龚六堂高级宏观03

Ramsey-Cass-Koopmans1.Ramsey-Cass-Koopmans..Solow.IS-LM...Ramsey(1928),.1965CassKoopmansRamsey-Cass-Koopmans.ttt)(tA)(tA)(tA)(tr)(trA)(tA)(trAt)(tw)(tL)()(tLtw()()()()rtAtwtLt+t)(tC()()()()()()AtwtLtrtAtCt=+−1t)(/)()(tLtAta=)(/)()(tLtCtc=tn1)()()()()()(tnatctatrtwta−−+=)(tA0lim{()exp[(())]}0ttatrvndv→∞−−≥∫3ntr−)()(tr++→RRtcu:))(((.)0,(.)0uu′′′≥≤510)(lim,)(lim0=′∞=′∞→→cucucc10ρ0(())nttUucteedρ∞−=∫t6nρ)(tc)(tk0)0(kk=()()[()]ntHucewrnacρλ−−=++−−λt()()0ntHucecρλ−−∂′=−=∂()Hrnaλλ∂=−=−−∂lim()0tatλ→∞=()(()cucrcucc)ρ′=−−′′()()ucccuccρ′′−′rtdctdt+t(())uctdc′rttdcdt+dcrdt)1(+dt+(())(1)uctdtrdtdc′++dtt+t(())(1)tuctdtrdtedcρ−′++tρ−dβ∞−=∫100000()()()()ttnttUuCedtucLedtucLedtucLeedtρρρ∞∞∞−−∞−====∫∫∫∫,.01L=,,0()tUucetβ.52(())(1)(())tuctdtrdtedcuctdcρ−′′++=(())(())(())uctdtuctuctcdt′′′′+≅+1teρdtρ−≅−((())(()))(1)(1)(())uctuctcdtrdtdtdcuctdcρ′′′′++−=dc{[(())()(())]()/}0uctructcodtdtdtρ′′′−++=dt()odt0lim()/0dtodtdt→=0dt→(())()(())0uctructcρ′′′−+=()()()ucccucσ′′=′1(()()uccucσ′−=−′′)cts(())/(())(()/())()/()((())/(()))ucsuctdcsctcsctducsuct′′−′′ts→(())/(())(()/())()()/()((())/(()))()ucsuctdcsctuccsctducsuctucc′′′−→′′′′−(())(())(())00()()(0)()TTrTnTrtntrtntaTectedtawtedt−−−−−−+=+∫∫()(1/)()torttrvdv=∫∞→T(())(())00()(0)()(0)(0)rtntrtntctedtawtedtaw∞∞−−−−=+=+∫∫rr=()()()ucccucσ′′=′1(())()(0)rttctceσρ−=(0)(0)[(0)(0)]cawµ=+)0(µ[()(1)//]01(0)rtntedσσρσµ∞−−+=∫t)(tr)(tr1σ)(tr(0)µ1σ53(0)µ1σ=()rt(0)µ)(tr(,)YFKL=KLLYy/=LKk/=(,)()[()()]FKLrKwLLfkrkwδδ−+−=−+−δrwf()(,1)KfkFL=,max(,)()KLFKLrKwLδ−+−(),()()fkrwfkkfkδ′′=+=−—ka=()()kfkcnkδ=−−+/cCL=()[()]()cucfkcuccρδ′′=−−′′−0limexp{(())}0ttkfkndvδ→∞′−−−=∫),(LKF(,)FKLCKKδ=++()()kfkcnkδ=−−+0()nttUuceeρ∞−=∫dt54(()){()()}Huctfkcnkµδ=+−−+()ucµ′=()Hnkµρµ∂=−−∂()lim0nttkeρµ−−→∞=()(())()cucfkcuccρδ′′=−−′′−cγkγ*cγ*kγ*1[()]cfkγρδσ′=−−*()()()kkfkcnkγδ=−−+*{()}kckfknδγ′=−−−*,ck***0ckγγ==*cγ*kγ0*kγk∞→k0)(→′kf0)(→′kf*0cγ*cγ*kγ0*kγk0→k∞→′)(kf0*cγ*cγ*kγ0**==kcγγ0**==kcγγ()()0fkcnkδ−−+=()0fkρδ′−−=**,kc**,kc55()()kfkcnkδ=−−+[()]cufkcucρδ′′′=−−−′**,kc**[()()]()()kfknkkccδ′=−+−−−*****()()()()uccfkkuc′′′=−−′′k1µ2µ120fnµµδ′+=−−120ufuµµ′′′=−′′1µ2µ**,kc.—cc.k,0=k0=0|()()kcfknkδ=∂′=−+∂0|0cck=∂=∂.kkggk0|0∂∂=kkcgkk0|0∂∂=kkc.3.1.10=k0=c..k=0=k0=ck*k*k0=ck...