Research InstituteDiscussion Paper Series№17
Title：The role of wage flexibility and optimal monetary policy
in a two-country modelDaisuke Ida /Mitsuhiro Okano
ida-dai@andrew.ac.jp

2021 年 4 月

http://www.andrew.ac.jp/soken/

The role of wage flexibility and optimal monetary policy
in a two-country model∗
Daisuke Ida†and Mitsuhiro Okano‡
April 21, 2021

Abstract
This paper examines wage flexibility and optimal monetary policy in a two-country New
Keynesian (NK) model. In contrast to the two-country model with no nominal wage rigidity,
we address the fact that in a two-country model with nominal wage rigidity, the dynamics
of the terms of trade significantly characterize the key mechanism of wage flexibility. In
particular, when nominal wages in both countries are perfectly flexible, the welfare gains
from a commitment policy are the greatest. When nominal wages in the foreign country
are stickier; however, the gains from a commitment policy are predominantly reduced. This
result is in contrast to the results obtained in previous studies.

JEL codes: E52; E58; F41
Keywords: Optimal monetary policy; Wage flexibility; Policy coordination; Commitment; Discretion∗

This paper was supported by JSPS KAKENHI Grant Numbers JP16H03618 and JP20K01784. All remaining

errors are our own.
†

Faculty of Economics, Momoyama Gakuin University, 1-1, Manabino, Izumi, Osaka 594-1198, Japan. Tel.:+81-725-54-3131. E-mail: ida-dai@andrew.ac.jp‡

Faculty of Economics, Osaka Gakuin University, 2-36-1 Kishibeminami, Suita-City, Osaka 564-8511, Japan.

Tel.: +81-6-6381-8434. E-mail: okano@ogu.ac.jp

1

Introduction

The purpose of this paper is to examine the effects of nominal wages and prices on optimal monetary policy in a two-country new Keynesian (NK) model. The importance of wage flexibility
and employment fluctuations have been discussed in many previous studies. The importanceof the impact of wage flexibility on monetary policy has also been pointed out by (Christiano,
Eichenbaum and Evans, 2005, Smets and Wouters, 2003). In addition, as noted by Christiano,
Eichenbaum and Evans (1999), the structural model that considers sticky prices and flexible
wages might be inconsistent with the fact that monetary tightening causes too sharp a decline
in real wages.1 The existence of nominal wage rigidity causes fluctuations in employment,
which, in turn, reduces social welfare. In this case, in addition to standard policy objectives,
optimal monetary policy is needed to stabilize fluctuations in wage inflation (Erceg, Hendersonand Levin, 2000, Galı́, 2013). These studies underscore the importance of considering the roleof wage rigidity in structural models. In addition, Palomino, Rodrı́guez and Sebastian (2020)
pointed out the effect of the lockdown and social distancing measures prompted by coronavirus
disease 2019 (COVID-19)-induced shocks on wage inequality in Europe.
The spillover mechanism of wage stickiness on international monetary policy has not been
sufficiently discussed, however. For example, one may focus on the fact that changes in UnitedStates (US) employment levels have led to changes in international asset prices. Indeed, previous studies have pointed out that changes in US employment affect asset prices, such as stock
prices and exchange rates (Ederington, Guan and Yang, 2019). Furthermore, since the 1980s,
income inequality has emerged in the US, and unemployment characterized by wage rigidity
has increased in Europe (Cahuc, Carcillo and Zylberberg, 2014). In fact, as noted by Galı́ andMonacelli (2016), a decline in the home country’s wage leads to a depreciation in the terms oftrade, which prompts increases in its output and employment. Recently, Iwasaki, Muto andShintani (2021) explored an international comparison of the role of wage rigidity by using a
non-linear dynamic stochastic general equilibrium model. In this study, we consider whether
the mechanism suggested by Galı́ and Monacelli (2016) still holds for the case of a two-country
economy model.
Why do policymakers and the private sector in countries such as the Eurozone and Japanpay so much attention to fluctuations in US employment? Needless to say, it is because policy1

Colciago (2011) considered the role of wage flexibility in an economy with liquidity constraints.

1

makers in other countries may be concerned about recessions caused by the exporting of a large
country’s unemployment, as long as there are international links between financial markets (Ederington et al., 2019). Several studies have explored the role of wage flexibility in the Eurozone(Branten, Lamo and Room, 2018, Fabiani, Kwapil, Rõõm, Galuscak and Lamo, 2010). In thesecases, there may be room for both home and foreign policymakers to consider the possibilityof international monetary policy coordination that might maximize global welfare losses byreducing the volatility in their respective employment levels. While this is an important topic,
the international spillover mechanism, and its implications for monetary policy have not beenadequately discussed in terms of optimal monetary policy.
In this paper, we construct a two-country NK model to examine the impact of wage flexibility on the international dimension of optimal monetary policy. Specifically, we construct a
two-country NK model facing both nominal wage and price rigidity. We simply incorporate
the idea suggested by Erceg et al. (2000) into the two-country NK model developed by Clarida,
Galı́ and Gertler (2002). Specifically, we construct a two-country NK model in which nominal
wage and price rigidity are asymmetric between the two countries. As pointed out by Rhee andTurdaliev (2013), the terms of trade depend on the real wage gap in an open economy. In ourmodel, the key understanding is to consider how the real wage gap plays a significant role in
the two-country model. This is because the terms of trade change in response to fluctuations
in domestic and foreign output in the two-country model. Finally, we derive the loss functionof the central bank by computing a quadratic approximation of the household utility function
when international policy coordination between the home and foreign central banks is considered. This derivation is a natural extension of an NK model that incorporates nominal wage
rigidity.
The main findings of our paper are summarized as follows. In contrast to the two-country
model with no nominal wage rigidity, we address the fact that the dynamics of the terms of tradesignificantly characterize the key mechanism underlying wage flexibility. Given wage flexibility
in the home country, we show that changes in the degree of foreign wage flexibility substantially
impact the worldwide welfare losses and gains from commitment policy. Specifically, whennominal wages in both countries are perfectly flexible, the welfare gains from a commitment
policy are the greatest. However, when nominal wages in the foreign country are stickier, thegains from commitment are predominantly reduced. This result is robust to any changes in2

several key parameters that play an important role in the two-country NK model. Therefore,
we emphasize that some of the results obtained in this paper contrast with those obtained inprevious studies.
The structure of this paper is as follows. Section 2 briefly reviews the related literature, andSection 3 provides a description of the two-country NK model in which nominal wage and pricerigidity coexist. Section 4 discusses optimal monetary policy in the model. Section 5 reports themain results of the paper. In Section 6, we explore the gains from a commitment policy in ourmodel. Section 7 briefly concludes. Appendix A provides a derivation of the macroeconomic
variables under flexible prices and wages in our two-country model. In the online appendix,
we drive the central bank’s loss function by calculating the second-order approximation of thehousehold utility function in our model.

2

Related literature

The objective of this section is to clarify previous papers related to this study. First, we
briefly review the effect of nominal wage flexibility on optimal monetary policy. Erceg et al.
(2000) introduced nominal wage rigidity into a standard NK model and examined its impact on
optimal monetary policy. Their study demonstrated that a stabilizing term in wage inflation
due to staggered changes in nominal wages added to the central bank’s policy objective as
well as stabilized inflation and output. Galı́ (2011) addressed the role of wage inflation in
explaining actual wage inflation dynamics in the US economy. Galı́ (2013) examined the impact
of nominal wage flexibility on optimal monetary policy in an NK model with nominal priceand wage rigidity based on the observation that the wage Phillips curve could explain wage
dynamics in the US.2 He showed that higher wage flexibility does not necessarily lead to higher
social welfare.3 Ascari, Colciago and Rossi (2017) examined the role of liquidity constraints
in an NK model with nominal price and wage rigidity. While these studies provide important2

Several studies have focused on the role of wage flexibility in examining monetary policy conducted under

the zero lower bound (ZLB) on nominal interest rates (Glover, 2019, Shen and Yang, 2018). The topic of the
ZLB is very important, but that investigation is beyond the scope of this paper. How does the ZLB affect theproperties of optimal monetary policy in a two-country NK model with nominal wage rigidity? We would liketo examine this issue as a future work.
3

This argument is based on the results of the calculation of welfare losses under several parameterizations of

nominal wage rigidity in the case of the Taylor rule.

3

contributions to the study of optimal monetary policy, we point out the limitation that they
all focused on wage flexibility and optimal monetary policy in a closed economy.
Second, we focus on a literature review on the effect of wage flexibility on international aspects of optimal monetary policy. How does wage flexibility affect the international dimensionof monetary policy? Several studies have emphasized the international dimension of optimalmonetary policy, but the role of wage flexibility has not yet been incorporated into a standard
two-country NK model (Clarida, Galı́ and Gertler, 2001, Clarida et al., 2002, De Paoli, 2009b,
Engel, 2011). Groll and Monacelli (2020) addressed the role of terms of trade dynamics undera commitment policy in a two-country model, but their insights were based on a fully flexible
model of nominal wages. While Monacelli (2003) showed that welfare losses due to a discretionary policy were substantially greater than those due to a commitment policy, their results
were based on those suggested for a small-open NK economy with flexible nominal wages.
Third, we address the contribution of this paper to previous studies. What role has been
considered by previous studies for wage flexibility in terms of an open economy? Several studies
have focused on wage flexibility in small-open NK models. Our study is related to Galı́ andMonacelli (2016), who examined the welfare gain from wage flexibility in a small-open NK
model. They showed that in contrast to conventional wisdom, increased wage flexibility often
worsens social welfare in a currency union. Our study is also related to Campolmi (2014)
and Rhee and Turdaliev (2013), who argued the important role of consumer price index (CPI)
inflation targeting when nominal wage rigidity was introduced in a small-open NK model. They
derived a central bank’s loss function for a small-open economy, but calculated it assumingthat both the intertemporal substitution of consumption and substitution between home andforeign goods was unity. In contrast to their work, we derive the central bank’s loss function
without assuming a unitary elasticity of intertemporal substitution of consumption.4 A key
to understanding this argument is to observe the impact of real wage changes on the terms oftrade, since a change in the terms of trade affects both producer price index (PPI) and CPI
inflation.
While these studies addressed the role of wage flexibility in a small-open NK model or a
4

The Cobb-Douglas consumption basket is assumed in our derivation of the central bank’s loss function. See

Pappa (2004) and Groll and Monacelli (2020) for a detailed derivation under a constant elasticity of substitution(CES)-type consumption basket.

