Working Paper Series E-2010-03

Optimal Monetary Policy and Nominal Exchange
Rate Volatility under Local Currency Pricing

Eiji Okano (Chiba Keizai University)©2010 Eiji Okano. All rights reserved. Short sections of text, not toexceed two paragraphs, may be quoted without explicit permission
provided that full credit is given to the source.

EK-115iv

Comments welcome.

Optimal Monetary Policy and Nominal Exchange
Rate Volatility under Local Currency Pricing∗
OKANO, Eiji†
This Draft: Dec., 2010
Abstract
We analyze the fluctuation in inflation and nominal exchange rate
under optimal monetary policy with local currency pricing, by developing a two-country model belonging to DSGE with local currency pricing
and comparing fluctuations under local currency pricing with fluctuations
under producer currency pricing. Although preceding DSGE literatures
assuming producer currency pricing show that stabilizing domestic inflation is optimal from the view point of minimizing welfare costs, weshow that completely stabilizing consumer price index inflation is optimal
from that view point. In addition, we show that completely stabilizing
consumer price index inflation is equivalent with completely stabilizing
nominal exchange rate.

Keywords: Local Currency Pricing, DSGE, Optimal monetary policy, Fixed
Exchange Rate Regime, CPI Inflation
JEL Classification: E52; E62; F41∗ I would like to thank Chikafumi Nakamura, Eiji Ogawa and seminar participants at Hitotsubashi University. All errors are my own. This research is financially assisted by The
Zengin Foundation for Studies on Economics and Finance.
† Correspondence to: Faculty of Economics, Chiba Keizai University, 59-5, Todorokicho
3-chome, Inage-ku, Chiba-shi, Chiba, 263-0021, JAPAN. Tel: +81-43-253-9738; Fax: +8143-254-6600; E-mail: e.okano@cku.ac.jp

1

Introduction

While new open economy macroeconomics (NOEM) try to analyze optimal exchange rate regime dynamic stochastic general equilibrium (DSGE) literatures
discuss optimal monetary policy. Roughly speaking, NOEM has the policy implication that flexible exchange rate regime is optimal from the view point of
maximizing social welfare which is equivalent to minimizing welfare costs, if
firms set their prices following producer currency pricing (PCP) while fixed exchange rate regime is optimal if firms set their prices following local currency
pricing (LCP). Some DSGE literatures assuming an open economy have policy
implication that stabilizing producer price index (PPI) inflation which is equivalent to domestic or GDP inflation is optimal from that view point. Some DSGE
literatures assuming an open economy assuming PCP and those do not haveattention to price setting behavior except for few papers. NOEM and DSGE
literatures have not still reconciled in policy implications in an open economy
and there is enough room to discuss optimal monetary policy under the LCP.
To analyze the sort of inflation rate which should be stabilized under the
LCP and reconcile policy implications derived by NOEM and DSGE literatures,
we develop a two-country economy model under the LCP belonging to DSGE
and study the fluctuation not only in inflation but also in nominal exchange
rate. Although some DSGE literatures assuming an open economy under the
PCP show that stabilizing PPI inflation is optimal from the view point of minimizing welfare costs, we show that completely stabilizing consumer price index(CPI) inflation is optimal from that view point. In addition, we show thatcompletely stabilizing CPI inflation is equivalent with completely stabilizing
nominal exchange rate under the LCP.
Now, we review some preceding papers to show the importance of our aim
to study the sort of inflation rate which should be stabilized under the LCP andreconcile policy implications derived by NOEM and DSGE literatures in thispaper. By developing not only PCP but also LCP model following NOEM, Deverereux and Engel[3] discuss the optimal exchange rate regime from the view
point of welfare maximization and show that fixed exchange rate regime is desirable under the LCP although floating exchange rate regime is desirable under
the PCP. Their finding is not trivial but important because conventional papers show that optimal monetary policy in an open economy requires exchange
rate flexibility. However, because of inwardness of NOEM, they cannot show
eﬀects on price stability with fixed exchange rate regime under the LCP. Hencethey do not provide what kind of inflation rate should be stabilized following
Woodford’s[10] motivation.
Gali and Monacelli[5] show that optimal monetary policy in a small open
economy is consistent with domestic price inflation targeting. Although they do
not mention explicitly, they assume the PCP. In addition, they compare threepolicy regimes, PPI inflation based and CPI inflation based Taylor rules andfixed exchange rate regime and show that PPI inflation based Taylor rule brings
the closest macroeconomic volatility from macroeconomic volatility brought
about by optimal monetary policy among those three regimes.1 Their policy implication is also important because their policy implication implies that
1 Correctly, Gali and Monacelli[5] dub not PPI inflation based Taylor rule but domestic

inflation based Taylor rule. However, the definition of their domestic inflation is consistent
with our definition of the PPI inflation.

1

outcome of optimal monetary policy is not fundamentally diﬀerent from theone of the closed economy. While they do not highlight the firms’ price setting
behavior, Gali and Monacelli[5] imply that PPI inflation targeting is optimalunder the PCP. In addition, they comply Woodford’s[10] motivation.
Somehow, some DSGE literatures do not focus on the firms’ price setting
behavior and those assume the PCP. There are few DSGE literatures focus
on the firms’ price setting behavior. Based on Gali and Monacelli[5]’s model,
Monacelli[7] introduces exporters whose price setting behavior can be regarded
as the LCP and analyze monetary policy in a low-pass through environment.
He can show that outcome of monetary policy is quite diﬀerent not only fromcanonical papers but also Gali and Monacelli[5] who implies that stabilization in
the PPI inflation achieves stabilization in output gap simultaneously. Becauseof low of one price (LOOP) gap, the analysis of monetary policy of an open
economy is fundamentally diﬀerent from the one of a closed economy. Whilehe focuses on important point, he cannot comply Woodford’s[10] motivation.
He does not reply what kind of inflation rate should be stabilized under such a
low-pass through environment stemming from LCP and cannot reconcile policy
implications derived by NOEM and DSGE literatures while he shows importance of commitment on the monetary policy. Another few DSGE author is
Okano[9] who shows that CPI inflation targeting stabilizes output to changes
in demand shock by utilizing a two-country economy model under the LCP.
Although his paper is insightful, he failures to show clear policy implication on
CPI inflation targeting to changes in productivity shock and he does not derive microfounded loss function which stems from second-order Taylor expanded
utility function alike with Monacelli[7]. Hence, it cannot be said that Okano
complies Woodford’s[10] motivation. In addition, he cannot derive clear policy
implication on fluctuations in the nominal exchange rate under the CPI inflation
targeting under the LCP and cannot reconcile with Deverereux and Engel[3]’s
policy implication. It is quite obvious that analyzing optimal monetary policy following DSGE under the LCP to comply Woodford’s[10] motivation andreconcile Deverereux and Engel[3]’s policy implication.
As mentioned above, our aims in this paper are finding the sort of inflation
rate which should be stabilized under the LCP and reconciling policy implications derived by NOEM with DSGE literatures. To achieve our aims, we develop both the LCP and the PCP model which assume a two-country. We derive
well microfounded loss function under both the LCP and the PCP, stemmingfrom second-order Taylor expanded utility function following Woodford[11] andGali[4]. We assume that central banks in two countries solve optimization problem under both the LCP and the PCP and impulse response functions (IRFs)
are calculated. We calculate IRFs under the special case in which the relativerisk aversion and the elasticity of substitution between goods produced in both
two countries are unity and under the general case in which the relative risk
aversion and the elasticity of substitution between goods produced in both twocountries are 3 and 4.5, respectively.2 Note that those elasticity settings in thespecial case is consistent with Gali and Monacelli’s[5] setting and those elasticity settings in the general case is consistent with Benigno and Benigno’s[1].
To compare with the result on Gali and Monacelli[5] and to discuss optimal
2 The relative risk aversion and the elasticity of substitution between goods produced inboth two countries are often dubbed the intertemporal elasticity of substitution and the intratemporal elasticity of substitution, respectively.

