Modern Macroeconomics II

Transcription

1 Modern Macroeconomics II Katsuya Takii OSIPP Katsuya Takii (Institute) Modern Macroeconomics II 1 / 461

2 Introduction Purpose: This lecture is aimed at providing students with standard methods in modern macroeconomics. In particular, the lecture extends Solow model, which was taught in Modern Macroeconomics I, into three directions. Firstly, I apply the dynamic optimization techniques to endogenize saving rate in Solow model. Secondly, I introduce stochastic shocks to analyze uncertainty in a dynamic context. Finally, I discuss how one can numerically analyze the model. In addition, I also discuss how one can analyze discrete choices in a dynamic context. These methods are useful not only for understanding macroeconomics, but also for understanding dynamic issues in any elds of economics, such as public economics and nancial economics. O ce hour: Room 602, 9:50-10:20, 16:20-16:50 on Monday. Appointment is required for other time. Address: takii@osipp.osaka-u.ac.jp Katsuya Takii (Institute) Modern Macroeconomics II 2 / 461

3 Introduction Grading Policy: 35% on assignments and 65% on a nal exam. 1 I will give you X assignments. Students must hand them in at the following lecture. If students turn an assignment in by the due date, I will give them 35/X points. If students turn an assignment in late, I will give them 14/X points. If students submit all assignments, you will receive 35 points. Students must write their answers with a pen. I don t allow the typed answers for this assignment. 2 The full score of nal exam is 65 points. I guarantee that more than 40 points out of 65 points will come from the assignment and the slide of lecture. Katsuya Takii (Institute) Modern Macroeconomics II 3 / 461

4 Introduction Remarks: 1 I assume that students have already taken Decision Theory, Modern Microeconomics 1, Modern Macroeconomics 1. 2 Because of the nature of the issues, the lectures are rather technical. Students are expected to prepare by themselves to understand the lecture. I highly recommend this course to the students who think of economics as a major discipline and go to the doctoral program. 3 I will teach this course in English unless all students prefer Japanese. I encourage students to make comments and questions in English. However, I will not prevent students from asking questions in Japanese. I can discuss your questions and comments in Japanese or English at my o ce hour. Katsuya Takii (Institute) Modern Macroeconomics II 4 / 461

5 Introduction Course Outline 1 Basic Dynamic Programming (6 lectures) Consumption, Optimal Growth Model and Recursive Competitive Equilibrium Appendix: Overlapping Generation Model 2 Continuous Time Dynamic Programing and Hamiltonian ( 2 lectures) Investment and Continuous Optimal Growth Model 3 Stochastic Dynamic Programming (3 lectures) Asset Pricing and Stochastic Optimal Growth Model (Real Business Cycle Model) 4 Dynamic Programing and Discrete Choice ( 3 lectures) Labor Search and Equilibrium Unemployment Model 5 Final Exam (1 lecture) Katsuya Takii (Institute) Modern Macroeconomics II 5 / 461

6 Introduction to Dynamic Programming Macroeconomics is a study to explain the behavior of aggregate data such as GDP per capita, in ation rate, and unemployment rate. For this purpose, we must infer the structure of the economy that brings the observable data. In order to infer the structure, we need a theory. current consensuses among macroeconomists There are two 1 Use the variants of dynamic stochastic general equilibrium models to analyze economy. 2 Developing a quantitative theory is useful. Theory can be used not only for conveying the idea, but also for extracting meaningful information from data. The foundation of current macroeconomics that brings two consensuses is the neoclassical growth model. This is the main subject of this lecture. Katsuya Takii (Institute) Modern Macroeconomics II 6 / 461

7 Introduction to Dynamic Programming Remember Kaldor s Stylized Facts (1963) 1 The growth rate of GDP per capita is nearly constant: 2 The growth rate of capital per capita is nearly constant:. 3 The growth rate of output per worker di ers substantially across countries. 4 The rate of return to capital is nearly constant: 5 The ratio of physical capital to output is nearly constant: 6 The shares of labor and physical capital are nearly constant: In order to meet this requirement, the growth model must exhibit the balanced growth path. Katsuya Takii (Institute) Modern Macroeconomics II 7 / 461

8 Introduction to Dynamic Programming The rst candidate is the extended Solow Model, which is summarized by the following 3 equations. with an assumption. K t+1 = F (K t, T t N t ) + (1 δ) K t C t T t+1 = (1 + g) T t N t+1 = (1 + n) N t C t = (1 s) F (K t, T t N t ) But, as Lucas (1976) critique suggests, this constant saving rate may not be a robust assumption. We would like to develop a model which allows us to analyze the impact of a policy change on real economy through changes in the saving rate. Katsuya Takii (Institute) Modern Macroeconomics II 8 / 461

9 Introduction to Dynamic Programming In order to endogenously determine the saving rate, we must know how consumers decide their consumption. One of a convenient assumption is that a representative consumer maximizes the following utility function. max fc t β t N t U (c t ), g τ t=0 K t+1 = F (K t, T t N t ) + (1 δ) K t c t N t, K 0 is given T t+1 = (1 + g) T t, T 0 is given N t+1 = (1 + n) N t, N 0 is given where c t = C t N t. This model is called the neoclassical growth model. In order to solve this problem, we need to know how to solve a dynamic optimization problem. Katsuya Takii (Institute) Modern Macroeconomics II 9 / 461

10 A Finite Horizon Problem Consider the following problem max fx t g s.t. S t+1 = G (X t, S t ), ( T 1 β (t τ) r (X t, S t ) + β (T τ) V T (S T ) t=τ S τ is given. The variable, fx t g, is called a control variable. It can be a vector. This is the variable which an agent tries to control in order to maximize his objective function. The variable fc t g is the control variable in the neoclassical growth model. The variable, fs t g, is called a state variable. It can be a vector. The state variable summarizes the current state of the economy. The vector f(k t, T t, N t )g is a state vector in the neoclassical growth model. ), Katsuya Takii (Institute) Modern Macroeconomics II 10 / 461

11 A Finite Horizon Problem The function r (X t, S t+1 ) is called the one period return function. One period return function summarizes the reward from current state variable and the current control variable. In the case of the neoclassical growth model, r (X t, S t+1 ) = N t U (C t ). The function G (X t, S t ) is called the transition function. It describes the dynamic behavior of the state variables. In the case of the neoclassical growth model 8 < F (K t, T t N t ) + (1 δ) K t C t 9 = G (C t, (K t, T t, N t )) = (1 + g) T t : (1 + n) N t ; The function, V T () is called the value function at the last period. Since the state variable summarizes the current state of economy. The value function indicates, how much value he expects to obtain in the future when the current state is S T. Katsuya Takii (Institute) Modern Macroeconomics II 11 / 461

12 A Finite Horizon Problem Given S τ, when the agent chooses X τ at date τ, S τ+1 is automatically determined. Given this S τ+1, the agent can choose X τ+1 at date τ + 1, and it determines S τ+2, and so on. In this way, the agent can recursively decide his decisions. Note that max fx t g = max fx t g = max fx s g s<t ( T 1 t=τ β (t τ) r (X t, S t ) + β (T τ) V T (S T ) ( )... + β (t 1 τ) r (X t 1, S t 1 ) + β (t τ) r (X t, S t ) +β (t+1 τ) r (X t+1, S t+1 ) +...β (T τ) V T (S T ) β (t 1 τ) 9 r (X t 1, S t 1 ) >< 2 3 β (t τ) >= r (X t, S t ) 6 + max fxx g >: x t 4 +β (t+1 τ) 7 r (X t+1, S t+1 ) β (T τ) >; V T (S T ) ) Katsuya Takii (Institute) Modern Macroeconomics II 12 / 461

13 A Finite Horizon Problem That is, once the agent knows, S t, he does not need to worry about other past variables f(x s, S s )g s<t because the state variable summarizes every important information of the economy at that time. Hence, the maximization problem at date T as follows: 1 can be simpli ed max fr (X T 1, S T 1 ) + βv T (G (X T 1, S T 1 ))g, X T 1 S T 1 is given. The solution to this problem depends on S T 1. That is, we can derive an optimal policy function x T 1 () as a function of S T 1 : x T 1 (S T 1 ) = arg max X T 1 fr (X T 1, S T 1 ) + βv T (G (X T 1, S T 1 ))g Katsuya Takii (Institute) Modern Macroeconomics II 13 / 461

14 A Finite Horizon Problem Since the policy function is an optimal strategy, we can de ne the value function at date T 1, V T 1 (), as follows: max fr (X T 1, S T 1 ) + βv T (G (X T 1, S T 1 ))g X T 1 = r (x T 1 (S T 1 ), S T 1 ) + βv T (G (x T 1 (S T 1 ), S T 1 )) V T 1 (S T 1 ). Using V T 1 (S T 1 ) we can rewrite the agent s problem at dates T 2 as follows: max fr (X T 2, S T 2 ) + βv T 1 (G (X T 2, S T 2 ))g, X T 2 S T 2 is given. This is the similar problem as before. Therefore, we can de ne V T 2 (S T 2 ) and using V T 2 (S T 2 ), we can describe the problem at date T 3 and so on. Katsuya Takii (Institute) Modern Macroeconomics II 14 / 461

15 A Finite Horizon Problem In general, an original problem can be rewritten as the sequence of a simple static problem: V t (S t ) max X t fr (X t, S t ) + βv t+1 (G (X t, S t ))g, This equation is called the Bellman Equation. Hence, a nite horizon problem can be solved by a sequence of policy functions and value functions, f(x t (), V t ())g T 1 t=0. For any initial state variable, S τ, the optimal policy function x τ (S τ ) determines X τ and the transition function G (X τ, S τ ) determines S τ+1. Given this S τ+1, the optimal policy function and the transition function determines X τ+1 and S τ+2 and so on. Katsuya Takii (Institute) Modern Macroeconomics II 15 / 461

16 A Finite Horizon Problem Assume that the rst order condition is valid. condition is The rst order 0 = r 1 (x t (S t ), S t ) + βv 0 t+1 (G (x t (S t ), S t )) G 1 (x t (S t ), S t ), (1) for any t. The rst term of the right hand side of equation (1) is the marginal return from changing X t. However, when the agent changes X t, it a ects not only the current return, but also the future returns by changing the future state variable. When the agent slightly changes X t, it will change S t+1 by G 1 (X t, S t ). If S t+1 slightly moves, it changes the present value of the future reward at date t + 1 by Vt+1 0 (S t+1). Since the agent discount the future by β, the future impact of changing X t is βvt+1 0 (G (X t, S t )) G 1 (X t, S t ). If the agent optimally chooses variables, these two e ects must be the same at any date t. Katsuya Takii (Institute) Modern Macroeconomics II 16 / 461

17 A Finite Horizon Problem It is useful to derive the envelope theorem: Vt 0 (S t ) = r 2 (x t (S t ), S t ) + βvt+1 0 (G (x t (S t ), S t )) G 2 (x t (S t ), S t ) + r1 (x t (S t ), S t ) + βvt+1 0 (G (x t (S t ), S t )) G 1 (x t (S t ), S t ) xt 0 (S t ) = r 2 (x t (S t ), S t ) + βvt+1 0 (G (x t (S t ), S t )) G 2 (x t (S t ), S t ) for any t. Katsuya Takii (Institute) Modern Macroeconomics II 17 / 461

18 A Finite Horizon Problem It shows the marginal bene t of changing the state variable, S t. When the state variable changes a little bit, it changes the current return by r 2 (x t (S t ), S t ). But since changing the state variable at date t will change the future state variable S t+1 by G 2 (X t, S t ), it also has a dynamic e ect. Since changing S t+1 a ects the future reword by Vt+1 0 (S t+1) and the agent discounts the future by β, the total e ect must be βvt+1 0 (G (X t, S t )) G 2 (X t, S t ). Katsuya Takii (Institute) Modern Macroeconomics II 18 / 461

19 A Finite Horizon Problem Example: Consider the following household problem, T 1 max fc t g t=τ s.t. a t+1 = (1 + ρ) a t + w c t given a τ β (t τ) U (c t ) + β (T τ) V T (a T ), We can de ne the Bellman equation as follows: V t (a t ) = max c t fu (c t ) + βv t+1 (a t+1 )g, s.t. a t+1 = (1 + ρ) a t + w c t, Katsuya Takii (Institute) Modern Macroeconomics II 19 / 461

20 A Finite Horizon Problem The rst order condition is Envelope theorem U 0 (c (a t )) = βv 0 t+1 ((1 + ρ) a t + w c (a t )) V 0 t (a t ) = βv 0 t+1 [(1 + ρ) a t + w c (a t )] [1 + ρ]. Combining Two equations U 0 (c (a t 1 )) β = U 0 (c (a t )) (1 + ρ) Katsuya Takii (Institute) Modern Macroeconomics II 20 / 461

21 A Finite Horizon Problem Euler equation: U 0 (c t ) βu 0 (c t+1 ) = 1 + ρ where c t+1 = c (a t+1 ) and c t = c (a t ). The left hand side is the marginal rate of substitution between consumption at date t and t + 1; the right hand side is the marginal gain from saving. Katsuya Takii (Institute) Modern Macroeconomics II 21 / 461

22 A Finite Horizon Problem Euler Equation Katsuya Takii (Institute) Modern Macroeconomics II 22 / 461

23 An In nite Horizon Problem Now let me consider the case, T goes in nite. problem is U (S τ, τ) = max fx t g ( t=τ β (t τ) r (X t, S t ) s.t. S t+1 = G (X t, S t ), S τ is given. That is, an original If T goes in nite, there is no last period and we cannot use the previous method. However, we can explain it by analogy between this model and the previous one. Assume that a policy function of this original problem is x u (S τ ). ), Katsuya Takii (Institute) Modern Macroeconomics II 23 / 461

24 An In nite Horizon Problem Remember that once we control a current state, the past decision does not in uence future maximization problem. Hence, U (S τ, τ) = max fx t g τ t=τ = max X τ ( β (t τ) r (X t, S t ), r (X τ, S τ ) + β max fx t g τ+1 = max X τ fr (X τ, S τ ) + βu (S τ+1, τ + 1)g ) t=τ+1 β (t (τ+1)) r (X t, S t ), Katsuya Takii (Institute) Modern Macroeconomics II 24 / 461

25 An In nite Horizon Problem Hence, we can consider the following recursive problem: V (S t ) = max X t fr (X t, S t ) + βv (S t+1 )g, (2) S t+1 = G (X t, S t ). Assume that the policy function of this problem is x (S t ). 1 With certain mild conditions (r (X, S) is bounded and continuous, f(s t+1, X t, S t ) js t+1 G (X t, S t )g is compact), we can show that there exists a unique value function V () and a policy function which solves equation (2) and that U (S τ, τ) = V (S τ ) and x u (S τ ) = x (S τ ). That is, Bellman equation (2) is equivalent to the original problem and the value function and policy function is time invariant. 2 With the further assumptions (r (X, S) is strict concave and f(s t+1, X t, S t ) js t+1 G (X t, S t )g is convex), V (S) is strictly concave and x (S τ ) is continuous and single valued. 3 With further mild condition ( r (X, S) is continuously di erentiable), it is known that V is continuously di erentiable. Katsuya Takii (Institute) Modern Macroeconomics II 25 / 461

26 An In nite Horizon Problem An economic interpretation of the Bellman equation (2) is that the agent maximizes the sum of current return, r (X t, S t ), and the discounted future values from S t+1, βv (S t+1 ). He is concerned about the trade-o between the current bene ts and the future bene ts when he chooses X t. Since he lives forever, it does not matter when he makes his decisions. Therefore, the value function and the policy function do not depend on time. His problem is stationary. Katsuya Takii (Institute) Modern Macroeconomics II 26 / 461

27 An In nite Horizon Problem There are several methods to analyze the Bellman equation (2): 1 Guess and Verify Method 2 Euler Equation 3 Numerical method. Katsuya Takii (Institute) Modern Macroeconomics II 27 / 461

28 Numerical Method The original problem can be approximated by a nite horizon problem: (" #) T 1 U (S τ, τ) = lim max T! fx t β (t τ) r (X t, S t ) + β (T τ) V T (S T ), g t=τ s.t. S t+1 = G (X t, S t ), S τ is given. It is shown that we can approximate the Bellman equation (2) by the corresponding nite horizon problem. V (S) = lim t! V t (S). V t (S) max X fr (X, S) + βv t+1 (G (X, S))g (3) Since there exist a unique value function V (), if we iterate equation (3) from an arbitrary initial value function V T (S) for any S, it must converge to V (S). Katsuya Takii (Institute) Modern Macroeconomics II 28 / 461

29 An In nite Horizon Problem Example: Suppose that S 2 f1g. Therefore, S t+1 = G (X t, S t ) = 1 for any t. Suppose that. x t (1) = arg max fr (X, 1) + βv t+1 (G (X, 1))g X fr (X, 1) + βv t+1 (1)g = arg max X = arg max X fr (X, 1)g x (1) Then V t (1) r (x (1), 1) + βv t+1 (1). Katsuya Takii (Institute) Modern Macroeconomics II 29 / 461

30 An In nite Horizon Problem The Existence of a Globally Stable Unique Value Function Katsuya Takii (Institute) Modern Macroeconomics II 30 / 461

31 Guess and Verify Method Guess and Verify method: One way to analyze the property of equation (2) is guess and very method. The rst, guess what would be the property of V () and assume that the property is true. Then verify that your guess satisfy equation (2). Since we know the value function is unique, if a property satis es equation (2), the value function must have the property. Katsuya Takii (Institute) Modern Macroeconomics II 31 / 461

32 Guess and Verify Method Example: Consider the following problem. Bellman Equation max fc t g t=0 β t u (c t ), s.t. a t+1 = (1 + ρ) a t + w c t V (a t ) = max c t fu (c t ) + βv ((1 + ρ) a t + w c t )g Katsuya Takii (Institute) Modern Macroeconomics II 32 / 461

33 Guess and Verify Method Guess V (a 0 ) V (a) if a 0 a. Suppose that Note that c (a t ) = arg max c t fu (c t ) + βv ((1 + ρ) a t + w c t )g u c at 0 + βv (1 + ρ) a 0 t + w c a 0 u (c (a t )) + βv (1 + ρ) at 0 + w c (a t ) u (c (a t )) + βv ((1 + ρ) a t + w c (a t )) Hence, our guess, V (a 0 ) V (a), is veri ed. Katsuya Takii (Institute) Modern Macroeconomics II 33 / 461

34 Guess and Verify Method Since many applied economists restrict their attention to the case to which we can apply the rst order condition, I explain this method by using the rst order condition. The policy function x () must satisfy the rst order condition and the bellman equation for any S as follows: 0 = r 1 (x (S), S) + βv 0 (G (x (S), S)) G 1 (x (S), S) (4) V (S) = r (x (S), S) + βv (G (x (S), S)). (5) Given the value function V (), equation (4) determines x (); given x (), equation (5) determines V (). That is, these are simultaneous equations. We can analyze our model by examining these two equations. In general we cannot solve a closed form solution. However, there is a special case in which we can nd a closed form solution. Katsuya Takii (Institute) Modern Macroeconomics II 34 / 461

35 Guess and Verify Method Example: Consider the following problem. max fc t g t=0 β t ln c t, s.t. a t+1 = (1 + ρ) a t c t We can de ne the Bellman equation as follows: s.t. a t+1 = (1 + ρ) a t c t V (a t ) = max c t fln c t + βv (a t+1 )g, Katsuya Takii (Institute) Modern Macroeconomics II 35 / 461

36 Guess and Verify Method Guess V (a t ) = f + g ln (a t ). Then consider the problem max c t fln c t + β [f + g ln (a t+1 )]g, s.t. a t+1 = (1 + ρ) a t c t, First Order Condition implies Policy Function 1 c (a t ) = βg (1 + ρ) a t c (a t ) c (a t ) = (1 + ρ) a t c (a t ) βg βg + 1 c (a t ) = (1 + ρ) a t βg βg c (a t ) = (1 + ρ) a t βg + 1 Katsuya Takii (Institute) Modern Macroeconomics II 36 / 461

37 Guess and Verify Method a t+1 a t+1 = (1 + ρ) a t c (a t ) = (1 + ρ) a t (1 + ρ) a t βg + 1 = βg (1 + ρ) a t βg + 1 Katsuya Takii (Institute) Modern Macroeconomics II 37 / 461

38 Guess and Verify Method The Maximized Value ln c (a t ) + β [f + g ln (a t+1 )] = ln (1 + ρ) a t βg (1 + ρ) βg β at f + g ln βg + 1 (1 + ρ) at = (1 + βg) ln + β [f + g ln βg] βg + 1 (1 + ρ) = (1 + βg) ln βg β [f + g ln βg] + (1 + βg) ln a t Hence, if we can nd f and g that satisfy f = (1 + βg) ln g = 1 + βg then we can verify our guess. (1 + ρ) + β [f + g ln βg] βg + 1 Katsuya Takii (Institute) Modern Macroeconomics II 38 / 461

