FDI in Post-Production Services and Product Market Competition∗
Jota Ishikawa†

Hodaka Morita

Hiroshi Mukunoki

Hitotsubashi Univ.

Univ. of New South Wales

Gakushuin Univ.

First Version: December, 2006
This Version: November 10, 2008

Abstract
Post-production services, such as sales, distribution, and maintenance, comprisea crucial element of business activity. A foreign firm faces a higher cost to perform
such services than its domestic rival because of the lack of proximity to customers.
We explore an international duopoly model in which a foreign firm can reduce its costfor post-production services by foreign direct investment (FDI), or alternatively canoutsource such services to its domestic rival. Trade liberalization, if not accompanied
by liberalization of service FDI, can hurt domestic consumers and decrease world
welfare, but the negative welfare impacts can be mitigated and eventually turned into
positive ones as service FDI is also liberalized. This finding yields important policy
implications, given the reality that the progress of liberalization in service sectors islimited compared to the substantial progress already made in trade liberalization.
Key words: post-production services, trade liberalization, FDI, outsourcing,international oligopoly
JEL classification numbers: F12, F13, F21, F23∗

We thank Pol Antrás, Richard Baldwin, Jay Pil Choi, Masahiro Endoh, Taiji Furusawa, Seiichi

Katayama, Kazuharu Kiyono, Hiroshi Kurata, Jim Markusen, Tomoya Mori, Toshihiro Matsumura,
Noriaki Matsushima, Dan Sasaki, participants of seminars at Hitotsubashi University, Kobe University,
Kyoto University, University of New South Wales, University of Tokyo, and participants at the APTS
2007 Meeting, the EARIE 2007 Meeting and the Otago Workshop 2008 for their helpful comments
and suggestions on earlier versions of this paper. The usual disclaimer applies. Ishikawa and Morita
acknowledge financial support from the Ministry of Education, Culture, Sports, Science and Technologyof Japan (MEXT) under the 21st Century Center of Excellence Project, and the Australian ResearchCouncil. Mukunoki acknowledges financial support by the Grant-in-Aid for Young Scientists (B), the
MEXT.
†
Corresponding author : Faculty of Economics, Hitotsubashi University, Kunitachi, Tokyo 186-8601,
Japan; Fax: +81-42-580-8882; E-mail: jota@econ.hit-u.ac.jp1

1

Introduction

Business activity does not end with the production of the final product. After production,a variety of business activities such as marketing, sales and distribution, and the provisionof maintenance and repair services should be effectively carried out to maximize thevalue of products that have been produced. This is a widely held view in the strategic
management literature. Porter (1985), for example, pointed out that firms’ primaryactivities can be divided into inbound logistics, operations, outbound logistics, marketingand sales, and service. In Porter’s classification, outbound logistics means activities
associated with collecting, storing, and physically distributing the product to buyers,
marketing and sales means activities associated with providing a means by which buyers
can purchase the product and inducing them to do so, and service means activities
associated with providing service to enhance or maintain the value of the product.
In the present paper, outbound logistics, marketing and sales, and service are together
referred to as “post-production services.” A crucial strategic decision that every producerof final products needs to make is whether to perform post-production services by itself or
outsource (some of) them to other firms. Since proximity to customers is a crucial element
for post-production services to be carried out effectively, this decision is particularly
important in the context of international trade. Foreign producers often outsource postproduction services to their domestic rivals. For example, automobiles manufactured byforeign auto-makers are often sold and distributed by their local rivals.1 Alternatively,
foreign producers can establish local affiliates in the domestic market and perform postproduction services by themselves (foreign direct investment (FDI) in post-production
services).2
The objective of this paper is to analyze the provision of post-production services
in the context of international trade and to explore its welfare consequences and policy
implications.3 To this end, we explore an international duopoly model in which two firms,
1

In the Japanese market, examples include Volkswagen-Toyota, Ford-Mazda, Volvo-Subaru, andPeugeot-Suzuki, among others. The following examples are also found in Japan. (i) Several pharmaceutical products produced by Bayer, a German firm, are sold and distributed by its Japanese rivals,
Meiji Seika and Kyorin Pharmaceutical. (ii) A Japanese liquor company, Suntory, sells wines, beers,
whiskies, brandies, liqueurs, and mineral waters made by foreign companies. (iii) Evian, a mineral water
made by French company Danone, is distributed in Japan by a Japanese beverage company, Calpis.
2
For example, in the late 1980s a number of foreign auto-makers such as BMW, Chrysler, andMercedes-Benz established their own distribution networks in Japan.
3
Our approach is fundamentally different from the incomplete contracting approach which has been
recently applied to the analysis of vertical structures in the context of international trade. For details,

2one domestic and the other foreign, produce differentiated products in their own countries
and compete in the domestic market. Post-production services must be performed before
a product is consumed. The foreign firm has the option of outsourcing post-production
services to its domestic rival by paying royalties or providing those services by itself inthe domestic market. In the latter case, however, the variable cost for services is highbecause of the lack of proximity to the domestic market. This variable cost can be
reduced if the foreign firm establishes its own service facilities in the domestic market byincurring a fixed cost for FDI. The connection between production and post-production
services, uniquely captured by our model, yields novel welfare consequences and policy
implications as outlined below.
Suppose that the tariff rate is initially high, and that the fixed cost for service FDI isalso initially high so that the foreign firm chooses to outsource post-production services to
its domestic rival. We find, contrary to the conventional result, that a tariff reduction may
hurt consumers and reduce world welfare. As in the standard analyses, the direct effect ofa tariff reduction is to benefit consumers and the foreign firm, and hurt the domestic firm.
In our framework, however, the domestic firm can mitigate the negative effect of a tariff
reduction by raising the price it charges the foreign firm for post-production services,and the higher service price hurts consumers. We demonstrate that, from the welfare
standpoint, the latter indirect effect can overshadow the former direct effect so thatthe tariff reduction actually hurts consumers and reduces world welfare in equilibrium.
Importantly, a reduction in the fixed cost for service FDI lowers the service price charged
by the domestic firm, which in turn mitigates the negative welfare effect of tariff reductionand eventually turns it into a positive effect.
Multilateral negotiations under GATT/WTO have greatly facilitated the liberalization of the trade in goods, and many countries have committed to maintain low levelsof tariff rates. However, with respect to the trade in services, although the General
Agreements on Trade in Services (GATS) came into effect in 1995 as a result of the
GATT Uruguay Round negotiations and has been contributing toward expanding trade
in services, the progress is still limited. For instance, only 52 WTO members have madecommitments to liberalizing distribution services under GATS (Roy, Marchetti, and Lim,
2006). The limited progress means that foreign firms may still have to incur substantial
extra costs for service FDI because of regulatory impediments.4 Melitz (2003), for exsee the second last paragraph of this section.
4
In his recent study on restrictiveness of FDI, Golub (2003) adopted “obligatory screening and approval procedures” and “restrictions on foreign ownership” as two main restrictions, and found that FDI

3

ample, pointed out that an exporting firm must set up new distribution channels in theforeign country and conform to all the shipping rules specified by the foreign customs
agency, and that, although some of these costs cannot be avoided, others are often manipulated by governments.5,6 When restrictions on service FDI are high, foreign firms
may have to rely on service outsourcing to perform post-production services in domestic
markets. In fact, according to OECD (2001), the number of the non-equity form of new
cross-border alliances in business services increased from 25 in 1989 to 1097 in 2000.
Our analysis uncovers a previously unnoticed importance of liberalization in the service sector. In our framework, the liberalization of the trade in services reduces the
fixed cost of service FDI, and the liberalization of the trade in goods reduces the tariff. We demonstrate that the liberalization of the trade in goods, if not accompaniedby the liberalization of service FDI, may hurt consumers and reduce world welfare, andthat the negative welfare effect is mitigated and eventually turned into a positive one
as the liberalization of service FDI makes progress. In other words, the liberalization ofservice FDI can convert a welfare-reducing trade liberalization into a welfare-enhancing
trade liberalization. Interestingly, the liberalization of service FDI improves welfare evenwhen it does not induce the foreign firm to actually undertake service FDI, because areduction of the fixed cost of service FDI lowers the service price that the domestic firm
can charge. We believe that these are important policy implications, given that postproduction services consist of an important subclass of services,7 and that foreign firms’
difficulties in undertaking post-production services in the domestic market have been
recently considered to be a serious non-tariff barrier.8
Cross-border transactions of services and FDI in services have been previously studrestrictions on business and distribution services are higher than those on manufacturing.
5
Melitz’s argument is based on a number of interviews with managers in Colombian firms making
export decisions conducted by Roberts and Tybout (1997).
6
In the late 80’s, Toysrus’ retail establishment was delayed in Japan because of the Large-scale RetailStore Low. The United States considered that its application was arbitrary and regarded the low as a
typical impediment against service FDI.
7
Browning and Singelmann (1975), for example, classified services into distribution services, producer
services, social services, and personal services, recognizing distribution services (transport, storage, retail,wholesale trade) as an important subclass of services.
8
For example, in the U.S.-Japan Auto Negotiation in 1995, the U.S. government required the Japanese
government to promote the dealership of imported cars by the domestic car producers. Foreign firms’
profitability will surely increase if the price they have to pay to outsource post-production services in
the local market is reduced. Our analysis, however, indicates that the liberalization of service FDI isequally or even more important not only for increasing foreign firms’ profitability but also for benefiting
domestic consumers and increasing world welfare.

4ied in the trade literature.9 Recently, several papers have considered market access anddistribution, an important example of post-production services, in the context of international trade. Richardson (2004) has shown in a spatial-economy model that thedomestic government has an incentive to open the access to retail distribution to foreign
manufacturers when tariffs can be used, but it may limit the access when trade policy is
not available. Francois and Wooton (2007) assume that sales of imported goods require
the domestic distribution services that are supplied under imperfect competition. They
have shown that trade volumes and the level of optimal tariff are positively related tothe degree of competitiveness in the service sector. In these previous models, production
and distribution of goods are assumed to be conducted in different industries. Qiu (2006)
has developed a model to study firms’ incentives to form cross-border strategic alliances
and their choice of entry modes in foreign markets. In his two-country, multi-firm model,
each firm’s cost of distributing its products in the foreign country is assumed to become
lower when the firm forms a strategic alliance with a firm in the foreign country. Itshould be noted that Qiu uses the term distribution costs to represent all costs incurred
after production, which are costs for post-production services in our terminology.
Our paper is related to the previous studies mentioned above in the sense that we
also investigate post-production services in the context of international trade. There
are, however, some fundamental differences. In our model, the foreign firm determines
whether it performs post-production services by itself or outsources them to its domestic
rival. This decision is made under the strategic interactions between the foreign firm
and the domestic firm, and their strategic interactions in the product market and theprovision of post-production services are linked in our model. This linkage, which is
uniquely explored in our analysis, in turn yields novel welfare and policy implications
for the liberalization of both the trade in goods and service FDI. To our knowledge,
our analysis is the first attempt to examine the linkage between FDI in post-production
services and product market competition.
Also, our analysis is distinctively different from the incomplete contracting approachthat has been recently applied to the analyses of vertical structures in the context ofinternational trade; see Antrás, 2003, 2005; Antrás and Helpman, 2004; Grossman andHelpman, 2004; and Feenstra and Hanson, 2005.10 Their analyses address the choice
between vertical integration and the purchase of a specialized input through contractual
9See, for example, Djajić and Kierzkowski (1989), Markusen (1989), Francois (1990), Konan and

Maskus (2006), and Wong, Wu, and Zhang (2006) for cross-border transactions of services, and Raff andvon der Ruhr (2001) and Markusen, Rutherford, and Tarr (2005) for FDI in services.
10
See also Spencer (2005) and Helpman (2006) for a recent survey of the literature.

5

outsourcing, where relationship-specific investments governed by incomplete contracts
play a central role. In contrast, as mentioned above, we focus on the connection between production and post-production services in the context of international trade,and examine its welfare and policy implications. Given our focus, we do not address
relation-specificity of investment and incompleteness of contracting.
The remainder of the paper is organized as follows: Section 2 develops an international duopoly model that captures the linkage between FDI in post-production services
and product market competition, and derives the equilibrium of the model. Section 3
investigates the effects of the liberalization of the trade in goods, the liberalization of FDI
for post-production services, and the connection between them. Section 4 elaborates on
the policy implications of our findings and explores the robustness of the results. Section
5 summarizes the paper and offers concluding remarks, which include a discussion on the
difference between post-production services and intermediate inputs in our framework.

2

The Model

Demands in the home country are characterized by a representative consumer who consumes non-numeráire goods as well as a numeráire good. The non-numeráire goods
consist of goods D and F which are imperfect substitutes. The numeráire good is competitively produced and freely traded between countries. The indirect utility function is
given by
V (pD , pF ) = V − a (pD + pF ) +(pF )2 (pD )2+− bpD pF + Y,
2
2

(1)

where pD and pF are the prices of good D and good F respectively, V is a positive
constant, and Y is the income in the domestic country.11 By using Roy’s identity, thedemand for each product i (∈ {D, F }) is given byxi (pi , pj ) = a − pi + bpj(i, j ∈ {D, F }, i 6= j)(2)where a (> 0) and b ∈ [0, 1) respectively represent the market size and the substitutabilityof the two products. As b gets closer to one, the two products become more similar.
We consider an international duopoly model in which the domestic firm (firm D) andthe foreign firm (firm F ) engage in Bertrand competition in the domestic market. Firm
D produces good D and firm F produces good F . The unit cost of producing goods is
11This indirect utility function is derived from a standard quasi-linear utility function given by

U (xD , xF , M ) = α (xD + xF ) − β{(xD )2 + (xF )2 }/2 − γxD xF + M where xD and xF denote the consumption of good D and that of good F respectively, and M is the consumption of the numeráire good.

