Linear Approximation to NeoClassical Growth. 1. Linearizing the neoclassical stochastic growth model

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1 Econ. 5b Sping 998 C. Sims Linea Appoximaion o NeoClassical Gowh. Lineaizing he neoclassical sochasic gowh model Conside he model in which a epesenaive agen maximizes max ; C, + = 0 subjec o = 0 γ µ µ / µ σ σ / σ θc + ( θ) I = A( αk + ( α) L ) E K = δk + I L = C β γ 8 () If we use λ and ν o denoe he Lagange muliplies fo he wo consains, he fis ode condiions fo an opimum (ohe han ansvesaliy) ae γ λθ C C Y : C Y : λ ( θ) ν I I = : K = σ Y+ βe λ+ A+ α ν δβe ν K = Y = θc + ( θ) I µ 8 µ µ µ µ σ $ + () I simplifies noaion and inepeaion if we inoduce special symbols fo he deivaive of oupu Y wih espec o invesmen goods I and consumpion C Q P Y C Y I = ( θ) = θ µ µ () and inoduce anohe special symbol fo he ae of eun on capial

2 σ A α Q Y K = + σ δ. (4) This allows ewiing he fis hee equaions of () as which can in un be solved o eliminae λ and ν and yield γ λ C C : = P : λ Q = ν, (5) I : βe λ+ Q+ + = λq K γ E C + Q + P + = β. (6) γ C Q P + Equaion (6) equaes he subjecive discoun faco o he expeced ae of eun on capial, coeced fo he ae of inflaion in capial goods pices in consumpion goods unis and fo he ineempoal maginal ae of subsiuion of consumpion goods. I is no had o see fom (6) ha in deeminisic seady sae, = $ β. (7) Of couse in his model Q, P, and, ae no acually ansacion pices, since hee is a single ype of epesenaive agen and heefoe no ade. Wih (7) in hand we do no need an explici soluion fo he seady sae in ode o lineaize, bu if he soluion is needed, i can be found by conveing (7) o he fom µ σ µ µ µ σ αy I ( θ) K σ α( δ) + δ = ( θ) Y δ β K + =. (8) Equaion (8) can be solved fo seady sae K explicily, once he igh hand side of () wih L= is subsiued fo Y. To keep ou sysem one in jus hee vaiables, we mus ewie (6) in ems of C, K, I alone, using he definiions of Q, P, and : γ µ µ γ µ µ + E C + I C I A + σ σ σ 8 µ + + µ µ α α + ( α) K 8 ( θ) θ ( C I ) + θ $ + δ = β (9)

3 I is convenien o lineaize wih espec o he naual logaihms of C, K and I ahe han he levels of he vaiables. We will fom a Taylo expansion in he logs of hese vaiables abou he seady sae of he sysem given by he equaions in () (wih L= subsiued ou) ogehe wih (8). We will use lowe case lees o efe o logs of vaiables and dx o efe, fo any vaiable X, o log( X) log( X), whee X is he seady sae value of X. Then he lineaized sysem is PCdc + QIdi = ( δ) Q Kdk (0) dk = δdk + ( δ) di () δ C δ γ µ θ ( µ ) dc µ θ ( µ ) Y + + = + δ σ γ µ µ σ α + δ ( ) dc ( ) di ( )( ) Y dk da + η µ µ C Y di + + In (), he η + em is an endogenous expecaional eo and saisfies E η + = 0. We can see fom (8) ha seady sae values of K, and hence (via he consains in ()) C and I, ae all independen of he paamee γ. Thus we can see fom () ha if γ is made lage while he ohe paamees emain fixed, he sysem has an equaion ha in he limi is appoximaely Edc + = dc, appaenly making he log of consumpion a maingale. While his uns ou o be coec, he agumen we have jus given is no igh, as making γ lage migh make he values of K along he soluion pah lage, so ha he fac ha he coefficiens on K ae small migh no imply ha hey have small influence on Edc +. To complee an analysis of he condiions unde which he Hall model's conclusion ha c is appoximaely a maingale is valid, we need o fom he sysem's chaaceisic polynomial. Calling he five coefficiens on choice vaiables in () ν i, i = 0,..., 4, we can lag () once and combine i wih (0) and () o obain he following sysem in maix noaion: PC QI 0 dc 0 0 ( δ) QK dc Y 0 + δ di 0 0 di 0 da 0 0$ dk$ = δ 4 $ dk $ + + δ ν ν ν ν ν $ $ () 0 0 η. () Noe ha because all he vaiables ae aken as deviaions fom seady sae, he lineaized sysem conains no consan ems. The lef-hand-side maix is non-singula fo all possible paamee values, because i can be shown ha ν0 and ν mus be of opposie sign (wih ν 0 < 0), so ha he deeminan = QIν0 PCν is always non-zeo. I heefoe has an invese we can calculae as

