Matlab Programming and Quantitative Economic Theory

Transcription

1 Matlab Programming and Quantitative Economic Theory Patrick Bunk and Hong Lan SFB C7 Humboldt University of Berlin June 4, 2010 Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

2 Quantitative Economic Theory one utility maximizing representative agent (HH) one profit maximizing firm market structure equilibrium, s.t. HH optimize given their BC firms maximize profits markets clear approximate the system around an equilibrium > find out the numerical solution to this class of models Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

3 A real business cycle model with labor choice max {c t,n t,k t+1 } t=0 E t t=0 β t [ log c t φn t ] s.t. y t = z t k α t n 1 α t (1) y t = c t + i t (2) k t+1 = (1 δ)k t + i t (3) z t+1 = ρz t + σɛ t+1 ɛ t i.i.d E t ɛ t+1 = 0 (4) k 0 is given Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

4 Bells and Whistles certain level of heterogeneity in a an economy goods capital stocks HH firms market structures production technologies matching markets (labor) idea: add complications might help to get it right hard to evaluate the quality of approximations Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

5 1 Recursive Form 2 The Curse of Dimensionality 3 Sparse Grids 4 Parallelization 5 Perturbation Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

6 Computational Difficulties Many economic models are of high dimension DSGE: multiple kinds of capital stocks, agents, firms, countries... Games: multiple players and states Bayesian analyses: compute high-dimensional integrals Bootstrapping: analyze many n-dimensional samples from n data points Simulation of large Markov processes - MCMC, Gibbs sampling... Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

7 Computational Difficulties If we are interested in the globally accurate solution of a high dimensional model Global methods are cursed by dimensionality or circumvent the curse by local approximation Perturbation method: Schmitt-Grohe and Uribe (2004) Linear methods: Blanchard and Kahn (1980) Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

8 Recursive Form A real business cycle model with labor choice max {c t,n t,k t+1 } t=0 E t t=0 β t [ log c t φn t ] s.t. y t = z t k α t n 1 α t (5) y t = c t + i t (6) k t+1 = (1 δ)k t + i t (7) z t+1 = ρz t + σɛ t+1 ɛ t i.i.d E t ɛ t+1 = 0 (8) k 0 is given Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

9 Recursive Form The recursive formulation of the optimization problem The stochastic case V (k, z) = max c,n {log(c) φn + βev (k, z )} (9) s.t. c + k = zk α n 1 α + (1 δ)k (10) z = ρz + σɛ (11) The deterministic case V (k) = max c,n {log(c) φn + βv (k )} (12) s.t. c + k = zk α n 1 α + (1 δ)k (13) Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

10 Recursive Form The deterministic Bellman equation { V (k) = max log(c) φn + βv (k ) } (14) c,n s.t. c + k = zk α n 1 α + (1 δ)k (15) The Bellman equation implicitly defines three policy functions c = c(k) (16) n = n(k) (17) k = k(k) (18) Computational task: Find out the parameterization of these policy functions Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

11 Recursive Form Result 20 policy function for capital k prime k Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

12 Recursive Form Characterization FOCs 1 c = βv k (k ) (19) φ = (1 α)β y n V k (k ) (20) Envelope condition one step forward of the envelope, V k (k) = 1 c R where R = α y k + 1 δ (21) V k (k ) = 1 c R (22) Euler equations [ ] c 1 = β c R φc = (1 α) y n (23) (24) Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

13 Recursive Form Solving for the stationary steady state Steady state represents the long-term equilibrium relationship of the model, It is the constant solution of Euler equations and all constraints, and it is the constant solution of the Bellman equation! Need to use parameter values, δ = α = 0.36 β = 0.99 φ = 2.5 z = 1 An example, the consumption Euler equation in steady state, 1 = βr, using β = 0.99, we obtain R = 1/0.99 The steady state values of the model are consumption c output y capital k labor supply n These values are the best guesses to initialize the value function iteration Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