4currency union, they did not consider the interaction of wage flexibility through the termsof trade in a two-country model with an exchange rate float. Considering the role of wage
flexibility in the international dimension of monetary policy is supported by the empirical facts
mentioned in the Introduction. Our paper is deeply related to Groll and Monacelli (2020), whoargued for the importance of endogenous inertia in the terms of trade in a two-country NK
model with wage flexibility. In their model, the adoption of a currency union created inherent
benefits for the social welfare of both countries. Ida (2021) considered the role of liquidity
constraints in a currency union with nominal wage rigidity. However, their model did notconsider the role of wage flexibility in a two-country NK model with an exchange rate float.
Our study addresses the impact of endogenous inertia in the terms of trade through the real
wage rigidity caused by staggered nominal wages. To the best of our knowledge, none of theprevious studies emphasized the role of wage flexibility in a two-country NK framework.
Although our model is a natural extension of the two-country model developed by Clarida
et al. (2002) to the case of nominal wage rigidity, we discuss the contribution of this paperin terms of several important results that were not obtained in previous studies. First, ourstudy demonstrates that foreign wage flexibility significantly impacts worldwide welfare losses,
depending on the degree of flexibility in the home country’s wages. For example, while Galı́ andMonacelli (2016) showed how the degree of wage flexibility affected welfare loss in a currency
union, this study focuses on how changes in wage flexibility in both countries affect worldwide
welfare loss in a two-country model with an exchange rate float. More concretely, while they
pointed out that wage flexibility negatively affects social welfare, our study shows that it does
not always do so, from the perspective of a two-country NK model. Second, we demonstrate
the welfare gains from commitment policy in a two-country NK model. Clarida et al. (2001),
Ida and Okano (2020), and Monacelli (2003) argued that commitment is more beneficial than
discretion in a small-open economy. While these studies focused on the gains from commitment,
they did not consider the role of wage flexibility in accounting for the delegation problem undera discretionary policy. Our model illustrates the substantial effect of wage flexibility on theperformance of commitment policy in both countries in a two-country model.

5

3

Model

We consider a two-country NK model with nominal wage rigidity. Consider an economy with
two large symmetric countries: home and foreign. The size of the economy for home andforeign is 1 − υ and υ, respectively. There are two production sectors in each country: a final
goods sector that is characterized by perfect competition, and an intermediate goods sector inwhich firms face monopolistic competition and Calvo (1983)-type nominal price rigidity. We
also assume that the degree of price stickiness differs in each country.
We assume that there are complete markets in both countries, and that only final goods
are traded. The number of final goods producers is equal to the number of households in eachcountry. Finally, unless otherwise noted, analogous equations hold for the foreign country. In
deriving several equations, we express the log-deviations from the steady state using lower caseletters. First, H represents the value of the steady state, and Htn is the value of the efficiency
level. We define ht = log(Ht /H) as the deviation of Ht from the steady state. Note that thevariables for the foreign country are denoted by an asterisk.
Except for the introduction of nominal wage rigidity, the model we adopt is based on thestandard two-country NK model developed by Clarida et al. (2002). Readers familiar with thetwo-country NK model can skip to Section 4.

3.1

Preferences

Preferences for consumption in the home country are given as follows:1−υ υ
Ct ≡ ΥCH,t
CF,t ,(1)

where Υ ≡ 1/ (1 − υ)1−υ υ υ , CH,t is the consumption of domestic goods, and CF,t is the consumption of foreign goods.
The household’s cost minimization yields the following:
PH,t CH,t = (1 − υ)Pt Ct ,(2)

PF,t CF,t = υPt Ct ,(3)

where the price index in the home country is given by the following:1−υ υ
Pt ≡ PH,t
PF,t = PH,t Stυ ,6(4)

where PH,t is the price of domestic goods, and PF,t is the price of foreign goods. St represents
the terms of trade, which are given as follows:
St ≡

3.2

PF,t
.
PH,t

(5)

Households

The intertemporal utility of an infinitely lived representative household is as follows:
 1−σ

Z 1
∞
X
1
1+ϕ
t Ct
Nt (j)dj ,β−E0
1−σ 1+ϕ 0

(6)

t=0

where β denotes the discount factor, Ct is consumption, and Nt (j) is the household’s type j’slabor supply.5 Parameter σ denotes the relative risk aversion coefficient for consumption, andϕ is the inverse of the Frisch elasticity of the labor supply.
The representative household maximizes the above utility function subject to the following
budget constraint as follows:
Z 1

Z 1
PH,t (i)CH,t (i)di + PF,t CF,t + Et {Qt,t+1 Dt+1 } ≤0

Wt (j)Nt (j)dj + Dt Γt − Tt

(7)0

where Dt is nominal bonds held for one period, Wt (j) denotes nominal wages, and Γt denotes
the nominal wage and dividend, respectively, earned from domestic firms. Tt denotes a lumpsum tax. We assume that the households located in each country can access state-contingent
bonds traded in a complete market both domestically and internationally and introduce thefollowing stochastic discount factor as follows:
Et (Qt,t+1 ) =1,
1 + it(8)where Qt,t+1 denotes a stochastic discount factor, and it is the nominal interest rate.
The first-order condition of the household’s utility maximization problem yields the following familiar consumption Euler equation as follows:
(
 
)
Pt
Ct 1/σ1 = β (1 + it ) Et
.
Ct+1
Pt+1

(9)

where it is the nominal interest rate.
5

We assume that there is a variety of labor j ∈ [0, 1]; we can also assume that each household has j ∈ [0, 1]

members of the labor force, as in Galı́ and Monacelli (2016).

7

Next, we consider a risk-sharing condition between countries. The Euler equation for foreign
consumption denominated in the domestic currency is as follows:
(
 
)
Pt /EtCt∗ 1/σ∗.
1 = β (1 + it ) Et∗
Ct+1
Pt+1 /Et+1(10)

where i∗t denotes the foreign nominal interest rate and Et is the nominal exchange rate. By
assuming that there exist state-contingent bonds that allow both domestic and foreign households to trade internationally, combining Equation (10) with the Euler equation for domestic
consumption and the definition of the real exchange rate, the real exchange rate becomes thefollowing:Ctσ = ϑCt∗σ

Et Pt∗
Pt


.

(11)

Since our model is based on the assumption of producer currency pricing, the law of one priceholds. 6
Et Pt∗= 1.
Pt
Therefore, taking ϑ = 1 without any loss of generality, as shown in Clarida et al. (2002),
international consumption risk-sharing leads to7
Ct = Ct∗ ,(12)

for all t.

3.3

Wage determination

In each country, households delegate the role of wage determination to a labor union. Following
Erceg et al. (2000), the wage setting is subject to Calvo-type staggered wage contracts. Thus,a fraction of 1 − θw can change nominal wages in its union, whereas the remaining fractionof θw cannot do so. Under this setting, the labor union for each country solves the following
6

When we assume local currency pricing, the following relationship does not hold. See Engel (2011) for a

detailed discussion of this issue.
7

In our model, we do not consider the presence of a consumption home bias. As shown in Pappa (2004)

and Engel (2011), the real exchange rate adjusts the difference between home and foreign consumption in thepresence of a home bias.

8maximization problem:
Et

∞
X


(βθw )k U Ct+k|t , Nt+k|t ,

(13)

k=0

subject to
Wto −w
Nt+k ,
Nt+k|t =Nt+k
Pt+k Ct+k + Et Qt+k,t+k+1 Dt+k|t = Wto Nt+k|t + Dt − Tt + Γt ,
(14)

where Ct+k|t and Ct+k|t denote consumption and labor, respectively, in period t + k for a unionthat last reset its nominal wage in period t. Equation (14) denotes the demand for labor inperiod t for t + k periods ahead, and w is the elasticity of substitution for individual labordemand. Wto denotes the optimal nominal wage that the labor union chooses in the current
period.
The first-order condition of this maximization problem yields the following:


 o
∞
X
Wt
k
w
− M M RSt+k|t= 0,(βθw ) Et Nt+k|t Uc,t+k
Pt+k

(15)

k=0

σ Nϕ
where M ≡ ww−1 and M RSt+1|t ≡ Ct+kt+k|t .

Log-linearizing this equation around the steady state yields the following:
wto = µw + (1 − βθw )∞X(βθw )k Et mrst+k|t + pt+k .

(16)

k=0

Z 1
Wt ≡

1−w

Wt (j)

1/(1−w )
dj

.

(17)

0

Log-linearization of the wage index is given as follows:
wt = θw wt−1 + (1 − θw ) wto ,(18)

where wt ≡ log Wt . Substituting Equation (18) into Equation (16), we obtain the following
wage new Keynesian Phillips curve (WNKPC) as follows:
 ww
πtw = βEt πt+1
− λw (µwt − µ ),

(19)

where πtw (≡ wt − wt−1 ) denotes wage inflation, and µw
t (≡ wt − pt − mrst ) denotes the average
wage markup, which is related to the marginal rate of substitution between consumption andthe labor supply. In addition,
λw =(1 − θw ) (1 − βθw )> 0.
θw (1 + w ϕ)
9

3.4

Firms

Each country has two production sectors. The first is the final goods sector, which produces
final goods using intermediate goods and is characterized by perfect competition. The second
is the intermediate goods sector, in which firms face monopolistic competition and Calvo-type
nominal price rigidity.
Final goods firms
The final goods sector is perfectly competitive and producers use inputs that are produced in
the intermediate goods sector. Specifically, final goods are produced according to the following
CES aggregate:
Z 1
Yt =

Yt (i)

p,t −1
p,t  p,t−1
p,t

di

,

(20)

0

where Yt is aggregate output, Yt (i) is the demand for intermediate goods produced by firm i,and p,t is the elasticity of substitution, which is time-varying, as assumed by Steinsson (2003).8
Note that both variables are normalized by 1 − ν.
Under the CES aggregate, the demand function is given by:PH,t (i) −p,tYt (i) =Yt ,PH,t
and the domestic price level is defined as follows: 1−1Z 1p,t1−p,tdiPH,t =
PH,t (i)
,

(21)

(22)

0

where PH,t (i) is the price for intermediate goods produced by firm i. Note that these variables
are also normalized by 1 − ν.
Intermediate goods firms
The goods sector is characterized by monopolistic competition, and each firm produces a differentiated intermediate good. Firm i’s production function is given by the following:
Yt (i) = At Nt (i)1−α ,8(23)Clarida et al. (2002) presumed the time-varying wage markup to introduce the exogenous cost-push shock.

In this study, instead of introducing the wage markup, a price markup derived from a time-varying elasticity
of substitution p,t is introduced to examine the effect of a cost-push shock on optimal monetary policy. See
Clarida et al. (2002) for a detailed discussion of the presence of a wage markup shock.

10

where α represents the parameter that indicates diminishing returns to scale in the production
function. At denotes an aggregate productivity disturbance, which follows an AR (1) process
given by log At = ρa log At−1 + at with 0 ≤ ρa < 1, where at is an independent and identically
distributed (i.i.d.) shock with constant variance σa2 .
In addition, firm i’s employment is characterized by the following CES index:
Z 1
Nt (i) ≡Nt (i, j)

w −1
w

  w−1
w

dj

.

