Working Paper Series E-2010-02

How Important is the Fiscal Policy Cooperation
in a Currency Union?

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

EK-112xvii

How Important is the Fiscal Policy Cooperation
in a Currency Union? ∗OKANO, Eiji†
This Draft: June, 2010

∗ I would like to thank Jacopo Cimadomo, Masaya Ishikawa, Futoshi Kurokawa, Chikafumi Nakamura, Masao Ogaki, Eiji Ogawa, Rui Ota, Nao Sudo, Takayuki Tsuruga, HughPatrick, Muhammad Rahman, Marcos P. Ribeiro, Etsuro Shioji, Casper George de Vries, Tsutomu Watanabe, David Weinstein, Taiyo Yoshimi and seminar and conference participants at
Columbia University, Federal Reserve Bank of Atlanta, Hitotsubashi University, Maastricht
University, Osaka University, Senshu University, University of Crete and University of Ottawa
for their useful comments. I have benefited from answers to my questions related to this paper
from Andrea Ferrero and Michael Woodford and from valuable comments from Jaromir Nosal.
Two anonymous referees’ critical comments improved this paper. All errors are my own. This
research was assisted by a grant from the Abe Fellowship Program administered by the Social
Science Research Council and the American Council of Learned Societies in cooperation withand with funds provided by the Japan Foundation Center for Global Partnership.
† Correspondence to: Fuculty of Economics, Chiba Keizai University, 59-5, Todorokicho
3-chome, Inage-ku, Chiba-shi, Chiba, 263-0021, JAPAN; Tel: +81-43-253-9738; Fax: +8143-254-6600; E-mail: e.okano@cku.ac.jp

Abstract
By constructing a dynamic stochastic general equilibrium model,
which assumes a currency union consisting of two countries with nontradables, we verify the importance of fiscal policy cooperation. As
shown in previous studies, we find that the role of fiscal policy isimportant in maximizing social welfare. However, we have a contrary result for fiscal policy cooperation. While the previous studies
highlight that fiscal policy cooperation has a nontrivial eﬀect in maximizing social welfare, we show that fiscal policy cooperation has nobenefits regardless of the share of nontradables. Self-oriented fiscal
policy can replicate social welfare under the cooperative setting.
Keywords: currency union, DSGE, policy cooperation, optimal monetary
policy, monetary and fiscal policy mix JEL Classification: E52; E62; F41

1Introduction

Although widely discussed in the literature on money and finance and international macroeconomics, currency unions still raise new issues in these fields. The
establishment of the European Monetary Union (EMU), which is the largest andmost important, provides researchers with important research agendas.1
For member countries of a currency union, who can no longer conduct theirown monetary policy, the role each country’s fiscal policy plays as a stabilization
or social welfare-maximizing tool is an important issue and has been discussed
in detail by many authors. Assuming a lump-sum tax, Benigno[5] analyzes optimal monetary policy in a two-country model with perfect risk sharing. He
implies that a central bank within a currency union can achieve welfare maximization not only union-wide but also in each country without support from afiscal authority. After introducing some frictions, this implication changes. Assuming a currency union consisting of an infinite number of countries, Gali andMonacelli[19] insist on a monetary and fiscal policy mix to maximize social welfare. Under this framework, the central bank can maximize union-wide welfare,
whereas it needs strong support from the fiscal authorities to maximize welfare
in each country. Introducing rule-of-thumb consumers, who are constrained tospending out of their current disposal income, Colciago, Ropel, Muscatelli and
Tirelli[14] find that fiscal policy plays a role not only as a surrogate for theloss of nominal exchange rate flexibility, but also as a stabilization tool for ruleof-thumb consumers’ consumption. Ferrero[16] analyzes optimal monetary andfiscal policy in a two-country currency union with a distorted steady state, andfinds that optimal fiscal policy is essential in a currency union to maximize social welfare. While these papers only assume tradables, Duarte and Wolman[15]introduce nontradables in their currency union model and show that inflation
diﬀerentials can be stabilized by adjusting taxation, although they did not consider optimization problems for policy authorities. To summarize the policy
implications of these previous studies, fiscal policy is important for stabilizing
an economy or enhancing social welfare in a currency union under assumptions
that take into account the real economy.
After accepting the importance of fiscal policy in a currency union, we now
focus on how to conduct fiscal policy. In particular, discussion on fiscal policy
cooperation is not trivial because the EMU consists of many countries. At
present, only Beetsma and Jensen[3] have clear policy implications on this topic;
fiscal policy cooperation is essential and important for enhancing social welfare
via avoiding Nash equilibria brought about by noncooperative fiscal authorities.2
It cannot be said that the importance of fiscal policy cooperation is anestablished policy implication with no opportunity for further discussion because
only Beetsma and Jensen[3] show the importance of fiscal policy cooperation.
However, the Maastricht Treaty and the Stability and Growth Pact provide the1 According to Rose[30], the EMU is the largest and most important currency union.
2 Duarte and Wolman[15], analyzing fiscal policy in a currency union with nontradables,
left an analysis of fiscal policy cooperation as a future research agenda.

1legal foundations for organizing fiscal cooperation in the EMU.3 According to
Beetsma and Jensen[3], there is pressure for more fiscal cooperation in Europe.
In addition, the fiscal crisis in Greece in March 2010 urges the member countriesof the EMU implementing cooperative financial assistance, and provides policy
authorities with motivation to discuss the necessity of fiscal policy cooperation.
There is room for fuller discussion of fiscal policy cooperation in a currency union
because so far only Beetsma and Jensen[3] have analyzed it clearly, and such a
discussion is an urgent task for researchers because of the current situation in
Europe.
How important is fiscal policy cooperation in a currency union? To answer
this question, we construct a dynamic stochastic general equilibrium (DSGE)
model with a currency union consisting of two countries with nontradables.
Introducing nontradables is not trivial because Beetsma and Jensen[3] assume
only tradables. Using this model, we analyze two policy regimes: (i) optimalmonetary policy alone, where the central bank conducts monetary policy toenhance social welfare; and (ii) the optimal monetary and fiscal policy mix,
where not only the central bank but also local governments in two countries
seek to enhance social welfare, under both cooperative fiscal authorities andself-oriented fiscal authorities. In addition, we solve an optimization problem
based on a well-microfounded loss function and conduct welfare analysis, which
Duarte and Wolman[15] who develop a currency union model with nontradables
to analyze fiscal policy, leave for future research.
The answer to our question is that there are no gains from fiscal policy
cooperation. Interestingly, our policy implication is not dependent on the shareof nontradables. Whether there are nontradables or not, self-oriented fiscal
policy can replicate the allocation brought about by a cooperative setting. Thisimplication is contrary to Beetsma and Jensen[3]. In addition, it is contrary to
Liu and Pappa[23], who show the importance of policy cooperation in a twocountry model with nontradables under a flexible exchange rate. Our policy
implication is novel compared with previous studies. In addition, our policy
implication does not necessarily depend on parameter values, which removes
fiscal authorities’ incentive to change the terms of trade (TOT), but may dependon the choice of stabilization tool for fiscal authorities. In a later section in thispaper, we discuss why our policy implications do not depend on parametervalues, and compare the results of our paper with those derived by previous
papers.
Before we analyze fiscal policy cooperation, we show that as the share of
nontradables increases, the role of optimal fiscal policy in enhancing social welfare becomes more important. Nontradables lead to consumption disparity between two countries, which stems from the consumer price index (CPI) disparity
3 Beetsma and Jensen[3] do not mention the plausibility of the Stability and Growth Pactitself. Canzoneri, Cumby and Diba[12] discuss fiscal discipline and exchange rate systems andconclude that to maintain a common currency union, the discipline of a Ricardian regime
is essential. They also point out that the fiscal constraint written into the Stability and
Growth Pact is suﬃcient for a Ricardian regime, although they do not mention fiscal policy
cooperation.

2

between the two countries. That nontradables causes the CPI disparity is consistent with Lipinska[22], who analyzes optimal monetary policy in countries
wanting to join the EMU, and shows that nontradables’ productivity shocks
lead to a stronger real exchange rate depreciation. This is the reason why fiscal
policy is important provided it is used to stabilize each economy by mitigating
the CPI disparity.
The paper is organized as follows. Section 2 constructs the model. Section
3 presents a log-linearized version of the model. Section 4 analyzes a role of
optimal fiscal policy under a cooperative setting. Section 5 considers the selforiented policy settings needed to secure the allocation brought about by acooperative solution. Section 6 concludes the paper. The technical details arederived in appendix.

2

The Model

We construct a closed-system currency union model belonging to the class of
DSGE models with nominal rigidities and imperfect competition, following Obstfeld and Rogoﬀ[26] and Gali and Monacelli[18]. The union-wide economy
consists of two equally sized countries, countries H and F . Country H produces an array of diﬀerentiated goods indexed by the interval h ∈ [0, 1], while
country F produces an array of diﬀerentiated goods indexed by f ∈ [1, 2].

2.1

Households

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

δ t Ut ,

(1)

t=0

1
Nt1+ϕ denotes the period utility in country H; Et dewhere Ut ≡ ln Ct − 1+ϕnotes the expectation, conditional on the information set at period t; δ ∈ (0, 1)
denotes the subjective discount factor; Ct denotes consumption in country H;
Nt ≡ NH,t + NN ,t denotes hours of work in country H; NH,t and NN ,t denotehours of work to produce tradables produced in country H and nontradables
produced in country H, respectively; and ϕ denotes the inverse of the labor supply elasticity. The preferences of the representative household in country F isdefined analogously. Quantities and prices particular to country F are denotedby asterisks while quantities and prices without asterisks are those in country
H.
More precisely, private consumption is a composite index defined by:
η
¸ η−1
∙
η−1
η−1
1
1
,
(2)
Ct ≡ γ η CT η,t + (1 − γ) η CNη,t
1

1

2
2
where CT ,t ≡ 2CH,tCF,tdenotes the consumption index for tradables; CH,t , CF,tand CN ,t denote Dixit—Stiglitz-type indices of consumption across the tradables3

produced in country H and produced in country F , and nontradables producedin country H, respectively; γ ∈ [0, 1] denotes the share of tradables in the
CPI; and η > 0 denotes the elasticity of substitution between tradables and∗nontradables. Note that Ct∗ is defined analogously to Eq.(2) whereas CN,t ,denoting the nontradables produced in country F replaces CN ,t .4
Total consumption expenditures by households in country H are given by
PH,t CH,t +PF,t CF,t +PN ,t CN ,t = Pt Ct , with PH,t and PF,t being Dixit—Stiglitztype indices of the price of tradables produced in countries H and F , respectively, and PN ,t being Dixit—Stiglitz-type indices of the price of nontradables
produced in country H. A sequence of budget constraints in country H is givenby:n
Dtn + Wt Nt + St ≥ Pt Ct + Et Qt,t+1 Dt+1,

(3)

where Qt,t+1 denotes the stochastic discount factor, Dtn denotes the nominal
payoﬀ of the bond portfolio purchased by households, Wt denotes the nominal
wage, and St denotes profits (net taxation) from ownership of the firms. The
budget constraint in country F is defined analogously. Furthermore:i 1h1−η 1−η,Pt ≡ γPT1−η
,t + (1 − γ) PN ,t
1

(4)

1

2
2
denotes the CPI, PT ,t ≡ PH,tPF,tdenotes the price index of tradables. Pt∗ is∗defined analogously to Eq.(4), whereas PN,t , denoting the price of nontradables
in country F , replaces PN ,t . We assume that the law of one price always holds,∗∗
and PF,t = PF,t
, implying that the prices of tradables arethus PH,t = PH,t∗equal in both countries. However, PN ,t and PN,t are not necessarily equal inboth countries because these represent the prices of diﬀerent goods. These facts
imply that purchasing power parity (PPP) does not necessarily hold. When all
goods are tradable, Eq.(4) implies Pt = Pt∗ ; namely, PPP always holds.
The optimal allocation of any given expenditure within each category of
goods implies the demand functions, as follows:
µ
µ
¶−1
¶−1
1 PH,t1 PF,t
CH,t =
CT ,t ; CF,t =CT ,t ,
2 PT ,t2 PT ,t
¶−η
¶−η
µ
µ
PT ,t
PN ,t
CT ,t = γ
Ct ; CN ,t = (1 − γ)
Ct .
(5)
Pt
Pt

The representative household maximizes Eq.(1) subject to Eq.(3). The optimality conditions are given by:
Ã
!
−1
Pt
Ct+1
1
Wt
=
δEt
; Ct Ntϕ =
,
(6)
−1
R
Pt
Ct Pt+1t
4 Following Stockman and Tesar[32], we assume that η is not necessarily unity, whereas
Obstfeld and Rogoﬀ[26] implicitly assume that η is unity. Obstfeld and Rogoﬀ[26] assume
C

γ

C

1−γ

,t N ,tCt ≡ γ γ T(1−γ)
1−γ . This implies η = 1 in our paper.

4

where Rt ≡ 1 + rt satisfying Rt−1 = Et Qt,t+1 denotes the gross nominal return on a riskless one-period discount bond paying oﬀ one unit of the common
currency (in short, the gross nominal interest rate), and rt denotes the net
nominal interest rate. The first equality in Eq.(6) is an intertemporal optimality condition, namely the Euler equation, and the second equality in Eq.(6) isan intratemporal optimality condition. Optimality conditions in country F aregiven analogously.
Combining and iterating the first equality in Eq.(6) with an initial condition,
we have the following optimal risk-sharing condition:
Ct = ϑCt∗ Qt ,

(7)

P∗

with Qt ≡ Ptt denoting the CPI diﬀerential between the two countries and ϑ∗= P−1 =
denoting a constant depending on the initial value. When C−1 = C−1∗
5∗P−1 = 1, we have ϑ = 1. In addition, we have Ct = Ct which implies that themarginal utility of consumption is equal between the two countries; namely, consumption is equal between the two countries because of the logarithmic utility
function when PPP is applied. Our paper allows the existence of nontradables
and if there are nontradables, Ct = Ct∗ no longer applies because Qt 6= 1. Inthat case, there is consumption disparity between the two countries and the single central bank can no longer stabilize both countries simultaneously. Hencethe necessity of using fiscal policy to stabilize the economy of each country if
there are nontradables. This is an intuitive explanation as to why fiscal policy
is important in a currency union with nontradables.
Eq.(7) is useful for understanding intuitively why Gali and Monacelli[19] insist on the importance of fiscal policy although they only assume tradables in acurrency union that consists of an infinite number of infinitesimally small countries. Because the home country is infinitesimally small, the share of the homecountry price index in the union-wide price index is negligible. The marginalutility of consumption between the home country and the union as a whole is
not equal and there is opportunity to conduct fiscal policy independently ineach country.6

5 See Chari, Kehoe and McGrattan[13] for details.
6 The reason why fiscal policy is important in a currency union is consistent with thispaper and Gali and Monacelli[19]; however, the stabilization tools and settings in these two
papers are clearly diﬀerent. We assume that the fiscal authorities adjust the quantity of
government bonds, while Gali and Monacelli[19] assume that the fiscal authorities adjust
government expenditure. In their setting, fiscal authorities’ funds are lump-sum taxes, whilein our setting, fiscal authorities’ funds are obtained by issuance of government bonds. Such
new bond issues change the distribution of private savings and that may change output andinflation. In Gali and Monacelli[19]’s lump-sum taxes setting, how the government secures
funds for spending is not clear because such a setting does not show an explicit government
budget constraint. In contrast, we show clearly how to manage government debt to maximizesocial welfare by introducing explicitly a government budget constraint.

5

2.2

Firms

Each producer uses a linear technology to produce a diﬀerentiated good as
follows:
YH,t (h) = AH,t NH,t (h) ,;YN ,t (h) = AN ,t NN ,t (h) ,(8)where YH,t (h) and YN ,t (h) denote the output of tradables h produced in country
H, and output of nontradables h produced in country H, respectively, andAH,t and AN ,t denote stochastic productivity shifters associated with tradables
produced in country H and nontradables produced in country H, respectively.
Each producer in country F uses a technology similar to that in country H.
Each firm produces a single diﬀerentiated good and prices its good to reflect the elasticity of substitution across goods produced given the CPI. Thisis because each firm plays an active part in the monopolistically competitivemarket. We assume that Calvo—Yun-style price-setting behavior applies, and,
therefore, that each firm resets its price with a probability of 1 − α in eachperiod, independently of the time elapsed since the last adjustment.
When setting a new price in period t, firms seek to maximize the expecteddiscounted value of profits. The first-order necessary conditions (FONCs) are
as follows:
#
"∞
³
´
X
k
−1
n
= 0,
Et
(αδ) (Pt+k Ct+k ) ỸH,t+k P̃H,t − ζM CH,t+k
k=0

Et

"∞
X

k

−1

(αδ) (Pt+k Ct+k )

k=0

³
´
n
ỸN,t+k P̃N ,t − ζM CN,t+k

#

=

0,

(9)

Wt
n
n
t
where M CH,t
≡ (1−τW)Aand M CN,t ≡ (1−τ )AN ,t denote the nominal marginalH,tcosts associated with tradables produced in country H and nontradables produced in country H, respectively; ỸH,t and ỸN ,t denote the total demands following changes in the prices of tradables produced in country H and nontradables
produced in country H, respectively; P̃H,t and P̃N ,t denote the adjusted pricesof tradables produced in country H and nontradables produced in country H,θrespectively; and ζ ≡ θ−1
is a constant markup, θ > 1 denotes the elasticity
of substitution across goods produced within a country and τ denotes the tax−1rate.7 Note that (Pt+k Ct+k ) is the marginal utility of nominal income.

2.3

Local Government

Whereas monetary frictions are omitted and the limitations of a “cashless economy” are considered following Woodford[35] throughout this paper, monetary
policy has important implications for fiscal decisions, as the level of the interest
rate determines the debt burden and the inflation rate aﬀects the real valueof debt. Fiscal policy consists of choosing one-period nominal risk-free debt to
7 Ferrero[16] regards it as a value-added tax rate.

6

finance an exogenous process of public spending.8 The flow government budget
constraint in country H is given by:n
Btn = Rt−1 Bt−1
− [PP,t τ (YH,t + YN ,t ) − PG,t (GH,t + GN ,t )] ,(10)

where Btn ≡ Pt Bt denotes the nominal risk-free rate on bonds issued by the
local government in country H; Bt denotes the real risk-free rate on bonds isPY+PN ,t YN ,tdenotessued by the local government in country H; PP,t ≡ H,t YH,tH,t +YN ,t
the weighted average price of goods produced in country H, namely, the proPG
+PN ,t GN ,tducer price index (PPI) in country H; and PG,t ≡ H,t GH,tdenotes
H,t +GN ,t
the average price of goods purchased by the government in country H. The
local government in country F has a budget constraint similar to that shownin Eq.(10). For simplicity, we assume that government purchases are fully allocated to a domestically produced good and that the total amount of these isexogenous. For any given level of public consumption, the government allocates
expenditures across goods in order to minimize total cost. This equation becomes the New Keynesian IS (NKIS) curve after being combined with Eq.(6)and log-linearized with the appropriate transversality condition.9

2.4

Market Clearing

The market for tradables and for nontradables in country H clears when domestic demand equals domestic supply, as follows:∗(h) + GH,t (h) ; YN ,t (h) = CN ,t (h) + GN ,t (h) , (11)YH,t (h) = CH,t (h) + CH,t∗(h) denotes country F ’s demand for generic tradables produced inwhere CH,tcountry H.
Finally, we define countrywide output and government expenditure as follows:

Yt ≡

PH,t
PN ,t
YH,t +
YN ,t
PP,t
PP,t

;

Gt ≡

PH,t
PN ,t
GH,t +
GN ,t ,
PG,t
PG,t

(12)

where Yt and Gt denote output and government expenditure in country H,
respectively.

8 We assume the tax rate is common in each country and constant over time for simplicity,whereas Ferrero[16], Schmitt-Grohe and Uribe[31] and Canzoneri, Cumby and Diba[12] assume
it can vary in each country and over time. In contrast, Gali and Monacelli[19] assume aconstant (negative) tax rate similar to our paper. A constant tax rate over time reflects
actual regimes, not only in the Euro area, but also in other countries, because adjustments ofthe tax rate are infrequent.
£¡¢¤
9 The transversality condition is given bylim Et Qt,k 12 Bkn + Bk∗n
= 0 and£lim Et δ k−t UC (C) RB

k→∞

¤

k→∞

= 0 with B = B ∗ > 0, which are consistent with Ferrero[16]where variables without a time subscript denote steady-state values.

