PERTURBATION METHODS

Kenneth L. Judd

Hoover Institution and NBER

June 28, 2006

1

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.

2

• 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

3

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

4

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)).

6

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)

8

• 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

9

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!

11

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

13

• 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.

14

• 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.

15

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.

21

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

23

• 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.

25

• 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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