39 Guess and Verify Method g g = 1 1 β f (1 + ρ) f = (1 + βg) ln + β [f + g ln βg] βg (1 + ρ) = (1 + βg) ln + βg ln βg 1 β βg = β (1 + ρ) ln 1 β 1 β β 1 β β 1 β ln β 5 1 β 1 = 1 + β ln (1 + ρ) (1 β) + β 1 β 1 β 1 β ln β 1 β 1 ln (1 + ρ) = + β ln β + ln (1 β) 1 β 1 β 1 β Katsuya Takii (Institute) Modern Macroeconomics II 39 / 461

40 Guess and Verify Method V (a t ) V (a t ) = 1 1 β ln (1 + ρ) 1 β + β 1 β ln β + ln (1 β) + ln a t Katsuya Takii (Institute) Modern Macroeconomics II 40 / 461

41 Guess and Verify Method The guess and verify method demands many calculations. If we are only interested in a policy function, but not a value function, there is one way to reduce calculation: the use of envelope theorem. The rst order condition and the envelope theorem of the Bellman equation (2) imply that 0 = r 1 (x (S), S) + βv 0 (G (x (S), S)) G 1 (x (S), S) (6) V 0 (S) = r 2 (x (S), S) + βv 0 (G (x (S), S)) G 2 (x (S), S) (7) for any S t. Again two unknown functions x () and V 0 () can be solved by two equations. Katsuya Takii (Institute) Modern Macroeconomics II 41 / 461

42 Guess and Verify Method Note that the rst order condition (6) is the same as the rst order condition (4); the envelope theorem (7) is not the same as the Bellman equation (5). Also note that the previous two equations determine x () and V (); the current two conditions determine x () and V 0 (). In general, there exists C such that Z V (S) = V 0 (S) ds + C. Since equation (6) and (7) cannot determine C, without a boundary condition, equation (6) and (7) cannot solve the value function. However, if we are only interested in a policy function, equations (6) and (7) can solve it. Katsuya Takii (Institute) Modern Macroeconomics II 42 / 461

43 Guess and Verify Method Example: Consider the previous problem again. s.t. a t+1 = (1 + ρ) a t c t V (a t ) = max c t fln c t + βv (a t+1 )g, The rst order condition and envelope theorem are 1 c (a t ) = βv 0 ((1 + ρ) a t c (a t )) V 0 (a t ) = β (1 + ρ) V 0 ((1 + ρ) a t c (a t )), Katsuya Takii (Institute) Modern Macroeconomics II 43 / 461

44 Guess and Verify Method Guess V 0 (a) = g a. Then the rst order condition is 1 c (a t ) = g β (1 + ρ) a t c (a t ) Policy function c (a t ) = (1 + ρ) a t c (a t ) βg βg + 1 c (a t ) = (1 + ρ) a t βg βg c (a t ) = (1 + ρ) a t βg + 1 Katsuya Takii (Institute) Modern Macroeconomics II 44 / 461

45 Guess and Verify Method Hence, = = β (1 + ρ) V 0 ((1 + ρ) a t c (a t )) (1 + ρ) c (a t ) (1 + ρ) = βg + 1 a t (1+ρ)a t βg +1 Hence if we nd g that satis es our guess is satis ed. g = βg + 1, Katsuya Takii (Institute) Modern Macroeconomics II 45 / 461

46 Guess and Verify Method g V 0 (a) g = 1 1 β. V 0 (a) = 1 (1 β) a Katsuya Takii (Institute) Modern Macroeconomics II 46 / 461

47 Guess and Verify Method Assignment: Suppose that r (I, K ) = 1 1+ρ [rk I C (I )], K 0 = G (I, K ) = I + (1 δ) K and β = 1 1+ρ where K is capital stock, I is investment, r is the rental price of capital, δ is the depreciation rate, ρ is an real interest rate and C () is the adjustment cost of investment, C 0 (I t ) > 0 and C 00 (I t ) > 0. A rm chooses I to maximizes the present value of the stream of r (I, K ). 1 De ne the Bellman Equation. 2 Show that the value function is an a ne in K: that is, there exist q and m such that V (K ) = qk + m. 3 Show that the investment I is a function of q. 4 Show that q must satisfy q = r ρ + δ 5 Suppose that C (I ) = A 2 I 2. Derive an investment function. Katsuya Takii (Institute) Modern Macroeconomics II 47 / 461

48 Guess and Verify Method Assignment: Suppose that r (I, K ) = 1 1+ρ [rk p (I + C (I, K ))], K 0 = G (I, K ) = I + (1 δ) K and β = 1 1+ρ where K is capital stock, I is investment, r is the rental price of capital, δ is the depreciation rate, p is the relative price of investment, ρ is an real interest rate, and C I (I t, K t ) > 0 C II (I t, K t ) > 0 and C K (I t, K t ) < 0, C (I, K ) is the constant returns to scale in I and K. A rm chooses I to maximizes the present value of the stream of r (I, K ). 1 De ne the Bellman Equation. 2 Show that the value function is linear in K: that is, there exists q such that V (K ) = qk. 3 Show that the investment to capital stock, I K, is a function of Tobin s Q: q Tobin = V (K ) pk = q p. Katsuya Takii (Institute) Modern Macroeconomics II 48 / 461

49 Euler Equation Euler Equation: In general, it is di cult to solve a closed form solution. However, it is possible to analyze the property of solutions for more general class of functions. Remember that the rst order condition and envelope theorem are 0 = r 1 (x (S), S) + βv 0 (G (x (S), S)) G 1 (x (S), S), V 0 (S) = r 2 (x (S), S) + βv 0 (G (x (S), S)) G 2 (x (S), S). Using two equations, we can eliminate V 0 (S) and derive the rst order nonlinear di erence equation. This is called Euler equation. r 1 (X t, S t ) βg 1 (X t, S t ) = r 1 (X t+1, S t+1 ) G 1 (X t+1, S t+1 ) G 2 (X t+1, S t+1 ) r 2 (X t+1, S t+1 ). (8) Katsuya Takii (Institute) Modern Macroeconomics II 49 / 461

50 Euler Equation Although the rst order condition and the envelope theorem can derive the Euler equation, the Euler equation cannot derive the rst order condition and the envelope theorem. That is, the elimination of V 0 (S t ) discards useful information. Therefore, the Euler equation is a necessary condition for the original problem, but not su cient. Note that we can derive the dynamics of X t from Euler equation: X t = Φ (S t+1, X t+1, S t ) We also have the transition equation for the dynamics of S. S t+1 = G (X t, S t ), Katsuya Takii (Institute) Modern Macroeconomics II 50 / 461

51 Euler Equation It means that X τ = Φ (S τ+1, X τ+1, S τ ).. X T 1 = Φ (S T, X T, S T 1 ).. S τ+1 = G (X τ, S τ ).. S T = G (X T 1, S T 1 ).. Katsuya Takii (Institute) Modern Macroeconomics II 51 / 461

52 Euler Equation These equations imply that from period τ to period T 1, the number of equations are 2 (T τ) and the number of X t and S t is 2 (T τ + 1). It means that we need 2 known variables to solve the sequence of X t and S t between τ and T 1. Now we have one initial condition S τ. In a nite horizon case, since the value function at the end period is given, the rst order condition, r 1 (X T, S T ) + βv 0 T +1 (G (X T, S T )) G 1 (X T, S T ) = 0, determines X T as a function of S T. This serves as a boundary condition. It gives us one more equation. Katsuya Takii (Institute) Modern Macroeconomics II 52 / 461

53 Euler Equation The di cult question is what would be the appropriate boundary condition in an in nite horizon problem. It is known that the following transversality condition is su cient for an optimal solution: lim t! βt V 0 (S t ) S t = lim β t 1 r 1 (X t 1, S t 1 ) t! G 1 (X t 1, S t 1 ) S t = 0. (9) Since V 0 (S t ) is the marginal value of S t, β t V 0 (S t ) S t is the present value of stock at t. The transversality condition implies that when t goes in nite, the present value of stock must be negligible. That is, we should not be concerned about an in nite period later. Although Euler equation is not su cient for the original problem, given usual technical conditions such as concavity etc, it is known that the Euler equation together with the transversality condition is su cient for the original problem. Katsuya Takii (Institute) Modern Macroeconomics II 53 / 461

54 Euler Equation Example: Consider the following problem. Bellman Equation max fc t g t=0 β t U (c t ), s.t. a t+1 = (1 + ρ t ) a t + w t c t V (a t, t) = max c t fu (c t ) + βv ((1 + ρ t ) a t + w t c t, t + 1)g Katsuya Takii (Institute) Modern Macroeconomics II 54 / 461

55 Euler Equation FOC Envelope Theorem U 0 (c t ) = βv a (a t+1, t + 1) V a (a t, t) = β (1 + ρ) V a (a t+1, t + 1) Katsuya Takii (Institute) Modern Macroeconomics II 55 / 461

56 Euler Equation Euler equation: U 0 (c (a t 1, t 1)) β U 0 (c t ) βu 0 (c t+1 ) = U 0 (c (a t, t)) (1 + ρ t ) = 1 + ρ t Transversality Condition lim t! βt V a (a t, t) a t = lim β t U 0 (c t 1 ) t! β a t = lim t! β t 1 U 0 (c t 1 ) a t Euler equation is the same as the nite horizon problem, but in nite horizon problem requires the transversality condition. Katsuya Takii (Institute) Modern Macroeconomics II 56 / 461

57 Euler Equation Assignment: Suppose that r (I t, K t ) = 1 1+ρ [r t K t p t (I t + C (I t, K t ))], t K t+1 = G (I t, K t ) = I t + (1 δ) K t and β t = 1 1+ρ where K t is t capital stock, I t is investment, r t is the rental price of capital, δ is the depreciation rate, p t is the relative price of investment, ρ t is an real interest rate, C I (I t, K t ) > 0 C II (I t, K t ) > 0 and C K (I t, K t ) < 0. Suppose that there exists B such that β t = 1 1+ρ B < 1. (Note t 1 that discount factor is time dependent and that 1+ρ = βu 0 (c t+1 ) t U 0 (c t on ) the equilibrium from Euler equation by consumers.) A rm chooses I to maximizes the present value of the stream of r (I, K ). 1 De ne the Bellman Equation. 2 Derive the Euler Equation. 3 Derive the Transversality Condition. Katsuya Takii (Institute) Modern Macroeconomics II 57 / 461

58 The Neoclassical Growth Model We apply Guess and Verify method and Euler Equation to the Neoclassical Growth Model. We also explain the numerical method by using the Neoclassical Growth Model. Katsuya Takii (Institute) Modern Macroeconomics II 58 / 461

59 The Neoclassical Growth Model Consider the previous neoclassical growth model: max fc t g t=0 β t N t U Ct N t, s.t. K t+1 = F (K t, T t N t ) + (1 δ) K t C t, K 0 is given T t+1 = (1 + g) T t, T 0 is given N t+1 = (1 + n) N t, N 0 is given First, we reformulate the model so that the analysis becomes simpler. Katsuya Takii (Institute) Modern Macroeconomics II 59 / 461

60 The Neoclassical Growth Model Note that K t+1 K t+1 = " F Kt T t N t, 1 + (1 δ) K t T t N t C t T t N t T t+1 N t+1 T t+1 N t+1 T t N t = f (k et ) + (1 δ) k et c et, k et+1 (1 + g) (1 + n) = f (k et ) + (1 δ) k et c et, k et+1 = f (k et) + (1 δ) k et c et (1 + g) (1 + n) # T t N t, Katsuya Takii (Institute) Modern Macroeconomics II 60 / 461

61 The Neoclassical Growth Model Note also that max fc t g = max fc t g t=0 t=0 fc et g t=0 = N 0 max β t N t U Ct N t β t (1 + n) t N 0 U Ct T t T t N t [β (1 + n)] t U (c et T t ) Katsuya Takii (Institute) Modern Macroeconomics II 61 / 461

62 The Neoclassical Growth Model Hence, the original problem is equivalent to max fc et g t=0 ˆβ t U (c et T t ) s.t. k et+1 = f (k et) + (1 δ) k et c et (1 + g) (1 + n) T t+1 = (1 + g) T t, given (k 0, T 0 ) We assume that ˆβ = β (1 + n) < 1 Katsuya Takii (Institute) Modern Macroeconomics II 62 / 461

63 The Neoclassical Growth Model Bellman Equation V (k et, T t ) = max c et U (cet T t ) + ˆβV (k et+1, T t+1 ) where ˆβ = β (1 + n) < 1. k et+1 = f (k et) + (1 δ) k et c et (1 + g) (1 + n) T t+1 = (1 + g) T t Katsuya Takii (Institute) Modern Macroeconomics II 63 / 461

64 The Neoclassical Growth Model First Order Condition Envelope Theorem U 0 (c et T t ) T t = ˆβV k (k et+1, T t+1 ) (1 + g) (1 + n) V k (k et, T t ) = ˆβV k (k et+1, T t+1 ) [f 0 (k et ) + (1 (1 + g) (1 + n) δ)] Katsuya Takii (Institute) Modern Macroeconomics II 64 / 461

65 The Neoclassical Growth Model Euler Equation Hence U 0 (c et T t ) T t (1 + g) (1 + n) ˆβ = U 0 (c et+1 T t+1 ) T t+1 f 0 (k et+1 ) + (1 δ) U 0 (c et T t ) (1 + n) U 0 (c et+1 T t+1 ) β (1 + n) = f 0 (k et+1 ) + (1 δ) U 0 (c et T t ) U 0 (c et+1 T t+1 ) β = f 0 (k et+1 ) + (1 δ) Katsuya Takii (Institute) Modern Macroeconomics II 65 / 461

66 The Neoclassical Growth Model Balanced Growth: one of Kaldor s Stylized Facts (1963) suggests that the rate of return to capital, f 0 (k et ), is constant, and therefore, k et = k et+1 = ke in the long run. Remember the dynamics of k et exhibits ke = f (k e ) + (1 δ) ke c et. (1 + g) (1 + n) This means c et = c et+1 = ce in the long run. Euler equation implies that U 0 (ce T t ) U 0 (ce T t+1 ) β = f 0 (ke ) + (1 δ) Hence, if we wish to construct a model that can be consistent with Kaldor s Stylized Facts (1963), must be constant U 0 (c e T t ) U 0 (c e T t+1 ) Katsuya Takii (Institute) Modern Macroeconomics II 66 / 461

67 The Neoclassical Growth Model In order to satisfy the empirical regularity that economy must exhibit a balanced growth, it is well-known that the following functional form is needed: U (c et T t ) = (c ett t ) (1 θ) 1, 0 θ 6= 1 1 θ This utility function is called the constant relative risk aversion utility function. Remark: It can be shown that Because U 0 (c e T t ) = (c e T t ) θ (c et T t ) (1 θ) 1 lim = ln (c et T t ). θ!1 1 θ U 0 (c e T t ) U 0 (c e T t+1 ) = (c e T t ) θ (c e (1 + g) T t ) θ = (1 + g)θ Katsuya Takii (Institute) Modern Macroeconomics II 67 / 461

68 The Neoclassical Growth Model Assignment: Suppose that U (c et T t ) = (c ett t ) (1 θ) 1, θ 0 1 θ Show that θ can be interpreted as the measure of the constant relative risk aversion: θ = U00 (c et T t ) c et T t U 0. (c et T t ) Katsuya Takii (Institute) Modern Macroeconomics II 68 / 461

69 The Neoclassical Growth Model It is also known that 1 θ measures the elasticity of substitution between consumption at any two points in time. To see this, consider the following two period problems. Suppose that U (C t, C t+1 ) = u (C t ) + βu (C t+1 ) and that the budget constraint is p t C t + p t+1 C t+1 = W, where p t and p t+1 are prices at t and t + 1, and W is the wealth. Since the rst order condition is p t+1 = U 2 (C t, C t+1 ) p t U 1 (C t, C t+1 ) = βu0 (C t+1 ) u 0, (C t ) If U (C t ) = (C t ) (1 θ) 1 1 θ, then u 0 (C t ) = C θ t. Hence, p t+1 p t = βu0 (C t+1 ) u 0 (C t ) = βc θ t+1 C θ t Katsuya Takii (Institute) Modern Macroeconomics II 69 / 461

70 The Neoclassical Growth Model It means that the elasticity of substitution, - must satisfy the following condition. d ln Ct+1 C t = d ln pt+1 p t It is shown that d ln Ct+1 C t = βct+1 d ln θ Ct θ = Ct+1 d Ct C t+1 Ct d( p t+1 = d ln Ct+1 Ct pt ) d ln( p t+1 p t+1 pt d ln Ct+1 C t βct+1 d ln θ Ct θ d [ln (C t+1 ) log (C t )] d fln β θ [ln (C t+1 ) ln (C t )]g d [ln (C t+1 ) ln (C t )] θd [ln (C t+1 ) ln (C t )] = 1 θ. pt ), Katsuya Takii (Institute) Modern Macroeconomics II 70 / 461

71 The Neoclassical Growth Model Suppose that U (c et T t ) = (c et T t ) (1 θ) 1 1 θ. Then our current objective function, max fcet g t=0 ˆβ t U (c et T t ) can be further rewritten by a simpler expression. Note that when θ 6= 1, max fc et g t=0 = max fc et g t=0 ( = max fc et g 8 < T 0 max fcet g = : [β (1 + n)] t (c et T t ) 1 θ 1 1 θ [β (1 + n)] t c et T 0 (1 + g) t 1 θ 1 θ t=0 [β (1 + n)] t (1 + g) (1 θ)t (c et T 0 ) 1 θ 1 1 θ ) t=τ [β (1 + n)] t N 0 1 θ hβ (1 + n) (1 + g) (1 θ)i t (cet ) 1 θ t=0 t=τ [β(1+n)] t 1 θ Katsuya Takii (Institute) Modern Macroeconomics II 71 / θ 9 = ;

72 The Neoclassical Growth Model Hence fc et g t=0 arg max = arg max fc et g ( t=0 [β (1 + n)] t (c et T t ) 1 θ 1 1 θ ) hβ (1 + n) (1 + g) (1 θ)i t (c et ) 1 θ 1 θ Katsuya Takii (Institute) Modern Macroeconomics II 72 / 461

73 The Neoclassical Growth Model On the other hand, when θ = 1, U (c et T t ) = ln c et T t. Hence fc et g t=0 fc et g t=0 N 0 max = N 0 max = N 0 [β (1 + n)] t ln c et T t [β (1 + n)] t ln c et + ln (1 + g) t + ln T 0 max fcet g t=0 [β (1 + n)] t ln c et + t=0 [β (1 + n)] t [t ln (1 + g) + ln T 0 ] Hence arg max fc et g N 0 t=0 fc et g t=0 [β (1 + n)] t ln c et T t = arg max [β (1 + n)] t ln c et Katsuya Takii (Institute) Modern Macroeconomics II 73 / 461

74 The Neoclassical Growth Model Hence, the original problem is equivalent to ( max (β ) t ũ (c et ) fc et g t=0 s.t. k et+1 = f (k et) + (1 δ) k et c et, given k 0 (1 + g) (1 + n) where ũ (c et ) = (c et ) 1 θ 1 θ if 0 θ 6= 1 and log (c et ), if θ = 1 and β = β (1 + n) (1 + g) (1 θ). We assume that β = β (1 + n) (1 + g) (1 θ) < 1. ) Katsuya Takii (Institute) Modern Macroeconomics II 74 / 461

75 The Neoclassical Growth Model We can de ne the Bellman equation as follows: V (k et ) = max c et fũ [c et ] + β V (k et+1 )g k et+1 = f (k et) + (1 δ) k et c et (1 + g) (1 + n) where ũ (c et ) = (c et ) 1 θ 1 θ if θ 6= 1 and log (c et ), if θ = 1 and β = β (1 + n) (1 + g) (1 θ). Katsuya Takii (Institute) Modern Macroeconomics II 75 / 461

76 The Neoclassical Growth Model First Order Condition and Envelope Theorem ũ 0 [c (k et )] = β V 0 (κ (k et )) (1 + g) (1 + n) V 0 (k et ) = β V 0 (κ (k et )) [f 0 (k et ) + (1 δ)] (1 + g) (1 + n) where κ (k et ) = f (k et) + (1 δ) k et c (k et ) (1 + g) (1 + n) Katsuya Takii (Institute) Modern Macroeconomics II 76 / 461

77 The Neoclassical Growth Model Assignment: Suppose that θ = 1 and therefore ũ (c) = ln c, δ = 1 and f (k e ) = ke α. Show that c (k et ) = (1 αβ ) k α et κ (k et ) = αβk α et (1 + g) θ Katsuya Takii (Institute) Modern Macroeconomics II 77 / 461

