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1、1.IntheUnitedStates,thecapitalshareofGDPisabout50%,theaveragegrowthintotaloutputis3%peryear,thedepreciaterateis5%peryear,thecapital-outputratiois4.SupposethattheproductionfunctionisCobb-Douglas,sothatthecapitalshareofoutputisconstant,andtheUnitedStateshasbeeninasteadystate
2、.(10points)a)Whatmustthesavingratebeintheinitialsteadystate?△Y/Y=0.03=>n=0.03sf(k)=(n+δ)ks=(n+δ)k/f(k)=(0.03+0.05)*4=0.32b)Whatisthemarginalproductofcapitalintheinitialsteadystate?MPK*K/Y=0.5=>MPK=0.5/(K/Y)=0.5/4=0.125c)Whatwillthecapital-outputratiobeattheGoldenRulesteady
3、state?MPK=(n+δ)=0.03+0.05=0.08MPK*K/Y=0.5=>K/Y=0.5/0.08=6.25d)WhatmustthesavingratebetoreachtheGoldenRatesteadystate?sf(k)=(n+δ)ks=(n+δ)k/f(k)=0.08*6.25=0.52.ProveeachofthefollowingstatementsaboutthesteadystateoftheSolowmodelwithpopulationgrowthandtechnologicalprogress.(10
4、points)a)Thecapital-outputratioisconstantInthesteadystate,weknowthatsy=(+n+g)k.Thisimpliesthatk/y=s/(+n+g).Sinces,δ,n,andgareconstant,thismeansthattheratiok/yisalsoconstant.Sincek/y=[K/(L*E)]/[Y/(L*E)]=K/Y,wecanconcludethatinthesteadystate,thecapital-outputratioisconstantb
5、)Capitalandlaboreachearnaconstantshareofaneconomy’sincome.Weknowthatcapital’sshareofincome=MPK(K/Y).Inthesteadystate,weknowfrompart(a)thatthecapital-outputratioK/Yisconstant.WealsoknowthattheMPKisafunctionofk,whichisconstantinthesteadystate;thereforetheMPKitselfmustbeconst
6、ant.Thus,capital’sshareofincomeisconstant.Labor’sshareofincomeis1-[capital’sshare].Hence,ifcapital’sshareisconstant,weseethatlabor’sshareofincomeisalsoconstanta)Totalcapitalincomeandtotallaborincomebothgrowattherateofpopulationgrowthplustherateoftechnologicalprogress,n+g.W
7、eknowthatinthesteadystate,totalincomegrowsatn+g--therateofpopulationgrowthplustherateoftechnologicalchange.Inpart(b)weshowedthatlabor’sandcapital’sshareofincomeisconstant.Ifthesharesareconstant,andtotalincomegrowsattheraten+g,thenlaborincomeandcapitalincomemustalsogrowatth
8、eraten+g.b)Therealrentalpriceofcapitalisconstant,andtherealwagegrowsattherateoftechnologi
