Numerical solution of dynamic equilibrium
models under Poisson uncertainty
Olaf Posch (Aarhus University and CREATES)
and Timo Trimborn (University of Hannover)
09.09.2011
Introduction
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Since their first appearance in the 1970s, dynamic stochastic
general equilibrium (DSGE) models became the workhorse in
dynamic macroeconomic theory.
I
successfully capture aggregate dynamics over the business
cycle, but surprisingly quiet on the effects of rare events:
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actual/potential rare events: economic events (Great
Depression, financial crises), wartime (world wars), natural
disasters (tsunamis, hurricanes, earthquakes, asteroid
collisions), epidemics (Black Death, avian flu)
(Barro 2006, Quart. J. Econ.)
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Introduction
Theory:
I
rare events present in quality ladder, endogenous growth and
matching models
(Grossman, Helpman 1991, MIT press; Aghion, Howitt 1992,
Econometrica; Wälde 2005, Int. Econ. Rev.; Lentz, Mortensen
2008, Econometrica)
Empirics:
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anecdotic and historical evidence for rare disasters
(Rietz 1988, J. Monet. Econ.; Barro 2006, Quart. J. Econ.)
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empirical evidence for Poisson jumps in US output data
(Posch 2009, J. Econometrics)
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Introduction
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Caveat: no analytical solutions for DSGE models, need
computational methods
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powerful computational methods such as perturbation and
projection methods
(Judd 1992, J. Econ. Theory, 1998 MIT press)
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highly accurate solving DSGE models under Normal
uncertainty (at least locally)
(Aruoba, Fernández-Villaverde, Rubio-Ramírez 2006, J. Econ.
Dynam. Control)
Open questions:
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I
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effects of large shocks on approximation errors are largely
unexplored
no clear answer has been given on the effects of rare events on
optimal decisions such as consumption or leisure (Barro-Rietz
rare disaster hypothesis)
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Introduction
This paper:
I
proposes a simple and powerful method for determining the
transition process in continuous-time DSGE models under
Poisson uncertainty numerically
I
shows how to extend DSGE models and existing solution
methods in order to allow for the possibilities of rare events
illustrates our approach for two popular methods computing
numerical solutions to dynamic general equilibrium models, by
solving
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I
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the neoclassical growth model with disasters (extending
backward integration)
(Brunner, Strulik 2002, J. Econ. Dynam. Control)
the Lucas’ model of endogenous growth with disasters
(Relaxation algorithm)
(Trimborn, Koch, Steger 2008, Macroecon. Dynam.)
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Introduction
I
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Our framework:
continuous-time formulation of DGSE models, which imply an
equilibrium system of controlled stochastic differential
equations (SDEs) under Poisson uncertainty
I
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relatively simple to obtain reduced forms using stochastic
calculus
closed-form solutions available for reasonable parametric
restrictions, which are important benchmark solutions to
explore broader classes of models
(Judd 1997, J. Econ. Dynam. Control)
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Our message: solution method works
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obtain highly accurate policy functions even for the complete
state space (globally) compared to the benchmark analytical
solutions
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Introduction
Plan of the talk
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The macroeconomics theory
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The numerical solution
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A neoclassical growth model with disasters
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Conclusion
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The macroeconomic theory
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Consider the autonomous infinite horizon stochastic control
problem
Z ∞
max E
e−ρt u(xt , ct )dt
0
s.t.
dxt = f (xt , ct )dt + g(xt− , ct− )dNt
given initial states x0 = x, N0 = z, (x, z) ∈ R2+ and the
control {ct }∞
t=0
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{Nt }∞
t=0 Poisson process with arrival rate λ = λ(xt , ct ),
we denote xt− ≡ lims→t xs as the left-limit of the variable at
date t
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(Controlled SDE) For illustration, consider the stochastic
control problem where g(x, c) = g(x), λ(x, c) = λ,
u(x, c) = u(c) and fc (x, c) = −1.
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The macroeconomic theory
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Suppose that u00 (c) 6= 0, then the reduced form system reads
u0 (c(xt + g(xt )))
(1 + gx (xt ))λ
dct = ρ − fx (xt , ct ) + λ −
u0 (c(xt ))
u0 (ct )
dt
u00 (ct )
+ (c(xt− + g(xt− )) − c(xt− )) dNt ,
dxt =f (xt , ct )dt + g(xt− )dNt ,
where ct = c(xt ) denotes the optimal policy function.