78 Guess and Verify Method for Growth Model The Neoclassical Growth Model Katsuya Takii (Institute) Modern Macroeconomics II 78 / 461

79 Euler Equation for Growth Model We cannot analyze guess and verify method for more general case. this case, we can derive Euler equation. Remember that the rst order condition and envelope theorem are In ũ 0 [c (k et )] = β V 0 (κ (k et )) (1 + g) (1 + n) V 0 (k et ) = β V 0 (κ (k et )) [f 0 (k et ) + (1 δ)] (1 + g) (1 + n) Katsuya Takii (Institute) Modern Macroeconomics II 79 / 461

80 Euler Equation for Growth Model Hence, the Euler equation is ũ 0 (c et 1 ) (1 + g) (1 + n) β = ũ 0 (c et ) f 0 (k et ) + (1 δ). (1 + g) (1 + n) ũ 0 (c et ) β (1 + n) (1 + g) (1 θ) = f 0 (k et+1 ) + (1 δ) ũ 0 (c et+1 ) ũ 0 (c et ) βũ 0 (c et+1 ) = f 0 (k et+1 ) + (1 δ) (1 + g) θ Katsuya Takii (Institute) Modern Macroeconomics II 80 / 461

81 Euler Equation for Growth Model Euler Equation 2 Katsuya Takii (Institute) Modern Macroeconomics II 81 / 461

82 Euler Equation for Growth Model Note that ũ (c et ) = (c et ) 1 θ 1 θ means ũ 0 (c et ) = (c et ) θ. Hence the growth rate of the consumption is (c et ) θ (c et+1 ) θ = β [f 0 (k et+1 ) + (1 δ)] (1 + g) θ θ cet+1 = β [f 0 (k et+1 ) + (1 δ)] c et c et+1 c et = (1 + g) θ " β [f 0 (k et+1 ) + (1 (1 + g) θ δ)] # 1 θ Katsuya Takii (Institute) Modern Macroeconomics II 82 / 461

83 Euler Equation for Growth Model One additional condition is the transversality condition: 0 = lim t! (β ) t V 0 (k et ) k et = lim t! (β ) t ũ 0 [c et 1 ] β k et (1 + g) (1 + n) 0 = lim t! [β ] t 1 [c et 1 ] θ k et Economic interpretation of this transversality condition is that it is not optimal to keep capital stock at the nal date when the marginal value of consumption is positive. If so, the agent can always lower capital stock and consume more. Katsuya Takii (Institute) Modern Macroeconomics II 83 / 461

84 Euler Equation for Growth Model Optimal Growth c et+1 c et = " β [f 0 (k et+1 ) + (1 (1 + g) θ δ)] # 1 θ k et+1 = f (k et) + (1 δ) k et c et, k 0 is given (1 + g) (1 + n) 0 = lim t! [β ] t 1 [c et 1 ] θ k et where β = β (1 + n) (1 + g) (1 θ) < 1. Katsuya Takii (Institute) Modern Macroeconomics II 84 / 461

85 Steady State Analysis Steady State Analysis: the steady state is a point such that c e = c et+1 = c et, k e = k et+1 = k et Note that if an economy reaches the steady state, the transversality condition is satis ed. 0 = lim t! [β ] t 1 [c e ] 0 = lim t! [β ] t 1 θ k e Katsuya Takii (Institute) Modern Macroeconomics II 85 / 461

86 Steady State Analysis Suppose that β = 1 steady state 1+ϱ where ϱ is the discount rate. Then, on the β f 0 (k e ) + (1 δ) = (1 + g) θ f 0 (k e ) = (1 + g)θ β (1 δ) = (1 + g) θ (1 + ϱ) (1 δ) and k e = f (k e ) + (1 δ) k e c e (1 + g) (1 + n) c e = f (k e ) + (1 δ) k e (1 + g) (1 + n) k e = f (k e ) [(1 + g) (1 + n) (1 δ)] k e Katsuya Takii (Institute) Modern Macroeconomics II 86 / 461

87 Steady State Analysis Golden Rule: Note dc e dk e = f 0 (k e ) [(1 + g) (1 + n) (1 δ)] d 2 c e d (k e ) 2 = f 00 (k e ) < 0 Hence Compare with f 0 ke GR = (1 + g) (1 + n) (1 δ) f 0 (k e ) = (1 + g)θ β (1 δ) Katsuya Takii (Institute) Modern Macroeconomics II 87 / 461

88 Steady State Analysis Because β = β (1 + g) 1 θ (1 + n) < 1, (1 + g) (1 + n) (1 δ) " (1 + g) θ (1 + g) θ = (1 + g) (1 + n) β h i = β (1 + g) 1 θ (1 + g) θ (1 + n) 1 < 0. β β (1 δ) # Hence f 0 ke GR < f 0 (ke ) ) ke < ke GR Because people discount future, the steady state value of capital stock is lower than the golden rule level of capital stock. The steady state level of capital stock is called the modi ed golden rule level of capital stock. Katsuya Takii (Institute) Modern Macroeconomics II 88 / 461

89 Steady State Analysis Saving rate: s = Y t C t Y t = [f (k e ) c e ] T t N t f (k e ) T t N t = f (k e ) [f (k e ) (δ + g + n + ng) k e ] f (k e ) = (δ + g + n + ng) k e f (k e ) Hence the constant saving rate is supported around the steady sate. Katsuya Takii (Institute) Modern Macroeconomics II 89 / 461

90 Steady State Analysis Assume f (k) = k α. Then f 0 (k) = αk α 1. Hence, s = (δ + g + n + ng) k e (ke ) α (ke ) 1 α s = δ + g + n + ng 1 ke s 1 α =. δ + g + n + ng This condition is the same as that in Solow model. Katsuya Takii (Institute) Modern Macroeconomics II 90 / 461

91 Steady State Analysis Now we can endogenize s. Note that on the steady state, f 0 (k e ) = (1 + g) θ (1 + ϱ) (1 δ) α (ke ) α 1 = (1 + g) θ (1 + ϱ) (1 δ), (ke ) 1 α α = (1 + g) θ (1 + ϱ) (1 δ), Hence s = (δ + g + n + ng) (ke ) 1 α α (δ + g + n + ng) = (1 + g) θ (1 + ϱ) (1 δ) α [(1 + g) (1 + n) (1 δ)] = (1 + g) θ (1 + ϱ) (1 δ) Katsuya Takii (Institute) Modern Macroeconomics II 91 / 461

92 Steady State Analysis Note that saving rate is larger when we have more population growth. When you expect to have more children in future, you will have move incentive to save. When ϱ is large, the agent largely discounts his future. more important than tomorrow, the agent saves less. Since today is Similarly, the larger the marginal rate of substitution 1 θ, the saving rate is. When θ is small, the marginal rate of substitution is large. The agent is willing to change his consumption in response to the changes in the return. When the economy is growing, the return is high. Therefore, this behavior implies that the agent saves more. Katsuya Takii (Institute) Modern Macroeconomics II 92 / 461

93 Steady State Analysis The steady state value of capital stop per unit of e ective labor is. k e = = Using that y e = (k e ) α " # 1 1 α 1 α (δ + g + n + ng) δ + g + n + ng (1 + g) θ, (1 + ϱ) (1 δ) " # 1 1 α α (1 + g) θ, (1 + ϱ) (1 δ) Y t N t = " α (1 + g) θ (1 + ϱ) (1 δ) # α 1 α T t. Katsuya Takii (Institute) Modern Macroeconomics II 93 / 461

94 Steady State Analysis Di erent from Solow model, population growth has no impact on the GDP per capita. Because the saving rate is an increasing function of population growth, the negative impacts of population growth on GDP per capita is canceled out. The more the agent discount the future, (large ϱ), the lower the saving rate and, therefore, the lower the steady state value of per capita GDP. Hence, the larger the marginal rate of substitution 1 θ, the larger per capita GDP is. Katsuya Takii (Institute) Modern Macroeconomics II 94 / 461

95 Steady State Analysis Assignment: Derive the consumption per capita on the steady state. Katsuya Takii (Institute) Modern Macroeconomics II 95 / 461

96 Numerical Analysis In order to obtain a numerical solution, many economists use Matlab. Although this may not be the fastest program, It is instructive to demonstrate how we can write Matlab code to obtain a numerical policy function and numerical value function of the neoclassical growth model. I rst explain the basic code of Matlab and later I use Matlab to derive a numerical policy function and numerical value function. Katsuya Takii (Institute) Modern Macroeconomics II 96 / 461

97 Numerical Analysis Turning on Matlab: To access Matlab, you can go to the network center. Getting Help in Matlab: If you have questions about the use of a particular command, you just type help [command] Example: the command "plot" plots variables. If you want help on how to use it, you would type help plot Katsuya Takii (Institute) Modern Macroeconomics II 97 / 461

98 Numerical Analysis Basic Math in Matlab: Symbols for basic operations are "+" for addition, "-" for subtraction, "*" for multiplication and "/" for division. "^" is "to the power of." If you want Matlab to perform a computation, just type it. Example: if you want Matlab to compute "2+2," just type into the command line 2+2 If you want to assign a value to a variable, just type it out. Example: if you want the variable "apple" to have the value 3 and the variable orange to have the value 5, then type apple=3 orange=5 Katsuya Takii (Institute) Modern Macroeconomics II 98 / 461

99 Numerical Analysis Basic Math in Matlab: Matlab will remember these values unless you change them. Computations with variables are just as simple as with numbers. Example: if you want to nd out what apple+orange is, you just type apple+orange You will notice that when you type something like "apple=3," the computer repeats it. To stop that from happening, just put a semi-colon at the end. Example: If you don t want the computer to repeat what you have done, type apple=3; Katsuya Takii (Institute) Modern Macroeconomics II 99 / 461

100 Numerical Analysis Vectors and Matrices: For the most part, in Matlab you will be programming with vectors and matrices, not just numbers. This is no big stretch, however. After all, a variable is just a 1 1 matrix. Matlab uses the notation X (i, j) to refer to the element in the ith row and the jth column of a matrix X. Example: if you have a matrix called "apple" and you want the 3rd element of the second row to equal 4.5, you would type apple(2,3)=4.5; If you haven t told Matlab that you want a matrix called apple that is at least 2x3, however, it may give you an error. Therefore, it is a good idea to initialize all matrices at the beginning of the program before you actually use them. This is called "declaring" a variable. One way to do this is to write apple=zeros(2,3); Matlab creates the 2x3 matrix apple, and sets all its elements to equal zero. Katsuya Takii (Institute) Modern Macroeconomics II 100 / 461

101 Numerical Analysis Loops in Matlab: All programming languages of which I am aware make extensive use of something called "loops". This is a way of repeating a certain task many times or under certain conditions. for... end: This is the classic loop. Example: Suppose you wanted to create a 101x1 vector called "orange", where the elements are the numbers from zero to 1 in increments of You would do the following: for i=1:101 orange(i)=(i-1)/100; end Matlab begins by setting i=1, then proceeds until it hits the word "end". Then it returns to the "for" line, sets i=2, and repeats this process until i=101. Katsuya Takii (Institute) Modern Macroeconomics II 101 / 461

102 Numerical Analysis You can have multiple concentric loops. Example: suppose that for some reason you want to have a 100x100 matrix X where X ij = i 2 + j 3. Then you would use: X=zeros(100,100); for i=1:100 for j=1:100 X(i,j)=i^2+j^3; end end Notice that what is inside each loop is indented. When you have a lot of loops it can be hard to keep track of things, and indentation is a way of telling where you are. Katsuya Takii (Institute) Modern Macroeconomics II 102 / 461

103 Numerical Analysis if... end: Another loop-like structure involves conditional statements. Example: Suppose you have generated a vector of 100 numbers called X. You want to see whether they are positive or negative, and store that information in another vector called Y. You could proceed as follows. Y=zeros(100,1); for i=1:100 if X(i)>0 Y(i)=1; end end When it reads "if", Matlab checks whether the condition is met (X(i)>0) and if so performs what comes afterwards until it hits "end." If the condition is not met, it just ignores everything until "end". Katsuya Takii (Institute) Modern Macroeconomics II 103 / 461

104 Numerical Analysis An alternative is Y=zeros(100,1); for i=1:100 if X(i)>0 Y(i)=1; else Y(i)=0; end end Here Matlab does the same as before, except that if the condition is not met it does not ignore everything, rather it executes the commands that lie after "else" instead. In this example, of course, that is redundant (since the default value of Y is zero). Katsuya Takii (Institute) Modern Macroeconomics II 104 / 461

105 Numerical Analysis Writing your own Programs and Functions in Matlab: Programs: A "program" is nothing but a list of commands (such as those above) that you save so that you can use them over and over again. How do you write programs? Just nd any text editor, and type out the sequence of commands that you desire. Then, save it to a disk Example: You can save your program to a le such as "macroii.m" If you want to run macroii.m, just type macroii Then Matlab will perform all the commands in the le, in order. There is a function in Matlab called "save." However, "save" does not save programmes, rather it saves the values of all the variables in the memory. Example: if you type: save katsuya Matlab will create a le called katsuya.mat. If for some reason you wanted to retrieve the values of the variables in that le, you would type: load katsuya Katsuya Takii (Institute) Modern Macroeconomics II 105 / 461

106 Numerical Analysis Functions: Sometimes you may nd it convenient to write down a function that does not exist in Matlab. Then you can use a command function. Example: This would be a le takii.m: function [z] = takii(x) z = (x+1)^2; After saving this le, if you input takii(2) into Matlab you should get 9 as output: takii.m is now a function. In that program, x represents the input and z represents the output. Katsuya Takii (Institute) Modern Macroeconomics II 106 / 461

107 Numerical Analysis Solving equations: Matlab can also help us to solve non-linear equations. Example: The following line embedded in another program nds the zeroes of takii, or fx : takii(x) = 0g: fsolve( takii, 10) Matlab starts at x = 10 and uses gradient methods to nd the value of x such that takii(x) is zero. Why would you want to do this? Many economic problems are solved by nding values of some variable x such that f (x) = 0 for some function f. For example, the deterministic steady state in the neoclassical growth model is de ned in this manner. Fsolve is one way to nd it.. Katsuya Takii (Institute) Modern Macroeconomics II 107 / 461

108 Numerical Analysis Comments: Good programming style follows certain conventions so that others can easily pick up your code and see what you are doing. For this purpose, it is important to write comments on your program. Example: If you want to make a comment, "This is where I compute the steady state." you would type: % Here I compute the steady state Matlab ignores everything in a line after it sees "%", so you can write anything there that you wish. Katsuya Takii (Institute) Modern Macroeconomics II 108 / 461

109 Numerical Analysis We are now ready to write a program to derive a policy function and a value function of the growth model expressed by the following Bellman equation. where ũ (c et ) = (c et ) 1 β = (1+n)(1+g )(1 θ) 1+ϱ. V (k et ) = max c et fũ [c et ] + β V (k et+1 )g k et+1 = k α et + (1 δ) k et c et (1 + g) (1 + n) θ 1 θ if θ 6= 1, log (c et ) if θ = 1, and Katsuya Takii (Institute) Modern Macroeconomics II 109 / 461

110 Numerical Analysis I explain the process with the help of the LECTURE TWO OF RECURSIVE MACROECONOMICS by MARTIN ELLISON. The following argument is based on his lecture note. There are four main parts to the code: 1 parameter value initialization, 2 state space de nitions, 3 value function iterations 4 output. Katsuya Takii (Institute) Modern Macroeconomics II 110 / 461

111 Numerical Analysis Parameter value initialization: Start new program by clearing the work space. Parameters are initialized with theta the coe cient of relative risk aversion, discount the discount rate, alpha the exponent on capital in the production function, g the growth rate of productivity, n the population growth rate, beta discount factor and delta the depreciation rate. clear all theta=1.2; discount=.05; alpha=.3; g=0.02; n=0.02; beta=((1+n)*((1+g)^(1-theta)))/(1+discount); delta=.05; Katsuya Takii (Institute) Modern Macroeconomics II 111 / 461

112 Numerical Analysis State space de nitions: We now construct a grid of possible capital stocks. The grid is centered on the steady-state capital stock per unit of e ective labor, with maximum ( ) and minimum ( ) values 105% and 95% of the steady-state value. There are N discrete points in the grid. Note that " # 1 1 α ke α = (1 + g) θ. (1 + ϱ) (1 δ) N=1000; Kstar=(alpha/(((1+g)^(theta))*(1+discount)-(1-delta)))^(1/(1- alpha)); Klo=Kstar*0.95; Khi=Kstar*1.05; step=(khi-klo)/n; K=Klo:step:Khi; l=length(k); Katsuya Takii (Institute) Modern Macroeconomics II 112 / 461

113 Numerical Analysis K is the vector. K = [Klo, Klo + step, Klo + 2step,..., Khi] {z } l Katsuya Takii (Institute) Modern Macroeconomics II 113 / 461

114 Numerical Analysis Value function iterations: Value function iteration requires 3 steps 1 Set initial guess for the value function. 2 Express one period return function as a function of current state variable and future state variable. 3 Iterations In this way, the Bellman equation can be expressed only by current state variable and future state variable. Add one period return function to the discounted initial guess and nd the optimal value. This value serves the initial guess in the next round. Iterate it until the initial guess meets the optimal value. Katsuya Takii (Institute) Modern Macroeconomics II 114 / 461

115 Numerical Analysis Initial Guess for the value function: initial guess is taken by assuming that all capital is always consumed immediately. In this case, v 0 = ln (ytot) if θ = 1 v 0 = (ytot)1 θ, if θ 6= 1. 1 θ ytot = ket α + (1 δ) k et. Katsuya Takii (Institute) Modern Macroeconomics II 115 / 461

116 Numerical Analysis Hence, our initial guess is written as ytot = K.^alpha+(1-delta)*K; ytot=ytot *ones(1,l); if theta==1 v=log(ytot); else v=ytot.^(1-theta)/(1-theta); end where ones(1,l).are 1 l vector of 1 and the syntax.^ de nes element-by-element operation, so each element of a vector or matrix is risen to the power of the exponent. Katsuya Takii (Institute) Modern Macroeconomics II 116 / 461

117 Numerical Analysis Note that v is l l matrix with every column being the same. example, if θ 6= 1, v = [(Klo) α +(1 δ)klo] 1 θ 1 θ [(Klo) α +(1 δ)klo] 1 θ 1 θ For [(Klo) α +(1 δ)klo] 1 θ 1 θ [(Khi) α +(1 δ)khi] 1 θ 1 θ [(Khi) α +(1 δ)khi] 1 θ 1 θ [(Khi) α +(1 δ)khi] 1 θ 1 θ Katsuya Takii (Institute) Modern Macroeconomics II 117 / 461

118 Numerical Analysis Express one period return function as a function of current state variable and future state variable. For use in the value function iterations, we de ne an l l matrix C 2 c e (1, 1)... 3 c e (1, l) C = c e (l, 1).. c e (l, l) c e (i, j) = k α et (j) + (1 δ) k et (j) (1 + g) (1 + n) k et+1 (i) k et (j) = Klo + (j 1) step, k et+1 (i) = Klo + (i 1) step Katsuya Takii (Institute) Modern Macroeconomics II 118 / 461

119 Numerical Analysis The program for this operation is for i=1:l for j=1:l C(i,j) = K(j).^alpha + (1-delta)*K(j) - (1+g)*(1+n)*K(i); end end Katsuya Takii (Institute) Modern Macroeconomics II 119 / 461

120 Numerical Analysis Next, we need to map consumption levels into utility levels by applying the suitable utility function. The syntax./ de nes element-by-element operation rather than matrix division implied by using the / operator. if theta==1 U=log(C); else U=(C.^(1-theta))/(1-theta); end Katsuya Takii (Institute) Modern Macroeconomics II 120 / 461

121 Numerical Analysis Iteration: Now we can use our rst guess of the value function to get the second estimate v1, which will be the start of our next T iterations. T=100 for j=1:t w=u+beta*v; v1=max(w); v=v1 *ones(1,l); end Katsuya Takii (Institute) Modern Macroeconomics II 121 / 461

122 Numerical Analysis Note that w is the l l matrix. Suppose θ 6= 1. Then w = [c e (1,1)] 1 θ [c e (1,l)] 1 θ 1 θ + β v (l, l) 1 θ + β v (1, 1) [c e (l,1)] 1 θ [c e (l,l)] 1 θ 1 θ + β v (l, 1).. 1 θ + β v (l, l) c e (i, j) = ket α (j) + (1 δ) k et (j) (1 + g) (1 + n) k et+1 (i) [ket+1 (i)] α + (1 δ) k et+1 (i) 1 θ v (i, j) = 1 θ k et (j) = Klo + (j 1) step, k et+1 (i) = Klo + (i 1) step Katsuya Takii (Institute) Modern Macroeconomics II 122 / 461

