞ¼maxuðc;DÞþbEy0;p0jy;pVðA;D;y;pÞD0;A0forallðA;D;y;pÞ;ð7:7Þwithcdefinedasin(7.3).Manipulationoftheconditionsforoptimalityimplies(7.5)and000000pucðc;DÞ¼½uDðc;DÞþbEy0;p0jy;ppð1
dÞucðc;DÞ:ð7:8ÞForconstantpricesðp¼p0Þ,theresultis0000uDðc;DÞ¼bREy0jyuDðc;DÞ:Theoptimalconditioncorrespondstoavariationwherethestockofdurablesisreducedbyeinthecurrentperiod,theresourcesaresavedandthenusedtopurchasedurablesinthenextperiod.6Asinnondurableconsumption,inthespecialcaseofbR¼1,themarginalutilityfromdurablesfollowsarandomwalk.Notethatregardlessofthetimingassumption,therearesomeconnectionsbetweenthetwoEulerequations,particularlyifutilityisnotseparablebetweendurablesandnondurablesðucD00Þ.Alsoshockstoincomewillinfluencebothdurableandnondurableexpenditures.7.2.2EstimationofaQuadraticUtilitySpecificationMankiw(1982)studiedthepatternofdurableexpenditureswhenuðc;D0Þisseparableandquadratic.InthiscaseMankiwfindsthatdurableexpendituresfollowsanARMAð1;1Þprocessgivenbyetþ1¼a0þa1etþetþ1
ð1
dÞet;wherea1¼bR.HeretheMApieceisparameterizedbytherateofdepreciation.InestimatingthemodelusingU.S.data,Mankiwfinds6.Thisconditiondoesnotobtainundertheprevioustimingbecauseofthetimetobuildaspectofdurablesassumedthere.\nDurableConsumption169Table7.1ARMAð1;1ÞestimatesonU.S.andFrenchdataNotrendLineartrendSpecificationa1da1dU.S.durableexpenditures1.00(0.03)1.5(0.15)0.76(0.12)1.42(0.17)U.S.carregistration0.36(0.29)1.34(0.30)0.33(0.30)1.35(0.31)Francedurableexpenditures0.98(0.04)1.20(0.2)0.56(0.24)1.2(0.36)Francecarexpenditures0.97(0.06)1.3(0.2)0.49(0.28)1.20(0.32)Francecarregistrations0.85(0.13)1.00(0.26)0.41(0.4)1.20(0.41)Notes:Annualdata.FortheUnitedStates:FREDdatabase,1959:1–1997:3.ForFrance:INSEE,1970:1–1997:2.U.S.registration:1968–1995.thatempirically,disquitecloseto1.Sodurablesmaynotbesodurableafterall.AddaandCooper(2000b)studytherobustnessofMankiw’sresultsacrossdifferenttimeperiods,fordifferentfrequencies,andacrosscountries(UnitedStatesandFrance).Theirresultsaresum-marizedintable7.1.Inthetabletherowspertaintobothaggregateddurableexpendi-turesandestimatesbasedoncars(forFrance,bothtotalexpendi-turesoncarsandnewcarregistrations).Themodelisestimatedwithandwithoutalineartrend.Therateofdepreciationiscloseto100percenteachyearformostofthespecifications.Mankiw’s‘‘puzzle’’turnsouttoberobustacrosscategoriesofdurables,countries,timeperiods,andthemethodofdetrending.OverthepastfewyearstherewasconsiderableeffortmadeintheliteraturetounderstandMankiw’sresult.Oneinterestingapproachwastoembellishthebasicrepresentationagentmodelbytheaddi-tionofadjustmentcostsandshocksotherthanvariationsinincome.Another,comingfromBar-IlanandBlinder(1992)andBertolaandCaballero(1990),wastorecognizethatatthehouseholdleveldura-bleexpendituresareoftendiscrete.Wedescribethesetwolinesofresearchinturnbelow.7.2.3QuadraticAdjustmentCostsBernanke(1985)goesbeyondthequadraticutilityformulationbyaddingpricevariationsandcostsofadjustment.Asheobserves,itisworthwhiletolookjointlyatthebehaviorofdurableandnon-\n170Chapter7durableexpendituresaswell.7Considerthedynamicoptimizationproblem00000VðA;D;y;pÞ¼maxuðc;D;DÞþbEy0jyVðA;D;y;pÞD0;A0forallðA;D;y;pÞ;ð7:9Þwherethefunctionalequationholdsforallvaluesofthestatevector.Bernankeassumesaquadraticutilityfunctionwithquadraticadjustmentcostsoftheform012a2d02uðc;D;DÞ¼
ðc
cÞ
ðD
DÞ
ðD
DÞ;222wherectisnondurableconsumptionandDtisthestockofdurables.Theadjustmentcostispartoftheutilityfunctionratherthanthebudgetconstraints,fortractabilityreasons.Becauseofthequadraticstructure,themodel(7.9)canbesolvedexplicitlyasa(nonlinear)functionoftheparameters.Currentnondurableconsumptionisafunctionoflaggednondurableconsumption,ofthecurrentandlaggedstockofdurables,andoftheinnovationtotheincomepro-cess.Durablescanbeexpressedasafunctionofthepaststockofdurablesandoftheinnovationtoincome.Thetwoequationswithanequationdescribingtheevolutionofincomeareestimatedjointlybynonlinearthree-stageleastsquares.Laggedmeasuresofprices,non-durableconsumption,durablestocksanddisposableincomecurrentincome,nondurableconsumptionandthestockofdurables,wereusedasinstrumentstocontrolforsimultaneityandmeasurementerror.Themodelisrejectedbecauseoverallitdoesnotsuitthedatawhentestingtheoveridentifyingrestrictions.Theestimationoftheadjustmentcostgivesconflictingresults(asdescribedinsomedetailbyBernanke1985).Thenonlinearfunctionofthisparameterimpliesasignificantadjustmentcost,whereastheparameteritselfisnotsta-tisticallydifferentfromzero.Bernanke(1984)teststhepermanenthypothesismodelatthemicrolevelbylookingatcarexpendituresforapanelofhouseholds.AlthoughBernankedoesnotrejectthemodelforthistypeofdata,theresultisatoddswithreal-lifeobservations(describedbelow)as7.SeealsoEichenbaumandHansen(1990).\nDurableConsumption171itpredictscontinuousadjustmentofthestockwhereascarexpendi-turesaretypicallylumpyattheindividuallevel.exercise7.1Writeaprogramtosolve(7.9).Givedecisionrulesbythehousehold.Usethesedecisionrulestocreateapaneldataset,allowinghouseholdstohavedifferentrealizationsofincome.ConsiderestimatingtheEulerequationsfromthehousehold’sopti-mizationproblem.Iftherearenonseparabilitiespresentinuðc;D;D0Þ,particularly,ucD00,thatwereignoredbytheresearcher,what‘‘incorrectinferences’’wouldbereached?7.3NonconvexAdjustmentCostsThemodelexploredintheprevioussectionisintendedtocapturethebehaviorofarepresentativeagent.Despiteitstheoreticalele-gancethemodeldoesnotmatchtwocharacteristicsofthedata.First,asnotedabove,Mankiw’sestimateofcloseto100percentdepre-ciationsuggestsrejectionofthemodel.Second,thereisevidenceatthehouseholdlevelthatadjustmentofthestockofdurablesisnotFigure7.1½s;Srule\n172Chapter7continuous.Rather,householdspurchasesofsomedurables,suchascarsasstudiedbyLam(1991),arerelativelyinfrequent.Thismayreflectirreversibilityduetoimperfectinformationaboutthequalityofuseddurablegood,thediscretenatureofsomedurablegoods,orthenatureofadjustmentcosts.Bar-IlanandBlinder(1992)andBar-IlanandBlinder(1988)presentasimplesettinginwhichafixedcostofadjustmentimpliesinactionfromtheagentwhenthestockofdurableisnottoofarfromtheoptimalone.Theyarguethattheoptimalconsumptionofdura-blesshouldfollowanðS;sÞpolicy.Whenthedurablestockdepreci-atestoalowervalues,theagentincreasesthestocktoatargetvalueSasdepictedinfigure7.1.7.3.1GeneralSettingTogainsomeinsightintotheimportanceofirreversibility,considerthefollowingformalizationofamodelinwhichirreversibilityisimportant.Bythiswemeanthatbecauseoffrictioninthemarketfordurables,householdsreceiveonlyafractionofthetruevalueofaproducttheywishtosell.ThismightbethoughtofasaversionofAkerlof’sfamouslemons’problem.8Inparticular,supposethatthepriceofdurablesisnormalizedto1whentheyarepurchasesðeÞbutthatthepriceofdurableswhentheyaresoldðsÞisgivenbyps<1.TheBellmanequationforthehouse-hold’soptimizationproblemisVðA;D;yÞ¼maxðVbðA;D;yÞ;VsðA;D;yÞ;ViðA;D;yÞÞ;ð7:10ÞwhereA0VbðA;D;yÞ¼maxuAþy

e;De;A0R00þbEy0jyVðA;Dð1
dÞþe;yÞ;ð7:11ÞA0VsðA;D;yÞ¼maxuAþy
þps;Dss;A0R00þbEy0jyVðA;Dð1
dÞ
s;yÞ;ð7:12Þ8.SeeHouseandLeahy(2000)foramodelofdurableswithanendogenouslemonspremium.\nDurableConsumption173A0ViðA;D;yÞ¼maxuAþy
;DA0R00þbEy0jyVðA;Dð1
dÞ;yÞforallðA;D;yÞ:ð7:13ÞThisisadmittedlyacomplexproblemasitincludeselementsofadiscretechoice(toadjustornot)andalsoanintensivemargin(givenadjustment,thelevelofdurablepurchases(sales)mustbedetermined).Thegapbetweenthebuyingandsellingpriceofdurableswillcreateinaction.Imagineahouseholdwithasubstantialstockofdurablesthatexperiencesanincomelossduetoalayoff.Intheabsenceofirreversibilityðps¼1Þ,thehouseholdmayoptimallyselloffsomedurables.Ifajobisfoundandtheincomeflowreturns,thenthestockofdurablescanberebuilt.However,inthepresenceofirreversibility,thesaleandsubsequentpurchaseofdurablesiscostlyduetothewedgebetweenthebuyingandsellingpriceofdurables.Thusinresponsetoanincomeshockthehouseholdmaybeinactiveandnotadjustitsstock.Thefunctionalequationin(7.10)cannotbesolvedbylinearizationtechniquesasthereisnosimpleEulerequationthattreatsthedis-cretechoicenatureoftheproblem.Forthatreasonvaluefunctioniterationtechniquesareneeded.Asinthedynamicdiscretechoiceproblemspecifiedinchapter3,wewouldstartwithinitialguessesofthevaluesofthethreeoptionsandtheninduceVðA;D;yÞthroughthemaxoperator.Fromtheseinitialsolutions,theiterationproce-durebegins.Asthereisalsoanintensivemargininthisproblem(givenadjustment,thestockofdurablesonecanchooseisacontin-uousvariable),astatespacefordurablesaswellasassetsmustbespecified.Thisisacomplexsettingbutonethatthevaluefunctioniterationapproachcanhandle.Sothepolicyfunctionscanbecreatedusingavectorofparametersthatdescribespreferencesandthestochasticprocesses.Inprinciple,inanestimationexercise,theseparameterscangeneratemomentsthatarematchedwithobservations.Thismethodisdescribedinsomedetail,foradifferentmodel,inthesubsequentsubsections.7.3.2IrreversibilityandDurablePurchasesGrossmanandLaroque(1990)developamodelofdurableconsump-tionandalsoconsideranoptimalportfoliochoice.Theyassumethat\n174Chapter7thedurablegoodisilliquidastheagentincursaproportionaltrans-actioncostwhensellingthegood.Theauthorsshowthatundertheassumptionofaconstantrelativeriskaversionutilityfunction,thestatevariableistheratioofwealthAoverthestockofdurablesD.Theoptimalbehavioroftheagentistofollowan½s;Srule,withatargetsA½s;S.TheagentdoesnotchangethestockofdurableiftheratioA=DiswithinthetwobandssandS.Iftheratiodriftsoutofthisinterval,theagentadjustsitbybuyingorsellingthegoodsuchthatA=D¼s.Eberly(1994)empiricallyinvestigatestherelevanceofsomeaspectsoftheGrossman-Laroquemodel.SheusesdatafromtheSurveyofConsumerFinanceswhichreportsinformationonassets,incomeandmajorpurchases.SheestimatesthebandssandS.ThesebandscanbecomputedbyobservingtheratioA=Dforindividualsjustbeforeanadjustmentismade.Thetargetscanbecomputedastheaverageratiojustafteradjustment.Eberly(1994)estimatesthebandwidthandinvestigatesitsdeterminants.Shefindsthattheyeartoyearincomevarianceandtheincomegrowthratearestrongpre-dictorsofthewidthoftheband.Attanasio(2000)developsamoreelaborateestimationstrategyforthesebands,allowingforunobservedheterogeneityattheindividuallevel.Thisheterogeneityisneededas,conditionalonhouseholdcharacteristicsandthevalueoftheratioofwealthtoconsumption,someareadjustingtheirstockandsomearenot.TheestimationisdonebymaximumlikelihoodondatadrawnfromtheConsumerExpenditureSurvey.Thewidthofthebandsarefunctionsofhouse-holdcharacteristicssuchasageandrace.Theestimatedmodelisthenaggregatedtostudytheaggregatedemandfordurables.Caballero(1993)usestheGrossmanandLaroque(1990)approachtoinvestigatetheaggregatebehaviorofdurablegoods.Theindivid-ualagentisassumedtofollowan½s;Sconsumptionrulebecauseoftransactioncosts.Intheabsenceoftransactioncosts,theagentwouldfollowaPIHtypebehaviorasdescribedinsection7.2.Caballeropostulatesthattheoptimalbehavioroftheagentcanbedescribedbythedistancebetweenthestockofdurablesheldbytheagentandthe‘‘target’’definedastheoptimalstockinthePIHmodel.Theagentadjuststhestockwhenthegapbetweenthereal-izedandthedesiredstockisbigenough.Inthissettingthestatevariablesarethestockofdurablesandthetarget.Thetargetstockisassumedtofollowaknownstochasticprocess.Henceinthismodel\nDurableConsumption175itisassumedthattheevolutionofthetargetisasufficientstatistictoinformofalltherelevanteconomicvariablessuchaspricesorincome.Theaggregatedemandfordurablesisthesumofallagentswhodecidetoadjusttheirstockinagivenperiod.HenceCaballerostressestheimportanceofthecrosssectionaldistributionofthegapbetweenthetargetandtherealizedstock.Whenthereisanaggre-gateshockonthetarget,theaggregateresponsedependsnotonlyonthesizeoftheshockbutalsoonthenumberofindividualsclosetotheadjustmentline.Theaggregatedemandfordurablescanthere-foredisplaycomplicateddynamicpatterns.ThemodelisestimatedonaggregateU.S.data.7.3.3ADynamicDiscreteChoiceModelSupposethatinsteadofirreversibility,thereisarestrictionthathouseholdscanhaveeithernocaroronecar.9Thus,byassumption,thehouseholdsolvesadynamicdiscretechoiceproblem.Wediscusssolutionsofthatproblem,estimationofparametersandaggregateimplicationsinthissection.10OptimalBehaviorWestartwiththedynamicprogrammingproblemasspecifiedinAddaandCooper(2000b).Atthestartofaperiodthehouseholdhasacarofaparticularage,alevelofincomeandarealizationofatasteshock.Formally,thehousehold’sstateisdescribedbytheageofitscar,i,avectorZ¼ðp;Y;eÞofaggregatevariables,andavectorz¼ðyÞofidiosyncraticvariables.Herepistherelativepriceofthe(new)durablegood.CurrentincomeisgivenbythesumYþy,whereYrepresentsaggregateincomeandyrepresentsidiosyncraticshockstonondurableconsumptionthatcouldreflectvariationsinhouseholdincomeorrequiredexpendituresoncarmaintenanceandothernecessities.11Thefinalelementinthestatevectorisatasteshock,e.9.Theassumptionthatonecaristhemaxisjustforconvenience.Whatisimportantisthatthecarchoicesetisnotcontinuous.10.ThispresentationreliesheavilyonAddaandCooper(2000b).11.AddaandCooper(2000b)explicitlyviewthisasahouseholdspecificincomeshock,butabroaderinterpretationisacceptable,particularlyinlightoftheiidas-sumptionassociatedwiththissourceofvariation.\n176Chapter7Ateverypointintimethehouseholddecideswhethertoretainacarofagei,tradeit,orscrapit.Ifthehouseholddecidestoscrapthecar,thenitreceivesthescrapvalueofpandhastheoptiontopur-chaseanewcar.Ifthehouseholdretainsthecar,thenitreceivestheflowofservicesfromthatcarandcannot,byassumption,purchaseanothercar.Thusthehouseholdisconstrainedtoownatmostasinglecar.Formally,letViðz;ZÞrepresentthevalueofhavingacarofageitoahouseholdinstateðz;ZÞ.FurtherletVkðz;ZÞandVrðz;ZÞrepresentiithevaluesfromkeepingandreplacinganageicarinstateðz;ZÞ.ThenVðz;ZÞ¼max½Vkðz;ZÞ;Vrðz;ZÞ;iiiwhereVkðz;ZÞ¼uðs;yþY;eÞþbð1