(24)

0

The demand for labor is given as follows:
Nt (i, j) =Wt (j)Wt−w
Nt (i),

(25)

and
Z 1
Nt (j) =
Nt (i, j) dj =0

where Nt ≡Wt (j)
Wt

−w
Nt ,

(26)

R1

0 Nt (i) di.

Following Calvo (1983), we assume that price rigidity is present in the goods sector. The
following explanation focuses on the home country, whereas we can consider the case for theforeign country. Thus, a fraction 1 − θp of all firms adjusts their price, while the remaining
fraction of firms θp do not. We now consider the firms that can adjust their price. Whenrevising their prices, these firms consider the uncertainty related to when they will next be able
to adjust prices. In this case, the firm’s optimization problem for the home country is given as
follows:Et∞
Xoθk Qt,t+k PH,t
Yt+k|t − Ψt+k Yt+k|t



,

k=0

subject toYt+k|t =oPH,tPH,t+k−p,t+k
Yt+k ,

where poH,t denotes optimal prices.
The first-order condition of this optimization problem yields the following:
∞
X



o
θpk Et Qt,t+k Yt+k|t PH,t
− Mp ψt+k|t = 0,

k=0

11

(27)

where Mp (≡ p /(p − 1)) denotes the price markup. The variable Ψt denotes the real marginal
cost, which is given as follows:
Ψt+k|t ≡Wt+k (1 + τt )−α .
(1 − α) At+k Nt+k|t(28)

We also defined average nominal marginal cost as follows:Ψt ≡Wt (1 + τt ).
(1 − α) At Nt−α(29)Log-linearizing Equation (27) leads to the following optimal price dynamics:
poH,t = µp + (1 − βθp )∞X(βθp )k Et ψt+k|t ,

(30)

k=0

where µp ≡ log Mp and ψt+k|t (≡ log Ψt+k|t ) denote the k-periods-ahead real marginal costin period t. The log-linearization of the aggregate price index evolves based on the following
equation.
pH,t = θp pH,t−1 + (1 − θp ) poH,t ,(31)

o . Substituting Equation (31) into Equation (30) and implementing sevwhere poH,t ≡ log PH,teral mathematical manipulations, we obtain the following price new Keynesian Phillips curve(PNKPC) expressed by the real marginal cost.
onpp− λp (µpt − µp ) + ut ,= βEt πH,t+1
πH,t

(32)

p
(≡ pH,t − pH,t−1 ) denotes the PPI inflation rate, pH,t = log PH,t . ut is a costwhere πH,tpush shock associated with the time-varying elasticity of substitution for individual goods. µpt(≡ pt − ψt ) represents the average price markup, which is related to the real marginal cost. In
addition,
λp ≡

3.5

(1 − θp ) (1 − βθp )1−α.
θp1 − α + αpEquilibrium

We now describe the equilibrium conditions in an open economy. The equilibrium conditions
for the goods market are given as follows:∗(1 − υ) Yt = (1 − υ) CH,t + υCH,t
,

(33)

∗
υYt∗ = (1 − υ) CF,t + υCF,t
.

(34)

12

Using these equations and Equations (2) and (3), we show that the trade balance within eachcountry is zero.
PH,t Yt = Pt Ct ,∗PF,tYt∗ = Pt∗ Ct∗ .

In this case, as shown in Clarida et al. (2002), home output can be rewritten as follows:
Yt = Ct Stυ ,From the zero trade balance in each country, we obtain the following:Ct∗ = Yt1−υ (Yt∗ )υ = Ct .

(35)

Equilibrium with sticky prices and wages
As noted earlier, we express log-deviations from the steady state with lower case letters. In
this section, we focus on the equations expressed by a gap term, which is denoted by lower caseletters with a tilde. The gap term implies the log-deviation from the equilibrium in which bothprices and wages are flexible. While we focus on the equilibrium with sticky prices and wages
in the main text, Appendix A provides the structural equations with flexible prices and wages.
First, consider the effect of wage flexibility on an open economy PNKPC. The open economy
PNKPC is again given by the following:
n
o
p
p
πH,t
= βEt πH,t+1− λp (µpt − µp ) + ut ,where the average price markup associated with the real marginal cost is given by the following:µpt − µt = −αỹt − ω̃t − υs̃t ,
1−α

(36)

p
πH,t
(≡ pH,t − pH,t−1 ) denotes the PPI inflation rate. ỹt is the output gap, w̃t is the real

wage gap, and s̃t is the terms of trade gap. Both the real wage gap and the terms of trade
gap negatively affect the real marginal cost, resulting in them being positively related to priceinflation. This mechanism is similar to one argued in Clarida et al. (2002). To do this, werewrite the PNKPC in the form of the gap variable. Using the definition of consumptionc̃t = (1 − υ)ỹt + υ ỹt∗ , the PNKPC expressed by the gap term is derived as follows:
n
o
λp α
p
p
πH,t
= βEt πH,t+1+
ỹt + λp ω̃t + λp υs̃t + ut ,
1−α
13

(37)

Except for the presence of the real wage, this new Keynesian Phillips curve (NKPC) is basedon the derivation of the standard two-country NK model (Clarida et al., 2002). Similarly, we
can obtain the same relationship for the foreign country as follows:noλ∗p α ∗
p,∗
p,∗
πF,t
= βEt πF,t+1+
ỹ + λ∗p ω̃t∗ − λ∗p υs̃t + u∗t ,
1−α t

(38)

p,∗
where πF,t(≡ p∗F,t −p∗F,t−1 ) denotes PPI inflation in the foreign country, and u∗t is the exogenous

cost-push shock in the foreign country.
In addition, we rewrite the WNKPC in terms of the gap variable as follows:
 w
λw ϕỹt + λw c̃t − λw ω̃t .
πtw = βEt πt+1
+
1−α

(39)

As in the closed economy model, wage inflation is affected by changes in the consumption
gap. However, our model addresses the impact of the terms of trade on wage inflation ina two-country model through international risk-sharing. Specifically, as shown in Equation(35), changes in the foreign output gap positively impact home consumption through international consumption risk-sharing. Accordingly, in contrast to Campolmi (2014) and Rhee
and Turdaliev (2013), home wage inflation increases in response to the increase in the foreign
output gap. However, the real wage gap has a negative impact on wage inflation. Therefore,
whether wage inflation rises or not depends on the change in the foreign output gap relative
to the change in the real wage gap. The above mechanism is important for understanding theanalysis in this paper. Similarly, for the foreign country, we obtain the following:
 w,∗
λ∗ ϕ
πtw,∗ = βEt πt+1
+ w ỹt∗ + λ∗w c̃∗t − λ∗w ω̃t∗ .
1−α

(40)

The real wage plays a significant role in examining optimal monetary policy in our model.
The dynamic behavior of the real wage in terms of the home country is characterized by thefollowing law of motion:ω̃t = ω̃t−1 + πtw − πt − ν∆s̃t − ∆ωtn .

(41)

In this paper, we emphasize the role of nominal wage stickiness in the two-country model.
Equation (41) shows that real wages are characterized by inertial changes in the terms oftrade. In contrast to a small-open economy, the foreign output gap endogenously affects the
dynamics of the terms of trade, leading to inertial changes in the real wage. Since changes in
the real wage have a direct impact on the NKPCs for both prices and wages, an inertial change
14

in the terms of trade also has the significant impact of an inertial change in real wages on boththe price and wage NKPCs. Thus, our model focuses on the interaction between the effects ofreal wages and the NKPCs of prices and wages through changes in the terms of trade. This
channel is in contrast to the cases in Campolmi (2014), Galı́ and Monacelli (2016), and Rhee
and Turdaliev (2013).

4

Optimal monetary policy in a two-country NK model withnominal wage rigidity

This section examines how the home and foreign central banks implement optimal monetary
policy in a two-country economy with nominal price and wage rigidity in both countries. In
Section 4.1, we derive the central bank’s loss function by implementing the second-order approximation of the household utility function. Section 4.2 explains the optimal monetary policy
problem under policy coordination.

4.1

The central bank’s loss function

We must derive a well-defined loss function with a micro-foundation to investigate optimalmonetary policy in an NK model. Rotemberg and Woodford (1997) and Woodford (2003)
showed that the second-order approximation of the household utility function corresponded tothe central bank’s loss function. In a two-country framework, Clarida et al. (2002) and Engel(2011) derived the central bank’s loss function by using a second-order approximation of theutility function.
To derive a well-defined loss function, we must perform a Taylor expansion around an efficient steady state. In this paper, an efficient steady state that removes the following distortions
is necessary for the derivation of the central bank’s loss function.
In our model, there is a distortion of the markups from monopolistic competition in intermediate goods firms and a distortion of the wage markups in the labor unions of each country.
As shown in the online appendix, the optimal subsidies that attain an efficient steady state are
given as follows:τ = τ∗ = 1 −151.
κµp µw

In deriving these subsidies, it is assumed that the government of each country will calculate theoptimal subsidy. These conditions are the same as in Clarida et al. (2002), who constructed
a two-country model with a monopolistically competitive labor market under flexible nominal
wages.
We derive the central bank’s loss function by implementing the second-order approximationof the following household utility function:∞X
β t [Wt − W̄ ]
t=0





∞
p 2 w (1 + w ϕ) w 2
uc C X tα+ϕ2≈−
β (1 − ν) σ +− ν(σ − 1) xt +
π +
(πt )
2
1−α
λp Θ t
λw
t=0



p
α+ϕ
w (1 + w ϕ) w,∗ 2
∗ 2
∗ 2
+ν σ+
− (1 − ν)(σ − 1) (xt ) + ∗ (πt ) +
(πt )
1−α
λp Θ
λ∗w

∗
− 2ν(1 − ν)(1 − σ)xt xt + t.i.p. + O(||ξ||3 ),
(42)
where
Θ=

p
(1 − θp )(1 − θp β) ∗ (1 − θp∗ )(1 − θp∗ β), λp =, λp =,
1 − α + αp
θp
θp∗

λw =

∗ )(1 − θ ∗ β)
(1 − θw )(1 − θw β) ∗(1 − θw
w, λw =.
∗ (1 +  ϕ)
θw (1 + w ϕ)θw
w

This loss function is expressed in terms of a consumption equivalence unit. The term t.i.p
includes terms that are independent of monetary policy, and O(||ξ||3 ) indicates terms of thirdor higher orders. The appendix B provides a detailed derivation of Equation (42). As in thestandard two-country NK model, our loss function contains several standard policy objectives.
Inflation and output gap objectives are assigned to both home and the foreign country. The
first objective implies the presence of price dispersion caused by nominal price rigidity. In
addition, the loss function for each country contains the wage inflation stabilization associatedwith wage stickiness. This objective is associated with the presence of the wage dispersion
caused by staggered wage contracts (Erceg et al., 2000).
Not surprisingly, one may think that this shape of the loss function is easily derived. However, in a small-open economy model, the derivation of the central bank’s loss function is quite
restrictive in that we require the assumption that both the intertemporal substitution of consumption and the elasticity of substitution between home and foreign goods must be unity(Galı́
and Monacelli, 2005). 9 In fact, as noted earlier, Campolmi (2014) and Rhee and Turdaliev
9

De Paoli (2009a) derived the central bank’s loss function under more generalized parameterizations. In this16

(2013) obtained the central bank’s loss function under the assumption of both σ = 1 anda Cobb-Douglas consumption basket. In contrast to their studies, we can derive the central
bank’s loss function in a two-country NK model with nominal wage rigidity even in the case ofσ 6= 1.10
In the following sections, we use this loss function to derive the optimal monetary policy
and evaluate the performance of a commitment policy.

4.2

Optimal monetary policy

This paper first considers optimal commitment policy under policy coordination. Since thepurpose of this paper is to simply evaluate the impact of nominal wage rigidity on optimalmonetary policy in a two-country NK model, we focus on the commitment solution under policy
coordination. Therefore, the issue of policy coordination (i.e., whether the home and foreign
central banks can jointly maximize global welfare without reneging on their commitment topolicy coordination) is not the main scope of this paper. To examine this issue, it is necessary
to explicitly derive the loss function of each central bank under a non-cooperative solution
when calculating the optimal monetary policy in a linear-quadratic (LQ) framework. However,in this case, the LQ problem is generally more complicated because the derivation of the loss
function requires a second-order approximation of the NKPC and the terms of trade.11
Under policy coordination, the benevolent central banks choose the interest rate path byintertemporally solving the minimization problem of the central bank’s loss function. We focus on the commitment solution as a benchmark case and, in the next subsection, we explore
a comparison of the commitment and discretionary solutions. Under a commitment policy,
central banks can commit to future monetary policy stances in the current period. This paderivation of the central bank’s loss function, we require the second-order approximation of the NKPC consideredin Benigno and Woodford (2005).
10

In this study, we focus only on the case of policy coordination. In this case, the loss function of the central

bank can be derived analytically by computing the second-order approximation of the household utility function.
However, when we consider both the cooperative and non-cooperative policy cases, it becomes more difficult toderive a well-defined loss function in our model. See Fujiwara and Wang (2017) for a detailed discussion of thecentral bank’s loss function in a two-country NK model.
11

If we depart from the problem based on the LQ problem, our model may be able to handle this problem.

