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Optimal Monetary Policy and Inflation Rate Volatility under
Local Currency Pricing
OKANO, Eiji*(Graduate School of Economics, Nagoya City University)

Abstract
We analyze the fluctuation in inflation and nominal exchange rate under optimalmonetary 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, we show thatcompletely stabilizing consumer price index inflation is optimal from that view point. In
addition, we show that completely stabilizing consumer price index inflation isequivalent with completely stabilizing nominal exchange rate.
Keywords: Local Currency Pricing, DSGE, Optimal Monetary Policy, CPI Inflation,
Exchange Rate
JEL Classification: E52; E62; F41

1.

Introduction

While new open economy macroeconomics (NOEM) try to analyze optimal exchange
rate regime, dynamic stochastic general equilibrium (DSGE) literatures discuss optimalmonetary policy. Roughly speaking, NOEM has the policy implication that flexible
exchange rate regime is optimal from the view point of maximizing social welfare whichis 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 theirprices 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. SomeDSGE 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*Correspondence to: Graduate School of Economics, Nagoya City University, 1,Aza Yamanobata, Mizuhocho, Mizuho-ku,
Nagoya-shi, 467-0851, JAPAN. Tel: +81-52-872-5721; Fax: +81-52-872-1531; E-mail: eiji_okano@econ.nagoya-cu.ac.jp-- 55 --

room to discuss optimal monetary policy under the LCP.
To analyze the sort of inflation rate which should be stabilized under the LCP andreconcile 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, weshow that completely 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 and reconcile policy
implications derived by NOEM and DSGE literatures in this paper. By developing not
only PCP but also LCP model following NOEM, Devereux and Engel (2003) discuss the
optimal exchange rate regime from the view point of welfare maximization and showthat 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. Hence they do
not provide what kind of inflation rate should be stabilized following Woodford’s (2001)
motivation.
Gali and Monacelli (2005) show that optimal monetary policy in a small open
economy is consistent with domestic price inflation targeting. Although they do notmention explicitly, they assume the PCP. In addition, they compare three policy regimes,
PPI inflation based and CPI inflation based Taylor rules and fixed 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 theirpolicy implication implies that outcome of optimal monetary policy is not fundamentallydifferent from the one of the closed economy. While they do not highlight the firms’
price setting behavior, Gali and Monacelli (2005) imply that PPI inflation targeting is
optimal under the PCP. In addition, they comply Woodford’s (2001) motivation.
Somehow, some DSGE literatures do not focus on the firms’ price setting behavior andthose assume the PCP. There are few DSGE literatures focus on the firms’ price setting
behavior. Based on Gali and Monacelli (2005)’s model, Monacelli (2005) 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 ofmonetary policy is quite different not only from canonical papers but also Gali andMonacelli (2005) who implies that stabilization in the PPI inflation achieves stabilizationin output gap simultaneously. Because of low of one price (LOOP) gap, the analysis ofmonetary policy of an open economy is fundamentally different from the one of a closed
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

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6 --

economy. While he focuses on important point, he cannot comply Woodford’s (2001)
motivation. He does not reply what kind of inflation rate should be stabilized under sucha low-pass through environment stemming from LCP and cannot reconcile policy
implications derived by NOEM and DSGE literatures while he shows importance ofcommitment on the monetary policy. Another few DSGE author is Okano (2007) who
shows that CPI inflation targeting stabilizes output to changes in demand shock byutilizing 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 fromsecond-order Taylor expanded utility function alike with Monacelli (2005). Hence, itcannot be said that Okano complies Woodford’s (2001) motivation. In addition, hecannot derive clear policy implication on fluctuations in the nominal exchange rate
under the CPI inflation targeting under the LCP and cannot reconcile with Devereux andEngel (2003)’s policy implication. It is quite obvious that analyzing optimal monetary
policy following DSGE under the LCP to comply Woodford’s (2001) motivation andreconcile Devereux and Engel (2003)’s policy implication.
As mentioned above, our aims in this paper are finding the sort of inflation rate whichshould 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, stemming from second-order Taylor expanded utility function
following Woodford (2003) and Gali (2008). 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 inwhich the relative risk aversion and the elasticity of substitution between goods
produced in both two countries are unity and under the general case in which therelative risk aversion and the elasticity of substitution between goods produced in both
two countries are 3 and 4.5, respectively. 2 Note that those elasticity settings in thespecial case is consistent with Gali and Monacelli’s (2005) setting and those elasticity
settings in the general case is consistent with Benigno and Benigno’s (2006). To
compare with the result on Gali and Monacelli (2005) and to discuss optimal monetary
policy on general parameterization, we analyze both two cases. Because we areinterested 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 in both two countries. Finally, we calculate
welfare costs varying the relative risk aversion and the elasticity of substitution between
goods produced in both two countries.
Now, we mention our results as follows. First of all, we show that optimal monetary
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
The relative risk aversion and the elasticity of substitution between goods produced in both two countries are often
dubbed the intertemporal elasticity of substitution and the intratemporal elasticity of substitution, respectively.

2

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inflation targeting. This result is quite different from the result on Gali and Monacelli(2005). 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 (2005) and Benigno and
Benigno (2006). Next, we can reconcile with Devereux and Engel (2003) because there
are no fluctuations on nominal exchange rate under the LCP. Roughly speaking, optimalmonetary policy under the LCP is consistent with fixed exchange rate regime and that isshown by Devereux and Engel (2003). In addition, this result is not depending on therelative risk aversion and the elasticity of substitution between goods produced in both
two countries. That there are no fluctuation on nominal exchange rate is consistentregardless of those preferences. Because Devereux and Engel (2003) analyzes undersome restriction which consistent with unitary elasticity of substitution between goods
produced in both two countries in our model, we can support their results and cangeneralize their policy implication. Summarizing our result, optimal monetary policy
under the LCP is not only consistent with CPI inflation targeting but also consistent with
fixed exchange 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 by deriving welfare
costs, FONCs for central bank with commitment and calibration. Section 4 analyzes
effect on macroeconomic volatility and welfare costs of varying 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 highlightfluctuations in inflation and nominal exchange rate.

2.

The Model

We construct a two-country model belonging to the class of DSGE models withnominal rigidities and imperfect competition, basically following Gali and Monacelli(2005) and Monacelli (2005). We alter Gali and Monacelli (2005)’s small open economy
model to two-country economy model following Obstfeld and Rogoff (2000) although
we assume all goods are tradables. The union-wide economy consists of two countries,
countries ‫ ܪ‬and ‫ܨ‬. Country ‫ ܪ‬produces an array of differentiated goods indexed by
the interval ݄ ‫[ א‬0,1], while country ‫ ܨ‬produces an array of differentiated goods
indexed by ݂ ‫[ א‬1,2]. In addition, we derive two models, one of them is assumed the LCP
and another one is assumed the PCP.
௏
Note that we take a definition ‫ݒ‬௧ ‫ ݈݊ ؠ‬ቀ ೟ ቁ if there are no provisions where ܸ௧௏denotes an arbitrary variable and ܸ denotes steady state value of ܸ௧ .

2.1. LCP Model

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8 --

Under the LCP, LOOP is not necessarily applied because firms can choose prices to sellgoods in countries ‫ ܪ‬and ‫ ܨ‬separately. Thus, ܲ௧ (݄) = ࣟ௧ ܲ௧‫ )݄( כ‬and ܲ௧ (݂) = ࣟ௧ ܲ௧‫)݂( כ‬‫כ‬‫כ‬hence ܲு,௧ = ࣟ௧ ܲு,௧and ܲி,௧ = ࣟ௧ ܲி,௧
do not necessarily hold where ܲ௧ (݄) and ܲ௧ (݂)
denote the price of a generic good produced in country ‫ ܪ‬in terms of country ‫’ܪ‬s
భ

భ

ଵ
భషഄ
ܲு,௧ ‫ ؠ‬ቂ‫׬‬଴ ܲ௧ (݄)ଵିఌ ݄݀ቃଶభషഄܲி,௧ ‫ ؠ‬ቂ‫׬‬ଵ ܲ௧ (݂)ଵିఌ ݄݀ቃ

anddenote indices ofcurrency,
the price of generic goods produced in countries ‫ ܪ‬and ‫ܨ‬, respectively, ࣟ௧ denotes
nominal exchange rate. 3 Note that quantities and prices particular to country ‫ ܨ‬aredenoted by asterisks while quantities and prices without asterisks are those in country‫ܪ‬.
2.1.1. Households
The preferences of the representative household in country ‫ ܪ‬are given by:௧࣯ ‫ܧ ؠ‬଴ σஶ
(1)௧ୀ଴ ߚ ܷ௧ ,
ଵ
ଵ
ଵାఝ
ଵିఙ
denotes the period utility, ‫ܧ‬௧ denotes thewhere ܷ௧ ‫ ؠ‬ଵିఙ ‫ܥ‬௧ െ ଵାఝ ܰ௧

expectation, conditional on the information set at period ‫ݐ‬, ߚ ‫( א‬0,1) denotes theଵ
subjective discount factor, ‫ܥ‬௧ denotes consumption, ܰ௧ ‫׬ ؠ‬଴ ܰ௧ (݄)݄݀ denotes hours
of work, ߪ denotes the relative risk aversion and ߮ denotes the inverse of the labor
supply elasticity. The preferences of the representative household in country ‫ ܨ‬isdefined analogously.
More precisely, private consumption is a composite index defined by:
ଵ

భ
ആ

ആషభ
ആ

ଵ

భ
ആ

ആ
ആషభ ആషభ
ആ‫ܥ‬௧ ‫ ؠ‬ቈቀଶቁ ‫ܥ‬ு,௧ + ቀଶቁ ‫ܥ‬ி,௧ ቉
where

(2)

