perturbation methods - ice homepageice.uchicago.edu/slides_2006/iceperturbationmethods1.pdf · •...
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PERTURBATION METHODS
Kenneth L. Judd
Hoover Institution and NBER
June 28, 2006
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Local Approximation Methods
• Use information about f : R → R only at a point, x0 ∈ R, to construct an approximation validnear x0
• Taylor Series Approximationf(x)
.=f(x0) + (x− x0) f
0(x0) +(x− x0)
2
2f 00(x0) + · · · + (x− x0)
n
n!f (n)(x0) +O(|x− x0|n+1)
=pn (x) +O(|x− x0|n+1)• Power series: P∞n=0 anzn— The radius of convergence is
r = sup|z| : |∞Xn=0
anzn| <∞,
—P∞
n=0 anzn converges for all |z| < r and diverges for all |z| > r.
• Complex analysis— f : Ω ⊂ C → C on the complex plane C is analytic on Ω iff
∀a ∈ Ω ∃r, ckÃ∀ kz − ak < r
Ãf(z) =
∞Xk=0
ck(z − a)k
!!— A singularity of f is any a s. t. f is analytic on Ω− a but not on Ω.
— If f or any derivative of f has a singularity at z ∈ C, then the radius of convergence in C ofP∞n=0
(x−x0)nn! f (n)(x0), is bounded above by k x0 − z k.
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• Example: f(x) = xα where 0 < α < 1.
— One singularity at x = 0
— Radius of convergence for power series around x = 1 is 1.
— Taylor series coefficients decline slowly:
ak =1
k!
dk
dxk(xα)|x=1 = α(α− 1) · · · (α− k + 1)
1 · 2 · · · · · k .
Table 6.1 (corrected): Taylor Series Approximation Errors for x1/4
Taylor series error x1/4
x N: 5 10 20 503.0 5(−1) 8(1) 3(3) 1(12) 1.31612.0 1(−2) 5(−3) 2(−3) 8(−4) 1.18921.8 4(−3) 5(−4) 2(−4) 9(−9) 1.15831.5 2(−4) 3(−6) 1(−9) 0(−12) 1.10671.2 1(−6) 2(−10) 0(−12) 0(−12) 1.0466.80 2(−6) 3(−10) 0(−12) 0(−12) .9457.50 6(−4) 9(−6) 4(−9) 0(−12) .8409.25 1(−2) 1(−3) 4(−5) 3(−9) .7071.10 6(−2) 2(−2) 4(−3) 6(−5) .5623.05 1(−1) 5(−2) 2(−2) 2(−3) .4729
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Log-Linearization and General Nonlinear COV
• Implicit differentiation implies
x =dx
x= − εfε
xfx
dε
ε= − εfε
xfxε,
• Since x = d(lnx), log-linearization implies log-linear approximation
lnx− lnx0 .= − ε0fε(x0, ε0)
x0fx(x0, ε0)(ln ε− ln ε0). (6.1.5)
• Generalization to nonlinear change of variables.— Take any monotonic h(·), and define x = h(X) and y = h(Y )
— Use the identity
f(Y,X) = f(h−1(h(Y )), h−1(h(X))) = f(h−1(y), h−1(x)) ≡ g(y, x).
to generate expansions
y(x).=y(x0) + y0(x)(x− x0) + ...
Y (X).=h−1 (y(h(X0)) + y0(h(X0))(h(X)− h(X0)) + ...)
— h(z) = ln z is commonly used by economists, but others may be better globally
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Implicit Function Theorem
• Suppose h : Rn → Rm is defined in H(x, h(x)) = 0, H : Rn ×Rm → Rm, and h(x0) = y0.
— Implicit differentiation shows
Hx(x, h(x)) +Hy(x, h(x))hx(x) = 0
— At x = x0, this implieshx(x0) = −Hy(x0, y0)
−1Hx(x0, y0)
if Hy(x0, y0) is nonsingular. More simply, we express this as
h0x = −¡H0
y
¢−1H0
x
— Linear approximation for h(x) is
hL(x).= h(x0) + hx(x0)(x− x0)
• To check on quality, we computeE = H(x, hL(x))
where H is a unit free equivalent of H. If E < ε, then we have an ε-solution.