That is, in the case of a non-linear, two-country NK model, we can use the method developed in Bodenstein,
Guerrieri and LaBriola (2019) to solve for both cooperative and non-cooperative policies.

17per regards the commitment policy as optimal from a timeless perspective, as suggested by
Woodford (2003).12 Since central banks that commit to an optimal policy can introduce policy inertia into the economy, this policy inertia enables central banks to influence the private
sector’s expectations. As pointed out in Pappa (2004), a commitment policy can eliminate thedeflationary bias induced by a discretionary policy, which is the source of the additional gains
from commitment that are associated with an open economy.
Combining the first-order conditions of the central bank’s optimization problem with thestructural equations, we obtain the following:
A0 Xt = A1 Xt−1 + B1 Rt + Ut ,

(43)

with
0

Xt = [X1t+1 Et X2t+1 ] ,0

X1t = [At A∗t ut u∗t φ1t φ2t φ3t φ4t ] ,0

X2t = [w̃t w̃t∗ πtw πtw,∗ πt πt∗ xt x∗t φ5t φ6t ] ,0

Rt = [rt rt∗ ] .
A0 , A1 , and B1 are coefficient matrices constructed by deep parameters. X1t denotes the vector
for the predetermined and state variables, X2t denotes the same for the jump variables, Rt is
the vector for the central bank’s instrumental variables, and Ut denotes the vector for structural
shocks. φ1t and φ2t denote the Lagrange multipliers for the PNKPCs, φ3t and φ4t represent
those for the WNKPCs. Finally, φ5t and φ6t are the Lagrange multipliers for the identitiesof the real wage. The properties of optimal monetary policy are simulated using the Dynare
software package.13

5

Quantitative results

In this section, we report the main results of the paper. Section 5.1 briefly explains the deepparameters used in the paper. Section 5.2 explores optimal monetary policy under commitment.
In Section 5.2, we demonstrate the effect of wage flexibility in the foreign country on the12

See Chapter 7 in Woodford (2003) for a detailed discussion of optimal monetary policy from a timelessperspective.
13

Dynare is available at http://www.dynare.org/.

18

impulse responses of the macroeconomic variables. Then, we calculate the effect of foreign
wage flexibility on the welfare losses under several key parameter restrictions.

5.1

Calibration

This section describes the deep parameters used in this paper. The parameters calibrated inthis paper are the standard ones used in previous studies. The discount factor is set to 0.99. In
addition, this paper assumes that parameter ϕ equals 1.0. The degree of nominal price rigidity
for each country is set to 0.75. Following Rotemberg and Woodford (1997), the elasticity of
substitution between goods p is set to 7.88. The elasticity of substitution for individual labor
supply w is set to 5.0. The elasticity of substitution among the labor varieties is set to 1.0 on
the basis of the calibration used in the standard NK model. Parameter υ, which denotes thedegree of openness, is set to 0.4 as a baseline parameter value.
The relative risk aversion coefficient for consumption, σ is 2.0, is a benchmark calibration.
This calibrated value is based on Pappa (2004). Thus, we assume a positive international
spillover effect. As pointed out in Clarida et al. (2002), international spillover effects disappear
in the case of σ = 1. The negative international spillover effect is also considered when σ < 1.
While we focus on the benchmark case σ > 1, robustness checks are also performed for σ = 1
and σ < 1.14
The degree of nominal wage rigidity for each country is set to 0.6 as a benchmark parametervalue. This parameter value is based on the standard NK model with nominal wage rigidity(Christiano et al., 2005, Erceg et al., 2000, Galı́, 2011). The value of the Calvo parameter for
nominal wage rigidity is assumed to be fixed at 0.6 in the home country, but is set to threedifferent values in the foreign country when conducting impulse response analysis in Section 5.2.
∗ = 0.01.15 The second case corresponds
The first case considers a fully flexible nominal wage, θw∗ = 0.6. In the third case, we set θ ∗ to 0.9, assuming a fairly
to our benchmark calibration, θw
whigh degree of nominal wage rigidity.
Finally, the standard deviations of productivity, σa , and price markup shocks, σu , are both
set to 0.02. The analogous values are assumed for the foreign country. Regarding autoregressive
14

See Pappa (2004) and Corsetti, Dedola and Leduc (2010) for a case of departure from the Cobb-Douglas

consumption aggregate.
15

Without loss of generality, we do not set it precisely to zero due to restrictions in the numerical computations.

19

coefficients for the structural shocks, we set ρa and ρu to 0.8 and 0, respectively. These
calibrated values for economic shocks are presumed to be the same in the foreign country. In
the subsequent sections, we focus on the case for the productivity and price markup shocks
generated in the foreign country.

5.2

Optimal commitment policy

Impulse response analysis
First, we present an impulse response analysis under a commitment policy as a benchmark
calibration. Figure 1 shows the impulse response function when a productivity shock occurs
in the foreign country. In this shock, the inflation rate decreases and output increases. An
increase in output due to a productivity shock will lead to an increase in the real wage and,
consequently, an increase in wage inflation. The results of these impulse responses would beconsistent with Erceg et al. (2000).
[ Figure 1 around here]
The following dynamics are specific to our two-country NK model. An increase in foreign
output lowers the relative price of foreign goods to home goods and improves the terms oftrade in terms of the home country. This change in the terms of trade leads to higher CPI
inflation in the foreign country and lower CPI inflation in the home country. A productivity
shock in a foreign country improves the terms of trade in terms of the home country, causing
an appreciation of the nominal exchange rate. Shocks in the foreign economy affect the home
economy through changes in the terms of trade, as long as σ is not unity.16 As a result,a productivity shock in the foreign country leads to lower inflation and output in the homecountry. As the terms of trade improve in the home country, the consumption of foreign goods
increases and the output of home goods decreases due to the labor burden. This can also impact
real wages through the substitution of labor for consumption, but it appears to be negligible,
at least in this simulation.17
16

See Clarida et al. (2002) and Pappa (2004) for a detailed explanation of the role of the coefficient of relativerisk aversion (CRRA) coefficient in a two-country NK model.
17∗
In the baseline calibration, we assume sticky wage settings of θw = θw
= 0.6.

20

In the two-country model, how does foreign wage flexibility affect the characteristics of acommitment policy? First, let us examine the foreign macroeconomic variables. If the nominal∗ = 0.01), then productivity shocks leadwage is fully flexible in the foreign country (i.e., θw

to a much greater increases in the real wage and wage inflation than when nominal wages
are sufficiently sticky in the foreign country. When the nominal wage is sticky in the foreigncountry, a productivity shock leads to even lower price inflation and higher output. Compared∗ = 0.01, a foreign productivity shock induces an attenuated response of theto the case of θw∗ = 0.6. The response of home real wages and home wage
nominal exchange rate when θw∗ = 0.6). In addition,
inflation is dampened when nominal wage rigidity is moderate (i.e., θw

the response of home inflation and output is weakened by the presence of the sticky nominal∗ = 0.9), the initial drop
wage. However, when nominal wage rigidity is extremely high (i.e., θw∗ = 0.9 than when θ ∗ = 0.01. This transmission mechanism isin output is greater when θw
w

specific to a two-country NK model with nominal wage rigidity.
Next, we demonstrate the properties of a commitment policy in response to a foreign costpush shock. Figure 2 shows the home and foreign responses to a foreign cost-push shock.
When the shock occurs, the foreign country faces a trade-off between output and inflation; acost-push shock increases price inflation and decreases output. In a commitment policy, theeconomy’s dynamic response is mitigated by further reducing output and generating a negative
output gap (Woodford, 2003). In the case of nominal wage rigidity, a cost-push shock generally
causes a reduction in the real wage and wage inflation. On the one hand, even in the case of∗ = 0.01, this reduction is clearly observed in Figure 2. On the other hand, higher nominal
θw

wage stickiness considerably mitigates the reduction in both the real wage and wage inflation.
Moreover, higher nominal wage stickiness appears to improve the policy trade-off between priceinflation and output.
[ Figure 2 around here]
This observation is consistent with the NK model of a closed economy with nominal wage
rigidity. This study emphasizes how changes in nominal wage rigidity in the foreign country
can affect macroeconomic variables in the home country. A decline in foreign output worsens
the terms of trade in terms of the home country. More precisely, this change affects thehome economy through the depreciation of the nominal exchange rate and an increase in CPI
inflation. This case corresponds to that under complete wage flexibility in the foreign country.
21

In the home country, output rises due to increased consumption in the foreign country, but theappropriate policy reaction attenuates the response to inflation.
Figure 2 shows that the higher the nominal wage stickiness, the stronger this tendency
becomes. In this case, the nominal exchange rate in terms of the home currency does notdepreciate, but appreciate, in response to a foreign cost-push shock. The higher the nominal
wage stickiness, the slower the appreciation in the nominal exchange rate to a cost-push shock in
the foreign country, and thus the more attenuated the change in the terms of trade. As pointed
out in Monacelli (2003), the responses of the nominal exchange rate and the terms of traderetain the stationary properties of the optimal commitment policy. As we will demonstrate, theresponse of the nominal exchange rate is non-stationary in the case of a discretionary policy.
Thus, even if there is moderate nominal wage stickiness in the home country, fluctuations inboth the home country’s real wage and the home country’s wage inflation rate are alleviated if
nominal wage stickiness in the foreign country is sufficiently high. Accordingly, the impact on
wage inflation in the home country appears to be negligible.
Welfare analysis
Given nominal wage rigidity in the home country, we find that changes in nominal wage rigidity
in the foreign country have a significant impact on the international transmission mechanismof monetary policy and therefore on the performance of commitment policy. Therefore, we
assess how changes in nominal wage rigidity in the foreign country affect global welfare undera commitment policy. Under several parameter restrictions, we calculate the welfare loss under∗ , σ, and υ. As
a commitment policy. Specifically, we focus on the parameter restrictions of θw

these parameters play an important role in our model, we examine whether they significantly
affect the performance of the optimal monetary policy.
Figure 3 shows the welfare loss in the world economy defined by Equation (42) as a function∗ , expressed as the percentage deviation from the value in theof foreign wage stickiness θwbaseline calibration. Consider first the case in which wages in both home and the foreigncountry are flexible, in the top panel of Figure 3. Each line corresponds to a case in which
the home country’s wages are flexible (θw = 0.01), somewhat sticky (θw = 0.4), the baseline(θw = 0.6), and highly sticky (θw = 0.8).
In the top panel, both central banks can achieve the least loss when both home and foreign

