Econ. 5 Spin 998 C. Sims inea Raional Expecaions odel Execise Answes (i) - (iii): The FC s in he eneal case ae C: C = A: λ = βe λ + + µ + λ [] [] wih µ = when (3) is no indin, and λ = when () is no indin. (Equaions in his answe shee ae in squae ackes, while hose in he oiinal polem se ae in odinay paenheses.) To answe (ii) fis, jus se λ in [] and [], o oain C = [3] µ =. [4] Because hee is no dynamic consain acive, hee is no convenional ansvesaliy condiion in his vesion of he polem. The equaion sysem in C, A, and µ, when [3] and [4] ae comined wih he indin consain (3), is C A = µ. [5] If we inoduce a vaiale X = Y o cas he full sysem ino fis-ode fom, he equaion desciin he evoluion of he exoenous andom income vaiale Y ecomes he wo-equaion sysem Y 5 +.. Y z. [6] X X = To fom he full 5x5 sysem in C, A, µ, Y, and X, we have o comine [5] and [6]. Since each involves a sepaae lis of vaiales, he Γ maices of coefficiens will e lock diaonal. In paicula, we will have Γ = I, Γ = 5.. [7]
Ψ =, Π =, C =. [8] Fo (i) he sysem is moe complicaed. The se of equaions we will use, in which λ akes he place of µ fom he sysem aove, is oained y sein µ o zeo in [] and [] and sackin up he esuls wih he acive consain and he exoenous vaiale dynamics, o aive a Γ = β + Ψ = Hee he vaiales ae odeed C, A, λ, Y, X. iv):, Γ, Π =,. 5 = + 5.., [9] C =. [] We don eally need he appaaus of maix decomposiions o check he soluion fo he case whee (3) always inds. This implies C is idenically one, which is he saiaion level of consumpion. o hihe value of he ojecive funcion is oainale han wha we e wih C idenically one (which is c βh. The only quesion is whehe his is feasile. Recall ha in his soluion we assume ha () neve inds. Is his possile? Wih A held a zeo y he indin consain, () simply asses C Y. Wih C always equal o one, his means ha Y mus always equal o exceed one. Unde hese cicumsances, when cuen income always suffices o suppo he saiaion level of consumpion, i is clealy feasile and opimal o accumulae no capial and o consume a he saiaion level a all imes. Bu is a level of Y ha always equals o exceeds one consisen wih he consain on exoenous vaiales iven y (4)? Jus aely. Takin uncondiional expecaions of all ems in ha equaion allows us o conclude ha EY =. If Y wih poailiy one, while also EY =, hen necessaily Y = wih poailiy one. Thus fo his soluion o e viale and consisen wih all he equaions of he sysem, i would have o e ue ha hee is no andom disuance in (4), o equivalenly ha z = wih poailiy one fo all. Fo he ohe case, whee () u no (3) inds, we do need o cay ou maix decomposiions. Fo a eneal 5x5 maix his would e impacical as a hand calculaion, u he. 5
maices in [9] ae lock ianula, wih a 3x3 lock in he uppe lef and a x in he lowe ih. This makes all he had calculaions x o 3x3, which is quie feasile y hand. The fis ask is o inve Γ. The invese of a lock ianula maix is lock ianula wih he same sucue and has he inveses of is diaonal sumaices on he diaonal. nce we have compued hese, he uppe ih lock of he invese is found diecly fom solvin he equaions in he uppe ih lock of Γ Γ = I. To e specific, he lowe ih lock of he invese is jus he ideniy. The invese of he uppe lef lock is found y saihfowad calculaion o e β + β + β +. [] If we le H sand fo he uppe ih cone of Γ, hen he condiion we solve o e H is which educes o β + β + + = H I, [] β H = The maix whose oos we need o analyze is Γ β + β + β + Γ, o. [3] + 5... [4] This maix muliplicaion is easy ecause evey column of he ih-hand maix excep he fouh has only a sinle elemen, so ha evey column of he poduc excep he fouh is jus a scala muliple of a column of he lef-hand maix. The esul is 3
β + + β + 5.. β + 5.. The chaaceisic polynomial of his maix is easily found as eβ + λj + λ λdλ 5. λ+. i. [6] 3 The oos of he quadaic faco in his expession ae complex and equal o.866e ± iπ. Thus hey ive ise o a componen of he soluion ha decays in maniude as.866 (which has a half life of 4.8 yeas) and oscillaes wih a peiod of 6 yeas. So lon as he inees ae is posiive, hee is a leas one unsale oo in he sysem, iven y +. The ohe non-zeo oo, β +, may o may no exceed one in asolue value. The convenional ansvesaliy condiion is β E λ A. [7] We know fom [] ha E λ ows a he ae β +. Theefoe we expec ha ansvesaliy will ule ou soluions wih componens ha ow as fas as +. Thus we will impose as a sailiy condiion ha he componen of he soluion coespondin o he + oo e suppessed. The ohe non-zeo oo, β +, may o may no equal o exceed +, accodin o whehe o no β + d i. So we have o conside some cases. Fis, suppose + is sicly he laes oo. To find a lef eienveco of [5] of he fom [ a c d] coespondin o he eienvalue ρ, we solve he equaion [ a c d]. β + + β + 5.. β + = ρ [ a c d] 5.. (Why no [ a c d] insead of [a c d ]? Because i uns ou he fis elemen has o e zeo, so nomalizin on he fis elemen of he veco only woks fo he ρ = oo.) Fo ρ=+, his ecomes he sysem [5] 4
