Hoover Institution, Stanford University. Department of Economics, Stanford University. Department of Economics, Santa Clara University

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1 Supplementary Material Supplement to How to solve dynamic stochastic models computing expectations just once : Appendices (Quantitative Economics, Vol. 8, No. 3, November 2017, ) Kenneth L. Judd Hoover Institution, Stanford University Lilia Maliar Department of Economics, Stanford University Serguei Maliar Department of Economics, Santa Clara University Inna Tsener Department of Applied Economics, University of the Balearic Islands Appendix A: Bellman and Euler equation algorithms for solving a growth model with inelastic labor supply In Algorithms A1 A6, we provide a description of three Bellman and two Euler equation algorithms, which we use to solve the neoclassical stochastic growth model with inelastic labor supply described in Section 2. Kenneth L. Judd: kennethjudd@mac.com Lilia Maliar: maliarl@stanford.edu Serguei Maliar: smaliar@scu.edu Inna Tsener: inna.tcener@uib.es Copyright 2017 The Authors. Quantitative Economics. The Econometric Society. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Availableat DOI: /QE329

2 2 Judd, Maliar, Maliar, and Tsener Supplementary Material Algorithm 1a. Value function iteration a. Choose an approximating function V( ; b) V. b. Choose integration nodes, ε j, and weights, ω j, j = 1 J. c. Construct a grid Γ ={k m z m } M m=1. d. Make an initial guess on b (1). At iteration i, perform the following steps: Step 1. Computation of values of V on the grid. a. Solve for k m satisfying u ((1 δ)k m + z m f(k m ) k m ) = β J j=1 ω j V 1 (k m zρ m exp(ε j ); b (i) ). b. Find c m satisfying c m = (1 δ)k m + z m f(k m ) k m. c. Find value function on the grid v m u(c m ) + β J j=1 ω j V(k m zρ m exp(ε j ); b (i) ). Step 2. Computation of b that fits value function on the grid. Run a regression to find b: b = arg minb v m V(k m z m ; b). Step 3. Convergence check and fixed-point iteration. a. Check for convergence for i 2: endstep2if 1 M (k m )(i) (k m )(i 1) (k < m) (i 1) b. Use damping with ξ = 1 to compute b (i+1) = (1 ξ)b (i) + ξ b. Algorithm 1b. Value function iteration with precomputation b. Precompute {I 0 I n } using (12). Step 1. Computation of values of V on the grid. At iteration i,form = 1 M: a. Given b (i),findb (i) from (11) and compute V 1 (k m zρ m; b (i) ). b. Solve for k m satisfying u ((1 δ)k m + z m f(k m ) k m ) = β V 1 (k m zρ m; b (i) ). d. Find value function on the grid v m u(c m ) + β V(k m zρ m; b (i) ).

3 Supplementary Material How to solve dynamic stochastic models 3 Algorithm 2a. Endogenous grid method a. Choose an approximating function V( ; b) V. b. Choose integration nodes, ε j, and weights, ω j, j = 1 J. c. Construct grid Γ ={k m z m} M m=1. d. Make an initial guess on b (1). At iteration i, perform the following steps: Step 1. Computation of values of V on the grid. a. Compute Ŵ(k m z m; b (i) ) J j=1 ω j V(k m zρ m exp(ε j ); b (i) ) and Ŵ 1 (k m z m; b (i) ) J j=1 ω j V 1 (k m zρ m exp(ε j ); b (i) ). b. Find c m = u 1 [βŵ 1 (k m z m; b (i) )]. c.useasolvertofindk m satisfying (1 δ)k m + z m f(k m ) = c m + k m. d. Find value function on the grid v m u(c m ) + βŵ(k m z m; b (i) ). Step 2. Computation of b that fits value function on the grid. Run a regression to find b: b = arg minb v m V(k m z m ; b). Step 3. Convergence check and fixed-point iteration. a. Check for convergence for i 2:endStep2if 1 M (k m) (i) (k m ) (i 1) < (k m ) (i 1) b. Use damping with ξ = 1 to compute b (i+1) = (1 ξ)b (i) + ξ b. Algorithm 2b. Endogenous grid method with precomputation b. Precompute {I 0 I n } using (12). Step 1. Computation of values of V on the grid. At iteration i, form = 1 M: a. Given b (i),findb (i) from (11) and compute Ŵ(k m zρ m; b (i) ) and Ŵ 1 (k m zρ m; b (i) ). b. Find c m = u 1 [βŵ 1 (k m zρ m; b (i) )]. d. Find value function on the grid v m u(c m ) + βŵ(k m zρ m; b (i) ).