,

ഇ

ഇ

ഇషభ
ഇషభ
ଵ
‫ܥ‬ு,௧ ‫ ؠ‬൤‫׬‬଴ ‫ܥ‬௧ (݄) ഇ ݄݀൨

ഇషభ
ഇషభ
ଶ
‫ܥ‬ி,௧ ‫ ؠ‬൤‫׬‬ଵ ‫ܥ‬௧ (݂) ഇ ݂݀൨

anddenoteDixit–

Stiglitz-type indices of consumption across the home goods and foreign goods,
respectively, and ߟ > 0 denotes the elasticity of substitution between tradables andnontradables. Note that ‫ܥ‬௧‫ כ‬is defined analogously to Eq.(2).
Total consumption expenditures by households in country ‫ ܪ‬are given byܲு,௧ ‫ܥ‬ு,௧ + ܲி,௧ ‫ܥ‬ி,௧ = ܲ௧ ‫ܥ‬௧ . A sequence of budget constraints in country ‫ ܪ‬is given by:‫ܤ‬௧ + ܹ௧ ܰ௧ െ ܶ௧ ൒ ܲ௧ ‫ܥ‬௧ + ‫ܧ‬௧ ൫ܳ௧,௧ାଵ ‫ܤ‬௧ାଵ ൯,(3)
where ܳ௧,௧ାଵ denotes the stochastic discount factor, ‫ܤ‬௧ denotes the nominal payoff ofthe bond portfolio purchased by households, ܹ௧ denotes the nominal wage, and ܶ௧
denotes lump-sum taxes. The budget constraint in country ‫ ܨ‬is given analogously.
Furthermore:భଵଵ ଵିఎ భషആଵିఎܲ௧ ‫ ؠ‬ቀଶ ܲு,௧ + ଶ ܲி,௧ ቁ ,(4)denotes the consumption price index (CPI).
is defined analogously to this equality.
ଵ
By log-linearizing this equality yields ‫݌‬௧ = ଶ ‫݌‬ு,௧ + ‫݌‬ி,௧ , which implies as follows:
3

ܲ௧‫כ‬

By citing Betts and Devereux[2], Mark[6] clearly explain the LCP.

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9 --

ଵ

ଵ

ߨ௧ = ଶ ߨு,௧ + ଶ ߨி,௧ ,(5)where ߨ௧ ‫݌ ؠ‬௧ െ ‫݌‬௧ିଵ denotes CPI inflation with ߨு,௧ = ‫݌‬ு,௧ െ ‫݌‬ு,௧ିଵ and ߨி,௧ = ‫݌‬ி,௧ െ‫݌‬ி,௧ିଵ .
The optimal allocation of any given expenditure within each category of goods implies
the demand functions, as follows:௉ (௛)‫ܥ‬௧ (݄) = ൬ ೟

ିఌ

൰

௉ಹ,೟

௉‫( כ‬௛)ିఌ‫ܥ‬௧‫ = )݄( כ‬൬ ೟ ‫ כ‬൰
௉ಹ,೟

௉ (௙)

‫ܥ‬ு,௧ ; ‫ܥ‬௧ (݂) = ൬ ೟

௉ಷ,೟

ିఌ

൰

௉‫( כ‬௙)‫ܥ‬ி,௧ିఌ‫כ‬‫ܥ‬ு,௧; ‫ܥ‬௧‫ = )݂( כ‬൬ ೟ ‫ כ‬൰
௉ಷ,೟

‫כ‬
‫ܥ‬ி,௧
.

(6)

The optimal allocation of expenditures between domestic and foreign goods is given
by:
ିఎ

ଵ ௉

‫ܥ‬ு,௧ = ଶ ቀ ௉ಹ,೟ ቁ
೟
‫כ‬
௉ಹ,೟
௉೟‫כ‬

ଵ

‫כ‬
‫ܥ‬ு,௧
= ଶቀ

ିఎ

ቁ

ିఎ

ଵ ௉

‫ܥ‬௧ ; ‫ܥ‬ி,௧ = ଶ ቀ ௉ಷ,೟ ቁ
೟

ଵ

‫כ‬
௉ಷ,೟
௉೟‫כ‬

‫כ‬
‫ܥ‬௧‫ܥ ; כ‬ி,௧= ଶቀቁ‫ܥ‬௧ ,

ିఎ‫ܥ‬௧‫ כ‬.

(7)

The representative household maximizes Eq.(1) subject to Eq.(3). The optimality
conditions are given by:஼ ష഑ ௉

(8)೟శభ ೟ܴ௧ ߚ‫ܧ‬௧ ቀ஼ ష഑
ቁ = 1,
௉೟శభ೟which is a conventional Euler equation andௐఝ‫ܥ‬௧ఙ ܰ௧ = ೟,
௉
೟

(9)

which is a standard intratemporal optimality condition where ܴ௧ ‫ ؠ‬1 + ‫ݎ‬௧ satisfyingܴ௧ିଵ = ‫ܧ‬௧ ܳ௧,௧ାଵ denotes the gross nominal return on a riskless one-period discount bondpaying off one unit of the common currency (in short, the gross nominal interest rate),and ‫ݎ‬௧ denotes the net nominal interest rate. Eq.(8) is an intertemporal optimality
condition, namely the Euler equation, and Eq.(9) is an intratemporal optimality
condition. Optimality conditions in country ‫ ܨ‬are given analogously.
Log-linearizing Eq.(8), we obtain:ଵଵܿ௧ = E௧ ܿ௧ାଵ െ ఙ ‫ݎ‬Ƹ௧ + ఙ E௧ ߨ௧ାଵ(10)
ோwith ‫ݎ‬Ƹ௧ ‫ ݈݊ ؠ‬ቀ ோ೟ቁ.
There is relationship on the gross nominal interest rate between countries ‫ ܪ‬and ‫ܨ‬which is uncovered interest rate parity (UIP) as follows:ܴࣟ௧ = ܴ௧‫ כ‬E௧ ቀ ࣟ೟శభ ቁ೟with ܴ௧‫ ؠ כ‬1 + ‫ݎ‬௧‫ כ‬. Log-linearizing the UIP, we have the familiar expression as follows:E௧ (ȟ݁௧ାଵ ) = ‫ݎ‬Ƹ௧ െ ‫ݎ‬Ƹ௧‫ כ‬,ࣟwith ȟ‫ݒ‬௧ ‫ݒ ؠ‬௧ െ ‫ݒ‬௧ିଵ and ݁௧ ‫ ؠ‬ln ቀ ࣟ೟ ቁ.
Combining Eq.(8) and the UIP and iterating with an initial condition, we have thefollowing optimal risk-sharing condition:‫ܥ‬௧ఙ = ߴ(‫ܥ‬௧‫) כ‬ఙ ࣫௧ ,ࣟ ௉‫כ‬

with ࣫௧ ‫ ؠ‬೟௉ ೟ denoting the real exchange rate and ߴ denoting a constant depending
೟on the initial value. Log-linearizing this equality, we have:

-- 10
10 --

ଵ

(11)

ܿ௧ = ܿ௧‫ כ‬+ ఙ ‫ݍ‬௧ .
2.1.2. Market Clearing

The market for tradables and for nontradables in country ‫ ܪ‬clears when domestic
demand equals domestic supply, as follows:(12)ܻ௧ (݄) = ‫ܥ‬௧ (݄) + ‫ܥ‬௧‫)݄( כ‬,where ܻ௧ (݄) denotes the output of good ݄, which is market clearing condition. Market
clearing condition in country ‫ ܨ‬is analogously. Plugging Eq.(7) into Eq.(12) yields:ଵ ௉ (௛)ܻ௧ (݄) = ଶ ൬ ೟

௉ಹ,೟

ഄ

Let

ିఌ

൰

ഄషభ
ഄషభ
ଵ
ܻ௧ ‫ ؠ‬ቂ‫׬‬଴ ܻ௧ (݄) ഄ ݄݀ቃିఎ௉ቀ ௉ಹ,೟ቁ

ିఎ

ଵ ௉

೟

ଵ ௉

= ଶ ቀ ௉ಹ,೟ቁ

ିఎ

೟

ିఌ

௉ಹ,೟

೟

௉‫כ‬

ቀ ௉ಹ,೟‫ כ‬ቁ೟ିఎ‫ܥ‬௧‫ כ‬.

(13)

represent index for aggregate output in country ‫ ܪ‬.

Under the LCP, we obtain:

ܻ௧ = ଶ ቀ ௉ಹ,೟ ቁଵ ௉‫( כ‬௛)‫ܥ‬௧ + ଶ ൬ ೟ ‫ כ‬൰

‫כ‬
ଵ ௉

‫ܥ‬௧ + ଶ ቀ ௉ಹ,೟
‫ כ‬ቁ
೟

ఎ

௉

ିఎ

‫ܥ‬௧‫ כ‬,

௉‫כ‬

ିఎ

‫ܥ‬௧ ቈ1 + ቀ ௉ಹ,೟ ቁ ቀ ௉ಹ,೟
‫ כ‬ቁ
೟

೟

ି

భ

࣫௧ ഑ ቉,(14)by combining Eqs.(13), Dixit–Stiglitz aggregators for output and prices, where we takeEq.(11) in the second lines in Eq.(14).
We define the terms of trade (TOT) as follows:௉࣭௧ ‫ ࣟ ؠ‬௉ಷ,೟‫ כ‬,
(15)
೟ ಹ,೟

where ࣭௧ is foreign TOT. The numerator is export price of goods produced in country‫ ܨ‬in terms of country ‫’ܪ‬s currency and the denominator is export price of goods
produced in country ‫ ܨ‬in terms of country ‫’ܪ‬s currency. Log-linearizing Eq.(15), we
have:
‫כ‬
.
(16)
‫ݏ‬௧ = ‫݌‬ி,௧ െ ݁௧ െ ‫݌‬ு,௧
Plugging Eq.(21) into log-linearized Eq.(14), we have:ఎଵଵ‫ݕ‬௧ = ܿ௧ + ଶ ‫ݏ‬௧ + ଶ ቀߟ െ ఙቁ ‫ݍ‬௧ ,which is log-linearized market clearing in country ‫ ܪ‬under the LCP. Although there isa difference between this equality and Eq.(42) because logarithmic real exchange rate‫ݍ‬௧ appears in this equality. However, this equality boils down to Eq.(42) because the
PPP is applied which implies that ‫ݍ‬௧ = 0 although we assume the LCP. We discuss
about the PPP under the LCP in section 2.3.
Combining Eq.(42) and its counterpart in country ‫ܨ‬, we have:ଵ‫ݏ‬௧ = ఎ (‫ݕ‬௧ െ ‫ݕ‬௧‫ ) כ‬െ ‫ݍ‬௧ ,which clarifies relationship between the TOT and relative output under the LCP. As
mentioned, ‫ݍ‬௧ = 0 is applied although we assume the LCP. Hence, this equality boils
down to Eq.(41).
2.1.3. Firms

