Problem Set 5. Graduate Macro II, Spring 2017 The University of Notre Dame Professor Sims
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1 Problem Se 5 Graduae Macro II, Spring 2017 The Universiy of Nore Dame Professor Sims Insrucions: You may consul wih oher members of he class, bu please make sure o urn in your own work. Where applicable, please prin ou figures and codes. This problem se is due in class on Thursday, March 30, (1) Preference Shocks and Variable Capial Uilizaion: Suppose we have a model wih variable capial uilizaion and a ime-varying disuiliy of labor. A represenaive household owns he capial sock, K, and makes capial accumulaion decisions. This agen also ges o pick how inensively o uilize capial, u, leasing capial services (he produc of uilizaion and physical capial), K u K, o a represenaive firm on a period-by-period basis. The cos of more inensive uilizaion of capial is faser depreciaion. The problem of he household can be wrien as follows: { } E 0 β ln C ν θ N 1+χ C,K +1,B +1,N,u =0 s.. C + K +1 (1 δ(u ))K + B +1 w N + R u K + Π + (1 + r 1 )B δ(u ) = δ 0 + φ 1 (u 1) + φ 2 2 (u 1) 2 ν is a ime-varying shock o preferences over labor, obeying a saionary AR(1) process ha is mean zero in he log: ln ν = ρ ν ln ν 1 + ε ν, (a) Derive he firs order condiions for he household problem. The firm produces oupu using labor and capial services, K u K, and akes facor prices, w and R, as given. Is problem is: N, K Π = A Kα N 1 α (b) Derive he firs order condiions for he firm problem. Produciviy obeys an AR(1) process in he log: w N R K ln A = ρ a ln A 1 + ε a, (c) Derive he aggregae resource consrain using he he definiion of profi and he bond marke-clearing condiion. (d) Provide a condiion on he parameer φ 1 o ensure ha seady sae capial uilizaion is 1. 1
2 (e) Derive a condiion giving he value of θ necessary o arge seady sae labor hours of N = 1/3. (f) Wrie a Dynare code o solve he model and compue impulse responses o boh produciviy and preference shocks using a firs order log-linear approximaion. Use he following parameer values: β = 0.99, δ = 0.02, α = 1/3, χ = 1, ρ a = 0.97, ρ ν = 0.95, s a = 0.01 (sandard deviaion of produciviy shock), and s ν = 0.02 (sandard deviaion of he preference shock). Use he value of θ consisen wih seady sae hours of 1/3 found in (e) above. Consider hree differen values of φ 2 : 100, 0.1, Graphically show impulse responses of oupu, hours, consumpion, invesmen, uilizaion, he real wage, and he real ineres rae o boh shocks. Commen on how he impulse responses o boh shocks vary wih he parameer φ 2, and provide some inuiion for your resuls. (g) Suppose ha you measure oal facor produciviy no aking ino accoun variable uilizaion. Tha is, le  = ln Y α ln K (1 α) ln N. Show impulse responses of measured TFP o boh shocks for he hree differen values of φ 2 from above. How does φ 2 affec he response of measured TFP o boh shocks? (h) How does he inclusion of he preference shock affec he relaive volailiy of HP filered hours (relaive o GDP) in he model? To see his, compue he relaive volailiy of hours o oupu wih boh shocks urned on and again wih he preference shock urned off (e.g. se he sandard deviaion of ha shock o zero). Use a value of φ 2 = 0.1 in doing his par. (2) GHH vs. Tradiional Preferences and he Effecs of Governmen Spending and Produciviy Shocks: Suppose you have a RBC model wih wo sochasic shocks: a shock o produciviy and a shock o governmen spending. We will consider wo differen preference specificaions: sandard separable preferences and GHH preferences. The household problem can be wrien: C,K +1,N,B +1 E 0 s.. β U(C, N ) C + K +1 (1 δ)k + B +1 w N + R K + Π T + (1 + r 1 )B (a) For he arbirary specificaion of household preferences, U(C, N ), find he firs order condiions for a soluion o he problem. The firm problem is he same in boh seups: N,K =0 Π = A K α N 1 α w N R K (b) Find he firs order condiions necessary for a soluion o he firm problem. (c) The governmen chooses is spending exogenously and balances is budge each period, G = T. Wha hen mus be rue abou bond-holding by households in equilibrium? Wrie down he aggregae resource consrain. Suppose ha preferences are given by he sandard separable form: U(C, N ) = ln C θ N 1+χ 2
3 (d) Suppose ha he non-sochasic seady sae value of A is A = 1, while he seady sae value of governmen spending is G = ωy, where Y is he seady sae value of oupu and 0 < ω < 1. Using hese preferences, derive an expression for he value of θ consisen wih seady sae hours of N = 1/3, and provide expressions for he seady sae values of Y, C, I, K, w, and R as a funcion of parameers. Suppose ha G and A follows saionary AR(1) processes in he log: ln A = ρ A ln A 1 + ε A, ln G = (1 ρ G ) ln G + ρ G ln G 1 + ε G, (e) Solve he model using a firs order log-linear approximaion in Dynare using he following parameer values: β = 0.99, δ = 0.02, α = 1/3, χ = 1, ρ A = 0.97, ρ G = 0.95, s A = 0.01 (sandard deviaion of produciviy shock), s G = 0.01 (sandard deviaion of governmen spending shock), ω = 0.20, and he value of θ consisen wih N = 1/3. Produce impulse response graphs of Y, C, I, N, w, and r o each shock over a 20 period horizon. Calculae he governmen spending muliplier, defined as he raio of he impac response of he level of oupu o he impac response of he level of governmen spending following a governmen spending shock. Now insead suppose ha preferences are given by he GHH