6

identical across firms and constant, and is denoted by c. An ad valorem tariff, t (≥ 0),
is imposed on imports of good F .12 Post-production services must be performed forgoods to be consumed. We capture this by assuming that one unit of post-production
services must be performed for one unit of goods to be consumed. Assume that firm D
has already established its facilities to perform post-production services for good D in
the domestic market, and its unit cost for post-production services is constant and given
by cS (> 0).
Also, firm D can perform post-production services for good F at the same unit costcS by incurring a fixed cost KD , which includes costs for suitably adjusting its facilities
and learning details on how to effectively perform services for firm F ’s product.13 We
assume that post-production services can be performed only by good producers (i.e.,
firms D and F ) because of the economy of scope. The qualitative nature of our results is
unchanged under an alternative model set-up in which a non-producer of the good canalso perform post-production services for good F , but it must incur substantially higher
fixed costs than firm D because it needs to learn the basics of the business from scratch.
See Subsection 4.2 for robustness of our results under an alternative set-up in which (i)independent service organizations can also perform post-production services for good F
without incurring high fixed costs, or (ii) more than one domestic firm exist and canperform post-production services for good F .
On the other hand, if firm F has not established service facilities in the domestic
market, its unit service cost is cS + m (m > 0) which is higher than cS because of
the lack of proximity to domestic customers. For example, without maintenance andrepair shops in the domestic market, firm F has to ship goods back and forth between
two countries to perform maintenance and repair services, and this requires substantial
costs for shipping and handling. Similarly, without a marketing and sales subsidiaryin the domestic market, firm F ’s sales representatives have to make frequent businesstrips to the domestic country, requiring substantial costs for travel time, transportation
and accommodation, and travel allowances. Firm F can reduce its unit service costby establishing local facilities for performing post-production services in the domestic
market. In particular, we assume that, if firm F undertakes FDI in post-production
services by incurring a fixed investment cost KF (> 0), its unit service cost is reduced
to cS . Note that the tariff on imports is still effective even if FDI in post-production
12
13

See Subsection 4.2 for the case of specific tariffs.
Although it seems natural to assume that firm D incurs the fixed cost KD , the qualitative nature of

our results remain mostly unchanged under an alternative assumption that firm F incurs the fixed cost
KD . See also Subsection 4.2.

7

services is made, which is in contrast to “tariff-jumping” FDI in production.
In sum, we consider the following three options for firm F to perform post-production
services in the domestic market: (i) No Local Facility: Firm F performs post-production
services at the unit cost cS + m without establishing its service facilities in the domestic
market; (ii) Service FDI: Firm F reduces the unit service cost to cS by incurring afixed cost KF to establish its service facilities; or (iii) Service Outsourcing: Firm F
outsources post-production services to firm D, which incurs a fixed cost KD and charges
a service price (or royalty) of r (> 0) per unit of services.
To simplify mathematical expressions, we set cS = 0.14 We can express the profits ofthe two firms as
πD (pD , pF ) = (pD − c)xD (pD , pF ) + µ [rxF (pF , pD ) − KD ] ,pF− c − (1 − λ) (1 − µ) m − µr xF (pF , pD ) − λKF ,πF (pF , pD ) =
1+t

(3)
(4)

where µ = 1 if firm F outsources post-production services to firm D and µ = 0 otherwise,and λ = 1 if firm F makes service FDI and takes λ = 0 otherwise. Note that when λ = 1,
µ = 0 always holds.
The timing of the game is as follows.
[Stage 1]: Firm D determines whether to offer a service price of r (> 0), to which firm D
must commit. If r is offered, firm F determines whether to accept the offer. We assume
that, if firm F accepts the offer, it commits to outsourcing all post-production services
for good F in the domestic market. Under this assumption, we can treat three options– no local facility, service FDI, and service outsourcing – as distinctive alternatives. If
firm F rejects the offer, or firm D does not make an offer, firm F determines whether
to make FDI in post-production services. See Figure 1 for a game tree that depicts theinteraction between firms D and F at Stage 1.
[Stage 2]: Firms D and F simultaneously set prices of their own products, and thenconsumers make purchase decisions.
[Insert Figure 1 around here]

2.1

Product market competition

In this subsection, we derive the equilibria of Stage 2 subgames. The game has three Stage
2 subgames depending on decisions made at Stage 1: (i) No Local Facility (NLF)
subgame: Firm F performs post-production services without establishing its service
14

The assumption does not change the qualitative nature of our results.

8

facilities at Stage 1; (ii) FDI subgame: Firm F makes FDI in post-production services
at Stage 1; (iii) Outsourcing (OS) subgame: Firm F outsources post-production
services to firm D at Stage 1.
Throughout our analysis, we assume that the value of a is large enough and the tariff
rate t is small enough so that each firm i (= D, F ) sells a strictly positive amount ofgood i in the domestic market in the equilibrium. In particular, we assume that thefollowing conditions hold:2)m.
Assumption 1: a > c and t < t ≡ (2+b){a−(1−b)c}−(2−b(2−b2 )(c+m)

Let us begin with the NLF subgame and the FDI subgame. In both subgames, µ = 0
applies in equations (3) and (4). In the second stage, each firm maximizes its own profits.
The first-order conditions are given by∂πD (pD , pF )∂pD
∂πF (pD , pF )∂pF

∂xD (pD , pF )= xD (pD , pF ) + (pD − c)
= 0,
∂pD


pF
∂xF (pD , pF )xF (pD , pF )=+− (1 − λ)m − c
= 0.
1+t
1+t
∂pF

By solving these two equations, we can derive the equilibrium prices as:peD (m, t; λ) =peF (m, t; λ) =a + c b{ct + (1 + t) (1 − λ)m}+,2−b
4 − b2a + c 2{ct + (1 + t) (1 − λ)m}+.
2−b
4 − b2(5)(6)

Since the cost of supplying the domestic market is (weakly) higher for firm F than forfirm D, we have peF (m, t; λ) ≥ peD (m, t; λ). The producer price of good F is given bypeF (m, t; λ) /(1 + t). The equilibrium sales are given byxeD (m, t; λ) =xeF (m, t; λ) =a − (1 − b) c b{ct + (1 − λ) (1 + t) m}+,
2−b
4 − b2
2 − b2 {ct + (1 − λ) (1 + t) m}a − (1 − b) c−.
2−b
4 − b2(7)(8)

Note that we have xeD (m, t; λ) > 0 and xeF (m, t; λ) > 0 by Assumption 1.
Since λ = 0 in the NLF subgame, the equilibrium prices, sales, and profits in that
LF ≡ p
LF ≡ pLF ≡subgame are respectively given by pNeD (m, t; 0), pN
eF (m, t; 0), xN
DF
DLF ≡ xN LF ≡ π (pN LF , pN LF ) = (pN LF − c)2 , π N LF ≡xeD (m, t; 0), xNeF (m, t; 0), πD
D D
F
F
D
F
LF , pN LF ) = {pN LF − (1 + t) (c + m)}2 / (1 + t). Similarly, the equilibrium prices,πF (pNF
D
F

sales, and profits in the FDI subgame are respectively given by pFDDI ≡ peD (m, t; 1),
F DI ≡ π (pF DI , pF DI ) =pFF DI ≡ peF (m, t; 1), xFDDI ≡ x
eD (m, t; 1), xFF DI ≡ x
eF (m, t; 1), πD
D DF(pFDDI − c)2 , πFF DI ≡ πF (pFF DI , pFDDI ) = {pFF DI − (1 + t) c}2 / (1 + t) − KF .
9

Also, consumer surplus, domestic welfare, and world welfare in the equilibrium ofk +these subgames are respectively given by CS k ≡ V (pkD , pkF ) − Y , W k ≡ CS k + πD

tpkF xkF /(1 + t), and W W k ≡ W k + πFk , where k = N LF or F DI.
Next turn to the OS subgame, where µ = 1 and λ = 0 apply. In the remainder ofthis subsection, we consider the case in which the service price r is small enough for eachfirm i (= D, F ) to sell a strictly positive amount of good i in the domestic market in
the equilibrium. In the second stage, each firm maximizes its profits with respect to its
price given the service price, r, set by firm D in the first stage. The first-order conditions
become
∂πD
∂pD
∂πF
∂pF

∂xD (pD , pF )∂xF (pF , pD )= xD (pD , pF ) + (pD − c)
+r
= 0,
∂pD
∂pD


xF (pD , pF )
pF
∂xF (pD , pF )
=
= 0.
+
−r−c
1+t
1+t
∂pF

By solving these equations, we find the equilibrium prices in the OS subgame as:pbD (r, t) =pbF (r, t) =a + c (3 + t) br
+
≡ pOS
D ,
2−b
4 − b2a + c {2 (1 + t) + b2 }r + 2ct
+
≡ pOS
F .
2−b
4 − b2(9)(10)

d (r, t) = V (pOS , pOS ) − Y ≡ CS OS , andThe equilibrium consumer surplus is given by CS
DF
it is decreasing in each good’s price. Since both prices are increasing in r, a rise of theservice price reduces the equilibrium consumer surplus with given t.
Lemma 1 In the equilibrium of the Outsourcing subgame, a rise in the service priceincreases the prices of both goods and reduces the consumer surplus, holding the tariff
rate fixed.
Proof. Since ∂pOS
j /∂r > 0 (j = {D, F }) and ∂V (pD , pF )/∂pj = −xj < 0, we have
∂CS OS /∂r < 0.
Also, we find that the equilibrium sales arexbD (r, t) =xbF (r, t) =a − (1 − b) c b{ 1 − b2 r − (c + r) t}
−
≡ xOS
D ,
2
2−b
4
−
b


a − (1 − b) c 2 1 − b2 r + 2 − b2 (c + r) t
−
≡ xOS
F .
2−b
4 − b2(11)(12)

OS > 0 hold when r is small enough.
Note that, given Assumption 1, xOS
D > 0 and xF

The effect of an increase in r on xOS
is always negative whereas the effect on xOS
F
D is
10

ambiguous: an increase in r shifts demands from good F to good D, but it gives firm D
an incentive to raise the price of good D. The first effect increases xOS
D , while the second
effect decreases it. Thus, the overall effect on xOS
D is ambiguous.
The equilibrium profits of firms D and F are respectively given by
2
OS
OS
OS
π
bD (r, t) = (pOSD − c) + r{pF − b(pD − c) − (1 + t) (c + r)} − KD ≡ πD , (13)2{pOS
F − (1 + t) (c + r)}≡ πFOS .
(14)πbF (r, t) =(1 + t)c (r, t) = CS OS +Also, domestic welfare and world welfare are respectively given by W[
π OS + tpOS xOS /(1 + t) ≡ W OS and W
W (r, t) = W OS + π OS ≡ W W OS .
D

F

F

F

Lemma 2 In the equilibrium of the Outsourcing subgame, a rise in the service pricereduces firm F ’s profit and world welfare, holding the tariff rate fixed.


2
2
2
Proof. By (14) and ∂{pOS
F −(1 + t) (c + r)}/∂r = −{ 2 − b t+2 1 − b }/(4−b ) < 0,
we have ∂πFOS /∂r < 0. We also have ∂W W OS /∂r = −[(1 − b2 ){(2 + b)2 (a − c + bc) + (4 +5b2 )r}+t (1 − b) {a (2 + b)2 +2(2+2b+b2 )bc+2 (1 + b) 4 + b2 r}+(4−3b2 ) (c + r) t2 ]/(4−b2 )2 < 0.
An increase in r increases firm F ’s service cost and thereby decreases its profit.
Although whether it increases or decreases firm D’s profit and domestic welfare is ambiguous, it necessarily decreases world welfare since its damages to consumers and firm
F always dominate the possible benefits of firm D as well as domestic welfare.

2.2

Equilibrium of the entire game

We now derive the equilibrium of the entire game. For expositional simplicity, we adopt
the following tie-breaking rules: (i) If firm F is indifferent between accepting and rejecting
a service price r offered by firm D at stage 1, firm F accepts it. (ii) If firm D is indifferent
between offering and not offering r, firm D does not offer it. (iii) If firm F is indifferent
between making and not making a service FDI, firm F makes it. Note that, since πFF DI
is strictly decreasing in KF while πFN LF is independent of KF and πFF DI > πFN LF holds
when KF = 0, there exists a unique value KF0 (> 0) such that πFF DI ≥ πFN LF if KF ≤ KF0
and πFN LF > πFF DI if KF > KF0 .
Recall that firm F ’s profit in the Outsourcing subgame, πbF (r, t), is strictly decreasing in the service price r. Also, if firm F does not accept r offered by firm D (or if
firm D does not offer any service price), then firm F ’s subsequent equilibrium profit is11max[πFN LF , πFF DI ], which is independent of r. This implies that firm F accepts service
price r at Stage 1 if and only if r ≤ r̄ holds, where r̄ (> 0) denotes the maximum acceptable service price. Note that r̄ is uniquely determined by πbF (r̄, t) = max[πFN LF , πFF DI ].
Assumption 1 implies max[πFN LF , πFF DI ] > 0, and hence πbF (r̄, t) > 0. This in turn implies that each firm i (= D, F ) sells a strictly positive amount of good i in the domestic
market in the equilibrium of the Outsourcing subgame for all r ∈ [0, r̄].
We have the effect of an increase in r on firm D’s equilibrium profit in the Outsourcing
OS , assubgame, πD
OS
∂πD
∂r

=

 OS

∂pOS
∂pF
∂pOS
OS
D
D
2(pD − c)+r−b− (1 + t)

∂r
∂r
∂r
 OS

OS
+ pF − b(pD − c) − (1 + t) (c + r) .

