Dynamic Stochastic General Equilibrium Models

Transcription

1 Dynamic Stochastic General Equilibrium Models Dr. Andrea Beccarini Willi Mutschler Summer 2012 A. Beccarini () DSGE Summer / 27

2 Note on Dynamic Programming TheStandardRecursiveMethod One considers a representative agent whose objective is to nd a control (or decision) sequence fu t gt=0 such that E 0 " t=0β t U(x t, u t ) # (1) is maximised, subject to x t+1 = F (x t, u t, z t ) (2) A. Beccarini (CQE) DSGE Summer / 27

3 where x t is a vector of m state variables at period t; u t is a vector of n control variables; z t is a vector of s exogenous variables whose dynamics does not depend on x t and u t ; β 2 (0, 1) denotes the discount factor. The uncertainty of the model only comes from the exogenous z t. Assume that z t follow an AR(1) process: z t+1 = a + bz t + ɛ t+1 (3) where ɛ is independently and identically distributed (i.i.d.). The initial condition (x 0, z 0 ) in this formulation is assumed to be given. A. Beccarini (CQE) DSGE Summer / 27

4 The problem to solve a dynamic decision problem is to seek a time-invariant policy function G mapping from the state and exogenous (x, z) into the control u. u t = G (x t, z t ) (4) With such a policy function (or control equation), the sequences of state fx t gt=0 and control fu tgt=0 can be generated by iterating the control equation as well as the state eq. (2), given the initial condition (x 0, z 0 ) and the exogenous sequence fz t gt=0 generated by eq. (3). A. Beccarini (CQE) DSGE Summer / 27

5 To nd the policy function G by the recursive method, one rst de nes a value function V: " V (x 0, z 0 ) = max 0 β fu t gt=0e t U(x t, u t ) t=0 # (5) Expression (5) could be transformed to unveil its recursive structure. A. Beccarini (CQE) DSGE Summer / 27

6 For this purpose, one rst rewrites eq. (5) as follows: V (x 0, z 0 ) = max fu t g t=0 ( U(x 0, u 0 ) + βe 0 " t=0β t U(x t+1, u t+1 ) #) (6) It is easy to nd that the second term in eq. (6) can be expressed as being β times the value V as de ned in eq. (5) with the initial condition (x 1, z 1 ). Therefore, we could rewrite eq. (5) as V (x 0, z 0 ) = max fu(x 0, u 0 ) + βe 0 [V (x 1, z 1 )]g (7) fu t gt=0 A. Beccarini (CQE) DSGE Summer / 27

7 The formulation of eq. (7) represents a dynamic programming problem which highlights the recursive structure of the decision problem. A. Beccarini (CQE) DSGE Summer / 27

8 The formulation of eq. (7) represents a dynamic programming problem which highlights the recursive structure of the decision problem. In every period t, the planner faces the same decision problem. A. Beccarini (CQE) DSGE Summer / 27

9 The formulation of eq. (7) represents a dynamic programming problem which highlights the recursive structure of the decision problem. In every period t, the planner faces the same decision problem. Choosing the control variable u t that maximizes the current return plus the discounted value of the optimum plan from period t + 1 onwards. A. Beccarini (CQE) DSGE Summer / 27

10 The formulation of eq. (7) represents a dynamic programming problem which highlights the recursive structure of the decision problem. In every period t, the planner faces the same decision problem. Choosing the control variable u t that maximizes the current return plus the discounted value of the optimum plan from period t + 1 onwards. Since the problem repeats itself every period the time subscripts become irrelevant. A. Beccarini (CQE) DSGE Summer / 27

11 One thus can write eq. (7) as V (x, z) = max u h nu(x, u) + βe 0 V ( x, io z ) (8) where the tilde () over x and z denotes the corresponding next period values. They are subject to eq. (2) and (3). Eq. (8) is said to be the Bellman equation, named after Richard Bellman (1957). If one knows the function V, we then can solve u via the Bellman equation but in generally, this is not the case. A. Beccarini (CQE) DSGE Summer / 27