dÞEVðz0;Z0Þiiiþ1þbdfEVðz0;Z0Þ
uðs;y0þY0;e0Þ11þuðs;y0þY0
p0þp;e0Þgð7:14Þ1andVrðz;ZÞ¼uðs;yþY
pþp;eÞþbð1
dÞEVðz0;Z0Þi12þbdfEVðz0;Z0Þ
uðs;y0þY0;e0Þ11þuðs;y0þY0
p0þp;e0Þg:1InthedefinitionofVkðz;ZÞ,thecarisassumedtobedestroyedi(fromaccidentsandbreakdowns)withprobabilitydleadingtheagenttopurchaseanewcarinthenextperiod.Thecostofanewcarinnume´rairetermsisp0
p,whichisstochasticsincethepriceofanewcarinthenextperiodisrandom.Further,sinceitisassumedthatthereisnoborrowingandlending,theutilitycostofthenewcarisgivenbyuðs;y0þY0;e0Þ
uðs;y0þY0
p0þp;e0Þwhichexceeds11p0
paslongasuð
Þisstrictlyconcaveinnondurableconsumption.Itispreciselyatthispointthattheborrowingrestrictionappearsasanadditionaltransactionscost.Addingineitherborrowingandlendingorthepurchaseandsaleofusedcarspresentsnomodelingdifficulties.Butaddinginwealthaswellasresalepricesasstatevariablescertainlyincreasesthedimensionalityoftheproblem.Thisremainsasworkinprogress.\nDurableConsumption177exercise7.2Reformulate(7.14)toallowthehouseholdtoborrow/lendandalsotoresellcarsinausedcarmarket.Whatadditionalstatevariableswouldyouhavetoaddwhenthesechoicesarein-cluded?Whatarethenewnecessaryconditionsforoptimalbehaviorofthehousehold?FurtherSpecificationFortheapplicationtheutilityfunctionisdefinedtobeadditivelyseparablebetweendurablesandnondurables:"#1
x
geðc=lÞuðsi;cÞ¼iþ;1
xwherecistheconsumptionofnondurablegoods,gisthecurvaturefortheserviceflowofcarownership,xthecurvatureforcon-sumption,andlisascalefactor.InthisspecificationthetasteshockðeÞinfluencesthecontemporaneousmarginalrateofsubstitutionbetweencarservicesandnondurables.Inorderfortheagent’soptimizationproblemtobesolved,asto-chasticprocessforincome,prices,andtheaggregatetasteshocksmustbespecified.Aggregateincome,prices,andtheunobservedpreferenceshockareassumedtofollowaVAR(1)processgivenby12Yt¼mYþrYYYt
1þrYppt
1þuYt;pt¼mpþrpYYt
1þrpppt
1þupt;et¼meþreYYt
1þreppt
1þuet:Thecovariancematrixoftheinnovationsu¼fuYt;upt;uetgis23oYoYp0W¼4opYop05:00oeAstheaggregatetasteshockisunobserved,weimposeablockdiagonalstructureontheVAR,whichenablesustoidentifyalltheparametersinvolvingpricesandaggregateincomeinasimplefirst-stepregression.Thisconsiderablyreducesthenumberofparameters12.Hereonlyasinglelagisassumedtoeconomizeonthestatespaceoftheagents’problem.\n178Chapter7tobeestimatedinthestructuralmodel.Weallowpricesandincometodependonlaggedincomeandlaggedprices.13Theaggregatetasteshockpotentiallydependsonlaggedpricesandincome.Thecoefficientsofthisprocessalongwithoeareesti-matedwithinthestructuralmodel.Byallowingapositivecorrela-tionbetweentheaggregatetasteshockandlaggedprices,giventhatpricesareseriallycorrelated,wecanreconcilethemodelwiththefactthatsalesandpricesarepositivelycorrelatedinthedata.Thisallowsustobettercapturesomeadditionaldynamicsofsalesandpricesinthestructuralestimation.Analternativewaywouldbetomodeljointlytheproducerandconsumersideoftheeconomy,togetanupward-sloppingsupplycurve.However,solvingfortheequi-libriumiscomputationallyverydemanding.SolvingtheModelThemodelissolvedbythevaluefunctioniterationmethod.StartingwithaninitialguessforViðz;ZÞ,thevaluefunctionisupdatedbybackwarditerationsuntilconvergence.Thepolicyfunctionsthataregeneratedfromthisoptimizationproblemareofanoptimalstoppingvariety.Thatis,giventhestateofthehousehold,thecarisscrappedandreplacedifandonlyifthecarisolderthanacriticalage.Lettinghkðzt;Zt;yÞrepresenttheprobabilitythatacarofagekisscrapped,thepolicyfunctionsimplythathkðzt;Zt;yÞ¼difk1ð7:16Þandftþ1ð1;Zt;yÞ¼StðZt;yÞ:Thus,foragivenyandagivendrawofTaggregateshocks,onecansimulatebothsalesandthecross-sectionaldistribution.ThiscanberepeatedNtimestoproduceNsimulateddatasetsoflengthT,whichcanbeusedintheestimation.DefineStnðZt;yÞ¼Stðpt;Yt;ent;yÞasthepredictedaggregatesalesgivenprices,aggregateincomeandPNunobservedtasteshockent.DefineStðZt;yÞ¼1=Nn¼1SntðZt;yÞastheaverageaggregatesalesconditionalonprices,aggregateincomeandperiodt
1cross-sectionaldistribution.EstimationMethodandResultsIntotalthereareeightparameterstoestimate:y¼fg;d;l;z;sy;reY;rec;oeg.TheestimationmethodfollowsAddaandCooper(2000b)andisamixbetweensimulatednon-linearleastsquaresandsimu-latedmethodofmoments.Thefirstpartofthecriterionmatchespredictedsalesofnewcarswiththeobservedones,conditionalonpricesandaggregateincome.Thesecondpartofthecriterionmatchesthepredictedshapeofthecross-sectionaldistributionofcarvintagestotheobservedone.Theobjectivefunctiontominimizeiswrittenasthesumofthetwocriteria:\n180Chapter7LðyÞ¼aL1ðyÞþL2ðyÞ;NNNwhereNisthenumberofsimulateddrawsfortheunobservedaggregatetasteshockent.Thetwocriteriaaredefinedby"#1XT1XN122LNðyÞ¼ðSt
StðyÞÞ
ðStnðyÞ
StðyÞÞ;TNðN
1Þt¼1n¼1XL2ðyÞ¼aðFi
FiðyÞÞ2;Nii¼f5;10;15;AR;MAgwhereSðyÞistheaverageFi,i¼5;10;15istheaveragefractionoftcarsofageiacrossallperiods,andFi,i¼AR;MAaretheautore-gressiveandmovingaveragecoefficientsfromanARMAð1;1Þesti-matedonaggregatesales.Theestimationusestwocriteriaforidentificationreasons.Match-ingaggregatesalesateachperiodextractsinformationontheeffectofpricesandincomeonbehaviorandhelpsidentifytheparameteroftheutilityfunctionaswellastheparametersdescribingthedistri-butionoftheaggregatetasteshock.However,themodelisabletomatchaggregatesalesunderdifferentvaluesfortheagent’soptimalstoppingtime.Inotherwords,therecanbedifferentcross-sectionaldistributionsthatproduceaggregatedsalesclosetotheobservedones.Inparticular,theparameterdispoorlyidentifiedbyusingonlythefirstcriterion.Thesecondcriterionpinsdowntheshapeofthecross-sectionaldistributionofcarvintages.ThedatacomefromFranceandtheUnitedStatesandincludebesidesthecross-sectionaldistributionofcarvintagesovertime,theaggregatesalesofnewcars,prices,andaggregateincome.Theesti-matedaggregatehazardfunctionsHtðZÞovertheperiod1972to1995forFranceand1981to1995fortheUnitedStatesareshowninfigures7.2and7.3.Notethattheprobabilityofreplacementforyoungcars,whichisequaltothed,isestimatedatalowvaluebetween5to10percent.Hence,incontrastwiththeestimatedPIHmodelsdescribedinsection7.2,themodelisabletoproduceasen-sibleestimateoftherateofdepreciation.Moreover,inestimatinganARMAð1;1Þ,asinsection7.2.2,onthepredictedaggregatesales,wefindthattheMAcoefficientisclosetozeroasintheobserveddata.Hence,fromaPIHperspective,themodelappearstosupporta100percentdepreciationrateattheaggregatelevel,butnotatthemicrolevelwherethedepreciationrateislow.\nDurableConsumption181Figure7.2Estimatedhazardfunction,FranceFigure7.3Estimatedhazardfunction,UnitedStates\n182Chapter7Oncethemodelisestimated,AddaandCooper(2000b)investigatetheabilityofthemodeltoreproduceanumberofotherfeaturessuchastheimpulseresponseofsalestoanincreaseinprices.Theyalsousetheestimatedmodeltodecomposethesourceofvariationinaggregatesales.Withinthemodeltherearetwomainsources,theendogenousevolutionofthecross-sectionaldistributionandtheeffectofaggregatevariablessuchaspricesorincome.Caballero(1993)seemstoimplythattheevolutionofthecross-sectionaldistri-butionisanimportantdeterminant.However,theempiricaldecom-positionshowsthatitsroleisrelativelyminor,comparedwiththeeffectofincomeandprices.TheImpactofScrappingSubsidiesAddaandCooper(2000a)usesthesameframeworktoanalyzetheimpactofscrappingsubsidiesintroducedfirstinFranceandlaterinanumberofEuropeancountriessuchasSpainorItaly.FromFebruary1994toJune1995theFrenchgovernmentofferedindividuals5,000francs(approximately5to10percentofthevalueofanewcar)forthescrappingofanoldcar(tenyearsorolder)andFigure7.4Salesofnewcars,inthousands,monthly\nDurableConsumption183thepurchaseofanewcar.Salesofnewcarswhichhadbeenlowintheprecedingperiod(seefigure7.4)increasedmarkedlyduringtheperiodthepolicywasinplace.InSeptember1995toSeptember1996,thegovernmentre-introducedthepolicy,withanagelimitofeightyears.AfterSeptember1996,thedemandfornewcarscollapsedatarecordlowlevel.Asisevidentinfigure7.4,thedemandforcarsiscyclicalandfol-lowsthebusinesscycle.Theincreaseddemandfornewcarsduringtheperiod1994to1996couldbedueeithertothepolicyortothecyclicalnatureofdemand.Ifthelatteristrue,theFrenchgovern-mentwaswastingmoneyoncarownerswhowouldhavereplacedtheircarsduringthatperiodanyway.Eveniftheincreaseddemandwasentirelyfueledbythescrappingsubsidies,thegovernmentwaspayingoutmoneytocarownerswhowouldhavereplacedtheircarintheperiodsaheadanyway.Theeffectofthepolicywasthentoanticipatenewsalesandcreatefuture,potentiallybiggercyclesofcardemand.Asahugenumberofnewcarsweresoldinthatperiod,demandfornewcarsdroppedwhenthepolicywasended.How-ever,apeakindemandisexpectedtoappearinabouttenyearsafterthecarsboughtin1995and1996arescrapped.AddaandCooper(2000a)estimatethemodelinsection7.3.3fortheprepolicyperiod.Thescrappingpricepwasaconstantlowvalue(around500Frenchfrancs)before1993.Withthepolicyinplace,thescrappingpriceincreasedandwasagespecific:pðiÞ¼500ifi<10;pðiÞ¼5;000ifib10:Intheestimatedmodeltheeffectofthepolicycanbesimulatedaswellasthecounterfactualwithoutthepolicyinplace.ThisisdoneStatetomorrow123410.010.010.010.9720.010.010.010.97Statetoday30.2250.2250.10.4540.010.010.010.97Figure7.5Transitionmatrixforp\n184Chapter7Figure7.6Expectedaggregatesales,relativetobaselineFigure7.7Expectedgovernmentrevenue,relativetobaseline\nDurableConsumption185conditionalonthecross-sectionaldistributionofcarsatthebegin-ningoftheperiodandconditionalontherealizedincomeandprices,asthepricesofnewcarsareassumedtobeindependentofthepolicy.(Thiseffectisdebatable,however,forempiricalevi-dencesuggeststhatpricesremainedstablethroughouttheperiodmainlybecausethegovernmentnegotiatedastablepricewithcarproducers.)Whilethefirstscrappingsubsidywaslargelyunexpectedbycon-sumers,thesecondwassomewhatanticipated,sinceafterthefirstsubsidy,therewasdiscussiononwhethertoimplementanothersuchsubsidy.ThisistakenintoaccountinthemodelbyaddingthescrappingpricepðiÞasastochasticstatevariable.Moreprecisely,pisassumedtofollowafirstorderMarkovprocesswithfourstates.Thesefourstatesareshowninfigure7.5.Thefirststatemodelsthe1994reformandthesecondonethe1995reform.State3isastatewithheighteneduncertainty,inwhichtherearenosubsidies.State4isthebaselinestate.Instate1,thescrapvalueissetat5,500Fforcarsolderthantenyears.Thisstateisnotassumedtobepermanent:thereisonlya1percentchancethatthesubsidywillbeineffectinthenextperiod,conditionalonbeinginforceinthecurrentperiod.Instate2,thescrapvalueisalso5,500Fbutforcarsolderthaneightyearsold.Figures7.6and7.7showthepredictedsalesandgovernmentrev-enuerelativetothebaseline.Themodelcapturesthepeakinsalesduringthetwopolicies,aswellasthedeclineinbetweenduetotheuncertainty.Thesalesarelowerforabouttenyears,withlittleevi-denceofasubsequentpeak.Thisresultisinlinewiththatdiscussedinsection7.3.3whereitwasfoundthatovertimethecross-sectionaldistributionhadlittleeffectonaggregatesales.Governmentrevenuesareloweroverthewholeperiod.Thegov-ernmentrevenueisformedbythevalue-addedtaxesfromthepur-chaseofnewcarsminusthepaid-outscrappingsubsidies.Fromtheperspectiveofgovernmentrevenues,thepolicyisclearlyundesirable.Thesubsidiesaccountedforabout8to10percentoftheincreaseddemandinsales.\n8Investment8.1OverviewandMotivationThischapterstudiescapitalaccumulation.Investmentexpendituresareoneofthemostvolatileelementsoftheaggregateeconomy.Fromtheperspectiveofpolicyintervention,investmentisalsoakeyissue.Thedependenceofinvestmentonrealinterestratesiscriticaltomanydiscussionsoftheimpactofmonetarypolicy.Furthermanyfiscalpolicyinstruments,suchasinvestmenttaxcreditsandaccel-erateddepreciationallowances,actdirectlythroughtheirinfluenceoncapitalaccumulation.Itshouldseemthenthatmacroeconomicswouldhavedevelopedandevaluatednumerousmodelstomeetthischallenge.Yet,relativetotheenormousworkdoneonconsumption,researchoninvestmentlagsbehind.AsnotedinCaballero(1999),thishaschangeddramat-icallyinthelast10orsoyears.1Partly,wenowhavetheabilitytocharacterizeinvestmentbehaviorinfairlyrichsettings.Combinedwithplant-leveldatasets,researchersareabletoconfrontarichsetofobservationswiththesesophisticatedmodels.Investment,withitsemphasisonuncertaintyandnonconvexitiesisaripeareaforapplicationsofdynamicprogrammingtechniques.Inthischapterwefirstanalyzeageneraldynamicoptimizationproblemandthenfocusonspecialcasesofconvexandnonconvexadjustmentcosts.Thissetsthestagefortheempiricalanalyzesthatfollow.Wealsodiscusstheuseoftheseestimatesfortheanalysisofpolicyinterventions.1.Therearenumeroussurveysofinvestment.SeeCaballero(1999)andChirinko(1993),andthereferencestherein,forfurthersummariesofexistingresearch.\n188Chapter88.2GeneralProblemTheunitofanalysiswillbetheplantthoughforsomeapplications(e.g.,considerationofborrowingconstraints)focusingonthefirmmaybemoreappropriate.The‘‘manager’’isassumedtomaximizethevalueoftheplant:therearenoincentiveproblemsbetweenthemanagerandtheowners.Theprobleminvolvesthechoiceoffactorsofproductionthatarerentedfortheproductionperiod,thehiringoflaborandtheaccumulationofcapital.Tofocusontheinvestmentdecision,weassumethatdemandforthevariableinputs(denotedbyx)isoptimallydeterminedgivenfactorprices(representedbythevectorw)andthestatevariablesoftheplant’soptimizationproblem,representedbyðA;KÞ.Herethevectorofflexiblefactorsofproduc-tionmightincludelabor,materials,andenergyinputsintothepro-ductionprocess.Theresultofthisoptimizationleavesaprofitfunction,denotedbyPðA;KÞ,thatdependssolelyonthestateoftheplant,wherePðA;KÞ¼maxRðAA^;K;xÞ