2monetary policy on general parameterization, we analyze both two cases. Because we are interested in macroeconomic volatility which aﬀects on the welfare
costs based on second-order approximated utility function and are interested
in nominal exchange rate volatility under the PCP and the LCP, we calculate
macroeconomic volatility including the nominal exchange rate varying the relative risk aversion and the elasticity of substitution between goods produced inboth two countries. Finally, we calculate welfare costs varying the relative risk
aversion and the elasticity of substitution between goods produced in both twocountries.
Now, we mention our results as follows. First of all, we show that optimalmonetary policy under the LCP brings no fluctuations not in the PPI inflation rate but in the CPI inflation rate. Roughly speaking, optimal monetary
policy under the LCP is the CPI inflation targeting. This result is quite diﬀerent from the result on Gali and Monacelli[5]. Our result is confirmed by IRFs,
volatility on the CPI inflation and loss function stemming from second-order approximated utility function. Interestingly, the quadratic terms of CPI inflation
rate appear our loss function and replace the quadratic terms of PPI inflation
under the LCP, although the quadratic terms of PPI inflation appear in ourloss function under the PCP alike with Gali and Monacelli[5] and Benigno and
Benigno[1]. Next, we can reconcile with Deverereux and Engel[3] because there
are no fluctuations on nominal exchange rate under the LCP. Roughly speaking,
optimal monetary policy under the LCP is consistent with fixed exchange rate
regime and that is shown by Deverereux and Engel[3]. In addition, this result
is not depending on the relative risk aversion and the elasticity of substitution
between goods produced in both two countries. That there are no fluctuation
on nominal exchange rate is consistent regardless of those preferences. Because
Deverereux and Engel[3] analyzes under some restriction which consistent withunitary elasticity of substitution between goods produced in both two countries
in our model, we can support their results and can generalize their policy implication. Summarizing our result, optimal monetary policy under the LCP is
not only consistent with CPI inflation targeting but also consistent with fixedexchange rate. Details on our results are discussed on the rest of this paper.
The rest of this paper is organized as follows. Section 2 derives two models,the LCP and the PCP model. Section 3 analyzes optimal monetary policy byderiving welfare costs, FONCs for central bank with commitment and calibration. Section 4 analyzes eﬀect on macroeconomic volatility and welfare costs ofvarying relative risk aversion and the elasticity of substitution between goods
produced in two countries. Section 5 concludes this paper. An appendix shows
analysis on international monetary policy cooperation between two countries,
which is omitted in the text because we highlight fluctuations in inflation andnominal exchange rate.

2

The Model

We construct a two-country model belonging to the class of DSGE models
with nominal rigidities and imperfect competition, basically following Gali andMonacelli[5] and Monacelli[7]. We alter Gali and Monacelli[5]’s small open economy model to two-country economy model following Obstfeld and Rogoﬀ[8] although we assume all goods are tradables. The union-wide economy consists3

of two countries, countries H and F . Country H produces an array of diﬀerentiated goods indexed by the interval h ∈ [0, 1], while country F produces anarray of diﬀerentiated goods indexed by f ∈ [1, 2]. In addition, we derive twomodels, one of them is assumed the LCP and¡ ¢another one is assumed the PCP.
Note that we take a definition vt ≡ ln VVt if there are no provisions whereVt denotes an arbitrary variable and V denotes steady state value of Vt .

2.1

LCP Model

Under the LCP, LOOP is not necessarily applied because firms can choose prices
to sell goods in countries H and F separately. Thus, Pt (h) = Et Pt∗ (h) and∗∗and PF,t = Et PF,t
do not necessarily hold
Pt (f ) = Et Pt∗ (f ) hence PH,t = Et PH,twhere Pt (h) and Pt (f ) denote the price of a generic good produced in country
1
hR
i 1−ε
1
1−ε
dh
and PF,t ≡H in terms of country H’s currency, PH,t ≡ 0 Pt (h)
1
i 1−ε
hR
2
1−ε
Pt (f )dhdenote indices of the price of generic goods produced in1

countries H and F , respectively, Et denotes nominal exchange rate.3 Note thatquantities and prices particular to country F are denoted by asterisks whilequantities and prices without asterisks are those in country H.
2.1.1

Households

The preferences of the representative household in country H are given by:
U ≡ E0∞X

β t Ut ,

(1)

t=0

1
1
Ct1−σ − 1+ϕ
Nt1+ϕ denotes the period utility, Et denotes thewhere Ut ≡ 1−σexpectation, conditional on the information set at period t, β ∈ (0, 1) denotes the
R1
subjective discount factor, Ct denotes consumption, Nt ≡ 0 Nt (h) dh denotes
hours of work, σ denotes the relative risk aversion and ϕ denotes the inverse of
the labor supply elasticity. The preferences of the representative household incountry F is defined analogously.
More precisely, private consumption is a composite index defined by:

Ct

≡

η
"µ ¶ 1µ ¶ η1 η−1 # η−1
1 η η−11ηηCH,t +
CF,t
,
2
2

(2)

θ
θ
hR
i θ−1
hR
i θ−1
θ−1
θ−1
1
2
where CH,t ≡ 0 Ct (h) θ dh
and CF,t ≡ 1 Ct (f ) θ df
denoteDixit—Stiglitz-type indices of consumption across the home goods and foreign
goods, respectively, and η > 0 denotes the elasticity of substitution between
tradables and nontradables. Note that Ct∗ is defined analogously to Eq.(2).
Total consumption expenditures by households in country H are given by
PH,t CH,t + PF,t CF,t = Pt Ct . A sequence of budget constraints in country H isgiven by:

Bt + Wt Nt − Tt ≥ Pt Ct + Et (Qt,t+1 Bt+1 ) ,3 By citing Betts and Devereux[2], Mark[6] clearly explanes the LCP.

4

(3)

where Qt,t+1 denotes the stochastic discount factor, Bt denotes the nominal
payoﬀ of the bond portfolio purchased by households, Wt denotes the nominal
wage, and Tt denotes lump-sum taxes. The budget constraint in country F isgiven analogously. Furthermore:Pt ≡µ1 1−η 1 1−η
P+ PF,t
2 H,t
2

1
¶ 1−η

,

(4)

denotes the consumption price index (CPI). Pt∗ is defined analogously to thisequality. By log-linearizing this equality yields pt = 12 pH,t + pF,t , which implies
as follows:
πt =

1
1
πH,t + πF,t ,
2
2

(5)

where πt ≡ pt − pt−1 denotes CPI inflation with πH,t = pH,t − pH,t−1 andπF,t = pF,t − pF,t−1 .
The optimal allocation of any given expenditure within each category of
goods implies the demand functions, as follows:
¶−ε
¶−ε
µ
µ
Pt (h)
Pt (f )Ct (h) =CH,t ; Ct (f ) =
CF,t
PH,t
PF,t
!−ε
!−ε
Ã
Ã
Pt∗ (h)Pt∗ (f )∗∗∗∗Ct (h) =CH,t ; Ct (f ) =
CF,t
.
(6)
∗
∗
PH,t
PF,t
The optimal allocation of expenditures between domestic and foreign goods
is given by:
µ
µ
¶−η
¶−η
1 PH,t1 PF,t
CH,t =
Ct ; CF,t =
Ct ,
2
Pt
2
Pt
µ ∗ ¶−ηµ ∗ ¶−η
1 PH,t1 PF,t
∗
∗
∗
=
C
;
C
=
Ct∗ .
(7)
CH,t
t
F,t
2 Pt∗
2 Pt∗
The representative household maximizes Eq.(1) subject to Eq.(3). The optimality conditions are given by:
Ã
!
−σ
Pt
Ct+1
= 1,
(8)
Rt βEtCt−σ Pt+1
which is a conventional Euler equation and
Ctσ Ntϕ =

Wt
,
Pt

(9)

which is a standard intratemporal optimality condition where Rt ≡ 1 + rt satisfying Rt−1 = Et Qt,t+1 denotes the gross nominal return on a riskless one-period
discount bond paying oﬀ one unit of the common currency (in short, the grossnominal interest rate), and rt denotes the net nominal interest rate. Eq.(8) isan intertemporal optimality condition, namely the Euler equation, and Eq.(9)is an intratemporal optimality condition. Optimality conditions in country F
are given analogously.
5

Log-linearizing Eq.(8), we obtain:
ct = Et ct+1 −11r̂t + Et πt+1
σ
σ

(10)

¡ ¢
with r̂t ≡ ln RRt .
There is relationship on the gross nominal interest rate between countries H
and F which is uncovered interest rate parity (UIP) as follows:
µ
¶
Et+1
∗
Rt = R t E t
Etwith Rt∗ ≡ 1 + rt∗ . Log-linearizing the UIP, we have the familiar expression as
follows:
Et (∆et+1 ) = r̂t − r̂t∗ ,¡ ¢with ∆vt ≡ vt − vt−1 and et ≡ ln EEt .
Combining Eq.(8) and the UIP and iterating with an initial condition, we
have the following optimal risk-sharing condition:σ

Ctσ = ϑ (Ct∗ ) Qt ,E P∗with Qt ≡ tPtt denoting the real exchange rate and ϑ denoting a constantdepending on the initial value. Log-linearizing this equality, we have:ct = c∗t +
2.1.2

1
qt .
σ

(11)

Market Clearing

The market for tradables and for nontradables in country H clears when domestic demand equals domestic supply, as follows:
Yt (h) = Ct (h) + Ct∗ (h) ,(12)where Yt (h) denotes the output of good h, which is market clearing condition.
Market clearing condition in country F is analogously. Plugging Eq.(7) into
Eq.(12) yields:1
Yt (h) =2µ

Pt (h)
PH,t

¶−ε µ

PH,t
Pt

¶−η

1
Ct +
2

Ã

Pt∗ (h)
∗
PH,t

!−ε µ

∗
PH,t
Pt∗

¶−η

Ct∗ .

