Working Paper Series E-2012-02

Sovereign Risk in the Euro Zone and Monetary Policy

Eiji Okano (Chiba Keizai University)
Eiji Ogawa (Hitotsubashi University)©2012 Eiji Okano and Eiji Ogawa. All rights reserved. Short sectionsof text, not to exceed two paragraphs, may be quoted without explicit
permission provided that full credit is given to the source.

EK-120i

Sovereign Risk in the Euro Zone and Monetary
Policy∗Eiji OKANO† and Eiji OGAWA‡
May, 2012
Abstract
We analyze monetary policy in a currency union with sovereign risk
using a three-country model including a two-country currency union andintroduce an ad hoc assumption that one of the two countries is exposed
to sovereign risk. In our model, if expected fiscal revenue is less thancurrent public debt, one country can default whereas the other country
cannot default. In a canonical setting that analyzes optimal monetary
policy, the optimal monetary policy is consistent with the Taylor rule.
However, the Taylor rule is not optimal and persistently causes defaults
in our model. In contrast, an interest rate peg, namely government bonds
swaps, terminates defaults after one period and is optimal in our model.

Keywords: Sovereign Risk; European Crisis; Fixed-interest-rate Rule; Fiscal
Theory of Price LevelJEL Classification: E52; E60; F41; F47∗ This paper is prepared for the APEA 8th Annual Conference that is held at Nanyang
Technological University, Singapore on June 28-29, 2012.
† Corresponding Author. Faculty of Economics, Chiba Keizai University, 59-5, Todorokicho
3-chome, Inage-ku, Chiba-shi, Chiba, 263-0021, Japan. Tel: +81-43-253-9725; Fax: +81-43254-6600; E-mail: e.okano@cku.ac.jp‡ Graduate School of Commerce and Management, Hitotsubashi University, Tokyo, Japan

1

Introduction

After the Lehman Crisis in Sep. 2008, inflation concerns appeared accompanied
by a slow economic recovery in the euro zone, including Germany, through adepreciation in the euro. However, the IMF and the member countries of theEuropean Monetary Union (EMU) faced the necessity of providing financial
support to Greece in Mar. 2010 because of the Greek fiscal crisis in late 2009.
To prevent contagion of this crisis to other member countries in the EMU, theEuropean Stability Mechanism (ESM) was established at that time. However,
unfortunately this crisis extended to other member countries in the EMU. Ireland faced a financial crisis stemming from the Greek crisis in autumn 2010 andit was decided that Ireland would be supported financially in Nov. 2010. In
2011, Portugal faced a fiscal crisis and financial support for Portugal was provided in Apr. 2011. In addition, fiscal crisis concerns also developed over Spain,
Italy and other member countries in the EMU and these concerns increased
government bond yields in these countries. To reverse increases in government
bond yields, the European Central Bank (ECB) intervened and supported thegovernment bond market in those countries. At present, some euro zone countries are facing inflation worries while other countries are facing sovereign risk
concerns, and therefore conducting monetary policy in the euro zone in the presence of both types of problems is diﬃcult because of the inconsistency between
stabilizing inflation and reducing sovereign risk. The ECB, as the only central
bank in the EMU, conducts a common monetary policy in the EMU that cannot
target multiple objectives.
The member countries in the EMU have debated these conflicting objectives.
The member countries held an urgent high-level meeting in Brussels in Jul.
2011, and developed a policy response to concern over the financial uncertainty
stemming from the Greek crisis. The policy includes expansion of the European
Financial System Facility (EFSF), purchase of Greek government bonds and a
20% debt waiver on these bonds. In addition, the private financial sector agreed
to roll over their 7.5-year government bonds for 15-year or 30-year government
bonds. Because of concern about domestic public sentiment, Angela Merkel,
the German Chancellor, furiously opposed the policy, in particular the issuingof euro zone bonds and increasing the EFSF’s funding. How can the euro zone
best cope with this problem? Our research agenda is to identify policies that
address not only the problems in the European economy but also those in theworld economy.
Our paper has policy implications for the current EU situation including the
support of government bond markets in Portugal, Spain and Greece by the EFSF
or ECB, even though such support may cause loosening of the fiscal disciplinein these countries. However, loosening of the fiscal discipline can be presumed
by selling operations if these countries subsequently face inflation concerns. In
short, combining buying operations for government bonds issued by countries
where the nominal interest rate is increasing and selling operations for thoseissued by countries facing inflation concerns, namely government bond swaps,
could dissolve the trade-oﬀ between inflation concerns and sovereign risk in acurrency union, especially the euro zone.
A notable feature of our model is that a country in a currency union may
go into default. Such a country issues government bonds with sovereign risk.
This feature makes it possible to examine monetary policy in the current euro1

zone context because it includes government issuance of bonds with sovereign
risk. Woodford[13] shows the importance of the relationship between the first
and the second moves in a game in conducting monetary policy. Our results
verify the results of Woodford[13]. In an economy with sovereign risk, Ricardian equivalence does not exist and fiscal policy is an active policy, namely
the first move policy. In such an economy, the Taylor rule, which is an active policy, namely the first move policy, should not be adopted, and instead afixed-interest-rate rule which is a passive policy, namely the second move policy,
should be adopted. The Taylor rule does not provide a unique solution in a rational expectations equilibrium (REE), whereas a fixed-interest-rate rule always
guarantees a unique solution in an REE in such an economy.
In many recent papers analyzing optimal monetary policy, including Woodford[15],Ricardian equivalence is implicitly assumed for simplicity. It is shown that
allocations brought about by optimal monetary policy can be replicated approximately by introduction of the Taylor rule. Discussions regarding optimalmonetary policy under this assumption are expanded to the situation of a currency union by Beetsma and Jensen[1], Benigno[4] and Ferrero[6]. However,needless to say, these authors discuss optimal monetary policy in a currency
union by assuming Ricardian equivalence.1
Uribe[12] presents important policy implications for this problem, although
his model is simplistic. His model features the fiscal theory of the price level(FTPL), which is advocated by Cochrane[5], Sims[11] and Woodford[13]. Uribe[12]analyzes monetary policy in an economy with sovereign risk and shows that the
Taylor rule prolongs the default period, whereas a monetary policy rule thatkeeps the interest rate on risky assets equal to the interest rate on nonrisky
assets and pegs them at their steady-state value, namely the fixed-interestrate rule results in an immediate convergence to default. However, exogenous
production, namely an endowment economy, is assumed in his model and themarginal utility of consumption is constant over time. Because of this, the GDP
gap and dynamics in inflation, which are focused on in the dynamic stochastic
general equilibrium (DSGE) literature, cannot be discussed in detail. In addition, his model has no foreign sector, and thus interaction with foreign economies
cannot be discussed. Hence, discussion on monetary policy in an economy withsovereign risk is only the first step.
We introduce Uribe[12]’s idea into our three-country model, which is basedon Gali and Monacelli[7]’s small open economy model, and we analyze monetary policy in a currency union with sovereign risk. As mentioned, we developa policy implication that the fixed-interest-rate rule results in an immediateconvergence to default as shown by Uribe[12]. In addition, we show that fixedinterest-rate rule is a suﬃcient condition for uniqueness of the REE solution.
Interestingly, we also show that volatilities in inflation and the foreign GDP gap
under the fixed-interest-rate rule are smaller than those under the Taylor rule.
The paper is organized as follows. Section 2 describes the trade-oﬀ faced
by ECB between inflation concerns and sovereign risk in countries undergoinga fiscal crisis. Section 3 derives our DSGE model with sovereign risk. Section
4 presents the macroeconomic dynamics numerically. Section 5 discusses thesolution uniqueness. Section 6 concludes.
1 Ricardian equivalence is not necessarily satisfied in Ferrero[6]. However, his log-linearized
model satisfies Ricardian equivalence around the steady state.

