Finance 400 A. Penati - G. Pennacchi. Recursive Utility

Transcription

1 Finance 400 A. Penai - G. Pennacchi Recursive Uiliy A class of non-ime-separable uiliy known as recursive uiliy has been sudied by a number of auhors. D. Kreps and E. Poreus (1978) Temporal Resoluion of Uncerainy and Dynamic Choice Theory, Economerica 46, p and L. Epsein and S. Zin (1989) Subsiuion, Risk Aversion, and he Temporal Behavior of Consumpion and Asse Reurns: A Theoreical Framework, Economerica 57, p analyze his ype of uiliy in a discree-ime seing, while D. Duffie and L. Epsein (1992) Asse Pricing wih Sochasic Differenial Uiliy, Review of Financial Sudies 5, p sudy he coninuous-ime limi. In coninuous-ime, recall ha sandard, ime-separable uiliy can be wrien as: Z # T V = E U[C(s), s] ds where U[C(s), s] is ofen aken o be of he form U[C(s), s]=e ρ(s ) u[c(s)]. Recursive uiliy, however, is specified as V Z # T = E f[c(s), V s ] ds where f is known as an aggregaor funcion. The specificaion is recursive in naure because curren uiliy, V, depends on expeced values of fuure uiliy, V s, s>.whenf has appropriae properies, D. Duffie and L. Epsein (1992) Sochasic Differenial Uiliy, Economerica 60, p show ha a Bellman-ype equaion can be derived which characerizes he opimal consumpion and porfolio choice policies for uiliy of his ype. For paricular funcional forms, hey have been able o work ou a number of asse pricing models. These noes consider general equilibrium in an economy where represenaive consumerinvesors have recursive uiliy. The specific model ha we will consider is ha of Maurice Obsfeld (1994) Risk-Taking, Global Diversificaion, and Growh American Economic Review 84, p Assumpions: A.1 Technology: 1

2 A single capial-consumpion good can be invesed in up o wo differen echnologies. The firs is a risk-free echnology whose oupu, V B (), follows he process dv B /V B = id (1) The second is a risky echnology whose oupu, V K (), follows he process dv K /V K = αd + σdz (2) Commen: Noe ha in his producion economy he specificaion of echnologies fixes he expeced raes of reurn and variances of he safe and risky invesmens. Individuals asse demands will deermine equilibrium quaniies of he asses supplied raher han asse prices. Since i, α, and σ are assumed o be consans, here is a consan invesmen opporuniy se. A.2 Preferences: Represenaive, infiniely-lived households mus choose beween consuming (a rae C s a dae s) and invesing in he single capial-consumpion good. The lifeime uiliy funcion a dae faced by each of hese households, denoed U,is where f is given by Commen: U = E f (C s,u s ) ds (3) ½ C 1 ( ¾ 1 ² ) s [(1 R) U s ] f(c s,u s )=½ ³ ¾ (4) 1 1 ² [(1 R) U s ] 1 Noe his specificaion s recursive naure in ha curren lifeime uiliy, U,dependson expeced values of fuure lifeime uiliy, U s, s>. f (C, U) is known as he aggregaor funcion. The form of equaion (4) is ordinally equivalen o he coninuous-ime limi of he discree-ime uiliy funcion specified in Obsfeld (1994). Recall ha uiliy funcions are ordinally equivalen, ha is, hey resul in he same consumer choices, if he uiliy funcions 2