123 Numerical Analysis Hence, v1 nd the maximum value for each column. h i v1 = [c e (i1,1)]1 θ 1 θ + β v (i1, 1).. [c e (il,l)] 1 θ 1 θ + β v (il, l) where ij is the maximum indices of the j th column. Hence, v is again, l l matrix with every column being the same. For example, if θ 6= 1, v = θ + β v (i1, 1).. [c e (i1,1)]1 θ 1 θ + β v (i1, 1) [c e (i 1,1)]1 θ [c e (il,l)] 1 θ 1 θ + β v (il, l).. [c e (il,l)] 1 θ 1 θ + β v (il, l) Katsuya Takii (Institute) Modern Macroeconomics II 123 / 461

124 Numerical Analysis Output: Value function iterations are now complete. The nal value function is stored in Val and the indices of the future k et+1 are stored in ind. These latter indices are converted into k et+1 values in optk. Using optk, we can create the optimal value of consumption, optc. [val,ind]=max(w); optk = K(ind); optc = K.^alpha + (1-delta)*K - (1+g)*(1+n)*optk; Katsuya Takii (Institute) Modern Macroeconomics II 124 / 461

125 Numerical Analysis To illustrate the results, I rst plot the value function. gure(1) plot(k,v1, LineWidth,2) xlabel( K ); ylabel( Value ); legend( Value function,4); Katsuya Takii (Institute) Modern Macroeconomics II 125 / 461

126 Numerical Analysis Katsuya Takii (Institute) Modern Macroeconomics II 126 / 461

127 Numerical Analysis Next I plot the policy function gure(2) plot(k,optc, LineWidth,2) xlabel( K ); ylabel( C ); legend( Policy function,4); Katsuya Takii (Institute) Modern Macroeconomics II 127 / 461

128 Numerical Analysis Katsuya Takii (Institute) Modern Macroeconomics II 128 / 461

129 Numerical Analysis It is also useful to plot how k et+1 varies as a function of k et. that an economy converges to the steady state. gure(3) plot(k,optk, LineWidth,2) hold on plot(k,k, r, LineWidth,2) xlabel( K ); ylabel( K ); legend( Policy Function 2, 45 degree line,4); It shows Katsuya Takii (Institute) Modern Macroeconomics II 129 / 461

130 Numerical Analysis Katsuya Takii (Institute) Modern Macroeconomics II 130 / 461

131 Numerical Analysis Another useful gure is showing net investment k et+1 k et as a function of k et. The gure con rms that net investment is positive when k et < ke and negative when k et > ke. gure(4) plot(k,(optk-k), LineWidth,2) xlabel( K ); ylabel( K - K ); Katsuya Takii (Institute) Modern Macroeconomics II 131 / 461

132 Numerical Analysis Katsuya Takii (Institute) Modern Macroeconomics II 132 / 461

133 Numerical Analysis To gain an insight into the dynamics of the adjustment process, it is useful to calculate the path of convergence of capital if initial capital is not equal to its steady-state value. p is the number of periods over which to follow convergence. The indices of the capital choices are stored in the p 1 vector mi and the capital stocks themselves are stored in the p 1 vector m. The initial index is taken as 1 so the initial capital stocks are at its maximum value. p=50; mi=zeros(p,1); m=zeros(p,1); mi(1)=n; m(1)=k(mi(1)); for i=2:p mi(i)=ind(mi(i-1)); m(i) = K(mi(i)); end Katsuya Takii (Institute) Modern Macroeconomics II 133 / 461

134 Numerical Analysis We can plot this dynamics by the following program. t=1:1:50 gure(5) plot(t,m) xlabel( t ); ylabel( k(t) ); Katsuya Takii (Institute) Modern Macroeconomics II 134 / 461

135 Numerical Analysis Katsuya Takii (Institute) Modern Macroeconomics II 135 / 461

136 Numerical Analysis Assignment: go to the following website of MARTIN ELLISON and download Matlab program for the deterministic growth model. Modify the program and replicate what I have done in this lecture. Katsuya Takii (Institute) Modern Macroeconomics II 136 / 461

137 Recursive Competitive Equilibrium Market Economy: This section introduces a recursive competitive equilibrium. Recursive competitive equilibrium is a particularly convenient because 1 it naturally ts the dynamic programing approach. 2 it is easily applied in a wide variety of settings, including those with distortions. 3 the equilibrium process can be computed and can be simulated to general equilibrium paths for the economy. Katsuya Takii (Institute) Modern Macroeconomics II 137 / 461

138 Recursive Competitive Equilibrium Then we ask how does this command economy relate to the market economy? In order to answer this question, remember that the rst and second welfare theorem. The rst welfare theorem says that the market economy is Pareto optimal. The second welfare theorem says that the Pareto optimum allocation can be supported by a market economy with a income transfer. Note that our command economy is Pareto optimum and since every agent is identical, there is no reason to consder a income transfer in our model. Hence, there must have prices which support our equilibrium. Katsuya Takii (Institute) Modern Macroeconomics II 138 / 461

139 Recursive Competitive Equilibrium In this section, we show that our economy can be supported by a market economy. In particular, we show that there is a price mechanism that can reproduce a following policy function and a value function that satisfy the rst order condition and envelope theorem of our problem. ũ 0 [c (k et )] = β V 0 (κ (k et )) (1 + g) (1 + n), V 0 (k et ) = β V 0 (κ (k et )) [f 0 (k et ) + (1 δ)], (1 + g) (1 + n) where κ (k et ) = f (k et) + (1 δ) k et c (k et ). (1 + g) (1 + n) Katsuya Takii (Institute) Modern Macroeconomics II 139 / 461

140 Recursive Competitive Equilibrium Let us rst consider the following representative household problem max fc t β t N t U(c t ) gt=0 t=0 s.t. A t+1 = (1 + ρ (k et )) A t + w (k et ) T t N t c t N t, A 0 = k e0 T 0 N 0 k et+1 = f (k et) + (1 δ) k et C m (k et ), k e0 is given (1 + g) (1 + n) T t+1 = (1 + g) T t, T 0 is given N t+1 = (1 + n) N t, N 0 is given where c t is consumption per capita, A t is the asset of this household, k et, T t and N t are aggregate state variables. Note that the interest rate ρ, and the wage per unit of e ective labor, w are function of the aggregate capital per unit of e ective labor, k et. We also assume that an initial individual asset is the same as the aggregate capital, A 0 = k e0 T 0 N 0. Katsuya Takii (Institute) Modern Macroeconomics II 140 / 461

141 Recursive Competitive Equilibrium Assignment: Suppose that U (c et T t ) = (c ett t ) (1 θ) 1, θ 0 1 θ Show that the original problem is equivalent to max fc et g t=0 (β ) t ũ (c et ) a et+1 = (1 + ρ (k et)) a et + w (k et ) c et, a 0 = k e0 (1 + g) (1 + n) s.t. k et+1 = f (k et) + (1 δ) k et C m (k et ), k e0 is given (1 + g) (1 + n) where ũ (c et ) = (c et ) 1 θ 1 θ if 0 θ 6= 1, log (c et ) if θ = 1, and β = β (1 + n) (1 + g) (1 θ). Katsuya Takii (Institute) Modern Macroeconomics II 141 / 461

142 Recursive Competitive Equilibrium Bellman Equation V h (a et, k et ) = max c et n o ũ (c et ) + β V h (a et+1, k et+1 ) a et+1 = (1 + ρ (k et)) a et + w (k et ) c et (1 + g) (1 + n) k et+1 = f (k et) + (1 δ) k et C m (k et ) (1 + g) (1 + n) Katsuya Takii (Institute) Modern Macroeconomics II 142 / 461

143 Recursive Competitive Equilibrium First order conditions ũ 0 (c (a et, k et )) = β V h 1 (a et+1, k et+1 ) (1 + g) (1 + n) where c (a et, k et ) is a policy function Envelope Theorem V h 1 (a et, k et ) = β V h 1 (a et+1, k et+1 ) [1 + ρ (k et )] (1 + g) (1 + n) Katsuya Takii (Institute) Modern Macroeconomics II 143 / 461

144 Recursive Competitive Equilibrium Assume that a representative rm maximizes capital and labor given a factor price functions n o max F Kt f, T t L t r (k et ) Kt f w (k et ) T t L t, K t,l t n o = T t max F ket, f 1 r (k et ) ket f w (k et ) L t k et,l t n o = T t max f ket f r (k et ) ket f w (k et ) L t k et,l t where ket f = K t f T t L t. The rental price r is also the function of the aggregate capital per unit of e ective labor. The rst order conditions are r (k et ) = f 0 ke f (k et ), w (k et ) = f ke f (k et ) r (k et ) ke f (k et ) Katsuya Takii (Institute) Modern Macroeconomics II 144 / 461

145 Recursive Competitive Equilibrium Arbitrage condition ρ (k et ) = r (k et ) δ Labor and Capital Market L t = N t, k f e (k et ) T t L t = a et T t N t We assume that the initial value of individual asset is the same as that of the aggregate asset. a et T t N t = k et T t N t Two market conditions with this initial condition imply k f e (k et ) = k et = a et. Katsuya Takii (Institute) Modern Macroeconomics II 145 / 461

146 Recursive Competitive Equilibrium De nition: Recursive competitive equilibrium consists of a value function V h 1 (a et, k et ), a policy function c (a et, k et ), a corresponding set of aggregate decision C m (k et ), production plans k f e (k et ), set of factor price functions, fr (k et ), w (k et ), ρ (k et )g such that these function satisfy Household ũ 0 (c (a et, k et )) = β V h 1 (a et+1, k et+1 ) (1 + g) (1 + n) V h 1 (a et, k et ) = β V h 1 (a et+1, k et+1 ) [1 + ρ (k et )] (1 + g) (1 + n) where a et+1 = (1 + ρ (k et)) a et + w (k et ) c (a et, k et ) (1 + g) (1 + n) k et+1 = f (k et) + (1 δ) k et C m (k et ) (1 + g) (1 + n) Katsuya Takii (Institute) Modern Macroeconomics II 146 / 461

147 Recursive Competitive Equilibrium Firm r (k et ) = f 0 ke f (k et ), w (k et ) = f ke f (k et ) r (k et ) k f e (k et ) Arbitrage condition ρ (k et ) = r (k et ) δ Capital and Labor Market with initial condition ke f (k et ) = k et = a et. the consistency of individual and aggregate decisions C m (k et ) = c (k et, k et ) Katsuya Takii (Institute) Modern Macroeconomics II 147 / 461

148 Recursive Competitive Equilibrium Note that there exist k f e (k et ), r (k et ), ρ (k et ) and w (k et ) that satisfy the de nition of RCE and ke f (k et ) = k et r (k et ) = f 0 (k et ) ρ (k et ) = f 0 (k et ) δ, w (k et ) = f (k et ) f 0 (k et ) k et. Katsuya Takii (Institute) Modern Macroeconomics II 148 / 461

149 Recursive Competitive Equilibrium From the de nition of the recursive competitive equilibrium a et+1 = (1 + r (k et) δ) k et + w (k et ) c (k et, k et ) (1 + g) (1 + n) = (1 + r (k et) δ) k et + f (k et ) r (k et ) k et c (k et, k et ) (1 + g) (1 + n) = f kf e (k et ) + (1 δ) k et c (k et, k et ) (1 + g) (1 + n) = f (k et) + (1 δ) k et C m (k et ) (1 + g) (1 + n) = k et+1 Katsuya Takii (Institute) Modern Macroeconomics II 149 / 461

150 Recursive Competitive Equilibrium Therefore, households rst order condition is Envelope theorem is ũ 0 (C m (k et )) = β V h 1 (k et+1, k et+1 ) (1 + g) (1 + n) V h 1 (k et, k et ) = β V h 1 (k et+1, k et+1 ) [f 0 (k et ) + 1 δ] (1 + g) (1 + n) Katsuya Takii (Institute) Modern Macroeconomics II 150 / 461

151 Recursive Competitive Equilibrium De ne V m0 (k et ) V h 1 (k et, k et ) ũ 0 (C m (k et )) = β V m0 (k et ) (1 + g) (1 + n) V m0 (k et ) = β V m0 (k et+1 ) [f 0 (k et ) + 1 δ] (1 + g) (1 + n) Katsuya Takii (Institute) Modern Macroeconomics II 151 / 461

152 Recursive Competitive Equilibrium Remember that Social Planner Problem is summarized by ũ 0 (C (k et )) = β V 0 (k et+1 ) (1 + g) (1 + n) V 0 (k et ) = β V 0 (k et+1 ) [f 0 (k et ) + (1 δ)] (1 + g) (1 + n) We know that there exists the set of policy functions and the marginal value function fc (k et ), V 0 (k et )g that satis es these equations exists and unique. It means that fc m (k et ), V m0 (k et )g exists and C m (k et ) = C (k et ), V m0 (k et ) = V 0 (k et ). Katsuya Takii (Institute) Modern Macroeconomics II 152 / 461

153 Recursive Competitive Equilibrium Given the existence of the aggregate consumption function C m (k et ) and the price functions ρ (k et ) and w (k et ), there exists c (aet, k et ), V1 h (a et, k et ) that satis es the de nition of RCE because these are the solutions to the standard Dynamic Programming given the price functions and the aggregate decisions. Katsuya Takii (Institute) Modern Macroeconomics II 153 / 461

154 Recursive Competitive Equilibrium We have proved that there is a price mechanism that can reproduce social optimal allocation. Obviously, we are interested in a decentralized economy, but not a planner problem. But, because the social planner problem is easier to deal with, if you are interested in a perfectly competitive economy as a benchmark, you can analyze an optimal growth model. If you are interested in a distorted economy, you cannot use an optimal growth model. However, you can still use a recursive competitive equilibrium, though the proof of existence might be more di cult. Although I mainly discuss an optimal growth model in this lecture, you can apply a recursive competitive equilibrium to varieties of issues. Katsuya Takii (Institute) Modern Macroeconomics II 154 / 461

155 Recursive Competitive Equilibrium Assignment: Consider the following social planner s problem. max fc t g t=0 β t U (C t ), s.t. K t+1 = F (K t, N) + (1 δ) K t C t, K 0 is given. where C t is consumption, N is population, K t is capital stock, β 2 (0, 1) is a discount factor, δ is the depreciation rate of capital stock. Population is assumed to be constant. Assume that the aggregate production function, F (, ), satis es usual assumptions: it exhibits the constant return to scale, the rst derivative is positive, the second is negative and it satis es Inada condition. No population growth and no technological growth. 1 De ne the recursive competitive equilibrium that can support the resource allocation of this problem. 2 Show that your equilibrium allocation is the same as the resource allocation of this problem. Katsuya Takii (Institute) Modern Macroeconomics II 155 / 461

156 Overlapping Generation Model Recursive competitive equilibrium assumes that agents behave as if they never die. Some of you may think that this assumption is extreme. In order to understand how restrictive this assumption is and when this assumption is justi ed, this section brie y discusses about an overlapping generation model (OGM). The main di erence from the recursive competitive equilibrium is that there is turnover in the population. New generation comes to society; old generation leaves. This model is especially useful when you are interested in interactions across generations. It is shown that (1) when we have turnover in the population, economy may not be Pareto optimal and (2) a recursive competitive equilibrium can be justi ed when parents care about their children very much. Katsuya Takii (Institute) Modern Macroeconomics II 156 / 461

157 Overlapping Generation Model Basic Model: For the simplest case, each agent lives in two periods; young and old. In each period, new young enters the economy and old leaves. Therefore there is turnover in the population in each period. The agents are identical in their generation, but di er across generation. Katsuya Takii (Institute) Modern Macroeconomics II 157 / 461

158 Overlapping Generation Model Katsuya Takii (Institute) Modern Macroeconomics II 158 / 461

159 Overlapping Generation Model Household: The agent with a constant risk aversion solves the following problem: max C yt,c ot+1 ( (1 θ) c yt θ 1 + ϱ (1 θ) c ot θ s.t.s n t = w t c yt (10) c ot+1 = 1 + ρ t+1 s n t (11) where st n is net saving, c yt and c ot are consumption when he is young and old, respectively. When the agent is young, he earns wage w t. He decides whether he consumes today or save it for the next period. When he gets age, he retires and receives income from his saving. Since he does not have the next period, he consumes all his wealth when he is old. ) Katsuya Takii (Institute) Modern Macroeconomics II 159 / 461

160 Overlapping Generation Model This model can be simpli ed by max s n t ( [w t st n ] (1 θ) θ 1 + ϱ The rst order condition is [c yt ] θ = 1 + ρ t+1 [cot+1 ] 1 + ϱ 1 + ρt+1 s n (1 θ) ) t 1 1 θ Hence, 1 c ot ρt+1 θ = (12) c yt 1 + ϱ This is OGM version of the Euler equation. The agent compares the interest rate and the discount rate. When the interest rate is higher than the discount rate, he saves more and increases his consumption tomorrow. When the discount rate is larger, he increases his consumption today. Katsuya Takii (Institute) Modern Macroeconomics II 160 / 461 θ

161 Overlapping Generation Model Combing two budget constraints (10) and (11), I can derive the intertemporal budget constraint: c yt + c ot+1 = w t. (13) 1 + ρ t+1 It shows that the present value of the stream of lifetime consumption is equal to the present value of lifetime income, which, in this case, wages during young. Substituting the Euler equation (12) into (13), ρt+1 1+ϱ 1 θ 1 + ρ t+1 Therefore, consumption is linear in wages. c yt = 3 (1 + ϱ) 1 θ 7 5 c yt = w t. (1 + ϱ) 1 1 θ θ θ ρ t+1 Katsuya Takii (Institute) Modern Macroeconomics II 161 / 461 w t

162 Overlapping Generation Model We can derive the saving rate, s ρ t+1, as the function of the interest rate: s ρ t+1 = w t c yt w t = 1 θ θ 1 + ρ t+1 (1 + ϱ) 1 1 θ θ θ ρ t+1 As you can see, the saving rate can be an increasing or decreasing function of the interest rate: s 0 ρ t+1 > 0, if 1 > θ s 0 ρ t+1 < 0, if 1 < θ s 0 ρ t+1 = 0, if 1 = θ Katsuya Takii (Institute) Modern Macroeconomics II 162 / 461

163 Overlapping Generation Model This is the result of a usual substitution e ect and income e ect. When the interest rate increases, the return to saving increases. Therefore the agent saves more (substitution e ect). On the other hand an increase in the interest rate implies an increases in lifetime income. Therefore, the agent can consumes more (income e ect). As I discussed in the representative agent model, the parameter 1 θ can be interpreted as the elasticity of substitution between consumption today and tomorrow. It measures the sensitivity of consumption to the change in price. When 1 θ is larger than 1, θ is smaller than 1, the agent is sensitive to the change in the return to saving. Therefore, substitution e ect dominates income e ect. The saving rate is an increasing function of the interest rate. Using this saving rate, the net saving is s n t = s ρ t+1 wt Katsuya Takii (Institute) Modern Macroeconomics II 163 / 461

164 Overlapping Generation Model Firm: Firms maximization conditions are the same as before. The following pro t maximization condition characterizes the rms behavior: r t = f 0 (k et ) w t = f (k et ) f 0 (k et ) k et Tt As usual, I express the rst order conditions by the e ciency unit term. Intermediation: The following arbitrage condition is also the same as before: ρ t = r t δ Katsuya Takii (Institute) Modern Macroeconomics II 164 / 461

165 Overlapping Generation Model Labor Market Clearing Conditions: Since only the young works in this economy, the demand for labor must be equal to the population of the young. L t = N yt where N yt is the population of the young. Katsuya Takii (Institute) Modern Macroeconomics II 165 / 461

166 Overlapping Generation Model Capital Market Clearing Conditions: We assume that current net saving by young increases the asset at the next period. But when the agent gets olds, he consumes every asset he has, which reduces total asset in the economy in the next period. Therefore, the following condition must be satis ed. A t+1 A t = st n N yt st n 1N yt 1 Because this relationship must be satis ed for any t, A t+1 = st n N yt Since the demand for capital must be equal to the supply of asset, K t+1 = A t+1 = st n N yt. Katsuya Takii (Institute) Modern Macroeconomics II 166 / 461

167 Overlapping Generation Model We assume that the growth rate of productivity and population is g and n, respectively: T t+1 = (1 + g) T t N yt+1 = (1 + n) N yt Using e ciency unit, both market clearing conditions are summarized by K t+1 T t+1 N yt+1 T t T t+1 N yt+1 T t N yt = st n k et+1 (1 + g) (1 + n) T t = st n. where k et+1 = K t+1 T t+1 N yt+1. Note that the denominator is not total population, but the number of young population because only young can work in this economy. Katsuya Takii (Institute) Modern Macroeconomics II 167 / 461