-- 11
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Each producer uses a linear technology to produce a differentiated good as follows:ܻ௧ (݄) = ‫ܣ‬௧ ܰ௧ (݄),(17)
where ‫ܣ‬௧ denotes stochastic productivity in country ‫ܪ‬. Firms in country ‫ ܨ‬have atechnology analogously to firms in country ‫ܪ‬.
Using Dixit–Stiglitz aggregators, Eq.(17) can be rewritten as:௒஽ܰ௧ = ೟ ೟,
(18)
஺
೟

with

ଵ ௒ (௛)‫ܦ‬௧ ‫׬ ؠ‬଴ ೟ ݄݀.
௒೟

Because ݀௧ is ‫צ ߦ צ(݋‬ଶ ), a first order approximation of thisequality is given by:‫ݕ‬௧ = ܽ௧ + ݊௧ ,(19)
which is consistent with Gali and Monacelli’s (2005) log-linearized production function.
Similar to many DSGE literatures including Gali and Monacelli (2005), we assume thatfirms set prices in Calvo–Yun-style price-setting behavior. Hence, a measure 1 െ ߠ
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 theelasticity of substitution across goods produced given the CPI. This is because each firmplays an active part in the monopolistically competitive market. In addition, we assume
that firms have the ability to engage in price discrimination by setting a domestic pricein terms of 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, the maximization
problems which producers in country ‫ ܪ‬face are as follows:
௉෨

ಹ,೟
௞
෨
൰
max‫ כ‬σஶ௞ୀ଴ ߠ ‫ܧ‬௧ ൜ܳ௧,௧ା௞ ൤ܲு,௧ ൬௉௉෨ಹ,೟ ,௉෨ಹ,೟

ିఌ

ಹ,೟శೖ

௉෨

௡
െ‫ܥܯ‬௧ା௞൬൬௉ ಹ,೟ ൰

ିఌ

ಹ,೟శೖ

෨‫כ‬

௉
‫כ‬
‫ܥ‬ு,௧ା௞ +  ࣟ௧ା௞ ܲ෨ு,௧൬௉‫כ‬ಹ,೟ ൰
ಹ,೟శೖ

ିఌ

௉෨ ‫כ‬‫ܥ‬ு,௧ା௞ + ൬௉‫כ‬ಹ,೟ ൰
ಹ,೟శೖ

‫כ‬
‫ܥ‬ு,௧ା௞
൰൨ൠ,

ିఌ

‫כ‬
‫ܥ‬ு,௧ା௞

(20)‫כ‬
where ܲ෨ு,௧ and ܲ෨ு,௧
are the prices chosen by firms when they obtain the chance tochange prices associated with goods produced and sold in country ‫ ܪ‬and goods
produced in country ‫ ܪ‬while sold in country ‫ܨ‬, respectively, ‫ܥܯ‬௧௡ ‫ܲ ؠ‬௉,௧ ‫ܥܯ‬௧ denotes

real marginal costs in country ‫ܪ‬, with ‫ܥܯ‬௧ ‫ؠ‬(ଵିఛ)ௐ೟஺೟ ௉ು,೟

and ܲ௉,௧ denotes producer priceindex (PPI) in country ‫ܪ‬, which are defined as follows:ܲ௉,௧ ‫ؠ‬‫כ‬‫כ‬௉ಹ,೟ ஼ಹ,೟ ାࣟ೟ ௉ಹ,೟஼ಹ,೟‫כ‬஼ಹ,೟ ା஼ಹ,೟

,which can be rewritten as ܲ௉,௧ = ܲு,௧ when the LOOP is applied. The PPI in country ‫ܨ‬ଵis defined analogously. By log-linearizing this equality, we have ‫݌‬௉,௧ = ଶ ‫݌‬ு,௧ +

ଵ
ଶ

‫כ‬
൫݁௧ + ‫݌‬ு,௧൯, which implies as follows:
ଵ

ଵ

‫כ‬
ߨ௉,௧ = ଶ ߨு,௧ + ଶ ൫ȟ݁௧ + ߨு,௧൯,
(21)
where ߨ௉,௧ denotes the PPI inflation in country ‫ ܪ‬and ߨ௉೟ = ߨு,௧ is applied whenthe LOOP is applied.

-- 12
12 --

Note that the maximization problems which producers in country ‫ ܨ‬face areanalogously to Eq.(20). Because of nominal rigidities, Eq.(20) looks complicated. When
there are no nominal rigidities, namely ߠ ՜ 0, Eq.(20) problems boil down to:
‫כ‬
‫כ‬
‫כ‬
max‫ܲ כ‬ு,௧ ‫ܥ‬ு,௧ + ࣟ௧ ܲு,௧‫ܥ‬ு,௧െ ‫ܥܯ‬௧௡ ൫‫ܥ‬ு,௧ + ‫ܥ‬ு,௧
൯,௉ಹ,೟ ,௉ಹ,೟

which 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.
‫כ‬
Under the LCP, we have multiple FONCs because firms can choose ܲ෨ு,௧ and ܲ෨ு,௧
separately. The FONCs for Eq.(20) are as follows:
ିఌ
௉෨ಹ,೟
௡
ஶ
௞
෨
‫ܧ‬௧ ൤σ௞ୀ଴ ߠ ܳ௧,௧ା௞ ൫ܲு,௧ െ ߞ‫ܥܯ‬௧ା௞ ൯ ൬൰ ‫ܥ‬ு,௧ା௞ ൨ = 0,௉ಹ,೟శೖ
ିఌ
‫כ‬
௉෨ಹ,೟
௡
‫כ‬
‫כ‬
௞
෨
‫ܧ‬௧ ൤σஶ
ߠ
ܳ
൫ܲ
ࣟ
െ
ߞ‫ܥܯ‬
൯
൬
൰
‫ܥ‬ு,௧ା௞
൨ = 0,
௧,௧ା௞ ு,௧ ௧ା௞
‫כ‬
௞ୀ଴
௧ା௞
௉ಹ,೟శೖ

which can be log-linearized as follows:௡௞‫݌‬෤ு,௧ = (1 െ ߚߠ) σஶ௞ୀ଴ (ߚߠ) E௧ (݉ܿ௧ା௞ ),௡‫כ‬௞= (1 െ ߚߠ) σஶ‫݌‬෤ு,௧௞ୀ଴ (ߚߠ) E௧ (݉ܿ௧ା௞ െ ݁௧ା௞ ),

(22)

ఏ

with ߞ ‫ ؠ‬ఏିଵ denoting a constant markup where we use the fact that ܳ௧,௧ା௞ =
ିఙ ௉
೟

஼

ߚ ௞ ቀ ೟శೖ
ቁ
஼
೟

௉೟శೖ

. Eq.(21) implies that firms set the price as a markup over a weighted

average of expected future marginal costs. Especially, the first equality in Eq.(22)
definitely corresponds to one derived by Gali and Monacelli (2005). The second equality
in Eq.(22) is not a familiar expression although it implies that firms set the price as amarkup over a weighted average of expected future nominal marginal costs. The second
equality in Eq.(22) is the log-linearized FONC for firms which produce goods in country‫ ܪ‬and sell them in country ‫ܨ‬. Those firms set the price in terms of country ‫’ܨ‬scurrency as a markup over a weighted average of expected future nominal marginal
costs in terms of country ‫’ܨ‬s 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 thepass-through. Now, we discuss the LOOP gap and the real exchange rate in our mode.
Following Monacelli (2005), we define the LOOP gap as follows:ࣟ೟ ௉ ‫כ‬ࣟ೟ ௉ ‫כ‬ಹ,೟ಷ,೟

Ȳு,௧ ‫ ؠ‬௉ ಹ,೟ ; Ȳி,௧ ‫ ؠ‬௉ ಷ,೟
where Ȳு,௧ and Ȳி,௧ denote the LOOP gap for goods produced in countries ‫ ܪ‬and ‫ܨ‬,respectively. When the LOOP is applied, we have Ȳு,௧ = Ȳி,௧ = 1.
Combining Eq.(7), the optimal risk-sharing condition and the definition of the TOT
yields:భ‫כ‬ିିଵ ିଵ ࣟ೟ ௉ಷ,೟Ȳு,௧ = Ȳி,௧࣭௧ ൬ ௉ ൰ ࣫௧ ഑ആ ,ಹ,೟which implies that the LOOP gap is a function of the TOT, the real exchange rate and theࣟ೟ ௉ ‫כ‬

relative price of goods consumed domestically. Because ࣭௧ିଵ ൬ ௉ ಷ,೟ ൰ = Ȳு,௧ Ȳி,௧ , that
ಹ,೟