22

wages are fully flexible. The loss is 18.1% smaller than the value in the baseline case. Galı́
(2011) argued that wage flexibility improves social welfare in a closed economy, and Galı́ andMonacelli (2016) argued that this holds true in a currency union. We address the fact, however,that this result is obtained in our two-country model.
[ Figure 3 around here]
For worldwide welfare, it is important whether wages in both home and the foreign country
are flexible or not. If wages in either country become somewhat stickier (e.g., from θw = 0.01
to θw = 0.4), the loss increases by as much as 10 percentage points. In other words, a further
increase in wage stickiness (from θw = 0.4 to θw = 0.6 or 0.8) does not cause such a sharpchange in welfare loss.
With home country wage stickiness as a given, the change in welfare loss is not monotonicallyincreasing in accordance with the change in foreign wage stickiness. In fact, in this simulation,∗ = 0.8 for all of the lines, and the loss becomes slightly
it appears that the loss peaks at θw

smaller as wage stickiness increases from there.
The non-monotonic relationship between wage stickiness and welfare loss has already been
pointed out by Galı́ (2013), Galı́ and Monacelli (2016), and Groll and Monacelli (2020).18
These previous studies have largely discussed the role of wage flexibility within one country
or a monetary union, but they did not focus on that mechanism in a two-country model.
Compared to previous studies, our results address the fact that the non-monotonicity of wage
stickiness and losses is not outstanding. This weak non-monotonicity implies that changes
in one country’s parameters alone do not have a significant impact on the global economy.
The welfare loss is noticeably smaller when wages in both countries are flexible. This result
cannot directly compare to the welfare loss under the closed economy model. However, wehighlight that when nominal wage stickiness differs between home and the foreign country,
it is important to take this difference into account to maximize social welfare in the global
economy. In particular, if the home country’s wage stickiness does not change, changes in the
18

Galı́ (2013) showed, in a standard NK model framework, that macroeconomic policies that increase wage

stickiness under initial high wage stickiness can lead to welfare losses and argued that “the desirability of more
wage flexibility are propositions that one should not take for granted (p.1002)”. Galı́ and Monacelli (2016)
showed similar results in a monetary union model. Groll and Monacelli (2020) analyzed how welfare losses
varied whether both prices and wages were flexible or sticky and found similar non-monotonicity.

23

foreign country’s wage stickiness will not affect welfare losses to the same extent as in a closed
economy. This means that in a closed economy or a small-open economy, changes in the real
wage gap in the home country are not affected by foreign endogenous variables. However, as
pointed out in Section 3, changes in the foreign country’s shocks affect the home real wage
gap through changes in the terms of trade in the two-country framework. Importantly, theminimum global welfare loss is attained when the terms of trade externality is fully exploitedin the case of complete wage flexibility in both countries. However, this gain is reduced in thecase of the presence of nominal wage stickiness in each country because central banks cannot
fully use the terms of trade externality. These results indicate the key mechanism underlying
the results of our non-monotonic welfare loss in our two-country model.
The middle panel of Figure 3 similarly plots varying intertemporal substitutions of consumption (σ, common between home and the foreign country). The baseline calibration, σ,
is set to 2.0. Given that home wages are sticky, a change in σ maintains the shape of the∗ ≥ 0.1), an increase in“loss curve” seen in the top panel. If foreign wages are sticky (i.e., θw

σ shifts the loss curve upward, causing welfare losses to increase continuously. As mentioned
above and also pointed out in Clarida et al. (2002), whether the impact of the foreign economy
positively or negatively affects the home country depends on σ. In our model, it has a positiveeffect when σ > 1. Why is the relationship between foreign wage stickiness and losses robust to
the positive or negative sign of the spillover effect? The reason is that the central bank, witha good understanding of the spillover effect, successfully induces fluctuations in the terms oftrade to bring about the optimal spillover.
Interestingly, only when the foreign wage is fully flexible, is the welfare loss minimized
when σ = 0.5. In contrast to this case, when the foreign wages are sticky, the welfare loss
curve increases as σ increases. Why does this inversion occur? As seen in the wage versionof the NKPC, when wages are sticky, the market distortions induced by slow wage changes
become greater as σ increases. Conversely, when the foreign wage is fully flexible, the market
distortions from the foreign wage disappear. Here, even if the home country’s wages are still
sticky, we can control the terms of trade to exploit the spillover effect. Since the spillover effectbecomes greater as σ increases, the reversal phenomenon described above occurs only when
foreign wages are fully flexible.
We confirm, therefore, that the relationship between foreign wage stickiness and welfare

24losses is robust, regardless of the intertemporal elasticity of consumption, with only a few
exceptions.
The bottom panel of Figure 3 shows the effect of the degree of economic openness υ onworldwide welfare loss. Our baseline calibration is υ = 0.4. When the economy is closed(υ = 0.01, blue line), foreign wage stickiness changes make little difference to global losses.
However, as the economy becomes open, the central bank can effectively manipulate foreign
wages to substantially improve economic welfare.
It is noteworthy that when the closed economy is considered, foreign wage stickiness, andthe welfare loss become unrelated. This result is not trivial, because flexible wages in theforeign country would improve the foreign county’s welfare, which, in turn, would improveglobal welfare. This mutual indifference can be explained as follows. As the degree of openness decreases, spillovers from the foreign country to the home country are muted, making it
impossible to improve welfare using terms of trade externalities, as previously discussed.
In other words, the non-monotonic relationship between foreign wage stickiness and the loss
curve becomes more distinct the more the degree of openness increases. This feature indicatesthat the more the degree of openness increases, the easier it is for the effects of fluctuations in
the foreign wage and the terms of trade to spill over into the home country, thus stabilizing the
global economy as a whole. This finding suggests that the degree of openness plays an importantrole in significantly improving the world economy when foreign wages are sufficiently flexible orhighly sticky. This result is a notable result for the two-country model in which both countries
are large, and it is not found in previous studies such as Galı́ and Monacelli (2016). Naturally,
this is because the role of economic openness cannot be discussed in a closed economy, and,
therefore, global welfare is also given in a small-open economy in which the foreign economy isgiven.
In summary, we find the following results obtained in Sections 5.2 and 5.3. First, the world
economy’s welfare loss is small when both home and foreign wages are flexible. Second, wheneither the wages in home or the foreign country are flexible, the welfare loss is also small,
although the welfare improvement will be half compared to the case of flexible wages in bothcountries. Third, the non-monotonicity between wage stickiness and welfare loss is not asapparent as observed in a closed economy and a monetary union. Fourth, regarding the second
finding, which states that global welfare improves when foreign wages are flexible, is robust25

to changes in the intertemporal elasticity in consumption. However, there is also a parameterthat would cancel the welfare-improving effect of flexible wages (e.g., the degree of openness,υ).

6

Welfare gains from commitment in a two-country model withsticky prices and wages

We assessed the performance of a commitment policy in a two-country model with nominal
wage rigidity in the previous section. This section explores the comparison of the performanceof commitment and discretionary policy. In Section 6.1, we calculate the gain from commitment
under several key parameter values. Section 6.2 confirms the performance of a commitment
policy using impulse response analysis.

6.1

Welfare analysis

In this section, we evaluate the gains from commitment policy in our model. While there areseveral studies on the effectiveness of commitment policy in economies in which nominal prices
and wages are sticky, the role of wage flexibility in a two-country model is an open question.
As noted earlier, in the standard NK model, a commitment policy generally leads to smaller
welfare losses than a discretionary policy (McCallum and Nelson, 2004).
The result that the welfare gains from a commitment policy are greater than those froma discretionary policy is carried over to the open economy model. Monacelli (2003) showedthat welfare losses due to discretionary policy are substantially greater than those due to acommitment policy in a small-open economy. As mentioned earlier, Pappa (2004) pointed outthat commitment policy can eliminate the deflationary bias caused by discretionary policy,
and thus create welfare gains from commitment in a two-country economy. In addition, thewelfare gain from a commitment policy still holds in the presence of sticky wages (Erceg et al.,2000, Galı́, 2011). Galı́ and Monacelli (2016) also pointed out the role of wage flexibility ina monetary union, but did not evaluate the performance of discretionary policy. While Groll
and Monacelli (2020) examined the benefits of commitment policy in the cases of both flexible
exchange rates and a monetary union, they did not focus on the role of wage flexibility.
It remains unclear whether the welfare losses under a discretionary policy are substantially26

greater than those under a commitment policy in a two-country NK model in which nominal
wages are sticky in both countries. In this paper, we show that the degree of wage flexibility inboth countries significantly impacts the performance of optimal commitment policy in the twocountry model with sticky prices and wages. However, it is not obvious whether commitment
policy are always better than discretionary policy in a two-country model with nominal priceand wage rigidity.
The purpose of this section is to quantitatively investigate the extent to which the losses
from discretionary policy are greater than those from commitment policy in our model. Figures
4 through 6 compare the results of Figure 3, the performance of commitment, with that ofdiscretion. The commitment case (solid blue line) is the same as each line in Figure 3. Each
∗ = 0.6, which is thevalue is expressed as a percentage deviation from the result at θw = θwbaseline parameter under the commitment policy.
First, Figure 4 shows the performance of both the commitment and discretionary policies onchanges in foreign wage stickiness, given the stickiness of the home country wage. We find thatwhen wages are flexible in the home country, welfare losses are much lower in the commitmentcase than in the baseline calibration. However, this is not the case for the discretionary policy.
Under the discretionary policy, although the welfare loss is reduced as the wage becomes more
flexible in the home country, the reduction is much smaller than in the commitment policy.
In other words, if nominal wages in both countries are completely flexible, the welfare loss ofthe discretionary policy will be greater than that of the commitment policy. Specifically, if
nominal wages are fully flexible in both countries, the gain from commitment policy (i.e., thepercentage difference from discretion) is roughly 20% points. Thus, this result indicates thatthe gain from a commitment policy is greater than the discretionary policy, as long as nominal
wages are completely flexible in both countries. We address that this result is not obtained
when we consider the two-country NK model excluding the role of nominal wage rigidity.
[ Figure 4 around here]
However, when the home country’s wage becomes stickier (θw ≥ 0.4), the welfare loss does
not change much under discretionary policy, while the reduction in welfare loss disappears undercommitment. Thus, when the degree of foreign wage stickiness increases, the performance of∗ = 0.6),
commitment policy deteriorates slightly. In fact, under realistic assumptions (θw = θw

the gain from commitment is only 1.7 percentage points. Importantly, when the nominal wage
27

in both countries become highly sticky, the difference between commitment, and discretionary
policies becomes negligible. In particular, the cost of discretionary policy is not necessarily
greater than that of commitment policy when considering the role of wage flexibility in a twocountry NK model. To the best of our knowledge, none of the previous studies has pointed out
this result.
Second, given a value of θw , we examine whether a change in the CRRA parameter, σ,∗ changes. Figure 5 computes the welfare losses
affects the gain from commitment policy as θwof both commitment and discretion, relative to the commitment case at the benchmark case,as the degree of foreign nominal wage rigidity changes under several parameter values of σ. As
shown in this figure, there appears to be no major differences between the four panels. Thus,
welfare losses under a discretionary policy are much greater than those under a commitment
policy, as long as nominal wages are completely flexible in the foreign country. In that sense,
the results indicate that the welfare loss is robust to σ, regardless of commitment, or discretion.
We observe that even in the case of σ < 1, the gain from commitment policy is not negligible
in the case of wage flexibility. This result does not appear to be obtained in the standard
two-country NK model considered by Clarida et al. (2002).
[ Figure 5 around here]
Third, we consider whether the degree of openness affects the performance of the commitment and discretion regimes. We do not need to consider the role of wage flexibility in a
two-country economy if the degree of openness never affects the performance of both regimes.
Figure 6 computes the welfare losses of both commitment and discretion, relative to the commitment case as the benchmark case, as the degree of foreign nominal wage rigidity changes
under several parameter values of υ. In this exercise, we focus on how the degree of openness
affects the performance of both the commitment and discretion regimes. The figure shows thatthe gains from a commitment policy remain around 2% points, in most cases, when the degree∗ ≥ 0.4). On the other hand, if
of foreign nominal wage rigidity is somewhat sticky (e.g., θw∗ falls below 0.3, foreign wage flexibility increases the gains from a commitment policy fromθw

2% to 7% points, as the degree of openness υ increases. We find that the degree of openness
ensures that flexible wages in both countries improves global welfare. In other words, if wages
in both countries are sticky, making wages more flexible will not significantly improve welfare.
Therefore, this result implies that wage flexibility in the foreign country increases the gains
28

from a commitment policy when the degree of openness is approximately equal in the twocountries.
[ Figure 6 around here]
In summary, when nominal wages are perfectly flexible in both countries, the welfare loss
under commitment policy is smaller than that under sticky wages, while the welfare loss under
discretionary policy is greater. However, when nominal wages are quite sticky in the foreigncountry, we find that there is no significant difference in the performance of the commitment
and discretionary policies. We can say that these results are found to be unaffected by changes
in several parameters that are important in a two-country model.

