The role of local currency pricing in the internationaltransmission effects of a government spending shock in an
economy with vertical production linkage and foreign direct
investment
Kohjiro Dohwa
May 2022
Discussion Paper No. 2210

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANThe role of local currency pricing in the internationaltransmission eﬀects of a government spending shock in
an economy with vertical production linkage and foreign
direct investment
Kohjiro Dohwa∗†

Abstract
By constructing a two-country model with asymmetry in price-setting behavior between home and foreign intermediate goods ﬁrms, vertical production and trade, andendogenous entry of three types of ﬁnal goods ﬁrms, this paper examines the eﬀects ofa home government spending shock. In particular, it focuses on the role of asymmetry
in price-setting behavior between home and foreign intermediate goods ﬁrms. A homegovernment spending shock is shown to result in the entry of multinational ﬁrms fromboth countries, an increase in the aggregate outputs of both countries, a deterioration
in home welfare, and an improvement in foreign welfare. In addition, with an increasein the ratio of home and/or foreign intermediate goods ﬁrms setting their export prices
in the local currency, the eﬀects of this shock on the entry of home multinational ﬁrms,
the increase in aggregate foreign output, the deterioration in home welfare and the improvement in foreign welfare are shown to be weakened, while the eﬀects of this shockon the entry of foreign multinational ﬁrms and the increase in aggregate home output
are intensiﬁed.
Keywords: Local currency pricing, Vertical production and trade, Firm entry, Foreign
direct investment, Government spending shock
JEL Classiﬁcation: F41, F42∗

Corresponding address: Faculty of Economics and Business Administration, Kyoto University of Advanced
Science, 18 Gotanda-cho, Yamanouchi, Ukyo-ku, Kyoto 615-8577, JAPAN; Tel: +81-75-406-9329; E-mailaddress: dohwa.kohjiro@kuas.ac.jp†
Research Fellow, Graduate School of Economics, Kobe University.

11

Introduction

Developed countries have been increasing their outward foreign direct investment (hereafter,
outward FDI) over the past 30 years or more. As Fig.1 shows, the U.S. has been increasing its
outward FDI the most rapidly by far, while Japan’s outward FDI has been increasing rapidly
since the 2010s. Meanwhile, as Fig.2 shows, inward foreign direct investment (hereafter,
inward FDI) has also been increasing in developed countries with the exception of Japan. Oneof the purposes of this paper, therefore, is to examine the macroeconomic eﬀects of ﬁrm entry,
including from outward and inward FDI.
[Insert Fig.1 around here][Insert Fig.2 around here]
Moreover, trade structures have been changing over the past few decades. A vertical trade
structure has prevailed in trade between Japan and Asia, for instance. This means that in
the current trade between Japan and China, intermediate goods produced in Japan are exported to China, where they are assembled into ﬁnal goods, which are then sold in China orexported to Japan and the U.S. This type of trade structure is not limited to between Japanand China, but has become the mainstream around the world. Using data from 10 OECDand four emerging economies, Hummels et al. (2001) ﬁnd that it is a feature of today’s global
economy.1 Another feature of current trade is that ﬁrms in many developed countries otherthan the U.S. are setting export prices in local currencies. In other words, they employ local
currency pricing (LCP). This has been discovered by many researchers (see Marston (1990),
Knetter (1993), Parsley (1993), Athukorala and Menon (1994), ECU Institute (1995), Gagnon
and Knetter (1995), and Gopinath and Rigobon (2008), for instance). Data from Japan’s
Ministry of Finance also shows that the ratio of Japanese ﬁrms exporting to the U.S. that set
export prices in local currency (LCP ﬁrms) is high at about 90 percent (2001 Jan-Jun: 87.4
percent, 2020 Jul-Dec: 87.4 percent), while the ratio of Japanese LCP ﬁrms exporting to the
EU is low but increasing year by year with the ratio exceeding 50 percent in 2020 (2001 JanJun: 42.6 percent, 2020 Jul-Dec: 55.8 percent). Further the data shows that the ratio of LCP
ﬁrms exporting from the EU to Japan mirrors the ratio of LCP ﬁrms exporting from Japanto the EU in terms of the trend (2001 Jan-Jun: 48.1 percent, 2020 Jul-Dec: 57.1 percent). On
the other hand, while the ratio of LCP ﬁrms exporting from the U.S. to Japan is low, it has
increased slightly in recent years (2001 Jan-Jun: 20.5 percent, 2020 Jul-Dec: 25.6 percent).
Despite the above characteristics of global goods markets, few researchers have theoretically examined the international transmission eﬀects of government spending by taking into
account all or some of these factors. In many developed countries, government spending has
played a major role in underpinning the economy following the collapse of Lehman Brothers,
when monetary policy was constrained by the zero lower bound. This makes it important toexamine the macroeconomic eﬀects of a government spending shock. In recent years, there
have been studies on the international transmission eﬀects of a government spending shock by
1

Feenstra (1998) and Yi (2003) also make the same point as Hammels et al. (2001).

2

the following researchers. Using the two-country New Keynesian (hereafter, NK) model in an
economy where monetary policy is constrained to the zero lower bound, Cook and Devereux(2011) mainly examine the ﬁscal spillover eﬀects. They show that the eﬀect of a home government spending shock is positive on home GDP, but negative on foreign GDP. Like Cookand Devereux (2011), Fujiwara and Ueda (2013) also examine ﬁscal multiplier and spillover
using the two-country NK model with a global liquidity trap. They show that ﬁscal spillovers
can be both positive and negative, depending on the magnitude of the intertemporal elasticity
of substitution in consumption. Additionally, they examine the ﬁscal spillover in an economy
with LCP, showing that it becomes positive under such a circumstance. Papers by theseresearchers are signiﬁcant in that they provide important insights into the eﬀects of recent
ﬁscal spending on home and foreign countries. However, they do not take into account all thecharacteristics of global goods markets mentioned above, though Fujiwara and Ueda (2013)
do examine scenarios of both LCP and producer currency pricing (PCP), in which ﬁrms set
export prices in their own currency.2
The paper by Johdo and Hashimoto (2005) exists as an open macroeconomic model withinternational ﬁrm relocation. Using a two-country model with ﬂexible prices, they examine
the eﬀects of an increase in the corporate tax rate on consumption and welfare in both countries. However, because their model does not assume sticky prices, it is unable to examine therole of LCP. In addition, they also perform their analyses using a two-country model without vertical trade. By constructing a two-country ﬂexible-price dynamic optimizing modelwith heterogeneity in ﬁrm productivity, Ghironi and Melitz (2005) examine the eﬀects of realshocks to both aggregate productivity and entry costs on the location patterns of ﬁrms. Using
model of Ghironi and Melitz (2005) in part, Corsetti et al. (2007) examine the eﬀects of ahome government spending shock on macroeconomic variables and welfare. By incorporating
free entry conditions into the model, they examine these eﬀects through the entry and exitof ﬁrms. However, despite constructing a framework that allows them to analyze ﬁrm entry
and exit in an open macroeconomic model diﬀerent from the model of Johdo and Hashimoto(2005), they also fail to incorporate the characteristics of recent global goods markets. Similarly, using the model of Ghironi and Melitz (2005) in part, Corsetti et al. (2013) examine theeﬀects of a transfer from home to foreign countries. And again, despite using a model with
useful features in that it deals with the ﬁrm entry of two types of tradable goods ﬁrms andone type of non-tradable goods ﬁrm, they too fail to take into account the characteristics ofrecent global goods markets. In particular, they are unable to address the role of LCP because
their analysis is not based on the NOEM model, which originally assumed sticky prices. Using an Obstfeld and Rogoﬀ (1995, 1996) type new open economy macroeconomics (hereafter,
NOEM) model with LCP and FDI, Cavallari (2010) examines the eﬀects of monetary andproductivity shocks on macroeconomic variables. She shows that endogenous ﬂuctuations inﬁrms, including multinational ﬁrms, amplify spillovers on consumption and employment in thetwo countries. However, she examines this in a model without vertical trade. By incorporating2

Clancy et al. (2016) and Boug et al. (2023) also examine the macroeconomic eﬀects of a government
spending shock occurring in one small open economy. However, they do not take into account the characteristics
of recent global goods markets mentioned above.

3

only the factor of international ﬁrm mobility into the model of Obstfeld and Rogoﬀ (1995,
1996), Johdo (2019) examines the eﬀects of a government spending shock on consumption
and nominal exchange rates. He shows that ﬁrm mobility weakens the eﬀects of this shock on
relative consumption and nominal exchange rates.345 As can be seen from the above studies,
there are many open macroeconomic models that deal with ﬁrm entry, including FDI. However, despite the fact that signiﬁcant FDI has been observed in the real economy, and thatdemand promotion policies may attract inward FDI, few of them deal with the relationshipbetween FDI and a government spending shock.
Studies by the following researchers exist with regard to vertical production and trade inopen macroeconomic models. Using the stochastic two-country NOEM model of the Obstfeld
and Rogoﬀ (1995, 1996) type, Huang and Liu (2006) examine the eﬀects of a home monetaryshock on the welfare of both countries. They show that not only in the case of PCP but also
in the case of LCP, an increase in the stage of production and trade tends to make a homemonetary shock proﬁtable for both home and foreign countries. By incorporating staggered
price-setting mainly into the Huang and Liu (2006) model, Huang and Liu (2007) examineinternational business cycles driven by monetary shocks. They ﬁnd that this model performs
much better than a single-stage model without vertical trade. Shi and Xu (2007) examine
the issue of non-cooperative optimal monetary policy in an economy with vertical production
and trade by incorporating two stages of production and trade into the stochastic two-country
NOEM model. They examine the issue by assuming either PCP or LCP for intermediate goods
ﬁrms. Gong et al. (2016) examine the issue of optimal monetary policy using a two-country
NK model with two stages of production and trade. They examine the issue by assuming
PCP in both intermediate and ﬁnal goods trade. Wei and Xie (2020) examine the issue of
optimal monetary policy using a small open economy NK model with multiple stages of production and trade. They show that a more desirable rule would be to target separate producer
price inﬂation in diﬀerent stages of production. Devereux et al. (2020) examine the eﬀects
of government spending shocks using a multi-country model with international production
networks. They show that network structures play a major role in ﬁscal spillover eﬀects whensticky wages are assumed in the model. Although the above studies other than the one by
Devereux et al. (2020) are associated with the design and eﬀects of monetary policy, they arevery signiﬁcant because they provide insights into the importance of a number of stages of
production and trade. The study by Devereux et al. (2020) is also interesting in that it uses
numerical analysis for a model with a large number of production structures and examines theﬁscal spillover eﬀects on the Eurozone of a government spending shock in Germany with and3This paper refers to the model that adopts a one-period-ahead price setting similar to the Obstfeld andRogoﬀ (1995, 1996) model as the NOEM model with respect to the nominal rigidities. On the other hand,
the model that adopts staggered price setting similar to the Calvo (1983) model is referred to as the New
Keynesian model.
4
Other related studies using the NOEM model include those by Caselli (2001), Lombardo and Sutherland(2004), Pierdzioch (2004), Chu (2005), Ganelli (2003, 2005a, 2005b), and Ganelli and Tervala (2010). However,these researchers use two-country models without ﬁrm entry.
5
A non-exhaustive list of the literature on ﬁrm entry, including outward FDI, is Corsetti et al. (2004),
Lewis (2006), Bilbiie et al. (2007), Russ (2007), Bergin and Corsetti (2008), Cavallari (2013), and Johdo(2013, 2015).

4

without a network structure. Their model points to the importance of using a macroeconomic
model with production linkages across countries to examine the international transmission
eﬀects of a government spending shock. However, None of these studies do not deal with therole of ﬁrm entry.
The purpose of this paper is to examine the eﬀects of a government spending shock on
macroeconomic variables and welfare using a two-country model that takes into account three
features of global goods markets.6 Speciﬁcally, by incorporating the factor of LCP for homeand foreign ﬁrms engaged in intermediate goods trade, this paper focuses its analysis on its
role on the eﬀects of a government spending shock.7 There are many studies by researchers on
the price-setting behavior of ﬁrms. Betts and Devereux (2000) were the ﬁrst to introduce LCP
into the two-country NOEM model to examine the eﬀects of a country’s government spending
shock on the welfare of both countries. However, they assume that the fraction of exporters
who set prices in local currency of sale is symmetric across countries. By incorporating theassumption of LCP adopted by Betts and Devereux (2000) into the model of Obstfeld andRogoﬀ (1995, 1996), Carré and Collard (2003) examine whether Europe should implementmonetary union from the perspective of welfare eﬀects in accordance with a permanent government spending shock in the home country. They show that home households will beneﬁt
more from a transition to monetary union when the degree of LCP rises. By incorporatingﬁrms’ asymmetric price-setting behavior into the model of Corsetti and Pesenti (2001), which
is a typical NOEM model, Michaelis (2006) examines the eﬀects of a home monetary shockon the welfare of both countries. He considers how the eﬀects of a home monetary shock on
the welfare of both countries changes based on the fraction of LCP ﬁrms. By incorporating
asymmetric price-setting behavior by ﬁrms and two stages of production and trade into themodel of Corsetti and Pesenti (2001), Dohwa (2014) examines the eﬀects of a home monetaryshock on the welfare of both countries. He incorporates the former factor for home and foreignﬁrms engaged in trade in intermediate goods. With the exception of Cavallari (2010), none of
the models presented in the previous paragraphs that factor in LCP take into account the existence of both PCP ﬁrms and LCP ﬁrms. They assume that all ﬁrms employ LCP. Devereux
and Yu (2019) examine the ﬁscal spillover eﬀects using a two-country NK model. They do
examine both PCP and LCP scenarios, but in the latter scenario, they assume that all ﬁrms
employ LCP. As can be seen from the above examples, not many studies examine the eﬀects
6

Dohwa (2018) examines the eﬀects of home corporate tax reduction by considering three characteristics
of global goods markets. The model in the current paper is based on that of Dohwa (2018).
7
The NOEM model with the element of LCP was constructed by Betts and Devereux (2000) and Devereux
and Engel (2003) based on the evidence that the law of one price does not hold. The NOEM model with thiselement assumes that prices are sticky in the currency of the destination market. The incorporation of thisassumption results in a reduction in the terms of trade as the price of imports relative to exports is reduced
when the nominal exchange rate depreciates. By contrast, Gopinath et al. (2020) proposed a dominantcurrency paradigm (DCP) approach, based on observations such as the reason the majority of trade is set in a
few dominant currencies (often the US dollar). Under DCP, ﬁrms set export prices in the dominant currency,
which is characterized by the fact that ﬁrms do not change it frequently. In addition, terms of trade are stable
under DCP, which is derived from the pricing of imports and exports in the common currency and the low
sensitivity of these prices to changes in the nominal exchange rate. This result for terms of trade under DCP
can be contrasted with the result for it under LCP.

