Research InstituteDiscussion Paper Series№ 7
Title：The role of money and optimal monetary policy in a
two-country economy

Daisuke Ida
ida-dai@andrew.ac.jp

2018 年 5 月

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

The role of money and optimal monetary policy
in a two-country economy
Daisuke Ida∗
May 9, 2018

Abstract
The importance of the role of money is evident from the monetary policies
of advanced countries. This paper examines the eﬀect of a non-separable utility
between consumption and real money balance on optimal monetary policy in a
two-country framework. It shows that relative real money balances between two
countries aﬀect the international risk-sharing condition directly when considering
the non-separable utility. Departure from the separable utility function changes
the structural equations in a two-country economy with nominal rigidities. In
addition, the shape of the loss function is diﬀerent from that of a standard loss
function. Therefore, the assumption of a non-separable utility aﬀects the propertiesof optimal monetary policy significantly. The presence of a non-separable utility
function between consumption and money crucially changes the termination dateof a zero interest rate policy for both countries.

Keywords: Optimal monetary policy; Money demand function; Two-country model; Zerolower bound on nominal interest rates
JEL classification: E52; E58; F41∗

Department 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

1

Introduction

The new Keynesian model (NKM) approach to recent monetary policy analysis is prevalent (Clarida, Gali and Gertler, 1999; Woodford, 2003; Walsh, 2017; Gali, 2015). The
standard NKM assumes that the central bank implements a monetary policy by manipulating its policy rate. As argued in a study by Woodford (2003), this theory is constructedin a cashless economy in which money does not play a significant role because the money
demand function indicates that a change in the nominal interest rate automatically leads
to determination of money aggregate. This implies the redundancy of the money demand
function when the model presumes separability between consumption and real money
balances in a household’s utility function.1
However, monetary policies in advanced countries reveal the important role of money.
Central banks in advanced country reduce their policy rates to mitigate the risk of
deflation arising from the Lehman crisis in the United States. Consequently, central
banks have faced with non-negativity constraints on nominal interest rates. Therefore,
the central banks increase the monetary base to combat deflation and recession when
faced with zero lower bound (ZLB). Such policy stances in advanced countries are referred
to as quantitative easing.
This paper focuses on the fact that a change in money supply in one country aﬀects
the monetary policy strategy of other countries. For instance, the quantitative easing
policy that the Federal Reserve Board (FRB) implemented in 2008 might have significantly aﬀected the monetary policy stance implemented by the Bank of Japan (BOJ).
In fact, a large-scale asset purchases (LSAPs) by FRB depreciated the exchange rate
in terms of dollar, whereas the BOJ faced a severe appreciation of the exchange rate.
However, the standard NKM that studies the eﬀect of ZLB on the economy does notconsider the role of money.
This paper addresses the role of money in a two-country economy. As mentioned1

Woodford (2003) examines the role of the non-separable utility between consumption and money

in a closed economy. Also, Kurozumi (2006) examines the determinacy condition in a model with thenon-separable utility.

1earlier, the role of money is limited, unless it is assumed that consumption and realmoney balances are not separable in a household’s utility function. A two-country modelwith sticky prices has been developed by several previous studies (Betss and Devereux,
2001; Clarida, Gali, and Gertler, 2002; Corsetti and Pesenti, 2001; Corsetti and Pesenti,
2005; Devereux and Engel 2003; Engel, 2009; Engel, 2011). These studies abstract therole of non-separable utility in the model. The present study considers the non-separable
utility between consumption and real money balance in a two-country framework. Morespecifically, this paper examines the role of money in a two-country model with homebias constructed by Engel (2009, 2011).2
Clarida, Gali, and Gertler (2002) shows that, under the assumption that both producercurrency pricing and home bias do not hold, consumption in the home and foreign countries is identical provided that a complete market exists in which home and foreign
households can trade state-contingent securities domestically and internationally. Thisis referred to as the consumption risk-sharing condition for open economies. Even if we
relax the assumption of no home bias, relative consumption between countries is adjusted by a movement of the real exchange rate (Chari, Kehoe, and McGrattan, 2002;
Pappa, 2004).3 However, this paper shows that relative real money balances between two
countries aﬀect the international risk-sharing condition directly when we consider a casewhere consumption and real money balances are not separable in the household’s utility
function.
How does such a departure from the separable utility function change the structural
equations in a two-country economy with nominal rigidities? Woodford (2003) considers
the role of the non-separable utility function in a closed economy model. It allows the new
Keynesian Phillips curve (NKPC) to be aﬀected by real money balances. Interestingly, ina two-country model, in which consumption and real money balances are not separable,
2

Engel (2009, 2011) examines how local currency pricing changes the properties of optimal monetary

policy in a two-country economy model with home bias. This paper introduces the presence of homebias in the model, whereas it abstracts that of pricing to market associated with local currency pricing.
3

See Chari, Kehoe, and McGrattan (2002) and Pappa (2004) for a detailed discussion about thisissue.

2

relative real money balances directly aﬀect not only the dynamic IS equation and themoney demand function, but also the NKPC. In addition, the non-separable utilitychanges the shape of the central bank’s loss function. Clarida, Gali, and Gertler (2002)
and Engel (2009, 2011) show, under policy coordination,4 the loss function contains
the squared output gaps and the producer price index (PPI) inflation rates of bothcountries in a two-country economy with producer-currency pricing. The shape of theloss function is diﬀerent from such a standard loss function when consumption and realmoney balances are not separable in the utility function. First, the central banks needto stabilize diﬀerences between home and foreign real money balances. Second, the loss
function includes the cross term for relative output gap and relative real money balances.
Hence, the assumption of the non-separable utility between consumption and realmoney balances significantly aﬀects the properties of optimal monetary policy. This paper regards optimal monetary policy in a timeless perspective as suggested by Woodford(2003). It focuses on commitment under policy coordination.5 Under optimal monetary
policy, a foreign natural interest rate shock induces higher inflation and output gap in
the foreign country. The foreign central bank increases its nominal interest rate gradually under optimal monetary policy. In particular, for higher substitutability, the foreign
policy rate is smaller initially, whereas the foreign central bank raises its policy rate
aggressively in subsequent periods. The home central bank maintains a positive policy
rate to stabilize the inflation rate. An increase in the policy rate leads to a negative
response of the output gap. For higher substitutability, the home central bank reinforces
increase in the policy rate under the optimal monetary policy. Therefore, a foreign natural interest rate shock generates a policy trade-oﬀ between inflation and the output gap
in the home country.
Optimal policy under a foreign money demand shock is diﬀerent from that under a4

As mentioned, policy coordination implies that both domestic and foreign central banks jointly

minimize a weighted sum of the loss function of domestic and foreign central banks.
5This paper does not cover gain from policy coordination because when optimal monetary policy

is considered under no policy coordination, one faces the complicated derivation of a well-defined loss
function of the central bank. For instance, see Engel (2009, 2011) for a detailed discussion of this issue.

3foreign natural interest rate shock. For a foreign country, higher substitutability leads tomore aggressive increase in the policy rate after the shock. Thus, the foreign central bankaccommodates a huge decline in the output gap in the initial period instead of increasing
the inflation rate for subsequent periods. On the other hand, the home central bankaggressively cuts its policy rate in response to the shock. Thus, under optimal monetary
policy, the home central bank cuts its policy rates aggressively. In contrast, a foreign
money demand shock leads to a boom in the output gap and an increase in inflation in
the home country.
Furthermore, this paper examines how the assumption of a non-separable utility
between consumption and money aﬀects the optimal monetary policy when the ZLB
simultaneously hits both countries. A new Keynesian model emphasizes the role of
ZLB (Eggertsson and Woodford, 2003; Jung, et al., 2005; Adam and Billi, 2006; Adamand Billi, 2007, Guerrieri and Iacoviello, 2015; McKay, Nakamura and Steinsson, 2016;
Nakata, 2017). Consider the case where a natural interest rate takes a negative value.
It cannot stimulate the real economy once it hits ZLB. As argued in Eggertsson andWoodford (2003) and Jung, Teranishi and Watanabe (2005), the central bank can insulate aggregate demand through history-dependent commitment policy even when thenominal interest rate reaches the zero bound.
Nakajima (2008) investigates optimal monetary policy in a two-country economy,
but the study considered only the domestic interest rate as zero. Ida (2013) derives
optimal monetary policy rules in a two-country economy in which both home and foreigncountries simultaneously face the ZLB. Cook and Devereux (2011) examine also the roleof fiscal policy under the ZLB in a two-country framework with Calvo pricing.6 Fujiwara
et al. (2010, 2014) examine also optimal monetary policy responses to non-negativity
constraints on nominal interest rate existing in both home and foreign countries.
The previous studies abstract the role of money in the NKM. However, the present
study addresses the recent evidence that money aggregate plays a significant role in

6

Their model is close to our model, however, this paper is diﬀerent from the study by Cook andDevereux (2011), in that the present paper focuses on the role of money in the model.

4non-conventional monetary policy. In unconventional monetary policy, central banks
anticipate the eﬀect of quantity of money on real economy in addition to a commitment
to a zero interest rate policy (ZIRP). Central banks, such as the FRB, the Bank ofEngland, and the BOJ, expect that quantitative easing would have an eﬀect of moneyon the real economy.7 It is well known that these central banks follow the quantitativeeasing policy and we show that the ZLB plays a significant role in a two-country modelwith the non-separable utility function between money and consumption. The previous
studies do not account for the eﬀect of money on the real economy in a two-country
model in which central banks simultaneously face the ZLB. Although this paper cannot
explicitly derive the policy rule associated with money aggregate, we can conjecture thatthe central bank can implicitly follow such a rule in terms of the micro-founded loss
function of the central banks.8This paper explores the case of a foreign negative natural interest rate shock under
optimal policy when ZLB hits both countries. The foreign central bank maintains a
ZIRP for a considerably long period even when the natural rate of interest takes apositive value. Accordingly, foreign inflation overshoots in response to such a monetary
policy stance, which introduces policy inertia into the foreign output gap. On the other
hand, a foreign negative interest rate decreases home inflation, whereas it increases thehome output gap. Therefore, the home central bank employs a ZIRP also. Since thehome country experiences a positive output gap, the termination date of a home ZIRP is
considerably shorter than that in the foreign country. After the termination of a ZIRP,
the home central bank sharply increases its policy rate. In particular, in contrast to
the foreign nominal interest rate following the decision to terminate a ZIRP, the home
central bank allows policy rate increase from a ZIRP termination date.
Also, I consider the case of a foreign money demand shock. Interestingly, it considerably changes the property of optimal monetary policy in contrast to a natural interest
7

See English et al. (2013) and Williamason (2015) for a detailed discussion of the eﬀect of quantitativeeasing policies.
8This paper focuses on the commitment policy under policy coordination. Therefore, the foreign

central bank can also implement the optimal rule based on money aggregate.

