Trade Liberalization and Aftermarket Services for Imports∗
Jota Ishikawa†

Hodaka Morita

Hitotsubashi University

University of New South Wales

Hiroshi Mukunoki
Gakushuin University
November 26, 2012

Abstract
By an international oligopoly model, this paper examines the effects of trade liberalizationin the presence of aftermarket services. After consumers purchase goods, a certain factionof the units fails, and repairs are necessary to make the broken units workable. Comparedto the case without the repair services for imports, the provision of the repair services by
the rival producer in the domestic country hurts consumers and worsens world welfare. By
contrast, the provision of the repair services by the original producer benefits consumers
and improves world welfare. When the fixed cost for FDI in aftermarket services is high,trade liberalization in goods may lead to the entry of the rival producer into the aftermarket
services for imports, and thereby hurt consumers and the foreign firm, benefit the domestic
firm, and worsen world welfare. The existence of independent service organizations (ISOs)
neither help consumers nor the foreign firm if there is the fixed cost of entry. The result
suggests that promoting service FDI is important to guarantee the conventional effects oftrade liberalization.
∗ We thank Kenzo Abe, Masahiro Ashiya, Taiji Furusawa, Jun-ichiro Ishida, Hiroshi Kinokuni, Kazuharu Kiy-ono, Kaz Miyagiwa, Ryosuke Okamoto, Makoto Okamura, Takao Ohkawa, Tetsuo Ono, Kentaro Tachi, Mike
Waldman and seminar participants at National Graduate Institute for Policy Studies (GRIPS), Osaka University,
the JEA Spring Meeting, and Hitotsubashi COE Conference on International Trade and FDI 2010 for their helpfulcomments and suggestions. The usual disclamer applies.
† 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

Aftermarket services, such as repair and maintenance services, refer to services used together
with durable equipment but purchased after the consumer had acquired the equipment. In thispaper, we focus on aftermarket services that are required by a certain fraction of durable units
after sales. Repair services are aftermarket services of this kind, given that durable goods often(but not always) break down and broken units require repair services for continued usage. One
might feel that repair services are not an activity of much economic significance. This, however, is
not at all the case. Eschenbach and Hoekman (2005), for example, reports that distribution andrepair services account for about 10% to 20% of the stock of inward service FDI in the Central
and Eastern European countries and the South East European countries. A number of antitrust
cases, including the influential 1992 US Supreme Court decision in Eastman Kodak Company
v. Image Technical Services, Inc, et al., concern the behavior of durable goods producers in themarket for repairs of their products (see, for example, Chen, Ross, and Stanbury, 1998; Waldman,2003).1
We analyze the provision of repair services in the context of international trade and explores
its welfare consequences and policy implications. To perform repair services effectively, proximity
between service providers and consumers is a critical element. In the context of international
trade, this implies that foreign durable-goods producers have a disadvantage in performing repair
services in the domestic country. Foreign producers can overcome this disadvantage by establishing local service facilities through foreign direct investments (FDIs), but FDI may be very
costly due not only to direct investment costs but also to a variety of regulatory impediments.
Foreign producers may therefore choose not to establish local service facilities, and their broken
units may remain unrepaired as a consequence.2
This is the context in which domestic durable goods producers often perform repair services
for their foreign rivals’ products, the practice often observed in reality. The repair services forcompetitors’ products are provided “voluntarily” in the sense that they are conducted without
1 The demands for repairs have been also increasing. For instance, it is reported that the number of the requests

for repairing which Panasonic receives is 130,000 in 1995 and 370,000 in 1999, and the ACT of Consumer Electronics
implemented in 2001 had been expected to increase the demand for repair service (Nikkei Ecology, April, 2001).
Canon receives 1,000,000 inquiries that are associated with repairing (Nikkei Joho Strategy, December, 2003).
Louis Vuitton Japan repaired 330,000 garments in 2006 (Nikkei Business, June 11, 2007).
2 For example, although some imported infrared heaters had a problem and they were subject to a productrecall in Japan, some foreign producers were not able to provide repair services for their own products in Japan.
Even though foreign producers perform repair services in their own countries, it involves significant inconvenienceand costs for consumers to ship broken units back and forth between different countries.

2the consent of the original producers.3
We analyze the provision of repair services through an international duopoly model in whicha domestic firm (firm D) and a foreign firm (firm F ) produce differentiated products (good D
and F ) in their own countries and compete in the domestic market. Both goods break downwith a certain probability, and broken units require repair services for usage. Firm D has alreadyestablished its facilities to perform repair services for good D. Firm F can establish its own
service facilities in the domestic country by incurring a fixed cost KF . Also, firm D can establish
facilities for repairing good F by incurring a fixed cost KD . Depending on parameterization, themodel exhibits one of the following three types of equilibrium: (i) Rival’s Repair equilibrium inwhich firm D performs repair services for good F , (ii) Own Repair equilibrium in which firm
F performs repair services for good F , and (iii) No Repair equilibrium in which neither firm
performs repair services for good F .
Through analyzing our model, we investigate the effect of trade liberalization in goods in its
connections to the effect of trade liberalization in services, with the following reality as a background. The Uruguay Round negotiations of General Agreement on Tariffs and Trade (GATT)
succeeded in establishing the framework of liberalizing cross-country transaction of services, thatis, the General Agreements on Trade in Services (GATS). The progress of liberalization in service sectors, however, has been limited compared to the degree of trade liberalization in goods.
For instance, People’s Republic of China has prohibited foreign firms’ provisions of after-sales
services including repair services until 2001. Indonesia and Thailand limit the foreign equity
ownership of maintenance and repair services up to 49%. Even if the provisions of repair services
by foreign companies are legally allowed, the foreign producer may not secure skilled workers by
the regulation on the posting of workers across borders.
Regulatory impediments such as the ones mentioned above would keep the costs for service
FDI high. This is captured by a high fixed cost for firm F to establish service facilities, KF , inour model, where the liberalization of service FDI reduces KF and the liberalization of goods
trade reduces the tariff rate.
We demonstrate that trade liberalization in goods may hurt domestic consumers and lower
world welfare, and that these negative effects are turned into positive ones if service FDI is also
3 For example, in Japan, Nidec Sankyo Service Engineering Corporation, which is a domestic subsidiary of

the Japanese machine-tool company, Nidec Sankyo Corporation, is providing maintenance and repair services forcompetitors’ products including imported products. Maruju Ironworks and Masuda Ironworks are providing themaintenance and repairs of machines, dies and molds produced by other companies. Fuji Xerox provides themaintenance and repair services of office equipment even if customers use the equipment of other firms. Several IT
companies including IBM and Fujitsu provide repair services for competitors’ network products. NEC supports
other companies’ products.