5

of a government spending shock taking both PCP and LCP ﬁrms into account in their model.
In this paper, we intend to incorporate into our model the existence of two types of tradable
goods ﬁrms and one type of non-tradable goods ﬁrm. In order to do so, taking into account thecharacteristics of recent global goods markets, we extend the two-country ﬂexible price modelof Corsetti et al. (2013) to create a two-country NOEM model. Additionally, as in the studies
by Michaelis (2006) and Dohwa (2014), this paper considers both PCP and LCP ﬁrms andassumes a diﬀerent fraction of these ﬁrms in home and foreign countries. To summarize, thispaper contributes to the literature by constructing a two-country NOEM model that incorporates ﬁrm entry, including FDI, a vertical structure of production and trade, and asymmetricprice-setting behavior between home and foreign ﬁrms engaged in intermediate goods trade,and then showing how it can be used to solve issues that cannot be addressed using moreconventional models, such as the international transmission eﬀects of a government spending
shock in the new framework and the role of LCP in how this shock impacts macroeconomic
variables and welfare.
The main results of this paper are as follows. In an economy with sticky prices, a positive
home government spending shock causes a trade deﬁcit in the home country through higher
imports, which in turn depreciates the nominal exchange rate to balance it. This nominal
exchange rate depreciation aﬀects the proﬁts and entry costs of ﬁnal goods ﬁrms, causing thenumber of these ﬁrms to ﬂuctuate. The number of home-located ﬁnal goods ﬁrms, including
foreign multinational ﬁrms, increases as entry costs increase, but proﬁts increase even more.
The reason for the increase in entry costs for these ﬁrms is that nominal exchange rate depreciation increases the home-currency prices of intermediate goods imported from abroad; andthe reason for the increase in their proﬁts is that nominal exchange rate depreciation causes
a demand shift from foreign ﬁnal goods to home ﬁnal goods, especially among domestic market and export ﬁrms. Similarly, the number of foreign-located ﬁnal goods ﬁrms, excluding
domestic market ﬁrms, also essentially increases. This is because the entry costs for theseﬁrms are lower, and their proﬁts exceed their entry costs. The reason for the lower entry costs
is that nominal exchange rate depreciation lowers the foreign-currency prices of intermediate
goods imported from the home country. The depreciation of the nominal exchange rate andthe increase in the number of ﬁnal goods ﬁrms selling in the foreign market push down theforeign price index and increase overall foreign consumption. This is because, with regard to
the former, a fall in the foreign-currency prices of foreign ﬁnal goods as well as imported homeﬁnal goods contribute to pushing down the foreign price index, while with regard to the latter, an increase in the supply of ﬁnal goods also contributes to pushing down the foreign priceindex. On the other hand, ﬂuctuations in overall home consumption through ﬂuctuations in
the home price index are ambiguous. This is because an increase in the number of ﬁnal goods
ﬁrms selling in the home market contributes to pushing down the home price index throughan increase in the supply of ﬁnal goods, while nominal exchange rate depreciation increases
the home-currency prices of home ﬁnal goods as well as imported foreign ﬁnal goods, thereby
pushing up the home price index. Although the number of ﬁnal goods ﬁrms in the world asa whole will increase, this will increase the demand for intermediate goods in both countries,
thereby causing an increase in employment levels in both home and foreign countries. How-6

ever, home employment increases at least as much as foreign employment if not more. One
reason for this is that the demand for foreign intermediate goods by home ﬁnal goods ﬁrms isessentially much lower in the foreign country than in the home country as a result of nominal
exchange rate depreciation. As regards aggregate outputs, both aggregate home output andaggregate foreign output essentially increase. Aggregate outputs in both countries increase
because a positive home government spending shock has a positive eﬀect on the total numberof ﬁnal goods ﬁrms located in each country, causing an increase in the supply of ﬁnal goodsin each country. Having considered an economy with sticky prices, this paper goes on to examine the persistent eﬀects of a positive home government spending shock on macroeconomic
variables in an economy with ﬂexible prices. This shock has persistent positive eﬀects, mainly
on aggregate outputs and employment levels in both countries. These are based on the fact
that, even in an economy with ﬂexible prices, an increase in the number of ﬁnal goods ﬁrms
increases the supply of goods with regard to aggregate output, and an increase in the numberof ﬁnal goods ﬁrms with regard to employment levels increases the demand for intermediate
goods in both countries. The eﬀects on the welfare of both countries are shown by the utility from consumption and the disutility from employment. Judging from the total of both
economies with sticky prices and ﬂexible prices, the welfare of the home country deteriorates
and that of the foreign country improves. In particular, the deterioration in home welfare isbased on lower utility from consumption and higher disutility from employment.
Finally, the present paper examines the role of LCP on the eﬀects of this shock on short-run
macroeconomic variables and welfare in both countries. We ﬁnd that an increase in the degreeof LCP weakens the degree of the home country’s trade deﬁcit, and therefore the degree of
nominal exchange rate depreciation. This weakening of the nominal exchange rate depreciation contributes to ﬂuctuations in the diﬀerent macroeconomic variables mentioned above.
In particular, an increase in the degree of LCP essentially enhances the new entry of foreign
multinational ﬁrms into the home market and the increase in aggregate home output, while
it essentially weakens the new entry of home multinational ﬁrms into the foreign market andthe increase in aggregate foreign output. The increase in new entries of foreign multinationalﬁrms is based on a weakening of the increase in entry costs due to a weakening of the nominal
exchange rate depreciation, while the weakening of new entry of home multinational ﬁrms
is based on a weakening of the decrease in entry costs due to the same. Further, when the
degree of LCP rises, it also weakens the deterioration in home welfare and the improvement inforeign welfare through its eﬀects on consumption and employment in each country. Overall
consumption and employment in both countries have the following characteristics. First, an
increase in foreign LCP weakens (or intensiﬁes) the decrease (or increase) in overall home consumption, while an increase in home LCP weakens the increase in overall foreign consumption.
The latter results from a weakening of the decrease in foreign CPI based on a weakening of(1) the increase in the supply of ﬁnal goods, (2) the decrease in the foreign-currency prices offoreign ﬁnal goods, and (3) the decrease in the foreign-currency prices of imported home ﬁnal
goods. Second, an increase in home LCP weakens the increase in home employment, while anincrease in foreign LCP intensiﬁes the increase in foreign employment.
The remainder of this paper is structured as follows. Section 2 presents the model. Section

7

3 examines the eﬀects of a home government spending shock on the macroeconomic variablesof both countries. Section 4 examines the eﬀects of a home government spending shock on thewelfare of both countries. The ﬁnal section summarizes the ﬁndings and concludes the paper.

2

The model

The world consists of two countries, one denoted as the home country and the otheras the foreign country. The foreign variables are denoted with an asterisk. Both countries
have the same population size, which is normalized to unity: Home households are deﬁned
over a continuum of unit mass and indexed by x ∈ [0, 1], foreign households by x∗ ∈ [0, 1].
Households are immobile across countries. They consume a composite of diﬀerentiated ﬁnal
goods available in their respective domestic markets. Our assumption regarding vertical tradeis based on that in Shi and Xu (2007). There are two types of ﬁrms in each country: ﬁnal
goods ﬁrms and intermediate goods ﬁrms. Final goods ﬁrms operate in either the tradable or
the non-tradable goods sectors. Final tradable goods are sold in the domestic markets, making
them import-competing goods, or exported. Final non-tradable goods are produced by themultinational ﬁrms in the trading partner. These ﬁrms produce diﬀerentiated ﬁnal goods
using a composite of domestically produced and imported intermediate inputs. On the other
hand, intermediate goods ﬁrms, which are broken down into either domestic market or exportﬁrms, produce diﬀerentiated products using labor. Both ﬁnal goods ﬁrms and intermediate
goods ﬁrms are monopolistically competitive producers. Domestic market and export ﬁrms
among the home-located ﬁnal goods ﬁrms are assumed to exist continuously in the interval[0, nD,t ] and the interval [0, nX,t ], respectively, and that foreign multinational ﬁrms among thehome-located ﬁnal goods ﬁrms exist continuously in the interval [0, n∗M N,t ], where nD,t , nX,t
and n∗M N,t are endogenous.8 There is free entry in the ﬁnal goods sector, but ﬁnal goods ﬁrms
face ﬁxed entry costs to start production of a particular good.9 Home and foreign intermediate
goods are the inputs required for the formulation of entry costs. With regard to the numberof intermediate goods ﬁrms in both countries, although it is assumed that domestic market
and export ﬁrms exist continuously in the intervals [0, 12 ] and the interval [ 21 , 1], respectively,10
it is also assumed that a fraction η of the export ﬁrms located in the home country and afraction η ∗ of the export ﬁrms located in the foreign country set their export prices in thelocal currency, i.e., they employ LCP. The remaining intermediate goods ﬁrms in the export
sector located in both countries set their export prices in their own currency, i.e., they employ
The domestic market ﬁrms among the home-located ﬁnal goods ﬁrms are indexed by zF |D ∈ [0, nD,t ].
Similarly, the export ﬁrms among the home-located ﬁnal goods ﬁrms, and the foreign multinational ﬁrms
among the home-located ﬁnal goods ﬁrms are indexed by zF |X ∈ [0, nX,t ] and zF |M N ∈ [0, n∗M N,t ], respectively.
A similar interpretation holds for zF∗ |D ∈ [0, n∗D,t ], zF∗ |X ∈ [0, n∗X,t ] and zF∗ |M N ∈ [0, nM N,t ].
9
As deﬁned above, although [0, nD,t ], [0, nX,t ] and [0, n∗M N,t ] represent intervals for home-located ﬁnal goods
ﬁrms, they can be also interpreted as intervals for the goods produced by home-located ﬁnal goods ﬁrms. A
similar interpretation holds for [0, n∗D,t ], [0, n∗X,t ] and [0, nM N,t ].
10
The home intermediate goods ﬁrms selling in the domestic and the export markets are indexed by zI|D ∈∗∗∈ [0, 21 ] and zI|X
∈ [ 12 , 1].
[0, 21 ] and zI|X ∈ [ 12 , 1], respectively. A similar interpretation holds for zI|D
8

8

PCP.11 This paper assumes that ﬁnal goods produced in each country are priced identicallyin all the periods to allow for ﬂexible prices. It also assumes that the law of one price does
not always hold for intermediate goods in all the periods.

2.1

Firms

2.1.1

Final goods ﬁrms

Each of the home-located ﬁnal goods ﬁrms uses home and foreign intermediate goods to
produce output according to the following production function:
1

1

Yt (zF |j ) = 2Yh,t (zF |j ) 2 Yf,t (zF |j ) 2 ,
where

ÃZ 1

j = D, X, M N
σ
! σ−1

2

σ−1

Yh,t (zF |j , zI|D ) σ dzI|D

Yh,t (zF |j ) =

(1)

,

(2)

0

ÃZ 1+η∗2

Yf,t (zF |j ) =

1
2

LCP
∗
Yf,t
(zF |j , zI|X
)

σ−1
σ

∗
dzI|X
+

Z 1
1+η ∗
2

σ
! σ−1

P CP
∗
Yf,t
(zF |j , zI|X
)

σ−1
σ

∗
dzI|X

.

(3)

In Eq.(1), Yt (zF |j ) is the output produced by home-located ﬁnal goods ﬁrm zF |j andYh,t (zF |j ) (Yf,t (zF |j )) is a composite of the home (foreign) intermediate inputs used by homelocated ﬁnal goods ﬁrm zF |j . Yh,t (zF |j ) and Yf,t (zF |j ) are given in Eqs.(2) and (3), where
Yh,t (zF |j , zI|D ) is the home intermediate input zI|D used by home-located ﬁnal goods ﬁrm zF |j ,
P CP
∗
LCP
∗
∗
Yf,t
(zF |j , zI|X
) (Yf,t(zF |j , zI|X)) is the foreign PCP (LCP) intermediate input zI|X
used byhome-located ﬁnal goods ﬁrm zF |j , and σ > 1 is the elasticity of substitution between any two
diﬀerentiated intermediate inputs. Based on the production functions deﬁned above, we candeﬁne these price indexes. We can also derive the demand of each of the home- and foreignlocated ﬁnal goods ﬁrms for intermediate inputs required in production activities. These aresummarized in Appendix A in the online appendices.
The resource constraints for each variety of goods produced by the home-located ﬁnal goods
ﬁrms are given as follows:
Z 1
Yt (zF |j 0 ) ≥Ch,t (zF |j 0 , x)dx + Gh,t (zF |j 0 ),
(4)
0

Z 1
Yt (zF |X ) ≥∗(zF |X , x∗ )dx∗ + G∗h,t (zF |X ),
Ch,t

(5)

0

where j 0 = (D, M N ), Ch,t (zF |j 0 , x) is the home household x’s consumption of goods produced∗by the home-located ﬁnal goods ﬁrm zF |j 0 , Ch,t(zF |X , x∗ ) is the foreign household x∗ ’s consumption of goods produced by the home-located ﬁnal goods ﬁrm zF |X , Gh,t (zF |j 0 ) is the home
As deﬁned above, although [0, 12 ] represents the interval for the home and foreign intermediate goods ﬁrms
sold in the domestic market, [0, 12 ] also represents the interval for the inputs produced by home and foreignintermediate goods ﬁrms selling in the domestic market. A similar interpretation holds for [ 12 , 1].
11

9

government’s consumption of goods produced by the home-located ﬁnal goods ﬁrm zF |j 0 andG∗h,t (zF |X ) is the foreign government’s consumption of goods produced by the home-locatedﬁnal goods ﬁrm zF |X . Using Eqs.(4) and (5), the home-located ﬁnal goods ﬁrms’ proﬁts are
given as follows:ΠF,t (zF |j 0 ) = (ph,t (zF |j 0 ) − Λt )Yt (zF |j 0 ),(6)
ΠF,t (zF |X ) = (εt p∗h,t (zF |X ) − Λt )Yt (zF |X ),(7)

where ph,t (zF |j 0 ) (p∗h,t (zF |X )) is the home (foreign) price, which corresponds to Yt (zF |j 0 ) (Yt (zF |X )),Λt is the unit cost to produce home ﬁnal goods, and εt is the nominal exchange rate, deﬁned
as the home-currency price of the foreign currency.
To start production, each of the ﬁnal goods ﬁrms must pay a ﬁxed cost. The entry cost foreach class of ﬁnal goods ﬁrms is assumed to be given in the following form:³´
qt (zF |j 00 ) = P̃h,t + P̃f,t nγj00 ,t ,(8)³´
qt (zF |M N ) = P̃h,t + P̃f,t n∗γ
M N,t ,³´
∗ ∗∗∗qt (zF |j 00 ) = P̃h,t + P̃f,t n∗γ
j 00 ,t ,´³∗∗
nγM N,t ,qt∗ (zF∗ |M N ) = P̃h,t+ P̃f,t

