Lecture 19. RBC and Sunspot Equilibria

Transcription

1 Lecure 9. RBC and Sunspo Equilibria In radiional RBC models, business cycles are propagaed by real echnological shocks. Thus he main sory comes from he supply side. In 994, a collecion of papers were published in a special volume in JET highlighing he possibiliy of muliple equilibria in endogneous growh models (my paper and he paper by Benhabib and Perli) as well as RBC models (Benhabib and Farmer, Farmer and Guo). In Farmer and Guo for example, hey show ha even if we compleely shu down he echnological shocks, he demand shocks, such as he animal spiris in he form of sunspos (classic paper on sunspo was by Cass and Shell, JPE 983), are able o produce business cycles in he magniude observed in he daa. They borrowed he seup from Benhabib and Farmer s Indeerminacy and Increasing Reurns (Benhabib and Farmer conains wo versions of increasing reurns: one resuls from monopolisic compeiion similar o Romer JPE 990, he oher resuls from exernaliy similar o Romer JPE 986). Farmer and Guo (994) conveys he Keynesian message, bu he approach is along he line of RBC. The criicism on Benhabib and Farmer (994) and Farmer and Guo (994) was ha in order o produce sunspo equilibria, one needs o have increasing reurns parameer ha is oo high o be empirically plausible. Wen (998) inroduced endogenous capaciy uilizaion ino he exernaliy version of Benhabib and Farmer (994) and was able o show ha he key parameer could be made empirically plausible. Hence Wen (998) was he sae-of-he-ar model in his camp opposie o he radiional RBC camp. Le me firs begin wih he model of Farmer and Guo in deail and hen briefly highligh he key poins in Wen (998). The Model of Farmer and Guo (JET 994) I am presening only he monopolisic compeiion version below. Consumer s Problem " # X max ρ E 0 log C A L γ γ =0 subjec o : K + w L +( δ + r )K C + Π K 0 given, where Π is he profi owned by he represenaive consumer. Technology a he monopolisically compeiive firm Level Y (i) =Z K α (i)l β (i), α+ β> Z = Z θ η, Z 0 is given

2 Aggregae Economy: µz /λ Y = Y (i) di λ, 0 <λ< 0 Le a = λα and b = λβ We need a + b o make sure ha he profi funcion is concave in K and L (Derive he demand curve for individual firm i s problem. Wrie down individual firm i s profi funcion. Show ha he condiion a + b is needed) FOCs: A C = b Y = E C ρ L γ C + L µ a Y + + δ K + TVC is: K + =( δ)k + Y C lim ρ K C =0as Re-arranging erms and we end up wih a se of difference equaions in K, C, Z: K + = BZ m K g C d +( δ)k C = E DZ C +K m g + Cd + + τ C + Z = Z θ η where φ =/(β + γ ), d= βφ, m = d, g = αm, B =(A/b) d,d= Baρ and τ = ρ( δ). Employmen is given by L = A b C Z K α φ Seps o solve he dynamic equaions: Find seady saes Linearize around seady saes and rewrie he equaions in redefined variables as percenage deviaions from he seady saes Solve he linearized sysem of difference equaions 2

3 where Seady Saes: h ω i /(χ ) K = v C = v (K ) χ Z = B( τ) ω = δ D /d τ v = D χ = g d Redefine variables as percenage deviaions away from seady sae ˆK K µ K K ln K K similarly for Ĉ and Ẑ Le he expecaion error o be given by, e + = E r ˆK+ ˆK + E r Ĉ + Ĉ+ E r Ẑ + Ẑ+ Then he differenial equaions can be rewrien in he redefined variables as ˆK ˆK + Ĉ = J Ĉ + ˆη+ + R e + Ẑ Ẑ + where J is 3 3 and R is 3 4. If all he eigenroos of J are ouside of he uni circle, he sysem is indeerminae because hen, he eigenroos of J will be all wihin he uni circle, making he sysem a sink. In radiional RBC model, i was shown ha one of he roos of J is inside he uni circle (again, please review Pengfei Wang s example). Hence Ĉ can be wrien as a linear combinaion of ˆK and Ẑ. Benhabib and Farmer showed ha when he increasing reurns parameer is high enough, all he eigenroos of J are ouside of he uni circle. Hence for given ˆK and Ẑ, hereareinfiniely many possible Ĉ, namely,hereare muliple equilibria (indeerminacy). 3

4 In he special case in which here is no fundamenal shock, namely ˆη =0 for all. Thenwehave, ˆK = a ˆK + a 2 Ĉ Ĉ = a 2 ˆK + a 22 Ĉ + b 2 ˆV where ˆV is an independen source of flucuaion (self-fulfilling beliefs, sunspos aciiviy,...) Calibraion and Conclusion λ b β a α Hansen RBC Benhabib and Farmer σ v is se o equal Resuls: US daa Hansen RBC Animal Spiris Y C I Hours Produciviy Animal spiris alone does a reasonably good job in maching he second momen of US ime series. Impulse Response Funcion: Animal Spiris model seems o capure he impulse response funcion beer han RBC models (see heir working paper version page 26). Wen (998): Capaciy Uilizaion Wen (998) inroduced capaciy uilizaion ino he exernaliy version of increasing reurns in Benhabib and Farmer (994) and he resource consrain becomes: c + k + ( δ )k = A e (u k ) α n α δ = uθ θ,θ> where e is exernaliy. The capaciy uilizaion is endogenously deermined: higher capaciy uilizaion increases oupu bu leads o faser depreciaion of capial goods. Thus, if A is low (a negaive produciviy shock), he machines do no have o run a full capaciy. The exernaliy is given by e =(u k ) αη n ( α)η 4

5 In Benhabib and Farmer, hey need η o be large o have indeerminacy (for example, η = 0.5). Wen shows ha wih capaciy uilizaion endogenously deermined, he reduced form of aggregae producion funcion becomes: where y = ba τ n τ k = k α(+η)τ k n ( α)(+η)τ n θ θ α( + η) θ τ n = θ α( + η) The degree of increasing reurns becomes: α( + η)τ k +( α)( + η)τ k which is more han +η. In oher words, o produce indeerminacy, we no longer need η o be oo large. In fac, Wen shows ha η =0. is enough when oher parameers are calibraed in a sandard fashion. Also, he impulse response funcions behave beer han he radiional RBC and beer han Farmer and Guo (994). The inuiion is provided in Fig 2 (he shifs of labor supply and labor demand curves) and in he ex on page This paper is an imporan piece in defending he business cycle models based on sunspos and indeermincy. 5