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• If hL(y) is not satisfactory, compute higher-order terms by repeated differentiation.— DxxH(x, h(x)) = 0 implies
Hxx + 2Hxyhx +Hyyhxhx +Hyhxx = 0
— At x = x0, this implies
h0xx = −¡H0
y
¢−1 ¡H0
xx + 2H0xyh
0x +H0
yyh0xh0x
¢— Construct the quadratic approximation
hQ(x).= h(x0) + h0x(x− x0) +
1
2(x− x0)
>h0xx(x− x0)
and check its quality by computing E = H(x, hQ(x)).
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Regular Perturbation: The Basic Idea
• Suppose x is an endogenous variable, ε a parameter— Want to find x(ε) such that f(x(ε), ε) = 0
— Suppose x(0) known.
• Use Implicit Function Theorem— Apply implicit differentiation:
fx(x(ε), ε)x0(ε) + fε(x(ε), ε) = 0 (13.1.5)
— At ε = 0, x(0) is known and (13.1.5) is linear in x0(0) with solution
x0(0) = −fx(x(0), 0)−1fε(x(0), 0)
— Well-defined only if fx 6= 0, a condition which can be checked at x = x(0).
— The linear approximation of x(ε) for ε near zero is
x(ε).= xL(ε) ≡ x(0)− fx(x(0), 0)
−1fε(x(0), 0)ε (13.1.6)
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• Can continue for higher-order derivatives of x(ε).— Differentiate (13.1.5) w.r.t. ε
fxx00 + fxx(x
0)2 + 2fxεx0 + fεε = 0 (13.1.7)
— At ε = 0, (13.1.7) implies that
x00(0)=−fx(x(0), 0)−1¡fxx(x(0), 0) (x
0(0))2
+2fxε(x(0), 0) x0(0) + fεε(x(0), 0))
— Quadratic approximation is
x(ε).= xQ(ε) ≡ x(0) + εx0(0) +
1
2ε2x00(0) (13.1.8)
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• General Perturbation Strategy— Find special (likely degenerate, uninteresting) case where one knows solution
∗ General relativity theory: begin with case of a universe with zero mass: ε is mass of universe∗ Quantum mechanics: begin with case where electrons do not repel each other: ε is force ofrepulsion
∗ Business cycle analysis: begin with case where there are no shocks: ε is measure of exogenousshocks
— Use local approximation theory to compute nearby cases
∗ Standard implicit function may be applicable∗ Sometimes standard implicit function theorem will not apply; use appropriate bifurcationor singularity method.
— Check to see if solution is good for problem of interest
∗ Use unit-free formulation of problem∗ Go to higher-order terms until error is reduced to acceptable level∗ Always check solution for range of validity
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Single-Sector, Deterministic Growth - canonical problem
• Consider dynamic programming problem
maxc(t)
Z ∞0
e−ρtu(c)dt
k = f(k)− c
• Ad-Hoc Method: Convert to a wrong LQ problem— McGrattan, JBES (1990)
∗ Replace u(c) and f(k) with approximations around c∗ and k∗∗ Solve linear-quadratic problem
maxcR∞0 e−ρt
¡u(c∗) + u0(c∗)(c− c∗) + 1
2u00(c∗)(c− c∗)2
¢dt
s.t. k = f(k∗) + f 0(k∗)(k∗ − k)− c
∗ Resulting approximate policy function isCMcG(k) = f(k∗) + ρ(k − k∗) 6= C(k∗) + C 0(k∗)(k − k∗)
∗ Local approximate law of motion is k = 0; add noise to getdk = 0 · dt + dz
∗ Approximation is random walk when theory says solution is stationary— Fallacy of McGrattan noted in Judd (1986, 1988); point repeated in Benigno-Woodford (2004).
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• Kydland-Prescott— Restate problem so that k is linear function of state and controls
— Replace u(c) with quadratic approximation
— Note 1: such transformation may not be easy
— Note 2: special case of Magill (JET 1977).
• Lesson— Kydland-Prescott, McGrattan provide no mathematical basis for method
— Formal calculations based on appropriate IFT should be used.
— Beware of ad hoc methods based on an intuitive story!