3 evaluaed a equivalen ses of decisions produce values ha are linear ransformaions of each oher. I can be shown (see Epsein and Zin (1989) and Duffie and Epsein (1992) ha > 0 is he coninuously compounded subjecive rae of ime preference, ²>0is he household s elasiciy of ineremporal subsiuion, and R>0 is he household s coefficien of relaive risk aversion. For he special case of R =1/², he uiliy funcion given in (3) and (4) is (ordinally) equivalen o he ime-separable, consan relaive risk-aversion case: U = E e s C1 R s ds (5) 1 R Le ω () be he proporion of each household s wealh invesed in he risky asse (echnology). The he ineremporal budge consrain is given by dw =[ω(α i)w + iw C] d + ωσwdz (6) When he aggregaor funcion, f, is pu in a paricular form by an ordinally equivalen change in variables, wha Duffie and Epsein (1992) Economerica refer o as a normalizaion hen a Bellman equaion can be used o solve he problem. The aggregaor in (4) is in normalized form. As before, le us define J (W ) as he maximized lifeime uiliy a dae J (W ) = max E {C,w} = max {C,w} E f (C s,u s ) ds (7) f (C s,j(w s )) ds Since his is an infinie horizon problem wih consan invesmen opporuniies, and he aggregaor funcion, f (C, U), is no an explici funcion of calendar ime, he only sae variable is W. Then he soluion o he consumer s consumpion - porfolio choice problem is given by he coninuous-ime sochasic Bellman equaion 0=max {C,w} f [C,J(W )] + L [J (W )] (8) 3

4 or 0 = max f [C, J (W)] + J W [ω W + iw C]+ 1 {C,w} 2 J WWω 2 σ 2 W 2 (9) ½ C 1 ( ¾ 1 ² ) [(1 R) J] = max ½ ³ ¾ + J {C,w} 1 1 W [ω W + iw C]+ 1 ² [(1 R) J] 1 2 J WWω 2 σ 2 W 2 Taking he firs order condiion wih respec o C, or C ( 1 ² ) J [(1 R) J] 1 W =0 (10) µ JW C = ² 1 R +² (10 0 ) Taking he firs order condiion wih respec o ω, J W W + J WW ωσ 2 W 2 =0 (11) or J W ω = J WW W σ 2 (11 0 ) Subsiuing he opimal values for C and ω given by (10 ) and (11 ) ino he Bellman equaion (9) we obain he differenial equaion: ³ JW 1 ² () 1 [(1 R) J] ³ 1 1 ² [(1 R) J] [(1 R) J] 1 1 ² (12) +J W J W 2 µ JW J WW σ 2 + iw # ² 1 R +² + 1 JW J WW σ 2 =0 or 4

5 ( µjw ) ² ² 1 R +² [(1 R) J] ² 1 (12 0 ) +J W J W 2 µ JW J WW σ 2 + iw # ² 1 R +² + 1 JW J WW σ 2 =0 If one guesses ha he soluion is of he form J (W) =(aw ) 1 R / (1 R) and subsiues his ino (12 0 ), one finds ha a = µ 1/(1 ²) where µ = ² 1 ² (² 1) ² i + # 2Rσ 2 (13) Thus, subsiuing his value for J ino (10 0 )wefind ha opimal consumpion is a fixed proporion of wealh and he opimal porfolio weigh of he risky asse is C = µw (14) ω = α i Rσ 2 (15) which is he same as for an individual wih sandard consan relaive risk-aversion and imeseperable uiliy. The resuls show ha he porfolio choice decision depends only on risk-aversion, R, bu ha he consumpion-saving decision depends on boh risk-aversion and he elasiciy of ineremporal subsiuion, ². We can sudy how he growh rae of he economy depends on he model s parameers. Assuming 0 < ω < 1 and subsiuing (14) and (15) ino (6) we have ha wealh follows he process dw/w = [ω +i µ] d + ω σdz (16) 5

6 = = 2 2Rσ 2 + i ² +(² 1) ² (i )+(1+²) 2Rσ 2 # i + d + ## 2Rσ 2 d + Rσ dz Rσ dz Thus, he expeced growh rae of wealh, as well as consumpion (since C = µw ), is g = ² (i )+(1+²) 2Rσ 2 (17) We see ha he growh rae is increasing in α, bu decreasing in R and σ. Obsfeld poins ou ha he inegraion of global financial markes ha allows residens o hold risky foreign, as well as domesic, invesmens increases diversificaion and effecively reduces individuals risky porfolio variance, σ 2. This will lead o an increase in naions expeced growh raes because individuals will allocae a greaer proporion of heir wealh in he (higher-yielding) risky asses. 6