-- 13
13 --

equality can be rewritten as follows:࣫௧ = 1,
which implies that the PPP is applied although the LOOP is not applied. 4 Log-linearizedversion of this equality is given by:‫ݍ‬௧ = 0.
(23)
In addition, plugging Eq.(23) into Eq.(11), we have ܿ௧ = ܿ௧‫ כ‬, which implies that themarginal utility of consumption between both countries are equal. In fact, households inboth countries consume same goods although there is price discrimination. 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 LOOPିଵ
inversely. In fact, plugging Eq.(23) into that equality, we have Ȳு,௧ = Ȳி,௧andlog-linearized version of this as follows:߰ு,௧ = െ߰ி,௧ ,which implies that gains from price discrimination corresponds to loses from pricediscrimination.
Log-linearized marlet clearing conditions in countries ‫ ܪ‬and ‫ ܨ‬clarifies 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:
݁௧ = ‫݌‬௧ െ ‫݌‬௧‫כ‬‫כ‬+ ‫ݏ‬௧ ,= ‫݌‬௉,௧ െ ‫݌‬௉,௧ଵ‫כ‬= ‫݌‬௉,௧ െ ‫݌‬௉,௧ + ఎ (‫ݕ‬௧ െ ‫ݕ‬௧‫) כ‬(24)
where we use Eq.(21) to derive the second line and Eq.(41) to derive the third line.
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:ଵିఉఏ ஶ
௞௞‫݌‬෤ு,௧ = ‫݌‬ு,௧ିଵ + σஶ௞ୀ଴ (ߚߠ) E௧ ൫ߨு,௧ା௞ ൯ + ଶ σ௞ୀ଴ (ߚߠ) E௧ ൫߰ு,௧ା௞ ൯௞
+(1 െ ߚߠ) σஶ௞ୀ଴ (ߚߠ) E௧ (݉ܿ௧ା௞ ),ଵିఉఏ ஶ
‫כ‬
‫כ‬
‫כ‬
௞
௞
‫݌‬෤ு,௧= ‫݌‬ு,௧ିଵ+ σஶ௞ୀ଴ (ߚߠ) E௧ ൫ߨு,௧ା௞ ൯ െ ଶ σ௞ୀ଴ (ߚߠ) E௧ ൫߰ு,௧ା௞ ൯௞
+(1 െ ߚߠ) σஶ(25)
௞ୀ଴ (ߚߠ) E௧ (݉ܿ௧ା௞ ),
‫כ‬
‫כ‬
‫כ‬
െ ‫݌‬ு,௧ିଵ
denotes inflation of goods bothwhere ߨு,௧ ‫݌ ؠ‬ு,௧ െ ‫݌‬ு,௧ିଵ and ߨு,௧ ‫݌ ؠ‬ு,௧
produced and sold in country ‫ ܪ‬and inflation of goods produced in country ‫ ܪ‬andsold in country ‫ܨ‬, respectively. As mentioned, firms set the price as a mark up over aweighted average of future marginal cost. In our LCP setting, those firms’ sales are notmeasured by the PPI, because it is the weighted average of both price of goods selling incountry ‫ ܪ‬and in country ‫ܨ‬. 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 ܲு,௧ and pay total costs measured by the ܲ௉,௧ and that gap is calculatedThis equality implies that the marginal utility of consumption in country ‫ ܪ‬is definitely same as it in country ‫ܨ‬.
Hence, the UIP can be derived by simply combining 8 and its counterpart in country ‫ ܨ‬without assuming, although
we describe that we assume the UIP in section 2.1.1.

4

-- 14
14 --

ଵ

by ‫݌‬௉,௧ െ ‫݌‬ு,௧ = ଶ ߰ு,௧ , which implies that that gap corresponds to the LOOP gap incountry ‫ܪ‬. Although the firms selling goods in country ‫ ܪ‬have no currency disparity insales and payment, the LCP behavior generate the LOOP gap. Thus, a weighted averageof expected future LOOP gap in country ‫ ܪ‬appears in the first equality in Eq.(25).
The price setting behavior of the firms selling goods in country ‫ ܨ‬generates the LOOP
gap, alike with another firms which sell goods in country ‫ܪ‬. Those firms, namely
exporters, obtain the sales of goods exported in terms of country ‫’ܨ‬s currency and pay
the total cost in terms of country ‫’ܪ‬s currency. Their sales are measured by country ‫’ܪ‬scurrency. Hence, their sales in terms of country ‫ ’ܨ‬currency is multiplied by nominal
exchange rate. They pay total costs which is measured by the PPI, alike with the firmsଵ‫כ‬+ ݁௧ ൯ = െ ଶ ߰ு,௧ . Thus, aselling goods in country ‫ܪ‬. The gap is calculated by ‫݌‬௉,௧ െ ൫‫݌‬ு,௧
weighted average of expected future LOOP gap in country ‫ ܪ‬appears in the second
equality in Eq.(25) although the sign is contrary to the first equality. Similar mechanism
works in firms in country ‫ ܨ‬not only for selling goods domestically but also forexporters. Although our LCP setting is different from Monacelli (2005), who assumes asmall open economy and importers, our LCP setting clearly generates the LOOP gap andthis 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:
஼ ഑ே

ക

ିଵ

௉

(26)

‫ܥܯ‬௧ = (1 െ ߬) ೟஺ ೟ ቀ ௉ು,೟ ቁ ,೟೟which is log-linearized as follows:ଵ݉ܿ௧ = ߪܿ௧ + ߮݊௧ + ଶ ‫ݏ‬௧ െ ܽ௧ ,(27)
which is consistent with Gali and Monacelli’s (2005) log-linearized marginal cost.
ଵ
Under the flexible price equilibrium, ‫ܥܯ‬௧ = ఍ implying that the real marginal cost is
constant and corresponds to inverse of a constant markup is applied. Using this fact andcombining Eqs.(14), (18) and Eq.(26), we have natural rate of output under the LCP incountry ‫ ܪ‬as follows:
భ

ିఎఙ

఍ షభ

ଵ ௉
ଵାఝ ௉
ܻത௧ = ଶ ቊ ௉ು,೟ ଵିఛ ‫ܣ‬௧ ቀ ௉ಹ,೟ቁ
೟

೟

௉ಹ,೟

ఎ

ቈ1 + ቀ ௉ ቁ ቀ
೟

‫כ‬
௉ಹ,೟
௉೟‫כ‬

ିఎ

ቁ

భ
ି
഑

ఙ കశ഑

࣫௧ ቉ ቋ

ି

‫ܦ‬௧

ക
കశ഑

,

with ܻത௧ denotes natural rate of output in country ‫ܪ‬, which implies that natural rate ofoutput is a function not only of productivity but also of relative prices because of anopen economy setting.
Before log-linearizing this equality, we define the output gap in country ‫ݔ ܪ‬௧ as thedeviation of percentage deviation of output in country ‫ݕ ܪ‬௧ from its natural level ‫ݕ‬ത௧ .
This relationship can be written as:‫ݔ‬௧ ‫ݕ ؠ‬௧ െ ‫ݕ‬ത௧ ,(28)
which is definitely consistent with Gali and Monacelli (2005)’s definition. The output gap
in country ‫ ܨ‬is defined analogously to Eq.(28)

-- 15
15 --

Now, we log-linearize that equality. Log-linearized natural rate of output under the
LCP is given by:(ఙఎିଵ)ఠమ ‫כ‬ఠ ఠ‫ݕ‬ത௧ = భ మ ܽ௧ െ
ܽ௧ ,
(29)
ఠ
ఠ
య

య

with ߱ଵ ‫ ߪ(ߟ ؠ‬+ 2߮) + 1, ߱ଶ ‫ ؠ‬2ߟ(1 + ߮) and ߱ଷ ‫߱ ؠ‬ଵଶ െ (ߪߟ െ 1)ଶ . While Gali andMonacelli (2005) regard foreign output is exogenous because of their small open
economy setting, foreign output, namely output in country ‫ ܨ‬is endogenous in our
two-country setting. Thus, productivity in country ‫ ܨ‬replaces foreign output in Eq.(29).
We turn to discuss Eq.(27), percentage deviation of marginal cost from its steady statevalue. Plugging Eqs.(42), (41), (19) and (28) into Eq.(27) yields:ఠఙఎିଵ݉ܿ௧ = ଶఎభ ‫ݔ‬௧ + ଶఎ ‫ݔ‬௧‫ כ‬,(30)
which implies that real marginal cost in country ‫ ܪ‬consists of output gap in both twocountries.
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:
ଶఎ
ଶఎ
ఙఎିଵ
ଶఎ
‫) כ‬
‫ݔ‬௧ = E௧ (‫ݔ‬௧ାଵ ) െ ఙ ‫ݎ‬Ƹ௧ + ఙ E௧ ൫ߨ௉,௧ାଵ ൯ + ఙ E௧ (ȟ‫ݔ‬௧ାଵ
(31)+ ఙ ‫ݎ‬ҧ௧ ,where ‫ݎ‬ҧ௧ ‫ ؠ‬െߪఈ(ଵିఘ)(ଵାఝ)ఠర
ఠయ

ഀ

ഀ

ܽ௧ െ ߪఈ(ଵିఘ)(ఙఎିଵ)(ଵାఝ)ఠఱ
ఠయ

ഀ

ഀ

ܽ௧‫ כ‬denotes the natural rate ofinterest in country ‫ ܪ‬with ߪఈ ‫ ߟߪ ؠ‬+ 1 , ߱ସ ‫߱ ؠ‬ଵ െ(ఙఎିଵ)మ
ఙഀఠ

and ߱ହ ‫ ؠ‬ఙ భ െ 1 . The
ഀNKIS in country ‫ܨ‬, 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. Because of the LCP,
Eq.(31) has some distinguished feature. Plugging Eq.(21) into Eq.(31), NKIS under LCP
can be rewritten as follows:
ଶఎ
ఎ
ఎ
ఎ
‫כ‬
‫ݔ‬௧ = E௧ (‫ݔ‬௧ାଵ ) െ ‫ݎ‬Ƹ௧ + E௧ ൫ߨு,௧ାଵ ൯ + E௧ ൫ߨு,௧ାଵ൯ + E௧ (ȟ݁௧ାଵ )
ఙഀ
ఙഀ
ఙഀ
ఙഀ
ଶఎ
‫כ‬
+ ఙ E௧ (ȟ‫ݔ‬௧ାଵ ) + ఙ ‫ݎ‬ҧ௧
ഀ
ഀ
ఎ
ఎ
ఎ
ఎ
‫כ‬
= E௧ (‫ݔ‬௧ାଵ ) െ ఙ ‫ݎ‬Ƹ௧ െ ఙ ‫ݎ‬Ƹ௧‫ כ‬+ ఙ E௧ ൫ߨு,௧ାଵ ൯ + ఙ E௧ ൫ߨு,௧ାଵ
൯
ഀ
ഀ
ഀ
ഀ
ఙఎିଵ
ଶఎ
‫) כ‬
+ ఙ E௧ (ȟ‫ݔ‬௧ାଵ+ ఙ ‫ݎ‬ҧ௧ ,
ఙఎିଵ