168 Overlapping Generation Model Equilibrium: The market equilibrium consists of the sequence of s n t, ρ t+1, w t, r t, k t, which satis es t=0 Consumer maximizes his utility: where s ρ t+1 Firm maximizes its pro ts: s n t = s ρ t+1 wt (14) = 1 θ θ 1 + ρ t+1 (1 + ϱ) 1 1 θ θ θ ρ t+1 r t = f 0 (k et ) (15) w t = f (k et ) f 0 (k et ) k et Tt (16) Katsuya Takii (Institute) Modern Macroeconomics II 168 / 461

169 Overlapping Generation Model The arbitrage condition: ρ t+1 = r t+1 δ (17) Market clearing condition: k et+1 (1 + g) (1 + n) T t = st n. (18) Katsuya Takii (Institute) Modern Macroeconomics II 169 / 461

170 Overlapping Generation Model Combining equilibrium conditions, we can derive k et+1 = = s n t (1 + g) (1 + n) T t s ρ t+1 wt (1 + g) (1 + n) T t = s (r t+1 δ) [f (k et ) f 0 (k et ) k et ] T t (1 + g) (1 + n) T t = s (f 0 (k et+1 ) δ) [f (k et ) f 0 (k et ) k et ] (1 + g) (1 + n) k et+1 = (f 0 (k et+1 ) + 1 δ) 1 θ θ (1 + ϱ) 1 θ + (f 0 (k et+1 ) + 1 δ) 1 θ θ [f (k et ) f 0 (k et ) k et ] (1 + g) (1 + n) (19) Katsuya Takii (Institute) Modern Macroeconomics II 170 / 461

171 Overlapping Generation Model As you can see, this is a nonlinear rst order di erence equation. Hence, potentially, we can solve it given an initial condition k 0. Unfortunately, we cannot say much about the property of this dynamic equation. It is well known that the OGM can produce the variety of dynamics. It is possible to have multiple steady states in the OGM. In another case, we cannot determine the dynamics of OGM. Moreover, OGM can yield a chaotic uctuation. The following gures are some examples. Katsuya Takii (Institute) Modern Macroeconomics II 171 / 461

172 Overlapping Generation Model Katsuya Takii (Institute) Modern Macroeconomics II 172 / 461

173 Overlapping Generation Model The dynamics of the OGM depends on the parameters of the production function and the utility function. This can be seen as the advantages and disadvantages of OGM. On one hand, it gives us possible explanations for several puzzling evidence including business cycle. On the other hand, it is di cult to tell where we stand and what would be a policy implication. Anything goes. Katsuya Takii (Institute) Modern Macroeconomics II 173 / 461

174 Overlapping Generation Model Two additional assumptions simplify the dynamics of our solution. Assume that θ = 1 and the production function is Cobb-Douglas f (k) = k α. Then equation (19) is k et+1 = (1 α) (k et ) α (2 + ϱ) (1 + g) (1 + n) A (k et ) α where A = (1 α) (2+ϱ)(1+g )(1+n) Katsuya Takii (Institute) Modern Macroeconomics II 174 / 461

175 Overlapping Generation Model Katsuya Takii (Institute) Modern Macroeconomics II 175 / 461

176 Overlapping Generation Model In this case, there is a unique steady state and economy globally and monotonically converges to the steady state. Let me derive the steady state value of k et. On the steady state k et = k et+1 = ke. Then 1 ke 1 α 1 α = (20) (2 + ϱ) (1 + g) (1 + n) Katsuya Takii (Institute) Modern Macroeconomics II 176 / 461

177 Dynamic Ine ciency Dynamic Ine ciency: In order to discuss the ine ciency of the OGM, I examine the golden rule level of capital stock, ke GR, which maximizes consumption per unit of e ective population when economy is stationary. Then I compare ke GR and ke. If ke is larger than ke GR, we can easily nd new allocation in which everybody is better o. To see this, lower ke to ke GR during period t. Since the agent can enjoy more consumption during period t, the agent is better o. From the next period, economy reaches ke GR. Since ke GR maximizes consumption per unit of e ective population, the following generations can enjoy higher consumption and they are better o. In other word, the steady state is not e cient. Remember that the optimal growth model always satisfy ke < k GR because β < 1. As the household in OGM does not need to worry about in nite period later, this condition does not need to be satis ed. Katsuya Takii (Institute) Modern Macroeconomics II 177 / 461

178 Dynamic Ine ciency Firstly, note that maximizing consumption per unit of e ective population is the same as maximizing consumption per unit of e ective labor: C t T t [N yt + N ot ] C t = h i, T t N yt + N yt 1+n = 1 + n C t, 2 + n T t N yt where C t = c yt N yt + c ot N ot. Hence, I nd the level of capital stock which maximizes consumption per workers given any level of technology. Katsuya Takii (Institute) Modern Macroeconomics II 178 / 461

179 Dynamic Ine ciency Social planner faces resource constraint any time. constraint is The resource This means K t+1 = F (K t, T t N yt ) C t + (1 δ) K t K t+1 T t+1 N yt+1 T t+1 N yt+1 T t N yt = F Kt, 1 T t N yt k et+1 (1 + g) (1 + n) = f (k et ) c et + (1 δ) k et where c et = C t T t N yt and k et = K t T t N yt. C t T t N yt + (1 δ) K t T t N yt Katsuya Takii (Institute) Modern Macroeconomics II 179 / 461

180 Dynamic Ine ciency On the steady state k et = k et+1 = k e and c et = c e. Hence, c e = f (ke ) + [(1 δ) (1 + g) (1 + n)] ke Hence, the level of capital stock which maximizes consumption per workers given the level of productivity is dc dk = f 0 ke GR [(1 + g) (1 + n) (1 δ)] = 0 Note that this golden rule is the same as the golden rule in the representative agent model. Katsuya Takii (Institute) Modern Macroeconomics II 180 / 461

181 Dynamic Ine ciency Note that ρ = r δ f 0 (k e ) = r = ρ + δ Hence k GR e < k e i (1 + g) (1 + n) (1 δ) > ρ + δ k GR e < k e i (1 + g) (1 + n) > 1 + ρ The growth rate of technology and population growth must be large. This is the condition that government debt per nominal GDP never explore as far as primary balance per nominal GDP does not explore. Katsuya Takii (Institute) Modern Macroeconomics II 181 / 461

182 Dynamic Ine ciency Assume that f (k) = k α, the golden rule level of capital stock per unit of e ective labor is α 1 α ke GR = (1 + g) (1 + n) (1 δ) 1 ke GR α 1 α = (21) (1 + g) (1 + n) (1 δ) Katsuya Takii (Institute) Modern Macroeconomics II 182 / 461

183 Dynamic Ine ciency Because I get ke = 1 α (2 + ϱ) (1 + g) (1 + n) 1 1 α k GR e < k e, i α (1 + g) (1 + n) (1 δ) < 1 α (2 + ϱ) (1 + g) (1 + n). Katsuya Takii (Institute) Modern Macroeconomics II 183 / 461

184 Dynamic Ine ciency Hence, there exists parameter values with which the steady state is not Pareto optimal. This result is contrasted with the one of the representative agent model in which the allocation is always Pareto optimal. This result sounds surprising since market is competitive and no externality in Overlapping generation model. The main reason is that we have the in nite number of agents in our economy. The social planner can transfer resources from young to old without a market. Obviously, the current old is better o and the current young is worse o. However, the social planner can compensate the current young when he becomes old by transferring resources from the next generation. Since we have the in nite number of generations, nobody may be worse o. This is the reason for ine ciency. This is called the dynamic ine ciency. Katsuya Takii (Institute) Modern Macroeconomics II 184 / 461

185 Dynamic Ine ciency Katsuya Takii (Institute) Modern Macroeconomics II 185 / 461

186 Dynamic Ine ciency Assignment: A household with a constant risk aversion solves the following problem: max log c yt + 1 c yt,c ot ϱ log c ot+1 s.t.s n t = (1 τ) w t c yt, c ot+1 = 1 + ρ t+1 s n t + B t+1 Government levies a lump sum tax, τ and uses the proceeds to pay bene ts to old individual by B. Suppose that Government balances budget constrained: B t = (1 + n) τw t. Assume that production function is Kt α (T t L t ) (1 α) and the maintain other assumptions in our lecture note. 1 Derive the di erence equation of k et = K t TN yt. How is the equation in uenced by τ? 2 Derive the steady state level of capital stock per unit of e ective workers, ke. How is it in uenced by τ? 3 Compare the derived ke and k GR. Discuss how τ a ects the welfare. Katsuya Takii (Institute) Modern Macroeconomics II 186 / 461

187 Altruism The above reasoning suggests that if one generation cares about the next generation, we may be able to recover e ciency. This motivates us to modify the model to include altruism. Katsuya Takii (Institute) Modern Macroeconomics II 187 / 461

188 Altruism Suppose that the agents cares about their children. More concretely, the young during period t maximizes the following utility function: U t = u (C yt ) + 1 u (c ot+1 ) n 1 + ϱ 1 + ϕ U t+1, where ϕ is the measure of sel shness. If ϕ = 0, parents treat their children like themselves. The variable U t is the total sum of discounted utility of the young during period t. During period t + 1, an agent expects to have 1 + n children. Since the parents are sel sh in that they care about themselves more than their children, the 1 bene ts from their children are discounted by 1+ϕ. Hence their utility from their children is 1+n 1+ϕ U t+1. Katsuya Takii (Institute) Modern Macroeconomics II 188 / 461

189 Altruism Assume that where β = 1+n (1+ϱ)(1+ϕ). 0 = lim s! β s U t+s. Then utility of generation t is U t = u (C yt ) ϱ u (c ot+1) + βu t+1 = u (C yt ) ϱ u (c ot+1) + β u (Cyt+1 ) ϱ u (c ot+2) +βu t+2 = u (C yt ) ϱ u (c ot+1) +β u (C yt+1 ) ϱ u (c ot+2) + β 2 U t+2 = β s u (c yt+s ) + 1 s=0 1 + ϱ u (c ot+s+1) + lim β s U t+s s! Katsuya Takii (Institute) Modern Macroeconomics II 189 / 461

190 Altruism U t U t = β s u (c yt+s ) + 1 s=0 1 + ϱ u (c ot+s+1) Assume that parents can transfer their income to their children. Then the budget constraint of each generation must change: c yt + s t = w t + m t (22) c ot+1 + (1 + n) m t+1 = 1 + ρ t+1 st (23) where m t is the transfer from the parents to their children. Hence, (1 + n) m t+1 = 1 + ρ t+1 [wt + m t c yt ] c ot+1 m t+1 = 1 + ρ t n [w t + m t c yt ] c ot n Katsuya Takii (Institute) Modern Macroeconomics II 190 / 461

191 Altruism This problem is equivalent to max fc yt+s,c ot+s+1 g s=0 β s u (c yt+s ) ϱ u (c ot+s+1) s.t m t+1 = 1 + ρ t n [w t + m t c yt ] c ot n 1+n where β = (1+ϱ)(1+ϕ) < 1. This is essentially the problem of representative agent. In the representative agent model, the market economy is Pareto optimal. (Remember that a social planner model can be attained by RCE.) The overlapping generation model with altruism can also attain Pareto optimal. Because parents care about children s utility and budget constraint is connected by bequests, any transfer from young to old cannot improve old s utility. Katsuya Takii (Institute) Modern Macroeconomics II 191 / 461

192 Altruism Note that this analysis implicitly assume that income transfer always occurs. When the parents are sel sh enough, they do not want to leave any bequests. To see this, we can de ne the Bellman equation of this problem as V (m t, t) = max u (c yt ) + 1 C yt,c ot 1 + ϱ u (c ot+1) + βv (m t+1, t + 1) s.t. m t+1 = 1 + ρ t n [w t + m t c yt ] where β = 1+n (1+ϱ)(1+ϕ) < 1. c ot n Katsuya Takii (Institute) Modern Macroeconomics II 192 / 461

193 Altruism The FOCs are u 0 (c yt ) = βv 1 (m t+1, t + 1) 1 + ρ t n 1 + ρ = t+1 (1 + ϱ) (1 + ϕ) V 1 (m t+1, t + 1) u 0 (c ot+1 ) = βv 1 (m t+1, t + 1) 1 + ϱ 1 + n 1 = (1 + ϕ) V 1 (m t+1, t + 1) Katsuya Takii (Institute) Modern Macroeconomics II 193 / 461

194 Altruism Although the equation presumes the FOCs are hold with equality, if ϕ is large enough, the rst order conditions may not be satis ed with equality. It is optimal for the parents to consume their all income. In this case, the intergenerational linkages are broken and the transfers from the next generation can improve the utility of the current generation. Hence, the equivalence of the representative agent model and the OGM occurs if the parents altruism is strong enough. Katsuya Takii (Institute) Modern Macroeconomics II 194 / 461

195 Continuous Time Problem We discusses how the discrete dynamic programming can be applied for the optimal growth model. However, this is not the only tool to analyze a dynamic optimization problem. Depending on the model, it is easier to analyze continuous time model. In particular, it is easier to analyze a transition dynamics using continuous time model. We will discuss the continuous time model as a limit of the discrete time model. Katsuya Takii (Institute) Modern Macroeconomics II 195 / 461

196 Continuous Time Problem In the previous model, time goes like t, t + 1, t + 2. In this section, I assume that time goes like t, t +, t Then I describe the continuous time model by taking a limit of :! 0. I assume that one period return between t and t + is constant, r (X t, S t ). Therefore, the sum of the returns is r (X τ, S τ ) + β r (X τ+, S τ+ ) + β 2 r (X τ+2, S τ+2 ) +..., where τ is an initial period. We can de ne at time t by t j = τ + j. Hence, j = t j τ. Therefore, the above sum can be rewritten as r (X τ, S τ ) + β t 1 τ r (Xt1, S t1 ) + β t 2 τ r (Xt2, S t2 )... Katsuya Takii (Institute) Modern Macroeconomics II 196 / 461

197 Continuous Time Problem Assume that where ϱ is the discount rate. 8 >< U (S τ ) = max fx t g >: lim!0 β = j= ϱ, Then the continuous version of the original model can be expressed as 9 (t j τ) 1 >= r X tj, S tj 1 + ϱ >;, s.t. S t+ = G (X t, S t ) = G c (X t, S t ) + S t, S τ is given. Katsuya Takii (Institute) Modern Macroeconomics II 197 / 461

198 Continuous Time Problem Since t e ϱt 1 = lim,!0 1 + ϱ the continuous version of the previous model is written as Z U (S τ ) = max e ϱ(t X t τ s.t. Ṡ t = G c (X t, S t ) S τ is given. τ) r (X t, S t ) dt Katsuya Takii (Institute) Modern Macroeconomics II 198 / 461

199 Continuous Time Problem I would like to derive the continuous version of the Bellman equation as the limit of the previous discrete model. Since β = 1 1+ ρ V (S t ) = max r (X t, S t ) + 1 X t 1 + ϱ V (S t+ ), S t+ S t = G c (X t, S t ). We can modify the value function as follows. (1 + ϱ) V (S t ) = max f(1 + ϱ) r (X t, S t ) + V (S t+ )g X t r (Xt, S ϱv (S t ) = max t ) + 2 ϱr (X t, S t ) X t + [V (S t+ ) V (S t )] ( ) r (Xt, S t ) + ϱr (X t, S t ) ϱv (S t ) = max X t + V (S t+ ) V (S t ) Katsuya Takii (Institute) Modern Macroeconomics II 199 / 461

200 Continuous Time Problem When goes 0, the continuous version of the Bellman equation is derived as follows. ϱv (S t ) = max r (Xt, S t ) + V 0 (S t ) Ṡ t X t = max r (Xt, S t ) + V 0 (S t ) G c (X t, S t ) X t Bellman Equation: ϱv (S t ) = max X t r (Xt, S t ) + V 0 (S t ) G c (X t, S t ) Katsuya Takii (Institute) Modern Macroeconomics II 200 / 461

201 Continuous Time Problem The rst order condition of this Bellman equation is 0 = r 1 (x (S t ), S t ) + V 0 (S t ) G1 c (x (S t ), S t ). (24) Substitute the policy function into the Bellman equation, ϱv (S t ) = r (x (S t ), S t ) + V 0 (S t ) G c (x (S t ), S t ). (25) Again, the rst order condition (24) and the Bellman equation (25) solves the value function V () and x (). We can apply the Guess and veri ed method to solve two equations. Katsuya Takii (Institute) Modern Macroeconomics II 201 / 461

202 Continuous Time Problem Example: Consider the following investment problem: Z max L t,i t 0 [F (K t, L t ) wl t p (C (I t, K t ) + I t )] e ρt dt s.t. K t = I t δk t, K 0 is given where F (K, L) and C (I, K ) are constant returns to scale in each factors. The Bellman Equation of this problem can be written as ρv (K ) = max F (K, L) wl p (C (I, K ) + I ) + V 0 (K ) (I δk ) I,L Katsuya Takii (Institute) Modern Macroeconomics II 202 / 461

203 Guess and Verify Method Guess V (K ) = QK. max I,L = max I,L = max g,l ff (K, L) wl p (C (I, K ) + I ) + Q (I δk )g F 1, L K w L K p C I K, 1 + I K + Q I δk K K fφ (l) wl p (c (g) + g) + Q [g δ]g K where φ (l) = F (1, l), c (g) = C (g, 1), l = L K and g = I K. If there exists Q that satis es ρq = max fφ (l) wl p (c (g) + g) + Q [g δ]g. g,l Then our guess is veri ed. Since the value function is unique, the linear must be the property of the value function. Katsuya Takii (Institute) Modern Macroeconomics II 203 / 461

204 Guess and Verify Method Note that Q = p 1 + c 0 (g) h i g = ψ Q tobin 1, Q tobin = Q p This is called the Q theory of investment. It says that the rm s growth rate is determined by Q tobin. Note that Q tobin can be estimated by Q tobin = V (K ) pk. This is the market value of a rm relative to the replacement cost of capital. Katsuya Takii (Institute) Modern Macroeconomics II 204 / 461

205 Guess and Verify Method Assume that c (g) = 1 2A (g B)2 and c 0 (g) = (g B ) A. Hence (g B) h i = Q tobin 1 A h i g = B + A Q tobin 1 = B + AQ tobin + ε where B + ε = B A. Empirical researchers often estimate this regression. Katsuya Takii (Institute) Modern Macroeconomics II 205 / 461

206 Continuous Time Problem Assignment: Consider the following household problem: Z max ln c t e ϱt dt, c t 0 s.t. ȧ t = ρa t c t, a 0 is given where a t is asset, w t is wage and c t is consumption. 1 De ne the Bellman equation. 2 Derive a Policy Function. Katsuya Takii (Institute) Modern Macroeconomics II 206 / 461

207 Continuous Time Problem The envelope theorem can be also derived. ϱv 0 (S t ) = r 2 (X t, S t ) + V 00 (S t ) G c (X t, S t ) + V 0 (S t ) G c 2 (X t, S t ). Note that V 00 (S t ) G c (X t, S t ) = V 00 (S t ) Ṡ t = dv 0 (S t ) dt. Hence the envelope theorem can be rewritten as dv 0 (S t ) dt = ϱv 0 (S t ) r2 (X t, S t ) + V 0 (S t ) G c 2 (X t, S t ). (26) This equation describes the dynamics of the marginal value of state variable V 0 (S t ). Katsuya Takii (Institute) Modern Macroeconomics II 207 / 461

208 Continuous Time Problem Optimal Control: De ning the costate variable, λ t, is equal to the marginal value of state variables, λ t V 0 (S t ), we can derive the result of alternative method to solve dynamic optimization, optimal control. Using λ t, the rst order condition (24) and the envelope theorem (26) and the transition equation can be rewritten as 0 = r 1 (x o (S t, λ t ), S t ) + λ t G c 1 (x o (S t, λ t ), S t ), (27) λ t = ϱλ t [r 2 (x o (S t, λ t ), S t ) + λ t G c 2 (x o (S t, λ t ), S t )],(28) Ṡ t = G c (x o (S t, λ t ), S t ), S τ is given. (29) where x o (S t, V 0 (S t )) = x (S t ). Katsuya Takii (Institute) Modern Macroeconomics II 208 / 461

209 Continuous Time Problem The rst order condition (27) determines a policy function x o (S t, λ t ). Given the policy function x o (S t, λ t ), equations (28) and (29) determine the path of the costate variable, λ t and the state variable, S t. Note that since we have one boundary condition, S τ, we need an additional boundary condition to solve the equations (28) and (29). For this purposes, we need the following transversality condition: lim t! e ϱt V 0 (S t ) S t = lim t! e ϱt λ t S t = 0. (30) Similar to the discrete time problem, given usual technical assumptions, concavity etc, it is shown that equations (27), (28), (29) and (30) are su cient conditions for the original problem. Katsuya Takii (Institute) Modern Macroeconomics II 209 / 461

