Suggested Solutions to Homework #6Econ 511b (Part I), Spring 20041.(a) Find the planner’s optimal decision rule in the stochastic one-sectorgro w th model without valued leisure b y linearizing the Euler equation.Let the production function tak e the form f (kt,nt,zt)=eztkαtn1−αt,lettheconsumer’s felicity function have a constan t elasticity of in tertemporalsubstitution γ−1,andletztfollo w an AR (1) process: zt+1= ρzt+ t+1,t+1˜iidN (0,σ2).Setα =0.36, ρ =0.95, σ =0.007, the discoun t factorβ =0.99, and the d epreciation r ate δ =0.025.Solvethemodelfortwodifferent values of γ: 1 (the log case) and 2.The recursiv e form ulation of cen tral planning problem isv (k, z)=maxk0c1−γ− 11 − γ+ βEzv (k0,z0)s.t.c + k0= ezkα+(1− δ) kz0= ρz + ,  ∼ N¡0,σ2¢orv (k, z)=maxk0(ezkα+(1− δ) k − k0)1−γ− 11 − γ+ βEzv (k0,z0)s.t.z0= ρz + ,  ∼ N¡0,σ2¢We can easily get the Stoc ha stic Eu ler eq ua tion asu0(ct)=βEz[u0(ct+1)(ezt+1f0(kt+1)+1− δ)]⇒ c−γt= βEz£c−γt+1¡αezt+1kα−1t+1+1− δ¢¤⇒ (eztkαt+(1− δ) kt− kt+1)−γ= βEzh¡ezt+1kαt+1+(1− δ) kt+1− kt+2¢−γ¡αezt+1kα−1t+1+1− δ¢iThe steady state for th e determ inistic m odel isu0(c∗)=β [u0(c∗)(f0(k∗)+1− δ)]⇒ β¡α (k∗)α−1+1− δ¢=1⇒ k∗=·1αµ1β− (1 − δ)¶¸1α−1⇒ k∗=µαβ1 − β (1 − δ)¶11−α1Now we start to linear ize the syste m . We deal with LHS and R H S separately.First, expandin g LH S ar ound stea d y-state leads todLHS =d(eztkαt+(1− δ) kt− kt+1)−γ= −γ (k∗α− δk∗)−γ−1k∗αbzt− γ (k∗α− δk∗)−γ−1¡αk∗α−1+1− δ¢bkt+γ (k∗α− δk∗)−γ−1bkt+1whereweusethefactz =0and define bzt= zt−z,bkt= kt−k∗,andbkt+1= kt+1−k∗.Second, RHS is equal todRHS =dβEth¡ezt+1kαt+1+(1− δ) kt+1− kt+2¢−γ¡αzt+1kα−1t+1+1− δ¢i= β (k∗α− δk∗)−γ−1k∗α£α (1 − γ) k∗α−1− αδ − γ (1 − δ)¤Et(bzt+1)+β (k∗α− δk∗)−γ−1h−γ¡αk∗α−1+1− δ¢2+ α (α − 1) k∗α−1¡k∗α−1− δ¢ibkt+1+βγ (k∗α− δk∗)−γ−1¡αk∗α−1+1− δ¢Et³bkt+2´LetdLHS =dRHS,wehavedLHS =dRHS⇒−γk∗αbzt− γ¡αk∗α−1+1− δ¢bkt+ γbkt+1= βk∗α£α (1 − γ) k∗α−1− αδ − γ (1 − δ)¤Et(bzt+1)+βh−γ¡αk∗α−1+1− δ¢2+ α (α − 1) k∗α−1¡k∗α−1− δ¢ibkt+1+βγ¡αk∗α−1+1− δ¢Et³bkt+2´which is a second-order linear difference equation with expectations. We cansimplify it a little bit by using the stead y-state equation, which leads todLHS =dRHS⇒−γk∗αbzt− γβ−1bkt+ γbkt+1= k∗α[(1 −γ) − β + β (1 − α) δ] Et(bzt+1)+£−γβ−1+¡1 − α−1¢¡β−1− (1 − δ)¢(1 − β)¤bkt+1+ γEt³bkt+2´(1)To save notation, now we express the equation asa1bzt+ a2bkt+ a3bkt+1= b1Et(bzt+1)+b2bkt+1+ b3Et³bkt+2´(2)wherea1= −γk∗αa2= −γβ−1a3= γb1= k∗α[(1 − γ) − β + β (1 − α) δ]b2= −γβ−1+¡1 − α−1¢¡β−1− (1 − δ)¢(1 − β)b3= γ2We solv e this equation using M eth od of Undetermined C oefficients as taugh t inthe lectures. Conjecture the form of solution as½bkt+1= abkt+ bbztbzt+1= ρbzt+ t+1Plug this in to the equation (1), we ha vea1bzt+ a2bkt+ a3³abkt+ bbzt´= b1Et(ρbzt+ t+1)+b2³abkt+ bbzt´+ b3Et³a³abkt+ bbzt´+ b (ρbzt+ t+1)´⇒ [b1ρ + b2b + b3b (a + ρ) − a1− a3b] bzt+£ab2+ b3a2− a2− a3a¤bkt=0⇒ [b1ρ − a1+(b2+ b3(a + ρ) − a3) b] bzt+£b3a2+(b2− a3) a − a2¤bkt=0To make this an identity, w e must have½b1ρ − a1+(b2+ b3(a + ρ) − a3) b =0b3a2+(b2− a3) a − a2=0Therefor e, the final solution is(a =(a3−b2)±√(b2−a3)2+4a2b32b3b =a1−b1ρ(b2+b3(a+ρ)−a3)where we need to keep only the root of a with absolute value smaller than 1.Wecan solve the numbers with computer (see attached code). For γ =1,wehave½bkt+1=0.9798bkt+3.4127bztbzt+1=0.95bzt+ t+1for γ =2,wehave½bkt+1=0.9868bkt+3.1331bztbzt+1=0.95bzt+ t+1(b) Using Matlab, compute impulse response functions for output, con-sumption, and inv e stm ent in response to an innovation in the lev el oftec hnology of size σ. That is, suppose that the econom y is at the deter-ministic steady state and the inno vation  suddenly jumps in the curren tperiod to and then returns to 0 in all subsequen t periods. Determine(n um erically) the values of output, consumption, and in v estm ent in thecurren t period and in the next, sa y, 10 periods. Express the impulse re-sponsesaspercentagedeviationsfromthe(deterministic)steady-statevalues. Compare the impulse response functions for the tw o values ofγ.Run the Matlab code accompanying this answ er.2. Consider a stoch astic neoclassical gro w th model with the follow ing struc-ture:3• Eac h consumer has preferences of the form:E0∞Xt=0βt³ct+ B(1−nt)1−ν−11−ν´1−σ− 11 − σwhere labor supply nt< 1.• The econom y’s tech nology is described by:ct+ kt+1− (1 − δ) kt= ztkαtn1−αtwhere log(zt) is s toc h astic and evolves accord in g to a stationar y AR (1)process:zt+1= ρzρtt+1with log(t)˜iidN (0,σ2).(a) Deriv e the Euler equation for the savings decision.To solv e this problem, we repeat the sam e steps as in the corresponding part inHW#4 (problem 2).A recursiv e competitiv e equilibrium for the stoc h astic neoclassical grow th model withvalued leisure is a set of function s:price function : r¡k, z¢,w¡k, z¢policy function : k0= gk¡k,k, z¢,n= gn¡k,k, z¢value function : v¡k,k, z¢aggregate state :k0= G¡k, z¢,z0= ρzρsuch that:(1) Given the evolu tion law of aggreg ate statek0= G¡k, z¢and z0= ρzρ,k0= gk¡k,k, z¢,n= gn¡k,k, z¢and v¡k, k, z¢solves consumer’s problem:v¡k,k, z¢=maxn,k0³c + B(1−n)1−ν−11−ν´1−σ− 11 − σ+ βEzv³k0, k0,z0´s.t.c + k0= r¡k, z¢k + w¡k, z¢nk0= G¡k, z¢z0= ρzρ, log(t) ∼ N¡0,σ2¢(2) Price is competitively determined:r¡k, z¢= F1¡k, gn¡k, k, z¢¢= αzÃkgn¡k, k, z¢!α−1+1− δw¡k, z¢= F2¡k, gn¡k, k, z¢¢=(1− α) zÃkgn¡k, k, z¢!α4(3) Consistency:G¡k, z¢= gk¡k, k, z¢The F.O.C. fo r k0is{k0} : U1(c, n)=βE³v1³k0, k0,z0´´Use the envelo pe conditionv1¡k,k, z¢= U1(c, n) r¡k, z¢w e get the Euler equation for savings decision:U1(ct,nt)=βE£U1(ct+1,nt+1) r¡kt+1,zt+1¢¤⇒Ãct+ B(1 − nt)1−ν− 11 − ν!−σ= βE"Ãct+1+ B(1 − nt+1)1−ν− 11 −