ljundquist and sargent solutions

Exercises in Recursive Macroeconomic Theory preliminary and incomplete Stijn Van Nieuwerburgh Pierre-Olivier Weill Lars Ljungqvist Thomas J. Sargent Introduction This is a first version of the solutions to the exercises in Recursive Macroeco- nomic Therory, First Edition, 2000, MIT press, by Lars Ljungqvist and Thomas J. Sargent. This solution manuscript is currently only available on the web. We in- vite the reader to bring typos and other corrections to our attention. Please email sargent@stanford.edu, poweill@stanford.edu or svnieuwe@stanford.edu. We will regularly update this manuscript during the following months. Some questions ask for computations in matlab. The program files can be downloaded from the ftp site zia.stanford.edu/pub/sargent/rmtex. The authors, Stanford University, March 15, 2003. Contents Introduction 2 List of Figures 5 Chapter 1. Time series 7 Chapter 2. Dynamic programming 33 Chapter 3. Practical dynamic programming 37 Chapter 4. Linear quadratic dynamic programming 43 Chapter 5. Search, matching, and unemployment 55 Chapter 6. Recursive (partial) equilibrium 83 Chapter 7. Competitive equilibrium with complete markets 95 Chapter 8. Overlapping generation models 109 Chapter 9. Ricardian equivalence 161 Chapter 10. Asset pricing 163 Chapter 11. Economic growth 175 Chapter 12. Optimal taxation with commitment 187 Chapter 13. Self-insurance 201 Chapter 14. Incomplete markets models 211 Chapter 15. Optimal social insurance 223 Chapter 16. Credible government policies 257 Chapter 17. Fiscal-monetary theories of inflation 267 Chapter 18. Credit and currency 283 Chapter 19. Equilibrium search and matching 307 Index 343 3 List of Figures 1 Exercise 1.7 a 21 2 Exercise 1.7 b 22 3 Exercise 1.7 c 22 4 Exercise 1.7 d 23 5 Exercise 1.7 e 23 6 Exercise 1.7 f 24 7 Exercise 1.7 g 24 1 Exercise 3.1 : Value Function Iteration VS Policy Improvement 41 1 Exercise 4.5 52 1 Exercise 8.1 113 1 Exercise 10.1 : Hansen-Jagannathan bounds 165 1 Exercise 13.2 205 1 Exercise 14.2 a 214 2 Exercise 14.2 b 214 3 Exercise 14.2 c 215 4 Exercise 14.5 : Cross-sectional Mean and Dispersion of Consumption and Assets 221 1 Exercise 15.2 226 2 Exercise 15.10 a : Consumption Distribution 238 3 Exercise 15.10 b : Consumption, Promised Utility, Profits and Bank Balance in Contract that Maximizes the Money Lender’s Profits 239 4 Exercise 15.10 c : Consumption, Promised Utility, Profits and Bank Balance in Contract that Gives Zero Profits to Money Lender 240 5 Exercise 15.11 a : Pareto Frontier, β = 0.95 241 6 Exercise 15.11 b : Pareto Frontier, β = 0.85 242 7 Exercise 15.11 c : Pareto Frontier, β = 0.99 243 5 6 LIST OF FIGURES 8 Exercise 15.12 a : Consumption, Promised Utility, Profits and Bank Balance in Contract that Maximizes the Money Lender’s Profits 245 9 Exercise 15.12 b : Consumption Distribution 246 10 Exercise 15.12 c : Wage-Tenure Profile 247 11 Exercise 15.14 a : Profits of Money Lender in Thomas-Worral Model 250 12 Exercise 15.14 b Evolution of Consumption Distribution over Time 251 1 Exercise 19.4 a: implicit equation for θ i 319 2 Exercise 19.4 b : Solving for unemployment level in each skill market 320 3 Exercise 19.4 b : Solving for the aggregate unemployment level321 4 Exercise 19.5 : Solving for equilibrium unemployment 323 5 Execise 19.6 : Solving for equilibrium unemployment 326 CHAPTER 1 Time series 7 8 1. TIME SERIES Exercise 1.1. Consider the Markov Chain (P, π 0 ) =  .9 .1 .3 .7  ,  .5 .5  , where the state space is x =  1 5  . Compute the likelihood of the following three histories for t = 0, 1, 2, 3, 4: a. 1,5,1,5,1. b. 1,1,1,1,1. c. 5,5,5,5,5. Solution The probability of observing a given history up to t = 4, say (x i 5 , x i 4 , x i 3 , x i 2 , x i 1 , x i 0 ), is given by P (x i 4 , x i 3 , x i 2 , x i 1 , x i 0 ) = P i 4 ,i 3 P i 3 ,i 2 P i 2 ,i 1 P i 1 ,i 0 π 0i 0 where P ij = Prob (x t+1 = x j |x t = x i ) and π 0i = Prob (x 0 = x i ). By applying this formula one obtains the following results: a. P (1, 5, 1, 5, 1) = P 21 P 12 P 21 P 21 π 01 = (.3) (.1) (.3) (.1) (.5) = .00045. b. P (1, 1, 1, 1, 1) = P 11 P 11 P 11 P 11 π 01 = (.9) 4 (.5) = .3281. c. P (5, 5, 5, 5, 5) = P 22 P 22 P 22 P 22 π 02 = (.7) 4 (.5) = .12. Exercise 1.2. A Markov chain has state space x =  1 5  . It is known that E (x t+1 |x t = x) =  1.8 3.4  and that E  x 2 t+1 |x t = x  =  5.8 