7

3

Log-linearized Model

This section describes the stochastic equilibrium that arises from perturbations
around the deterministic equilibrium. Lowercase letters denote percentage deviations of steady-state values for the respective uppercase letters when there ist, where Vt denotes an arbitrary variable
no note to the contrary (i.e., vt ≡ dV
V
and V denotes the steady-state value of Vt ). Lowercase letters accompanied by
R as a superscript indicate the logarithmic diﬀerence between the two countries
for the respective uppercase letters (i.e., vtR ≡ vt − vt∗ ), while lowercase lettersaccompanied by W as a superscript indicate the logarithmic weighted sum of
the two countries for the respective uppercase letters (i.e., vtW ≡ 12 (vt + vt∗ )).
Combining Eqs.(4), (5) and (10)—(12) and the log-linear approximation, we
have NKISs as follows:
ỹtW

=

ỹtR

=

βW
βW
βW W
W
W
Et ỹt+1− βW r̂t + βW Et πt+1+
r̂t−1 − βW bWb ,t +1 − σGδδ t−1
γ β̄βT(1 − γ) β̄βN
γ β̄βTβW W−
π −aH,t −
aN ,t −
aF,t ,δ t22
2(1 − γ) β̄βN ∗(13)−
aN ,t + σG ςW gtW ,2R−βR δbRt + βR (1 − γ) υnt − βR (1 − γ) nt−1 + βR bt−1 − β̄γaH,t ,+ β̄γaF,t − β̄ (1 − γ) aN ,t + β̄ (1 − γ) a∗N ,t + ςR σG gtR ,(14)where ỹt ≡ yt − ȳt denotes the logarithmic output gap measured from its naturallevel in country H and ȳt denotes the logarithmic natural output level in countryt
H, which becomes ỹt = 0 under the long run equilibrium; r̂t ≡ dR
R denotes thedeviation of the nominal interest rate from its steady-state value; πt denotes
the CPI inflation rate in country H; nt denotes the logarithmic nontradables
P ∗ ,tprice disparity between countries H and F (NPD) with Nt ≡ PN
, βW ≡
N ,t(1−σG )σB
σB (1−σG )σB +(1−σG )τ , βR ≡ (1−σG )δτ −(1−δ)σB , υ ≡ 1−(1 − δ) $, $ ≡ 1+(η − 1) γ, νW ≡[σB (1−ρG )+1−σG ]
βW ρT
βW ρN
G )δ−(1−δ)σB ], νR ≡ [(1−σσB +(1−σG )τ
(1−σG )δτ −(1−δ)σB , βT ≡ 1 − 1−σG , βN ≡ 1 − 1−σG ,βW ρGG− λ1 and ςR ≡ νR − λ1 , σB ≡ Bβ̄ ≡ (1−σGλ)(1+ϕ) , ςW ≡ νW + (1−σY and σG ≡ YG )λbeing the steady-state ratio of government bonds to output and the steadystate ratio of government expenditure to output, respectively; and ρG < 1,
ρT < 1 and ρN < 1 being the coeﬃcient associated with exogenous processes
on government expenditure, on the productivity shifter of tradables and on the
productivity of nontradables, respectively.10 When all goods are tradables, γ =RR
1 is applied and Eq.(14) equals: ỹtR = −βR δbRt +βR bt−1 −γaH,t +γaF,t +ςR σG gt ,in which not only the productivity of nontradables aN ,t and a∗N ,t , but also
the NPD nt disappears. The appearance of the NPD in the equality implies
that nontradables magnifies the output gap disparity between the two countries
although risk sharing is perfect internationally. This is the reason why a higher
10 We assume that the government expenditure and productivity shifters follow AR(1) processes (see Subsection 4.2.)

8

share of nontradables increases the importance of the role of fiscal policy inmaximizing social welfare. In addition, we can understand why Ferrero[16]highlights the importance of optimal fiscal policy in a currency union, although
he does not assume nontradables. When the steady-state value of the quantityof government bonds and expenditure are zero, σB = σG = 0, which implies an
eﬃcient steady state and this is assumed by Benigno[5]. In this case, Eq.(14)becomes: ỹtR = −γaH,t + γaF,t , which implies that only productivity aﬀects
the output gap disparity.11 Following Ferrero[16], we assume a distorted steadystate where σB = σG = 0 is not applied. In this case, relative government
expenditure aﬀects the output gap disparity and the role of fiscal policy is larger
than the case of σB = σG = 0, although there are no nontradables.
Combining Eqs.(5)—(9), (11) and (12) and taking a log-linear approximation,
the NKPCs in terms of the output gap are given by:
πP,t = δEt πP,t+1 +

κλ
ψκ
ỹt −
nt ,
1 − σG
2(15),λ≡where πP,t denotes the PPI inflation rate in country H with κ ≡ (1−α)(1−αδ)α1+(1 − σG ) ϕ and ψ ≡ (1 − γ) γ (η − 1) along with its counterpart in country F .
κλỹt , which corresponds
When γ = 1, Eq.(15) is written as πP,t = δEt πP,t+1 + 1−σGwith that derived by Gali and Monacelli[18], who insist that inflation—output
tradeoﬀs can be dissolved simultaneously in a small open economy under strong
parameter restrictions by inflation targeting. Indeed, when inflation targeting,∗such as πP,t = πP,t
= 0 for all t, is introduced in our currency union withspecial restrictions, i.e., γ = 1 and σB = σG = 0, these equalities imply thatỹt = ỹt∗ = 0 for all t and that the output gap is eliminated.
Combining Eqs.(5)—(9), (11) and (12) and taking a log-linear approximation,
we have the New Keynesian real exchange rate determiner (NKRD), which is
our version of the real exchange rate determiner, as follows:
¡
¢
¡
¢
R
R
R
πN
,t = δEt πN,t+1 + κϕỹt + κnt − κϕγ 1 − β̄ aH,t + κϕγ 1 − β̄ aF,t ,
£
¡
¢¤
£
¡
¢¤
− κ 1 + ϕ (1 − γ) 1 − β̄ aN ,t + κ 1 + ϕ (1 − γ) 1 − β̄ a∗N ,t ,
κσG ³
ϕ´ R
g ,
−
1−
(16)
1 − σG
λ t
Rwhere πN,t ≡ − (nt − nt−1 ) denotes relative nontradables inflation. Eq.(16)equals qt = 0 which implies that the CPI disparity disappears between thetwo countries when the currency union has no nontradables; i.e., as γ = 1.12
In that case, the problem with the CPI disparity is resolved, because eachcountry has the same CPI. This implies that PPP holds. Furthermore, because
R
log-linearized Eq.(7) is given by qt = cRt , ct = 0, which means there is no
consumption disparity between both countries when qt = 0.
Benigno[5] and Gali and Monacelli[19] assume that all goods are tradable andthat the law of one price holds. However, PPP does not necessarily hold in Gali
11 In addition, y R = 0, which implies that there is no output disparity in that case.
t
12 We obtain this result by substituting log-linearized Eq.(4) into Eq.(16).

9and Monacelli[19]. They assume a currency union that consists of an infinite
number of countries, whereas Benigno[5] assumes a currency union consisting of
two countries. The settings in Gali and Monacelli[18] make a distinction in the
CPI between that of an infinitesimally small country’s economy and that of theunion-wide economy because the CPI in the former country does not aﬀect theunion-wide CPI. This stems from the small open economy assumption. Thus,
qt = 0 is not applied in Gali and Monacelli[19], although all goods are tradable.
In a later section, we suggest that optimal fiscal policy is very important forstabilizing both inflation and the output gap simultaneously because of theinfluence of the nontradables. The assumption of a small open economy does
not permit us to apply qt = 0. Thus, the policy implications of Gali andMonacelli[19] and this paper are very similar, whereas the policy implicationsof Gali and Monacelli[19] and Benigno[5] are contrary.
Eq.(16) depicts the real exchange rate anomaly, which is reported by Benigno and Thoenissen[8] and Canzoneri, Cumby and Diba[11], who find thatthe actual direction of changes in the real exchange rate cannot be explained by
the Balassa—Samuelson theorem. Eq.(16) well reflects their findings in a wellfounded micro setting. However, it cannot be easily understood because Eq.(16)has correctly assumed nominal rigidities. To understand the nature of Eq.(16),
we inspect Eq.(16) without nominal rigidities. Under such a condition, Eq.(16)can be rewritten as:
© ¡
¢
£
¡
¢¤
¡
¢
qt = (1 − γ) ϕγ 1 − β̄ aH,t + 1 + ϕ (1 − γ) 1 − β̄ aN ,t − ϕγ 1 − β̄ aF,t
¾
£
¡
¢¤ ∗
ϕσG R
− 1 + ϕ (1 − γ) 1 − β̄ aN ,t −
g
,
(17)
1+ϕ t
because α = 0 and ỹt = ỹt∗ = 0 hold. Eq.(17) implies that an increase in the
productivity of tradables produced in country H causes an increase (depreciation) in the real exchange rate. An increase in the productivity of tradables
produced in country H causes a decrease in the CPI in coutry H via a decreasein the real marginal cost in the tradables and the nontradables sectors. Finally,
the real exchange rate increases (depreciates).
In addition, our NKRD is consistent with Altissimo, Benigno and Palenzuela[1]’s equality, which determines the real exchange rate and corresponds to our
NKRD. Their equality shows that the productivity shifter of tradables does not
aﬀect the real exchange rate and only the productivity shifter of nontradables
aﬀects the real exchange rate, under their special setting where the relative risk
aversion, the elasticity of substitution between tradables and nontradables, andthe elasticity of substitution between tradables produced in countries H and
F are all unity. Their special setting is consistent with our model, because
we assume Eq.(1) and an Armington form of the definition of the consumption
index for tradables and the elasticity of substitution between tradables andnontradables does not appear in Eq.(16). Altissimo, Benigno and Palenzuela [1]assume zero steady-state government expenditure, which corresponds to G = 0.

10

Substituting G = 0 into Eq.(17) yields:"
qt = (1 − γ)aN ,t − a∗N ,t −

ϕ
1+ϕ

µ

dGt
Y

¶R #

,

(18)

which is consistent with the equilibrium condition on the CPI diﬀerential under
the special case corresponding to our setting derived by Altissimo, Benigno andPalenzuela [1], who said that the Balassa—Samuelson eﬀect disappears in thatspecial setting because the productivity shifter of tradables disappears as inEq.(18).13

4

Optimal Cooperative Solution

In this section, we analyze the macroeconomic implications of an alternative
policy regime for the Euro area: an optimal monetary policy alone regime andan optimal monetary and fiscal policy regime under a cooperative setting. Furthermore, we assume that each policy authority is responsible for minimizing
social losses. Under an optimal monetary policy alone regime, the central bankis the only policy authority, whereas the central bank and local governmentsin the two countries are both authorities under an optimal monetary and fiscal
policy regime. Policy authorities seek to minimize the social loss function subject to our structural model.14 Hereafter, let us assume η = 1, implying thatthe elasticity of substitution between tradables and nontradables is unity, whichis assumed implicitly by Obstfeld and Rogoﬀ[26], for simplicity.
The period loss function is derived by the welfare criterion. The welfare
criterion is derived by subtracting a second-order approximated FONC for firms
Eq.(9) MW from a second-order approximated utility function Eq.(1), which is
given by:
W W − ΦMW

=

½

h
i
1
2
ω̃1 ỹt2 + (ỹt∗ ) − ω̃2 (ỹt at + ỹt∗ a∗t )(1 − σG ) 4
t=0¾¡ ∗ ¢2 i Φ (1 − κλ) W
1 h 2
,
+
+ πP,tỹt−ω̃3 (ỹt gt + ỹt∗ gt∗ ) + ω̃4 πP,t4
1 − σG³´+ t.i.p. + o kξk3 ,

E0

∞
X

δt

13 Altissimo, Benigno and Palenzuela [1] show that an increase in the productivity of nontradables in the home (foreign) country increases (decreases) the real exchange rate, while anincrease in government expenditure in the home (foreign) country decreases (increases) thereal exchange rate when relative risk aversion, the elasticity of substitution between tradables
and nontradables, and the elasticity of substitution between tradables produced in countries
H and F are all unity. Because we consider a currency union, the logarithmic real exchange
rate corresponds to the CPI disparity qt . Thus, Eq.(18) is consistent with the equation derived
by Altissimo, Benigno and Palenzuela [1].
14 Our structural model consists of Eqs.(13)—(16) and a counterpart of Eq.(15) in countryF.

11with W W ≡ E0

∞
P

t=0

¡
¢
δ t UtW − U W denoteing the sum of the discounted value

denotof the deviation of utility from its steady-state value, Φ ≡ 1 − 1−τζ
ing the steady-state wedge between the marginal rate of substitution between
consumption and leisure and the marginalof labor, t.i.p. denoteing´³ product
3
denoting the terms that are
the terms of the independent policy, o kξkhigher than third order and ω̃1 , ω̃2 , ω̃3 and ω̃4 being complicated coeﬃcientsthat consists of structural parameters in our model. Plugging the fact thatP∞WW
E0 t=0 δ t Φ(1−κλ)= Γ0 into this equality, we have:
1−σG ỹt + ΦM
WW

=

∙
¸
∞
1 X t Λy 2 Λy ∗ 2 Λπ 2
Λπ ¡ ∗ ¢2+ Γ0 + t.i.p.,
− E0
δ
ŷt +
(ŷt ) +πP,t +
πP,t
2 t=0
2
2
2
2
³
´
3
+ o kξk ,which is the welfare criterion without linear terms where; ŷt ≡ yt − yte denotes
the welfare-relevant output gap; yte ≡ Ω1 γaH,t + Ω1 (1 − γ) aN ,t + Ω2 gt denotes
Φ W
πP,0 denotes
the logarithmic eﬃcient level of output in country H; and Γ0 ≡ κλχ+κΦςa transitory component which is predetermined;, with Λy ≡ (1−σ )2 , Λπ ≡G22(1+Φ)θ
ΦΘ2(1−σG )κ + 2 , χ ≡ (1 + ϕ) (1 + Φ) (1 − σG ), ς ≡ (1 + ϕ) (1 − σG ) − σG andΘ ≡ α (4θ − 1) − 3; and Ω1 and Ω2 are complicated coeﬃcients that consist of
structural parameters in our model.
Because the transitory component is predetermined over the set of admissible
policies, the higher values of this welfare criterion correspond to lower values ofthe loss function, which is given by:
L W ≡ E0∞X

δ t LW
t ,

(19)

t=0

with:
∙

¸
Λy 2 Λy ∗ 2 Λπ 2
Λπ ¡ ∗ ¢2
πP,t
,
ŷ +
(ŷt ) +
π +
2 2 t
2
2 P,t
2

1
LW
t ≡

(20)

15
where LWt denotes the union-wide period loss function.
We now consider the diﬀerence in welfare loss between our paper and theDSGE literature for an open economy. Our period welfare loss Eq.(20) has two
features that do not appear in other period welfare functions. First, Eq.(20)
implies that policy authorities should not minimize the output gap ỹt , but should
minimize the welfare-relevant output gap ŷt because of the distorted steady state
15 Sutherland[33] and Benigno and Woodford[9] derive the second-order approximated utility
function without the presence of a linear term under the distorted steady state. Thus, wefollow Benigno and Woodford[10] to derive Eq.(20) because of the distorted steady state in
our model. See Appendix D for details on the derivation. Note that Woodford[35] discusses
how the presence of linear terms generally leads to an incorrect evaluation of welfare. A simple
example of this result is proposed by Kim and Kim[21].

12and the nonzero steady state value of government expenditure. When the steadystate is eﬃcient and government expenditure is zero in the steady state, namely,
both Φ = 0 and σG = 0 are applied, Eq.(20) can be rewritten as follows:
LWt =(1 + ϕ) 2 (1 + ϕ) ∗ 2
θ 2
θ ¡ ∗ ¢2,ỹt +(ỹt ) +πP,t +π444κ
4κ P,twhich implies that the policy authorities have to minimize the output gap. This
expression is similar to Gali and Monacelli[18] and Gali[17]. Note that Ω1 = 1,
Ω2 = 0 and yte = ȳt = γaH,t + (1 − γ) aN ,t when Φ = 0 and σG = 0.
Second, by rewriting Eq.(20), we can see that nontradables aﬀect the form of
the period loss. Using log-linearized Eqs.(5) and (11), Eq.(20) can be rewritten
as follows:
LW
t

=

2
¡ ¢2 Λy [γ (1 − σG )]2
Λy [(1 − γ) $ (1 − σG )](tt − t̄t )2 +(nt − n̄t )2
Λy ŷtW +442Λy 2γ (1 − σG ) (1 − γ) $
Λπ 2
Λπ ¡ ∗ ¢2
πP,t ,+
tt n t +
π +
4
2 P,t
2
P

F,t
, t̄t and n̄t denote thewhere tt denotes the logarithmic TOT with Tt ≡ PH,tlogarithmic target level of the TOT and the NPD, respectively, with t̄t = n̄t ≡Ω1 (1−γ)Ω1 (1−γ) ∗Ω1Ω1Ω2 −σG R
1−σG aH,t + γ(1−σG ) aN ,t − 1−σG aF,t − γ(1−σG ) aN ,t + γ(1−σG ) gt implying thatboth the target level of the logarithmic TOT and the NPD are equal. As shownin this equality, not only minimization of the distance of the TOT from its
target level, but also minimization of the distance of the NPD from its target
level are imposed policy objectives. Furthermore, a cross-term of the TOT and
the NPD appears in this equality. These facts imply that stabilizing the NPD
is essential for minimizing social loss in a currency union with nontradables.
However, when all goods are tradable, namely γ = 1, this equality boils downto:

¡ W ¢2 Λy [γ (1 − σG )]2
Λπ 2
Λπ ¡ ∗ ¢2
2
LW
=
Λ
+
(tt − t̄t ) +πP,t +
πP,t ,
y ŷt
t
4
2
2

which is a familiar expression in the open economy version DSGE literature,such as Benigno and Benigno[7], assuming all goods are tradable except for thewelfare-relevant output gap replacing the output gap. Because our model allows
for nontradables, our welfare loss does not necessarily correspond to the welfare
loss in other DSGE studies for an open economy.
Note that the linear constraints implied by our model’s structural equationsin terms of the welfare-relevant output gap diﬀer from the constraints impliedby the equations in terms of the output gap. By using the definition of thewelfare-relevant output gap, the NKPC in country H Eq.(15) can be written as
follows:
πP,t = δEt πP,t+1 +κλŷt + εt ,1 − σG

13(21)

where εt ≡ κ (1 + ϕ) Ω3 γaH,t + κ (1 + ϕ) Ω3 (1 − γ) aN ,t − κσG Ω4 gt is a composite cost-push term with Ω3 and Ω4 denoting complicated coeﬃcients. ThisNKPC corresponds to the one derived by Benigno and Woodford[10]. A composite cost-push term indicates the degree to which the exogenous disturbances preclude simultaneous dissolution of the tradeoﬀ between inflation and the welfarerelevant output gap. Thus, the inflation—output gap tradeoﬀs can no longer bedissolved completely, even if all goods are tradable.

4.1

Role of Optimal Fiscal Policy

We next investigate the role of optimal fiscal policy by comparing FONCs,
which clarify the relationship between PPI inflation and the output gap under
the optimal monetary policy alone, and under the optimal monetary and fiscal
policy. Under both regimes, policy authorities minimize the sum of the discounted value of social losses in Eq.(19), subject to the structural model withcommitment.16 Under the optimal monetary policy
alone, only the centralª∞ bank©
∗, ỹt , ỹt∗ , nt , r̂t t=0 , while
minimizes Eq.(19) by choosing the sequence πP,t , πP,tboth the central
local governments
cooperatively minimize Eq.(19)ª∞© bank∗ and two, ỹt , ỹt∗ , nt , r̂t , bt , b∗t t=0 under the optimal monetary andby choosing πP,t , πP,tfiscal policy.
4.1.1

Optimal Monetary Policy Alone

Under the optimal monetary policy alone, the FONC for union-wide inflationand the output gap is given by:
πtW = −¢
Λy (1 − σG ) ¡ W
Wŷt − ŷt−1
,
Λπ κλ

(22)

which is a familiar expression in papers on optimal monetary policy in an open
economy. This implies that local government does not need to dissolve unionwide inflation—output tradeoﬀs. A solitary central bank can stabilize both inflation and the welfare-relevant output gap simultaneously, even though nontradables exist.17
Next, we investigate the relative block of the FONC. We are interested in
the eﬀects of nontradables. Thus, we analyze the relative block of the FONC in
16 Because of commitment, lagged Lagrange multipliers appear in the FONCs for policy
authorities. This means that policy authorities are able to aﬀect the private sector’s inflation
expectations. See Appendix E.
17 Because of the distorted steady state, the NKPC in terms of the welfare-relevant output
gap includes a cost-push term as shown in Eq.(21). This term indicates the degree to which
the exogenous disturbances preclude the simultaneous dissolution of the tradeoﬀ between
inflation and the welfare-relevant output gap. While we have the FONC Eq.(22), the inflation—
output gap tradeoﬀs cannot be completely dissolved but they are close to being dissolved.
Under our parameterization, which is shown in Section 4.2.1, the standard deviations of theunion-wide inflation rate and the welfare-relevant output gap are 6.3215e-005 and 0.0014,
respectively. These standard deviations are equivalent under both optimal monetary policy
alone and optimal monetary and fiscal policy.

14both cases: namely, the case where all goods are tradable and the case where
there are nontradables. When all goods are tradable, namely, γ = 1, the relative
block of the FONC is given by:
R
=−
πP,t

¢ 2 (1 − σG )Λy (1 − σG ) ¡ R
Rŷt − ŷt−1−(μ2,t − μ2,t−1 ) ,
Λπ κλ
Λπ κλ

(23)

where μ2,t denotes the Lagrange multiplier associated with Eq.(14); that is, therelative block of the NKIS. This case corresponds to Ferrero’s (2009) setting.
Eq.(23) implies that the inflation—output tradeoﬀs no longer disappear simultaneously. Because of this, Ferrero[16] insists that fiscal policy is needed toenhance social welfare. Next, we abandon the assumption that all goods are
tradable. In this case, the relative block of the FONC is given by:
R
πP,t

=

μ5,t

=

¢ 2 (1 − σG )Λy (1 − σG ) ¡ R
Rŷt − ŷt−1−(μ2,t − μ2,t−1 ) ,
Λπ κλ
Λπ κλ
4 (1 − σG ) κϕ
(μ5,t − μ5,t−1 ) ,−Λπ κλ (1 + δ + κ)
1
(1 − γ) βR υμ2,t +
μ5,t−1 ,
1+δ+κ

−

(24)

where μ5,t denotes the Lagrange multiplier associated with Eq.(16), the NKRD.
The two equalities in Eq.(24) imply not only that the inflation—output tradeoﬀs
do not disappear simultaneously, but also that the relationship between inflationand the output gap is weakened.
4.1.2

Optimal Monetary Policy and Fiscal Policy

Under the optimal monetary and fiscal policy, the FONC for union-wide inflationand the output gap is given by Eq.(22). Thus, the union-wide inflation and thewelfare -relevant output gap are stabilized by optimal monetary policy and fiscal
policy although nontradables exist in a currency union.
The FONC for relative block inflation and the welfare-relevant output gap
is given by:
R
=−
πP,t

¢
Λy (1 − σG ) ¡ R
Rŷt − ŷt−1
.
Λπ κλ

(25)

This equality also implies that relative inflation and the welfare-relevant output
gap are stabilized by optimal monetary policy and fiscal policy. Both Eqs.(22)
and (25) imply that inflation—output tradeoﬀs disappear, not only at the unionwide level, but also in each country under the optimal monetary and fiscal policy
regime. Note that we have Eq.(25), even though nontradables exist.
Why do we have Eq.(25) rather than Eq.(24)? Under the optimal monetaryand fiscal policy, we have μ2,t = 0 as the optimality condition along with optimality conditions derived under the optimal monetary policy alone, because wenot only have the nominal interest rate, but also government bonds in the policy

15

function. We obtain Eq.(25) by substituting μ2,t = 0 along with the initial condition μ5,−1 = 0 into Eq.(24). Because of this, we have μ5,t = 0, although thisequality is not obtained directly by implementation of fiscal policy. This fact
implies that optimal fiscal policy removes the eﬀects of the CPI disparity, which
introduces the consumption disparity between both countries by removing the
disparity in the demand block.