22Inflation Concerns and Sovereign Risk

The ECB defines its role as maintaining the purchasing power of the euro andprice stability in the euro zone, and conducts monetary policy to keep inflation at
2% or less in the short term and around 2% on average over the midterm. DSGE
models imply that inflation is stabilized by increasing the nominal interest rate.
An increase in the nominal interest rate involves an increase in the naturalinterest rate and prevents an increase in the GDP gap. That is, the nominalinterest rate removes inflationary pressure and stabilizes the inflation rate. In
fact, as shown in Fig. 11, the policy interest rate, namely the nominal interest
rate, increases when inflation increases and vice versa. After the Lehman Crisis
in Sep. 2008, the policy interest rate was suddenly reduced because of a liquidity
shortage in the euro zone, although it had been raised in Apr. 2011. This
increase in the policy interest rate resulted from a 12-month inflation figure of
2.6% announced in Mar. 2011, which was the highest figure in the previous 29
months. Following on from this, a new 12-month inflation rate of 2.6 % wasannounced in Jun. 2011, and the policy rate was increased in Jul. 2011.
In contrast, there are several countries, including Greece, where a fiscal crisis
emerged. Although the announced Greek fiscal deficit was 3.6% of GDP, it waslater revealed that the actual deficit was 13.6%. Greek government bonds weresold in large quantities. As shown in Fig. 22, long-term interest rates in Greece
increased significantly. The yield on Greek 10-year government bonds increased
to 17.83% in Jun. 2011 and the spread in yields between Greek government
bonds and German federal bonds widened to 14.81%. Following the discovery
of Greece’s huge fiscal deficit, other European countries’ fiscal deficits received
the attention of investors. For example, Irish and Portuguese fiscal deficits were
14.7% and 8% of GDP, and long-term interest rates in those countries increased
suddenly when the yield on Greek government bonds increase relative to theyield on German federal bonds. Furthermore, the government bond yields for
those countries were increasing relative to the yield on German federal bonds,
such that the yield on Portuguese 10-year government bonds reached 8.5% in
Apr. 2011 and the yield on Irish 10-year government bonds reached 8.5% in
Jul. 2011. In such a situation, an increase in the policy interest rate increases
funding costs in those countries and may increase the possibility of default.
Hence, monetary easing is necessary to avoid default on government bonds.
Monetary easing involves lowering the policy interest rate to stabilize inflationin the euro zone. At present, the ECB faces a trade-oﬀ between a restrictive
monetary policy to stabilize inflation and a monetary easing policy to avoid
default on government bonds.

3

The Model

We derive a three-country model including a currency union based on Gali andMonacelli[7]’s small economy model. The currency union consists of countries H
and F , which organize a monetary union. We assume that there is a default risk
in country F and the mechanism of default follows Uribe[12]. The households on
the interval [0, α) belong to country H while those on the interval [α, 1] belong
to country F . Country A is outside the currency union where the households
line up on the interval [1, 2].

3

3.1

Households

A representative household’s preferences are given by:
Ã∞
!
Ã∞
!
Ã∞
!
X
X
X
U H ≡ E0
β t UH,t ; UF ≡ E0
β t UF,t ; U ∗ ≡ E0
β t Ut∗
t=0

t=0

(1)

t=0

1+ϕ
1+ϕ
1
1
1
1
where UH,t ≡ 1−σ
Ct1−σ − 1+ϕNH,t, UF,t ≡ 1−σ
Ct1−σ − 1+ϕNF,tand Ut∗ ≡1+ϕ1
1
C 1−σ − 1+ϕ(NF,t )
denotes the period utility in countries H, F and
1−σ t
A, respectively, Et denotes the expectation conditional on the information set

at period t, β ∈ (0, 1) denotes the subjective discount factor, Ct denotes the
RαR1
consumption index, NH,t ≡ 0 NH,t (h) dh, NF,t ≡ α NH,t (f ) df and Nt∗ ≡R2 ∗Nt (a) da denote hours of labor in countries H, F and A, respectively, and1
ϕ denotes the inverse of the elasticity of labor supply. Note that variables incountry A are accompanied by an asterisk while those in the currency union arenot. The consumption index is defined as follows:
1

1

2
2
Ct ≡ 2CE,t
CA,t
,

(2)

1−α
1
α
denotes consumption in the currency union,
where CE,t ≡ αα (1−α)
1−α CH,t CF,t
ε
∙³
¸ ε−1
ε
´ 1ε Rh¡ ¢ 1 R
i
ε−1
ε−1
ε−1
1
1 ε α
1
ε
ε
CH,t ≡Ct (h)dh, CF,t ≡C (f )
df
and
α
1−α
0
α t
ε
hR
i
ε−1
ε−1
2
denotes consumption subindexes of the continuum
CA,t ≡ 1 Ct (a) ε daof diﬀerentiated goods produced respectively in countries H, F and A, Ct (h),Ct (f ) and Ct (a) denote generic goods produced in countries H, F and A,
respectively, σ > 1 denotes the degree of relative risk aversion and ε > 1 denotes
the elasticity of substitution across goods. Consumption in country A is defined
analogously to Eq.(2).
Total consumption expenditures are given by PH,t CH,t +PF,t CF,t +PA,t CA,t =11h¡ ¢ R
i 1−ε
h³
i 1−ε
´R
α
1
1−ε
1−ε
1
Pt Ct , where PH,t ≡ α1 0 Pt (h)dh, PF,t ≡ 1−α
P
(f
)
df
α t
1
hR
i 1−ε
2
1−ε
and PA,t ≡ 1 Pt (a)dadenote the producer price indexes (PPIs) incountries H and F , respectively. In addition:
1−α
α
PF,t
,
Pt ≡ PH,t(3)

denotes the consumer price index (CPI) and:1−ααPE,t ≡ PH,t
PF,t

(4)

denotes the price index of goods produced in the currency union. The CPI incountry A is defined analogously to Eq.(3).
By log-linearizing (3) and taking a first diﬀerential, we have:
πt =

1
1
πE,t + πA,t ,
2
2

(5)

where πt ≡ pt − pt−1 denotes CPI inflation and πE,t ≡ pE,t − pE,t−1 andπA,t ≡ pA,t − pA,t−1 denote PPI inflation in a currency union and country A,
4t
respectively. Note that we define vt ≡ dV
V , which is the percentage deviationfrom an arbitrary variable’s steady-state value where Vt is an arbitrary variable
and V is the arbitrary variable’s steady-state value. Similar to Eq.(5), we have:

πE,t = απH,t + (1 − α) πF,t(6)

by log-linearizing Eq.(4), where πH,t ≡ pH,t − pH,t−1 and πF,t ≡ pF,t − pF,t−1
denote PPI inflation in countries H and F , respectively.
By solving cost minimization problems for households, we have the optimal
allocation of expenditures as follows:
µ
µ
¶−ε
¶−ε
Pt (f )
1 Pt (h)1
Ct (h) =CH,t ; Ct (f ) =CF,t .
(7)
α PH,t1−αPF,tHence, total demand for goods produced in countries H and F is given by:
µ
µ
¶−1
¶−1
PH,t
PF,t
CH,t = αCt ; CF,t = (1 − α)
Ct .
(8)
Pt
Pt
In addition, total demand for goods produced in a currency union and country
A is given by:
µ
µ
¶−1
¶−1
1 PE,t
1 PA,t
CE,t =
Ct ; CA,t =
Ct .
(9)
2
Pt
2
Pt
Similar to Eq.(7), we have:
Ct (a) =µPt (a)
PA,t¶−ε