(9)
(10)
(11)

where j 00 = (D, X), P̃h,t (P̃f,t ) is the home price index that corresponds to Yh,t (Yf,t ), which is
a composite of the inputs produced by domestic market (export) ﬁrms in the home (foreign)intermediate goods sector, and γ > 0 is a measure of the concavity of the cost function. For
example, Eq.(8) shows that each of the ﬁrms that belongs to the home-located ﬁnal goods
tradable sector requires both nγj00 ,t units of the composite of home intermediate inputs and nγj00 ,t
units of that of foreign intermediate inputs to create a new ﬁnal good. Based on Eqs.(8)−(11),
we can represent the resource constraints in home and foreign intermediate inputs used byhome and foreign ﬁnal goods ﬁrms. These are also summarized in Appendix A in the online
appendices.
2.1.2Intermediate goods ﬁrms

As shown in more detail below, the three types of home intermediate goods ﬁrms produce
diﬀerentiated goods using a continuum of labor inputs provided by the home households:µZ 1
Yh,t (zI|D ) =`t (zI|D , x)

ξ−1
ξ

ξ
¶ ξ−1
dx
,

0

∗LCP
∗P CP
(zI|X ) =(zI|X ) = Yh,t
Yh,t

µZ 1
`t (zI|X , x)

ξ−1
ξ

ξ
¶ ξ−1

dx

,

0

where Yh,t (zI|D ) is the output of goods produced by home intermediate goods ﬁrm zI|D to∗P CP∗LCP
ward three types of home-located ﬁnal goods ﬁrms, Yh,t(zI|X ) (Yh,t(zI|X )) is the output

10

of goods produced by home PCP (LCP) intermediate goods ﬁrm zI|X toward three types offoreign-located ﬁnal goods ﬁrms, `t (zI|D , x) (`t (zI|X , x)) is the labor of home household x employed by home intermediate goods ﬁrm zI|D (zI|X ), and ξ > 1 is the elasticity of substitution
among labor varieties. First, the proﬁt of a home intermediate goods ﬁrm zI|D is given as
follows:
ΠI,t (zI|D ) = (p̃h,t (zI|D ) − Wt )Yh,t (zI|D ),where p̃h,t (zI|D ) is the home price, which corresponds to Yh,t (zI|D ), and Wt is the aggregate
wage index (shown below). Assuming that nominal wages are ﬂexible, and given the demand
function expressed in Eq.(A.10), the optimal price is determined as follows:12
p̃h,t (zI|D ) =σ
Wt ≡ p̃h,t .
σ−1

(12)

Eq.(12) shows that the home intermediate goods ﬁrm zI|D sets its product’s price at themarginal cost (Wt ) multiplied by the mark-up ratio (σ/(σ − 1)).
Next, the proﬁts of a home PCP intermediate goods ﬁrm zI|X and a home LCP intermediate
goods ﬁrm zI|X are given as follows:∗P CP
ΠPI,tCP (zI|X ) = (p̃Ph,tCP (zI|X ) − Wt )Yh,t(zI|X ),
∗LCP
∗LCP
ΠLCP
(zI|X ) − Wt )Yh,t(zI|X ),I,t (zI|X ) = (εt p̃h,t∗P CP
where p̃Ph,tCP (zI|X ) (p̃∗LCP(zI|X )) is the home (foreign) price, which corresponds to Yh,t(zI|X )h,t
∗LCP
(Yh,t (zI|X )).
As per the process of analysis adopted for proﬁt maximization of the home intermediate
goods ﬁrm zI|D , the sales prices of these ﬁrms can be expressed in the following equation whennominal wages are ﬂexible:σp̃Ph,tCP (zI|X ) = εt p̃∗LCP(zI|X ) =Wt ≡ p̃h,t .
(13)
h,t
σ−1

Eq.(13) shows that the sales price of the PCP intermediate goods ﬁrm zI|X is equal to thatof the LCP intermediate goods ﬁrm zI|X . Therefore, even if intermediate goods ﬁrms set theirexport prices in diﬀerent currencies, the law of one price holds for every intermediate goodunder ﬂexible wages.
On the other hand, as mentioned in Section 3, our model takes into account nominal wage
rigidity in the short run. Under sticky prices, which are established on the assumption of sticky
wages, the law of one price does not hold for the inputs produced by LCP intermediate goods
ﬁrms. This is because LCP intermediate goods ﬁrms do not pass on changes in exchange rates
to export prices denominated in the local currency. Focusing on a symmetric equilibrium, as
shown in Corsetti and Pesenti (2005), the prices of the intermediate inputs sold in the export
market, taking into account the incomplete pass through of the nominal exchange rate, are as
follows:∗
P̃ˆh,t∗∗,(14)
P̃h,t = 1−η , P̃f,t = εt1−η P̃ˆf,tεt
12

For Eq.(A.10), see Appendix A in the online appendices.

11

³
´
³
´
∗
∗
where P̃ˆh,t P̃ˆf,t
is the predetermined component of P̃h,tP̃f,t , which is the foreign (home)currency price that corresponds to a composite of the inputs produced by home (foreign)export ﬁrms in the intermediate goods sector.

2.2Households and government

We deﬁne the utility function for the home household x as follows:
Ut (x) =

∞
X

µ
β

s−t

s=t

¶
Ms (x)− κ`s (x) ,lnCs (x) + χln
Ps

(15)

where β ∈ (0, 1) is the subjective discount factor, C(x) is the aggregate consumption index
of the home household x, M (x) is the home household x’s holdings of the home country’s
currency, P is the price, which corresponds to C(x) (CPI), `(x) is the home household x’slabor service, and the other Greek letters are positive parameters. This utility function implies
that the home household x gains utility by consuming ﬁnal goods and holding real money, andsuﬀers disutility by supplying labor13 . The aggregate consumption index of home household
x is given by:
1−δ
δ
CT,t
(x)CN,t(x)
Ct (x) = δ,(16)δ (1 − δ)1−δ
where CT,t (x) and CN,t (x) are tradable and non-tradable composites, respectively, of diﬀerentiated ﬁnal goods consumed by the home household x, and δ ∈ (0, 1) is the share of tradablesin the ﬁnal goods consumed by the home household x. These variables are given by:³

1ρCT,t (x) = (nD,t ) (Ch,t (x))

ρ−1
ρ

1
+ (n∗X,t ) ρ (Cf,t (x))

ρ−1
ρ

ρ
´ ρ−1

,

(17)

θ
´ θ−1
³
θ−1
1
,
CN,t (x) = (n∗M N,t ) θ Ch|f 0 sM N,t (x) θ

where

µ
Ch,t (x) =(nD,t )

− θ1

Z nD,t
Ch,t (zF |D , x)

θ−1
θ

(18)

θ
¶ θ−1

dzF |D

,

(19)

,

(20)

0

µ
1
(n∗X,t )− θCf,t (x) =Z n∗X,t0µCh|f 0 sM N,t (x) =− θ1(n∗M N,t )θ−1
Cf,t (zF∗ |X , x) θ dzF∗ |X

Z n∗M N,t
Ch,t (zF |M N , x)

θ−1
θ

θ
¶ θ−1

θ
¶ θ−1

dzF |M N

.

(21)

0
13

The utility function in this paper assumes a form that takes a linear disutility from working. By assuming
this form, home and foreign wages are uniquely determined based on a ﬁrst-order necessary condition onlabor supply. This makes it possible to explain the eﬀects of a positive home government spending shock on
macroeconomic variables from an analytical investigation perspective.

12

In Eqs.(17) and (18), Ch,t (x) (Cf,t (x)) and Ch|f 0 sM N,t (x) are Dixit and Stiglitz (1977) typeindexes of consumption by home household x across the tradables produced in the home(foreign) country and non-tradables produced in the home country, respectively. ρ > 0 is
the elasticity of substitution between the ﬁnal tradable goods produced in the home andforeign countries and θ > 1 is the elasticity of substitution between any two diﬀerentiatedﬁnal goods. In Eqs.(19)−(21), Ch,t (zF |D , x) is the consumption of the home ﬁnal good zF |D
by home household x, Cf,t (zF∗ |X , x) is the consumption of the foreign ﬁnal good zF∗ |X by home
household x, and Ch,t (zF |M N , x) is the consumption of the home ﬁnal good zF |M N by home
household x. We can deﬁne the CPI. In addition, we can also derive the demand function
for each household in the home country. These are summarized in Appendix B in the online
appendices.
The home household x maximizes utility subject to the following budget constraint:∗(x) Mt (x)Bt+1 (x) εt Bt+1
It (x) Tt (x)+++ Ct (x) +
+
Pt
Pt
Pt
Pt
Pt

=

(1 + it )Bt (x) εt (1 + i∗t )Bt∗ (x) Mt−1 (x) wt (x)`t (x) ΠF,t (x) ΠI,t (x)
+
+
+
+
+
,
Pt
Pt
Pt
Pt
Pt
Pt

(22)

where Bt (x) (Bt∗ (x)) is the stock of home (foreign) currency denominated bonds that the home
household x holds at the beginning of period t, It (x) is the home household x’s ‘investment’in ﬁnal goods ﬁrms (ﬁnancing entry costs), it (i∗t ) is the nominal interest rate between periodst − 1 and t evaluated in home (foreign) currency terms, wt (x) is the nominal wage, whichcorresponds to `t (x), Tt (x) is nominal taxes to the home government, and ΠF,t (x) and ΠI,t (x)are dividend revenues from the ﬁnal and intermediate goods ﬁrms, respectively, that the home
household x owns.
As mentioned in Corsetti et al. (2004, 2013), this paper assumes that households areendowed with a well-diversiﬁed international portfolio of claims on ﬁnal goods ﬁrms’ proﬁts,
so that they ﬁnance the same fraction of the cost of creating new ﬁnal goods in each country.
Thus, the investment of the home household x in a diversiﬁed portfolio of ﬁnal goods ﬁrms isdeﬁned as follows:Ã
Z n∗M N,t
X Z nj00 ,t1
qt (zF |j 00 )dzF |j 00 +qt (zF |M N )dzF |M N
It (x) =2 j 00 =D,X 00Ã+εt

X Z n∗j00 ,t
j 00 =D,X

0qt∗ (zF∗ |j 00 )dzF∗ |j 00 +Z nM N,t0!!
qt∗ (zF∗ |M N )dzF∗ |M N.

We assume that, in return, each of the home households receives an equal share of the
proﬁts of all ﬁnal goods ﬁrms located in the home and foreign countries:Ã
Z n∗M N,t
X Z nj00 ,t
1
ΠF,t (x) =ΠF,t (zF |j 00 )dzF |j 00 +
ΠF,t (zF |M N )dzF |M N
2 j 00 =D,X 0
0

13

Ã
+εt

X Z n∗j00 ,t
j 00 =D,X

0Π∗F,t (zF∗ |j 00 )dzF∗ |j 00 +Z nM N,t0!!
Π∗F,t (zF∗ |M N )dzF∗ |M N.

In addition, the household is a monopolistic supplier of a diﬀerentiated labor service andfaces the following labor-demand curve:!
¶−ξ Ã Z 1
µ
Z 1
Z 1+η
2
2
wt (x)∗LCP∗P CP
Yh,t (zI|D )dzI|D +Yh,t(zI|X )dzI|X +Yh,t(zI|X )dzI|X ,`t (x) =
1+η
1
Wt
0
2
2
(23)
1
´ 1−ξ
³R
1
where Wt = 0 wt (x)1−ξ dx
is the constant-elasticity-of-substitution (CES) wage index.
We now turn to the intertemporal maximization problem. Subject to Eq.(22), the home
household x maximizes Eq.(15). Thus, the ﬁrst-order necessary conditions for Ct (x), Mt (x)and `t (x) are derived as follows:
Ct+1 (x)Pt= β(1 + it+1 ),Ct (x)
Pt+1(24)

Mt (x)
1 + it+1=χ
Ct (x),
Pt
it+1

(25)

wt (x)ξκ=Ct (x).
Pt
ξ−1

(26)

Eq.(24) is the Euler equation, Eq.(25) is the real money demand function, and Eq.(26) showsthat the real wage rate is equal to a constant markup over the marginal rate of substitution
between consumption and leisure.
From now, we denote the ﬁrst-order necessary conditions for home households as a whole.
For example, we deﬁne the aggregate consumption of home households in period t as theintegral of Ct (x) over all x. We denote such a variable as Ct . We also deﬁne Mt , BH,t and BF,tin analogous ways for money holdings and two types of bond holdings, respectively. Then, by
focusing on symmetric equilibrium, where all home households are identical within the homecountry, we can derive the following relationships for all t:
Ct = Ct (x),Mt = Mt (x),BH,t = Bt (x),BF,t = Bt∗ (x).

(27)

Taking into account Eqs.(24)−(27) and assuming a symmetric equilibrium, the ﬁrst-ordernecessary conditions for Ct (x), Mt (x) and `t (x) are corrected as follows, respectively:Pt
Ct+1= β(1 + it+1 )
,
Ct
Pt+1
Mt
1 + it+1
=χ
Ct ,
Pt
it+1
ξκ
Wt
=
Ct .
Pt
ξ−1

14

(28)
(29)
(30)

The budget constraint for the home government can be given as follows:
Mt − Mt−1 Tt+= Gt .
Pt
Pt
Here, since the governments are assumed to purchase goods from home- and foreign-locatedﬁnal goods ﬁrms, Gt , the aggregate consumption index of the home government, is given by:
Gt =GδT,t G1−δN,t,δ δ (1 − δ)1−δ

where GT,t and GN,t are tradable and non-tradable composites of diﬀerentiated ﬁnal goods,
respectively, that are consumed by the home government. These variables are given by:14µGT,t =(nD,t ) Gh,t³

GN,t =
where(nD,t )

ρ−11
+ (n∗X,t ) ρ Gf,tρθ−11(n∗M N,t ) θ Gh|fθ 0 sM N,tµ
Gh,t =

ρ−1
ρ

1
ρ

− θ1

Z nD,t
Gh,t (zF |D )

ρ
¶ ρ−1

,

θ
´ θ−1

θ−1
θ

,
θ
¶ θ−1

dzF |D

,

0

µ
− θ1

(n∗X,t )Gf,t =Z n∗X,t0µGh|f 0 sM N,t =1(n∗M N,t )− θ

Gf,t (zF∗ |X )θ−1
θdzF∗ |X

Z n∗M N,t
Gh,t (zF |M N )

θ−1
θ

θ
¶ θ−1

,
θ
¶ θ−1

dzF |M N.