3

liberalized. The main logic behind this result can be explained as follows. Suppose that initiallyboth the tariff rate and firm F ’s fixed cost KF are high. Under the high tariff rate, neither firm
establishes service facilities in the equilibrium (the No Repair equilibrium). Because of the hightariff, the domestic country’s imports of good F is small. Hence the number of broken units ofgood F that requires repair is also small, implying that neither firm can recover the fixed costby performing repair services for good F .
Now suppose that the liberalization of the trade in goods gradually reduces the tariff rate. The
reduction of the tariff rate eventually switches the equilibrium from the No Repair equilibrium
to the Rivals’ Repair equilibrium. Lower tariff increases imports of good F . When the tariff rate
becomes sufficiently low, firm D can recover the fixed cost since there is sufficient demand for
repairing good F , whereas firm F cannot recover the fixed cost KF which is assumed to be high.
Once firm D establishes service facilities for repairing good F , firm D can earn profit not only
from selling good D but also from performing repair services for good F . Then, an increase in thesales of good D reduces consumers’ willingness to pay for good F , which in turn decreases firm
D’s profit from performing repair services for good F . Hence, firm D is less willing to increases
the sales of good D in the Rival’s Repair equilibrium than in the No Repair equilibrium, implyingthat competition between firms D and F in the product market is weaker in the Rival’s Repair
equilibrium than in the No Repair equilibrium. We call this effect as collusive effect of the rival
firm’s provision of repair services.
We find that, because of the collusive effect, the switch from the No Repair equilibrium to
the Rival’s Repair equilibrium hurts consumers and firm F , and reduces world welfare, although
it benefits firm D. It is worth emphasizing that a time inconsistency problem on the side of
consumers is a key driving force of the negative effects on consumers and world welfare. To see
this, suppose that consumers can collectively commit not to repair broken units of good F . If thiscommitment is credible, neither firm establishes service facilities, and hence the collusive effectnever arises in the equilibrium. Hence, tariff reduction always benefit consumers and increase
welfare. Such a collective commitment, however, is difficult to be made credible because, once
service facilities are established, it is each consumer’s benefit to renege on the commitment andrepair their broken units.
We find that, in the presence of the time inconsistency problem, the negative effects of tradeliberalization in goods just mentioned are turned into positive ones if service FDI is also liberalized. This is because the liberalization of service FDI reduces KF , which in turn induces firm F
to establish its own service facilities. Having its own facilities, firm F can earn profits not only
from selling good F but also from repairing it. Firm F can prevent firm D from establishing4

service facilities for good F by providing a full warranty when it sells good F . The full warrantymeans that the repair price of good F is zero. Hence, firm D cannot make positive profits from
repairing good F , implying that firm D would not establish service facilities. From firm F ’s
standpoint, although the full warranty generates negative profits in the aftermarket, it increases
firm F ’s profit in the product market because it increases consumers’ willingness to pay for goodF , implying that firm F ’s overall profitability is unaffected by the provision of the full warranty.
Hence, when KF is sufficiently low, it is firm F , not firm D, that establishes service facilities
when the tariff rate is sufficiently reduced, and so trade liberalization in goods now switches the
No Repair equilibrium into the Own Repair equilibrium. The collusive effect does not arise in
the Own Repair equilibrium because firm D cannot earn any profit by repairing good F , andhence the positive welfare effects of trade liberalization in goods are preserved.

1.1

Contributions and relationship to the literature

There have been several studies that investigate the connection between international trade andservice sectors.4 However, to our best knowledge, the present paper and another paper of ours
(Ishikawa, Morita, and Mukunoki, 2010) are the only studies that investigate the link between
trade liberalization in goods and liberalization of FDI for services that are performed after production of goods (referred to as “post-production services” in what follows).
The present paper is a companion paper of Ishikawa, Morita, and Mukunoki (2010), henceforth
IMM, in the sense that these two papers analyze two different types of post-production services.
IMM have analyzed an international duopoly model in which post-production services (such as
sales and distribution) must be performed before goods are purchased. The foreign firm has
an option of outsourcing post-production services to its domestic rival by paying royalties orproviding those services by itself in the domestic market.
A fundamental difference between the present paper and IMM is that the present paper
analyzes the provision of post-production services that are required only by a certain fraction ofunits after sales of goods (aftermarket services), whereas IMM analyzes the provision of postproduction services that must be performed for all units before sales of goods. Despite thedifference, we obtain the main policy implication that is similar to the one obtained in IMM.
That is, these two papers together identify the importance of the liberalization of service FDI
by showing, for both types of post-production services, that trade liberalization in goods may
hurt domestic consumers and lower world welfare, and that these negative effects are turned into
4 See Djajić and Kierzkowski (1989) , Markusen (1989), Francois (1990), Markusen, Rutherford and Tarr (2005),

Wong, Wu and Zhang (2006), and Francois and Wooton (2010), among others.

5positive ones if service FDI is also liberalized. This is the first contribution of the present paper.
Our second contribution is that we identify time inconsistency problem on the side of consumers as a key driving force of the main results. Time inconsistency has been a central issue inthe microeconomic theory of durable goods since the seminal work by Coase (1972).5 The problem arises because durable goods sold in the future affect the future value of units sold today,
and in the absence of the ability to commit, the monopolist does not internalize this externality (Waldman, 2003). This is problematic for the monopolist, because the lack of commitment
reduces its overall profitability.6
Time inconsistency is an important issue when we analyze durable-goods producers’ behavior
in aftermarkets. Consider situations in which consumers are locked in once they purchase a newunit from a durable-goods producer. Borenstein, Mackie-Mason and Netz (1995) argue that the
durable-goods producer will charge a price for its aftermarket products and services in excess
of marginal cost to exploit consumers’ locked-in positions. The higher aftermarket price createsan efficiency loss because it reduces consumption of aftermarket products and services. The
efficiency loss, in turn, reduces the producer’s overall profitability because consumers anticipate
this behavior and pay less for a new unit. The producer will be better off if it can crediblycommitment to charge lower aftermarket prices, but Borenstein, Mackie-Mason and Netz argue
that such commitments are often difficult in reality due to time inconsistency.
However, it seems that at least under some circumstances, durable-goods producers can commit their behavior in aftermarkets through long-term contracts as pointed out by Shapiro (1995),
Chen and Ross (1998), and Waldman (2003).7 Chen and Ross (1998) address the commitment
problem focusing on an aftermarket of repair services. They consider a durable-goods monopolist
who produces durable units that break down with a certain probability, where repaired units andunbroken units are perfect substitute.8 Chen and Ross show that the durable-goods monopolist’s
5 See also, for example, Stockey (1981), Bulow (1982), Gul, Sonnenschein and Wilson (1986), and Ausubel and