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Perturbation Method for Dynamic Programming
• Formalize problem as a system of functional equations— Bellman equation:
ρV (k) = maxc
u(c) + V 0(k)(f(k)− c) (1)
— C(k): policy function defined by
0=u0(C(k))− V 0(k) (2)
ρV (k)=u(C(k)) + V 0(k)(f(k)−C(k))
— Apply envelope theorem to (1) to get
ρV 0(k) = V 00(k)(f(k)− C(k)) + V 0(k)f 0(k) (1k)
— Steady-state equations
c∗ = f(k∗) ρV (k∗) = u(c∗) + V 0(k∗)(f(k∗)− c∗)0 = u0(c∗)− V 0(k∗) ρV 0(k) = V 00(k)(f(k∗)− c∗) + V 0(k)f 0(k)
— Steady State: We know k∗, V (k∗), C(k∗), f 0(k∗), V 0(k∗):
ρ = f 0(k∗), C(k∗) = f(k∗), V (k∗) = ρ−1u(c∗), V 0(k∗) = u0(c∗)
— Want Taylor expansion:
C(k).=C(k∗) + C 0(k∗)(k − k∗) + C 00(k∗)(k − k∗)2/2 + ...
V (k).=V (k∗) + V 0(k∗)(k − k∗) + V 00(k∗)(k − k∗)2/2 + ...
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• Linear approximation around a steady state— Differentiate (1k, 2) w.r.t. k:
ρV 00=V 000(f − C) + V 00(f 0 − C 0) + V 00f 0 + V 0f 00 (1kk)
0=u00C 0 − V 00 (2k)
— At the steady state
0 = −V 00(k∗)C 0(k∗) + V 00(k∗)f 0(k∗) + V 0(k∗)f 00(k∗) (1∗k)
— Substituting (2k) into (1∗k) yields
0 = −u00(C 0)2 + u00C 0f 0 + V 0f 00
— Two solutions
C 0(k∗) =ρ
2
Ã1±
s1 +
4u0(C(k∗))f 00(k∗)u00(C 0(k∗))f 0(k∗)f 0(k∗)
!— However, we know C 0(k∗) > 0; hence, take positive solution
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• Higher-Order Expansions— Conventional perception in macroeconomics: “perturbation methods of order higher than oneare considerably more complicated than the traditional linear-quadratic case ...” — Marcet(1994, p. 111)
— Mathematics literature: No problem (See, e.g., Bensoussan, Fleming, Souganides, etc.)
• Compute C 00(k∗) and V 000(k∗).
— Differentiate (1kk, 2k):
ρV 000=V 0000(f − C) + 2V 000(f 0 −C 0) + V 00(f 00 − C 00) (1kkk)
+V 000f 0 + 2V 00f 00 + V 0f 000
0=u000(C 0)2 + u00C 00 − V 000 (2kk)
— At k∗, (1kkk) reduces to
0 = 2V 000(f 0 − C 0) + 3V 00f 00 − V 00C 00 + V 0f 000 (1∗kkk)
— Equations (1∗kkk,2∗kk) are LINEAR in unknowns C
00(k∗) and V 000(k∗):Ãu00 −1V 00−2(f 0 − C 0)
!ÃC 00
V 000
!=
ÃA1A2
!— Unique solution since determinant −2u00(f 0 −C 0) + V 00 < 0.
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• Compute C(n)(k∗) and V (n+1)(k∗).
— Linear system for order n is, for some A1 and A2,Ãu00 −1V 00−n(f 0 − C 0)
!ÃC(n)
V (n+1)
!=
ÃA1A2
!— Higher-order terms are produced by solving linear systems
— The linear system is always determinate since −nu00(f 0 −C 0) + V 00 < 0
• Conclusion:— Computing first-order terms involves solving quadratic equations
— Computing higher-order terms involves solving linear equations
— Computing higher-order terms is easier than computing the linear term.
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Accuracy MeasureConsider the one-period relative Euler equation error:
E(k) = 1− V 0(k)u0(C(k))
• Equilibrium requires it to be zero.• E(k) is measure of optimization error— 1 is unacceptably large
— Values such as .00001 is a limit for people.
— E(k) is unit-free.
• Define the Lp, 1 ≤ p <∞, bounded rationality accuracy to belog10 k E(k) kp
• The L∞ error is the maximum value of E(k).