(15)

Since an increase in r raises pOS
and thereby raises pOSF
D , it increases firm D’s profits
in the product market. The first term of (15) represents this strategic effect and it ispositive. Although an increase in r increases firm D’s per unit profit from performing
post-production services for firm F , it also decreases the imports of good F . Hence, thechange in firm D’s profit from performing post-production services for firm F , represented
in the sum of the second and the third terms, has an ambiguous sign.
In what follows, we focus our analysis on the range of parameterizations in whichOS /∂r > 0 holds for all r ≤ r̄. Under this condition, if firm D offers its service price∂πD

so that the offer is accepted by firm F , firm D offers the maximum acceptable pricer = r̄. This condition is satisfied when the market size, a, is sufficiently large, or thesubstitutability of products, b, is sufficiently high.15
We classify the equilibrium of the entire game as follows: (i) No Local Facility(NLF) equilibrium: Firm F performs post-production services without establishing its
service facilities in the domestic market at Stage 1; (ii) FDI equilibrium: Firm F makes
FDI in post-production services in the domestic market at Stage 1; (iii) Outsourcing(OS) equilibrium: Firm F outsources post-production services to firm D at Stage 1.
Proposition 1 characterizes the equilibrium, and Figure 2 provides a diagrammatic
representation of the proposition.
Proposition 1 The game has a unique equilibrium for any given parameterization.
0 such that the equilibrium is characterized by (i) and (ii)
There exists a threshold KD

below.
15

See the Appendix for details. We have found that the qualitative nature of the results is mostly

unchanged without imposing this condition. Without this condition, however, the analysis becomes
substantially complex without adding new insights.

120(i) Suppose KF ≤ KF0 . Then the equilibrium is an Outsourcing equilibrium if KD ≤ KD
0 , where K 0 is strictly increasing in K
and an FDI equilibrium if KD > KD
F for allD

KF ∈ (0, KF0 ].
0(ii) Suppose KF > KF0 . Then the equilibrium is an Outsourcing equilibrium if KD ≤ KD
0 , where K 0 is independent of K for
and a No Local Facility equilibrium if KD > KDFD

all KF > KF0 .
Proof: See Appendix.
[Insert Figure 2 around here]
Proposition 1 and Figure 2 can be explained as follows: First consider the case of
KF ≤ KF0 , which implies πFF DI ≥ πFN LF . The maximum acceptable service price r̄ isthen determined by πbF (r̄, t) = πFF DI , which givesqα − α2 − (4 − b2 )2 (1 + t) KF
(16)
r̄ =
(2 − b2 ) t + 2 (1 − b2 )
F DI , firm D offers r = r̄ atwhere α ≡ (2 + b) {a − (1 − b) c} − 2 − b2 ct. If π̂D (r̄, t) > πDStage 1, and the equilibrium of the entire game is an Outsourcing equilibrium. On the
F DI , firm D does not offer r at Stage 1, and the equilibriumother hand, if π̂D (r̄, t) ≤ πD
F DI isis an FDI equilibrium. Given that πbD (r̄, t) is strictly decreasing in KD while πD
0 , such that π̂ (r̄, t) > π F DI if
independent of KD , there exists a threshold, denoted KDDD
0 , while π̂ (r̄, t) ≤ π F DI otherwise. We have that the threshold K 0 is strictly
KD < K D
DDD

increasing in KF for all KF ∈ (0, KF0 ]: as the fixed cost of service FDI, KF , increases, themaximum acceptable service price r̄ increases. This in turn increases π̂D (r̄, t), resulting
0 .
in an increase in the threshold KD

Next consider the case of KF > KF0 . This implies πFF DI < πFN LF , and hence r̄ isdetermined by πbF (r̄, t) = πFN LF , which givesr̄ =(2 − b2 ) (1 + t) m≡ r̂(m, t).
(2 − b2 ) t + 2 (1 − b2 )

(17)

0 , π
N LF holds
Through an analogous procedure, we find that, if KD < KDbD (r̄, t) > πD
0 , π
N LF
and the equilibrium is an Outsourcing equilibrium, while, if KD ≥ KD
bD (r̄, t) ≤ πD

holds and the equilibrium is a No Local Facility equilibrium. In this case, r̄ is independent
0 isof KF (because πFN LF is independent of KF ), which implies that the threshold KD
0 , when it is viewed as a function of
independent of KF . Note that the threshold KD

KF , is discontinuous at KF = KF0 (see Figure 2). This is because the value of firm D’s
13

F DI to π N LF while its gains from service outsourcing
outside option jumps up from πDDincrease continuously as KF increases (see Appendix for details).
The following Corollary is useful for the analysis in the next section.
0 |00
00 , there exists a value K
Corollary 1 For any given KD < KD
KF =KFF ∈ (0, KF ) suchthat the equilibrium is an Outsourcing equilibrium if KF ∈ (KF00 , KF0 ) while it is an FDI
equilibrium if KF ∈ (0, KF00 ].

3

Liberalization of goods trade and service FDI

This section investigates the effects of the liberalization of the trade in goods, liberalization of FDI for post-production services, and the connection between them. In our
analysis, trade liberalization is represented by a reduction in the tariff rate, t, and liberalization of service FDI is represented by a reduction in the fixed cost of service FDI,KF . Let t0 ∈ (0, t̄] denote the tariff rate before trade liberalization, and KF0 (> 0) denotethe fixed cost of service FDI before liberalization of service FDI. Assume that KF0 > KF0
for all t ∈ [0, t0 ].16 That is, the pre-liberalization level of KF is so high that, if firm D
does not offer service price r at Stage 1, firm F does not invest in service FDI in thesubsequent equilibrium for any given t ∈ [0, t0 ].
As mentioned in the Introduction, the trade liberalization of goods has recently madesubstantial progress through multilateral negotiations under GATT/WTO, while theprogress of liberalization in service sectors has been slow so far. Given this, we firstinvestigate the effects of tariff reduction, holding KF fixed at the pre-liberalization level
KF0 . We then investigate the effects of the liberalization of service FDI, showing thatthe liberalization of service FDI can convert a welfare-reducing trade liberalization intoa welfare-enhancing trade liberalization. Also, in Subsection 3.3 we analyze the effectsof tariff reduction when KF is endogenously determined, and consider the effects of theliberalization of service FDI in this setup.

3.1

Outsourcing equilibrium as the pre-liberalization equilibrium

0 |In this subsection, we consider the case of KD < KD
t=t0 , so that the pre-liberalization
0 is strictly decreasing inequilibrium is an Outsourcing equilibrium. We find that KD
16

We find that KF0 is strictly decreasing in t for all t ∈ [0, t̄] (see the proof of Proposition 1 in the

Appendix). Given this, we assume KF0 > KF0 |t=0 .

14

0 |
0
t for all t ∈ [0, t̄] if KF > KF0 .17 Hence, KD < KD
t=t0 implies that KD < KD for

all t ∈ [0, t0 ]. That is, holding KF fixed at KF0 , the equilibrium of the game is an
Outsourcing equilibrium for all t ∈ [0, t0 ]. In Figure 3, Point A represents the pre0 andliberalization equilibrium that we consider in this subsection, where the shifts of KD0 |KF0 depict effects of tariff reduction.18 The case of KD > KD
t=t0 (Point B in Figure 3)

will be discussed in the next subsection.
[Insert Figure 3 around here]
First, we fix KF at KF0 and investigate the effects of tariff reduction. Since theequilibrium of the game with KF = KF0 is an Outsourcing equilibrium for all t ∈ [0, t0 ],OS respectively, and firm D
the equilibrium prices of goods D and F are pOS
D and pF

charges r = r̄ = r̂(m, t) in the equilibrium. We find that{(2 − b2 )b2 m∂ r̂(m, t)=−<0∂t{(2 − b2 ) t + 2 (1 − b2 )}2
for all t ∈ [0, t0 ]. That is, holding KF = KF0 fixed, the trade liberalization of goods
increases the equilibrium service price charged by firm D.
The logic behind this result can be explained as follows. Consider an Outsourcing
subgame. Suppose good D’s price is lowered, holding good F ’s price as fixed. Firm D’s
profit from performing post-production services for firm F then becomes lower, because
firm F sells less due to good D’s lower price. Because of this effect, firm D’s incentive
to increase its sales by lowering good D’s price is weaker in the Outsourcing subgame
than in the No Local Facility subgame. This implies that product market competition
becomes weaker if firm F outsources post-production services by accepting a service pricer offered by firm D if r is sufficiently high.19
Now suppose firm D offers r = m; that is, the service price r is equal to firm F ’sunit cost for performing post-production services without establishing local facilities.
If firm F rejects this offer, its unit service cost is m since firm F does not invest inservice FDI given KF = KF0 > KF0 . Firm F is then strictly better off by accepting theoffer r = m, because service outsourcing at r ≥ m weakens product market competition.
Consequently, firm F accepts r = r̂(m, t) > m in the equilibrium, and this in turn implies
170
We find that if KF > KF0 , KD
is decreasing in t for all t ∈ [0, t0 ] (see the proof of Proposition 1 in0
the Appendix). If KF < KF0 , on the other hand, KD
is either decreasing or increasing in t.
18
0
Figure 3 depicts the case in which KD is decreasing in t for KF < KF0 . The qualitative nature of the0
results would be unchanged if KD
is increasing in t for KF < KF0 .
19
For product market competition to become weaker, r ≥ m is sufficient but not necessary. That is,even if r < m, product market competition can still become weaker when r is sufficiently close to m.

15

that the equilibrium price of good F , pOS
F , is higher than the price firm F would charge in
LF . That is, we have pOS > pN LF
the equilibrium of the No Local Facility subgame, pNFFF

because service outsourcing at the service price r = r̂(m, t) weakens product market
competition.
Let us now turn to the effect of a tariff reduction from t to t − ∆ on the equilibrium
service price. In general, the burden of ad valorem tariff increases with the price level.
N LF /(1 + t), a tariff reduction increases firm F ’s producer
That is, given pOS
F /(1 + t) > pF

price, and consequently firm F ’s profit as well, more in the OS equilibrium than in the
NLF equilibrium, holding the service price r = r̂(m, t) fixed. In other words, a tariff
reduction benefits firm F more in the Outsourcing equilibrium than in the No Local
Facility equilibrium. This increases the maximum acceptable service price, and firm D
can increase the service price r from r = r̂(m, t) to r = r̂(m, t − ∆) in order to absorb apart of firm F ’s gain from the tariff reduction.
We call this effect the profit-absorbing motive of a service price change, which resultsin ∂ r̂(m, t)/∂t < 0. Note that r̂(m, t) is convex in t, so the degree of an increase inr̂(m, t) by a tariff reduction gets larger as the initial tariff rate gets smaller.
Concerning the effects of tariff reduction on equilibrium consumer prices pOS
D and
pOS
F , we have
∂ pOS
D r=b
r (m,t)
∂t
∂ pOS
F r=b
r (m,t)∂t∂ r̂(m, t)
∂t



=

br̂(m, t) (3 + t) b+
4 − b24 − b2=2 (c + r̂(m, t)) 2 (1 + t) + b2+
4 − b24 − b2,∂ r̂(m, t)∂t.

Since ∂ r̂(m, t)/∂t < 0, the signs of the above equations are ambiguous. Although a tariff
reduction directly reduces the commodity prices by reducing costs of firm F , it raises
them through the indirect effect caused by an increase in the service price. If the latter
effect dominates the former, a tariff reduction increases equilibrium consumer prices.
This can indeed be the case, as shown by Lemma 3.
Lemma 3 Holding KF fixed at KF0 , there exist critical values t0 (> 0), b0 (> 0), and ζ 0000(> 0) such that ∂pOS
D /∂t < 0 holds if and only if t < t , b > b , and c/m < ζ . Therealso exist critical values t00 (< t0 ), b00 (> b0 ), and ζ 00 (< ζ 0 ) such that ∂pOS
D /∂t < 0 and
00
00
00
∂pOS
F /∂t < 0 hold if and only if t < t , b > b , and c/m < ζ .