210 Continuous Time Problem The above conditions are nicely summarized by de ning Hamiltonian H (X t, S t, λ t ): H (X t, S t, λ t ) = r (X t, S t ) + λ t G c (X t, S t ). Then above conditions are expressed as 0 = H X (X t, S t, λ t ) λ t = ϱλ t H S (X t, S t, λ t ) Ṡ t = H λ (X t, S t, λ t ), S τ is given. 0 = lim t! e ϱt λ t S t These are the useful conditions for analyzing the continuous version of the dynamic optimization problem. Katsuya Takii (Institute) Modern Macroeconomics II 210 / 461

211 Continuous Time Problem Hamiltonian can be nicely interpreted. When we choose a dynamic optimization problem, we know that current decision a ects not only the current return, but also the future returns. The second term of Hamiltonian summarizes the impact on the future returns. Remember, λ t = V 0 (S t ). That is, the costate variable can be interpreted as the marginal impact of the state variable on the present value of the discounted future returns. Since Ṡ t = G c (X t, S t ), the second term is interpreted as the impact of a change in S t on the future returns. Therefore, Hamiltonian summarizes the important trade o of the dynamic optimization problem: the impact on the current return and the future returns. The rst order condition, 0 = H X, implies that the agent chooses the control variable X t so that he maximizes Hamiltonian. Katsuya Takii (Institute) Modern Macroeconomics II 211 / 461

212 Continuous Time Problem Example: Consider the following investment problem: s.t. K t = I t δk t Hamiltonian Z max L t,i t 0 [F (K t, L t ) wl t p (C (I t, K t ) + I t )] e ρt dt H (L t, I t, K t, q t ) = F (K t, L t ) wl t p (C (I t, K t ) + I t ) + q t [I t δk t ]. Katsuya Takii (Institute) Modern Macroeconomics II 212 / 461

213 Continuous Time Problem Derive the necessary conditions and transversality condition for this problem. 0 = H L (L t, I t, K t, q t ) = F L (K t, L t ) w 0 = H I (L t, I t, K t, q t ) = q t p (1 + C I (I t, K t )) q t = ρq t H K (L t, I t, K t, q t ) = ρq t [F K (K t, L t ) pc K (I t, K t ) δq t ] K t = H q (L t, I t, K t, q t ) = I t δk t, K 0 is given. 0 = lim t! e ρt q t K t Katsuya Takii (Institute) Modern Macroeconomics II 213 / 461

214 Continuous Time Problem Marginal Productivity of Labor is equal to wage. F L (K t, L t ) = w The value of an additional unit of installed capital, q t, is the marginal cost, which is equal to the price of investment good plus the marginal adjustment cost. q t = [1 + C I (I t, K t )] p Note that if C I (I t, K t ) = 0, q t = p. Katsuya Takii (Institute) Modern Macroeconomics II 214 / 461

215 Continuous Time Problem The interpretation of q t : De ne π K = F K (K, L) marginal cash ow attributed to a unit of capital. pc K (I, K ) is the q t = ρq t π K,t + δq t ρq t = π K δq t + q t ρ = π K δq t + q t q t The shadow price of capital, q t, is the price at which a marginal unit of installed capital could be bought or sold. Katsuya Takii (Institute) Modern Macroeconomics II 215 / 461

216 Continuous Time Problem Note that if C K (I, K ) = 0, π K = F K (K, L). Then, we can de ne the user cost (=rental price) r t = π K,t. Hence, ρq t = r t δq t + q t r t δq t + q t+1 q t (1 + ρ) q t r t + (1 δ) q t+1 r t = π K,t = F K (K, L) This is the dynamics of q t which I have derived in Modern Macroeconomics 1. Katsuya Takii (Institute) Modern Macroeconomics II 216 / 461

217 Continuous Time Problem Suppose that not only C K (I, K ) = 0, but also C I (I, K ) = 0. q t = p t. Hence Then ρ = r t δp t + ṗ t p t r t = π K,t = F K (K, L) Furthermore, Suppose p t = 1 over time. ρ = r t δ This is the arbitrage condition in Modern Macroeconomics 1. In order to obtain an alternative interpretation of q t, we need to know how to solve a di erential equation. Katsuya Takii (Institute) Modern Macroeconomics II 217 / 461

218 Continuous Time Problem Lemma Suppose that x t follows a di erential equation: dx t dt = a t + b t x t 1 Suppose that x τ is given. Then the solution to this di erential equation is R T Z T x T = e τ b R t s ds x τ + a t e τ b s dt ds τ 2 Suppose that 0 = lim T! x T e R T t b s ds. Then the solution to this di erential equation is x τ = Z τ a t e R t τ b s ds dt Katsuya Takii (Institute) Modern Macroeconomics II 218 / 461

219 Continuous Time Problem Proof: Note that the di erential equation can be rewritten as Z T τ R t a t e τ b s ds = dx R t dt e t τ b s ds hx = d t e R t dt R t Z T d hx t e R t a t e τ b s ds dt = dt τ = x T e R T τ τ b s ds i b t x t e τ b s ds i b s ds x τ dt R t τ b s ds Katsuya Takii (Institute) Modern Macroeconomics II 219 / 461

220 Continuous Time Problem Proof: Suppose that x τ is given. R T x T = e τ b s ds Z T x τ + a t e τ R t τ b s ds dt Suppose that 0 = lim T! x T e R T t b s ds. x τ = Z τ a t e R t τ b s ds dt Katsuya Takii (Institute) Modern Macroeconomics II 220 / 461

221 Continuous Time Problem Alternative interpretation of q t : Applying the lemma in this context. q t = ρq t π K,t + δq t q t (ρ + δ) q t = π K,t q t e (ρ+δ)t (ρ + δ) q t e (ρ+δ)t = π K,t e (ρ+δ)t dq t e (ρ+δ)t dt hq i t e (ρ+δ)t τ = π K,t e (ρ+δ)t = lim q te (ρ+δ)t q τ e (ρ+δ)τ = t! Z τ Z τ π K,t e (ρ+δ)t dt π K,t e (ρ+δ)t dt Katsuya Takii (Institute) Modern Macroeconomics II 221 / 461

222 Continuous Time Problem Hence, q τ e (ρ+δ)τ = Z τ π K,t e (ρ+δ)t dt + lim t! q t e (ρ+δ)t Note that TVC implies 0 = lim t! e ρt q t K t. Suppose that there exists B such that K t B <. Then 0 = lim t! e ρt q t = lim t! q t e (ρ+δ)t. Hence q τ e (ρ+δ)τ = q τ = = Z τ Z τ Z τ π K,t e (ρ+δ)t dt π K,t e (ρ+δ)t e (ρ+δ)τ dt π K,t e (ρ+δ)(t τ) dt The shadow price of capital, q t, is equal to the present discounted value of the stream of marginal cash ow attributed to a unit of capital installed at time t. Katsuya Takii (Institute) Modern Macroeconomics II 222 / 461

223 Continuous Time Problem Assignment: Consider the following household problem: Z max c t 0 ln c t e ϱt dt, s.t. ȧ t = ρa t + w t c t, a 0 is given where a t is asset, w t is wage and c t is consumption. 1 De ne Hamiltonian. 2 Derive the rst order conditions. Katsuya Takii (Institute) Modern Macroeconomics II 223 / 461

224 Continuous Time Problem Assignment: Consider the following budget constraint ȧ t = ρ t a t + w t τ t c t, a 0 is given where c t is consumption, a t is asset, w t is wage payment, τ t is tax payment and ρ t is interest rate. Suppose that the following condition is satis ed R 0 = lim a T T e 0 ρ s ds. T! This is called no Ponzi game condition, which means that debt cannot increase faster than the interest rate. Derive the following intertemporal budget constraint: Z 0 c t e R t 0 ρ s ds dt = h 0 + a 0 Z 0 τ t e R t 0 ρ s ds dt, where h 0 = R 0 w te R t 0 ρ s ds dt is human wealth at date 0. Katsuya Takii (Institute) Modern Macroeconomics II 224 / 461

225 Continuous Time Problem Assignment: Consider the following government budget constraint ḃ t = g t τ t + ρ t b t, b 0 is given where b t is government bond, τ t is tax revenue, ρ t is interest rate and g t is government expenditure. Suppose that the following condition is satis ed 0 = lim T! b T e R T 0 ρ s ds. This is called no Ponzi game condition for government. following intertemporal budget constraint: Derive the Z b 0 + g t e 0 R t Z 0 ρ s ds dt = τ t e 0 R t 0 ρ s ds dt Katsuya Takii (Institute) Modern Macroeconomics II 225 / 461

226 Continuous Time Problem Assignment: Using results in the previous two assignments and the following capital market clearing condition, Derive the following equation Z 0 c t e R t a t = k t + b t, for any t Z 0 ρ s ds dt = h 0 + k 0 g t e 0 R t 0 ρ s ds dt Note that this budget constraint does not include neither debt, b t, and tax, τ t. Therefore, the method of nancing does not change the budget constraint of the representative consumer. This is called Ricardian Equivalence Katsuya Takii (Institute) Modern Macroeconomics II 226 / 461

227 Continuous Time Growth Model In this section I apply optimal control to the neoclassical growth model and analyze the neoclassical growth model. Consider the following problem. Z max e ϱt N t U (c t ) dt c t 0 K t = F (K t, T t N t ) δk t c t N t, Ṫ t = gt t, Ṅ t = nn t, where K 0, T 0 and N 0 are given. Katsuya Takii (Institute) Modern Macroeconomics II 227 / 461

228 Continuous Time Growth Model Note that Hence g ket = g K TN = g K (g T + g N ) k et k et = F (K t, T t N t ) δk t c t N t K t Ṫt + Ṅt, T t N t = [F (k et, 1) δk et c et ] T t N t K t (g + n), = [f (k et) δk et c et ] k et (g + n), k et = f (k et ) c et (g + n + δ) k et, where f (k t ) = F (k t, 1). From the second equation to the third, I use the assumption on the production function, the constant return to scale. Katsuya Takii (Institute) Modern Macroeconomics II 228 / 461

229 Continuous Time Growth Model Assignment: 1 Prove that if Ẋ t = gx t, X t = X 0 e gt. 2 Suppose that U (c et T t ) = (c et T t ) (1 θ) 1 1 θ, θ 0. Show that the original problem is equivalent to Z max e ϱtũ (c et ) dt, c et 0 k et = f (k et ) c et (g + n + δ) k et, k e0 is given where ũ (c et ) = (c (1 θ) et ) 1 θ if 0 θ 6= 1 and ũ (c et ) = ln c et if θ = 1, and ϱ = ϱ n (1 θ) g. We assume that ϱ > 0. [Hint: the above statement implies N t = N 0 e nt and T t = T 0 e gt. ] Katsuya Takii (Institute) Modern Macroeconomics II 229 / 461

230 Continuous Time Growth Model De ne Hamiltonian of this problem: H (c t, k t, λ t ) = (c (1 θ) et) + λ t [f (k et ) c et (g + n + δ) k et ] 1 θ Katsuya Takii (Institute) Modern Macroeconomics II 230 / 461

231 Continuous Time Growth Model The rst order conditions are λ t = c θ et λ t = ϱ λ t λ t f 0 (k et ) (g + n + δ), 0 = lim t! λ t k et e ρt k et = f (k et ) c et (g + n + δ) k et, k e0 is given Katsuya Takii (Institute) Modern Macroeconomics II 231 / 461

232 Continuous Time Growth Model Note that Hence g λ = g c θ e = θg ce θ ċet = λ t = ϱ f 0 (k et ) (g + n + δ) c et λ t = [ϱ n (1 θ) g] f 0 (k et ) (g + n + δ) = (ϱ + δ + θg) f 0 (k et ) ċ et c et = 1 f 0 (k et ) θ (ϱ + δ + θg) (31) Equation (31) is the continuous version of Euler equation. Katsuya Takii (Institute) Modern Macroeconomics II 232 / 461

233 Continuous Time Growth Model The corresponding transversality condition is 0 = lim t! λ t k et e ϱ t 0 = lim t! (c et ) θ k et e [ϱ n (1 θ)g ]t Katsuya Takii (Institute) Modern Macroeconomics II 233 / 461

234 Continuous Time Growth Model Together with the transition equation of k t, the following two di erential equations and two boundary conditions solve the optimal growth path: ċ et = 1 f 0 (k et ) (ϱ + δ + θg), (32) c et θ k et = f (k et ) c et (g + n + δ) k et, k e0 is given. (33) 0 = lim t! (c et ) θ k et e [ϱ n (1 θ)g ]t, (34) Katsuya Takii (Institute) Modern Macroeconomics II 234 / 461

235 Phase Diagram The Phase Diagram: One of the merits of working with a continuous model is that we can use the phase diagram. Let me describe what the phase diagram is. The rst, let me de ne the steady state. De nition On the steady state the path of (c e, k e ) satis es ċ e = k e = 0. Therefore, the following equation must be satis ed on the steady sate: ċ e = 0 : f 0 (k e ) = ϱ + δ + θg (35) k e = 0 : c e = f (k e ) (g + n + δ) k e (36) Katsuya Takii (Institute) Modern Macroeconomics II 235 / 461

236 Phase Diagram ċ e = 0: Note that k e is uniquely determined by equation (35). 1 When k et = ke, ċe = 0. 2 When k et < ke, f 0 (k et ) > ϱ + δ + θg. ) ċ et > 0. 3 When k et > ke, f 0 (k et ) < ϱ + δ + θg. ) ċ et < 0. Katsuya Takii (Institute) Modern Macroeconomics II 236 / 461

237 Phase Diagram Phase Diagram 1 Katsuya Takii (Institute) Modern Macroeconomics II 237 / 461

238 Phase Diagram Equation (36) implies that dc e dk e = f 0 (k e ) (g + n + δ), d 2 c e d (k e ) 2 = f 00 (k e ) < 0 Hence, ce has the maximum value at f 0 ke GR = g + n + δ. (37) k e = 0: Capital stock, ke GR is called the golden rule level of the capital stock. If you want to maximize c e when k e = 0, equation (37) must be satis ed. 1 2 When ce = f (ket ) When c et > f (ket ) (g + n + δ) ket, k et = 0 (g + n + δ) ket, k et < 0. 3 When c et < f (ket ) (g + n + δ) ket, k et > 0. Katsuya Takii (Institute) Modern Macroeconomics II 238 / 461

239 Phase Diagram Phase Diagram 2 Katsuya Takii (Institute) Modern Macroeconomics II 239 / 461

240 Phase Diagram Compare Note that f 0 (ke ) = ϱ + δ + θg, and f 0 ke GR = g + n + δ. g + n + δ < ϱ + δ + θg i (ϱ n) > (1 θ) g. Because we have assumed ρ n > (1 θ) g, f 0 (ke ) > f 0 ke GR, ke < ke GR. Since the agent discount future, it is not optimal to reduce current consumption to reach the golden rule level of the capital stock. The steady state value of capital stock, ke, is called the modi ed golden rule level of capital stock. Note that given the assumption of ρ n > (1 θ) g, 0 = lim t! (c e ) θ k e e [ϱ n (1 θ)g ]t. Katsuya Takii (Institute) Modern Macroeconomics II 240 / 461

241 Phase Diagram Phase Diagram 3 Katsuya Takii (Institute) Modern Macroeconomics II 241 / 461

242 Phase Diagram The saddle point path: Note that given k e0, there exists c (k e0 ) which converges to the steady state. If c e0 = c (k e0 ), economy will eventually reach the steady state. Since the steady state satis es the transversality condition (34), this path is optimal. This path is called the saddle point path. Katsuya Takii (Institute) Modern Macroeconomics II 242 / 461

243 Phase Diagram Phase Diagram 4 Katsuya Takii (Institute) Modern Macroeconomics II 243 / 461

244 Phase Diagram If c e0 > c (k e0 ), it is known that it hits k et = 0 line in a nite time. But when k et = 0, c et must jump to 0 from equation (33). Otherwise, k et becomes negative. However, the Euler equation (32) does not allow the jump. Hence this path violates the Euler equation (32). If c e0 < c (k e0 ), the phase diagram says that this path eventually converges to the point c et = 0 and k et <. We show that this path violate the transversality condition (34). Katsuya Takii (Institute) Modern Macroeconomics II 244 / 461

245 Phase Diagram Proof: Because for large t, k et > ket GR, f 0 (k et ) < f 0 ke GR. ċ et = 1 f 0 (k et ) (ϱ + δ + θg) c et θ < 1 h i f 0 ke GR (ϱ + δ + θg) θ Hence = 1 [g + n + δ (ϱ + δ + θg)] θ = 1 ϱ [(1 θ) g + n ϱ] = θ θ Katsuya Takii (Institute) Modern Macroeconomics II 245 / 461

246 Phase Diagram Proof: Hence, for large t This implies that Hence, (c et ) θ k et e ϱt > c et < c e0 e ϱ θ t. c e0 e ϱ θ t θ k et e ϱt = c e0 k et > 0 0 < lim t! (c et ) θ k et e ϱt. It proves that this path violates the transversality condition. Katsuya Takii (Institute) Modern Macroeconomics II 246 / 461

247 Phase Diagram In sum, the unique globally stable optimal path is characterized by the saddle point path. On this path, for any given k e0, the agent optimally chooses c (k e0 ) with his expectation that economy will converge to the steady state. Although I only explain the basic idea about how to use Phase Diagram in this lecture, you can apply this method to the varieties of economic issues. Katsuya Takii (Institute) Modern Macroeconomics II 247 / 461

248 Phase Diagram Assignment: Consider the following social planner s problem given government expenditure G: Z max e ϱt U (c t ) dt c t 0 k t = f (k t ) δk t c t g, where c t is consumption, g is constant government expenditure, k t is capital stock, ϱ is the discount rate, δ is the depreciation rate of capital stock. Assume that the aggregate production function, f (), satis es usual assumptions: f 0 () > 0, f 00 () < 0, f (0) = 0, f 0 (0) =, f 0 ( ) = 0. 1 Draw and explain the Phase Diagram which summarizes the dynamics of k t and c t 2 Discuss how government expenditure g in uences the steady state value of k t and c t. Katsuya Takii (Institute) Modern Macroeconomics II 248 / 461

249 Stochastic Dynamic Optimization So far, we don t have any uncertainty in the model. Introducing uncertainty in our model greatly increases the applicability of our method to several other issues such as a nancial market, business cycle and labor search. In this section, we simply add uncertainty to the basic dynamic programming. Katsuya Takii (Institute) Modern Macroeconomics II 249 / 461

250 Stochastic Dynamic Optimization We introduce random factors into the optimization problem. ) ( U (S τ, τ) = max E τ β (t τ) r (X t, S t ) fx t g τ t=τ s.t. S t+1 = G (X t, S t, ε t+1 ), S τ is given, where ε t+1 has a distribution function Q (ε t+1 )., Katsuya Takii (Institute) Modern Macroeconomics II 250 / 461

251 Stochastic Dynamic Optimization Note that U (S τ, τ) = max fx t g τ = max E τ X τ E τ ( ( t=τ β (t τ) r (X t, S t ) ) r (X τ, S τ ) + β max fxt g τ+1 E τ+1 t=τ+1 β (t (τ+1)) r (X t, S t ) = max X τ fr (X τ, S τ ) + βe τ U (S τ+1, τ + 1)g Katsuya Takii (Institute) Modern Macroeconomics II 251 / 461

252 Stochastic Dynamic Optimization The Bellman equation can be written as Z V (S t ) = max r (X t, S t ) + β V (G (X t, S t, ε t+1 )) dq (ε t+1 ) X τ where and = = Z Z V (G (X t, S t, ε t+1 )) dq (ε t+1 ) V (G (X t, S t, ε t+1 )) Q 0 (ε t+1 ) dε t n V (G (X t, S t, ε i )) Pr (ε t+1 = ε i ) i=1 Pr (ε t+1 = ε i ) = Q (ε i ) Q (ε i 1 ) We can analyze stochastic dynamic optimization problem using the method similar to the deterministic case. Katsuya Takii (Institute) Modern Macroeconomics II 252 / 461 or

253 Stochastic Dynamic Optimization Example: Consider the following Lucas (1978) model of asset prices: 1 Consider an economy consisting of a large number of identical agents. 2 Each agent is born with one tree. Each period, each tree yields fruits (=dividends) in the amount d t into its owner at the beginning of period t. 3 Let p (d t ) is the price of a tree in period t, which is measured in units of consumption goods per tree. Katsuya Takii (Institute) Modern Macroeconomics II 253 / 461

