随机Learning-by-doing经济增长模型

200499:100026788(2004)0920009206Learning2by2doing,(,430074):Learning2by2doing,2,.:;;:O29;F224:AStochasticLearning2by2doingEconomicGrowthModelZHOUShao2bo,HUShi2geng(DeptofApplMath,HuazhongUniversityofScienceandTechnology,Wuhan430074,China)Abstract:Weinvestigateastochasticeconomicgrowthmodelwithlearning2by2doingproductionfunction.Byusingstochasticoptimalmethods,weobtainexpectedgrowthrate,risk2freerateandpropensitytoconsumeatthesteadystate.Finally,weanalyzeelaboratelytheimpactoftaxationandtechnologyparameterandtechnologyvolatilityongrowthrate,risk2freeandconsume2wealthratio.Keywords:learning2by2doing;volatility;economicgrowthrate:2003209222:(1971-),,,,,,Email:[e-mail]1R&D(ResearchandDevelopment),,,,,,Learning2by2doing.Romer1986[1]1990[2],,,.,,Bellman[3-6].,,,,Clemens[7-9]Hamiltonian,Learning2by2doing,2,.Cobb2Douglas:dY(t)=F(K(t),A(t)L(t))(dt+dz(t))=CK(t)A(A(t)L(t))1-A(dt+dz(t)),(1)0A1,C0,K(t),L(t),1,A(t),,,,A(t)=G(F(K(t),A(t)L(t))),(2)dz(t),0,R2dt.B(t),dB(t)=r0B(t)dt,(3)©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.W(t),W(t)=K(t)+B(t).S,T(t)dT(t)=SF(K(t),A(t)L(t))(dt+dz(t)).(4),(1)-(4),dW(t)=[r0(W-K)+(1-S)(r(t)K(t)+e(t))-C(t)]dt+dX(t),(5)r(t),e(t)(),C(t),(5),,:dX=(1-S)(r(t)K(t)+e(t))dz(t).maxC,KV(0)=E00C(t)1-H1-He-Btdt,s.t.(5),B0,H0.1Hamiltonian[7]HC,K,W,K,5K5W=C(t)1-H1-He-Bt+K[r0(W-K)+(1-S)(r(t)K(t)+e(t))-C(t)]+R2W25K5W,(6)K,R2W=E[dX]2dt=(1-S)2(r(t)K(t)+e(t))2R2.:5H5C=C(t)-He-Bt-K=0,(7)5H5K=K[-r0+(1-S)r(t)]+125R2W5K5K5W=0,(8)dK=-5H5Wdt+5K5WdX=-Kr0dt+5K5WdX.(9)V(W(t),t),V(W,t)=EtsC(s)1-H1-He-Bsds,Bellman-5V5t=maxC,KE[dW]dt5V5W+12R2W52V5W2+e-BtC(t)1-H1-H.(10)-VtC,K,BellmanC,K,:5(-Vvt)5C=C(t)-He-Bt-5V5W=0,(11)5(-Vvt)5K=5V5W[-r0+(1-S)r(t)]+125R2W5K52V5W2=0.(12)Bellman(10)W,:VtW+VWWE[dW]dt+VW55WE[dW]dt+12V=0,VtW=-VWWE[dW]dt+VW55WE[dW]dt+12V=VWtdt+VWWdW+12V©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.=-VWWE[dW]dt+VW55WE[dW]dt+12V=VWW(dW-E[dW])-VW55WE[dW]dtdt=VWWdX-VWr0dt.(12)K=VW,H=-Vt,(10)-(13)(6)-(9)..,limtEt[H(t)]=0.(7)Ito,dK=e-Bt-BC-Hdt-HC-H-1dC+H(H+1)2C-H-2(dC)2=K-Bdt-HdCC+H(H+1)2(dC)2C2,(8)(9)5K5W,dK,,K,E[dC]Cdt=r0-BH+H+12E[dC]2C2dt.(14)Ramsey[10],W(t)C(W),Ito’sdC(W)=5C5WdW+52C5W2(dW)2.,Merton[11],,C(W)=MW,M,.,,,,dWW=dBB=dKK.(15)2,,.,Y,,,,,A(t)=GF(K(t),A(t)L(t)),G0,G,,.(1)A=(CG)1AL1-AAK,F=(C)1A(GL)1-AAK.