5

rate shock. The money demand shock leads to a negative response of both inflationand the output gap in the foreign country, whereas the home country experiences surge
in both inflation and the output gap. Nevertheless, the home central bank employs a
ZIRP for a few periods. After the termination of a ZIRP, the home central bank sharply
increases its policy rate and maintains for some time, whereas the ZIRP termination isshorter than that for the foreign country. In contrast to a movement of nominal interest
rate in the home country, the foreign central bank raises its policy rate gradually after
the termination of a ZIRP. Such a policy stance leads to increase in inflation and theoutput gap in the home country.
The present paper is structured as follows. Section 2 derives a two-country new Keynesian model with a non-separable utility function between consumption and real money
balances. Section 3 describes equilibrium conditions in the model. Section 4 provides
the log-linearized equations of the model around the eﬃcient steady state. Section 5
derives the central bank’s loss function by calculating the second-order approximationof the household’s utility function in both countries. Then I examine optimal monetary
policy in a two-country model with the non-separable utility function. It demonstrates
the properties of optimal monetary policy. Section 6 considers the properties of optimalmonetary policy when the ZLB hits both countries simultaneously. Section 7 briefly concludes. Appendix A provides the detailed derivation of the second-order approximationof the household’s utility function. Appendix B derives the optimal monetary policy
with commitment under policy coordination.

2

The model

A standard two-country model, home and foreign, with nominal price rigidities developedby Clarida, Gali and Gertler (2002) and Engel (2009, 2011) is adopted. Based on a study
by Engel (2011), the model employs diﬀerent preferences in the two countries. This
paper allows for home bias. In each country, households obtain utility from consumption
and real money balances and disutility from labor supply. This model assumes that
consumption and real money balances are not separable in a household’s utility function.
6

Households in each country have access to a complete set of state-contingent securities
that are traded both domestically and internationally.
Each country has firms that face a monopolistically competitive environment. According to Calvo (1983), such firms set staggered nominal prices. Since firms set theirprices in their own currency, this implies that the present paper assumes the producercurrency pricing (PCP) model. Finally, unless otherwise noted, analogous equations holdfor a foreign country.

2.1

Households

The intertemporal utility of an infinitely lived representative household subscripted by
h is

Ut = E0∞∑[]
β u(Ct (h), zt (h)) − v(Nt (h)) ,
t

(1)

t=0

where Ct denotes the consumption, Zt denotes the real money balances, and Nt denotes
the household’s labor supply. In addition, the parameter β is discount factor. The
utility function u(·) is strictly concave and continuously diﬀerentiable with respect toboth Ct and Zt . The disutility of labor supply v(·) is strictly convex and continuously
diﬀerentiable. Moreover, unlike Clarida, Gali and Gertler (2002) and Engel (2011), theutility function is not separable between consumption and real money balances in thispaper. Thus, I assume ucm (·) ≥ 09 . This paper assumes the following Cobb-Douglas
preferences for consumption:
Ct = (CH,t (h))a/2 (CF,t (h))(1−a/2) ,(2)

where CH,t (h) represents consumption of home goods and CF,t (h) represents the consumption of foreign goods. The parameter a denotes the degree of home bias. When theparameter a takes unity, home and foreign preferences are the same as those in the study

9

As in Benhabib, Schmitt-Gohe and Uribe (2001), here the case of ucm (·) ≤ 0 is not considered.

7

by Clarida, Gali, and Gertler (2002). Here, CH,t (h) and CF,t (h) are CES aggregates overa continuum of goods produced in each country:[∫ 1]θ/(θ−1)(θ−1)/θ
CH,t =
CH,t (h, j)
dj
,
0

[∫ 1

]θ/(θ−1)(θ−1)/θ

CF,t =(3)

CF,t (h, j)

dj

,

(4)

0

where the parameter θ is the elasticity of substitution for individual goods.10
The labor supply Nt (h) is an aggregate of the labor supply that the household sells
to each of a continuum of firms located in the home country:∫ 1
Nt (h) =Nt (h, j)dj,(5)0

The representative household maximizes Equation (1) subject to the following budget
constraint:
Pt Ct (h) + Mt (h) + Et [Qt,t+1 Bt+1 (h)] = Bt (h) + Mt−1 (h) + Wt (h)Nt (h) + Πt + Tt ,(6)
where Bt (h) is the nominal bond, Mt is the nominal money supply, and Tt denotes thenominal lump-sum tax. Also, Wt (h) and Πt are the nominal wage and the dividend fromintermediate goods firms. The variable Qt,t+1 denotes the stochastic discount factor,
which is defined as follows:
Et (Qt,t+1 ) =1,
1 + it(7)

where it is the nominal interest rate. In addition, Pt is the aggregate price index for
consumption, which is given as follows:
Pt = κ−1 (PH,t )a/2 (PF,t )(1−a/2) ,(8)

where κ = (1 − a/2)(1−a/2) a/2a/2 . PH,t is the price of domestic goods, and PF,t is theprice of foreign goods. Equation (8) are derived from the household’s cost minimization
10

In this model, this parameter is time-invariant. As in Steinsson (2003), we can introduce a time-varying elasticity of substitution if we examine the case of a price mark-up shock.

8

problem. Both PH,t and PF,t follow the CES aggregate over prices of individual varieties
j:
[∫ 1

]1−θ
1−θ

PH,t =

PH,t (h, j)

dj

0

[∫ 1

(9)

.

(10)

]1−θ
1−θ

PF,t =,

PF,t (h, j)

dj

0Note that the foreign households also have analogous preferences and face an analogous budget constraint.
Similar to Engel (2011), we can drop subscript h for the household and use the factthat the aggregation per capita consumption of each good is equal to consumption ofeach good by each household. From the first order conditions of the utility maximization
problem, the following equations are obtained:()−1
a PH,tCt ,CH,t =
2 Pt
(
)(
)−1
a
PF,t
CF,t = 1 −
Ct ,
2
Pt
(
)−θ
PH,t (j)CH,t (j) =CH,t ,PH,t()−θ
PF,t (j)CF,t (j) =CH,t ,PF,tituz (Ct , Zt ),=uc (Ct , Zt )1 + itvn (Nt )Wt−=,
uc (Ct , Zt )Pt
uc (Ct+1 , Zt+1 ) Pt.
Qt,t+1 = β
uc (Ct , Zt ) Pt+1

(11)
(12)
(13)
(14)
(15)
(16)
(17)

Equations (11) and (12) are derived from the household’s cost minimization problem
under the assumption of the Cobb-Douglas preferences for consumption. Equations (13)and (14) represent demand equations for both CH,t and CF,t . Equation (15) is the money
demand function. This equation states that the marginal rate of substitution between
consumption and real money balances is equal to the opportunity cost of holding money.
Equation (16) states the intratemporal optimality condition that the marginal rate of
substitution between consumption and household labor supply is equal to the real wage.
9

As noted earlier, there are complete markets in each country. Therefore, by taking the
expectations of each side of Equation (17) and using the definition of the stochastic
discount factor, the following familiar Euler equation is derived:
βEt (1 + it )uc (Ct+1 , Zt+1 ) Pt
= 1.
uc (Ct , Zt ) Pt+1

(18)

Equation (18) requires that in equilibrium, the marginal utility of consumption intertemporally equalizes through the adjustment of the real interest rate. Note that the analogous Euler equation is applied for the foreign country. In particular, by the assumptionof the international tradability of state-contingent securities, the intertemporal eﬃcientcondition for the foreign country is given byβEt (1 + i∗t )∗∗uc (Ct+1, Zt+1) ε t Pt
= 1.
∗∗
uc (Ct , Zt ) εt+1 Pt+1

(19)

where εt is the nominal exchange rate in terms of the home currency price. In addition,
an asterisk is used to denote variables associated with foreign countries.
Finally, as in Engel (2011), this paper presumes at this stage that labor supply ofhouseholds is the same, that is, Nt = Nt (h).

2.2

Firms

The firms in each country are characterized by monopolistic competition and each firm
produces diﬀerentiated goods. Each home good Yt (j) is produced by the following production function:
Yt (j) = At Nt (j),(20)

where At is an aggregate productivity disturbance.
The firm’s profit function is given by∗∗Πt (j) = PH,t (j)CH,t (j) + εt PH,t(j)CH,t(j) − (1 − τ )Wt Nt (j),(21)

where PH,t (j) the home-currency price of the goods in the case where it is sold in the∗home country. On the other hand, PH,t(j) denotes the foreign-currency price of the goods

10when it is sold in the foreign country. The parameter τ denotes optimal subsidy. Finally,
CH,t (j) is the aggregate sales of the goods in the home country:∫ 1
CH,t (j) =CH,t (h, j)dj.

(22)

0

Analogous equations hold good in the foreign country. Note the following:∗
Yt (j) = CH,t (j) + CH
(j),
t

(23)

Yt∗ (j) = CF,t (j) + CF∗t (j).