Lemma 3 indicates that, if KF is fixed at KF0 , a tariff reduction may hurt consumers
and reduce world welfare by increasing equilibrium consumer prices. Proposition 2 formalizes this by investigating how the trade liberalization of goods, if not accompanied16

by the liberalization of FDI for post-production services, affects consumers, world welfare, and firms’ profitability. In what follows, let CS(KF , t), W W (KF , t), πD (KF , t), andπF (KF , t), respectively, denote consumer surplus, world welfare, firm D’s profit, and firm
F ’s profit in the equilibrium of the entire game. Note that, since both πFN LF and πFF DI
are decreasing in t and πF (KF , t) = max[πFN LF , πFF DI ] in equilibrium, ∂πF (KF , t)/∂t < 0always holds.
Proposition 2 Holding KF fixed at KF0 , there exists a range of parameterizations inwhich a tariff reduction hurts consumers, decreases world welfare, and benefits firm D.
More precisely, ∂CS(KF , t)/∂t > 0, ∂W W (KF , t)/∂t > 0, and ∂πD (KF , t)/∂t < 0 holdif t < t00 , b > b00 , and c/m < ζ 00 .
Holding KF fixed at KF0 , a tariff reduction induces firm D to increase its service pricedriven by the profit-absorbing motive, and Lemma 3 tells us that this can in turn increase
equilibrium consumer prices. This is the driving force of the negative welfare effects ofthe trade liberalization of goods when it is not accompanied by the liberalization ofservice FDI. In particular, Proposition 2 tells us that a tariff reduction necessarily hurts
consumers, decreases world welfare, and benefits firm D, if it increases equilibrium prices
of both goods D and F (i.e., if t < t00 , b > b00 , and c/m < ζ 00 ).20 Note that this conditionis sufficient but not necessary. The negative welfare effects may persist when a tariff
reduction increases good D’s price but decreases good F ’s price. That is, consumers areworse off when the negative effect of the tariff reduction due to an increase in good D’s
price outweighs its positive effects because of a decrease in good F ’s price.21
Next we consider the effects of the liberalization of service FDI, represented by a
0 for all t ∈ [0, t ],reduction in KF from KF0 , holding the tariff rate fixed. Since KD < KD
0

Corollary 1 tells us (see also Figures 2 and 3) that for any given t ∈ [0, t0 ], there exists
a value KF00 ∈ [0, KF0 ) such that the equilibrium of the entire game is an Outsourcing
equilibrium if KF > KF00 and an FDI equilibrium if KF ≤ KF00 .
Proposition 3 tells us that, holding the tariff rate fixed, liberalization of service FDI
benefits consumers, increases world welfare, and hurts firm D, at least weakly. See Figure
4 for a diagrammatic representation of the proposition.
20

In international oligopoly models, import tariffs have strategic effects that cause rent-shifting, andhence the welfare of domestic country, W (KF , t), can be either increasing or decreasing in t even if∂CS(KF , t)/∂t > 0 holds. Since ∂πF (KF , t)/∂t < 0 is always satisfied, however, ∂W (KF , t)/∂t > 0
holds whenever ∂W W (KF , t)/∂t > 0 holds.
21
Also, even when a tariff reduction decreases the prices of both goods D and F , firm D’s profit may
still increase because of the higher service price it can charge.

17

Proposition 3 For any given t ∈ [0, t0 ], CS(KF , t), W W (KF , t), and πF (KF , t) aredecreasing in KF while πD (KF , t) is increasing in KF for all KF ∈ (0, KF0 ], with thefollowing properties:(i) ∂CS(KF , t)/∂KF = 0 for all KF < KF00 , ∂CS(KF , t)/∂KF < 0 for all KF ∈(KF00 , KF0 ), and ∂CS(KF , t)/∂KF = 0 for all KF > KF0 .
(ii) ∂W W (KF , t)/∂KF < 0 for all KF < KF00 , ∂W W (KF , t)/∂KF < 0 for all KF ∈(KF00 , KF0 ), and ∂W W (KF , t)/∂KF = 0 for all KF > KF0 .
(iii) ∂πF (KF , t)/∂KF < 0 for all KF < KF00 , ∂πF (KF , t)/∂KF < 0 for all KF ∈(KF00 , KF0 ), and ∂πF (KF , t)/∂KF = 0 for all KF > KF0 .
(iv) ∂πD (KF , t)/∂KF = 0 for all KF < KF00 , ∂πD (KF , t)/∂KF > 0 for all KF ∈(KF00 , KF0 ), and ∂πD (KF , t)/∂KF = 0 for all KF > KF0 .
[Insert Figure 4 around here]
A reduction in KF has pro-competitive effects when it reduces the equilibrium service price r̄ charged by firm D because a reduction in the service price intensifies thecompetition between firms D and F in the product market. Recall that πFF DI ≥ πFN LF
if and only if KF ≤ KF0 . If KF ∈ (KF00 , KF0 ], a reduction in KF increases πFF DI , which
in turn decreases the maximum acceptable service price r̄ because r̄ is determined byπbF (r̄, t) = πFF DI in this case. Hence a reduction in KF benefits consumers and increases
world welfare in the interval (KF00 , KF0 ].22 It is interesting to note that, in this interval, areduction in KF yields pro-competitive consequences even though the reduction in KF
does not induce firm F to actually invest in service FDI in the equilibrium. At KF = KF00 ,firm F invests in service FDI, and this increases the consumer surplus and world welfare
in a discontinuous manner (see Figure 4). Beyond this point, further reduction in KF
in the interval (0, KF00 ) has no effects on firm D’s profit and consumer surplus, though it
increases world welfare by increasing firm F ’s profit.
We now explore the connection between trade liberalization of goods and liberalization of service FDI. Proposition 2 tells us that there exists a range of parameterizations in which CS(KF0 , t1 ) < CS(KF0 , t0 ) and W W (KF0 , t1 ) < W W (KF0 , t0 ) hold, where0 ≤ t1 < t0 ≤ t̄. That is, holding KF fixed at KF0 , trade liberalization represented by
the reduction in tariff rate from t0 to t1 can hurt consumers and reduce world welfare.
Proposition 4 below tells us that the negative welfare effects of the tariff reduction from22

In the interval (KF0 , KF0 ], a reduction in KF has no impacts, because r̄ is determined by πbF (r̄, t) =πFN LF , where a change in KF has no effect on this equation.

18

t0 to t1 disappear and turn into positive ones when KF is reduced to a sufficiently low
level by the liberalization of service FDI.
Proposition 4 Take any parameterization in which CS(KF0 , t1 ) < CS(KF0 , t0 ) andW W (KF0 , t1 ) < W W (KF0 , t0 ) hold, where 0 ≤ t1 < t0 ≤ t̄. Note that KF0 and KF00
are evaluated at t = t1 in (i) and (ii) below.
(i) There exists a unique K̃FCS ∈ [KF00 , KF0 ) such that CS(KF , t1 ) > CS(KF , t0 ) for all
KF ∈ (0, K̃FCS ), where KF00 < K̃FCS holds under a range of parameterizations.
(ii) There exists a unique K̃FW W ∈ [KF00 , KF0 ) such that W W (KF , t1 ) > W W (KF , t0 ) for
all KF ∈ (0, K̃FW W ), where KF00 < K̃FW W holds under a range of parameterizations.

The logic behind Proposition 4 can be explained as follows: Let us consider what happens when liberalization of service FDI reduces KF from KF = KF0 . The reduction does
not affect the equilibrium service price and has no welfare effects, as long as KF > KF0
holds. Once KF becomes smaller than KF0 , further reduction in KF reduces the maximum acceptable service price r̄, resulting in a lower equilibrium service price. The lower
service price intensifies the competition between firms D and F in the product market,
mitigating the negative welfare effects of tariff reduction. Interestingly, the liberalizationof service FDI can convert a welfare-reducing tariff reduction into a welfare-enhancing
tariff reduction even if the reduction of KF does not induce firm F to actually make service FDI. In particular, if KF is reduced from KF = KF0 to KF ∈ (KF00 , min[K̃FCS , K̃FW W ]),then the tariff reduction from t0 to t1 benefits consumers and increase world welfare eventhough KF is not low enough for firm F to make service FDI.
As Proposition 5 below tells us, when the liberalization of service FDI reduces KF
to a sufficiently low level so that firm F actually makes service FDI in the equilibrium,
any tariff reduction has positive welfare effects.
Proposition 5 Under an FDI equilibrium, a tariff reduction necessarily benefits consumers, hurts firm D, and increases world welfare. More precisely, ∂CS (KF , t) /∂t < 0,
∂πD (KF , t) /∂t > 0, and ∂W W (KF , t) /∂t < 0 hold for all KF < KF00 .

3.2

No Local Facility equilibrium as the pre-liberalization equilibrium

0 |This subsection discusses the case of KD > KD
t=t0 , so that the pre-liberalization equilib-

rium is a No Local Facility equilibrium. As in the previous section, consider a reduction
19

in tariff from t0 to t1 (0 ≤ t1 < t0 ≤ t̄), holding KF fixed at KF0 . There are two sub0 |0cases. One subcase is KD > KD
t=t1 (> KD |t=t0 ), where trade liberalization does notchange the nature of the equilibrium. That is, the equilibrium of the game is a No Local
Facility equilibrium not only before but also after the tariff reduction. In the No Local
Facility equilibrium, firm F does not outsource post-production services to firm D, andhence the profit-absorbing motive of a service price change, which was identified in theprevious subsection as the source of the negative welfare effects of trade liberalization, is
not relevant here. This implies that trade liberalization has standard welfare effects ofbenefiting consumers and increasing world welfare in this subcase.
0 |0The other subcase is KD
t=t1 > KD > KD |t=t0 , where trade liberalization changes

the nature of the equilibrium from a No Local Facility equilibrium to an Outsourcing
equilibrium (see Point B in Figure 3). Recall that outsourcing of post-production services
weakens the competition between firms D and F in the product market. Because ofthis effect, we find that the tariff reduction from t0 to t1 can have negative welfare
effects as depicted in Figure 5. As in Proposition 4, the negative welfare effects oftrade liberalization disappear and turn into positive ones when trade liberalization isaccompanied by the liberalization of service FDI.
[Insert Figure 5 around here]

3.3

Effects of trade liberalization with endogenous FDI costs

Thus far we have explored the effects of liberalization of goods trade and service FDI bytreating the tariff rate and the fixed cost of service FDI as exogenous variables. As discussed in the Introduction, many countries have committed to maintain low levels of tariff
rates under GATT/WTO multilateral agreements. Consequently, for many countries it
is no longer possible to use tariffs as flexible policy instruments to enhance domestic
welfare. In contrast, concerning service FDI, the limited progress of GATS means thatcountries can still manipulate the inflows of service FDI by raising the levels of regulatory impediments. In this subsection, we investigate the effects of trade liberalization (areduction of tariff rate from t0 to t1 ) with endogenous FDI costs by assuming that the domestic government chooses KF to maximize the domestic welfare under the exogenously
given level of tariff rate t. We shall then consider the effects of the liberalization of service
FDI, and demonstrate that the qualitative nature of our results remains unchanged inthis setup.
Suppose that, before Stage 1, the domestic government chooses the level of KF at

20

Stage 0, taking the tariff rate t as given. For any given t ∈ (0, t̄], there exists a unique levelof KF , denoted by KF∗ (t), that the domestic government chooses to maximize domestic
welfare in the equilibrium. Given the pre-liberalization level of tariff rate t0 ∈ (0, t̄],the domestic government chooses KF = KF∗ (t0 ) at Stage 0. Parallel to our analysis
in Subsection 3.1, in what follows we consider the case in which the pre-liberalization
equilibrium (that is, the equilibrium under t = t0 and KF = KF∗ (t0 )) is an Outsourcing
equilibrium. The qualitative nature of our results are mostly unchanged under cases inwhich the pre-liberalization equilibrium is an FDI equilibrium or an NLF equilibrium.
Proposition 6 There exists a range of parameterizations in which(i) the pre-liberalization equilibrium is an Outsourcing equilibrium, and(ii) CS(KF∗ (t1 ), t1 ) < CS(KF∗ (t0 ), t0 ) and W W (KF∗ (t1 ), t1 ) < W W (KF∗ (t0 ), t0 ) holdwhere 0 ≤ t1 < t0 ≤ t̄.
Proposition 6 tells us that liberalization of goods trade can have negative welfare
effects. Suppose that the tariff rate is reduced from t0 to t1 . If KF is fixed at KF∗ (t0 ),a result similar to Proposition 2 holds; that is, the tariff reduction can hurt consumers
and decrease world welfare under a range of parameterizations. In response to the tariff
reduction, the domestic government changes KF from KF∗ (t0 ) to KF∗ (t1 ) to maximize
domestic welfare. Proposition 3 tells us that, holding t = t1 fixed, equilibrium consumer
surplus is decreasing in KF while firm D’s equilibrium profit is increasing in KF . We
also find that the equilibrium tariff revenue can be either increasing or decreasing in (or anon-monotone function of) KF . Hence the relationship between domestic welfare and KF
is ambiguous, and it depends on parameterizations.23 We have found that the domestic
government may increase KF in response to the tariff reduction, or may decrease it but
not to a low enough level that induces firm F to actually invest in service FDI. In suchcases, the tariff reduction can hurt consumers and decrease world welfare.
Proposition 7 Take any parameterization in which CS(KF∗ (t1 ), t1 ) < CS(KF∗ (t0 ), t0 )and W W (KF∗ (t1 ), t1 ) < W W (KF∗ (t0 ), t0 ) hold, where 0 ≤ t1 < t0 ≤ t̄. Note that KF00 is
evaluated at t = t1 in (i) and (ii) below.
(i) There exists a unique K̂FCS ∈ [KF00 , KF∗ (t1 )) such that CS(KF , t1 ) > CS(KF∗ (t0 ), t0 )for all KF ∈ (0, K̂FCS ), where KF00 < K̂FCS holds under a range of parameterizations.
23

In international oligopoly models, any policy that increases foreign firms’ operation costs generates

strategic effects that cause rent-shifting, and hence such a policy tends to result in ambiguous welfare
effects in each country.

21

(ii) There exists a unique K̂FW W ∈ [KF00 , KF∗ (t1 )) such that W W (KF , t1 ) > W W (KF∗ (t0 ), t0 )for all KF ∈ (0, K̂FW W ), where KF00 < K̂FW W holds under a range of parameterizations.
Proposition 7 is parallel to Proposition 4, telling us that the negative welfare effectsof trade liberalization disappear and turn into positive ones when trade liberalizationis accompanied by the liberalization of service FDI. A necessary condition for the tariff
reduction to hurt consumers and decrease world welfare is KF∗ (t1 ) > KF00 ; that is, thedomestic government sets the level of KF at a relatively high level so that firm F does notinvest in service FDI. Proposition 7 tells us that, by forcing the domestic government to
reduce KF , multilateral negotiations such as GATS can increase world welfare as well as
consumer surplus (see Proposition 3), and convert a welfare-reducing trade liberalization
into a welfare-enhancing trade liberalization.