(13)

ε
hR
i ε−1
ε−1
1
represent index for aggregate output in counLet Yt ≡ 0 Yt (h) ε dhtry H. Under the LCP, we obtain:

Yt

=
=

µ ∗ ¶−η
1 PH,t
Ct +
Ct∗ ,
2 Pt∗
#
µ
µ
¶−η "
¶η µ ∗ ¶−η
1
PH,t
PH,t
1 PH,t−σ,
Ct 1 +
Qt
2
Pt
Pt
Pt∗
1
2

µ

PH,t
Pt

¶−η

6

(14)

by combining Eqs.(13), Dixit—Stiglitz aggregators for output and prices, where
we take Eq.(11) in the second lines in Eq.(14).
We define the terms of trade (TOT) as follows:
St ≡

PF,t
∗ ,
Et PH,t(15)

where St is foreign TOT. The numerator is export price of goods produced incountry F in terms of country H’s currency and the denominator is export
price of goods produced in country F in terms of country H’s currency. Loglinearizing Eq.(15), we have:st = pF,t − et − p∗H,t .

(16)

Plugging Eq.(21) into log-linearized Eq.(14), we have:
µ
¶
1
η
1
η−
qt ,
y t = ct + st +22
σwhich is log-linearized market clearing in country H under the LCP. Although
there is a diﬀerence between this equality and Eq.(42) because logarithmic realexchange rate qt appears in this equality. However, this equality boils down
to Eq.(42) because the PPP is applied which implies that qt = 0 although weassume the LCP. We discuss about the PPP under the LCP in section 2.3.
Combining Eq.(42) and its counterpart in country F , we have:st =1(yt − yt∗ ) − qt ,ηwhich clarifies relationship between the TOT and relative output under the
LCP. As mentioned, qt = 0 is applied although we assume the LCP. Hence, thisequality boils down to Eq.(41).
2.1.3

Firms

Each producer uses a linear technology to produce a diﬀerentiated good as
follows:
Yt (h) = At Nt (h) ,(17)where At denotes stochastic productivity in country H. Firms in country F
have a technology analogously to firms in country H.
Using Dixit—Stiglitz aggregators, Eq.(17) can be rewritten as:
Yt Dt
,
(18)
At
¡
¢
R1
2
dh.
Because
d
is
o
k
ξ
k
with Dt ≡ 0 YtY(h), a first order approximation oftt
this equality is given by:
Nt =yt = at + nt ,(19)

which is consistent with Gali and Monacelli’s[5] log-linearized production function.
7

Similar to many DSGE literatures including Gali and Monacelli[5], we assume that firms set prices in Calvo—Yun-style price-setting behavior. Hence,a measure 1 − θ firms sets new prices each period, with an individual firm’s
probability of re-optimizing in any given period being independent of the time
elapsed since it last set its prices. Each producer produces a single diﬀerentiated good and prices its good to reflect the elasticity of substitution across goods
produced given the CPI. This is because each firm plays an active part in themonopolistically competitive market. In addition, we assume that firms havethe ability to engage in price discrimination by setting a domestic price in termsof domestic currency for domestic sales that diﬀers from the price that it setsfor exports. This is the LCP behavior. Under the Calvo—Yun-style price-setting
behavior and the LCP behavior in a monopolistically competitive market, themaximization problems which producers in country H face are as follows:
⎧
⎡
Ã
!−ε
Ã
!−ε
∞
∗
⎨
X
P̃
P̃
H,t
H,t
∗
∗
θk Et Qt,t+k ⎣P̃H,t
CH,t+k + Et+k P̃H,t
CH,t+kmax∗∗⎩
PH,t+kPH,t+kP̃H,t ,P̃H,t
k=0
⎛Ã
⎞⎤⎫
!−ε
!−ε
Ã
∗
⎬
P̃H,t
P̃H,t
n ⎝
∗
⎠⎦ , (20)−M Ct+kCH,t+k +CH,t+k∗⎭PH,t+kPH,t+k∗are the prices chosen by firms when they obtain the chance
where P̃H,t and P̃H,tto change prices associated with goods produced and sold in country H andgoods produced in country H while sold in country F , respectively, M Ctn ≡)Wt
PP,t M Ct denotes real marginal costs in country H, with M Ct ≡ (1−τAt PP,t andPP,t denotes producer price index (PPI) in country H, which are defined asfollows:

PP,t ≡∗∗
CH,t
PH,t CH,t + Et PH,t,∗
CH,t + CH,twhich can be rewritten as PP,t = PH,t when the LOOP is applied. The PPI
in country F is defined
analogously.
By log-linearizing this equality, we have¡¢
pP,t = 12 pH,t + 12 et + p∗H,t , which implies as follows:
πP,t =¢
1
1¡∗∆et + πH,t,
πH,t +
2
2(21)

where πP,t denotes the PPI inflation in country H and πPt = πH,t is applied
when the LOOP is applied.
Note that the maximization problems which producers in country F faceare analogously to Eq.(20). Because of nominal rigidities, Eq.(20) looks complicated. When there are no nominal rigidities, namely θ → 0, Eq.(20) problems
boil down to:
¡
¢
∗
∗
∗
max∗ PH,t CH,t + Et PH,tCH,t
− M Ctn CH,t + CH,t,
PH,t ,PH,twhich implies that each firm sets its price in terms of local currency in which
each firm’s good is sold and pay costs to produce in terms of producer currency.

8

Under the LCP, we have multiple FONCs because firms can choose P̃H,t and∗P̃H,tseparately. The FONCs for Eq.(20) are as follows:

⎡

⎡

Et ⎣

Et ⎣

∞
X

k=0

∞
X

k=0

³
´
n
θk Qt,t+k P̃H,t − ζM Ct+k³´∗
n
θ k Qt,t+k P̃H,t
Et+k − ζM Ct+k

Ã

P̃H,t
PH,t+k

!−ε

Ã

∗
P̃H,t
∗
PH,t+k!−εwhich can be log-linearized as follows:p̃H,tp̃∗H,t==(1 − βθ)(1 − βθ)

∞
X

k=0
∞
X
k=0

⎤

CH,t+k ⎦ =

0,

∗
⎦ =
CH,t+k

0,

⎤

¡
¢
k
(βθ) Et mcnt+k ,¡¢k(βθ) Et mcnt+k − et+k ,(22)θdenoting a constant markup where we use the fact that Qt,t+k =with ζ ≡ θ−1
³
´−σ
C
Pt
β k Ct+k
Pt+k . Eq.(21) implies that firms set the price as a markup over atweighted average of expected future marginal costs. Especially, the first equality
in Eq.(22) definitely corresponds to one derived by Gali and Monacelli[5]. The
second equality in Eq.(22) is not a familiar expression although it implies thatfirms set the price as a markup over a weighted average of expected futurenominal marginal costs. The second equality in Eq.(22) is the log-linearized
FONC for firms which produce goods in country H and sell them in country
F . Those firms set the price in terms of country F ’s currency as a markup
over a weighted average of expected future nominal marginal costs in terms ofcountry F ’s currency. We learn further the character of Eq.(22) after discuss
some identities including the relative prices which is peculiar to LCP behavior.
Under the LCP, the LOOP is not necessarily applied because of Eqs.(20)
and (22), which imply that firms set their price of goods in terms of local
currency, namely the LCP. Because of that setting, there is the LOOP gap,
which measures the degree of the pass-through. Now, we discuss the LOOP gap
and the real exchange rate in our mode. Following Monacelli[7], we define theLOOP gap as follows:ΨH,t ≡∗∗
Et PH,tEt PF,t; ΨF,t ≡PH,tPF,twhere ΨH,t and ΨF,t denote the LOOP gap for goods produced in countries H
and F , respectively. When the LOOP is applied, we have ΨH,t = ΨF,t = 1.
Combining Eq.(7), the optimal risk-sharing condition and the definition of
the TOT yields:µ∗ ¶Et PF,t− 1−1
Qt ση ,
ΨH,t = Ψ−1
S
F,t t
PH,t
which implies that the LOOP gap is a function of the TOT, the real exchange rate and the relative price of goods consumed domestically. Because
9

St−1

³

∗
Et PF,tPH,t´= ΨH,t ΨF,t , that equality can be rewritten as follows:
Qt = 1,

which implies that the PPP is applied although the LOOP is not applied.4
Log-linearized version of this equality is given by:
qt = 0.