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Global Quality of Asymptotic Approximations
Graph of log10 |E(k|
• Linear approximation is very poor even for k close to steady state• Order 2 is better but still not acceptable for even k = .9, 1.1
• Order 10 is excellent for k ∈ [.6, 1.4]
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Stochastic, Discrete-Time Growth
maxct E©P∞
t=0 βt u(ct)
ªs.t. kt+1 = (1 + εz)F (kt − ct)
(13.7.19)
• New state variable:— kt is capital stock at the beginning of period t
— consumption comes out of k
— the remaining capital, kt − ct, is used in production
— resulting output is (1 + εz)F (kt − ct) = kt+1
— perturbation parameter is ε, the standard deviation, not the variance.
• Do deterministic perturbation analysis.— Solution when ε = 0 is C(k) solving
u0 (C(k)) = β u0 (C (F (k − C(k)))) F 0 (k −C(k)) . (13.7.20)
— At the steady state, k∗, F (k∗ −C(k∗)) = k∗, and 1 = β F 0(k∗ −C(k∗))
— Derivative of (13.7.20) with respect to k implies
u00 (C(k)) C 0(k)= βu00 (C (F (k −C(k)))) C 0 (F (k −C(k)))
×F 0(k −C(k))[1−C 0(k)]F 0(k − C(k))
+β u0 (C (F (k −C(k)))) F 00 (k − C(k)) [1− C 0(k)](13.7.21)
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— At k = k∗, (13.7.21) reduces to (drop all arguments)
u00C 0 = βu00C 0F 0[1−C 0]F 0 + βu0F 00[1− C 0]. (13.7.22)
with stable solution
C 0 =1
2
⎛⎝1− β − β2u0
u00F 00 +
sµ1− β − β2
u0
u00F 00¶2+ 4
u0
u00β2 F 00
⎞⎠— Take another derivative of (13.7.21) and set k = k∗ to find
u00C 00 + u000C 0C 0 =βu000 (C 0F 0(1−C 0))2 F 0 + βu00C 00 (F 0(1−C 0))2 F 0
+2βu00C 0F 0(1−C 0)2 F 00 + βu0F 000(1−C 0)2
+βu0F 00(−C 00),which is a linear equation in the unknown C 00(k∗).
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• Stochastic problem:— Euler equation is
u0 (C(k)) = β E u0 (g(ε, k, z)) R(ε, k, z) , (13.7.23)
whereg(ε, k, z) ≡ C((1 + εz)F (k −C(k))),
R(ε, k, z)≡ (1 + ε z)F 0 (k − C(k)) .(13.7.24)
— Compute Cε
∗ Differentiate (13.7.24) with respect to ε yields (we drop arguments of F and C)gε=Cε + C 0 (zF − (1 + εz)F 0Cε) , (13.7.25)
gεε=Cεε + 2C0ε (zF − (1 + εz)F 0Cε) + C 00 (zF − (1 + εz)F 0Cε)
2,
+C 0¡−zF 0Cε 2 + (1 + εz)F 00C2ε − (1 + εz)F 0Cεε
¢.
∗ At ε = 0, (13.7.25) implies thatgε=Cε + C 0(zF − F 0Cε), (13.7.26)
gεε=Cεε + 2C0ε(zF − F 0Cε) + C 00(zF − F 0Cε)
2,
+C 0(−2zF 0Cε + F 00C2ε − F 0Cεε).
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∗ Differentiate (13.7.23) with respect to εu00Cε = β E u00gε(1 + εz)F 0 + u0F 0z − u0(1 + εz)F 00Cε (13.7.27)
u000C2ε+ u00Cεε = β Eu000g2ε(1 + εz)F 0 + 2u00gεF 0z (13.7.28)
−2u00gε(1 + εz)F 00Cε + u00gεε(1 + εz)F 0
−2u0zF 00Cε + u0(1 + εz)F 000C2ε − u0(1 + εz)F 00Cεε∗ Since Ez = 0, (13.7.27) says that Cε = 0, which in turn implies that
gε=C0zF,
gεε=Cεε + 2C0ε z F + C 00(zF )2 −C 0F 0Cεε.
— Compute Cεε
∗ Second-order terms in (13.7.28), we find that at ε = 0,u000C2ε + u00Cεε=β E
©u000g2ε F
0 + 2u00gε F 0 z − 2u00gε F 00Cε
+u00gεε F 0 − 2u0 z F 00Cε +u0F 000C2ε − u0F 00Cεε
ª∗ Using the normalization Ez2 = 1, we find that
u00Cεε = β£u000C 0C 0F 2F 0 + 2u00C 0FF 0 +u00(Cεε + C 00F 2 −C 0F 0Cεε)F
0 − u0F 00Cεε
¤∗ Solving for Cεε yields
Cεε =u000C 0C 0F 2 + 2u00C 0F + u00C 00F 2
u00C 0F 0 + βu0F 00
• This exercise demonstrates that perturbation methods can also be applied to the discrete-timestochastic growth model.
