Interest Rate Uncertainty as a Policy Tool

⇤

Fabio Ghironi†

University of Washington,

CEPR, EABCN, and NBER

Galip Kemal Ozhan‡

Bank of Canada

April 16, 2020

(click here for the latest version)

Abstract

We study a novel policy tool–interest rate uncertainty–that can be used to discourage in-

e�cient capital inflows and to adjust the composition of external accounts between short-term

securities and foreign direct investment (FDI). We identify the trade-o↵s that are faced in nav-

igating external balance and price stability. The interest rate uncertainty policy discourages

short-term inflows mainly through portfolio risk and precautionary saving channels. A markup

channel generates net FDI inflows under imperfect exchange rate pass-through. We further

investigate new channels under di↵erent assumptions about the irreversibility of FDI, currency

of export invoicing, risk aversion of outside agents, and e↵ective-lower-bound in the rest of the

world. Under every scenario, uncertainty policy is inflationary.

JEL codes: E32, F21, F32, F38, G15.

Keywords: International Financial Policy, Stochastic Volatility, Short-Term and Long-Term

Capital Movements, Unconventional Monetary Policy.

⇤First version: January 8, 2018. We would like to thank our discussants Jean Barthelemy, Nicolas Caramp, JavierGarcıa-Cicco, Robert Kollmann, Alberto Martin, Dmitry Mukhin and Vic Valcarcel for their comments and sugges-tions on various versions of this paper. We also benefitted from discussions with Paul Beaudry, Brent Bundick, DmitryBrizhatyuk, Matteo Cacciatore, Giancarlo Corsetti, Kemal Dervis, Galina Hale, Gita Gopinath, Refet Gurkaynak,Juan Carlos Hatchondo, Hande Kucuk, Gulcin Ozkan, and Ricardo Reis as well as participants at various seminarsand conferences. All remaining errors are ours.

†Department of Economics, University of Washington, Savery Hall, Box 353330, Seattle, WA 98195, U.S.A. orfabio.ghironi.1@gmail.com URL: http://faculty.washington.edu/ghiro

‡Bank of Canada, 234 Wellington Street, Ottawa, ON K1A 0G9, Canada or gozhan@gmail.com URL:http://galipkemalozhan.com

n 1 − n

,

h

Ct(h)

Lt(h)

E0

∞∑

t=0

βtU(Ct(h), Lt(h)),

U(Ct(h), Lt(h)) =Ct(h)1−γ−1

1−γ − χLt(h)1+ϕ

1+ϕ γ,χ,ϕ ≥ 0 β ∈ (0, 1)

K K∗

I I∗

K∗∗

K∗

Kt+1(h) = (1− δ)Kt(h) + It(h),

K∗,t+1(h) = (1− δ)K∗,t(h) + I∗,t(h),

δ ∈ (0, 1)

Lt ≡[∫ 1

0 Lt(h)ϵW−1ϵW dh

] ϵWϵW−1

ϵW > 1

Wt ≡[∫ 1

0 Wt(h)1−ϵW dh] 1

1−ϵW , Wt(h)

h h

Lt(h) =

(Wt(h)

Wt

)−ϵW

Lt.

h Wt(h)

t− 1 t

κW

2

(Wt(h)

Wt−1(h)− 1

)2

Wt(h)Lt(h),

κW ≥ 0 κW = 0

B B∗ S

PtCt(h) +Bt+1(h) + StB∗,t+1(h) +η2Pt(

Bt+1(h)Pt

)2 + η2StP ∗

t (B∗,t+1(h)

P ∗t

)2 + PtIt(h) + StP ∗t I∗,t(h)

= RtBt(h) + StR∗tB∗,t(h) + PtrK,tKt(h) + StP ∗

t rK∗,tK∗,t(h) +Wt(h)Lt(h)

−κW

2

(Wt(h)

