Problem Set 4. Graduate Macro II, Spring 2011 The University of Notre Dame Professor Sims

Transcription

1 Problem Set 4 Graduate Macro II, Spring 2011 The University of Notre Dame Professor Sims Instructions: You may consult with other members of the class, but please make sure to turn in your own work. Where applicable, please print out figures and codes. This problem set is due by 12:00 noon on Friday, March 25. (1) Government Spending in a RBC Model: Download data on real government spending (the series is titled government consumption expenditures and gross investment ) for the period (quarterly). (a) Compute the average share of government spending in total output for this period. (b) Linearly detrend the log of the government spending series. Then estimate an AR(1) process on the detrended series: ln ĝ t = a 0 + ρ g ln ĝ t 1 + ε g,t Report your estimates of ρ g and the standard deviation of the innovation, σ εe,g. Now suppose that we have a standard RBC model, but augmented with government spending. The government finances its expenditures, g t, by lump sum taxes on households, T t, and issuing bonds, d t. d t < 0 means that the government is is in debt; d t > 0 means that the government is saving. Households consume, work, accumulate capital, and save in government bonds, b t. The household problem is: k t+1,b t+1,n t,c t E 0 β t (ln c t + θ ln(1 n t )) t=0 s.t. c t + k t+1 (1 δ)k t + (b t+1 b t ) = w t n t + R t k t + Π t T t + r t b t (c) Write down the first order conditions necessary for an interior solution of the household problem. The government budget constraint is given by: g t + (d t+1 d t ) = T t + r t d t (d) Assuming that firms do not operate in the bond market, what is the condition required for bond market-clearing? 1

2 (e) Use this condition, in conjunction with the government and household budget constraints, to show that (i) bond holdings disappear from the household budget constraint and (ii) the household behaves as though g t = T t each period. The firm problem is standard and static, given that households own the capital stock and firms are assumed to not operate in debt markets: n t,k t Π t = a t k α t n 1 α t w t n t R t k t (f) Write down the first order condition for the firm s profit imization problem. Suppose that log government spending obeys the following process: ln g t = (1 ρ g )(ln ψ + ln y ) + ρ g ln g t 1 + ε t ψ is the steady state share of government spending in output. Assume that aggregate technology follows a mean zero, AR(1) in the log: ln a t = ρ ln a t 1 + ε a,t (g) Solve for the non-stochastic steady state of the model, assuming that n = 1. Provide an expression for the θ that yields this steady state. What is the relationship between the required θ to generate n = 1 and ψ, the steady state share of government spending? Provide intuition for your answer. (h) Use DYNARE to solve for the linearized policy functions of this model. Use your answers from (b) to calibrate ρ g and the standard deviation of the government spending shock. Use ρ = for the persistence parameter of technology, and use standard deviation of the technology shock of Set α = 0., δ = 0.025, and β = Use the value of θ you found in part (g). Show impulse response functions of the endogenous variables of the model to a government spending shock. Comment on how the inclusion of government spending shocks improves the fit of the model (in terms of second moments) relative to the one shock world with just technology shocks. (i) Re-do (h) for two different values of ρ g : ρ g = 0.5 and ρ g = Comment on the differences in the impulse responses to a spending shock with these different parameters, relative to what you found in part (h). Provide some intuition for any differences you find. 2) Money and Welfare: Consider the basic RBC model as above, appended to include a cash in advance constraint. The household problem is as follows: c t,k t+1,n t,b t+1,m t+1 E 0 β t (ln c t + θ ln(1 n t )) t=0 s.t. 2

3 ( ) ( ) Bt+1 B t Mt+1 M t c t + k t+1 + (1 δ)k t + + (a) Solve for the first order conditions of the household problem. The firm solves a static profit imization problem: k t,n t Π t = a t k α t n 1 α t (b) Write down the firm s first order conditions. w t n t R t k t Suppose that the Fed follows a money growth rule of the form: = w t n t + R t k t + i t B t + Π t M t c t ln M t+1 ln M t = (1 ρ m )π + ρ m (ln M t ln M t 1 ) + ε m,t (c) Define m t = M t+1 and 1 + π t = pt 1. Re-write this process in terms of m t and π t. Suppose that log aggregate technology obeys the following stochastic process: ln a t = ρ ln a t 1 + ε a,t Define β = 1 1+ρ p.calibrate the parameters of the model as follows: α = 0., θ = 1.75, ρ p = 0.01, δ = 0.025, ρ = 0.974, σ εa = 0.009, ρ m = 0.75, π = 0.01, and σ εm = (d) Solve for the linearized policy functions in DYNARE using these parameter values. Be sure to re-write the model so that it is stationary (steady state inflation is the source of nonstationarity). Show impulse responses of the endogenous variables of the model plus the log level of the nominal money supply and the log price level to a one standard deviation monetary shock. Note: you will have to construct the IRFs of the nominal money supply and price level outside of the Dynare code given the responses of real money balances and inflation. (e) What is the Friedman rule? What is the basic logic for it? Steady state welfare can be defined as: U = 1 1 β (ln c + θ ln(1 n )) Create a grid of values of π, ranging from the π that would implement the Friedman rule plus to The reason you need to not let π get all the way to the Friedman rule is that this will cause a numerical issue in Dynare. Have a step in the grid of You can implement this easily by having a Matlab source m-file that contains the following lines of code:

4 1 for j = 1:N 2 pistar = ps(1,j); save parameterfile ps4 cia pistar beta 4 dynare ps4 cia noclearall 5 cs(j,1) = oo.steady state(cpos,1); 6 ns(j,1) = oo.steady state(npos,1); 7 wf(j,1) = (1/(1 beta))*(log(exp(cs(j,1))) + theta*log(1 exp(ns(j,1)))); 8 end Here N is the size of the grid for possible steady state inflation values. By including the line save parameterfile ps4 cia pistar beta you can save the value of steady state inflation. Within the Dynare.mod file, include the following set of code right after the parameter declaration: 1 parameters alpha beta theta...; 2 load parameterfile ps4 cia; set param value('pistar',pistar) 4 set param value('beta',beta) You specify the other parameters within the.mod file just as you normally would. (f) Create plots of steady state consumption, hours, and welfare as a function of π. Verify that all are decreasing in π. Provide some intuition as to why. () News Shocks: Consider the same basic RBC model as above, but without government spending or money. Use the parameterization given above in part (1). Alter the process for technology as follows: ln a t = ρ ln a t 1 + ε t + v t q v t q is called a news shock : it provides news about an expected shift in future TFP q periods out into the future. It is uncorrelated with ε t. Set the standard deviations of both the regular technology shock and the news shock equal to (a) Compute impulse responses to a news shock using the benchmark parameterization and q = 4 (i.e. four quarters of foresight). Set the standard deviation of news shocks equal to the standard deviation of the regular contemporaneous technology shock. To code this in Dynare you can create auxiliary state variables as follows: ln a t = ρ ln a t 1 + news t 1 + ε t news t = news2 t 1 news2 t = news1 t 1 news1 t = v t Defining auxiliary state variables like this allows you to keep everything dated t, t 1, or t+1, which Dynare requires to work. Do news shocks cause business cycle-like movements in aggregate variables before the change in TFP occurs? Provide intuition why or why not. 4

5 (b) Does the inclusion of news shocks helhe model better fit the data in an unconditional sense (i.e. in terms of volatilities, correlations, and autocorrelations)? If so, along what dimensions and why? Does your answer for (b) mesh with your conclusions on (a)? Discuss. 5