多恩布什宏观经济学第十版课后习题答案03.doc

多恩布什宏观经济学第十版课后习题答案03.doc

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CHAPTER3

SolutionstotheProblemsintheTextbook

ConceptualProblems:

1.Theproductionfunctionprovidesaquantitativelinkbetweeninputsandoutput.Forexample,theCobb-Douglasproductionfunctionmentionedinthetextisoftheform:

Y=F（N,K）=AN1-qKq,

whereYrepresentsthelevelofoutput.（1-q）andqareweightsequaltothesharesoflabor（N）andcapital（K）inproduction,whileAisoftenusedasameasurefortheleveloftechnology.Itcanbeeasilyshownthatlaborandcapitaleachcontributetoeconomicgrowthbyanamountthatisequaltotheirindividualgrowthratesmultipliedbytheirrespectiveshareinincome.

2. TheSolowmodelpredictsconvergence,thatis,countrieswiththesameproductionfunction,savingsrate,andpopulationgrowthwilleventuallyreachthesamelevelofincomepercapita.Inotherwords,apoorcountrymayeventuallycatchuptoaricheronebysavingatthesamerateandmakingtechnologicalinnovations.However,ifthesecountrieshavedifferentsavingsrates,theywillreachdifferentlevelsofincomepercapita,eventhoughtheirlong-termgrowthrateswillbethesame.

3. Aproductionfunctionthatomitsthestockofnaturalresourcescannotadequatelypredicttheimpactofasignificantchangeintheexistingstockofnaturalresourcesontheeconomicperformanceofacountry.Forexample,thediscoveryofnewoilreservesoranentirelynewresourcewouldhaveasignificanteffectonthelevelofoutputthatcouldnotbepredictedbysuchaproductionfunction.

4. InterpretingtheSolowresidualpurelyastechnologicalprogresswouldignore,forexample,theimpactthathumancapitalhasonthelevelofoutput.Inotherwords,thisresidualnotonlycapturestheeffectoftechnologicalprogressbutalsotheeffectofchangesinhumancapital（H）onthegrowthrateofoutput.Toeliminatethisproblemwecanexplicitlyincludehumancapitalintheproductionfunction,suchthat

Y=F（K,N,H）=ANaKbHcwitha+b+c=1.

Thenthegrowthrateofoutputcanbecalculatedas

DY/Y=DA/A+a（DN/N）+b（DK/K）+c（DH/H）.

5. Thesavingsfunctionsy=sf（k）assumesthataconstantfractionofoutputissaved.Theinvestmentrequirement,thatis,the（n+d）k-line,representstheamountofinvestmentneededtomaintainaconstantcapital-laborratio（k）.Asteady-stateequilibriumisreachedwhensavingisequaltotheinvestmentrequirement,thatis,whensy=（n+d）k.Atthispointthecapital-laborratiok=K/Nisnotchanging,socapital（K）,labor（N）,andoutput（Y）allmustbegrowingatthesamerate,thatis,therateofpopulationgrowthn=（DN/N）.

6. Inthelongrun,therateofpopulationgrowthn=（DN/N）determinesthegrowthrateofthesteady-stateoutputpercapita.Intheshortrun,however,thesavingsrate,technologicalprogress,andtherateofdepreciationcanallaffectthegrowthrate.

7. LaborproductivityisdefinedasY/N,thatis,theratioofoutput（Y）tolaborinput（N）.Asurgeinlaborproductivitythereforeoccursifoutputgrowsatafasterratethanlaborinput.IntheU.S.wehaveexperiencedsuchasurgeinlaborproductivitysincethemid-1990sduetotheenormousgrowthinGDP.Thissurgecanbeexplainedfromtheintroductionofnewtechnologiesandmoreefficientuseofexistingtechnologies.Manyclaimthattheincreasedinvestmentinanduseofcomputertechnologyhasstimulatedeconomicgrowth.Furthermore,increasedglobalcompetitionhasforcedmanyfirmstocutcostsbyreorganizingproductionandeliminatingsomejobs.Thus,withlargeincreasesinoutputandaslowerrateofjobcreationweshouldexpectlaborproductivitytoincrease.（Oneshouldalsonotethatahigher-skilledlaborforcealsocancontributetoanincreaseinlaborproductivity,sincethesamenumberofworkerscanproducemoreoutputifworkersaremorehighlyskilled.）

TechnicalProblems:

1.a. AccordingtoEquation

（2）,thegrowthofoutputisequaltothegrowthinlabortimesthelaborshareplusthegrowthofcapitaltimesthecapitalshareplustherateoftechnicalprogress,thatis,

DY/Y=（1-q）（DN/N）+q（DK/K）+DA/A,where

1-qistheshareoflabor（N）andqistheshareofcapital（K）.Thusifweassumethattherateoftechnologicalprogress（DA/A）iszero,thenoutputgrowsatanannualrateof3.6percent,since

DY/Y=（0.6）（2%）+（0.4）（6%）+0%=1.2%+2.4%=+3.6%,

1.b. Theso-called"Ruleof70"suggeststhatthelengthoftimeittakesforoutputtodoublecanbecalculatedbydividing70bythegrowthrateofoutput.Since70/3.6=19.44,itwilltakejustunder20yearsforoutputtodoubleatanannualgrowthrateof3.6%,

1.c. NowthatDA/A=2%,wecancalculateeconomicgrowthas

DY/Y=（0.6）（2%）+（0.4）（6%）+2%=1.2%+2.4%+2%=+5.6%.

Thusitwilltake70/5.6=12.5yearsforoutputtodoubleatthisnewgrowthrateof5.6%.