(23)
c∗t ,

In addition, plugging Eq.(23) into Eq.(11), we have ct =which implies
that the marginal utility of consumption between both countries are equal. In
fact, households in both countries consume same goods although there is pricediscrimination. As mentioned, the LOOP is not necessarily applied although
Eq.(23) implies that the PPP is definitely applied. This sounds inconsistent at
glance. However, although a price of one goods violate the LOOP, the PPP
is applied when another goods violate the LOOP inversely. In fact, plugging
Eq.(23) into that equality, we have ΨH,t = Ψ−1
F,t and log-linearized version ofthis as follows:ψH,t = −ψF,t ,which implies that gains from price discrimination corresponds to loses fromprice discrimination.
Log-linearized marlet clearing conditions in countries H and F clarifies therelationship among nominal exchange rate, the price level and the TOT. Plugging log-linearized definition of the CPI into log-linearized marlet clearing conditions yields:
et

=
=
=

pt − p∗t
pP,t − p∗P,t + st ,1
pP,t − p∗P,t + (yt − yt∗ )η(24)

where we use Eq.(21) to derive the second line and Eq.(41) to derive the thirdline. Eq.(24) implies that output diﬀerential between both countries aﬀects
nominal exchange rate.
In turn, we discuss the character of Eq.(22), log-linearized FONCs for firms
under the LCP. By taking the definition of the LOOP gap, Eq.(22) can be
rewritten as follows:∞∞X
1 − βθ X(βθ)k Et (πH,t+k ) +(βθ)k Et (ψH,t+k )p̃H,t = pH,t−1 +
2
k=0
∞
X

+ (1 − βθ)p̃∗H,t=p∗H,t−1 +(βθ)k Et (mct+k ) ,

k=0

∞
X

∞
¡ ∗
¢ 1 − βθ X−(βθ)k Et πH,t+k(βθ)k Et (ψH,t+k )
2

k=0
∞
X

+ (1 − βθ)

k=0

k=0

(βθ)k Et (mct+k ) ,

(25)

k=0

4 This equality implies that the marginal utility of consumption in country H is definitely
same as it in country F . Hence, the UIP can be derived by simply combining 8 and its
counterpart in country F without assuming, although we describe that we assume the UIP in
section 2.1.1.

10∗
where πH,t ≡ pH,t − pH,t−1 and πH,t
≡ p∗H,t − p∗H,t−1 denotes inflation of goods
both produced and sold in country H and inflation of goods produced in country
H and sold in country F , respectively. As mentioned, firms set the price as amark up over a weighted average of future marginal cost. In our LCP setting,those firms’ sales are not measured by the PPI, because it is the weighted averageof both price of goods selling in country H and in country F . However, real
marginal cost is measured by the PPI, as shown in the definition of nominal
marginal cost. That is, those firms obtain sales measured by PH,t and pay
total costs measured by the PP,t and that gap is calculated by pP,t − pH,t =1
2 ψH,t , which implies that that gap corresponds to the LOOP gap in country
H. Although the firms selling goods in country H have no currency disparity insales and payment, the LCP behavior generate the LOOP gap. Thus, a weighted
average of expected future LOOP gap in country H appears in the first equality
in Eq.(25).
The price setting behavior of the firms selling goods in country F generates
the LOOP gap, alike with another firms which sell goods in country H. Thosefirms, namely exporters, obtain the sales of goods exported in terms of country
F ’s currency and pay the total cost in terms of country H’s currency. Their sales
are measured by country H’s currency. Hence, their sales in terms of country
F ’ currency is multiplied by nominal exchange rate. They pay total costs whichis measured by the PPI, alike¡ with the¢ firms selling goods in country H. The
gap is calculated by pP,t − p∗H,t + et = − 12 ψH,t . Thus, a weighted averageof expected future LOOP gap in country H appears in the second equality inEq.(25) although the sign is contrary to the first equality. Similar mechanism
works in firms in country F not only for selling goods domestically but also
for exporters. Although our LCP setting is diﬀerent from Monacelli[7], who
assumes a small open economy and importers, our LCP setting clearly generates
the LOOP gap and this setting aﬀects the forms of New Keynesian Philips Curve(NKPC) and social welfare stemming from a second-order approximated utility
function.

2.1.4

Marginal Cost and Natural Rate of Output

Plugging Eq.(9) into the definition of the marginal cost, we obtain as follows:
M Ct = (1 − τ )Ctσ Ntϕ
At

µ

PP,t
Pt

¶−1

,

(26)

which is log-linearized as follows:1
mct = σct + ϕnt + st − at ,2

(27)

which is consistent with Gali and Monacelli’s[5] log-linearized marginal cost.
Under the flexible price equilibrium, M Ct = ζ1 implying that the real marginal
cost is constant and corresponds to inverse of a constant markup is applied. Using this fact and combining Eqs.(14), (18) and Eq.(26), we have natural rate ofoutput under the LCP in country H as follows:
1
Ȳt =
2

(

PP,t ζ −1 1+ϕA
Pt 1 − τ t

µ

PH,t
Pt

¶−ησ "

1+

11

µ

PH,t
Pt

¶η µ

∗
PH,t
Pt∗

¶−η

1
−σ

Qt

1
#σ ) ϕ+σ−ϕ

Dt ϕ+σ ,with Ȳt denotes natural rate of output in country H, which implies that naturalrate of output is a function not only of productivity but also of relative prices
because of an open economy setting.
Before log-linearizing this equality, we define the output gap in country H
xt as the deviation of percentage deviation of output in country H yt from its
natural level ȳt . This relationship can be written as:
xt ≡ yt − ȳt ,(28)which is definitely consistent with Gali and Monacelli[5]’s definition. The output
gap in country F is defined analogously to Eq.(28)Now, we log-linearize that equality. Log-linearized natural rate of output
under the LCP is given by:
ȳt =

ω1 ω2
(ση − 1) ω2 ∗
at −
at ,
ω3
ω3

(29)
2

with ω1 ≡ η (σ + 2ϕ) + 1, ω2 ≡ 2η (1 + ϕ) and ω3 ≡ ω12 − (ση − 1) . While Gali
and Monacelli[5] regard foreign output is exogenous because of their small open
economy setting, foreign output, namely output in country F is endogenous
in our two-country setting. Thus, productivity in country F replaces foreign
output in Eq.(29).
We turn to discuss Eq.(27), percentage deviation of marginal cost from its
steady state value. Plugging Eqs.(42), (41), (19) and (28) into Eq.(27) yields:
mct =

ω1
ση − 1 ∗
xt +
xt ,
2η
2η

(30)

which implies that real marginal cost in country H consists of output gap inboth two countries.
2.1.5

The Demand and Supply Sides

Plugging Eqs.(21), (23), (42) and (41) into Eq.(10) yields New Keynesian IS
Curve (NKIS) as follows:2η2η
ση − 1 ¡ ∗ ¢ 2ηr̂t +Et (πP,t+1 ) +Et ∆xt+1 +
r̄t ,
(31)
xt = Et (xt+1 ) −
σα
σα
σα
σα