4.2

Sensitivity Analysis

4.2.1

Parameterization

We run a series of dynamic simulations and adopt the following benchmark
parameterization. We set the values of the inverse of the labor supply elasticity
ϕ, the elasticity of substitution across goods θ, the elasticity of substitution
between tradables and nontradables η, the subjective discount factor δ, thesteady-state share of government bonds to output σB , the steady-state share ofgovernment expenditure to output σG and the tax rate τ equal to 3, 11, 0.75,
0.5, 1, 0.99, 2.4, 0.276 and 0.3, respectively, which is consistent with quarterly
time periods in the model.18 Except for α, γ, ϕ and η, these parameterizations
are used in Ferrero[16].19 According to our calculation, nontradables accountfor 50.3% of all goods in the Euro area; thus, we set the share of nontradables in
the CPI as γ = 0.5.20 Following Obstfeld and Rogoﬀ[26], we set η = 1. We also
assume that the government expenditure and productivity shifters are describedaccording to the following AR(1) processes:
aH,t = ρT aH,t−1 + ξH,t ; aF,t = ρT aF,t−1 + ξF,t , ; aN ,t = ρN aN ,t−1 + ξN ,t
∗
W
W
W
R
R
R
a∗N ,t = ρN a∗N ,t−1 + ξN,t ; gt = ρG gt−1 + ξG,t ; gt = ρG gt−1 + ξG,t ,∗W
R
where ξH,t , ξF,t , ξN ,t , ξN,t , ξG,t and ξG,t denote the i.i.d. shocks. We set ρT , ρNand ρG equal to 0.705, 0.784 and 0.8, following Batini, Harrison and Millard[2]and Ribeiro[28].21
18 σ

B = 2.4 implies that the steady-state debt—output annual ratio is 0.6.
19 Many DSGE studies use the parameter values in Rotemberg and Woodford[29]. However,to compare our results with those derived by Ferrero[16] and to analyze the Euro area, wemainly use his parameter values for the Euro area. More precisely, the parameter values ofθ, δ, σB , σG are set as in Ferrero[16]. Ferrero[16], however, sets a diﬀerent degree of pricerigidity in countries H and F . We set α equal to 0.75, which is assumed by Beetsma andJensen[3]. Because ϕ does not appear in Ferrero[16], we set it equal to 3, which is adopted by
Gali and Monacelli[18].
20 Following the definition that regards goods produced in the manufacturing industry, agriculture, forestry, fishery and mining as tradables and regards goods produced in other industries as nontradables, as used by Canzoneri, Cumby and Diba[11], nontradables, in termsof both current and purchaser’s prices, accounted for 50.3% of the sum of nontradables andtradables in major Euro area countries such as Belgium, Germany, France, Greece, Italy, theNetherlands, Portugal and Spain in 1999.
21 There are few papers that estimate AR(1) parameters associated with the productivity ofthe tradables and nontradables sectors separately. Following Benigno and Thoenissen[8], we
adopt the result of Batini, Harrison and Millard[2], who estimate AR(1) parameters associatedwith the productivity of tradables and nontradables sectors separately. Note that we recognize

16

To examine the impulse response functions (IRFs), we consider one percent
changes in the productivity shifter of tradables in country H, aH,t , and the
productivity shifter of nontradables in country H, aN ,t , to investigate the eﬀects
of the existence of nontradables. Additionally, to compare our results with thoseof Ferrero[16], we consider one percent changes in the union-wide government
expenditure shifter, gtW , and the relative government expenditure shifter, gtR .
4.2.2

Optimal Monetary Policy Alone

Figures 1 and 2 depict the IRFs under optimal monetary policy alone withcommitment, in the case where all goods are tradable, namely γ = 1, and thebenchmark case, namely γ = 0.5, respectively.22 First, we consider changes in
the productivity shifter of tradables in country H. In both cases, an increase in
the productivity shifter of tradables in country H causes a decrease in the PPI
inflation rate in country H through a decrease in the marginal cost of tradables
in country H (17th panel in Figures 1 and 2). When all goods are tradables,
this change simply decreases the welfare-relevant output gap in country H (9thpanel in Figure 1). Furthermore, when half of the goods are tradables, a decreasein the marginal cost of tradables in country H stemming from an increase in
the productivity of tradables produced in country H causes a decrease in themarginal cost of nontradables produced in country H. This is the cause of a
relative decrease in the CPI in country H (29th panel in Figure 2). This is incontrast with the case where all goods are tradables where the CPI disparity does
not change (29th panel in Figure 1). A relative decrease in the CPI in country
H causes an increase in the NPD. A decrease in the price of nontradables incountry H boosts demand for nontradables in country H. Thus, the decrease in
the welfare-relevant output gap in country H in the benchmark case is smaller
than that in the case where all goods are tradables (9th panel in Figures 1 and 2).
This increase in the CPI disparity is inconsistent with the Balassa—Samuelson
theorem, which implies a decrease in the CPI disparity. This increase in the
CPI disparity is consistent with Benigno and Thoenissen[8]’s real exchange rateanomally.
The volatility of the PPI inflation rate to changes in the productivity of
tradables in country H under the case where all goods are tradables and thebenchmark case, are 0.0080 and 0.0081, respectively (cells in the 11th row and3rd column in Tables 1 and 2). Intuitively, the volatility of the PPI inflation
rate in the benchmark case is smaller than that in the case where all goods
are tradables, because just half of the goods are tradable. In the benchmark
case, however, a decrease in the marginal cost in the tradables sector decreases
the marginal cost in the nontradables sector. Thus, the eﬀect of the changes
that their estimated parameters are smaller than those used in most RBC studies. Ribeiro[28]is one of the few papers that estimate autoregressive processes of government expenditure inEurope. We adopt his estimation result.
22 Given the benchmark parameterization, the four eigenvalues are larger than 1 in value
for the four forward-looking variables under the optimal monetary policy alone regime. The
Blanchard—Kahn conditions are met.

17

in the productivity of tradables in country H to the PPI inflation rate is larger
in the benchmark case. Clearly, existence of nontradables makes the PPI inflation rate more volatile. In addition, as mentioned in the latter section, thisdecreases social welfare and makes fiscal policy more important when there are
nontradables.
The dynamics brought about by changes in the productivity shifter of tradables in country H can be confirmed by investigating the model, especially the
relative block. Eq.(14) implies that an increase in the productivity shifter of
tradables in country H decreases the welfare-relevant output gap disparity between both countries. The definition of NPD inflation and Eq.(16) shows thata decrease in the welfare-relevant output gap disparity between both countries
increases the CPI disparity although a decrease in the welfare-relevant output
gap disparity does not change the CPI disparity in the case where all goods are
tradables. Monetary policy alone can simultaneously stabilize both the welfarerelevant output gap and the union-wide inflation rate by increasing the nominalinterest rate.23 However, the country-level welfare-relevant output gap and inflation rate cannot simultaneously be stabilized in both cases. In the case where
all goods are tradables, a distorted steady state avoids simultaneous stabilization of both the welfare-relevant output gap and the inflation rate, as shown inEq.(14). In the benchmark case, both a distorted steady state and the existenceof nontradables avoids simultaneous stabilization of both the welfare-relevant
output gap and the inflation rate, as shown in Eqs.(14) and (16). Because these
two equalities help avoid simultaneous stabilization, fluctuations of PPI inflation in the benchmark case are larger than that in the case where all goods are
tradables.
Regarding the benchmark case, we can explain the result of changes in the
productivity shifter of nontradables in country H in the same manner as we can
explain changes in the productivity shifter of tradables in country H. However,the coeﬃcient of the productivity shifter of nontradables in the NKRD Eq.(16) islarger than the coeﬃcient of the productivity shifter of tradables. Thus, the CPI
disparity increases and the volatility of other macroeconomic variables is higher
following changes in the productivity of nontradables than following changes inthat of tradables (30th panel in Figure 2 and 7th, 9th, 11th, 13th, 15th and 17throws in Table 2). As mentioned, an increase in the CPI disparity is consistent
with Lipinska’s (2008) finding that nontradables’ productivity shocks lead toa stronger real exchange rate depreciation. In the case where all goods are
tradables, any macroeconomic variables do not fluctuate when the productivity
shifter of nontradables in country H changes because only tradables exist.
Now, we consider the innovation in union-wide government expenditure.
Both IRFs and macroeconomic volatilities are the same in both cases because
the union-wide government expenditure does not aﬀect the relative block of themodel. To secure funds for government expenditure, the nominal interest rate
decreases and there is pressure for the welfare-relevant output gap to increase
23 Both the union-wide inflation rate and the union-wide welfare-relevant output gap deviateslightly from their steady state, because of Eq.(21). See footnote 17 for a quantitative measure.

18

(27th panel in Figures 1 and 2). As implied in Eq.(13), however, a decreasein the lagged nominal interest rate decreases the union-wide welfare-relevant
output gap. Thus, the union-wide welfare-relevant output gap is stabilized,
although it increases at first (3rd panel in Figures 1 and 2).24 The union-wide
inflation rate is stabilized despite an increase in the union-wide welfare-relevant
output gap, because an increase in government expenditure decreases the PPI
inflation rate through Eq.(21) (7th panel in Figures 1 and 2). Note that becauseof Eq.(21), inflation—output gap tradeoﬀs are no longer dissolved completely,even though the optimality condition in Eq.(22) is applied when union-wide
government expenditure increases. Along with the union-wide welfare-relevant
output gap, the welfare-relevant output gap in each country is stabilized (11thand 15th panels in Figures 1 and 2). Furthermore, the PPI inflation rate incountries H and F is stabilized (19th and 23rd panels in Figures 1 and 2).
Note that Ferrero[16] shows that the nominal interest rate increases to a unit
innovation in union-wide government expenditure because the tax rate increases
simultaneously to secure funds to finance government expenditure, whereas thetax rate is constant over time in our setting.
When relative government expenditure changes, the welfare-relevant output
gap disparity between countries and the welfare-relevant output gap and inflation rate cannot be stabilized at the country level in both cases (12th, 16th,
20th and 24th panels in Figures 1 and 2).
Benigno[5], who assumes all goods are tradable, implies that monetary policy
alone can simultaneously stabilize both the output gap and the inflation rate ineach country even though asymmetric shocks hit the economy. In our model,
when asymmetric shocks hit the economy, the PPI inflation rate and welfarerelevant output gap in each country are no longer stabilized regardless of theshare of tradables γ. This fact stems from the distorted steady state, whichis assumed by both us and Ferrero[16]. In addition, the volatilities of the PPI
inflation rate in response to each shock in the benchmark case are larger thanthose in the case where all goods are tradables, except for the volatility of the
PPI inflation rate to the union-wide government expenditure shock (11th and13th rows in Tables 1 and 2). This fact reflects that nontradables amplifies theeﬀects of asymmetric shocks.
4.2.3

Optimal Monetary Policy and Fiscal Policy Mix

Figures 3 and 4 depict the IRFs under the optimal monetary policy and fiscal
policy mix with commitment in the cases where all goods are tradables and thebenchmark, respectively.25 To eliminate the eﬀects of changes in the productivity shifter of tradables in country H, the fiscal authorities in both countries
24 As mentioned in Section 2.3, all interest-bearing assets held by households are in the formof government bonds. Lowering the interest rate in the previous period reduces the nominal
payoﬀ of the bond portfolio purchased by households. Because of the budget constraint, thisreduces output via a reduction in consumption.
25 Given the benchmark parameterization, five eigenvalues are larger than 1 for the five
forward-looking variables under the optimal monetary and fiscal policy regime. Thus, theBlanchard—Kahn conditions are met.

19

increase government bonds in both cases (33rd and 37th panels in Figures 3
and 4). An increase in government bonds during the previous period increases
the nominal payoﬀ of the bond portfolio purchased by households. Thus, consumption in both countries increases. Because an increase in government bonds
in country H is higher than that in country F , an increase in consumption ishigher than that in country F . This eliminates the pressure to decrease thewelfare-relevant output gap in country H and eliminates the pressure to increase the welfare-relevant output gap in country F . Thus, the welfare-relevant
output gap in both countries is stabilized. Under the optimal monetary policy
and fiscal policy mix, the volatility of the welfare-relevant output gap in bothcountries dramatically decreases compared with the volatility under monetary
policy alone in both cases (7th to 10th rows in Tables 1 and 2). Because thewelfare-relevant output gap is well stabilized, the PPI inflation rate in eachcountry is also well stabilized in both cases (17th and 21st panels in Figures
3 and 4). The volatility of the PPI inflation rate in both countries decreases
dramatically compared with the volatility under monetary policy alone in bothcases (cells in the 11th to 14th rows and 3rd column in Tables 1 and 2). In thecase where all goods are tradables, CPI disparity does not fluctuate because all
goods are tradables, which means that PPP applies (29th panel in Figure 3 andthe cell at the 16th row and 3rd column in Table 1). In the benchmark case,because the PPI inflation rate in country H increases only in the first period,
the CPI disparity decreases (29th panel in Figure 4). However, the volatilityof the CPI disparity decreases dramatically compared with the volatility undermonetary policy alone (cells in the 15th and 16th rows and 3rd column in Table
2).
Now, we consider changes in the productivity shifter of nontradables. Herethe macroeconomic variables do not fluctuate in response to changes in the
productivity of nontradables in the case where all goods are tradables. Thus,
we focus only on macroeconomic volatility and fluctuations in the benchmark
case. An increase in the quantity of government bonds in the previous period
drastically stabilizes the welfare-relevant output gap (10th and 14th panels in
Figure 4). Because of this, the PPI inflation also stabilizes (18th and 22nd panels
in Figure 4). The volatility of the PPI inflation rate and the welfare-relevant
output gap is less than that under optimal monetary policy alone (cells in the
7th to 14th rows and 4th column in Table 2). As shown in Eq.(17), the CPI
disparity decreases (30th panel in Figure 4). The CPI disaprity is dramaticallystabilized compared with that under the optimal monetary policy alone (cells
in the 15th and 16th rows and 4th column in Table 2).
Next, we consider changes in union-wide government expenditure. Apartfrom government bonds, the mechanism of the macroeconomic variables’ behavior is similar to that under optimal monetary policy alone in both cases.
Although both fiscal authorities aﬃliate with the central bank, the IRFs andthe macroeconomic volatilities behave the same as those under optimal monetary policy alone regarding the welfare-relevant output gap, the PPI inflation
rate and the CPI disparity (3rd, 7th, 11th, 15th, 19th and 23rd panels in Figures
3 and 4 and cells in the 3rd to 16th rows and 5th column in Tables 1 and 2)20

in each case. This implies that there is no notable role for fiscal policy, which
could replace monetary policy to stabilize the economy. When a union-wide
shock, such as union-wide government expenditure, hits the economy, however,the central bank and the fiscal authorities cooperate against a union-wide shock.
This result is the same as the one derived by Ferrero[16].
When the relative government expenditure shifter shocks the economy, thefiscal authority in country H decreases the quantity of government bonds while
the fiscal authority in country F increases the quantity of government bonds
in both cases (36th and 40th panels in Figures 3 and 4). As Eq.(14) implies,
an increase in relative government expenditure increases the welfare-relevant
output gap disparity. However, a decrease in the quantity of government bonds
relative to the previous period decreases the current welfare-relevant output
gap disparity. This eliminates the pressure to increase the PPI inflation rate incountry H and to decrease it in country F . Thus, the PPI inflation rates inboth countries are stabilized immediately in both cases (20th and 24th panels in
Figures 3 and 4). The volatility of the PPI inflation rate is dramatically reduced
in both cases (cells in the 11th to 14th rows and 6th column in Tables 1 and 2).
The role of the fiscal authorities is larger than that of the central bank instabilizing both inflation and the welfare-relevant output gap in both cases.
This fact strongly supports the implication derived by Ferrero[16], who insistson the importance of optimal monetary and fiscal policy in a currency union.
In addition, we insist that the role of optimal fiscal policy as a stabilization tool
when there are nontradables is more important than that when there are only
tradables, which has already been mentioned above. The volatility of the PPI
inflation rate decreases from 0.0121 to 1.4184e-005 in country H and 2.2170e006 in country F , and from 0.0408 to 3.6616e-004 in the benchmark case, byadding optimal fiscal policy for changes in the productivity of nontradables andin the changes in the relative government expenditures (cells in the 11th to 14throws and 4th and 6th columns in Table 2). In contrast, the volatility of the
PPI inflation rate is unchanged at zero and falls from 0.0301 to 3.6616e-004
in the case where all goods are tradables, for changes in the productivity of
nontradables and in changes in relative government expenditure, respectively(cells in the 11th to 14th rows and 4th and 6th columns in Table 1). Clearly,
the degree of improvement in the volatility in the benchmark case is larger thanthat in the case where all goods are tradables. A more formal discussion of thisfrom the viewpoint of welfare appears in Section 4.3.
Behind the stabilization of the volatility of the PPI inflation rate in thebenchmark case for changes in productivity, there is stabilization in the CPI
disparity. By comparing the 29th and 30th panels in Figure 2 with those in
Figure 4, we confirm that the CPI disparity is stabilized. In fact, the volatilityof the CPI disparity for changes in the productivity in the benchmark case under
the optimal monetary and fiscal policy are smaller than that under the optimalmonetary policy alone and are close to zero, which is the volatility for optimalmonetary policy alone in the case where all goods are tradables (cells in the
15th and 16th rows and 3rd and 4th columns in Tables 1 and 2). Optimal fiscal
policy stabilizes the PPI inflation rate by stabilizing the CPI disparity: namely,21

minimizing the consumption disparity between the two countries.
Finally, we point out that the volatility of the nominal interest rate decreases
under the optimal monetary policy and fiscal policy. Furthermore, this tendency
does not depend on the share of tradables (17th and 18th rows in Tables 1 and
2).
The results of this sensitivity analysis prove the policy implications that are
mentioned by McKinnon[24]. Cooperative fiscal authorities have a certain rolein stabilizing both the welfare-relevant output gap and inflation simultaneously
when nontradables exist.

4.3

Welfare Analysis

In this section, we analyze the social welfare associated with both regimes,
focusing on the share of nontradables. This paper finds that the role of optimal
fiscal policy under the assumption that there are nontradables is more important
than the one under the assumption that all goods are tradable. To clarify therole of optimal fiscal policy, we compare our result to that of Ferrero[16], whofinds that such a policy is essential for the minimization of social losses when
all goods are tradable.
Now, we define the welfare criteria. Setting δ → 1 on Eq.(19), the expectedwelfare losses of any policy can be written in terms of the variances of inflationand the welfare-relevant output gap as follows:L̃W ≡¡ ∗ ¢ ΛyΛπΛπΛy+var (πP,t ) +var πP,tvar (ŷt ) +var (ŷt∗ ) ,44
44

with L̃W being the welfare loss under the setting δ → 1.
Figure 5 depicts social losses associated with the two regimes analyzed in theprevious section: optimal monetary policy alone and optimal monetary policy
and fiscal policy. As noted above, both regimes are fully committed. Under optimal monetary policy alone, when the share of nontradables increases, welfare
losses increase. However, optimal monetary and fiscal policy bring about approximately zero welfare losses, independent of the share of nontradables.26 The
necessity of optimal fiscal policy is clear from this analysis. Note that optimalmonetary policy alone cannot result in a zero welfare loss when all goods are
tradable, while the welfare losses are minimized among the losses brought about
by the optimal monetary policy alone. As Benigno[5] mentions, when all goods
are tradable, optimal monetary policy alone can eliminate the inflation—output
tradeoﬀs approximately and simultaneously.27 However, Ferrero[16] insists that
fiscal policy is needed, even though all goods are tradable. This discrepancy
stems from the assumption of the steady-state behavior of the fiscal authority.
Using DSGE analysis, Benigno[5] assumes a zero steady-state value of government expenditure and the quantity of bonds. As in our setting, Ferrero[16] does
26 Because of Eq.(21), zero welfare losses definitely cannot be obtained although optimal
fiscal policy is conducted.
27 In addition, it is attained when price stickiness is the same in both countries.

22not assume a zero steady-state value of government expenditure and bonds.
Government expenditure and bonds have nonzero values in the steady state.
This results in additional elasticities, the steady-state share of government expenditure with respect to output, and the steady-state share of government
bonds with respect to output, σG and σB , respectively. These elasticities change
the format of the demand block of the economy, which inhibits perfect risk sharing. Thus, the study of Ferrero[16] corresponds with our study in the case where
all goods are tradable.
Our results insist that the existence of nontradables creates acute losses
in the Euro economy, not only because of the assumption of a steady state,
but also because of the real exchange rate anomaly. In our benchmark setting
where γ = 0.5, the welfare loss–the percentage deviation of utility from its
steady state, brought about by optimal monetary policy alone–is 0.26%, while
it is 0.13% when all goods are tradable, γ = 1. As noted above, approximately
50.3% of goods are nontradable; thus, the role of optimal fiscal policy is greaterthan that suggested by Ferrero[16].