4

Discussion

We have shown that the liberalization of trade in goods could hurt consumers, reduceworld welfare, and benefit the domestic firm when the foreign firm outsources postproduction services to the domestic rival firm, but that a reduction in the fixed cost ofservice FDI can mitigate and eventually eliminate these anti-competitive effects of tradeliberalization. In this section, we first discuss the policy implications of our results, andthen explore the robustness of our results under several alternative modelling choices.24

4.1

Policy implications

Through multilateral negotiations under GATT/WTO, countries have been lowering thebarriers for the trade in goods. Growing attention is now being paid to the market accessof foreign firms in the service sector. The GATT Uruguay Round negotiations succeededin establishing the framework of liberalizing cross-country transactions of services, that is,
the GATS. The actual degree of liberalization, however, has been relatively small. For
instance, only 52 WTO members have made commitments to liberalizing distribution
services under GATS (Roy, Marchetti, and Lim, 2006). Under the limited progress ofliberalization in the service sector, many foreign firms still face significantly high costs
for service FDI, which prevent them from establishing local service facilities to perform
post-production services by themselves in the local market.
24

Detailed analyses of the robustness of our results are available upon request.

22

In our theoretical framework, the current state of the world corresponds to a situationin which the tariff rate t is reduced to a reasonably low level but the fixed cost for service
FDI, KF , is still high. Our comparative statics results then suggest that further progressof trade liberalization under GATT/WTO may hurt consumers and decrease world welfare, if it is not accompanied by the liberalization of service FDI. Our analysis uncovers apreviously unnoticed importance of liberalization of service FDI in its connection to tradeliberalization. That is, the liberalization of service FDI is important not only because it
reduces per-unit costs of post-production services but also because it recovers the gains
from the trade liberalization of goods for both consumers and world welfare. We have also
found that the qualitative nature of the results remains unchanged under an alternative
setup in which, without the liberalization of service FDI, KF is endogenously determinedby the domestic government. Therefore, making progress on the liberalization of service
FDI under GATS is crucial to secure positive welfare consequences of trade liberalization
under GATT/WTO. Since the anti-competitive effects of trade liberalization are more
likely as the tariff becomes lower (see Proposition 2), the liberalization of service FDI
becomes more important as trade liberalization proceeds.
Recently, many regional trade agreements (RTAs) have established codes for the liberalization in the service sector in addition to those for the liberalization of the trade ingoods. In these RTAs, some countries have undertaken further commitments on the liberalization of FDI in post-production services on top of the existing GATS commitments.
For instance, in its RTA with Australia, Thailand allows Australian firms 100% foreign
equity ownership for distribution of their products, even though it limits foreign equity
ownership up to 49% in its GATS distribution commitments.25 Singapore made broader
commitments on the retailing of certain goods in its RTAs with the US, Australia, andKorea. Our analysis indicates that RTAs with deeper commitments towards the liberalization of service FDI are more likely to make trade liberalization pro-competitive,
suggesting that the recent proliferation of RTAs may be superior to multilateral liberalization under GATT/WTO.
Our model also yields a new policy implication regarding foreign producers’ accessibility to service outsourcing in the local market. Suppose that the values of KF and KD
are initially high so that the model exhibits a No Local Facility equilibrium, and that25

Oman has also undertaken similar commitments in its RTA with the US. Many countries which had

no GATS commitments in distribution services, such as Bahrain, Chile, Colombia, Costa Rica, Dominican
Republic, El Salvador, Guatemala, Honduras, Morocco, and Nicaragua have undertaken commitments
in distribution services in their RTAs with the US. See Roy, Marchetti, and Lim (2006) for details.

23the domestic government implements a policy to reduce KD , firm D’s fixed cost for performing post-production services for firm F . A sufficient reduction of KD switches theequilibrium to an Outsourcing equilibrium, in which firm D charges a service price thatis higher than m, product-market competition becomes weaker, and consumer surplus
and world welfare both become lower than in the No Local Facility equilibrium. Ourmodel therefore offers a warning to the government on the possibility that such a policy
may result in negative welfare consequences unless it is accompanied by the liberalizationof service FDI.
We end this subsection by commenting on horizontal FDI. Since horizontal FDI inproduction to serve the local market ‘jumps’ tariffs, it has the same effect as a tariffelimination. In our model, the tariff elimination may hurt consumers and reduce world
welfare if it is not accompanied by the liberalization of service FDI. Our findings therefore
indicate that, to secure its positive welfare consequences, the liberalization of FDI inproduction should be accompanied by the liberalization of service FDI.

4.2

Robustness

Bargaining power: The assumption that firm D makes a take-it-or-leave-it offer gives
firm D all the bargaining power to set the service price. The qualitative nature of ourresults would remain unchanged under more general bargaining procedures in which thetwo firms share the surplus from service outsourcing, as long as firm D has sufficiently
strong bargaining power. Recall that, in the Outsourcing equilibrium, firm D offers
the maximum acceptable service price r = r̄ at which firm F is indifferent between
outsourcing the post-production services and performing the services by itself. If r̄ issufficiently high, a tariff reduction can increase the equilibrium service price through thelogic presented in Subsection 3.1 (see the third to fifth paragraph of the subsection), andthis effect is the driving force of our main comparative results presented in Propositions
2 to 5. If the two firms share the surplus through bargaining, the equilibrium service
price, r̂, is less than r̄ and decreasing in firm F ’s bargaining power. However, as long as
firm D’s bargaining power is sufficiently strong, r̂ takes a sufficiently high value so that atariff reduction still increases the equilibrium service price. Then, our main results holdunder a more general bargaining set-up in a range of parameterizations.
More than one domestic firms: We have analyzed the strategic interaction between
firms D and F by assuming that only one firm can produce the final good in the domestic
country. One can consider an alternative set-up in which N (≥ 2) symmetric domestic

24

firms, indexed by D1, D2, ..., DN , produce differentiated products. In the presenceof more than one domestic firm, the qualitative nature of our main results remains
unchanged if firm F negotiates prices for service outsourcing with one domestic firm at atime in a sequential fashion. In particular, suppose that firm F first negotiates with firm
D1 on service prices, and, if firm F decides not to outsource services to firm D1, thenfirm F performs services by itself (No Local Facility or Service FDI) or negotiates withfirm D2, and so on.26 In the product-market competition stage, N + 1 firms compete
in a Bertrand fashion under differentiated oligopoly. We find that the alternative model
has a unique pure-strategy equilibrium, which is a No Local Facility equilibrium, an FDI
equilibrium, or an Outsourcing equilibrium, depending on parameterizations. In the
Outsourcing equilibrium, the equilibrium service price r̂ can increase as tariff is reduced,
and our main comparative statics results hold under a range of parameterizations.
Alternatively, suppose that domestic firms simultaneously offer service prices, andfirm F accepts one offer or rejects all offers. There exists no pure strategy equilibriumin this case since domestic firms incur KD in providing services (see Sharkey and Sibley,1993). Each firm chooses the probability of offering service prices in a mixed strategy
equilibrium, and the case in which only one firm offers the maximum acceptable service
price r = r remains an equilibrium with a positive probability. Also, if domestic firms
have different per-unit costs for performing post-production services for firm F , there
exists a pure-strategy equilibrium in which a domestic firm offers r = r̄ when the domestic
firm’s cost is substantially lower than other domestic firms’ costs.
Independent service organizations: Our model assumes that post-production services can be performed only by goods producers (firms D and F ) because of economy ofscope. Alternatively, suppose that several independent service organizations (ISOs) can
also perform post-production services for firm F at constant marginal cost k (≥ 0), andfor simplicity assume that their fixed costs are zero. In this alternative set-up, firm F
has an option of outsourcing services to an ISO at the per-unit service price of k. The
logical structure of this alternative set-up is identical to the one of the base model if
k = m, and they are similar as long as k > 0, implying that the qualitative nature of
our results remains unchanged in this alternative set-up as long as k > 0. Also, the free26

The sequential set-up can be regarded as approximating the following scenario, which we feel is fairly

realistic: When a foreign firm attempts to outsource post-production services to one of its domestic rivals,
the foreign firm needs to identify a candidate firm by incurring search costs, and negotiate the terms ofservice outsourcing with the candidate. If the negotiation is unsuccessful, the firm will identify anothercandidate to negotiate with.

25

entry of ISOs does not change the results.
Two-part tariff for service outsourcing: We have focused on per-unit royalties forservice outsourcing by assuming that firm D offers a per-unit service price r to perform
post-production services for firm F . The qualitative nature of our results would remainunchanged under an alternative set-up in which firm D can offer a two-part tariff (R, r)(R ≥ 0, r ≥ 0), where R denotes a fixed fee and r denotes a per-unit royalty. In the
Outsourcing equilibrium, firm D chooses (R, r) to maximize its profit πbD (r, t)+R subject
to π
bF (r, t) − R = max[πFN LF , πFF DI ]. That is, firm D chooses (R, r) so that firm F isindifferent between outsourcing services and performing them by itself. We have foundthat R = 0 and r = r̄ holds under a range of parameterizations.27 That is, given that
product market competition becomes weaker as the royalty rate r increases, there isa range of parameterizations in which firm D offers the maximum acceptable royalty
rate r = r̄ and the zero fixed fee even when a two-part tariff is allowed. Also, even if
r = r̄ does not hold and r < r and R > 0 holds in the equilibrium, a decrease in t canstill increase r and the qualitative nature of our results is unchanged under a range ofparameterizations.28
Cournot competition: Consider an alternative set-up in which firms compete against
each other by choosing quantities. Suppose that, in an Outsourcing subgame, firm D
increases the quantity of good D, holding the quantity of good F fixed. This does notaffect firm D’s profit for performing post-production services for firm F since the quantity of good F is fixed. That is, unlike Bertrand competition, Cournot competition does
not capture the idea that, although firm D can increase the sales of its own productby adopting a more aggressive strategy, such a strategy also reduces its profit from performing services for its rival firm. This in turn implies that, under Cournot competition,
service outsourcing does not weaken product market competition. Since this effect isthe driving force of our main comparative statics results, our findings do not hold under
Cournot competition.
Specific tariff : We have considered ad valorem tariffs, given their prevalence in the realworld. In the case of specific tariffs, it can be shown that the equilibrium service pricebecomes independent of the tariff rate with linear demands. With non-linear demand
functions, however, it can be shown that the qualitative nature of our results remains
27

If the market size represented by a is large enough, then ∂{bπD (r, t) + πbF (r, t)}/∂r > 0 holds, which

in turn implies that R = 0 and r = r̄ hold in the Outsourcing equilibrium.
28
Firm D always offers r > 0 because under Bertrand competition with differentiated products,∂{b
πD (r, t) + πbF (r, t)}/∂r > 0 necessarily holds at r = 0.

26unchanged in the case of specific tariffs.
The fixed cost for service outsourcing: We have assumed that firm D incurs a fixed
cost for service outsourcing, KD . If firm F incurs KD instead, its profit in the Outsourcing equilibrium (πFOS ) is decreasing in KD and hence the maximum acceptable service
price, r, is also decreasing in KD . Although this changes the quantitative details of theresults, their qualitative nature remains unchanged. That is, product market competition becomes weaker in the Outsourcing equilibrium when r̄ is sufficiently high, and thisin turn leads to results analogous to Propositions 2 to 5 in a range of parameterizations.
Non-tariff barriers: We can interpret t as a proxy for non-tariff barriers rather thana tariff. Then a reduction in t results in the saving of real costs, which works in favor ofworld welfare. We have found that W W (KF0 , t1 ) < W W (KF0 , t0 ) (where 0 ≤ t1 < t0 ≤ t̄)
can hold even in this case under a range of parameterizations, and hence Propositions
4 and 5 hold under the alternative interpretation of t. Also, domestic welfare can stilldecrease when KF is reduced, and hence Propositions 6 and 7 hold even though thedomestic country does not earn tariff revenue in this case.