4
5 ∗
at −σα (1−ρ)(ση−1)(1+ϕ)ω
at denotes the natural rate
where r̄t ≡ −σα (1−ρ)(1+ϕ)ω
ω3
ω3
2

of interest in country H with σα ≡ ση + 1, ω4 ≡ ω1 − (ση−1)and ω5 ≡ σωα1 − 1.
σαThe NKIS in country F , which is analogous to Eq.(31), can be derived by using
Eqs.(41) and (23) and counterparts of Eqs.(10), (42) and(21).
Eq.(31) looks like ordinary NKIS in the DSGE literature at glance. Becauseof the LCP, Eq.(31) has some distinguished feature. Plugging Eq.(21) into
Eq.(31), NKIS under LCP can be rewritten as follows:
¡ ∗
¢
2η
η
η
η
r̂t +
Et (πH,t+1 ) +Et πH,t+1
Et (∆et+1 )xt = Et (xt+1 ) −
+
σα
σα
σα
σα
ση − 1 ¡ ∗ ¢ 2η+Et ∆xt+1 +
r̄t
σα
σα
¡ ∗
¢
η
η ∗
η
η
= Et (xt+1 ) −
r̂t −
r̂ +
Et (πH,t+1 ) +Et πH,t+1
σα
σα t
σα
σα
ση − 1 ¡ ∗ ¢ 2ηEt ∆xt+1 +
r̄t ,
(32)
+
σα
σα
12

where we take log-linearized UIP to derive second line. As shown in the firstline, changes in expected nominal exchange rate aﬀects the NKIS. Second line
shows that not only domestic nominal interest rate, but also foreign nominalinterest rate appears the NKIS.
Plugging log-linearized Calvo’s pricing rule and Eq.(30) to Eq.(25), we haveequalities which determines the dynamics of inflation as follows:
πH,t

=

∗
πH,t

=

λ
λω1
λ (ση − 1) ∗
ψH,t +
xt +
xt ,
2
2η
2η
¡ ∗
¢ λ
λω1
λ (ση − 1) ∗− ψH,t +xt +xt ,βEt πH,t+122η
2η
βEt (πH,t+1 ) +(33). The first equality is inflation dynamics for goods soldwith λ ≡ (1−βθ)(1−θ)θ
domestically and the second equality is inflation dynamics for goods exported.
Because Eq.(33) derived from Eq.(25), the FONCs for firms in country H, thethird and the fourth terms in the RHS, which stem from real marginal costin country H, are consistent between both equalities. The signs of the second
terms in the RHS are inverse between both equalities. The reason is that thelosses from price discrimination are compensated by the gains from price discrimination, and vice versa. Counter part of Eq.(33) are derived from counterpart of Eq.(25).
Plugging Eq.(33) into Eq.(21), we have New Keynesian Philips Curve (NKPC)
in country H as follows:
πP,t = βEt (πP,t+1 ) +λω1λ (ση − 1) ∗ β
1xt +xt − Et (∆et+1 ) + ∆et ,
2η
2η
2
2

and plugging counter part of Eq.(33) into Eq.(21) yields counter part of thisequality in country F . This NKPC is featured by appearance of changes innominal exchange rate. Gali and Monacelli[5] mention that full stabilizationof domestic prices coincides with full stabilization of output gap, namely xt =πH,t = 0 for all t. In our model, their domestic prices correspond to the PPI andthey assume fully-exogenous foreign output which implies that the percentage
deviation of marginal cost from its steady state value is not aﬀected by thepercentage deviation of foreign output from its steady state value. That is, theyclaim that full stabilization of PPI implies that output conforms its natural rate
if we ignore foreign output gap or assume ση = 1 in this equality. Even if we
ignore foreign output gap or assume ση = 1 in this equality, full stabilization of
PPI does not necessarily imply that output conforms its natural rate because ofchanges in nominal exchange rate, as shown in the forth and the fifth terms in
the RHS. Changes in nominal exchange rate as if work cost push shocks under
the LCP. Thus, full stabilization of PPI no longer implies that output conforms
its natural rate if we ignore foreign output gap or assume ση = 1. Plugging
Eqs.(24), (29) and (28) into that equality, we can eliminate changes in nominal
exchange rate and obtain as follows:
πP,t

=

¡ ∗
¢ β
¢
β ¡
βEt (πP,t+1 ) + βEt πP,t+1− Et (xt+1 ) + Et x∗t+1 + κ$ xt
η
η
1
1
∗
− κς x∗t − πP,t
− xt−1 + x∗t−1 + ω6 at − ω6 at−1 − ω6 a∗t + ω6 a∗t−1
η
η
(34)
13

1
with κ$ ≡ 1+β+λω, κς ≡ 1+β−λ(ση−1), ω6 ≡ ω2 $3ω(σ+ϕ)
and $3 ≡ 1+β (1 − ρ).
ηη
3
Exogenous shocks appear in Eq.(34) which shows that exogenous productivity
aﬀects PPI inflation.
Monacelli[7] derives CPI based NKPC. Following Monacelli[7], we derive
CPI based NKPC. Plugging the first equality in Eq.(33) and its counterpart incountry F into Eq.(5) yields:

πt = βEt (πt+1 ) +

κα
κα ∗
xt +
x
2
2 t

(35)

with κα ≡ λ (σ + ϕ). As mentioned by Gali and Monacelli[5], κα is consistent
with the slope coeﬃcient of standard closed economy NKPC. A full stabilization not of PPI inflation but of CPI inflation implies that output conforms its
natural rate when the nominal interest rate in both countries absorbs the eﬀects
from changes in productivity in NKISs. Gali and Monacelli[5] mention that afull stabilization of PPI inflation implies output conforms its natural level andthere is no output gap, in their non-LCP setting under a small open economy,
as mentioned. However, our CPI based NKPC Eq.(35) implies that a full stabilization of CPI inflation implies output conforms its natural level and there isno output gap, in our LCP setting under a two-country. This can be understoodalternatively and intuitively by comparing Eqs.(5) and (21). To derive Eq.(34),
we use Eq.(21) implying that the PPI inflation is aﬀected by changes in nominal
exchange rate while we use Eq.(5) to derive Eq.(35).
In addition, Eq.(35) contrasts CPI based NKPC in Monacelli[7]. In his LCP
setting, imports purchase foreign goods at the costs in terms of foreign currency
while they sell foreign goods at the costs in terms of domestic currency. Because
importers maximize their profits, the LOOP gap appears in CPI based NKPC
in Monacelli[7]. Our LCP setting is quite diﬀerent from Monacelli’s[7] setting.
Goods markets are fully partitioned, there are no importers and each producer
prices their goods in terms of consumer’s currency. As mentioned in section
2.1.3, the LOOP gap does not appear in Eq.(35), diﬀerent from Monacelli[7].

2.2

PCP Model

Under the PCP, the LOOP is applied which is given by Pt (h) = Et Pt∗ (h) andPt (f ) = Et Pt∗ (f ) hence:∗∗; PF,t = Et PF,t
PH,t = Et PH,t

(36)

pH,t = et + p∗H,t ; pF,t = et + p∗F,t ,

(37)

and

are applied.
2.2.1

Households

The preference of the representative household, private consumption index, Consumption index, the optimal allocation of any given expenditure within eachcategory of goods and the optimal allocation of expenditures between domestic
and foreign goods are given by Eqs.(1), (2), (3), (4), (6) and (7), alike with the
LCP model. Because households face same optimization problem, intertemporal
14

and intratemporal optimality conditions are given by Eqs.(8) and (9). The UIP
is applied in the PCP model, hence optimal risk-sharing condition is appliedin PCP model. Log-linearized definition of the CPI, intertemporal optimality
condition and the risk-sharing condition are also given by Eqs.(5), (10) and (11).
2.2.2Market Clearing

Market clearing condition is given by Eq.(12) alike with the LCP model. Plugging Eqs.(6) and (7), we have Eq.(13). Because of LOOP, Eq.(13) can be rewritten as:
¶−ε µ
¶−η
µ
PH,t
Pt (h)Ct .
Yt (h) =PH,tPt
by utilizing Eq.(36). Plugging Dixit-Stiglitz aggregator of output into this equality yields:
Yt =

µ

PH,t
Pt

¶−η

Ct ,

(38)

which is demand function consistent with Benigno and Benigno’s[1].5
Definition of the TOT is given by Eq.(15). Plugging Eq.(36) into Eq.(15)yields:
St =

PF,t
,
PH,t

(39)

which is applicable only to PCP model because Eq.(36) is not applicable to LCP
model.
Log-linearizing Eq.(39), we have:st = pF,t − pH,t(40)

with st ≡ lnSt . Eq.(40) is only if applicable to PCP model because Eq.(37) is
not applied under the LCP, alike with Eq.(39).
Plugging Eq.(37) into this equality, we have:st =1(yt − yt∗ ) ,η(41)which clarifies relationship between the TOT and relative output under the
PCP. Gali and Monacelli[5] and Benigno and Benigno[1] who assume the PCP
derive same equality.
Log-linearizing Eq.(38) yields:yt =η
st + ct ,2(42)where we use Eq.(37). As mentioned, Eq.(42) is final form of log-linearizedmarket clearing under the LCP in country H. The diﬀerence in price setting
behavior between the LCP and the PCP does not aﬀect market clearing.
5 We do not assume government expenditure. Thus, government expenditure does notappear in those equalities although it appears in Benigno and Benigno[1].