14 Recursive Form The discretized state space algorithm (Deterministic case) Make grids for both state variable (k): k 1 < k 2 <... < k kg, and for the control variable (n): n 1 < n 2 <... < n hg Calculate u(n i, k j, k n) for all state-control variable pairs (3 dimensional combinations), (n i, k j, k n) Find the new iteration of the value function as, { } V 1 (k n) = max u(n i, k j, k n) + βv 0 (k j ) k j,n Check for convergence, i.e. check norm(v 1 (k n) V 0 (k n)) tol for all grid values k n If not, go back to step 2 If so, done Use the indices from the maximum step 4 to recover the policy function Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

15 Recursive Form Discretized state space - domain of the value function labor in period t capital in period t capital in period t 20 Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

16 Recursive Form Discretized state space - domain of the value function Matlab Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

17 Recursive Form 1 Recursive Form 2 The Curse of Dimensionality 3 Sparse Grids 4 Parallelization 5 Perturbation Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

18 The Curse of Dimensionality Curse of dimensionality The computational cost and storage requirements for a n-dimensional value function approximation with a prescribed tolerance of error V n ˆV n = O(n) depends exponentially on the dimension n In the stochastic version of the model, the computational cost and storage requirements of integrating conditional expectation of q states depends exponentially on the number of states q Increasing the number of points in the discretized state space will dramatically increase computational burden, even on the most powerful machine There is some good news! Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

19 The Curse of Dimensionality Mitigate the Curse Judd (1992), (1998), Gaspar and Judd (1997) a large domain creates a high computational burden carefully choose the domain of the value function choose the shape of the domain Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

20 The Curse of Dimensionality Judd s suggestion I Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

21 The Curse of Dimensionality Judd s suggestion I Get rid of kinks in the model (smoothness improves computation) Use finite-dimensional states Use continuous time formulation > reduces state space by proper modeling Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

22 The Curse of Dimensionality Judd s suggestion II Reduce state space a bit further > Use spheres instead of cubes Spheres are much more compact hyper sphere and circumscribed hypercube as dimension gets large, most of the mass is in the corners Ratio of n-sphere to n-cube volume for n even ( π 2 )n ((n/2)!) 1 Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

23 The Curse of Dimensionality Judd s suggestion II Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

24 The Curse of Dimensionality 1 Recursive Form 2 The Curse of Dimensionality 3 Sparse Grids 4 Parallelization 5 Perturbation Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

25 Sparse Grids Sparse Grid Sparse grids have been used in several context Numerical integration Projection methods for DSGE models Barthelmann, Novak and Ritter (2000), Computational Mathematics Kubler and Krueger (2004), JEDC Malin, Krueger and Kubler (2007), JEDC project report Solution to partial differential equations Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

26 Sparse Grids A sparse grid Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

27 Sparse Grids Sparse Grid To construct the sparse grid for a DSGE model, we need to Specify the basis function, i.e. Chebyshev nodes or Gauss-Lobotto nodes, both of them are defined on [ 1, 1] Construct the sparse grid using Smolyak s method Convert the domain into the actual domain of the state variables in the model Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

28 Sparse Grids An example from Malin, Krueger and Kubler (2007) The first level, three dimensional sparse grid compute the basis for the grid of points G 1 = {0} For n = 2,..., ) G n = {ζ 1,...ζ n } where ζ j = cos j = 1,...n ( π(j 1) n 1 Define a sequence of positive integers by m(1) = 1 and m(i) = 2 i for i = 2, 3,... This leads to G m(1) = G 1 = {0} and G m(2) = G 3 = { 1, 0, 1} Key property of this Smolyak s construction: G m(i) G m(i+1) Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

29 Sparse Grids An example from Malin, Krueger and Kubler (2007) The first level, three dimensional sparse grid The grid: H 3,1 = G m(2) G m(1) G m(1) G m(1) G m(2) G m(1) G m(1) G m(1) G m(2) The first level grid consists of 7 points:x 1,..., x 7 ( 1, 0, 0), (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 1, 0), (0, 0, 1) and (0, 0, 1) Use (once per dimension): x i,1 = 2 s i,1 s i,1 s i,1 s i,1 1 to convert the standard sparse grid into the state variable sparse grid Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

30 Sparse Grids An example from Malin, Krueger and Kubler (2007) State Sparse Grid Standard Sparse Grid labor in period t capital in period t capital in period t Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

31 Sparse Grids 1 Recursive Form 2 The Curse of Dimensionality 3 Sparse Grids 4 Parallelization 5 Perturbation Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

