Perturbation Methods I: Basic Results

Transcription

1 Perturbation Methods I: Basic Results (Lectures on Solution Methods for Economists V) Jesús Fernández-Villaverde 1 and Pablo Guerrón 2 March 19, University of Pennsylvania 2 Boston College

2 Introduction

3 Introduction Remember that we want to solve a functional equations of the form: H (d) = 0 for an unknown decision rule d. Perturbation solves the problem by specifying: n d n (x, θ) = θ i (x x 0 ) i i=0 We use implicit-function theorems to find coefficients θ i s. Inherently local approximation. Often good global properties. 1

4 Motivation Many complicated mathematical problems have: 1. either a particular case 2. or a related problem. that is easy to solve. Often, we can use the solution of the simpler problem as a building block of the general solution. Very successful in physics. Sometimes perturbation is known as asymptotic methods. 2

5 A simple example Imagine we want to compute 26 by hand. We do not remember how to do it. But, we note that 26 = = = = 5.1 Exact solution: 26 = More in general: x = y 2 (1 + ε) = y (1 + ε) y (1 + θ) Accuracy depends on how big ε is. 3

6 Applications in economics Judd and Guu (1993) showed how to apply it to economic problems. Recently, perturbation methods have been gaining much popularity. In particular, second- and third-order approximations are easy to compute and notably improve accuracy. Perturbation theory is the generalization of the well-known linearization strategy. Hence, we can use much of what we already know about linearization. 4

7 Regular versus singular perturbations Regular perturbation: a small change in the problem induces a small change in the solution. Singular perturbation: a small change in the problem induces a large change in the solution. Example: excess demand function. Most problems in economics involve regular perturbations. Sometimes, however, we can have singularities. Example: introducing a new asset in an incomplete market model. 5

8 References General: 1. A First Look at Perturbation Theory by James G. Simmonds and James E. Mann Jr. 2. Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory by Carl M. Bender, Steven A. Orszag. Economics: 1. Perturbation Methods for General Dynamic Stochastic Models by Hehui Jin and Kenneth Judd. 2. Perturbation Methods with Nonlinear Changes of Variables by Kenneth Judd. 3. A gentle introduction: Solving Dynamic General Equilibrium Models Using a Second-Order Approximation to the Policy Function by Martín Uribe and Stephanie Schmitt-Grohe. 6

9 An Economics Application

10 Stochastic neoclassical growth model max E 0 t=0 β t log c t s.t. c t + k t+1 = e zt kt α, t > 0 z t = ρz t 1 + σε t, ε t N (0, 1) Note: full depreciation. Equilibrium conditions: 1 1 = βe t αe zt+1 k α 1 t+1 c t c t+1 c t + k t+1 = e zt k α t z t = ρz t 1 + σε t 7

11 Solution and steady state Exact solution (found by guess and verify ): c t = (1 αβ) e zt k α t k t+1 = αβe zt k α t Steady state is also easy to find: k = (αβ) 1 1 α c = (αβ) α 1 α (αβ) 1 1 α z = 0 Steady state in more general models. 8

12 The goal We are searching for decision rules: { c t = c (k t, z t ) d = k t+1 = k (k t, z t ) Then, we have: 1 c (k t, z t ) = βe αe ρzt+σεt+1 k (k t, z t ) α 1 t c (k (k t, z t ), ρz t + σε t+1 ) c (k t, z t ) + k (k t, z t ) = e zt kt α This is a system of functional equations. 9

13 A perturbation solution Rewrite the problem in terms of perturbation parameter λ. Different possibilities for λ. For this case, I pick: z t = ρz t 1 + λσε t, ε t N (0, 1) 1. When λ = 1, stochastic case. 2. When λ = 0, deterministic case (with z 0 = 0 and then e zt = 1). Now we are searching for the decision rules: c t = c (k t, z t ; λ) k t+1 = k (k t, z t ; λ) 10

14 Taylor s theorem We will build a local approximation around (k, 0; 0). Given equilibrium conditions: ( ) 1 E t c (k t, z t ; λ) β αe ρzt+λσεt+1 k (k t, z t ; λ) α 1 = 0 c (k (k t, z t ; λ), ρz t + λσε t+1 ; λ) c (k t, z t ; λ) + k (k t, z t ; λ) e zt kt α = 0 We will take derivatives with respect to k t, z t, and λ and evaluate them around (k, 0; 0). Why? Apply Taylor s theorem and a version of the implicit-function theorem. 11