152.2.3

Firms

Firms technology is given by Eq.(17) which can be rewritten as Eq.(18). Thus,
log-linearized technology is given by Eq.(19) alike with LCP model.
We assume Calvo-Yun-style price setting behavior alike with LCP model.
However, maximization problem which is faced by firms under the PCP is quite∗and Eq.(36), Eq.(20) can be rewritten as:
simple. Because of P̃H,t = Et P̃H,t
max
P̃H,t

∞
X

k=0

θ k Et

⎧
⎨
⎩

⎡

´
³
n
Qt,t+k ⎣ P̃H,t − M Ct+k

Ã

P̃H,t
PH,t+k

!−ε

⎤⎫
⎬
¡
¢
∗
⎦ , (43)
CH,t+k + CH,t+k⎭

which is familiar expression in literatures assuming Calvo pricing. Plugging
Eq.(36) into the PPI definition, we have PP,t = PH,t and plugging Eq.(37) into
Eq.(21) yields:
πP,t = πH,t .

(44)

The FONC of Eq.(43) is given by:
⎤
⎡
!−ε
Ã
∞
³
´
X
¡
¢
P̃
H,t
n
∗
⎦ = 0,
CH,t+k + CH,t+k
θ k Qt,t+k P̃H,t − ζM Ct+k
Et ⎣PH,t+kk=0

which is familiar expression in literatures assuming the PCP. Log-linearizing
this equality, we have:p̃H,t = (1 − βθ)

∞
X

k=0

¡
¢
(βθ)k Et mcnt+k ,which corresponds to the first equality in Eq.(22). Terms related to the LOOP
gap disappear because the LOOP is definitely applied in the PCP model. Thisequality can be rewritten as follows:p̃H,t = pH,t−1 +

∞
X

k=0

(βθ)k Et (πH,t+k ) + (1 − βθ)∞X(βθ)k Et (mct+k ) ,

(45)

k=0

which corresponds to one derived by Gali and Monacelli[5]. Because of theLOOP, LOOP gap disappears in Eq.(45), although LOOP gap appears in thefirst equality in Eq.(25).
2.2.4

Marginal Cost and Natural Rate of Output

Plugging Eq.(9) into the definition of the marginal cost, we obtain Eq.(26) andits log-linearized equality Eq.(27). However, the natural rate of output under
the PCP is quite diﬀerent from one under the LCP at glance. Combining not
only Eqs.(14), (18) and Eq.(26) but also PP,t = PH,t , we have:"ζ −1 1+ϕ
Ȳt =
A
1−τ t

µ

PH,t
Pt

1
¶−(ση−1) # σ+ϕ

16

−

ϕ

Dt σ+ϕ ,which can be log-linearized as follows:
ȳt =

ω1 ω2
(ση − 1) ω2 ∗
at −
at .
ω3
ω3

This equality is consistent with log-linearized natural rate of output under the
LCP Eq.(29) although natural rate of output is quite diﬀerent between the PCP
and the LCP before log-linearizing. This implies that diﬀerences in price setting
behavior do not aﬀect the natural rate of output.
That natural rate of output under the PCP is consistent with one under the
LCP implies that there is same relationship between marginal cost and output
gap. In fact, plugging Eqs.(42), (41), (19) and (28) into Eq.(27) yields:
mct =

ω1
ση − 1 ∗
xt +
xt ,
2η
2η

which is consistent with Eq.(30). Diﬀerence between the PCP and the LCP
models is price setting behavior. Because the marginal cost has no relationshipwith price setting behavior, Eq.(30) is applied under both the PCP and the LCP.
Note that Gali and Monacelli[5] show that real marginal cost has relationshipwith just domestic output gap and their result is diﬀerent from Eq.(30). Thisdiﬀerence stems from our two-country setting. As mentioned, foreign output is
not exogenous in our setting and productivity in country F appears in Eq.(29),while foreign output appears in their expression in terms of percentage deviationfrom its steady state value. In their setting, not foreign productivity but foreign
output aﬀects domestic natural rate of output. Foreign output gap no longer
aﬀects domestic output gap which stems from percentage deviation of domestic
real marginal cost from its steady state value. Because percentage deviationof domestic real marginal cost from its steady state value corresponds to its
deviation from its flexible price equilibrium value, foreign output gap disappears
in Gali and Monacelli[5]. In fact we have mct = ω2η1 xt if we regard output incountry F as exogenous.
2.2.5

The Demand and Supply Sides

Plugging Eqs.(21), (23), (41) and (42) into Eq.(10) yields NKIS as follows:
xt = Et (xt+1 ) −2η2ηση − 1 ¡ ∗ ¢ 2ηr̂t +Et (πP,t+1 ) +Et ∆xt+1 +
r̄t ,
σα
σα
σα
σα

(46)

which is consistent with NKIS under the LCP Eq.(31). While the LOOP is
not applied in LCP model, the LOOP is applied in PCP model. Hence, NKISs
are not completely same between both models although those are quite same atglance. Plugging Eq.(44) into Eq.(31), we have:
xt = Et (xt+1 ) −2η2ηση − 1 ¡ ∗ ¢ 2ηr̂t +Et (πH,t+1 ) +Et ∆xt+1 +
r̄t ,
σα
σα
σα
σα

which is applicable only if to PCP model and πH,t replaces πP,t in this equality.
Because the LOOP is definitely applied in PCP model, neither changes in expected nominal exchange rate nor foreign nominal interest rate appear in NKIS
under the PCP.

17

By rearranging Eq.(45), we have NKPC in country H under the PCP as
follows:
πP,t = βEt (πP,t+1 ) +λω1λ (ση − 1) ∗
xt +
xt ,
2η
2η

(47)

which is two-country version NKPC derived by Gali and Monacelli[5]. Whileforeign output gap appears in Eq.(47), that does not appear in the NKPC
derived by Gali and Monacelli[5] who assume small open economy where foreign
variables are exogenous. Because our model is a two-country model where theforeign variables are endogenous, foreign output gap appears in our NKPC under
the PCP. In fact, if we regard output in country F as exogenous, we have:
πP,t = βEt (πP,t+1 ) +

λω1
xt ,
2η

which is quite similar to NKPC derived by Gali and Monacelli[5] and can bederived alternatively only if ση = 1 in our two-country model under the PCP
because foreign output gap disappears in such a case. Gali and Monacelli[5]mention that full stabilization of PPI implies that xt = πH,t = 0 which isplausible if output gap in country F disappears in Eq.(47). Because of twocountry setting, foreign output gap does not disappears as long as we do notassume ση = 1. Hence, full stabilization of PPI does not necessarily imply
xt = πH,t = 0 in our two-country setting.

3

Optimal Monetary Policy under the LCP and
the PCP

3.1

Welfare Costs

We assume central banks conduct optimal monetary policy. Central banks minimizes welfare costs. Welfare costs consist of the period loss function which isderived by the welfare criterion. Following Woodford[11] and Gali[4], we havesecond-order approximated utility function as follows:
¡
¡
¢
¢
W
3
3
WLCP
= −LW
; WPWCP = −LW(48)
LCP + t.i.p. + o k ξ k
P CP + t.i.p. + o k ξ k
P
P
∞
∞
t W
W
t W
where LW
LCP ≡ E0t=0 β LLCP,t and LP CP ≡ E0
t=0 β LP CP,t denote theW∗loss function in LCP and PCP models, respectively, WLCP = 12 (WLCP + WLCP)
1
W∗
and WP CP = 2 (WP CP + WP CP ) denote average welfare criteria in LCP andW
and WPWCP denote welfarePcriteria in counPCP models, respectively, WLCP∞
try H in LCP and PCP models, respectively with W ≡t=0 E0 (Wt ) and
Ut −U
Wt ≡ UC C . Further:∙¸¡ W ¢2 (1 + ϕ) η 2 2
1 ε 2
ε
∗ 2
≡
+
)
+
(σ
+
ϕ)
x
+
π
(π
z
(49)
LW
LCP,t
t
t ,
2 2λ t2λ t
4∙¸¡ W ¢2 (1 + ϕ) η 2 2
1 ε 2
ε ¡ ∗ ¢2
π
LW
≡
+
+
(σ
+
ϕ)
x
+
π
z
P CP
t
t ,(50)2 2λ H,t 2λ F,t4

are period loss function in countries H and F , respectively, , zt ≡ st − s̄t beingR
the deviation of the TOT from its eﬃcient level, s̄t ≡ η21+ϕη(1+ϕ) ȳt being theeﬃcient level of TOT. Note that we define vtW ≡ 12 (vt + vt∗ ) and vtR ≡ vt − vt∗ .
18