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The appearance of c(xt + g(xt )) means that at date τ1 the
solution at another date τ ∈ R influences the slopes dct and
dxt .
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Economically: consumers are aware of the possibility of jumps
in their optimal consumption and take into account the level
of consumption if such events occur
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The numerical solution
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Technically, we solve
dct = f1 (ct , xt , ~c, ~x) dt + g1 (ct− , xt− )dNt ,
dxt = f2 (ct , xt , ~c, ~x) dt + g2 (ct− , xt− )dNt ,
augmented with boundary conditions for the beginning and
the end of the horizon.
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~c and ~x are the unknown optimal pairs for the entire solution
of the system above, need to compute on the solution on the
entire domain of xt , usually x ∈ (0, ∞)
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solve using the idea of Waveform Relaxation
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The numerical solution
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Step 1 (pathwise continuous):
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compute the policy function implied by the (conditional)
deterministic system
dct = f1 (ct , xt , ~c, ~x) dt
dxt
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= f2 (ct , xt , ~c, ~x) dt
Step 2 (discontinuities):
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the stochastic paths can be obtained by augmenting the
solution by realizations of the stochastic process Nt , making
use of the entire solution ~c
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The numerical solution
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Step 1a (initial guess):
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provide an initial guess of optimal pairs (~c0 , ~x0 ), the system
reduces to
dct = f˜1 (ct , xt ) dt
dxt = f˜2 (ct , xt ) dt
as the feedback is neglected and solve by standard algorithms
to obtain (~c1 , ~x1 )
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Step 1b (update guess and iterate):
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take the trial solution (~c0 , ~x0 ) as given and define for each
iteration i = 1, ..., n
dci = f1 (ci (t), xi (t), ~ci−1 , ~xi−1 ) dt
dki
=
f2 (ci (t), xi (t), ~ci−1 , ~xi−1 ) dt
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The numerical solution
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Technically: we construct a fix-point iteration for the operator
N such that the desired solution z : R → R2 is a fix point of
this operator: N (z) = z
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we start with a trial solution z0 and iterate by evaluating N
until kzi − zi−1 k is sufficiently small
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Remark: cubic spline interpolation,
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A neoclassical growth model with disasters
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One-good production economy with constant returns to scale
technology
(Merton 1975, Rev. Econ. Stud.)
Yt = AKtα L1−α
where
dKt = (It − δKt ) dt − γKt− dNt
Nt number of (natural) disasters up to time t at arrival rate
λ≥0
Kt capital stock, It investment, Ct consumption
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Closed economy
Yt = Ct + It
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A neoclassical growth model with disasters
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Benevolent planner maximizes
U0 ≡ E
Z ∞
e−ρt u(Ct )dt,
u0 > 0, u00 < 0
0
subject to
dKt = (Yt − Ct − δKt ) dt − γKt− dNt
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Preferences
u(Ct ) =
Ct1−θ
1−θ
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A neoclassical growth model with disasters
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Reduced form
dCt =
rt − ρ − δ − λ + λ(1 − γ)C̃(Kt )−θ
Ct dt
θ
− 1 − C̃(Kt− ) Ct− dNt
dKt =(AKtα L1−α − Ct − δKt )dt − γKt− dNt
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defining 1 − C̃(Kt− ) ≡ (C(Kt ) − C((1 − γ)Kt ))/C(Kt )
percentage change of optimal consumption after a disaster
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A neoclassical growth model with disasters
Closed-form solutions
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constant-saving-function