32 Parallelization Bound the State Space What if all this is still not enough? Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

33 Parallelization Getting Things Done Standard strategy > 1. calculate computing time (back on the envelope) 2. optimize the code 3. Throw money at the problem (buy a faster computer) 3a. wait for better computers Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

34 Parallelization Calculate the State Space Size and Computing Time State Space: choices: capital k p t, k t h consumption c t labor l t investment k p t+1, k h t+1 payoff: flow utility function u(c,l), discount factor β Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

35 Parallelization Calculate the State Space Size and Computing Time domains and discretization: [0, 1000] for k p, k h, [0, 100] for l # choices: = 10 8 # states: = 10 6 full enumeration: (#choices) #states = (10 8 ) (106 ) DP: #choices (#States) 2 = = Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

36 Parallelization 3 Curses of Dimensionality state space choice space domains and discretization steps Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

37 Parallelization Calculate the State Space Size and Computing Time How long does it take? how many computations are needed per point? = GFLOPS (10 9 operations per second) 3171years CPU GFLOPS Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

38 Parallelization Superfluous space k t+1 y t (k t ) + (1 δ)k t, k t+1 (1 δ)k t Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

39 Parallelization Judd s suggestion I Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

40 Parallelization Calculate the State Space Size and Computing Time How long does it take? how many computations are needed per point? = GFLOPS (10 9 operations per second) 3171years CPU GFLOPS superfluous space (-90%) Judd I (-99%) (hard choice) Judd II (-70%) > down to 347 days/ 23 GFLOPS 15 days on a decent PC Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

41 Parallelization Calculate the State Space Size and Computing Time How long does it take? how many computations are needed per point? = GFLOPS (10 9 operations per second) 3171years CPU GFLOPS superfluous space (-90%) Judd I (-99%) (hard choice) Judd II (-70%) > down to 347 days/ 23 GFLOPS 15 days on a decent PC atrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

42 Parallelization Getting Things Done Standard strategy 1. calculate computing time (back on the envelope) Is the problem large? > 2. optimize the code (parallelize) 3. Throw money at the problem (buy a faster computer) Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

43 Parallelization Getting Things Done parallelize code one of the hardest tasks in computer science one of the main challenges in the past 20 years sequential processing reached some physical limits (6 GHz?) speed of light limiting factor cm s > 5cm/calculation prize just got bigger! Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

44 Parallelization Getting Things Done parallelize code one of the hardest tasks in computer science one of the main challenges in the past 20 years sequential processing reached some physical limits (6 GHz?) speed of light limiting factor cm s > 5cm/calculation prize just got bigger! Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

45 Parallelization Divide & Conquer algorithm design paradigm 1. break down a problem recursively 2. keep track of branches beautiful source code (recursion) applications: sorting (quicksort) multiplication (Karatsuba) Fourier transformation DFT/FFT (Cooley-Tukey) Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

46 Parallelization Example of D&C - Value Iteration V (s) := max a { s P(s, s a)[r a (s, s ) + βv (s )] } (25) Bellman equation as an update rule take any starting guess: V 0 (s) := 0, s S O(S) update V 1 (s) := max a { s P(s, s a)[r a (s, s ) + βv 0 (s )] } s S O(S 2 C) or O(SC) compute t := max s S V t (s) V t 1 (s) O(S) loop until t ɛ O(P(S)) Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

47 Parallelization Example of D&C - Value Iteration V (s) := max a { s P(s, s a)[r a (s, s ) + βv (s )] } (25) Bellman equation as an update rule take any starting guess: V 0 (s) := 0, s S O(S) update V 1 (s) := max a { s P(s, s a)[r a (s, s ) + βv 0 (s )] } s S O(S 2 C) or O(SC) compute t := max s S V t (s) V t 1 (s) O(S) loop until t ɛ O(P(S)) Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

48 Parallelization Example of D&C - Value Iteration V (s) := max a { s P(s, s a)[r a (s, s ) + βv (s )] } (25) Bellman equation as an update rule take any starting guess: V 0 (s) := 0, s S O(S) update V 1 (s) := max a { s P(s, s a)[r a (s, s ) + βv 0 (s )] } s S O(S 2 C) or O(SC) compute t := max s S V t (s) V t 1 (s) O(S) loop until t ɛ O(P(S)) Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