Deneckere (1989).
6 Based on the analysis of Bulow (1982), Waldman (2003) explains Coase’s argument as follows. Bulow considersa monopolist of a perfectly durable good who sells output in each of two periods. In this model, if the firm in thefirst period can commit to a production level for the second period, then the firm’s profit maximizing first-period
choices are to commit to selling zero in the second period and to produce the monopoly output level and sell itat the monopoly price. But what if commitment is not possible? Then, if the firm tries to sell the monopoly
output in the first period, consumers will be unwilling to pay the monopoly price. The reason is that consumers
fear the firm will collect the monopoly price in the first period and then produce additional units in the second
period, thus driving down the second-period value of the used units that the consumers own. The result is eachconsumer’s willingness to pay in the first period is reduced, and overall monopoly profitability falls.
7 Shapiro (1995) has shown that the efficiency loss associated with durable-goods producers’ inability to commit
to their behavior in aftermarkets can be “far smaller” than traditional monopoly deadweight losses.
8 Chen and Ross (1994) also consider durable products that break down with a certain probability. A monopolist

6inability to commit to its behavior in the aftermarket creates no efficiency loss and no reductionin the monopolist’s overall profit, because all broken units are repaired in equilibrium whether
or not the monopolist can commit.9
We contribute to the literature by showing that time inconsistency of consumers can be an
important issue, whereas previous analyses in the literature have focused on time inconsistencyof producers. Following Chen and Ross (1998), we consider a product that breaks down witha certain probability and requires repair services for usage. We find that all broken units arerepaired in Rival’s Repair equilibrium as well as Own Repair equilibrium, and hence, as inChen and Ross (1998), producers’ inability to commit does not cause efficiency losses in ourmodel. Time inconsistency problem on the side of consumers arises because we consider durable
goods producers’ strategic interactions in the context of international trade. In our model, firm
D provides repair services for good F unless consumers can collectively commit not to repair
their broken units of good F , if firm F ’s fixed cost for establishing service facilities, KF , ishigh. And the resulting collusive effect hurts consumers and reduces world welfare in the Rival’s
Repair equilibrium. This problem is resolved when liberalization of service FDI reduces KF to a
sufficiently low level.10
Time inconsistency problem of consumers seem much more difficult to be resolved comparedto time inconsistency problem of producers. This is because numerous consumers must collectively commit to their behavior in the aftermarket to resolve the former problem, whereas such a
produces such a durable product, and its repair services require the monopolist’s proprietary parts. It is assumed
that the monopolist can commit to its repair price at the time of original purchase. Chen and Ross show that the
durable-goods monopolist can effectively price discriminate between high-intensity users and low-intensity users
by monopolizing the repair market. See Chen and Ross (1999) for a related analysis in which the primary market
is competitive rather than monopolistic.
9 The demand for repairs becomes inelastic below the threshold price of repairs under which all broken units arerepaired, because the amount of broken units is fixed in the aftermarket. Even if the aftermarket is monopolized
by a single firm, the monopolist would set the threshold price because the marginal revenue of repairs always
exceeds the marginal cost of them. This means that the monopolist’s inability to commit in the aftermarket does
not generate any efficiency loss. By the same reason, the competition in the aftermarket does not directly lead toefficiency gains.
10 In contrast, post-production services in IMM do not cause time inconsistency problems because they mustbe performed for all units before sales. In IMM, the direct effect of a tariff reduction is beneficial for consumers
and the foreign firm, and is harmful for the domestic firm. 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 thehigher service price works in the direction of raising goods prices. IMM show that, from the welfare standpoint,
the latter effect can overshadow the former effect so that the tariff reduction actually hurts consumers and reduces
world welfare in equilibrium. Importantly, if the fixed cost for service FDI is also reduced, the domestic firm hasless room to increase the service price in response to the tariff reduction, and a sufficient reduction of the fixed
cost converts the negative welfare effect of tariff reduction into a positive effect.

7

collective commitment is not required for the latter problem and hence long-term contracts could
enable producers to make commitment credible. This observation in turn suggests that liberalization of service FDI is an important way to resolve the time inconsistency problem identified
through our analysis.
The reminder of the paper is organized as follows. Section 2 previews the model and briefly
explains the mechanism behind the main results. Section 2 formally develops an international
duopoly model of durable-goods producers and aftermarket services, and derives the equilibriumof the model. Section 3 investigates the effects of the liberalization of trade in goods, the liberalization of FDI for repair services, and their connection. Section 4 discuss the robustness of theresults under alternative setups. Section 5 summarizes the paper and offers concluding remarks.
The Appendix contains Proofs of Lemmas and Propositions.

2

The model and equilibrium

A domestic firm (firm D) and a foreign firm (firm F ) engage in Cournot competition in thedomestic market by producing horizontally differentiated products.11 Let xi (i = D, F ) denotethe amount of good i produced by firm i. The domestic government levies a specific tariff, t, in
the imports of good F . The utility of a representative consumer is given by U (dD , dF , Z) =V (dD , dF ) + Z where di and Z denote the consumption of good i and a numeráire good,
respectively.

Define VF (dD , dF ) ≡ ∂V (dD , dF )/∂di and Vij (dD , dF ) ≡ ∂ 2 V (dD , dF )/∂di ∂dj(i, j ∈ {D, F }). We assume Vi (dD , dF ) > 0 and Vij (dD , dF ) < 0 hold.12
Following Chen and Ross (1998), we assume that, after consumers purchase the goods, afraction (1 − q) (q ∈ (0, 1)) of each good i breaks down immediately because of imperfect quality
control. That is, qxi units of good i works correctly whereas (1 − q)xi units require repair.
Unbroken units and repaired broken units are perfect substitutes. Without repair, broken units
are useless with zero scarp values.
In order to perform repair services for good i, service facilities for good i must be establishedin the domestic country. We suppose that firm D has already established its facilities to perform
repair services for good D and provides a full warranty when it sells good D. Firm F can establish
its own service facilities by incurring a fixed cost KF and provide a full warranty for good F .
Also, firm D can establish facilities for repairing good F by incurring a fixed cost KD and charge
r to repair a broken unit of good F . The full-warranty assumption is for simplifying the analysis
11 The main results of our paper would be preserved even if the firms engage in Bertrand competition.
12 To ensure that the marginal revenue of each firm is decreasing in its sales, we also assume 2V(∂Vii (dD , dF )/∂di ) di < 0 holds.