4 4 Judd, Maliar, Maliar, and Tsener Supplementary Material Algorithm 3a. Envelope condition method a. Choose an approximating function V( ; b) V. b. Choose integration nodes, ε j, and weights, ω j, j = 1 J. c. Construct grid Γ ={k m z m } M m=1. d. Make an initial guess on b (1). At iteration i, perform the following steps: Step 1. Computation of values of V on the grid. a. Use b (i) to compute V 1 (k m z m ; b (i) ). b. Compute the corresponding values of c m using c m = u 1 [ V 1 (k m z m ; b (i) )(1 δ + z m f (k m )) 1 ]. c. Find k m using k m = (1 δ)k m + z m f(k m ) c m. d. Find value function on the grid v m u(c m ) + β J j=1 ω j V(k m zρ m exp(ε j ); b (i) ). Step 2. Computation of b that fits the value function on the grid. Run a regression to find b: b = arg minb v m V(k m z m ; b). Step 3. Convergence check and fixed-point iteration. a. Check for convergence for i 2: endstep2if 1 M (k m )(i) (k m )(i 1) (k < m) (i 1) b. Use damping with ξ = 1 to compute b (i+1) = (1 ξ)b (i) + ξ b. Algorithm 3b. Envelope condition method with precomputation b. Precompute {I 0 I n } using (12). Step 1. Computation of values of V on the grid. At iteration i,form = 1 M: a. Given b (i),findb (i) from (11); use b (i) to compute V 1 (k m z m ; b (i) ) and use b (i) to compute V(k m zρ m; b (i) ). d. Find value function on the grid v m u(c m ) + β V(k m zρ m; b (i) ).

5 Supplementary Material How to solve dynamic stochastic models 5 Algorithm 4a. Euler equation algorithm parameterizing Q a. Choose an approximating function Q( ; b) Q. b. Choose integration nodes, ε j, and weights, ω j, j = 1 J. c. Construct grid Γ ={k m z m } M m=1. d. Make an initial guess on b (1). At iteration i, perform the following steps: Step 1. Computation of values of Q on the grid. a. Use b (i) to compute Q(k m z m ; b (i) ). b. Find k m using k m = (1 δ)k m + z m f(k m ) u 1 ( Q(k m z m ;b (i) ) 1 δ+z m f(k m ) ). c. Find the values of q m on the grid q m β J j=1 ω j Q(k m zρ m exp(ε j ); b (i) )[1 δ + zf (k m )]. Step 2. Computation of b that fits the Q function on the grid. Run a regression to find b: b = arg minb q m Q(k m z m ; b). Step 3. Convergence check and fixed-point iteration. a. Check for convergence for i 2:endStep2if 1 M (k m )(i) (k m )(i 1) (k < m) (i 1) b. Use damping with ξ = 1 to compute b (i+1) = (1 ξ)b (i) + ξ b. Algorithm 4b. Euler equation algorithm parameterizing Q with precomputation b. Precompute {I 0 I n } using (12). Step 1. Computation of values of Q on the grid. At iteration i,form = 1 M: a. Given b (i),findb (i) from (11); compute Q(k m z m ; b (i) ) and Q(k m zρ m; b (i) ). c. Find the values of q m on the grid q m β Q(k m zρ m; b (i) )[1 δ + zf (k m )].

6 6 Judd, Maliar, Maliar, and Tsener Supplementary Material Algorithm 5a. Euler equation algorithm parameterizing Q and K a. Choose approximating functions K( ; v) K and Q( ; b) Q. b. Choose integration nodes, ε j, and weights, ω j, j = 1 J. c. Construct grid Γ ={k m z m } M m=1. d. Make an initial guess on v (1). At iteration i, perform the following steps: Step 1. Computation of values of Q on the grid. a. Use v (i) to compute K(k m z m ; v (i) ). b. Find c m using c m = (1 δ)k m + z m f(k m ) K(k m z m ; v (i) ). c. Find the values of q m on the grid q m u (c m )[1 δ + z m f (k m )]. Step 2. Computation of v that fits the capital function on the grid. a. Run a regression to find b: b = arg Mm=1 minb q m Q(k m z m ; b), and compute Q(k m z m ; b). b. Compute the values of next-period capital on the grid Jj=1 ω j Q( K(k m z m ;v (i) ) z ρ m exp(ε j ); b) k m β (1 δ + zf Q(k (k m z m ; b) m )) K(k m z m ; v (i) ). c. Run a regression to find v: v = arg min Mm=1 v k m K(k m z m ; v (i) ). Step 3. Convergence check and fixed-point iteration. a. Check for convergence for i 2: endstep2if 1 M (k m )(i) (k m )(i 1) (k < m) (i 1) b. Use damping with ξ = 0 15 to compute v (i+1) = (1 ξ)v (i) + ξ v.