ഀ

ഀ

(32)

where we take log-linearized UIP to derive second line. As shown in the first line,
changes in expected nominal exchange rate affects the NKIS. Second line shows that not
only domestic nominal interest rate, but also foreign nominal interest rate appears the
NKIS.
Plugging log-linearized Calvo’s pricing rule and Eq.(30) to Eq.(25), we have equalities
which determines the dynamics of inflation as follows:ఒఒఠఒ(ఙఎିଵ) ‫כ‬ߨு,௧ = ߚE௧ ൫ߨு,௧ାଵ ൯ + ଶ ߰ு,௧ + ଶఎభ ‫ݔ‬௧ +‫ݔ‬௧ ,ଶఎఒఒఠభ
ఒ(ఙఎିଵ) ‫כ‬‫כ‬‫כ‬ߨு,௧ = ߚE௧ ൫ߨு,௧ାଵ ൯ െ ଶ ߰ு,௧ + ଶఎ ‫ݔ‬௧ +‫ݔ‬௧ ,
ଶఎ

-- 16
16 --

(33)

(ଵିఉఏ)(ଵିఏ)

with ߣ ‫ؠ‬. The first equality is inflation dynamics for goods sold domesticallyఏand the second equality is inflation dynamics for goods exported. Because Eq.(33)
derived from Eq.(25), the FONCs for firms in country ‫ܪ‬, the third and the fourth terms
in the RHS, which stem from real marginal cost in country ‫ܪ‬, are consistent between
both equalities. The signs of the second terms in the RHS are inverse between both
equalities. The reason is that the losses from price discrimination are compensated by
the gains from price discrimination, and vice versa. Counter part of Eq.(33) are derived
from counter part of Eq.(25).
Plugging Eq.(33) into Eq.(21), we have New Keynesian Philips Curve (NKPC) in
country ‫ ܪ‬as follows:ఒఠఒ(ఙఎିଵ) ‫כ‬ఉଵߨ௉,௧ = ߚE௧ ൫ߨ௉,௧ାଵ ൯ + ଶఎభ ‫ݔ‬௧ +‫ݔ‬௧ െ ଶ E௧ (ȟ݁௧ାଵ ) + ଶ ȟ݁௧ ,ଶఎ

and plugging counter part of Eq.(33) into Eq.(21) yields counter part of this equality incountry ‫ܨ‬. This NKPC is featured by appearance of changes in nominal exchange rate.
Gali and Monacelli (2005) mention that full stabilization of domestic prices coincides
with full stabilization of output gap, namely ‫ݔ‬௧ = ߨு,௧ = 0 for all ‫ݐ‬. In our model, theirdomestic prices correspond to the PPI and they assume fully-exogenous foreign output
which implies that the percentage deviation of marginal cost from its steady state value
is not affected by the percentage deviation of foreign output from its steady state value.
That is, they claim that full stabilization of PPI implies that output conforms its naturalrate if we ignore foreign output gap or assume ߪߟ = 1 in this equality. Even if we
ignore foreign output gap or assume ߪߟ = 1 in this equality, full stabilization of PPI
does not necessarily imply that output conforms its natural rate because of changes innominal exchange rate, as shown in the forth and the fifth terms in the RHS. Changes innominal exchange rate as if work cost push shocks under the LCP. Thus, full stabilizationof PPI no longer implies that output conforms its natural rate if we ignore foreign output
gap or assume ߪߟ = 1. Plugging Eqs.(24), (29) and (28) into that equality, we caneliminate changes in nominal exchange rate and obtain as follows:
ఉ
ఉ
‫כ‬
‫) כ‬
ߨ௉,௧ = ߚE௧ ൫ߨ௉,௧ାଵ ൯ + ߚE௧ ൫ߨ௉,௧ାଵ൯ െ ఎ E௧ (‫ݔ‬௧ାଵ ) + ఎ E௧ (‫ݔ‬௧ାଵ+ ߢధ ‫ݔ‬௧
ଵ

ଵ

‫כ‬
‫כ‬
‫כ‬
െߢచ ‫ݔ‬௧‫ כ‬െ ߨ௉,௧െ ఎ ‫ݔ‬௧ିଵ + ఎ ‫ݔ‬௧ିଵ+ ߱଺ ܽ௧ െ ߱଺ ܽ௧ିଵ െ ߱଺ ܽ௧‫ כ‬+ ߱଺ ܽ௧ିଵwith ߢధ ‫ؠ‬ଵାఉାఒఠభఎ, ߢచ ‫ؠ‬ଵାఉିఒ(ఙఎିଵ)ఎ, ߱଺ ‫ؠ‬ఠమ ధయ (ఙାఝ)ఠయ(34)
and ߸ଷ ‫ ؠ‬1 + ߚ(1 െ ߩ) .

Exogenous shocks appear in Eq.(34) which shows that exogenous productivity affects
PPI inflation.
Monacelli (2005) derives CPI based NKPC. Following Monacelli (2005), we derive CPI
based NKPC. Plugging the first equality in Eq.(33) and its counterpart in country ‫ ܨ‬into
Eq.(5) yields:఑఑ߨ௧ = ߚE௧ (ߨ௧ାଵ ) + ଶഀ ‫ݔ‬௧ + ଶഀ ‫ݔ‬௧‫כ‬(35)
with ߢఈ ‫ ߪ(ߣ ؠ‬+ ߮). As mentioned by Gali and Monacelli (2005), ߢఈ is consistent withthe 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 thenominal interest rate in both countries absorbs the effects from changes in productivity

-- 17
17 --

in NKISs. Gali and Monacelli (2005) mention that a full stabilization of PPI inflation
implies output conforms its natural level and there 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 is no output gap, in our LCP setting under a two-country. This canbe understood alternatively 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 (2005). In his LCP setting,imports purchase foreign goods at the costs in terms of foreign currency while they sellforeign goods at the costs in terms of domestic currency. Because importers maximize
their profits, the LOOP gap appears in CPI based NKPC in Monacelli (2005). Our LCP
setting is quite different from Monacelli’s (2005) 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 inEq.(35), different from Monacelli (2005).

2.2. PCP Model
Under the PCP, the LOOP is applied which is given by ܲ௧ (݄) = ࣟ௧ ܲ௧‫ )݄( כ‬and ܲ௧ (݂) =ࣟ௧ ܲ௧‫ )݂( כ‬hence:‫כ‬‫כ‬ܲு,௧ = ࣟ௧ ܲு,௧; ܲி,௧ = ࣟ௧ ܲி,௧
(36)
and
‫כ‬
‫כ‬
‫݌‬ு,௧ = ݁௧ + ‫݌‬ு,௧; ‫݌‬ி,௧ = ݁௧ + ‫݌‬ி,௧
,(37)
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 andforeign goods are given by Eqs.(1), (2), (3), (4), (6) and (7), alike with the LCP model.
Because households face same optimization problem, intertemporal 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 applied in PCP model. Log-linearized definitionof the CPI, intertemporal optimality condition and the risk-sharing condition are also
given by Eqs.(5), (10) and (11).
2.2.2. Market 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:
௉ (௛)ܻ௧ (݄) = ൬ ೟

௉ಹ,೟

൰

ିఌ

ିఎ

௉

ቀ ௉ಹ,೟ ቁ೟‫ܥ‬௧ .

-- 18
18 --

by utilizing Eq.(36). Plugging Dixit-Stiglitz aggregator of output into this equality
yields:ିఎ௉ܻ௧ = ቀ ௉ಹ,೟ቁ೟(38)‫ܥ‬௧ ,

which is demand function consistent with Benigno and Benigno’s (2006). 5
Definition of the TOT is given by Eq.(15). Plugging Eq.(36) into Eq.(15) yields:௉࣭௧ = ௉ ಷ,೟ ,
ಹ,೟

(39)

which is applicable only to PCP model because Eq.(36) is not applicable to LCP model.
Log-linearizing Eq.(39), we have:(40)‫ݏ‬௧ = ‫݌‬ி,௧ െ ‫݌‬ு,௧
with ‫ݏ‬௧ ‫ ؠ‬ln࣭௧ . 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:ଵ‫ݏ‬௧ = ఎ (‫ݕ‬௧ െ ‫ݕ‬௧‫) כ‬,(41)
which clarifies relationship between the TOT and relative output under the PCP. Gali andMonacelli (2005) and Benigno and Benigno (2006) who assume the PCP derive sameequality.
Log-linearizing Eq.(38) yields:ఎ(42)‫ݕ‬௧ = ଶ ‫ݏ‬௧ + ܿ௧ ,where we use Eq.(37). As mentioned, Eq.(42) is final form of log-linearized market
clearing under the LCP in country ‫ܪ‬. The difference in price setting behavior between
the LCP and the PCP does not affect market clearing.
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 simple. Because of‫כ‬ܲ෨ு,௧ = ࣟ௧ ܲ෨ு,௧
and Eq.(36), Eq.(20) can be rewritten as:
௉෨

ಹ,೟
௡
௞
෨
max σஶ
൰௞ୀ଴ ߠ ‫ܧ‬௧ ൜ܳ௧,௧ା௞ ൤൫ܲு,௧ െ ‫ܥܯ‬௧ା௞ ൯ ൬௉

௉෨ಹ,೟

ିఌ

ಹ,೟శೖ

‫כ‬
൫‫ܥ‬ு,௧ା௞ + ‫ܥ‬ு,௧ା௞
൯൨ൠ,

(43)

which is familiar expression in literatures assuming Calvo pricing. Plugging Eq.(36) into
the PPI definition, we have ܲ௉,௧ = ܲு,௧ and plugging Eq.(37) into Eq.(21) yields:
ߨ௉,௧ = ߨு,௧ .
(44)
The FONC of Eq.(43) is given by:
௉෨

ಹ,೟
௡
௞
෨
‫ܧ‬௧ ൤σஶ൰௞ୀ଴ ߠ ܳ௧,௧ା௞ ൫ܲு,௧ െ ߞ‫ܥܯ‬௧ା௞ ൯ ൬௉
ಹ,೟శೖ

ିఌ

‫כ‬
൫‫ܥ‬ு,௧ା௞ + ‫ܥ‬ு,௧ା௞൯൨ = 0,which is familiar expression in literatures assuming the PCP. Log-linearizing thisequality, we have:
We do not assume government expenditure. Thus, government expenditure does not appear in those equalities
although it appears in Benigno and Benigno (2006).

