Exercises in Recursive Macroeconomic Theory preliminary and incomplete Stijn Van Nieuwerburgh Pierre-Olivier Weill Lars Ljungqvist Thomas J. Sargent Introduction This is a ¯rst 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 [email protected], [email protected] or [email protected]. We will regularly update this manuscript during the following months. Some questions ask for computations in matlab. The program ¯les 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, Pro¯ts and Bank Balance in Contract that Maximizes the Money Lender's Pro¯ts 239 4 Exercise 15.10 c : Consumption, Promised Utility, Pro¯ts and Bank Balance in Contract that Gives Zero Pro¯ts to Money Lender 240 5 Exercise 15.11 a : Pareto Frontier, ¯ = 0:95 241 6 Exercise 15.11 b : Pareto Frontier, ¯ = 0:85 242 7 Exercise 15.11 c : Pareto Frontier, ¯ = 0:99 243 5 6 LIST OF FIGURES 8 Exercise 15.12 a : Consumption, Promised Utility, Pro¯ts and Bank Balance in Contract that Maximizes the Money Lender's Pro¯ts 245 9 Exercise 15.12 b : Consumption Distribution 246 10 Exercise 15.12 c : Wage-Tenure Pro¯le 247 11 Exercise 15.14 a : Pro¯ts 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 θi 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. :9 :1 :5 Consider the Markov Chain (P; ¼ ) = ; ; where the state 0 :3 :7 :5 µ· ¸ · ¸¶ 1 space is x = : Compute the likelihood of the following three histories for 5 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 (xi5 ; xi4 ; xi3 ; xi2 ; xi1 ; xi0 ); is given by P (xi4 ; xi3 ; xi2 ; xi1 ; xi0 ) = Pi4;i3 Pi3;i2 Pi2;i1 Pi1;i0 ¼0i0 where P = Prob (x = x x = x ) and ¼ = Prob (x = x ). ij t+1 jj t i 0i 0 i By applying this formula one obtains the following results: a. P (1; 5; 1; 5; 1) = P21P12P21P21¼01 = (:3) (:1) (:3) (:1) (:5) = :00045. 4 b. P (1; 1; 1; 1; 1) = P11P11P11P11¼01 = (:9) (:5) = :3281. 4 c. P (5; 5; 5; 5; 5) = P22P22P22P22¼02 = (:7) (:5) = :12. Exercise 1.2. 1 A Markov chain has state space x = : It is known that E (x x = x) = 5 t+1j t · ¸ 1:8 5:8 and that E x2 x = x = : Find a transition matrix consistent 3:4 t+1j t 15:4 · ¸ · ¸ with these conditional¡ expectations.¢ Is this transition matrix unique (i.e., can you ¯nd 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 ) x = x) = P h; t+1 j t where h(x) is a function of the state represented by an n 1 vector h: Applying this formula yields: £ E (x x = x) = P x and E x2 x = x = P x2: t+1j t t+1j t This yields a set of 4 linear equations: ¡ ¢ 1. TIME SERIES 9 1:8 1 5:8 1 = P and = P ; 3:4 5 15:4 25 · ¸ · ¸ · ¸ · ¸ which can be solved for the 4 unknowns. Alternatively, using matrix notation, we can rewrite this as e = P h, where e = [e1; e2]; e1 = E (xt+1 xt = x) ; e2 = E x2 x = x and h = [h ; h ]; where h = x and h = x2 : j t+1j t 1 2 1 2 ¡ ¢ 1:8 5:8 1 1 = P : 3:4 15:4 5 25 · ¸ · ¸ Then P is uniquely determined as P = eh¡1: Uniqueness follows from the fact that h1 and h2 are linearly independent. After some algebra we obtain a well- de¯ned stochastic matrix: :8 :2 P = : :4 :6 · ¸ Exercise 1.3. Consumption is governed by an n state Markov chain P; ¼0 where P is a stochastic matrix and ¼0 is an initial probability distribution. Consumption takes one of the values in the n 1 vector c¹. A consumer ranks stochastic processes of consumption t = 0; 1 : : : according£ to 1 t E ¯ u(ct); t=0 X c1¡γ where E is the mathematical expectation and u(c) = 1¡γ for some parameter 1 t γ 1. Let ui = u(c¹i). Let vi = E[ t=0 ¯ u(ct) c0 = c¹i] and V = Ev, where ¯ ¸(0; 1) is a discount factor. j 2 P a. Let u and v be the n 1 vectors whose ith components are ui and vi, re- spectively. Verify the follo£wing formulas for v and V : v = (I ¯P )¡1u; and ¡ V = i ¼0;ivi. b. ConsiderP the following two Markov processes: :5 1 0 Process 1: ¼ = , P = . 0 :5 0 1 · ¸ · ¸ :5 :5 :5 Process 2: ¼ = , P = . 0 :5 :5 :5 · ¸ · ¸ 1 For both Markov processes, c¹ = . Assume that γ = 2:5; ¯ = :95. Compute 5 · ¸ 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 = 1; 2. j i 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(data Model ) Prob(Model ) Prob(Model data) = j i ¢ i : ij Prob(data Model ) Prob(Model ) j j j ¢ j The denominator of this expressionP 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 1 t a. Given that vi = E [ t=0 ¯ u(ct) c0 = ci] ; we can apply the usual vector nota- tion (by stacking ): j P 1 v = E ¯tu(c ) c = c : t j 0 " t=0 # X To apply the forecasting function formula in the notes: 1 E ¯k (h(x ) x = x) = (I ¯P )¡1 h: t+k j t ¡ Xk=0 Let h(x) = u(c). Then it follows immediately that: 1 v = E ¯tu(c ) c = c = (I ¯P )¡1 u: t j 0 ¡ " t=0 # X Second, to compute V = Ev, simply note that in general the unconditional expec- n tation at time 0 of a foreasting function h is given by: E(h(x0)) = i=1 hi¼0;i = ¼0 h, or, in particular: 0 P n V = vi¼0;i: i=1 X 1 t Also, you should be able to verify that V = E [ t=0 ¯ u(ct)] by applying the law of iterated expectations. b. the matlab program exer0103.m computes Pthe solutions. Process1 and Process 2: V = 7:2630 for γ = 2:5 Process1 and Process 2: V = ¡3:36 for γ = 4 ¡ 1.