Growth model with Epstein-Zin preferencesand stochastic volatility

H̊akon Tretvoll

July 8, 2011

1 Introduction

This document goes through a method of solving a growth model withEpstein-Zin preferences and stochastic volatility through a log-linear approx-imation.

2 Model

2.1 Household preferences

Recursive preferences:

Ut = V [ct, µt(Ut+1)] = [(1− β)cρt + βµt(Ut+1)ρ]1/ρ (1)

where the certainty equivalent µ is specified by

µt(Ut+1) = [Et(Uαt+1)]1/α

Conventional interpretation: ρ < 1 captures time preference and α < 1captures risk aversion.

Intertemporal elasticity of substitution =1

1− ρCoefficient of relative risk aversion = 1− α

Note that the household has no disutility of labor and provides a constantlabor supply.

1

2.2 Firm production and capital evolution

Production a function of capital and labor: yt = f(kt, atn) where f is homo-geneous of degree 1. Since household labor supply is constant we normalizen = 1.For now, we do not consider capital adjustment costs, and we have thefollowing equation for the evolution of capital:

kt+1 = (1− δ)kt + it (2)

and the following overall resource constraint:

f(kt, atn) = ct + it (3)

[By setting δ = 1 (full depreciation, and setting α = ρ = 0 we obtain theBrock-Mirman model which has an analytical solution.]

2.3 Technology

Labor augmenting technology growing at rate g:

log gt = log

(atat−1

)= log g + e>xt (4)

xt+1 = Axt + v1/2t Bw1,t+1 (5)

vt+1 = (1− φv)v + φvvt + τw2,t+1 (6)

where w1,t+1 ∼ N(0, I), w2,t+1 ∼ N(0, 1), and these are assumed to be un-correlated.

We call xt the “predictable component” of consumption growth, and vt+1is a process for the volatility of this predictable component (this is potentiallya model with stochastic volatility).

In appendix A we discuss the process for technology used by Kaltenbrun-ner and Lochstoer (2010, RFS) and how it compares to the one we use. Evenwith constant volatility in equation (6) we cannot map our process into theprocess used by KL (2010).

2

2.4 Rescale problem

We rescale the problem with at1, and define the rescaled variables as:

k̃t =ktat−1

; c̃t =ctat

; ĩt =itat

Ũt =Utat

The rescaled specification of utility is:

Ũt =[(1− β)c̃ρt + βµt

(gt+1Ũt+1

)ρ]1/ρ(7)

The rescaled capital evolution equation is:

k̃t+1 = (1− δ)k̃tgt

+ ĩt (8)

The rescaled resource constraint is:

ỹt = f

(k̃tgt, 1

)= c̃t + ĩt (9)

2.5 First-order conditions

The problem we solve is then written

Ũt(k̃t, xt, vt) = maxc̃t

[(1− β)c̃ρt + βµt

(gt+1Ũt+1(k̃t+1, xt+1, vt+1)

)ρ]1/ρ(10)

s.t. k̃t+1 = (1− δ)k̃tgt

+ ĩt (11)

f

(k̃tgt, 1

)= c̃t + ĩt (12)

1This corresponds with the scaling in KL when φ = 1. When φ < 1 so their processfor technology is trend-stationary, they rescale with the time trend exp(µt) only (which inour case is gt).

3

The first-order condition w.r.t. c̃t is:

(1− β)c̃ρ−1t = βµρ−αt Et

(gαt+1Ũ

α−1t+1

∂Ũt+1

∂k̃t+1

)(13)

The envelope condition is:

∂Ũt

∂k̃t= βŨ1−ρt µ

ρ−αt Et

(gαt+1Ũ

α−1t+1

∂Ũt+1

∂k̃t+1

)(1− δgt

+ f1

(k̃tgt, 1

)1

gt

)(14)

The derivations of these are in appendix B.

3 A log-linear approximation

To obtain log-linear approximations we start by making the following guess∂Ũt∂k̃t

:

Ũk,t =∂Ũt

∂k̃t≈ exp(p1)k̃pk−1t exp

(p>x xt + pvvt

)This implies the following log-linear approximation for the derivative of

the value function:

log Ũk,t ≈ p1 + (pk − 1) log k̃t + p>x xt + pvvt (15)

3.1 A second log-linear approximation

The guess for the derivative of the value function also implies an expressionfor the value function itself:

Ũt ≈ p0 +exp(p1)

pkk̃pkt exp

(p>x xt + pvvt

)⇒ log Ũt ≈ log

(p0 +

exp(p1)

pkexp

(pk log k̃t

)exp

(p>x xt + pvvt

))(16)

To get a log-linear expression for the value function we take a log-linearapproximation of the above expression around the steady state values oflog k̃t, xt, and vt.