variey: U(C, N ) = ln ( ) C θ N 1+χ (f) Using hese preferences, derive an expression for he value of θ consisen wih seady sae hours of N = 1/3, and provide expressions for he seady sae values of Y, C, I, K, w, and R as a funcion of parameers. (g) Solve he model using a firs order log-linear approximaion using he same parameer values as above. Produce impulse response graphs of Y, C, I, N, w, and r o each shock over a 20 period horizon. Calculae he governmen spending muliplier, defined as he raio of he impac response of he level of oupu o he impac response of he level of governmen spending following a governmen spending shock. (h) Compare and conras your impulse responses o he wo shocks wih he wo differen preference specificaions. Provide some inuiion for your findings. (3) News Shocks: This problem asks you o work ou a business cycle model where here are anicipaed shocks o produciviy, which he lieraure has called news shocks. In paricular, suppose ha he process for exogenous produciviy, A, obeys he following sochasic process: ln A = ρ ln A 1 + ε 4 This is he sandard AR(1) in he log, bu he shock is observed 4 periods in advance of when produciviy changes. In his way, a posiive (or negaive) realizaion of ε provides news abou he level of A in 4 periods. The model ha we will consider is a varian of Jaimovich and Rebelo (2009, American Economic Review): Can News Abou he Fuure Drive he Business Cycle? I is a real model (nohing nominal) wih hree main wiss relaive o a sandard RBC model: invesmen adjusmen coss, variable capial uilizaion, and GHH ype preferences. For now, le s assume a generic preference specificaion, where flow uiliy 3
4 is is U = U(C, N ) and is increasing in consumpion and decreasing in labor hours. A represenaive household owns he capial sock, makes invesmen decisions, and also makes uilizaion decisions. I leases capial services (he produc of uilizaion and he physical capial sock) o a represenaive firm. The household problem can be wrien: C,K +1,I,N,u E 0 s.. β U(C, N ) =0 C + I w N + R u K + Π [ K +1 = 1 κ ( ) ] 2 I 1 I + (1 δ(u ))K 2 I 1 δ(u ) = δ 0 + δ 1 (u 1) + δ 2 2 (u 1) 2 (a) Derive he firs order condiions for he household problem. I is bes o wrie a Lagrangian wih wo consrains; e.g. le λ be he muliplier on he flow budge consrain and µ he muliplier on he capial accumulaion equaion. The firm problem is sandard, where i picks capial services, K individually: = u K, no capial or uilizaion N,K Π = A Kα N 1 α (b) Derive he firs order condiions for he firm problem. w N R K (c) Wrie down he definiion of a compeiive equilibrium and derive he aggregae resource consrain. (d) Derive a resricion on he parameer δ 2 necessary o ensure ha seady sae uilizaion equals 1 (you do no need o know he specific form of preferences o do his). Suppose ha preferences are given by he sandard addiively separable form: U(C, N ) = ln C θ N 1+χ (e) Wrie a Dynare file o solve his model using a firs order log-linear approximaion. Use parameer values α = 1/3, β = 0.99, δ 0 = 0.025, χ = 1, he value of δ 1 you derived above, and a value of θ consisen wih seady sae labor hours of 1/3. Use values of ρ = 0.95 and he sandard deviaion of he shock of I is sraighforward o include he news shock in Dynare jus wrie e(-4) where you would normally wrie e in he shock process. For now, assume ha δ 2 = 1000 (effecively, no variable uilizaion) and κ = 0 (no invesmen adjusmen coss). Produce and prin ou impulse responses of oupu, consumpion, hours, and invesmen o he news shock. Wha happens o hese variables in he period beween he news his (on impac ) and when he shock ranslaes ino higher produciviy four periods laer? Can you provide some inuiion for his? (f) Re-do he exercise in (e), bu his ime se δ 2 = 0.01, effecively urning on variable uilizaion (bu keep κ = 0). Commen on how he impulse responses are differen, and ry o provide some inuiion. (g) Repea he exercise in (e), bu insead urn on invesmen adjusmen coss, seing κ = 3 (bu se δ 2 = 1000). Commen on how he impulse responses differ from he baseline, and ry o provide some 4
5 inuiion. (h) Now urn boh of hese feaures on simulaneously, seing δ 2 = 0.01 and κ = 3. Produce he impulse responses o he news shock. Knowing wha you know abou co-movemens among aggregae variables in he acual ime series daa, can news shocks be an imporan driving force of he business cycle in his model? Now insead consider he GHH preference specificaion, where: ( ) U(C, N ) = ln C θ N 1+χ (i) Use he same parameer values you did in par (e) (e.g. urn off variable uilizaion and he invesmen adjusmen cos), compue impulse responses o he news shock when he household has hese preferences. Use a value of θ consisen wih seady sae labor hours of 1/3 (noe, his value of θ will be differen han wha you found above). Commen on how he impulse responses look differen from wha you found in par (e), and ry o provide some inuiion. (j) Coninue o use GHH preferences, bu now urn on boh variable uilizaion and he invesmen adjusmen cos (δ 2 = 0.01 and κ = 3). Produce impulse response o he news shock, and compare hem wha you found for he base RBC model wih separable preferences, no uilizaion, and no adjusmen coss. Commen on he differences, and ry o provide some inuiion for how each of he hree differen changes relaive o he baseline (preferences, uilizaion, and adjusmen coss) help accoun for he differences. 5
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