wx:xHereRðAA^;K;xÞdenotesrevenuesgiventheinputsofcapital(K),thevariablefactors(x),andashocktorevenuesand/orproductivity,denotedbyAA^.ThereducedformprofitfunctionthusdependsonthestochasticvariableAthatencompassesbothAA^andw,andthestockofphysicalcapital(K).ThusweoftenrefertoAasaprofit-abilityshocksinceitreflectsvariationsintechnology,demandandfactorprices.Takingthisprofitfunctionasgiven,weconsidervariationsofthefollowingstationarydynamicprogrammingproblem:VðA;K;pÞ¼maxPðA;KÞ
CðK0;A;KÞ
pðK0
ð1
dÞKÞK0000þbEA0;p0jA;pVðA;K;pÞforallðA;K;pÞ;ð8:1ÞwhereK0¼Kð1
dÞþIisthecapitalaccumulationequationandIisinvestment.Hereunprimedvariablesarecurrentvaluesandprimedvariablesrefertofuturevalues.InthisproblemthemanagerchoosesthelevelofthefuturecapitalstockdenotedK0.Thetimingassump-tionisthatnewinvestmentbecomesproductivewithaone-period\nInvestment189lag.TherateofdepreciationofthecapitalstockisdenotedbydA½0;1.Themanagerdiscountsthefutureatafixedrateofb.2exercise8.1Supposethatincontrastto(8.1),investmentinperiodtisproductiveinthatperiod.Comparethesetwoformulationsoftheinvestmentproblem.Assumingthatallfunctionsaredifferentiable,createEulerequationsforeachspecification.Explainanydifferences.exercise8.2Howwouldyoumodify(8.1)toallowthemanager’sdiscountfactortobeinfluencedbyvariationsintherealinterestrate?Therearenoborrowingrestrictionsinthisframework.Sothechoiceofinvestmentandthusfuturecapitalisnotconstrainedbycurrentprofitsorretainedearnings.Wereturntothisissuelaterinthechapterwhenwediscusstheimplicationsofcapitalmarketimperfections.Therearetwocostsofobtainingnewcapital.Thefirstisthedirectpurchaseprice,denotedbyp.Noticethatthispriceispartofthestatevectorasitisasourceofvariationinthiseconomy.3Second,therearecostsofadjustmentgivenbythefunctionCðK0;A;KÞ.Thesecostsareassumedtobeinternaltotheplantandmightincludeinstallationcosts,disruptionofproductiveactivitiesintheplant,theneedtoretrainworkers,theneedtoreconfigureotheraspectsoftheproductionprocess,andsoon.Thisfunctionisgeneralenoughtohavecomponentsofbothconvexandnonconvexcostsofadjustmentaswellasavarietyoftransactionscosts.8.3NoAdjustmentCostsTomakeclearthecontributionofadjustmentcosts,itisusefultostartwithabenchmarkcaseinwhichthesecostsareabsent:CðK0;A;KÞ10forallðK0;A;KÞ.Note,though,thatthereisstillatimetobuildaspectofinvestmentsothatcapitalaccumulationremainsforwardlooking.Thefirst-orderconditionfortheoptimalinvest-mentpolicyisgivenby2.Thisiscorrespondstotheoutcomeofastochasticgrowthmodelifthereareriskneutralconsumers.Otherwise,aformulationwithvariablerealinterestratesmaybewarranted.3.Inmanyeconomiesitisalsoinfluencedbypolicyvariationsintheformofinvest-menttaxcredits.\n190Chapter8000bEA0;p0jA;pVkðA;K;pÞ¼p;ð8:2Þwheresubscriptsonthefunctionsdenotepartialderivatives.Thisconditionimpliesthattheoptimalcapitalstockdependsonthereal-izedvalueofprofitability,A,onlythroughanexpectationsmecha-nism:giventhetimetobuild,currentprofitabilityisnotrelevantforinvestmentexceptasasignaloffutureprofitability.Furthertheoptimalcapitalstockdoesnotdependonthecurrentstockofcapital.000Using(8.1)tosolveforEðA0;p0jA;pÞVkðA;K;pÞyields000bEðA0;p0jA;pÞ½PkðA;KÞþð1
dÞp¼p:ð8:3ÞThisconditionhasanaturalinterpretation.ThecostofanadditionalunitofcapitaltodayðpÞisequatedtothemarginalreturnoncapital.Thismarginalreturnhastwopieces:themarginalprofitsfromthecapital,PðA0;K0Þ,andtheresalevalueofundepreciatedcapitalatkthefutureprice,ð1
dÞp0.Substitutingforthefuturepriceofcapitalanditeratingforward,wefindthatXytpt¼b½bð1
dÞEAtþtjAtPKðKtþtþ1;Atþtþ1Þ;t¼0whereptisthepriceofcapitalinperiodt.Sothefirm’sinvestmentpolicyequatesthepurchasepriceofcapitaltodaywiththedis-countedpresentvalueofmarginalprofitsinthefuture.Notethatinstatingthiscondition,weareassumingthatthefirmwillbeopti-mallyresettingitscapitalstockinthefuturesothat(8.3)holdsinallsubsequentperiods.Whilesimple,themodelwithoutadjustmentcostsdoesnotfitthedatawell.CooperandHaltiwanger(2000)arguethatrelativetoobservations,thismodelwithoutadjustmentcostsimpliesexces-sivesensitivityofinvestmenttovariationsinprofitability.Sooneoftheempiricalmotivationsfortheintroductionofadjustmentcostsistotempertheotherwiseexcessivelyvolatilemovementsininvest-ment.Furtherthismodelisunabletomatchtheobservationofinac-tionincapitaladjustmentseen(anddiscussedbelow)inplant-leveldata.Forthesereasonsvariousmodelsofadjustmentcostsareconsidered.44.Moreoverthespecialcaseofnoadjustmentcostsisgenerallynestedintheseothermodels.\nInvestment1918.4ConvexAdjustmentCostsInthissectionweassumethatCðK0;A;KÞisastrictlyincreas-ing,strictlyconvexfunctionoffuturecapital,K0.5Thefirmchoosestomorrow’scapitalðK0Þusingitsconditionalexpectationsoffutureprofitability,A0.Ofcourse,totheextentthatA0iscorrelatedwithA,currentprofitswillbecorrelatedwithfutureprofits.AssumingthatVðK;A;pÞexists,anoptimalpolicy,obtainedbysolvingthemaximizationproblemin(8.1),mustsatisfy0000CK0ðK;A;KÞþp¼bEðA0;p0jA;pÞVK0ðA;K;pÞ:ð8:4ÞTheleftsideofthisconditionisameasureofthemarginalcostofcapitalaccumulationandincludesthedirectcostofnewcapitalaswellasthemarginaladjustmentcost.Therightsideofthisexpres-sionmeasurestheexpectedmarginalgainsofmorecapitalthroughthederivativeofthevaluefunction.Thisisconventionallytermed‘‘marginalQ’’anddenotedbyq.Notethetiming:theappropriatemeasureofmarginalQistheexpecteddiscountedvalueforthefol-lowingperiodduetotheone-periodinvestmentdelay.000Using(8.1)tosolveforEðA0;p0jA;pÞVK0ðA;K;pÞ,wecansimplify(8.4)toaEulerequation:0CK0ðK;A;KÞþp0000000¼bEðA0;p0jA;pÞfPKðK;AÞþpð1
dÞ
CK0ðK;A;KÞg:ð8:5ÞTointerpretthisnecessaryconditionforanoptimalsolution,con-siderincreasingcurrentinvestmentbyasmallamount.Thecostofthisinvestmentismeasuredontheleftsideofthisexpression:thereisthedirectcostofthecapitalðpÞaswellasthemarginaladjustmentcost.Thegaincomesinthefollowingperiod.Theadditionalcapitalincreasesprofits.Further,asthemanager‘‘returns’’totheoptimalpathfollowingthisdeviation,theundepreciatedcapitalisvaluedatthefuturemarketpricep0andadjustmentcostsarereduced.exercise8.3Supposethattheproblemhadbeenwritten,moreconventionally,withthechoiceofinvestmentratherthanthefuturecapitalstock.DeriveandanalyzetheresultingEulerequation.5.InsomeapplicationsthecostofadjustmentfunctiondependsoninvestmentandiswrittenCðI;KÞwhereI¼K0
ð1
dÞK.\n192Chapter88.4.1QTheory:ModelsOneofthedifficultaspectsofinvestmenttheorywithadjustmentcostsisempiricalimplementation.Asthevaluefunctionandhenceitsderivativeisnotobservable,(8.4)cannotbedirectlyestimated.ThusthetheoryistestedeitherbyfindingasuitableproxyforthederivativeofVðA;K;pÞorbyestimatingtheEulerequation,(8.5).Wefocushereonthedevelopmentofatheorythatfacilitatesestimationbasedonusingtheaveragevalueofthefirmasasubstituteforthemarginalvalueofanadditionalunitofcapital.Thisapproach,calledQtheory,imposesadditionalstructureon(8.1).Inparticular,followingHayashi(1982),weassumethatPðK;AÞisproportionaltoKandthatthecostofadjustmentfunctionisquadratic.6Furtherweassumethatthepriceofcapitalisconstant.Thereforewehave02gK
ð1
dÞKVðA;KÞ¼maxAK
KK02K000
pðK
ð1
dÞKÞþbEA0jAVðA;KÞ:ð8:6ÞAsalways,Bellman’sequationmustbetrueforallðA;KÞ.Supposethattheshocktoprofitability,A,followsanautoregressiveprocessgivenbyA0¼rAþe0;wherejrj<1ande0iswhitenoise.Thefirst-orderconditionforthechoiceoftheinvestmentlevelimpliesthattheinvestmentrateinði1I=KÞisgivenby100i¼ðbEA0jAVKðA;KÞ
pÞ:ð8:7Þg00HereEA0jAVKðA;KÞisagaintheexpectedvalueofthederivativeofthevaluefunction,atermwecalled‘‘marginalQ.’’Tosolvethisdynamicprogrammingproblem,wecanguessatasolutionandverifythatitworks.Giventhelinear-quadraticstructureoftheproblem,itisnaturaltoguessthatVðA;KÞ¼fðAÞK;6.AbelandEberly(1994)containfurtherdiscussionoftheapplicabilityofQtheoryformoregeneraladjustmentcostandprofitfunctions.\nInvestment193wherefðAÞissomeunknownfunction.ThisguessallowsustowritetheexpectedmarginalQasafunctionofA:000EA0jAVKðA;KÞ¼EA0jAfðAÞ1ff~ðAÞ:NotethatinthiscasetheexpectedvaluesofmarginalandaverageQ(definedasVðA;KÞ=K¼fðAÞ)arethesame.7Usingthisvaluefunc-tionintheEulerequation,wewrite1i¼ðbff~ðAÞ
pÞ1zðAÞ:gThisexpressionimpliesthattheinvestmentrateisindependentofthecurrentlevelofthecapitalstock.Toverifyourguess,wesubstitutethisinvestmentpolicyfunctionintotheoriginalfunctionalequation,whichimpliesthatg2fðAÞK¼AK
ðzðAÞÞK
pzðAÞKþbff~ðAÞK½ð1
dÞþzðAÞ2mustholdforallðA;KÞ.Clearly,theguessthatthevaluefunctionisproportionaltoKiscorrect:thevalueofKcancelsout.So,fromourconjecturethatVðA;KÞisproportionaltoK,wefindanoptimalinvestmentpolicythatconfirmsthesuggestedproportionality.TheremainingpartoftheunknownvaluefunctionfðAÞisgivenimplicitybytheexpressionabove.8Theresultthatthevaluefunctionisproportionaltothestockofcapitalis,atthispoint,anicepropertyofthelinear-quadraticfor-mulationofthecapitalaccumulationproblem.Inthediscussionofempiricalevidenceitformsthebasisforawiderangeofapplica-tions,sinceitallowstheresearchertosubstitutetheaveragevalueofQ(observablefromthestockmarket)formarginalQ(unobservable).8.4.2QTheory:EvidenceDuetoitsrelativelysimplestructure,theconvexadjustmentcostmodelisoneoftheleadingmodelsofinvestment.Infact,asdis-cussedabove,theconvexmodelisoftensimplifiedfurthersothat7.Hayashi(1982)wasthefirsttopointoutthatinthiscaseaverageandmarginalQcoincide,thoughhisformulationwasnonstochastic.8.InterestinglythenaturalconjecturethatfðAÞ¼Adoesnotsatisfythefunctionalequation.\n194Chapter8adjustmentcostsarequadratic,asin(8.6).Necessaryconditionsforoptimalityforthismodelareexpressedintwoways.First,fromthefirst-orderconditions,theinvestmentrateislinearlyrelatedtothedifferencebetweenthefuturemarginalvalueofnewcapitalandthecurrentpriceofcapital,asin(8.7).Usingtheargu-mentsfromabove,thismarginalvalueofcapitalcanundersomeconditionsbereplacedbytheaveragevalueofcapital.ThissetsthebasisfortheQtheoryempiricalapproachdiscussedbelow.Second,onecanbaseanempiricalanalysisontheEulerequationthatemergesfrom(8.6).ItnaturallyleadstoestimationusingGMMasdiscussedbelow.ThediscussionofestimationbasedonQtheorydrawsheavilyontwopapers.ThefirstbyGilchristandHimmelberg(1995)providesacleanandclearpresentationofthebasicapproachandevidenceonQtheorybasedestimationofcapitaladjustmentmodels.AthemeinthisandrelatedpapersisthatempiricallyinvestmentdependsonvariablesotherthanaverageQ,particularlymeasuresofcashflow.ThesecondbyCooperandEjarque(2001)worksfromGilchristandHimmelberg(1995)toexplorethesignificanceofimperfectcompetitionandcreditmarketfrictions.9Thispaperillustratestheuseofindirectinference.TestsofQtheoryonpaneldataarefrequentlyconductedusinganempiricalspecificationofXitðI=KÞit¼ai0þa1bEqitþ1þa2þuit:ð8:8ÞKitHeretheisubscriptreferstofirmorplantiandthetsubscriptrep-resentstime.From(8.7),a1shouldequal1=g.Thisisaninterestingaspectofthisspecification:underthenullhypothesisonecaninfertheadjustmentcostparameterfromthisregression.Thereisacon-stanttermintheregressionthatisplantspecific.Thiscomesfromamodificationofthequadraticcostofadjustmentto020gK
ð1
dÞKCðK;KÞ¼
aiK2KasinGilchristandHimmelberg(1995).109.WearegratefultoJoaoEjarqueforallowingustousethismaterial.10.Theerrortermin(8.8)isoftenascribedtostochasticelementsinthecostofadjustmentfunction;thenaiismodifiedtobecomeait¼aiþeit.\nInvestment195Finally,thisregressionincludesathirdterm,Xit=Kit.InfactQtheorydoesnotsuggesttheinclusionofothervariablesin(8.8),sinceallrelevantinformationisincorporatedinaverageQ.Rather,thesevariablesareincludedasameansoftestingthetheory,wherethetheorypredictsthatthesevariablesfromtheinformationsetshouldbeinsignificant.Henceresearchersfocusonthestatisticalandeco-nomicsignificanceofa2.Inparticular,XitoftenincludesfinancialvariablesasawayofevaluatinganalternativehypothesisinwhichtheeffectsoffinancialconstraintsarenotincludedinaverageQ.Theresultsobtainedusingthisapproachhavebeenmixed.Esti-matesoflargeadjustmentcostsarenotuncommon.Hayashi(1982)estimatesa1¼0:0423andthusgofabout25.GilchristandHimmel-berg(1995)estimatea1at0.033.Furthermanystudiesestimateapositivevaluefora2whenXitisameasureofprofitsand/orcashflow.11ThisistakenasarejectionoftheQtheory,whichofcourseimpliesthattheinferencedrawnaboutgfromtheestimateofa1maynotbevalid.Moreoverthesignificanceofthefinancialvariableshasleadresearcherstoconcludethatcapitalmarketimperfectionsmustbepresent.CooperandEjarque(2001)arguethattheapparentfailureofQtheorystemsfrommisspecificationofthefirm’soptimizationproblem:marketpowerisignored.AsshownbyHayashi(1982),iffirmshavemarketpower,thenaverageandmarginalQdiverge.ConsequentlythesubstitutionofmarginalforaverageQinthestan-dardinvestmentregressioninducesmeasurementerrorthatmaybepositivelycorrelatedwithprofits.12CooperandEjarque(2001)askwhetheronemightfindpositiveandsignificanta2in(8.8)inamodelwithoutanycapitalmarketimperfections.TheirmethodologyfollowstheindirectinferenceproceduresdescribedinGourierouxandMonfort(1996)andGourierouxetal.(1993).Thisapproachtoestimationwasdiscussedinchapter4.Thisisaminimumdistanceestimationroutineinwhichthestructuralparametersoftheoptimizationproblemarechosentobringthereducedformcoefficientsfromtheregressiononthesimulateddataclosetothosefromtheactualdata.Thekeyisthatthesamereducedformregressionisrunonboththeactualandsimulateddata.11.Hubbard(1994)reviewsthesefindings.12.CooperandEjarque(2001)donotattempttocharacterizethismeasurementerroranalyticallybutusetheirsimulatedenvironmenttounderstanditsimplications.SeeEricksonandWhited(2000)foradetailedandprecisediscussionofthesignificanceofmeasurementerrorintheQregressions.\n196Chapter8CooperandEjarque(2001)usetheparameterestimatesofGilchristandHimmelberg(1995)for(8.8)asrepresentativeoftheQtheorybasedinvestmentliterature.Denotetheseestimatesfromtheirpooledpanelsampleusingtheaverage(Tobin’s)Qmeasurebyða;aÞ¼ð0:03;0:24Þ.13CooperandEjarque(2001)addthreeother12momentsreportedbyGilchristandHimmelberg(1995):theserialcorrelationofinvestmentrates(0.4),thestandarddeviationofprofitrates(0.3),andtheaveragevalueofaverageQ(3).LetCddenotethevectormomentsfromthedata.IntheCooperandEjarque(2001)study,dC¼½0:030:240:40:33:Theestimationfocusesontwokeyparameters:thecurvatureoftheprofitfunctionðaÞandtheleveloftheadjustmentcostsðgÞ.Sootherparametersaresetatlevelsfoundinpreviousstudies:d¼0:15andb¼0:95.Thisleavesða;gÞandthestochasticprocessforthefirm-specificshockstoprofitabilityastheparametersremainingtobeestimated.CooperandEjarque(2001)estimatetheserialcorrela-tionðrÞandthestandarddeviationðsÞoftheprofitabilityshockswhiletheaggregateshockprocessisrepresentedprocessasatwo-stateMarkovprocesswithasymmetrictransitionmatrixinwhichtheprobabilityofremainingineitherofthetwoaggregatestatesis0.8.14Asdescribedinchapter4,theindirectinferenceprocedurepro-ceeds,inthisapplication,asfollows:0Givenavectorofparameters,Y1ða;g;r;sÞ,solvethefirm’sdynamicprogrammingproblemof02agK
