高级宏观经济学笔记

XG’s(2e)©1(v1.0)[e-mail]’s(2e)©21Solow())()(),()(tLtAtKFtY=(1.1))(kfy=2)()(tnLtL=&,)()(tgAtA=&,)()()(tKtsYtKδ−=&(1.2)()())()()(tkgntksftkδ++−=&(1.3)sf(k)(n+g+)kk*k1.1Solow()***)(kgnkfcδ++−=(1.4)()()[]sgnskgngnskfsc∂∂++−′=∂∂),,,(),,,(***δδδ*GRk()δδ++=′gngnskfGR),,,(*(1.5)()()),,,(),,,(**δδδgnskgngnsksf++=⇒()skgnkfskkfs∂∂++=+∂∂′****)()(δ2ALKk=,ALYy=,()1,)(KFkf=)(kfInada∞=′→)(lim0kfk0)(lim=′∞→kfkXG’s(2e)©3⇒())()(***kfsgnkfsk′−++=∂∂δ(1.6)⇒())()()()(******kfsgnkfkfskkfsy′−++′=∂∂′=∂∂δ(1.7))()()(****kfkfkkK′=α*kk=*kk=()**)(kgnksfδ++=()[])(1)()()(1)()()()()()(**************kkkfkfkkfkfkkfsgnkfkfkfssyysKKααδ−=′−′=′−++′=∂∂(1.8))(kkk&&=*k()**)(kkkkkkkk−∂∂≈=&&(1.9)()*kkkkk=∂∂−=&λ()()[]*ktktk−−=λ&(1.10)()()[]**0kkektkt−+≈−λ()[]()())()()()(*****kfkfkgngngnkfskkkkk′++−++=++−′−=∂∂−≡=δδδλ&[]()δα++−=gnkK)(1*(1.11)GrowthAccounting)()()()()()()()()()(tAtAtYtLtLtYtKtKtYtY&&&&∂∂+∂∂+∂∂=(1.12))()()()()()()()()()()()()()()()()()()()(tAtAtAtYtYtAtLtLtLtYtYtLtKtKtKtYtYtKtYtY&&&&∂∂+∂∂+∂∂=)()()()()()()(tRtLtLttKtKtLK++≡&&αα(1.13)XG’s(2e)©4)()()()()()()()()()(tRtLtLtKtKttLtLtYtYK+−=−&&&&α3(1.14)Solow2Ramsey-Cass-koopmans2.1Y=F(K,AL)Solow(2.2)(2.3)1dtHtLtCueUtt)())((0∫∞=−=ρ(2.1)θθ−=−1)())((1tCtCu40θ()01−−−gnθρ(2.2))(tC)(tLHρ2.2))((),()(tkfKALKFtr′=∂∂=(2.3)[]))(()())(()(),()(tkftktkftALALKFtW′−=∂∂=(2.4)(effectivelabor)))(()())(()()()(tkftktkftAtWtw′−==(2.5)∫∫∞=−∞=−+≤0)(0)()()()0()()(ttRttRdtHtLtWeHKdtHtLtCe(2.6)∫==tdrtR0)()(τττs[]∫∞=−−+=0)()()()()()()0()(ttRsRsRdtHtLtCtWeHKeHsK(2.7)0)(lim)(≥−∞→HsKesRs(2.8)31)()(=+ttKLαα4Constant-relative-risk-aversionCRRAXG’s(2e)©5)(tc)()()(tctAtC=[][]()θθθθθθθθθθθ−=−=−=−−−−−−−−1)()0(1)()0(1)()(1)(1111111tceAtceAtctAtCgtgt(2.9)(2.1)()∫∫∞=−−−−∞=−−−=−=0111011)()0()0()(1)(tntgttttdttceeeHLAdtHtLtCeUθθθθρθθρ∫∞=−−−=011)(ttdttceθθβ(2.10)HLAB)0()0(1θ−≡()gnθρβ−−−≡1(2.5)HLA)0()0(()()∫∫∞=+−∞=+−+≤0)(0)()()0()(ttgntRttgntRdtetwekdtetce(2.11))(tc(2.11)(2.10)Lagrangian∫∞=−−−=011)(ttdttceBθθβL()()−++∫∫∞=+−∞=+−0)(0)()()()0(ttgntRttgntRdtetcedtetwekλ(2.12)5()tgntRteetcBetcL+−−−=⇒=∂∂)()(0)(λθβ(2.13)(2.13)()()tgndrtgntRtctBt++−=++−=−−∫=0)(ln)(ln)(lnlnτττλλθβ(2.14)(2.14)t()gntrtctc++−=−−)()()(&θβ(2.15)θθρθβgtrgntrtctc−−=−−−=)()()()(&(2.16)5XG’s(2e)©62.3c(2.16)(2.3)(2.16)θθρgtkftctc−−′=))(()()(&(2.17)kk(n+g)k())()())(()(tkgntctkftk+−−=&(2.18)dc/dt=0dk/dt=0Ekck(0)k*E1E2saddlepathk*12.1kc2.10=c&0=k&ckk(0)Eck2.4:Ramsey-Cass-KoopmansPareto-efficientXG’s(2e)©7Ramsey-Cass-KoopmansESolow(n+g)gGolden-RuleLevelofCapitalRamsey-Cass