5

Conclusion

Post-production services such as sales, distribution, and maintenance consist of an important subclass of services. Although the liberalization of the trade in goods has madesubstantial progress through multilateral negotiations under GATT/WTO, the progressof the liberalization in the service sector has been limited so far. In this paper, we haveuncovered a previously unnoticed importance of liberalization in the service sector by exploring an international duopoly model that captures the linkage between product market
competition and provision of post-production services. That is, we have found that thetrade liberalization of goods may have negative welfare effects if it is not accompaniedby the liberalization of service FDI.
Trade liberalization reduces trade costs, and this intensifies competition between aforeign firm and a domestic firm in the product market. At the same time, when the foreign firm outsources post-production services, trade liberalization induces the domestic
firm to charge a higher service price to absorb a part of the foreign firm’s incremental
profit due to lower trade costs. We have demonstrated that, if the foreign firm’s fixed
cost of service FDI is relatively high, the latter negative welfare effect overshadows theformer positive one so that trade liberalization hurts consumers and reduces world welfare in a range of parameterizations. Importantly, this negative welfare effect of trade27liberalization is mitigated and eventually turned into a positive one as service FDI isalso liberalized. This is because a reduction in the fixed cost of service FDI decreases
the price of service outsourcing that the foreign firm would accept. We have found thatthe qualitative nature of the results remains unchanged under an alternative setup inwhich, without the liberalization of service FDI, KF is endogenously determined by thedomestic government.
Our analysis has therefore indicated that the liberalization of service FDI is important
not only because it reduces per-unit costs of post-production services but also because it
recovers gains from trade liberalization in goods for both consumers and world welfare.
Making progress on the liberalization of service FDI under GATS is crucial to secure
positive welfare consequences of trade liberalization under GATT/WTO.
We offer two final remarks to conclude the paper. First, we comment on the difference
between post-production services and intermediate inputs in our framework. In our international duopoly model, the foreign firm has an option of outsourcing post-production
services to its domestic rival or performing the services by itself in the domestic market.
It is possible to consider a model with an analogous logical structure in which postproduction services are replaced by intermediate inputs. For example, one can consider
a foreign firm that does not have the facilities to produce an intermediate input, andcan suppose that the foreign firm determines whether it procures the intermediate input
from its domestic rival or produces the input by building its own production facilities.
Since intermediate inputs are not services but goods, service FDI has no direct effects on the foreign firm’s make-or-buy decision. In contrast, the liberalization of service
FDI plays a critical role in our framework. That is, in order to perform post-production
services effectively, the foreign firm needs to undertake service FDI and establish its
own service facilities in the domestic market because of the importance of proximity
to customers. In our analysis, the connection between production and post-production
services has yielded a novel policy implication that trade liberalization should be accompanied by the liberalization of service FDI to secure its positive welfare effects. The
recent progress of trade liberalization is not yet accompanied by the sufficient progressof liberalization of service FDI, and this reality has motivated us to study the connection
between production and post-production services in international contexts.29
29

Also, since the liberalization of the trade in goods affects the intermediate-goods market as well as the

final-goods market, its policy implications may be different between the model with intermediate inputs
and the model with post-production services. To the best of our knowledge, the model of intermediate
inputs as mentioned above has not been previously explored (see Chen, Ishikawa, and Yu (2004) for a
related analysis) and its investigation is left to future research.

28

Second, given that imperfect competition in the product market is an important element of our analysis, one might argue that strengthening product market competition
could be a substitute for liberalizing service FDI. Suppose that the domestic government has stimulated competition in the product market by inducing firms’ entry, andconsequently there are N (≥ 2) domestic firms. As we discussed in Subsection 4.2, theliberalization of service FDI can still be critical in recovering gains from the trade liberalization of goods, as long as the price for service outsourcing is determined by one-to-one
negotiations. Also, although the domestic government could induce the entry of someindependent service organizations, the liberalization of service FDI can still be criticalas discussed in Subsection 4.2.

Appendix
Conditions for ∂ π̂D (r, t)/∂t > 0 for r ∈ [0, r̄]By (13),a (2 + b) {(1 + b) 4 − 2b + b2 + 2bt} −{ 1 − b2  (2 + b) 4 − 2b + b2  + 2t (1 − b) 4 + 6b + b2  + 2b2 t2 }c −2β (b, t) r∂b
πD (r, t)=∂r(4 − b2 )2where β (b, t) = 8(1 − b2 ) (1 + t) + b2 (1 − b2 − t2 ). When b is large and t > 0, β (b, t) < 0
and then ∂bπD (r, t)/∂r > 0 always holds. When β (b, t) > 0, ∂bπD (r, t)/∂r is concave in r.
When it is evaluated at r = rb(m, t), we have"∂bπD (r, t)=∂r
r=br(m,t)#
a (2 + b) {(1 + b) 4 − 2b + b2 + 2bt}−{ 1 − b2 (2 + b) 4 − 2b + b2 + 2t (1 − b) 4 + 6b + b2 + 2b2 t2 }c(4 − b2 )2−

2m 2 − b2(1 + t) β (b, t)

2 2(4 − b ) {(2 − b2 ) t + 2 (1 − b2 )}and it is positive if a is large enough, and c and m are small enough to satisfy [a(2 +b){(1 + b) 4 − 2b + b2 + 2bt} − {(1 − b2 )(2 + b)(4 − 2b + b2 ) + 2t(1 − b)(4 + 6b + b2 ) +2b2 t2 }c]{(2 − b2 )t + 2(1 − b2 )}/2(2 − b2 )(1 + t)β (b, t) > m. Since rb0 (KF , t) is increasing
0

in KF with rb0 (KF , t) = rb(m, t), rb0 (KF , t) < rb(m, t) is satisfied and so ∂bπD (r, t)/∂r > 0
also holds for KF ∈ [0, KF0 ). Thus, ∂bπD (r, t)/∂r|r=br0 (KF ,t) > 0 is also satisfied as long as∂b
πD (r, t)/∂r|r=br(m,t) > 0 holds.
29

Proof of Proposition 1
By comparing πFF DI with πFN LF , we have
2 − b2 [2 (2 + b) {a − (1 + b) c} − (2 − b2 ){2ct + (1 + t) m}]m
F DI
N LF
≡ KF0 T KF
πF −πFT 0 ⇐⇒
2
2(4 − b )where KF0 is the cut-off value of the fixed cost of service FDI. We can verify that KF0 isstrictly decreasing in t.
OS > π F DI holds. By
Under KF ≤ KF0 , πFF DI ≥ πFN LF holds and firm D offers r if πDD
OS with π F DI , we havecomparing πDDqα − α2 − (4 − b2 )2 (1 + t) KF Z1OSF DI
e 0 T KD
≡K
(A1)
πD
− πD
T 0 ⇐⇒
D(4 − b2 )2 {(2 − b2 ) t + 2 (1 − b2 )}2where Z1 ≡ b 1 − b2 (2 + b) 4 − 3b + 2b2 {a − (1 + b) c} + b (1 + b) {a (2 + b) (8 − 6b +
2b2 − b3 ) − (1 − b)2 16 + 2b + b3 c}t + b{a (2 + b) 4 + b − 2b2 − c (1 − b) (8 − 7b2 + 2b −q2b3 )}t2 + 2 − b2 b2 ct3 + {8 − 7b2 − b4 + 8(1 − b)2 t − b2 t2 } α2 − (4 − b2 )2 (1 + t) KF .
0 represents firm D’s gains from service outsourcing gross of the fixed cost
Note that KD
F DI holdsof service outsourcing. Since πbD (r, t) is increasing in r, πbD (r, t) + KD = πD0 =πF DI + K is always positive (i.e.,evaluated at r = 0, and r > 0 holds, KD
bD (r, t) − πDD0e
Z1 > 0). Note that K is either increasing or decreasing in t.

D0OS > π N LF holds. By
Under KF > KF , πFN LF > πFF DI holds and firm D offers r if πD
D
OS
N
LF
comparing πD with πD , we have
OS
N LF
πD
− πD
T 0 ⇐⇒(1 + t) (1 + b) mZ2
0
bD
=K
T KD
2
(4 − b ) {(2 − b2 ) t + 2 (1 − b2 )}2where Z2 ≡ { 2 − b2 t + 2 1 − b2 }[(2 + b) 2 − 2b + b2 {a − (1 − b) c} − 4 (1 − b) ct] −(1 − b) (1 + t) {4 2 − b2 (1 + t)−b4 }m. Since πbD (r, t) is increasing in r, πbD (r, t)+KD >N LF holds evaluated at r = m, and r = r
0 =π
N LF + K
πDb(m, t) > m holds, KD
bD (r, t) − πD
D
0
0
OS
e
is always positive (i.e., Z2 > 0). Note that K> K holds because π
−

D
D
D
0
KF =KF
N
LF
OS
F
DI
0
b with respect to t, we
πD< (πD − πD ) K =K 0 is satisfied. By differentiating K
D
F
F



0
2
2
2
b
have ∂ KD /∂t = − (1 + b) m[4t (1 − b) 2 − b {6 1 − b + 2 − b2 t} (c + m) + tZ3 +





2 (1 − b) Z4 ]/[ 4 − b2 { 2 − b2 t+2 1 − b2 }3 ] where Z3 ≡ ab2 (2 + b) 2 − b2 2 − 2b + b2 +(1 − b) 4 − 4b2 − b3
12 − 12b2 + 4b3 − b4 c+2 (1 − b) 24 − 48b2 + 18b4 + b6 m and Z4 ≡ab2 (2 + b) (1 + b) 2 − 2b + b2 + 1 − b2 8 − 12b2 + 2b3 − b5 c+ 8 − 20b2 + 8b4 + b6 m.
Since Z3 and Z4 are increasing in a and a > c+m must hold, Z3 > Z3 |a=c+m = (48−48b−
88b2 +96b3 +36b4 −44b5 +4b6 −b8 )c+ 48 − 48b − 88b2 + 92b3 + 32b4 − 32b5 + 2b6 − 3b7 m >30
0 and Z4 > Z4 |a=c+m = (1 + b) 8 − 8b − 8b2 + 12b3 − 2b4 + b6 c+(8−16b2 +2b3 +6b4 +b 0 /∂t < 0 holds for all t ∈ [0, t].
b5 + 2b6 )m > 0 are satisfied. Hence, ∂ K
D

Putting them all together, we have(e 0 if KF ≤ K 0
K
D
F
0
KD =
.
0
b
KD if KF > KF0
0 hold so that π F DI ≤ π N LF and π OS > π N LF are(i) If KF ≥ KF0 and KD < KDFF
D
Dsatisfied, firm D offers service outsourcing and firm F accepts the offer. In this case,the equilibrium is an OS equilibrium where the service price is set at r = rb(m, t) and it
is uniquely determined given m and t. Once the service price is determined, there is a
00 hold so thatunique equilibrium in the product market. (ii) If KF < KF0 and KD < KD
OS > π F DI are satisfied, firm D offers service outsourcing by setting
πFF DI > πFN LF and πDD

the service price at r = r and firm F accepts the offer. In this case, the equilibrium
is an OS equilibrium where the service price and the equilibrium prices of products are
0 hold so that π F DI > π N LF anduniquely determined. (iii) If KF < KF0 and KD ≥ KD
FF
F DI ≥ π OS are satisfied, firm D does not offer a service price and firm F makes service
πD
D

FDI. In this case, the equilibrium is an FDI equilibrium. (iv) Otherwise, πFF DI ≤ πFN LF
OS ≤ π N LF hold and the equilibrium is an NLF equilibrium since firm D does notand πD
D

offer a service price and firm F does not make service FDI. In all cases, the equilibrium
service price, the equilibrium prices of products, and the other endogenous variables are
uniquely determined for any given parameterization.
0
at KF = KF0
The discontinuity of KD

Suppose KF = KF0 so that πFF DI = πFN LF holds. Then, if firm D does not offer a service
price r at stage 1, firm F makes service FDI, and hence firm D’s subsequent equilibrium
F DI . Now suppose K increases from K = K 0 to K = K 0 +  ( > 0)
profit is πD
F
F
F
F
F

so that πFF DI < πFN LF holds. Then, if firm D does not offer a service price r, firm F
N LF .
does not make service FDI, and hence firm D’s subsequent equilibrium profit is πD
F DI < π N LF . That is, service
Comparing firm D’s profits in these two cases, we have πDD

FDI reduces firm F ’s per unit service cost, and this intensifies the competition between
the two firms, lowering firm D’s profit. Note that, since πFN LF is independent of KF , themaximum acceptable service price r̄, determined by πbF (r̄, t) = max[πFN LF , πFF DI ], is also
0 is determined by π
independent of KF for all KF ≥ KF0 . Then, since KD
bD (r̄, t)|KD =KD0 =F DI if K = K 0 and π
N LF if K > K 0 where ππDbD (r̄, t)|KD =KD0 = πDbD (r̄, t) is continuous
F
F
FF
F DI < π N LF implies that K 0 |0
0 > K |K =K 0 +and strictly decreasing in KD , πD
D
D KF =KF
D F
F

31

when  approaches to zero. This results in the discontinuity.

Proof of Corollary 1
0 and K0
Suppose 0 < KD < KDF ≥ KF are satisfied given t. By Proposition 1, theequilibrium is an OS equilibrium. Since we restrict our attention to the case where∂bπD (r, t)/∂r > 0 is satisfied, and ∂br0 (KF , t) /∂KF > 0 holds by (16), ∂πD /∂KF > 0
holds for 0 < KF < KF0 . Besides that, since πFF DI = πFN LF holds at KF = KF0 , we have
OS = πrb(m, t) = rb0 (KF , t) at KF = KF0 , and thereby πDbD (br(m, t), t) = πbD (br0 (KF0 , t) , t) is
N LF > π F DI , π OS − π F DI > π OS − π N LF holds at K = K 0
satisfied. Since we have πD
F
D
D
D
D
D
F
0 < K 00 |00
00
which means KD
D KF =K 0 . Since KD is increasing in KF by (A1) and KD |KF =0 =
F

00 |
00
00
00
0
0, given KD < KD
KF =K 0 , there exists KF such that KD < KD for KF < KF < KF
F

00 < K for 0 ≤ K ≤ K 00 .
and KD
D
F
F

Proof of Lemma 3
0 and so r = rb(m, t),By (9), (10), and (17), given KF ≥ KF0 and KD < KD

"
∂ pOS
D r=b
r (m,t)
∂t
∂ pOS
F r=b
r (m,t)
∂t

b
=

=

#


{ 2 − b2 t + 2 1 − b2 }2 c+{2 − 5b2 + 4 1 − b2 t + 2 − b2 t2 }(2 − b2 )m,(4 − b2 ) {(2 − b2 ) t + 2 (1 − b2 )}2"#
2{ 2 − b2 t + 2 1 − b2 }2 c+{ 4 − 6b2 − b4 + 8 1 − b2 t + 2 2 − b2 t2 } 2 − b2 m
(4 − b2 ) {(2 − b2 ) t + 2 (1 − b2 )}2.