0

We make the following assumption about consumption substitutability. To begin with, we
assume that the elasticity of substitution among the ﬁnal goods produced in one country is
likely to be higher than the elasticity of substitution across the ﬁnal tradable goods produced
in diﬀerent countries, namely ρ ≤ θ. We also assume that ﬁnal tradable goods produced
in diﬀerent countries are close substitutes, namely, ρ > 1. Therefore, we get the following
magnitude relationship for consumption substitutability: 1 < ρ ≤ θ.
Here, although our analytical purpose is to examine the eﬀects of an unanticipated permanent government spending shock of the home country, in our model, it is analyticallyconvenient to introduce a measure of monetary stance µt ≡ Pt Ct .15 Using this measure, we
14

See Appendix C in the online appendices for public demand for goods produced by the three types of ﬁnal
goods ﬁrms selling in the home country.
15
Our deﬁnition of the variables of monetary policy is based on those in Corsetti and Pesenti (2005) andCorsetti and Dedola (2005). In this deﬁnition, the government controls nominal consumption. Further, as∗∗
mentioned in footnotes 20 and 21, we use the relationship BH,t+1 = BH,t+1= BF,t+1 = BF,t+1= BH,t =∗∗BH,t = BF,t = BF,t = 0. Therefore, a temporary home monetary easing at period t, associated with a higherµt , leads to a lower it+1 .

15

can rewrite Eqs.(28) and (29) as follows:11= β(1 + it+1 )
,
µt
µt+1

Mt = χ

1 + it+1µt .
it+1

Now, we deﬁne the equilibrium conditions for the asset market. The worldwide net supply
of bonds has to be equal to zero. Therefore, the equilibrium conditions for the asset market
are given as follows:∗∗= 0.
= 0, BF,t + BF,t
BH,t + BH,t

2.3

Final goods prices, price indexes of ﬁnal goods and CPIs

This subsection derives the prices of ﬁnal goods and uses them to rewrite price indexes ofﬁnal goods and CPIs. Using Eqs.(4), (6), (B.23) and (C.1), we can easily derive the optimal
price for the goods sold by domestic market ﬁrm zF |D , which belongs to the home-locatedtradable sector ﬁnal goods ﬁrms as follows:16
ph,t (zF |D ) =θΛt ≡ ph,t .
θ−1

(31)

Similarly, we can also derive the optimal prices for the goods sold by export ﬁrm zF∗ |X , which
belongs to the foreign-located tradable sector ﬁnal goods ﬁrms, and foreign multinational ﬁrm
zF |M N , which belongs to the home-located non-tradable sector ﬁnal goods ﬁrms as follows:
pf,t (zF∗ |X )εtθΛ∗t ≡ p∗f,t ,
θ−1

=

(32)

θ
Λt ≡ ph,t .
(33)
θ−1With regard to the ﬁnal goods ﬁrms selling in the foreign market, we can derive the optimal
prices for the goods of these ﬁrms as follows:
ph,t (zF |M N ) =p∗f,t (zF∗ |D ) = p∗f,t (zF∗ |M N ) =θΛ∗t ≡ p∗f,t ,
θ−1

(34)

θ
Λt ≡ ph,t .
(35)
θ−1Here, using Eqs.(12)−(14) and (A.5), the unit costs of producing home and foreign ﬁnal
εt p∗h,t (zF |X ) =goods, which are given in Eqs.(A.1) and (A.2), can be represented as Λt = 2
1
σ−1

η−1
2

1
σ−1

1−η ∗

σ
ε 2
σ−1 t

Wt

σ
and Λ∗t = 2
ε Wt∗ .17 Therefore, from these two expressions and Eqs.(31)−(35), ph,tσ−1 t∗and pf,t can be rewritten as follows:
1

ph,t = 2 σ−1
16
17

1−η ∗
θ
σ
εt 2 Wt ,θ−1σ−1For Eqs.(B.23) and (C.1), see Appendix B and Appendix C in the online appedices, respectively.
For Eqs.(A.1), (A.2), and (A.5), see Appendix A in the online appendices.

16

η−1
θ
σ
εt 2 Wt∗ .
θ−1σ−1
Also from Eqs.(B.3)−(B.6), PT,t and PN,t , which correspond to the price indexes for tradable
and non-tradable composites of diﬀerentiated ﬁnal goods consumed by home households and∗∗PT,tand PN,t, which correspond to the price indexes for tradable and non-tradable compositesof diﬀerentiated ﬁnal goods consumed by foreign households are given as:18
1

p∗f,t = 2 σ−1

1∗

PT,t = ph,t Φt1−ρ ,∗1PN,t = ph,t nM1−θ
N,t ,

1

(36)

1

∗
PT,t
= p∗f,t Φt 1−ρ ,

1−θ
∗
PN,t
= p∗f,t nM
N,t ,

(37)

where Φt ≡ nD,t + n∗X,t T OTt1−ρ and Φ∗t ≡ n∗D,t + nX,t T OTtρ−1 .19
From Eqs.(36) and (37), the CPIs of both countries are given as:
δ

∗ 1−δ

1−ρ
Pt = ph,t nM1−θ
,N,t Φt
1−δ

∗

δ

1−ρ
1−θ
Pt∗ = p∗f,t nM.
N,t Φt2.4

Free entry and the balance of tradeThis subsection mainly presents the conditions that hold under a situation of free entry
and the balance of payments of the home country. To begin with, using Eqs.(4)−(7), (31),(33) and (35), the proﬁts earned by the home-located ﬁnal goods ﬁrms can be given as follows:ph,t Yt (zF |D )
δ µt
ph,t Gt
ΠF,t (zF |D ) =(38)=  + 1−δ ρ−(1−δ)  ,
∗ θ−1
θ
θ Φt
ρ−1
nM N,t Φtph,t Yt (zF |X )
δ εt µ∗ T OT ρ−1
ΠF,t (zF |X ) ==  t ∗ t
+
θ
θ
Φt

ΠF,t (zF |M N ) =T OTtρ ph,t G∗t∗nM N,t Φt1−δθ−1

ph,t Yt (zF |M N )1 − δ  µt=+θθn∗M N,tρ−(1−δ)ρ−1ph,t Gt
δ
∗ θ−δ
ρ−1
nMθ−1
Φ
N,t t


,

(39)


.

(40)

Similarly, the proﬁts earned by the foreign-located ﬁnal goods ﬁrms can be given as follows:


∗
∗ ∗
∗
∗
∗
pf,t Yt (zF |D )pf,t Gt
δ µ,=  t∗ + 1−δ(41)Π∗F,t (zF∗ |D ) =ρ−(1−δ)
∗ ρ−1
θ
θ Φt
θ−1
nM N,t Φt18

For Eqs.(B.3)−(B.6), see Appendix B in the online appendices.

19

The terms of trade under trade in ﬁnal goods are deﬁned as T OTt ≡

17

εt p∗
f,t
ph,t .

Π∗F,t (zF∗ |X ) =p∗f,t Yt∗ (zF∗ |X )θΠ∗F,t (zF∗ |M N ) =

³ ´
=

δ
θ

p∗f,t Yt∗ (zF∗ |M N )
θ

µt
εt

T OTt1−ρ
Φt


=

+

T OTt−ρ p∗f,t Gtρ−(1−δ)
ρ−1

∗ 1−δ
nMθ−1
N,t Φtp∗f,t G∗t

µ∗t

1−δ 
+
θ
nM N,t∗ δ
nM N,t Φt ρ−1
θ−δ
θ−1


,

(42)


.

(43)

Other things being equal, Eqs.(38)−(43) show that a higher number of ﬁnal goods ﬁrms ina sector reduces the proﬁts of each ﬁnal goods ﬁrm in that sector.
With free entry, optimal investment in new ﬁnal goods of the home country implies thatthe value of a home-located ﬁnal goods ﬁrm is equal to the cost of creating a home ﬁnal good,and in equilibrium this must be equal to the value of the proﬁts of a home-located ﬁnal goodsﬁrm. Therefore, the following relationships are derived:qt (zF |D ) = (P̃h,t + P̃f,t )nγD,t = ΠF,t (zF |D ),(44)

qt (zF |X ) = (P̃h,t + P̃f,t )nγX,t = ΠF,t (zF |X ),(45)

qt (zF |M N ) = (P̃h,t + P̃f,t )n∗γ
M N,t = ΠF,t (zF |M N ).

(46)

We deﬁne these relationships as the free entry conditions of the home country. Similarly,
the free entry conditions of the foreign country can be given as follows:∗∗∗∗qt∗ (zF∗ |D ) = (P̃h,t+ P̃f,t)n∗γD,t = ΠF,t (zF |D ),

(47)

∗
∗
∗
∗
qt∗ (zF∗ |X ) = (P̃h,t+ P̃f,t)n∗γX,t = ΠF,t (zF |X ),

(48)

∗
∗
qt∗ (zF∗ |M N ) = (P̃h,t+ P̃f,t)nγM N,t = Π∗F,t (zF∗ |M N ).

(49)

Next, aggregating the budget constraints of households in the home country and using thegovernment budget constraint and the relationship BH,t+1 = BH,t = BF,t+1 = BF,t = 0, thebalance of trade of the home country can be given as follows:20


1−ρ
ρ−1
ρ
∗
∗
∗
nX,t T OTt
nX,t G ph,t T OT
nX,t (εt µt )T OTt
h,t Gt δµt + δp
= 0.
(50)
δ
+ δ 1−δ t ρ−(1−δ)t −
δ
1−δ
∗
∗ θ−1
∗ ρ−1
Φt
Φt
ρ−1
θ−1
nM N,t ΦtnM N,t Φt
On the left-hand side of Eq.(50), the ﬁrst and second terms represent home exports, while
the third and fourth terms represent home imports.

20

With regard to the relationship BH,t+1 = BH,t = BF,t+1 = BF,t = 0, refer to footnote 21.

18

3Transmission mechanism in an economy without trade
in international bonds

This section examines the eﬀects of an unanticipated permanent government spending
shock in the home country.21 We distinguish between three periods. In the initial period, theeconomy is in a symmetric steady state where no country has any net claims on the other.
In period t, a home government spending shock occurs, and a short-run equilibrium is seen,
which assumes that nominal wages are ﬁxed, before the shock can be observed. In the longrun (from period t + 1 onward), nominal wages are determined based on Eq.(30) and its
foreign analog, and all variables reach their new steady-state values. Variables in the initialsteady-state are hereafter represented without a time subscript. Variables in the short-run
and long-run equilibriums are represented in terms of deviations³from the initial steady-state
´
dXt+1
b
dXt
∗
∗
∗
∗
b
with G = G = 0, BH = BH = BF = BF = 0 and µ = µ = 1 X ≡ X , X ≡ X .22 In
this steady state, our model has the solutions such as nD = nX = n∗D = n∗X , n∗M N = nM N ,
2σ−1
σ
∗1+γ
` = `∗ = (2 σ−1 θ/δ)n1+γ= (2 σ−1 θ/(1 − δ))nM
D
N and ε = 1 (see Appendix D in the online
appendices).
This section examines both the short-run and long-run eﬀects of the shock. The short-run
eﬀects are examined, in particular, by focusing on the degree of LCP, while the long-run eﬀects
are examined by focusing on the persistence of the shock.23

3.1

The relationships between various macroeconomic variables inthe short-run and long-run equilibriums

The next subsection will examine the eﬀects of a home government spending shock on themacroeconomic variables. The eﬀects will be examined especially by focusing on the degreeof LCP and the diﬀerence between the short-run and long-run equilibriums.
Before the above analysis, however, this subsection ﬁrst linearizes each of Eqs.(44)−(50)and considers the relationships between various macroeconomic variables. From Eqs.(44) and(45), we obtain the following equations:• The short-run equilibrium11c(ρ − 1)(η + η ∗ )b − q(z∗\nc
εb + G
D = − nX +F |D ),
2
2
4

(51)

21

This paper focuses on analytical investigation as much as possible. Therefore, the eﬀects of a positive
home government spending shock are examined by ruling out trade in international bonds. This assumption
is based on Corsetti et al. (2004) and Cavallari (2010).
¡¢
22
b = dGt
With regard to home government spending, its short-run and long-run deviations are denoted by GC³´b = dGt+1 , respectively, since its initial value is zero.
and G
C

23

This paper assumes that the nominal money balances are always constant at the initial steady state level
and that, therefore, unanticipated permanent government expenditures in the home country are ﬁnanced bypermanent tax increases.