(24)

Following Calvo (1983), this paper assumes that firms set their nominal prices on
the basis of a staggered manner. Thus, a fraction 1 − ω of all firms adjusts their price,whereas the remaining fraction of firms ω do not. Therefore, a fraction ω of prices
remains unchanged from the previous period. These firms consider uncertainty during
price revision until the next price adjustment.
We now consider the firms that can adjust their prices. As in Engel (2011), when thefirm can reset its price, it will be able to change its sales price in both the countries. In
the PCP model, firms set their nominal prices in its own currency. The home firm that∗∗∗can reset its price sets PH,t (j) and PH,t(j) = εt PH,t(j) in terms of home currency. In thiscase, the firm’s optimization problem is given by[]∞∑
k uc (Ct+k , Zt+k )
o
o∗∗
∗
Et
(ωβ)
PH,t (j)CH,t+k (j) + εt PH,t (j)CH,t+k (j) − (1 − τ )Wt+k Nt+k (j) ,uc (Ct , Zt )
k=0
(25)
o
o∗∗
where PH,t(j) is the optimal home price index and PH,t(j) is the optimal price for foreignoo∗∗sales. The firms that reset their prices select its reset prices PH,t(j) and PH,t(j) tomaximize the firm’s objective function (25), subject to the sequence of demand curves
given by Equation (13) and the corresponding foreign demand function for home goods.

11The first order conditions of this profit maximization problem are as follows:11
∑
k
θ
Et ∞
k=0 (ωβ) uc (Ct+k , Zt+k )(1 − τ )Wt+k PH,t+k CH,t+k /At+ko∑∞PH,t =,
θEt k=0 (ωβ)k uc (Ct+k , Zt+k )PH,t+k
CH,t+k∑kθ ∗Et ∞k=0 (ωβ) uc (Ct+k , Zt+k )(1 − τ )Wt+k (εt+k PH,t+k ) CH,t+k /At+ko∗∗∑∞PH,t =.
∗
Et k=0 (ωβ)k uc (Ct+k , Zt+k )(εt+k PH,t+k )θ CH,t+k
Under Calvo pricing, price levels are defined as follows:
]
[
1−θ
o 1−θ
+ ω(PH,t−1 ),PH,t = (1 − ω)(PH,t )[]∗o∗ 1−θ
∗1−θPH,t = (1 − ω)(PH,t )+ ω(PH,t−1 )
.

3

(26)
(27)

(28)
(29)

Equilibrium

This section describes the equilibrium conditions in a two-country model. The equilibrium conditions for the goods market in each country are given as follows:∗
Yt = CH,t + CH
,
t

(30)

Yt∗ = CF,t + CF∗t .

(31)

Substituting Equation (11) and the corresponding equation for the foreign country,
Equation (30) can be rewritten as follows:[()]
a 1−a/2a∗ −a/2 ∗Yt = κ S t
Ct + 1 −
(St )
Ct ,
2
2
)
[
(
]
a ∗ −a/2 ∗a−a/2∗
Yt = κ (St )Ct + 1 −
(St )
Ct ,
2
2

(32)
(33)

where St and St∗ are the home terms of trade and the foreign terms of trade, respectively:
PF,t
,
PH,t
∗
PH,t
∗
St = ∗ .
PF,t
St =

11

(34)
(35)

Under the PCP model, this equation is redundant because the law of one price holds. However,under the local-currency pricing model, this equation is required since the law of one price no longer
holds. See Corsetti, Dedola and Leduc, (2010) and Engel (2011) for a detailed discussion of this issue.

12Note that since the law of one prices hold under the PCP model, we have St = (St∗ )−1 .
Here, in this model purchasing power parity (PPP) does not hold because of thepresence of home bias. Even when PPP does not hold, since households in each country can trade a state-contingent security internationally, combining Equation (18) with
Equation (19) and imposing a suitable normalization of initial condition, we obtain thefollowing equation:
uc (Ct∗ , Zt∗ )ε t Pt(a−1)= ∗ = St.
uc (Ct , Zt )
Pt

(36)

When the PPP condition holds in the absence of home bias, the relationship uc (Ct , Zt ) =uc (Ct∗ , Zt∗ ) holds. The standard two-country NKM assumes a cashless-limit case or thecase that consumption and real money balances are separable in the household’s utility
function. As shown in Clarida, Gali and Gertler (2002), uc (Ct ) = uc (Ct∗ ) in such a case.
This implies that consumption in the home country completely co-moves one in theforeign country. When home bias is present, as shown in Chari, Kehoe, and McGrattan(2002), the relative consumption is aﬀected by the real exchange rate. In contrast to theirmodel, our model allows that both home and foreign real money balances change therelative consumption under the assumption of a non-separable utility function between
consumption and real money balances.
Under flexible prices, all firms set their prices equal to a constant markup overmarginal cost. In addition, this paper assumes that the government can set a constant optimal subsidy rate for monopolistic firms. Such an optimal subsidy allows thegovernment to achieve an eﬃcient allocation in the deterministic steady state.
n
n∗
PH,t
= εnt PH,t
= (1 − τ )µ

Wtn
,
At

(37)

where a variable with (n) denotes one in the flexible price equilibrium. In addition, theparameter µ is the firm’s price markup, which is given by θ/(θ − 1). When optimal
subsidies are imposed in place, the following equation holds∗
P̄H,t = ε̄t P̄H,t= (1 − τ )µ

13

W̄t
,
At

(38)

where a variable with (−) denotes one in the eﬃcient level equilibrium. Thus, it follows
from Equations (37) and (38) that an optimal subsidy rate is given by(1 − τ )µ = 1.

4

Log-linearization

The system derived in the previous section is now log-linearized around the steady state.
Here lower case variables are used to denote a log-deviation from the steady state. Specifically, a log-linearized variable around the steady state is expressed by ht = log(Ht /H̄),
where H̄ represents a steady-state value.
First, log-linearization of Equation (32) is given byyt =a(2 − a)a(2 − a) ∗s t + ct +
ct .
2
2
2

(39)

Similarly, log-linearization of Equation (33) is given byyt∗ = −a(2 − a)a(2 − a)st + c∗t +
ct .
2
2
2

(40)

Next, log-linearizing Equation (36) gives the following
σ(ct − c∗t ) = χ(zt∗ − zt ) + (a − 1)st ,(41)

where σ = −ucc C̄/uc and χ = ucz Z̄/uz . In the case of the non-separable utility function,
as noted earlier, relative real money balances aﬀect relative consumption, inducing achange in the terms of trade. If the utility function is separable, the parameter χ is zero.
Hence, the first-term of the right-hand side is zero. Therefore, relative consumption between the two countries is only adjusted by the change in the terms of trade as providing
there is home bias. In particular, if we make an assumption of both the separable utility
and no home bias, a movement of home consumption completely coincides with thatof foreign consumption due to the presence of an international complete market (e.g,Clarida, Gali and Gertler, 2002).
Subtracting Equation (40) from Equation (39) results in the following equation:yt − yt∗ = a(2 − a)st + (a − 1)(ct − c∗t ).
14

(42)

Substituting Equation (42) into Equation (41), the following relationship is derivedst =1[σ(yt − yt∗ ) + (a − 1)χ(zt − zt∗ )],Θ

(43)

where Θ = σa(2 − a) + (a − 1)2 .
At this point, inserting Equation (41) into Equation (43), the following equation is
obtained:
ct = ψyt + (1 − ψ)yt∗ + ζ(zt∗ − zt ),

(44)

where
Θ + (a − 1),2Θχa(2 − a)
.
ζ=
2Θ
ψ=

It is assumed that an increase in real money balances leads to an increase in consumption
in the closed economy model. However, this inference might not be correct in an open
economy model. Equation (36) indicates that this is because the diﬀerence between
home and foreign real money balances directly changes relative consumption. Equation(43) shows that it turns out that home consumption is negatively aﬀected by home realmoney balances only if the household’s utility function in each country is not separable
between consumption and money. However, home consumption co-moves with a change
in foreign real money balances.12 Similarly, an analogous equation holds in the foreigncountry as follows:c∗t = ψyt∗ + (1 − ψ)yt − ζ(zt∗ − zt ).

(45)

Next, log-linearization of the firm’s optimization condition leads to the following
NKPC:
πt = βπt+1 + δ(wt − pH,t − at ),12

If the parameter χ takes a negative value, a polar result is obtained.

15

(46)

whereδ=(1 − ω)(1 − ωβ).
ωAlso, πt = log(PH,t /PH,t−1 ) is the PPI inflation rate. An analogous equation holds forthe foreign country.
Log-linearization of Equation (16) leads to
wt − pH,t = ηnt + σct − χzt +(2 − a)
st ,
2

(47)

where η = vN N N̄ /vn . Substituting the log-linearized production function, Equations(44) and (47) leads to[]Θ+1
σ(Θ + 1)σ(Θ − 1) ∗Θ−1
wt − pH,t =+ η yt +yt − χ
zt + χ
zt + ηat .
2Θ
2Θ
2Θ
2Θ
Under eﬃcient price equilibrium, Equation (48) becomes[]σ(Θ + 1)σ(Θ − 1) n∗
Θ+1 n
Θ − 1 n∗nn
wt − pH,t =+ η ytn +yt − χ
zt + χ
z + ηat .
2Θ
2Θ
2Θ
2Θ t

(48)

(49)

where a variable with subscript n denotes one under eﬃcient price equilibrium. Subtracting Equation (49) from Equation (48), we obtain[]σ(Θ + 1)σ(Θ − 1) ∗Θ+1
Θ−1w̃t − p̃H,t =+ η ỹt +ỹt − χz̃t + χ
z̃t .
2Θ
2Θ
2Θ
2Θ

(50)

where a variable with a tilde denotes one in terms of deviation of a variable from one ofits flexible price equilibrium. Finally, substituting Equation (50) into Equation (46), we
obtain the NKPC expressed in terms of the output gap.
πt = βEt πt+1 + κ1 ỹt + κ2 ỹt∗ − κ3 z̃t + κ4 z̃t∗

(51)

where
[

]
σ(Θ + 1)σ(Θ − 1)
κ1 = δ+ η , κ2 = δ
,
2Θ
2Θ
Θ+1
Θ−1
κ3 = χ
, κ4 = χ.
2Θ
2Θ
In contrast to the previous studies that examine monetary policy in a two-country economy, this paper shows that foreign real money balances directly aﬀect home inflation
16

because of the presence of a non-separable utility. Thus, the money demand function
plays a significant role in this model. For instance, given the level of real money balances in the home country, a decline in the policy rate in the foreign country induces anincrease in foreign real money balances, thus increasing home inflation.
An analogous equation holds for the foreign country:∗+ κ1 ỹt∗ + κ2 ỹt − κ3 z̃t∗ + κ4 z̃t ,πt∗ = βEt πt+1(52)Here, the home money demand function expressed by the gap term is given byz̃t = µ1 ỹt + µ2 ỹt∗ + µ3 z̃t∗ − µ4 it + ςt ,(53)

where ςt denotes the money demand shock. In addition, coeﬃcients in Equation (53) aredefined asηc χ[Θ + (a − 1)]2Θ + χa(2 − a)ηc χ[Θ − (a − 1)], µ1 =, µ2 =
,
2Θ
2Θµ0
2Θµ0
ηc χa(2 − a)ηiµ3 =, µ4 = ,
2Θµ0
µ0