15 Asymptotic expansion I c t = c (k t, z t ; 1) k,0,0 = c (k, 0; 0) +c k (k, 0; 0) (k t k) + c z (k, 0; 0) z t + c λ (k, 0; 0) c kk (k, 0; 0) (k t k) c kz (k, 0; 0) (k t k) z t c kλ (k, 0; 0) (k t k) c zk (k, 0; 0) z t (k t k) c zz (k, 0; 0) z 2 t c zλ (k, 0; 0) z t c λk (k, 0; 0) (k t k) c λz (k, 0; 0) λz t c λ2 (k, 0; 0)

16 Asymptotic expansion II k t+1 = k (k t, z t ; 1) k,0,0 = k (k, 0; 0) +k k (k, 0; 0) (k t k) + k z (k, 0; 0) z t + k λ (k, 0; 0) k kk (k, 0; 0) (k t k) k kz (k, 0; 0) (k t k) z t k kλ (k, 0; 0) (k t k) k zk (k, 0; 0) z t (k t k) k zz (k, 0; 0) z 2 t k zλ (k, 0; 0) z t k λk (k, 0; 0) (k t k) k λz (k, 0; 0) z t k λ2 (k, 0; 0)

17 Comment on notation From now on, to save on notation, we will write [ 1 F (k t, z t ; λ) = E c(k β αeρzt +λσε t+1 k(kt,z t;λ) α 1 t t,z t;λ) c(k(k t,z t;λ),ρz t+λσε t+1;σ) c (k t, z t ; λ) + k (k t, z t ; λ) e zt kt α ] = [ 0 0 ] Note that: F (k t, z t ; λ) = H (c t, c t+1, k t, k t+1, z t ; λ) = H (c (k t, z t ; λ), c (k (k t, z t ; λ), z t+1 ; λ), k t, k (k t, z t ; λ), z t ; λ) I will use H i to represent the partial derivative of H with respect to the i component and drop the evaluation at the steady state of the functions when we do not need it. 14

18 First-order approximation We take derivatives of F (k t, z t ; λ) around k, 0, and 0. With respect to k t : F k (k, 0; 0) = 0 With respect to z t : F z (k, 0; 0) = 0 With respect to λ: F λ (k, 0; 0) = 0 15

19 Solving the system I Remember that: F (k t, z t ; λ) = H (c (k t, z t ; λ), c (k (k t, z t ; λ), z t+1 ; λ), k t, k (k t, z t ; λ), z t ; λ) = 0 Because F (k t, z t ; λ) must be equal to zero for any possible values of k t, z t, and λ, the derivatives of any order of F must also be zero. Then: F k (k, 0; 0) = H 1 c k + H 2 c k k k + H 3 + H 4 k k = 0 F z (k, 0; 0) = H 1 c z + H 2 (c k k z + c z ρ) + H 4 k z + H 5 = 0 F λ (k, 0; 0) = H 1 c λ + H 2 (c k k λ + c λ ) + H 4 k λ + H 6 = 0 16

20 Solving the system II Note that: F k (k, 0; 0) = H 1 c k + H 2 c k k k + H 3 + H 4 k k = 0 F z (k, 0; 0) = H 1 c z + H 2 (c k k z + c z ρ) + H 4 k z + H 5 = 0 is a quadratic system of four equations on four unknowns: c k, c z, k k, and k z. Procedures to solve quadratic systems: 1. Blanchard and Kahn (1980). 2. Uhlig (1999). 3. Sims (2000). 4. Klein (2000). All of them equivalent. Why quadratic? Stable and unstable manifold. 17

21 Solving the system III Also, note that: F λ (k, 0; 0) = H 1 c λ + H 2 (c k k λ + c λ ) + H 4 k λ + H 6 = 0 is a linear and homogeneous system in c λ and k λ. Hence: c λ = k λ = 0 This means the system is certainty equivalent. Interpretation no precautionary behavior. Difference between risk-aversion and precautionary behavior. Leland (1968), Kimball (1990). Risk-aversion depends on the second derivative (concave utility). Precautionary behavior depends on the third derivative (convex marginal utility). 18

22 Comparison with linearization After Kydland and Prescott (1982) a popular method to solve economic models has been the use of a LQ approximation of the objective function of the agents. Close relative: linearization of equilibrium conditions. When properly implemented linearization, LQ, and first-order perturbation are equivalent. Advantages of linearization: 1. Theorems. 2. Higher order terms. 19