8ii (dD , dF ) +but not essential for the results (see Section 4.2 for details).
We consider a three-stage game. In stage 1, the two firms simultaneously decide whether they
provide the repair services for good F . When firm i (i ∈ {D, F }) provides the services, it must
incur a fixed set-up cost Ki . The fixed cost is a sunk cost. The fixed cost for firm D, KD (≥ 0),
should represent the costs of establishing additional facilities, those of learning the details of thecompetitor’s product, and those of preparing the proper parts and components for repairing goodF . Meanwhile, KF includes the costs of establishing the facilities by undertaking FDI in repair
services. We assume KD ≤ KF , which reflects the presumption that the costs of establishing new
facilities outside the home country are higher than the costs of expanding the existing facilities
in the home country.
In stage 2, the two firms produce and supply the goods to the domestic market and thedomestic consumers purchase them. The two firms have the identical marginal cost of production,
which is denoted by c. The domestic government levies a non-negative, specific tariff, t, on theimports of good F .13
In stage 3, consumers find some units of the purchased goods are broken down. If the repair
services are provided by the original producer, broken units are subject to free repairs because of
the full warranty. If firm D provides the repair services for good F , it charges the service price r
per-unit of repairs. The level of r is chosen by firm D. Given this, consumers choose whether theyorder the repairs of the broken units if the repair services are provided. Let RF (∈ [0, (1 − q)xF ])denote the amount of repaired broken units of good F . Due to the economies of scope between
the repair services and the production activities, each firm has a cost advantage over its rival in
the repairs of its own product. Specifically, the marginal cost of repairing its own product, mL , isno higher than the cost of repairing the rival’s product, mH . We assume mL ≤ mH ≤ c holds sothat the costs of repairs are no higher than the production cost. We also assume the producers’
cost advantage in aftermarket services is large enough to exclude the entries of ISOs.14
The operating profits of each firm (i.e., the profits gross of the fixed costs of establishing
service facilities) are denoted by Πi (i ∈ {D, F }). We assume the second-order conditions of
13 By regarding t as the degree of cost disadvantages of the foreign firm, we can interpret the situation as if thetwo firms are heterogenous in the production cost. The main results of this paper would be mostly unchanged
with this alternative set-up. However, the welfare property of the model need to be slightly modified because thehigher cost of the foreign firm no longer works as a transfer from the foreign country to the domestic country as
the tariff does.
14 Main results of the paper would be unchanged even if we assume the producers of goods can exploit the
profits of ISOs by selling parts and other components that are indispensable to provide repair services. If ISOs
are completely free from the producers’ influences, however, there are some results specific to ISOs. See Section
4.1 for details.

9profit maximizations, ∂ 2 Πi /∂xi ∂xi < 0 holds. We use the backward induction to derive thesub-game perfect equilibrium.

2.1

Repair services and product market competition

Here, we analyze the market equilibrium given the firms’ decisions made in stage 1. We should
first mention that all broken units of good F are repaired by firm F in equilibrium if firm F
undertakes service FDI, even if firm D also establishes the service facilities for repairing good F .
Suppose both firms have established the service facilities for good F . Since firm F provides afull warranty for good F , firm D cannot attract consumers in stage 3 unless the service price isnegative, r < 0. Since the marginal cost of providing services is positive, firm D has no incentives
to set a negative price for repairing good F .
Firm F that undertook service FDI, on the other hand, always has an incentive to provide
a full warranty when it sells good F . While a full warranty generates negative profits in theaftermarket, it increases firm F ’s profits in the product market because it increases the consumer’s
willingness to pay for good F in the product market and so does the price of good F that firm F
can charge. As long as firm F provides the repair services for good F , the loss in the aftermarket
coincides with the gain in the product market, and the full warranty has no effect on firm F ’s
overall profits. As we will see below, firm D’s repairs of good F hurts firm F . Therefore, firm F
is willing to use the full warranty as a strong tool to get firm D out of the aftermarket for good
F .15
The strong position of firm F in the service market implies that the following three cases
are possible equilibrium outcomes: (i) Rival’s Repair (RR) equilibrium where only firm D
provides the repair services for good F , (ii) Own Repair (OR) equilibrium where only firm
F provides repair services for good F , and (iii) No Repair (NR) equilibrium where no repair
services are provided for good F .
2.1.1

The RR equilibrium

First, let us consider the RR subgame. In stage 3, each consumer maximizes V (xD , qxF + RF ) −rRF with respect to RF given RF ≤ (1 − q) xF , which determines the demand for repairs regarding good F . If VF (xD , qxF +RF ) ≥ r holds at RF = (1 − q) xF , which means that VF (xD , xF ) ≥ rholds, the consumer orders firm D to repair all broken units. If VF (xD , qxF ) ≥ r > VF (xD , xF )holds, on the other hand, a certain fraction of the broken units remains unrepaired. In this case,
15 By the same reason, the equilbrium properties of our model are unchaged even if firms cannot provide a full

warranty, and the two firms engage in price competition in the aftermarket. See Section 4.2 for details.

10the inverse demand for repairs is given by r = VF (xD , qxF + RF ). Since VF F < 0 holds, thedemand for repairs is decreasing in the repair price for RF ∈ (0, (1 − q)xF ).
Given the demand for repairs, firm D determines r so that it maximizes the profit fromproviding the repair services for good F .16 Firm D’s maximization problem is written bymax (r − mH ) RFrs.t.

RF ≤ (1 − q) xF .