ρ = ((1 − γ)1−αθ − 1)λ − (1 − αθ)δ
⇒
Ct = C(Kt ) = (1 − s)AKtα L1−α
calibration (α, θ, δ, λ, γ) = (0.5,2.5,0.05,0.2,0.1) ⇒ ρ =
0.0178, s = 0.4
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linear-policy-function
α=θ
⇒
Ct = C(Kt ) = bKt
calibration (α, θ, δ, λ, γ) = (0.5,0.5,0.05,0.2,0.1), ρ = 0.0178
⇒ b = 0.1
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A neoclassical growth model with disasters
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Step 1a (initial guess):
I
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solve the deterministic system
rt − ρ − δ
Ct dt
dCt =
θ
dKt = (AKtα L1−α − Ct − δKt )dt
Step 1b (update guess and iterate):
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define for each iteration i = 1, . . . , n
ri − ρ − δ − λ + λ(1 − γ)C̃i−1 (Ki )−θ
dCi =
Ci dt
θ
dKi = (AKiα L1−α − Ci − δKi )dt
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A neoclassical growth model with disasters (θ = 2.5)
7
6
consumption
5
4
3
2
1
0
0
20
40
60
80
capital
Optimal policy functions: deterministic (dashed) vs. stochastic (solid)
compared to the analytical benchmark solution (dotted)
19 / 28
A neoclassical growth model with disasters (θ = 2.5)
0.9488
0.9487
0.9486
0.9485
ctilde
0.9484
0.9483
0.9482
0.9481
0.948
0.9479
0
20
40
60
80
capital
Optimal jump functions: numerical calculated solution (solid) compared to the
analytical benchmark solution (dotted)
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A neoclassical growth model with disasters (θ = 2.5)
−3
−3
10
10
−4
−4
10
−5
relative error (log scale)
absolute error (log scale)
10
10
−6
10
−7
10
−8
10
−9
−6
10
−7
10
−8
10
−9
10
10
−10
10
−5
10
0
−10
20
40
60
10
80
0
20
40
capital
60
80
10
relative change last iteration (log scale)
absolute change last iteration (log scale)
80
−3
10
−4
10
−5
10
−6
10
−7
10
−8
10
−9
10
−10
10
60
capital
−3
0
−4
10
−5
10
−6
10
−7
10
−8
10
−9
10
−10
20
40
60
capital
80
10
0
20
40
capital
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A neoclassical growth model with disasters (θ = 1)
6
consumption
5
4
3
2
1
0
0
10
20
30
40
capital
50
60
70
80
Optimal policy functions: deterministic (dashed) vs. stochastic (solid)
22 / 28
A neoclassical growth model with disasters (θ = 1)
0.945
ctilde
0.94
0.935
0.93
0.925
0.92
0
10
20
30
40
capital
50
60
70
80
Optimal jump functions: numerical calculated solution (solid)
23 / 28
A neoclassical growth model with disasters (θ = 1)
−3
−3
10
10
−4
−4
10
−5
relative error (log scale)
absolute error (log scale)
10
10
−6
10
−7
10
−8
10
−9
0
−8
10
−10
10
20
30
40
capital
50
60
70
10
80
−3
0
10
20
30
40
capital
50
60
70
80
10
20
30
40
capital
50
60
70
80
−3
10
relative change last iteration (log scale)
10
absolute change last iteration (log scale)
−7
10
10
−10
−4
10
−5
10
−6
10
−7
10
−8
10
−9
10
−10
10
−6
10
−9
10
10
−5
10
0
−4
10
−5
10
−6
10
−7
10
−8
10
−9
10
−10
10
20
30
40
capital
50
60
70
80
10
0
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A neoclassical growth model with disasters (θ = 0.5)
6
consumption
5
4
3
2
1
0
0
10
20
30
40
capital
50
60
70
80
Optimal policy functions: deterministic (dashed) vs. stochastic (solid)
compared to the analytical benchmark solution (dotted)
25 / 28
A neoclassical growth model with disasters (θ = 0.5)
0.9
0.9
0.9
ctilde
0.9
0.9
0.9
0.9
0.9
0.9
0
10
20
30
40
capital
50
60
70
80
Optimal jump functions: numerical calculated solution (solid) compared to the
analytical benchmark solution (dotted)
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A neoclassical growth model with disasters (θ = 0.5)
−5
−5
10
relative error (log scale)
absolute error (log scale)
10
−10
10
−15
−15
10
10
10
20
30
40
capital
50
60
70
80
−5
10
−10
10
−15
10
0
0
relative change last iteration (log scale)
absolute change last iteration (log scale)
0
−10
10
10
20
30
40
capital
50
60
70
80
10
20
30
40
capital
50
60
10
20
30
40
capital
50
60
70
80
−5
10
−10
10
−15
10
0
70
80
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Conclusion
This paper
I
proposes a simple and powerful method for determining the
transition process in continuous-time DSGE models under
Poisson uncertainty numerically
I
shows how to extend DSGE models and existing solution
methods in order to allow for the possibilities of rare events
I
illustrates our approach for two popular methods computing
numerical solutions to dynamic general equilibrium models
28 / 28