Wt−1(h)− 1)2

Wt(h)Lt(h) + dt(h) + Tt(h),

η2ξt(B∗,t+1)2 η > 0

Tt(h) Rt+1 R∗t+1

t t+ 1 dt(h)

rK,t rK∗,t

1 + ηbt+1 = Rt+1Et

[βt,t+1

Πt+1

],

1 + ηb∗t+1 = R∗t+1Et

[βt,t+1

Π∗t+1

rert+1

rert

],

βt,t+s ≡βUC,t+s

UC,tUC,t

t Πt Π∗t t− 1 t

bt+1 ≡ Bt+1(h)Pt

b∗t+1 ≡ B∗,t+1(h)P ∗t

rert ≡ StP ∗t

Pt

Rt+1

R∗t+1

=(1 + ηbt+1)Et[

βt,t+1

Π∗t+1

rert+1

rert]

(1 + ηb∗,t+1)Et[βt,t+1

Πt+1]

.

η = 0

η

1 = Et [βt,t+1 (rK,t+1 + 1− δ)] ,

1 = Et

[βt,t+1

rert+1

rert(rK∗,t+1 + 1− δ)

],

qt = 1,

q∗t = rert.

K∗,t+1

Wt(h) wt ≡ WtPt

wt = µWt

(χLϕ

t

C−γt

),

Wt(h) = Wt µWt

µWt ≡ ϵW

(ϵW − 1)(1− κW

2 (ΠWt − 1)2

)+ κW

(ΠW

t (ΠWt − 1)− Et

[βt,t+1

Πt+1(ΠW

t+1 − 1)(ΠWt+1)

2Lt+1

Lt

]) ,

ΠWt ≡ wt

wt−1Πt

Yt

i ∈ [0, 1]

j ∈ [0, 1]

Yt =

(a

1ω Y

ω−1ω

E,t + (1− a)1ω Y

ω−1ω

R,t

) ωω−1

,

YE,t =(∫ 1

0 YE,t(i)ϵ−1ϵ di

) ϵϵ−1

YR,t =(∫ 1

0 YR,t(j)ϵ−1ϵ dj

) ϵϵ−1

a ∈

(0, 1) a

a > 12

YE,t = a

(PE,t

Pt

)−ω

Yt,

YR,t = (1− a)

(PR,t

Pt

)−ω

Yt,

PE,t PR,t

Pt

Pt =(aP 1−ω

E,t + (1− a)P 1−ωR,t

) 11−ω

.

i ∈ [0, 1]

Kt(i) K∗t (i)

Lt(i)

YE,t(i) +

(1− n

n

)Y ∗E,t(i) = Kt(i)

α1K∗t (i)

α2Lt(i)1−α1−α2 ,

1−nn Y ∗

E,t(i) i α1,α2

α1 + α2 ∈ (0, 1)

YE,t(i) =(PE,t(i)PE,t

)−ϵYE,t

Y ∗E,t(i) =

(P ∗eE,t(i)

StP ∗E,t

)−ϵY ∗E,t PE,t(i)

i P ∗eE,t(i) i

P ∗E,t(i) =

P ∗eE,t(i)

St. PE,t =

(∫ 10 PE,t(i)1−ϵdi

) 11−ϵ

P ∗E,t =

(∫ 10 P ∗

E,t(i)1−ϵdi

) 11−ϵ

rK∗,t

mct =w1−α1−α2t rα1

K,t (rK∗,t)α2

(1− α1 − α2)1−α1−α2 αα1

1 αα22

.

i PE,t(i) P ∗E,t(i)

Et

⎡

⎢⎢⎢⎢⎢⎣

∞∑

s=t

βt,t+s

⎛

⎜⎜⎜⎜⎜⎝+

(1− κ

2

(PE,t+s(i)

PE,t+s−1(i)− 1)2) PE,t+s(i)

Pt+sYE,t+s(i)

(1−nn

)(1− κ∗

2

(P ∗E,t+s(i)

P ∗E,t+s−1(i)

− 1)2) St+sP ∗

E,t+s(i)

Pt+sY ∗E,t+s(i)

−mct(YE,t+s(i) +

(1−nn

)Y ∗E,t+s(i)

)