49 Parallelization Example of D&C - Value Iteration V (s) := max a { s P(s, s a)[r a (s, s ) + βv (s )] } (25) Bellman equation as an update rule take any starting guess: V 0 (s) := 0, s S O(S) update V 1 (s) := max a { s P(s, s a)[r a (s, s ) + βv 0 (s )] } s S O(S 2 C) or O(SC) compute t := max s S V t (s) V t 1 (s) O(S) loop until t ɛ O(P(S)) Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

50 Parallelization Example of D&C - Value Iteration V (s) := max a { s P(s, s a)[r a (s, s ) + βv (s )] } (25) Bellman equation as an update rule take any starting guess: V 0 (s) := 0, s S O(S) update V 1 (s) := max a { s P(s, s a)[r a (s, s ) + βv 0 (s )] } s S O(S 2 C) or O(SC) compute t := max s S V t (s) V t 1 (s) O(S) loop until t ɛ O(P(S)) Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

51 Parallelization Getting Things Done Standard strategy 1. calculate computing time (back on the envelope) 2. optimize the code (parallelize) > 3. Throw money at the problem (buy a faster computer) 3a. wait for better computers Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

52 Parallelization Speed things up Name CPU Price GFLOPS ratio Top Dell OptiPlex780 C2D Dell Workstation 2xXeon Table: Currently available computers Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

53 Parallelization Speed things up II new trend: General Purpose processing with GPUs GPUs calculate parts of the image > branches specialized early in multi-core processing last years development of interface to tap into that power CUDA by Nvidia 2007 OpenCL Standard 2009 Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

54 Parallelization Speed things up III Name CPU Price GFLOPS ratio Top Dell OptiPlex780 C2D Dell Workstation 2xXeon EUR PC NVIDIA GT Table: Currently available processing power Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

55 Parallelization Speed things up III Name CPU Price GFLOPS ratio Top Dell OptiPlex780 C2D Dell Workstation 2xXeon EUR PC NVIDIA GT EUR PC NVIDIA GTX Table: Currently available processing power Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

56 Parallelization Speed things up III Name CPU Price GFLOPS ratio Top Dell OptiPlex780 C2D Dell Workstation 2xXeon EUR PC NVIDIA GT EUR PC NVIDIA GTX EUR PC ATI EUR PC 3xATI Table: Currently available processing power Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

57 Parallelization net performance gains Test N CPU time GPU time Speed Up exp(a) 1600x A.*B 1600x Black-Scholes A*B 1600x FFT(A) 1600x Table: average performance gains GTX8800 (512) vs C2D (23) Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

58 Parallelization CUDA CUDA is a C SDK CUDA Framework is written and accessed in C works on all supported graphics cards without code changes Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

59 Parallelization CUDA - How does it look? Look at the code for vector addition Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

60 Parallelization Why Matlab? C (90 % of code on how to do things, 10% on what to do) > CS are better at this than Economists! Matlab (0% of code on how, 100% on what to do) my requirements: hide CUDA-API take care of low level stuff Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

61 Parallelization Why Matlab? Matlab GPU plugins requirements: tap into GPU power w/o GPU knowledge execute Matlab code on the GPU transparently ability to port code construct new functions that work on the GPU w/o deep CUDA knowledge Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

62 Parallelization GPUmat and Alternatives MathWorks is working on it (closed registration) AcceleratorEyes Jacket - closed source (hard to fix) ($350+) GPUlib - open source, tedious memory management GPUmat - free for academic use, early development stage Mathematica CUDA plugin (alpha) R (R+GPU) plugin (beta) Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

63 Parallelization How to tap into the GPU-Power adjust your algorithms (sequential to parallel) check your graphics card (5min) install CUDA Framework (5min) install CUDA Matlab Plugin (3min) unzip GPUmat Framework (2min) (private beta, public end of June) run GPUstart.m Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

64 Parallelization How to use GPUmat Matlab Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

65 Parallelization Laptop CPU Core i5m 21 GFLOPS GPU GT330M 182 GFLOPS Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