µ0 =

where ηc is the income elasticity of money demand and ηi is the interest elasticity ofmoney demand. In contrast to the standard money demand function, Equation (53)includes foreign real money balances, which are positively related to home real money
balances. Thus, other things being equal, a decline in the policy rate in the foreigncountry increases foreign money balances, which leads to a decrease in the home inflation
rate. Similarly, for the foreign country the money demand function is given byz̃t∗ = µ1 ỹt∗ + µ2 ỹt + µ3 z̃t − µ4 i∗t + ςt ,(54)

Finally, a dynamic IS equation is given as follows:∗∗,¯ t ) + ϑ2 Et ∆z̃t+1 − ϑ3 ∆z̃t+1− σ0−1 (it − Et πt+1 − rrỹt = Et ỹt+1 + ϑ1 Et ∆ỹt+1(55)

where rr
¯ t denotes the natural rate of interest rate that holds the real interest rate in theeﬃcient price equilibrium, which is given by][(Θ − 1)Θ+1nn∗
Et ∆yt+1 +Et ∆yt+1 .
rr¯ t = σ0
2Θ
2Θ
17

(56)

In addition, the coeﬃcients for the dynamic IS curve are defined as follows:
σ0 = σΘ+1
Θ−1
Θ−1, ϑ1 = σ −1
, ϑ2 = σ −1 χ, ϑ3 = σ −1 χ
.
2Θ
Θ+1
Θ+1

Similarly, the corresponding IS equation for the foreign country is obtained, given by∗∗∗ỹt∗ = Et ỹt+1+ ϑ1 Et ∆ỹt+1 − σ0−1 (i∗t − Et πt+1
− rr
¯ ∗t ) + ϑ2 Et ∆z̃t+1− ϑ3 ∆z̃t+1 ,(57)

where rr
¯ ∗t denotes the natural rate of interest for the foreign country.

5

Optimal Monetary Policy

This section examines how a non-separable utility function between consumption andreal money balances changes the properties of optimal monetary policy in a two-country
economy model. Section 5.1 derives the loss function of the central bank in a twocountry model in which consumption and money are not separable in the household’sutility function. In contrast to the previous studies, the present paper addresses the roleof a non-separable utility in a two-country model. Section 5.2 presents an argument on
optimal monetary policy. Section 5.2.1 derives optimal monetary policy with commitment, and Section 5.2.2 shows the eﬀect of a non-separable utility between consumption
and real money balance in a two-country economy by illustrating the results of impulse
response functions.

5.1

The derivation of the central bank’s loss function

To investigate optimal monetary policy in an NKM, we must derive a well-defined loss
function with a micro-foundation. Rotemberg and Woodford (1997) and Woodford(2003) show that the second-order approximation of the household’s utility function
corresponds to the central bank’s loss function. In a two-country framework, Clarida,
Gali and Gertler (2002) and Engel (2011) derive the central bank’s loss function by using the second-order approximation of the utility function. In a two-country framework,18

we can consider an optimal policy under both policy and no policy coordination.13 As
mentioned earlier, this paper focuses on the case of policy coordination.
The utility function of the planner is given by
Wt = u(Ct , Zt ) + u(Ct∗ , Zt∗ ) − v(Nt ) − v(Nt∗ ),(58)

To obtain the well-defined loss function, we need to eliminate distortions created bymonopolistic competition and real money balances. The first distortion is eliminated by
an optimal subsidy rate that eliminates price markup created by monopolistic competition. Fiscal authorities choose an optimal subsidy rate that restores the natural level of
output to the eﬃcient level of one in a zero-inflation steady state. As noted earlier, suchan optimal subsidy is given as follows:(1 − τ )µ = 1.
The second distortion is a result of an opportunity cost of holding money. As shown
in Woodford (2003), this opportunity cost should be considerably small in steady state
to obtain a well-defined loss function of the central bank. In particular, Woodford (2003)
argues that real money balances are suﬃciently close to being satiated in the optimal
steady state. To do so, we can eliminate the distortion produced by the opportunity costof money.14
Calculating the second-order approximation of this planner’s objective function (58)around the nondistorted steady state, we can obtain the following central bank’s loss
function:∞∑
t=0

Wt ≈ −Ω∞∑

β t Lt + t.i.p. + O(||ξ||3 ),

(59)

t=0

where t.i.p includes the terms that are independent of monetary policy and O(||ξ||3 )indicates the terms of third or higher orders. Here, the periodic loss function Lt in
13

As discussed in Clarida, Gali and Gertler (2002), an optimal subsidy rate is diﬀerent between policy

coordination and no policy coordination. Engel (2011) argues that we may have diﬃculty in having thederivation of the central bank’s loss function under no policy coordination in the LCP model.
14

See Chapter 6 in Woodford (2003) for a detailed discussion of this issue.

19Equation (59) is given by
Lt = πt2 + (πt∗ )2 + λ1 [ỹt2 + (ỹt∗ )2 ] + λ2 ỹt ỹt∗ + λ3 [i2t + (i∗t )2 ] + λ4 (z̃t∗ − z̃t )2+ λ5 (z̃t − z̃t∗ )(ỹt − ỹt∗ ),

(60)

where
Ω=

1
ωθ
uc C̄,
2 (1 − ω)(1 − ωβ)ϱ = σ − χηc ,(1 − ω)(1 − ωβ) ϱ[2ψ(ψ − 1)(1 − ϱ−1 ) + 1] + η,ωθ
4(1 − ω)(1 − ωβ) (ϱ − 1)ψ(ψ − 1)
,
λ2 =
ω
θ
(1 − ω)(1 − ωβ)ηi
,
λ3 =
ωθv̄
2(1 − ω)(1 − ωβ) (ϱ − 1)ζ 2
λ4 =
,
ω
θ
(1 − ω)(1 − ωβ) (ϱ − 1)(ψ − 1)ζ
,
λ5 =
ω
θ
λ1 =

where v̄ denotes the velocity of money demand. Appendix A provides a detailed derivation of the loss function. As in the case of the standard two-country NKM, the loss
function (60) includes the squared output gaps and the PPI inflation rates of both countries. Stabilization of the PPI inflation rates in each country captures the welfare costs
associated with nominal price rigidities.
Importantly, this loss function indicates several comments in contrast to the standard
loss function derived in a two-country model with nominal price rigidities. This paperemphasizes that the shape of the loss function is diﬀerent from the standard loss function
derived in Clarida, Gali and Gertler (2002) and Engel (2009, 2011) when a non-separable
utility between consumption and real money balances are presumed in the model.15
First, in contrast to the loss function derived in Clarida, Gali and Gertler (2002) andEngel (2011), the central banks need to stabilize the variation between home and foreign
15

Woodford (2003) derives the central bank’s loss function in a closed economy with a non-separable

utility function between consumption and real money balances.

20

real money balances, because such a variability of relative real money balances directly
aﬀects relative consumption through the condition (41). As in Clarida, Gali and Gertler(2002), complete international risk sharing leads to Ct = Ct∗ under the assumption ofboth the separable utility and no home bias. In contrast, in our model, the relative realmoney balances due to a non-separable utility between money and consumption does notcreate Ct = Ct∗ . Therefore, a fluctuation in relative real money balances worsens social
welfare in both countries.
Second, the loss function includes the cross term for relative output gap and relative
real money balances. To understand this term intuitively, this term is expanded asfollows:(z̃t − z̃t∗ )(ỹt − ỹt∗ ) = z̃t ỹt − z̃t∗ ỹt − z̃t ỹt∗ + z̃t∗ ỹt∗ ,(61)

When we assume a non-separable utility, an increase in real money balance leads
to an increase in consumption in both countries. This is reflected by the first and thefourth term in Equation (60). However, according to the home (foreign) money demand
function, an increase in foreign (home) real money balance increases home (foreign) realbalances, resulting in increases in home (foreign) consumption. Thus, a rise in realmoney balances in the foreign (home) country indirectly creates a positive eﬀect of home(foreign) consumption. Both the second and the third terms in Equation (61) reflect
this.
Note that when money and consumption are separable in the utility function, theloss function set coeﬃcients for both the fourth and the fifth terms as zero. Therefore,
Equation (60) reduces to16
Lt = πt2 + (πt∗ )2 + λ1 [ỹt2 + (ỹt∗ )2 ] + λ2 ỹt ỹt∗ + λ3 [i2t + (i∗t )2 ].

(62)

In particular, if real money balances are absent in the model, the central bank’s loss
function corresponds to the case of the result in study by Engel (2011).
16

Ida (2013) derives optimal monetary policy rules in a two-country model with the assumption of

the separable utility assumption. The loss function in his model corresponds to Equation (62) in thispaper.

21

5.2

The optimization problem of central banks

In this paper, both home and foreign central banks implement their monetary policies
under policy coordination. Under a coordination regime, the central banks minimize
their loss function, Equation (60), subject to Equations (51), (53) and (55) and thecorresponding constraints in the foreign country. Policy coordination implies that both
domestic and foreign central banks jointly minimize a weighted sum of the loss functionof domestic and foreign central banks.
We note that both home and foreign money demand functions are additional constraints on the optimal monetary policy of both home and foreign central banks if consumption and real money balances are not separable in the utility function. If we postulate a separable utility function, the central banks no longer regard the money demand
function as a constraint on optimal monetary policy because real money balances areautomatically determined if the central bank sets its interest rate in accordance with its
policy function.
This paper focuses on a commitment policy, in which central banks can commit to
future monetary policy stances at current period. This paper regards the commitment
policy as optimal from a timeless perspective as suggested by Woodford (2003).1718 Whenthe central banks commit to a policy, they can introduce policy inertia into the economy.
This policy inertia enables the central banks to influence the private-sector expectations.
A discretionary policy is often subject to commitment; that is, the discretionary policy
cannot use private-sector expectations. The welfare loss under a commitment policy issmaller than that under a discretionary policy (Woodford, 2003; McCallum and Nelson,
2004; Walsh, 2017).19 Moreover, as pointed out by Pappa (2004), a commitment policy
can eliminate a deflationary bias induced by a discretionary policy, which is the source17

See Chapter 7 in Woodford (2003) for a detailed discussion of optimal monetary policy from atimeless perspective.
18

Fujiwara, Kam, and Sunakawa (2016) consider the sustainable plan for optimal monetary policy

under the case of both policy coordination and no policy coordination in a two-country model.
19

If we introduce endogenous persistence into the forward-looking model, it is possible that the gainfrom commitment decreases (e.g., Steinsson, 2003).