⎞

⎟⎟⎟⎟⎟⎠

⎤

⎥⎥⎥⎥⎥⎦,

YE,t(i) =(PE,t(i)PE,t

)−ϵYE,t Y ∗

E,t(i) =(P ∗eE,t(i)

StP ∗E,t

)−ϵY ∗E,t

PE,t+s(i) P ∗E,t+s(i)

i.e. rpE ≡ PEP

µE,t

rpE,t = µE,tmct,

i.e. rp∗E ≡ P ∗E

P ∗

µ∗E,t

rp∗E,t = µ∗E,t

mctrert

,

µE,t ≡ϵ

(ϵ− 1)(1− κ

2 (ΠE,t − 1)2)+ κ

(ΠE,t(ΠE,t − 1)− Et

[βt,t+1

Πt+1(ΠE,t+1 − 1)(ΠE,t+1)2

YE,t+1

YE,t

]) ,

µ∗E,t ≡

ϵ

(ϵ− 1)(1− κ∗

2 (Π∗E,t − 1)2

)+ κ∗

(Π∗

E,t(Π∗E,t − 1)− Et

[βt,t+1

Π∗t+1

(Π∗E,t+1 − 1)(Π∗

E,t+1)2 rert+1

rert

Y ∗E,t+1

Y ∗E,t

]) .

ΠE,t ≡rpE,t

rpE,t−1Πt Π∗

E,t ≡rp∗E,t

rp∗E,t−1Πt

YE,t +

(1− n

n

)Y ∗E,t = Kt

α1K∗tα2Lt

1−α1−α2 ,

Kt =∫ 10 Kt(i)di K∗

t =∫ 10 K∗

t (i)di Lt =∫ 10 Lt(i)di

α1wtLt = (1− α1 − α2) rK,tKt,

α2rK,tKt = α1rK∗,tK∗t .

Yt = Ct+It+I∗t +κW

2

(ΠW

t − 1)2

wtLt+κ

2(ΠE,t − 1)2 rpE,tYE,t+

(1− n

n

)κ∗

2

(Π∗

E,t − 1)2

rp∗E,tY∗E,t.

bt+1 + b∗t+1 = 0 b∗∗,t+1 + b∗,t+1 =

0 Tt =

η2

[Pt(

Bt+1(h)Pt

)2 + StP ∗t (

B∗,t+1(h)P ∗t

)2].

bt+1 + rertb∗,t+1 +(1−nn

)rertK∗,t+1 −K∗

t+1

= RtΠtbt +

R∗t

Π∗trertb∗,t +

(1−nn

)rert (rK,∗,t + 1− δ)K∗,t −

(r∗K,t + 1− δ

)K∗

t + TBt,

TBt ≡(1−nn

)µ∗E,tmctY ∗

E,t − rertµR,tmc∗tYR,t

t t + 1

CAt

(bt+1 − bt) + rert (b∗,t+1 − b∗,t)︸ ︷︷ ︸+

(1− n

n

)rert (K∗,t+1 −K∗,t)−

(K∗

t+1 −K∗t

)

︸ ︷︷ ︸≡ CAt,

Rt+1

R=

(Rt

R

)ρ(Πt

Π

)(1−ρ)ρΠ (YtY

)(1−ρ)ρY

eut ,

ρ

ρΠ ρY

ut

ut AR(1)

ut = ρuut−1 + eσtεt,

εt

σt AR(1)

σt = (1− ρσ)σ + ρσσt−1 + εσt ,

εσt σ

Yt Ct It I∗t Kt K∗,t Lt YE,t Y ∗E.t mct rpE,t rp∗E,t wt rK,t rK,∗,t bt

Rt Πt rert

β

γ

χ, ϕ

η

κW

ϵW ,

α1 α2

κ κ∗

ϵ

ρ ρΠ

ρY σ

ut

1 + ηbt+1 =Rt+1

euSWt

Et

[βt,t+1

Πt+1

],

1 + ηb∗t+1 =R∗

t+1

euSW∗t

Et

[βt,t+1

Π∗t+1

rert+1

rert

],

1 + ηb∗∗,t+1 =R∗

t+1

euSW∗t

Et

[β∗t,t+1

Π∗t+1

],

1 + ηb∗t+1 =Rt+1

euSWt

Et

[β∗t,t+1

Πt+1

rertrert+1

].

uSWt uSW∗t

AR(1)

uxt = ρxuxt−1 + eσxt εxt ,

x ∈ {SW,SW∗} σxt

AR(1)

σxt = (1− ρσx)σx + ρσ

xσxt−1 + εσ

x

t .

