MalthusandSolow(中级宏观经济学,香港中文大学)

Chapter6EconomicGrowth:MalthusandSolowTheMalthusianModelofEconomicGrowth•Malthusargued:advancesinthetechnologyforproducingfoodincreasedpopulationgrowthnoincreaseinthestandardoflivingunlessthereweresomelimitsonpopulationgrowth•Adynamicmodelwithmanyperiods•Confineattentiontowhathappensinthecurrentperiodandthefutureperiod•Aggregateproductionfunctionwithconstantreturnstoscale:(1)whereiscurrentaggregateoutputiscurrentfixedsupplyoflandiscurrentlabor•Thereisnoinvestmentandnogovernmentspending•Assumeeachpersoniswillingtoworkatanywageandhasoneunitoflabortosupply(anormalization),sothatinEquation(1),isboththepopulationandthelaborinputY),(NLzFYLNN•Supposepopulationgrowthdependsonthequantityofconsumptionperworker(2)whereisthepopulationinthefuture(next)periodisanincreasingfunctionisaggregateconsumptionsothatiscurrentconsumptionperworker•,mainlyduetothefactthathigherfoodconsumptionperworkerreducesdeathratesthroughbetternutrition)('NCgNN'NgCNC0)/(/NCddg•Allgoodsproducedareconsumed,soC=Y.•Hence,(3)•ThenuseEquation(3)tosubstituteforCinEquation(2):(4)•Theconstant-returns-to-scalepropertyoftheproductionfunctionimpliesthat•AftermultiplyingeachsidebyN,Equation(4)canberewrittenas(5)),(NLzFYC)),(('NNLzFgNN)1,(),(NLzFNNLzFNYNNLzFgN))1,(('•PopulationgrowthdependsonconsumptionperworkerintheMalthusianmodel•isthesteadystateforthepopulation•Ifthenpopulationincreases•Ifthenpopulationdecreases*N*NNNN'*NNNN'SteadyStateAnalysisoftheMalthusianModel•Recall•Letting,and•ThenfromEquation(2):•Inthesteadystate,sothencanbedetermined)1,(NLzFNY)(lzfyc)(/'cgNNNYyNLlNCc*'NNN1'NN*c•Inpanel(b),isdetermined•Inpanel(a),isdeterminedfromtheper-workerproductionfunction•Steadystatepopulationisgivenby*c*l**lLNTheSteadyStateEffectsofanIncreaseinz•Supposetheeconomyisinitiallyinasteadystate,with,whichthenincreasesonceandforalltimeto•unchanged•falls•Steadystatepopulationincreases1z2z*c*l*N•Populationincreasesovertimetoitssteadystatevalue•increasesattimeTconsumptionperworkerincreasesthendeclinetoitssteadystatevalue•Anotherexample:populationcontrolbothc*andl*increase.zHowUsefulistheMalthusianModelofEconomicGrowth?•BeforetheIndustrialRevolutioninabout1800:economicgrowthconsistentwiththeMalthusianModel•Fromtheperspectiveoftheearly21stcentury:MalthuswaswrongWhy?-Didnotallowfortheeffectofincreasesinthecapitalstockonproduction,and-DidnotaccountforalltheeffectsofeconomicforcesonpopulationgrowthTheSolowModel:ExogenousGrowthConsumers:•Thepopulationgrowsovertime:whereisthepopulationinthefutureperiodistherateofgrowthinthepopulation•Assumethatconsumersconsumeaconstantfractionofincomeineachperiod:whereiscurrentconsumptionistheaggregatesavingsrateNnN)1('YsC)1('NnCsTheRepresentativeFirm:•Constantreturnsproductionfunctionwithcapitalandlaborinputs:whereisoutputperworkeriscapitalperworker•whereisconstantdepreciationrateand)(kzfyykd10dIKdK)1(CompetitiveEquilibrium:•Equilibriumcondition:ICYKdNKszFK)1(),(nkdnkszfk1)1(1)(),(NKszFI•Quantityofcapitalperworkerconvergestoaconstant,quantityofoutputperworkerconvergestoaconstant,•Ifs,nandzareconstant,realincomeperworkercannotgrownobettermentinlivingstandards*k)(**kzfySteadyStateAnalysis•Inthelongrun,whentheeconomyconvergestothesteadystatequantityofcapitalperworker,k*,allaggregatequantities(K,Y,IandC)growattheraten.AnalysisoftheSteadyState•Inthesteadystate,byrearranging,nkdnkszfk1)1(1)(*****)()(kdnkszfComparativeStatics:AnIncreaseintheSavingsRate•Thecurve,,shiftsuplevelsofcapitalperworkerandoutputperworkerarehigherBUTthereisnoeffectonthegrowthratesofaggregatevariables)(*kszf•BeforetimeT,aggregateoutputisgrowingattheconstantrate,•SavingsrateincreasesattimeT•AftertimeT,outputthenconvergesinthelongruntoanewhighersteadystategrowthpathnConsumptionperWorkerandGoldenRuleCapitalAccumulation•Consumptionperworkerinthesteadystateis•(goldenrulequantityofcapitalperworker)givesthemaximumconsumptionperworker,•Propertyofthegoldenrule:***)()(kdnkzfc*grk**cdnMPkComparativeStatics:AnIncreaseinLaborForceGrowth•Anincreaseinthelaborforcegrowthrate(n)causesadecreaseinthequantityofcapitalperworkerandoutputperworker•Growthratesinaggregateoutput,aggregateconsumptionandaggregateinvestmentincreaseComparativeStatics:AnIncreaseinTotalFactorProductivity•TheSolowModelpredictsthatacountry’sstandardoflivingcancontinuetoincreaseinthelongrunonlyiftherearecontinuingincreasesintotalfactorproductivity(z)MathematicalSolution(Appendixp.636)•Inthesteadystate,•Hence,•Thesignisambiguous,sothatconsumptionperworkercouldincreaseordecreasewithanincreaseinthesavingsrate***)()(kdnkzfcdsdkdnkzfdsdc***])('[•Thegoldenrulesteadystatequantityofcapitalperworkersolvestheproblem•solvesor•As,])()([max***kdnkzfk*grk0)()(**kdnkszf)()(**grgrgrkzfkdns0)(*dnkfzgr0])()([***kkdnkzf•Recallinthesteadystate,•TotallydifferentiatingEquation(A.1),weget•Hence,solvingfortheappropriatederivatives,0)()(**kdnkszf0)(')(***dnkszfkzfdsdk0)('***dnkszfkdndk0)(')(***dnkszfksfdzdk0)()(])([*****dzksfdnkdskzfdk