5Implementing a Cooperative Solution by Selforiented Fiscal Authorities

Some studies, such as Benigno[4], Obstfeld and Rogoﬀ[27] and Benigno and
Benigno[7], show that self-oriented monetary authorities can replicate the cooperative outcome in a decentralized framework so that there is no need for international monetary policy cooperation. Following their context, we investigate
whether it is possible that fiscal policies set in a noncooperative environment
can implement the optimal cooperative solution in this section.
While the central bank commits to minimizing the union-wide social loss LW
subject to the structural model, we assume that each fiscal authority commits
to minimizing its respective losses as follows:
LN C ≡ E0

∞
X
t=0

C
δ t LN; LN C∗ ≡ E0
t

∞
X

C∗
δ t LN,tt=0

subject to the structural model with:
C
≡
LN
t

Λπ 2
Λy 2
Λπ ¡ ∗ ¢2 Λy ∗ 2C∗≡π +ŷ ; LN
πP,t +(ŷt ) ,
t
2 P,t
2 t
2
2

(26)

C
where LN
denotes the social loss assigned to the fiscal authority in country Htunder the self-oriented setting.28
Next, we calculate the union-wide social loss under the cooperative setting
LW , and the union-wide social loss£ brought about¤ by self-oriented fiscal authorities in both countries LN CW ≡ 12 LN C + LN C∗ . For simplicity without loss of
28 Following Beetsma and Jensen[3], we split the per period union-wide social loss function¡

1∗
Eq.(20) as follows: LW
t = 2 Lt + Lt
Λy2

¡ ∗ ¢2
ŷt

.

¢

2 +
with Lt ≡ Λ2π πP,t

23

Λy 2
ŷ and L∗t ≡ Λ2π
2 t¡ ∗ ¢2πP,t+generality, we assume ρT = ρN = ρG = 0 and each shock has constant variance.
After tedious calculations, we have the following:
LW

=
=

LN CW

∙
¸n
Λy Ω8 Ψ1
Ψ1
2
2
(1
−
Ψ
)
+
Λ
[κ(1 + ϕ)Ω3 ] γ 2 [var (ξH,t )π1(1 − δ) 2 (1 − Ψ21 )2¡ ∗ ¢¤
2
2£+var (ξF,t )] + [κ(1 + ϕ)Ω3 ] (1 − γ) var (ξN ,t ) + var ξN,t¡ ∗ ¢¤o2£+ (κσG Ω4 ) var (ξG,t ) + var ξG,t,where Ψ1 < 1 denotes one of the solutions to the characteristic equation ofEq.(21) and its counterpart in country F and Ω8 denotes a complicated coefficient. This implies that a self-oriented fiscal authority can achieve the cooperative allocation in the Nash equilibrium without imposing a complicated loss
function on fiscal authorities. That is to say, there are no gains from fiscal policy
cooperation to maximize social welfare. Interestingly, LW = LN CW is applied
independent of the share of tradables. Our policy implication is applicable toboth the cases where there either are nontradables or not. Thus, our policy implication is diﬀerent from the result of not only Beetsma and Jensen[3] who show
the necessity of fiscal policy cooperation in a currency union where all goods are
tradable, but also by McKinnon[24] who insists on the necessity of moving fiscal
policy control from the national government to the central government in a currency union with nontradables. Furthermore, our policy implication diﬀers from
Liu and Pappa[23], whose model is a flexible exchange rate two-country modelwith nontradables, and who highlight the importance of policy cooperation.
Benigno and Benigno[6][7] and Liu and Pappa[23] show the importance ofpolicy cooperation by comparing the cooperative setting with the self-oriented
setting, although they do not assume a currency union. Under special cases,
however, Benigno and Benigno[6][7] and Liu and Pappa[23] imply that there
is no or only a quantitatively small role for policy cooperation. Benigno and
Benigno[6][7] show that when the inverse of relative risk aversion times theelasticity of substitution between domestic and foreign tradables is unity, theallocation brought about by self-oriented policy makers corresponds to the cooperative allocation. Under such a case, each policy authority can control theirown output gap and inflation independently of the TOT. Liu and Pappa[23],whose model implies that the inverse of relative risk aversion, the elasticity of
substitution between domestic tradables and foreign tradables and the elasticity of substitution between tradables and nontradables, are unity, show thatthe gains from policy cooperation are quantitatively small when two countries
have symmetric trading structures, which implies that the ratio of tradables tothe sum of tradables and nontradables are equal in the two countries. In sucha case, the TOT externality disappears, because output gap and inflation are
independent of the TOT, as mentioned by Benigno and Benigno[6][7]. Once thesymmetric trade structure is applied in Liu and Pappa[23], their parameterization is consistent with Benigno and Benigno[6][7]’s special case and Liu andPappa[23] show that there are small gains from policy cooperation in such a24

situation.
Our model implies that the inverse of relative risk aversion and the elasticity
of substitution between domestic tradables and foreign tradables are unity, as
in Liu and Pappa[23]. Furthermore, the share of tradables γ is equal in the twocountries and the elasticity of substitution between tradables and nontradables
η is assumed to be unity after Section 4. The assumption of η = 1 is also consistent with Liu and Pappa[23]’s setting. Thus, our settings correspond to Liu andPappa[23]’s special case in which the gains from policy cooperation are quantitatively small. Although we should consider that Liu and Pappa[23] do notassume a currency union, our policy implication is not necessarily inconsistent
with theirs.
It is important and interesting to compare our results with the policy implication derived by Beetsma and Jensen[3], who develop a two-country currency
union model without nontradables, implying that fiscal policy cooperation is important. While the diﬀerence in the policy implication between us and Benigno
and Benigno[6][7] and Liu and Pappa[23] depends on parameter values, the difference in the policy implication between us and Beetsma and Jensen[3] does
not depend on parameter values. In fact, although Beetsma and Jensen[3] assume an Armington form of the consumption index, which consists of tradables
produced in two countries, as do we, they do not assume a logarithmic utility
function of consumption as in Eq.(1).29 Beetsma and Jensen[3]’s parameterization is not consistent with Benigno and Benigno[6][7]’s special case in which
the allocation brought about by self-oriented policy makers corresponds to thecooperative allocation. However, it cannot be said that the reason we havea diﬀerent policy implication on fiscal policy cooperation results from our parameterization assumptions. Beetsma and Jensen[3] assume a household utility
function in which government expenditure appears and fiscal authorities manage government expenditure, although we do not assume such a utility function,
as shown in Eq.(1), and we assume that fiscal authorities manage government
debt. By second-order approximating such a utility function including government expenditure, the cross-term of the logarithmic TOT and the relative
government expenditure gap, which gives additional welfare benefits from the
TOT externality, appears in their union-wide loss function. They set the relativerisk aversion larger than unity in their benchmark case, which is not consistent
with us. However, the coeﬃcient of that cross-term does not depend on relative risk aversion and does not disappear if relative risk aversion becomes unity,
which corresponds to our parameterization. This implies that whether the TOT
externality appears or not depends on the stabilization tools used by the fiscal
authorities because adopting the utility function including government expenditure makes it possible to solve an optimization problem for fiscal authorities
that control government expenditure. They assume that the fiscal authorities’
loss functions in a noncooperative setting are domestically oriented and do notinclude such a cross-term. In our setting, the union-wide loss function is sim29 Our consumption index is given by Eq.(2).

If there are only tradables, however, Eq.(2)
1

1

2
2
reduces to a consumption index of tradables: namely, Ct = 2CH,tCF,tis applied.

25

ply the weighted sum of each country’s loss function, as shown in Eqs.(20) and(26) and there is no TOT externality. In Beetsma and Jensen[3]’s setting, however, the union-wide-level loss function is not simply the sum of each country’s
loss function, and there is a TOT externality, which gives each fiscal authority
opportunity to change the TOT through government expenditure to enhanceeach country’s social welfare. Clearly, policy cooperation is important in their
setting.
As mentioned, the diﬀerence between the policy implication of Beetsma andJensen[3] and ours does not depend on parameter values. That diﬀerence depends on the stabilization tools used by the fiscal authorities. Although further
detailed discussion is needed, the diﬀerence in stabilization tools may aﬀect thenecessity or importance of policy cooperation. Inferring from a comparison between the results of Beetsma and Jensen[3] and us, managing government debt
as a stabilization tool may be less costly than managing government expenditure, because managing government debt does not need policy cooperation
which involves nonnegligible costs.30

6Conclusion

We discussed optimal monetary and fiscal policy in a currency union. Our mostimportant and interesting finding is that self-oriented fiscal policy can replicate
the allocation brought about by a cooperative setting and there are no gains fromfiscal policy cooperation, either when there are nontradables or when there arenot. This policy implication contrasts with Beetsma and Jensen[3], who show
the importance of policy cooperation. Taking into account the costs of policy
cooperation, our main finding implies that cooperation on fiscal policy in theEuro zone may be futile, because there are no gains from policy cooperation.
Our results imply that recent pressure for more fiscal policy cooperation in
Europe is not necessarily beneficial.
Comparing our main finding with Beetsma and Jensen[3]’s finding, there isa possibility that the necessity or importance of policy cooperation depends onthe choice of stabilization tools. The necessity or importance of cooperation has
been discussed in preceding papers, focusing on the parameter values that results in a TOT externality. Beetsma and Jensen[3] assume that fiscal authorities
manage government debt and they show the necessity of fiscal policy cooperation. As mentioned, the diﬀerence in the policy implication between this paper
and Beetsma and Jensen[3] does not depend on parameter values, but rather
depends on the stabilization tools used. Further discussion on the choice ofstabilization tools to avoid welfare losses associated with policy cooperation is
an important objective for future research.

30 There are costs to forcing players to comply with the cooperation framework and to share
their understanding of the game, according to Kawai[20].

26A

Details on Derivation of the Model

A.1

Households

Preferences of the representative household in countries H and F are given by:
µ
¶
∞
X
1
δ t ln Ct −Nt1+ϕ ,U ≡ E0
1+ϕ
t=0
µ
¶
∞
X
1
∗
t
∗
∗ 1+ϕ
,
(A.1)
≡ E0
δ ln Ct −
(N )
U
1+ϕ t
t=0

∗
where Ct∗ denotes consumption in country F , Nt∗ ≡ NF,t + NN,t denotes hours
R2
R
2
∗
∗
≡
N
of work in country F , NF,t ≡ 1 NF,t (f ) df and NN,t
N ,t (f ) df denote1
hours of work to produce tradables produced in country F and nontradables
produced in country F , respectively. The first equality in Eq.(A.1) is Eq.(1) inthe text.
More precisely, private consumption is a composite index defined by:
η
¸ η−1
∙
η−1
η−1
1
1
η
η
Ct ≡
γ η CT ,t + (1 − γ) η CN ,t
,

Ct∗

≡

η
¸ η−1
∙
¢ η−1
¢ η−1
1 ¡
1 ¡
∗
∗
η
η
η
γ η CT ,t+ (1 − γ) CN ,t
,

(A.2)

θ
θ
hR
i θ−1
hR
i θ−1
θ−1
θ−1
1
2 ∗
∗
θ
with CN ,t ≡ 0 CN ,t (h) θ dh
, CN
≡
C
(f
)
dh
CH,t ≡
,t
N ,t
1
θ
θ
hR
i
h
i
θ−1
θ−1
R2
θ−1
θ−1
1
CH,t (h) θ dh
and CF,t ≡CF,t (f ) θ df
, where the index
01{h, f } denotes a variable that is specific to agents h and f , CT∗ ,t denotes the∗consumption index for tradables in country F , and CN,t denotes an index of
consumption across the nontradable goods produced in country F . The firstequality in Eq.(A.2) is Eq.(2) in the text.
A sequence of budget constraints is given by:
Z 1
Z 2
PH,t (h) CH,t (h) dh +PF,t (f ) CF,t (f ) df
Dtn + Wt Nt − St ≥0+
Dtn∗ + Wt∗ Nt∗ − St∗

≥

Z 1

Z 1
0

+

0

1

n
PN ,t (h) CN ,t (h) dh + Et Qt,t+1 Dt+1,∗
PH,t (h) CH,t(h) df +

Z 1
0

Z 2

∗
PF,t (f ) CF,t(f ) df

1

∗
∗
n∗
PN
,t (f ) CN ,t (f ) df + Et Qt,t+1 Dt+1 ,

(A.3)

1
1
hR
i 1−θ
hR
i 1−θ
1
2
1−θ
1−θ
dh
, PF,t ≡ 1 PF,t (f )df
and PN ,t ≡with PH,t ≡ 0 PH,t (h)
1
1
hR
i 1−θ
hR
i 1−θ
1
2 ∗
∗
PN ,t (h)1−θ dh, where PN
PN ,t (f )1−θ df
denotes the price
,t ≡
0
1

27

index of nontradables produced in country F and St∗ denotes the lump sum
taxes in country F .
The optimal allocation of any given expenditure within each category of
goods yields the following demand functions:
¶−θ
¶−θ
µ
µ
PH,t (h)PF,t (f )CH,t (h) =CH,t , ; CF,t (f ) =CF,t ,
PH,t
PF,t
¶−θ
¶−θ
µ
µ
PH,t (h)PF,t (f )∗∗∗∗CH,t (h) =CH,t , ; CF,t (f ) =
CF,t
,
PH,t
PF,t
!−θ
Ã
¶−θ
µ
∗
PN
PN ,t (h),t (f )∗∗
CN ,t (h) =CN ,t , ; CN ,t (f ) =CN,t(. A.4)
∗
PN ,t
PN
,t
R1
R2
These equalities imply that 0 PH,t (h) CH,t (h) dh = PH,t CH,t , 1 PF,t (f ) CF,t (f ) df =
R1
R2 ∗
∗
∗
∗
PF,t CF,t , 0 PN ,t (h) CN ,t (h) dh = PN ,t CN ,t and 1 PN ,t (f ) CN,t (f ) df = PN ,t CN ,t .
Total consumption expenditures by households in countries H and F aregiven by:PH,t CH,t + PF,t CF,t + PN ,t CN ,t∗∗∗∗
PF,t CF,t+ PH,t CH,t+ PN
,t CN ,t==Pt Ct ,
Pt∗ Ct∗ .

Combining Eq.(A.3) and these equalities, we obtain:
Dtn + Wt Nt + Stn∗
Dt + Wt∗ Nt∗ + St∗≥≥n
Pt Ct + Et Qt,t+1 Dt+1,
∗ ∗n∗
Pt Ct + Et Qt,t+1 Dt+1
,

(A.5)

where the first equality in Eq.(A.5) is Eq.(3) in the text.
Combining Eq.(A.4) and aggregators, we have:1
CH,t =
2

µ
µ

PH,t
PT ,t

¶−1

CT ,t ,¶−1
PH,tCT∗ ,t ,
PT ,t
¶−η
µ
PT ,t
CT ,t = γ
Ct ,
Pt
¶−η
µ
PT ,t
Ct∗ ,
CT∗ ,t = γ
Pt∗

∗
CH,t
=

1
2

;

1
CF,t =
2

;

∗
CF,t
=

µ
µ

PF,t
PT ,t

¶−1

CT ,t ,¶−1
PF,tCT∗ ,t ,
PT ,t
¶−η
µ
PN ,t
CN ,t = (1 − γ)Ct ,Pt
µ ∗ ¶−η
PN ,t
∗
CN
=
(1
−
γ)
Ct∗ .
,t
Pt∗

;
;

1
2

(A.6)

The first, second, fifth and sixth equalities in Eq.(A.6) are Eq.(5) in the text.
CPIs are given by:
Pt

≡

Pt∗

≡

h
h

1−η
γPT1−η
,t + (1 − γ) PN ,t

1
i 1−η

¡ ∗ ¢1−η
γPT1−η
,t + (1 − γ) PN ,t
28

,
1
i 1−η

,

(A.7)

where Pt∗ denotes the CPI in country F . The first equality in Eq.(A.7) is Eq.(4)in the text.
The representative household maximizes Eq.(A.1) subject to Eq.(A.5). Optimality conditions are given by:Ã!
"¡¢−1 ∗ #
−1
∗
Ct+1
Pt
Pt
Ct+1
1
1
=
=
, ; δEt
,
(A.8)
δEt
−1 ∗
∗
Rt
R
Ct−1 Pt+1
t(Ct ) Pt+1
Ct Ntϕ =Wt,
Pt

Ct∗ (Nt∗ )ϕ =

;

Wt∗
.
Pt∗

(A.9)

The RHS of Eq.(A.8) is an intertemporal optimality condition in country F ,whereas the RHS of Eq.(A.9) is an intratemporal optimality condition in country
F . The LHS of both Eqs.(A.8) and (A.9) are Eq.(6) in the text.
Combining and iterating Eq.(A.8) with an initial condition, we have thefollowing optimal risk-sharing condition:
Ct = ϑCt∗ Qt ,(A.10)∗∗
which is Eq.(7) in the text. When C−1 = C−1= P−1 = P−1= 1, we have
ϑ = 1.

A.2

Firms

Each producer can use a linear technology to produce a diﬀerentiated good as
follows:
YH,t (h) = AH,t NH,t (h) ,YF,t (f ) = AF,t NF,t (f ) ,;;YN ,t (h) = AN ,t NN ,t (h) ,∗YN∗ ,t (f ) = A∗N ,t NN,t (f ) ,(A.11)

θ
θ
³R
´ θ−1
³R
´ θ−1
θ−1
θ−1
1
2
θ
θ
Y
(h)
dh
,
Y
≡
Y
(f
)
df
, YN ,t ≡with YH,t ≡
F,t
0 H,t
1 F,t
θ
θ
³R
´
³
´
θ−1
θ−1
R2 ∗
θ−1
θ−1
1
Y
(h) θ dhand YN∗ ,t ≡Y(f ) θ df, where AF,t and
0 N ,t
1 N ,tA∗N ,t denote stochastic productivity shifters associated with tradables and nontradables produced in country F , respectively. The first equalities in Eq.(A.11)
are Eq.(8) in the text.
Using Dixit—Stiglitz aggregators, Eq.(A.11) can be rewritten as:

AH,t NH,t
YH,t = R 1 Y (h)
H,t
dh
YH,t
0
AF,t NF,t
YF,t = R 2 Y (f )F,t
YF,t df
1;;

AN ,t NN ,t
YN ,t = R 1 Y (h) ,
N ,t
dh
YN ,t
0

YN∗ ,t = R∗A∗N ,t NN,t∗ (f )
2 YN
,t
df
∗
YN
1
,t

.