e = + a + = + 5. + c + d = + c. + c = + d + a β + = +. [8] j Clealy a = and = + β. The coefficiens c and d can hen e solved fo fom he las wo equaions of [8] and ae found o e 3 + 6 c = 4 + + The sailiy condiion has he fom, hen, d = 3 + 4 + + The exisence condiion is ha he column space of e spanned y ha of,. [9] e j λ = + β A + cy + dy.. [] c d Γ Ψ = + c [] [ c d] Γ Π = + + β. [] This will always e saisfied, ecause as lon as β and + ae posiive, >. (f couse in his simple example and c ae scalas, so his spannin condiion jus equies ha he ihhand side of [] e non-zeo if he ih-hand side of [] is. Fom [9] i is easy o check ha + c, so ha he sailiy condiion fixes he value of η and heey uaanees hee is no possiiliy of non-uniqueness. So ou conclusion is ha hee is in his case a unique soluion o all he FC s includin convenional ansvesaliy. When + β, hen + is no lone he laes oo, and oh oos, + and + β, would eneae violaions of ansvesaliy if pesen. We expec eneally ha when hee is only one endoenous eo, no moe han one sailiy condiion can e imposed, houh hee will e excepions. The second sailiy condiion has coefficiens found fom = β + a + β = + β + a β + = + β 5. + c + d = β + c. + c = β + d. [3] 5
This sysem s soluion has a Checkin fo exisence, we fom = =, so i implies as a sailiy condiion A = cy dy. +. 5c. [4] e + β c d j c d + c Γ Ψ =, [5] + c whee c j, fo example is he c componen of he eienveco associaed wih he j h oo, and + β c d e j e c d j + + β Γ Π =. [6] β + As is usual in cases like his wih moe unsale oos han η s, we find hee ha he ih-hand side of [5] is no in he space spanned y (i.e., in his x case, is no a scala muliple of) he ih-hand side of [6]. (The alea is a lile edious. oe ha in his case he uppe em on he ih-hand side of [6] is neaive, and he lowe em posiive. I is no oo had o show ha oh componens of he ih-hand side of [5] ae posiive. Thus no soluion exiss in his case. Havin done all his wok, we mus now noe an unpleasan fac. o only has he second case iven us no equiliium, u he fis case, which seemed o poduce a unique soluion, has acually iven us a spuious soluion. In he second case, he polem is jus ha he ue soluion eneally has () indin fo a while, hen (3) indin. Tha is, in his case, whee discounin of he fuue is vey heavy, opimal ehavio involves delieaely consumin availale wealh, unil i is one and (3) ecomes indin. Thus he messae ha ou linea sysem canno poduce a soluion is coec. The fis case, houh, seems o saisfy evey condiion fo an opimum. I has a concave ojecive funcion, a convex consain se, and saisfies all FC s, includin convenional ansvesaliy. Howeve, his is a case whee he convenional ansvesaliy condiion is inadequae. I may e easies o see ha he poposed soluion is no a soluion y a diec + β > he soluion will make A explode a ha ae, and o keep (3) non- aumen. If indin i will have o explode in a posiive diecion. Bu hen fom he fom of he sailiy condiion, which makes C depend posiively on A, his implies C also evenually explodes upwad exceedin, evenually. Bu we can ceainly impove on any soluion ha implies C eve exceeds one. To do so, a evey dae whee C exceeds one we se C=, while keepin he A ime pah unchaned. This of couse makes () non-indin a hese daes, u his is ceainly feasile. Since C is he saiaion level of consumpion, his aleed policy impoves uiliy in compaison wih ou appaen soluion o he polem. If convees exponenially owad zeo in expecaion. If C eve ecomes eae han one, hen we can impove on he soluion as aove jus y swichin o C= on he daes whee he supposed soluion says o have C>. If no, hen we have fom [] ha EC + > C, all. This allows us o apply a useful esul known as he mainale conveence heoem: + β <, he FC [] implies ha λ 6
If X is a sochasic pocess saisfyin E X + X, all (which makes i a sumainale), and if hee is a andom vaiale (o a consan) Y such ha E Y < and X < Y, all, hen X convees wih poailiy one. Conveence of X wih poailiy one means ha he poailiy of an X sequence ha fails o convee as an odinay sequence of eal numes is zeo. The heoem clealy implies ha, if C emains always elow one, i mus convee o somehin. Bu he decision ule, oehe wih he consain, implies ha C always depends on he ealizaion of Y wih a fixed coefficien. Hence he unceainy aou is value one sep ahead is always he same, and i canno possily e convein o anyhin. The ue soluion o he polem can e found only numeically. I will e well appoximaed y he decision ule we have deived fo he linea sysem in he ane whee i implies C considealy less han one. As he linea decision ule sas o imply C close o one, he ue ule sas o imply lowe C (hence hihe savin and moe apid owh in A), and when A es lae enouh ha inees on A alone exceeds one, C ecomes exacly one. Wha s won wih ansvesaliy hee? The polem is ha he A pah a he soluion is no epesenaive of he full ane of A s in he feasile se. Fo he sepaain hypeplane heoem o apply, we need ha E λ β A fo all feasile A pahs, no jus fo he A pah coespondin o he poposed soluion. (Hee we conside vayin A while holdin he sochasic pocess fo λ fixed, ecause his condiion is deived fom he need o check ha E λ DA da + DA da d + + i convees, as a funcion of he da s.) We know fom [] ha E λ β has a ime pah popoional o β +. I is feasile o choose an A pah ha is deeminisic and popoional o +. Fo such a pah, clealy E A λ β convees o a nonzeo consan, violain he ue ansvesaliy condiion. 7