5

-- 19
19 --

௡
௞
‫݌‬෤ு,௧ = (1 െ ߚߠ) σஶ௞ୀ଴ (ߚߠ) E௧ (݉ܿ௧ା௞ ),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. This equality can be
rewritten as follows:ஶ௞௞‫݌‬෤ு,௧ = ‫݌‬ு,௧ିଵ + σஶ (45)
௞ୀ଴ (ߚߠ) E௧ ൫ߨு,௧ା௞ ൯ + (1 െ ߚߠ) σ௞ୀ଴ (ߚߠ) E௧ (݉ܿ௧ା௞ ),which corresponds to one derived by Gali and Monacelli (2005). Because of the LOOP,
LOOP gap disappears in Eq.(45), although LOOP gap appears in the first equality inEq.(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) and its
log-linearized equality Eq.(27). However, the natural rate of output under the PCP isquite different from one under the LCP at glance. Combining not only Eqs.(14), (18) andEq.(26) but also ܲ௉,௧ = ܲு,௧ , we have:భകି(ఙఎିଵ) ഑శക ିଵାఝ ௉
ܻത௧ = ൤ ‫ܣ‬௧ ቀ ಹ,೟ቁ൨‫ܦ‬௧ ഑శക ,
ଵିఛ
௉೟
఍ షభ

which can be log-linearized as follows:(ఙఎିଵ)ఠఠ ఠ‫ݕ‬ത௧ = ఠభ మ ܽ௧ െ ఠ మ ܽ௧‫ כ‬.
యయThis 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 notaffect 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:ఠఙఎିଵ݉ܿ௧ = ଶఎభ ‫ݔ‬௧ + ଶఎ ‫ݔ‬௧‫ כ‬,

which is consistent with Eq.(30). Difference between the PCP and the LCP models is
price setting behavior. Because the marginal cost has no relationship with price setting
behavior, Eq.(30) is applied under both the PCP and the LCP. Note that Gali andMonacelli (2005) show that real marginal cost has relationship with just domestic
output gap and their result is different from Eq.(30). This difference stems from our
two-country setting. As mentioned, foreign output is not exogenous in our setting andproductivity in country ‫ ܨ‬appears in Eq.(29), while foreign output appears in theirexpression in terms of percentage deviation from 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
deviation of domestic real marginal cost from its steady state value corresponds to its
deviation from its flexible price equilibrium value, foreign output gap disappears in Galiఠand Monacelli (2005). In fact we have ݉ܿ௧ = ଶఎభ ‫ݔ‬௧ if we regard output in country ‫ ܨ‬asexogenous.

-- 20
20 --

2.2.5. The Demand and Supply Sides
Plugging Eqs.(21), (23), (41) and (42) into Eq.(10) yields NKIS as follows:
ଶఎ
ଶఎ
ఙఎିଵ
ଶఎ
‫) כ‬
‫ݔ‬௧ = E௧ (‫ݔ‬௧ାଵ ) െ ఙ ‫ݎ‬Ƹ௧ + ఙ E௧ ൫ߨ௉,௧ାଵ ൯ + ఙ E௧ (ȟ‫ݔ‬௧ାଵ+ ఙ ‫ݎ‬ҧ௧ ,
ഀ

ഀ

ഀ

ഀ

ഀ

ഀ

ഀ

ഀ

(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 at glance. Plugging Eq.(44) into
Eq.(31), we have:
ଶఎ
ଶఎ
ఙఎିଵ
ଶఎ
‫) כ‬
‫ݔ‬௧ = E௧ (‫ݔ‬௧ାଵ ) െ ఙ ‫ݎ‬Ƹ௧ + ఙ E௧ ൫ߨு,௧ାଵ ൯ + ఙ E௧ (ȟ‫ݔ‬௧ାଵ+ ఙ ‫ݎ‬ҧ௧ ,which is applicable only if to PCP model and ߨு,௧ replaces ߨ௉,௧ 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.
By rearranging Eq.(45), we have NKPC in country ‫ ܪ‬under the PCP as follows:ఒఠఒ(ఙఎିଵ) ‫כ‬ߨ௉,௧ = ߚE௧ ൫ߨ௉,௧ାଵ ൯ + ଶఎభ ‫ݔ‬௧ +‫ݔ‬௧ ,(47)
ଶఎ

which is two-country version NKPC derived by Gali and Monacelli (2005). Whileforeign output gap appears in Eq.(47), that does not appear in the NKPC derived by Gali
and Monacelli (2005) who assume small open economy where foreign variables are
exogenous. Because our model is a two-country model where the foreign variables areendogenous, foreign output gap appears in our NKPC under the PCP. In fact, if we regard
output in country ‫ ܨ‬as exogenous, we have:ఒఠߨ௉,௧ = ߚE௧ ൫ߨ௉,௧ାଵ ൯ + భ ‫ݔ‬௧ ,ଶఎwhich is quite similar to NKPC derived by Gali and Monacelli (2005) and can be derived
alternatively only if ߪߟ = 1 in our two-country model under the PCP because foreign
output gap disappears in such a case. Gali and Monacelli (2005) mention that fullstabilization of PPI implies that ‫ݔ‬௧ = ߨு,௧ = 0 which is plausible if output gap incountry ‫ ܨ‬disappears in Eq.(47). Because of two-country setting, foreign output gap
does not disappears as long as we do not assume ߪߟ = 1. Hence, full stabilization of PPI
does not necessarily imply ‫ݔ‬௧ = ߨு,௧ = 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 is derived by thewelfare criterion. Following Woodford (2003) and Gali (2008), we have second-orderapproximated utility function as follows:
ௐ
ௐ
ௐ
ௐ
ࣱ௅஼௉
= െࣦ௅஼௉+ ‫ݐ‬. ݅. ‫݌‬. +‫צ ߦ צ(݋‬ଷ ) ; ࣱ௉஼௉= െࣦ௉஼௉+ ‫ݐ‬. ݅. ‫݌‬. +‫צ ߦ צ(݋‬ଷ ),
(48)
ௐ
ௐ
ௐ
ௐ
ஶ
ஶ
௧
௧
where ࣦ௅஼௉ ‫ܧ ؠ‬଴ σ௧ୀ଴ ߚ ‫ܮ‬௅஼௉,௧ and ࣦ௉஼௉ ‫ܧ ؠ‬଴ σ௧ୀ଴ ߚ ‫ܮ‬௉஼௉,௧ denote the loss function

-- 21
21 --

ଵ

ଵ

‫) כ‬
ௐ
ௐ
in LCP and PCP models, respectively, ࣱ௅஼௉= ଶ (ࣱ௅஼௉ + ࣱ௅஼௉and ࣱ௉஼௉= ଶ (ࣱ௉஼௉ +‫) כ‬ௐdenote average welfare criteria in LCP and PCP models, respectively, ܹ௅஼௉
and
ࣱ௉஼௉
ௐ
ܹ௉஼௉ denote welfare criteria in country ‫ ܪ‬in LCP and PCP models, respectively with௎೟ ି௎ࣱ ‫ ؠ‬σஶ௧ୀ଴ E଴ (ࣱ௧ ) and ࣱ௧ ‫ ؠ‬௎ ஼ . Further:಴(ଵାఝ)ఎ మ ଶ
ଵ ఌ ଶ
ఌ
ௐ )ଶ
‫) כ‬ଶ
(ߨ
(ߪ
‫ܮ‬ௐ
‫ؠ‬
ቂ
ߨ
+
+
+
߮)(‫ݔ‬+‫ݖ‬௧ ቃ,
௧
௧
௧
௅஼௉,௧
ଶ ଶఒ
ଶఒ
ସ
ଶ
(ଵାఝ)ఎ మ ଶ
ଵ ఌ ଶ
ఌ
‫כ‬
ௐ ଶ
‫ܮ‬ௐ
‫ݖ‬௧ ቃ,௉஼௉ ‫ ؠ‬ଶ ቂଶఒ ߨு,௧ + ଶఒ ൫ߨி,௧ ൯ + (ߪ + ߮)(‫ݔ‬௧ ) +
ସ

(49)

(50)
are period loss function in countries ‫ ܪ‬and ‫ܨ‬, respectively, , ‫ݖ‬௧ ‫ݏ ؠ‬௧ െ ‫ݏ‬ҧ௧ being theଵାఝఎdeviation of the TOT from its efficient level, ‫ݏ‬ҧ௧ ‫ ؠ‬ఎమ (ଵାఝ) ‫ݕ‬ത௧ோ being the efficient level ofଵTOT. Note that we define ‫ݒ‬௧ௐ ‫ ؠ‬ଶ (‫ݒ‬௧ + ‫ݒ‬௧‫ ) כ‬and ‫ݒ‬௧ோ ‫ݒ ؠ‬௧ െ ‫ݒ‬௧‫ כ‬.

3.2. FONCs for Central Banks
We next briefly mention the FONCs for the central bank. We assume that the central
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:ଵௐ ),ߨ௧ௐ = െ ఌ (‫ݔ‬௧ௐ െ ‫ݔ‬௧ିଵ‫ݖ‬௧ = 0,
where ߤଷ,௧ is a Lagrange multiplier associated with the average block of the NKPC.
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:ଵௐ ),ߨ௧ௐ = െ (‫ݔ‬௧ௐ െ ‫ݔ‬௧ିଵఌ(ଵାఝ)ఎ మோߨ௉,௧ = െ ఌ(ଵାఎఝ) (‫ݖ‬௧ െ ‫ݖ‬௧ିଵ ).