(A.12)

Under Calvo—Yun-style price-setting behavior, the pricing rules are given by:
PH,t

=

i 1
h
1−θ
1−θ 1−θαPH,t−1+ (1 − α) P̃H,t,29

PN ,t

=

PF,t

=

∗
PN
,t

=

i 1
h
1−θ
1−θ 1−θαPN,t−1+ (1 − α) P̃N
,
,t
i 1
h
1−θ
1−θ 1−θαPF,t−1+ (1 − α) P̃F,t,1∙³´1−θ ¸ 1−θ
¡ ∗
¢1−θ
∗
α PN,t−1+ (1 − α) P̃N ,t,(A.13)∗
where P̃F,t and P̃N,t are the prices chosen by firms when they obtain the chance
to change prices associated with tradables and nontradables produced in country
F , respectively.
The maximization problems faced by firms are as follows:
"∞
#
³
´
X
k
−1
n
(αδ) (Pt+k Ct+k ) ỸH,t+k P̃H,t − M CH,t+k ,max EtP̃H,tk=0

max EtP̃N ,tmax EtP̃F,tmax Et
∗
P̃N
,t

"∞
X

k=0
"∞
X
k=0
"∞
X
k=0

´−θ

k

(αδ) (Pt+k Ct+k )
k¡

(αδ)

¢−1
∗
∗
ỸF,t+k
Ct+k
Pt+k

k¡

(αδ)

#
³
´
n
ỸN ,t+k P̃N ,t − M CN ,t+k ,

−1

∗
∗
Ct+k
Pt+k

¢−1

#
³
´
n
P̃F,t − M CF,t+k ,

#
³
´
∗
∗n
,
ỸN∗ ,t+k P̃N,t − M CN ,t+k

´−θ
³
P̃H,t
P̃ ,t
ỸH,t+k ≡ PH,t+k
YH,t+k and ỸN ,t+k ≡ PNN,t+k
YN ,t+k , where ỸF,t+k ≡µ ∗ ¶−θ
³
´−θ
P̃
P̃F,t
YF,t+k and ỸN∗ ,t+k ≡ Y ∗N ,t
YN∗ ,t+k denote the total demands
PF,t+k³

N ,t+k

W∗

W∗

n
∗n
t
t
≡ (1−τ )Aand M CNdewhen the prices change, and M CF,t,t ≡ (1−τ )A∗F,t
N ,tnote the marginal costs associated with tradables and nontradables producedin country F , respectively.
The FONCs are as follows:
#
"∞
³
´
X
k
−1
n
= 0,
(αδ) (Pt+k Ct+k ) ỸH,t+k P̃H,t − ζM CH,t+k
Et
k=0

"∞
X

(αδ) (Pt+k Ct+k )

Et

(αδ)

Et

Et

k=0
"∞
X

k=0
"∞
X

(αδ)

k=0

−1

k

k¡

k¡

#
³
´
n
ỸN ,t+k P̃N ,t − ζM CN
,t+k

¢−1
∗
∗
Pt+k
ỸF,t+k
Ct+k

¢−1 ∗
∗
∗
Pt+k
ỸN ,t+k
Ct+k

30

#
³
´
n
P̃F,t − ζM CF,t+k

#
³
´
∗
n∗
P̃N ,t − ζM CN ,t+k

=

0,

=

0,

=

0.
(A.14)

The first and the second equalities in Eq.(A.14) are Eq.(9) in the text.
We define the real marginal costs as:with PP∗ ≡M CH,t ≡n
M CH,tPP,t;M CN ,t ≡M CF,t ≡n
M CF,t
∗
PP,t

;

∗
M CN
,t ≡

n
M CN
,t
,
PP,t
n
(M C ∗ )N ,t
∗
PP,t

,

(A.15)∗∗
PF,t YF,t +PN,t YN ,t.
∗
YF,t +YN,t

Combining the first equalities of Eqs.(A.14) and (A.15) yields:
(∞
)
h
i
X
−(θ−1) −(η−1)
k
−η
−θ
−1
Et
= 0,
(αδ) X̃H,t+k XT ,t+k − ζ X̃H,t+k XH,t+k XT ,t+k XP,t+k M CH,t+k
k=0(A.16)

P

P̃

P

P

H,t
,t+k
with X̃H,t+k ≡ PH,t+k, XH,t+k ≡ PH,t+k, XT ,t+k ≡ PTt+kand XP,t+k ≡ PP,t+k.
T ,t+kt+kCombining the definition of the marginal cost and Eq.(A.9), we have:

Ct Ntϕ Pt,(1 − τ ) PP,t AH,tCt∗ (Nt∗ )ϕ Pt∗,
M CF,t =∗ A(1 − τ ) PP,tF,tM CH,t =;;Ct Ntϕ Pt,(1 − τ ) PP,t AN ,tCt∗ (Nt∗ )ϕ Pt∗
∗
M CN
.
,t =
∗ A∗
(1 − τ ) PP,tN ,tM CN ,t =(A.17)We define the country-wide real marginal cost as:

A.3

M Ct

≡

M Ct∗

≡

M CH,t YH,t + M CN ,t YN ,t,
YH,t + YN ,t∗∗
M CF,t YF,t + M CN
,t YN ,t.
YH,t + YN ,tLocal Government

The government expenditure index is given by:
GH,t ≡
GF,t ≡µZ 1

GH,t (h)

θ−1
θ

dh

0

µZ 2
1

GF,t (f )

θ−1
θ

dh

θ
¶ θ−1
θ
¶ θ−1

,
,

;
;

GN ,t ≡G∗N ,t ≡

µZ 1
0

µZ 2
1

GN ,t (h)G∗N ,t (f )

θ−1
θ

θ−1
θ

df
df

θ
¶ θ−1

θ
¶ θ−1

,
,

where GF,t and G∗N ,t denote government expenditure on tradables and nontradables produced in country F , respectively. For simplicity, we assume thatgovernment purchases are fully allocated to a domestically produced good. For
any given level of public consumption, the government allocates expenditures
across goods in order to minimize total cost. This yields the following set of31

government demand schedules, analogous to those associated with private consumption.
¶−θPH,t (h)GH,tGH,t (h) =
PH,t
¶−θ
µ
PF,t (f )GF,t (f ) =
GF,t
PF,t
µ

;
;

¶−θ
PN ,t (h)GN ,t (h) =GN ,t ,
PN ,t
!−θ
Ã
∗
PN
,t (f )∗
GN ,t =G∗N ,t . (A.18)
∗
PN
,t
µ

The flow government budget constraints are given by:½Z 1nn
Bt = Rt−1 Bt−1 −PH,t (h) [τ YH,t (h) − GH,t (h)] dh
+

Z 1
0

Btn∗

=

0

¾
PN ,t (h) [τ YN ,t (h) − GN ,t (h)] dhn∗Rt−1 Bt−1
−

+

Z 2
1

½Z 2
1

PF,t (h) [τ YF,t (f ) − GF,t (h)] dh

£ ∗
¤
∗
∗
PN
,t (h) τ YN ,t (h) − GN ,t (h) dh¾(A.19)

where Btn∗ ≡ Pt∗ Bt∗ denote the nominal risk-free bonds issued by local government in country F and Bt∗ denote the real risk-free bonds issued by local
government in country F , respectively.
Combining the definition of prices and output, we have:¶−θPH,t (h)YH,t ,
YH,t (h) =
PH,t
¶−θ
µ
PF,t (f )YF,t (f ) =YF,t ,
PF,t
µ

;
;

¶−θ
PN ,t (h)YN ,t (h) =YN ,t ,PN ,t¶−θµ
PH,t (h)
∗
∗
YH,t
(h) =
YH,t
.(A.20)PH,tµSubstituting Eqs.(A.18) and (A.20) into Eq.(A.19), we have:
Btn
Btn∗

=
=

n
Rt−1 Bt−1− [τ (PH,t YH,t + PN ,t YN ,t ) − (PH,t GH,t + PN ,t GN ,t )] ,
¢ ¡
¢¤
£ ¡
n∗
∗
∗
∗
∗
.
Rt−1 Bt−1 − τ PF,t YF,t + PN
,t YN ,t − PF,t GF,t + PN ,t GN ,t

These equalities can be rewritten as:
Btn
Btn∗

∗
≡
with PG,t
text.

=
=

n
Rt−1 Bt−1− [τ PP,t (YH,t + YN ,t ) − PG,t (GH,t + GN ,t )] ,
£ ∗ ¡
¢
¡
¢¤
n∗
∗
Rt−1 Bt−1− τ PP,t
YF,t + YN∗ ,t − PG,t
GF,t + G∗N ,t ,(A.21)∗∗
PF,t GF,t +PN,t GN ,t. The first equality in Eq.(A.21) is Eq.(10) in the
GF,t +G∗
N ,t32

Eq.(A.21) yields the consolidated government budget constraint which is
given by:
1 n
(B + Btn∗ ) =
2 t

¢ 1
1¡ n
n∗
− {[τ PP,t (YH,t + YN ,t ) − PG,t (GH,t + GN ,t )] ,Rt−1 Bt−1+ Bt−12£ ∗ ¡¢ 2 ∗ ¡¢¤ª∗+ τ PP,t YF,t + YN ,t − PG,t GF,t + G∗N ,t.
(A.22)

The appropriate transversality conditions for government assets are givenby:1
lim Et Qt,k (Bkn + Bk∗n ) = 0,2k→∞which appears in footnote 9 in the text.
Starting from Eq.(A.22) with the appropriate transversality condition, theresulting consolidated intertemporal budget constraint can be written as:
∙ −1
Ct
1
Bt−1
Rt−1
2
Πt
#
(∞
−1
X ∙ PP,t+k τ (YH,t+k + YN ,t+k ) − PG,t+k (GH,t+k + GN ,t+k )1(Ct∗ )
∗
=
+
Bt−1
δk
Et
Π∗t
2
Ct+k Pt+k
k=0
³
´ ⎤⎫
∗
∗
GF,t+k + G∗N ,t+k ⎬PP,t+kτ (YF,t+k + YN ,t+k ) − PG,t+k
⎦ ,
+
∗ P∗
⎭
Ct+k
t+k
P∗

with Π∗t ≡ P ∗t being the gross CPI inflation rate in country F . This can bet−1
rewritten as:
"
#
−1
Ct−1
PP,t τ (YH,t + YN ,t ) − PG,t (GH,t + GN ,t )1
(Ct∗ )∗=
Bt−1 +
Bt−1
Rt−1
2
Πt
Π∗t
Pt Ct
¡
¢
¡
¢
∗
∗
τ YF,t + YN∗ ,t − PG,tPP,t
GF,t + G∗N ,t+Pt∗ Ct∗Ã¡ ∗ ¢−1 !
−1
Ct+1
Ct+1
+ δEt Rt
Bt +Bt∗ .(A.23)
Πt+1
Π∗t+1

A.4

Market Clearing

Market clearing conditions for tradables are given by:
YH,t (h)=∗CH,t (h) + CH,t(h) + GH,t (h) ,YF,t (f )=∗CF,t (f ) + CF,t(f ) + GF,t (f ) .

The first equality in Eq.(A.24) is the LHS of Eq.(11) in the text.

33(A.24)

As for nontradables, equilibrium requires that:
YN ,t (h)YN∗ ,t (f )==CN ,t (h) + GN ,t (h) ,∗∗CN,t (f ) + GN ,t (f ) .

(A.25)

The first equality in Eq.(A.25) is the RHS of Eq.(11) in the text.
The market in country H for tradables clears when domestic demand is given
by Eq.(A.24). As for nontradables, equilibrium requires Eq.(A.25).
Using Eqs.(A.4), (A.10) and (A.18), Eq.(A.24) can be rewritten as:#)¶−θ ( µ¶−1 "µ¶−η µ¶−ηµγ PH,tPT ,tPH,t (h)PT ,t+ GH,t ,YH,t (h) =
Ct
+
Q−1
t
PH,t
2 PT ,tPtPt∗#)¶−θ ( µ¶−1 "µ¶−η µ¶−ηµγ PF,tPT ,tPF,t (f )PT ,t−1
YF,t (f ) =+ GF,t ,
Ct
+
Qt
PF,t
2 PT ,tPt
Pt∗twhere we use the fact that Ct∗ = C
Qt , which is derived from Eq.(A.10). Combining these equalities and Eqs.(A.4), (A.10) and (A.18), Eq.(A.24) can be
rewritten as:#µ¶−1 "µ
¶−η µ
¶−η
PT ,t
γ PH,tPT ,t−1+ GH,t ,YH,t =Ct+
Qt
2 PT ,tPtPt∗#µ¶−1 "µ
¶−η µ
¶−η
PT ,t
γ PF,tPT ,t+ GF,t . (A.26)
YF,t =Ct+
Q−1
t
2 PT ,tPt
Pt∗

Using Eqs.(A.4), (A.10) and (A.18), Eq.(A.25) can be rewritten as:
#
µ
¶−θ "
¶−η
µ
PN ,t
PN ,t (h)(1 − γ)Ct + GN ,t ,YN ,t (h) =
PN ,t
Pt
#
µ ∗ ¶−η
¶−θ "
µ
P
(h)
P
N
,t
N
,t
−1
YN∗ ,t (f ) =(1 − γ)Ct Qt + G∗N ,t .
PN ,t
Pt∗
Combining these equalities and Eqs.(A.4), (A.10) and (A.18), Eq.(A.25) can be
rewritten as:
¶−η
µ
PN ,t
YN ,t = (1 − γ)
Ct + GN ,t ,Pt
µ ∗ ¶−η
PN ,t∗∗Ct Q−1(A.27)
YN ,t = (1 − γ)t + GN ,t .
Pt∗
Eq.(A.26) implies that:
YH,t − GH,t
= Tt ,
YF,t − GF,t
34

P

F,t
where Tt ≡ PH,tdenotes the terms of trade (TOT).
Eq.(A.27) implies that:

YN ,t − GN ,t−(η−1)= Nηt Qt,
YN∗ ,t − G∗N ,tP∗,twhere Nt ≡ PN
denotes the nontradables price diﬀerence between countries H
N ,tand F (NPD).
Finally, we define country-wide output and government expenditure as:

Yt ≡
Gt ≡

PH,t
PN ,t
YH,t +
YN ,t
PP,t
PP,t

PH,t
PN ,t
GH,t +
GN ,t
PG,t
PG,t

;
;

Yt∗ ≡

PF,t
PN ,t
YF,t + ∗ YN ,t ,
∗
PP,t
PP,t

(A.28)

G∗t ≡

∗
PN
PF,t
,t
GF,t + ∗ GN ,t .
PG,t
PG,t

(A.29)The LHS equalities in Eqs.(A.28) and (A.29) are Eq.(12) in the text.

A.5

Net Exports

Following Gali and Monacelli[18], we define net exports in country H as follows:
N Xt ≡ Yt −

Pt
PG,t
Ct −
Gt ,
PP,t
PP,t

(A.30)

where N Xt denotes net exports in country H.

B

Nonstochastic Steady State

We focus on equilibria where the state variables follow paths that are close toa deterministic stationary equilibrium, in which ΠH,t = ΠN = ΠF,t = Π∗N ,t = 1
P

P

P∗

P

H,t
,t
F,t
with ΠH,t ≡ PH,t−1, ΠN ,t ≡ PNN,t−1, ΠF,t ≡ PF,t−1
and Π∗N ,t ≡ P ∗N ,t where
N ,t−1variables without the subscript indicating the period denote their nonstochastic
steady state value. These imply that the PPI inflation rate is zero in this steadystate. Note that X̃H = X̃N = X̃F = X̃∗N = 1 is applied in this steady state with

P̃

P̃

P̃ ∗

P̃

,t
, X̃N ,t ≡ PN
, X̃F,t ≡ PF,tand X̃∗N ,t ≡ PN∗ ,t . Because this steadyX̃H,t ≡ PH,t
H,t
N ,t
F,t
N ,t
state is nonstochastic, all productivities are unit values, i.e., AH = AN = AF =A∗N = 1. In addition, we assume that GH = GF , GN = G∗N and B = B ∗ inthis steady state.
In this steady state, the gross nominal interest rate is equal to the inverseof the subjective discount factor, as follows:R = δ −1 .

35

Eq.(A.14) can be rewritten as:
Ã
!
Ã
!
KH,t
KN ,t
; P̃N ,t = Et
P̃H,t = Et
−1
−1
PH,t
FH,t
PN
,t FN ,t
Ã
!
Ã
!
∗
KN
KF,t
,t
∗
; P̃N,t = Et,
P̃F,t = Et
−1
∗ F∗
PN
PF,t
FF,t
,t N ,t

(B.1)

with:
KH,t ≡ ζ
KN ,t ≡ ζ

∞
X

k=0
∞
X

−1

(Pt+k Ct+k )(Pt+k Ct+k )

k=0
∞
X

KF,t ≡ ζ

k=0

−1

n
ỸH,t+k M CH,t+knỸN ,t+k M CN
,t+k

¡ ∗
¢−1
∗
n∗
ỸF,t+k M CF,t+k
Pt+k Ct+k

∞
X
¡ ∗
¢−1 ∗
∗
∗
n∗
Pt+k Ct+kỸN ,t+k M CH,t+k
KN,t ≡ ζ

;
;
;
;

k=0

FH,t ≡ PH,t
FN ,t ≡ PN ,t
FF,t ≡ PF,t
∗
FN
,t ≡

∞
X

k=0
∞
X

k=0

ỸH,t+k−1

ỸN ,t+k(Pt+k Ct+k )

k=0
∞
X

∞
X
¡

−1

(Pt+k Ct+k )

¡ ∗
¢−1
∗
ỸF,t+k
Pt+k Ct+k

∗
∗
Pt+k
Ct+k

k=0

¢−1

ỸN∗ ,t+k .

Eq.(B.2) implies that:nζYH M CH
(1 − αδ) (P C)nζYN M CNKN =(1 − αδ) (P C)
ζYF M CFn
KF =(1 − αδ) (P ∗ C ∗ )n∗ζYN∗ M CN∗=
KN(1 − αδ) (P ∗ C ∗ )

KH =

;
;
;
;

PH YH
(1 − αδ) (P C)
PN Y NFN =(1 − αδ) (P C)
PF YF
FF =
(1 − αδ) (P ∗ C ∗ )
∗
PN
YN∗
FN =
.
(1 − αδ) (P ∗ C ∗ )FH =These equalities and Eq.(B.1) imply that:∗= ζM C n∗ ,PH = ζM C n ; PN = ζM C n ; PF = ζM C n∗ ; PN

where we use the equalities as follows:
n
n
n∗
M CH
= M CN≡ M C n ; M CFn = M CN
≡ M C n∗ ,which are implied by Eq.(A.17). These equalities imply that:PH = PN ,∗PF = PN
.
36

(B.3)

(B.2)

∗
∗
Combining these equalities and the definition of PP,t , PP,t
, PG,t and PG,t
, we
have:

PP = PH = PN = PG ,∗= PG∗ .
PP∗ = PF = PN

(B.4)

Following Gali and Monacelli[18], we assume that PPP (purchasing power
parity) holds in the steady state, which means that:Q = 1.

(B.5)

Eqs.(B.4) and (B.5) imply the following:∗
P = P ∗ = PT = PN = PN
= PH = PF = PP = PP∗ = PG = PG∗ .

(B.6)

∗
, we have:
Note that because PF = PH and PN = PN

T = N = 1.

(B.7)

Because of Eqs.(B.3), Eq.(B.4) can be rewritten as:
M C n = M C n∗ .
Thus, we have:
M C = M C ∗ = ζ −1 ,nn∗.
with M C ≡ MPC and M C ∗ ≡ M C
PFurthermore, Eqs.(A.17) and (B.4) imply the following:ϕ

CN ϕ = C ∗ (N ∗ ) =

1−τ
.
ζ

(B.8)

Eq.(B.8) implies the familiar expression:(1 − τ ) UC (C) =(1 − τ ) UC (C ∗ ) =ζUN (N ) ,ζUN (N ∗ ) .

(B.9)

Note that because τ ∈ (0, 1) and θ > 1, this steady state is distorted.
Eq.(A.26) can be rewritten as:
YH = γC + GH;

YF = γC + GF ,(B.10)

by using Eq.(B.6). Because GH = GF , YH = YF . As with Eq.(B.6), Eq.(A.29)
can be rewritten as:
YN = (1 − γ) C + GN;

Because GN = G∗N , YN = YN∗ .
37

YN∗ = (1 − γ) C + G∗N .

(B.11)

Eq.(B.6) and Eq.(18) in the text imply the following:
Y = YH + YN
G = GH + GN

Y ∗ = YF + YN∗ ,G∗ = GF + G∗N .

;;(B.12)

Combining Eqs.(B.11) and (B.12), we have:
Y =C +G;Y ∗ = C + G∗ .

(B.13)

Because GH = GF and GN = G∗N , Eq.(B.12) implies G = G∗ . Thus,
Y = Y ∗.
Eqs.(A.10) and (B.5) imply that:C = C ∗.

(B.14)

Eqs.(B.8) and (B.14) imply the following:
N = N ∗.
Eq.(A.21) yields the following:¶µ1−δ= τ Y − G,Bδ

(B.15)

nwith B ≡ BP . This equality implies B = B ∗ .
We assume B > 0; thus, another transversality condition for local government is given by:£¤(B.16)
lim Et δ k−t UC (C) RB = 0,k→∞which appears in footnote 9 in the text.

C

Log-linearization of the Model

C.1

Aggregate Demand and Output

Log-linearizing Eq.(7) in the text, we obtain the following:cR
t = qt ,(C.1)

where qt denotes the logarithmic CPI diﬀerential between the two countries.
Log-linearizing Eq.(A.7) and rearranging yields:
qt = (1 − γ) nt .

(C.2)

Log-linearizing and manipulating Eq.(A.7), we obtain:πt = γπT ,t + (1 − γ) πN ,t ,
38

(C.3)

with πT ,t = 12 πH,t + 12 πF,t which is derived by log-linearizing the definition of
the price index of tradables, where πt denotes the CPI inflation rate in country
H, πT ,t denotes the tradable goods price inflation rate, πH,t and πF,t denotethe inflation rates of tradables produced in countries H and F , respectively, andπN ,t denotes the inflation rate of nontradables produced in country H.
Log-linearizing the definition of PPI, we have:
pP,t = γpH,t + (1 − γ) pN ,t .

(C.4)

This equality implies that:
πP,t = γπH,t + (1 − γ) πN ,t ,(C.5)

where πP,t denotes the PPI inflation rate in country H.
Log-linearizing Eq.(A.28), we have:yt==

γyH,t + γpH,t − γpP,t + (1 − γ) yN ,t + (1 − γ) pN ,t − (1 − γ) pP,t
γyH,t + (1 − γ) yN ,t + γpH,t + (1 − γ) pN ,t − pP,t .

Substituting Eq.(C.4) into this equality, we have:yt==

γyH,t + (1 − γ) yN ,t + pP,t − pP,t ,γyH,t + (1 − γ) yN ,t .

(C.6)

Log-linearizing the definition of the average price of goods purchased by thegovernment in country H yields:
pG,t = γpH,t + (1 − γ) pN ,t ,(C.7)

which implies that pP,t = pG,t .
Combining the log-linearized LHS of Eq.(A.29) and Eq.(C.7), we have:
gt = γgH,t + (1 − γ) gN ,t .

(C.8)

Log-linearizing the first equalities of Eqs.(A.26) and (A.27) and substituting
these equalities into Eq.(C.6), we have:
yt = (1 − σG ) ct +(1 − σG ) γ(1 − σG ) ψtt +nt + σG gt .
2
2

(C.9)

Subtracting the counterpart of Eq.(C.9) in country F from Eq.(C.9), we
have:ytR = γ (1 − σG ) tt + (1 − γ) $ (1 − σG ) nt + σG gtR .

(C.10)

Log-linearizing Eq.(A.23), we have:
bt

=

1
1
1
Et ct+1 − ct − πt + Et πt+1 + r̂t−1 − r̂t + bt−1 +
δ
δ
δ
τ
σG
−
yt +
gt .
σB
σB
39

µ

1−δ
δ

¶

γ
tt
2
(C.11)

Combining Eqs.(C.3), (C.6), (C.8), (C.10), (C.11) and the counterpart ofEq.(C.11), we have:
ytW

=

ytR

=

βW
βW
W
W
Et yt+1+ βW Et πt+1
− βW r̂t +r̂t−1 − βW bWt
1 − σG
δ
βW W
βW W
+
(C.12)
bt−1 −
π + σG νW gtW ,δδ tR−βR δbRt + βR (1 − γ) υnt − βR (1 − γ) nt−1 + βR bt−1
+ σG νR gtR .

