01--高级宏观：拉姆齐模型

1FrankPlumptonRamseyFrankPlumptonRamsey(1903-1930),Britishmathematicianandphilosopher,bestknownforhisworkonthefoundationsofmathematics.ButRamseyalsomaderemarkablecontributionstoepistemology,semantics,logic,philosophyofscience,mathematics,statistics,probabilityanddecisiontheory,economicsandmetaphysics.10n0t=t()ntLte=Yt(1.1)()((),()())FKtAtLt=()gtAte=(1.2).()((),()())()()KtFKtAtLtKtCtδ=−−()()()CtLt=ct()()()KtLt=kt()()()()eCtAtLt=ctBlanchardFischer1989BarroSala-I-Martin1995ZilibottiDirk,kruger.Ramsey(1928)Cass1965Koopmas19651()()()ektAt=())()eektct=−0(),()maxtctkt∞−∫..()(ektfk(0)0ee=kt11()ln(),ccθθθθ−−=kt(1.3).()(()()eefngktδ−++(())eUctdt.())()()()eeesttctngktδ=−−++kkCRRA,11Uc≠=θ11((1))(())()(())(())11gttttgteectectcteeeUctθθρρρρθθθ−−−−−−−−===−−eU(1.4)((1)g)γρθ=−−0(),()max(())eetectkteUctdt∞−∫...()(())()()()eeeestktfktctngktδ=−−++kk(0)0ee=0γ2(1.5)(,,;)(())()[(())()()()]teeeeeHckteUcttfktctngktλλδ−=+−−++e1c()et22(,,;)0eeeHcktcλ∂=∂(1.6)2.(,,;)()eeeHckttkλλ∂=−∂(1.7)3TVClim()()0ettktλ→∞=(1.8)'(())()teeUcttγλ−=(1.9).()/()'(())()ettfktngλλδ=−++(1.10)lim()()0ettktλ→∞=(1.11)1(1.9)(1.10)()tλ(1.9)t:..()''(())()'(())teeteUctcteUctγγλγ−−=−te(1.12)1.9..''(())()()()'(())eeeUctctttUctλγλ=−(1.13)1.10()tλ.''(())()[(())()]'(())eeeeUctctfktngUctγ−=−−++δ(1.14)1.14.()''(())()()['(())()]'(())eeeeeectUctctctfktngUctδγ=−−+++(1.15)()''(())'(())eeectUctUctθ=−1/σθ=1/['()/'()]['()/'()(/)ststststccducucucucdccσ]−=−(1.16)/stcc3s→tσ='()''()uccuc−1.15()ctχ.()t()kt'((efk()ct.'(())()()()()eecefktngcttctδγχθ−+++==(1.17)t2(,)eeck.'(())()()()()eecefktngcttctδγχθ−+++==(1.18)()(())()()()0eeektfktngktctδ=−++−e=(1.19).'(())()()0()eecefktngctctδγχσ−+++===.(())()()0eefktngktδγ=−+++=(1.20)(1.21)e))tngδγ=+++(1.22)(())()()eefktngkteδ=−++(1.23)4'ec0ec''0c0ek*ekgoldek**ek.0ek=.0ec=ecekDDB12()ect,()ekt^()ct,^()ktgγ()ct,()kttc,tk1n1tNtLnttLe=21tCt()()/()ctCtLt=53Uu22()00()nttceρ∞−−=∫dtρuu'(0)=∞'()0∞=nρ4c0UaAratrtwtA/ttL=tt]0tt.ttttawracna=+−−5232240()[limtvrndvttae−−→∞∫≥trn−tr01ttrrt=vdv∫24()[()]0limttrntate−−→∞≥24’3U025()()[()]nHuxewrnacρµ−−=++−−µ()0'()ntHucecρµ−−∂=⇒=∂26..()Hrnaµµ∂=−⇒=−−∂µ27lim[]0tttaµ→∞=282728263ρBenthamitefelicityfunctionnρρ=45.tttttAwLrAC=+−t.....2()/tttttttttttttAALALALAaddtwracnaLLLLL−===−=+−−629..()()''()()'()nuccenuceρµρ−−−−=−−ntρuc26e292729()/'()ntρµ−−=.''()[]('()ucccruccρ=−)210''()'()uccuc−1/['()/'()]['()/'()(/)ststststccducucucucdccσ]−=−211/stcc6210.1()ccρσ=+r.