(1)

Let rb denote the solution to this maximization problem. Then, the equilibrium amount of repairedbF (∈ [0, (1 − q)xF ]) is given by solving rb = VF (xD , qxF + R
bF ).
units, R
bF ) + Z with respect to xD and xF , subject
In stage 2, the consumer maximizes V (xD , qxF + R
bF , where I denotes the income of the representative consumer. The
to pD xD + pF xF ≤ I − rbRbF )inverse demand for good D and that for good F are respectively given by pD = VD (xD , qxF + R
bF ). Given these demand functions, the two firms respectively maximize
and pF = qVF (xD , qxF +R
bF and ΠF = {pF − (c + t)}xF with respect toΠD = [pD − {c + (1 − q) mL }]xD + (br − mH )R
xD and xF . The two firms’ profit maximizations constitute the equilibrium sales of each good.
bF , and the
The equilibrium sales in turn determine the equilibrium amount of repaired units, R
equilibrium repair price, rb. We have the following lemma.
Lemma 1 Even if the repair services for good F are provided by firm D, all broken units of thegood are repaired in equilibrium.
Intuitive explanation is as follows.17 Because an unbroken unit and a repaired unit of thesame good are perfect substitutes, the firm D’s repair of a broken unit of good F is regarded as
if firm D sells an extra unit of good F . Because we have assumed that mH ≤ c holds, mH ≤ c + t
always holds so that firm D’s unit cost of repairing good F is lower than firm F ’s unit cost of
selling good F . Besides that, the ”quality” of repaired unit of good F is higher than that of thepurchased unit of good F in the sense that the latter can break down after the purchase. Thismeans that consumer’s willingness to pay for an extra unit of good F is higher for the repaired
bF = 0, firm D’sunit than for the originally purchased unit. Therefore, if evaluated at the R
marginal revenue from repairing an extra unit is higher than firm F ’s marginal revenue fromselling an extra unit of good F . Because of these properties, the marginal revenue of firm D from
repairing an extra unit of good F is always larger than the marginal cost.
16 Because firm D monopolizes the repair market for good F , the equilibrium repair price and the equilibrium

level of RF should not be changed even if firm D chooses the service price to maximize the profit from the repair
services.
17 This equilibrium property is the same as that of Chen and Ross (1998), though the logic behind our model isslightly different because the rival producer, rather than the original producer, provides the repair services.

11

Meanwhile, firm D anticipates that its repairs of good F in stage 3 increases the attractivenessof good F in the product market and thereby decreases its profits from selling good D in stage
2. However, we find that the positive effect of repairs on profits in the aftermarket always
dominates the negative effect in the product market. Furthermore, it is not profitable for firm
F to manipulate xF so that it prevents the full-repairs of good F by firm D. Consequently, allbroken units of good F are repaired in the RR equilibrium.
bF = (1 − q)xF holds, the equilibrium service price is given by rb = VF (xD , xF ). Then,
Since R
the first-order conditions of the firms’ profit maximizations, ∂ΠD /∂xD = 0 and ∂ΠF /∂xF = 0,
can be written as
VD (xD , xF ) + VDD (xD , xF )xD + (1 − q)VF D (xD , xF )xF
q [VF (xD , xF ) + VF F (xD , xF )xF ]= c + (1 − q) mL ,(2)

= c + t.

(3)

RR
The equilibrium sales, (xRR
D , xF ), are derived by solving the above two equations and theequilibrium prices of the two goods and the equilibrium repair-price are respectively given by
RR
RR
RR
RR
RR
RR
RR
pRR
= VF (xRR
D = VD (xD , xF ), pF = qVF (xD , xF ), and r
D , xF ).

2.1.2

The OR equilibrium

Suppose firm F undertakes FDI in services to provide the repair services for good F by itself
and gives a full-warranty to each buyer of good F . In stage 2, the consumer anticipates thatall broken units of good F are freely repaired in stage 3. This means that RF = (1 − q) xF
holds, and the consumer maximizes V (xD , xF ) + Z subject to pD xD + pF xF ≤ I in stage 2. The
first-order condition yields the inverse demand-function for each good as pD = VD (xD , xF ) andpF = VF (xD , xF ). The two firms’ problems become:max ΠD = [pD − {c + (1 − q) mL }]xD = [VD (xD , xF ) − {c + (1 − q) mL }]xD ,xD

max ΠF = [pF − {c + t + (1 − q) mL }]xF = [VF (xD , xF ) − {c + t + (1 − q) mL }]xF .
xF

Then, the first-order conditions are given by
VD (xD , xF ) + VDD (xD , xF )xD= c + (1 − q) mL ,(4)

VF (xD , xF ) + VF F (xD , xF )xF= c + t + (1 − q) mL .

(5)

OR
The equilibrium sales are derived by solving (4) and (5), which are denoted by (xOR
D , xF ).
OR
OR
ORThe equilibrium prices of the goods are respectively given by pOR=
D = VD (xD , xF ) and pF
OR
VF (xORD , xF ).

12

2.1.3

The NR equilibrium

Suppose neither firm D nor firm F establishes the repair facilities for good F in Stage 1. In thiscase, all broken units of good F remain unrepaired (RF = 0), which means that dF = qxF holds.18
In Stage 2, the consumer maximizes V (xD , qxF ) +Z subject to pD xD +pF xF ≤ I. The first-ordercondition yields the demand for each good, which is respectively given by pD = VD (xD , qxF ) andpF = qVF (xD , qxF ).
Given these inverse demand functions, the firms’ maximization problems are given by[VD (xD , qxF ) − {c + (1 − q) mL }]xD ,max ΠD=max ΠF= {qVF (xD , qxF ) − (c + t)}xF .

xDxF

The first-order conditions of profit maximizations are as follows:
VD (xD , qxF ) + VDD (xD , qxF )xD
q [VF (xD , qxF ) + qVF F (xD , qxF )xF ]= c + (1 − q) mL ,(6)

= c + t.

(7)

By solving these equations, we obtain the equilibrium sales of goods, which are denoted by
R
NR
NR
NR
NR
NR
=
(xN
D , xF ). The equilibrium prices are respectively given by pD = VD (xD , xF ) and pF
RNR
qVF (xN
D , qxF ).

2.2

The effects of repair services

Here, we explore the effects of repair services for good F on the product market competition
by comparing the three sub-game equilibria just mentioned. The comparisons are essential forunderstanding the effects of trade liberalization in goods discussed in the next section, because ashift of the regime in the aftermarket, which is induced by trade liberalization, is an importantfactor to distinguish between welfare-reducing trade liberalization and welfare-improving tradeliberalization.
We first examine how the repair services change the demand for good F in the product market,
holding xD constant. Then, we discuss how they change the strategic interaction between thetwo firms.
2.2.1

Market-contraction effect and valuation effect

In the NR subgame, if the consumer were to consume deF of good F , she must purchase deF /qunit in the product market because a fraction of good F fails and remains unrepaired. When
18 The outcomes of the NR equilibrium would be unchanged even if we allow the repurchase of good F

refund by firm F after consumers find the broken units. See Section 4.3 for details.