σSW∗,

1 = µUIPt+1

UC∗,t

UC,trert,

µUIPt+1 ≡ 1+ηb∗t+1

1+ηbt+1

Et

!UC,t+1Pt+1

"

Et

#UC∗,t+1

Pt+1rert+1

$ .

µUIPt+1

µUIPt+1

µUIPt+1

µUIPt+1

µUIPt+1

µUIPt+1

i.e. κ = κ∗ = 0

i.e., εσSW∗

t

t = 1

i.e., εσt t = 2

t = 2

t = 2

xB∗,B∗

∗t+1 ≡ rt+1 − r∗t+1 − st+1 + st,

xK∗,K∗

∗t+1 ≡ rK∗,t+1 − rK∗

∗ ,t+1 − ˆrert+1 + ˆrert,

xB∗,K∗

t+1 ≡ rt+1 − πt+1 − rK∗,t+1.

Et

[xB

∗,B∗∗

t+1

]= Et

[xK

∗,K∗∗

t+1

]=

Et

[xB

∗,K∗

t+1

]= 0.

rK,t+1 ≡ RK,t+1−RK

RKRK,t+1 ≡ rK,t+1 + 1− δ

q∗t = 1rert

Et

[xB

∗,B∗∗

t+1

]≈ −1

2V art(∆st+1) + Covt(m

∗t+1,∆st+1),

Et

[xK

∗,K∗∗

t+1

]≈ −1

2

(V art∆ ˜rert+1 + V artrK∗,t+1 − V artrK∗

∗ ,t+1)

+Covt (∆ ˜rert+1, rK∗,t+1)− Covt(logβ∗

t,t+1, rK∗,t+1 −∆ ˜rert+1 − rK∗∗ ,t+1

),

Et

[xB

∗,K∗

t+1

]≈ −1

2V artπt+1 +

1

2V artrK∗,t+1 + Covt

(rK∗,t+1 + πt+1, logβ

∗t,t+1 −∆ ˜rert+1

).

i.e. κ = κ∗ = 0

rer

(P ∗H

P

)ϵ(P ∗H(i)

P ∗

)1−ϵ

Y ∗H −

(ϵ− 1

ϵ

)(P ∗H

P ∗

)ϵ(P ∗H(i)

P ∗

)−ϵ

Y ∗H .

K∗,t+1(h) = (1− δ)K∗,t(h) + I∗,1,t(h),

I∗,j−1,t+1(h) = I∗,j,t(h); j = 2, ..., J.

I∗,t(h) =∑J

j=11J I∗,j,t(h),

1J

j I∗,j,t(h) t

j

J

q∗,t+J−1 = Et+J−1[βt+J−1,t+J

(rert+JrK,∗,t+J + q∗,t+J (1− δ)

)],

Et [βt,t+J−1q∗,t+J−1] =1

J(rert + Et [βt,t+1rert+1] + ...+ Et [βt,t+J−1rert+J−1]) .

(bt+1 − bt) + rert (b∗,t+1 − b∗,t)︸ ︷︷ ︸+1

J

[(1− n

n

)rert (K∗,t+J −K∗,t)−

(K∗

t+J −K∗t

)]

︸ ︷︷ ︸≡ CAt

(rer

P ∗H

P

)ϵ(P ∗hH (i)

P

)1−ϵ

Y ∗H −

(ϵ− 1

ϵ

)(rer

P ∗H

P

)ϵ(P ∗hH (i)

P

)−ϵ

Y ∗H .