CA,t ,(10)

by solving the cost minimization problem for households in country A.
By aggregating households’ budget constraints, we have:
DH,t + WH,t NH,t + ΓH,t
DF,t + Wt NF,t + Γt≥≥

Pt Ct + Et (Qt,t+1 DH,t+1 ) ,Pt Ct + Et (Qt,t+1 DF,t+1 ) ,(11)

where Et (Qt,t+1 ) denotes the stochastic discount factor, DH,t and DF,t denotethe nominal payoﬀ in period t of the portfolio held at the end of period t − 1
in countries H and F , respectively, WH,t and WF,t denote the nominal wage incountries H and F , respectively, and ΓH,t and ΓF,t denote profits from ownershipof the firms in countries H and F , respectively. The first and second equalities
are the aggregated budget constraints in countries H and F , respectively.
Similar to Eq.(11), the household budget constraint in country A is given
by:
¡
¢
∗
Dt∗ + Wt∗ NF,t + Γ∗t ≥ Pt Ct + Et Qt,t+1 Dt+1.
(12)
The representative household in a currency union maximizes Eq.(1) subject
to Eq.(11). The optimality conditions are given by:
Ã
!
−σ
Ct+1
Pt
1
= f,
βEt
(13)
Ct−σ Pt+1
Rt
ϕ=
Ctσ NH,tWt
W∗
ϕ; Ctσ NF,t
= t .
Pt
Pt
5

(14)

Eq.(13) is the intertemporal optimality condition and Eq.(14) is the intratemporal optimality condition, where Rtf ≡ 1 + rtf denotes the gross risk-free nominalinterest rate that satisfies 1f = Et (Qt,t+1 ) and rtf denotes the net interest rate.
Rt

Log-linearizing Eq.(13) yields:
ct = Et (ct+1 ) −1 f
1r̂ + Et (πt+1 ) ,
σ t
σ

(15)

dRf

with r̂tf = Rt .
The representative household in country A maximizes Eq.(1) subject toEq.(12). The optimality conditions are given by:Ã¡¢−σ ∗ !
∗
Ct+1
Pt
1
= ∗,
βEt
(16)
−σ ∗
∗
Rt
(Ct ) Pt+1
(Ct∗ )σ (Nt∗ )ϕ =

Wt∗
.
Pt∗

(17)

Eq.(16) is the intertemporal optimality condition and Eq.(17) is the intratemporal optimality condition, where Rt∗ ≡ ¡1+rt∗ denotes
the gross nominal interest¢
rate in country A that satisfies R1∗ = Et Q∗t,t+1 . Log-linearizing Eq.(16) yields:t¡¢ 1
1 ¡ ∗ ¢,
c∗t = Et c∗t+1 − r̂t∗ + Et πt+1
σ
σ

dR∗

(18)

with r̂t∗ = Rt and πt∗ ≡ p∗t − p∗t−1 .
Combining Eqs.(13) and (16) with an appropriate initial condition yields:
Ct = Ct∗ ,(19)which implies that marginal utility of consumption is no diﬀerent between thecurrency union and country A.

3.2

Firms

A typical firm in each country produces a diﬀerentiated good with a linear
technology represented by the production function:
Yt (h) = AH,t NH,t (h) ,;Yt (f ) = AF,t NF,t (f ) ,where YH,t (h) and YF,t (f ) denote the output of a generic good in countries H
and F , respectively, and AH,t and AF,t denote the productivity in countries H
and F , respectively. Firms in country A have a technology similar to firms in
the currency union. That is, Yt (a) = A∗t Nt∗ (a) holds.
εh¡ ¢ Ri ε−1ε−1αAnalogous to consumption indexes, we define YH,t ≡ α1 0 Yt (h) ε dh
ε
´R
h³
i ε−1
ε−1
1
1
ε
and YF,t ≡
Y
(f
)
df
. Combining these definitions and thet1−αα
PPI indexes, we have:
Yt (h) =µPt (h)PH,t¶−ε

YH,t ; Yt (f ) =6µPt (f )PF,t¶−ε

YF,t .

(20)

In addition, we define Yt∗ ≡hR
2

Yt∗ (a) =ε−1

Y ∗ (a) ε dh
1 t

ÃPt∗ (a)
∗
PA,t

!−ε

ε
i ε−1

and we have:

Yt∗ .

(21)

By combining the production technology in the currency union and Eq.(20),
we have an aggregate production function related to aggregate employment asfollows:
NH,t =
where ZH,t ≡YH,t ZH,t
YF,t ZF,t; NF,t =
,
AH,t
AF,t

R α ³ Pt (h) ´−εPH,t0dh and ZF,t ≡R 1 ³ Pt (f ) ´−ε
α

PF,t

(22)
df denote the pricedispersions in countries H and F , respectively. By log-linearizing Eq(22), we
have:
yH,t = nH,t − aH,t ; yF,t = nF,t − aF,t .

(23)

´
³
2
Notice that ZH,t and ZF,t disappear in Eq.(23) because of o kξk .
By combining the production technology in country A and Eq.(21), we havean aggregate production function related to aggregate employment as follows:Nt∗ =with Zt∗ ≡R 2 ³ Pt∗ (a) ´−ε
1

∗
PA,t

Yt∗ Zt∗
,
A∗t

(24)

da being the price dispersion in country A. Log-linearizing Eq.(24) yields:yt∗ = n∗t − a∗t .

(25)

Each firm is a monopolistic producer of one of the diﬀerentiated goods. Eachfirm sets their prices Pt (h), Pt (f ) and Pt∗ (a) taking as given Pt , Pt∗ , PH,t , PF,t ,∗, Ct and Ct∗ . We assume that firms set prices in a staggered fashion in thePA,tCalvo—Yun style, according to which each seller has the opportunity to change
its price with a given probability 1 − θ, where an individual firm’s probability of
re-optimizing in any given period is independent of the time elapsed since it lastreset its price. When a firm has the opportunity to set a new price in period t,
it does so in order to maximize the expected discounted value of its net profits.
The FONC for firms are given by:
"∞
µ
¶#
X
ε
k
= 0,
θ Qt,t+k ỸH,t+k P̃H,t −Et
PH,t+k M CH,t+k
ε−1
k=0
"∞
µ
¶#
X
ε
k
Et
= 0,
θ Qt,t+k ỸF,t+k P̃F,t −PF,t+k M CF,t+k
ε−1
k=0
"∞
µ
¶#
X
ε
∗
∗
∗
Et
P̃A,t
= 0, (26)θ k Q∗t,t+k Ỹt+k−
M Ct+kP∗
ε − 1 A,t+k
k=0

7

W

W

H,t
F,t
where M CH,t ≡ (1−τH,t )P, M CF,t ≡ (1−τF,t )Pand M Ct∗ ≡H,t AH,tF,t AF,t

Wt∗∗ A∗(1−τt∗ )PA,ttdenote the real marginal costs in countries H, F and A, respectively, ỸH,t+k ≡³³ ∗ ´−ε
´−ε
´−ε
³
P̃H,t
P̃F,t
P̃
∗
∗
YH,t+k , ỸF,t+k ≡ PF,t+k
YF,t+k and Ỹt+k≡ P ∗tYt+kdePH,t+k
t+knote demand for goods produced in countries H, F and A, respectively, when∗firms choose a new price, P̃H,t , P̃F,t and P̃A,tdenote newly set prices in countries H, F and A, respectively, and τH,t , τF,t and τA,t denote the tax rate incountries H, F and A, respectively.
By log-linearizing Eq.(26), the inflation dynamics can be written as:

πH,t
πF,t
∗
πA,t

=
=
=

βEt (πH,t+1 ) + κmcH,t ,βEt (πF,t+1 ) + κmcF,t ,¡ ∗¢βEt πA,t+1+ κmc∗t ,(27). Eq.(27) shows that not only an increase in current
with κ ≡ (1−θ)(1−θβ)θmarginal cost but also an increase in future marginal cost increases current
inflation because expected inflation appears on the RHS in Eq.(27).
Substituting Eq.(14) and Eq.(17) into the definition of the real marginal costyields:1−αM CH,t=M CF,t

=

M Ct∗

=

1

2
SE,t2 SH,tϕ
Ctσ NH,t,(1 − τH,t ) AH,tϕα
1
Ctσ NF,t2
St2 SF,t,(1 − τF,t ) AF,t∗ σ∗ ϕ
− α − 1−α (Ct ) (Nt )SH,t2 SF,t 2,(1 − τt∗ ) A∗t

(28)

P

F,t
denotes the terms of trade (TOT) in the currency union,
where SE,t ≡ PH,t

P

P

A,t
SH,t ≡ PH,tdenotes the TOT between countries H and A and SF,t ≡ PA,tF,tdenotes the TOT between countries F and A. Log-linearizing Eq.(28) yields:

mcH,t

=

mcF,t

=

mc∗t

=

with τ̂H,t ≡

3.3

1−α
1
τ
sE,t + sH,t + σct + ϕnH,t − aH,t +τ̂H,t ,
2
2
1−τ
α
1
τ
− sE,t + sF,t + σct + ϕnF,t − aF,t +τ̂F,t ,
2
2
1−τ
α
1−α
τ
sF,t + σc∗t + ϕnF,t − a∗t +τ̂ ∗ ,− sH,t −
2
2
1−τ t

dτH,t
dτF,t
τ , τ̂F,t ≡τand τ̂t∗ ≡

(29)

dτt∗
τ .

Government

Similar to private consumption, government expenditure is a Dixit—Stiglitz aggregator defined by:εε¶Z 1∙µ ¶ Z α
¸ ε−1
∙µ
¸ ε−1
ε−1
ε−1
1
1
ε
ε
Gt (h)dh, ; GF,t ≡Gt (f )
df,
GH,t ≡α1−α0αwhere GH,t and GF,t denote indexes of government expenditure in countries H
and F , respectively. For simplicity, we assume that each government allocates a8level of government expenditure only among domestic goods. Each government
implies the following demands for the generic goods h and f :Gt (h) =µPt (h)
PH,t¶−εGH,t ; Gt (f ) =µPt (f )PF,t¶−ε

GF,t .

(30)

Similar to that in a currency union, government expenditure in country A is
ε
hR
i ε−1
ε−1
2
a Dixit—Stiglitz aggregator defined by G∗t ≡ 1 G∗t (a) ε dh
. Hence, we
have:
!−ε
Ã
Pt∗ (a)∗Gt (a) =
G∗t .
(31)
∗
PA,t
The flow government budget constraints are given by:
Z αf
Pt BH,t = Rt−1
Pt−1 BH,t−1 −Pt (h) [τH,t Yt (h) − Gt (h)] dh,0

Pt BF,t=Rt−1 Pt−1 BF,t−1 (1 − δt ) −Z 1
αPt (f ) [τF,t Yt (f ) − GF,t (h)] dh,(32)

where BH,t and BF,t denote the per capita value of bond issuance of government
bonds in real terms in countries H and F , respectively, Rt ≡ 1 + rt denotes thegross nominal interest rate for risky assets, rt denotes the net nominal interest
rate for risky assets and δt denotes the default rate. We assume that government
bonds issued by country H’s government are safety assets, although ones issuedby country F ’s government are risky assets. Hence, δt and Rt appear in thesecond line in Eq.(32). The first and the second equalities in Eq.(32) are thegovernment budget constraints in countries H and F , respectively.
Substituting Eq.(30) and Eq.(21) into Eq.(32)yields:
Pt BH,tPt BF,t

=
=

f
Rt−1
Pt−1 BH,t−1 − PH,t (τH,t YH,t − GH,t ) ,Rt−1 Pt−1 BF,t−1 (1 − δt ) − PF,t (τF,t YF,t − GF,t ) .

(33)

The first and the second equalities in Eq.(33) are the government budget constraints in countries H and F , respectively. Dividing both sides of Eq.(33) by
Pt−1 yields:
Πt BH,t=Πt BF,t=−(1−α)

fRt−1
BH,t−1 − Πt St(τH,t YH,t − GH,t ) ,Rt−1 BF,t−1 (1 − δt ) − Πt Stα (τF,t YF,t − GF,t ) .

(34)

By log-linearizing Eq.(34), we have:
bH,t

=

1
1 f
1
(1 − α) (1 + β)
α − (1 − β)− πH,t +bH,t−1 + r̂t−1
sE,t −
sH,t
β
β
β
2β
2β
1−α
1−α
α
1−α
τ
−
yH,t
sF,t +
sE,t−1 +sH,t−1 +sF,t−1 −
2β
β
2β
2β
ςB
ςG
τ
+
gH,t −
τH,t
ςB
ςB
9

bF,t

=

1
1
δ
1
α (1 + β)δ̂t − πF,t +bF,t−1 + r̂t−1 −sE,tβββ (1 − δ)β2β
1 − α − (1 − β)
α
α
1−α
α
−
sF,t −
sH,t − sE,t−1 +sF,t−1 +sH,t−1
2β
2β
β
2β
2β
τ
ςG
τ
−
yF,t +
gF,t −
τF,t
(35)
ςB
ςB
ςB

with δ̂t ≡ dδδ t where ςG ≡ YG denotes the steady-state ratio of government
expenditure to GDP and ςB ≡ YB denotes the steady-state share of government
debt to GDP.
Similar to Eq.(33), the government budget constraint in country A is given
by:
∗
∗
∗
∗
Pt∗ Bt∗ = Rt−1
Pt−1
Bt−1
− PA,t(τt∗ Yt∗ − G∗t ) ,which can be rewritten similar to Eq.(34) as follows:
α

1−α

∗
∗
2
Bt−1
− Π∗t SH,tSF,t2 (τt∗ Yt∗ − G∗t ) .
Π∗t Bt∗ = Rt−1

(36)

Log-linearizing Eq.(36) yields the following:
b∗t

=

1 ∗
1
1
α
1−α
α
+ r̂ ∗ − π ∗ + sH,t +bsF,t −
sH,t−1
β t−1 β t−1 β A,t 222β1−ατ ∗ ςG ∗
τ ∗
−
yt +
gt −
τ .
sF,t−1 −
2β
ςB
ςB
ςB t

(37)

。

Next, we discuss the government budget constraint in country F where there
is a possibility of default and the default dynamics follow Uribe[12]. The appropriate transversality condition for government bonds issued by country F ’s
government is given by:∙¸
Pt+j BF,t+j
t+j+1
= 0.
(38)
lim β
Et Rt+j (1 − δt+j+1 )j→∞
Pt+j+1
Optimizing households must be indiﬀerent between holding government bonds
and holding state contingent claims. This implies the following intertemporal
optimality condition:
Ã
!
−σ
Pt
Ct+1
1
βEt
=
.
(39)
−σ
R
E
(1
− δt+1 )Ct Pt+1
t t
In Eq.(39), Et (1 − δt+1 ) Rt replaces Rtf in Eq.(13). By log-linearizing Eq.(39),
we have:
³
´
1
δ
(40)
ct = Et (ct+1 ) − Et (πt+1 ) +Et δ̂t+1 ,σσ (1 − δ)with δ̂t ≡ dδδ t . By subtracting Eq.(40) from Eq.(15), we have:r̂t = r̂tf +³´δ
Et δ̂t+1 .
1−δ
10

(41)

