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1、TheSolowgrowthmodel1.1AssumptionsòTheproductionfunction:Y(t)=F(K(t),A(t)L(t))ALiseffectivelabor.Labor-augmentingHarrod-neutralY(t)=F(A(t)K(t),L(t))Capital-augmentingY(t)=A(t)F(K(t),L(t))Hicks-neutralòConstantreturnstoscale.F(cK(t),cA(t)L(t))=cF(K(t),A(t)L(t))∀c≥0Nospecializatio
2、n:theeconomyisbigenough.Nootherimportantfactors,e.g.Landandothernaturalresources⎛K⎞1⇒F⎜,1⎟=F(K,AL)⎝AL⎠AL⇔y=f(k)kistheamountofcapitalperunitofeffectivelabor.òf(0)=0,f'(k)>0,f"(k)<0òInada(1964)conditions:limf'(k)=∞,limf'(k)=0k→0k→∞Toensurethatthepathoftheeconomydoesnotdivergeα1−α
3、òExample:Cobb-DouglasproductionfunctionF(K,AL)=K(AL)0<α<1toverifyitsconstantreturntofindtheintensiveformtocheckallconditionstoseethatalltechnicalprogressesaresameòTheevolutionoftheinputsintoproductiondL(t)L&(t)≡=nL(t)dtA&(t)=gA(t)K&(t)=sY(t)−δK(t)1Weassumen+g+δ>0òWhatcanweexpec
4、tinsuchasimplemodel?1.2Thedynamicsofthemodelòk&(t)=sf()(k()t−n+g+δ)k(t)InvestmentperunitofeffectivelaborkActualandbreak-eveninvestmentInadaconditionensuresthatthetwolinesmusteventuallycrossf"(k)<0ensuresthattheyintersectonlyonceòThephasediagramforkintheSolowmodelk&k∗k2òThebalan
5、cedgrowthpath:asituationwhereeachvariableofthemodelisgrowingataconstantrate.“thesteadystate”(BarroandSala-i-MartinEconomicGrowth1999)∗*sf(k)=()n+g+δk*(∗())y=fks,n,g,δK&Y&==n+gKYK/LY/Laregrowingatrateg.1.3TheimpactofachangeinthesavingrateòThesavingrateisexogenousintheSolowmodelò
6、Theimpactonoutput:òThelong-runeffectInvestmentperunitofeffectivelaborkTheeffectofanincreaseinthesavingrateoninvestment3òTheshort-runeffectskGrowthrateofY/LLn(Y/L)timeòTheleveleffect:Apermanentincreaseinthesavingrateproducesatemporaryincreaseinthegrowthrateofoutputperworker.òQua
7、ntitativeImplicationsapproximationsaroundthelong-runequilibrium∗∗()∂y()∗()∂ks,n,g,δ=f'ks,n,g,δ∂s∂s(∗())()∗()sfks,n,g,δ=n+g+δks,n,g,δ∀s∗∗()∗∂k()∗()∂k⇒sf'k+fk=n+g+δ∂s∂s∗()∗∂kfk⇒=()()∗∂sn+g+δ−sf'k∗()()∗∗∂yf'kfk⇒=()()∗∂sn+g+δ−sf'k∗(∗)kf'k∗∗()∗∂ysf()kαKk⇒==∗∗()∗∗∂sykf'k1−α()k1−K()∗f
8、k4∗1òEmpiricalevidenceshows:α(k)isapproximately.Hence,