13or thedeF /q units of good F are sold, the corresponding market price is given by pF = qVF (xD , deF ) (seepoint A in Figure 1). The RN curve in Figure 1 represents the relationship between the priceand the sales of good F in the NR subgame, holding xD constant.
[Figure 1 around here]
In the OR subgame, the consumption of good F , deF , coincides with the sales of good F
because firm F repairs good F without charge. In this case, the price of good F is given by
pF = VF (xD , deF ) (see point C in Figure 1). Holding xD constant, the relationship between theprice and the sales of good F in the OR subgame is depicted as OR curve in Figure 1.
Starting from no repairs for good F , the repair services by firm F shift the demand for good
F from the RN curve to the OR curve. Accordingly, given that the consumption of good F is deF ,the price and the sales of good F moves from point A to point C in Figure 1. To decompose themovement into two effects, let point B in Figure 1 be the intersection of deF and qVF (xD , deF ).
As is mentioned above, the consumer makes a precautionary purchase of good F in NR
subgame because she anticipates that a fraction of the purchased units of good F will fail andremain unworkable. Given deF and the price of good F , the amount of the extra purchase of good
F is given by deF /q − deF = (1 − q)deF /q. The increased durability of good F with repair services
for good F eliminates this precautionary purchase of good F . From the viewpoint of firm F , thismeans that its sales of good F decreases. We call the effect the market-contraction effect of repair
services. In Figure 1, the movement from point A to point B represents the market-contraction
effect.
The increased durability of good F also raises the attractiveness of good F for the consumer
and her willingness to pay for the good. Therefore, repair services increase the price of good Fthat realize deF units of consumption from qVF (xD , deF ) to VF (xD , deF ). We call this effect thevaluation effect. The difference in the price, (1 − q) VF (xD , deF ), captures the valuation effect. In
Figure 1, the movement from point B to point C corresponds to the valuation effect.
The valuation effect increases the price that can be charged while the market-contraction effectdecreases the amount of sales to achieve deF . Although the overall effect on the total revenue andthe profit captured by firm F seems to be ambiguous, we can easily verify that the shift from the
NR subgame to the OR subgame does not change the total revenue (and so the total expenditureof consumers) while it always increases the profit of firm F , if xD and deF are kept constant.
2.2.2“Business-stealing” by the rival firm

Next, we consider the effect of the shift from the OR subgame to the RR subgame on the price andthe sales of good F . In the OR subgame, firm F repairs good F and it captures the benefit from14

the valuation effect in the product market. If firm D repairs good F , however, the benefit fromthe valuation effect is captured by firm D in the aftermarket. In the RR subgame, each consumer
anticipates at the time of purchasing good F that she will pay (1 − q) rbxF in the aftermarket.
Given that the consumption of good F is deF , the consumer’s marginal willingness to pay for good
F , VF (xD , deF ), should be equal to the sum of pF and (1 − q) rb. Since firm D sets rb = VF (xD , deF )in stage 3, the price of good F in the product market becomes pF = VF (xD , deF ) − (1 − q) rb =qVF (xD , deF ). The expected price of repair services in the aftermarket makes the price of good F
in the product market smaller than the price of good F in the OR subgame, VF (xD , deF ).
Note that the expected repair price, (1 − q) rb = (1 − q) VF (xD , xF ), coincides with the magnitudes of the valuation effect. This means that the benefit from the valuation effect is completely
stolen by firm D. As a result, the price of good F declines from point C to point B in Figure 1,
given deF . The RR curve, which is shifted downward from the OR curve by the same amount as
the valuation effect, represents the relationship between the price and the sales of good F in the
product market in the RR subgame.
It is apparent that, holding xD and deF constant, the shift from the OR subgame to the RR
subgame reduces both the total revenue and the profit captured by firm F , while it does notchange the total expenditure of consumers.
Up to this point, we have investigated the demands for good F in each subgame holding xD isconstant. In the equilibrium analysis, however, firm D strategically chooses xD in each subgame.
If we consider changes in xD , there emerges a collusive effect in the RR subgame. The detailed
explanation will be made in the following subsection, which discusses the strategic interaction
between the two firms.
2.2.3

Interaction between firms and collusive effect

Bearing the above-mentioned effects in mind, now we compare the equilibrium outcomes withtaking into account the changes in firm D’s incentives and the strategic interactions between thetwo firms. As have mentioned earlier, only the workable units of each good are substitutes for
consumers. Therefore, the product market competition in the NR subgame can be regarded asif the two firms compete in xD and qxF . Firm D’s reaction curve is derived from (6) and it isdepicted as the dd line in Figure 2. Similarly, firm F ’s reaction curve is derived from (7) and itRNR
is depicted as the f f line in Figure 2. The equilibrium amounts of workable units, (xN
D , qxF ),are determined at the intersection of the dd line and the f f line.
[Figure 2 around here]

15

How the equilibrium amount changes in the RR subgame? The firm F ’s reaction curve in
the RR subgame is the same as that in the NR subgame. In the RR subgame, firm F cannot
capture the valuation effect of repairs due to the business-stealing by firm D. Therefore, given
xD and the amount of the workable units of good F , dF , the price of good F in the productmarket is given by qVF (xD , dF ) in both subgames (see Figure 1 and the discussion of Sections
2.2.1 and 2.2.2). As a result, given dF = xF holds in the RR subgame and dF = qxF holds in
the NR subgame, firm F ’s reaction function with respect to dF become the same (see (3) and(7)). Therefore, the f f line also represents the firm F ’s reaction curve in the RR subgame.
The firm D’s reaction curve in the RR subgame, on the other hand, shifts to the Dd line which
locates inside the dd line. In the RR subgame, an increase in the sales of good D reduces firm
D’s profits from the repair services since it decreases both imports of good F and the equilibrium
repair price it will change in the next stage. Hence, firm D becomes less willing to increase xD .
The effect, which we call the collusive effect, decreases firm D’s optimal supply of good D given
xF . More specifically, the Dd line is derived from (2) and the third-term of the right-hand sideof the equation, which is negative because VF D < 0, represents the collusive effect.
The equilibrium workable units of each good which should be equal to the equilibrium sales,
RR(xRR
D , xF ), are determined at the intersection of the Dd line and the f f line. As is seen in theRNR
hold in equilibrium. Note
> qxN
and xRR
figure, the collusive effect makes both xRRFF
D < xDthat if the collusive effect were absent, the repair services by firm D would have no effect on theequilibrium amount of workable units. Also, note that even if the amount of the workable unitsof good F remain unchanged, the market contraction effect of repair services makes the amountof the sales of good F decline. We can confirm that, although the collusive effect makes theworkable units of good F larger, the market contraction effect dominates it and the equilibriumNR
sales of good F actually declines in equilibrium: xRR
F < xF .