(crc)σρ=−212CIES11()1cθucθ−−=−0,1θθ≠uc()logc=1θ=213θ−1/σθ=212.crcρθ−=21427(()(0)ttrnteµµ−−=)(0)0µ2152861'()1'()1ttucrucρ++=+ρr11(1)[](1)ttccrθρ+=++1/θσ=7()m[]0ttrnttae−−→∞=li2160ta21624216trn−trn−trn−244ar.()()()[]/()rntrntrntdaedtaearne−−−−−−=−−.()ttnawttc−−=−(rne−−)t()()()000([]/)TTTrtntrtntrtntttdaedtdtwedtcedt−−−−−−=−∫∫∫t()()()00(0)TTrtnTrtntrtntTttecedtwedta−−−−−−+=∫∫a217−T216→∞~()()00(0)(0)(0)rtntrtntttcedtwedtawaW∞∞−−−−=+=+=∫∫218214(1/)[]0trtetccθρ−=219219218[(1)//]0(0)/()rtntedθθρθ∞−−+=∫cW220tt5^(,)(,)tttttYFKALFKL==221()tAteγ=^ttLAL=ttYL222^^^^()tttttyLfk==^()(/,1)tttfkFKAL=77YYLKKL8^'()ttYfkK∂=∂^^^[()'()]ttttttYfkkfkeLγ∂=−∂223(0)0f='(0)f=∞'()0f∞=Inada^^^[(,)]ttttttttLFKLRkweγ−−−Π=224tRttRrδ=+^tL^'()ttfkrδ=[phone][(^^^)'()]tttttfkkfkewγ−=22661cc^teγ−=^.^^.'()/()/tttttfkcccc()δρθγγθ−−−=−=2272ttak=kw228.ttttrkcnk=+−−tkk^ttteγ−=kw229.^^()tttternkcγγ−=+−−−^t^tc225226229kf230.^^^()()tttknkγδ=−++−3kk^tteγ−=^'()ttfkrδ=+216TVCli^0^('())m[]0tvfkndvtkeδγ−−−−→∞∫=231227230(,^^)tck4cc.^^)/()0tt=.^^()/()0ttkk=^'()tfkδρθγ=++2329^'()tfkδρθγ=++cf233^^()()ttknkγδ=−++^tγ232233TVC231rf232^**'()knδγ=−+^0'c^0c^0''c^0k^*k^goldk^**k.^0k=.^0c=^c^kDDB12(1)nρθγ+−334enρρθγγ=++eρ8333^'()tfknγδ=++335334^^*goldkk8γ10ρθγ+ρθ^0nγ+(=−(n+ρc232cTVC^0'^''c01k,DD^0''c(trtntee)γ−+()trtntteekγ−+()limtrtntteeγ−+→∞tk=∞BlanchardFischer19892BlanchardFischer19892DD230kf.^^^())tttknkγδ++−^tc^..2^^2['())]0tttdkfkkcdtγδ=−+−^'()()tfknγδ++.^0tc^tk^t0BB212^tk^tc.^^(crc)σ=−r212212^0kt→→∞7θθ11^k^*lowkθ^*khighθlowθhighθθ228CD^^()fkAkα=sn*()/(x)αγδρθ=++++δ336BarroSala-I-Martin1995P89,AppendixB1/SolowSwan*sθ=1/*sθ*s1/*sθ*s9227.^^()/()ttcc12().^^/(tkk)tdt..^^^^^^^()/()['()/()].[()/()]tttttttyykfkfkkk=10.Uu^^()[(1)]000()()ntttntttcedtuceργθγργθ∞∞−−+−−−−−==∫∫ρθγ+.^^^^()()ttttkfknkcγδ=−++−ma0xUsjkf...^^^()()tttknkγδ^tc=−++−237Romer1986Lucas198811convergenceFAkα=0k11cUσσ−=−.()ttkAknαδ=−ttkc+−1()0x1nttcemaσρσ−∞−−−∫st...()tttkAknkcαδ=−+−tk01()[()1tnttcHeAknkcθραµδθ−−−=+−+−]tt−)237238-(0ntHcceθρµ−−=⇔=13239.1[(ktHAkαµµαδ−=−=−+)]nTVClim[]0tttkµ→∞=240238.1[(tAknαµαδµ−=−−+)].1(tcAkcα)αδρθ−−−=241.0cc=.0kk=24110tAkααδρ−−−=1*1()Akααρδ−=+242*.*0()(kcAknkkαδ=⇔=−+*)2432310γ=*1()(tAknαα−+)δ*1()tAkααρδ−=+nρ234TVC9n0=0δ=kA.ttkα=−tc1*1()Akααρ−=2441.()tAkcαcαδρθ−−−=*1()AcAαααρ−=245BarroSala-I-Martin1995109BlanchardFischer1989BZilibotti10***^.^***^^*^^^**.(1)1'()()1''()0''()'()ttttttttttnfknkkkkkfkcccccfkcfkcρθγγδδρθγθθθ+−−−−++−−−==−−−−−0,0,0nγδ===24614**1.*2**.()1(1)()0ttttttAkkkkAkccccαααααθ−−−−=−−246244245*1()tAkαρα−=.*2*.1(1)0ttttkkkcccρρααθ−