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%.

2.a. AccordingtoEquation

（2）,thegrowthofoutputisequaltothegrowthinlabortimesthelaborshareplusthegrowthofcapitaltimesthecapitalshareplusthegrowthrateoftotalfactorproductivity（TFP）,thatis,

DY/Y=（1-q）（DN/N）+q（DK/K）+DA/A,where

1-qistheshareoflabor（N）andqistheshareofcapital（K）.Inthisexampleq=0.3;therefore,ifoutputgrowsat3%andlaborandcapitalgrowat1%each,thenwecancalculatethechangeinTFPinthefollowingway

3%=（0.3）（1%）+（0.7）（1%）+DA/A==>DA/A=3%-1%=2%,

thatis,thegrowthrateoftotalfactorproductivityis2%.

2.b. Ifbothlaborandthecapitalstockarefixedandoutputgrowsat3%,thenallthisgrowthhastobecontributedtothegrowthinfactorproductivity,thatis,DA/A=3%.

3.a. IfthecapitalstockgrowsbyDK/K=10%,theeffectonoutputwouldbeanadditionalgrowthrateof

DY/Y=（.3）（10%）=3%.

3.b. IflaborgrowsbyDN/N=10%,theeffectonoutputwouldbeanadditionalgrowthrateof

DY/Y=（.7）（10%）=7%.

3.c. IfoutputgrowsatDY/Y=7%duetoanincreaseinlaborbyDN/N=10%,andthisincreaseinlaborisentirelyduetopopulationgrowth,thenpercapitaincomewoulddecreaseandpeople’swelfarewoulddecrease,since

Dy/y=DY/Y-DN/N=7%-10%=-3%.

3.d. Ifthisincreaseinlaborisduetoaninfluxofwomenintothelaborforce,theoverallpopulationdoesnotincreaseandincomepercapitawouldincreasebyDy/y=7%.Thereforepeople'swelfarewouldincrease.

4.Figure3-4showsoutputperheadasafunctionofthecapital-laborratio,thatis,y=f（k）.Thesavingsfunctionissy=sf（k）,anditintersectsthestraight（n+d）k-line,representingtheinvestmentrequirement.Atthisintersection,theeconomyisinasteady-stateequilibrium.Nowletusassumethattheeconomyisinasteady-stateequilibriumbeforetheearthquakehits,thatis,thesteady-statecapital-laborratioiscurrentlyk*.Assumefurther,forsimplicity,thattheearthquakedoesnotaffectpeople'ssavingsbehavior.

Iftheearthquakedestroysonequarterofthecapitalstockbutlessthanonequarterofthelaborforce,thenthecapital-laborratiofallsfromk*tok1andper-capitaoutputfallsfromy*toy1.Nowsavingisgreaterthantheinvestmentrequirement,thatis,sy1>（d+n）k1,andthecapitalstockandthelevelofoutputpercapitawillgrowuntilthesteadystateatk*isreachedagain.

However,iftheearthquakedestroysonequarterofthecapitalstockbutmorethanonequarterofthelaborforce,thenthecapital-laborratioincreasesfromk*tok2.Savingnowwillbelessthantheinvestmentrequirementandthusthecapital-laborratioandthelevelofoutputpercapitawillfalluntilthesteadystateatk*isreachedagain.

Ifexactlyonequarterofboththecapitalstockandthelaborstockaredestroyed,thenthesteadystateismaintained,thatis,thecapital-laborratioandtheoutputpercapitadonotchange.

Iftheseverityoftheearthquakehasaneffectonpeoples’savingsbehavior,thenthesavingsfunctionsy=sf（k）willmoveeitherupordown,dependingonwhetherthesavingsrate（s）increases（ifpeoplesavemore,somorecanbeinvestedinanefforttorebuild）ordecreases（ifpeoplesaveless,sincetheydecidethatlifeistooshortnottoliveitup）.

y

y=f（k）

y2

y* （n+d）k

y1

sy

0

k1k*k2 k

5.a.Anincreaseinthepopulationgrowthrate（n）affectstheinvestmentrequirement,andthe（n+d）k-linegetssteeper.Asthepopulationgrows,moresavingmustbeusedtoequipnewworkerswiththesameamountofcapitalthattheexistingworkersalreadyhave.Thereforeoutputpercapita（y）willdecreaseaswillthenewoptimalcapital-laborratio,whichisdeterminedbytheintersectionofthesy-curveandthe（n1+d）k-line.Sinceper-capitaoutputwillfall,wewillhaveanegativegrowthrateintheshortrun.However,thesteady-stategrowthrateofoutputwillincreaseinthelongrun,sinceitwillbedeterminedbythenewandhigherrateofpopulationgrowth.

y （n1+d）k

y=f（k）

yo （no+d）k

y1 sy

3

0

k1 ko k

5.b.Startingfromaninitialsteady-stateequilibriumatalevelofper-capitaoutputy*,theincreaseinthepopulationgrowthrate（n）willcausethecapital-laborratiotodeclinefromk*tok1.Outputpercapitawillalsodecline,aprocessthatwillcontinueatadiminishingrateuntilanewsteady-statelevelisreachedaty1.Thegrowthrateofoutputwillgradually