(C.13)

Log-linearizing Eq.(A.30) and substituting Eq.(C.9) yields:ncxt =(1 − σG ) ψnt ,2with ncxt ≡ dNYXt denoting the percentage deviation of the net exports in country
H from the steady-state value of output. Note that this equality becomes ncxt =0 which implies that balanced trade is definitely applied, under our benchmark
parameterization, η = 1.

C.2

Aggregate Supply and Inflation

Log-linearizing Eq.(A.16), we have:"∞#Xk(αδ) (x̃H,t+k + xH,t+k + xT ,t+k − xP,t+k − mcH,t+k ) = 0,Et
k=0with x̃H,t+k ≡ ln X̃H,t+k , xH,t+k ≡ ln XH,t+k , xT ,t+k ≡ ln XT ,t+k and xP,t+k ≡ln XP,t+k .
PkUsing the fact that x̃H,t+k = xH,t − s=1 πH,t+s , this can be rewritten as:
"∞
Ã
!#
k
X
X
k
= 0.
Et
(αδ) x̃H,t −πH,t+s + xH,t+k + xT ,t+k − xP +k,t − mcH,t+k
s=1

k=0

Furthermore, using the fact that
this can be rewritten as:∞P∞

k=0 (αδ)

P∞
k Pk
k
1
s=1 πH,t+s = 1−αδ
k=1 (αδ) πH,t+k ,

∞

∞

X
X
X
1
1
k
k
k
(αδ) πH,t+k + Et(αδ) xH,t+k + Et(αδ) xT ,t+k ,x̃H,t −
Et
1 − αδ
1 − αδ
k=1

k=0

−Et

∞
X

k=0

k

(αδ) xP,t+k − Et

k=0

∞
X

k

(αδ) mcH,t+k = 0.

k=0

Rearranging this, we have:
x̃H,t

=

∞
X

k=1

k

(αδ) πH,t+k − (1 − αδ)

∞
X

k=0

40

k

(αδ) xH,t+k − (1 − αδ)

∞
X

k=0

k

(αδ) xT ,t+k ,+ (1 − αδ)
=

∞
X

k=0

k

(αδ) xP +k,t + (1 − αδ)∞Xk(αδ) mcH,t+k ,k=0αδπH,t+1 − (1 − αδ) xH,t − (1 − αδ) xT ,t + (1 − αδ) xP,t ,+ (1 − αδ) mcH,t + αδx̃H,t+1 .

(C.14)

Log-linearizing the first equality of Eq.(A.13), we have:x̃H,t =

απH,t .
1−α(C.15)

Combining Eqs.(C.14) and (C.15) yields:
πH,t

δπH,t+1 − κxH,t − κxT ,t + κxP,t + κmcH,t ,=δπH,t+1 + (1 − γ) κpN ,t − (1 − γ) κpH,t + κmcH,t .

=Taking the conditional expectation at t, the second equality can be rewritten
as:
πH,t

=

δEt πH,t+1 + κ (1 − γ) pN ,t − κ (1 − γ) pH,t + κmcH,t . (C.16)

Similar to Eq.(C.16), the log-linearized second equality of Eq.(A.14) is given by:
πN ,t

=

δEt πN,t+1 − κγpN ,t + κγpH,t + κmcN ,t .

(C.17)

Other FONCs for firms can be log-linearized similarly.
Substituting Eqs.(C.17) and (C.16) into Eq.(C.5), we have a PPI-based inflation dynamics equation as follows:
πP,t = δEt πP,t+1 + κmct ,(C.18)

where we use mct = γmcH,t + (1 − γ) mcN ,t which is derived by log-linearizing
the definition of country-wide marginal cost.
Combining Eq.(C.17) and its counterpart for country F , the nontradables
inflation diﬀerential is given by:R
RR
πN,t = δEt πN,t+1 + κγnt − κγtt + κmcN ,t ,(C.19)

R
πN,t ≡ − (nt − nt−1 ) ,(C.20)

withbeing relative nontradables inflation.
By log-linearizing the first equalities in Eq.(A.12) and combining it with
Eq.(C.6), we have:
yt = γaH,t + (1 − γ) aN ,t + nt ,(C.21)

where we also use the log-linearized definition of hours of work, nt = γnH,t +(1 − γ) nN ,t .
41

Combining log-linearized Eq.(A.17), and Eqs.(C.9) and (C.21), we have:
mcH,t

=

mcN ,t=λψσGyt − nt − (1 + ϕγ) aH,t − (1 − γ) ϕaN ,t −gt ,1 − σG2
1 − σGλψσGyt − nt − ϕγaH,t − [1 + (1 − γ) ϕ] aN ,t −gt .
1 − σG2
1 − σG(C.22)

Substituting Eq.(C.22) into the log-linearized definition of the marginal cost
mct = γmcH,t + (1 − γ) mcN ,t , we have:
mct =

λ
ψ
σG
yt − nt − (1 + ϕ) γaH,t − (1 + ϕ) (1 − γ) aN ,t −gt .
1 − σG2
1 − σG(C.23)

Combining the second equality in Eq.(C.22) and its counterpart for country
F , the logarithmic marginal cost diﬀerential associated with nontradables isgiven by:
mcR
N ,t

C.3

=

λ
y R − ψnt − ϕγaH,t + ϕγaF,t − [1 + (1 − γ) ϕ] aN ,t ,1 − σG t
σGgR .
(C.24)+ [1 + (1 − γ) ϕ] a∗N ,t −1 − σG t

Marginal Cost and Output Gap

Following Gali[18], we define the relationship between output, its natural level
and the output gap as:
yt ≡ ȳt + ỹt ,(C.25)

where ỹt denotes the logarithmic output gap measured from its natural level, andȳt denotes the logarithmic natural output level. Under the long-run equilibrium,ỹt = 0 must hold.31
When the fiscal authorities design their policies to reduce the distortion
generated by monopolistically competitive markets, real marginal costs under
the long-run equilibrium are constant, and their logarithm is given by mct =0. In addition, under the long-run equilibrium, PPP is applied.32 Thus, thelogarithmic NPD under the long-run equilibrium is given by nt = 0.
Combining these facts, Eq.(C.23) implies that:ȳt = β̄γaH,t + β̄ (1 − γ) aN ,t +

σG
gt .
λ

(C.26)

Combining Eqs.(C.12), (C.13), (C.25) and (C.26) can be rewritten as:
ỹtW

=

βW
βW
βW W
W
W
Et ỹt+1− βW r̂t + βW Et πt+1+
r̂t−1 − βW bWb ,t +1 − σGδδ t−1

31 Following Gali[18], nominal rigidities disappear in the long-run equilibrium.
32 Following Gali[18], we assume a steady state where PPP is applied.

42

βW W
γ β̄βT(1 − γ) β̄βN
γ β̄βTπ −aH,t −
aN ,t −
aF,t ,δ t22
2(1 − γ) β̄βN ∗aN ,t + σG ςW gtW ,(C.27)−
2R−βR δbRt + βR (1 − γ) υnt − βR (1 − γ) nt−1 + βR bt−1 − β̄γaH,t ,
−

ỹtR

=

+ β̄γaF,t − β̄ (1 − γ) aN ,t + β̄ (1 − γ) a∗N ,t + ςR σG gtR ,(C.28)

which are Eqs.(13) and (14) in the text, respectively.
Combining Eqs.(C.18), (C.23), (C.25) and (C.26) we have:
πP,t = δEt πP,t+1 +

κλ
ψκ
ỹt −
nt ,
1 − σG2(C.29)

which is Eq.(15) in the text. Similar to Eq.(C.29), we have the counterpart ofEq.(C.29) in country F .

C.4

NKRD

Combining Eqs.(C.19), (C.23), (C.25) and (C.26), we have:
¡
¢
¡
¢
R
R
R
πN
,t = δEt πN,t+1 + κϕỹt + κnt − κϕγ 1 − β̄ aH,t + κϕγ 1 − β̄ aF,t ,
£
¡
¢¤
£
¡
¢¤
− κ 1 + ϕ (1 − γ) 1 − β̄ aN ,t + κ 1 + ϕ (1 − γ) 1 − β̄ a∗N ,t ,
κσG ³
ϕ´ R
−
(C.30)1−g ,1 − σGλ twhich is Eq.(16) in the text.

D

Welfare Criterion

Following Gali and Monacelli[18], Gali[17] and Benigno and Woodford[10], weshow the derivation of the welfare criterion in the text based on the secondorder approximated utility function of Eq.(A.1) in the present appendix. η = 1
is assumed through the present appendix.
This section consists of four subsections. Subsection D.1 presents the secondorder Taylor expansion of the utility function. Subsection D.2 presents thesecond-order approximation of the FONCs for firms. Subsection D.3 eliminates
the linear term and completes the derivation of the welfare criterion. Subsection
D.4 discusses other details regarding the coeﬃcients and the NKPC in terms of
the welfare-relevant output gap.

D.1

Step 1: The Second-order Taylor Expansion of the
Utility Function

The second-order Taylor expansion of the period utility function in Eq.(1) inthe text is given by:
∙ µ
¶
¶
¸
µ
1 UCC 2 2 UN
1 UN N 2 2
Ut − U1 2
1 2−1
C ct + ct +C ct −
N nt + nt +
N nt
= C
UC C
2
2 UC
UC
2
2 UC
43

=

´
³
3
+ o kξk1
1 UCC 2 UN N
ct + c2t +
Cct −
2
2 UC
UC C
´
³
3
+ o kξkµ1
nt + n2t2¶+

UN N N 2 2n
UC C t(D.1)

where we assume that utility is separable by consumption and hours of work, i.e.,
UCN = 0. Plugging UC = C −1 , UCC = −C −2 , UN = N ϕ and UN N = ϕN ϕ−1
into Eq.(D.1), we have:µ¶³´
Ut − UUN N
1+ϕ 2= ct −
nt +nt + o kξk3
UC C
UC C
2
∙
¸
³
´
N
N
1+ϕ
3
2
nt +
= Φ nt + ct −(1 + Φ) nt + o kξk , (D.2)
CC2
where we use the fact that 1 − Φ = 1−τζ and Eq.(B.9).
Likewise, we have:∙¸³´Ut∗ − U
N
1+ϕ
N ∗
3
∗
∗
∗ 2
nt +
= Φ n t + ct −(1 + Φ) (nt ) + o kξk .
UC C
C
C
2

(D.3)

Eq.(A.12) can be rewritten as:
NH,t =

YH,t DH,t
YN ,t DN ,t
NN ,t =
,
AN ,t
AN ,t

R 1 ³ PH,t (h) ´−θ
R 1 ³ PN ,t (h) ´−θ
dh
and
D
≡
dh where we use thewith DH,t ≡ 0
N
,t
PH,t
PN ,t
0
R1
R1
YH,t (h)dh
YN ,t (h)dhfact that 0 YH,t
= DH,t and 0 YN ,t= DN ,t .
Log-linearizing these equalities, we obtain:
nH,t = yH,t + dH,t + t.i.p,;

nN ,t = yN ,t + dN ,t + t.i.p.

Combining these equalities with Eq.(C.6) and the log-linearized definition ofcountry level hours of work, nt = γnH,t + (1 − γ) nN ,t yields:
nt = yt + γdH,t + (1 − γ) dN,t + t.i.p.

(D.4)
∗

∗

PF,t (f )YF,t (f )+PN ,t (f )YN ,t (f )PH,t (h)YH,t (h)+PN ,t (h)YN ,t (h)∗and PP,t(f ) ≡.
∗ (h)YH,t (h)+YN ,t (h)YF,t (h)+YN,t∗∗
These yield pP,t (h) = γpH,t +(1 − γ) pN ,t (h) and pP,t (f ) = γpf,t +(1 − γ) pN ,t (f )Let PP,t (h) ≡by log-linearizing. Taking these equalities, Eq.(D.4) can be rewritten as:
nt

=
=

µ
¶−θ
µ
¶−θ
PH,t (h)PN ,t (h)yt + γ ln Eh+ (1 − γ) ln Eh+ t.i.p,
PH,t
PN ,t
∙
µ
¶
µ
¶¸
PH,t (h)PN ,t (h)+ ln (1 − γ)
+ t.i.p,yt − θEh γ ln
PH,t
PN ,t
44

yt − θEh [γ (pH,t (h) − pH,t ) + (1 − γ) (pN ,t (h) − pN ,t )] + at ,µ¶PP,t (h)+ t.i.p,= yt − θ ln Eh
PP,t
¶−θ
Z 1µ
PP,t (h)= yt + ln
dh + t.i.p,
PP,t
0
(D.5)
= yt + dt + t.i.p,
R 1 ³ PP,t (h) ´−θ
dh. Likewise, we have:with Dt ≡ 0PP,t=n∗t = yt∗ + d∗t + t.i.p.

(D.6)

Substituting Eqs.(D.5) and (D.6) into Eqs.(D.2) and (D.3), we have:¸∙¢
1(1 + ϕ) (1 + Φ) ¡ 2
Ut − UΦyt + (1 + Φ) dt +yt + ct −=yt − 2yt at ,UC C
1 − σG
1 − σG2³´
3+ t.i.p. + o kξk ,½
i¾∗
Ut − U1(1 + ϕ) (1 + Φ) h ∗ 2
Φ
∗
∗
∗
∗
∗ ∗
(yt ) − 2yt at
yt + (1 + Φ) dt +,y + ct −=UC C
1 − σG t
1 − σG2³´+ t.i.p. + o kξk3 ,(D.7)

with at ≡ γaH,t + (1 − γ) aN ,t and a∗t ≡ γaF,t + (1 − γ) a∗N ,t where we use the−1because N = Y .
fact that N
C = (1 − σG )Combining Eqs.(C.1), (C.2), (C.9) and (C.10), we have:
ct =

1
1
2σG W
yt +
y ∗ − c∗t +g .
1 − σG
1 − σG t
1 − σG t

Combining Eq.(D.7) and this equality yields:½
1(1 + ϕ) (1 + Φ) £ 2
UtW − UΦW
(1 + Φ) (dt + d∗t ) +
yt ,
yt −
=
UC C
1 − σG(1 − σG ) 2
2
io
³
´
2
+ t.i.p. + o kξk3 .
(D.8)−2yt at + (yt∗ ) − 2yt∗ a∗t

Let p̂P,t (h) ≡ pP,t (h) − pP,t . As derived by Gali and Monacelli[18], note
that:
µ
¶1−θ
PP,t (h)= exp [(1 − θ) p̂P,t (h)] ,PP,t=

1 − (1 − θ) p̂P,t (h) +In the symmetric equilibrium, we haveEhµPP,t (h)PP,tPP,t (h)PP,t = 1. This implies:

¶1−θ

45

¢
¡
(1 − θ)2p̂P,t (h) + o kξk3 (. D.9)
2

= 1.

(D.10)

Combining Eqs.(D.9) and (D.10), we have:
θ−12
Eh p̂P,t (h) .
2´−θ³PP,t (h)yields:In addition, the second-order approximation toPP,t
Eh p̂P,t (h) =µPP,t (h)PP,t¶−θ= 1 − θ p̂P,t (h) +This equality implies:
Dt = 1 − θEh p̂P,t (h) +(D.11)¡¢
θ22
p̂P,t (h) + o kξk3 .
2

¢
¡
θ2
2
Eh p̂P,t (h) + o kξk3 .
2

Substituting Eq.(D.11) into this equality, we have:
Dt==This equality implies:
dt =

¢
¡
θ
2
1 + Eh p̂P,t (h) + o kξk3
2¡¢
θ
1 + varh (p̂P,t (h)) + o kξk3 .
2
´
³
θ
3
varh (pP,t (h)) + o kξk ,2(D.12)

which clearly corresponds to the equality derived by Gali and Monacelli[18].
Lemma 1∞X

δ t varh (pP,t (h))

t=0

=

∞

1X t 2
δ πP,tκ t=0

Proof : See Woodford[35], p 399—400.
Substituting Lemma 1, Eqs.(D.12) and (D.8) into the definition of welfare
in the text, we have:
W

W=

Note that E0

∞
X

∙

Φ
(1 + Φ) θ 2(1 + Φ) θ ¡ ∗ ¢2πỹtW −
πP,t −
1
−
σ
(1
−
σ
)
4
(1
− σG ) 4 P,t
G
G
t=0
¸
(1 + ϕ) (1 + Φ)
(1 + ϕ) (1 + Φ) ∗22−
(yt − at ) −(yt − a∗t )(1 − σG ) 4(1 − σG ) 4³´
3+ t.i.p. + o kξk .
(D.13)

E0

P∞

k=0 δ

δ

t

W

t Ut −UW
UC C = W

because UC C = 1 and U = U ∗ .

46

D.2

Step 2: The Second-order Approximation of the FONCs
for Firms

Substituting the first and the second equalities in Eq.(B.1) into the first andthe second equalities in Eq.(A.13), we have:´ µ F ¶θ−1
1 ³
H,t
θ−1
1 − αΠH,t =
1−α
KH,t
¶θ−1
´ µF
1 ³
N ,t
1 − αΠθ−1
=
N ,t
1−α
KN ,t

(D.14)Taking logarithms on both sides in Eq.(D.14), we have:
¶
µ
1
α
θ−1
= (θ − 1) (log KH,t − log FH,t )
− log
−
Π
1 − α 1 − α H,t¶µ1
α= (θ − 1) (log KN ,t − log FN ,t ) (D.15)
− log
−
Πθ−1
1 − α 1 − α N ,t
The first-order approximation of the LHS in Eq.(D.15) is given by:¶µ´³(θ − 1) α
1
α
2
=
− log
−
Πθ−1
πH,t + o kξkH,t1−α 1−α
1−α
µ
¶
´
³
1
(θ − 1) α
α
2
− log
=
(D.16)−
Πθ−1
πN ,t + o kξkN ,t1−α 1−α1−αA weighted average of the two equalities in Eq.(D.16) is given by:(θ − 1) α(θ − 1) α(θ − 1) αγπH,t +
(1 − γ) πN ,t =πP,t ,1−α1−α1−αwhere we use Eq.(C.5). The second-order approximation of the RHS of thisequality yields:³´(θ − 1) α(θ − 1) α3 2(θ − 1) α3
πP,t =
πP,t +
πP,t + o kξk .
1−α1−α(1 − α) 4
Combining this equality and Eq.(D.15), we have:³´(θ − 1) α3 2(θ − 1) α3
πP,t +
πP,t = (θ − 1) (kt − ft ) + o kξk ,1−α(1 − α) 4(D.17)

with kt ≡ γkH,t + (1 − γ) kN ,t and ft ≡ γfH,t + (1 − γ) fN ,t³ where´ we use thefact that K = F . Note that KH,t , KN ,t , FH,t and FN ,t are kξk2 .

Log-linearizing the first and the second equality in the LHS in Eq.(B.2) andcombining them, we have:
kt = k̃t −θαπP,t ,
1−α47(D.18)

P∞k
G
with k̃t ≡ (1 − αδ) Et k=0 (αδ) k̃t,t+k and k̃t,t+k ≡ (1 + ϕ) yt+k − (1+ϕ)σat+k +λPk(1+ϕ)σG
gt+k + θ s=1 πP,t+s .
λLog-linearizing the first and the second equality on the RHS in Eq.(B.2) andcombining them, we have:
ft = f˜t −P∞

θαπP,t ,
1−α(D.19)k ˜(1+ϕ)σGσG˜
at+k +
k=0 (αδ) ft,t+k and ft,t+k ≡ − 1−σG yt+k −λPk(1+ϕ)σG
gt+k + (θ − 1) s=1 πP,t+s .
λwith f˜t ≡ (1 − αδ) Et

Subtracting Eq.(D.19) from Eq.(D.18) yields:
kt − ft = k̃t − f˜t .

(D.20)

Substituting Eq.(D.20) into Eq.(D.17) yields:³´³´(θ − 1) α(θ − 1) α3 23πP,t +
πP,t = (θ − 1) k̃t − f˜t + o kξk .
1−α(1 − α) 4(D.21)An arbitrary variable Vt can be approximated as:
Vt

=
=
=

eln Vt³´
1
2
3
eln V + eln V (ln Vt − ln V ) + eln V (ln Vt − ln V ) + o kξk
2
¶
µ
³
´
1 2
3
V 1 + vt + vt + o kξk .
2Thus, we have the second-order approximation of k̃t and f˜t as follows:
k̃t

f˜t

=

1
k̃t + k̃t2
2=

(1 − αδ) Et==

1f˜t + f˜t2
2
(1 − αδ) Et

∞
X

µ
¶
1 2
k̃t,t+k + k̃t,t+k
2(D.22)µ¶1 ˜2˜(αδ) ft,t+k + ft,t+k
2(D.23)

k

(αδ)

k=0

∞
X

k

k=0

Subtracting Eq.(D.23) from Eq.(D.22) yields:k̃t − f˜t

=

∙³

´ 1³
´¸
2
2
− f˜t,t+kk̃t,t+kk̃t,t+k − f˜t,t+k +
2
k=0
³
´
(1 − αδ) α3−(D.24)
πP,t Zt + o kξk
2 (1 − α)(1 − αδ) Et

∞
X

(αδ)k

48

with
Z t ≡ Et

∞
X

k=0

³
´
(αδ)k k̃t,t+k + f˜t,t+k .