12 The typical method in this case is to construct a sequence of value functions by iterating the following equation: h V j+1 (x, z) = max nu(x, u) + βe 0 V j ( x, io z ) (9) u In terms of an algorithm, the method can be described as follows: A. Beccarini (CQE) DSGE Summer / 27

13 Step 1 Guess a di erentiable and concave candidate value function, V j A. Beccarini (CQE) DSGE Summer / 27

14 Step 1 Guess a di erentiable and concave candidate value function, V j Step 2. Use the Bellman equation to nd the optimum u and then compute V j+1 according to eq. (9). A. Beccarini (CQE) DSGE Summer / 27

15 Step 1 Guess a di erentiable and concave candidate value function, V j Step 2. Use the Bellman equation to nd the optimum u and then compute V j+1 according to eq. (9). Step 3. If V j+1 = V j, stop. Otherwise, update V j and go to step 1. A. Beccarini (CQE) DSGE Summer / 27

16 Under some regularity conditions regarding the function U and F, the convergence of this algorithm is warranted. However, the di culty of this algorithm is that in each Step 2, we need to nd the optimum u that maximize the right side of eq. (9). This task makes it di cult to write a closed form algorithm for iterating the Bellman equation. A. Beccarini (CQE) DSGE Summer / 27

17 The Euler Equation Alternatevely, one starts from the Bellman equation (8). The rst-order condition for maximizing the right side of the equation takes the form: U(x, u) u + βe " F (x, u, z) u V ( x, # z ) = 0 (10) x The objective here is to nd V / x. Assume V is di erentiable and thus from eq. (8) and exploiting eq. (10) it satis es V (x, z) x = U(x, G (x, z)) x + βe " F (x, G (x, z), z) x V ( x, # z ) F (11) This equation is often called the Benveniste-Scheinkman formula. A. Beccarini (CQE) DSGE Summer / 27

18 Assume F / x = 0. The above formula becomes V (x, z) x = U(x, G (x, z)) x (12) Substituting this formula into eq (10) gives rise to the Euler equation: " U(x, u) F (x, u, z) U( x, # z ) + βe u u = 0 x In economic analysis, one often encounters models, after some transformation, in which x does not appear in the transition law so that F / x = 0 is satis ed. However, there are still models in which such transformation is not feasible. A. Beccarini (CQE) DSGE Summer / 27

19 Deriving the First-Order Condition from the Lagrangian Suppose for the dynamic optimization problem as represented by eq.(1) and (2), we can de ne the Lagrangian L: L = E 0 ( t=0β t U(x t, u t ) β t+1 λ 0 t+1 [x t+1 F (x t, u t, z t )] ) (13) where λ t, the Lagrangian multiplier, is a m 1 vector. A. Beccarini (CQE) DSGE Summer / 27

20 Setting the partial derivatives of L to zero with respect to λ t, x t and u t will yield eq. (2) as well as U(x t, u t ) F (xt+1, u t+1, z t+1 ) + βe t λ t+1 = λ t x t x t+1 U(x t, u t ) u t + E t λ t+1 F (x t, u t, z t) u t = 0 (14) In comparison with the Euler equation, we nd that there is an unobservable variable λ t appearing in the system. Yet, using eq. (13) and (14), one does not have to transform the model into the setting that F / x = 0. This is an important advantage over the Euler equation. A. Beccarini (CQE) DSGE Summer / 27

21 The Log-linear Approximation Method Solving nonlinear dynamic optimization model with log-linear approximation has been proposed in particular by King et al. (1988) and Campbell (1994) in the context of Real Business Cycle models. In principle, this approximation method can be applied to the rst-order condition either in terms of the Euler equation or derived from the Langrangean. _ X Formally, let X t be the variables, the corresponding steady state. Then, _ x t lnx t lnx is regarded to be the log-deviation of X t. In particular, 100x t is the _ percentage of X t that it deviates from X. A. Beccarini (CQE) DSGE Summer / 27