23 Second-order approximation We take second-order derivatives of F (k t, z t ; λ) around k, 0, and 0: F kk (k, 0; 0) = 0 F kz (k, 0; 0) = 0 F kλ (k, 0; 0) = 0 F zz (k, 0; 0) = 0 F zλ (k, 0; 0) = 0 F λλ (k, 0; 0) = 0 We substitute the coefficients that we already know. A linear system of 12 equations on 12 unknowns (remember Young s theorem!). Why linear? Cross-terms on kλ and zλ are zero. More general result: all the terms in odd derivatives of λ are zero. 20

24 Correction for risk We have the term 1 2 c λ2 (k, 0; 0). Captures precautionary behavior. We do not have certainty equivalence any more! Important advantage of second order approximation. Changes ergodic distribution of states. 21

25 Higher-order terms We can continue the iteration for as long as we want. Great advantage of procedure: it is recursive! Often, a few iterations will be enough. The level of accuracy depends on the goal of the exercise: 1. Welfare analysis: Kim and Kim (2001). 2. Empirical strategies: Fernández-Villaverde, Rubio-Ramírez, and Santos (2006). 22

26 A Numerical Example

27 A numerical example Steady State: Parameter β α ρ σ Value c = k = First-order components: Second-order components: c k (k, 0; 0) = k k (k, 0; 0) = 0.33 c z (k, 0; 0) = k z (k, 0; 0) = c kk (k, 0; 0) = k kk (k, 0; 0) = c kz (k, 0; 0) = k kz (k, 0; 0) = 0.33 c zz (k, 0; 0) = k zz (k, 0; 0) = c λ 2 (k, 0; 0) = 0 k λ 2 (k, 0; 0) = 0 c λ (k, 0; 0) = k λ (k, 0; 0) = c kλ (k, 0; 0) = k kλ (k, 0; 0) = c zλ (k, 0; 0) = k zλ (k, 0; 0) = 0. 23

28 Comparison c t = e zt k 0.33 t and: c t (k t k) z t (k t k) (k t k) z t zt k t+1 = e zt k 0.33 t k t (k t k) z t (k t k) (k t k) z t zt

29 Capital Next Period Capital Next Period Comparison of exact and first-order solution Exact First-order Capital Current Period Comparison of exact and second-order solution Exact Second-order Capital Current Period 25

30 Computer In practice you do all this approximations with a computer: 1. First-, second-, and third- order: Dynare. 2. Higher order: Mathematica, Dynare++. Burden: analytical derivatives. Why are numerical derivatives a bad idea? Alternatives: automatic differentiation? 26

31 Local properties of the solution I Perturbation is a local method. It approximates the solution around the deterministic steady state of the problem. It is valid within a radius of convergence. 27

32 Local properties of the solution II What is the radius of convergence of a power series around x? An r R + that x, x z < r, the power series of x will converge. such A Remarkable Result from Complex Analysis The radius of convergence is always equal to the distance from the center to the nearest point where the decision rule has a (non-removable) singularity. If no such point exists then the radius of convergence is infinite. Singularity here refers to poles, fractional powers, and other branch powers or discontinuities of the functional or its derivatives. 28

33 Local properties of the solution III Holomorphic functions are analytic: 1. A function is holomorphic at a point x if it is differentiable at every point within some open disk centered at x. 2. A function is analytic at x if in some open disk centered at x it can be expanded as a convergent power series: f (z) = θ n (z x) n Distance is in the complex plane. n=0 Often, we can check numerically that perturbations have good non-local behavior. However: problem with boundaries. 29

34 Non-local accuracy test Proposed by Judd (1992) and Judd and Guu (1997). Given the Euler equation: 1 c i (k t, z t ) = E t ( αe z t+1 k i (k t, z t ) α 1 ) c i (k i (k t, z t ), z t+1 ) we can define: ( αe EE i (k t, z t ) 1 c i z t+1 k i (k t, z t ) α 1 ) (k t, z t ) E t c i (k i (k t, z t ), z t+1 ) Units of reporting. Interpretation. 30

35 Figure 8: Euler Equation Errors at z = 0, τ = 2 / σ = Perturbation 1: Log-Linear -4 Perturbation 1: Linear Log10 Euler Equation Error Perturbation 2: Quadratic -8 Perturbation Capital 31

36 Figure 9: Euler Equation Errors at z = 0, τ = 2 / σ = Perturbation 1: Linear -4 Value Function Iteration -5 Log10 Euler Equation Error Finite Elements -9 Chebyshev Polynomials Capital 32