(D.25)

The first term on the RHS in Eq.(D.24) can be rewritten as:
∞
X

Et

k

(αδ)

k=0

∞
³
´
X
k
(αδ)
k̃t,t+k − f˜t,t+k = Et1
λỹt+k +
PP,t (D.26)
1 − σG
1 − αδ

k=0

P∞
kwith PP,t ≡ Et k=0 (αδ) πP,t+k .
The second term on the RHS in Eq.(D.24) can be rewritten as:
³
´
1 X
2
2
=
(αδ)k k̃t,t+k− f˜t,t+k
Et
2
∞

k=0

³ 2
´
1 X
ff 2
f
(αδ)k kk
−
f
Et
t,t+k
t,t+k
2
∞

k=0

+ Et

∞
X

k

(αδ)

s=1

k=0

1
+ Et
2

∞
X

k=0

k
X

k

(αδ)

h
i
f t,t+k − (θ − 1) fff
πP,t+k θ kk
t,t+k

Ã k
X

πP,t+s

s=1

!2

h
i
θ 2 − (θ − 1)2 (D.27)

with
f t,t+kkk

fff t,t+k≡≡

σG (1 + ϕ)σG (1 + ϕ)at+k +
gt+k ,λλσGσG (1 + ϕ)σG (1 + ϕ)−ỹt+k −
at+k +
gt+k . (D.28)
1 − σGλλ(1 + ϕ) ỹt+k −The last term on the RHS in Eq.(D.27) can be rewritten as:
∞

1 X
k
(αδ)
Et
2
k=0

Ã k
X

πP,t+s

s=1

!2

h
i
2
θ 2 − (θ − 1) =∞X
2θ − 1k(αδ) πP,t+k (πP,t+k + 2PP,t+k ) .
Et(1 − αδ) 2
k=0(D.29)

Furthermore, the second term on the RHS in Eq.(D.29) can be rewritten as:
Et

∞
X

k=0

(αδ)

k

k
X
s=1

∞
h
i
X
k
f t,t+k − (θ − 1) fff
πP,t+s θ kk(αδ) πP,t+k Jt+k ,
=
E
t
t,t+k
k=0

(D.30)

withJt ≡ E t

∞
X

k=0

k

(αδ)

h

i
f t,t+k − (θ − 1) fff
θkkt,t+k .
49

(D.31)

Substituting Eqs.(D.29) and (D.30) into Eq.(D.28) yields:
³
´
1 X
k
2
2
=
(αδ) k̃t,t+k− f˜t,t+k
Et
2
∞

k=0

³
´
2
1 X
k f2
f
(αδ) kk
Ett,t+k − f f t,t+k
2
∞

k=0
∞
X

+ Et

k

(αδ) πP,t+k Jt+k(D.32)

k=0

+

∞

X
2θ − 1kEt(αδ) πP,t+k (πP,t+k + 2PP,t+k ) .
(1 − αδ) 2
k=0

Substituting Eqs.(D.25) and (D.31) into Eq.(D.24), we have:k̃t − f˜t

=

Et

∞
X

(αδ)

k

k=0

+ Et

∞
X

k=1

+

½

∙³
´ 1³ 2
´¸¾
2
f
f
f
f
(1 − αδ) kkt,t+k − f f t,t+k +kkt,t+k − f f t,t+k2(αδ)k πP,t+k + (1 − αδ) Et

2θ − 1
Et
2

∞
X

k=0

k

∞
X

(αδ)k πP,t+k Jt+kk=0(αδ) (πP,t+k + 2PP,t+k ) −´³(1 − αδ) α3πP,t Zt + o kξk .
2 (1 − α)(D.33)

Substituting Eq.(D.21) into Eq.(D.33) to eliminate the term k̃t − f˜t in the
LHS in Eq.(D.32) yields:
3 2
1 − αδ
πP,t + πP,t+
πP,t Zt
42=

Eq.(D.31) can be rewritten as:
Jt =³´ (1 − α) (1 − αδ) ³ 2´
2
f t,t+k − fff
ff
f
κ kk
−
f
+
kk
t,t+k
t,t+k
t,t+k
2α
+ (1 − α) δEt πP,t+1 + (1 − α) (1 − αδ) Et πP,t+1 Jt+1
(1 − α) (2θ − 1)+
δEt πP,t+1 (πP,t+1 + 2PP,t+1 )µ2¶
3 2
1 − αδ
+ αδEt πP,t+1 + πP,t+1+
πP,t+1 Zt+1
4
2
³
´
3
+ o kξk .
(D.34)

h
´i
³
1 X
k f
f
f
f
. (D.35)(αδ) kk
Ett,t+k − f f t,t+k + (2θ − 1) kk t,t+k − f f t,t+k
2
∞

k=0

Substituting Eqs.(D.18), (D.19) and (D.28) into Eq.(D.25), we have:
Et

∞
X

k=0

(αδ)

k

³

´
2θ − 1
f t,t+k − fff
PP,t .
kk
t,t+k = Zt −1 − αδ
50(D.36)

Substituting Eq.(D.28) into Eq.(D.26) yields:
Et

∞
X

k

(αδ)

k=0

³

f t,t+k − fff
kk
t,t+k

´

=

Et

∞
X

(αδ)

k

k=0

−

³
´
k̃t,t+k − f˜t,t+k

1
PP,t ,
1 − αδ

(D.37)

where we use the fact that
f t,t − fff =
kk
t,t

λ
ỹt .
1 − σG(D.38)

Substituting Eqs.(D.36) and (D.37) into Eq.(D.35) yields:
Jt =

∙
¸
∞
³
´ 2 (θ − 1)
1 X
k
˜
Z
(αδ)
+
(2θ
−
1)
k̃
−
f
Et
PP,t . (D.39)
tt,t+kt,t+k −
2
1 − αδ
k=0

In the first order, Eq.(D.21) can be rewritten as:
πP,t = κEt

∞
X

k=0

(αδ)

k

³

´
³
´
2
k̃t,t+k − f˜t,t+k + o kξk .

Substituting this equality into Eq.(D.39), we have:Jt =1
α (2θ − 1)
2θ − 1Zt +πP,t −
PP,t .
22κ
1 − αδ

Substituting this equality into Eq.(D.34) yields:
¸
∙
2 ´
3 2
1 − αδ
1 ³f2
fff
πP,t Zt = κ kkt,t − f f t,t +kk t,t − f f t,t
πP,t + πP,t +42
2(1 − αδ) δ+ δEt πP,t+1 +Et πP,t+1 Zt+1
2
3δ
δΘ
2
2
+ Et πP,t+1+
Et πP,t+1
4³
4´+ o kξk3.

2
Adding Θ
4 πP,t to both sides in this equality, we have:

¸
2 ´
1 ³f2
Θ 2
f
f
f
Mt = κ kk t,t − f f t,t +,kkt,t − f f t,t + δEt Mt+1 + πP,t2
4∙(D.40)Θ 22+ 1−αδ
with Mt ≡ πP,t + 34 πP,t2 πP,t Zt + 4 πP,t . Substituting Eq.(D.38) into
Eq.(D.40), we have:

πP,t = δπP,t+1 +´³λỹt + o kξk2
1 − σG51

in the first order. Thus, Eq.(D.40) corresponds to the second-order approximated NKPC in Benigno and Woodford[10]. Iterating Eq.(D.40) forward, we
have:
∙
¸
∞
∞
X
X
2 ´
1 ³f2
Θ 2
t f
f
f
δ kk t,t − f f t,t +δ t πP,t, (D.41)
M = κE0
kkt,t − f f t,t + E0
2
4
t=0
t=0

with M ≡ M0 where we take an expectation in period zero.
Eq.(D.28) implies:22

f − fff
kk
t,t
t,t

=

2σG (1 + ϕ)
2σG (1 + ϕ)ỹt at +ỹt gt + t.i.p.
(1 − σG ) λ(1 − σG ) λhi2
σG= ω̃1 yt − ω̃4 ω̃2 at −
ω̃3 gt ,λω̃1 ỹt2 −with ω̃1 ≡ (1−σς )2 , ω̃2 ≡ 1 + σςG , ω̃3 ≡ 1 − (1−σGς)(1+ϕ) and ω̃4 ≡ (1−σGλ)(1+ϕ) .
G
Substituting this equality and Eq.(D.38) into Eq.(D.41), we have:
½
¾
∞
i2 Θ
X
λ
ω̃1 h
σG
t
2
yt − ω̃4 ω̃2 at −M = κE0δỹt +ω̃3 gt + πP,t1 − σG
2
λ
4
t=0
´
³
(D.42)+ t.i.p. + o kξk3 .
The counterpart of Eq.(D.42) in country F is derived similarly as:½¾∞
i2 Θ ¡X¢2λω̃1 h ∗
σG
∗
t
∗
∗
∗
∗
M = κE0δỹ +yt − ω̃4 ω̃2 at −
ω̃3 gt +π
1 − σG t
2
λ
4 P,t
t=0
´
³
+ t.i.p. + o kξk3 .

Combining this equality and Eq.(D.42) yields:
∙
½
∞
´2
X
λ
ω̃1 ³
σG
W
t
W
yt − ω̃4 ω̃2 at −
M
= κE0
δ
ỹt +
ω̃3 gt + (yt∗1 − σG
4
λ
t=0
¸
¾
h
´
i
2
¡ ∗ ¢2σGΘ 2−ω̃4 ω̃2 a∗t −ω̃3 gt∗π + πP,t
+
λ
8 P,t
³
´
3
+ t.i.p. + o kξk .
(D.43)

D.3

Step 3: Elimination of the Linear Term and Completing the Derivation

Multiplying Φ by both sides in Eq.(D.43), and subtracting this from Eq.(D.13),
we have:
½
∞
h
i
X
1
2
W
W
t
W − ΦM= E0
δ
[(1 + ϕ) (1 + Φ) + κΦω̃1 ] yt2 + (yt∗ )(1 − σG ) 4
t=0
52

¸
2κΦ (1 − σG )1+Φ(yt at + yt∗ a∗t )ω̃2 +λ(1 − σG ) 2∙¸¡ ∗ ¢2 i
2κΦσG
1 (1 + Φ) θΦΘ h 2−ω̃3 (yt gt + yt∗ gt∗ ) ++πP,t + πP,tλ
4 (1 − σG ) κ2¾³´
Φ (1 − κλ) W3ỹ++ t.i.p. + o kξk .
1 − σG t− (1 + ϕ)∙Rearranging this equality, we have:
½
∞
i
h
X
1
2
W
t
= −E0
δ
W
[(1 + ϕ) (1 + Φ) + κΦω̃1 ] yt2 + (yt∗ )(1 − σG ) 4
t=0¸∙
2κΦ (1 − σG )1+Φ(yt at + yt∗ a∗t )− (1 + ϕ)ω̃2 +λ(1 − σG ) 2∙¸¾¡ ∗ ¢2 i
2κΦσG
1 (1 + Φ) θΦΘ h 2∗ ∗πP,t + πP,t−ω̃3 (yt gt + yt gt ) ++λ
4 (1 − σG ) κ
2
∞
´
³
X
Φ (1 − κλ) W
3+ E0(D.44)δtỹt + ΦMW + t.i.p. + o kξk .
1 − σG
t=0
´
³
P
2
t κλ
W
Note that MW = E0 ∞because of Eq.(D.43). Thus,
δ
y
+
o
kξk
t=0
1−σG tthe second and the third terms in Eq.(D.44) can be rewritten as:
E0

∞
X
t=0

δt

Φ (1 − κλ) W
ỹ + ΦMW
1 − σG t

=
=
=
=

∞
X

Φ (1 − κλ) WΦΦỹ + ΦMW +MW −MW
1 − σG t
κλ
κλ
k=0
µ
¶
∞
X
1
Φ (1 − κλ) WΦE0δtỹt − Φ− 1 MW +
MW
1
−
σ
κλ
κλ
G
t=0
Et

δk

Φ
MW
κλ
Γ0 .

We use this fact to derive the above equality as follows:
W
πP,0

=
=

E0

∞
X

t=0
W

M ,

δt

´
³
κλ
ytW + o kξk2 ,1 − σG

which can be derived by iterating the second-order³approximatedperiod FONCs
´
2
λ
.
ỹ
+
o
kξk
for firms, namely, πP,t = δEt πP,t+1 + 1−σtG

Substituting Eq.(D.45) into Eq.(D.44) and rearranging, we have:
½
∞
h
i
X
1
2
W
t
W
= −E0
δ
ω1 yt2 + (yt∗ ) − ω2 (yt at + yt∗ a∗t ) ,(1 − σG ) 4
t=0
¾
´
³
¡ ∗ ¢2 i
1 h 2∗ ∗+ Γ0 + t.i.p. + o kξk3 ,−ω3 (yt gt + yt gt ) + ω4 πP,t + πP,t4
53

(D.45)

with ω1 ≡ (1 + ϕ) (1 + Φ) + κΦω̃1 , ω2 ≡ (1 + ϕ)h2κΦ(1−σG )ω̃2
1+Φ+ (1−σλG )2i, ω3 ≡(1+Φ)θ
and ω4 ≡ (1−σ+ ΦΘ2 . As shown in this equality, the linear termG )κdisappears. By arranging this equality, we have:2κΦσG ω̃3
λ

W

W

=

∙
¸
∞
1 X t Λy 2 Λy ∗ 2 Λπ 2
Λπ ¡ ∗ ¢2
− E0
δ
ŷ +
(ŷt ) +π +πP,t+ Γ0 + t.i.p.
2 t=0
2 t
2
2 P,t
2
³
´
+ o kξk3 ,which corresponds to the second-order approximated welfare function in the
text.

D.4

Other Details on Coeﬃcients and the NKPC in Termsof the Welfare-relevant Output Gap

Note that complicated coeﬃcients associated with the target level of output are
as follows:hi(1 − σG ) (1 + ϕ) 4κΦ (1 − σG )2 (ς + σG ) + λς (1 + Φ)
Ω1 ≡(χ + κΦ + ς) λςΩ2≡(1 − σG )2 4κΦσG [ς − (1 − σG ) (1 + ϕ)].
(χ + κΦς) λς

The NKPCs in terms of the welfare-relevant output gap are diﬀerent to theNKPCs in terms of the output gap. Eq.(38) can be rewritten as:
πP,t

=
=
=
=
=

κλ
ỹt
1 − σGκλδEt πP,t+1 +(ŷt + yte − ȳt )1 − σG
¢
κλ
κλ ¡
κλ ³
σG ´
gt
Ω2 −
Ω1 − β̄ at +
δEt πP,t+1 +ŷt +1 − σG
1 − σG
1 − σGλκλδEt πP,t+1 +ŷt + κ (1 + ϕ) Ω3 at + κσG Ω4 gt
1 − σGκλδEt πP,t+1 +ŷt + εt ,(D.46)
1 − σGδEt πP,t+1 +2) (ς+σG )+λς(1+Φ)
1
G )(1+ϕ])with Ω3 ≡ 4κΦ(1−σG(χ+κΦ+ς)ς
and Ω4 ≡ (1−σG )4κΦ[ς−(1−σ− 1−σ.
(χ+κΦ+ς)ς
G
This equality corresponds to Eq.(21) in the text.

54

E

Lagrangian and its FONCs

E.1

Optimal Monetary Policy Alone

The Lagrangian is given by:
(∞ ∙
µ
X
W
δ t LW+μ$ = E01,t ỹt −
t
t=0

βW
βW
W
ỹ W + βW r̂t − βW πt+1−r̂t−1
1 − σG t+1
δ

¶
£
¤
βW W
W
+ μ2,t ỹtR − βR (1 − γ) υnt + βR (1 − γ) nt−1+
πt + βW bt
δ
µ
¶
µ
¶
κλ
κλ
∗
∗
∗
+ μ3,t πP,t − δπP,t+1 −ỹt + μ4,t πP,t − δπP,t+1 −ỹ
1 − σG
1 − σG t
µ
¶¸¾
δ
κϕ
1
+μ5,t nt −,nt+1 +ỹtR −
nt−1
1+δ+κ
1+δ+κ
1+δ+κ

because bt = b∗t = 0 for all t.
The FONCs are as follows:
ΛπβW
πP,t +(μ1,t − μ1,t−1 ) + (μ3,t − μ3,t−1 )
2
2δ
Λπ ∗
βW
π +
(μ1,t − μ1,t−1 ) + (μ4,t − μ4,t−1 )
2 P,t
2δ
Λy
1
λκ
κϕ
ŷt + μ1,t + μ2,t −μ3,t +μ5,t2
2(1 − σG )
1+δ+κ
βW
−
μ1,t−1(1 − σG ) 2δΛy ∗ 1λκκϕỹ + μ1,t − μ2,t −μ4,t −
μ5,t2 t
2(1 − σG )
1+δ+κ
βW
−
μ1,t−1(1 − σG ) 2δ1
−βR (1 − γ) υμ2,t − μ5,t −
μ5,t−1
1+δ+κ
μ1,t

=

0

=

0

=

0

=

0

=

0

=

0

(E.1)

Note that the fifth equality in Eq.(E.1) corresponds to the second equality inEq.(24) in the text. Because of commitment, a lagged Lagrangian multiplierappears.
Combining the first to the fourth and the sixth equalities in Eq.(E.1), wehave:Λπ πtW + (μ3,t − μ3,t−1 ) + (μ4,t − μ4,t−1 ) = 0,
1+ϕ W
κλ
κλ
ŷ −
μ3,t −
μ4,t = 0.
1 − σG t(1 − σG ) 2(1 − σG ) 2

Combining both equalities in Eq.(E.2) yields:
πtW = −¢
Λy (1 − σG ) ¡ W
Wŷt − ŷt−1
,
Λπ κλ
55

(E.2)

which is Eq.(22) in the text.
Combining the first to the fourth and the sixth equalities in Eq.(E.1), we
have:
Λπ R
π + (μ3,t − μ3,t−1 ) − (μ4,t − μ4,t−1 )
2 P,t
Λy R
κλ
κλ
2κϕ
μ3,t +
μ4,t +
ŷt + μ2,t −μ5,t2
1 − σG
1 − σG
1+δ+κ

=

0,

=

0.

(E.3)

Combining both equalities in Eq.(E.3) yields:
R
πP,t

=

¢ 2 (1 − σG )Λy (1 − σG ) ¡ R
R−ŷt − ŷt−1(μ2,t − μ2,t−1 )
Λπ κλ
Λπ κλ
(1 − σG ) 4κϕ
(μ5,t − μ5,t−1 ) ,−Λπ κλ (1 + δ + κ)

−

(E.4)

which is the first equality in Eq.(24) in the text.

E.2

Optimal Monetary and Fiscal Policy

The Lagrangian is given by:
(∞ ∙
µ
X
t
W
δ Lt + μ1,t ỹtW −$ = E0

βW
βW
W
W
ỹt+1
+ βW r̂t − βW πt+1−r̂t−1
1 − σG
δ
t=0
¶
£
βW W
βW W
W
+ μ2,t ỹtR + βR δbR+
π + βW bt −bt − βR (1 − γ) υnt
δ t
δ t−1
µ
¶
¡ ∗
κλ
+βR (1 − γ) nt−1 ] + μ3,t πP,t − δπP,t+1 −ỹt + μ4,t πP,t1 − σG
¶
µ
κλ
δ
κϕ
∗
∗
−δπP,t+1 −ỹ + μ5,t nt −nt+1 +ỹ R
1 − σG t
1+δ+κ
1+δ+κ t
¶¸¾
1
.
−
nt−1
1+δ+κ

The FONCs of the Lagrangian are given by Eq.(E.1) and the following equalities:
βW
μ1,t + βR δμ2,t2
βW
μ1,t − βR δμ2,t
2

=

0,

=

0.

(E.5)

Combining both equalities in Eq.(E.5), we have:
μ2,t = 0.

(E.6)

Substituting Eq.(E.6) into Eq.(E.4), we have:
R
πP,t
=−

¢
Λy (1 − σG ) ¡ R(1 − σG ) 4κϕRŷt − ŷt−1−(μ5,t − μ5,t−1 ) . (E.7)Λπ κλ
Λπ κλ (1 + δ + κ)
56

Substituting Eq.(E.6) and the initial condition μ5,−1 = 0 into the fifth equality in Eq.(E.1), we have:
μ5,t = 0.

(E.8)

Substituting Eq.(E.8) into Eq.(E.7) yields:
R
πP,t
=−

which is Eq.(25) in the text.

F¢Λy (1 − σG ) ¡ R
R,ŷt − ŷt−1Λπ κλDerivation of Social Loss

Using the stable roots obtained by analyzing the determinacy, this section calculates social loss analytically.33 We assume that the model includes the priceshocks that forbid the central bank from being able to stabilize inflation andthe output gap simultaneously.
Similar to Eq.(D.46), we have the NKPC in terms of the welfare-relevant
output gap in country F as follows:∗πP,t= δEt πP,t+1 +κλŷ ∗ + ε∗t ,1 − σG t(F.1)

with ε∗t ≡ κ (1 + ϕ) Ω3 a∗t + κσG Ω4 gt∗ . Combining this equality and Eq.(D.46),
we have:W+
πtW = δEt πt+1κλŷ W + εWt .
1 − σG t∗
W
Note that εW
t = Ω5 γξH,t + Ω5 γξF,t + Ω5 (1 − γ) ξN ,t + Ω5 (1 − γ) ξN ,t − Ω6 ξG,t3
with Ω5 ≡ κ(1+ϕ)Ω
and Ω6 ≡ κσG Ω4 .
2Firstly, we calculate the system of the average block. Substituting Eq.(22)in the text into this equality, we have:WW= Ω7 δ −1 ŷtW − δ −1 ŷt−1+ Ω8 δ −1 εW(F.2)
Et ŷt+1
t ,
³
´
Ω8 κλ
G
π κλ
with Ω7 ≡ 1−σand Ω8 ≡ ΛyΛ(1−σ. Its vector form is givenΛπ κλ 1 + δ + 1−σG
G)
by:
∙
¸
∙ W ¸ ∙¸W
Et ŷt+1ŷtΩ8 δ −1
=M
+
εW
W
t
ŷtW
ŷt−1
0
¸
∙
Ω7 δ −1 −δ −1.
with M ≡1
0
33 See Monacelli[25] and Walsh[34].