6.2

Impulse response analysis

The results in Section 6.1 show that there is a significant difference in the performance ofcommitment and discretion with respect to changes in wage flexibility in each country. Why
did we focus on the difference between commitment and discretion in a two-country NK modelwith nominal wage rigidity? To understand this mechanism, we provide an impulse response
analysis of both regimes. Figures 7 and 8 show the impulse responses of the macroeconomic
variables in both countries.
Figure 7 illustrates the impact of foreign productivity shocks on macroeconomic variables
in both countries under commitment and discretion. From this figure, it appears that there
is no difference between commitment and discretion, except for the response of wage inflationin both countries. In addition, we observe that the response of the nominal exchange rate tothese shocks under both regimes seems to be stationary. This seems to be in stark contrast
with the cases presented by Clarida et al. (2002), Galı́ (2013), and Galı́ and Monacelli (2016).
An intuitive mechanism for such a stationary response of the nominal exchange rate may be
explained by the dynamics of the terms of trade. Sluggish changes in the terms of trade are
driven by gradual movements in the nominal wage in both countries. Thus, as long as thenominal wage is sticky in each country, the nominal exchange rate is characterized by inertial
movements in the terms of trade. This is why the nominal exchange rate is stationary. Groll
and Monacelli (2020) focused on the role of the terms of trade in explaining the performance of29

optimal monetary policy in a two-country NK model,19 , but it differed from Groll and Monacelli(2020) in that it addressed the impact of wage flexibility on the terms of trade in a two-country
model.
[ Figure 7 around here]
In contrast to the case in Figure 7, Figure 8 illustrates that there is a difference between
the commitment and discretionary policies when a foreign cost-push shock occurs, but thedifference appears to be quantitatively small. Thus, as pointed out in Woodford (2003), ourmodel indicates that the commitment policy succeeds in stabilizing the macroeconomic variables
in both countries by introducing policy inertia into the economy. This is in stark contrast withthe result in the case of a foreign productivity shock. For each response to a foreign cost-pushshock, the response under the commitment policy is smaller than under a discretionary policy
regime. The response of the nominal exchange rate is also stationary under both regimes,
but the response under discretion is much more depressed and persistent than that undercommitment.
[ Figure 8 around here]
Figures 9 and 10 show the impulse responses under the both regimes when the foreigncountry’s nominal wages are flexible.20 First, Figure 9 plots the impulse response to a foreign
productivity shock. At first glance, the results appear similar to those in Figure 7, but the
response of the home and foreign inflation rates differ in the commitment and discretionary
cases. In addition, the response of the nominal exchange rate is stationary, even when there is
full wage flexibility in the foreign country. Thus, the nominal exchange rate remains stationary
as long as the source of the shock is related to a foreign productivity shock. In other words, incontrast to previous studies, our model shows that wage flexibility in the foreign country does
not affect the stationarity issue with respect to the nominal exchange rate.
[ Figure 9 around here]
19

Rhee and Turdaliev (2013) and Campolmi (2014) analyzed the role of wage flexibility in considering optimalmonetary policy, but these studies focused only on small-open economies.
20

In Figures 9 and 10, nominal wages are fixed to the benchmark value in the home country.

30

Next, Figure 10 plots the impulse responses to foreign cost-push shocks under commitment
and discretionary policies when foreign nominal wages are flexible. The results in Figure 10
differ from those in Figure 8. The first thing to note is that the response of the nominal
exchange rate under discretionary policy is non-stationary. Monacelli (2003) argued that thestationarity of the nominal exchange rate is related to that of the terms of trade. While thenominal exchange rate is non-stationary under a discretionary policy, the terms of trade retain
their stationary property. This result is consistent with the discussion by Monacelli (2003).
In addition, the response of foreign CPI inflation is asymmetric between the discretionary andcommitment policies. Furthermore, when compared to the case in Figure 8, discretionary policy
provides the same policy inertia to the economy as commitment policy, but the latter provides
stronger policy inertia than the former. This difference is reflected by the costs from discretion,
as shown in the top panel of Figure 3 and the top panel of Figure 4. However, while thesefigures show the costs from discretion shrink, this observation seems to be consistent with theimpulse response obtained in Figure 8.
[ Figure 10 around here]

7

Concluding remarks

This paper examined wage flexibility and optimal monetary policy in a two-country NK model.
In contrast to a two-country model with no nominal wage rigidity, we addressed the fact thatin a two-country model with nominal wage rigidity between the two countries, the dynamics ofthe terms of trade significantly characterize the key mechanism of wage flexibility.
We showed that in this situation, given wage flexibility in the home country, the degree of
foreign wage flexibility substantially impacts worldwide welfare losses and the gain from commitment policy. Specifically, when the nominal wages in both countries are perfectly flexible,
the welfare gains from the commitment policy are the greatest. However, when nominal wages
in the foreign country are stickier, the gains from commitment are predominantly reduced.
This result is robust to any change in several key parameters that play an important role in
the two-country NK model.
Finally, we mention the limitations of this study. In this paper, we focus only on the caseof international policy coordination. However, if either central bank has an incentive to renege

31

on international policy coordination, the degree of nominal wage rigidity in each country may
have a significant impact on the cost of an uncoordinated resolution. This is a very important
topic, but it is beyond the scope of this paper. We would like to explore this topic as a future
work.

A

Appendix A: Equilibrium with flexible prices and wages

In this appendix, we provide the expression of the natural level of the macroeconomic variables
in terms of the case in which prices and wages are completely flexible in both countries. These
expressions are used to define the gap variables shown in main manuscript. Again, unlessotherwise noted, analogous equations hold for the foreign country.
First, the price markup is the surplus of the marginal product relative to the wage costof hiring a worker, and the two are equal under perfect competition. Under price and wage
flexibility, the price markup is given by the following:µpt = mpnnt − wtn − pnH,t= log (1 − α) + at − αnnt − ωtn − υsnt .
Next, the wage markup is the gap between the real wage and the marginal rate of substitution(again, both are equal under perfect competition).
n
n
n
n
n
µw
t = wt − mrst = wt − (σct + ϕnt ),Defining total markup, µ, as the sum of µp and µw , we obtain the following equation:nµpt + µw
t = log (1 − α) + ((1 − σ) (1 − υ)) at + (υ (1 − σ) − σ (1 + α) − ϕ) nt− τt + υ (1 − σ) yt∗ .
Now, we provide the variables under flexible wage and price equilibrium. First, the naturallevel of employment is given as follows:
nnt = n − ψa at + ψy∗ yt∗,n ,where
ψn ≡ (1 + α) + ϕ − υ (1 − σ) , n ≡log (1 − α) − µυ(1 − σ), ψa ≡ ((1 − σ) (1 − υ)) , ψy∗ ≡
.
ψn
ψn
32

Output, terms of trade, and consumption, at their natural levels, are given as follows:
ytn = at + (1 − α) nnt ,snt = ytn − yt∗ ,cnt = ytn − υsnt .
Finally, the natural level of real wages is given by the following:
wtn = µw + σcnt + ϕnnt
= log (1 − α) − µp + at − αnnt − υsnt .
These equations are used to define the gap variables in the main text.

B

Appendix B: Derivation of the central bank’s loss function

B.1

Preliminaries

We derive the loss function approximated around a steady state. Before doing so, I define
relevant notation. First, H̄ denotes the value of the steady state, and Htn is the value of theefficient level. Second, as in the main text, we define ht = log(Ht /H̄) as the deviation of Htfrom the steady state. To implement the second-order approximation of the planner’s objectivefunction, we introduce the following equation:
Ht1
Ht − H̄ = H̄
− 1 ≈ ht + h2t .
2
H̄

B.2

(A.1)

Optimal subsidies

The planner’s objective function under policy coordination is given by
Wt = u(Ct ) − (1 − ν)v(Nt ) − νv(Nt∗ ),(A.2)

The planner maximizes this objective function, subject to the following resource constraint:
C = Y 1−ν (Y ∗ )ν = (N 1−α )1−ν [(N ∗ )1−α ]ν
From the first-order condition, we obtainuc C = (1 − α)vn N
33

(A.3)

We now seek for the optimal subsidies that can eliminate distortions arose from the presenceof price and wage mark-ups. From the optimal condition for labor supply,
vn N
w W
=
uc C
w − 1 P
Then this equation can be rewritten as follows:
W
vn N
= µw
= µw M RSP
uc C
It follows from the real marginal cost in the steady state that
(1 − τ )µp µw M RS = 1,
κM P L
where MPL denotes the marginal product of labor. In the efficient steady state, the following
condition must hold:
M RS
=1
MPL
Therefore, using these conditions, we find that the optimal subsidies are given byτ =1−1
κµp µw(A.4)

Similarly, the optimal subsidies for foreign country are given byτ∗ = 1 −B.3

1κµp µw(A.5)

Derivation of the central bank’s loss function

We now provide the second-order approximation of the following household utility function.
The derivation of the loss function in the main text is based on the idea of Clarida et al.
(2002). We implement the second-order approximation of the following planner’s objectivefunction around their efficient price equilibrium. The following step of the derivation of theloss function is mainly based on that of Clarida et al. (2002) and Engel (2011).
The second-order approximation of first-term of the right-hand side of the planner’s objec-

34

tive function are follows:ucc(Ct − C)2 + O(||ξ||3 ),2
(Ct − C) ucc C 2 Ct − C 2= u(C) + uc C+
+ O(||ξ||3 ),
C
2
C




1 2 2
ucc C 21 2
ct + ct+ O(||ξ||3 ),= u(C) + uc C ct + ct +22
2
uc C 2 ucc C 2 2= u(C) + uc Cct +c +ct + O(||ξ||3 ),2 t
2u(Ct ) ≈ u(C) + uc (Ct − C) +Defining σ = −ucc C/uc , we obtain(1 − σ) 2
U (Ct ) ≈ u(C) + uc C ct +ct + t.i.p. + O(||ξ||3 ),
2

(A.6)

The second-order approximation of second-term of the right-hand side of the planner’s objectivefunction are follows:vnn(Nt − N )2 + O(||ξ||3 ),2
(Nt − N ) vnn N 2 Nt − N 2++ O(||ξ||3 ),= v(N ) + vn N
N
2
N




1 2
vnn N 21 2 2= v(N ) + vn N nt + nt +nt + nt+ O(||ξ||3 ),2
2
2
vn N 2 vnn N 2 2= v(N ) + vn N nt +n +nt + O(||ξ||3 ),2 t
2v(Ct ) ≈ v(N ) + vn (Nt − N ) +Defining ϕ = −vnn N/vn , we obtain(1 + ϕ) 2
nt + t.i.p. + O(||ξ||3 ).
v(Nt ) ≈ v(N ) + vn N nt +2
Similarly, the approximation of the third-term of the right-hand side is given by(1 + ϕ) ∗ 2∗∗∗v(Nt ) ≈ v(N ) + vn N nt +(nt ) + t.i.p. + O(||ξ||3 ).
2Here, price dispersion terms are derived as follows:
Z 1
Nt =Nt (i)di0
Z 1Z 1=
Nt (j)djdi
0
0

Z 1
Z 1
Nt (j)=Nt (i)dj di
Nt (i)
0
0

Z 1
Z 1
Wt (j) −w=
Nt (i)
djdi
Wt
0
0
Z 1
=
Nt (i)di∆w,t ,
0

35

(A.7)

(A.8)

(A.9)

where ∆w,t denotes the wage dispersion, which is defined as follows:∆w,t ≡Wt (j)Wt

−w.

