Introduction to Numerical Methods

Transcription

1 Introduction to Numerical Methods Wouter J. Den Haan London School of Economics c by Wouter J. Den Haan

2 "D", "S", & "GE" Dynamic Stochastic General Equilibrium

3 What is missing in the abbreviation? DSGE models include some form of forward looking behavior Typically rational expectations Does forward looking makes these models diffi cult to solve? Yes. How the economy behaves today depends on how agents think the economy will behave in the future for all possible outcomes Agent-based models in which agents predict according to some of pre-specified rules are easier to solve

4 Recursive problems Recursive problem same state variables = same choices Numerous problems are not recursive Some non-recursive problems can be rewritten as a recursive model by adding state variables e.g., recursive contracts of Marcet & Marrimon

5 State variables are... the variables that determine the outcomes in period t predetermined variables like the capital stock realizations of exogenous variables, like the productivity shock not other endogenous variables to be determined within the period like prices Not always trivial to know what the state variables are

6 Example economy max {ct+j,k t+j } j=0 E t j=0 s.t. c1 ν t+j 1 βj 1 ν c t+j + k t+j = z t+j k α t+j 1 + (1 δ)k t+j 1 z t+j = (1 ρ) + ρz t+j 1 + ε t k t 1 given E t+j [ε t+j+1 ] = 0 & E t+j [ε 2 t+j+1 ] = σ2

7 Alternative notation c t + k t = z t k α t 1 + (1 δ)k t 1 k t is the end-of-period t capital stock, chosen in t and productive in period t + 1 It is also possible that k t stands for the beginning-of-period t capital stock. Then the budget constraint is written as c t + k t+1 = z t k α t + (1 δ)k t This is just a change in notation, not in the model

8 Bellman equation If the problem is recursive it can be rewritten using the Bellman equation v(k 1, z) = max c,k c 1 ν 1 1 ν + E [βv(k, z +1)] s.t. c + k = zk α 1 + (1 δ)k 1

9 First-order conditions c + k = zk 1 α + (1 δ)k 1 [ ( )] c ν = βe c ν +1 αz +1 k α δ

10 Solution have the following form: c = c(k 1, z) k = k(k 1, z) Why is it diffi cult to find these solutions?

11 What should the solutions satisfy? c(k 1, z) + k(k 1, z) = zk α 1 + (1 δ)k 1 [ ( )] c(k 1, z) ν = βe c(k(k 1, z), z +1 ) ν αz +1 k(k 1, z) α δ with z +1 = (1 ρ) + ρz + ε +1

12 Example with analytical solution If δ = ν = 1 then we know the analytical solution. It is or k t = αβ exp(z t )k a t 1 c t = (1 αβ) exp(z t )k a t 1 ln k t = ln(αβ) + α ln k t 1 + z t ln c t = ln(1 αβ) + α ln k t 1 + z t Solution does not depend on law of motion for z t. Why not?

13 State space We are going to approximate the policy functions c(k 1, z) and k(k 1, z) This requires specifying the domain That domain better be bounded = numerical techniques are applied to models that either have no growth or have no growth after transformation of variables

14 Different ways to numerically solve these models 1 Methods that focus on the first-order conditions 1 projection methods 2 perturbation methods 2 Methods that are based on the Bellman equation similar to projection methods often slower, but can handle more complex models (e.g. discontinuities)

15 References Den Haan, W.J., Dynamic optimization problems basic stuff on getting first-order conditions, transversality conditions, state variables, and dynamic programming Den Haan, W.J., Equilibrium models continuation but now using equilibrium infinite-horizon (mainly monetary models) and OLG models Ljungqvist, L. and T.J. Sargent, 2004, Recursive Macroeconomic Theory describe many DSGE models and their properties Stockey, N.L, R.E. Lucas Jr, with E.C. Prescott, 1989, Recursive methods in economic dynamics