Eq.(41) shows that the interest rate on risky assets is equal to the interest rate
on nonrisky assets with an expected default rate. That is, the interest rate on
risky assets includes a risk premium.
Multiplying Rt (1 − δt+1 ) by both sides of Eq.(34) and iterating by substituting j times forward yields:
" j
#
Y
n
(1 − δt+k+1 ) Rt−1 Bt−1 (1 − δt )Rt+j BF,t+j (1 − δt+j+1 ) =
k=0

−

" j
j
X
Y

k=0

h=k

#

Rt+h (1 − δt+h+1 ) PF,t+k (τF,t+k YF,t+k−GF,t+k ) ,tdenotes gross CPI inflation in a currency union. By dividing
where Πt ≡ PPt−1
both sides of this equality Pt+j+1 , we have:
#
" j
Y
Bt+j
BF,t−1−1(1 − δt+j+1 ) =Rt+k (1 − δt+k+1 ) Πt+j+1 Rt−1
(1 − δt )Rt+j
Pt+j+1
Pt
k=0
" j
#
j
X
Y
−1
Rt+h (1 − δt+h+1 ) Πt+h+1 SPF,t+k ,
+

k=0

P

(τ

Y

−G

h=k

)

F,t
where SPF,t ≡ F,t F,t PF,tdenotes the nominal fiscal surplus in termstof the CPI. Using the conditional expectation operator on both sides of thisequality, substituting Eq.(39) into this equality and repeatedly applying the
expected value operator, we have:∙¸nn
BF,t+j
BF,t−1C −σ
Et Rt+j(1 − δt+j+1 )= β −(j+1) ¡ t −σ ¢ Rt−1
(1 − δt )Pt+j+1Pt
Et Ct+1
Ã!
jX1
k−j−1 −σ− Et
βCt+k SPF,t+k .
−σ
Ct+j+1k=0

By multiplying both sides of this equality by β j , taking the limit of j → ∞ andsubstituting Eq.(38), we have:Ct−σ Rt−1

∞
n
X
¡ −σ
¢
BF,t−1(1 − δt ) =β k Et Ct+k
SPF,t+k .
Pt
k=0This equality is the central equation of FTPL where we assume δt = 0, as
pointed out by Uribe[12]. While Uribe[12] assumes an endowment economy
implying that the marginal utility of consumption is constant over time, we introduce production activity explicitly. Introducing production activity implies
that the marginal utility of consumption is not necessarily constant. Hence, incontrast to the equality derived by Uribe[12], the marginal utility of consumption appears on both sides of this equality. By rearranging this equality, we
have the default rate in equilibrium as follows:¢
P∞ k ¡ −σ
k=0 β Et Ct+k SPF,t+k
δt = 1 −.
(42)
B
Ct−σ Rt−1 F,t−1
Pt
11

As mentioned by Uribe[12], Eq.(42) shows that if the expected net present
value of the primary balance in terms of the marginal utility of consumption isequal to the real balance of government debt in terms of the marginal utility of
consumption at the beginning of a period, the default rate reaches zero.2
Eq.(42) can be rewritten as a second-order diﬀerential equation as follows:
Ã
!
−σ
Ct+1
−1
−1
Rt [1 − Et (δt+1 )] BF,t . (43)Π
Rt−1 Πt BF,t−1 (1 − δt ) = SPF,t + βEt
Ct−σ t+1
Eq.(43) shows that an increase in inflation decreases the default rate because
inflation reduces the real debt burden for the government in the next period.
By log-linearizing Eq.(43), we get:
=

δ̂t

3.4

1−β
1 − δβ (2 − δ)
1 − δβ (2 − δ)
1 − δβ (2 − δ)
spF,t +
r̂t−1 −
πt +bF,t−1(1 − δ) δδδδβσ (1 − δ)+Et (ct+1 ) − βσ (1 − δ) ct + β (1 − δ) Et (πt+1 ) − β (1 − δ) r̂tδ− β (1 − δ) bF,t .
(44)−

Market Clearing Conditions

The market clearing conditions in a currency union are given by:
Yt (h) =Yt (f ) =Ct (h) + Ct∗ (h) + Gt (h) ,Ct (f ) + Ct∗ (f ) + Gt (f ) ,

(45)

∗
∗
(h) and CF,t(f ) denote demands for good a produced in country A.
where CH,t
By combining Eqs.(7), (8), (7) and (15), we can rewrite Eq.(20) as follows:#¶−ε "µ¶−1µ
PH,tPt (h)Yt (h) =Ct + GH,t ,
PH,t
Pt
#
µ
¶−ε "µ¶−1
Pt (f )PF,t
Yt (f ) =(46)
Ct + GF,t .
PF,tPt

Substituting Eq.(46) into Eq.(45) yields:
YH,t

=

YF,t

=

µ
µ

PH,t
Pt
PF,t
Pt

¶−1

¶−1

Ct + GH,t ,Ct + GF,t .

(47)

Similar to Eq.(45), country A’s market clearing condition is given by:Yt∗ (a) = Ct (a) + Ct∗ (a) + G∗t (a) ,2 In other words,

Eq.(42) shows that if the expected net present value of the primarybalance in terms of the nominal marginal utility of consumption is equal to the real balanceof government debt at the beginning of a period, the default rate reaches zero.

12

which can be rewritten as follows by substituting in the definition for government
expenditure:µ ∗ ¶−1
PA,t∗Ct∗ + G∗t .
(48)
Yt =
Pt∗
By substituting Eqs.(3) and (3) into Eq.(46), we have:
1−α

P

− 1−α
2

1

YH,t

=

2
SE,t2 SH,t
Ct + GH,t ,

YF,t

=

2
SE,t2 SF,t
Ct + GF,t ,

− 12

P

−α

1

α
2

(49)

− 12

= SE,t SH,t and PF,t
= SE,t SF,t are used to derive Eq.(49). By
where PH,tttlog-linearizing Eq.(49), we have:1−α1sE,t + sH,t + ct + ςG gH,t ,2
2
α1
yF,t = − sE,t + sF,t + ct + ςG gF,t .
(50)
2
2
Similar to Eq.(49), substituting an equality corresponding to Eq.(3) in country A and Eq.(3) into Eq.(48) yields:
yH,t

=

−α

− 1−α

Yt∗ = SH,t2 SF,t 2 Ct∗ + G∗t ,
P

α

1−α

2
= SH,tSF,t2where PA,tt
we have:is used to derive Eq.(51). By log-linearizing Eq.(51),α1−αyt∗ = − sH,t −sF,t + c∗t + ςG gt∗ .
2
2

3.5

(51)

(52)

Natural Rate of Output and GDP Gap

In a flexible price equilibrium, the real marginal cost is constant and it corresponds to the inverse of the constant markup over time. Namely, M CH,t =M CF,t = M Ct∗ = ζ −1 . Substituting these conditions into Eq.(28) and loglinearizing yields:
ȳH,t

=

ȳF,t

=

ȳt∗

=

(1 − α) (σ − 1)
σ−1
σςG
τ
sE,t +
sH,t +
τ̂H,tgH,t −
2$1
2$1$1 (1 − ςG )(1 − τ ) $1
1+ϕ
+
aH,t
$1
σ−1
α (σ − 1)
σςG
τ
−
sE,t
sF,t +
gF,t −
τ̂F,t
2$1
2$1
(1 − ςG ) $1
(1 − τ ) $1
1+ϕ
+
aF,t
$1
α (σ − 1)(1 − α) (σ − 1)σςGτ−sH,t −
sF,t +
g∗ −
τ̂ ∗
2$1
2$1
(1 − ςG ) $1 t
(1 − τ ) $1 t
1+ϕ ∗
+
a ,
(53)
$1 t

where ȳH,t ≡ yH,t − xH,t 、ȳF,t ≡ yF,t − xF,t and ȳt∗ ≡ yt∗ − x∗t denote the naturalrate of output in countries H, F and A, respectively, and xH,t , xF,t and x∗tdenote the GDP gap in countries H, F and A, respectively.
13