3.3. Calibration
We run a series of dynamic simulations and adopt the following benchmark
parameterization. We set the price stickiness ߠ, the subjective discount factor ߚ, theelasticity of substitution across goods ߝ and the inverse of the labor supply elasticity ߮
equal to 0.75, 11, 0.99, 3, respectively, which are consistent with quarterly time periodsin the model. 6 We compare two cases. One of them is special case in which ߪ = ߟ = 1
and another one is general case in which ߪ = 3 and ߟ = 4.5. Note that ߪ = ߟ = 1 isassumed by Gali and Monacelli (2005) while ߪ = 3 and ߟ = 4.5 are assumed by
Benigno and Benigno (2006). We assume that productivity shifters are describedaccording to the following AR(1) processes:‫כ‬+ ߦ௧‫ כ‬,ܽ௧ = ߩܽ௧ିଵ + ߦ௧ ; ܽ௧‫ܽߩ = כ‬௧ିଵwhere ߦ௧ and ߦ௧‫ כ‬denote the i.i.d. shocks. We set ߩ equal to 0.9. To examine theimpulse response functions (IRFs), we consider one percent changes in the productivity6

ߠ = 0.75 implies that average length of price contracts equal to 4.

-- 22
22 --

shifter in country ‫ܽ ܪ‬௧ , and the productivity shifter in country ‫ܨ‬, ܽ௧‫ כ‬.
Impulse responses to one percent increase in productivity in country ‫ ܪ‬in special
case are shown in Figure 1 while macroeconomic volatilities are shown in 3rd and 4thcolumns in Table 1 7. 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 ‫ ܪ‬and ‫ ܨ‬are completely stabilized. Under the
PCP, PPI inflation rate in countries ‫ ܪ‬and ‫ ܨ‬is completely stabilized (Panels 3 and 4).
This result is consistent with Gali and Monacelli (2005) who imply that PPI inflation
targeting brings zero output gap. This result can be understood by paying attention toEq.(47). Plugging ߪ = ߟ = 1 into Eq.(47) yields:ఒఠߨ௉,௧ = ߚE௧ ൫ߨ௉,௧ାଵ ൯ + ଶఎభ ‫ݔ‬௧ ,which is NKPC in the special case under the PCP. This NKPC implies that stabilizing PPI
inflation brings stabilizing output gap simultaneously and is consistent with one derived
by Gali and Monacelli (2005) although the slope of our NKPC is slightly different fromtheirs 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 (2005) but also other DSGE literatures
assuming an open economy (Panels 5 and 6). This can be understood by paying
attention to Eq.(35) which implies that CPI inflation becomes zero when output gap incountries ‫ ܪ‬and ‫ ܨ‬are stabilized. Hence, it may be said that CPI inflation targeting
brings completely stabilizing output gap. Interestingly, nominal exchange rate iscompletely stabilized under the LCP which is consistent with Devereux and Engel (2003)
developing NOEM model, assuming the LCP and showing that fixed exchange rate is
optimal regime from the viewpoint of maximizing welfare. This result stems fromstabilizing CPI inflation rate. Perfect stabilization in CPI inflation is consistent withperfect stabilization in the CPI level. 8 In our model, PPP is always applied hence݁௧ + ‫݌‬௧‫݌ = כ‬௧ is applied. Perfect stabilization in CPI inflation implies ‫݌‬௧ = ‫݌‬௧‫ = כ‬0 whichis consistent with ݁௧ = 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 ‫ ܪ‬in thegeneral 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(2005) show that that trade-offs dissolved simultaneously (Panels 1 to 4). Because ourmodel is a two-country model, foreign output is endogenous while it is exogenous in Gali
and Monacelli (2005)’s small open economy model. In a small economy setting, foreign
output gap disappears in the NKIS although that appears in the NKIS, as shown inEq.(47). foreign output gap disappears in Eq.(47) only if ߪߟ = 1. Hence, neither output
gap nor PPI inflation stabilized simultaneously.
However, although output gap in countries ‫ ܪ‬and ‫ ܨ‬is not stabilized, CPI inflation is
There are 2 eigenvalues larger than 1 in modulus for 2 forward-looking variables. Hence the rank condition isverified
8 We assume zero inflation deterministic steady state.
9 There are 2 eigenvalues larger than 1 in modulus for 2 forward-looking variables. Hence the rank condition isverified
7

-- 23
23 --

completely stabilized under the LCP. This can be understood by paying attention toEq.(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) canbe rewritten as:ߨ௧ = ߚE௧ (ߨ௧ାଵ ) + ߢఈ ‫ݔ‬௧ௐ .
In this NKPC, the slope is not affected by ߪ and ߟ. Thus, result on volatility is notdifferent between the special and the general cases. In addition, CPI inflation iscompletely stabilized, there is no fluctuation in the nominal exchange rate, alike with thespecial case (Panel 13). As mentioned, our result that there is no fluctuation in thenominal exchange rate under the LCP is consistent with the result of Devereux and Engel(2003). Devereux and Engel (2003) assume Arminton Form of consumption which
implies ߟ = 1. 10 Now, we apply more general setting such as ߟ = 4 while our result on
fluctuations in nominal exchange rate is consistent with their result. This implies that
Devereux and Engel (2003)’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 ߪ and the elasticity of substitution between goods
produced in countries ‫ ܪ‬and ‫ߟ ܨ‬. There are many macroeconomic variables in ourmodel 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 relative risk
aversion ߪ ‫[ א‬1,10] and the elasticity of substitution between goods produced incountries ‫ ܪ‬and ‫[ א ߟ ܨ‬1,10]. Under the PCP, volatility of PPI inflation is definitely
zero when ߟ = 1 although the higher the ߟ the higher the volatility (Panel 2). When
ߟ = 1, one of FONCs related to relative inflation for central banks under the PCP can be
rewritten as:
ଵ
ோ
ோ )
ߨ௉,௧
= െ ఌ (‫ݔ‬௧ோ െ ‫ݔ‬௧ିଵ
(51)
ோbecause ‫ݖ‬௧ = ‫ݔ‬௧ is applied when ߟ = 1. Along with another FONC related to average
output gap for central banks under the PCP, those FONCs imply that stabilization in PPI
inflation strictly consistent with stabilization in output gap. Hence, volatility of PPI
inflation is definitely zero when ߟ = 1. Equally, this implies that ߪ = ߟ = 1 is not a
sufficient condition to dissolve inflation–output gap trade-offs but just ߟ = 1 is asufficient condition to dissolve that trade-offs under the PCP. As implied by calibration
in former section, volatility of CPI inflation is definitely zero regardless of ߪ and ߟ
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 ߪ and ߟ. This immediately implies that there is no fluctuation in CPI
inflation countries ‫ ܪ‬and ‫ ܨ‬regardless of ߪ and ߟ under the LCP.
1

10

1

In that case, Eq.(2) is rewritten by ‫ = ݐܥ‬2‫ܥ‬2‫ܪ‬,‫ܥ ݐ‬2‫ܨ‬,‫ݐ‬

-- 24
24 --

Average output gap is completely stabilized under both the LCP and the PCP
regardless of ߪ and ߟ (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 deviationfrom efficient level is definitely zero regardless of ߪ and ߟ under the LCP (Panel 7).
This stems from the FONC for central banks under the LCP related to TOT deviationfrom efficient level. However, TOT deviation from efficient level is definitely zero under
the PCP only if ߟ = 1 (Panel 8). In that case, ‫ݖ‬௧ = ‫ݔ‬௧ோ is applied and there is nofluctuation in output gap in countries ‫ ܪ‬and ‫ܨ‬, as implied by Eq.(51). Both PPI
inflation and output gap are completely stabilized when ߟ = 1. Hence, TOT deviationfrom efficient level is definitely zero through complete stabilization in output gap incountries ‫ ܪ‬and ‫ܨ‬. However, complete stabilization in TOT deviation from efficientlevel is no longer achieved when ߟ = 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 is definitely zero regardless of ߪ andߟ under the LCP. Even if the LCP is assumed, the PPP is applied which implies that‫ݏ‬௧ = ‫݌‬௧ െ ‫݌‬௧‫ כ‬. Because of optimal monetary policy, CPI inflation is definitely stabilized
which is consistent with zero fluctuation in the CPI level under the LCP. Thus, thenominal exchange rate is definitely stabilized regardless of ߪ and ߟ under the LCP. As
mentioned in former section, our result is consistent with Devereux and Engel (2003)’s
result which shows that optimal monetary policy under the LCP is consistent with fixedexchange rate regime although they assume Armington form of consumption whichcorresponds to ߟ = 1. Our model does not assume Armington form of consumption andthere is no parametric restriction in ߟ. Thus, our result implies that Devereux and Engel(2003)’s policy implication is not applied in special parameterization but applied ingeneral setting. In addition, we derive more important policy implication. Optimalmonetary policy under the LCP stabilizes the CPI inflation definitely and coincides withcomplete 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 targetingand fixed exchange rate regime. However, it can be said that CPI inflation targeting andfixed exchange rate regime are optimal and equivalent under the LCP although there issome room to discuss precisely.
Finally, we discuss effects on welfare costs varying the relative risk aversion ߪ andthe elasticity of substitution between goods produced in countries ‫ ܪ‬and ‫ߟ ܨ‬. When
we introduce ߚ ՜ 1 in Eq.(48), we have welfare criteria as follows:(1 + ߮)ߟଶ
1 ߝߝௐࣦ௅஼௉,௧‫ ؠ‬ቈ ‫ߨ(ݎܽݒ‬௧ ) + ‫ߨ(ݎܽݒ‬௧‫ ) כ‬+ (ߪ + ߮)‫ݔ(ݎܽݒ‬௧ௐ ) +‫ݖ(ݎܽݒ‬௧ )቉,
2 2ߣ
2ߣ
4
(1 + ߮)ߟଶ
1 ߝ
ߝ
ௐ
‫כ‬
ࣦ௉஼௉,௧‫ ؠ‬ቈ ‫ݎܽݒ‬൫ߨ௉,௧ ൯ + ‫ݎܽݒ‬൫ߨ௉,௧൯ + (ߪ + ߮)‫ݔ(ݎܽݒ‬௧ௐ ) +‫ݖ(ݎܽݒ‬௧ )቉,
2 2ߣ
2ߣ
4
and we utilize these equalities to calculate welfare costs.
Figure 4 depicts effects on welfare costs varying the relative risk aversion ߪ and theelasticity of substitution between goods produced in countries ‫ ܪ‬and ‫ߟ ܨ‬. We have