4

log Ũt ≈ q1 + qk log k̃t + q>x xt + qvvt (17)

The expressions relating the qi coefficients to the pi coefficients of the orig-inal guess are in appendix C.1. Note however that a log-linear approximationof Ũt (about the deterministic steady state) has the form:

log Ũt ≈ log Ũ −Ũkk̃

Ũlog k̃ − Ũv

Ũv +

Ũkk̃

Ũlog k̃t +

Ũx

Ũxt +

Ũv

Ũvt

Since we know the values of Ũt, Ũk,t, and k̃t in steady state we can imme-diately pin down the coefficient qk:

qk =Ũkk̃

Ũ=Ũ − p0Ũ

pk (18)

where the last equation uses equation (41) to relate this expression to the picoefficients. As we can see we have an equation relating p0 and pk.

p0 = Ũ −Ũkk̃

pk(19)

4 Investigating the shapes of Ũt and Ũk,t

To get an idea of the shapes of the functions we are trying to approximate, weuse Dynare to solve for a third-order approximation to the model. We startby considering the model without stochastic volatility and with iid shocks tothe growth rate. That is, we set vt = v ∀t, e = B = 1, and A = 0.

We’ll also assume the following functional form for the production func-tion f(kt, atn):

f(kt, atn) = kθt (atn)

1−θ

Note that here we define the rescaled variables: c̃t =ctat

and k̃t =ktat−1

.

Rescaling in this way ⇒ k̃t+1 determined at time t.

5

The equations that describe our model are then:

Ũt =[(1− β)c̃ρt + βµt

(gt+1Ũt+1

)ρ]1/ρµt

(gt+1Ũt+1

)=(Et[(gt+1Ũt+1

)α])1/αk̃t+1 = (1− δ)

k̃tgt

+

(k̃tgt

)θ− c̃t

log gt+1 = log g + v1/2w1,t+1

(1− β)c̃ρ−1t = βµρ−αt Et

(gαt+1Ũ

α−1t+1

∂Ũt+1

∂k̃t+1

)

∂Ũt

∂k̃t= (1− β)c̃ρ−1t Ũ

1−ρt

1− δgt

+ θ

(k̃tgt

)θ−11

gt

In this version the steady-state relationships are:

k̃ = g

(βθ

g1−ρ − β(1− δ)

) 11−θ

c̃ = (1− δ) k̃g

+

(k̃

g

)θ− k̃

Ũ =

(1− β

1− βgρ

)1/ρc̃ Ũk =

1− ββ

(Ũ

c̃

)1−ρ1

gρ

µ = gŨ

In this version of the model, the state variables are k̃t and w1,t.

4.1 Euler equation

Combining the FOC w.r.t. c and the envelope condition yields:

∂Ũt

∂k̃t= Ũ1−ρt (1− β)c̃

ρ−1t

1− δgt

+ θ

(k̃tgt

)θ−11

gt

Using this equation at t+1 to substitute out for ∂Ũt+1

∂k̃t+1in the FOC w.r.t. c

gives the Euler equation:

6

c̃ρ−1t = βµρ−αt Et

gα−1t+1 Ũα−ρt+1 c̃ρ−1t+1(1− δ) + θ( k̃t+1

gt+1

)θ−1 (20)where

c̃t = (1− δ)k̃tgt

+

(k̃tgt

)θ− k̃t+1

4.1.1 Special case: α = ρ

When agents have additive preferences (α = ρ) the Euler equation becomes:(1− δ) k̃tgt

+

(k̃tgt

)θ− k̃t+1

ρ−1 =βEt

gρ−1t+1(1− δ) k̃t+1

gt+1+

(k̃t+1gt+1

)θ− k̃t+2

ρ−1(1− δ) + θ( k̃t+1gt+1

)θ−1

4.2 Dynare++

Using Dynare++ we calculate third-order approximations to Ũt and Ũk,t.Dynare++ calculates approximations around the deterministic steady stateof the form:

7

log Ũt − log Ũ = g1,U(1)(

log k̃t − log k̃)