Vt ≡ (1− β)U(Ct(h), Lt(h))− β[Et (−Vt+1)

1−α]1/(1−α)

,

α ∈ R α = 0 U ≤ 0

α

γ = 2 U ≤ 0U ≥ 0

Vt ≡ (1− β)U(Ct(h), Lt(h)) + β!Et (Vt+1)

1−α"1/(1−α).

β∗t,t+1,

β∗t,t+1 ≡βU∗

C,t+1

U∗C,t

(−V ∗

t+1(Et[−V ∗1−α∗

t+1

])1/(1−α∗)

)−α∗

.

α∗ < 0

Rt+1

R∗t+1

=

(1 + ηb∗t+1)Et[βU∗

C,t+1

U∗C,t

(−V ∗

t+1%Et

!−V ∗1−α∗

t+1

"&1/(1−α∗)

)−α∗

1Π∗

t+1]

(1 + ηb∗∗,t+1)Et[βU∗

C,t+1

U∗C,t

(−V ∗

t+1%Et

!−V ∗1−α∗

t+1

"&1/(1−α∗)

)−α∗

rertΠt+1rert+1

]

.

α∗ α

a

PtCt(h) +Bt+1(h) + StB∗,t+1(h) +η2Pt(

Bt+1(h)Pt

)2 + η2StP ∗

t (B∗,t+1(h)

P ∗t

)2 + PtIt(h) + StP ∗t I∗,t(h)

= RtBt(h) + StR∗t (1 + τt−1)B∗,t(h) + PtrK,tKt(h) + StP ∗

t rK,∗,tK∗,t(h) +Wt(h)Lt(h)

−κW

2

(Wt(h)

Wt−1(h)− 1)2

Wt(h)Lt(h) + dt(h) + Tt(h) + T τt (h).

τ

1 + b∗,t+1 = (1 + τt)R∗t+1Et

[β∗t,t+1

Π∗t+1

rert+1

rert

].

Rt+1 R∗t+1 ≡ (1 + τt)R∗

t+1

1+τt = eut , ut = ρuut−1+eσtεt, σt AR(1)

σt = (1 − ρσ)σ + ρσσt−1 + εσt .

V it = Et

∞∑

j=0

βj(Cit+j

1−ρ − 1

1− ρ− χ

Lit+j

1+ϕ

1 + ϕ

)i ∈ {IRUPT,CCU}.

λ

V CCUt = Et

∞∑

j=0

βj

((1 + λ)CIRUPT

t+j

)1−ρ − 1

1− ρ− Et

∞∑

j=0

βjχLIRUPTt+j

1+ϕ

1 + ϕ.

1 + ηbt+1 = Rt+1Et

[βt,t+1

Πt+1

]

1 + ηb∗t+1 = R∗t+1Et

[βt,t+1

Π∗t+1

rert+1

rert

]

Kt+1(h) = (1− δ)Kt(h) + It(h)

K∗,t+1(h) = (1− δ)K∗,t(h) + I∗,t(h)

1 = Et [βt,t+1 (rK,t+1 + 1− δ)]

1 = Et

[βt,t+1

rert+1

rert(rK,∗,t+1 + 1− δ)

]

wt = µWt

(χLϕ

t

C−ρt

)

YE,t = a(

PE,t

Pt

)−ωYt

YR,t = (1− a)(

PR,t

Pt

)−ωYt

1 =(a · rp1−ω

E,t + (1− a)rp1−ωR,t

) 11−ω

mct =w

1−α1−α2t r

α1K,t(r

∗K,t)