ð1
dÞK0VðA;KÞ¼maxAK
K
pðK
ð1
dÞKÞK02K00þbEA0jAVðA;KÞforallðA;KÞð8:9Þusingvaluefunctioniteration.ThemethodoutlinedinTauchen(1986)isusedtocreateadiscretestatespacerepresentationofthe13.CooperandEjarque(2001)havenounobservedheterogeneityinthemodelsothattheconstantfromtheregressionaswellasthefixedeffectsareignored.Theremainingcoefficientsaretakentobecommonacrossallfirms.14.Theestimatesarenotsensitivetoaggregateshocks.Themodelisessentiallyesti-matedfromtherichcross-sectionalvariationasinthepanelstudyofGilchristandHimmelberg(1995).\nInvestment197Table8.1EstimatedstructuralparametersStructuralparametersagrsyGH95CE0.689(0.011)0.149(0.016)0.106(0.008)0.855(0.04)2Table8.2RegressionresultsandmomentsReducedformcoefficientestimates/momentsIpa1a2scstdqKKGH950.030.240.40.253CE0.0410.2370.0270.2512.95shockprocessgivenðr;sÞ.Usethisintheconditionalexpectationoftheoptimization.0Giventhepolicyfunctionsobtainedbysolvingthedynamicpro-grammingproblem,createapaneldatasetbysimulation.0EstimatetheQtheorymodel,asin(8.8),onthesimulatedmodel,andcalculaterelevantmoments.LetCsðYÞdenotethecorrespondingmomentsfromthesimulateddata.0ComputeJðYÞdefinedasJðYÞ¼ðCd
CsðYÞÞ0WðCd
CsðYÞÞ;ð8:10ÞwhereWisanestimateoftheinverseofthevariance-covariancematrixofCd.0FindtheestimatorofY,YY^,thatsolvesminJðYÞ:YThesecondrowoftable8.1presentstheestimatesofstruc-turalparametersandstandarderrorsreportedinCooperandEjar-que(2001).15Table8.2reportstheresultingregressionresultsand15.Thecomputationofstandarderrorsfollowsthedescriptioninchapter4ofGour-ierouxandMonfort(1996).\n198Chapter8moments.HeretherowlabeledGH95representstheregressionresultsandmomentsreportedbyGilchristandHimmelberg(1995).Themodel,withitsfourparameters,doesagoodjobofmatchingfourofthefiveestimates/momentsbutisunabletoreproducethehighlevelofserialcorrelationinplant-levelinvestmentrates.Thisappearstobeaconsequenceofthefairlylowlevelofgwhichimpliesthatadjustmentcostsarenotverylarge.Raisingtheadjustmentcostswillincreasetheserialcorrelationofinvestment.Theestimatedcurvatureoftheprofitfunctionof0.689impliesamarkupofabout15percent.16Thisestimateofa,andhencethemarkup,isnotatvariancewithresultsreportedintheliterature.Theotherinterestingparameteristheestimateofthelevelasso-ciatedwiththequadraticcostofadjustment,g.Relativetootherstudies,thisappearsquitelow.However,aninterestingpointfromtheseresultsisthattheesti-mateofgisnotidentifiedfromtheregressioncoefficientonaverageQ.Fromtable8.1,theestimatedvalueofg¼0:149isfarfromtheinverseofthecoefficientonaverageQ(about4).Soclearlytheiden-tificationofthequadraticcostofadjustmentparameterfroma2ismisleadinginthepresenceofmarketpower.exercise8.4Writeaprogamtosolve02agK
ð1
dÞKVðA;KÞ¼maxAK
KK02K000
pðK
ð1
dÞKÞþbEA0jAVðA;KÞð8:11Þusingavaluefunctioniterationroutinegivenaparameterizationoftheproblem.UsetheresultstoexploretherelationshipofinvestmenttoaverageQ.Isthereanonlinearityinthisrelationship?Howisinvestmentrelatedtoprofitabilityinyoursimulateddataset?8.4.3EulerEquationEstimationThisapproachtoestimationshareswiththeconsumptionapplica-tionspresentedinchapter6asimplebutpowerfullogic.TheEuler16.CooperandEjarque(2001)showthatifp¼y
histhedemandcurveandy¼Akflð1
fÞtheproductionfunction,maximizationofprofitovertheflexiblefactorlleadstoareducedformprofitfunctionwheretheexponentoncapitalisfðh
1Þ=½ð1
fÞð1
hÞ
1.Heref¼0:33andh¼0:1315,implyingamarkupofabout15percent.\nInvestment199equationgivenin(8.5)isanecessaryconditionforoptimality.Inthequadraticcostofadjustmentmodelcasethissimplifiesto1g2it¼bEtpKðAtþ1;Ktþ1Þþptþ1ð1
dÞþitþ1þgð1
dÞitþ1
pt:g2Letetþ1bedefinedfromrealizedvaluesofthesevariables:1etþ1¼it
bpKðAtþ1;Ktþ1Þþptþ1ð1
dÞgg2þitþ1þgð1
dÞitþ1
pt:ð8:12Þ2ThentherestrictionimposedbythetheoryisthatEtetþ1¼0.ItispreciselythisorthogonalityconditionthattheGMMprocedureexploitsintheestimationofunderlyingstructuralparameters,y¼ðb;g;d;aÞ.Toillustrate,wehavesolvedandsimulatedamodelwithqua-draticadjustmentcostsðg¼2Þwithconstantinvestmentgoodprices.Thatdatasetallowsustoestimatetheparametersofthefirm’sproblemusingGMM.Tomakethisastransparentaspossible,assumethattheresearcherknowsthevaluesofallparametersexceptforg.Thuswecanrelyonasingleorthogonalityconditiontodetermineg.Supposethatweusethelaggedprofitabilityshockastheinstrument.Define1XWðgÞ¼etþ1ðgÞAt:ð8:13ÞTtTheGMMestimateofgisobtainedfromtheminimizationofWðgÞ.Thisfunctionisshowninfigure8.1.Clearly,thisfunctionismini-mizednearg¼2.17Whited(1998)containsathoroughreviewandanalysisofexistingevidenceonEulerequationestimationofinvestmentmodels.AsWhitednotes,theEulerequationapproachcertainlyhasavirtueovertheQtheorybasedmodel:thereisnoneedtotrytomeasure17.Theprogramtoestimatethismodelisverysimple.OnceWðgÞisprogrammed,itissimplyabasicroutinetominimizethisfunction.ObtainingWðgÞiseasytoo,usingtheinformationonparametersplusobservationsinthedatasetoninvestmentratesandtheratioofoutputtocapital(todeterminemarginalprofitrates).Theminimizationmaynotoccurexactlyatg¼2becauseofasamplingerror.Theinterestedreadercanextendthisanalysistocreateadistributionofestimatesbyredrawingshocks,simu-lating,andthenre-estimatinggfromtheGMMprocedure.\n200Chapter8Figure8.1FunctionWðgÞmarginalQ.Thussomeoftherestrictionsimposedontheestima-tion,suchastheconditionsspecifiedbyHayashi,donothavetobeimposed.EstimationbasedonaninvestmentEulerequationgener-allyleadstorejectionoftheoveridentifyingrestrictions,andasintheQtheorybasedempiricalwork,theinclusionoffinancialcon-straintsimprovestheperformanceofthemodel.ThepointofWhited(1998)istodigfurtherintotheseresults.Importantly,heranalysisbringsfixedadjustmentcostsintotheevaluationoftheEulerequationestimation.Asnotedearlieranddiscussedatsomelengthbelow,investmentstudieshavebeenbroadenedtogobeyondconvexadjustmentcoststomatchtheobservationsofnonadjustmentinthecapitalstock.Whited(1998)takesthisintoaccountbydividinghersampleintothesetoffirmsthatundertakespositiveinvestment.EstimationoftheEulerequa-tionforthissubsetismuchmoresuccessful.FurtherWhited(1998)findsthatwhilefinancialvariablesareimportantoverall,theyarealsoweaklyrelevantforthefirmswithongoinginvestment.\nInvestment201Theseresultsareprovocative.Theyforceustothinkjointlyaboutthepresenceofnonconvexadjustmentcostsandfinancialvariables.Wenowturntotheseimportanttopics.8.4.4BorrowingRestrictionsThusfarwehaveignoredthepotentialpresenceofborrowingrestrictions.Thesehavealonghistoryinempiricalinvestmentanal-ysis.AsinourdiscussionoftheempiricalQtheoryliterature,finan-cialfrictionsareoftenviewedasthesourceofthesignificanceofprofitratesand/orcashflowininvestmentregressions.Thereisnothingparticularlydifficultaboutintroducingborrowingrestrictionsintothecapitalaccumulationproblem.Consider02agK
ð1
dÞKVðA;KÞ¼maxAK
Kð8:14ÞK0AGðA;KÞ2K000
pðK
ð1
dÞKÞþbEA0jAVðA;KÞforallðA;KÞ;ð8:15ÞwhereGðA;KÞconstrainsthechoicesetforthefuturecapitalstock.Forexample,ifcapitalpurchaseshadtobefinancedoutofcurrentprofits,thenthefinancialrestrictionisK0
ð1
dÞKaAKað8:16ÞsothatGðA;KÞ¼½0;AKaþð1
dÞK:ð8:17ÞThedynamicoptimizationproblemwitharestrictionof(8.17)cancertainlybeevaluatedusingvaluefunctioniterationtechniques.Theproblemofthefirmcanbebroadenedtoincluderetainedearningsasastatevariableandtoincludeotherfinancialvariablesinthestatevector.Thereareanumberofunresolvedissuesthoughthathavelimitedresearchinthisarea:0WhataretheGðA;KÞfunctionssuggestedbytheory?0ForwhatGðA;KÞfunctionsisthereawedgebetweenaverageandmarginalQ?Thefirstpointisworthyofnote:whilewehavemanymodelsofcapitalaccumulationwithoutborrowingrestrictions,thealternativemodelofinvestmentwithborrowingrestrictionsisnotonthetable.Thustherejectionofthemodelwithoutconstraintsinfavorofonewithconstraintsisnotasconvincingasitcouldbe.\n202Chapter8Table8.3Descriptivestatistics,LRDVariableLRDAverageinvestmentrate12.2%Inactionrate:Investment8.1Fractionofobservationswithnegative10.4investmentSpikerate:Positiveinvestment18Spikerate:Negativeinvestment1.4Thesecondpoint,relatedtoworkbyChirinko(1993)andGomes(2001),returnstotheevidencediscussedearlieronQtheorybasedempiricalmodelsofinvestment.ThevaluefunctionVðA;KÞthatsolves(8.15)containsalltheinformationabouttheconstrainedopti-mizationproblem.Aslongasthisfunctionisdifferentiable(whichrestrictstheGðA;KÞfunction),marginalQwillstillmeasurethereturntoanextraunitofcapital.TheissueiswhethertheborrowingfrictionintroducesawedgebetweenmarginalandaverageQ.18EmpiricallytheissueiswhetherthiswedgebetweenmarginalandaverageQcancreatetheregressionresultssuchasthosereportedinGilchristandHimmelberg(1995).8.5NonconvexAdjustment:TheoryEmpiricallyonefindsthatattheplantleveltherearefrequentperi-odsofinvestmentinactivityandalsoburstsofinvestmentactivity.Table8.3,takenfromCooperandHaltiwanger(2000),documentsthenatureofcapitaladjustmentintheLongitudinalResearchData-base(LRD),aplant-levelU.S.manufacturingdataset.19Hereinactionisdefinedasaplant-levelinvestmentratelessthan0.01andaspikeisaninvestmentrateinexcessof20percent.Clearly,thedataexhibitbothinactionaswellaslargeburstsofinvestment.AsarguedbyCaballeroetal.(1995),Cooperetal.(1999),andCooperandHaltiwanger(2000),itisdifficulttomatchthistypeofevidencewithaquadraticcostofadjustmentmodel.Thusweturntoalternativemodelswhichcanproduceinaction.Inthefirsttypeof18.If,intheexampleabove,a¼1,thentheconstraintisproportionaltoK.InthiscaseitappearsthataverageandmarginalQareequal.19.CooperandHaltiwangerprovideafulldescriptionofthedata.\nInvestment203modelwerelaxtheconvexadjustmentcoststructureandassumethatthecostsofadjustmentdependonlyonwhetherinvestmenthasbeenundertaken,andnotitsmagnitude.Wethenconsiderasecondtypeofmodelinwhichthereissometypeofirreversibility.Thenextsectionreportsonestimationofthesemodels.8.5.1NonconvexAdjustmentCostsForthisformulationofadjustmentcosts,wefollowCooperandHaltiwanger(1993)andCooperetal.(1999)andconsideradynamicprogrammingproblemspecifiedattheplantlevelasVðA;K;pÞ¼maxfViðA;K;pÞ;VaðA;K;pÞgforallðA;K;pÞ;ð8:18Þwherethesuperscriptsrefertoactiveinvestmentaandinactivityi.Theseoptions,inturn,aredefinedbyi00VðA;K;pÞ¼PðA;KÞþbEA0;p0jA;pVðA;Kð1
dÞ;pÞandVaðA;K;pÞ¼maxPðA;KÞl
FK
pðK0
ð1
dÞKÞK0000þbEA0;p0jA;pVðA;K;pÞ:Heretherearetwocostsofadjustmentthatareindependentofthelevelofinvestmentactivity.Thefirstisalossofprofitflowequalto1