+ g1,U(2)w1,t

+ g2,U(1)(

log k̃t − log k̃)2

+ g2,U(2)(

log k̃t − log k̃)w1,t + g2,U(3)w

21,t

+ g3,U(1)(

log k̃t − log k̃)3

+ g3,U(2)(

log k̃t − log k̃)2w1,t

+ g3,U(3)(

log k̃t − log k̃)w21,t + g3,U(4)w

31,t

log Ũk,t − log Ũk = g1,Uk(1)(

log k̃t − log k̃)

+ g1,Uk(2)w1,t

+ g2,Uk(1)(

log k̃t − log k̃)2

+ g2,Uk(2)(

log k̃t − log k̃)w1,t + g2,Uk(3)w

21,t

+ g3,Uk(1)(

log k̃t − log k̃)3

+ g3,Uk(2)(

log k̃t − log k̃)2w1,t

+ g3,Uk(3)(

log k̃t − log k̃)w21,t + g3,Uk(4)w

31,t

where gi,x(j) is the coefficient on the j-th term of order i in the equation forvariable x.

4.3 Log-linearisation

A log-linear approximation of Ũt in this version of the model is of the form:

log Ũt ≈ log Ũ +Ũkk̃

Ũ

(log k̃t − log k̃

)+Ũw

Ũwt

= q1 + qk log k̃t + qwwt

Hence, in this case we can solve directly for some of the coefficients fromsteady-state quantities:

qk =Ũkk̃

Ũ; and q1 = log Ũ − qk log k̃

With this approximation for the level of the value function, the impliedapproximation for the derivative Ũk,t is

log Ũk,t ≈ q1 + log qk + (qk − 1) log k̃t + qwwt

8

We cannot solve for qw from steady-state values as we don’t know Ũw.We can however compare our approximation to the 3rd order approximationcalculated by Dynare++.

4.4 Comparison

Figures 1- 4 makes various comparisons between the output from Dynare++and a log-linear approximation to the function involved. Figure 1 plots log Ũtvs. log k̃t while figure 2 plots the same relationship in levels. Similarly,figure 3 plots log Ũkt vs. log k̃t while figure 4 plots the relationship in levels.

Note that with the log-linear approximation above we can only approxi-mate the value funciton and its derivative for the case where w1,t = 0, sincewe have not yet solved for qw.

−2 −1.9 −1.8 −1.7 −1.6 −1.5 −1.4−0.419

−0.4185

−0.418

−0.4175

−0.417

−0.4165

−0.416

−0.4155

log kt

log

Ut

Shape of log−value function

Figure 1: Blue: solution from Dynare++. Red: log-linear approximation

4.5 More log-linearisations

4.5.1 log ct

The first-order conditions in section 4 can be combined into the followingequation:

9

0.15 0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.240.6575

0.658

0.6585

0.659

0.6595

0.66

kt

Ut

Shape of value function

Figure 2: Blue: solution from Dynare++. Red: log-linear approximation

−2 −1.9 −1.8 −1.7 −1.6 −1.5 −1.4−4.5

−4.4

−4.3

−4.2

−4.1

−4

−3.9

−3.8

log kt

log

Ukt

Shape of log−derivative of value function

Figure 3: Blue: solution from Dynare++. Red: log-linear approximation

10

0.15 0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.240.01

0.011

0.012

0.013

0.014

0.015

0.016

0.017

0.018

0.019

0.02

kt

Ukt

Shape of derivative of value function

Figure 4: Blue: solution from Dynare++. Red: log-linear approximation

(1− β)c̃ρ−1t = Ũ1−ρt θ

(k̃tgt

)θ−11

gt

Taking logs and rearranging we get

log c̃t =log(1− β)

1− ρ− log Ũt −

log θ

1− ρ+

1− θ1− ρ

log k̃t +θ

1− ρlog gt

Plugging in the approximation for log Ũt, we have

log c̃t =log(1− β)− log θ + θ log g

1− ρ−q1+

(1− θ1− ρ

− qk)

log k̃t+

(θv1/2

1− ρ− qw

)wt

4.5.2 log k̃t+1

The capital evolution equation implies:

log k̃t+1 = log

( k̃tgt

)θ− c̃t

11

Taking a first-order approximation of the RHS we get:

log

( k̃tgt

)θ− c̃t

= log( k̃

g

)θ− c̃

− c̃(k̃g

)θ− c̃

(log c̃t − log c̃)