α2

(1−α1−α2)1−α1−α2α

α11 α

α22

rpE,t = µE,tmct

rp∗E,t =µ∗E,tmctrert

YE,t +(1−nn

)Y ∗E,t = Kt

α1K∗tα2Lt

1−α1−α2

α1wtLt = (1− α1 − α2) rK,tKt

α2rK,tKt = α1r∗K,tK∗t

Yt = Ct + It + I∗t + κW

2

(ΠW

t − 1)2

wtLt

+κ2 (ΠE,t − 1)2 rpE,tYE,t

+(1−nn

)κ∗

2

(Π∗

E,t − 1)2

rp∗E,tY∗E,t

bt+1 + rertb∗,t+1 +(1−nn

)rertK∗,t+1 −K∗

t+1

= RtΠt

bt +R∗

tΠ∗

trertb∗,t +

(1−nn

)rert (rK,∗,t + 1− δ)K∗,t

−(r∗K,t + 1− δ

)K∗

t + TBt

RtR =

(Rt−1

R

)ρ (ΠtΠ

)(1−ρ)ρΠ(YtY

)(1−ρ)ρYeut

β

ρ

χ

ϕ

ψ

κW

ϵW

a

α1

α2

κ

ω

κ∗

ϵ

ρR

ρΠ

ρY

λ

κ = κ∗ = 0

κW = 0

a = 0.95

Pt Tt =

η2

!Pt(

Bt+1(h)Pt

)2 + StP ∗t (

B∗,t+1(h)P ∗t

)2"

bt+1 + rertb∗,t+1 +#1−nn

$rertI∗,t =

RtΠt

bt +R∗

tΠ∗

trerrb∗,t + wtLt + rK,tKt +

#1−nn

$rertrK,∗,tK∗,t + I∗t

+(µE,t − 1)mctYE,t +#1−nn

$ %µ∗E,t − 1

&mctY ∗

E,t − Yt.

wtLt + rK,tKt = mct%YE,t +

#1−nn

$Y ∗E,t

&− r∗K,tK

∗t

bt+1 + rertb∗,t+1 +#1−nn

$rertI∗,t − I∗t =

%RtΠt

&bt +

%R∗

tΠ∗

t

&rertb∗,t

+#1−nn

$rertrK,∗,tK∗,t − r∗K,tK

∗t +

#1−nn

$µ∗E,tmctY ∗

E,t − rertµR,tmc∗tYR,t.

b∗∗,t+1 +b∗t+1

rert+%

nn−1

&I∗trert

− I∗,t =%R∗

tΠ∗

t

&b∗∗,t +

%Rt

rertΠt

&b∗,t

−rK,∗,tK∗,t +%

n1−n

&%r∗K,tK

∗t

rert

&+%

n1−n

&mc∗tµR,tYR,t − mct

rertµ∗E,tY

∗E,t.

rert(1 − n)

nbt+1 + (1− n)b∗t+1 = 0 nb∗,t+1 + (1− n)b∗∗,t+1 = 0

2n(bt+1 + rertb∗,t+1) + 2((1− n)rertI∗,t − nI∗t ) = 2n%%

RtΠt

&bt +

%R∗

tΠ∗

&rertb∗,t

&

+2(1− n)rertrK,∗,tK∗,t − 2nr∗K,tK∗t + 2(1− n)µ∗

E,tmctY ∗E,t − 2nrertµR,tmc∗tYR,t.

2n K∗ K∗

bt+1 + rertb∗,t+1 +!1−nn

"rertK∗,t+1 −K∗

t+1

= RtΠtbt +

R∗t

Π∗trertb∗,t +

!1−nn

"rert (rK,∗,t + 1− δ)K∗,t −

#r∗K,t + 1− δ

$K∗

t + TBt,

TBt ≡!1−nn

"µ∗E,tmctY ∗

E,t − rertµR,tmc∗tYR,t

1 + ηb∗∗,t+1 = R∗t+1Et

[β∗t,t+1

Π∗t+1

],

1 + ηb∗t+1 = Rt+1Et

[β∗t,t+1

Πt+1

rertrert+1

],

1 = Et

⎡

⎢⎢⎣β∗t,t+1

⎛

⎜⎜⎝rK∗∗ ,t+1 + 1− δ

︸ ︷︷ ︸≡RK∗∗ ,t+1

⎞

⎟⎟⎠

⎤

⎥⎥⎦ ,

1 = Et

⎡

⎢⎣β∗t,t+1rertrert+1

⎛

⎜⎝rK∗,t+1 + 1− δ︸ ︷︷ ︸

≡RK∗,t+1

⎞

⎟⎠

⎤

⎥⎦ .