22

of an additional gain from commitment associated with an open economy.
Suppose ϕ1t and ϕ2t are Lagrange multipliers associated with both home and foreign
NKPCs. Let ϕ3t and ϕ4t be the Lagrange multiplier associated with both home andforeign IS equations. Finally, I denote ϕ5t and ϕ6t as the Lagrange multiplier with respect
to both home and foreign money demand equations. In this instance, the economic
system can be written as follows:
A0 Xt = A1 Xt−1 + B1 Rt + Γt ,

(63)

where
Xt = [X1t X2t ]′′X1t = [rr¯ t rr¯ ∗t ςt ςt∗ ϕ1t ϕ2t ϕ3t ϕ4t ϕ5t ϕ6t ] ,′X2t = [z̃t z̃t∗ πt πt∗ ỹt ỹt∗ ] ,′Rt = [it i∗t ] .
In addition, A0 , A1 , and B1 are matrices constructed by deep parameters, and Γt denotes
a shock vector. I simulate the property of optimal monetary policy by using the Dynare
software package.20

5.3

Calibration

This section describes deep parameters used in this paper. The discount factor is set to0.99. The relative risk aversion coeﬃcient for consumption σ is 5.0 based on the valueof Steinsson (2003). In addition, following Steinsson (2003), this paper assumes that theparameter η equals 2.0. Following Rotemberg and Woodford (1997) and Pappa (2004),
the elasticity of substitution between goods θ is set as 7.88. The income elasticity ofmoney demand ηi is set to 1.0, following Woodford (2003). As in Woodford (2003), I
assume that the interest elasticity of money demand ηc takes a value 28. With regard
to the parameter a, that denotes the degree of home bias, this parameter is set to 1.2.

20

The dynare is available on the web page: http://www.dynare.org/.

23

Thus, we allow the presence of home bias for home goods. The velocity of money v̄ is
set to 1.0.
Next, set the parameter χ that denotes the degree of substitutability between consumption and real money balances in the household’s utility function. In Woodford(2003), this parameter takes a value of 0.02. Kurozumi (2008) shows that a value of theparameter χ crucially aﬀects the determinacy condition, which uniquely determines therational expectations equilibrium. In this paper, the parameter χ takes a range from 0
to 0.08 as a benchmark case.21 This paper set the parameter χ to 0.01 as a benchmark
calibration.
Finally, the standard deviations of the natural interest rate σr and the money demand
shock σς is set to 1.0 and 1.0, respectively. With regard to autoregressive coeﬃcientsfor structural shocks, we choose 0.7, 0.5 for ρr and ρς , respectively. These calibrated
values for economic shocks are presumed to be the same in the foreign country. Table 1
summarizes the baseline parameters used in this paper.
[Table 1 around here]

5.4

Results

This section provides the simulation results from impulse response analyses and reports
the impulse response analysis when a foreign natural interest rate shock occurs. It
examines the case where a money demand shock occurs in the foreign country.
5.4.1

A foreign natural interest rate shock

Figure 1 shows the impulse response to a foreign natural interest rate shock under optimalmonetary policy. Both inflation and the output gap increase in the foreign country. The
foreign central bank raises its nominal interest rate gradually under optimal monetary
policy. In particular, for the case in which the parameter χ takes a higher value, an

21This paper confirms that the Blanchard-Kahn condition, which is required to achieve a uniquerational expectations equilibrium, is satisfied under these ranges of the parameter χ.

24initial increase in the foreign policy rate is smaller, whereas the foreign central bank moreaggressively raises its policy rate subsequently. Since a change in real money balances
induces a co-movement in consumption, the central bank employs a smaller response of
the interest rate to the shock in order to restrain an undesirable fluctuation of inflationand the output gap. A higher value of the parameter χ weakens the ability of the foreign
central bank to impart policy inertia into the inflation rate. Accordingly, the central
bank cannot introduce policy inertia into the output gap. Real money balances are only
aﬀected by a change in the foreign policy rate. Thus, in the case of the natural interest
rate shock, the parameter χ may not play a significant role for the foreign country.
[Figure 1 around here]
In contrast to a change in the foreign nominal interest rate, the home central banksets its policy rate to a positive value. In contrast to the case of the foreign country, achange in the parameter χ significantly aﬀects endogenous variables in the home country.
For a higher value of the parameter χ the home central bank reinforces such a rise in
the policy rate under optimal monetary policy. In particular, a foreign natural interest
rate shock generates a policy trade-oﬀ between inflation and the output gap in the homecountry. Such a trade-oﬀ is more severe as the parameter χ takes a higher value because
a higher value of the parameter χ leads to a larger decline in the output gap in the initialperiod. On the other hand, the home central bank implements monetary tightening
aggressively to combat a situation where a higher value of the parameter χ generates anincrease in inflation. In addition, as in the case of the foreign country, a higher value ofthe parameter χ counteracts policy inertia into both home inflation and the home output
gap. A change in the home policy rate aﬀects home real money balances. Finally, thereal exchange rate remains unchanged even if the substitutability between consumption
and money increases.
5.4.2

A foreign money demand shock

Next, this section reports the impulse response analysis for a foreign money demand
shock. The impulse response of a foreign money demand shock is diﬀerent from that of25

a foreign natural rate shock. This impulse response is shown in Figure 2. For a money
demand shock, the eﬀect on the real economy is negligible as long as the parameter χ
takes a much smaller value, which is close to the case of the separable utility. Thisimplies that the money demand shock has little eﬀect on the real economy. However, alarger value of the parameter χ significantly aﬀects the real economy. In fact, in the caseof the money demand shock, endogenous variables of both countries are clearly aﬀectedby any change in the parameter χ.
For the foreign country, a money demand shock induces a decline in both inflationand the output gap. On the other hand, the foreign interest rate remains unchanged
in response to the shock in the case of the parameter χ = 0.01. However, when theparameter χ takes higher values, the foreign central bank accommodates a huge decline
in both inflation and the output gap, whereas it increases the interest rate. This isbecause the foreign central bank reacts aggressively to the endogenous variables since
the output gap co-moves an increase in the money demand in the presence of the nonseparable utility function between money and consumption.
[Figure 2 around here]
In contrast to the case of a natural interest rate shock, a change in the parameter χaﬀects the impulse responses for the home country. In addition, compared to the caseof the foreign country, an increase in the foreign money demand leads to a boom in thehome country. This boom is created by a reduction of home central bank’s monetaryeasing. In particular, for a larger value of the parameter χ, the optimal monetary policy
requires the home central bank to cut its policy rates more aggressively. Such a reductionof the home policy rate induces a further increase in both inflation and the output gap
in the home country.
Finally, in contrast to the natural interest rate shock, the real exchange rate depreciates in the initial period. Then, the money demand shock induces an appreciation of
the real exchange rate in the subsequent periods. As shown in Figure 2, a larger valueof the parameter χ produces a larger depreciation of the real exchange rate initially anda persistent appreciation subsequently. It follows from Equation (41) that an increase26

in foreign money produces a depreciation of the real exchange rate in terms of homecurrency.
Intuitively, a foreign money demand shock increases foreign real money balances.
Such an increase in the real money balances produces a decline in foreign inflation throughthe foreign NKPC. The foreign output gap declines also through the foreign IS equation.
A decline in both foreign inflation and the foreign output gap induces a depreciation of
the real exchange rate, which generates an increase in home inflation through the home
NKPC. In addition, as an increase in foreign real money balances leads to an increasein the home money demand. This induces an increase in the output gap in the homecountry. Therefore, a foreign money demand shock produces a recession in the foreigncountry, whereas the boom is created by that shock in the home country.

6

The role of money and the zero lower bound on
nominal interest rates in a two-country economy

This section examines the eﬀect of the ZLB on the real economy when the non-separable
utility function between consumption and real money balances in a two-country framework is assumed. Section 6.1 reviews briefly the role of the ZLB in a standard NKM.
Section 6.2 reports the impulse response of a foreign negative natural interest rate shock
under the ZLB. Section 6.3 considers how a foreign negative money demand shock aﬀects
the real economy of each country under the ZLB.

6.1

The zero lower bound on nominal interest rates

The importance of the presence of the ZLB in a NKM has been argued in literature(Eggertsson and Woodford, 2003; Jung, et al., 2005; Adam and Billi, 2006; McKay,
Nakamura and Steinsson, 2016; Nakata, 2017). Consider the case where a natural interest
rate temporarily takes a negative value. Since the central bank cannot reduce its policy
rate to a negative value, it cannot stimulate the real economy once the ZLB hits the
economy. As argued in Eggertsson and Woodford (2003) and Jung et al. (2005), the27

central bank can insulate the aggregate demand through the history dependence of acommitment policy even when the nominal interest rate reaches the zero bound.
However, the aforementioned studies examine optimal monetary policy with the zerobound in closed economies and have limited bearing on the current global situation.
Nakajima (2008) investigates optimal monetary policy under the zero bound in a twocountry economy, but in his work only the domestic interest rate is zero. Ida (2013)derives optimal monetary policy rules in a two-country economy in which both homeand foreign country simultaneously face the ZLB. He shows that optimal policy rulesthat excludes the information from past policy rates only retains the optimality when
two countries simultaneously face the ZLB. Fujiwara et al. (2010, 2014) examine also
optimal monetary policy in the case where nonnegativity constraints on nominal interest
rate exist in both home and foreign countries. They point out that the global liquidity
trap induces its international dependence, and show that price level targeting can achieve
higher welfare than inflation targeting in such a case. According to their studies, when
the ZLB is binding in both the countries, there are gains from a commitment policy
when both domestic and foreign central banks coordinate monetary policies.
As noted earlier, the previous studies do not stress the role of money in the NKM.
However, recent evidence indicates that money aggregate plays a significant role in unconventional monetary policy, in which central banks would expect the eﬀect of thequantity of money on the real economy in addition to a commitment to a ZIRP. Central
banks, such as the FRB, the Bank of England, and the BOJ, address the eﬀect of moneyon the real economy as quantitative easing. Since these central banks have employed
quantitative easing policies, the presence of the ZLB should play a significant role ina two-country model that the role of money is included. Importantly, previous studies
do not include the eﬀect of money on the real economy in a two-country model even inwhich central banks simultaneously face the ZLB.
As in Fujiwara et al. (2014), I consider a situation where both home and foreigncountry simultaneously face the ZLB. Thus, both home and foreign central banks face28

the following nonnegativity constraints on nominal interest rates:22
it ≥ 0, i∗t ≥ 0.