+θ(

k̃g

)θ− c̃

(k̃

g

)θ [log k̃t − log k̃ − (log gt − log g)

]

By rearranging this equation we get that

log k̃t+1 = κ̂0k + κ̂ck log c̃t + κ̂kk log k̃t + κ̂gk log gt

where

κ̂ck = −c̃(

k̃g

)θ− c̃

κ̂kk =θ(

k̃g

)θ− c̃

(k̃

g

)θ

κ̂gk = −θ(

k̃g

)θ− c̃

(k̃

g

)θ

κ̂0k = log

( k̃g

)θ− c̃

κ̂ck log c̃− κ̂kk log k̃ − κ̂gk log g

Inserting the approximation for log c̃t and the law of motion for log gt, wehave:

12

.

13

5 Solving for the approximations

TODO: REREAD,UPDATE!!! EVERYTHING BELOW THIS IS OLD!!Given the guesses for log Ũk,t and log Ũt, we have expressions in ap-

pendix C.1 that allow us to solve for the qi coefficients as functions of the picoefficients. Next we derive an approximation for log Ũ of the form of (17)and equate coefficients to obtain more equations that will allow us to solvefor the pi’s. After this step we have 3+dim(xt) equations from appendix C.1and 3 + dim(xt) equations from equating coefficients. We have 7 + dim(xt)variables however, so we need one more equation (to pin down p0). Forthis we’ll use the steady state values and impose that in steady state ourapproximation for Ũ must be exact. An outline of the steps involved is asfollows:

• Obtain an approximation for the Bellman equation.

• Obtain an approximation for log c̃t by approximating the first-orderand envelope conditions.

• Obtain an approximation for log µt by using the guess for log Ũt, theassumed law of motion for technology, and an approximation for thecapital accumulation equation.

• By inserting the approximations for log c̃t and log µt into the approxi-mate Bellman equation we can equate coefficients with the log-linearisation (17).

To simplify notation in the next couple of sections, we define new coef-ficients when necessary. These take the form κterm,eqn to indicate the termand equation this coefficient corresponds to. All coefficients are defined inappendix C.

5.1 Approximating the Bellman equation

First we rewrite equation (7):

Ũρt =[(1− β) exp (ρ log c̃t) + β exp

(ρ log µt

(gt+1Ũt+1

))]Then we take a first-order approximation in log c̃t and log µt around their

non-stochastic steady-state values c̃ and µ:

14

ρ log Ũt ≈ ρ log Ũ +(1− β)ρc̃ρ

(1− β)c̃ρ + βµρ(log c̃t − log c̃) (21)

+βρµρ

(1− β)c̃ρ + βµρ(

log µt

(gt+1Ũt+1

)− log µ

)⇒ log Ũt ≈ κ0U + κcU log c̃t + κµU log µt

(gt+1Ũt+1

)(22)

From equation (22) we see that we need log-linear approximations to log c̃t

and log µt

(gt+1Ũt+1

). The κiU ’s are defined in appendix C.2.

5.2 Approximation for log c̃t

Combining equations (13) and (14) we get

(1− β)

(Ũtc̃t

)1−ρ=Ũk,tfk,t

Taking logs and rearranging:

log c̃t =log(1− β)

1− ρ+ log Ũt −

1

1− ρlog Ũk,t +

1

1− ρlog fk,t (23)

Now to eliminate log fk,t from this expression, we take a first-order ap-proximation around the steady state value k̃:

log fk,t

(exp(log k̃t)

)≈ log fk +

fkkk̃

fk

(log k̃t − log k̃

)⇒ log fk,t ≈ κ0f + κkf log k̃t (24)

We insert (24) and equations (15) and (17) into equation (23):

⇒ log c̃t ≈ κ0c + κkc log k̃t + κ>xcxt + κvcvt (25)

The κif ’s and κic’s are defined in appendix C.3.