15.4  . Find a transition matrix consistent with these conditional expectations. Is this transition matrix unique (i.e., can you find another one that is consistent with these conditional expectations)? Solution From the formulas for forecasting functions of a Markov chain, we know that E (h(x t+1 )|x t = x) = Ph, where h(x) is a function of the state represented by an n × 1 vector h. Applying this formula yields: E (x t+1 |x t = x) = P x and E  x 2 t+1 |x t = x  = P x 2 . This yields a set of 4 linear equations: 1. TIME SERIES 9  1.8 3.4  = P  1 5  and  5.8 15.4  = P  1 25  , which can be solved for the 4 unknowns. Alternatively, using matrix notation, we can rewrite this as e = P h, where e = [e 1 , e 2 ], e 1 = E (x t+1 |x t = x) , e 2 = E  x 2 t+1 |x t = x  and h = [h 1 , h 2 ], where h 1 = x and h 2 = x 2 :  1.8 5.8 3.4 15.4  = P  1 1 5 25  . Then P is uniquely determined as P = eh −1 . Uniqueness follows from the fact that h 1 and h 2 are linearly independent. After some algebra we obtain a well- defined stochastic matrix: P =  .8 .2 .4 .6  . Exercise 1.3. Consumption is governed by an n state Markov chain P, π 0 where P is a stochastic matrix and π 0 is an initial probability distribution. Consumption takes one of the values in the n×1 vector ¯c. A consumer ranks stochastic processes of consumption t = 0, 1 . . . according to E ∞  t=0 β t u(c t ), where E is the mathematical expectation and u(c) = c 1−γ 1−γ for some parameter γ ≥ 1. Let u i = u(¯c i ). Let v i = E[  ∞ t=0 β t u(c t )|c 0 = ¯c i ] and V = Ev, where β ∈ (0, 1) is a discount factor. a. Let u and v be the n × 1 vectors whose ith components are u i and v i , re- spectively. Verify the following formulas for v and V : v = (I − βP ) −1 u, and V =  i π 0,i v i . b. Consider the following two Markov processes: Process 1: π 0 =  .5 .5  , P =  1 0 0 1  . Process 2: π 0 =  .5 .5  , P =  .5 .5 .5 .5  . For both Markov processes, ¯c =  1 5  . Assume that γ = 2.5, β = .95. Compute unconditional discounted expected utility V for each of these processes. Which of the two processes does the consumer prefer? Redo the calculations for γ = 4. Now which process does the consumer prefer? c. An econometrician observes a sample of 10 observations of consumption rates 10 1. TIME SERIES for our consumer. He knows that one of the two preceding Markov processes generates the data, but not which one. He assigns equal “prior probability” to the two chains. Suppose that the 10 successive observations on consumption are as follows: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1. Compute the likehood of this sample under process 1 and under process 2. Denote the likelihood function Prob(data|Model i ), i = 1, 2. d. Suppose that the econometrician uses Bayes’ law to revise his initial proba- bility estimates for the two models, where in this context Bayes’ law states: Prob(Model i |data) = Prob(data|Model i ) · Prob(Model i )  j Prob(data|Model j ) · Prob(Model j ) . The denominator of this expression is the unconditional probability of the data. After observing the data sample, what probabilities does the econometrician place on the two possible models? e. Repeat the calculation in part d, but now assume that the data sample is 1, 5, 5, 1, 5, 5, 1, 5, 1, 5. Solution a. Given that v i = E [  ∞ t=0 β t u(c t )|c 0 = c i ] , we can apply the usual vector nota- tion (by stacking ): v = E  ∞  t=0 β t u(c t )|c 0 = c  . To apply the forecasting function formula in the notes: E ∞  k=0 β k (h(x t+k )|x t = x) = (I −βP) −1 h. Let h(x) = u(c). Then it follows immediately that: v = E  ∞  t=0 β t u(c t )|c 0 = c  = (I − βP ) −1 u. Second, to compute V = Ev, simply note that in general the unconditional expec- tation at time 0 of a foreasting function h is given by: E(h(x 0 )) =  n i=1 h i π 0,i = π  0 h, or, in particular: V = n  i=1 v i π 0,i . Also, you should be able to verify that V = E [  ∞ t=0 β t u(c t )] by applying the law of iterated expectations. b. the matlab program exer0103.m computes the solutions. Process1 and Process 2: V = −7.2630 for γ = 2.5 Process1 and Process 2: V = −3.36 for γ = 4 [...]... 