*k*knkδ=−+0cc..1.0=kc0=c0=k*kgkkcf**()()()goldfknδ′=+()goldfkδ′−.Ramsey-Cass-Koppmansnδρ+=′)(*kf56δ−′)ˆ(kfρnρ1()csfk=−11(),()1cfkAkucσασ−−==−10α0σ0)(=+−−kncAkδα01=−−−δρααkA***()1()cnsfkαδρδ+=−=+nρα*s—()cfk()czfk=()()zzcfkkckzcfkckγα′==−=−*1()[]()()zfkzsσγδ1ρσσ−′=−++−*1sσ=1()[]zfkzσγσ−′=−1zσσ−=1zσσ−0zγ1zσσ−0zγ0zγ=*1sσ1zσσ−*1sσ1zσσ−1()[]()zzfkkzfkzσγγσ−′′′=−+*1sσ1zσσ−0zγ0zγ0zγ0s*1sσ0s*1sσ0s57*1sσ=1()stσ=*1sσ1()stσ0s*1sσ1()stσ0ssss*s*s*sttt——σα=—*(cnkρδδσ+⎛⎞)=−+⎜⎟⎝⎠—σα—*(cnkρδδσ+⎛⎞)−+⎜⎟⎝⎠—σα—*(cnkρδδσ+⎛⎞−+⎜⎟⎝⎠)+—0kkk−0kk=()ktk=0t≤≤∞0kk≠lim(),lim()ttktkctc→∞→∞==()()0ktct0t≤≤∞,kk+−()0fknk−=58,ck0=k0=ck*k*kk=0=c*kk0cc..1.0=k(0)c(0)c0=k0=k(0)0f=(0)c0c=0k=()fkδ′−n*(0)kkckkrδ−′))0((kfρr/cccγ=/cccγ=kγyγ(0)kk()()kfkcnkkγδ=−−+()/kkwcddkkγγ=−−t()()wfkkfk′=−1[(())kkwcwcfknkkkkσργγδσσ−′=−++−+−−]1(())0wcfknkkσρδσσ−′+−+−−≤0kγ0kγ1(())0wcfknkkσρδσσ−′+−+−−2()11[(())][(())]kwfkwcfknfknkkδρσρσργδδσσσσσ′−−−−−′′−+−+−+−+−591(())0fknσρδσσ−′−+−≤0kγ0kγ1(())0fknσρδσσ−′−+−22[()]/[(()]0kwfkfknkγδρδ′′−−−−−*(0)kk0kγ0kγ*(0)kk0kγ*(0)kk0kγ()kt()yt()ct*k*y*cαAkkf=)(()()kfkcnkδ=−−+()[()]()cucfkcuccρδ′′=−−′′−(1)loglog/[log]()kckdkAeendtαδ−−=−−+(1)log[log]1[]kdcAedtααρδσ−−=−−(1)loglog/kckAeenαδ−−−=+(1)logkAeααρδ−−=+**[log]log[log](1)()0logkdknndtkdccdtcρδρδααρδα+⎡⎤⎡⎤⎡⎤−+−⎢⎥⎢⎥⎢⎥=⎢⎥⎢⎥⎢⎥−+⎢⎥⎢⎥⎢⎥−⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎣⎦12212[4()()[()]]nαρδεξξρδδσα−+=±++−+nξρ=−e*loglog(constant)tkkε=+ε()()yykfkkyfkγk′==603.1.23.1.10..),,(tLkFY=KLtχ(,)tYFkLeχ=ˆtYyLeχ=ˆtKkLeχ=(,)()[()()]tFKLrKwLLfkrkweχδδ−−+−=−+−δrw,ˆmax(,)()KLFKLrKwLδ−+−(),[()()]tfkrwfkkfkeχδ′=+=−′ka=knckfkˆ)(ˆ)ˆ(ˆδχ++−−=tLeCLCcχ−==/ˆ/ˆ()ˆ[()]()cucfkcuccρδσχ′′′=−−−−′0}))ˆ((exp{ˆlim0=−−−′−∫∞→ttdvnkfkχδ**ˆ,ˆkcˆˆ0kc==ˆ()()0ˆ()0fkcnkfkχδρδσχ−−++=′−−−=**ˆ,ˆkc1.2***ˆ)()ˆ(ˆknkfcδχ++−=2Sidrauski61δχ++=′nkfgold)ˆ(δ−′)ˆ(goldkfn+χσχδρ−−=′)ˆ(*kfδ−′)ˆ(kfσχρ+χσρ)1(−+n——kLK=/)(/kfLY=)()(kfkkfw′−=—kkfkfrw−′=)(/)(/χ..AdamSmithJohnRae1937SamuelsonSamuelson..Samuelson.Samuelson.Uzawa(1968)BeckerMulligan(1997)Laibson(19972001)MarshallTakashiKamihigashi(2002)GongZou(2002b)..Uzawa.2.1++→RRtcu:))(()(s∆∫=∆sdttus0))(()(β0)0(=∆0u0)(,0)(,0)(′′′uuuβββ0)()(′−uuuββ∫∞∆−0)(maxdtecut)(kfy=623.2.6()kfkcnk=−−∫∞∆−0)(maxdtecut3.2.60(0)kk=.dssusd))(()(β=∆0()(())ucUeucβ∞−∆d=∆∫0(0)kk=()(())dkfkcnkducβ−−=∆()()(())(())ucfkcnkHucucλββ−−=+λ{()}()()(())ufkcnkucucucλβλβ+−−′′′−='()(())dHfkdkuλλλλβnc∂−=−=−∆∂lim0kλ∆→∞={()}()()(())ufkcnkucucucλβλβ+−−′=′′+))3.2.8Euler3.2.8((nfkλλβ′=−−3.2.103.2.63.2.10.(,)ck(,)kλ.3.2.7(,)ckλλ=()(())ccuuufkcnkββλββ′−=′+−−222()()(())((())((()))ccccccuuuuuufkcnkuuufkcnkufkcnkβββββββλββββ′′′′′′′′′−−−−−+=−′′+−−+−−)cc22()(()((()ckcuufknufkcnkβββλββ′′′−−=′+−−))).()kfkcnk=−−(())(()kcccnfkfkcλλβλλ′=−−+−−)nk2.2,**(,)ck0==ck63*()0fkcnk−−=**()(())fknu