15

5.3 Approximation for log µt

(gt+1Ũt+1

)To obtain the a log-linear approximation for log µt

(gt+1Ũt+1

)we plug in the

law of motion for gt+1 from equation (4) and the log-linear approximation ofthe value function (17).

log µt

(gt+1Ũt+1

)=

1

αlogEt

[exp

(α[log gt+1 + log Ũt+1

])](26)

First we work out the sum log gt+1 + log Ũt+1:

log gt+1 + log Ũt+1 ≈ log gt+1 + q1 + qk log k̃t+1 + q>x xt+1 + qvvt+1= log gt+1 + q1 + qk

(log(f(k̃t, 1)− c̃t

)− log gt+1

)+ q>x xt+1 + qv ((1− φv)v + φvvt + τw2,t+1)

Before we proceed further, we obtain an approximation to the term

log(f(k̃t, 1)− c̃t

):

log(f(k̃t, 1)− c̃t

)≈ log

(f(k̃, 1)− c̃

)+

fkk̃

f(k̃, 1)− c̃

(log k̃t − log k̃

)− c̃f(k̃, 1)− c̃

(log c̃t − log c̃)

⇒ log(f(k̃t, 1)− c̃t

)≈ κ0fc + κkfc log k̃t + κcfc log c̃t (27)

Inserting this into the above and collecting some terms, we have:

log gt+1 + log Ũt+1 ≈ q1 + qkκ0fc + qv(1− φv)v + (1− qk) log gt+1+ qkκkfc log k̃t + qkκcfc log c̃t + q

>x xt+1 + qvφvvt + qvτw2,t+1

= q1 + qkκ0fc + qv(1− φv)v + (1− qk) log g

+ ((1− qk)e+ qx)>(Axt + v

1/2t Bw1,t+1

)+ qkκkfc log k̃t

+ qkκcfc log c̃t + qvφvvt + qvτw2,t+1

⇒ log gt+1 + log Ũt+1 ≈ κ0gU + κkgU log k̃t + κ>xgUxt + κvgUvt+ v

1/2t ((1− qk)e+ qx)>Bw1,t+1 + qvτw2,t+1 (28)

16

Now it is useful to calculate the conditional expectation and conditional

variance of α(

log gt+1 + log Ũt+1

):

Et[α(

log gt+1 + log Ũt+1

)]= α

(κ0gU + κkgU log k̃t + κ

>xgUxt + κvgUvt

)Vt[α(

log gt+1 + log Ũt+1

)]= α2

(vt((1− qk)e+ qx)>BB>((1− qk)e+ qx) + q2vτ 2

)With these expressions we can evaluate the expectations in equation (26):

log µt

(gt+1Ũt+1

)≈ κ0µ + κkµ log k̃t + κ>xµxt + κvµvt (29)

The κifc’s, the κigU ’s, and the κiµ’s are defined in appendix C.4.

5.4 Equating coefficients for log Ũt

We insert the approximations for log c̃t and log µt

(gt+1Ũt+1

), equations (25)

and (29), into our approximation for the Bellman equation, equation (22).

log Ũt ≈ κ0U + κcUκ0c + κµUκ0µ + (κcUκkc + κµUκkµ) log k̃t+(κcUκ

>xc + κµUκ

>xµ

)xt + (κcUκvc + κµUκvµ) vt (30)

Equating coefficients between equations (17) and (30) gives us the follow-ing equations to solve:

q1 = κ0U + κcUκ0c + κµUκ0µ (31)

qk = κcUκkc + κµUκkµ (32)

q>x = κcUκ>xc + κµUκ

>xµ (33)

qv = κcUκvc + κµUκvµ (34)

With these equations and the equations linking the qi’s to the pi’s inappendix C.1 we have 6+2×dim(xt) equations, but 7+2×dim(xt) unknowns.So to solve this system we need one more equation.

17

6 Solving for pi’s

In this section we undertake the somewhat ambitious task of trying to solvethe set of equations for the set of coefficients {p0, p1, pk, p>x , pv}. We breakthis down and solve for one coefficient at a time.

6.1 Solving for p1

Assume we have solved for {p0, pk, p>x , pv}. Then we see from equations (40)-(43) that we can solve for {q1, qk, q>x , qv}. Hence we can solve for p1 fromequation (31):

q1 = κ0U + κcU

(log(1− β)

1− ρ− p1

1− ρ+ q1 +

κ0f1− ρ

)+ κµU

(q1 + qkκ0fc + qv(1− φv)v + (1− qk) log g

+ qkκcfc

[log(1− β)

1− ρ− p1

1− ρ+ q1 +

κ0f1− ρ

]+α

2q2vτ

2)

⇒ p11− ρ

(κcU + κµUqkκcfc) = κ0U + (κcU + κµUqkκcfc)

(log(1− β)

1− ρ+ q1 +

κ0f1− ρ

)+ κµU

(q1 + qkκ0fc + qv(1− φv)v + (1− qk) log g +

α

2q2vτ

2)

We have an equation for p1 given {p0, pk, p>x , pv}.

