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1、1.IntheUnitedStates,thecapitalshareofGDPisabout50%,theaveragegrowthintotaloutputis3%peryear,thedepreciaterateis5%peryear,thecapital-outputratiois4.SupposethattheproductionfunctionisCobb-Douglas,sothatthecapitalshareofoutputisconstant,andtheUnitedStateshasbeeninasteadystate.(lOpoints)a)Wh
2、atmustthesavingratebeintheinitialsteadystate?AY/Y=0.03=>n=0.03sf(k)=(n+》)ks=(n+》)k/f(k)=(0.03+0.05)*4=0.32b)Whatisthemarginalproductofcapitalintheinitialsteadystate?MPK*K/Y二0.5=>MPK=0.5/(K/Y)=0.5/4二0」25c)Whatwillthecapital-outputratiobeattheGoldenRulesteadystate?MPK=(n+d)=0.03+0.05=0.08M
3、PK*K/Y=0.5=>K/Y=0.5/0.08=6.25d)WhatmustthesavingratebetoreachtheGoldenRatesteadystate?sf(k)=(n+》)ks=(n+5)k/f(k)=0.08*6.25=0.52.ProveeachofthefollowingstatementsaboutthesteadystateoftheSolowmodelwithpopulationgrowthandtechnologicalprogress・(10points)a)Thecapital-outputratioisconstantInthe
4、steadystate,weknowthatsy=(+n+g)k.Thisimpliesthatk/y=s/(+n+g)・Sinces,&n,andgareconstant,thismeansthattheratiok/yisalsoconstant.Sincek/y=[K/(L*E)]/[Y/(L*E)]=K/Y,wecanconcludethatinthesteadystate,thecapital-outputratioisconstantb)Capitalandlaboreachearnaconstantshareofaneconomy?sincome.Wekn
5、owthatcapital'sshareofincome=MPK(K/Y).Inthesteadystate,weknowfrompart(a)thatthecapital・outputratioK/Yisconstant.WealsoknowthattheMPKisafunctionofk,whichisconstantinthesteadystate;thereforetheMPKitselfmustbeconstant.Thus,capitafsshareofincomeisconstant:.Labor，sshareofincomeis1・[capital'ss
6、hare].Hence,ifcapital9sshareisconstant,weseethatlaborsshareofincomeisalsoconstanta)Totalcapitalincomeandtotallaborincomebothgrowattherateofpopulationgrowthplustherateoftechnologicalprogress,n+g・Weknowthatinthesteadystate,totalincomegrowsatn+g—therateofpopulationgrowthplustherateoftechnol
7、ogicalchange.Inpart(b)weshowedthatlabor，sandcapital'sshareofincomeisconstant.Ifthesharesareconstant,andtotalincomegrowsattheraten+g,thenlaborincomeandcapitalincomemustalsogrowattheraten+g.b)Therealrentalpriceofcapitalisconstant,andtherealwagegrowsattherateoftechnologicalp