7 Supplementary Material How to solve dynamic stochastic models 7 Algorithm 5b. Euler equation algorithm parameterizing Q and K with precomputation b. Precompute {I 0 I n } using (12). Step 2. Computation of v that fits the capital function on the grid. a. Run a regression to find b, b = arg Mm=1 minb q m Q(k m z m ; b), and compute Q(k m z m ; b). Given b, find b from (11), and compute Q( K(k m z m ; v (i) ) zm; b ρ ). b. Compute the value of next-period capital on the grid k m β Q( K(k m z m ;v (i) ) zm; b ρ ) (1 δ + zf Q(k (k m z m ; b) m )) K(k m z m ; v (i) ). Algorithm 6. Euler equation algorithm parameterizing K (not compatible with precomputation) a. Choose approximating functions K( ; v) K. Step 1. Computation of values of k on the grid. a. Use v (i) to compute K(k m z m ; v (i) ) and k m j = K( K(k m z m ; v (i) ) zm ρ exp(ɛ j ); v (i) ), j = 1 J. b. Find c m j using c m j = (1 δ) K(k m z m ; v (i) ) + z m f( K(k m z m ; v (i) )) k m j. c. Find the values of c m on the grid u (c m ) = β J j=1 ω j u (c m j )[1 δ + zρ m exp(ɛ j )f ( K(k m z m ; v (i) ))]. d. Find the values of k m on the grid k m = (1 δ)k m + z m f(k m ) c m. Step 2. Computation of v that fits the capital function on the grid. a. Run a regression to find v: v = arg min v k m K(k m z m ; v (i) ).

8 8 Judd, Maliar, Maliar, and Tsener Supplementary Material Appendix B: ECM and Euler equation algorithms for solving a growth model with elastic labor supply In Algorithms 7 and 8, we provide a description of ECM and Euler equation algorithms, which we use to solve the neoclassical stochastic growth model with valued leisure described in Section 2. Algorithm 7a. Envelope condition method a. Choose an approximating function V( ; b) V. b. Choose integration nodes, ε j, and weights, ω j, j = 1 J. c. Construct grid Γ ={k m z m } M m=1. d. Make an initial guess on b (1). At iteration i, perform the following steps: Step 1. Computation of values of V on the grid. a. Use b (i) to compute V 1 (k m z m ; b (i) ). b. Solve for l m that satisfies B(1 l m ) μ [1 δ + z m f 1 (k m l m )]=V 1 (k m z m ; b (i) )z m f 2 (k m l m ). c. Compute the corresponding values of c m using c m = u 1 1 [ B(1 l m) μ z m f 2 (k m l m ) ]. d. Find k m using k m = (1 δ)k m + z m f(k m l m ) c m. e. Find value function on the grid v m u(c m l m ) + β J j=1 ω j V(k m zρ m exp(ε j ); b (i) ). Step 2. Computation of b that fits the value function on the grid. Run a regression to find b: b = arg minb v m V(k m z m ; b). Step 3. Convergence check and fixed-point iteration. a. Check for convergence for i 2:endStep2if 1 M (k m )(i) (k m )(i 1) (k < m) (i 1) b. Use damping with ξ = 1 to compute b (i+1) = (1 ξ)b (i) + ξ b.

9 Supplementary Material How to solve dynamic stochastic models 9 Algorithm 7b. Envelope condition method with precomputation b. Precompute {I 0 I n } using (12). Step 1. Computation of values of V on the grid. At iteration i,form = 1 M: a. Given b (i),findb (i) from (11); use b (i) to compute V 1 (k m z m ; b (i) ) and use b (i) to compute V(k m zρ m; b (i) ). e. Find value function on the grid v m u(c m l m ) + β V(k m zρ m; b (i) ).