(64)

Fujiwara et al. (2014) argue that the termination date of a ZIRP for the home country is diﬀerent from that for the foreign country when both home and foreign countries
simultaneously face the ZLB.23 Their results still hold in this model regardless of a deterministic shock case. Importantly, the substitutability between consumption and moneyin the household’s utility function generates the diﬀerence between the termination dateof a ZIRP for the home country and that for the foreign country.

6.2

A foreign natural interest rate shock

Figure 3 portrays the responses of endogenous variables to a foreign negative naturalrate of interest shock. The negative shock for the foreign natural interest rate induces arecession in the foreign country. The foreign central bank employs a ZIRP and maintains
that policy for long term. Indeed, as shown in Figure 3, the ZIRP terminates at period
33. After the termination of the ZIRP, the foreign central bank gradually increases its
policy rate. This optimal response of the foreign central bank is consistent with theassertion of Eggertsson and Woodford (2003).
[Figure 3 around here]
On the other hand, a foreign negative interest rate shock generates a trade-oﬀ between
inflation and the output gap in the home country. The home central bank conducts a
ZIRP also in response to a decline in the rate of inflation. Since the output gap is apositive response to the shock, the home central bank terminates the ZIRP at period 15.
22

Unlike Fujiwara et al. (2010, 2014), this paper examines the role of money in a two-country modelin which both home and foreign country face the ZLB under the assumption that a deterministic shock
occurs in both country. The simulations in this paper are implemented by the Dynare. See also Adamand Billi (2006, 2007) and Nakov (2008) a detailed discussion about a stochastic shock case.
23

Their simulation is based on two-state Markov switching that the natural interest rates of bothcountries follow the Makov process.

29This termination date is shorter than that by roughly 15 periods compared to the caseof the foreign country.
Surprisingly, unlike the argument in Eggertsson and Woodford (2003), the home
central bank raises its policy rate aggressively when it decides to depart from the ZIRP.
In particular, it accommodates considerable overshooting of the nominal interest rate.
Although deflation in the home country is caused by a decline in the foreign naturalinterest rate, the natural interest rate in the home country still takes a positive value.
The home central bank needs to prevent an undesirable boom created by introducing a
ZIRP induced by a foreign natural interest rate shock. Therefore, the home central bankraises its policy rate immediately and aggressively when it overcomes the deflationarypressure.
[Figure 4 around here]
How does the parameter χ aﬀect the eﬀect of the ZLB on both countries? Figure
4 shows how a change of the parameter χ aﬀects the response of nominal interest rates
of both the countries. It implies that the termination date of the foreign ZIRP in thecase of χ = 0.04 is slightly longer than that in the case of χ = 0.01. Moreover, for thehome country, the central bank implements a more overshooting reaction of the policy
rate than one in the case of χ = 0.01.

6.3

A foreign money demand shock

Next, consider the eﬀect of the ZLB in the case of a foreign money demand shock.
Interestingly, the money demand shock considerably changes the property of optimalmonetary policy in contrast to the case of a natural interest rate shock. A money
demand shock, which increases money demand in the foreign country, creates a decline
in both inflation and the output gap. While the foreign central bank initially decides
on an increase in the policy rate in response to the shock, it attempts to employ the
ZIRP also because the shock worsens the real economy. Indeed, the foreign central bankcontinues to maintain its zero interest rate for a long time. It terminates the ZIRP atperiod 17, and then employs the gradualism for the policy rate stance.
30

On the contrary, the home country experiences an increase in both inflation and theoutput gap. However, under policy coordination, the home central bank, which considers
the foreign ZIRP eﬀect on the home country, maintains the nominal interest rate at zerofor shorter periods. Indeed, the home central bank decides to terminate a ZIRP atperiod 3. As in the case of the foreign natural rate shock, the home central bank that
implements optimal monetary policy allows the policy rate to overshoot immediately
once the central bank determines the termination of the ZIRP.
[Figure 5 around here]
How does a change in the parameter χ aﬀect the path of the nominal interest rates
in both the countries in the case of the money demand shock? Figure 6 illustrates thepath of the nominal interest rates of both the countries when the money demand shock
occurs in the foreign country.24 According to this figure, a larger value of the parameterχ requires the foreign central bank to increase its policy rate more aggressively after
the shock. However, after such an aggressive monetary tightening, as in the case of theχ = 0.02, the central bank decides to set its nominal interest rate zero. The termination
date of the ZIRP in the case of χ = 0.04 is longer than that of the case of χ = 0.02. In
fact, as shown in Figure 6, the foreign central bank keeps the nominal interest rate atzero by period 22 in the case of χ = 0.04. On the contrary, the termination date of the
ZIRP of the home country in the case of χ = 0.04 is also longer than that in the caseof χ = 0.02. In addition, an overshooting response of the home nominal interest rate is
amplified by a larger value of the parameter χ.
[Figure 6 around here][Figure 7 around here]
Finally, the paper checks how the degree of persistence of the foreign money demand
shock aﬀects the interest rate path of both of the countries. Figure 7 shows the interest
24Here, the interest rate path for the case of χ = 0.01 is not reported. This is because it is confirmedthat the home nominal interest rate increases in response to the money demand shock. Thus, in thecase of χ = 0.01, the home nominal interest rate does not face the ZLB.

31

rate path of both of the countries when persistence of the foreign money demand shock(ρ∗ζ ) changes. The foreign central bank raises its policy rate more aggressively as theparameter ρ∗ζ takes a larger value. According to this figure, a larger value of the parameterρ∗ζ forces the foreign central bank to retain the ZIRP for a longer period. Indeed, thetermination date of the ZIRP of the foreign central bank for the case of ρ∗ζ = 0.8 isabout twice the duration of the case of ρ∗ζ = 0.3. A larger value of the parameter ρ∗ζ also
prolongs the termination date of the ZIRP in the home country. The nominal interest
rate of the home country overshoots more predominately as the eﬀect of foreign money
demand shock is persistent.
These results imply that the money demand function significantly aﬀects the path ofendogenous variables when both the countries face the ZLB simultaneously. Therefore,
from a global perspective, this paper addresses that the model should consider the casethat money aggregate significantly influences the real economy in the presence of thenonnegativity constraints on the nominal interest rates.

7

Conclusions

The NKM has been used in recent monetary policy analysis. In the case of the standard NKM, the money demand function may be redundant because the model presumes
the separation between consumption and real money balances in the household’s utility
function. However, recent evidences from monetary policy in advanced countries reveal
the importance of role of money since central banks in advanced country reduce theirpolicy rates to combat the risk of deflation derived from the Lehman crisis that occurredin the United States. As a result, they have faced nonnegativity constraints on nominalinterest rates. Therefore, the central banks decided to increase a quantity of monetarybase to combat deflation and recession once they face the ZLB. The standard NKM thatstudies the ZLB eﬀect on the economy does not consider the role of money.
This paper addresses the role of money in a two-country economy. The present study
considered the non-separable utility between consumption and real money balance ina two-country framework. This paper examined the role of money in a two-country
32

model constructed by Engel (2009, 2011), who constructed a two-country model withhome bias. Unlike Clarida, Gali, and Gertler (2002) and Engel (2009, 2011), this paper
showed that relative real money balances between two countries directly aﬀect the risksharing condition for consumption when one allows the case where consumption and realmoney balances are not separable in the household’s utility function.
Such a departure of the separable utility function changes the structural equations
in a two-country economy with nominal rigidities. Interestingly, in a two-country modelin which consumption and real money balances are not separable, relative real money
balances between two countries directly aﬀect not only the dynamic IS equation andthe money demand function, but also the NKPC. In addition, the non-separable utilitychanges the shape of the central bank’s loss function. In contrast to Clarida, Gali,
and Gertler (2002) and Engel (2009, 2011), the shape of the loss function is diﬀerent
from such a standard loss function when consumption and real money balances are notseparable in the utility function. First, the central banks need to stabilize a variability ofdiﬀerence between home real money balances and foreign real money balances. Second,
the loss function includes the cross term for relative output gap and relative real money
balances.
The assumption of the non-separable utility between consumption and real money
balances significantly aﬀects the properties of optimal monetary policy. Furthermore,this paper attempts to examine how the assumption of a non-separable utility between
consumption and money aﬀects the optimal monetary policy when the ZLB simultaneously hits both countries. Several studies that explore the properties of optimal monetary
policy in a two-country model do not account for the eﬀect of money on the real economy
in a two-country model in which central banks simultaneously face the ZLB. This paper
showed that the presence of a non-separable utility function between consumption andmoney crucially changes the termination date of a ZIRP for both countries.
Finally, there are possible further extensions of the work in this paper. First, as in
Deverex and Engel (2003), Monacelli (2005), and Engel (2009, 2011), it is interesting
how the relaxation of the assumption of producer-currency pricing changes the properties33

of optimal monetary policy in the case of a non-separable utility function. Second, thispaper provided the log-linearized structural equations around the zero-inflation steadystate. However, as argued by Ascari and Ropele (2007) and Ascari and Sbordone (2014),
it might be important whether the presence of a non-separable utility function changes
optimal monetary policy in the case where a positive trend inflation exists in both countries. Third, this paper simply assumed a deterministic economic shock when examining
the eﬀect of the ZLB in a two-country economy. This is because the two-country model
may make the numerical algorithm for solving the problem of the ZLB more complicatedin the case of a stochastic shock. Indeed, as argued in the existing literature, we may
often have diﬃculty in numerically solving optimal policy due to the presence of statevariables (e.g., Adam and Billi, 2006; Adam and Billi, 2007; Nakov, 2008; Guerrieri and
Iacoviello, 2015). Accordingly, the numerical solution might be more complicated asstate variables associated with commitment increase in a two-country model. As in thestudies by Adam and Billi (2006, 2007) and Nakov (2008), however, the properties of
optimal monetary policy obtained in this paper may be changed by the introduction of astochastic shock. For instance, as the extension of this study, it may be interesting how
the non-linear eﬀects of the ZLB are introduced in this paper.