1 Exercise 1.7 a Exercise 1.7 Get the Matlab programs bigshow.m and freq.m Use bigshow to compute and display a simulation of length 80, an impulse response function, and a spectrum for each of the following scalar stochastic processes yt In each of the following, wt is a scalar martingale difference sequence adapted to its own history and the initial values of lagged y’s a yt = wt b yt = (1 + 5L)wt... Figure 7 25 26 1 TIME SERIES Exercise 1.8 This exercise deals with Cagan’s money demand under rational expectations A version of Cagan’s (1956) demand function for money is (1) mt − pt = −α(pt+1 − pt ), α > 0, t ≥ 0, where mt is the log of the nominal money supply and pt is the price level at t Equation (1) states that the demand for real balances varies inversely with the expected rate of inflation, (pt+1... = 0 for all t? d Briefly tell where, if anywhere, condition (4) plays a role in your answer to part a e For the parameter values α = 1, ρ = 1, compute and display all the equilibria Solution a First, consider the money demand equation and rewrite the demand for money as a function of the future time path of prices: (1 + α) − αL−1 pt α = (1 + α) 1 − L−1 pt 1+α mt = (11) We know that in equilibrium: ms... In equilibrium, −2 −1 mt ≡ ms ∀t ≥ 0 t (3) (i.e., the demand for money equals the supply) For now assume that (4) |ρα/(1 + α)| < 1 An equilibrium is a {pt }∞ that satisfies equations (1), (2), and (3) for all t t=0 a Find an expression an equilibrium pt of the form n (5) pt = wj mt−j + ft j=0 Please tell how to get formulas for the wj for all j and the ft for all t b How many equilibria are there? c... BA x0 + tr (BCC )] + βd = x0 Gx0 + β [x0 A BA x0 + tr (BCC ) + d] Collecting terms, this yields two equations: (9) B = G + β [A BA] , and (10) d(1 − β) = βtr (C BC) i.Use ex0105.m to compare your solutions Make sure not√ forget the discount to factor β = 95 The command to compute B is doublej ( 95A, G), which produces:    B = 104 ∗    −1.3284 −1.2803 0 0 −0.6690 d = −2.5240e + 006 −1.2803 −1.2620... conditions Then compute the covariance stationary mean and variance of yt assuming the following parameter sets of parameter values: i ρ = 1.2 −.3 0 0 , µ = 10, c = 1 ii ρ = 1.2 −.3 0 0 , µ = 10, c = 2 iii ρ = 9 0 0 0 , µ = 5, c = 1 iv ρ = 2 0 0 5 , µ = 5, c = 1 v ρ = 8 3 0 0 , µ = 5, c = 1 Hint 1: The Matlab program doublej.m , in particular, the command X=doublej(A,C*C’) computes the solution of the... {ct , zt } is covariance stationary a Compute the initial mean and covariance matrix that make the process covariance stationary b For the initial conditions in part a, compute numerical values of the following population linear regression: ct+2 = α0 + α1 zt + α2 zt−4 + where E t t 1 zt zt−4 = 0 0 0 Solution a Use ex0105.m to compare your solutions  1.97 1.24 0.24  1.24 1.97 07  Cx (0) =  0.24 0.07... 1.24 0.24  1.24 1.97 07  Cx (0) =  0.24 0.07 1.57   0.48 0.24 92 0 0 0 0.48 24 92 1.57 0 0 0 0 0 0    ,   and µ = 666 666 0 0 3333 × 3 = 2 2 0 0 1 b Following the same line of reasoning as before, derive the orthogonality conditions: where X = 1 zt zt−4 EX (Y − XB) = 0, and Y = ct+2 Solving for β : β = (E(XX )) where −1 E (XY ) ,    1 1 1 1 1 1 2 −.0336 1  cov(zt , zt−4 ) 1  =  1.57... , for t = 0, 1, 2 ,where xt = yt+1 yt yt−1 yt−2 1 condition, A is a 5 × 5 matrix and C is an 5 × 1 matrix c 0 0 0 0     wt+1 ,   , x0 is a given initial b Assume that the initial conditions are such that yt is covariance stationary Consider the initial vector x0 as being drawn from a distibution with mean µ0 and covariance matrix Σ0 Given stationarity, we can derive the unconditional mean... that all the eigenthe one associated with the constant 4.26 4.73 5.26 4.73 0 3.83 4.26 4.73 5.26 0 0 0 0 0 0    ,   and µ = 5 5 5 5 1 In order for the sequence {xt } to satisfy stationarity, the intitial value x0 needs to be drawn from the stationary distribution with µ and Cx (0) as the unconditional moments iv ρ = 2 0 0 5 , µ = 5, c = 1,   2 0 0 5 1.5  1 0 0 0 0    A =  0 1 0 0 0  . Ljungqvist and Thomas J. Sargent. This solution manuscript is currently only available on the web. We in- vite the reader to bring typos and other corrections to our attention. Please email sargent@ stanford.edu,. Macroeconomic Theory preliminary and incomplete Stijn Van Nieuwerburgh Pierre-Olivier Weill Lars Ljungqvist Thomas J. Sargent Introduction This is a first version of the solutions to the exercises in. Cross-sectional Mean and Dispersion of Consumption and Assets 221 1 Exercise 15.2 226 2 Exercise 15.10 a : Consumption Distribution 238 3 Exercise 15.10 b : Consumption, Promised Utility, Profits and Bank