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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. [email protected] URL: http://faculty.washington.edu/ghiro
‡Bank of Canada, 234 Wellington Street, Ottawa, ON K1A 0G9, Canada or [email protected] URL:http://galipkemalozhan.com
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n 1 − n
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,
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∗
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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
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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
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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
]) ,
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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)
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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)
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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 .
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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,
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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
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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
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ρ ρΠ
ρY σ
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ut
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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
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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
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µUIPt+1
µUIPt+1
µUIPt+1
µUIPt+1
i.e. κ = κ∗ = 0
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i.e., εσSW∗
t
t = 1
i.e., εσt t = 2
t = 2
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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
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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
),
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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
).
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i.e. κ = κ∗ = 0
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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 .
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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]) .
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(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
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(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 .
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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).
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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
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a
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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 .
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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.
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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
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β
ρ
χ
ϕ
ψ
κW
ϵW
a
α1
α2
κ
ω
κ∗
ϵ
ρR
ρΠ
ρY
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λ
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κ = κ∗ = 0
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κW = 0
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a = 0.95
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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∗
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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
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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
))].
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− 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
,
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−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
$+ Covt
!log rert
rert+1, logRK∗,t+1
".
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I∗,t =1J [(K∗,t+1 − (1− δ)K∗,t) + ...+ (K∗,t+J − (1− δ)K∗,t+J−1)]
I∗t = 1J
!"K∗
t+1 − (1− δ)K∗t
#+ ...+
"K∗
t+J − (1− δ)K∗t+J−1
#$
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,
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(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.
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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.