19

1
1c
(ρ − 1)(η + η ∗ ) + 4∗\nc
εb − q(zX = − nD +F |X ),
2
2
4

(52)

• The long-run equilibrium
1c
1 ∗
\
b
nD = − nc(53)
X + G − q(zF |D ),
2
2
1c
1 ∗ b
\
nX = − nc(54)
D + ε − q(zF |X ).
22
Eqs.(51) and (52) have the following characteristics. To begin with, when the number³ ´∗∗of export (domestic market) ﬁrms among foreign-located ﬁnal goods ﬁrms, i.e., ncncX
D ,increases, the home (foreign) household’s consumption of ﬁnal goods produced in the homecountry decreases, which causes a decrease in the sales revenues of domestic market (export)ﬁrms among home-located ﬁnal goods ﬁrms, thereby decreasing proﬁts for these domestic
market (export) ﬁrms. Consequently, it leads to their exit from the home market. These
relationships are also shown in the long-run equilibrium (see Eqs.(53) and (54)). Next, when
the nominal exchange rate depreciates (bε > 0), it increases the sales revenues of domestic
market and export ﬁrms among home-located ﬁnal goods ﬁrms. The increase in the sales
revenues of domestic market ﬁrms among home-located ﬁnal goods ﬁrms is solely due toa deterioration of the home country’s terms of trade caused by the nominal exchange rate
depreciation. The increase in the sales revenues of export ﬁrms among home-located ﬁnal
goods ﬁrms is due to the deterioration of the home country’s terms of trade caused by thenominal exchange rate depreciation as well as an increase in sales revenues in terms of thehome country’s currency due to the nominal exchange rate depreciation. This increase in thesales revenues of domestic market and export ﬁrms among home-located ﬁnal goods ﬁrms
causes an increase in the proﬁts of these two types of ﬁrms. Consequently, there is an increasein the number of these two types of ﬁrms. Additionally, from the perspective of LCP in bothcountries, we can show that these results have the following properties. When the value of η
and/or η ∗ rises, the increase in the number of domestic market and export ﬁrms among homelocated ﬁnal goods ﬁrms gets steeper. This is because a rise in η and/or η ∗ causes a steeper
increase in proﬁts for these two types of ﬁrms through a sharper deterioration in the homecountry’s terms of trade under trade in ﬁnal goods.24 Note that these parameters disappear
in the long-run equilibrium (see Eqs.(53) and (54)). This is because the sales price of the
LCP intermediate goods ﬁrm is equal to that of the PCP intermediate goods ﬁrm. Further,from the perspective of the elasticity of substitution between ﬁnal tradable goods producedin the home and foreign countries, we can also show that these results have the following
properties. When the value of ρ rises, the increase in the number of these two types of ﬁrms
gets steeper. This is because the rise in this value causes an increasing shift in demand fromthe ﬁnal tradable goods produced in the foreign country to the ﬁnal tradable goods produced
24

The terms of trade under trade in ﬁnal goods is deﬁned as T OTt =∗for T OTt can be given as T[
OT = η+η εb.
2

20

εt p∗
f,t
ph,t . Thus, a ﬁrst-order approximation

³´
b > 0 , the home government’s spending
in the home country. Finally, when Gt increases G
toward ﬁnal goods sold in the home country increases, which causes an increase in the sales
revenues of domestic market ﬁrms among home-located ﬁnal goods ﬁrms. Consequently, itleads to ﬁrm entry into the home market. This relationship is also shown in the long-rum
equilibrium (see Eq.(53)).
From Eqs.(47) and (48), we obtain the following equations:• The short-run equilibrium11c(ρ − 1)(η + η ∗ )n∗D = − nc−εb − q ∗\(zF∗ |D ),
X
2
2
4
1c
1
(ρ − 1)(η + η ∗ ) + 4
b − q ∗\n∗X = − nc−εb + G(zF∗ |X ),
D
2
2
4

(55)
(56)

• The long-run equilibrium
1c
1
∗
∗\
n∗D = − nc
(57)
X − q (zF |D ),
2
2
1c
1
∗
∗\
b b
n∗X = − nc
(58)
D − ε + G − q (zF |X ).
22
Eqs.(55) and (56) have the following characteristics. To begin with, when the number ofexport (domestic market) ﬁrms among home-located ﬁnal goods ﬁrms, i.e., ncc
X (nD ), increases,
the foreign (home) household’s consumption of ﬁnal goods produced in the foreign country
decreases, which causes a decrease in the sales revenues of domestic market (export) ﬁrms
among foreign-located ﬁnal goods ﬁrms, thereby decreasing proﬁts for these domestic market(export) ﬁrms. Consequently, it leads to their exit from the foreign market. These relationships are also shown in the long-run equilibrium (see Eqs.(57) and (58)). Next, when thenominal exchange rate depreciates (bε > 0), it decreases the sales revenues of domestic market
and export ﬁrms among foreign-located ﬁnal goods ﬁrms. The decrease in sales revenues ofdomestic market ﬁrms among foreign-located ﬁnal goods ﬁrms is solely due to an improvementof the foreign country’s terms of trade caused by the nominal exchange rate depreciation. The
decrease in sales revenues of export ﬁrms among foreign-located ﬁnal goods ﬁrms is due toan improvement in the foreign country’s terms of trade caused by the nominal exchange rate
depreciation as well as a decrease in sales revenues in terms of the foreign country’s currency
due to the nominal exchange rate depreciation. This decrease in sales revenues of domestic
market and export ﬁrms among foreign-located ﬁnal goods ﬁrms causes a decline in the proﬁts
of these two types of ﬁrms. Consequently, there is a decline in the number of these two typesof ﬁrms. In addition, from the perspective of LCP in both countries, we can show that theseresults have the following properties. When the value of η and/or η ∗ rises, the decrease in
the number of domestic market and export ﬁrms among foreign-located ﬁnal goods ﬁrms gets
steeper. This is because a rise in η and/or η ∗ causes a sharper reduction in proﬁts for these
two types of ﬁrms through a steeper improvement in the foreign country’s terms of trade under21

trade in ﬁnal goods. Note that these parameters also disappear in the long-run equilibrium(see Eqs.(57) and (58)). Further, from the perspective of the elasticity of substitution between
the ﬁnal tradable goods produced in the home and foreign countries, we can also show thatthe rise in the value of ρ intensiﬁes³´ the decrease in the number of these two types of ﬁrms.
bFinally, when Gt increases G > 0 , the home government’s spending toward ﬁnal goods soldin the home country increases, which causes an increase in the sales revenues of export ﬁrms
among foreign-located ﬁnal goods ﬁrms. Consequently, it leads to ﬁrm entry into the foreign
market. This relationship is also shown in the long-run equilibrium (see Eq.(58)).
From Eqs.(46) and (49), we obtain the following equations:• The short-run equilibrium
b
∗
\
[
n
M N = G − q(zF |M N ),

(59)

∗
∗\
n
[
M N = −q (zF |M N ),

(60)

\
b
[
∗
n
M N = G − q(zF |M N ),(61)

• The long-run equilibrium

∗
∗\
[
n
M N = −q (zF |M N ).

³´b > 0 increases proﬁts forEq.(59) has the following characteristics. An increase in Gt G
foreign multinational ﬁrms among home-located ﬁnal goods ﬁrms, since it causes an increasein the sales revenues of this ﬁrm. Consequently, it encourages new entry into the home market.
This relationship is also shown in the long-run equilibrium (see Eq.(61)).
From Eq.(50), we obtain the following equations:• The short-run equilibrium´ 1³´(ρ − 1)(η + η ∗ ) + 2
1³
b
∗
∗
c
c
εb = − nc
−
n
+
n
−
n
c
D
X + G,
X
2
2
2 D

(62)

• The long-run equilibrium
´ 1³
´
1³
b
c
c
∗
∗
b
c
ε = − nc
−
n
+
n
−
n
(63)
D
X + G.
X
D
2
2
Eq.(62) has the following characteristics. To begin with,³ an increase´ in the relative number∗cof ﬁnal tradable goods ﬁrms selling in the home country nc
leads to an appreciation
D − nXof the nominal exchange rate. The balance of trade is restored via the appreciation, since suchan increase leads to an increase in the net export of ﬁnal tradable goods. Next, an increase

22

´
∗
c
in the relative number of ﬁnal tradable goods ﬁrms selling in the foreign country nD − ncXleads to a depreciation of the nominal exchange rate. Unlike in the previous case, the balanceof trade is restored via the depreciation, since such an increaseleads³´ to a decrease in the netbexport of ﬁnal tradable goods. Finally, an increase in Gt G > 0 leads to a decrease in the
net export of ﬁnal tradable goods. Consequently, it leads to a depreciation of the nominal
exchange rate. These relationships are also shown in the long-run equilibrium (see Eq.(63)).
³

3.2

Eﬀects of a government spending shock on macroeconomic variables

This subsection analyzes the eﬀects of a government spending shock.
3.2.1

Eﬀects on the nominal exchange rate

This subsubsection examines the eﬀect of a positive home government spending shock on
nominal exchange rates. The short-run nominal exchange rate is obtained using the following
steps.
To begin with, from Eqs.(51) and (56), we derive:∗ncnc
ρ(η + η ∗ ) εb
D
− X =
.
b
b
b
2γ
G
G
G

(64)

Next, from Eqs.(52) and (55), we derive:∗ncnc
ρ(η + η ∗ ) εb
X
D
−
=−
.
b
b
b
2γ
G
G
G

(65)

Therefore, from Eqs.(62), (64) and (65), we derive:
εb
2γ
=
> 0,
b
∆
G

(66)

where ∆ ≡ ρ(η + η ∗ ) + γ{2 + (ρ − 1)(η + η ∗ )} > 0. The long-run nominal exchange rate isalso obtained in a manner similar to that used to derive the short-run nominal exchange rate
as follows.
bε= 1 > 0.
(67)bG
Eqs.(66) and (67) show that this shock unambiguously leads to a depreciation of the nominal
exchange rate in both the short-run and long-run equilibriums. Since the mechanism of thisresult is the same in both short-run and long-run equilibriums, we can use the equations forthe short-run equilibrium to explain it as follows. Eqs.(51) and (56) show that, when thisshock occurs, the number of domestic market ﬁrms among home-located ﬁnal goods ﬁrms andexport ﬁrms among foreign-located ﬁnal goods ﬁrms increases equally. Although the prices of23

export ﬁrms among foreign-located ﬁnal goods ﬁrms are the same as the prices of domestic
market ﬁrms among home-located ﬁnal goods ﬁrms, the prices of the former ﬁrms are lower
than they were before the shock, and thus the value of home imports increases. This causes a
trade deﬁcit in the home country, which in turn causes the nominal exchange rate to depreciate
to balance out the trade deﬁcit. In addition, Eq.(66) also shows that the degree of depreciation
decreases in response to the rise in η and/or η ∗ . This can be explained as follows. To begin
with, from Eqs.(64) and (65), when the value of η and/or η ∗ rises, the relative number of ﬁnal
tradable goods ﬁrms selling in the home country increases, and the relative number of ﬁnal
tradable goods ﬁrms selling in the foreign country decreases. These arise from a deteriorationin the terms of trade of the home country, and hence reduce the magnitude of the trade deﬁcits
in the home country. In addition, the magnitude of the trade deﬁcit also decreases with anincrease in the value of η and/or η ∗ , which is the component of coeﬃcient of dεt in Eq.(62).
Consequently, from these two perspectives, the degree of depreciation, which is required tocorrect the resulting trade deﬁcit, decreases. Here, Eq.(67) shows that, except in a full PCP
situation, the degree of long-run depreciation exceeds the degree of short-run depreciation andis equal to the magnitude of this shock. This can be explained as follows. Even though there
are LCP ﬁrms in the intermediate goods sector, the situation in the long-run equilibrium willbe the same as if there were only PCP ﬁrms in the intermediate goods sector. Therefore, from
c
c
∗
∗
c
Eqs.(64) and (65), the values of nc
D and nX are equal, and the values of nD and nX are equal.
Consequently, from Eq.(63), the degree of the long-run depreciation is equal to the magnitudeof this shock.
3.2.2

Eﬀects on the number of ﬁnal goods ﬁrms

This subsubsection examines the eﬀects of a positive home government spending shock on
the number of diﬀerent types of ﬁnal goods ﬁrms.25 To begin with, from Eqs.(51), (52), (55),(56) and (66), the number of domestic market and export ﬁrms among home-located ﬁnal
goods ﬁrms in the short-run equilibrium are as follows:nc
1ρ(1 + γ)(η + η ∗ ) − γ(2 + η − η ∗ )
D
=
+
> 0,
b
1+γ
2(1 + γ)∆
G

(68)

nc
ρ(1 + γ)(η + η ∗ ) + γ(2 + η ∗ − η)
X
=
> 0.
b
2(1
+
γ)∆
G

(69)

Eqs.(68) and (69) show that the eﬀects of this shock on the number of domestic market andexport ﬁrms among home-located ﬁnal goods ﬁrms in the short-run equilibrium are positive.26
25

The ﬂuctuation in the number of various ﬁnal goods ﬁrms ought ideally to be explained by movements inall the variables represented by each of the required free entry conditions. However, the explanations here arein terms of the direct eﬀect of a change in the policy variable, the eﬀect of changes in the nominal exchange
rate, and the eﬀect of changes in entry costs due to the depreciation of the nominal exchange rate, based on
the implications derived from the initial response to a positive home government spending shock.
26
Our model does not include the features of the so-called Melitz (2003) model. However, if included, apositive home government spending shock would aﬀect both the thresholds of both production and exporting
decisions, so the total eﬀect on the number of exporters is ambiguous.

24

This result can be explained intuitively as follows. When the shock occurs, the proﬁts forthese two types of ﬁrms increase. Speciﬁcally, the proﬁts for domestic market ﬁrms among
home-located ﬁnal goods ﬁrms increase as a direct eﬀect of the shock and nominal exchange
rate depreciation. The proﬁts for export ﬁrms among home-located ﬁnal goods ﬁrms increase
through nominal exchange rate depreciation. At the same time, it increases entry costs forthese two types of ﬁrms through a depreciation of the nominal exchange rate. However, theproﬁts are greater than the entry costs for both types of ﬁrms. Consequently, there is an increase in the number of these two types of ﬁrms as the shock increases the diﬀerence between
proﬁts and entry costs for both (see Eqs.(51), (52) and (66)). Here, diﬀerentiating Eq.(68)with respect to η and η ∗ , we can show that the rise in η and/or η ∗ intensiﬁes the increasein the number of domestic market ﬁrms among home-located ﬁnal goods ﬁrms. This can be
explained as follows. The larger the value of η and/or η ∗ , the greater the increase in diﬀerence
between proﬁts and entry costs for domestic market ﬁrms among home-located ﬁnal goods
ﬁrms. This means that the steeper increase in proﬁts due to the sharper deterioration in
the home country’s terms of trade and the weaker increase in entry costs due to a weakernominal exchange rate depreciation contribute signiﬁcantly to a steeper increase in this difference. Consequently, the degree of the increase in the number of these domestic market
ﬁrms intensiﬁes. On the other hand, diﬀerentiating Eq.(69) with respect to η and η ∗ , we canshow that a rise in η essentially weakens the increase in the number of export ﬁrms among
home-located ﬁnal goods ﬁrms, while a rise in η ∗ intensiﬁes the increase in the number ofthese export ﬁrms. In particular, the latter result can be explained based on the intensiﬁcation of the increase in diﬀerence between proﬁts and entry costs for these export ﬁrms. Thisis based on an acceleration of the weakening of the increase in entry costs caused by both
the weakening of the nominal exchange rate depreciation and the weakening of the nominal
exchange rate pass-through, although the increase in proﬁts due to this depreciation will also
weaken.27 In another analysis, comparing scenarios with and without vertical trade, we may
be able to show that the number of domestic market ﬁrms and exporters among home-locatedﬁnal goods ﬁrms increases signiﬁcantly in the latter. This is because, in the scenario withvertical trade, nominal exchange rate depreciation increases entry costs for these two types ofﬁnal goods ﬁrms, whereas, in the scenario without vertical trade, entry costs for these ﬁrms
remain constant despite nominal exchange rate depreciation due to these ﬁrms not requiring
imported foreign intermediate goods for market entry.
Next, from Eqs.(51), (52), (55), (56) and (66), the number of domestic market and exportﬁrms among foreign-located ﬁnal goods ﬁrms in the short-run equilibrium are as follows:

27

∗
nc
γ(2 + η ∗ − η) − ρ(1 + γ)(η + η ∗ )D,=b2(1 + γ)∆
G

(70)

∗
nc
ρ(1 + γ)(η + η ∗ ) + γ(2 + η − η ∗ )
1
X
−
> 0.
=
b
1+γ
2(1 + γ)∆G

(71)

The eﬀect of this shock on qt (zF |X ) can be given as

25

\
∗
q(z
F |X )
nc
ε
b
X
= 1−η
b
b +γ Gb .
2 GGEq.(70) shows that the eﬀect of this shock on the number of domestic market ﬁrms among
foreign-located ﬁnal goods ﬁrms is ambiguous, while Eq.(71) shows that the eﬀect of thisshock on the number of export ﬁrms among foreign-located ﬁnal goods ﬁrms is positive. These
results can be explained intuitively as follows. To begin with, when this shock occurs, theproﬁts for domestic market ﬁrms among foreign-located ﬁnal goods ﬁrms decrease. Speciﬁcally,
under conditions other than full PCP, the proﬁts for these ﬁrms decrease as a result of thedepreciation of the nominal exchange rate. At the same time, it also causes a decrease in entry
costs for these ﬁrms through the depreciation of the nominal exchange rate. The magnituderelation of these two decreases is determined by the size of two types of LCP parameters.
Consequently, the eﬀect on the number of these ﬁrms is ambiguous. On the other hand, while
this shock lowers the entry costs for export ﬁrms among foreign-located ﬁnal goods ﬁrms
through nominal exchange rate depreciation, it also essentially increases the proﬁts for theseﬁrms. A signiﬁcant portion of the increase in the proﬁts for these ﬁrms is as a direct result of
this shock. Consequently, there is an increase in the number of these ﬁrms (see Eqs.(55), (56)and (66)). Here, diﬀerentiating Eq.(70) with respect to η and η ∗ , we can show that a rise in η
and/or η ∗ weakens (or intensiﬁes) the increase in (or the decrease in) the number of domestic
market ﬁrms among foreign-located ﬁnal goods ﬁrms. On the other hand, diﬀerentiating
Eq.(71) with respect to η and η ∗ , we can show that a rise in η weakens the increase in thenumber of export ﬁrms among foreign-located ﬁnal goods ﬁrms, while a rise in η ∗ essentiallyintensiﬁes the increase in the number of these export ﬁrms. These mechanisms can also be
explained using the same logic as adopted for the result obtained from Eq.(69). For example,
the latter result can be explained based on an intensiﬁcation in the increase in diﬀerence
between proﬁts and entry costs for these export ﬁrms. This is based on a deceleration of
the weakening of the decrease in entry costs caused solely by the weakening of the nominal
exchange rate depreciation, although the increase in proﬁts due to this depreciation will also
intensify.28 In another analysis, comparing the scenarios with and without vertical trade, we
may be able to show that the number of domestic market ﬁrms and exporters among foreignlocated ﬁnal goods ﬁrms is signiﬁcantly reduced in the latter. This is because, in the scenario
with vertical trade, nominal exchange rate depreciation decreases entry costs for these two
types of ﬁnal goods ﬁrms, whereas, in the scenario without vertical trade, entry costs forthese ﬁrms remain constant despite nominal exchange rate depreciation due to these ﬁrms not
requiring imported home intermediate goods for market entry.
Further, from Eqs.(59), (60) and (66), the number of foreign multinational ﬁrms among
home-located ﬁnal goods ﬁrms in the short-run equilibrium, and that of home multinationalﬁrms among foreign-located ﬁnal goods ﬁrms in the short-run equilibrium are as follows:

28

∗
[
n
1
γ(1 − η ∗ )
MN
=
−
> 0,
b
1+γ
(1 + γ)∆
G

(72)

n
[
γ(1 − η)
MN
=
≥ 0.
b
(1 + γ)∆G(73)

The eﬀect of this shock on qt∗ (zF∗ |X ) can be given as

26

∗
∗
q ∗\
(zF
)
nc
|X
ε
b
X
= − 1−ηbb +γ Gb .
2 GGEq.(72) shows that the eﬀect of this shock on the number of foreign multinational ﬁrms
among home-located ﬁnal goods ﬁrms is positive, while Eq.(73) shows that the eﬀect of thisshock on the number of home multinational ﬁrms among foreign-located ﬁnal goods ﬁrms is
non-negative. This is because, for the former, this shock brings a larger increase in proﬁts as
a direct eﬀect of the shock than it does an increase in entry costs through nominal exchange
rate depreciation, while, for the latter, in a foreign country, it prevents the entry costs fromincreasing through nominal exchange rate depreciation (see Eqs.(59), (60) and (66)). Here,diﬀerentiating both Eqs.(72) and (73) with respect to η and η ∗ , we can show that a rise in η
and/or η ∗ essentially intensiﬁes the increase in the number of foreign multinational ﬁrms among
home-located ﬁnal goods ﬁrms, while a rise in η and/or η ∗ essentially weakens the increase in
the number of home multinational ﬁrms among foreign-located ﬁnal goods ﬁrms. These results
are based on the fact that both the increase in entry costs in the home market and the decreasein entry costs in the foreign market are weakened through the weakening of nominal exchange
rate depreciation. In another analysis, comparing the scenarios with and without verticaltrade, we may be able to show that the number of foreign multinational ﬁrms among homelocated ﬁnal goods ﬁrms signiﬁcantly increases, while the number of home multinational ﬁrms
among foreign-located ﬁnal goods ﬁrms signiﬁcantly decreases in the latter. The explanation
for the former is similar to the explanation for the fact that the number of domestic market
and export ﬁrms among home-located ﬁnal goods ﬁrms may increase signiﬁcantly, while theexplanation for the latter is similar to the explanation for the fact that the number of domestic
market and export ﬁrms among foreign-located ﬁnal goods ﬁrms may decrease signiﬁcantly.
Finally, the number of diﬀerent types of ﬁnal goods ﬁrms in the long-run equilibrium are
as follows:
[
∗
∗
∗
[
nc
nc
nc
nc
n
n
1
D
X
MN
D
X
=
=
=
=
= MN =
> 0.
(74)
b
b
b
b
b
b
2(1 + γ)
G
G
G
G
G
G
Here, taking Eq.(67) into account, we obtain a relationship whereby the sum of the freeentry conditions of two types of tradable goods ﬁrms selling in the home market is equal tothat of the free entry conditions of two types of tradable goods ﬁrms selling in the foreignc∗market. In addition, by combining this relationship with the two conditions of nc
D = nX and∗c 29 we also obtain that the number of four types of tradable goods ﬁrms is equal.
nc
D = nX ,Further, taking the free entry conditions of two types of multinational ﬁrms into account, we
obtain Eq.(74) shown above. It is important to note that the number of domestic market ﬁrms
among foreign-located ﬁnal goods ﬁrms is certainly positive. In the short-run equilibrium, theentry costs of these ﬁrms have the potential to exceed their proﬁts. However, in the long-run
equilibrium, the proﬁts of these ﬁrms certainly exceed their entry costs. Hence, there is anincrease in the number of these ﬁrms. Comparing the scenarios with and without verticaltrade, the number of six types of ﬁnal goods ﬁrms will probably not be equal in the scenario
without vertical trade: the number of three types of home-located ﬁnal goods ﬁrms may
increase signiﬁcantly and the number of three types of foreign-located ﬁnal goods ﬁrms may
decrease signiﬁcantly. The mechanism by which this outcome arises can be explained by the29

From Eqs.(64) and (65), these conditions are derived.

27

mechanism used in the analysis of short-run equilibrium.
3.2.3

Eﬀects on CPI and overall consumption

This subsubsection examines the eﬀects of a positive home government spending shock on
the CPIs of both the countries and overall consumption in both countries. Let us ﬁrst consider
the eﬀects of this shock on the CPIs of both the countries in the short-run equilibrium, which
can be given as follows:
Pb
δ
1−δ
γΩ1
=−
−
+
,
b
(1 + γ)(ρ − 1) (1 + γ)(θ − 1) 2(1 + γ)(θ − 1)(ρ − 1)∆
G

(75)

c∗
P
γΩ2
=−
< 0,
b
2(1 + γ)(θ − 1)(ρ − 1)∆G(76)

where Ω1 ≡ 2(1 + γ)(1 − η ∗ )(ρ − 1)(θ − 1) + δ(θ − 1)(2 + η − η ∗ + (η + η ∗ )(1 + γ)(ρ − 1)) +2(1 − δ)(ρ − 1)(1 − η ∗ ) > 0 and Ω2 ≡ 2(1 + γ)(1 − η)(ρ − 1)(θ − 1) + δ(θ − 1)(2 + η ∗ − η +(η + η ∗ )(1 + γ)(ρ − 1)) + 2(1 − δ)(ρ − 1)(1 − η) > 0. Eq.(75) shows that the eﬀect of thisshock on home CPI in the short-run equilibrium is ambiguous, while Eq.(76) shows that theeﬀect of this shock on foreign CPI in the short-run equilibrium is negative. Using the variables
representing the nominal exchange rate and the number of diﬀerent types of ﬁnal goods ﬁrms,
Eqs.(75) and (76) can be rewritten as follows:
Ã
!
!
Ã
∗
∗
[
Pb
δ
nc
1−δn
2(1 − η ∗ ) + δ(η + η ∗ ) εb
nc
D
X
MN
=−
+
+
+
,
(77)
b
b
b
b
b
2(ρ − 1) G
θ−1 G
4
G
G
G
c∗
P
=−
b
G

Ã

δ
2(ρ − 1)

Ã

∗
nc
nc
X
D
+
b
b
G
G

!

1−δn
[
MN
+
b
θ−1 G

!

2(1 − η) + δ(η + η ∗ ) εb
−
.
b
4
G

(78)

From Eqs.(77) and (78), Eqs.(75) and (76) can be explained as follows. To begin with, thehome country result can be explained by way of two channels: the increase in the weighted
sum of the number of three types of ﬁnal goods ﬁrms that sell in the home market; and thedepreciation of the nominal exchange rate. The eﬀect of the ﬁrst channel on home CPI in the
short-run equilibrium is brought about by an increase in the supply of ﬁnal goods sold in thehome country. The eﬀect of the second channel on home CPI in the short-run equilibrium isbrought about by an increase in the home-currency prices of home ﬁnal goods sold in the homecountry and an increase in the home price index of ﬁnal tradable goods due to a deteriorationin the terms of trade of the home country. Consequently, the overall eﬀect of this shock onhome CPI in the short-run equilibrium is ambiguous. Next, the foreign country result can bealso explained by way of two channels: the increase in the weighted sum of the number of three
types of ﬁnal goods ﬁrms that sell in the foreign market; and the depreciation of the nominal
exchange rate. The eﬀect of the ﬁrst channel on foreign CPI in the short-run equilibrium isbrought about by an increase in the supply of ﬁnal goods sold in the foreign country. The
eﬀect of the second channel on foreign CPI in the short-run equilibrium is brought about by a28

decrease in the foreign-currency prices of foreign ﬁnal goods sold in the foreign country and adecrease in the foreign price index of ﬁnal tradable goods due to an improvement in the termsof trade of the foreign country. Consequently, the overall eﬀect of this shock on foreign CPI
in the short-run equilibrium is negative. This is determined independent of home and foreign
LCP parameters. Here, a rise in η ∗ weakens the increase in (or intensiﬁes the decrease in)home CPI, and a rise in η weakens the decrease in foreign CPI. In particular, the latter result
can be explained by the decline in the degree of the decrease in the two channels on foreign
CPI due to the rise in η. In addition, our model can also handle ﬂuctuations in home-currency(foreign-currency) prices of home (foreign) ﬁnal goods sold in the home (foreign) country. Thisis a key feature of our model, which introduces vertical trade. Furthermore, comparing the
scenarios with and without vertical trade, we may be able to show an increase in downward
pressure on home CPI, while we may be able to show a decrease in downward pressure on
foreign CPI in the latter. This is based on the fact that, with regard to home CPI, there
may be a greater increase in the number of domestic market ﬁrms and foreign multinationalﬁrms among home-located ﬁnal goods ﬁrms, while with regard to foreign CPI, there may be agreater decrease in the number of domestic market ﬁrms and home multinational ﬁrms among
foreign-located ﬁnal goods ﬁrms.
In the long-run equilibrium, the eﬀects of this shock on the CPIs of both the countries are
as follows:
µ
¶
b
P
1
δ
1−δ
=
1−
−
,
(79)
b
2
(ρ − 1)(1 + γ) (θ − 1)(1 + γ)
G
µ
¶
c∗
P
1
δ
1−δ
=−
1+
+
< 0.
(80)
b
2
(ρ − 1)(1 + γ) (θ − 1)(1 + γ)
GEq.(79) shows that the eﬀect of this shock on home CPI is ambiguous, while Eq.(80) showsthat its eﬀect on foreign CPI is negative. The eﬀect of this shock on home CPI is brought
about by an increase in the supply of ﬁnal goods sold in the home country and the increase in
the home-currency prices of home ﬁnal goods sold in the home country. On the other hand,the eﬀect of this shock on foreign CPI is brought about by an increase in the supply of ﬁnal
goods sold in the foreign country and the decrease in the foreign-currency prices of foreign ﬁnal
goods sold in the foreign country. Compared with the CPIs in the short-run equilibrium, theCPIs in both countries are characterized by the disappearance of the terms of trade channel.
The interpretation of the analysis comparing scenarios with and without vertical trade in thelong-run equilibrium is the same as that for the short-run equilibrium analysis.
Next, let us consider the eﬀects of a positive home government spending shock on overallb −Pb) and overall foreign consumption Cc∗ (= −Pc∗ ). The eﬀect of thishome consumption C(=
b is ambiguous since its eﬀect on home CPI is ambiguous. On the other hand, theshock on Cc∗ is positive since its eﬀect on foreign CPI is always negative. Here,eﬀect of this shock on C
we consider the ﬂuctuations in overall consumption in both countries from the perspective
b when the increase (or decrease) in home CPI weakens (orof LCP. From the deﬁnition of C,b weakens (or intensiﬁes). From the deﬁnition of Cc∗ ,intensiﬁes), the decrease (or increase) in Cc∗ weakens. Moreover, we consider
when the decrease in foreign CPI weakens, the increase in C29