3.2

FONCs for Central Banks

We next briefly mention the FONCs for the central bank. We assume that thecentral bank in each country conducts optimal monetary policy with commitment cooperatively. Under the LCP, central banks minimizes Eq.(49) and the
FONCs for them are given by:
πtW

=

−

zt

=

0,

¢
1¡ W
x − xWt−1 ,ε twhere μ3,t is a Lagrange multiplier associated with the average block of theNKPC. Because of commitment, lagged multiplies appear in the FONCs.
Under the PCP, central banks minimizes Eq.(50) and the FONCs for them
are given by:

3.3

πtW

=

R
πP,t

=

¢
1¡ W
x − xWt−1 ,ε t(1 + ϕ) η 2−(zt − zt−1 ) .
ε (1 + ηϕ)−Calibration

We run a series of dynamic simulations and adopt the following benchmark
parameterization. We set the price stickiness θ, the subjective discount factorβ, the elasticity of substitution across goods ε and the inverse of the labor
supply elasticity ϕ equal to 0.75, 11, 0.99, 3, respectively, which are consistent
with quarterly time periods in the model.6 We compare two cases. One of themis special case in which σ = η = 1 and another one is general case in which σ = 3and η = 4.5. Note that σ = η = 1 is assumed by Gali and Monacelli[5] while
σ = 3 and η = 4.5 are assumed by Benigno and Benigno[1]. We assume that
productivity shifters are described according to the following AR(1) processes:
at = ρat−1 + ξt ; a∗t = ρa∗t−1 + ξt∗ ,where ξt and ξt∗ denote the i.i.d. shocks. We set ρ equal to 0.9. To examine
the impulse response functions (IRFs), we consider one percent changes in the
productivity shifter in country H at , and the productivity shifter in country F ,a∗t .
Impulse responses to one percent increase in productivity in country H in
special case are shown in Figure 1 while macroeconomic volatilities are shownin 3rd and 4th columns in Table 17 . To eliminate the eﬀect of changes in
productivity, central banks decreases nominal interest rate under both the LCP
and the PCP (Panels 7 and 8). Because of this, output gap in countries H and
F are completely stabilized. Under the PCP, PPI inflation rate in countries H
and F is completely stabilized (Panels 3 and 4). This result is consistent with
Gali and Monacelli[5] who imply that PPI inflation targeting brings zero output

6 θ = 0.75 implies that average length of price contracts equal to 4.
7 There are 2 eigenvalues larger than 1 in modulus for 2 forward-looking variables. Hencethe rank condotion is verified

19

gap. This result can be understood by paying attention to Eq.(47). Plugging
σ = η = 1 into Eq.(47) yields:
πP,t = βEt (πP,t+1 ) +

λω1
xt ,
2η

which is NKPC in the special case under the PCP. This NKPC implies thatstabilizing PPI inflation brings stabilizing output gap simultaneously and isconsistent with one derived by Gali and Monacelli[5] although the slope of our
NKPC is slightly diﬀerent from theirs because we assume a two-country economy.
Under the LCP, not PPI inflation but CPI inflation is stabilized and this result is quite diﬀerent not only from Gali and Monacelli[5] but also other DSGE
literatures assuming an open economy (Panels 5 and 6). This can be understoodby paying attention to Eq.(35) which implies that CPI inflation becomes zero
when output gap in countries H and F are stabilized. Hence, it may be said
that CPI inflation targeting brings completely stabilizing output gap. Interestingly, nominal exchange rate is completely stabilized under the LCP which isconsistent with Devereux and Engel[3] developing NOEM model, assuming the
LCP and showing that fixed exchange rate is optimal regime from the viewpoint of maximizing welfare. This result stems from stabilizing CPI inflation
rate. Perfect stabilization in CPI inflation is consistent with perfect stabilization in the CPI level.8 In our model, PPP is always applied hence et + p∗t = pt
is applied. Perfect stabilization in CPI inflation implies pt = p∗t = 0 which isconsistent with et = 0. Thus, under the LCP, there are neither changes in the
CPI inflation nor nominal exchange rate.
Impulse responses to one percent increase in productivity in country H in
the general case are shown in Figure 2.9 In the general case under the PCP,
inflation—output gap trade-oﬀs are no longer dissolved simultaneously although
Gali and Monacelli[5] show that that trade-oﬀs dissolved simultaneously (Panels
1 to 4). Because our model is a two-country model, foreign output is endogenous
while it is exogenous in Gali and Monacelli[5]’s small open economy model. In
a small economy setting, foreign output gap disappears in the NKIS althoughthat appears in the NKIS, as shown in Eq.(47). foreign output gap disappears
in Eq.(47) only if ση = 1. Hence, neither output gap nor PPI inflation stabilized
simultaneously.
However, although output gap in countries H and F is not stabilized, CPI
inflation is completely stabilized under the LCP. This can be understood bypaying attention to Eq.(35). Average output gap is always stabilized not only
under the PCP but also under the LCP (Rows 3 and 4 in Table 1). Hence, CPI
inflation is stabilized because Eq.(35) can be rewritten as:
πt = βEt (πt+1 ) + κα xWt .
In this NKPC, the slope is not aﬀected by σ and η. Thus, result on volatility
is not diﬀerent between the special and the general cases. In addition, CPI
inflation is completely stabilized, there is no fluctuation in the nominal exchange
rate, alike with the special case (Panel 13). As mentioned, our result that there
8 We assume zero inflation deterministic steady state.
9 There are 2 eigenvalues larger than 1 in modulus for 2 forward-looking variables. Hencethe rank condotion is verified20

is no fluctuation in the nominal exchange rate under the LCP is consistent withthe result of Devereux and Engel[3]. Devereux and Engel[3] assume Arminton
Form of consumption which implies η = 1.10 Now, we apply more general
setting such as η = 4 while our result on fluctuations in nominal exchange rateis consistent with their result. This implies that Devereux and Engel[3]’s finding
can be applied in general parameterization. We have further discussion on this
topic in next section.

4

Macroeconomic Volatilities and Welfare Costs

In this section, we focus on macroeconomic volatilities and welfare costs undervarying the relative risk aversion σ and the elasticity of substitution between
goods produced in countries H and F η. There are many macroeconomic variables in our model and we focus on some important variables which is related
to our loss functions Eqs.(49) and (50) and the nominal exchange rate.
Figure 3 shows eﬀects on macroeconomic volatilities of varying the relativerisk aversion σ ∈ [1, 10] and the elasticity of substitution between goods produced in countries H and F η ∈ [1, 10]. Under the PCP, volatility of PPI
inflation is definitely zero when η = 1 although the higher the η the higher thevolatility (Panel 2). When η = 1, one of FONCs related to relative inflation for
central banks under the PCP can be rewritten as:
R
=−
πP,t

¢
1¡ R
xt − xR
t−1
ε

(51)

because zt = xR
t is applied when η = 1. Along with another FONC related toaverage output gap for central banks under the PCP, those FONCs imply thatstabilization in PPI inflation strictly consistent with stabilization in output gap.
Hence, volatility of PPI inflation is definitely zero when η = 1. Equally, this
implies that σ = η = 1 is not a suﬃcient condition to dissolve inflation—output
gap trade-oﬀs but just η = 1 is a suﬃcient condition to dissolve that tradeoﬀs under the PCP. As implied by calibration in former section, volatility of
CPI inflation is definitely zero regardless of σ and η under the LCP (Panel 3).
This stems from NKIS under the LCP Eq.(35). Because of FONCs for central
bank related to average inflation, there is no fluctuation in average output gap
regardless of σ and η. This immediately implies that there is no fluctuation in
CPI inflation countries H and F regardless of σ and η under the LCP.
Average output gap is completely stabilized under both the LCP and the
PCP regardless of σ and η (Panels 5 and 6). This stems from the FONC
for central banks related to average inflation which are common under both the
LCP and the PCP and imply that average inflation and output gap are stabilized
simultaneously. TOT deviation from eﬃcient level is definitely zero regardlessof σ and η under the LCP (Panel 7). This stems from the FONC for central
banks under the LCP related to TOT deviation from eﬃcient level. However,TOT deviation from eﬃcient level is definitely zero under the PCP only if η = 1(Panel 8). In that case, zt = xR
t is applied and there is no fluctuation in output
gap in countries H and F , as implied by Eq.(51). Both PPI inflation and output
gap are completely stabilized when η = 1. Hence, TOT deviation from eﬃcient1