57

These roots are the solution to the characteristic equations as follows:Ψ2 − tr (M) + det (M) = 0,
with tr (M) = Ω7 δ −1 = Ψ1 + Ψ2 and det (M) = δ −1 = Ψ1 Ψ2 > 1.
Let us suppose | Ψ1 |< 1. Ψ2 satisfies:Ψ1 < 1 < δ −1 < Ψ2 = δ −1 Ψ−1
1 .
The pair of solutions to the characteristic equation is as follows:
s
Ã
!
Ω7
4δ
1± 1− 2 ,
Ψ1,2 =
2δΩ7
with Ψ1,2 denoting the pair of solutions to the characteristic equation.
Eq.(F.2) can be rewritten as:
¶
µ
Ω8 W
Ω7
1 2 W
1−
L + L ŷt =
ε ,
δ
δ
δ t−1

(F.3)

where L is the lag operator. The coeﬃcient on the LHS in Eq.(F.3) can be
rewritten as:
1−Ω7
1
L + L2 = (1 − Ψ1 L) (1 − Ψ2 L) .
δδ

Substituting this into Eq.(F.3), we have:(1 − Ψ1 L) (1 − Ψ2 L) ŷtW =Because (1 − Ψ2 L)−1 = −P∞

k=1 (Ψ2 L)

−k

Ω8 W
ε .
δ t

(F.4)

, this can be rewritten as:(1 − Ψ1 L) ŷtW = −Ω8 Ψ1 εW
t ,where we use the fact that Ψ2 = δ −1 Ψ−1
1 . Thus, the final form of the solution
is given by:
ŷtW = −Ω8 Ψ1∞X

Ψk1 εW
t−k .

(F.5)

k=0

Secondly, we calculate the system of the relative block. Subtracting its
counterpart in country F from Eq.(D.46), we have:R+
πtR = δEt πt+1κλŷ R + εRt ,1 − σG t∗
R
with εRt = 2Ω5 γξH,t − 2Ω5 γξF,t + 2Ω5 (1 − γ) ξN ,t − 2Ω5 (1 − γ) ξN ,t − Ω6 ξG,t .
Substituting Eq.(25) in the text into this equality and using a similar procedure
to derive Eq.(F.5), we have:ŷtR = −Ω8 Ψ1

∞
X

k=0

58

Ψk1 εR
t−k .

(F.6)

Note that vt = vtW + 12 vtR . Thus, combining Eqs.(F.5) and (F.6), we have:ŷt = −Ω8 Ψ1∞X

Ψk1 εt−k .

(F.7)

k=0

Subtracting Eq.(F.7) with a one period lag from Eq.(F.7), we have:(ŷt − ŷt−1 ) = Ω8 Ψ1 (1 − Ψ1 )

∞
X

k=0

Ψ1k−1 εt−k − Ω8 εt .

(F.8)

The FONCs under the optimal monetary and policy regime imply:
πP,t = −Ω−1
8 (ŷt − ŷt−1 ) .
Combining Eqs.(F.7) and (F.8), we have:
#
"
∞
X
Ψk−1
εt−k − εt .
πP,t = − Ψ1 (1 − Ψ1 )
1

(F.9)

(F.10)

k=0

Note that vt∗ = vtW − 12 vtR . Thus, combining Eqs.(F.5) and (F.6), we have:ŷt∗ = −Ω8 Ψ1∞X

Ψk1 ε∗t−k .

(F.11)

k=0

Subtracting Eq.(F.7) with a one period lag from Eq.(F.7) yields:
∞
X
¡ ∗
¢
∗
ŷt − ŷt−1= Ω8 Ψ1 (1 − Ψ1 )
Ψ1k−1 ε∗t−k − Ω8 ε∗t .

(F.12)

k=0

The FONCs under the optimal monetary and policy regime imply:
¡ ∗
¢
∗
∗
= −Ω−1ŷt − ŷt−1.
πP,t8Combining Eqs.(F.7) and (F.8), we have:
#
"
∞
X
k−1 ∗
∗
∗
Ψ1 εt−k − εt .
πP,t = − Ψ1 (1 − Ψ1 )(F.13)(F.14)

k=0

Eq.(F.7) implies:ŷt2 = (Ω8 Ψ1 )

2

∞
X

2
Ψ2k
1 εt−k .

(F.15)

k=0

Eq.(F.10) implies:22(πP,t − εt ) = Ψ21 (1 − Ψ1 )
59

∞
X

k=0

2(k−1) 2
εt−k .

Ψ1(F.16)

The LHS in Eq.(F.16) can be rewritten as:22(πP,t − εt ) = πP,t
− 2πP,t εt + ε2t .

(F.17)

Multiplying by εt on both sides of Eq.(F.10), we have:
πP,t εt = Ψ1 ε2t ,(F.18)

because the serial correlation of shocks is zero. Combining Eqs.(F.17) and (F.18)
yields:2(πP,t − εt )2 = πP,t
+ (1 − 2Ψ1 ) ε2t .

(F.19)

Combining Eqs.(F.16) and (F.19), we have:2
πP,t= Ψ21 (1 − Ψ1 )2∞
X

2(k−1) 2
εt−k − (1 − 2Ψ1 ) ε2t .

Ψ1

k=0

(F.20)

By using a similar procedure to derive Eqs.(F.15) and (F.20), we have:(ŷt∗ )2=(Ω8 Ψ1 )2

∞
X

k=0

¡ ∗ ¢2πP,t=¡ ∗ ¢2εt−k ,
Ψ2k1

Ψ21 (1 − Ψ1 )2

∞
X

k=0

(F.21)2(k−1) ¡ ∗ ¢2εt−k − (1 − 2Ψ1 ) (ε∗t )2 . (F.22)

Ψ1

Under the self-oriented setting, the Lagrangian for country H is given by:
(∞ ∙
µ
X
βW
βW
t
W
W
δ Lt + μ1,t ỹtW −ỹt+1+ βW r̂t − βW πt+1−
$ = E0
r̂t−1
1
−
σ
δ
G
t=0
¶
£
βW W
βW W
W
+ μ2,t ỹtR + βR δbR+
π + βW bt −bt − βR (1 − γ) υnt
δ t
δ t−1
µ
¶
¡ ∗
κλ
+βR (1 − γ) nt−1 ] + μ3,t πP,t − δπP,t+1 −ỹt + μ4,t πP,t1 − σG
¶
µ
κλ
δ
κϕ
∗
nt+1 +ỹ R−δπP,t+1−ỹ ∗ + μ5,t nt −1 − σG t
1+δ+κ
1+δ+κ t¶¸∙¸
1(1 − σG ) Λy W
(1 − σG ) Λy W
−
ŷt −
ŷt−1
nt−1
+ μ6,t πtW +
1+δ+κ
κλΛπ
κλΛπ
∙
(1 − σG ) Λy R (1 − σG ) Λy R
2 (1 − σG )
R
+
ŷt −
ŷt−1 +
μ2,t
+ μ7,t πP,t
κλΛπ
κλΛπ
Λπ κλ
¸
2 (1 − σG )4 (1 − σG ) κϕ
4 (1 − σG ) κϕ−μ2,t−1 +μ5,t −
μ5,t−1Λπ κλ
Λπ κλ (1 + δ + κ)Λπ κλ (1 + δ + κ)∙¸¾
1
+μ8,t μ5,t − (1 − γ) βR υμ2,t −
μ5,t−1 .
1+δ+κ
60

Because the central bank conducts optimal monetary policy, the FONCs under the optimal monetary policy alone, Eq.(24) in the text, appear in this
Lagrangian as constraints. A similar Lagrangian is given for the government
in country F although L∗t replaces Lt . Note that any exogenous shifters areomitted in this Lagrangian.
∞The government in country H chooses the sequence {πP,t , ŷt , nt , bt }t=0 while© ∗ª∞the government in country F chooses the sequence πP,t , ŷt∗ , nt , b∗t t=0 undercommitment. The FONCs are given by:
βW
βW
1
μ1,t + μ3,t −
μ1,t−1 − μ3,t−1 + μ6,t + μ7,t
2δ
2δ
2
1
κλ
κϕ
βW
Λy ŷt + μ1,t + μ2,t −μ3,t +
μ5,t −
μ1,t−12
1 − σG
1+δ+κ
(1 − σG ) 2δ
(1 − σG ) Λy
(1 − σG ) Λyμ6,t ++μ7,tκλΛπ 2κλΛπ
1−βR (1 − γ) υμ2,t + μ5,t −
μ5,t−11+δ+κ
βW
μ1,t + βR δμ2,t
2
βW
βW
1
∗
Λπ πP,t+μ1,t + μ4,t −
μ1,t−1 − μ4,t−1 + μ6,t − μ7,t
2δ
2δ
2
1
κλ
κϕ
βW
∗
Λy ŷt + μ1,t + μ2,t −μ4,t −
μ5,t −
μ1,t−12
1 − σG
1+δ+κ
(1 − σG ) 2δ
(1 − σG ) Λy
(1 − σG ) Λy+μ7,t
μ6,t −
κλΛπ 2κλΛπβW
μ1,t − βR δμ2,t2
Λπ πP,t +

=

0

=

0

=

0

=

0

=

0

=

0

=

0.

(F.23)
The fourth and the seventh equalities in Eq.(F.23) imply:
μ1,t = 0 ; μ2,t = 0.

(F.24)The third equality in Eq.(F.23) and Eq.(F.24) imply:
μ5,t = 0,(F.25)

given the initial condition μ5,−1 = 0. Substituting Eqs.(F.24) and (F.25) into
the first, the second, the fifth and the sixth equalities in Eq.(F.23) yields:1Λπ πP,t + (μ3,t − μ3,t−1 ) + μ6,t + μ7,t2
κλ(1 − σG ) Λy
(1 − σG ) Λy
Λy ŷt −μ3,t +
μ7,t
μ6,t +
1 − σGκλΛπ 2κλΛπ
1∗Λπ πP,t+ (μ4,t − μ4,t−1 ) + μ6,t − μ7,t
2
κλ
(1
−
σ
(1
−
σG ) Λy)ΛGyΛy ŷt∗ −μ4,t +
μ7,t
μ6,t −
1 − σGκλΛπ 2κλΛπ61=

0 (F.26)=0 (F.27)=0 (F.28)=0 (F.29)

Eqs.(F.26) and (F.28) imply the following:
1
μ6,t
2

=

μ7,t

=

1
1
(μ3,t − μ3,t−1 ) − (μ4,t − μ4,t−1 )
2
2
Λπ R
1
1
− πP,t − (μ3,t − μ3,t−1 ) + (μ4,t − μ4,t−1 )222−Λπ πtW −(F.30)(F.31)

Combining Eq.(F.30) and Eq.(22) in the text, we have:¢ 1
1(1 − σG ) Λy ¡ W
1Wμ6,t =
ŷt − ŷt−1− (μ3,t − μ3,t−1 ) − μ4,t .
2
κλ
2
2

(F.32)

Combining Eqs.(F.27) and (F.29), we have:2¢ (1 − σG ) Λy1
1
(1 − σG ) Λy ¡ WW− (μ3,t − μ3,t−1 ) − (μ4,t − μ4,t−1 ) = −(μ6,t − μ6,t−1 ) .
ŷt − yt−1−22
κλ(κλ)2 Λπ
Substituting this equality into Eq.(F.32) yields:"#2
1 (1 − σG )2 Λy(1 − σG ) Λyμ=μ6,t−1 .
+
6,t
2
2
2
(κλ) Λπ(κλ) ΛπThis equality implies the following:
μ6,t = 0,(F.33)

given the initial condition μ6,−1 = 0.
Substituting Eqs.(F.24) and (F.25) into Eq.(24) in the text, we have:
R
=−
πP,t

(1 − σG ) Λy R (1 − σG ) Λy R
ŷt +ŷt−1 .
κλΛπ
κλΛπ

Combining this equality and Eq.(F.31) yields:
μ7,t =¢ 1(1 − σG ) Λy ¡ R
1R
ŷt − ŷt−1− (μ3,t − μ3,t−1 ) + (μ4,t − μ4,t−1 ) . (F.34)
κλ2
2
2

Combining Eqs.(F.27) and (F.29), we have:¢ (1 − σG )2 Λy1(1 − σG ) Λy ¡ R
1R
ŷt − ŷt−1−(μ7,t − μ7,t−1 ) .
− (μ3,t − μ3,t−1 ) + (μ4,t − μ4,t−1 ) = −222κμ4,t λκλΛπ
Combining this equality and Eq.(F.34), we have:2

μ7,t = −(1 − σG ) Λy(μ7,t − μ7,t−1 ) ,κλΛπwhich implies the following
μ7,t = 0,62(F.35)

given the initial condition μ7,−1 = 0.
Substituting Eqs.(F.33) and (F.35) into Eqs.(F.26)—(F.29) yields:
πP,t=(μ3,t − μ3,t−1 )

=

∗
πP,t

=

(μ4,t − μ4,t−1 )=1(μ3,t − μ3,t−1 )Λπ(1 − σG ) Λy(ŷt − ŷt−1 )κλ1−(μ4,t − μ4,t−1 )Λπ¢(1 − σG ) Λy ¡ ∗∗ŷt − ŷt−1.
κλ−(F.36)

Combining the first and the second equalities in Eq.(F.36) yields:
πP,t = −(1 − σG ) Λy(ŷt − ŷt−1 ) .
κλΛπ(F.37)

Combining the third and the fourth equalities in Eq.(F.36) yields:
∗
πP,t
=−

¢
(1 − σG ) Λy ¡ ∗∗ŷt − ŷt−1.
κλΛπ(F.38)

Combining Eqs.(F.37) and (F.38) yields:
πtW

=

πtR

=

¢
(1 − σG ) Λy ¡ WWŷt − ŷt−1,κλΛπ¢(1 − σG ) Λy ¡ RR−ŷt − ŷt−1
,
κλΛπ
−

which correspond to Eqs.(22) and (25) in the text, respectively. Thus, theoptimality conditions for self-oriented fiscal authorities are the same as the one
under the optimal monetary and fiscal policy regime. This implies that thesocial loss is the same between the optimal monetary and fiscal policy regime
under the cooperative setting and the self-oriented fiscal authorities with optimalmonetary policy.
The definitions of the composite cost push terms imply the following:
ε2t

=

2

=

(ε∗t )

22
22Ω9 ξH,t+ Ω10 ξN,t + Ω11 ξG,t¡¢¡ ∗ ¢22
2∗Ω9 ξF,t+ Ω10 ξN+ Ω11 ξG,t
,
,t
2

2

with Ω9 ≡ [κ (1 + ϕ) Ω3 γ] , Ω10 ≡ [κ (1 + ϕ) Ω3 (1 − γ)] and Ω11 ≡ (κσG Ω4 ) .
Substituting these equalities into Eqs.(F.15), (F.20) to (F.22) yields:ŷt2=(Ω8 Ψ1 )2

∞
X

k=0
2
πP,t

=

Ψ21 (1 − Ψ1 )

¡
¢
2
2
2
Ψ2k
+ Ω10 ξN
Ω9 ξH,t−k1,t−k + Ω11 ξG,t−k

2

∞
X

k=0

2(k−1) ¡Ψ12
2
2+ Ω10 ξN
Ω9 ξH,t−k,t−k + Ω11 ξG,t−k

63

¢

2

(ŷt∗ )

=

¢
¡
2
2
2
− (1 − 2Ψ1 ) Ω9 ξH,t+ Ω10 ξN,t + Ω11 ξG,t
∞
h
X
¡ ∗
¢2
¡ ∗
¢2 i
2
2
Ω
ξ
ξ
(Ω8 Ψ1 )
Ψ2k
ξ
+
Ω
+
Ω
9 F,t−k10111
N ,t−kG,t−k
k=0¡ ∗ ¢2πP,t=Ψ21 (1 − Ψ1 )2

∞
X

k=0

2(k−1)

Ψ1

h
¡ ∗
¢2
¡ ∗
¢2 i
2
Ω9 ξF,t−k+ Ω10 ξN+ Ω11 ξG,t−k,t−k¡ ∗ ¢2¡ ∗ ¢2 i2+ Ω10 ξN+ Ω11 ξG,t
− (1 − 2Ψ1 ) Ω9 ξF,t,th(F.39)

Substituting Eq.(F.39) into Eq.(20) in the text, we have:∙¸Λy Ω8 Ψ1
Ψ1W
+ Λπ (1 − Ψ1 ) {Ω9 [var (ξH,t ) + var (ξF,t )]Lt=
2 (1 − Ψ21 )2¡ ∗ ¢¤¡ ∗ ¢¤ª££+Ω10 var (ξN ,t ) + var ξN+ Ω11 var (ξG,t ) + var ξG,t,,twhere we take the expectation in period zero on both sides. Substituting thisequality into Eq.(19) in the text yields:∙¸Λy Ω8 Ψ1
Ψ1W=
+ Λπ (1 − Ψ1 ) {Ω9 [var (ξH,t ) + var (ξF,t )]L(1 − δ) 2 (1 − Ψ21 )2¡ ∗ ¢¤¡ ∗ ¢¤ª££+Ω10 var (ξN ,t ) + var ξN ,t + Ω11 var (ξG,t ) + var ξG,t. (F.40)
Substituting Eq.(F.39) into Eq.(26) in the text, we have:∙¸Λy Ω8 Ψ1
Ψ1NC=
+ Λπ (1 − Ψ1 ) [Ω9 var (ξH,t ) + Ω10 var (ξN ,t )Lt
2 (1 − Ψ21 )2+Ω11 var (ξG,t )] ,where we take the expectation in period zero on both sides. Substituting thisequality into the definition of the respective loss in country H in the text yields:∙¸Λy Ω8 Ψ1
Ψ1
C
=
(1
−
Ψ
)
[Ω9 var (ξH,t ) + Ω10 var (ξN ,t )
LN
+
Λ
π
1
t
(1 − δ) 2 (1 − Ψ21 )2+Ω11 var (ξG,t )] .
Substituting this equality and its counterpart in country F into the definitionof the union-wide social loss brought about by self-oriented fiscal authorities inboth countries, we have:∙¸Λy Ω8 Ψ1
Ψ1
N CW
L
=
+ Λπ (1 − Ψ1 ) {Ω9 [var (ξH,t ) + var (ξF,t )](1 − δ) 2 (1 − Ψ21 )2¡ ∗ ¢¤¡ ∗ ¢¤ª££+Ω10 var (ξN ,t ) + var ξN ,t + Ω11 var (ξG,t ) + var ξG,t,which implies that LW = LN CW . Furthermore, this equality and Eq.(F.40)
correspond to the equality on page 24 of the text.

64

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67

Table 1: Macroeconomic Volatility in the Case where All Goods are Tradables(γ = 1)
Variable
ŷtW
πtW
ŷt
ŷt∗
πP,t
∗
πP,t

qt
r̂t
bt
b∗t

Regime
OMP
OMFP
OMP
OMFP
OMP
OMFP
OMP
OMFP
OMP
OMFP
OMP
OMFP
OMP
OMFP
OMP
OMFP
OMP
OMFP
OMP
OMFP

aH,t
2.5471e-004
2.5471e-004
1.3760e-005
1.3760e-005
0.00705.0941e-004
0.00650.00000.00802.7784e-005
0.00805.3753e-006
0.00000.00000.01196.8816e-004
0.00000.01350.00000.0088aN ,t0.00000.00000.00000.00000.00000.00130.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000ShocksgtW
0.00130.00136.0145e-005
6.0145e-005
0.00130.00130.00130.00136.0145e-005
6.0145e-005
6.0145e-005
6.0145e-005
0.00000.00000.01780.00160.00000.01620.00000.0162

Notes:
OMP: Optimal monetary policy alone
OMFP: Optimal monetary and fiscal policy

68

gtR
0.00000.00000.00000.00000.00830.00780.00830.00780.0301
3.6616e-004
0.0301
3.6616e-004
0.00000.00000.00000.00000.00000.00880.00000.0088

Table 2: Macroeconomic Volatility in the Benchmark Case (γ = 0.5)
Variable
ŷtW
πtW
ŷt
ŷt∗
πP,t
∗
πP,t

qt
r̂t
bt
b∗t

Regime
OMP
OMFP
OMP
OMFP
OMP
OMFP
OMP
OMFP
OMP
OMFP
OMP
OMFP
OMP
OMFP
OMP
OMFP
OMP
OMFP
OMP
OMFP

aH,t
1.2655e-004
1.2655e-004
6.8366e-006
6.8366e-006
0.00182.5391e-004
0.00168.0096e-007
0.00811.4054e-005
0.00812.4640e-006
0.00355.4188e-006
0.00593.4418e-004
0.00000.00680.00000.0044

Shocks
aN ,tgtW
1.5016e-004
0.00131.5016e-004
0.00137.0097e-006 6.0145e-005
7.0097e-006 6.0145e-005
0.00220.00132.9940e-004
0.00130.00200.00139.2692e-007
0.00130.0121 6.0145e-005
1.4184e-005 6.0145e-005
0.0121 6.0145e-005
2.2170e-006 6.0145e-005
0.00480.00005.8689e-006
0.00000.00740.01783.8585e-004
0.00160.00000.00000.00880.01620.00000.00000.00520.0162

Notes:
OMP: Optimal monetary policy alone
OMFP: Optimal monetary and fiscal policy

69

gtR
0.00000.00000.00000.00000.00470.00780.00470.00780.04083.6616e-004
0.04083.6616e-004
0.01050.01280.00000.00000.00000.00660.00000.0066

Figure 1: IRFs under Optimal Monetary Policy Alone in the Case where All
Goods Are Tradable

70Figure 2: IRFs under Optimal Monetary Policy Alone in the Benchmark Case71

Figure 3: IRFs under Optimal Monetary Policy and Fiscal Policy in the Case
where All Goods Are Tradable

72

Figure 4: IRFs under Optimal Monetary Policy and Fiscal Policy in the Benchmark Case73

Figure 5: Eﬀect on Welfare of Varying Shares of Tradable Goods

74