22 The general idea of this method is to replace all the necessary equations in the model by approximations, which are linear in the log-deviation form. A. Beccarini (CQE) DSGE Summer / 27

23 The general idea of this method is to replace all the necessary equations in the model by approximations, which are linear in the log-deviation form. Given the approximate log-linear system, one then uses the method of undetermined coe cients to solve for the decision rule, which is also in the form of log-linear deviations. A. Beccarini (CQE) DSGE Summer / 27

24 The general idea of this method is to replace all the necessary equations in the model by approximations, which are linear in the log-deviation form. Given the approximate log-linear system, one then uses the method of undetermined coe cients to solve for the decision rule, which is also in the form of log-linear deviations. The general procedure involves the following steps: A. Beccarini (CQE) DSGE Summer / 27

25 The general idea of this method is to replace all the necessary equations in the model by approximations, which are linear in the log-deviation form. Given the approximate log-linear system, one then uses the method of undetermined coe cients to solve for the decision rule, which is also in the form of log-linear deviations. The general procedure involves the following steps: Step 1. Find the necessary equations characterizing the equilibrium law of motion of the system. These necessary equations should include the state equation (2), the exogenous equation (3) and the rst-order condition derived either as Euler equation (??) or from the Lagrangian (13) and (14). A. Beccarini (CQE) DSGE Summer / 27

26 The general idea of this method is to replace all the necessary equations in the model by approximations, which are linear in the log-deviation form. Given the approximate log-linear system, one then uses the method of undetermined coe cients to solve for the decision rule, which is also in the form of log-linear deviations. The general procedure involves the following steps: Step 1. Find the necessary equations characterizing the equilibrium law of motion of the system. These necessary equations should include the state equation (2), the exogenous equation (3) and the rst-order condition derived either as Euler equation (??) or from the Lagrangian (13) and (14). Step 2. Derive the steady state of the model. This requires rst parameterizing the model and then evaluating the model at its certainty equivalence form. A. Beccarini (CQE) DSGE Summer / 27

27 The general idea of this method is to replace all the necessary equations in the model by approximations, which are linear in the log-deviation form. Given the approximate log-linear system, one then uses the method of undetermined coe cients to solve for the decision rule, which is also in the form of log-linear deviations. The general procedure involves the following steps: Step 1. Find the necessary equations characterizing the equilibrium law of motion of the system. These necessary equations should include the state equation (2), the exogenous equation (3) and the rst-order condition derived either as Euler equation (??) or from the Lagrangian (13) and (14). Step 2. Derive the steady state of the model. This requires rst parameterizing the model and then evaluating the model at its certainty equivalence form. Step 3. Log-linearize the necessary equations characterizing the equilibrium law of motion of the system A. Beccarini (CQE) DSGE Summer / 27

28 The following building block for such log-linearization can be used: X t _ X e x t (15) e x t +ay t 1 + x t + ay t (16) x t y t 0 (17) A. Beccarini (CQE) DSGE Summer / 27

29 The following building block for such log-linearization can be used: X t _ X e x t (15) e x t +ay t 1 + x t + ay t (16) x t y t 0 (17) Step 4. Solve the log-linearized system for the decision rule (which is also in log-linear form) with the method of undetermined coe cients. A. Beccarini (CQE) DSGE Summer / 27

30 The Ramsey Problem The Model Ramsey (1928) posed a problem of optimal resource allocation, which is now often used as a prototype model of dynamic optimization. Let C t denote consumption, Y t output and K t the capital stock. Assume that the output is produced by capital stock and it is either consumed or invested, that is, added to the capital stock. Formally, Y t = A t K α t (18) Y t = C t + K t+1 K t (19) A. Beccarini (CQE) DSGE Summer / 27