66 Parallelization net performance gains Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

67 Parallelization Backup Slides Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

68 Perturbation Perturbation Assume we know the solution, plug the assumed solution back into the system, then perturbate the system using total differentials. The previous step will impose some restrictions on the system Use those restrictions to parameterize the assumed solution. Done! Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

69 Perturbation The model in equilibrium [ 1 Consumption Euler equation: c t = βe t Labor Euler equation: c t = 1 α φ z tkt α n α t ( 1 c t+1 αz t+1 k α 1 Budget constraint: c t + k t+1 = z t kt α n 1 α t + (1 δ)k t Productivity shock: z t+1 = ρz t + σɛ t+1 t+1 n1 α Endogenous state variable, k t, exogenous state variable, z t, endogenous variables, c t, n t There are four equations, four unknowns t δ )] Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

70 Perturbation The assumed solution Assume the following solution c t = c(k t, z t, σ) n t = n(k t, z t, σ) k t+1 = k(k t, z t, σ) The productivity shock implies z t+1 = z(z t, σ), so that c t+1 = c(k(k t, z t, σ), z(z t, σ), σ) n t+1 = n(k(k t, z t, σ), z(z t, σ), σ) }{{}}{{} k t+1 z t+1 Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

71 Perturbation Solving for steady state The assumed solution in steady state takes form of c = c(k, z, 0) n = n(k, z, 0) k = k(k, z, 0) They can be solved using given parameter values, all Euler equations and constraints in steady state σ = 0 in steady state, meaning there is no uncertainty Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

72 Perturbation Solving for dynamics Example: c t + k t+1 z t k α t n 1 α t (1 δ)k t = 0 Plug in the solution c(k t, z t, σ) + k(k t, z t, σ) z t k α t n(k t, z t, σ) 1 α (1 δ)k t = 0 In general F (k t, z t, σ) = 0 First order total differential DF = D0 = 0, or F k dk t + F zdz t + F σdσ = 0 dk ] t Equivalently, [F k F z F σ dz t = 0 dσ To allow arbitrary change in k t, z t, σ (equivalently, to allow dk t, dz t, dσ to take any value), we require F k = F z = F σ = 0 Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

73 Perturbation Solving for dynamics Euqation c(k t, z t, σ) + k(k t, z t, σ) z t k α t n(k t, z t, σ) 1 α (1 δ)k t = 0 To find F k, first find F kt, then evaluate F kt at steady state, F kt = c k (k t, z t, σ) + k k (k t, z t, σ) αz t k α 1 t n(k t, z t, σ) 1 α (1 α)z t k α t n(k t, z t, σ) α n k (k t, z t, σ) (1 δ) Evaluate F kt at steady state, using F k = 0, F k = c k + k k αzk α 1 n 1 α (1 α)zk α n α n k (1 δ) = 0 Similarly, for F z and F σ, we have F z = c z + k z k α n 1 α (1 α)zk α n α n z = 0 F σ = c σ + k σ (1 α)zk α n α n σ = 0 Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

74 Perturbation Solving for dynamics Apply the same procedure to consumption and labor Euler equations, we can altogether obtain 9 equations for 9 unknowns 9 unknowns are (c k, c z, c σ, k k, k z, k σ, n k, n z, n σ) In first order perturbation, (c σ, k σ, n σ) are always zero, because by requiring F k = F z = F σ = 0, we only allow changes to happen in these three basic directions, σ has no impact until we allow changes to happen along some composite directions, i.e. F kk, F zz. The 9 unknowns can be used to construct the first order approximation of the assumed solution c t = c + c k (c t c) + c z(z t z) n t = n + n k (c t c) + n z(z t z) k t+1 = k + k k (c t c) + k z(z t z) Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69

75 Perturbation A short summary Perturbation is still a local approximation method Using total differential and derivatives means we observe the model s behavior in the neighborhood of the equilibrium Using higher order perturbation, we can observe the model s behavior when changes happen along all directions around the equilibrium Schmitt-Grohe and Uribe (2004) s code of perturbation is efficient, because it takes analytic derivatives first Early version of Dynare takes numerical derivatives, that is why it crashes often Patrick Bunk and Hong Lan (SFB C7 Humboldt Matlab University Programming of Berlin) and Quantitative Economic Theory June 4, / 69