−log(R∗t+1) ≈ Etlog

(β∗t,t+1

Π∗t+1

)

︸ ︷︷ ︸≡M∗

t+1

+1

2V art

(β∗t,t+1

Π∗t+1

),

−log(Rt+1) ≈ EtlogM∗t+1 + Etlog

(St

St+1

)

+12

[V artlog(M∗

t+1) + V artlog(

StSt+1

)+ 2Covt

(logM∗

t+1, log(

StSt+1

))].

− log(Rt+1)! "# $rt+1

≈ EtlogM∗t+1! "# $

≡Etm∗t+1

+ log (St)! "# $≡st

−Etlog (St+1)

+12

%V artm∗

t+1 + V art (st − st+1)&+ Covt

'm∗

t+1, st − st+1(.

rt+1 − r∗t+1 ≈ Etst+1 − st −1

2V art (st − st+1)− Covt

'm∗

t+1, st − st+1(.

K∗ K∗∗

0 = Etlogβ∗t,t+1 + EtlogRK∗∗ ,t+1

+12

%V artlogβ∗t,t+1 + V artlogRK∗

∗ ,t+1 + 2Covt(logβ∗t,t+1, logRK∗∗ ,t+1)

&,

0 = Etlogβ∗t,t+1 + Etlogrert

rert+1RK∗,t+1

+12V artlogβ∗t,t+1 +

12 V artlog

rertrert+1

RK∗,t+1

! "# $

= V artlogrert

rert+1+ V artlogRK∗,t+1

+2Covt)log rert

rert+1, logRK∗,t+1

*

+Covt(logβ∗t,t+1, log

rertrert+1

RK∗,t+1)! "# $

= Covt(logβ∗t,t+1, logrert

rert+1)

+Covt(logβ∗t,t+1, logRK∗,t+1)

.

Etlogrert

rert+1+ EtlogRK∗,t+1 − EtlogRK∗

∗ ,t+1 =

−12

)V artlog

rertrert+1

+ V artlogRK∗,t+1 − V artlogRK∗∗ ,t+1

*− Covt

)log rert

rert+1, logRK∗,t+1

*

−Covt)log rert

rert+1, logβ∗t,t+1

*− Covt

'logβ∗t,t+1, logRK∗,t+1

(+ Covt

'logβ∗t,t+1, logRK∗

∗ ,t+1(.

B∗ K∗

−rt+1 ≈ Etlogβ∗t,t+1

Πt+1

rertrert+1

+1

2V artlog

β∗t,t+1

Πt+1

rertrert+1

,

−rt+1 = Etlogrert

rert+1− EtlogΠt+1 + Etlogβ∗t,t+1

−12

!V artlog

rertrert+1

+ V artlogβ∗t,t+1 + V artlogΠt+1

"− Covt

#logβ∗t,t+1, logΠt+1

$

+Covt!log rert

rert+1, logβ∗t,t+1

"− Covt

!log rert

rert+1, logΠt+1

".

0 ≈ Etlogrert

rert+1+ EtlogRK∗,t+1 + Etlogβ∗t,t+1

+1

2

%V art

&log

rertrert+1

+ logβ∗t,t+1 + logRK∗,t+1

'(

) *+ ,

= 12

!V artlog

rertrert+1

+ V artlogβ∗t,t+1 + V artlogRK∗,t+1

"

+Covt!log rert

rert+1, logβ∗t,t+1

"+ Covt

!log rert

rert+1, logRK∗,t+1

"+ Covt

#logβ∗t,t+1, logRK∗,t+1

$

.

rt+1

rt+1 − EtlogΠt+1 − EtlogRK∗,t+1 ≈ −12V artlogΠt+1 +

12V artlogRK∗,t+1

+Covt#logβ∗t,t+1, logΠt+1

$+ Covt

!log rert

rert+1, logΠt+1

"

+Covt#logβ∗t,t+1, logRK∗,t+1

$+ Covt

!log rert

rert+1, logRK∗,t+1

".