(A.10)

Moreover, using production function, we can rewrite the above equation as follows:
Yt (i) Yt −1/(1−α)
Nt (i)di =
di
Yt At
0
0
 −1/(1−α) Z 1 Yt
Yt (i) −1/(1−α)
=
di,
At
Yt
0

 −1/(1−α) Z 1 PH,t (i) −p /(1−α)
Yt
=
di
At
PH,t
0
 −1/(1−α)
Yt
∆p,t
=
At
Z 1

Z 1

(A.11)

where price dispersion, which is defined as follows:
Z 1
∆p,t ≡
0

PH,t (i)PH,t−p /(1−α)
di(A.12)∆p,t ∆w,t ,(A.13)

Hence,

Nt =

Yt
At

−1/(1−α)Lemma 1 (Gali, 2015) The following properties hold:pvari (PH,t )2Θw
log ∆w,t ≈varj (Wt )2
log ∆p,t ≈(A.14)(A.15)

Proof. Following Gali (2015), we derive the above two conditions. Let p̂H,t (i) = pH,t (i) − pH,t .
Here, pH,t (i) = log(PH,t (i)) and pH,t = log(PH,t ). Then,Z 1
0PH,t (i)
PH,t

1−p



di = exp (1 − p )p̂H,t (i)Implementing the McLaughlin expansion of the right-hand side leads to(1 − p )2
exp (1 − p )p̂H,t (i) ≈ 1 + (1 − p )p̂H,t (i) +p̂H,t (i)2 + O(||ξ||3 ).
2Here, using the definition of the price index for PH,t , we know
Z 1
1=
0

PH,t (i)
PH,t
36

1−p
di

(A.16)

Then, integrating Equation (A.16) in the interval [0,1], we obtain
Z 1
0PH,t (i)
PH,t

1−p

Z 1

(1 − p )2di ≈ 1 + (1 − p )p̂H,t (i) di +
2
0

Z 1

p̂H,t (i)2 di + O(||ξ||3 ).

0(A.17)
DefineZ 1
Ei p̂H,t (i) =p̂H,t (i)di.
0

Then we can rewrite Equation (A.17) as follows:
Ei p̂H,t (i) =(p − 1)Ei p̂H,t (i)22

(A.18)The objective of this calculation is to derive the second-order approximation of
Z 1
0PH,t (i)PH,t−p /(1−α)
di.

The second-order approximation of the above equation becomesPH,t (i)PH,t−p /(1−α)


≈1−



2
p
p
1
p̂H,t (i) +p̂H,t (i)2
1−α
2 1−α

Integrating in the interval [0,1], we obtain
Z 1
0PH,t (i)PH,t−p /(1−α)

di ≈ 1 −



2
p
p
1
p̂H,t (i) +p̂H,t (i)2
1−α
2 1−α

Substituting Equation (A.17) into Equation (A.19), we obtainPH,t (i) −p /(1−α)∆p,t ≡
di
PH,t
0



2

p − 1pp
1
2
Ei p̂H,t (i) +Ei p̂H,t (i)2
=1−
1−α
2
2 1−α


1 1 − α − (1 − α)p + p
=1+
Ei p̂H,t (i)2
2
1−α


1 1 − α + αp=1+
Ei p̂H,t (i)2
2
1−α


p
1
=1+
Ei p̂H,t (i)2
1 − α 2Θ
Z 1

Here
2

Z 1

Ei p̂H,t (i) =(pH,t (i) − pH,t )2 di ≡ vari (PH,t )037(A.19)

Then we finally obtain∆p,t = 1 +p1vari (PH,t )1 − α 2ΘTaking the logarithm of this equation, we can also obtain
δp,t ≡ log ∆p,t ≈pvari (PH,t )2Θ

Similarly, we also the second-order approximation of the wage dispersion as follows:log ∆w,t ≈wvarj (Wt )2Using ct = (1 − ν)yt + νyt∗ in Equation (A.6), using nt = (1/(1 − α)(yt − at ) + δp,t + δw,t
and nt = (1/(1 − α)(yt − at ) + δp,t + δw,t in Equations (A.7) and (A.8), respectively, we obtain



2 
1−σ
∗
∗
Wt ≈ u(C) + uc C
(1 − ν)yt + νyt +(1 − ν)yt + νyt
2



1
− (1 − ν) v(N ) + vn N
(yt − at ) + δp,t + δw,t
1−α

2 
1+ϕ
1
+
(yt − at ) + δp,t + δw,t
2
1−α



1
∗
∗
∗
∗
∗
−ν v(N ) + vn N
(y − at ) + δp,t + δw,t
1−α t

2 
1
1+ϕ
∗
∗
∗
∗
(y − at ) + δp,t + δw,t
+ O(||ξ||3 ).
+
2
1−α t
Since we obtain uc C = vn N/(1 − α) in the efficient steady state, the above equation can berearranged as follows:
uc C
Wt − W̄ ≈(1 − σ)(1 − ν)2 yt2 + (1 − σ)ν 2 (yt∗ )2 + 2ν(1 − ν)(1 − σ)yt yt∗
2(1 − ν)(1 + ϕ) 2 ν(1 + ϕ) ∗ 2 (1 − ν)(1 + ϕ)
ν(1 + ϕ) ∗ ∗−yt −(y ) +
a t yt +a y(1 − α)(1 − α) t(1 − α)(1 − α) t t∗∗− (1 − ν)(δp,t + δw,t ) − ν(δp,t+ δw,t) + t.i.p. + O(||ξ||3 ).
where W̄ = u(C) − (1 − ν)v(N ) − νv(N ∗ ). Further calculation leads touc C
Wt − W̄ ≈(1 − ν)[1 − σ − ν(σ − 1)]yt2 + (1 − ν)[1 − σ − (1 − ν)(σ − 1)](yt∗ )22(1 − ν)(1 + ϕ) 2 ν(1 + ϕ) ∗ 2− 2ν(1 − ν)(1 − σ)yt yt∗ −
yt −
(y )
(1 − α)(1 − α) t(1 − ν)(1 + ϕ)
ν(1 + ϕ) ∗ ∗∗∗+at yt +
a y − (1 − ν)(δp,t + δw,t ) − ν(δp,t + δw,t )(1 − α)(1 − α) t t+ t.i.p. + O(||ξ||3 ).

(A.20)
38

The natural rate of output for both countries is given by





α+ϕ
1+ϕ
∗ ∗
(1 − ν)at yt + νat yt = (1 − ν) σ +− ν(σ − 1) ytn yt
1−α
1−α


α+ϕ
+ν σ+
− (1 − ν)(σ − 1) ytn,∗ yt∗1−α+ ν(1 − ν)ytn yt + ν(1 − ν)ytn,∗ yt∗
Substituting this equation into Equation (A.20), we obtain




p
α+ϕ
uc C
2
(1 − ν) σ +− ν(σ − 1) xt + vari (PH,t ) + w varj (Wt )Wt − W̄ ≈ −
2
1−α
Θ



p
α+ϕ
∗ 2
∗
∗
+ν σ+
− (1 − ν)(σ − 1) (xt ) + vari (PF,t ) + w varj (Wt )1−αΘ− 2ν(1 − ν)(1 − σ)xt x∗t + t.i.p. + O(||ξ||3 ).
(A.21)
Here, following the proposition of Woodford (2003, Chap 6), we can obtain the following
conditions for price dispersion and wage dispersion of both countries. We only derive thesecond-order approximation for price dispersion for home country.
δp,t = vari [log(pt (j)) − P̄t−1 ]= Ei [log(pH,t (i)) − P̄H,t−1 ]2 − [Ei log pH,t (i) − P̄H,t−1 ]2= θp Ei [log(pH,t−1 (i)) − P̄H,t−1 ]2 + (1 − θp )(log poH,t − P̄H,t−1 )2 − (P̄H,t − P̄H,t−1 )2 (A.22)
HereoPH,t = (1 − θp ) log PH,t+ θp P̄H,t−1oP̄H,t − P̄H,t−1 = (1 − θp ) log PH,t
− (1 − θp )P̄H,t−1olog PH,t− P̄H,t−1 =1(P̄H,t − P̄H,t−1 )1 − θp

Then from Equation (A.22), we obtain
1(P̄H,t − P̄H,t−1 )2 − (P̄H,t − P̄H,t−1 )2
1 − θp
θp= θp δp,t−1 +(P̄H,t − P̄H,t−1 )2
1−θδp,t = θp δp,t−1 +Since P̄t = log Pt + +o(kξk)2 ,δp,t = θp δp,t−1 +θp
π 2 + O(||ξ||3 )1 − θp t
39

Then

P∞

∞
X

t
t=0 β δp,t becomes





θp
θp
θp
2
2
2
2
β δp,t = θp δp,−1 +π + β θp δp,0 +π + β θp δp,1 +π + ···
1 − θp 0
1 − θp 1
1 − θp 2
t=0


θp
2
2
2 2
2
π + βπ1 + β π2 + · · ·= θp δp,−1 + θp β(δp,0 + βδp,1 + β δp,2 + · · · ) +1 − θp 0∞∞
X
θp X t 2t
β πt + O(||ξ||3 )= θp δp,−1 + θp ββ δp,t +1 − θp
t

t=0

t=0

Then
(1 − θp β)

∞
X
t=0

⇒

∞
X

∞

θp X t 2β δp,t = θp δp,−1 +β πt + O(||ξ||3 )1 − θptβ t δp,t =

t=0

t=0

θp
(1 − θp )(1 − θp β)∞Xβ t πt2 + t.i.p. + O(||ξ||3 )(A.23)

t=0

The analogous equations hold for πt∗ , πtw , and πtw,∗ .
∞
X
t=0
∞
X
t=0
∞
X

∗
β t δp,t=β t δw,t =∗β t δw,t
=

t=0

∞
X
θp∗
β t (πt∗ )2 + t.i.p. + O(||ξ||3 ).
(1 − θp∗ )(1 − θp∗ β)(A.24)

θw(1 − θw )(1 − θw β)

β t (πtw )2 + t.i.p. + O(||ξ||3 ),(A.25)β t (πtw,∗ )2 + t.i.p. + O(||ξ||3 ).

