Bank of Canada staff working papers provide a forum for staff to publish work-in-progress research independently from the Bank’s Governing Council. This work may support or challenge prevailing policy orthodoxy. Therefore, the views expressed in this note are solely those of the authors and may differ from official Bank of Canada views. No responsibility for them should be attributed to the Bank. ISSN 1701-9397 ©2020 Bank of Canada

Staff Working Paper/Document de travail du personnel — 2020-13

Last updated: April 16, 2020

Interest Rate Uncertainty as a Policy Tool by Fabio Ghironi1 and Galip Kemal Ozhan2

1Department of Economics University of Washington Seattle, WA 98195, U.S.A. ghiro@uw.edu 2Canadian Economic Analysis Department Bank of Canada Ottawa, ON K1A 0G9, Canada kozhan@bankofcanada.ca

Page ii

Acknowledgements We thank our discussants Jean Barthélemy, Nicolas Caramp, Javier García-Cicco, Robert Kollmann, Alberto Martin, Dmitry Mukhin and Vic Valcarcel for their comments and suggestions on various versions of this paper. We also benefitted from discussions with Paul Beaudry, Brent Bundick, Dmitry Brizhatyuk, Matteo Cacciatore, Giancarlo Corsetti, Kemal Derviş, Galina Hale, Gita Gopinath, Refet Gürkaynak, Juan Carlos Hatchondo, Hande Küçük, Gülçin Özkan and Ricardo Reis, as well as participants at various seminars and conferences. All remaining errors are ours.

Page iii

Abstract We study a novel policy tool—interest rate uncertainty—that can be used to discourage inefficient capital inflows and to adjust the composition of external account between short-term securities and foreign direct investment (FDI). We identify the trade-offs faced in navigating between 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 different assumptions about the irreversibility of FDI, the currency of export invoicing, risk aversion of outside agents, and effective lower bound in the rest of the world. Under every scenario, uncertainty policy is inflationary.

Bank topics: International financial markets; Uncertainty and monetary policy; Monetary policy

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

Résumé Nous étudions un nouvel instrument de politique, soit l’incertitude quant à la trajectoire future des taux d’intérêt, qui peut servir à décourager les entrées de capitaux inefficientes et à ajuster la répartition des investissements entre titres à court terme et investissements directs étrangers dans le compte extérieur. Nous déterminons les arbitrages à faire entre la balance extérieure et la stabilité des prix. La politique d’incertitude quant à la trajectoire future des taux d’intérêt décourage les entrées de capitaux à court terme principalement par deux canaux : les risques de portefeuille et l’épargne de précaution. Par ailleurs, un mécanisme de majoration des prix entraîne des entrées nettes d’investissements directs étrangers lorsque les variations du taux de change ne sont pas entièrement répercutées sur les prix. Nous examinons également de nouveaux mécanismes selon diverses hypothèses sur l’irréversibilité des investissements directs étrangers, la monnaie de facturation des exportations, l’aversion pour le risque des agents extérieurs et la valeur plancher des taux d’intérêt à l’étranger. Dans chaque scénario, la politique d’incertitude est inflationniste.

Sujets : Marchés financiers internationaux; Incertitude et politique monétaire; Cadre de politique monétaire Codes JEL : E32, F21, F32, F38, G15

Page iv

Non-technical summary We examine the hypothesis that using interest rate uncertainty as an unconventional policy tool is effective in dampening inefficient capital inflows and adjusting the composition of the external account between short-term securities and foreign direct investment (FDI).

We develop a two-region New Keynesian model (the emerging market economy [EM]) and the rest of the world [RoW]) that separates international flows in short-term bonds from international investment in physical capital, which we interpret as FDI. The model allows the central bank to manipulate the variance of the domestic policy interest rate in discretionary fashion, which its central bank uses to discourage short-term capital flows and attract FDI while trying to preserve inflation and output stabilization. International financial markets are incomplete, which implies a deviation from perfect risk sharing.

First, we show that using interest rate uncertainty as a policy tool is effective in discouraging inefficient capital inflows. An increase in demand uncertainty for the RoW’s short-term bonds generates capital inflows into the EME. We show that an increase in the EME’s interest rate uncertainty can discourage these capital inflows, shifting their composition towards FDI, and it induces a counteracting effect on the risk-sharing wedge.

Second, we show that the interest rate uncertainty policy discourages bond inflows mainly through investor portfolio risk in the RoW and the EME’s precautionary savings channels. In response to increased uncertainty in the EME’s policy rate, the EME’s bond returns do not compensate for the additional risk created by the central bank, and RoW investors shift away from holding EME debt. In addition, EME households cut consumption in response to uncertainty, but savings go to the RoW, where the risk is lower. The combined effect of these two channels is an outflow in the bond component of the capital account.

Third, in our model a price-markup channel operates to affect FDI dynamics. Under sticky prices, firms cannot effectively adjust to changes in demand and this inability causes markups to move. Furthermore, under imperfect exchange-rate pass-through, exporting firms respond to rising uncertainty by decreasing their prices and by increasing their demand for inward FDI to expand production. We show that this precautionary pricing behavior depends heavily on the level of exchange rate pass-through.

We further compare the interest rate uncertainty policy with a more tailored instrument: capital control uncertainty. We calculate a consumption compensating variation under two policy options. When evaluated during capital inflow periods and when both policies generate the same magnitude of uncertainty on the respective tool, the EME’s households prefer to live under the interest rate uncertainty regime, instead of the capital control uncertainty regime.

We think of this policy as fully discretionary tool to be deployed only in special circumstances. With this perspective, our analysis provides a roadmap for how this unconventional tool might work.

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.