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Bifurcation Methods
• Suppose H(h(ε), ε) = 0 but H(x, 0) = 0 for all x.— IFT says
h0(0) = −Hε(x0, 0)
Hx(x0, 0)
— H(x, 0) = 0 implies Hx(x0, 0) = 0, and h0(0) has the form 0/0 at x = x0.
— l’Hospital’s rule implies, if which is well-defined if Hεx(x0, 0) 6= 0,
h0(0) = −Hεε(x0, 0)
Hεx(x0, 0).
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Example: Portfolio Choices for Small Risks
• Simple asset demand model:— safe asset yields R per dollar invested and risky asset yields Z per dollar invested
— If final value is Y =W ((1− ω)R + ωZ), then portfolio problem is
maxω
Eu(Y )
• Small Risk Analysis— Parameterize cases
Z = R + εz + ε2π (1)
— Compute ω(ε) .= ω(0) + εω0(0) + ε2
2 ω00(0).around the deterministic case of ε = 0.
— Failure of IFT: at ε = 0, Z = R, and ω( ε) is indeterminate, but we know that ω( ε) is uniquefor ε 6= 0
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• Bifurcation analysis— The first-order condition for ω
0 = Eu0 ¡WR + ωW (εz + ε2π)¢(z + επ) ≡ G(ω, ε) (2)
0 = G(ω, 0), ∀ω. (3)
— Solve for ω(ε) .= ω(0) + εω0(0) + ε2
2 ω00(0). Implicit differentiation implies
0 = Gωω0 +Gε (4)
Gε=Eu00(Y )W (ωz + 2ωεπ)W (z + επ) + u0(Y )π (5)
Gω=Eu00(Y )(z + επ)2ε (6)
— At ε = 0, G(ω, 0) = Gω(ω, 0) = 0 for all ω.
— No point (ω, 0) for application of IFT to (3) to solve for ω0(0).
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• We want ω0 = limε→0 ω(ε).
— Bifurcation theorem keys on ω0 satisfying
0=Gε(ω0, 0)
=u00(RW )ω0σ2zW + u0(RW )π (7)
which implies
ω0 = − π
σ2z
u0(WR)
Wu00(WR)(8)
— (8) is asymptotic portfolio rule
∗ same as mean-variance rule∗ ω0 is product of risk tolerance and the risk premium per unit variance.∗ ω0 is the limiting portfolio share as the variance vanishes.∗ ω0 is not first-order approximation.
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• To calculate ω0(0):— differentiate (2.4) with respect to ε
0 = Gωωω0ω0 + 2Gωεω
0 +Gωω00 +Gεε (9)
where (without loss of generality, we assume W = 1)
Gεε = Eu000(Y )(ωz + 2ωεπ)2(z + επ) + u00(Y )2ωπ(z + επ)
+2u00(Y )(ωz + 2ωεπ)πGωω =Eu000(Y )(z + επ)3εGωε = Eu000(Y )(ωz + 2ωεπ)(z + επ)2ε+ u00(Y )(z + επ)2πε
+u00(Y )(z + επ)2— At ε = 0,
Gεε = u000(R)ω20Ez3 Gωω = 0
Gωε = u00(R)Ez2 6= 0 Gεεε 6= 0— Therefore,
ω0 = −12
u000(R)u00(R)
Ez3Ez2ω
20. (10)
— Equation (10) is a simple formula.
∗ ω0(0) proportional to u000/u00∗ ω0(0) proportional to ratio of skewness to variance.∗ If u is quadratic or z is symmetric, ω does not change to a first order.
— We could continue this and compute more derivatives of ω(ε) as long as u is sufficiently differ-entiable.
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• Other applications - see Judd and Guu (ET, 2001)— Equilibrium: add other agents, make π endogenous
— Add assets
— Produce a mean-variance-skewness-kurtosis-etc. theory of asset markets
— More intuitive approach to market incompleteness then counting states and assets
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