6.2 Solving for pv

Assume we have solved for {p0, pk, p>x }. Then we see from equations (41)and (42) that we also have {qk, q>x }. Hence, we can solve for pv from equa-tion (34):

qv = κcUκvc + κµU

(qkκcfcκvc + qvφv

+α

2((1− qk)e+ qx)>BB> ((1− qk)e+ qx)

)= (κcU + κµUqkκcfc)

(− pv

1− ρ+ qv

)+ κµUφvqv

+ κµUα

2((1− qk)e+ qx)>BB> ((1− qk)e+ qx)

18

Inserting for qv from equation (43) and rearranging gives the followingexpression for pv:

pv =κµU

α2

((1− qk)e+ qx)>BB> ((1− qk)e+ qx)

(1− κµUφv)(Ũ−p0Ũ

)+ (κcU + κµUqkκcfc)

(1

1−ρ −Ũ−p0Ũ

) (35)We have an equation for pv given {p0, pk, p>x }.

6.3 Solving for p>x

Assume we have solved for {p0, pk}. Then we see from equations (41) and (42)that we also have {qk, q>x }. Hence, we can solve for p>x from equation (33):

q>x = κcUκ>xc + κµU

(((1− qk)e+ qx)>A+ qkκcfcκ>xc

)= (κcU + κµUqkκcfc)

(− 1

1− ρp>x + q

>x

)+ κµU(1− qk)e>A+ κµUq>x A

Inserting for q>x from equation (42) and rearranging gives the followingequation that can be solved for p>x :

[Ũ − p0Ũ

+ (κcU + κµUqkκcfc)

(1

1− ρ− Ũ − p0

Ũ

)]p>x

− κµU

(Ũ − p0Ũ

)p>xA = κµU(1− qk)e>A (36)

We have an equation we can solve for p>x given {p0, pk}.

6.3.1 When xt is a scalar. . .

In the special case where xt is a scalar, so is px, e and A. Equation (36) thenbecomes:

px =κµU(1− qk)eA

Ũ−p0Ũ

+ (κcU + κµUqkκcfc)(

11−ρ −

Ũ−p0Ũ

)− κµU

(Ũ−p0Ũ

)A

19

6.4 Solving for pk

Assume we have solved for p0. Then we can solve for pk from equation (32):

qk = κcUκkc + κµUqk(κkfc + κcfcκkc)

⇒ (1− κµUκkfc)qk = (κcU + κµUκcfcqk)κkc

Inserting for qk from equation (41) and for κkc gives:

(1− κµUκkfc)Ũ − p0Ũ

pk =

(κcU + κµUκcfc

Ũ − p0Ũ

pk

)

×

([Ũ − p0Ũ

− 11− ρ

]pk +

1

1− ρ+

κkf1− ρ

)

This is a quadratic in pk: 0 = ap2k + bpk + c

a =

(κµUκcfc

Ũ − p0Ũ

)(Ũ − p0Ũ

− 11− ρ

)

b =

(κµUκcfc

Ũ − p0Ũ

)1

1− ρ(1 + κkf )

+ κcU

(Ũ − p0Ũ

− 11− ρ

)− (1− κµUκkfc)

Ũ − p0Ũ

c =κcU

1− ρ(1 + κkf )

Given a value for p0 we can solve for the roots of this equation

6.5 Solving for p0

We have shown that given a value for p0 we can solve for {pk, p>x , pv, p1}. Tosolve for p0 given {pk, p>x , pv, p1}, we can use equation (16), which must holdin the non-stochastic steady state. Then we have

20

p0 = Ũ −exp(p1)

pkk̃pk exp (pvv) (37)

Hence, we have an iterative procedure to solve for the coefficients {p0, p1, pk, p>x , pv}.

