Problem Set 2

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1、14.452:EconomicGrowthProblemSet2Duedate:November18,2011inrecitation.Exercise1:ConsidertheSolowgrowthmodelincontinuoustimewithconstantsavingrates,depreciationrate,populationgrowthratenandlabor-augmentingtechnologicalprogressatrateg>0,thatis,supposethe

2、pro-ductionfunctionfunctiontakestheformY(t)=F(K(t);A(t)L(t))whereA_(t)=A(t)=g>0.Recallthatthee¤ectivecapital-laborratiok(t)=K(t)=(A(t)L(t))ischaracterizedbythedi¤erentialequationk_(t)sf(k(t))=(+g+n),(1)k(t)k(t)wheref(k(t))=F(k(t);1)istheproductionfu

3、nctionnormalizedbye¤ectivelabor,andtheoutputper-capitaisgivenbyy(t)=A(t)f(k(t)):(2)Thisexercisesconcernsanapproximationforthegrowthrateofoutputper-capitaaroundthesteady-state.1.Lety(t)=A(t)f(k)denotethesteady-statelevelofoutputpercapita,thatis,thele

4、velofoutputpercapitathatwouldapplyifthee¤ectivecapital-laborratiowereatitssteady-statelevelandtechnologywereatitstimetlevel.Showthat,inaneighborhoodofthesteadystate,outputpercapitay(t)canbeapproximatedbythefollowingdi¤erentialequation:y_(t)'g(1"k(

5、k))(+g+n)(logy(t)logy(t));(3)y(t)0where"(k)=f(k)kistheelasticityoftheproductionfunctionf(k)kf(k)evaluatedatk.(Hint:First,considertheTaylorexpansionoftheright-handsideofEq.(1)withrespecttolog(k(t))aroundlogkandderiveanapproximationfork_(t)=k(t)

6、.SecondusethisapproximationalongwithEq.(2)toderiveanapproximationfory_(t)=y(t)asafunctionofthedistanceofe¤ectivecapital-laborratiofromitssteady-state,logk(t)logk.Third,1considertheTaylorexpansionofEq.(2)withrespecttolog(k(t))aroundlogktoderiveanapp

7、roximationforthedistanceofoutputper-capita,logy(t)logy(t),intermsofthedistanceofe¤ectivecapital-laborratio,logk(t)logk.CombinethelasttwostepstoderiveEq.(3).)2.InterpretEq.(3).Inparticular,whatdoesthisequationimplyaboutthegrowthrateofcountriesawayf

8、romtheirsteady-states?Whathappenstothegrowthrateasthecountriesapproachtheirsteady-states?Explainwhatdeterminesthespeedofconvergencetothesteady-stateandprovideanintuition.Exercise2:ConsideraneconomywithN<1goods,denotedbyj2f1;::;Ng,andasetHofhou