By these equations, we have
∂ pOS
D r=b
r (m,t)
∂t
∂ pOS
F r=b
r (m,t)∂t(2 − b2 ){2 − 5b2 + 4 1 − b2 t + 2 − b2 t2 }c
S 0 ⇐⇒ −= ζ0 T ,222{(2 − b ) t + 2 (1 − b )}m
(2 − b2 ){ 4 − 6b2 − b4 + 8 1 − b2 t + 2 2 − b2 t2 }c
S 0 ⇐⇒ −= ζ 00 T .
22
2
2{(2 − b ) t + 2 (1 − b )}m

It can be verified that ∂ζ 0 /∂b > 0, ∂ζ 00 /∂b > 0, ∂ζ 0 /∂t < 0, and ∂ζ 00 /∂t < 0, and thereby
the sign of ∂pOS
i /∂t is more likely to be negative when b is large and t is small, and c/m
is relatively small. Beside that, since
2 2 − b22b4−b
ζ 0 − ζ 00 => 0,
2{(2 − b2 ) t + 2 (1 − b2 )}200
a decrease in t is less likely to increase pOS
than pOS
D . Since ζ |t=0 > 0 ⇐⇒ b > b ≡
p√ F
√
√
√
10/5, ζ 00 |t=0 > 0 ⇐⇒ b > b00 ≡13 − 3 (> 10/5), ζ 0 |b=1 > 0 ⇐⇒ t < t0 ≡ 3,32and ζ 00 |b=1 > 0 ⇐⇒ t < t00 ≡√6/2 (<√

3), pOSD increases with a tariff reduction if andonly if t < t0 , b > b0 , and c/m < ζ 0 hold, and the prices of both goods increase with atariff reduction if and only if t < t00 , b > b00 , and c/m < ζ 00 hold.

Proof of Proposition 2
OS /∂p =First note that consumer surplus is decreasing in both pD and pF . By (13), ∂πD
D
2
OS
2(pOS
D − c) − br = 2 (2 + b) {a − (1 − b) c} + b{2 (1 + t) + b }r > 0 and ∂πD /∂pF = r > 0

r(m, t)/∂t <hold. Since we focus on the case where ∂ π̂D (r, t)/∂t > 0 holds and ∂r/∂t = ∂b
0 is satisfied at KF = KF0 , ∂πD (KF , t)/∂t < 0 holds whenever both ∂pD /∂t < 0 and∂pD /∂t < 0 hold. Beside that, since tariff revenue is a transfer from firm F to thedomestic country, and a decrease in consumer surplus always outweighs firms’ gains
from increases in goods’ prices, the world welfare is also decreasing in pD and pF .
By Lemma 3, if t < t00 , b > b00 , and c/m < ζ 00 are satisfied, ∂pD /∂t < ∂pF /∂t < 0holds and thereby ∂CS(KF , t)/∂t > 0, ∂CS(KF , t)/∂t > 0, and ∂πD (KF , t)/∂t < 0necessarily hold. Besides that, Lemma 3 suggests that ∂pD /∂t < 0 holds if and only if
t < t0 , b > b0 , and c/m < ζ 0 are satisfied, and ∂pD /∂t < ∂pF /∂t < 0 holds only if t < t0 ,b > b0 , and c/m < ζ 0 are satisfied. Since ∂CS(KF , t)/∂t > 0 and ∂W W (KF , t)/∂t > 0
can hold only if ∂pD /∂t < 0 is satisfied, they hold only if t < t0 , b > b0 , and c/m < ζ 0 are
satisfied. As for ∂πD (KF , t)/∂t, it can be negative even if t < t0 , b > b0 , and c/m < ζ 0
are not satisfied and ∂pF /∂t > ∂pD /∂t > 0 holds, since an increase in the service priceby a tariff reduction directly increases the profits of firm D.

Proof of Proposition 3
When KF ≤ KF00 holds, the equilibrium is an FDI equilibrium where the prices of goods,
consumer surplus, and the profits of firm D are independent of KF . An increase in
KF reduces world welfare, because KF is incurred by firm F under the FDI equilibrium. Hence, ∂CS(KF , t)/∂KF = 0, ∂W W (KF , t)/∂KF = ∂πF (KF , t)/∂KF < 0 and∂πD (KF , t)/∂KF = 0 hold for all KF < KF00 . When KF ∈ (KF00 , KF0 ) holds, the equilibrium is an OS equilibrium where the service price is set at r = rb0 (KF , t) and it
is increasing in KF . Given t, an increase in r necessarily raises the prices of both
goods, and so ∂CS(KF , t)/∂KF < 0, ∂W W (KF , t)/∂KF < 0, ∂πF (KF , t)/∂KF < 0,and ∂πD (KF , t)/∂KF > 0 hold for all KF ∈ (KF00 , KF0 ) where the last inequality is
due to ∂bπD (r, t)/∂r > 0. When KF > KF0 holds, the equilibrium is an OS equilibrium where the service price is set at r = rb(m, t) and it is independent of KF .

33

Hence, ∂CS(KF , t)/∂KF = 0, ∂W W (KF , t)/∂KF = 0, ∂πF (KF , t)/∂KF = 0, and∂πD (KF , t)/∂KF = 0 hold for all KF > KF0 .
By comparing the equilibrium prices between an FDI equilibrium and an OS equilibrium, we have pbD (r, t) − pFDDI = (3 + t) br/(4 − b2 ) > 0, pbF (r, t) − pFF DI = [{2 (1 + t) +b2 }r + 2ct]/(4 − b2 ) > 0. In addition, rb (KF , t) ≤ rb(m, t) holds for KF ∈ (KF00 , KF0 ]with equality at KF = KF0 . Hence, for given t and any KF 1 , KF 2 , and KF 3 such that
0 < KF 3 < KF00 < KF 2 < KF0 < KF 1 < KF0 , CS(KF 3 , t) > CS(KF 2 , t) > CS(KF 1 , t),
W W (KF 3 , t) > W W (KF 2 , t) > W W (KF 2 , t), πF (KF 3 , t) > πF (KF 2 , t) > πF (KF 2 , t),and πD (KF 3 , t) < πD (KF 2 , t) < πD (KF 1 , t) hold. In sum, for any given t ∈ [0, t̄],CS(KF , t), W W (KF , t), and πF (KF , t) are decreasing in KF while πD (KF , t) is increasing in KF for all KF ∈ (0, KF0 ].

Proof of Proposition 4Firstly, we should note that pOS> pFi DI (i ∈ {D, F }) always holds and thereby CS F DI >i
CS OS is also satisfied given t. As for the world welfare, W W F DI − W W OS = (CS F DI −F DI −π OS )+(π F DI −π OS ) = (CS F DI −CS OS )+(T RF DI −CS OS )+(T RF DI −T ROS )+(πD
DFF
F DI = π OS and
T ROS ) holds at KF = KF0 and given t. The last equality is because both πD
D

πFF DI = πFOS hold at KF = KF00 . We have ∂{(CS F DI − CS OS ) + (T RF DI − T ROS )}/∂a =[r{(1 + b) (2 + b) + 2 2 + 2b − b2 t + 2 + b − b2 t2 }]/{(2 − b)2 (2 + b) (t + 1)} > 0. By
using this and a > c + r, the following inequality holds: (CS F DI − CS OS ) + (T RF DI −T ROS ) > (CS F DI − CS OS ) a=c+r + (T RF DI − T ROS ) a=c+r = r(Z3 +Z4 r)/{2(1+t)(4−b2 )2 } > 0 where Z3 = 2(1 + b)(2 + b)2 bc + 2(4 + 8b + 5b2 + b4 )ct + 2(8 + 4b − b3 )ct2 +2(4 − b2 )t3 c > 0 and Z4 = (1 + b)(4 + 12b − 3b2 + 5b3 ) + (12 + 24b + b2 − 4b3 + 3b4 )t +(12 + 8b − b2 − 2b3 )t2 + (4 − b2 )t3 > 0. As a result, W W F DI > W W OS is satisfied at
KF = KF0 and given t.
Secondly, we should examine the cut-off values of the fixed investment cost of service
FDI before and after the trade liberalization. Since both KF0 and KF00 depend on t, wedenote them by KF0 (t) and KF00 (t) respectively for expositional convenience from here on.
0

While KF0 is decreasing in t and K F := max[K 0 (t0 ), KF0 (t1 )] = KF0 (t1 ) always holds (seethe proof of Proposition 1), it is ambiguous whether KF00 (t1 ) is larger or smaller thanOS − π F DI can be either increasing or decreasing in t.
KF00 (t0 ) since πDD

Now we explore the existence of the unique cut-off values K̃FCS and K̃FW W suchthat CS(KF , t1 ) > CS(KF , t0 ) holds if KF < K̃FCS and W W (KF , t1 ) > W W (KF , t0 )holds if KF < K̃FW W . When KF ≤ K 00F := min[KF00 (t0 ), KF00 (t1 )] holds, the equilibrium
becomes an FDI equilibrium at both t = t0 and t = t1 . Since we have KF00 (t) > 0 given34

KD > 0, we can always find some KF such that this inequality holds irrespective of theparameter values. As long as KF ≤ K 00F , CS(KF , t1 ) > CS(KF , t0 ) and W W (KF , t1 ) >W W (KF , t0 ) always hold (see Proposition 5). Thus, the cut-off values must satisfy
K̃FCS ≥ K 00F and K̃FW W ≥ K 00F .
00

Next consider the case where K 00F < KF < K F := max[KF00 (t0 ), KF00 (t1 )]. There are
two cases: (i) KF00 (t1 ) < KF00 (t0 ) and (ii) KF00 (t0 ) < KF00 (t1 ). (i) Suppose KF00 (t1 ) < KF00 (t0 )holds. If KF increases from KF = KF00 (t1 ) to KF = KF00 (t1 ) + ε (≤ KF00 (t0 )) where
ε is an infinitesimal value (ε > 0), the equilibrium at t = t1 changes from an FDI
equilibrium to an OS equilibrium, and the change necessarily lowers consumer surplus
and world welfare for given t. Note that under an OS equilibrium with KF < KF0 ,any increase in KF reduces consumer surplus as well as world welfare (see Proposition 3). There are two sub-cases: (a) If CS(KF00 (t1 ) + ε, t1 ) < CS(KF00 (t1 ) + ε, t0 )holds, the cut off value is determined by K̃FCS = KF00 (t1 ). Similarly, we have K̃FW W =KF00 (t1 ) if W W (KF00 (t1 ) + ε, t1 ) < W W (KF00 (t1 ) + ε, t0 ) holds, (b) if CS(KF00 (t1 ) + ε, t1 ) ≥CS(KF00 (t1 ) + ε, t0 ) and CS(KF00 (t0 ), t1 ) < CS(KF00 (t0 ), t0 ) hold, on the other hand, there
exists K̃FCS ∈ (KF00 (t1 ), KF00 (t0 )) such that CS(K̃FCS , t1 ) = CS(K̃FCS , t0 ) holds. Similarly,
there exists K̃FW W ∈ (KF00 (t1 ), KF00 (t0 )) such that W W (K̃FW W , t1 ) = W W (K̃FW W , t0 ) holds
if W W (KF00 (t1 ) + ε, t1 ) ≥ W W (KF00 (t1 ) + ε, t0 ) and W W (KF00 (t0 ), t1 ) < W W (KF00 (t0 ), t0 )are satisfied. (c) Otherwise, neither K̃FCS nor K̃FW W exists in KF ∈ (KF00 (t1 ), KF00 (t0 )).
(ii) Suppose KF00 (t1 ) > KF00 (t0 ) holds. Since CS(KF , t1 ) > CS(KF , t0 ) and W W (KF , t1 )> W W (KF , t0 ) necessarily hold for KF ∈ (KF00 (t0 ), KF00 (t1 )), neither K̃FCS nor K̃FW W exists in KF ∈ (KF00 (t0 ), KF00 (t1 )). By (i) and (ii), we can confirm that K̃FCS ≥ KF00 (t1 ) andK̃FW W ≥ KF00 (t1 ) must hold.
00

We have seen that K̃FCS does not exist in KF ∈ [0, K F ) if both CS(K 00F , t1 ) >
00

00

00

CS(K 00F , t0 ) and CS(K F , t1 ) > CS(K F , t0 ) hold. Then, let us consider the case for K F ≤00

00

KF . Suppose KF increases from KF = K F to KF = K F + ε (≤ KF0 (t0 )). Then the equi00

librium is an OS equilibrium whether t = t0 or t = t1 . (i) Suppose K F = KF00 (t0 ) holds.
Since CS(KF00 (t0 ), t0 ) > CS(KF00 (t0 ) + ε, t0 ) and CS(KF00 (t0 ), t1 ) ≈ CS(KF00 (t0 ) + ε, t1 ) are
00

00

00

satisfied for small ε, CS(K F +ε, t1 ) > CS(K F +ε, t0 ) necessarily holds. Thus, K̃FCS > K F
00

holds in this case. By the same reason, K̃FW W > K F must hold in this case. (ii) Suppose
00

00

00

K F = KF00 (t1 ) holds. In this case, CS(K F + ε, t1 ) < CS(K F + ε, t0 ) can be satisfied
since CS(KF00 (t0 ), t1 ) > CS(KF00 (t0 ) + ε, t1 ) and CS(KF00 (t0 ), t0 ) ≈ CS(KF00 (t0 ) + ε, t0 ).
00

00

If it is satisfied, then we obtain K̃FCS = K F . By the same reason, K̃FW W = K F if
00

00

00

00

W W (K F +ε, t1 ) < W W (K F +ε, t0 ) is satisfied. Otherwise, K̃FCS > K F and K̃FW W > K F
must hold.