Acknowlegements
I thank Kohei Hasui, Tomohiro Hirano, and Taisuke Nakata for the helpful comments
and suggestions. This study has been greatly improved by the helpful comments andsuggestions made by seminar participants at Okayama University and Kansai University. This paper was supported by Grant-in-Aid for Scientific Research (17K13766). Allremaining errors are my own.

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Princeton University Press, Princeton.

A

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

This appendix provides a detailed derivation of the central bank’s loss function in a
two-country model with a non-separable utility between consumption and real money
balances. As shown in Woodford (2001, 2003), the loss function corresponds to thesecond-order approximation of the household’s utility function.

A.1

Preliminaries

I 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 ofthe eﬃcient level. Second, as in the main text, I define ht = log(Ht /H̄) as the deviation ofHt from the steady state. To implement the second-order approximation of the planner’s
objective function, I introduce the following equation:()
Ht1
Ht − H̄ = H̄
− 1 ≈ ht + h2t .
2
H̄

(A.1)

To derive the second-order approximation of the planner’s objective function, I introduce several useful equations. First, expressing Equation (41) in terms of log-deviationfrom their eﬃcient price equilibrium counterparts, we obtain the following equations:c̃t = ψ ỹt + (1 − ψ)ỹt∗ + ζ(z̃t∗ − z̃t ),(A.2)

c̃∗t = ψ ỹt∗ + (1 − ψ)ỹt − ζ(z̃t∗ − z̃t ).

(A.3)

Also, expressing production functions for both home and foreign countries in termsof log-deviation from their eﬃcient price equilibrium counterparts, we obtainθñt = ỹt + var(pH,t ),2θ∗∗ñt = ỹt + var(p∗F,t ).
2
38

(A.4)
(A.5)

Furthermore, home and foreign money demand functions, which are expressed bylog-deviation from their eﬃcient price equilibrium counterparts are as follows:

A.2z̃t = ηc c̃t − ηi it ,(A.6)

z̃t∗ = ηc c̃∗t − ηi i∗t .

(A.7)

Second-order approximation of the planner’s objective function

I 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,
Gali, and Gertler (2002). I implement the second-order approximation of the following
planner’s objective function around their eﬃcient price equilibrium. The following stepof the derivation of the loss function is mainly based on that of Clarida, Gali and Gertler(2002) and Engel (2009, 2011).
The planner’s objective function under policy coordination is given by
Wt = u(Ct , Zt ) + u(Ct∗ , Zt∗ ) − v(Nt ) − v(Nt∗ ),(A.8)

The second-order approximation of first and second terms of the right-hand side of
the planner’s objective function are follows:
(
)
1 2
n
2
u(Ct , Zt ) ≈ uc ct c̃t + c̃t + sz z̃t + sz (1 − σz )z̃t + χc̃t z̃t + t.i.p. + +O(||ξ||3 ), (A.9)
2
(
)
1 ∗ 2
∗
∗
∗n
∗
∗
∗ 2
∗ ∗
u(Ct , Zt ) ≈ uc ct c̃t + (c̃t ) + sz z̃t + sz (1 − σz )(z̃t ) + χc̃t z̃t + t.i.p. + +O(||ξ||3 ),2(A.10)
whereσz =uzz Z̄
uz Z̄
.
, sz = −uz
uc C̄The term uc cnt and uc Ctn∗ of the second-order approximation of Equations (A.9 ) (A.10 ) are given byuc cnt = uc C̄ + uc C̄(1 − σ)cnt + O(||ξ||3 ),(A.11)

n∗3
uc cn∗
t = uc C̄ + uc C̄(1 − σ)ct + O(||ξ|| ).

(A.12)

39

As assumed in Woodford (2003), this paper presumes that the real money balances
satiate at some finite level, and that the variation in interest rates in eﬃcient price equilibrium is quite small. Since we can make use of Woodford (2003, chapter 6) assumption
6.1, therefore, we obtain the following useful relationships:25
sz σz = −(v̄ηi )−1(A.13)

ηc = v̄χηi .

(A.14)

Using Equations (A.7 ), (A.8 ), (A.11 ), (A.12 ), (A.13 ), and (A.14 ), we can rewrite
Equations (A.11 ) and (A.12 ) as follows:
[
]
1
1 −1
2
T 2
n
u(Ct , Zt ) ≈ uc C̄ (1 + sz )c̃t + (1 − σ + χηc )c̃t − (v̄) ηi (it − it ) + (1 − σ)ct c̃t2
2+ t.i.p. + O(||ξ||3 ),(A.15)
[
]
1 −1
1
∗ 2
∗
∗
T∗ 2
∗
∗n ∗
∗
u(Ct , Zt ) ≈ uc C̄ (1 + sz )c̃t + (1 − σ + χηc )(c̃t ) − (v̄) ηi (it − it ) + (1 − σ)ct c̃t2
2+ t.i.p. + O(||ξ||3 ),(A.16)

where I used the relationship sz = ∆v̄. ∆ = ī/(1 + ī) denotes opportunity costs of money
holdings in the steady state. Also, note that iT = iT ∗ = ηi v̄∆.
Next, the second-order approximation of both third and fourth terms of the righthand side are given by
[
]
1
2
n
V (Nt ) = V (N̄ )¯(N ) ñt + ñt + (1 + η)nt ñt + t.i.p. + O(||ξ||3 ),
2
[
]
1 ∗ 2
′
∗
∗
∗n ∗
¯
V (Nt ) = V (N̄ )(N ) ñt + (ñt ) + (1 + η)nt ñt + t.i.p. + O(||ξ||3 ),
2
′

′

′

(A.17)(A.18)′

I used the relationship V (N̄ ) = V (N̄ )+V (N̄ )(1+η)nnt and the corresponding equation
for the foreign country in the derivation of the above equations. Substituting Equation(A.5 ) and using the relationship nnt ñt = nnt ỹt + O(||ξ||2 ), we obtain
[
]
θ
1
′
n
2
V (Nt ) = V (N̄ )¯(N ) ỹt + ỹt + (1 + η)nt ỹt + var(pH,t ) + t.i.p. + O(||ξ||3 ). (A.19)
2
2
25

See Woodford (2003) for a detailed discussion of this relationship.

40

Similarly,[′
V (Nt∗ ) = V (N̄ )¯(N )

]

1
θ
∗
∗
ỹt∗ + (ỹt∗ )2 + (1 + η)nn∗t ỹt + var(pF,t )2

2+ t.i.p. + O(||ξ||3 ).
(A.20)

Also, substituting Equation (A.2 ) into Equation (A.15 ) and rearranging, we obtainthe following expression[1
1
u(Ct , Zt ) = uc C̄ ỹt + (1 − σ + χηc )(ψ ỹt + (1 − ψ)ỹt∗ + ζ(z̃t∗ − z̃t ))2 − (v̄)−1 ηi (it − iTt )22
2](1 + η)nnt (ψ ỹt + (1 − ψ)ỹt∗ + ζ(z̃t − z̃t∗ )) + t.i.p. + O(||ξ||3 ).
(A.21)
Similarly, we can obtain the corresponding equation for the foreign country:
[
1
1
∗
∗
u(Ct , Zt ) = uc C̄ ỹt∗ + (1 − σ + χηc )(ψ ỹt∗ + (1 − ψ)ỹt − ζ(z̃t∗ − z̃t ))2 − (v̄)−1 ηi (i∗t − iTt ∗ )2
2
2
]
∗
∗
3
(1 + η)nn∗(A.22)t (ψ ỹt + (1 − ψ)ỹt − ζ(z̃t − z̃t )) + t.i.p. + O(||ξ|| ),Combining (A.21 ) with (A.22 ), we obtain[1∗∗u(Ct , Zt ) + u(Ct , Zt ) = uc C̄ ỹt + ỹt∗ + (1 − σ + χηc )[(2ψ 2 − 2ψ + 1)(ỹt2 − (ỹt∗ )2 )2
+ 4ψ(1 − ψ)ỹt ỹt∗ + ζ 2 (z̃t∗ − z̃t )2 ] − 2ζ(2ψ − 1)(ỹt − ỹt∗ )(z̃t − z̃t∗ )
]
1 −1
∗ 2
n
n∗ ∗
− (v̄) ηi (it + it ) + (1 + η)nt ỹt + (1 + η)nt ỹt + t.i.p. + O(||ξ||3 ).
2(A.23)
I assumed that iT = iT ∗ = 0 in this derivation.
′

Since the relationship uc C̄ = V (N̄ )N̄ holds in the eﬃcient level of the steady state,taking into the above relationship consideration and combining Equation (23) with Equations (A.19 ) and (A.20 ), we obtain[σ + χηc
Wt = −uc C̄
(1 + 2ψ(ψ − 1)(1 − ϱ−1 )(ỹt2 + (ỹt∗ )2 ) + 2(σ + χηc − 1)ζ 2 (z̃t∗ − z̃t )22
1+ 4ψ(1 − ψ)ỹt ỹt∗ + 2ζ(2ψ − 1)(ỹt − ỹt∗ )(z̃t − z̃t∗ ) + (v̄)−1 ηi (it + i∗t )2 + η ỹt2 + η(ỹt∗ )22]
θ∗+ (var(pH,t ) + var(pF,t)) + t.i.p. + O(||ξ||3 ).
(A.24)
241

Furthermore, from the firm j’s demand function, we obtain the following equations:26
varj log yt (j) = θ2 varj log PH,t (j),varj log yt∗ (j) = θ2 varj log(PF,t∗ )(j),Now define∆H,t = varj log PH,t ,∗∆F,t = varj log PF,t.