l.Thisisintendedtocaptureanopportunitycostofinvestmentinwhichtheplantmustbeshutdownduringaperiodofinvestmentactivity.ThesecondnonconvexcostissimplysubtractedfromtheflowofprofitsasFK.TheinclusionofKhereisintendedtocapturetheideathatthesefixedcosts,whileindependentofthecurrentlevelofinvestmentactivity,mayhavesomescaleaspectstothem.20InthisformulationtherelativepriceofcapitalðpÞisallowedtovaryaswell.Beforeproceedingtoadiscussionofresults,itmightbeusefultorecallfromchapter3howonemightobtainasolutiontoaproblem20.SeeAbelandEberly(1994)foramodelinwhichfixedcostsareproportionaltoK.Ifthesecostswereindependentofsize,thenlargeplantswouldfaceloweradjustmentcosts(relativetotheircapitalstock)andthusmightadjustmorefrequently.So,asinthequadraticspecification,thecostsarescaledbysize.Thisisneverthelessanassumption,andtherelationshipbetweenplantsizeandinvestmentactivityisstillanopenissue.\n204Chapter8suchas(8.18).21Thefirststepistospecifyaprofitfunction,sayPðA;KÞ¼AKa,forwhichwesettheparametersðF;b;l;a;dÞaswellasthestochasticprocessesfortherandomvariablesðA;pÞ.DenotethisparametervectorbyY.ThesecondstepistospecifyaspaceforthestatevariablesðA;K;pÞandthusforcontrolvariableK0.Oncethesestepsarecomplete,thevaluefunctioniterationlogic(sub-scriptsdenoteiterationsofthemapping)takesover:0ProvideaninitialguessforVðA;K;pÞ,suchastheone-period1solution.0Usingthisinitialguess,computethevaluesforthetwooptions,VaðA;K;pÞandViðA;K;pÞ.110Usingthesevalues,solveforthenextguessofthevaluefunction:VðA;K;pÞ¼maxfVaðA;K;pÞ;ViðA;K;pÞg.2110Continuethisprocessuntilconvergence.0Afterthevaluefunctionisfound,computethesetofstatevariablessuchthattheaction(inaction)isoptimalandthattheinvestmentlevelintheeventadjustmentisoptimal.0Fromthesepolicyfunctions,simulatethemodelandcreateeitherapaneloratimeseriesdataset.Thepolicyfunctionforthisproblemwillhavetwoimportantdimensions.First,thereisthedeterminationofwhethertheplantwilladjustitscapitalstockornot.Second,conditionalonadjust-ment,theplantmustdetermineitslevelofinvestment.Asusual,theoptimalchoiceofinvestmentdependsonthemarginalvalueofcap-italinthenextperiod.However,incontrastto,say,thequadraticcostofadjustmentmodel,thefuturevalueofadditionalcapitaldependsonfuturechoicewithrespecttoadjustment.ThusthereisnosimpleEulerequationlinkingthemarginalcostofadditionalcapitaltodaywithfuturemarginalbenefit,asin(8.5),sincethereisnoguaranteethatthisplantwillbeadjustingitscapitalstockinthenextperiod.Notethatthetwotypesofcostshaveverydifferentimplicationsforthecyclicalpropertiesofinvestment.Inparticular,whenadjust-mentcostsinterferewiththeflowofprofitsðl<1Þ,thenitismoreexpensivetoinvestinperiodsofhighprofitability.Yet,iftheshocks21.Recalltheoutlineofthebasicvaluefunctioniterationprogramforthenon-stochasticgrowthmodelandthemodificationofthatfornonconvexadjustmentcostsinchapter3.\nInvestment205aresufficientlycorrelated,thereisagaintoinvestingingoodtimes.Incontrast,ifcostsarelargelylumpsum,thengiventhetime-to-buildaspectoftheaccumulationdecision,thebesttimetoinvestiswhenitisprofitabletodoso(Aishigh)assumingthattheseshocksareseriallycorrelated.Thuswhetherinvestmentisprocyclicalorcountercyclicaldependsonboththenatureoftheadjustmentcostsandthepersistenceofshocks.Wewilldiscussthepolicyfunctionsforanestimatedversionofthismodellater.Fornowwelookatasimpleexampletobuildintuition.MachineReplacementExampleForanexampleofasimplemodelofmachinereplacement,weturntoamodifiedversionstudiedbyCooperandHaltiwanger(1993).Herethereisnochoiceofthesizeoftheinvestmentexpenditure.Investmentmeansthepurchaseofanewmachineatanetpriceofp.Byassumption,theoldmachineisscrapped.Thesizeofthenewmachineisnormalizedto1.22Tofurthersimplifytheargument,weassumethatnewcapitalbecomesproductiveimmediately.Inadditionthepriceofnewcapi-talgoodisassumedtobeconstantandcanbeinterpretedasinclud-ingthefixedcostofadjustingthecapitalstock.InthiscasewecanwritetheBellmanequationasVðA;KÞ¼maxfViðA;KÞ;VaðA;KÞgforallðA;KÞ;wherethesuperscriptsrefertoactiveinvestmentaandinactivityi.Theseoptions,inturn,aredefinedbyi0VðA;KÞ¼PðA;KÞþbEA0jAVðA;Kð1
dÞÞanda0VðA;KÞ¼PðA;1Þl
pþbEA0jAVðA;ð1
dÞÞ:Here‘‘action’’meansthatanewmachineisboughtandisimmedi-atelyproductive.Thecostofthisisthenetpriceofthenewcapitalandthedisruptioncausedbytheadjustmentprocess.LetDðA;KÞbetherelativegainstoaction,so22.AsdiscussedbyCooperandHaltiwanger(1993)andCooperetal.(1999),thisassumptionthatanewmachinehasfixedsizecanbederivedfromamodelwithembodiedtechnologicalprogressthatisrenderedstationarybydividingthroughbytheproductivityofthenewmachine.Inthiscasetherateofdepreciationmeasuresbothphysicaldeteriorationandobsolescence.\n206Chapter8DðA;KÞ1VaðA;KÞ
ViðA;KÞ¼PðA;1Þl
PðA;KÞ
p00þbðEA0jAVðA;ð1
dÞÞ
EA0jAVðA;Kð1
dÞÞÞ:Theproblemposedinthisfashionisclearlyoneoftheoptimalstoppingvariety.GiventhestateofprofitabilityðAÞ,thereisacriti-calsizeofthecapitalstockðKðAÞÞsuchthatmachinereplacementoccursifandonlyifK0.Further,withthecostsofacquiringnewcapitalðp>0;l<1Þlargeenoughandtherateofdepreciationlowenough,capitalwillnotbereplacedeachperiod:K<1.Thustherewillbea‘‘replacementcycle’’inwhichthereisaburstofinvestmentactivityfollowedbyinactivityuntilthecapitalagesenoughtowarrantreplacement.23ThepolicyfunctionisthengivenbyzðA;KÞAf0;1g,wherezðA;KÞ¼0meansinactionandzðA;KÞ¼1meansreplacement.Fromtheargumentabove,foreachAthereexistsKðAÞsuchthatzðA;KÞ¼1ifandonlyifKaKðAÞ.Withtheassumptionthatcapitalbecomesproductivelyimmedi-ately,theresponseofKðAÞtovariationsinAcanbeanalyzed.24Suppose,forexample,thatl¼1andAisiid.Inthiscasethedepen-denceofDðA;KÞonAissolelythroughcurrentprofits.ThusDðA;KÞisincreasinginAaslongasthemarginalproductivityofcapitalisincreasinginA,PðA;KÞ>0.SoKðAÞwillbeincreasinginAandAKreplacementwillbemorelikelyingoodtimes.Alternatively,supposethatl<1.Inthiscase,duringperiodsofhighproductivity,itisdesirabletohavenewcapital,butitisalsocostlytoinstallit.IfAispositivelyseriallycorrelated,thentheeffectofAonDðA;KÞwillreflectboththedirecteffectoncurrentprofitsandtheeffectsonthefuturevalues.Iftheopportunitycostislarge(asmalll)andshocksarenotpersistentenough,thenmachinereplacementwillbedelayeduntilcapitalislessproductive.23.CooperandHaltiwanger(2000)andCooperetal.(1999)arguethatthesefeaturesalsoholdwhenthereisaone-periodlagintheinstallationprocess.24.Cooperetal.(1999)analyzethemorecomplicatedcaseofaone-periodlagintheinstallationofnewcapital.\nInvestment207AggregateImplicationsofMachineReplacementThismodelofcapitaladjustmentattheplantlevelcanbeusedtogenerateaggregateimplications.LetftðKÞbethecurrentdistributionofcapitalacrossafixedpopulationofplants.Supposethattheshockinperiodt,At,hastwocomponents,At¼atet.Thefirstisaggregateandthesecondisplantspecific.FollowingCooperetal.(1999),assumethattheaggregateshocktakesontwovaluesandtheplantspecificshocktakesontwentyvalues.Furtherassumethattheidio-syncraticshocksareiid.Withthisdecomposition,writethepolicyfunctionaszðat;et;KtÞ,wherezðat;et;KtÞ¼1signifiesactionandzðat;et;KtÞ¼0indicatesinaction.Clearly,thedecisiononreplace-mentwillgenerallydependdifferentiallyonthetwotypesofshocks,sincetheymaybedrawnfromdifferentstochasticproperties.Forexample,iftheaggregateshockismorepersistentthantheplant-specificone,theresponsetoavariationinatwillbelargerthantheresponsetoaninnovationinet.DefineðHðat;KÞ¼zðat;et;KÞdGtðeÞ;ewhereGtðeÞistheperiodtcumulativedistributionfunctionoftheplant-specificshocks.HereHðat;KÞisahazardfunctionrepresentingtheprobabilityofadjustmentforallplantswithcapitalKinaggre-gatestateat.Totheextentthattheresearchermaybeabletoobserveaggregatebutnotplant-specificshocks,Hðat;KÞrepresentsahazardthataveragesoverthef0;1gchoicesoftheindividualplantssothatHðat;KÞA½0;1.Usingthisformulation,letIðat;ftðKÞÞbetherateofinvestmentinstateatgiventhedistributionofcapitalholdingsftðKÞacrossplants.AggregateinvestmentisdefinedasXIðat;ftðKÞÞ¼Hðat;KÞftðKÞ:ð8:19ÞKThustotalinvestmentreflectstheinteractionbetweentheaverageadjustmenthazardandthecross-sectionaldistributionofcapitalholdings.Theevolutionofthecross-sectionaldistributionofcapitalisgivenbygtþ1ðð1
dÞKÞ¼ð1
Hðat;KÞÞgtðKÞ:ð8:20Þ\n208Chapter8Expressionssuchasthesearecommoninaggregatemodelsofdis-creteadjustment;see,forexample,Rust(1985)andCaballeroetal.(1995).Givenaninitialcross-sectionaldistributionandahazardfunction,asequenceofshockswillthusgenerateasequenceofaggregateinvestmentlevelsfrom(8.19)andasequenceofcross-sectionaldistributionsfrom(8.20).Thusthemachinereplacementproblemcangenerateapaneldatasetand,throughaggregation,timeseriesaswell.Inprinciple,esti-mationfromaggregatedatasupplementstheperhapsmoredirectrouteofestimatingamodelsuchasthisfromapanel.exercise8.5Useavaluefunctioniterationroutinetosolvethedynamicoptimizationproblemwithafirmwhentherearenon-convexadjustmentcosts.Supposethatthereisapanelofsuchfirms.Usetheresultingpolicyfunctionstosimulatethetimeseriesofaggregateinvestment.Thenuseavaluefunctioniterationroutinetosolvethedynamicoptimizationproblemwithafirmwhentherearequadraticadjustmentcosts.Createatimeseriesfromthesimulatedpanel.Howwellcanaquadraticadjustmentcostmodelapproximatetheaggregateinvestmenttimeseriescreatedbythemodelwithnonconvexadjustmentcosts?8.5.2IrreversibilityThespecificationsconsideredthusfardonotdistinguishbetweenthebuyingandsellingpricesofcapital.However,therearegoodreasonstothinkthatinvestmentisatleastpartiallyirreversiblesothatthesellingpriceofaunitofusedcapitalislessthanthecostofaunitofnewcapital.Thisreflectsfrictionsinthemarketforusedcapitalaswellasspecificaspectsofcapitalequipmentthatmaymakethemimperfectlysuitableforusesatotherproductionsites.Toallowforthis,wealterouroptimizationproblemtodistinguishthebuyingandsellingpricesofcapital.ThevaluefunctionforthisspecificationisgivenbyVðA;KÞ¼maxfVbðA;KÞ;VsðA;KÞ;ViðA;KÞgforallðA;KÞ;wherethesuperscriptsrefertotheactofbuyingcapitalb,sellingcapitalsandinactioni.Theseoptions,inturn,aredefinedbyb0VðA;KÞ¼maxPðA;KÞ
IþbEA0jAVðA;Kð1
dÞþIÞ;I\nInvestment209s0VðA;KÞ¼maxPðA;KÞþpsRþbEA0jAVðA;Kð1
dÞ
RÞ;Randi0VðA;KÞ¼PðA;KÞþbEA0jAVðA;Kð1
dÞÞ:Underthebuyoption,theplantobtainscapitalatacostnormal-izedtoone.Undertheselloption,theplantretiresRunitsofcapitalatapriceps.Thethirdoptionisinaction,sothecapitalstockdepre-ciatesatarateofd.Intuitivelythegapbetweenthebuyingandsell-ingpriceofcapitalwillproduceinaction.Supposethatthereisanadverseshocktotheprofitabilityoftheplant.Ifthisshockwasknowntobetemporary,thensellingcapitalandrepurchasingitinthenearfuturewouldnotbeprofitablefortheplantaslongasps<1.Thusinactionmaybeoptimal.Clearly,though,theamountofinactionthatthismodelcanproducewilldependonboththesizeofpsrelativeto1andtheserialcorrelationoftheshocks.258.6EstimationofaRichModelofAdjustmentCostsUsingthisdynamicprogrammingstructuretounderstandtheopti-malcapitaldecisionattheplant(firm)level,weconfrontthedataoninvestmentdecisionsallowingforarichstructureofadjustmentcosts.26Todoso,wefollowCooperandHaltiwanger(2000)andconsideramodelwithquadraticadjustmentcosts,nonconvexad-justmentcostsandirreversibility.WedescribetheoptimizationproblemandthentheestimationresultsobtainedbyCooperandHaltiwanger.8.6.1GeneralModelThedynamicprogrammingproblemforaplantisgivenbyVðA;KÞ¼maxfVbðA;KÞ;VsðA;KÞ;ViðA;KÞgforallðA;KÞ;ð8:21Þwhere,asabove,thesuperscriptsrefertotheactofbuyingcapitalb,sellingcapitalsandinactioni.Theseoptions,inturn,aredefinedby25.Aninterestingextensionofthemodelwouldmakethisgapendogenous.26.ThedatasetisdescribedbyCooperandHaltiwanger(2000)andisforabalancedpanelofU.S.manufacturingplants.Comparabledatasetsareavailableinothercoun-tries.Similarestimationexercisesusingthesedatasetswouldbeofconsiderableinterest.\n210Chapter8bg2VðA;KÞ¼maxPðA;KÞ
FK
I
½I=KKI20þbEA0jAVðA;Kð1
dÞþIÞ;sg2VðA;KÞ¼maxPðA;KÞþpsR
FK
½R=KKR20þbEA0jAVðA;Kð1
dÞ
RÞ;andi0VðA;KÞ¼PðA;KÞþbEA0jAVðA;Kð1
dÞÞ:CooperandHaltiwanger(2000)estimatethreeparameters,Y1ðF;g;psÞandassumethatb¼0:95,d¼0:069.FurthertheyspecifyaprofitfunctionofPðA;KÞ¼AKywithy¼0:50estimatedfromapaneldatasetofmanufacturingplants.27Notethattheadjustmentcostsin(8.21)excludeanydisruptionstotheproductionprocesssothatthePðA;KÞcanbeestimatedandtheshockprocessinferredindependentlyoftheestimationofadjustmentcosts.Iftheseaddi-tionaladjustmentcostswereadded,thentheprofitfunctionandtheshockswouldhavetobeestimatedalongwiththeparametersoftheadjustmentcostfunction.Theseparametersareestimatedusinganindirectinferencerou-tine.Thereducedformregressionusedintheanalysisis2iit¼aiþc0þc1aitþc2ðaitÞþuit;ð8:22Þwhereiitistheinvestmentrateatplantiinperiodt,aitisthelogofaprofitabilityshockatplantiinperiodt,andaiisafixedeffect.28Thisspecificationwaschosenasitcapturesinaparsimoniouswaythenonlinearrelationshipbetweeninvestmentratesandfundamentals.Theprofitabilityshocksareinferredfromtheplant-leveldatausingtheestimatedprofitfunction.29CooperandHaltiwangerdocumenttheextentofthenonlinearresponseofinvestmenttoshocks.27.SeethediscussionbyCooperandHaltiwanger(2000)oftheestimationofthisprofitfunction.28.MorerecentversionsoftheCooper-Haltiwangerpaperexploreaddinglaggedinvestmentratestothisreducedformtopickupsomeofthedynamicsoftheadjust-mentprocess.29.Thisisanimportantstepintheanalysis.Determiningthenatureofadjustmentcostswilldependonthecharacterizationoftheunderlyingprofitabilityshocks.Forexample,ifaresearcheristryingtoidentifynonconvexadjustmentcostsfromburstsofinvestment,thengettingthedistributionofshocksrightiscritical.\nInvestment211Table8.4ParameterestimatesStructuralparameterParameterestimateestimates(s.e.)for(8.22)Specifi-cationgFpsc0c1c2LRD
0.0130.2650.20All0.0430.000390.967
0.0130.2550.171(0.00224)(0.0000549)(0.00112)Fonly00.03331
0.020.3170.268(0.0000155)gonly0.12501
0.0070.2410.103(0.000105)psonly000.93
0.0160.2660.223(0.000312)Table8.4reportsCooperandHaltiwanger’sresultsforfourdif-ferentmodelsalongwithstandarderrors.Thefirstrowshowstheestimatedparametersforthemostgeneralmodel.TheparametervectorY¼½0:043;0:00039;0:967impliesthepresenceofstatisticallysignificantconvexandnonconvexadjustmentcosts(butnonzero)andarelativelysubstantialtransactioncost.Restrictedversionsofthemodelarealsoreportedforpurposesofcomparison.Clearly,themixedmodeldoesbetterthananyoftherestrictedmodels.CooperandHaltiwangerarguethattheseresultsarereasonable.30First,asnotedabovealowlevelfortheconvexcostofadjustmentparameterisconsistentwiththeestimatesobtainedfromtheQtheorybasedmodelsduetothepresenceofimperfectcompetition.Furthertheestimationimpliesthatthefixedcostofadjustmentisabout0.04percentofaverageplant-levelprofits.CooperandHal-tiwangerfindthatthiscostissignificantrelativetothedifferencebetweenadjustingandnotadjustingthecapitalstock.Soinfacttheestimatedfixedcostofadjustment,alongwiththeirreversibility,producesalargeamountofinaction.FinallytheestimatedsellingpriceofcapitalismuchhigherthantheestimatereportinRameyandShapiro(2001)forsomeplantsintheaerospaceindustry.CooperandHaltiwanger(2000)alsoexploretheaggregateimpli-cationsoftheirmodel.Theycontrastthetimeseriesbehaviorofthe30.Theresultsarerobusttoallowingthediscountfactortovarywiththeaggregateshockinordertomimictherelationshipbetweenrealinterestratesandconsumptiongrowthfromahousehold’sEulerequation.\n212Chapter8estimatedmodelwithbothconvexandnonconvexadjustmentcostsagainstoneinwhichthereareonlyconvexadjustmentcosts.Eventhoughthemodelwithonlyconvexadjustmentcostsdoesrelativelypoorlyontheplant-leveldata,itdoesreasonablywellintermsofmatchingtimeseries.Inparticular,CooperandHaltiwanger(2000)findthatover90percentofthetimeseriesvariationininvestmentcreatedbyasimulationoftheestimatedmodelcanbeaccountedforbyaquadraticadjustmentmodel.Ofcourse,thisalsoimpliesthatthequadraticmodelmisses10percentofthevariation.Notetoothatthisframeworkforaggregationcapturesthesmooth-ingbyaggregatingoverheterogeneousplantsbutmissessmoothingcreatedbyvariationsinrelativeprices.FromThomas(2002)andKahnandThomas(2001)weknowthatthisadditionalsourceofsmoothingcanbequitepowerfulaswell.8.6.2MaximumLikelihoodEstimationAfinalapproachtoestimationfollowstheapproachinRust(1987).Consideragain,forexample,thestochasticmachinereplacementproblemgivenbyVðA;K;FÞ¼maxfViðA;K;FÞ;VaðA;K;FÞgforallðA;K;FÞ;ð8:23Þwherei00VðA;K;FÞ¼PðA;KÞþbEA0jAVðA;Kð1