10 10 Judd, Maliar, Maliar, and Tsener Supplementary Material Algorithm 8a. Euler equation algorithm parameterizing Q a. Choose an approximating function Q( ; b) Q. b. Choose integration nodes, ε j, and weights, ω j, j = 1 J. c. Construct grid Γ ={k m z m } M m=1. d. Make an initial guess on b (1). At iteration i, perform the following steps: Step 1. Computation of values of Q on the grid. a. Use b (i) to compute Q(k m z m ; b (i) ). b. Solve for l m that satisfies B(1 l m ) μ [1 δ + z m f 1 (k m l m )]= Q(k m z m ; b (i) )z m f 2 (k m l m ). c. Compute the corresponding values of c m using c m = u 1 1 [ B(1 l m) μ z m f 2 (k m l m ) ]. d. Find k m using k m = (1 δ)k m + z m f(k m l m ) c m. e. Find the values of q m on the grid q m β J j=1 ω j Q(k m zρ m exp(ε j ); b (i) )[1 δ + zf 1 (k m l m )]. Step 2. Computation of b that fits the Q function on the grid. Run a regression to find b: b = arg minb q m Q(k m z m ; b). Step 3. Convergence check and fixed-point iteration. a. Check for convergence for i 2:endStep2if 1 M (k m )(i) (k m )(i 1) (k < m) (i 1) b. Use damping with ξ = 1 to compute b (i+1) = (1 ξ)b (i) + ξ b.

11 Supplementary Material How to solve dynamic stochastic models 11 Algorithm 8b. Euler equation algorithm parameterizing Q with precomputation b. Precompute {I 0 I n } using (12). Step 1. Computation of values of Q on the grid. At iteration i,form = 1 M: a. Given b (i),findb (i) from (11); compute Q(k m z m ; b (i) ) and Q(k m zρ m; b (i) ). e. Find the values of q m on the grid q m β Q(k m zρ m; b (i) )[1 δ + zf 2 (k m l m )].

12 12 Judd, Maliar, Maliar, and Tsener Supplementary Material Appendix C: Euler equation algorithm for solving the multicountry model In this section, we describe the Euler equation methods that we use to analyze the multicountry model in Section 3. Algorithm 9a. Euler equation algorithm parameterizing Q h and K h a. Choose approximating functions K h ( ; v h ) K h, h = 1 N,andQ h ( ; b h ) Q h. b. Choose integration nodes, ε j = (εj 1 εn j ), and weights, ω j, j = 1 J. c. Fix simulation length T = 2000 and (k 0 z 0 ) = (1 1), where1 (1 1) R N. d. Draw and fix a sequence of productivity levels {z t } t=1 T using (40). e. Construct integration nodes, z t+1 j = (zt+1 j 1 zn t+1 j ) with zh t+1 j = (zh t )ρ exp(εj h). f. Make an initial guess on (v 1 ) (1) (v h ) (1). At iteration i,given(v 1 ) (i) (v h ) (i) : Step 1. Computation of values of Q on the simulated points. For t = 1 T: a. Use k h t+1 = K h (k t z t ; (v h ) (i) ), h = 1 N, to recursively calculate {k t+1 } t=0 T. b. Compute {c t } t=0 T satisfying c h t = ( N h=1 c h t )/N. c. Compute q h t from q h t u (c h t )[1 δ + zh t f (k h t )]. Step 2. Computation of v h that fits the values of capital on the simulated points. a. Run a regression to find b h, bh = arg min Mm=1 b h q t h Q h (k t z t ; b h ), and compute Q h (k t z t ; b h ). b. Compute the values of next-period capital on the simulated points Jj=1 ω j Qh (k t+1 z t+1 j ; b h ) kh t+1 β [1 δ + z Q h h (k t z t ; b h ) t f (k h t )]Kh (k t z t ; v h ), h = 1 N. c. Run a regression to find v h : v h arg min Tt=1 v h k h t+1 K h (k t z t ; (v h ) (i) ). Step 3. Convergence check and fixed-point iteration. a. Check for convergence for i 2;endStep2if 1 Tt=1 Nh=1 TN (kh t+1 )(i) (k h t+1 )(i 1) < (k h t+1 )(i 1) b. Use damping with ξ = 0 1 to compute (v h ) (i+1) = (1 ξ)(v h ) (i) + ξ v h.

13 Supplementary Material How to solve dynamic stochastic models 13 Algorithm 9b. Euler equation algorithm parameterizing Q h and K h with precomputation b. Precompute {I 0 I n } using (53). Step 2. Computation of v h that fits the values of capital on the simulated points. a. Run a regression to find b h, bh = arg min Mm=1 b q t h Q h (k t z t ; b h ), and compute Q h (k t z t ; b h ). Given b,find b from (47), and compute Q h (k t+1 z ρ t ; ( b h ) ). b. Compute the values of next-period capital on the simulated points kh t+1 β Q h (k t+1 z ρ t ;( b h ) ) [1 δ + z Q h h (k t z t ; b h ) t f (k h t )]Kh (k t z t ; v h ), h = 1 N. Algorithm 10. Euler equation algorithm parameterizing K h (not compatible with precomputation) a. Choose approximating functions K h ( ; v h ) K h, h = 1 N. Step 1. Computation of values of k h t+1 on the simulated points. For t = 1 T: a. Use k h t+1 = K h (k t z t ; (v h ) (i) ), h = 1 N, to recursively calculate {k t+1 } t=0 T. b. Compute {c t } t=0 T satisfying ct h = ( N h=1 ct h)/n. c. Compute k h t+2 j = K h (k t+1 z ρ t exp(ɛ j); (v h ) (i) ). d. Find {c t+1 j } t=0 T satisfying ct+1 j h = ( N h=1 ct+1 j h )/N. Step 2. Computation of v h that fits the values of capital on the simulated points. a. Compute the values of next-period capital on the simulated points for h = 1 N: kh t+1 β J j=1 ω j u (ct+1 j h )(u (ct h)) 1 [1 δ + (zt h)ρ exp(ɛ h j )f (k h t+1 )]Kh (k t z t ; v h ). b. Run a regression to find v h : v h arg min Tt=1 v h k h t+1 K h (k t z t ; (v h ) (i) ).