31 where α 2 (0, 1) and A t is the technology which may follow an AR(1) process: A t+1 = a 0 + a 1 A t + ɛ t+1 (20) Assume ɛ t to be i.i.d. Equation (18) and (19) indicates that we could write the transition law of capital stock as K t+1 = A t K α t C t (21) Note that one has assumed here that the depreciation rate of capital stock is equal to 1. This is a simpli ed assumption by which the exact solution is computable. A. Beccarini (CQE) DSGE Summer / 27

32 The representative agent is assumed to nd the control sequence fc t gt=0 such that " # maxe 0 β t ln C t t=0 (22) given the initial condition (K 0, A 0 ). A. Beccarini (CQE) DSGE Summer / 27

33 The Ramsey Problem The Euler Equation The rst task is to transform the model into a setting so that the state variable K t does not appear in F (). This can be done by assuming K t+1 (instead of C t ) as model s decision variable. To achieve a notational consistency in the time subscript, one may denote the decision variable as Z t. Therefore the model can be rewritten as maxe 0 " t=0β t ln A t K α t K t+1# (23) subject to K t+1 = Z t A. Beccarini (CQE) DSGE Summer / 27

34 Note that here one has used (21) to express C t in the utility function. Also note that in this formulation the state variable in period t is still K t. Therefore F / x = 0 and F / u = 1. The Bellman equation in this case can be written as V (K t, A t ) = max K t+1 fln(a t K α t K t+1 ) + βe t [V (K t+1, A t+1 )]g (24) The necessary condition for maximizing the right hand side of the Bellman equation (2.12) is given by 1 V (Kt+1, A t+1 ) A t Kt α + βe t = 0 (25) K t+1 K t+1 A. Beccarini (CQE) DSGE Summer / 27

35 Meanwhile applying the Benveniste-Scheinkman formula, V (K t, A t ) = αa tk α 1 t K t A t Kt α (26) K t+1 Substituting eq. (26) into (25) allows us to obtain the Euler equation: " # 1 αa t+1 Kt+1 α 1 A t Kt α + βe t K t+1 A t+1 Kt+1 α = 0 K t+2 which can further be written as " # 1 αa t+1 Kt+1 α 1 + βe t C t A t+1 Kt+1 α = 0 (27) K t+2 This Euler equations (27) along with (21) and (20) determine the transition sequences of {K t+1 } t=1, {A t+1} t=1 and {C t} t=0 given the initial condition K 0 and A 0. A. Beccarini (CQE) DSGE Summer / 27

36 The First-Order Condition Derived from the Lagrangian Next, derive the rst-order condition from the Lagrangian. De ne the Lagrangian: ( L = E 0 β t ln C t E t β t+1 λt+1(k 0 t+1 A t, Kt α + C t ) ) t=0 t=0 Setting to zero the derivatives of L with respect to λ t, C t and K t, one obtains (21) as well as 1/C t βe t λ t+1 = 0 (28) βe t λ t+1 αa t K α 1 t = λ t (29) These are the rst-order conditions derived from the Lagrangian. Substituting out for λ one obtains the Euler equation (27). A. Beccarini (CQE) DSGE Summer / 27

37 The Ramsey Problem The Exact Solution and the Steady States The exact solution for this model which could be derived from the standard methods described below can be written as K t+1 = αβa t K α t (30) This further implies from (21) that C t = (1 αβ)a t K α t (31) Given the solution paths for C t and K t+1, one is then able to derive the steady state. One steady state is on the boundary, that is K = 0 and C = 0. A. Beccarini (CQE) DSGE Summer / 27

38 To obtain a more meaningful interior steady state, we take logarithm for both sides of (30) and evaluate the equation at its certainty equivalence form: logk t+1 = log(αβa t ) + αlogk t (32) At the steady state, K t+1 = K t = _ K. Solving (32) for logk, we _ obtain logk = Therefore, log (αβa) 1 α. K = (αβa) 1/(1 α) (33) Given K, C is resolved from (21): _ C = _ A _ K α _ K (34) _ A. Beccarini (CQE) DSGE Summer / 27