4 ν 0 QI ν0 0 PC δν QIν PCν δpc Muliplying he maix in (4) imes he igh-hand-side squae maix in () yields νqi νqi ν4qi ν ( δ) QK νpc νpc ν4pc + ν0 ( δ) QK ν PC δ ν PC δ ν δ QI + δ ν PC δ 0 4 $. (4) ( ) ( ) ( ) 6$ As γ +, ν 0 and ν go o in such a way ha ν ν 0 bounded. Thus (5) conveges in his case o he limiing fom 0 0 PC δ 0 QI δ δ 6PC 0 QI. (5), while he emaining ν i emain $, (6) which has eal eigenvalues of and. We conclude ha in his limiing case of exeme isk avesion he Hall conclusion ha consumpion is appoximaely a andom walk is coec, since he model has one explosive oo,, ha will be suppessed in he soluion, and one oo of. Anohe ineesing case is µ =, wih no movemen in he elaive pice of C and I. This makes ν = ν = 0. Then we can see fom (5) and he definiion of ha we will be back o he fom (6) if ν 4 0. This occus, as we can see by efeing back o () (whee ν 4 is he coefficien on dk ), when eihe σ o α, boh of which delive in he limi a linea poducion echnology. Bu noice ha lineaiy of Y in K is no enough. Geneally if µ, he sable oo of he sysem is no close o one even if σ =. Noe, hough, ha he discussion above has no checked whehe a seady sae exiss in he limiing cases consideed. In fac, hey ofen do no. As a check on hese analyical mehods, heefoe, we pesen below gaphs of esuls of some numeical calculaions. We will see ha hee is ofen no seady sae when σ o α is nea one. The fis figue below shows conous fo he single sable oo of he lineaized sysem as a funcion of µ and σ wih γ =, δ =. 9, β =. 95, θ =, α =. 5. These paamees ae chosen o be moe o less ealisic fo an annual ime uni. Noe ha hee is a egion in which he oo is above.95 (even above ove pa of he egion). Mos of he egion is wihin he ecangle in which σ lies beween -. and +., wih he ange of σ values in he egion slighly widening as µ 4

5 inceases. This esul is quie sensiive o he choice of α. The second plo shows he effec of changing oα =.. In his plo oos dop away fom one moe apidly as σ deceases. The band in which oos exceed.95 equies σ >.. If we inepe acual income shaes as eflecing compeiive make deeminaion of faco pices, he α value of. is moe ealisic. Values of α lage han he obseved make shae of capial ae someimes jusified by a claim ha some of obseved labo income is acually a eun o human capial. Of couse if his is so, hen a singlecapial good model like his one will be a bad appoximaion, unless human and physical capial ae nealy pefec subsiues. In he emaining figues ohe paamees ae changed, in diecions ha make he maingale-like behavio of consumpion an inceasingly good appoximaion. By he las figue, we have made nealy he enie egion whee µ < 5. o σ >.5 display uni-oo-like behavio. Bu o do his we have had o make γ highe han is usually hough o be ealisic. I seems fai o conclude fom hese plos ha, wih convenionally easonable assumpions abou isk avesion and he discoun faco, he maingale-like behavio of consumpion pediced by he pemanen income hypohesis is a good appoximaion fo aggegae daa only when capial and labo ae highly subsiuable. 5

6 Sable Roo as a Funcion of µ and σ, α=.5.5 µ In his egion hee is no seady sae σ 6

7 Sable Roo as a Funcion of µ and σ, α= In his egion hee is no seady sae. µ σ 7

8 Sable Roo as a Funcion of µ and σ, α=.5, γ= In his egion hee is no seady sae

9 Sable Roo as a Funcion of µ and σ, α=., γ=4, A= In his egion hee is no seady sae

10 Sable Roo as a Funcion of µ and σ, α=., γ=, A= In his egion hee is no seady sae..5 In his egion hee is no seady sae

11 Sable Roo as a Funcion of µ and σ, α=., γ=, A= In his egion hee is no seady sae