dÞ;FÞandVaðA;K;FÞ¼maxPðA;KÞl
FK
pðK0
ð1
dÞKÞK0000þbEA0jAVðA;K;FÞ:Herewehaveaddedthefixedcostofadjustmentintothestatevectorasweassumethattheadjustmentcostsarerandomattheplantlevel.LetGðFÞrepresentthecumulativedistributionfunctionfortheseadjustmentcosts.Assumethattheseareiidshocks.Then,givenaguessforthefunctionsfVðA;K;FÞ;ViðA;K;FÞ;VaðA;K;FÞg,thelikelihoodofinactioncanbecomputeddirectlyfromthecumu-lativedistributionfunctionGð
Þ.Thusalikelihoodfunctioncanbeconstructedthatdependsontheparametersofthedistributionofadjustmentcostsandthoseunderlyingthedynamicoptimization\nInvestment213problem.Fromthere,amaximumlikelihoodestimatecanbeobtained.318.7ConclusionThethemeofthischapterhasbeenthedynamicsofcapitalaccumu-lation.Fromtheplant-levelperspective,theinvestmentprocessisquiterichandentailsperiodsofintenseactivityfollowedbytimesofinaction.Thishasbeendocumentedattheplantlevel.Usingthetechniquesoftheestimationofdynamicprogrammingmodels,thischapterhaspresentedevidenceonthenatureofadjustmentcosts.Manyopenissuesremain.First,thetimeseriesimplicationsofnonconvexitiesisstillnotclear.Howmuchdoesthelumpinessattheplant-levelmatterforaggregatebehavior?Putdifferently,howmuchsmoothingobtainsfromtheaggregateacrossheterogeneousplantsaswellasthroughvariationsinrelativeprices?Second,thereareahostofpolicyexperimentstobeconsidered.What,forexample,aretheimplicationsofinvestmenttaxcreditsgiventheestimatesofadjustmentcostparameters?exercise8.6Addinvariationsinthepriceofnewcapitalintotheoptimizationproblemgivenin(8.21).Howwouldyouusethistostudytheimpactof,say,aninvestmenttaxcredit?31.TheinterestedreadershouldreadcloselythediscussionofRust(1987)andthepapersthatfollowedthislineofwork.NotethatoftenassumptionsaremadeonGð
Þtoeasethecomputationofthelikelihoodfunction.\n9DynamicsofEmploymentAdjustment9.1MotivationThischapterstudieslabordemand.Theusualtextbookmodeloflabordemanddepictsafirmaschoosingthenumberofworkersandtheirhoursgivenawagerate.Butthedeterminationofwages,employment,andhoursismuchmorecomplexthanthis.Thekeyistorecognizethattheadjustmentofmanyfactorsofproduction,includinglabor,isnotcostless.Westudythedynamicsofcapitalaccumulationelsewhereinthisbookandinthischapterfocusatten-tiononlabordemand.Understandingthenatureofadjustmentcostsandthusthefactorsdetermininglabordemandisimportantforanumberofreasons.First,manycompetingmodelsofthebusinesscycledependcruciallyontheoperationoflabormarkets.AsemphasizedinSargent(1978),acriticalpointindistinguishingcompetingtheoriesofthebusinesscycleiswhetherlabormarketobservationscouldplausiblybetheoutcomeofadynamicmarket-clearingmodel.Second,attemptstoforecastmacroeconomicconditionsoftenresorttoconsiderationofobservedmovementsinhoursandemploymenttoinferthestateofeconomicactivity.Finally,policyinterventionsinthelabormarketarenumerousandwidespread.Theseincluderestrictionsonwages,restrictionsonhours,costsoffiringworkers,andsoforth.Policyevaluaterequiresamodeloflabordemand.Webeginthechapterwiththesimplestmodelsofdynamiclabordemandwhereadjustmentcostsareassumedtobeconvexandcon-tinuouslydifferentiable.Thesemodelsareanalyticallytractableaswecanoftenestimatetheirparametersdirectlyfromfirst-ordercon-ditions.However,theyhaveimplicationsofconstantadjustment\n216Chapter9thatarenotconsistentwithmicroeconomicobservations.Nickell(1978)argues:Onepointworthnotingisthatthereseemslittlereasontosupposecostsperworkerassociatedwitheitherhiringorfiringincreasewiththerateatwhichemployeesflowinorout.Indeed,giventhelargefixedcostsassociatedwithpersonnelandlegaldepartments,itmayevenbemorereasonabletosup-posethattheaveragecostofadjustingtheworkforcediminishesratherthanincreaseswiththespeedofadjustment.ThisquoteissupportedbyrecentevidenceinHamermesh(1989)andCaballeroetal.(1997)thatlaboradjustmentisrathererraticattheplantlevelwithperiodsofinactivitypunctuatedbylargeadjust-ments.Thusthischaptergoesbeyondtheconvexcaseandconsidersmodelsofadjustmentwhichcanmimicthesemicroeconomicfacts.9.2GeneralModelofDynamicLaborDemandInthischapterweconsidervariantsofthefollowingdynamicpro-grammingproblem:VðA;e
1Þ¼maxRðA;e;hÞ
oðe;h;AÞ
Cðe;e
1Þh;e0þbEA0jAVðA;eÞforallðA;e
1Þ:ð9:1ÞHereArepresentsashocktotheprofitabilityoftheplantand/orfirm.Asinourdiscussionoftheinvestmentproblem,thisshockcouldreflectvariationsinproductdemandsorvariationsinthepro-ductivityofinputs.Generally,Awillhaveacomponentthatiscom-monacrossplants,denoteda,andonethatisplantspecific,denotede.1ThefunctionRðA;e;hÞrepresentstherevenuesthatdependonthehoursworkedðhÞandthenumberofworkersðeÞaswellastheprofitabilityshock.Otherfactorsofproduction,suchascapital,areassumedtoberentedandoptimizationovertheseinputsareincor-poratedintoRðA;e;hÞ.2Thefunctionoðe;h;AÞisthetotalcostofhiringeworkerswheneachsupplieshunitsoflabortime.Thisgeneralspecification1.Herewearealsoassumingthatthediscountfactorisfixed.Ingeneral,thediscountwoulddependonaanda0.2.Incontrasttothechapteroncapitaladjustment,hereweassumethattherearenocoststoadjustingthestockofcapital.Thisisforconvenienceonly,andacompletemodelwouldincorporatebothformsofadjustmentcosts.\nDynamicsofEmploymentAdjustment217allowsforovertimepayandotherprovisions.Assumethatthiscompensationfunctionisincreasinginbothofitsargumentsandisconvexwithrespecttohours.Furtherweallowthiscompensationfunctiontobestatedependent.Thismayreflectacovariancewiththeidiosyncraticprofitabilityshocks(due,perhaps,toprofit-sharingarrangements)oranexogenousstochasticcomponentinaggregatewages.ThefunctionCðe;e
1Þisthecostofadjustingthenumberofworkers.Hamermesh(1993)andHamermeshandPfann(1996)pro-videalengthydiscussionofvariousinterpretationsandmotivationsforadjustmentcosts.Thisfunctionismeanttocovercostsassociatedwiththefollowing:0Searchandrecruiting0Training0Explicitfiringcosts0Variationsincomplementaryactivities(capitalaccumulation,reor-ganizationofproductionactivities,etc.)Itisimportanttonotethetimingimplicitinthestatementoftheoptimizationproblem.Thestatevectorincludesthestockofworkersinthepreviousperiod,e
1.Incontrasttothecapitalaccumulationproblem,thenumberofworkersinthecurrentperiodisnotpre-determined.Instead,workershiredinthecurrentperiodareimme-diatelyutilizedintheproductionprocess:thereisno‘‘timetobuild.’’Thenextsectionofthechapterisdevotedtothestudyofadjust-mentcostfunctionssuchthatthemarginalcostofadjustmentispositiveandincreasinginegivene
1.Wethenturntomoregeneraladjustmentcostfunctionsthatallowformorenonlinearanddiscon-tinuousbehavior.9.3QuadraticAdjustmentCostsWithoutputtingadditionalstructureontheproblem,particularlythenatureofadjustmentcosts,itisdifficulttosaymuchaboutdynamiclabordemand.Asastartingpoint,supposethatthecostofadjustmentisgivenbyh2Cðe;e
1Þ¼ðe
ð1
qÞe
1Þ;ð9:2Þ2\n218Chapter9soCðe;e
1Þisconvexineandcontinuouslydifferentiable.Here,qisanexogenousquitrate.Inthisspecificationofadjustmentcosts,theplant/firmincursacostofchangingthelevelofemploymentrelativetothestockofworkersðð1
qÞe
1Þthatremainonthejobfromthepreviousperiod.Ofcourse,thisisamodelingchoice:onecanalsoconsiderthecasewheretheadjustmentcostisbasedonnetratherthangrosshires.3Thefirst-orderconditionsfor(9.1)using(9.2)areRhðA;e;hÞ¼ohðe;h;AÞ;ð9:3ÞRðA;e;hÞ
oðe;h;AÞ
hðe
ð1
qÞeÞþbEVðA0;eÞ¼0:ð9:4Þee
1eHerethechoiceofhours,givenin(9.3),isstatic:thefirmweighsthegainstotheincreasinglaborinputagainstthemarginalcost(assumedtobeincreasinginhours)ofincreasinghours.Incontrast,(9.4)isadynamicrelationshipsincethenumberofemployeesisastatevariable.Assumingthatthevaluefunctionisdifferentiable,EVðA0;e0Þcanbeevaluatedusing(9.1),leadingtoeReðA;e;hÞ
oeðe;h;AÞ
hðe
ð1
qÞe
1ÞþbE½hðe0
ð1
qÞeÞð1
qÞ¼0:ð9:5ÞThesolutiontothisproblemwillyieldpolicyfunctionsforhoursandemploymentgiventhestatevector.Lete¼fðA;e
1Þdenotetheemploymentpolicyfunctionandh¼HðA;e
1Þdenotethehourspolicyfunction.Thesefunctionsjointlysatisfy(9.3)and(9.5).Asabenchmark,supposetherewerenoadjustmentcosts,h10,andthecompensationfunctionisgivenbyoðe;h;AÞ¼eoo~ðhÞ:Herecompensationperworkerdependsonlyonhoursworked.Further,supposethatrevenuesdependontheproductehsothatonlytotalhoursmattersfortheproductionprocess.Specifically,RðA;e;hÞ¼ARR~ðehÞð9:6ÞwithRR~ðehÞstrictlyincreasingandstrictlyconcave.Inthisspecialcase,thetwofirst-orderconditionscanbemanipu-latedtoimplythat3.Wecanstudytheimplicationsofthatspecificationbysettingq¼0in(9.2)tostudythealternative.\nDynamicsofEmploymentAdjustment219oo~0ðhÞ1¼h:oo~ðhÞSo,intheabsenceofadjustmentcostsandwiththefunctionalformsgivenabove,hoursareindependentofbotheandA.Consequentlyallvariationsinthelaborinputarisefromvariationsinthenumberofworkersratherthanhours.Thisisefficientgiventhatthemarginalcostofhoursisincreasinginthenumberofhoursworkedwhiletherearenoadjustmentcostsassociatedwithvaryingthenumberofworkers.Atanotherextreme,supposethereareadjustmentcostsðh00Þ.Further,supposethatcompensationissimplyoðe;h;AÞ¼eh;sotherearenocoststohoursvariation.Inthiscase(9.3)impliesARR~0ðehÞ¼1.Usingthis,weseethat(9.5)isclearlysatisfiedatacon-stantlevelofe.Hencethevariationinthelaborinputwouldbeonlyintermsofhours,andwewouldneverobserveemploymentvariations.Ofcourse,inthepresenceofadjustmentcostsandastrictlyconvex(inh)compensationfunction,theplant/firmwilloptimallybalancethecostsofadjustinghoursagainstthoseofadjustingthelaborforce.Thisisempiricallyrelevantsinceinthedatabothemploymentandhoursvariationareobserved.Note,though,thatitisonlyadjust-mentinthenumberofworkersthatcontainsadynamicelement.Thedynamicsinhoursisderivedfromthedynamicadjustmentofemployees.4Itisthistrade-offbetweenhoursandworkeradjustmentthatliesattheheartoftheoptimizationproblem.Givenfunctionalforms,thesefirst-orderconditionscanbeusedinanestimationroutinethatexploitstheimpliedorthogonalitycon-ditions.Alternatively,avaluefunctioniterationroutinecanbeusedtoapproximatethesolutionto(9.1)using(9.2).Weconsiderbelowsomespecifications.ASimulatedExampleHerewefollowCooperandWillis(2001)andstudythepolicyfunc-tionsgeneratedbyaquadraticadjustmentcostmodelwithsomeparticularfunctionalformassumptions.5Supposethatoutputis4.Addtothisthedynamicadjustmentofotherfactors,suchascapital.5.Asdiscussedlaterinthischapter,thismodelisusedbyCooperandWillis(2001)fortheirquantitativeanalysisofthegapapproach.\n220Chapter9aCobb-DouglasfunctionoftotallaborinputðehÞandcapital,andassumethefirmhasmarketpowerasaseller.InthiscaseconsideraRðA;e;hÞ¼AðehÞ;ð9:7Þwhereareflectslabor’sshareintheproductionfunctionaswellastheelasticityofthedemandcurvefacedbythefirm.Further,imposeacompensationschedulethatfollowsBils(1987):2oðe;hÞ¼we½w0þhþw1ðh
40Þþw2ðh
40Þ;ð9:8Þwherewisthestraight-timewage.Insteadofworkingwith(9.5),CooperandWillis(2001)solvethedynamicprogrammingproblem(9.1)withthefunctionalformsabove,usingvaluefunctioniteration.Thefunctionalequationfortheproblemis2ahðe
e
1ÞVðA;e
1Þ¼maxAðehÞ
oðe;hÞ
h;e2e
10þbEA0jAVðA;eÞforallðA;e
1Þ:ð9:9ÞInthisanalysis,decisionsareassumedtobemadeattheplantlevel.Accordinglyweassumethattheprofitabilityshockshavetwocomponents:apiecethatiscommonacrossplants(anaggregateshock)andapiecethatisplantspecific.Bothtypesofshocksareassumedtofollowfirst-orderMarkovprocesses.Theseareembeddedintheconditionalexpectationin(9.9).Inthisformulationtheadjustmentcostsarepaidonnetchangesinemployment.Furthertheadjustmentcostsdependontherateofadjustmentratherthantheabsolutechangealone.6Thepolicyfunctionthatsolves(9.9)isgivenbye¼fðA;e
1Þ.Thispolicyfunctioncanbecharacterizedgivenaparameterizationof(9.9).CooperandWillis(2001)assumethefollowing:0Labor’sshareis0.65andthemarkupis25percentsothatain(9.7)is0.72.0ThecompensationfunctionusestheestimatesofBils(1987)andShapiro(1986):fw0;w1;w2g¼f1:5;0:19;0:03g;thestraight-timewage6.Theliteratureonlaboradjustmentcostscontainsbothspecifications.CooperandWillis(2001)findthattheirresultsarenotsensitivetothispartofthespecification.\nDynamicsofEmploymentAdjustment221Figure9.1Employmentpolicyfunctions:Quadraticcostswisnormalizedto0.05forconvenience.Theelasticityofthewagewithrespecttohoursiscloseto1,onaverage.0Theprofitabilityshocksarerepresentedbyafirst-orderMarkovprocessandaredecomposedintoaggregateðAÞandidiosyncraticcomponentsðeÞ.AAf0:9;1:1gandetakeson15possiblevalues.Theserialcorrelationfortheplant-levelshocksis0.83andis0.8fortheaggregateshocks.7Thisspecificationleavesopentheparameterizationofhinthecostofadjustmentfunction.Intheliteraturethisisakeyparametertoestimate.ThepolicyfunctionscomputedfortwovaluesofAattheseparameterchoicesaredepictedinfigure9.1.Herewehaveseth¼1whichisatthelowendofestimatesintheliterature.Thesepolicyfunctionshavetwoimportantcharacteristics:7.Alternatively,theparametersoftheseprocessescouldbepartofanestimationexercise.\n222Chapter90fðA;eÞisincreasinginðeÞ.