3.6

Monetary Policy

As mentioned above, Uribe[12] shows that the Taylor rule prolongs the default
period, whereas a fixed interest rate reduces the default period to just one period
in an economy subject to default. In our paper, we discuss whether Uribe[12]’s
policy implication is applicable or not in our more complex model. To satisfy ourobjective, we define two policy rules, the Taylor rule and the fixed-interest-rate
rule.
The Taylor rule is defined as follows:r̂tf = φπt ,where φ denotes the reaction coeﬃcient of the nominal interest rate to inflation.
In contrast, we define the fixed interest rate as follows:r̂tf = r̂t =1.
βThe Taylor rule implies that the nominal interest rate is hiked following an
increase in inflation and vice versa. φ > 1 is the Taylor principle and the nominalinterest rate is hiked by more than the increase in inflation if this condition
holds. Inflation is immediately stabilized and the GDP gap is also stabilized via
stabilizing inflation. Many papers introducing Calvo pricing into their DSGE
models depict this mechanism. In contrast, the nominal interest rate is constantand corresponds to its steady-state value over time under the fixed-interestrate rule. In addition, the nominal interest rate for nonrisky assets, which isthe policy instrument, corresponds to the nominal interest rate for risky assets
under the fixed-interest-rate rule. The fixed interest-rate rule is not complex
and was adopted in the US. The nominal interest rate had been fixed from Apr.
1942 until the end of WWII in the US under the bond price support regime
and it has been fixed since Feb. 1999 in Japan. A zero-interest-rate policy has
been adopted in the US since Dec. 2006, which means that interest rates havebeen fixed.3 Furthermore, the policy that equalizes the nominal interest rate
for both nonrisky and risky assets corresponds to purchasing Exchange Traded
Fund or Real Estate Investment Trust by the Bank of Japan.
The fixed-interest-rate rule is a policy that converges two interest rates with
diﬀerent levels of risk and pegs this interest rate to its steady-state value. As
mentioned in Section 2, in Jun. 2011 the yield on Greek government bonds was
17.83%, while the yield on German federal bonds was 2.9%. If the steady-statevalue of the interest rate is 4 to 6%, Greek government bonds are purchasedby the central bank while Germany federal bonds are sold by the central bankunder the fixed-interest-rate rule. Hence, we designate the fixed-interest-rate
rule as a government bond swap arrangement.
Finally, we assume that the Taylor rule is always adopted in country A
independently from the currency union’s monetary policy as follows:∗r̂t∗ = φ∗ πA,t.

3 The zero-interest-rate policy was removed temporarily between Aug. 2000 and Mar. 2006.

14

4

Macroeconomic Dynamics

4.1

Parameterization

Following previous DSGE studies, we solve the model numerically because it
is too diﬃcult to solve the DSGE model analytically. Following Ferrero[6],in analyzing the monetary and fiscal policy mix in a currency union, we set
the subjective discount factor, steady-state tax rate, steady-state government
expenditure ratio to GDP, steady-state government debt ratio to GDP and
degree of relative risk aversion to 0.99, 0.3, 0.276, 0.24 and 3, respectively. The
elasticity of labor supply is set to 3 following Gali and Monacelli[7]. Following
Beetsma and Jensen[1], who analyze the monetary and fiscal policy mix in acurrency union similar to Ferrero[6], we set the degree of price stickiness to 0.75.
Price stickiness is asymmetric in Beetsma and Jensen[1]’s two-country model.
The degree of price stickiness is set equal to 0.75 in one of the countries in his
two-country model. We set the reaction coeﬃcient of the nominal interest rate
to inflation equal to 1.5 not only in the currency union but also in country A. We
assume that productivity, tax rate and government expenditure are exogenous
AR(1) processes and their AR(1) coeﬃcients are set equal to 0.9. The timing of
the model is quarterly. Hence, we set the annual steady-state government debt
to GDP ratio equal to 60%, which suggests a quarterly ratio of 24%. This isequal to the upper limit of the Stability and Growth Pact. Our setting on pricestickiness implies that prices are constant for four quarters, namely one year.

4.2

Macroeconomic Dynamics

Although our model includes productivity, government expenditure and the tax
rate in each country, we only focus on changes in these stemming from exogenous
shocks in country F . Tab. 1 shows the macroeconomic volatilities for onepercent changes in each shock. Generally speaking, the nominal interest rate is
hiked and inflation stabilized immediately in response to inflation pressure under
the Taylor rule, and this scenario is common in DSGE models. In our model,
however, this mechanism is not necessarily applied. For example, in response to
an increase in government expenditure, government bond swaps make inflation
more stable than under the Taylor rule. There is no notable diﬀerence between
the volatilities of PPI inflation to an increase in the tax rate in country F underboth rules. As for the GDP gap in country F , a fixed-interest-rate rule makes
inflation more stable than under the Taylor rule for increases in government
expenditure and the tax rate in country F . The volatility of the GDP gap incountry H to an increase in government expenditure in country F under a fixedinterest rate is smaller than under the Taylor rule. We do not derive the welfare
criterion stemming from a second-order approximated utility function, and wecannot discuss the welfare associated with the introduction of both rules. In
many DSGE studies, however, the welfare criteria stemming from a secondorder approximated utility function includes variances of PPI inflation and theoutput gap, and these welfare criteria imply that stabilizing PPI inflation andthe output gap minimizes welfare costs. Our results show that the Taylor rule is
not the best way to stabilize PPI inflation and the output gap, and our results
imply that the allocation under the Taylor rule is not close to the macroeconomic
outcome under optimal monetary policy, which has been verified by Rotemberg

15and Woodford[10]. Hence, our results, which are derived using a DSGE modelwith sovereign risk, may bring Rotemberg and Woodford’s[10] established theory
into question. In addition, it is noteworthy that government bond swaps always
stabilize PPI inflation and the GDP gap in country A, except for an increase inproductivity in country F .
Figs. 1, 2 and 3 show impulse responses to a one-percent negative productivity shock, a one-percent positive shock in government expenditure and aone-percent negative tax rate shock in country F . Such changes immediately
decrease tax revenue in country F . Following such changes in exogenous variables, government debt, not only in country F but also in countries H and A,
increases infinitely under the Taylor rule. Government debt in each country,
however, converges to the steady state in 20 to 23 quarters after changes inthose exogenous variables under the fixed-interest-rate rule. The default rate
is remarkable. Although the default rate increases to infinite changes in eachof the exogenous variables under the Taylor rule, it converges in two quarters
following the³ shocks´ under the fixed-interest-rate rule. This result is obvious
because Et δ̂t+1 = 0, which means the expected default rate is zero, and isapplied under government bond swaps. The prices of government bonds in eachcountry are the same, and the prices of risky and nonrisky assets are also thesame under government bond swaps. This means that the central bank purchases government bonds in country F or risky assets when the prices of thebonds or prices of the assets decrease. In such a case, fundraising in country
F is still easy and the country avoids a revenue shortfall when the threat of
one emerges. This is the reason why the government debt converges in 20 to 23
quarters after the shocks. As shown by Uribe[12], who analyzes sovereign risk
and price stability in a closed economy and shows that default is inevitable forprice stability under the Taylor rule, where the fixed interest rate immediately
converges to the default interest rate. Our results are consistent with Uribe’s[12]results.
Finally, we consider the dynamics of the nominal exchange rate. For eachshock, it depreciates by 0.08 to 1.28% in the first quarter under the Taylor rule,
while it depreciates 0.05 to 0.7% under government bond swaps. The volatility
under government bond swaps is smaller than the volatility under the Taylor
rule for each shock.