Next, we discuss how the shift from the NR subgame to the OR subgame changes the equilibrium amount. In both subgames, firm D cannot capture any rents from the repair services ofgood F . Therefore, firm D’s reaction function in the OR subgame becomes the same as that in
the NR subgame, which has been depicted as the dd curve in Figure 3.
[Figure 3 around here]
Firm F ’s reaction function, on the other hand, changes because now she can capture the rents
from the valuation effect of repair services. The firm F ’s reaction curve which is derived from
(5) is depicted as the F F line in Figure 3. Although the unit cost of supplying good F is also
increased from c + t to c + t + (1 − q)mL , we can confirm that an increase in the marginal revenue
due to the valuation effect always dominates the increase in the unit cost. Therefore, the F F
16

Rline locates outside the f f line. The shift of firm F ’s reaction curve makes xOR
< xN
and
D
D
R
xOR
> qxN
hold.
F
F

Note that the ranking between xRR
and xORand between xRR
and xORare ambiguousFF
D
D
and they depend on the relative magnitudes of the valuation effect in the OR subgame andthe collusive effect in the RR subgame. In sum, we have the following proposition as to theequilibrium amount of the working units and the equilibrium amount of the sales of the twogoods.19
R
OR
RR
OR
NR
Proposition 1 Given the tariff level, (i) qxN< min[xRR
F
F , xF ], (ii) max[xD , xD ] < xD ,NR
and (iii) xRRhold.
F < xF

This proposition implies that the repair services for good F always: (i) increase the equilibrium
workable unit of good F , (ii) decrease the equilibrium sales as well as the equilibrium workable
unit of good D, and (iii) decrease the equilibrium sales of good F if firm D provides the repair
services.
2.2.4

Consumer surplus and firms’ profits

Having analyzed the effects of repair services on the equilibrium sales and the equilibrium workable units, we now turn to the effects on consumers and the firms. Let CS k denote the consumer
surplus in the k (k ∈ {RR, OR, N R}) equilibrium. We have the following proposition.
Proposition 2 Given the tariff level, CS RR < CS N R < CS OR holds.
Compared to the NR subgame, the valuation effect increases firm F ’s marginal gains fromselling a workable unit of good F in the OR subgame. Therefore, the product market competition
becomes more intense in the OR subgame and consumers prefer the OR equilibrium to the NR
equilibrium.
On the contrary, the collusive effect generated by the rival’s repairs weakens the productmarket competition in the RR subgame. This effect raises the equilibrium price of good D, while
it also increases the equilibrium workable units of good F . The latter positive effect, however, isrelatively small because it is the second-order effect and an import tariff makes the market-size ofgood F being weakly smaller than that of good D. Therefore, the former negative effect dominates
the latter positive effect and consumers prefer the NR equilibrium to the RR equilibrium.
19 To make the proof of the following propositions as simple as possible, we use a standard quadratic form as

the sub-utility function from here on: V (dD , dF ) = a (dD + dF ) − (d2D + d2F )/2 − bdD dF where b represents thesubstitutability of the two products and we assume b ∈ (0, 1). Note that VF F = VDD = −1 and VF D = VDF = −bhold under this form. Even if we consider the other forms of the sub-utility function, the basic results of our paper
would be unchanged.

17Regarding the firms’ profits, let Πki denote the operating profits of firm i (i ∈ {D, F }) in thek (k ∈ {RR, OR, N R}) equilibrium. We have the following proposition.
NR
NR
RR
Proposition 3 Given the tariff level, ΠRR< ΠOR
and ΠORF < ΠFF
D < ΠD < ΠD hold.

If we move from the NR equilibrium to the RR equilibrium, the rent from the valuation effectis captured by Firm D, and only the market-contraction effect matters for firm F , which has
a negative effect on the profit of Firm F by reducing the sales of good F without affecting theworkable unit of good F . Although the collusive effect works in favor of firm F , it is the secondorder effect and dominated by the market-contraction effect. By contrast, the market-contraction
effect does not matter for firm D since it is unrelated to the equilibrium workable units of goodF , while the valuation effect and the collusive effect work in favor of firm D. Hence, we have
R
NR
ΠN
> ΠRR
and ΠRRFF
D > ΠD .

In the OR equilibrium, on the other hand, firm F can capture the rents associated with thevaluation effect, and its positive effect on the profits always outweighs the negative effect due tothe market-contraction effect. Besides that, the increase in the workable units of good F by its
R
own repairs (qxN< xOR
FF ) has a strategic effect in the product market which shifts rents fromORR
R
and ΠN> ΠN
firm D to firm F . As a result, we have ΠOR
D > ΠD .
F
F

2.2.5

A time inconsistency problem of consumers

In Section 3.2.4, we have shown that the repairs of the imported good by the domestic rival firmcause a collusive effect and hurt consumers, compared to the case without repair services. The
consumer’s loss is due to a time inconsistency problem on the side of the consumers.
In Stage 3, given that consumers have already purchased the goods and hold the broken unitsof good F , purchasing the repair services from firm D benefits them. However, the consumers’
repairs in Stage 3 causes the collusive effect and reduces firm D’s supply of good D in Stage 2,
and the weaker product market competition hurts consumers.
Since the latter effect dominates the former effect, if consumers could pre-commit whether theyutilize the repair services for good F provided by firm D in the RR subgame, they would commit
not to utilize them in order to avoid the collusive effect. In this case, the equilibrium outcomes of
the RR equilibrium coincides with those of the NR equilibrium. However, it seems more realisticthat consumers are unable to make such a commitment credibly before they purchase the goods.
Therefore, we assume that consumers cannot pre-commit the utilization of repair services.
In the absence of pre-commitment, even if consumers ex-ante anticipate that the repairs ofimports by the domestic rival firm cause the collusive effect which increases the price of good18

D and eventually hurts them, they cannot refrain from ordering repairs to the domestic firm in
the aftermarket because the repairs ex-post benefit consumers given the prices of the goods. Dueto this time-inconsistency problem, consumers cannot avoid the negative welfare effect of repair
services conducted by firm D.