1
10fðA;eÞisincreasinginA.Asprofitabilityincreases,sodoesthe
1marginalgaintoadjustment,andthuseishigher.Thequadraticadjustmentcostmodelcanbeestimatedeitherfromplant(firm)dataoraggregatedata.Toillustratethis,wenextdiscusstheapproachofSargent(1978).Wethendiscussamoregeneralapproachtoestimationinamodelwitharicherspecificationofadjustmentcosts.exercise9.1Writedownthenecessaryconditionsfortheoptimalchoicesofhoursandemploymentin(9.9).Provideaninterpretationoftheseconditions.Sargent:LinearQuadraticSpecificationAleadingexampleofbringingthequadraticadjustmentcostmodeldirectlytothedataisSargent(1978).Inthatapplication,Sargentassumestherearetwotypesoflaborinput:straight-timeandover-timeworkers.Theproductionfunctionislinear-quadraticineachofthetwoinputs,andthecostsofadjustmentarequadraticandsepa-rableacrossthetypesoflabor.Asthetwotypesoflaborinputsdonotinteractineithertheproductionfunctionortheadjustmentcostfunction,wewillfocusonthemodelofstraight-timeemploymentinisolation.FollowingSargent,assumethatrevenuefromstraight-timeemploymentisgivenbyR12RðA;eÞ¼ðR0þAÞe
e:ð9:10Þ2HereAisaproductivityshockandfollowsanAR(1)process.Sar-gentdoesnotincludehoursvariationinhismodelexceptthroughtheuseofovertimelabor.Accordinglythereisnodirectdependenceofthewagebillonhours.Instead,heassumesthatthewageratefollowsanexogenousprocess(withrespecttoemployment)givenbyXi¼nwt¼n0þniwt
iþzt:ð9:11Þi¼1Inprinciple,theinnovationtowagescanbecorrelatedwiththeshockstorevenues.88.Thefactorsthathelpthefirmforecastfuturewagesarethenincludedinthestatespaceoftheproblem;thatis,theyareintheaggregatecomponentofA.\nDynamicsofEmploymentAdjustment223Withthisstructurethefirm’sfirst-orderconditionwithrespecttoemploymentisgivenbyR11bEtetþ1
etþð1þbÞþet
1¼ðwt
R0
AtÞ:ð9:12ÞhhFromthisEulerequation,currentemploymentwilldependonthelaggedlevelofemployment(throughthecostofadjustment)andon(expected)futurevaluesofthestochasticvariables,productivity,andwages,asthesevariablesinfluencethefuturelevelofemploy-ment.AsdescribedbySargent,thesolutiontothisEulerequationcanbeobtainedsothatemploymentinagivenperioddependsonlaggedemployment,currentand(conditionalexpectationsof)futurewages,andcurrentand(conditionalexpectationsof)futurepro-ductivityshocks.Giventhedrivingprocessforwagesandproduc-tivityshocks,thisconditionalexpectationcanbeevaluatedsothatemploymentinperiodtissolelyafunctionoflaggedemployment,currentandpastwages.Thepastwagesarerelevantforpredictingfuturewages.SargentestimatestheresultingVARmodelofwagesemploymentusingmaximumlikelihoodtechniques.9Theparametersheesti-matedincludedðR1;h;rÞ,whereristheserialcorrelationofthepro-ductivityshocks.InadditionSargentestimatedtheparametersofthewageprocess.ThemodelisestimatedusingquarterlydataontotalU.S.civilianemployment.Interestinglyhealsodecidedtouseseasonallyunad-justeddataforsomeoftheestimation,arguingineffectthatthereisnoreasontoseparatetheresponsestoseasonalandnonseasonalvariations.Thedataaredetrendedtocorrespondtothestationarityofthemodel.Hefindsevidenceofadjustmentcostsinsofarashissignificantlydifferentfromzero.10Sargentarguesthattheseresults‘‘...aremod-eratelycomfortingtotheviewthattheemployment-real-wageobservationsliealongademandscheduleforemployment’’(p.1041).9.Sargent(1978)estimatesamodelwithbothregularandovertimeemployment.Forsimplicitywepresentedthemodelofregularemploymentalone.10.Healsodiscussesatlengththeissueofidentificationandfindsmultiplepeaksinthelikelihoodfunction.Informallytheissueisdistinguishingbetweentheserialcor-relationinemploymentinducedbylaggedemploymentfromthatinducedbytheserialcorrelationoftheproductivityshocks.\n224Chapter9exercise9.2Thereareanumberofexercisestoconsiderworkingfromthissimplemodel.1.Writeaprogramtosolve(9.9)fortheemploymentandhourspolicyfunctionsusingvaluefunctioniteration.Whataretheprop-ertiesofthesepolicyfunctions?Howdothesefunctionschangeasyouvarytheelasticityofthecompensationfunctionandthecostofadjustmentparameter?2.Solve(9.9)usingalog-linearizationtechnique.Compareyourresultswiththoseobtainedbythevaluefunctioniterationapproach.3.Considersomemomentssuchastherelativevariabilityofhoursandemploymentandtheserialcorrelationsofthesetwovariables.Calculatethesemomentsfromasimulatedpanelandalsofromatimeseriesconstructedfromthepanel.Lookforstudiesthatcharac-terizethesemomentsatthemicroand/oraggregatelevels.Or,betteryet,calculatethemyourself.Constructanestimationexerciseusingthesemoments.4.Supposethatyouwantedtoestimatetheparametersof(9.9)usingGMM.Howwouldyouproceed?9.4RicherModelsofAdjustmentInpart,thepopularityofthequadraticadjustmentcoststruc-turereflectsit’stractability.Buttheimplicationsofthisspecifica-tionofadjustmentcostsconflictwithevidenceofinactivityandburstsattheplantlevel.Thusresearchershavebeenmotivatedtoconsiderarichersetofmodels.Thosearestudiedhereandthenareusedforestimationpurposesbelow.Forthesemodelsofadjustmentwediscussthedynamicoptimizationproblemandpresentpolicyfunctions.9.4.1PiecewiseLinearAdjustmentCostsOneofthecriticismsofthequadraticadjustmentcostspecificationistheimplicationsofcontinuousadjustment.Attheplantlevel,asmentionedearlier,thereisevidencethatadjustmentismuchmoreerraticthanthepatternimpliedbythequadraticmodel.Piecewiselinearadjustmentcostscanproduceinaction.\nDynamicsofEmploymentAdjustment225ForthiscasethecostofadjustmentfunctionisgþDeifDe>0;Cðe;e
1Þ¼
ð9:13ÞgDeifDe<0:Theoptimalpolicyrulesarethendeterminedbysolving(9.1)usingthisspecificationoftheadjustmentcostfunction.Theoptimalpolicyrulewilllookquitedifferentfromtheoneproducedwithquadraticadjustmentcosts.Thisdifferenceisacon-sequenceofthelackofdifferentiabilityintheneighborhoodofzeroadjustment.Consequentlysmalladjustmentswillnotoccursincethemarginalcostofadjustmentdoesnotgotozeroasthesizeoftheadjustmentgoestozero.Furtherthisspecificiationofadjust-mentcostsimpliesthereisnopartialadjustment.Sincethemar-ginalcostofadjustmentisconstant,thereisnobasisforsmoothingadjustment.Theoptimalpolicyischaracterizedbytwoboundaries:e
ðAÞandeþðAÞ.IfeA½e
ðAÞ;eþðAÞ;thenthereisnoadjustment.Intheevent
1ofadjustment,theoptimaladjustmentistoe
ðAÞifeeþðAÞ.
1FollowingCooperandWillis(2001)andusingthesamebasicparametersasdescribedabove,wecanstudytheoptimalpolicyfunctionforthistypeofadjustmentcost.Assumethatgþ¼g
¼0:05,whichproducesinactionattheplantlevelin23percentoftheobservations.11Then(9.1)alongwith(9.13)canbesolvedusingvaluefunctioniterationandtheresultingpolicyfunctionsevaluated.Theseareshowninfigure9.2.Notethatthereisnoadjustmentforvaluesofe
1inaninterval:theemploymentpolicyfunctioncoincideswiththe45-degreeline.Outsideofthatinternaltherearetwotargets:e
ðAÞandeþðAÞ.Again,thispolicyfunctionisindexedbythevaluesofgþandg
.Sotheseparameterscanbeestimatedbymatchingtheimplicationsofthemodelagainstobservationsofemploymentadjustmentattheplantand/oraggregatelevels.Wewillreturntothispointbelow.exercise9.3Specifythedynamicprogrammingproblemforlaboradjustmentusingapiecewiselinearadjustmentcoststructure.Whatdeterminestheregionofinaction?Studythismodelnumericallybysolvingthedynamicprogrammingproblemandobtainingpolicyfunctions.11.Thisinactionrateistoohighrelativetoobservation:theparameterizationisforillustrationonly.\n226Chapter9Figure9.2Employmentpolicyfunctions:Piecewiselinearadjustmentcosts9.4.2NonconvexAdjustmentCostsTheobservationsofinactivityattheplantlevelthatmotivatethepiecewiselinearspecificationarealsousedtomotivateconsiderationoffixedcostsintheadjustmentprocess.AsnotedbyHamermeshandPfann(1996),theannualrecruitingactivitiesofeconomicsdepartmentsprovideafamiliarexampleoftheroleoffixedcosts.IntheUnitedStates,hiringrequirespostingofanadvertisementofvacancies,extensivereviewofmaterialprovidedbycandidates,travelofarecruitingteamtoaconventionsite,interviewsofleadingcandidates,universityvisits,andfinallyavotetoselectamongthecandidates.Clearly,therearefixedcostcomponentstomanyoftheseactivitiesthatcomprisethehiringofnewemployees.1212.Thisdepictionmotivatesconsiderationofasearchmodelastheprimitivethatunderliesamodelofadjustmentcosts.SeethediscussionofYashiv(2000)inchapter10.\nDynamicsofEmploymentAdjustment227Asaformalmodelofthis,considerVðA;eÞ¼max½VaðA;eÞ;VnðA;eÞforallðA;eÞ;ð9:14Þ
1
1
1
1whereVaðA;eÞrepresentsthevalueofadjustingemploymentand
1VnðA;eÞrepresentsthevalueofnotadjustingemployment.These
1aregivenbya0VðA;e
1Þ¼maxRðA;e;hÞ
oðe;hÞ
FþbEA0jAVðA;eÞ;ð9:15Þh;en0VðA;e
1Þ¼maxRðA;e
1;hÞ
oðe
1;hÞþbEA0jAVðA;e
1Þ:ð9:16ÞhInthisspecificationthefirmcaneitheradjustthenumberofemployeesornot.ThesetwooptionsarelabeledactionðVaðA;eÞÞ
1andinactionðVnðA;eÞÞ.Ineithercase,hoursareassumedtobe
1freelyadjustedandthuswillrespondtovariationsinprofitabilityevenifthereisnoadjustmentinthenumberofworkers.Notetoothatthisspecificationassumesadjustmentcostsdependongrosschangesinthenumberofworkers.Inthiswaythemodelcanpotentiallymatchtheinactioninemploymentadjustmentattheplantleveldefinedbyzerochangesinthenumberofworkers.Theoptimalpolicyhasthreedimensions.First,thereisthechoiceofwhethertoadjustornot.LetzðA;e
1ÞAf0;1gindicatethischoicewherezðA;e
1Þ¼1ifandonlyifthereisadjustment.Second,thereisthechoiceofemploymentintheeventofadjustment.LetfðA;e
1ÞdenotethatchoicewherefðA;e
1Þ¼e
1ifzðA;e
1Þ¼0.Finally,thereisthechoiceofhours,hðA;e
1Þ,whichwillreflectthedecisionofthefirmwhetherornottoadjustemployment.AstheseemploymentadjustmentsdependonðA;e
1Þthroughe¼fðA;e
1Þ,onecanalwaysconsiderhourstobeafunctionofthestatevectoralone.Therearesomerichtrade-offsbetweenhoursandemploymentvariationsembeddedinthismodel.Supposethatthereisapositiveshocktoprofitability:Arises.Ifthisvariationislargeandperma-nent,thentheoptimalresponseofthefirmwillbetoadjustemploy-ment.Hourswillvaryonlyslightly.Iftheshocktoprofitabilityisnotlargeorpermanentenoughtotriggeradjustment,thenbydefi-nition,employmentwillremainfixed.Inthatcasethemainvariationwillbeinworkerhours.Thesevariationsinhoursandemploymentareshowninfigure9.3.Thepolicyfunctionsunderlyingthisfigurewerecreatedusinga\n228Chapter9Figure9.3Employmentpolicyfunctions:Nonconvexadjustmentcostsbaselineparameterswithfixedcostsat0.1ofthesteadystateprofits.13exercise9.4Specifythedynamicprogrammingproblemforlaboradjustmentusinganonconvexadjustmentcoststructure.Whatdeterminesthefrequencyofinaction?Whatcomovementofhoursandemploymentispredictedbythemodel?Whatfeaturesofthepolicyfunctionsdistinguishthismodelfromtheonewithpiecewiselinearadjustmentcosts?Studythismodelnumericallybysolvingthedynamicprogrammingproblemandobtainingpolicyfunctions.9.4.3AsymmetriesAsdiscussedinHamermeshandPfann(1996),thereiscertainlyevi-denceinfavorofasymmetriesintheadjustmentcosts.Forexample,13.Atthisleveloffixedcosts,thereisabout50percentemploymentinaction.Again,theparameterizationisjustforillustration.\nDynamicsofEmploymentAdjustment229theremaybeacostofadvertisingandevaluationthatispropor-tionaltothenumberofworkershiredbutnocostsoffiringworkers.Alternatively,itmaybeofinteresttoevaluatetheeffectsoffiringcostsonhiringpoliciesasdiscussedinthecontextofsomeEuropeaneconomies.Itisrelativelystraightforwardtointroduceasymmetriesintothemodel.Giventheapproachtoobtainingpolicyfunctionsbysolving(9.1)throughavaluefunctioniterationroutine,asymmetriesdonotpresentanyadditionaldifficulties.Aswiththeotherparameter-izationsofadjustmentcosts,thesemodelcanbeestimatedusingavarietyoftechniques.PfannandPalm(1993)provideaniceexampleofthisapproach.TheyspecifyanadjustmentcostfunctionofgDe12Cðe;e
1Þ¼
1þe
gDeþhðDeÞ;ð9:17Þ2whereDe1ðe
e
