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
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