Macroeconomic Sol_midterm-2014

MacroeconomicAnalysisECON6022Fall2014Oct25,2014INSTRUCTIONS:Timingandpoints:Theexamlastsfor150minutes.Themaximalnumberofpointstobeattainedforthisexamis100points.Language:PleaseformulateyouranswersinEnglish.Identication:PleasewriteyourNAMEandUniversityNUMBERonthecoveroftheanswerbookthatyouuse.Pleasewriteinanintelligibleway,andwriteallyouranswersintheanswerbook.1CapitalShare[20Points]Ingrowthaccounting,basedontheproductionfunctionY=A
K
L1,theparameterisdeterminedbysettingitequaltotheaveragecapitalshareofacountry.a.Explainthejusticationforthisprocedure.[Hint:Youmaywanttousesomesimpleequationstoexplainit.]Solution:SinceMPK=@Y@K=
A
K1
L1,thenMPK
K=
Y.Thus,=MPK
KY.Asweknow,MPKisthemarginalproductofcapital.Incompetitiveeconomy,itequalstherentalrateofcapital.Therefore,=MPK
KYrepresentstheaveragecapitalshareofacountry.b.Whichparticularassumptionsareneeded?[Hint:Ashortanswerwithone(oratmosttwo)sentence(s)issucient.]Solution:Twoassumptionsareneededforthisprocedure:First,theproductionfunctiontakestheCobb-Douglasform;Second,theeconomyiscompetitive,whichensurestherentalrateofcapitalequalsitsmarginalproduct.2TheSolowModelwithProductivityGrowth[40Points]ConsideraSolowmodelinwhichtheproductionfunctionisY=F(K;AN)=K(AN)1,whereYistheaggregateoutput,Kcapital,Nlaborforce,andAlaborproductivity.Thisformoftechnologyissaidtobe1\labor-augmenting.AssumethatAisgrowingataconstantrate:
A=A=g.ThecapitalaccumulationequationisKt+1Kt=
Kt=s
Ytd
Kt,wheresavingrates,capitaldepreciationratedandpopulationgrowthratenareconstants.Wedeneanewvariable,capitalpereectivelabor,asfollows,~k=K=(AN).1.Derivethemarginalproductofcapital(MPK)fortheproductionfunctiongivenabove.Doesthefunctionexhibitdiminishingmarginalproductofcapital?Solution:Themarginalproductofcapitalis@Y@K=K1(AN)1:TakesecondderivativewithrespecttoK,wehave@2Y@K2=(1)K2(AN)10:Sotheproductionfunctionexhibitsdiminishingmarginalproduct.2.Pleaseshowthefollowingdynamicsfor~k:
~kt=s~kt(n+d+g)~kt:Solution:
Kt=sYtdKt,pluginYt=Kt(AtNt)1,wehave
Kt=s
Kt(AtNt)1dKt.DividebothsidesbyAtNt,wehave
KtAtNt=sKt
(AtNt)dKtAtNt=s~ktd
~kt:(1)Noticethat
~kt~kt=
KtKt
NtNt
AtAt
~kt=
KtKt
~kt
NtNt
~kt
AtAt
~kt
KtAtNt=
~kt+n~kt+g~kt:(2)Equation(1)and(2)thenshow
~kt=s~kt(n+d+g)~kt:(3)3.Solveforthesteadystatelevelofcapitalpereectivelabor,~k.Solution:Let
~kt=0,wehave~k=(sn+g+d)1=(1).4.Similarly,dene~y=Y=(AN)tobetheoutputpereectivelaborandy=Y=Ntheoutputperlabor.Solveforboththesteadystatelevelofoutputpereectivelabor~yandoutputperlabor,y.Solution:~y=(sn+g+d)=(1)andyt=At(sn+g+d)=(1).5.SupposetheeconomystaysinsteadystatebeforePeriodT.AtPeriodT,theproductivitygrowthratefallspermanentlyfromgto0.2a.ShowhowthiseconomyconvergestothenewsteadystateinaSolow-diagram.Solution:Thefollowinggureshowshowtheeconomyconvergestoanewsteady-state:(n+d+g)k!(n+d)k!y!1*y!2*f(k!)sf(k!)k!2*k!1*Inthecasewithanpermanentdecreaseinproductivitygrowthrate,theeconomywillconvergetoanewsteadystatewithhigher~kand~y.BeforePeriodt=T,break-eveninvestmentequalsactualinvestmentandtheeconomyisatthesteadystate.AtPeriodt=T,theproductivitygrowthratedecreasestozero,thereforethebreak-eveninvestmentjumpsdown,whichimpliesthat~khastoincrease.Itinceasesuntilthepointwhereitreachesthenewsteadystate.b.Plotthetrajectoryoflog(y)againsttime,beforeandafterPeriodT.Solution:Thefollowinggureshowsthetrajectoryoflog(y):tTlog(yt)Bythepropertyoflogfunction,theslopeofthetrajectoryshouldbe4ytyt.Wehaveyt=~yt
ATakelogofbothsides:ln(yt)=ln(~yt)+ln(A)3Thentaketimederivativesofbothsides,thegrowthrateofoutputpercapitacanbewrittenas4ytyt=4~yt~yt+gBeforePeriodT,outputperlaborgrowsatarateofg,because4~yt~yt=0.AfterPeriodTthecapitalpereectivelaborisincreasingtowardsanewsteadystate,buttheproductivitygrowthdropstozero,sooutputperlaborwillstillincreaseattheconvergencespeedof~yt,whichisdepictedaslessthanginthegraph.Aftertheeconomyrestsinthenewsteadystate,thegrowthrateofoutputperlaboriszero.c.Plotthetrajectoryof
y=yagainsttime,beforeandafterPeriodT.Solution:Thefollowinggureshowsthetrajectoryof
y=y,whichisthegrowthrateofoutputperlabor:tΔyyT3ConsumptionandSavingswithBorrowingConstraint[40Points]Considerarepresentativeconsumerwholivesforthreeperiods,heorshehasanexogenousendowmentstreamgivenbyy1;y2andy3andcanborrowandlendatagiveninterestrater,whichistheinterestrate.Assumethathe/shestartsoutwithnowealth(thatis,b1=0).Thediscountfactoris=1andutilityfunctionisincreasingandconcaveinconsumption.Theconsumer'sproblemis:maxc1;c2;c3u(c1)+u(c2)+u(c3)subjecttoc1+b2=y1c2+b3=y2+(1+r)
b2c3=y3+(1+r)
b3wherebtisthenancialwealthinPeriodt.1.Writedowntheinter-temporalbudgetconstraint.Solution:Theinter-temporalbudgetconstraintfortheconsumerisc1+c21+r+c3(1+r)2=y1+y21+r+y3(1+r)242.DerivetheEulerequations.Solution:Wecantransformtheconsumer'soptimizationproblemasmaxc1;c2u(c1)+u(c2)+u[(1+r)2(y1c1)+(1+r)(y2c2)+y3]FirstOrderConditionsare:FOC(c1):u0(c1)+(1)(1+r)2u0(c3)=0FOC(c2):u0(c2)+(1)(1+r)u0(c3)=0Rearrangetheterms,wehavethefollowingEulerequations,u0(c1