254 Stochastic Dynamic Optimization Household max E τ fc t β (t τ) U (c t ), g t=τ s.t. c t + p (d t ) s t+1 = [p (d t ) + d t ] s t d t+1 = G (d t, ε t+1 ), ε t+1 ~Q (ε t+1 ) where s t is the number of trees owned at the beginning of period, t. Capital market clearing condition: Because everything is identical, nobody sells and buys in an equilibrium. s t = 1 Katsuya Takii (Institute) Modern Macroeconomics II 254 / 461

255 Stochastic Dynamic Optimization Bellman Equation V (s t, d t ) = max c t U (c t ) + β R V (s t+1, d t+1 ) dq (ε t+1 ) s.t. s t+1 = [p (d t) + d t ] s t c t p (d t ) d t+1 = G (d t, ε t+1 ), ε t+1 ~Q (ε t+1 ) Katsuya Takii (Institute) Modern Macroeconomics II 255 / 461

256 Stochastic Dynamic Optimization First Order Conditions: Envelope Theorem: U 0 (c t ) = β V 1 (s t, d t ) = β Z V1 (s t+1, d t+1 ) dq (ε t+1 ) p (d t ) Z [p (dt ) + d t ] V 1 (s t+1, d t+1 ) dq (ε t+1 ) p (d t ) Katsuya Takii (Institute) Modern Macroeconomics II 256 / 461

257 Stochastic Dynamic Optimization Euler Equation: 1 Combining the rst order condition and the envelope theorem [p (d t ) + d t ] U 0 (c t ) = V 1 (s t, d t ) 2 Using envelope theorem, Euler equation can be derived as [p (d t ) + d t ] U 0 (c t ) Z [p (dt ) + d t ] = β [p (d p (d t ) t+1 ) + d t+1 ] U 0 (c t+1 ) dq (ε t+1 ) Hence Z U 0 [p (dt+1 ) + d (c t ) = β t+1 ] U 0 (c p (d t ) t+1 ) dq (ε t+1 ) Katsuya Takii (Institute) Modern Macroeconomics II 257 / 461

258 Stochastic Dynamic Optimization TVC: 0 = lim β t E [V 1 (s t, d t ) s t jd τ ] t! = lim β t E τ [p (dt ) + d t ] U 0 (c t ) s t t! The operator E t [] is the conditional expectation given the information available at date t. Hence, stochastic version of the transversality condition demands an expectation conditioning on the information available at date τ. Katsuya Takii (Institute) Modern Macroeconomics II 258 / 461

259 Stochastic Dynamic Optimization Equilibrium Condition implies Euler equation and TVC are s t = 1 ) c t = d t Z U 0 [p (dt+1 ) + d t+1 ] (d t ) = β U 0 (d t+1 ) dq (ε t+1 ) p (d t ) 0 = lim β t E τ [p (dt ) + d t ] U 0 (d t ) t! Katsuya Takii (Institute) Modern Macroeconomics II 259 / 461

260 Stochastic Dynamic Optimization Euler Equation can be rewritten as Z p (d t ) = M t+1 [p (d t+1 ) + d t+1 ] dq (ε t+1 ) = E t [M t+1 [p (d t+1 ) + d t+1 ]] = E [M t+1 [p (d t+1 ) + d t+1 ] jd t ] where M t+1 = β U 0 (c t+1 ) U 0 (c t ) = β U 0 (d t+1 ) U 0 (d t ) is the intertemporal marginal rate of substitution also known as the stochastic discount factor or pricing kernel. Because information needed to predict future sequences of d t+i is summarized by d t, the last equality must be satis ed. Katsuya Takii (Institute) Modern Macroeconomics II 260 / 461

261 Stochastic Dynamic Optimization Note that p (d t ) = E t [M t+1 [p (d t+1 ) + d t+1 ]] = E t [M t+1 d t+1 ] + E t [M t+1 p (d t+1 )] = E t [M t+1 d t+1 ] + E t [M t+1 E t+1 [M t+2 [p (d t+2 ) + d t+2 ]]] = E t [M t+1 d t+1 ] + E t [M t+1 M t+2 d t+2 ] + E t [M t+1 M t+2 p (d t+2 )] = E t [M t+1 d t+1 ] + E t [M t+1 M t+2 d t+2 ] +E t [M t+1 M t+2 E t+2 [M t+3 [p (d t+3 ) + d t+3 ]]] = E t [M t+1 d t+1 ] + E t [M t+1 M t+2 d t+2 ] = +E t [M t+1 M t+2 M t+3 [p (d t+3 ) + d t+3 ]] E t Π i x =1 M t+x d t+i + lim E t Π i x =1 M t+x [p (d t+i ) + d t+i ] i! i=1 Katsuya Takii (Institute) Modern Macroeconomics II 261 / 461

262 Stochastic Dynamic Optimization Note that Π i x =1M t+x = β U0 (c t+1 ) U 0 (c t ) βu0 (c t+2 ) U 0 (c t+1 ) βu0 (c t+3 ) U 0 (c t+2 )..β U0 (c t+i ) U 0 (c t+i 1 ) = β i U 0 (c t+i ) U 0 (c t ) Hence lim E Π i x =1M t+x [p (d t+i ) + d t+i ] jd t i! = lim E β i U 0 (c t+i ) i! U 0 (c t ) [p (d t+i ) + d t+i ] jd t = lim i! β i E [U 0 (c t+i ) [p (d t+i ) + d t+i ] jd t ] U 0 (c t ) = 0 Katsuya Takii (Institute) Modern Macroeconomics II 262 / 461

263 Stochastic Dynamic Optimization The share price is an expected discounted stream of dividends with a stochastic discount factor.. p (d t ) = E Π i x =1M t+x d t+i jd t i=1 where Π i x =1M t+x = β i U 0 (d t+i ) U 0 (d t ) Katsuya Takii (Institute) Modern Macroeconomics II 263 / 461

264 Stochastic Dynamic Optimization Euler Equation can be also rewritten as Z 1 = M t (1 + ρ r t ) dq (ε t) = E [M t (1 + ρ r t )] where ρ r t = d t +p(d t ) p(d t 1 ) p(d t 1 ) is the return from this asset. Katsuya Takii (Institute) Modern Macroeconomics II 264 / 461

265 Stochastic Dynamic Optimization Lemma Cov (X t, Y t ) = E [X t Y t ] E [X t ] E [Y t ] Proof. Cov (X t, Y t ) = E [(X t E [X t ]) (Y t E [Y t ])] = E [X t Y t E [X t ] Y t X t E [Y t ] + E [X t ] E [Y t ]] = E [X t Y t ] E [X t ] E [Y t ] E [X t ] E [Y t ] + E [X t ] E [Y t ] = E [X t Y t ] E [X t ] E [Y t ] Katsuya Takii (Institute) Modern Macroeconomics II 265 / 461

266 Stochastic Dynamic Optimization Using lemma, 1 = E [M t (1 + ρ r t )] = E [M t] E [1 + ρ r t ] + Cov(M t, 1 + ρ r t ) Hence we can get E [1 + ρ r t ] = 1 E [M t ] [1 Cov(M t, 1 + ρ r t )] E [ρ r t ] = 1 E [M t ] [1 Cov(M t, 1 + ρ r t )] 1 (38) Katsuya Takii (Institute) Modern Macroeconomics II 266 / 461

267 Stochastic Dynamic Optimization Risk free rate of return: If there is a risk free asset, the return on this asset ρ must satisfy Hence, equation (38) implies Cov(M t, 1 + ρ) = ρ = 1 E [M t ] Return on the Risky Asset: Substituting the risk free rate into equation (38), E [1 + ρ r t ] = (1 + ρ) [1 Cov(M t, 1 + ρ r t )] E [ρ r t ] = ρ (1 + ρ) Cov(M t, 1 + ρ r t ) Katsuya Takii (Institute) Modern Macroeconomics II 267 / 461

268 Stochastic Dynamic Optimization A stock return must be equal to risk free rate of return +Risk premium. The risk premium is now decreasing function of covariance between a return and M t. Why? Note that because M t = β U 0 (c t ) U 0 (c t 1 ), c t "= U 0 (c t ) #= M t # So when covariance is low, we will expect a high return when M t is low, that is, your consumption is high, and a low return when your consumption is low. So you cannot smooth your consumption with this asset much. This model is called Consumption Capital Asset Pricing Model (Consumption CAPM). Katsuya Takii (Institute) Modern Macroeconomics II 268 / 461

269 Stochastic Dynamic Optimization Assignment: Consider the following investment problem: max E 0 I t,l t Π t x =0M x [z t F (K t, L t ) wl t C (I t, K t ) I t ] t=0 K t = I t + (1 δ) K t z t+1 = z t + ε t+1, ε t+1 ~Q (ε t+1 ) where F (K t, L t ) and C (I t, K t ) are constant returns to scale, M t < β 1 is the stochastic discount factor. (For this exercise, the sequence of M t is taken as given. ) 1 De ne the Bellman Equation of this problem. 2 Show that the value function is linear in K t. 3 Derive Euler Equation of this problem. Katsuya Takii (Institute) Modern Macroeconomics II 269 / 461

270 Rational Expectation and GMM Euler equation often gives us E t (X t+s (ψ x )) = Y t ψ y, where X t+s (ψ x ) and Y t ψ y are a (K 1) column vector which depends on ψ = ψ x, ψ y. Example, suppose that U (c) = (c)(1 θ) 1 1 θ. Then, U 0 (c) = c θ. Hence, Euler equation of consumption CAPM can be summarized by 1 = E t Mt ρ r t+1 where M t+1 = β U 0 (c t+1 ) U 0 (c t ) K = 1 = β (c t+1) θ = β ct θ. (c t ) θ c t+1 θ ct X t+s (ψ x ) = β c t+1 Y t ψ y = 1, ψ x = (β, θ) 1 + ρ r t+1 In this case, Katsuya Takii (Institute) Modern Macroeconomics II 270 / 461

271 Rational Expectation and GMM De ne u t+s (ψ) = Y t ψ y X t+s (ψ x ). Then E t [u t+s (ψ)] = E t h Y t ψ y = Y t ψ y = 0. i X t+s (ψ x ) E t [X t+s (ψ x )] We can interpret u t+s (ψ) as a prediction error. variable at date t, z j, where j = 1,.., J. Then Take any known E (z j u t+s (ψ)) = E [z j E t [u t+s (ψ)]] = 0 8j This is called an orthogonality condition. Katsuya Takii (Institute) Modern Macroeconomics II 271 / 461

272 Rational Expectation and GMM Given the orthogonality condition, for large N N i zj i lim ui t+s (ψ) = E [z j u t+s (ψ)] = 0, 8j N! N If the number of orthogonality conditions is equal to the number of ψ, GMM estimate ψ by 0 = N i z i j ui t+s (ψ ) N This is equivalent to a non-linear IV estimation: if there is 1 orthogonality condition and 1 parameter ψ x with ut+s i (ψ) = yt i ψ x xt+s i, 0 = N i zj i yt i ψx xi t+s N ψx N i z i x i t+s N = N i z i y i t N ) ψ x = " N i z i x i t+s # 1 N Katsuya Takii (Institute) Modern Macroeconomics II 272 / 461 i z i y i t

273 Rational Expectation and GMM In general, the number of orthogonality conditions is not equal to the number of parameters, ψ. 1 If the number of orthogonality condition is smaller than the number of parameters, we cannot identify the parameters. 2 If the number of orthogonality condition is greater than the number of parameters, we have the overidenti cation of the parameters. De ne a (KJ 1) vector, g (N : ψ) = 0 N i N i z1 i ui t+s (ψ) Ṇ. zj i ui t+s (ψ) N parameters are overidenti ed, GMM estimates ψ by ψ = arg min g (N : ψ) 0 W 1 ψ N g (N : ψ) 1 C A. When the where WN 1 is a (KJ KJ) optimal weighting matrix, which can be estimated from data. Katsuya Takii (Institute) Modern Macroeconomics II 273 / 461

274 Stochastic Growth Model In this section, we introduce a stochastic shock into the neoclassical growth model. In addition to the shock, we allow that household can value leisure. These framework plays the benchmark for studying business cycle. Katsuya Takii (Institute) Modern Macroeconomics II 274 / 461

275 Stochastic Growth Model The general problem is as follows: max E 0 fc t,l t g β t N t U(c t, 1 l t ) t=0 t=0 s.t. K t+1 = z t F (K t, T t l t N t ) + (1 δ)k t c t N t, K 0 is given T t+1 = (1 + g) T t, T 0 is given N t+1 = (1 + n) N t, N 0 is given ln z t+1 = ζ ln z t + ε t+1, z 0 is given, ε t+1 ~Q (ε t+1 ) Katsuya Takii (Institute) Modern Macroeconomics II 275 / 461

276 Stochastic Growth Model Assignment: Assume that the utility exhibits CRRA: U (c et T t, l t ) = [c ett t v (1 l t )] 1 θ 1, θ 0 1 θ Show that the original problem can be rewritten as max E 0 fc et,l t (β ) t ũ (c et, 1 l t ) g t=0 s.t. k et+1 = z tf (k et, l t ) + (1 δ) k et c et. k e0 is given (1 + g) (1 + n) ln z t+1 = ζ ln z t + ε t+1, z 0 is given, ε t+1 ~Q (ε t+1 ) where ũ (c et, 1 l t ) = [c et v (l t )] 1 θ 1 θ if 0 θ 6= 1, ũ (c et, 1 l t ) = ln c et + ln v (1 l t ) if θ = 1 and β = β (1 + n) (1 + g) (1 θ). Katsuya Takii (Institute) Modern Macroeconomics II 276 / 461

277 Stochastic Growth Model Assuming that β < 1, the following Bellman Equation can be de ned. Z V (k et, z t ) = max ũ(c et, 1 l t ) + β V (k et+1, z t+1 ) dq (ε t+1 ) fc et,l t g s.t. k et+1 = z tf (k et, l t ) + (1 δ) k et c et (1 + g) (1 + n) ln z t+1 = ζ ln z t + ε t+1, Katsuya Takii (Institute) Modern Macroeconomics II 277 / 461

278 Stochastic Growth Model First Order Conditions ũ 1 (c (k et, z t ), 1 l (k et, z t )) = β R V 1 (κ (k et, z t ), z t+1 ) dq (ε t+1 ) (1 + g) (1 + n) ũ 2 (c (k et, z t ), 1 l (k et, z t )) = β R V 1 (κ (k et, z t ), z t+1 ) dq (ε t+1 ) z t F 2 (k et, l (k et, z t )) (1 + g) (1 + n) where κ (k et, z t ) = z t F (k et,l(k et,z t ))+(1 δ)k et c(k et,z t ) (1+g )(1+n). Hence, consumption and leisure relationship is in uenced by ũ 2 (c et, 1 l t ) ũ 1 (c et, 1 l t ) = z tf 2 (k et, l t ) Katsuya Takii (Institute) Modern Macroeconomics II 278 / 461

279 Stochastic Growth Model Katsuya Takii (Institute) Modern Macroeconomics II 279 / 461

280 Stochastic Growth Model Envelope theorem V 1 (k et, z t ) = β R V 1 (κ (k et, z t ), z t+1 ) dq (ε t+1 ) [z t F 1 (k et, l (k et, z t )) + 1 δ] (1 + g) (1 + n) Katsuya Takii (Institute) Modern Macroeconomics II 280 / 461

281 Stochastic Growth Model Euler Equation 1 Combining the rst order condition and envelope theorem, V 1 (k et, z t ) = ũ 1 (c et, 1 l t ) [z t F 1 (k et, l t ) + 1 δ] 2 Using envelope theorem = Hence, = = ũ 1 (c et, 1 l t ) [z t F 1 (k et, l t ) + 1 δ] β R ũ 1 (c et+1, 1 l t+1 ) [z t+1 F 1 (k et+1, l t+1 ) + 1 δ] dq (ε t+1 ) [z t F 1 (k et, l t ) + 1 δ] (1 + g) (1 + n) ũ 1 (c et, 1 l t ) Z β ũ 1 (c et+1, 1 l t+1 ) [z t+1f 1 (k et+1, l t+1 ) + 1 δ] dq (ε (1 + g) (1 + n) t+1 ) Z βũ 1 (c et+1, 1 l t+1 ) [z t+1f 1 (k et+1, l t+1 ) + 1 δ] (1 + g) θ dq (ε t+1 ) Katsuya Takii (Institute) Modern Macroeconomics II 281 / 461

282 Stochastic Growth Model Euler Equation = ũ 1 (c et, 1 l t ) Z βũ 1 (c et+1, 1 l t+1 ) [z t+1f 1 (k et+1, l t+1 ) + 1 δ] (1 + g) θ dq (ε t+1 ) TVC 0 = lim t! (β ) t E 0 [V 1 (k et, z t ) k et ] = lim t! (β ) t E 0 [ũ 1 (c et, 1 l t ) [z t F 1 (k et, l t ) + 1 δ] k et ] Katsuya Takii (Institute) Modern Macroeconomics II 282 / 461

283 Stochastic Growth Model Summary Dynamics (c et, l t, k et, z t ) Intratemporal substitution between leisure and consumption Euler Equation and TVC = ũ 2 (c et, 1 l t ) ũ 1 (c et, 1 l t ) = z tf 2 (k et, l t ) ũ 1 (c et, 1 l t ) Z βũ 1 (c et+1, 1 l t+1 ) [z t+1f 1 (k et+1, l t+1 ) + 1 δ] (1 + g) θ dq (ε t+1 ) 0 = lim t! (β ) t E 0 [ũ 1 (c et, l t ) [z t F 1 (k et, 1 l t ) + 1 δ] k et ] where β = β (1 + n) (1 + g) (1 θ). Katsuya Takii (Institute) Modern Macroeconomics II 283 / 461

284 Stochastic Growth Model Summary Dynamics (c et, l t, k et, z t ) Capital accumulation and the dynamics of productivity shocks. k et+1 = z tf (k et, l t ) + (1 δ) k et c et. k e0 is given (1 + g) (1 + n) ln z t+1 = ζ ln z t + ε t+1, z 0 is given, ε t+1 ~Q (ε t+1 ) Katsuya Takii (Institute) Modern Macroeconomics II 284 / 461

285 Numerical Analysis With their seminal analysis of business cycle, Kidland and Prescott (1982) develops a new method of empirical work, calibration, in macroeconomics. In order to understand its spirits, let Kidland and Prescott (1996) speak how computational experiment can be implemented, in particular with a stochastic growth model. According to Kidland and Prescott (1996), an economic computational experiment has four steps. 1 Pose a Question 2 Use a well-tested theory 3 Construct a model economy 4 Calibrate the model economy 5 Run the experiment Katsuya Takii (Institute) Modern Macroeconomics II 285 / 461

286 Numerical Analysis Pose a question: The purpose of a computational experiment is to derive a quantitative answer to some well-posed question. Thus, the rst step in carrying out a computational experiment is to pose such a question. Kidland and Prescott (1996) Question may involve 1 Assessments of the theoretical implications of changes in policy. 2 the assessments of the ability of a model mimic features of the actual economy. In case of a basic real business cycle model, Kidland and Prescott (1996) poses Does a model designed to be consistent with long-term economic growth produce the sort of uctuation that we associate with the business cycle? How much would the U.S. postwar economy have uctuated if technology shocks had been the only source of uctuations? Katsuya Takii (Institute) Modern Macroeconomics II 286 / 461

287 Numerical Analysis Use a well-tested theory: With a particular question in mind, a researcher needs some strong theory to carry out a computational experiment: that is, a researcher needs a theory that has been tested through use and found to provide reliable answers to a class of questions. Here, by theory we do not mean a set of assertions about the actual economy. Rather, following Lucas (1980), economic theory is de ned to be an explicit set of instructions for building... a mechanical imitation system to answer a question Kidland and Prescott (1996) The basic theory used in the modern study of business cycles is the neoclassical growth model. Kidland and Prescott (1996) Katsuya Takii (Institute) Modern Macroeconomics II 287 / 461

288 Numerical Analysis Construct a Model Economy: With a particular theoretical framework in mind, the third step in conducting a computational experiment is to construct a model economy. Here, key issues are the amount of detail and the feasibility of computing the equilibrium process. Kidland and Prescott (1996)..an abstraction can be judged only relative to some given question. To criticize or reject a model because it is an abstraction is foolish: all models are necessarily abstractions. A model environment must be selected based on the question being addressed. Kidland and Prescott (1996) The selection and construction of a particular model economy should not depend on th answer provided. In fact, searching within some parametric class of economies for the one that best ts a set of aggregate time series makes little sense, because it isn t likely to answer an interesting question. Kidland and Prescott (1996) Katsuya Takii (Institute) Modern Macroeconomics II 288 / 461

289 Numerical Analysis In case of a basic real business cycle model, they add a productivity shock and the choice of leisure to the neoclassical growth model because two-thirds of aggregate output are attributable to uctuations of labor. Katsuya Takii (Institute) Modern Macroeconomics II 289 / 461