5

Determinacy

Generally speaking, φ > 1 is a necessary condition to guarantee uniqueness ofan REE in a standard DSGE model.4 That is, the reaction coeﬃcient of thenominal interest rate to inflation must be larger than one for uniqueness ofan REE. Woodford[?] shows that the Taylor rule stabilizes inflation as long as
this condition and Ricardian equivalence are applied and the Taylor rule is a
desirable policy rule from the viewpoint of price stability. This condition, whichassures uniqueness of an REE, varies depending on the model assumptions,as shown in Benhabib, Schmitt-Grohe and Uribe[8]. We have to verify the
4 Blanchard and Khan[2] show that the uniqueness of an REE is assured when the numberof jumpers corresponds to the number of characteristic roots inside the unit circle. In a basic
closed economy DSGE model or a DSGE model assuming Ricardian equivalence, Blanchard
and Khan’s[2] condition is guaranteed if φ > 1. See Monacelli[9] for details.

16

uniqueness of an REE because our model has a distinguishing feature; it is athree-country model consisting of two countries in a currency union and one
other country with sovereign risk. Of course, our solution under the Taylor
rule and the fixed-interest-rate rule assure the uniqueness of an REE given
our parameterization. Our parameterization follows notable preceding studies,
although not all of the empirical papers have results that are consistent with ourparameterization. In particular, the empirical result of price stickiness varies.
Rotemberg and Woodford[10] report it to be 0.66, while Benigno and LopezSalid[3] assume that it is 0.875. Hence, in this section, we discuss values of thereaction coeﬃcient of the nominal interest rate to inflation φ that guarantee
uniqueness of an REE under varying degrees of price stickiness θ.
Fig. 4 shows values of the reaction coeﬃcient of the nominal interest rate toinflation that guarantee uniqueness of an REE under varying degrees of pricestickiness. The vertical axis is price stickiness and the horizontal axis is thereaction coeﬃcient of the nominal interest rate to inflation. Uniqueness of an
REE is assured in the area below the solid line. Note that we set the reaction
coeﬃcient of the nominal interest rate in country A to 1.5. As shown by Fig.
4, uniqueness depends on the degree of price stickiness, and uniqueness is guaranteed even if the reaction coeﬃcient of the nominal interest rate to inflationis less than one. In addition, Fig. 4 shows that uniqueness is not guaranteed
when price stickiness exceeds 0.8 in many cases. Although the empirical results
are not consistent among previous studies, the Taylor rule may severely harman economy if the degree of price stickiness exceeds 0.8 and the Taylor rule
no longer contributes to stabilizing inflation. While the Taylor rule does notnecessarily assure uniqueness of an REE, uniqueness is guaranteed if θ ∈ [0, 1]
under government bond swaps.
Our result in this section supports Woodford’s[14] implication. Following
stable prices in WWII under a bond price support regime in the US and harshinflation in the 1980s in Brazil under the Taylor rule, Woodford[14] advocatesthat the relationship between the first and the second moves in monetary andfiscal policy is important. The first and the second moves mean a combination ofactive monetary policy, namely the Taylor rule, and passive fiscal policy, namely
Ricardian equivalence or a combination of passive monetary policy and active
fiscal policy. There is a point in common between the US in WWII and Brazil
in the 1980s in terms of huge government debt to finance war expenditure andpublic-works spending. Although it cannot be determined with certainty, if Ricardian equivalence is not applicable to the US in WWII and Brazil in the 1980s,
fiscal policy in these countries was most probably active. In that case, monetary
policy must be passive, namely a bond price support regime or the fixed-interestrate rule that corresponds to government bonds swap. Woodford[14] points outthat price stability is the role of the central bank, and the central bank adopted
the Taylor rule in Brazil in the 1980s. Under the Taylor rule, people expected
price stability and low inflation. This expectation did not decrease real government debt, and the revenue shortfall became chronic. Finally, Brazil issued
public debt continuously, and eventually defaulted. This scenario was replicatedin a previous section in our paper. If Ricardian equivalence is not applicable,
the Taylor rule is perilous monetary policy.
Default is an ongoing problem, as illustrated by the recent financial crisis
stemming from the Greek fiscal deficits in Mar. 2008. The yield of Greek government bonds reached 17.83%, an all-time high, and its spread with the yield17

on German federal bonds widened to 14.81percentage points. The default rate
in our model is analogous to sovereign risk reflecting a risk premium. Internalizing default is a cause of the nonexistence of Ricardian equivalence.5 Hence, ourmost practical policy implication for the current euro zone situation is that the
ECB should purchase not only Greek government bonds but also Portuguese
and Spanish government bonds and sell German federal bonds.

6

Conclusion

This paper developed a three-country model consisting of one country and acurrency union with two countries, into which we introduce sovereign risk andanalyze monetary policy. The Taylor rule is not a powerful enough policy tostabilize both inflation and the GDP gap in an economy with default. In addition, there is no uniqueness of an REE under the Taylor rule if price stickinessexceeds a certain level. In contrast, there is uniqueness of an REE under government bond swaps independent of price stickiness. Hence, the ECB should adopt
government bond swaps, which is a policy involving the purchase of government
bonds with sovereign risk and selling government bonds issued by countries with
inflation concerns.
Because of fiscal crisis concerns, the yields on long-term government bonds
are beginning to increase not only in Portugal, Ireland, Italy, Greece and Spain,
namely the PIIGS, but also in other euro zone countries, including France.
As mentioned above, the ECB faces a trade-oﬀ on monetary policy amid anincrease in long-term interest rates, although inflation is gradually increasing.
Because of sudden increases in interest rates stemming from the fiscal crises,
the ECB is now starting to purchase Italian and Spanish government bonds. In
Italy and Spain, there is a notable increase in interest rates stemming from therisk premium associated with sovereign risk. The EFSF is now being used topurchase government debt in these countries. Because inflation concerns andsovereign risk coexist in the euro zone, introducing government bond swaps,
which is a policy involving purchasing government bonds with sovereign risk
and selling government bonds issued by countries with inflation concerns, isnecessary. In the portfolio balance model, government bonds in two countries
are imperfect substitutes when there is sovereign risk in one country. In such acase, government bond swaps make it possible for the interest rates in the twocountries to move in opposite directions to each other. Our paper also showedthat government bond swaps are more desirable than the Taylor rule.

Appendix
A

Nonstochastic Steady StateIn this section, we show the paths of state variables under a deterministic
PH,t,equilibrium that guarantees ΠH,t = ΠF,t = Π∗A,t = 1 with ΠH,t ≡ PH,t−1
P

P∗

F,t
ΠF,t ≡ PF,t−1、Π∗A,t ≡ P ∗t . In the equilibrium, we assume zero inflation.
t−1

5 Ricardian equivalence is not applied in our model. As mentioned by Ferrero[6], our loglinearized model guarantees Ricardian equivalence around equilibrium.