(A.26)

t=0
∞
X

∗
θw
∗ )(1 − θ ∗ β)
(1 − θw
w

t=0
∞
X
t=0

Finally, taking the discounted sum of Equation (A.26), we obtain the following the central
bank’s loss function.

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43

10-4

Home inflation

Foreign inflation

00*= 0.01w*= 0.60w*= 0.90
w

-5
-10

-0.01
-0.02

-15

-0.03

-20

-0.04
2

4

6

8

10

12

14

16

18

20

2

4

6

Home output

0

8

10

12

14

16

18

20

14

16

18

20

16

18

20

14

16

18

20

14

16

18

20

16

18

20

Foreigin output
0.4
0.3

-0.02

0.2
-0.04
0.1
-0.06

0
2

4

6

8

10

12

14

16

18

20

2

Home CPI inflation

4

6

8

10

12

Foreign CPI inflation

0.3

0
0.2
-0.05
0.1

-0.1

0

-0.15
-0.2

-0.1
2

4

10

4

6

-4

8

10

12

14

16

18

20

2

Home wage inflation

4

6

8

10

12

14

Foreign wage inflation

0.2
0.15

2
0.1
0.05

0

0
-2
2
8

10

4

6

-3

8

10

12

14

16

18

20

2

Home real wage

4

6

0.3

4

0.2

2

0.1

0

10

12

Foreign real wage

0.4

6

8

0
2

4

6

8

10

12

14

16

18

20

2

4

6

Home terms of trade

0

8

10

12

Nominal exchange rate
0

-0.1

-0.1

-0.2

-0.2

-0.3

-0.3

-0.4
-0.4
2

4

6

8

10

12

14

16

18

20

2

4

6

8

10

12

Figure 1: Impulse responses of a foreign productivity shock

44

14

20

10-4

Home inflation

Foreign inflation
0.4

15
*
= 0.01w*= 0.60w*= 0.90
w

10
5

0.2
0

0
-0.2
2

4

6

8

10

12

14

16

18

20

2

Home output

4

6

8

10

12

14

16

18

20

14

16

18

20

16

18

20

14

16

18

20

14

16

18

20

16

18

20

Foreigin output

0

0.06
-0.2
0.04
-0.4

0.02
0

-0.6
2

4

6

8

10

12

14

16

18

20

2

4

6

Home CPI inflation

0.3

8

10

12

Foreign CPI inflation
0.2

0.2
0.1
0.1

0

0

-0.1

-0.1

-0.2
2
10

4

6

-4

8

10

12

14

16

18

20

2

Home wage inflation

4

6

10

12

14

Foreign wage inflation

0.5

5

8

0
0
-0.5
-5
-1
-10

-1.5
2

0

10

4

6

-3

8

10

12

14

16

18

20

2

Home real wage

4

6

8

10

12

Foreign real wage

0
-0.5

-2

-1
-4
-1.5
-6

-2
2

4

6

8

10

12

14

16

18

20

2

Home terms of trade

4

6

8

10

12

Nominal exchange rate

0.6

0.6
0.4
0.4
0.2
0.2

0

0

-0.2
2

4

6

8

10

12

14

16

18

20

2

4

6

8

10

Figure 2: Impulse responses of a foreign cost-push shock

45

12

14

Worldwide welfare losses
0
-5

w
w

-10

w
w

-15

= 0.01= 0.40= 0.60= 0.80

-20
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

*
w

Worldwide welfare losses
0-2-4= 0.50= 1.00= 2.00= 6.00

-6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

*
w

Worldwide welfare losses0-2-4= 0.01= 0.20= 0.40

-6
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

*
w

Figure 3: Relative welfare loss (1): Various parameter settings
Note: Each value is expressed as a percentage deviation from the result at the baseline parameter setting of∗θw = θw= 0.6.

461

w= 0.010-10commitment
discretion

-20
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.6

0.7

0.8

0.9

1

0.6

0.7

0.8

0.9

1

0.6

0.7

0.8

0.9

1

= 0.40
w
0

-10

-20
0

0.1

0.2

0.3

0.4

0.5

= 0.60
w
0

-10

-20
0

0.1

0.2

0.3

0.4

0.5

= 0.80
w
0

-10

-20
0

0.1

0.2

0.3

0.4

0.5
*
w

Figure 4: Commitment vs. Discretion (1): Domestic wage stickiness∗Note: Each value is expressed as a percentage deviation from the result at θw = θw
= 0.6, which is the settingof the baseline parameter under the commitment policy.

47

= 0.5050-5commitment
discretion

-10
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.6

0.7

0.8

0.9

1

0.6

0.7

0.8

0.9

1

0.6

0.7

0.8

0.9

1

= 1.00
5
0
-5
-10
0

0.1

0.2

0.3

0.4

0.5

= 2.00
5
0
-5
-10
0

0.1

0.2

0.3

0.4

0.5

= 6.00
5
0
-5
-10
0

0.1

0.2

0.3

0.4

0.5
*
w

Figure 5: Commitment vs. Discretion (2): Intertemporal substitution of consumption∗
Note: Each value is expressed as a percentage deviation from the result at θw = θw
= 0.6, which is the settingof the baseline parameters under the commitment policy.

48

= 0.0120
commitment
discretion

-2
-4
-6
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.6

0.7

0.8

0.9

1

0.6

0.7

0.8

0.9

1

= 0.20
2
0
-2
-4
-6
0

0.1

0.2

0.3

0.4

0.5

= 0.40
2
0
-2
-4
-6
0

0.1

0.2

0.3

0.4

0.5
*
w

Figure 6: Commitment vs. Discretion (3): Degree of openness∗
Note: Each value is expressed as a percentage deviation from the result at θw = θw
= 0.6, which is the settingof the baseline parameters under the commitment policy.

49

10-4

Home inflation

Foreign inflation

0

0

-5

-0.01
commitment
discretion

-10

-0.02

-15

-0.03

-20

-0.04
2

4

6

8

10

12

14

16

18

20

2

4

6

Home output

0

8

10

12

14

16

18

20

14

16

18

20

16

18

20

14

16

18

20

14

16

18

20

16

18

20

Foreigin output
0.3

-0.01
0.2
-0.02
0.1

-0.03
-0.04

0
2

4

6

8

10

12

14

16

18

20

2

Home CPI inflation

0.05

4

6

8

10

12

Foreign CPI inflation

0.2

0
0.1
-0.05
0
-0.1
-0.15

-0.1
2

2

10

4

6

-4

8

10

12

14

16

18

20

2

Home wage inflation

10

10

4

6

-3

8

10

12

14

Foreign wage inflation

1
5

0
-1

0

-2
2
10

4

6

-3

8

10

12

14

16

18

20

2

4

6

Home real wage

6

8

10

12

Foreign real wage
0.15

4

0.1

2

0.05

0

0
2

4

6

8

10

12

14

16

18

20

2

Home terms of trade

0

4

6

-0.1

-0.2

-0.2

-0.3

-0.3

-0.4

10

12

Nominal exchange rate

0

-0.1

8

-0.4
2

4

6

8

10

12

14

16

18

20

2

4

6

8

10

12

14

Figure 7: Impulse responses of a foreign productivity shock: Commitment vs. Discretion

50

10

10-4

Home inflation

Foreign inflation
0.4

commitment
discretion

0.3

5

0.2
0.1

0

0
2

4

6

8

10

12

14

16

18

20

2

Home output

0.06

4

6

8

10

12

14

16

18

20

14

16

18

20

16

18

20

14

16

18

20

14

16

18

20

16

18

20

Foreigin output

0
-0.1

0.04
-0.2
0.02
-0.3
0

-0.4
2

4

6

8

10

12

14

16

18

20

2

Home CPI inflation

0.2

4

6

8

10

12

Foreign CPI inflation

0.3
0.2

0.1

0.1
0
0
-0.1

-0.1
2
10

4

6

-4

8

10

12

14

16

18

20

2

Home wage inflation

4

6

10

12

14

Foreign wage inflation

0.02

5

8

0.01
0

0
-0.01

-5
-0.02
2
0

10

4

6

-3

8

10

12

14

16

18

20

2

Home real wage

-0.1

-1

-0.2

-1.5

-0.3

-2

-0.4
4

6

8

10

12

6

14

16

18

20

8

10

12

Foreign real wage

0

-0.5

2

4

2

4

6

Home terms of trade

8

10

12

Nominal exchange rate
0

0.4
0.3

-0.1

0.2
-0.2
0.1
0

-0.3
2

4

6

8

10

12

14

16

18

20

2

4

6

8

10

12

14

Figure 8: Impulse responses of a foreign cost-push shock: Commitment vs. Discretion

51

10-4

Home inflation

5

10-3

Foreign inflation

00commitment
discretion

-5

-5
-10

-10

-15
2

4

6

8

10

12

14

16

18

20

2

Home output

0

4

6

0.3

-0.02

0.2

-0.03

10

12

14

16

18

20

14

16

18

20

16

18

20

14

16

18

20

14

16

18

20

16

18

20

Foreigin output

0.4

-0.01

8

0.1

-0.04

0
2

4

6

8

10

12

14

16

18

20

2

4

6

Home CPI inflation

8

10

12

Foreign CPI inflation

0

0.2
0.15

-0.05

0.1
-0.1

0.05
0

-0.15
2
6

10

4

6

-3

8

10

12

14

16

18

20

2

4

6

Home wage inflation

8

10

12

14

Foreign wage inflation
0.2
0.15

4

0.1

2

0.05
0
0
-2
2

4

6

8

10

12

14

16

18

20

2

Home real wage

0.02

4

6

0.3

0.01

0.2

0.005

0.1

0

10

12

Foreign real wage

0.4

0.015

8

0
2

4

6

8

10

12

14

16

18

20

2

Home terms of trade

0

4

6

-0.1

-0.2

-0.2

-0.3

-0.3

-0.4

10

12

Nominal exchange rate

0

-0.1

8

-0.4
2

4

6

8

10

12

14

16

18

20

2

4

6

8

10

12

14

Figure 9: Impulse responses of a foreign productivity shock: In the case of flexible wages

52

5

10-4

Home inflation

Foreign inflation
0.4

0
0.2
commitment
discretion

-5

0
-10
-15

-0.2
2

4

6

8

10

12

14

16

18

20

2

Home output

0.1

4

6

8

10

12

14

16

18

20

14

16

18

20

16

18

20

14

16

18

20

14

16

18

20

16

18

20

Foreigin output

0
-0.2

0.05

-0.4
-0.6

0
2

4

6

8

10

12

14

16

18

20

2

4

6

Home CPI inflation

0.4

8

10

12

Foreign CPI inflation
0.2
0.1

0.2

0
0
-0.1
-0.2

-0.2
2

4

6

8

10

12

14

16

18

20

2

Home wage inflation

0.02

4

6

0

-0.02

-1

-0.04

10

12

14

Foreign wage inflation

1

0

8

-2
2

4

6

8

10

12

14

16

18

20

2

Home real wage

4

6

8

10

12

Foreign real wage

0

0
-0.01

-0.5

-0.02

-1

-0.03

-1.5

-0.04
-2
2

4

6

8

10

12

14

16

18

20

2

Home terms of trade

1

4

6

10

12

Nominal exchange rate

0.5

0.5

8

0

0

-0.5
2

4

6

8

10

12

14

16

18

20

2

4

6

8

10

12

14

Figure 10: Impulse responses of a foreign cost-push shock: In the case of flexible wages

53