21

7 The Equations

The equations we have to solve are:

• For the qi’s:

q1 = log Ũ −Ũ − p0Ũ

(pk log k̃ + pvv

)qk =

Ũ − p0Ũ

pk

q>x =Ũ − p0Ũ

p>x

qv =Ũ − p0Ũ

pv

• For pk: 0 = ap2k + bpk + c where

a =

(κµUκcfc

Ũ − p0Ũ

)(Ũ − p0Ũ

− 11− ρ

)

b =

(κµUκcfc

Ũ − p0Ũ

)1

1− ρ(1 + κkf )

+ κcU

(Ũ − p0Ũ

− 11− ρ

)− (1− κµUκkfc)

Ũ − p0Ũ

c =κcU

1− ρ(1 + κkf )

• For px (when x is a scalar):

px =κµU(1− qk)eA

Ũ−p0Ũ

+ (κcU + κµUqkκcfc)(

11−ρ −

Ũ−p0Ũ

)− κµU

(Ũ−p0Ũ

)A

• For pv:

pv =κµU

α2

((1− qk)e+ qx)>BB> ((1− qk)e+ qx)

(1− κµUφv)(Ũ−p0Ũ

)+ (κcU + κµUqkκcfc)

(1

1−ρ −Ũ−p0Ũ

)22

• For p1:

p11− ρ

(κcU + κµUqkκcfc) = κ0U + (κcU + κµUqkκcfc)

(log(1− β)

1− ρ+ q1 +

κ0f1− ρ

)+ κµU

(q1 + qkκ0fc + qv(1− φv)v + (1− qk) log g +

α

2q2vτ

2)

• Steady-state relationships:

Ũ ≈ p0 +exp(p1)

pkk̃pk exp (pvv)

Ũk ≈ exp(p1)k̃pk−1 exp (pvv)

23

A Mapping to Kaltenbrunner and Lochstoer

(2010, RFS)

We can map the process for technology specified above to the one used in KL(sort of. . . ). Their process for technology is specified as:

Zt = exp(µt+ zt) (38)

zt = φzt−1 + εt (39)

where εt ∼ N(0, σ2).We can obtain a similar process by setting the parameters in section 2.3

as follows:

at = Zt

log g = µ

e> = 1

xt = zt − zt−1A = φ

φv = τ = 0

v = 1

Bw1,t+1 = εt+1 − εt

Note the difference in the shocks however. The time t + 1 shocks in KLhave conditional expectation of 0 and conditional variance of σ2 (conditioningon time t information), while our shocks have conditional expectation−εt andconditional variance σ2. In KL the unconditional expectation and varianceare equal to the conditional ones, while for our shocks the unconditionalexpectation is 0 and the unconditional variance is 2σ2.

So in some sense we cannot map between our specification and the oneused in KL.

Note that to get a stationary problem, KL use a different rescaling of thevalue function depending on whether φ = 1 or φ < 1 (see their technicalappendix).

24

B Deriving equations (13) and (14)

To derive equations (13) and (14) it is useful to work out some derivatives indetail:

∂µt∂c̃t

=1

αµ1−αt

∂Et(

(gt+1Ũt+1)α)

∂c̃t

∂µt

∂k̃t=

1

αµ1−αt

∂Et(

(gt+1Ũt+1)α)

∂k̃t

∂Et(

(gt+1Ũt+1)α)

∂c̃t= Et

(gαt+1αŨ

α−1t+1

∂Ũt+1

∂k̃t+1

∂k̃t+1∂c̃t

)

= −αEt

((gt+1Ũt+1)

α−1∂Ũt+1

∂k̃t+1

)∂Et

((gt+1Ũt+1)

α)

∂k̃t= Et

(gαt+1αŨ

α−1t+1

∂Ũt+1

∂k̃t+1

∂k̃t+1

∂k̃t

)

= αEt

((gt+1Ũt+1)

α−1∂Ũt+1

∂k̃t+1

)∂f

∂k̃t

Combining these we get expressions for the derivatives of the certaintyequivalent:

∂µt∂c̃t

= −µ1−αt Et

((gt+1Ũt+1)

α−1∂Ũt+1

∂k̃t+1

)∂µt

∂k̃t= µ1−αt Et

((gt+1Ũt+1)

α−1∂Ũt+1

∂k̃t+1

)∂f

∂k̃t

The first-order condition w.r.t. c̃t gives:

1

ρŨ1−ρt

((1− β)ρc̃ρ−1t + βρµ

ρ−1t

∂µt∂c̃t

)= 0

And the envelope condition gives:

∂Ũt

∂k̃t= βŨ1−ρt µ

ρ−1t

∂µt

∂k̃t

25

Inserting the expressions for the derivatives w.r.t. the certainty equiva-lents into the above, gives equations (13) and (14).

C Definition of coefficients

This appendix keeps track of definitions of the coefficients defined througout.

