《高级宏观经济学(2)》课件-7.pdf

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1、IIIEconomicGrowth(Continued)D.EndogenousGrowth:Romer's(1986)Model1BasicIdea²AvariantofAKmodel:Usinglearning-by-doing(investing)toeliminatethetendencyfordiminishingreturnsbyassumingthatknowledgecreationisasideproductofinvestment.2Firms²Thelabour-augmentingneoclassicalproductionfunctionfor¯rm

2、iYi=F(Ki;AiLi)PwhereL=iLiisconstant.AssumethatproductivityAigrowthcomesfromlearning-by-doingthatworksthrougheach¯rm'sinvestmentandthateach¯rm'sknowledgeisapublicgoodthatother¯rmscanaccessatzerocost.²TheassumptionsaboutproductivitygrowthimpliesthatAicanbewrittenasAi=KandYi=F(Ki;KLi)whereeach

3、¯rmfacesdiminishingreturnstocapitalKi:@F=@Ki>0;@2F=@K2<0.i²Letf(ki;K)´F(Ki=Li;K)andyi´Yi=Li=f(ki;K),thena¯rm'spro¯tisLi[f(ki;K)¡(r+±)ki¡w]:51Pro¯tmaximizationgivesf1(ki;K)=@yi=@ki=r+±f(ki;K)¡kif1(ki;K)=@Yi=@Li=w:²Inequilibrium,all¯rmschoosethesamelevelofcapital,sothatwehaveki=kandK=kL.Since

4、f(ki;K)ishomogeneousofdegreeonein(ki;K),wehavef(ki;K)=ki=f~(K=ki)=f~(L);wheref(ki;K)=kiorf~(L)istheaverageproductofcapital(APK).f~(L)satis¯esf~0(L)>0andf~00(L)<0.NoteLisconstantandsoisAPKf~(L)becausethelearning-by-doingelim-inatesdecreasingreturnstocapital.²Di®erentiatingf(ki;K)=ki=f~(K=ki)

5、withrespecttoki,weget202f1(ki;K)=ki¡f(ki;K)=ki=f~(K=ki)(¡K=ki);whichimplies0f1(ki;K)=f~(L)¡Lf~(L)orequivalently,0r=f1(k1;K)¡±=f~(L)¡Lf~(L)¡±:Sothemarginalproductofcapital(MPK)f1isbelowAPKandMPKisincreasinginL(becausef~00(L)<0).523HouseholdsThehouseholds'problemisgivenby01Zc1¡µ¡11¡½tBCmaxe@A

6、dtc01¡µsubjecttoa_=ra+w¡candR¡tlimae0r(v)dv¸0:t!1Theoptimalconditionis°c=_c=c=(r¡½)=µandthetransversalityconditionisR¡tlimae0r(v)dv=0:t!14Equilibirum²Equilibriumconditions:(i)a=k(hence_a=k_);(ii).ki=k:²FromSections2and3,wehave0°c=_c=c=(1=µ)[f~(L)¡Lf~(L)¡±¡½]where°cisconstantaslongasLisconst

7、antand°cisincreasinginL(thescalee®ect).²Toobtain°k=k=k_,werewritethehouseholdbudgetconstraintk_=rk+w¡c:53Since0w=f(k;K)¡kf1(k;K)=f~(L)k¡k[f~(L)¡Lf~(L)]and0r=f~(L)¡Lf~(L)¡±,wehavek_=f~(L)k¡c¡±kor°k=k=k_=f~(L)¡±¡c=k:Notethat_a=a=k=k_=_c=candthereisnotransi