1Þ.Ifg10,thenthisreducesto(9.2)withq¼0.AsPfannandPalm(1993)illustrate,theasymmetryinadjustmentcostsiscontrolledbyg.Forexample,ifg<0,thenfiringcostsexceedhiringcosts.Usingthismodelofadjustmentcosts,PfannandPalm(1993)esti-mateparametersusingaGMMapproachondataformanufacturingintheNetherlands(quarterly,seasonallyunadjusteddata,1971.I–1984.IV)andannualdataforU.K.manufacturing.Theyhavedataonbothproductionandnonproductionworkers,andtheemploymentchoicesareinterdependent,giventheproductionfunction.Forbothcountriestheyfindevidenceofthestandardquadraticadjustmentcostmodel:hispositiveandsignificantlydifferentfromzeroforbothtypesofworkers.Moreoverthereisevidenceofasymmetry.Theyreportthatthecostsoffiringproductionworkersarelowerthanthehiringcosts.But,theoppositeistruefornon-productionworkers.9.5TheGapApproachTheworkinCaballeroandEngel(1993b)andCaballeroetal.(1997)pursuesanalternativeapproachtostudyingdynamiclaboradjust-ment.Insteadofsolvinganexplicitdynamicoptimizationproblem,theypostulatethatlaboradjustmentwillrespondtoagapbetweentheactualanddesiredemploymentlevelataplant.Theythentestfornonlinearitiesinthisrelationship.\n230Chapter9Thethemeofcreatinganemploymenttargettodefineanemploy-mentgapasaproxyforthecurrentstateisquiteintuitiveandpow-erful.Asnotedinourdiscussionofnonconvexadjustmentcosts,whenafirmishitbyaprofitabilityshock,agapnaturallyemergesbetweenthecurrentlevelofemploymentandthelevelofemploy-mentthefirmwouldchooseiftherewerenocostsofadjustment.Thisgapshouldthenbeagoodproxyforthegainstoadjustment.Thesegains,ofcourse,arethencomparedtothecostsofadjust-ment,whichdependonthespecificationoftheadjustmentcostfunction.Thissectionstudiessomeattemptstocharacterizethenatureofadjustmentcostsusingthisapproach.14Thepowerofthegapapproachisthesimplificationofthedynamicoptimizationproblemasthetargetlevelofemploymentsummarizesthecurrentstate.However,aswewillsee,thesegainsmaybedifficulttorealize.Theproblemarisesfromthefactthatthetargetlevelofemployment,andthusthegap,isnotobservabletotheresearcher.Tounderstandthisapproach,itisusefultobeginwithadiscus-sionofthepartialadjustmentmodel.Wethenreturntoevidenceonadjustmentcostsfromthisapproach.9.5.1PartialAdjustmentModelResearchersoftenspecifyapartialadjustmentmodelinwhichthefirmisassumedtoadjustthelevelofemploymenttoatarget.15Theassumedmodeloflaboradjustmentwouldbee
e¼lðe
eÞ:ð9:18Þtt
1t
1Soherethechangeinemploymentet
et
1isproportionaltothedifferencebetweenthepreviouslevelofemploymentandatarget,e,wherelparameterizeshowquicklythegapisclosed.Wheredoesthispartialadjustmentstructurecomefrom?Whatdoesthetargetrepresent?CooperandWillis(2001)consideradynamicprogrammingproblemgivenbyðe
eÞ2k20Lðe;e
1Þ¼minþðe
e
1ÞþbEe0jeLðe;eÞ;ð9:19Þe2214.ThispresentationdrawsheavilyonCooperandWillis(2001).WearegratefultoJohnHaltiwangerandJonWillisforhelpfuldiscussionsonthistopic.15.Thestructureisusedtostudyadjustmentofcapitalaswell.\nDynamicsofEmploymentAdjustment231wherethelossdependsonthegapbetweenthecurrentstockofworkersðeÞandthetargetðeÞ.Thetargetistakenasanexogenousprocess,thoughingeneralitreflectstheunderlyingshockstoprofit-abilitythatareexplicitintheoptimizingmodel.Inparticular,sup-posethatefollowsanAR(1)processwithserialcorrelationofr.Furtherassumethattherearequadraticadjustmentcosts,para-meterizedbyk.Thefirst-orderconditiontotheoptimizationproblemisðe
eÞþkðe
eÞ
bkEðe0
eÞ¼0;ð9:20Þ
1wherethelasttermwasobtainedfromusing(9.19)tosolveforqL=qe.Giventhattheproblemisquadratic,itisnaturaltoconjectureapolicyfunctioninwhichthecontrolvariableðeÞislinearlyrelatedtothetwoelementsofthestatevectorðe;eÞ:
1e¼leþle:ð9:21Þ12
1Usingthisconjecturein(9.20)andtakingexpectationsofthefuturevalueofeyieldsðe
eÞþkðe
eÞ
bkðlreþðl
1ÞeÞ¼0:ð9:22Þ
112Thiscanbeusedtosolveforeasalinearfunctionofðe;eÞwith
1coefficientsgivenby1þbkl1rl1¼ð9:23Þ1þk
bkðl2
1Þandkl2¼:ð9:24Þð1þk
bkðl2
1ÞÞClearly,iftheshocksfollowarandomwalkðr¼1Þ,thenpartialadjustmentisoptimalðl1þl2¼1Þ.Otherwise,theoptimalpolicycreatedbyminimizationofthequadraticlossislinearbutdoesnotdictatepartialadjustment.9.5.2MeasuringtheTargetandtheGapTakingthistypeofmodeldirectlytothedataisproblematicasthetargeteisnotobservable.Intheliterature(e.g.,seethediscussioninCaballeroandEngel1993b)thetargetisintendedtorepresentthe\n232Chapter9destinationoftheadjustmentprocess.Therearetworepresentationsofthetarget.One,termedastatictarget,treatseasthesolutionofastaticoptimizationproblem,asifadjustmentcostsdidnotexist.Thusesolves(9.5)withh10andhourssetoptimally.Asecondapproachtreatseasthelevelofemploymentthatthefirmwouldchooseiftherewerenoadjustmentcostsforasingleperiod.Thisistermedthefrictionlesstarget.Thislevelofemploy-mentsolvese¼fðA;eÞ,wherefðA;e
1Þisthepolicyfunctionforemploymentforthequadraticadjustmentcostmodel.Thusthetar-getisthelevelofemploymentwherethepolicyfunction,contingentontheprofitabilityshock,crossesthe45-degreelineasinfigure9.1.FollowingCaballeroetal.(1997),definethegapasthedifferencebetweendesiredðeÞandactualemploymentlevels(inlogs):i;t~zz1e
e:ð9:25Þi;ti;ti;t
1Hereei;t
1isnumberofworkersinheritedfromthepreviousperiod.So~zzi;tmeasuresthegapbetweenthedesiredandactuallevelsofemploymentinperiodt,priortoanyadjustments,butafteranyrel-evantperiodtrandomvariablesarerealizedastheseshocksareembeddedinthetargetandthusthegap.Thepolicyfunctionforthefirmisassumedtobe16Dei;t¼fð~zzi;tÞ:ð9:26ÞThekeyoftheempiricalworkistoestimatethefunctionfð
Þ.Unfortunately,estimationof(9.26)isnotfeasibleasthetarget,andthusthegaparenotobservable.Sothebasictheorymustbeaugmentedwithatechniquetomeasurethegap.Therearetwoapproachesintheliteraturecorrespondingtothetwonotionsofatargetlevelofemployment,describedearlier.Caballeroetal.(1997)pursuethethemeofafrictionlesstarget.Toimplementthis,theypostulateasecondrelationshipbetweenanother(closelyrelated)measureofthegap,ð~zz1Þandplant-specifici;tdeviationsinhours:~zz1¼yðh
hÞ:ð9:27Þi;ti;tHere~zz1isthegapinperiodtafteradjustmentsinthelevelofehavei;tbeenmade:~zz1¼~zz
De.i;ti;ti;t16.Basedondiscussionsabove,thepolicyfunctionofthefirmshoulddependjointlyonðA;e
1Þandnotonthegapalone.\nDynamicsofEmploymentAdjustment233Theargumentinfavorofthisapproachagainreturnstoourdis-cussionofthechoicebetweenemploymentandhoursvariationinthepresenceofadjustmentcosts.Inthatcasewesawthatthefirmchosebetweenthesetwoformsofincreasingoutputwhenprofit-abilityrose.Thus,ifhoursaremeasuredtobeaboveaverage,thiswillreflectagapbetweenactualanddesiredworkers.Iftherewasnocostofadjustment,thefirmwouldchoosetohiremoreworkers.Butinthepresenceofthesecoststhefirmmaintainsapositivegap,andhoursworkedareaboveaverage.Thekeyto(9.27)isy.Sincetheleft-handsideof(9.27)isalsonotobservable,theanalysisisfurtheramendedtogenerateanestimateofy.Caballeroetal.(1997)estimateyfromDei;t¼a
yDhi;tþei;t;ð9:28ÞwheretheerrortermincludesunobservedchangesinthetargetlevelofemploymentðDeÞaswellasmeasurementerror.Caballeroetal.i;t(1997)notethattheequationmayhaveomittedvariablebiasasthechangeinthetargetmaybecorrelatedwithchangesinhours.FromthediscussioninCooperandWillis(2001),thisomittedvariablebiascanbequiteimportant.Onceyisestimated,Caballeroetal.(1997)canconstructplantspecificgapmeasuresusingobservedhoursvariations.Inprinciple,themodelofemploymentadjustmentusingthesegapmeasurescanbeestimatedfromplant-leveldata.Instead,Caballeroetal.(1997)focusontheaggregatetimeseriesimplicationsoftheirmodel.Inparticular,thegrowthrateofaggregateemploymentisgivenbyðDEt¼zFðzÞftðzÞ;ð9:29ÞzwhereFðzÞistheadjustmentrateorhazardfunctioncharacterizingthefractionofthegapthatisclosedbyemploymentadjustment.FromaggregatedatathisexpressioncanbeusedtoestimateFðzÞ.AsdiscussedinCaballeroetal.(1997),ifFðzÞis,say,aquadratic,then(9.29)canbeexpandedimplyingthatemploymentgrowthwilldependonthefirstandthirdmomentsofthecross-sectionaldistri-butionofgaps.ThefindingsofCaballeroetal.(1997)canbesummarizedasfollows:0Using(9.28),yisestimatedat1.26.\n234Chapter90Therelationshipbetweentheaverageadjustmentrateandthegapisnonlinear.0Thereissomeevidenceofinactioninemploymentadjustment.0Aggregateemploymentgrowthdependsonthesecondmomentofthedistributionofemploymentgaps.Incontrast,CaballeroandEngel(1993b)donotestimatey.Instead,theycalibrateitfromastructuralmodelofstaticoptimizationbyafirmwithmarketpower.Indoingso,theyareadoptingatargetthatignoresthedynamicsofadjustment.Fromtheirperspective,thegapisdefinedusing(9.25)whereecorrespondstothesolutionofi;tastaticoptimizationproblemoverbothhoursandemploymentwithoutanyadjustmentcosts.Theyarguethatthisstatictargetwillapproximatethefrictionlesstargetquitewellifshocksarerandomwalks.AswithCaballeroetal.(1997),oncethetargetisdetermined,ameasureofthegapcanbecreated.Thisapproachtoapproximatingthedynamicoptimizationprob-lemisappliedextensivelybecauseitissoeasytocharacterize.Fur-theritisanaturalextensionofthepartialadjustmentmodel.But,asarguedinCooperandWillis(2001),theapproachmayplaceexces-siveemphasisonstaticoptimization.17CaballeroandEngel(1993b)estimatetheirmodelusingaggre-gateobservationsonnetandgrossflowsforU.S.manufacturingemployment.Theyfindthataquadratichazardspecificationfitstheaggregatedatabetterthantheflathazardspecification.Thekeypointinbothofthesepapersistherejectionoftheflathazardmodel.BothCaballeroetal.(1997)andCaballeroandEngel(1993b)arguethattheestimatesofthehazardfunctionfromaggre-gatedataimplythatthecross-sectionaldistribution‘‘matters’’foraggregatedynamics.Putdifferently,bothstudiesrejectaflathazardspecificationinwhichaconstantfractionofthegapisclosedeachperiod.Giventhatthisevidenceisobtainedfromtimeseries,thisimpliesthatthenonconvexitiesattheplant-levelhaveaggregateimplications.Thisisanimportantfindingintermsofthewaymac-roeconomistsbuildmodelsoflaboradjustment.17.Thispointwasmadesomeyearsago.Nickell(1978)says:‘‘...themajorityofexistingmodelsoffactordemandsimplyanalyzetheoptimaladjustmentofthefirmtowardsastaticequilibriumanditisverydifficulttodeducefromthisanythingwhateveraboutoptimalbehaviorwhenthereisno‘equilibrium’toaimat.’’\nDynamicsofEmploymentAdjustment235Totheextentthattheflathazardmodelistheoutcomeofaqua-draticadjustmentcostmodel,bothpapersrejectthatspecificationinfavorofamodelthatgeneratessomenonlinearitiesintheadjust-mentprocess.But,asthesepapersdonotconsiderexplicitmodelsofadjustment,onecannotinferfromtheseresultsanythingabouttheunderlyingadjustmentcoststructure.Further,asarguedbyCooperandWillis(2001),themethodologyofthesestudiesmayitselfinducethenonlinearrelationshipbetweenemploymentadjustmentandthegap.CooperandWillis(2001)con-structamodeleconomywithquadraticadjustmentcosts.Theyas-sumethatshocksfollowafirst-orderMarkovprocess,withserialcorrelationlessthanunity.18TheyfindthatinusingeithertheCab-alleroetal.(1997)orCaballeroandEngel(1993b)measurementsofthegap,thecross-sectionaldistributionofemploymentgapsmaybesignificantinatimeseriesregressionofemploymentgrowth.9.6EstimationofaRichModelofAdjustmentCostsThusfarwehavediscussedsomeevidenceassociatedwiththequadraticadjustmentcostmodelsandprovidedsomeinsightsintotheoptimalpolicyfunctionsfrommorecomplexadjustmentcostmodels.Inthissectionwegoastepfurtheranddiscussattemptstoevaluatemodelsthatmayhavebothconvexandnonconvexadjust-mentcosts.Aswithotherdynamicoptimizationproblemsstudiedinthisbook,thereis,ofcourse,adirectwaytoestimatetheparametersoflaboradjustmentcosts.Thisrequiresthespecificationofamodelofadjustmentthatneststhevarietyofspecialcasesdescribedabovealongwithatechniquetoestimatetheparameters.Inthissubsectionweoutlinethisapproach.19LettingArepresenttheprofitabilityofaproductionunit(e.g.,aplant),weconsiderthefollowingdynamicprogrammingproblem:0VðA;e
1Þ¼maxRðA;e;hÞ
oðe;h;AÞ
CðA;e
1;eÞþbEA0jAVðA;eÞ:h;eð9:30Þ18.TheprocessistakenfromtheCooperandHaltiwanger(2000)studyofcapitaladjustment.Astheseshocksweremeasuredusingstaticlaborfirst-ordercondition,CooperandWillis(2001)studytherobustnessoftheirresultstovariationsintheseMarkovprocesses.19.ThisdiscussionparallelstheapproachinCooperandHaltiwanger(2000).\n236Chapter9Asabove,letaRðA;e;hÞ¼AðehÞ;ð9:31Þwheretheparameteraisagaindeterminedbythesharesofcapitalandlaborintheproductionfunctionaswellastheelasticityofdemand.Thefunctionoðe;h;AÞrepresentstotalcompensationtoworkersasafunctionofthenumberofworkersandtheiraveragehours.Asbefore,thiscompensationfunctioncouldbetakenfromotherstudiesorperhapsaconstantelasticityformulationmightbeadequate:w¼wþwhz.01Thecostsofadjustmentfunctionnestsquadraticandnonconvexadjustmentcostsofchangingemployment82>>>>Hne
e
1e
1;2e
1CðA;e
1;eÞ¼ð9:32Þ>>2>>Fne
e
1:Fþe
1ifee
1;2e
1VFðA;eÞ¼maxRðA;e;hÞ
oðe;h;AÞ
FF
1h;e2ne
e
10
e
1þbEA0jAVðA;eÞife