C.1 Relating qi’s to pi’s

Steady state values are denoted Ũ , k̃, x, and v. The equation we are approx-imating is

log Ũt ≈ log(p0 +

exp(p1)

pkexp

(pk log k̃t

)exp

(p>x xt + pvvt

))Some useful derivatives:

∂ log Ũt

∂ log k̃t

∣∣∣∣∣k̃t=k̃

=exp(p1)k̃

pk exp (pvv)

p0 +exp(p1)pk

k̃pk exp(pvv)

=Ũ − p0Ũ

pk

∇ log Ũt∇ log xt

∣∣∣∣∣xt=x

=

exp(p1)pk

k̃pk exp (pvv)

p0 +exp(p1)pk

k̃pk exp(pvv)p>x

=Ũ − p0Ũ

p>x

∂ log Ũt∂vt

∣∣∣∣∣vt=v

=Ũ − p0Ũ

pv

The log-linear approximation of log Ũt is then given by

26

log Ũt ≈ log Ũ +Ũ − p0Ũ

(pk

(log k̃t − log k̃

)+ p>x xt + pv (vt − v)

)⇒ log Ũt = log Ũ −

Ũ − p0Ũ

(pk log k̃ + pvv

)+Ũ − p0Ũ

(pk log k̃t + p

>x xt + pvvt

)

log Ũt ≈ q1 + qk log k̃t + q>x xt + qvvt

This gives the following relationship between the qi’s and the pi’s:

q1 = log Ũ −Ũ − p0Ũ

(pk log k̃ + pvv

)(40)

qk =Ũ − p0Ũ

pk (41)

q>x =Ũ − p0Ũ

p>x (42)

qv =Ũ − p0Ũ

pv (43)

C.2 From section 5.1

κ0U = ρ log Ũ − κcU log c̃− κµU log µ

κcU =(1− β)ρc̃ρ

(1− β)c̃ρ + βµρ

κµU =βρµρ

(1− β)c̃ρ + βµρ

27

C.3 From section 5.2

κ0f = log fk − κkf log k̃

κkf =fkkk̃

fk

κ0c =log(1− β)

1− ρ+ q1 −

p11− ρ

+κ0f

1− ρ

κkc = qk −pk − 11− ρ

+κkf

1− ρ

κ>xc = q>x −

p>x1− ρ

κvc = qv −pv

1− ρ

28

C.4 From section 5.3

κ0fc = log(f(k̃, 1)− c̃

)− κkfc log k̃ − κcfc log c̃

κkfc =fkk̃

f(k̃, 1)− c̃

κcfc = −c̃

f(k̃, 1)− c̃

κ0gU = q1 + qkκ0fc + qv(1− φv)v + (1− qk) log g + qkκcfcκ0cκkgU = qk(κkfc + κcfcκkc)

κ>xgU =[((1− qk)e+ qx)>A+ qkκcfcκ>xc

]κvgU = qkκcfcκvc + qvφv

κ0µ = κ0gU +α

2q2vτ

2

κkµ = κkgU

κxµ = κxgU

κvµ = κvgU +α

2((1− qk)e+ qx)>BB>((1− qk)e+ qx)

D Steady-state calculations

We need expressions for the steady state values of Ũ , k̃, c̃, and µ since theseshow up in the equations for the coefficients. We’ll use the non-stochasticsteady state, so we set w1,t+1 = w2,t+1 = 0 in equations (5) and (6).

• From the definition of the certainty equivalent we have

µ(gŨ)

= gŨ

• From the captial accumulation equation (2) we have

c̃ = f(k̃, 1)− k̃g

29

• From the rescaled utility equation (7) we have

Ũ =

(1− β

1− βgρ

)1/ρc̃

• From the first-order condition (13) we can solve for Ũk:

Ũk =1− ββ

(gŨ

c̃

)1−ρ• From the envelope condition (14) we can solve for k̃:

∂f

∂k̃=g1−ρ

β

so if f is Cobb-Douglas with capital share θ, we have

k̃ =

(θβ

g1−ρ

) 11−θ

Hence, when solving for the pi’s, we have expressions for all the steadystate terms involved. By inspecting the various coefficients we have definedabove, we see that some coefficients are simply functions of parameters andsteady state values. The following coefficients can be solved independentlyof the approximation coefficients qi’s and pi’s:

κ0U , κcU , κµU , κ0f , κkf , κ0fc, κkfc, κcfc

30