Then, we obtain the following equations:27∞∑
ω∆H,t =
β t πt2 + t.i.p. + O(||ξ||3 ),
(1
−
ω)(1
−
ωβ)
t=0
t=0

(A.25)∞∑
ω∆F,t =β t (πt∗ )2 + t.i.p. + O(||ξ||3 ).
(1
−
ω)(1
−
ωβ)
t=0
t=0

(A.26)∞∑∞∑

Substituting Equations (A.25 ) and (A.26 ) into the discounted sum of Equation(A.24 ), we obtain the loss function (60) in the main text.

B

Appendix B: Optimization problem of optimal monetary policy under policy coordination

In this appendix, I describe the detailed derivation of the central bank’s optimization
problem28 . As noted in the main text, under a coordination regime, the central banks
minimize their loss function, Equation (60), subject to Equations (51), (53) and (55)and the corresponding constraints in the foreign country. Policy coordination implies
that both domestic and foreign central banks jointly minimize a weighted sum of theloss function of domestic and foreign central banks.
To implement optimal monetary policy under policy coordination, we define the Lagrangian as follows:
26

See Walsh (2017) for a detailed derivation of these equations.

27

See Woodford (2003) for a detailed derivation of these equations.

28

This appendix provides the optimal monetary policy when the ZLB is present in both countries.

42

L = E0∞∑{β t πt2 + (πt∗ )2 + λ1 [ỹt2 + (ỹt∗ )2 ] + λ2 ỹt ỹt∗ + λ3 [i2t + (i∗t )2 ] + λ4 (z̃t∗ − z̃t )2

t=0

(
)
+ λ5 (z̃t − z̃t∗ )(ỹt − ỹt∗ ) − 2ϕ1t βπt+1 + κ1 ỹt + κ2 ỹt∗ − κ3 z̃t κ4 z̃t∗ − πt
( ∗)− 2ϕ2t βπt+1+ κ1 ỹt∗ + κ2 ỹt − κ3 z̃t∗ κ4 z̃t − πt∗[]∗∗− 2ϕ3t ỹt+1 + ϑ1 ∆ỹt+1− σ0−1 (it − πt+1 − rr¯ t ) + ϑ2 ∆z̃t+1 − ϑ3 ∆z̃t+1
− ỹt
]
[ ∗
∗
∗
− ϑ3 ∆z̃t+1 − ỹt∗
− rr
¯ ∗t ) + ϑ2 ∆z̃t+1+ ϑ1 ∆ỹt+1 − σ0−1 (i∗t − πt+1
− 2ϕ4t ỹt+1()− 2ϕ5t µ1 ỹt + µ2 ỹt∗ + µ3 z̃t∗ − µ4 it + ςt − z̃t)}(− 2ϕ6t µ1 ỹt∗ + µ2 ỹt + µ3 z̃t − µ4 i∗t + ςt∗ − z̃t∗

(B.1)

First order conditions of this optimization problem are as follows:
πt − ϕ1t−1 + ϕ1t − (σ0 β)−1 ϕ3t−1 = 0,(B.2)πt∗ − ϕ2t−1 + ϕ2t − (σ0 β)−1 ϕ4t−1 = 0,

(B.3)

λ1 ỹt + λ2 ỹt∗ − λ5 (z̃t∗ − z̃t ) − κ1 ϕ1t − κ2 ϕ2t − β −1 ϕ3t−1 + ϕ3t + ϑ1 (ϕ4t − β −1 ϕ4t−1 )− µ1 ϕ5t − µ2 ϕ6t = 0,

(B.4)

λ1 ỹt∗ + λ2 ỹt + λ5 (z̃t∗ − z̃t ) − κ2 ϕ1t − κ1 ϕ2t − β −1 ϕ4t−1 + ϕ4t + ϑ1 (ϕ3t − β −1 ϕ3t−1 )− µ2 ϕ5t − µ1 ϕ5t = 0,

(B.5)

λ3 it + σ0−1 ϕ3t + µ4 ϕ5t = 0,(B.6)

λ3 i∗t + σ0−1 ϕ4t + µ4 ϕ6t = 0,

(B.7)

Combining these optimal conditions with structural equations of both countries, we
obtain the economic system (63).

43

Table 1: Parameter Values

Parameters Values Explanation
β

0.99

Discount factor

σ

5.0

Risk aversion coeﬃcient for consumptionη

2.0

Inverse of elasticity of labor supplyt

ω

0.75

Price stickiness

θ

7.88

Elasticity of substitution for individual goods

ηy

1.0

Income elasticity of money demandηr

28Interest elasticity of money demand

χ

0.01

Substitutability between consumption and real money balancesa1.2

Degree of home bias

ρr

0.7

Auto-regressive coeﬃcient for the home natural interest rate shock

ρς

0.5

Auto-regressive coeﬃcient for the home money demand shock

ρ∗r

0.7

Auto-regressive coeﬃcient for the foreign natural interest rate shock

ρ∗ς

0.5

Auto-regressive coeﬃcient for the foreign money demand shock

σr

1.0

Standard deviation of natural rate shock for home country

σς

1.0

Standard deviation of money demand shock for home country

σr∗

1.0

Standard deviation of natural rate shock for foreign country

σς∗

1.0

Standard deviation of money demand shock for foreign country

44

0.15

Foreign inflation

50Home inflation
0.1

0.06

0.05

0.15

-10

0.04

20

0.02

15

-5

Home policy rate

10

0

5

-0.05
20

0

15

-0.02
10

-0.1
5

0.15

-0.04

Foreign output gap

×10-3

×10-3

5

5

5

Home output gap

10

15

15

Foreign policy rate

10

15

Real exchange rate

10

20

20

20

45

0.03

0

0.1

0

0.1

20

0.02

15

0

10

0.05

5

0.05

20

-0.05

15

-0.05
10

0.01

5

-0.01

0

2

0.05

-2

Foreign real money balances

0.05

0

Home real money balances

0

-4

20

-0.05
-0.05

15

-6
10

-0.1

5

-0.15
20

-0.1

15

-8
10

-0.2
5

χ=0.04

χ=0.08-0.15

χ=0.01Figure 1: Optimal monetary policy under a foreign natural interest rate.

0.1
0.05
0
-0.05

5

-3

×10

5

5

Home inflation

10

15

15

Foreign output gap

10

10

15

Home real money balances

5

20

20

20

Foreign inflation

15
10

0.05
0

5

20

0

15

-0.1

Home policy rate

10

15

10

15

χ=0.08

20

20

-2

0

2

4

6

-0.02

0

0.02

0.04

0.06

-5
10

-0.05

5

5

Foreign real money balances

5

-0.15

0.05
0
-0.05
-0.1

1.5
1
0.5
0

χ=0.04

×10-3

×10-3

5

5

5

Home output gap

10

15

15

Foreign policy rate

10

15

Real exchange rate

10

20

20

20

46

0
-5
-10
-15

0.1

0.15

0.05
0
-0.05

χ=0.01Figure 2: Optimal monetary policy under a foreign money demand shock.

0.3

0.02
0.01

0.2

0
0.1

-0.01
-0.02

0

-0.03

-0.1

-0.04

-0.2

Home output gap
Foreign output gap-0.05

Home inflation
Foreign inflation

-0.3

-0.06
-0.07

-0.4
0

5

10

15

20

25

30

35

40

0

5

10

15

Quarters

20

25

30

35

40

Quarters

0.014

0.01
Home nominal interest rate
Foreign nominal interest rate

0.012

0.005

0.01

0

0.008

-0.005

0.006

-0.01

0.004

-0.015

0.002

-0.02

0

Home real money balances
Foreign real money balances-0.025
0

5

10

15

20

25

30

35

40

0

Quarters

5

10

15

20

25

30

35

40

Quarters

Figure 3: Optimal monetary policy under the ZLB when a foreign natural interest rateoccurs.

47-3(a) Home nominal interest rate

0.015

2.5

(b) Foreign nominal interest rate

×10

2

0.01
1.5

1
0.005

0.5

0

0
0

5

10

15

20

25

30

35

40

0

Quarters

5

10

15

20

25

30

35

40

Quarters
χ=0.01

χ=0.04

Figure 4: Optimal monetary policy under the ZLB when a foreign natural interest rateoccurs.

48

3

×10-3

0.03
Home output gap
Foreign output gap

2

Home inflation
Foreign inflation

0.02
0.01

1

0
0
-0.01
-1

-0.02

-2

-0.03

-3

-0.04
0

5

10

15

20

25

30

35

40

0

5

10

15

Quarters

20

25

30

35

40

Quarters

-3

6

×10

1

5

Home real money balances
Foreign real money balances

0.8Home nominal interest rate
Foreign nominal interest rate

4

0.6

3

0.4

2

0.2

1

0

0

-0.2
0

5

10

15

20

25

30

35

40

0

Quarters

5

10

15

20

25

30

35

40

Quarters

Figure 5: Optimal monetary policy under the ZLB when a foreign money demand shock
occurs in the case of χ = 0.02.

49

-3

8

(a) Home nominal interest rate×10(b) Foreign nominal interest rate

0.016

7

0.014

6

0.012

5

0.01

4

0.008

3

0.006

2

0.004

1

0.002

0

0
0

5

10

15

20

25

30

35

40

0

Quarters

5

10

15

20

25

30

35

40

Quarters
χ=0.02

χ=0.04

Figure 6: Optimal monetary policy under the ZLB when a foreign money demand shock
occurs.

50

-3

8

-3

(a) Home nominal interest rate

×10

8

7

7

6

6

5

5

4

4

3

3

2

2

1

1

0

(b) Foreign nominal interest rate

×10

0
0

5

10

15

20

25

30

35

40

0

5

Quarters

10

15

20

25

30

35

40

Quartersρ*ζ =0.3

ρ*ζ =0.5

ρ*ζ =0.8

Figure 7: Optimal monetary policy and persistence of a foreign money demand shock
under the ZLB in the case of χ = 0.02.

51