-Koopmans(1.5)gngnkfGR+=++=′δ)(6(2.19)0=c&(2.17)gkfθρ+=′)(*(2.20)(2.20)(2.19)(2.2)()01)()(*−−−=′−′gnkfkfGRθρ(2.21))(•′fGRkk*2.22Ramsey-Cass-Koopmans*kmodifiedgolden-rulecapitalstock2.5(2.17)ρ*k2.10=c&G(t)())()()())(()(tkgntGtctkftk+−−−=&(2.23)()[]()∫∫∞=+−∞=+−−+≤0)(0)()()()0()(ttgntRttgntRdtetGtwekdtetce(2.24)G(t)2.10=k&3Diamond3.1DiamondRamsey-Cass-Koopmans6Ramsey-Cass-Koopmans0=δXG’s(2e)©8Diamondt=0,1,2,….Lttn()11−+=ttLnLtC1tC2ttθρθθθ−++−=−+−111111211tttCCU0θ1−ρ(3.1)),(ttttLAKFY=()11−+=ttAgA)(ttkfr′=)()(ttttkfkkfw′−=3.2t()()tttttCAwrC11121−+=++(3.2)tttttwACrC=++++121111(3.3)(3.3)(3.1)++−+−++−=++−+−121111211111111tttttttCrCwACCλθρθθθL(3.4)λθ=⇒=∂∂−ttCC110L(3.5)λρθ1121211110+−+++=+⇒=∂∂tttrCCL(3.6)θρ1111211++=++tttrCC(3.7)(3.7)(3.3)()()()()ttttwArCθθθθρρ−+++++=11111111(3.8)()()()()θθθθρρ−++++=111111)(rrs(3.9)XG’s(2e)©9[]ttttwArsC)(111+−=(3.10)3.3ktttttwALrsK)(11++=(3.11)11++ttLA()()trtwrsgnk)(11111++++=(3.12)1+trtw()()[])()())((11111tttttkfkkfkfsgnk′−′++=++(3.13)(3.13)1+tktktkttkk=+1kCobb-Douglasρθ+==⇒=21)(1srs()αααkwkkf−=⇒=1)(()()()ααρttkgnk−+++=+1211111(3.14)2.2XG’s(2e)©10k0k1k1k2k*Oldktkt+145k*New1k*New22.2k3.42.244.1R&D7()[]()[]αα−−−=1)(1)()(1)(tLatAtKatYLK10α(4.1)[][]θγβ)()()()(tAtLatKaBtALK=&0B0≥β0≥γ(4.2))()(tsYtK=&(4.3))()(tnLtL=&0≥n(4.4)7Cobb-DouglassaKaLXG’s(2e)©114.2(4.1)(4.3)()()ααααα−−−−−=111)()()(11)(tLtAtKaastKLK&(4.5)K(t)()()αα−−−≡111LKKaascα−=≡1)()()()()()(tKtLtActKtKtgKK&(4.6)()[])()(1)()(tgntgtgtgKAK−+−=α&(4.7)(4.2)A(t)γβLKAaBac≡1)()()()()()(−=≡θγβtAtLtKctAtAtgAA&(4.8)())(1)()()(tgntgtgtgAKAA−++=θγβ&(4.9)0)(=tgK&0)(=tgA&(4.7)(4.9)nggAK+=βγβθnggAK−−=1(4.10)(dgK/dt0)dgK/dt=0(dgK/dt0)n0gAgKgKgA0-n/dgA/dt=0(dgA/dt0)(dgA/dt0)=1=1-)/(a)(b)4.1gKgA(4.2)KA()θβ+4.10)(=tgK&XG’s(2e)©120)(=tgA&1+θβ4.2a()11−βθEE()ngAβθγβ+−+=1*nggAK+=**(4.11)sR&DaKaLgKgKgAgA00nndgA/dt=0dgA/dt=0dgK/dt=0dgK/dt=0-n/-n/g*Kg*AE(a)+1(b)+14.21+θβ1+θβ1+θβ4.2(b)s1=+θβn4.3a1+θβn4.3bXG’s(2e)©13gKgKgAgA00ndgA/dt=0dgK/dt=dgA/dt=0dgK/dt=0-n/(a)(b)4.31=+θβ4.3AK[]αα−=1)()()()(tLtAtKtY(4.12)φ)()(tBKtA=0B0φ(4.13)(4.12)(4.13)(4.3)(4.4)(4.13)(4.12)()ααφαα−−−=111)()()()(tLtKBtKtY(4.14)()ααφαα−−−=111)()()()(tLtKtKsBtK&(4.15)4.2()αφα−+11φ11φn1φ1=φn0n=04.14)()(tbKtY=αα−−≡11LBb(4.16))()(tsbKtK=&(4.17)sbAY=AKAKXG’s(2e)©1455.1Solow[]αα−=1)()()()(tHtAtKtY(5.1)YKASolowHlaborservicerawlaborhumancapital)()()(tKtsYtKδ−=&(5.2))()(tgAtA=&)()(tnLtL=&(5.3)E)()()(EGtLtH=(5.4))(•GE8EeEGφ=)(0φ(5.5)ky)()()()()(tLEGtAtKtk=(5.6))()()()())(()(tLEGtAtYtkfty==(5.7)()()())()(