1

10 In that case, Eq.(2) is rewritten by C = 2C 2 C 2tH,t F,t21

level is definitely zero through complete stabilization in output gap in countries
H and F . However, complete stabilization in TOT deviation from eﬃcient levelis no longer achieved when η = 1 is not applied.
Now, we discuss on volatility of the nominal exchange rate. As many literatures show, we show that optimal monetary policy is flexible exchange rate
regime (Panel 10) under the PCP. On contrary, the nominal exchange rate isdefinitely zero regardless of σ and η under the LCP. Even if the LCP is assumed, the PPP is applied which implies that st = pt − p∗t . Because of optimalmonetary policy, CPI inflation is definitely stabilized which is consistent withzero fluctuation in the CPI level under the LCP. Thus, the nominal exchange
rate is definitely stabilized regardless of σ and η under the LCP. As mentioned
in former section, our result is consistent with Devereux and Engel[3]’s result
which shows that optimal monetary policy under the LCP is consistent with
fixed exchange rate regime although they assume Armington form of consumption which corresponds to η = 1. Our model does not assume Armington formof consumption and there is no parametric restriction in η. Thus, our result
implies that Devereux and Engel[3]’s policy implication is not applied in special
parameterization but applied in general setting. In addition, we derive more
important policy implication. Optimal monetary policy under the LCP stabilizes the CPI inflation definitely and coincides with complete stabilization in
the nominal exchange rate. Furthermore, monetary policy which stabilizes the
CPI inflation or the nominal exchange rate is optimal under the LCP. We do
not analyze an explicit targeting rule or a regime such as CPI inflation targeting and fixed exchange rate regime. However, it can be said that CPI inflation
targeting and fixed exchange rate regime are optimal and equivalent under the
LCP although there is some room to discuss precisely.
Finally, we discuss eﬀects on welfare costs varying the relative risk aversion
σ and the elasticity of substitution between goods produced in countries H andF η. When we introduce β → 1 in Eq.(48), we have welfare criteria as follows:∙¸¡ W ¢ (1 + ϕ) η 2
1 ε
ε
∗
LW
+
≡
)
+
)
+
(σ
+
ϕ)
var
x
)
,
var
(π
var
(π
var
(z
t
t
LCP,t
t
t
2 2λ
2λ
4
∙
¸
¡ ∗ ¢
¡ W ¢ (1 + ϕ) η 2
1 ε
ε
≡
)
+
)
,
LW
var
(π
var
π
+
(σ
+
ϕ)
var
x
+
var
(z
P,t
t
P CP,t
P,t
t
2 2λ
2λ
4

and we utilize these equalities to calculate welfare costs.
Figure 4 depicts eﬀects on welfare costs varying the relative risk aversion σ
and the elasticity of substitution between goods produced in countries H and F
η. We have already discussed some macroeconomic volatilities which comprisewelfare costs. Thus, we can understand eﬀects on welfare losses varying σ andη. Because there are no fluctuations in average output gap, CPI inflation incountries H and F and the TOT deviation from eﬃcient level regardless of σand η, there are no welfare losses regardless of σ and η under the LCP. However,there are no welfare costs only if η = 1 because there are no fluctuations in PPI
inflation in countries H and F and the TOT deviation from eﬃcient level under
the PCP. Except for η = 1, there are some welfare costs because there are
fluctuations in PPI inflation in countries H and F and the TOT deviation fromeﬃcient level under the PCP even if optimal monetary policy is conducted.

22

5Conclusion

We analyze optimal monetary policy under the LCP model by comparing with
the PCP model. We have two main findings as follows. We insist that optimalmonetary policy under the LCP brings no fluctuations not in the PPI inflation
rate but in the CPI inflation rate. Roughly speaking, optimal monetary policy
under the LCP is the CPI inflation targeting. This result is quite diﬀerent fromthe result on Gali and Monacelli[5]. We show that there are no fluctuations in
the nominal exchange rate under the LCP. Roughly speaking, optimal monetary policy under the LCP is consistent with fixed exchange rate regime andthat is shown by Deverereux and Engel[3]. We can reconcile with Deverereux
and Engel[3] and we derive our policy implication complying Woodford’s[10]motivation.
Our finding shed light on Mussa’s puzzle which focuses on the fact that comovement of nominal exchange rate and real exchange rate along with Betts
and Devereux[2] although they do not analyze optimal monetary policy. Because
complete stabilization in CPI inflation rate coincides with complete stabilizationin the nominal exchange rate under the LCP, one of answers to Mussa’s puzzle
may be optimal monetary under the LCP. Solving Mussa’s puzzle along withthe result on this paper is one of future research agenda.

AppendixA

Gains from International Monetary Cooperation

To be added.

References[1] Benigno, Pierpaolo, and Gianluca Benigno (2006), “Designing Targeting
Rules for International Monetary Policy Cooperation,” Journal of Monetary
Economics, 53, 473—506.
[2] Betts, Caroline and Michael B. Devereux (2000), “Exchange Rate Dynamics
in a Model of Pricing-to-market,” Journal of International Economics, 50,
215-244.
[3] Devereux, Michael B. and Charles Engel (2003), “Monetary Policy in theOpen Economy Revisited: Price Setting and Exchange Rate Flexibility”, Review of Economic Studies, 70, 765-84.
[4] Gali, Jordi (2008), “Monetary Policy, Inflation and the Business Cycle,”
Princeton University Press, Princeton, NJ.
[5] Gali, Jordi and Tommaso Monacelli (2005), “Monetary Policy and Exchange
Rate Volatility in a Small Open Economy,” Review of Economic Studies, 72,
707—734.

23

[6] Mark, Nelson (2001), “International Macroeconomics and Finance,” Blackwell, Oxford, UK.
[7] Monacelli, Tomasso (2005), “Monetary Policy in a Low Pass-Through Environment,” Journal of Money, Credit and Banking, 37, 1047-1066.
[8] Obstfeld, Maurice, and Kenneth Rogoﬀ (2000), “New Directions for Stochastic Open Economy Models,” Journal of International Economics 50, 117—153.
[9] Okano, Eiji (2007), “The Choice of the Inflation Rate as a Target in an
Economy with Pricing-to-market,” Japan and the World Economy, 19, 48—
67.
[10] Woodford, Michael (2001), “Inflation Stabilization and Welfare,” NBER
Working Paper, No. 8071.
[11] Woodford, Michael (2003), “Interest and Prices,” Princeton University
Press, Princeton, NJ.

24

Table 1: Macroeconomic Volatility to One Percent Increase in Productivity
Variables

Pricing

xW
t

LCP
PCP
LCP
PCP
LCP
PCP
LCP
PCP
LCP
PCP
LCP
PCP
LCP
PCP
LCP
PCP
LCP
PCP
LCP
PCP
LCP
PCP
LCP
PCP
LCP
PCP
LCP
PCP
LCP
PCP

πtW
xt
x∗t
πP,t
∗
πP,t

πt
πt∗
r̂t
r̂t∗
yt
yt∗
st
zt
et

Special (σ = η = 1)ata∗t
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00130.00000.00000.00000.00000.00510.00510.00000.00000.00510.00510.00000.00000.00000.00000.00510.00510.00000.00000.00510.00510.00110.00110.0021 1.1460e-006
0.00110.0011
1.1460e-006
0.0021
0.02290.00000.02290.00000.00000.02290.00000.02290.02290.02290.02290.02290.00000.00000.00000.00000.00000.00000.02290.022925

General (σ = 3, η = 4.5)ata∗t
0.00000.00000.00000.00000.00000.00000.00000.00000.00280.00283.2172e-004 3.2172e-004
0.00280.00283.2172e-004 3.2172e-004
0.00110.0011
1.0486e-004 1.0486e-004
0.00110.0011
1.0486e-004 1.0486e-004
0.00000.00000.00120.00120.00000.00000.00120.00120.00230.00230.00240.00200.00230.00230.00200.00240.01910.00380.02170.00640.00380.01910.00640.02170.00510.00510.00630.00630.00000.00000.00120.00120.00000.00000.00570.0057

Figure 1: IRFs to Productivity in Country H in the Special Case (σ = η = 1)26

Figure 2: IRFs to Productivity in Country H in the General Case (σ = 3,
η = 4.5)27

Figure 3: Eﬀects on Macroeconomic Volatilities of Varying Relative Risk Aversion σ and Elasticity of Substitution between Goods Produced in Countries H
and F η28

Figure 4: Eﬀects on Welfare Costs of Varying Relative Risk Aversion σ andElasticity of Substitution between Goods Produced in Countries H and F η29