35

00

00

00

Given CS(K F + ε, t1 ) > CS(K F + ε, t0 ), a further increase in KF from KF = K F + ε00continuously reduces both CS(KF , t0 ) and CS(KF , t1 ) for KF ∈ (K F , KF0 (t0 )], andreduces only CS(KF , t1 ) and does not affect CS(KF , t0 ) for KF ∈ (KF0 (t0 ), KF0 (t1 )].
Because we have considered the parameterizations in which CS(KF0 , t1 ) < CS(KF0 , t0 )00

hold, there necessarily exists K̃FCS in the range K F < KF < min[KF0 (t1 ), KF0 ]. Note thatK̃FCS cannot exceed KF0 (t1 ) since both CS(KF , t1 ) and CS(KF , t0 ) are independent of00

KF in this range. By the same procedure, we can obtain K̃FW W ∈ (K F , min[KF0 (t1 ), KF0 ])0000

if W W (K F + ε, t1 ) > W W (K F + ε, t0 ) holds.
Thus, given CS(KF0 , t1 ) < CS(KF0 , t0 ) and W W (KF0 , t1 ) < W W (KF0 , t0 ) hold for0 ≤ t1 < t0 ≤ t̄, we can always find a unique K̃FCS ∈ [KF00 (t1 ), KF0 (t1 )) such that
CS(KF , t1 ) > CS(KF , t0 ) for all KF ∈ (0, K̃FCS ) and a unique K̃FW W ∈ [KF00 (t1 ), KF0 (t1 ))such that W W (KF , t1 ) > W W (KF , t0 ) for all KF ∈ (0, K̃FW W ).
To prove KF00 (t1 ) < K̃FCS and KF00 (t1 ) < K̃FW W can hold under a range of parameterizations, we provide a numerical example. Parameters are set at a = 5, b = 0.9,
c = 0.5, m = 1.5, KD = 0.01, and KF0 = 5. We consider trade liberalization in which
the import tariff is reduced from t0 = 0.8 to t1 = 0. The parameterization is consistent
with ∂b
πD (r, t)/∂r > 0 for r ∈ [0, r] and positive sales of the two firms in any equilibrium.
Under these parameterizations, we have KF0 (t0 ) = 4.3055 < KF0 (t1 ) = 4.7229. Since0 |0
KF0 = 5 > max[KF0 (t0 ), KF0 (t1 )] holds, we have KD
t=t0 = 15.375 < KD |t=t1 = 30.279.
0 |0Given KD = 0.01 < min[ KDt=t0 , KD |t=t1 ], the pre-liberalization equilibrium is an OS

equilibrium with KF > KF0 . It is calculated that KF00 (t1 ) = 0.01324 < KF00 (t0 ) = 0.01915.
We can calculate that (i) CS(KF , t1 )−CS(KF , t0 ) = −7.934 < 0 holds, (ii) CS(KF , t1 ) =CS(KF , t0 ) holds at KF = 0.92121 (< KF0 (t0 )), (iii) CS(KF , t1 ) − CS(KF , t0 ) = 5.5131√√
206.07 − 10.176KF − 5.156 192.63 − 18.317KF − (5.9482 + 3.2686KF ) > 0 holds for√
KF ∈ (KF00 (t1 ), 0.92121), (iv) CS(KF , t1 )−CS(KF , t0 ) = 0.52044( 206.07 − 10.176KF −√14.355)2 + 20.455 206.07 − 10.176KF − 292.0 > 0 holds for KF ∈ (KF00 (t1 ), KF00 (t0 )],(v) CS(KF , t1 ) − CS(KF , t0 ) = 1.624 holds for KF ∈ [0, KF00 (t0 )].

Thus, we haveK̃FCS = 0.92121 which satisfies K̃FCS > KF00 (t1 ).
Similarly, we can calculate that (i) W W (KF0 , t1 ) − W W (KF0 , t0 ) = −1.1207 < 0,(ii) W W (KF , t1 ) = W W (KF , t0 ) at KF = 1.1896 (< KF0 (t0 )), (iii) W W (KF , t1 ) −√√W W (KF , t0 ) = −3.9653 192.63 − 18.317KF + 16.987 206.07 − 10.176KF − 188.64 +3.2689KF > 0 for KF ∈ (KF00 (t0 ), 1.1896), (iv) W W (KF , t1 ) − W W (KF , t0 ) = −243.77 +√6.296KF + 16.987 206.07 − 10.176KF > 0 holds for KF ∈ (KF00 (t1 ), KF00 (t0 )], and (v)W W (KF , t1 ) − W W (KF , t0 ) = 0.17598 holds for KF ∈ [0, KF00 (t1 )]. Thus, we have
K̃FW W = 1.1896 which satisfies K̃FW W > KF00 (t1 ).
36

Proof of Proposition 5
By differentiating CS F DI with respect to t, we have dCS F DI /dt = −c[(2 + b)2 {a −
2(1 − b) c} − 4 − 3b2 ct]/ 4 − b2 , and it is positive by Assumption 1. Similarly, by difF DI , and W W F DI with respect to t, we obtain dπ F DI /dt = 2bc[(2 + b) {a−ferentiating πD
D
2
2
F
DI
(1 − b) c} + bct]/ 4 − b> 0 and dW W
/dt = −c[(1 − b) (2 + b)2 {a − (1 − b) c} +2
4 − 3b2 ct]/ 4 − b2 < 0.

Proof of Proposition 6
To prove the proposition, the basic parameters are set at the same values used in the
proof of Proposition 5: a = 5, b = 0.9, c = 0.5, m = 1.5. In addition, we set V = 100 and
KD = 1. The value of V does not affect the ranking of each surplus nor the optimum levelof KF . We consider trade liberalization in which the import tariff is reduced from t0 = 0.5
to t1 = 0.4. Under the parameterizations, we have KF0 (t0 ) = 4.3055 < KF0 (t1 ) = 4.7229.
0
0 |For KF > KF0 (t), KD = 1 < KD
t=t0 = 17.734 < KD |t=t1 = 18.915 holds and thereby
OS = π F DI for K , wethe equilibrium is an OS equilibrium. For KF ≤ KF0 , by solving πDFDhave KF00 (t1 ) = 0.18697 < KF00 (t0 ) = 0.19246. Hence, the equilibrium is an OS equilibrium
if KF00 (t) < KF ≤ KF0 (t) and an FDI equilibrium if 0 ≤ KF ≤ KF00 (t).
Given t, the domestic government maximizes W (KF , t) with respect to KF . Suppose√t = t0 . We have W (KF , t0 ) = 7.2581 + 3.2373KF + 5.170 9 197.61 − 15.264KF for
KF ∈ (KF00 (t0 ), KF0 (t0 )]. We can verify that W (KF , t0 ) is an inverse U-shaped curve in
KF for this range, which takes the maximum at K̂F0 = 3.2108. Thus, the maximized levelof the domestic welfare under KF ∈ (KF00 (t0 ), KF0 (t0 )] becomes W (K̂F0 , t0 ) = 80.686. Inother cases, the domestic welfare is independent of KF and it is given by W (KF , t0 ) =76.805 (< W (K̂F0 , t0 )) for KF ∈ (KF0 (t0 ), +∞) and W (KF , t0 ) = 79.947 (< W (K̂F0 , t0 ))for KF ∈ [0, KF00 (t0 )]. Accordingly, W (KF , t0 ) is maximized at KF = K̂F0 and thereby
KF∗ (t0 ) = K̂F0 .
√Similarly, suppose t = t1 . We have W (KF , t1 ) = −4.2261 + 3.3998KF + 5.8901
199.29 − 14.247KF for KF ∈ (KF00 (t1 ), KF0 (t1 )], which is an inverse U-shaped curve in

KF for this range. It takes the maximum at K̂F1 = 3.2964 and the maximized level of thedomestic welfare under KF ∈ (KF00 (t1 ), KF0 (t1 )] becomes W (K̂F1 , t1 ) = 79.678. In othercases, the domestic welfare is independent of KF and it is given by W (KF , t1 ) = 79.553(< W (K̂F1 , t1 )) for KF ∈ (KF0 (t1 ), +∞) and W (KF , t1 ) = 78.926 (< W (K̂F1 , t1 )) for
KF ∈ [0, KF00 (t1 )]. Accordingly, W (KF , t1 ) is maximized at KF = K̂F1 and thereby
KF∗ (t1 ) = K̂F1 . Note that KF∗ (t1 ) > KF∗ (t0 ) means the tariff reduction increases, rather

37

than decreases, the optimum level of KF in this case.
By substituting KF = KF∗ (t0 ) and KF = KF∗ (t1 ) into CS(KF , t) and W W (KF , t),
we have CS(KF∗ (t1 ), t1 ) − CS(KF∗ (t0 ), t0 ) = −0.50163 < 0 and W W (KF∗ (t1 ), t1 ) −W W (KF∗ (t0 ), t0 ) = −0.051217 < 0.

Proof of Proposition 7
Suppose CS(KF∗ (t1 ), t1 ) < CS(KF∗ (t0 ), t0 ) and W W (KF∗ (t1 ), t1 ) < W W (KF∗ (t0 ), t0 ) aresatisfied. These inequalities hold only if the equilibrium under KF = KF∗ (t1 ) is an
Outsourcing equilibrium. Since ∂CS(KF , t1 )/∂KF ≤ 0 and ∂W W (KF , t1 )/∂KF ≤ 0hold (see Proposition 3), we have CS(KF , t1 ) < CS(KF∗ (t0 ), t0 ) and W W (KF , t1 ) <W W (KF∗ (t0 ), t0 ) for all KF ≥ KF∗ (t1 ). For KF ∈ [0, KF00 (t1 )], the post-liberalization
equilibrium becomes an FDI equilibrium and thereby CS(KF , t1 ) > CS(KF∗ (t0 ), t0 )and W W (KF , t1 ) > W W (KF∗ (t0 ), t0 ) necessarily hold. If KF increases from KF =KF00 (t1 ) to KF = KF00 (t1 ) + ε and CS(KF00 (t1 ) + ε, t1 ) < CS(KF∗ (t0 ), t0 ) holds where εis an infinitesimal value (ε > 0), we have K̂FCS = KF00 (t1 ). If CS(KF00 (t1 ) + ε, t1 ) ≥CS(KF∗ (t0 ), t0 ) holds, on the other hand, there exists K̂FCS ∈ (KF00 (t1 ), KF∗ (t1 )) such that
CS(KF , t1 ) = CS(KF∗ (t0 ), t0 ) holds at KF = K̂FCS and CS(KF , t1 ) > CS(KF∗ (t0 ), t0 )holds for all KF ∈ (KF00 (t1 ), K̂FCS ). By the same procedure, we have K̂FW W = KF00 (t1 ) if
W W (KF00 (t1 ) + ε, t1 ) < W W (KF∗ (t0 ), t0 ) holds and K̂FW W ∈ (KF00 (t1 ), KF∗ (t1 )) otherwise.
To prove KF00 (t1 ) < K̂FCS and KF00 (t1 ) < K̂FW W can hold under a range of parameterizations, we use the same parameterizations as those used in the proof of Proposition 6.
√
We have CS(KF .t1 )−CS(KF∗ (t0 ), t0 ) = 0.16843( 199.29 − 14.247KF −14.117)2 +10.957√
199.29 − 14.247KF − 136.26 and W W (KF , t1 ) − W W (KF∗ (t0 ), t0 ) = 2.3997KF + 5.8902√−14.247KF + 199.29 − 80.660. We have K̂FCS = 3.2131 > KF00 (t1 ) = 0.18697 whereCS(KF , t1 ) = CS(KF∗ (t0 ), t0 ) holds at KF = K̂FCS and CS(KF .t1 ) > CS(KF∗ (t0 ), t0 )holds for all KF ∈ [0, K̂FCS ). Similarly, we have K̂FW W = 3.2446 > KF00 (t1 ) where
W W (KF , t1 ) = W W (KF∗ (t0 ), t0 ) holds at KF = K̂FW W and W W (KF , t1 ) > W W (KF∗ (t0 ), t0 )holds for all KF ∈ [0, K̂FW W ).

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40

Figure 1
Choice of Service Mode
Firm D
Oﬀer a Service Contract(commits to service price)

Not Oﬀer

Firm F
Accept(commits to outsource
all services)

RejectFirm F
Firm F
Service FDI

No LocalFacility

Service FDI

0

m

0

Per-unit service cost of ﬁrm F

No LocalFacilitymr

Figure 2
Equilibrium Service Sheme

KD

FDI

NLF

′
KD
(KF )

OS
KF

KF′

Figure 3
Tariﬀ Reduction and Equilibrium Service Sheme

KD

FDI

NLF

B

KD

A

OS
KF′′

′
KD
(KF )

KF′

KF0

KF

Figure 4′
Eﬀects of Liberalization in Service FDI with KD < KD

CS(KF , t)
FDI

OS

KF′

KF′′
W W (KF , t)
FDI

KF′′

KF

OS

KF′

KF

Figure 4 (continued)

π F (KF , t)
FDI

OS

KF′′

KF′

KF′′

KF′

KF

π D (KF , t)

KF

Figure 5
Anti-Competitive Scheme Switch by Trade Liberalization

CS(KF , t)

OS

NLF

CS(KF , t0 )CS(KF , t1 )

t1

t̂

t0

W W (KF , t)

OS

t̄

t

NLF

W W (KF , t0 )W W (KF , t1 )

t1

t̂

t0

t̄

t