(16)1,(16)r(t)=FK=ACKA-1(AL)1-A=AC1AG1-AA,(17)e(t)=FL=(1-A)CKAA1-AL-A=(1-A)C1AG1-AAK,(18)(17)(18),,AC1AG1-AAdz(1-A)C1AG1-AAKdz.,,0,B(t)=0,K=W,(7),(17)(18),(8)r0=A(1-S)C1AG1-AA[1-(1-S)C1AG1-AAQR2]=AS[1-HR2S],(19)S=(1-S)C1AG1-AA,,,,.dCC=dKK,dC=MdW=MdK,(1),(17)(18),=r0-BH+H+12E[dC]2C2dt=r0-BH+H+12E[dK]2K2dt119Learning2by2doing©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.=r0-BH+H+12R2(1-S)2(rK+e)2K2=r0-BH+H+12R2(1-S)2(C1AG1-AA)2=AS-BH+R2S2H+12-A.(20)(16),A(t)L,,,(20)L:,,.,,dK=dY-dT-dC,dT=SdY,dC=MdK,,K,dKK=(1-S)dYK-dCK,M=(1-S)C1AG1-AA-=BH+H-AHS+R2S2A-H+12.(21)2,.,2.(20)(21),2,H+12-A,.H1,H+12-A0,,2.,,,,,,.,H1,H+12-A0,,-,,,,,,.Ito,E0[K1-H]=E0[K(0)1-He(1-H)[(-12RK2)t+RK(z(t)-z(0))]]=K(0)1-He(1-H)(-12RK2H)t,(22)R2KK.V(0)=E00(MK)1-H1-He-Btdt,,V(0)=K(0)1-HM1-H(1-H)[B-(1-H)(-12RK2H)].:Et[H(t)]=EtC1-H1-He-Bt+K[(1-S)(rK+e)-C]+RX225K5W=EtM1-H1-H+(S-M)M-H+12S2R2M-He-BtK1-H=M1-H1-H+(S-M)M-H+12S2R2M-HK(0)1-He-B-(1-H)-12R2KHt,B(1-H)(-12R2KH),limtEt[H(t)]=0,.2120049©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.3,2.2()H+12A,1)S31=1+A(1-S)[2H2R2S(H+12-A)]-1,S31,.2)G31=12AC-1A(1-S)-1R-2H-1A-H+12-1A1-A,G31,.3)R2S2(H+12-A)R2,R2BL(20)SG,55S=AH5S5S+2R2S5S5S(H+12-A)=0,S31=1+A(1-S)2H2R2SH+12-A-1,S=S31,525S2=2H+12-AR2(C1AG1-AA)20.,S31S,S31.1).55G=AH5S5G+2R2S5S5GH+12-A=0,G31=12AC-1A(1-S)-1R-2H-1(A-H+12)-1A1-A,G=G31,525G2=2(H+12-A)R2(5S5G)20,,G31G,G31.2)L3).H+12A1,H1,,,,,H1,H+121,2,,,L3()1)S32=1-12H-1R-2C-1AG-1-AA,r0S32,.2)G32=2HR2(1-S)C1AAA-1,r0G32,.3)R2S2HAHAGR2,S.r0HR2L(19)SG,319Learning2by2doing©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.5r05S=A5S5S1-2HR2S=0,S32=1-12H-1R-2C-1AG-1-AA,S=S32,52r05S2=-2AHR2(5S5S)20,,S32r0S,r0S32.1).5r05G=A5S5G1-2HR2S=0,G32=2HR2(1-S)C1AAA-1,G=G32,52r05G2=-2AHR2(5S5G)20,,G32r0G,r0G32.2).3).4(-)H+12A.1)S31=1+A(1-S)2H2R2S(H+12-A)-1,-MS31,.2)G31=12AC-1A(1-S)-1R-2H-1(A-H+12)-1A1-A,2MG31,.3)R2S2(A-H+12)R2,2R2B.2.24,H+12A1)SS31,2,,,,,,,,L2)GG31,2,,,,,,,,L3)H+12A,H1,2R2,H1,H+12A,-R2,,,,,LChristianeClemens!(144)4120049©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.4,,,t1,t2,t3,,,,L,,L,,,,,,24,L:[1].[J].,1999,5(7):32-36.(14):[1]RomerPM.Increasingreturnsandlong2rungrowth[J].JournalofPoliticalEconomy,1986,94:1002-1037.[2]RomerPaulM.Endogenoustechnologicalchange[J].JournalofPoliticalEconomy,1998,(5)partÊ:S71-S102.[3]GongLT,ZouHF.Directpreferencesforwealth,theriskpremiumpuzzle,growth,andpolicyeffectiveness[J].JournalofEconomicDynamicandControl,2002,26:247-270.[4]GokanY.Alternativegovernmentfinancingandstochasticendogenousgrowth[J].JournalofEconomicDynamicsandControl,2002,26:681-706.[5]ObstfeldM.Evaluatingrisk