ﬂuctuations in overall consumption in both countries by comparing the scenarios with andwithout vertical trade. In the scenario without vertical trade, there is likely to be greater
downward pressure on home CPI, so overall home consumption is more likely to increase thanin the scenario with vertical trade. On the other hand, in the scenario without vertical trade,
there is likely to be smaller downward pressure on foreign CPI, so overall foreign consumption
is more likely to decrease than in the scenario with vertical trade.
c∗ , which are the long-run eﬀects of this shock on overall consumptionb and CThe variables C
in the home and foreign countries, also ﬂuctuate in the same direction as short-run eﬀects.
These ﬂuctuations are determined based on the deﬁnition of consumption and the ﬂuctuationof the CPI. The interpretation of the analysis comparing scenarios with and without verticaltrade in the long-run equilibrium is the same as that for the short-run equilibrium analysis.
3.2.4

Eﬀects on employment

This subsubsection examines the eﬀects of a positive home government spending shock on
the employment levels of both countries. Let us ﬁrst consider the eﬀects of this shock on theemployment levels of both countries in the short-run equilibrium, which are as follows:µ¶b̀
1γ(2θ(1 − η) − (2 − η − η ∗ ))
=
1+
> 0,
(81)
b
2
θ∆
G
µ
¶
`b∗
1
γ(2θ(1 − η ∗ ) − (2 − η − η ∗ ))
=
1−
> 0.
(82)
b
2
θ∆
G
Eqs.(81) and (82) show that the eﬀect of this shock on employment levels of both countries
are positive. Using the variables that represent the number of diﬀerent types of ﬁnal goods
ﬁrms, their proﬁts, and the price indexes of intermediate goods imported by them, Eqs.(81)
and (82) can be rewritten as follows:
Ã Ã
!
Ã
!
∗
∗
X nc
X nc
b̀
[
1 δ
n
n
[
j 00
M
N
j 00
MN
=
+
+ (1 − δ)
+
b
b
b
b
b
2
2
00
00
G
G
G
G
G


j =D,Xj =D,X




∗\
∗
∗\
∗
\
\
X
X
Π
(z
)
Π
(z
)
00
00
Π
(z
)
Π
(z
)
δ
F F |jF F |j + (1 − δ)  F F |M N + F F |M N 
+ 
+
b
b
b
b
2 j 00 =D,X
G
G
G
G
j 00 =D,X
Ã
!!
c̃
c̃∗
1 P
P
f
−
+ (2θ − 1) h
,
(83)
b
b
2θ G
G
Ã Ã
!
Ã
!
∗
∗
X nc
X nc
[
`b∗
1 δ
n
[
n
j 00
M
N
j 00
MN
=
+
+
+ (1 − δ)
b
b
b
b
b
2
2
G
G
G
j 00 =D,X G
j 00 =D,X G




∗\
∗
∗\
∗
\
\
X
X
Π
(z
)
Π
(z
)
00
00
ΠF (zF |j )Π (z)δ
F F |j + (1 − δ)  F F |M N + F F |M N 
+
+ 
b
b
b
b
2 j 00 =D,X
G
G
G
G
j 00 =D,X

30

1
−
2θ

Ã

c̃ P
c̃∗
P
f
(2θ − 1)
+ h
b
b
G
G

!!
.

(84)

As shown in Eqs.(83) and (84), the eﬀects of this shock on the employment levels of bothcountries can be separated into three channels: the weighted sum of the number of three
types of ﬁnal goods ﬁrms located in both countries; the weighted sum of the proﬁts of three
types of ﬁnal goods ﬁrms located in both countries; and the sum of the price indexes ofintermediate goods imported by ﬁnal goods ﬁrms located in both countries. From Eqs.(83)
and (84), the mechanisms of these three channels can be explained as follows. To begin with,when the ﬁrst and the second channels are positive, they cause an increase in intermediate
goods required for the production of ﬁnal goods. Therefore, they have a positive eﬀect on theemployment levels of both countries. On the other hand, when the third channel is negative,
this is based on an increase in production costs of ﬁnal goods. It, therefore, has a negativeeﬀect on the employment levels of both countries since it causes a decrease in intermediate
goods used by ﬁnal goods ﬁrms. Now, in both countries, the sum of the ﬁrst and the second
channels is positive. Additionally, the third channel in Eq.(83) shows that the demand forhome intermediate goods by foreign ﬁnal goods ﬁrms increases signiﬁcantly due to the nominal
exchange rate depreciation caused by this shock, while the third channel in Eq.(84) shows thatthe demand for foreign intermediate goods by home ﬁnal goods ﬁrms decreases signiﬁcantly
for the same reason. Here, the third channel in Eq.(83) has a strong positive side. Therefore,
the overall eﬀect of this shock on the employment level of the home country is positive. On
the other hand, although the third channel in Eq.(84) has a strong negative side, the positiveeﬀect due to the sum of the ﬁrst and second channels dominates the whole. Therefore, theoverall eﬀect of this shock on the employment level of the foreign country is also positive.
This is determined independently of home and foreign LCP parameters. Here, a rise in η
weakens the increase in home employment, and a rise in η ∗ intensiﬁes the increase in foreign
employment.
In the long-run equilibrium with ﬂexible prices in both the ﬁnal and intermediate goods
sectors, the eﬀects of this shock on the employment levels of both countries can be given as
follows:b̀
2θ − 1
=
> 0,
(85)
b
2θ
G
`b∗
1
> 0.
=
b
2θ
G

(86)

Eqs.(85) and (86) show that, as in the short-run equilibrium, the eﬀects of this shock on
the employment levels of both countries are positive. In addition, unlike in the case of theshort-run equilibrium, the home country’s third channel is clearly positive while the foreigncountry’s third channel is clearly negative. Hence, the eﬀect of this shock on home employment
is greater than that on foreign employment.
Based on the results pertaining to short-run and long-run equilibriums, this shock has apersistent positive eﬀect on employment levels in both countries over two periods.

31

3.2.5Eﬀects on aggregate output

This subsubsection examines the eﬀects of a positive home government spending shock on
the aggregate outputs of both countries. Let us ﬁrst consider the eﬀects of this shock on theaggregate outputs of both countries in the short-run equilibrium, which are as follows:Yb
γ + γη ∗ + ρ(η + η ∗ ) + γ(ρ − 1)(η + η ∗ )
> 0,
=
b
∆
G

(87)

Yc∗
γ(1 − η)
=
≥ 0.
b
∆
G

(88)

Eq.(87) shows that the eﬀect of this shock on aggregate home output is positive, while Eq.
(88) shows that its eﬀect on aggregate foreign output is non-negative. Using the variables
that represent the numbers of diﬀerent types of ﬁnal goods ﬁrms, Eqs.(87) and (88) can be
rewritten as follows:
Ã µ
!
¶
∗
[
Yb
δ nc
nc
n
D
X
(89)
= (1 + γ)++ (1 − δ) M N > 0,
b
b
b
b
2
G
G
G
G
Ã Ã
!
!
∗
∗
Yc∗
δ nc
nc
n
[
MN
X
D
= (1 + γ)++ (1 − δ)
≥ 0.
(90)
b
b
b
b
2 G
G
G
G
As shown in Eq.(89), the eﬀect of this shock on the weighted sum of the number of three
types of home-located ﬁnal goods ﬁrms is positive. This means that these ﬁrms increase their
output of goods. On the other hand, as shown in Eq.(90), the eﬀect of this shock on theweighted sum of the number of three types of foreign-located ﬁnal goods ﬁrms is non-negative.
This means that these ﬁrms, especially foreign ﬁnal goods exporters and home multinationalﬁrms, also essentially increase their output of goods. However, when η = 1, this shock has
no eﬀect on aggregate foreign output. This is because it has no eﬀect on the weighted sumof the number of three types of foreign-located ﬁnal goods ﬁrms. Here, a rise in η and/or η ∗essentially intensiﬁes the increase in aggregate home output while weakening the increase inaggregate foreign output. These results can be explained by the ﬂuctuation in the number ofdiﬀerent types of ﬁnal goods ﬁrms due to the rise in η and/or η ∗ . For example, a rise in η ∗intensiﬁes the increase in the number of each of the three types of home-located ﬁnal goods
ﬁrms. Therefore, the rise in this value intensiﬁes the increase in aggregate home output.
By the way, the factor of ﬁrm entry plays a special role in the ﬂuctuations in aggregate
outputs in both countries. For example, a rise in the value of γ essentially weakens the increasein aggregate home output while intensifying the increase in aggregate foreign output. Thesecan be explained as follows. When the value of γ rises, the three types of home-located ﬁnal
goods ﬁrms are less likely to enter the home market because their entry costs increase, while
the three types of foreign-located ﬁnal goods ﬁrms are more likely to enter foreign markets
because their entry costs decrease.30 These lower the output of ﬁnal goods produced in the30

In accordance with what is noted in footnote 25, with regard to ﬂuctuations in the number of ﬁnal goods
ﬁrms, the explanation here of an increase or a decrease in the entry costs for these ﬁrms due to a rise in γ isalso based on a higher degree of nominal exchange rate depreciation due to the same.

32

home market and increase the output of ﬁnal goods produced in the foreign market, thus
weakening the increase in aggregate home output and intensifying the increase in aggregate
foreign output. In addition, the factor of vertical trade also plays a special role in ﬂuctuations
in aggregate output in both countries. The ﬂuctuation in aggregate home output may besigniﬁcantly lower than in the scenario without vertical trade. As shown in Section 3.2.2, thisis because, in the scenario with vertical trade, nominal exchange rate depreciation increases
entry costs for the three types of home-located ﬁnal goods ﬁrms, whereas, in the scenario
without vertical trade, entry costs for these ﬁrms remain constant despite nominal exchange
rate depreciation due to these ﬁrms not requiring imported foreign intermediate goods formarket entry. That is, comparing the scenarios with and without vertical trade, the former
has a greater potential to discourage the new entry of these ﬁrms into the home market thanthe latter, and hence also a greater potential to reduce the supply of goods by these ﬁrms.
On the other hand, the ﬂuctuation in aggregate foreign output may be signiﬁcantly higher
than in the scenario without vertical trade. As shown in Section 3.2.2, this is because, in the
scenario with vertical trade, nominal exchange rate depreciation decreases entry costs for thethree types of foreign-located ﬁnal goods ﬁrms, whereas, in the scenario without vertical trade,entry costs for these ﬁrms also remain constant despite nominal exchange rate depreciation
due to these ﬁrms not requiring imported home intermediate goods for market entry. That is,comparing the scenarios with and without vertical trade, the former has a greater potential
to facilitate the new entry of these ﬁrms into the foreign market than the latter, and hence
also a greater potential to increase the supply of goods by these ﬁrms.
Next, the long-run eﬀects of this shock on aggregate outputs in the home and foreigncountries are as follows:
Yb
Yc∗
1
=
= > 0.
bb
2
G
G
From Eq.(74), the number of diﬀerent types of ﬁnal goods ﬁrms located in the home andforeign countries are all equally positive. Hence, from the long-run versions of Eqs.(89) and(90), the aggregate home output in the long-run equilibrium increases by the same levels as
the aggregate foreign output in the long-run equilibrium.
Based on the results pertaining to short-run and long-run equilibriums, this shock essentiallyhas a persistent positive eﬀect on aggregate outputs in both countries over two periods. In
particular, unlike the result of Cook and Devereux (2011), the ﬁscal spillover eﬀect on aggregate
foreign output in our analysis is shown to be positive, as in Devereux and Yu (2019) and
Devereux et al. (2020).

4

WelfareThis section examines the eﬀects of a positive home government spending shock on thewelfare of both countries. Following Obstfeld and Rogoﬀ (1995, 1996) and others, this paper focuses on the real parts of a household’s utility and assume that the eﬀect of real

33

balances on utility is small enough to be neglected.31 Taking the ﬁrst-order approximation of the household’s utility under such anµ assumption, we examine the³ eﬀects of such´bRb
ξ−1 b̀βC
σ−1 ξ−1 b̀a shock on the welfare of both countries dUb = Cb − σ−1
+
−
b
b
b ,
σ
ξ G
1−β G
σ
ξ G
G
G
¶
³c
´
c∗
β
dU ∗R
C
σ−1 ξ−1 `b∗C∗
σ−1 ξ−1 `b∗= b − σ ξ b + 1−β b − σ ξ b. Eﬀects on welfare are diﬃcult to evalub
G

G

G

G

G

ate from an analytical perspective, so this paper evaluate them numerically, speciﬁcally byfocusing on the degree of LCP. To perform an analysis using a numerical example, the valuesof seven parameters need to be speciﬁed. The parameters that measure the elasticities of substitution between any two diﬀerentiated ﬁnal goods at the two stages of production are ﬁrst
set at θ = σ = 6 in according with Gong et al. (2016).32 The elasticity of substitution between
goods produced in the home and foreign countries is set at ρ = 4.5 in accordance with Benigno
and Benigno (2006). The elasticity of substitution among labor varieties is set at ξ = 4.52 inaccordance with Gali (2013). The subjective discount factor is set at β = 0.99 in accordancewith Ferrero (2009). The share of the tradable composite of diﬀerentiated ﬁnal goods is set atδ = 0.25 in accordance with Obstfeld and Rogoﬀ (2005). Finally, γ is set somewhat arbitrarily
at 1. In what follows, some speculations from the perspective of analytical investigation are
introduced. This is followed by an analysis using a numerical example.

4.1

Analytical investigation

This subsection examines the eﬀects of a positive home government spending shock on
short-run welfare in both countries from the perspective of analytical analysis, since the LCP
parameters only appear in the short-run equilibrium. Moreover, it is essentially unclear what
the eﬀect of this shock is on the short-run welfare of either country. Having said that, let
us consider under what circumstances this shock raises or does not raise the levels of homeand foreign short-run welfare. To begin with, it follows that this shock raises home short-run
welfare as long as η and η ∗ meet the following condition:
µ ½
¾
½
¾¶
θΓ 2σξ{(θ − 1)δ + (ρ − 1)(1 − δ)} − Φ1 − γ(θ − 1) θσξδ(1 + (ρ − 1)(1 + γ)) − Φ2
η
µ ½
¾
½
+ θΓ 2σξ{(θ − 1)δ + (ρ − 1)(1 − δ)} − Φ1 + θσξγ 2(ρ − 1){(θ − 1)(1 + γ) + (1 − δ)}
¾
¶
µ
½
¾
∗
+δ(θ−1)(1−(ρ−1)(1+γ)) −γΦ1 η + 2γθσξ (ρ−1)(1−δ)−(ρ−1)(θ−1)(1+γ)−δ(θ−1)
31

By abstracting from the utility of real balances, this paper follows the formulation of Obstfeld and Rogoﬀ(1995, 1996). Many literatures pertaining to the NOEM model use this formulation. For e.g., see Betts andDevereux (2000), Corsetti et al. (2000), Obstfeld and Rogoﬀ (2000, 2002), Tille (2001), Sutherland (2004),
Corsetti and Pesenti (2005), Berger (2006), Shi and Xu (2007), and Dohwa (2008, 2014, 2018).
32
A non-exhaustive list of contributions on estimates of these parameters, which correspond to a mark-up
of 20 percent in the steady state, include Gali and Monacelli (2005, 2008), Gali (2008), Corsetti et al. (2011),
and Farhi and Werning (2012).