14 14 Judd, Maliar, Maliar, and Tsener Supplementary Material Table C.1. Accuracy and cost of the Euler equation algorithm for the multicountry model: GSSA parameterizing K. Polynomial Precomputation M1 M2 GH(2) Degree L 1 L CPU L 1 L CPU L 1 L CPU L 1 L CPU 1st nd rd th th N = 5 1st nd rd ,221 4th ,640 N = 10 1st ,263 2nd ,610 3rd , ,232 N = 20 1st nd ,204 N = 30 1st nd ,543 Note: The main result columns correspond to variants of GSSA that evaluate integrals by using the precomputation method, the monomial integration methods with 2N and 2N nodes, and the Gauss Hermite quadrature method with two nodes, respectively; the statistics L 1 and L are, respectively, the average and maximum of absolute residuals across optimality condition and test points (in log 10 units) on a stochastic simulation of 10,000 observations; CPU is the time necessary for computing a solution (in seconds).

15 Supplementary Material How to solve dynamic stochastic models 15 Appendix D: Euler equation algorithm for solving the Aiyagari model In this section, we describe a solution algorithm for Aiyagari s (1994) model analyzed in Section 4. We focus on the algorithm for solving the individual problem in which the precomputation results appear. The rest of the algorithm is standard and relies on stochastic simulation and a bisection technique as in Aiyagari (1994). Algorithm 11a. Euler equation algorithm parameterizing Q and K a. Given K, compute R = αk α 1 δ and W = (1 α)(k) α. b. Choose an approximating function K( ; b l ) K and Q( ; v l ) Q for l {1 J}. c. Construct grid Γ ={k m z m } M m=1. d. Make an initial guess on K( ; b l ) (1). At iteration i, perform the following steps: Step 1. Computation of values of K on the grid. a. Find c m l using c m l = Wz l + (1 + R)k m K( ; b l ). (1 + R). c. Run a regression to find v l for l {1 J}: v l = arg min Mm=1 v l q m l Q(k m ; v). d. Find k m l using k m l = Wz l + (1 + R)k m u 1 [β j π jl Q(k m l ; vj )]. b. Compute q m l = c γ m l Step 2. Convergence check and fixed-point iteration. a. Check for convergence for i 2; endstep1if max m l (c m l) (i 1) (c m l ) (i) (c m l ) (i 1) < b. Use damping with ξ = 0 5 to compute ( K( ; b l )) (i+1) = (1 ξ)( K( ; b l )) (i) + ξ( k m l )(i).

16 16 Judd, Maliar, Maliar, and Tsener Supplementary Material Algorithm 11b. Euler equation algorithm parameterizing Q with precomputation Step 1. Computation of values of K on the grid. At iteration i,form = 1 M: d. Given v l, l = 1 J,find( v l ) from (67). e. Find k m l using k m l = Wz l + (1 + R)k m u 1 [β Q(k m l ; ( vl ) )]. Algorithm 12. Euler equation algorithm for Aiyagari s (1994) model parameterizing K (not compatible with precomputation) b. Choose an approximating function K( ; b l ) K for l {1 J}. Step 1. Computation of values of K on the grid. At iteration i,form = 1 M: For l ={1 J}: b. Compute k m j = K(k m l ; bj ) for j ={1 J}. c. Find c m j for j ={1 J} using c m j = Wz j + (1 + R)k m l k m j. d. Find k m l using k m l = Wz l + (1 + R)k m u 1 [β j π jl(c m j ) γ [1 + R]]. Co-editor José-Víctor Ríos-Rull handled this manuscript. Manuscript received 9 November, 2012; final version accepted 6 December, 2016; available online 2 June, 2017.