Perturbation Methods, Lecture notes of Economics

Download Perturbation Methods and more Lecture notes Economics in PDF only on Docsity! Perturbation Methods Jesús Fernández-Villaverde University of Pennsylvania May 28, 2015 Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 1 / 91 Introduction Introduction Remember that we want to solve a functional equation of the form: H (d) = 0 for an unknown decision rule d . Perturbation solves the problem by specifying: dn (x , θ) = n ∑ i=0 θi (x − x0)i We use implicit-function theorems to find coeffi cients θi’s. Inherently local approximation. However, often good global properties. Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 2 / 91 Introduction Applications to 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. A first-order perturbation theory and linearization deliver the same output. Hence, we can use much of what we already know about linearization. Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 5 / 91 Introduction 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 markets model. Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 6 / 91 Introduction 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. Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 7 / 91 A Baby Example Another Way to Solve the Problem Now let us suppose that you missed the lecture when “guess and verify”was explained. You need to compute the RBC. What you are searching for? A decision rule for consumption: ct = c (kt , zt ) and another one for capital: kt+1 = k (kt , zt ) Note that our d is just the stack of c (kt , zt ) and k (kt , zt ). Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 10 / 91 A Baby Example Equilibrium Conditions We substitute in the equilibrium conditions the budget constraint and the law of motion for technology. And we write the decision rules explicitly as function of the states. Then: 1 c (kt , zt ) = βEt αeρzt+σεt+1k (kt , zt ) α−1 c (k (kt , zt ) , ρzt + σεt+1) c (kt , zt ) + k (kt , zt ) = eztkα t System of functional equations. Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 11 / 91 A Baby Example Main Idea Transform the problem rewriting it in terms of a small perturbation parameter. Solve the new problem for a particular choice of the perturbation parameter. This step is usually ambiguous since there are different ways to do so. Use the previous solution to approximate the solution of original the problem. Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 12 / 91 A Baby Example Taylor’s Theorem Equilibrium conditions: Et ( 1 c (kt , zt ; σ) − β αeρzt+σεt+1k (kt , zt ; σ) α−1 c (k (kt , zt ; σ) , ρzt + σεt+1; σ) ) = 0 c (kt , zt ; σ) + k (kt , zt ; σ)− eztkα t = 0 We will take derivatives with respect to kt , zt , and σ. Apply Taylor’s theorem to build solution around deterministic steady state. How? Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 15 / 91 A Baby Example Asymptotic Expansion I ct = c (kt , zt ; σ)|k ,0,0 = c (k, 0; 0) +ck (k , 0; 0) (kt − k) + cz (k, 0; 0) zt + cσ (k, 0; 0) σ + 1 2 ckk (k, 0; 0) (kt − k)2 + 1 2 ckz (k, 0; 0) (kt − k) zt + 1 2 ckσ (k, 0; 0) (kt − k) σ+ 1 2 czk (k, 0; 0) zt (kt − k) + 1 2 czz (k, 0; 0) z2t + 1 2 czσ (k, 0; 0) ztσ + 1 2 cσk (k, 0; 0) σ (kt − k) + 1 2 cσz (k, 0; 0) σzt + 1 2 cσ2 (k, 0; 0) σ2 + ... Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 16 / 91 A Baby Example Asymptotic Expansion II kt+1 = k (kt , zt ; σ)|k ,0,0 = k (k, 0; 0) +kk (k, 0; 0) (kt − k) + kz (k, 0; 0) zt + kσ (k , 0; 0) σ + 1 2 kkk (k, 0; 0) (kt − k)2 + 1 2 kkz (k, 0; 0) (kt − k) zt + 1 2 kkσ (k, 0; 0) (kt − k) σ+ 1 2 kzk (k, 0; 0) zt (kt − k) + 1 2 kzz (k, 0; 0) z2t + 1 2 kzσ (k, 0; 0) ztσ + 1 2 kσk (k, 0; 0) σ (kt − k) + 1 2 kσz (k, 0; 0) σzt + 1 2 kσ2 (k, 0; 0) σ2 + ... Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 17 / 91 A Baby Example First-Order Approximation We take derivatives of F (kt , zt ; σ) around k, 0, and 0. With respect to kt : Fk (k, 0; 0) = 0 With respect to zt : Fz (k, 0; 0) = 0 With respect to σ: Fσ (k, 0; 0) = 0 Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 20 / 91 A Baby Example Solving the System I Remember that: F (kt , zt ; σ) = H (c (kt , zt ; σ) , c (k (kt , zt ; σ) , zt+1; σ) , kt , k (kt , zt ; σ) , zt ; σ) = 0 Because F (kt , zt ; σ) must be equal to zero for any possible values of kt , zt , and σ, the derivatives of any order of F must also be zero. Then: Fk (k, 0; 0) = H1ck +H2ckkk +H3 +H4kk = 0 Fz (k, 0; 0) = H1cz +H2 (ckkz + ckρ) +H4kz +H5 = 0 Fσ (k, 0; 0) = H1cσ +H2 (ckkσ + cσ) +H4kσ +H6 = 0 Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 21 / 91 A Baby Example Solving the System II A quadratic system: Fk (k, 0; 0) = H1ck +H2ckkk +H3 +H4kk = 0 Fz (k, 0; 0) = H1cz +H2 (ckkz + ckρ) +H4kz +H5 = 0 of 4 equations on 4 unknowns: ck , cz , kk , and kz . 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. Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 22 / 91 A Baby Example Some Further Comments Note how we have used a version of the implicit-function theorem. Important tool in economics. Also, we are using the Taylor theorem to approximate the policy function. Alternatives? Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 25 / 91 A Baby Example Second-Order Approximation We take second-order derivatives of F (kt , zt ; σ) around k, 0, and 0: Fkk (k, 0; 0) = 0 Fkz (k, 0; 0) = 0 Fkσ (k, 0; 0) = 0 Fzz (k, 0; 0) = 0 Fzσ (k, 0; 0) = 0 Fσσ (k, 0; 0) = 0 Remember Young’s theorem! We substitute the coeffi cients that we already know. A linear system of 12 equations on 12 unknowns. Why linear? Cross-terms kσ and zσ are zero. Conjecture on all the terms with odd powers of σ. Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 26 / 91 A Baby Example Correction for Risk We have a term in σ2. Captures precautionary behavior. We do not have certainty equivalence any more! Important advantage of second-order approximation. Changes ergodic distribution of states. Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 27 / 91 A Numerical Example Comparison ct = 0.6733eztk0.33t ct ' 0.388069+ 0.680101 (kt − k) + 0.388069zt −2.41990 2 (kt − k)2 + 0.680099 (kt − k) zt + 0.388064 2 z2t and: kt+1 = 0.3267eztk0.33t kt+1 ' 0.1883+ 0.33 (kt − k) + 0.1883zt −1.1742 2 (kt − k)2 + 0.33 (kt − k) zt + 0.1883 2 z2t Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 30 / 91 A Numerical Example Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 31 / 91 A Numerical Example A Computer In practice you do all this approximations with a computer: 1 First-, second-, and third-order: Matlab and Dynare. 2 Higher-order: Mathematica, Dynare++, Fortran code by Jinn and Judd. Burden: analytical derivatives. Why are numerical derivatives a bad idea? Alternatives: automatic differentiation? Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 32 / 91 A Numerical Example Non Local Accuracy Test Proposed by Judd (1992) and Judd and Guu (1997). Given the Euler equation: 1 c i (kt , zt ) = Et ( αezt+1k i (kt , zt )α−1 c i (k i (kt , zt ), zt+1) ) we can define: EE i (kt , zt ) ≡ 1− c i (kt , zt )Et ( αezt+1k i (kt , zt )α−1 c i (k i (kt , zt ), zt+1) ) Units of reporting. Interpretation. Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 35 / 91 A Numerical Example Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 36 / 91 The General Case The General Case Most of previous argument can be easily generalized. The set of equilibrium conditions of many DSGE models can be written as (note recursive notation) EtH(y , y ′, x , x ′) = 0, where yt is a ny × 1 vector of controls and xt is a nx × 1 vector of states. Define n = nx + ny . Then H maps Rny × Rny × Rnx × Rnx into Rn. Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 37 / 91 The General Case The Perturbation Parameter The scalar σ ≥ 0 is the perturbation parameter. If we set σ = 0 we have a deterministic model. Important: there is only ONE perturbation parameter. The matrix ηε takes account of relative sizes of different shocks. Why bounded support? Samuelson (1970) and Jin and Judd (2002). Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 40 / 91 The General Case Solution of the Model The solution to the model is of the form: y = g(x ; σ) x ′ = h(x ; σ) + σηε′ where g maps Rnx × R+ into Rny and h maps Rnx × R+ into Rnx . The matrix η is of order nx × nε and is given by: η = [ ∅ ηε ] Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 41 / 91 The General Case Perturbation We wish to find a perturbation approximation of the functions g and h around the non-stochastic steady state, xt = x̄ and σ = 0. We define the non-stochastic steady state as vectors (x̄ , ȳ) such that: H(ȳ , ȳ , x̄ , x̄) = 0. Note that ȳ = g(x̄ ; 0) and x̄ = h(x̄ ; 0). This is because, if σ = 0, then EtH = H. Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 42 / 91 The General Case Tensors General trick from physics. An nth-rank tensor in a m-dimensional space is an operator that has n indices and mn components and obeys certain transformation rules. [Hy ]iα is the (i , α) element of the derivative of H with respect to y : 1 The derivative of H with respect to y is an n× ny matrix. 2 Thus, [Hy ]iα is the element of this matrix located at the intersection of the i-th row and α-th column. 3 Thus, [Hy ]iα[gx ]αβ[hx ] β j = ∑ ny α=1 ∑nxβ=1 ∂Hi ∂y α ∂g α ∂x β ∂hβ ∂x j . [Hy ′y ′ ]iαγ: 1 Hy ′y ′ is a three dimensional array with n rows, ny columns, and ny pages. 2 Then [Hy ′y ′ ]iαγ denotes the element of Hy ′y ′ located at the intersection of row i , column α and page γ. Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 45 / 91 The General Case Solving the System I gx and hx can be found as the solution to the system: [Fx (x̄ ; 0)]ij = [Hy ′ ]iα[gx ]αβ[hx ] β j + [Hy ]iα[gx ]αj + [Hx ′ ]iβ[hx ] β j + [Hx ]ij = 0; i = 1, . . . , n; j , β = 1, . . . , nx ; α = 1, . . . , ny Note that the derivatives of H evaluated at (y , y ′, x , x ′) = (ȳ , ȳ , x̄ , x̄) are known. Then, we have a system of n× nx quadratic equations in the n× nx unknowns given by the elements of gx and hx . We can solve with a standard quadratic matrix equation solver. Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 46 / 91 The General Case Solving the System II gσ and hσ are identified as the solution to the following n equations: [Fσ(x̄ ; 0)]i = Et{[Hy ′ ]iα[gx ]αβ[hσ] β + [Hy ′ ]iα[gx ]αβ[η] β φ[ε ′]φ + [Hy ′ ]iα[gσ] α +[Hy ]iα[gσ] α + [Hx ′ ]iβ[hσ] β + [Hx ′ ]iβ[η] β φ[ε ′]φ} i = 1, . . . , n; α = 1, . . . , ny ; β = 1, . . . , nx ; φ = 1, . . . , nε. Then: [Fσ(x̄ ; 0)]i = [Hy ′ ]iα[gx ]αβ[hσ] β + [Hy ′ ]iα[gσ] α + [Hy ]iα[gσ] α + [fx ′ ] i β[hσ] β = 0; i = 1, . . . , n; α = 1, . . . , ny ; β = 1, . . . , nx ; φ = 1, . . . , nε. Certainty equivalence: this equation is linear and homogeneous in gσ and hσ. Thus, if a unique solution exists, it must satisfy: hσ 6= 0 gσ = 0 Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 47 / 91 The General Case Second-Order Approximation III The unknowns of these expansions are [gxx ]iab , [gxσ]ia, [gσx ]ia, [gσσ]i , [hxx ] j ab , [hxσ] j a, [hσx ] j a, [hσσ]j . These coeffi cients can be identified by taking the derivative of F (x ; σ) with respect to x and σ twice and evaluating them at (x ; σ) = (x̄ ; 0). By the arguments provided earlier, these derivatives must be zero. Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 50 / 91 The General Case Solving the System I We use Fxx (x̄ ; 0) to identify gxx (x̄ ; 0) and hxx (x̄ ; 0): [Fxx (x̄ ; 0)]ijk =( [Hy ′y ′ ]iαγ[gx ] γ δ [hx ] δ k + [Hy ′y ]iαγ[gx ] γ k + [Hy ′x ′ ] i αδ[hx ] δ k + [Hy ′x ]iαk ) [gx ]αβ[hx ] β j +[Hy ′ ]iα[gxx ]αβδ[hx ] δ k [hx ] β j + [Hy ′ ]iα[gx ]αβ[hxx ] β jk + ( [Hyy ′ ]iαγ[gx ] γ δ [hx ] δ k + [Hyy ]iαγ[gx ] γ k + [Hyx ′ ] i αδ[hx ] δ k + [Hyx ]iαk ) [gx ]αj +[Hy ]iα[gxx ]αjk + ( [Hx ′y ′ ]iβγ[gx ] γ δ [hx ] δ k + [Hx ′y ]iβγ[gx ] γ k + [Hx ′x ′ ] i βδ[hx ] δ k + [Hx ′x ]iβk ) [hx ] β j +[Hx ′ ]iβ[hxx ] β jk +[Hxy ′ ]ijγ[gx ] γ δ [hx ] δ k + [Hxy ]ijγ[gx ] γ k + [Hxx ′ ] i jδ[hx ] δ k + [Hxx ]ijk = 0; i = 1, . . . n, j , k, β, δ = 1, . . . nx ; α,γ = 1, . . . ny . Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 51 / 91 The General Case Solving the System II We know the derivatives of H. We also know the first derivatives of g and h evaluated at (y , y ′, x , x ′) = (ȳ , ȳ , x̄ , x̄). Hence, the above expression represents a system of n× nx × nx linear equations in then n× nx × nx unknowns elements of gxx and hxx . Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 52 / 91 The General Case Structure of the Solution The perturbation solution of the model satisfies: gσ(x̄ ; 0) = 0 hσ(x̄ ; 0) = 0 gxσ(x̄ ; 0) = 0 hxσ(x̄ ; 0) = 0 Standard deviation only appears in: 1 A constant term given by 12gσσσ2 for the control vector yt . 2 The first nx − nε elements of 12hσσσ2. Correction for risk. Quadratic terms in endogenous state vector x1. Those terms capture non-linear behavior. Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 55 / 91 The General Case Higher-Order Approximations We can iterate this procedure as many times as we want. We can obtain n-th order approximations. Problems: 1 Existence of higher order derivatives (Santos, 1992). 2 Numerical instabilities. 3 Computational costs. Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 56 / 91 Change of Variables Erik Eady It is not the process of linearization that limits insight. It is the nature of the state that we choose to linearize about. Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 57 / 91 Change of Variables Theory The first-order solution can be written as f (x) ' f (a) + (x − a) f ′ (a) Expand g(y) = h (f (X (y))) around b = Y (a), where X (y) is the inverse of Y (x). Then: g (y) = h (f (X (y))) = g (b) + gα (b) (Y α (x)− bα) where gα = hAf Ai X i α comes from the application of the chain rule. From this expression it is easy to see that if we have computed the values of f Ai , then it is straightforward to find gα. Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 60 / 91 Change of Variables Coeffi cients Relation Remember that the linear solution is:( k ′ − k0 ) = a1 (k − k0) + b1z (l − l0) = c1 (k − k0) + d1z Then we show that: a3 = γ ζ k γ−ζ 0 a1 b3 = γkγ−1 0 b1 c3 = µ ζ l µ−1 0 k1−ζ 0 c1 d3 = µlµ−10 d1 Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 61 / 91 Change of Variables Finding the Parameters Minimize over a grid the Euler Error. Some optimal results Euler Equation Errors γ ζ µ SEE 1 1 1 0.0856279 0.986534 0.991673 2.47856 0.0279944 Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 62 / 91 Change of Variables A Quasi-Optimal Approximation Sensitivity analysis reveals that for different parametrizations γ ' ζ This suggests the quasi-optimal approximation: k ′γ − kγ 0 = a3 ( kγ − kγ 0 ) + b3z lµ − lµ0 = c3 ( kγ − kγ 0 ) + d3z If we define k̂ = kγ − kγ 0 and l̂ = l µ − lµ0 we get: k̂ ′ = a3k̂ + b3z l̂ = c3k̂ + d3z Linear system: 1 Use for analytical study. 2 Use for estimation with a Kalman Filter. Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 65 / 91 Perturbing the Value Function Perturbing the Value Function We worked with the equilibrium conditions of the model. Sometimes we may want to perform a perturbation on the value function formulation of the problem. Possible reasons: 1 Gain insight. 2 Diffi culty in using equilibrium conditions. 3 Evaluate welfare. 4 Initial guess for VFI. More general point: we can perturb any operator problem that we find useful. Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 66 / 91 Perturbing the Value Function Basic Problem Imagine that we have: V (kt , zt ) = max ct [ (1− β) c1−γ t 1− γ + βEtV (kt+1, zt+1) ] s.t. ct + kt+1 = eztkθ t + (1− δ) kt zt = λzt−1 + σεt , εt ∼ N (0, 1) Write it as: V (kt , zt ;χ) = max ct [ (1− β) c1−γ t 1− γ + βEtV (kt+1, zt+1;χ) ] s.t. ct + kt+1 = eztkθ t + (1− δ) kt zt = λzt−1 + χσεt , εt ∼ N (0, 1) Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 67 / 91 Perturbing the Value Function Expanding the Value Function By certainty equivalence, we will show below that: V3,ss = V13,ss = V23,ss = 0 Taking advantage of the equality of cross-derivatives, and setting χ = 1, which is just a normalization: V (kt , zt ; 1) ' Vss + V1,ss (kt − kss ) + V2,sszt + 1 2 V11,ss (kt − kss )2 + 1 2 V22,ssz2tt +V12,ss (kt − kss ) z + 1 2 V33,ss Note that V33,ss 6= 0, a difference from the standard linear-quadratic approximation to the utility functions. Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 70 / 91 Perturbing the Value Function Expanding the Consumption Function The policy function for consumption can be expanded as: ct = c (kt , zt ;χ) ' css + c1,ss (kt − kss ) + c2,sszt + c3,ssχ where: c1,ss = c1 (kss , 0; 0) c2,ss = c2 (kss , 0; 0) c3,ss = c3 (kss , 0; 0) Since the first derivatives of the consumption function only depend on the first and second derivatives of the value function, we must have c3,ss = 0 (precautionary consumption depends on the third derivative of the value function, Kimball, 1990). Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 71 / 91 Perturbing the Value Function Linear Components of the Value Function To find the linear approximation to the value function, we take derivatives of the value function with respect to controls (ct), states (kt , zt), and the perturbation parameter χ. Notation: 1 Vi ,t : derivative of the value function with respect to its i-th argument, evaluated in (kt , zt ;χ) . 2 Vi ,ss : derivative evaluated in the steady state, (kss , 0; 0). 3 We follow the same notation for higher-order (cross-) derivatives. Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 72 / 91 Perturbing the Value Function System of Equations II If we set χ = 0 and compute the steady state, we get a system of six equations on six unknowns, css , kss , Vss , V1,ss , V2,ss , and V3,ss : css + δkss = kθ ss Vss = (1− β) c1−γ ss 1− γ + βVss (1− β) c−γ ss − βV1,ss = 0 V1,ss = βV1,ss ( θkθ−1 ss + 1− δ ) V2,ss = β [ V1,sskθ ss + V2,ssλ ] V3,ss = βV3,ss From the last equation: V3,ss = 0. From the second equation: Vss = c1−γ ss 1−γ . From the third equation: V1,ss = 1−β β c−γ ss . Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 75 / 91 Perturbing the Value Function System of Equations III After cancelling redundant terms: css + δkss = kθ ss 1 = β ( θkθ−1 ss + 1− δ ) V2,ss = β [ V1,sskθ ss + V2,ssλ ] Then: kss = [ 1 θ ( 1 β − 1+ δ )] 1 θ−1 css = kθ ss − δkss V2,ss = 1− β 1− βλ kθ ssc −γ ss V1,ss > 0 and V2,ss > 0, as predicted by theory. Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 76 / 91 Perturbing the Value Function Quadratic Components of the Value Function From the previous derivations, we have: (1− β) c (kt , zt ;χ) −γ − βEtV1,t+1 = 0 V1,t = βEtV1,t+1 ( θeztkθ−1 t + 1− δ ) V2,t = βEt [ V1,t+1eztkθ t + V2,t+1λ ] V3,t = βEt [V2,t+1σεt+1 + V3,t+1] where: kt+1 = eztkθ t + (1− δ) kt − c (kt , zt ;χ) zt = λzt−1 + χσεt , εt ∼ N (0, 1) We take derivatives of each of the four equations w.t.r. kt , zt , and χ. We take advantage of the equality of cross derivatives. The envelope theorem does not hold anymore (we are taking derivatives of the derivatives of the value function). Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 77 / 91 Perturbing the Value Function First Equation III Derivative with respect to χ: − (1− β) γc (kt , zt ;χ) −γ−1 c3,t −βEt (−V11,t+1c3,t + V12,t+1σεt+1 + V13,t+1) = 0 In steady state:( βV11,ss − (1− β) γc−γ−1 ss ) c3,ss = βV13,ss or c3,ss = β( βV11,ss − (1− β) γc−γ−1 ss )V13,ss Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 80 / 91 Perturbing the Value Function Second Equation I We have: V1,t = βEtV1,t+1 ( θeztkθ−1 t + 1− δ ) Derivative with respect to kt : V11,t = βEt [ V11,t+1 ( θeztkθ−1 t + 1− δ− c1,t ) ( θeztkθ−1 t + 1− δ ) +V1,t+1θ (θ − 1) eztkθ−2 t ] In steady state: V11,ss = [ V11,ss ( 1 β − c1,ss ) + βV1,ssθ (θ − 1) kθ−2 ss ] or V11,ss = β 1− 1 β + c1,ss V1,ssθ (θ − 1) kθ−2 ss Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 81 / 91 Perturbing the Value Function Second Equation II Derivative with respect to zt : V12,t = βEt  V11,t+1 ( eztkθ t − c2,t ) ( θeztkθ−1 t + 1− δ ) +V12,t+1λ ( θeztkθ−1 t + 1− δ ) + V1,t+1θeztkθ−1 t  In steady state: V12,ss = V11,ss ( kθ ss − c2,ss ) + V12,ssλ+ βV1,ssθkθ−1 t or V12,ss = 1 1− λ [ V11,ss ( kθ ss − c2,ss ) + βV1,ssθkθ−1 ss ] Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 82 / 91 Perturbing the Value Function Third Equation II Derivative with respect to χ: V23,t = βEt [ −V11,t+1eztkθ t c3,t + V12,t+1e ztkθ t σεt+1 + V13,t+1eztkθ t −V21,t+1λc3,t + V22,t+1λσεt+1 + V23,t+1λ ] In steady state: V23,ss = 0 Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 85 / 91 Perturbing the Value Function Fourth Equation We have V3,t = βEt [V2,t+1σεt+1 + V3,t+1] . Derivative with respect to χ: V33,t = βEt [ −V21,t+1c3,tσεt+1 + V22,t+1σ2ε2t+1 + V23,t+1σεt+1 −V31,t+1c3,t + V32,t+1σεt+1 + V33,t+1 ] In steady state: V33,ss = β 1− β V22,ss Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 86 / 91 Perturbing the Value Function System I c1,ss = V11,ss βV11,ss − (1− β) γc−γ−1 ss c2,ss = β βV11,ss − (1− β) γc−γ−1 ss ( V11,sskθ ss + V12,ssλ ) V11,ss = β 1− 1 β + c1,ss V1,ssθ (θ − 1) kθ−2 ss V12,ss = 1 1− λ [ V11,ss ( kθ ss − c2,ss ) + βV1,ssθkθ−1 ss ] V22,ss = β 1− βλ2 [ V11,ss ( kθ t − c2,ss ) kθ ss + 2V12,ssλk θ ss +V1,sskθ ss − V12,ssλc2,ss ] V33,ss = β 1− β σ2V22,ss plus c3,ss = V13,ss = V23,ss = 0. Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 87 / 91 Perturbing the Value Function Our Example We know that Vss = c1−γ ss 1−γ . We can compute the decrease in consumption τ that will make the household indifferent between consuming (1− τ) css units per period with certainty or ct units with uncertainty. Thus: c1−γ ss 1− γ + 1 2 V33,ss = (css (1− τ))1−γ 1− γ ⇒( (1− τ)1−γ − 1 ) c1−γ ss = (1− γ) 1 2 V33,ss or τ = 1− [ 1+ 1− γ c1−γ ss 1 2 V33,ss ] 1 1−γ Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 90 / 91 Perturbing the Value Function A Numerical Example We pick standard parameter values by setting β = 0.99,γ = 2, δ = 0.0294, θ = 0.3, and λ = 0.95. We get: V (kt , zt ; 1) ' −0.54000+ 0.00295 (kt − kss ) + 0.11684zt −0.00007 (kt − kss )2 − 0.00985z2t −0.97508σ2 − 0.00225 (kt − kss ) zt c (kt , zt ;χ) ' 1.85193+ 0.04220 (kt − kss ) + 0.74318zt DYNARE produces the same policy function by linearizing the equilibrium conditions of the problem. The welfare cost of the business cycle (in consumption terms) is 8.8475e-005, lower than in Lucas (1987) because of the smoothing possibilities allowed by capital. Use as an initial guess for VFI. Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 91 / 91