34¶−2γ(2θ − 1)Φ1 + 4γθσξ(θ − 1)δ

> 0,

(91)

where Γ ≡ γ(ρ − 1) + ρ > 0, Φ1 ≡ (θ − 1)(ρ − 1)(σ − 1)(ξ − 1)(1 + γ) > 0 and Φ2 ≡(2θ − 1)(ρ − 1)(σ − 1)(ξ − 1)(1 + γ) > 0.
On the other hand, when η and η ∗ satisfy the following condition, this shock lowers homeshort-run welfare:
¾
½
¾¶
µ ½
θΓ 2σξ{(θ − 1)δ + (ρ − 1)(1 − δ)} − Φ1 − γ(θ − 1) θσξδ(1 + (ρ − 1)(1 + γ)) − Φ2
η
µ ½
¾
½
+ θΓ 2σξ{(θ − 1)δ + (ρ − 1)(1 − δ)} − Φ1 + θσξγ 2(ρ − 1){(θ − 1)(1 + γ) + (1 − δ)}
¾
¶
µ
½
¾
∗
+δ(θ−1)(1−(ρ−1)(1+γ)) −γΦ1 η + 2γθσξ (ρ−1)(1−δ)−(ρ−1)(θ−1)(1+γ)−δ(θ−1)¶−2γ(2θ − 1)Φ1 + 4γθσξ(θ − 1)δ

< 0.

(92)

Condition (91) shows that this shock causes the positive eﬀect on short-run welfare fromconsumption of ﬁnal goods to dominate the negative eﬀect on short-run welfare from employment. On the other hand, condition (92) shows that this shock causes the negative eﬀect on
short-run welfare from employment to dominate the positive eﬀect on short-run welfare fromconsumption of ﬁnal goods. When η = η ∗ = 0, there is a high likelihood that the condition(92) will hold. This is because while the value of η ∗ is small, so that overall home consumption
decreases signiﬁcantly, or in the case it increases, the degree of the increase is small, the valueof η is also small, so that home employment increases signiﬁcantly. Now, when η = η ∗ = 0,
condition (92) can be rewritten as follows:1+γ >θσξ(δ(θ − 1) + (1 − δ)(ρ − 1)).
(θ − 1)(ρ − 1)((2θ − 1)(σ − 1)(ξ − 1) + θσξ)If this condition is satisﬁed and γ takes an extremely high value, this shock will greatly
lower the short-run welfare of the home country. This can be explained as follows. When thevalue of γ is large, the entry costs for the three types of ﬁnal goods ﬁrms selling in the homemarket become high. Hence, it becomes diﬃcult for these ﬁrms to enter the ﬁnal goods marketof their respective countries. This causes a decline in the supply of goods by these ﬁrms in thehome market, which intensiﬁes (or weakens) the increase (or decrease) in home CPI. Therefore,
the decrease (or increase) in overall home consumption intensiﬁes (or weakens). Consequently,
this shock greatly lowers the short-run utility of the home country.33 On the other hand, if
33

When the value of γ increases, the decrease (or increase) in overall home consumption in the long-run
equilibrium also intensiﬁes (or weakens). This can be explained by the same mechanism as that for an
intensiﬁcation in the decrease (or a weakening in the increase) in overall home consumption in the short-run
equilibrium. Therefore, the mechanism for the eﬀect of a change in the value of γ on the long-run welfare of
the home country can be explained in the same way as in the short-run equilibrium.

35

we do not introduce vertical trade into our model, as shown in Section 3.2.3, a positive homegovernment spending shock has the potential to raise overall home consumption to a greater
degree than when we introduce vertical trade into the model, not only when η = η ∗ = 0.
Consequently, this shock has the potential to signiﬁcantly increase the short-run utility of thehome country.34
Next, this shock raises foreign short-run welfare when η and η ∗ satisfy the following condition:
µ
¶
µ ½
∗
− Φ1 (θΓ + γ(2θ − 1)) + γ(θ − 1)θσξδΓ η + − θ ΓΦ1 + σξγ{2(ρ − 1){(1 − δ) +(1+ γ)(θ − 1)}
¾¶ µ
½
¾
¶
+δ(θ−1)(2−Γ)} η+ 2θσξγ (ρ−1){(1+γ)(θ−1)+(1−δ)}+δ(θ−1) −2γθΦ1 > 0. (93)On the other hand, when η and η ∗ satisfy the following condition, this shock lowers foreign
short-run welfare:
¶
µ ½
µ
∗
− Φ1 (θΓ + γ(2θ − 1)) + γ(θ − 1)θσξδΓ η + − θ ΓΦ1 + σξγ{2(ρ − 1){(1 − δ) +(1+ γ)(θ − 1)}
¾¶ µ
½
¾
¶
+δ(θ−1)(2−Γ)} η+ 2θσξγ (ρ−1){(1+γ)(θ−1)+(1−δ)}+δ(θ−1) −2γθΦ1 < 0. (94)
Condition (93) shows that this shock causes the positive eﬀect on short-run welfare fromconsumption of ﬁnal goods to dominate the negative eﬀect on short-run welfare from employment. On the other hand, condition (94) shows that this shock causes the negative eﬀect on
short-run welfare from employment to dominate the positive eﬀect on short-run welfare fromconsumption of ﬁnal goods. When η = η ∗ = 0, there is high likelihood that condition (93)will hold. This is because, while the value of η is small, so that overall foreign consumption
increases signiﬁcantly, the value of η ∗ is also small, so that foreign employment increases only
slightly.
Here, judging only by the short-run equilibrium for now, under the condition (92), thisshock can be regarded as a beggar-thyself policy in the sense that it lowers home welfare.

4.2

Numerical example

This subsection examines the eﬀects of a positive home government spending shock. Before
examining the eﬀects of this shock on the welfare of both countries, let us examine the eﬀects
of this shock on the overall consumptions of both countries in the short-run and long-run
b and Cc∗ ) and on the employment levels of both countries in the shortb C
c∗ , Cequilibriums (C,run and long-run equilibriums ( b̀, `b∗ , b̀ and `b∗ ). These analyses adopt scenario (a) in Table 1
as the benchmark scenario.
34

If we do not introduce vertical trade into our model, overall home consumption in the long-run equilibriumis also likely to be higher than if we introduce vertical trade into the model. This can be explained by the samemechanism as the possible increase in overall home consumption in the short-run equilibrium. Therefore, thepossibility of an increase in the long-run utility of the home country can be explained in the same way as thepossibility of an increase in the short-run utility of the home country.

36

[Insert Table 1 around here]b and
To begin with, the ﬁrst and second rows in Table 1 show the eﬀect of this shock on C∗∗
c , respectively. In all scenarios, the eﬀect of this shock on C
c is positive. On the otherCb is negative. In scenario
hand, in scenarios other than (c) and (d), the eﬀect of this shock on Cc∗ is largest. This is because all of the intermediate goods ﬁrms(a), the positive eﬀect on C
employ PCP. When the value of η rises, this eﬀect is signiﬁcantly weakened compared with thebenchmark scenario (compare scenario (a) with scenarios (b) and (d)). Similarly, in scenariob is largest. When the value of η ∗ rises, this eﬀect is weakened(a), the negative eﬀect on C
(compare scenario (a) with scenarios (c) and (d)). Next, the third and fourth rows in Table 1
show the eﬀect of this shock on b̀ and `b∗ , respectively. In all scenarios, the eﬀects of this shockon b̀ and `b∗ are positive. In addition, in scenario (a), the eﬀect of this shock is largest on b̀
and smallest on `b∗ . This is also because all the intermediate goods ﬁrms employ PCP. Whenthe value of η rises, the eﬀect on b̀ is signiﬁcantly weakened compared with the benchmark
scenario (compare scenario (a) with scenarios (b) and (d)). On the other hand, when the valueof η ∗ rises, the eﬀect on `b∗ is signiﬁcantly intensiﬁed compared with the benchmark scenario(compare scenario (a) with scenarios (c) and (d)). The ﬁfth and sixth rows in Table 1 showb and Cc∗ , respectively, while the seventh and eighth rows show its
the eﬀect of this shock on C
eﬀect on b̀ and `b∗ , respectively. Each of these variables, having reached its steady-state value,
takes the same value in all scenarios.
Let us examine the eﬀects of a positive home government spending shock on the welfare ofboth countries. The nineth and tenth rows in Table 1 show the eﬀect of this shock on homecountry utility and foreign country utility, respectively. In all scenarios in Table 1, the eﬀect
on the home country’s utility is negative, but that on the foreign country’s utility is positive.
Therefore, all scenarios show that this shock has a beggar-thyself and prosper-thy-neighboreﬀect. These eﬀects are weakened in scenarios (b)−(d) compared with the benchmark scenario,
albeit only slightly.

5Conclusions

By incorporating the three factors of LCP, vertical production and trade, and endogenous entry by ﬁnal goods ﬁrms into the standard NOEM model with nominal wage and pricerigidities, this study has examined how a positive home government spending shock aﬀects
macroeconomic variables and welfare. Its main ﬁndings can be summarized as follows. First,
an increase in the degree of LCP weakens the depreciation of the nominal exchange rate causedby this shock. Second, an increase in the degree of LCP magniﬁes the eﬀect of this shock on
the number of ﬁnal goods ﬁrms located in the home and foreign countries. In particular,
an increase in the degree of LCP essentially weakens the increase in the number of homemultinational ﬁrms among foreign-located ﬁnal goods ﬁrms and intensiﬁes the increase in thenumber of foreign multinational ﬁrms among home-located ﬁnal goods ﬁrms. Third, the effects of this shock on the employment levels of both countries are positive. When the degree37

of LCP increases, the eﬀect of this shock on home employment is essentially weakened, while
its eﬀect on foreign employment is essentially strengthened. Fourth, the eﬀects of this shockon the aggregate outputs of both countries are essentially positive. When the degree of LCP
increases, the eﬀect of this shock on aggregate home output is essentially strengthened, while
its eﬀect on aggregate foreign output is essentially weakened. Finally, the eﬀect of this shockon home welfare is negative, while that on foreign welfare is positive. Therefore, this shock
has a beggar-thyself and prosper-thy-neighbor eﬀect.
Based on the various above ﬁndings, the following points can be suggested. First, it can besaid, based on the second ﬁnding above that, when the government’s objective is to preventthe promotion of outward FDI, government spending should be made when the pricing ofintermediate goods ﬁrms is largely based on LCP. Similarly, when the government’s objective
is to accelerate the promotion of inward FDI, government spending should be made under thesame circumstances as above. Next, the fourth ﬁnding shows that the international transmission eﬀect of a home government spending shock on aggregate foreign output is largely absent
when the pricing of intermediate goods ﬁrms is largely based on LCP, so if the home government is only interested in expanding its own output, it should do so under this circumstance.
Finally, the ﬁfth ﬁnding shows that a government spending shock has a beggar-thyself eﬀect
in this model, so the implementation of such a policy must be carefully considered. However,such an eﬀect is weakened when many intermediate goods ﬁrms employ LCP, so, other things
being equal, a higher LCP ratio is preferable. Therefore, if it is necessary to implement such a
policy, it is preferable to do so under such circumstances. Bacchetta and Wincoop (2005) pointout that when the nominal exchange rate ﬂuctuates, home ﬁrms are better oﬀ choosing PCPover LCP if the goods produced in the home country are fairly diﬀerentiated.35 Therefore,
when many intermediate goods ﬁrms employ PCP, such an eﬀect is intensiﬁed. Therefore, insuch an environment it is advisable to implement other economic policies, such as corporate
tax reduction, as shown by Dohwa (2018).
This paper obtained the above ﬁndings by making some strong assumptions. It would be
more desirable to ﬁnd the various results by relaxing these assumptions. First, the study may
yield more interesting results, for instance, if it were to adopt a model that accepts a tradebalance imbalance. Second, like Corsetti et al. (2007), this study might yield diﬀerent results
if it were modiﬁed to assume that the government spends only on goods produced by domestic
market ﬁnal goods ﬁrms. These issues remain for future research.

Acknowledgements
In writing this paper, I received consistent encouragement from Shigeto Kitano, Ryuzo
Miyao, and Masahiko Shibamoto, for which I would like to express my special thanks. I am
also grateful to Masataka Eguchi, Katsufumi Fukuda, Taiji Hagiwara, Kohei Hasui, Daisuke
Ida, Shingo Iokibe, Yutaka Kurihara, Eiji Okano, Kobe University seminar participants, andparticipants of the 2022 Japanese Society of Monetary Economics Autumn Meeting, Kobe
35

In other words, they point out that if the goods produced in the home country are fairly homogeneous,home ﬁrms are better oﬀ choosing LCP over PCP.

38

University, for their helpful comments and suggestions. All the errors in this paper are my
own.

39

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45

Fig. 1 Outward FDI from the U.S., Japan, the UK, and Germany.

1 billion dollar
10000
9000
8000
7000
6000
5000
4000
3000
2000
1000
0
1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018 2020
U.S.

Japan

Source: International Financial Statistics, IMF.

UK

GermanyFig. 2 Inward FDI to the U.S., Japan, the UK, and Germany.

1 billion dollar
14000
12000
10000
8000
6000
4000
2000
0
1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018 2020
U.S.

Japan

Source: International Financial Statistics, IMF.

UK

Germany

Table 1: Eﬀects of a positive home government spending shock.

b
C
c∗
C
b̀
`b∗
b
C
c∗
C
b̀
`b∗
dU R
dU ∗R

(a) η = η ∗ = 0(b) η = 1, η ∗ = 0(c) η = 0, η ∗ = 1(d) η = η ∗ = 1−0.445
0.555
0.917
0.083

−0.015
0.014
0.492
0.408

0.096
0.125
0.592
0.508

0.095
0.016
0.500
0.500

−0.445−0.445−0.445−0.445

0.555

0.555

0.555

0.555

0.917

0.917

0.917

0.917

0.083
−103.953
50.1280.083−103.247
49.3760.083−103.201
49.4220.083−103.14349.31846