290 Numerical Analysis Calibrate the Model Economy: Now that a model has been constructed, the fourth step in carrying out a computational experiment is to calibrate that model. Originally, in physical sciences, calibration referred to the graduation of measuring instrument. Kidland and Prescott (1996) Generally, some economic questions have known answers, and the model should give an approximately correct answer to them if we are to have any con dence in the answer given to the question with unknown answer. Thus, data are used to calibrate the model economy so that it mimics the world as closely as possible along a limited, but clearly speci ed, number of dimensions. Kidland and Prescott (1996) Katsuya Takii (Institute) Modern Macroeconomics II 290 / 461

291 Numerical Analysis Calibrate the Model Economy: In case of a standard business cycle model, 1 the parameters must meet the long run averages of certain statistics such as the share of output paid to labor and the fraction of available hours worked per household, both of which have been remarkably stable over time. 2 In addition, empirical results obtained in micro studies conducted at the individual or household level are often used as a means of pinning down certain parameters. Katsuya Takii (Institute) Modern Macroeconomics II 291 / 461

292 Numerical Analysis Calibrate the Model Economy: The di erence between estimation and calibration is also emphasized. Note that calibration is not an attempt at assessing the size of something: it is not estimation, It is important to emphasize that the parameter values selected are not the ones that provide the best t in some statistical sense. In some cases, the presence of a particular discrepancy between the data and the model economy is a test of the theory being used. Kidland and Prescott (1996) Prescott (1986) also claims that The models constructed within this theoretical framework are necessarily highly abstract. Consequently, they are necessarily false, and statistical hypothesis testing will reject them. This does not imply, however, that nothing can be learned from such quantitative theoretical exercises. Katsuya Takii (Institute) Modern Macroeconomics II 292 / 461

293 Numerical Analysis Run the Experiment: The fth and nal step in conducting a computational experiment is to run the experiment. Kidland and Prescott (1996) If the model economy has no aggregate uncertainty, then it is simply a matter of comparing the equilibrium path of the model economy with the path of the actual economy. If the model economy has aggregate uncertainty, rst a set of statistics summarize relevant aspects of the behavior of the actual economy is selected. Then the computational experiment is used to generate many independent realizations of the equilibrium process for the model economy. In this way, the sampling distribution of this set of statistics can be determined to any degree of accuracy for the model economy with the value of the set of statistics for the actual economy. Kidland and Prescott (1996) A standard business cycle exercise typically compares the second moment such as the variance and the covariance of variables. Katsuya Takii (Institute) Modern Macroeconomics II 293 / 461

294 Numerical Analysis Cooley and Prescott (1995) describes the process of calibration more precisely. They identify three steps. 1 The rst step is to restrict a model to a parametric class. 2 to construct a set of measurements that are consistent with the parametric class of models. 3 The third step is to assign values to the parameters of our models. Let us conduct these steps using our models. Katsuya Takii (Institute) Modern Macroeconomics II 294 / 461

295 Numerical Analysis Restricting the model: we need to restrict technology and preference. We have already restricting a part of our model to meet the balanced growth path: ũ (c et, 1 l t ) = [c et v (1 l t )] 1 θ 1 θ if θ 6= 1, ũ (c et, l t ) = ln c et + ln v (1 l t ) if θ = 1. Cooley and Prescott (1995) assumes that θ = 1 and ln v (1 l) = η ln (1 l). Therefore, ũ (c et, 1 l t ) = ln c et + η ln (1 l t ). Knowing that labor share and capital share is stable, it is common to assume F (k e, l) = ke α l 1 α. Because it is known that α = rk Y = 1 wl Y, even if the economy is on the transition path, the share must be stable. We assume that Q (ε t+1 ) is normally distributed with the mean 0 and the standard deviation of σ ε. Katsuya Takii (Institute) Modern Macroeconomics II 295 / 461

296 Numerical Analysis Note that ũ (c et, 1 l t ) = ln c et + η ln (1 l t ) and y et = z t F (k e, l) = z t ke α l 1 α implies ũ 1 (c et, 1 l t ) = 1 c et, ũ 2 (c et, 1 l t ) = η 1 l t, z t F 1 (k et, l t ) = αz t ket α 1 lt 1 α = α y et k et and z t F 1 (k et, l t ) = (1 α) z t ketl α α t = (1 α) y et l t. Therefore, our economy is summarized by the following equations Intratemporal substitution between leisure and consumption where y et = z t k α etl 1 ũ 2 (c et, 1 l t ) = z t F 2 (k et, l t ) ũ 1 (c et, 1 l t ) ηc et = (1 α) y et 1 l t l t t α. Katsuya Takii (Institute) Modern Macroeconomics II 296 / 461

297 Numerical Analysis Euler Equation and TVC = 1 c et = ũ 1 (c et, 1 l t ) Z βũ 1 (c et+1, 1 l t+1 ) [z t+1f 1 (k et+1, l t+1 ) + 1 δ] (1 + g) θ dq (ε t+1 ) h i Z β α y et+1 k et δ dq (ε t+1 ) 1 + g c et+1 0 = lim (β ) t E 0 [ũ 1 (c et, l t ) [z t F 1 (k et, 1 l t ) + 1 δ] k et ] t! 2 h i 3 α y et = lim [β (1 + n)] t k E 0 4 et + 1 δ k et 5 t! where y et = z t k α etl 1 t α. Katsuya Takii (Institute) Modern Macroeconomics II 297 / 461 c et

298 Numerical Analysis Capital accumulation and the dynamics of productivity shocks. k et+1 = y et + (1 δ) k et c et. k e0 is given (1 + g) (1 + n) ln z t+1 = ζ ln z t + ε t+1, z 0 is given, ε t+1 ~N 0, σ 2 ε where y et = z t k α etl 1 t α. Katsuya Takii (Institute) Modern Macroeconomics II 298 / 461

299 Numerical Analysis De ning Consistent Measurements: Cooley and Prescott (1995) was a quite careful about the measurement of capital stock, and output. Because the model is abstract, there is no treatment of government sector, household production, foreign sector and inventory. Therefore, the model economy s capital K includes capital used in all these sectors plus the stock of inventory. Similarly, output includes the output productivity by all of this capital. They did some imputation. I discuss how they impute the values. 1 Estimate the return on asset ρ t from private capital stock. 2 They estimate depreciation rate δ i separately for consumer durables and government sector. 3 Using the estimated ρ and δ i, they estimate the service ow for consumer durables and government sector, r i K i t = (ρ t + δ i ) K i t Katsuya Takii (Institute) Modern Macroeconomics II 299 / 461

300 Numerical Analysis They estimate the return on asset ρ t from private capital stock. Note that ρ t = rk p K p δk p where K p is private capital stock. K p is estimated by (1) the net stock of xed reproducible private capital (not including the stock of consumer durables) + (2) the stock of inventories and (3) the stock of land where (1) is from Musgrave (1992), (2) is from the NIPA and (3) is from the Flow of Funds Accounts. rk p is estimated by unambiguous capital income + α p ambiguous capital income +δk where α p is estimated to satisfy α p = the estimated rk p GDP,.δK is estimated by GNP-NNP (consumption of xed capital), unambiguous capital income is the sum of rental income, corporate pro ts and net interest, and ambiguous capital income is sum of proprietors income + net national product - national income. Katsuya Takii (Institute) Modern Macroeconomics II 300 / 461

301 Numerical Analysis Next they estimate depreciation rate for government sector and consumer durable δ i by the following equation Kt+1 i Y t+1 = (1 δ i ) K t i + I t i Y t+1 Y t Y t where i is consumer durables or government sector and Y t is GNP. For consumer durables, It i is consumption of consumer durable as reported in the NIPA. For government investment It i is also from NIPA. The capital stock Kt i for consumer durable and government investment are taken from Musgrave (1992). Finally, they estimate the service ows of capital from the following equation. r i K i t = (ρ t + δ i ) K i t Y t Katsuya Takii (Institute) Modern Macroeconomics II 301 / 461

302 Numerical Analysis Calibrating a Speci c Model Economy: Remember that our economy is summarized by 4 equations with suitable boundary conditions ηc et 1 l t = (1 α) y et 1 c et = Z β c et+1 l t h α y et+1 k et g i δ dq (ε t+1 ) k et+1 = y et + (1 δ) k et c et, given k e0 (1 + g) (1 + n) ln z t+1 = ζ ln z t + ε t, given z 0 ε t+1 ~N 0, σ 2 ε where y et = z t ketl α t 1 α and 0 = lim t! [β (1 + n)] t [α yet ket E +1 δ]k et 0 c et. Note that y et, c et and k et are interpreted as the detrended GNP, Consumption and Capital per capita. We need to calibrate α, β, δ, η, n, g, ζ and σ ε. Katsuya Takii (Institute) Modern Macroeconomics II 302 / 461

303 Numerical Analysis g = measured by the time series average of rate of growth of real GNP per capita (On the balanced growth, g Y = g). N n = average population growth. α = capital share is calibrated by the time series average of α = r tk t Y t = r pk p t + r c K c t + r g K g t GNP + r c K c t + r g K g t Katsuya Takii (Institute) Modern Macroeconomics II 303 / 461

304 Numerical Analysis ζ = 0.95 and σ ε = : Note that It means that ln z t = ln Y t a ln K t (1 α) ln l t T t N t ln z t+1 ln z t = (ln Y t+1 ln Y t ) a (ln K t+1 ln K t ) (1 α) (ln l t+1 N t+1 ln l t N t ) (1 α) (ln T t+1 ln T t ) (ln Y t+1 ln Y t ) a (ln K t+1 K t ) (1 α) (ln l t+1 N t+1 ln l t N t ) (1 α) g So given α and g we can generate a series of ln z t+1 ln z t and therefore ln z t given ln z 0. (Cooley and Prescott (1995) actually report that they measure it under the assumption of g = 0, though.) These sequences are used to calibrate ζ and σ ε of ln z t+1 = ζ ln z t + ε t+1, ε t+1 ~N 0, σ 2 ε Katsuya Takii (Institute) Modern Macroeconomics II 304 / 461

305 Numerical Analysis In order to calibrate the remaining parameters β, δ, and η. more theory and stylized facts on the balanced growth. δ = : On the steady state, capital accumulation equation implies k e = y e + (1 δ) k e c e (1 + g) (1 + n) (1 + g) (1 + n) k e = (1 δ) k e + i e (1 δ) k e = (1 + g) (1 + n) k e i e We use 1 δ = (1 + g) (1 + n) ie ke δ = I K + 1 (1 + g) (1 + n) Given the measured values of g and n, Cooley and Prescott (1995) calibrate δ using this equation. Katsuya Takii (Institute) Modern Macroeconomics II 305 / 461

306 Numerical Analysis β = 0.987: If there is no uncertainty, Euler equation becomes On the steady state, 1 c et = β c et+1 h α y et+1 k et y e 1 + g 1 β = α k + e 1 δ 1 + g = α Y K + 1 δ 1 + g Given the measured value of δ, Cooley and Prescott (1995) calibrate β using this equation. i δ Katsuya Takii (Institute) Modern Macroeconomics II 306 / 461

307 Numerical Analysis η = 1.78: Intratemporal substitution between consumption and leisure implies ηc et = (1 α) y et 1 l t l t η = (1 α) y et 1 c et l t l t = (1 α) Y t 1 l t. C t l t Given the knowledge of α, Cooley and Prescott (1995) calibrate η using this equation. For l t, they use micro evidence from time allocation studies by Ghez and Becker (1975) and Juster and Sta ord (1991) that shows one third of their discretionary time is allocated to market activities. Katsuya Takii (Institute) Modern Macroeconomics II 307 / 461

308 Numerical Analysis Final task is to run experiment. For this purpose, we need to compute an equilibrium. Note that our economy is summarized by 4 equations with suitable boundary conditions ηc et 1 l t = (1 α) y et 1 c et = Z β c et+1 l t h α y et+1 k et g i δ dq (ε t+1 ) k et+1 = y et + (1 δ) k et c et, given k e0 (1 + g) (1 + n) ln z t+1 = ζ ln z t + ε t+1, given z 0 ε t+1 ~N 0, σ 2 ε where y et = z t k α etl 1 t α and 0 = lim t! [β (1 + n)] t E 0 [α yet ket +1 δ]k et c et. Katsuya Takii (Institute) Modern Macroeconomics II 308 / 461

309 Numerical Analysis If the set of possible states of nature were small, one could nd a numerical value function and policy function. The advantages of this method would be 1 It would preserve the curvature of the model. 2 It would allow study of the model in an equilibrium that is far from the steady sate. If the set of possible states of nature were large, we need to rely on a approximation of the model. Katsuya Takii (Institute) Modern Macroeconomics II 309 / 461

310 Numerical Analysis One way to approximate model is to conduct the approximation of the derived dynamics around the steady state. This method has four step processes. 1 To nd the steady states without stochastic shocks. 2 To construct a approximation of the model around steady sate. 3 To derive the solution of the approximation of the model. 4 Analyze the solution. You can write your program to conduct this process using Matlab. Katsuya Takii (Institute) Modern Macroeconomics II 310 / 461

311 Dynare Dynare can greatly help writing this program. What is Dynare? Dynare is a powerful and highly customizable engine, with an intuitive front-end interface, to solve, simulate and estimate DSGE models. (Dynare User Guide)..., it is a pre-processor, and a collection of Matlab routines that has the great advantages of reading DSGE model equations written almost as in an academic paper. (Dynare User Guide) Basically, the model and its related attributes, like a shock structure for instance, is written equation by equation in an editor of your choice. The resulting le will be called the.mod le. This le is the called from Matlab. This initiates for the Matlab routines used to either solve or estimate the model. (Dynare User Guide) Katsuya Takii (Institute) Modern Macroeconomics II 311 / 461

312 Dynare Installing Dynare 1 Go to 2 Download Dynare 4.1 to your holder at the network center. 3 It is also useful to download the user guide from Katsuya Takii (Institute) Modern Macroeconomics II 312 / 461

313 Dynare The Procedure: 1 You write a lename.mod le, which contains the essential information on your model. Then, using this information, Dynare constructs a lename.m le that Matlab can then run. 2 Save your lename.mod le somewhere that Matlab can nd it. 3 In Matlab, dynare lename 1 Set a path to Dynare folder including sub-folder. 2 Change the directory to the folder that contains lename.mod. 3 Type Katsuya Takii (Institute) Modern Macroeconomics II 313 / 461

314 Dynare Basic Instructions for the program by Dynare. 1 Each instruction must be terminated by a semicolon, ;. 2 You can comment out any line by starting the line with two forward slashes, //, or comment out an entire section by starting with /* and ending with */. Katsuya Takii (Institute) Modern Macroeconomics II 314 / 461

315 Dynare The program contains the following 4 blocks. 1 Preamble: lists variables and parameters. 2 Declaration of the model: spells out the model. 3 Set initial conditions: Specify initial conditions and gives indication to nd the steady state of a model. 4 Specifying Shocks: de nes the shocks to the system. 5 Computation: instructs Dynare to undertake speci c operations (e.g. forecasting, estimating impulse response functions, dx t+j d ε t where x t+j = c et+j, y et+j, k et+j, l and ln z t+j and providing various descriptive statistics.) Katsuya Takii (Institute) Modern Macroeconomics II 315 / 461

316 Dynare Preamble consists of the some setup information: the endogenous and exogenous variables, the parameters and assign values to these parameters. 1 var speci es the lists of endogenous variables, to be separated by commas. In our example, endogenous variables are detrended consumption per capita c e, detrended capital per capita k e, detrended GNP per capita (y e ), hours of work (l), and the technology (z). 2 varexo speci es the list of exogenous variables that will be chocked. In our example, an exogenous variable is a stochastic term, ε. 3 parameters speci es the list of parameters of the model and assigns values: Katsuya Takii (Institute) Modern Macroeconomics II 316 / 461

317 Dynare var c, k, l, y, lnz; varexo epsilon; parameters beta, alpha, eta, delta, g, n, zeta; beta=0.987; alpha=0.4; eta=1.78; delta=0.012; g=0.0156; n=0.012; zeta=0.95; Katsuya Takii (Institute) Modern Macroeconomics II 317 / 461

318 Dynare Declaration of the model: Now it is time to specify the model itself. You must write model; describing your model; end; Katsuya Takii (Institute) Modern Macroeconomics II 318 / 461

319 Dynare There are several rules for describing your model. 1 There need to be as many equation as you declared endogenous variables. 2 Time subscripts: "x" means x t, "x(-n)" means x t n and "x(+n)" means x t+n. 3 In dynare, the timing of each variable re ects when that variable is decided. Because state variables are in general determined at past, a transition equation should be written as S t = G (X t 1, S t 1, ε t ), but not S t+1 = G (X t, S t, ε t+1 ). On the other hand, the variable shown as x (+1) tells Dynare to count that variable as a forward-looking variable. These distinctions are important to insure saddle point stability. 4 You can ignore the expectation terms when you write a program. Katsuya Takii (Institute) Modern Macroeconomics II 319 / 461

320 Dynare Remember that our economy is summarized by the following 5 equations. ηc et 1 l t = (1 α) y et 1 c et = Z β c et+1 l t h α y et+1 k et g i δ dq (ε t+1 ) k et+1 = y et + (1 δ) k et c et, given k e0 (1 + g) (1 + n) ln z t+1 = ζ ln z t + ε t, given z 0 ε t+1 ~N 0, σ 2 ε y et = z t ketl α t 1 α and 0 = lim t! [β (1 + n)] t E 0 [α yet ket +1 δ]k et c et. Katsuya Takii (Institute) Modern Macroeconomics II 320 / 461

321 Dynare model; (eta*c)/(1-l)=(1-alpha)*(y/l); 1/c=(beta/c(+1))*((alpha*(y(+1)/k)+1-delta)/(1+g)); k=(y+(1-delta)*k(-1)-c)/((1+g)*(1+n)); lnz=zeta*lnz(-1)+epsilon; y=exp(lnz)*(k(-1)^alpha)*(l^(1-alpha)); end; Katsuya Takii (Institute) Modern Macroeconomics II 321 / 461

322 Dynare Set initial conditions: Next, you must tell Dynare initial conditions. initval; Listing the initial conditions end; Dynare can help nding your model s steady sate by calling the appropriate Matlab functions. But it is usually only successful if the initial values you entered are closed to the true steady state. Hence it is good idea that you can solve your steady state and nd the steady state values by yourself. Katsuya Takii (Institute) Modern Macroeconomics II 322 / 461

323 Dynare Assignment: Compute the steady state value of (c e, y e, k e, l) when β = 0.987, α = 0.4, η = 1.78, δ = 0.012, g = , n = 0.012, and ζ = 0.95 using Matlab. You have Two ways of doing so. Choose one of them. 1 Derive an analytical solution of (c e, y e, k e, l) on the steady state when σ 2 ε = 0. Then compute (c e, y e, k e, l) using Matlab or Excel. 2 Use the command fsolve and function in Matlab to nd the steady state value of (c e, y e, k e, l) when σ 2 ε = 0. Katsuya Takii (Institute) Modern Macroeconomics II 323 / 461

324 Dynare steady; In addition, if you start your analysis from your steady state, you can add Then Dynare recognizes that your initial conditions are just approximations and you wish to start your analysis from your steady sate. Katsuya Takii (Institute) Modern Macroeconomics II 324 / 461

325 Dynare initval; k= ; c= ; l= ; y= ; lnz=0; epsilon=0; end; steady; Katsuya Takii (Institute) Modern Macroeconomics II 325 / 461

326 Dynare Adding Shocks: We then specify the innovations and their matrix of variance covariance. This is done by writing: shocks; var [the name of stochastic term] =[number]; end; In our program, shocks; var epsilon=0.0007^2; end; Katsuya Takii (Institute) Modern Macroeconomics II 326 / 461

327 Dynare Computation: Finally, you tell Dynare that you are done by typing stoch_simul; This commands instructs Dynare to compute a Taylor approximation of the decision and transition function for the model, impulse response functions, dx t+j d ε t where x t+j = c et+j, y et+j, k et+j, l and ln z t+j, and various descriptive statistics (moments, variance decomposition, correlation and autocorrelation coe cients.) There are several options which you can learn later. For example, by default, Dynare drops the rst 100 values, but you can change that using the appropriate option. Katsuya Takii (Institute) Modern Macroeconomics II 327 / 461

328 Dynare Katsuya Takii (Institute) Modern Macroeconomics II 328 / 461

329 Dynare Katsuya Takii (Institute) Modern Macroeconomics II 329 / 461

330 Dynare Katsuya Takii (Institute) Modern Macroeconomics II 330 / 461

331 Dynare Katsuya Takii (Institute) Modern Macroeconomics II 331 / 461