18

In addition, X̃H = X̃F = X̃A∗ = 1 is applied under this equilibrium with
P̃

P̃ ∗

P̃

∗
X̃H,t ≡ PH,t
, X̃F,t ≡ PF,tand X̃A,t≡ PA,t∗ . Because this equilibrium is nonH,tF,tA,tstochastic, AH = AF = A∗ = 1. In addition, we assume GH = GF = G∗ andτH = τF = τ ∗ . Following Ferrero[6], we assume SE,−1 = SH,−1 = SF,−1 = 1.
The gross interest rate is given by:
Rf = R∗ = β −1 .
Eqs.(13) and (24) imply:
Rf = R (1 − δ) ,(54)

because we assume a positive default rate.
Eq.(17) can be rewritten as:
Ã

!
KH,t
P̃H,t = Et
−1
PH,t
FH,t
Ã
!
KF,t
P̃F,t = Et
−1
PF,t
FF,t
⎛
⎞

∗
KA,t
⎜
⎟
∗
P̃A,t
= Et ⎝ ³
⎠,
´−1
∗
∗
PA,t
FA,t

where:
KH,t ≡ ζ∞Xnθ k Qt,t+k ỸH,t+k M CH,t+k

k=0
∞
X

KF,t ≡ ζnθ k Qt,t+k ỸF,t+k M CF,t+k

k=0
∞
X

Kt∗ ≡ ζ

;

k=0

;

n∗
θ k Q∗t,t+k ỸN∗ ,t+k M Ct+k

;

(55)

FH,t ≡ PH,t
FF,t ≡ PF,t∗Ft∗ ≡ PA,t∞X

θk Q−1t,t+k ỸH,t+k

k=0

∞
X

θ k Qt,t+k ỸF,t+k

k=0

∞
X

∗
θ k Q∗t,t+k Ỹt+k
.

k=0

(56)

Eq.(56) implies the following:nζYH M CH1 − θβζYF M CFn
KF =1 − θβζY ∗ M C n∗∗K =1 − θβ

;

KH =

;
;

PH YH
1 − θβPF YF
FF =1 − θβ
P∗Y ∗
∗
F = A
.
1 − θβFH =Combining these equalities and Eq.(55) yields:n; PF = ζM CFn ; PA∗ = ζM C n∗ .
PH = ζM CH(57)

Hence:
M CH = M CF = M C ∗ =
19

1
ζ

(58)

is applied. We define M C ≡ ζ1 .
Eq.(48) implies the following:
1−τ
ζ
1−τ
ζ
1−τ
ζ

1−α

1

−α

1

−α

− 1−α
2

=

SE 2 SH2 C σ YHϕ=SE 2 SF2 C σ YFϕ=SH 2 SFϕ

C σ (Y ∗ ) .

(59)

Substituting the initial condition for the TOT into Eq.(59) yields:
YH = YF = Y ∗ ≡ Y.

(60)

Eqs.(49) and (51) imply the following:
1

1

YH

=

2
2
SH,t
SE,t
C + G,

YF=

SE,t2 SF,t2 C + G,Y∗=SH 2 SF

−α

−1

−α

− 1−α
2

C + G.

(61)

Substituting the initial condition for the TOT and Eq.(60) into Eq.(61)yields:Y = C + G.

(62)

Eq.(59) can be rewritten as:(1 − τ ) UC (C) = ζUN (N ) ,(63)

which is a familiar expression. Because of τ ∈ (0, 1) and θ > 1, this steady stateis ineﬃcient.
Eqs.(27) and (47) imply:¶µ
1−β= τ Y − G,(64)Bβwith B ≡ BH = BF = B ∗ .

References[1] Beetsma, Roel M.W.J. and Henrik Jensen (2005), “Monetary and Fiscal
Policy Interactions in a Micro-Founded Model of a Monetary Union,” Journalof International Economics, 67, 320—352.
[2] Blanchard, Olivier J. and Charles Kahn (1980), “The Solution of Linear
Diﬀerence Equations under Rational Expectations,” Econometrica, 48, 1305—
1311.
[3] Benigno, Pierpaolo and J. David Lopez-Salid (2006), “Inflation Persistence
and Optimal Monetary Policy in the Euro Area,” Journal of Money, Credit
and Banking, 38, 587—614.
20

[4] Benigno, Pierpaolo (2004), “Optimal Monetary Policy in a Currency Area,”
Journal of International Economics, 63, 293—320.
[5] Cochrane, John. H. (1998), “A Frictionless View of U.S. Inflation,” NBER
Macroeconomic Annual, 13, 323—384.
[6] Ferrero, Andrea (2009), “Fiscal and Monetary Rules for a Currency Union,”
Journal of International Economics, 77, 1—10.
[7] Gali, Jordi and Tommaso Monacelli (2005), “Monetary Policy and Exchange
Rate Volatility in a Small Open Economy,” Review of Economic Studies, 72,
707—734.
[8] Jess Benhabib, Stephanie Schmitt-Grohe and Martin Uribe (2001), “Monetary Policy and Multiple Equilibria,” American Economic Review, 91, 167—
186.
[9] Monacelli, Tommaso (2008), “A New Keynesian Optimizing Framework with
Nominal Rigidities,” Lecture note.
[10] Rotemberg, Julio J., and Michael Woodford (1997), “An OptimizationBased Econometric Framework for the Evaluation of Monetary Policy,” In
NBER Macroeconomic Annual 1997, edited by Ben S. Bernanke, and Julio
J. Rotemberg, pp 297—346. Cambridge, MA: MIT Press.
[11] Sims, Christopher (1994), “A Simple Model for the Study of the Determination of the Price Level and the Interaction of Monetary and Fiscal Policy,”
Economic Theory, 4, 381—389.
[12] Uribe, Martin (2006), “A Fiscal Theory of Sovereign Risk,” Journal of
Monetary Economics 53, 1857—1875.
[13] Woodford, Michael (1994), “Monetary Policy and Price Level Determinacy
in a Cash-in-Advance Economy,” Economic Theory, 4, 345—380.
[14] Woodford, Michael (2001), “Fiscal Requirements for Price Stability,” Journal of Money, Credit and Banking, 33, 669—728.
[15] Woodford, Michael (2003), “Interest Rate and Prices: Foundations of a
Theory of Monetary Policy,” Princeton University Press, Princeton and Oxford.

21

Table 1: Macroeconomic Volatility
Variables
GDP gap
in country H
GDP gap
in country F
GDP gap
in currency union
GDP gap
in country A
PPI inflationin country H
PPI inflationin country F
Inflationin currency union
PPI inflationin country A
Gov. debt
in country H
Gov. debt
in country F
Gov. debt
in country A
Default
rate
Interest rate
for Nonrisky assets
Interest rate
for risky assets
Interest rate
in country A
Exchangerate

Monetary policy
Productivity
0.00300.00600.00310.00550.00120.00570.00444.3432e-004
0.00210.00580.00310.00410.00220.00493.3698e-004
0.00511.79350.00480.58200.00350.00260.02293.79510.00460.00320.00004.00300.00000.03385.0105e-004
0.16758.3369e-004

TaylorGBS
TaylorGBS
TaylorGBS
TaylorGBS
TaylorGBS
TaylorGBS
TaylorGBS
TaylorGBS
TaylorGBS
TaylorGBS
TaylorGBS
TaylorGBS
TaylorGBS
TaylorGBS
TaylorGBS
TaylorGBS

22

Shocks in country F
Gov. expenditure
0.00200.00130.00180.00112.5775e-004
0.00129.3797e-004
9.2430e-005
8.4394e-004
0.00120.00128.4492e-004
4.6207e-004
0.00100.0047
7.1391e-005
0.31510.00100.2286
7.4479e-004
0.14955.5011e-004
1.50052.0480e-004
6.6163e-004
0.00001.58250.00000.00700.02290.00185.2489e-004

Tax Rate
0.00130.00250.00674.6771e-004
0.00350.00120.00201.7856e-004
9.2156e-004
0.00240.00140.00189.8619e-004
0.00210.01011.3893e-004
0.79550.00200.29880.00140.32730.0011
1.94740.00180.00140.00002.05410.00000.01502.0632e-004
0.00370.0010Figure 1: IRF to Negative Productivity Shock in Country F

23

Figure 2: IRF to Positive Government Expenditure Shock in Country F

24

Figure 3: IRF to Negative Tax Rate Shock in Country F

25

Figure 4: Determinacy and Price Stickiness26