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already discussed some macroeconomic volatilities which comprise welfare costs. Thus,
we can understand effects on welfare losses varying ߪ and ߟ. Because there are nofluctuations in average output gap, CPI inflation in countries ‫ ܪ‬and ‫ ܨ‬and the TOT
deviation from efficient level regardless of ߪ and ߟ, there are no welfare losses
regardless of ߪ and ߟ under the LCP. However, there are no welfare costs only if
ߟ = 1 because there are no fluctuations in PPI inflation in countries ‫ ܪ‬and ‫ ܨ‬and the
TOT deviation from efficient level under the PCP. Except for ߟ = 1, there are somewelfare costs because there are fluctuations in PPI inflation in countries ‫ ܪ‬and ‫ ܨ‬andthe TOT deviation from efficient level under the PCP even if optimal monetary policy isconducted.
Because we focus on optimal choice of inflation rate as a target, we omit detailed
discussion on gains from policy cooperation in the text. As shown in Appendix A, there
are some gains from opricy cooperation under both the LCP and the PCP, as long asߟ ് 1. When ߟ = 1, there are no gains from policy cooperation because the TOT
externality disappears. 115.

Conclusion

We analyze optimal monetary policy under the LCP model by comparing with the PCPmodel. We have two main findings as follows. We insist that optimal monetary 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 (2005). 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 and that is shown by Devereux and Engel (2003). We can reconcile with
Devereux and Engel (2003) and we derive our policy implication complying Woodford’s(2001) motivation.
Our finding shed light on Mussa’s puzzle which focuses on the fact that co-movementof nominal exchange rate and real exchange rate along with Betts and Devereux (2000)
although they do not analyze optimal monetary policy. Because complete stabilization in
CPI inflation rate coincides with complete stabilization in 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 with the result on this paper is one of future research
agenda.

AppendixA

Gains from International Monetary Cooperation

Devereux and Engel (2003) assuming Armington form of consumption, which corresponds to ߟ = 1 in our model,
imply that there are no gains from policy cooperation under both the LCP and PCP. As mentioned, there are no gains
from policy cooperation when ߟ = 1 in our model. Hence, our result on policy cooperation is consistent with
Devereux and Engel (2003).

11

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We calculate social losses analitically. Under the PCP, gains from policy coopetarion is
given by:ଶ(ଵାఝ)୻మబ (ఙାఝ)మ
ఒఌ(ଵାఎఝ)మ
ቂ1െ
ቃ [var(ܽ௧ ) + var(ܽ௧‫]) כ‬,మ(ଵିఉ)ఠయ୻భȞ଴ ‫ߟ ؠ‬ଶ (1 + ߟ߮)ଶ െ (1 + ߟ߮) and Ȟଵ ‫( ؠ‬1 + ߮)ߟଶ + ߣߝ(1 + ߟ߮)ଶ > ߣߝ(1 + ߟ߮)ଶఒఌ(ଵାఎఝ)మ
ே஼ௐ
1െ
> 0 where ࣦ௉஼௉denotes welfare costs in the PCP model under
୻భ
ே஼ௐ
ௐ
ࣦ௉஼௉
െ ࣦ௉஼௉
=

with
hence

non-cooperative setting. Under non-cooperative setting, period loss in country ‫ ܪ‬isఌଵାఝ ଶ
given by ‫ܮ‬ே஼௉஼௉,௧ = ଶఒ ߨ௉,௧ + ଶ ‫ݔ‬௧ and its counterpart in country ‫ ܨ‬is givenே஼ௐௐanalogously. Note that Ȟ଴ = 0 when ߟ = 1. In that case, we have ࣦ௉஼௉െ ࣦ௉஼௉
=0which implies that there are no gains from policy cooperation.
Under the LCP, gains from policy coopetarion is given by:
ே஼ௐ
ௐ
ࣦ௅஼௉
െ ࣦ௅஼௉
=ଶ(ଵାఝ)(ఙାఝ)మ ୻మబ[var(ܽ௧ ) + var(ܽ௧‫]) כ‬,(ଵିఉ)ఠయమ

ே஼ௐwhere ࣦ௅஼௉denotes welfare costs in the LCP model under non-cooperative setting.
ఌ
Under non-cooperative setting, period loss in country ‫ ܪ‬is given by ‫ܮ‬ே஼௅஼௉,௧ = ଶఒ ߨ௧ +ଵାఝ‫ݔ‬௧ଶ and its counterpart in country ‫ ܨ‬is given analogously. When ߟ = 1, we have
ଶ
ே஼ௐ
ௐ
ࣦ௅஼௉
െ ࣦ௅஼௉
= 0 which implies that there are no gains from policy cooperation. Notethat we assume discretionary setting and ߩ = 0 for simplicity in this section.

References
Benigno, Pierpaolo, and Gianluca Benigno (2006), “Designing Targeting Rules forInternational Monetary Policy Cooperation,” Journal of Monetary Economics, 53, 473–
506.
Betts, Caroline and Michael B. Devereux (2000), “Exchange Rate Dynamics in a Model ofPricing-to-market,” Journal of International Economics, 50, 215-244.
Devereux, Michael B. and Charles Engel (2003), “Monetary Policy in the Open Economy
Revisited: Price Setting and Exchange Rate Flexibility”, Review of Economic Studies, 70,
765-84.
Gali, Jordi (2008), “Monetary Policy, Inflation and the Business Cycle,” Princeton
University Press, Princeton, NJ.
Gali, Jordi and Tommaso Monacelli (2005), “Monetary Policy and Exchange Rate
Volatility in a Small Open Economy,” Review of Economic Studies, 72, 707–734.
Mark, Nelson (2001), “International Macroeconomics and Finance,” Blackwell, Oxford,
UK.
Monacelli, Tomasso (2005), “Monetary Policy in a Low Pass-Through Environment,”
Journal of Money, Credit and Banking, 37, 1047-1066.
Obstfeld, Maurice, and Kenneth Rogoff (2000), “New Directions for Stochastic Open
Economy Models,” Journal of International Economics 50, 117–153.
Okano, Eiji (2007), “The Choice of the Inflation Rate as a Target in an Economy with

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Pricing-to-market,” Japan and the World Economy, 19, 48–67.
Woodford, Michael (2001), “Inflation Stabilization and Welfare,” NBER Working Paper,
No. 8071.
Woodford, Michael (2003), “Interest and Prices,” Princeton University Press, Princeton,
NJ.

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Table 1: Macroeconomic Volatility to One Percent Increase in Productivity
Variables
PricingSpecial (ߪ = ߟ = 1)
General (ߪ = 3, ߟ = 4.5)
ܽ௧
ܽ௧‫כ‬
ܽ௧
ܽ௧‫כ‬
LCP
0.00000.00000.00000.0000‫ݔ‬௧ௐPCP
0.00000.00000.00000.0000LCP
0.00000.00000.00000.0000ߨ௧ௐ
PCP
0.00000.00000.00000.0000‫ݔ‬௧
LCP
0.00000.00000.00280.0028PCP
0.00000.00133.2172e-004 3.2172e-004
‫כ‬
‫ݔ‬௧
LCP
0.00000.00000.00280.0028PCP
0.00000.00003.2172e-004 3.2172e-004
ߨ௉,௧
LCP
0.00510.00510.00110.0011
PCP
0.00000.00001.0486e-004 1.0486e-004
‫כ‬
ߨ௉,௧
LCP
0.00510.00510.00110.0011
PCP
0.00000.00001.0486e-004 1.0486e-004
ߨ௧LCP
0.00000.00000.00000.0000PCP
0.00510.00510.00120.0012
ߨ௧‫כ‬
LCP
0.00000.00000.00000.0000PCP
0.00510.00510.00120.0012‫ݎ‬Ƹ௧
LCP
0.00110.00110.00230.0023PCP
0.0021
1.1460e-006
0.00240.0020
‫כ‬
‫ݎ‬Ƹ௧
LCP
0.00110.00110.00230.0023PCP
1.1460e-006
0.00210.00200.0024‫ݕ‬௧
LCP
0.02290.00000.01910.0038PCP
0.02290.00000.02170.0064‫ݕ‬௧‫כ‬LCP
0.00000.02290.00380.0191
PCP
0.00000.02290.00640.0217‫ݏ‬௧
LCP
0.02290.02290.00510.0051PCP
0.02290.02290.00630.0063‫ݖ‬௧
LCP
0.00000.00000.00000.0000PCP
0.00000.00000.00120.0012
݁௧
LCP
0.00000.00000.00000.0000PCP
0.02290.02290.00570.0057

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Figure1: IRFs to Productivity in Country ‫ ܪ‬in the Special Case (ߪ = ߟ = 1)

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Figure2: IRFs to Productivity in Country ‫ ܪ‬in the General Case (ߪ = 3, ߟ = 4.5)

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Figure 3: Effects on Macroeconomic Volatilities of Varying Relative Risk Aversion ߪ
and Elasticity of Substitution between Goods Produced in Countries ‫ ܪ‬and ‫ߟ ܨ‬

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Figure 4: Effects on Welfare Costs of Varying Relative Risk Aversion ߪ and Elasticity
of Substitution between Goods Produced in Countries ‫ ܪ‬and ‫ߟ ܨ‬

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