I∗,t =1J [(K∗,t+1 − (1− δ)K∗,t) + ...+ (K∗,t+J − (1− δ)K∗,t+J−1)]

I∗t = 1J

!"K∗

t+1 − (1− δ)K∗t

#+ ...+

"K∗

t+J − (1− δ)K∗t+J−1

#$

bt+1 + rertb∗,t+1 +"1−nn

#rert

1J (K∗,t+J + δK∗,t+J−1 + ...+ δK∗,t+1)− 1

J

"K∗

t+J + δK∗t+J−1 + ...+ δK∗

t+1

#

= RtΠtbt +

R∗t

Π∗trertb∗,t +

"1−nn

#rert

"rK,∗,t +

1J (1− δ)

#K∗,t −

%r∗K,t +

1J (1− δ)

&K∗

t + TBt,

(1− n

n

)κ∗

2

(P ∗eE,t+s(i)

P ∗eE,t+s−1(i)

− 1

)2P ∗eE,t+s(i)

Pt+sY ∗E,t+s(i).

i (PE,t(i), P ∗eE,t(i), YE,t(i), Y ∗

E,t(i))

Et

⎡

⎢⎢⎢⎢⎢⎣

∞∑

s=t

βt,t+s

⎛

⎜⎜⎜⎜⎜⎝+

(1− κ

2

(PE,t+s(i)

PE,t+s−1(i)− 1)2) PE,t+s(i)

Pt+sYE,t+s(i)

(1−nn

)(1− κ∗

2

(P ∗eE,t+s(i)

P ∗eE,t+s−1(i)

− 1)2) P ∗e

E,t+s(i)

Pt+sY ∗E,t+s(i)

−mct(YE,t+s(i) +

(1−nn

)Y ∗E,t+s(i)

)

⎞

⎟⎟⎟⎟⎟⎠

⎤

⎥⎥⎥⎥⎥⎦.

PE,t+s(i) P ∗eE,t+s(i)

i.e. rpE ≡ PEP

µE,t

rpE,t = µE,tmct,

µ∗E,t

rp∗E,t =µ∗E,tmct

rert,

µE,t ≡ϵ

(ϵ− 1)(1− κ

2 (ΠE,t − 1)2)+ κ

(ΠE,t(ΠE,t − 1)− Et

[βt,t+1

Πt+1(ΠE,t+1 − 1)(ΠE,t+1)2

YE,t+1

YE,t

]) ,

µ∗E,t ≡

ϵ

(ϵ− 1)(1− κ∗

2 (Π∗eE,t − 1)2

)+ κ∗

(Π∗e

E,t(Π∗eE,t − 1)− Et

[βt,t+1

Πt+1(Π∗e

E,t+1 − 1)(Π∗eE,t+1)

2 Y∗E,t+1

Y ∗E,t

]) ,

Π∗eE,t ≡

rp∗E,t

rp∗E,t−1

rertrert−1

Πt.

V i,Ct = Et

∞∑

j=0

βj(Ci

t+j)1−ρ − 1,

1− ρV i,Lt = −Et

∞∑

j=0

βj(Li

t+j)1+ϕ

1 + ϕ.

V CCUt = Et

∞∑

j=0

βj

((1 + λcond)CIRUPT

t+j

)1−ρ− 1

1− ρ− Et

∞∑

j=0

βjχ

(LIRUPTt+j

)1+ϕ

1 + ϕ.

V CCUt = (1 + λcond)1−ρ

[V IRUPT,Ct +

1

(1− β)(1− ρ)

]− 1

(1− β)(1− ρ)+ V IRUPT,N

t .

λcond =

⎛

⎝V CCUt − V IRUPT,N

t + 1(1−β)(1−ρ)

V IRUPT,Ct + 1

(1−β)(1−ρ)

⎞

⎠

11−ρ

− 1.

λuncond =

⎛

⎝E[V CCUt

]− E

[V IRUPT,Nt

]+ 1

(1−β)(1−ρ)

E[V IRUPT,Ct

]+ 1

(1−β)(1−ρ)

⎞

⎠

11−ρ

− 1.