2.3

Entry into the repair services for good F

To derive the equilibrium of the entire game, we examine the firms’ entry decisions in stage 1.
The two firms simultaneously decide whether they provide the repair services for good F by
incurring the cost of entry, Ki (i ∈ {D, F }), given the choice of the rival firm. If firm F provides
the service by undertaking a service FDI, firm D does not provide it because she cannot earn
positive operating profits in the repair market for good F to cover the fixed cost. Hence, Firm
D enters the repair market only if firm F does not enter.
Given that firm F does not undertake a service FDI, firm D’s gains in operating profits fromNR
providing the repair services for good F is given by ∆ΠD = ΠRR
D − ΠD . Since we can confirm

that ∂{∆ΠD }/∂t < 0 holds, we have the following lemma.
Lemma 2 If firm F does not provide the repair services for good F , a tariff reduction increases
firm D’s gains from providing the repair services.
A tariff reduction increases the imports of good F . This increases the amount of the broken
units of good F , making it more attractive for firm D to earn profits from repairing the rival’s
product. The lemma means that, as long as KD < K D ≡ ∆ΠD |t=0 holds, there exists a thresholdvalue of tariff, tD > 0, such that ∆ΠD > KD holds if and only if t < tD .20 For expositional
simplicity, we set tD = 0 if KD ≥ K D holds.
Regarding firm F ’s gains from the entry, let ∆ΠF denote its gains in operating profits fromundertaking a service FDI and providing the repair services. Because firm F provides a fullwarranty if she undertook a service FDI, its strong position in the repair services of good F
means that she always enters the repair market if ∆ΠF > KF holds, regardless of firm D’s entry
decisions. We have the following proposition.
Proposition 4 The equilibrium of the entry game becomes: (i) the OR equilibrium if ∆ΠF > KF
holds, (ii) the RR equilibrium if ∆ΠF ≤ KF and t < tD hold, and (iii) the NR equilibriumotherwise.
20 Lemma 2 implies that ∆ΠD is maximized at t = 0.

Let t denote the minimum level of tariff that eliminates

the imports of good F under the RR equilibrium and the NR equilibrium. Clearly, ∆ΠD = 0 holds if t = t.
Therefore, if KD satisfies KD < K D , there exists a threshold value of tariff, tD , such that ∆ΠD < KD holds fort ∈ (tD , t), ∆ΠD = KD holds for t = tD , and ∆ΠD > KD holds for for t ∈ [0, tD ).

19

Given KD < K D holds, Figure 4 depicts the possible equilibrium outcomes in the (t, KF )space.2122 When KF is high, if is unprofitable for firm F to undertake a service FDI. Under thissituation, if t is high, the imports of good F are small and an increase in ΠD by providing repair
services for good F cannot exceed the fixed cost, KD . Therefore, repair services for good F are
not provided in equilibrium when both t and KF are high (the region “NR” in the figure). WhenKF is high while t is low, it becomes profitable for firm D to provide repair services for good F ,and the aftermarket for good F is monopolized by firm D (the region “RR”). When both KF andt are low, however, ∆ΠF > KF holds and Firm F can increase its overall profit by establishing
service facilities and selling its product with full warranty. The full warranty ensures that firm
D never establish service facilities whenever firm D anticipates that firm F will establish service
facilities.23 Therefore, the repairs for good F are sorely provided by firm F in equilibrium if∆ΠF > KF holds (the region “OR”).
[Figure 4 around here]

3

Liberalization of goods trade and service FDI

In this section, we examine the welfare effects of trade liberalization in goods and its connection
to the structure of the aftermarket service. Trade liberalization, represented by a decline in t,affects welfare within each regime of the aftermarket services, and it may also affect welfare byinducing a switch of the regime.24 Here, we will show that the overall effects of trade liberalization
drastically differ depending on the extent to which service FDI is liberalized.
21 We can see that ∆Π jumps up at t = t . If firm F does not provide the services, the equilibrium of the entire
F
D
R
game becomes the NR equilibrium for t ≥ tD and the RR equilibrum for t < tD . Hence, ∆ΠF = ΠOR− ΠNFFR > ΠRR holds which implies
holds for t ≥ tD and ∆ΠF = ΠOR− ΠRRholds for t < tD . By Proposition 3, ΠN
F
F
F
F

that firm F has a stronger incentive to undertake a service FDI if she faces a potential entry of the rival firm.
22 It is ambiguous whether ∆ΠF is decreasing or an inverse-U shaped curve in t. The increased imports froma tariff reduction increase firm F ’s gains from the entry, but there is an additional effect. In the RR subgame
and the NR subgame, because firm F cannot capture the rents associated with the broken units, the demand
curves are flatter than those in the OR case (see Figure 1). Hence, the tariff reduction increases xF less in the
OR subgame than it does in the RR and the NR subgame. If the cost of providing services (mL ) is sufficiently
large and that of supplying the goods (c and t) is sufficiently small, the latter effect dominates the former effectand trade liberalization undermines firm F ’s entry. See the Appendix for details. In Figure 4, we depict the casewhere ∆ΠF is an inverse U-shaped curve in t. The shape of ∆ΠF does not affect the main results of the paper.
23 As long as m
H ≥ mL holds, the entry decisions of the two firm does not depend on the full-warranty
assumption. If mH < mL holds and firm F does not provide a full warranty, it is possible that both firms enterin equilibrum. In this case, however, firm F will offer a full warranty to exclude firm F from the aftermarket if
she can choose it. See Section 4.2 for details.
24 See the Appendix for the detailed calculations of the effects of trade liberalization within each regime.

20

To describe the different effects of trade liberalization, we first compare the two specific cases:(i) the fixed cost of service FDI is high enough so that trade liberalization induces firm D’s entry,(ii) it is low enough so that trade liberalization induces firm F ’s entry. Then, we explain a generalproperty of lowering the fixed cost.

3.1

Trade liberalization when the fixed-cost of service FDI is highLet KF0 and t0 respectively denote the initial level of the fixed costs for FDI in services and theinitial level of tariff. Suppose ∆ΠF < KF0 and t0 > tD hold so that the entry into the repair
services for good F are unprofitable for both firm D and firm F . In this case, the equilibrium of
the entire game is initially the NR equilibrium (see Point A of Figure 4). Starting from t0 , if thetariff is gradually reduced, we have the following welfare effect within the NR equilibrium.
Within the NR equilibrium, trade liberalization has standard effects which increase the imports, benefits consumers and firm F , and hurts firm D. However, it may worsen world welfare
because the “quality” of good F is inferior to that of good D. The quality of good F is inferior
in the sense that a fraction of good F fails and remains unrepaired. Trade liberalization has thesubstitution effect which increases the consumption of good F and decreases that of good D,
and the effect reduces consumers’ gains from trade liberalization. As a result, firm D’s profit
loss can outweigh the consumers’ gains. As tariff is reduced, the imports of good F increase
and the gains from entry into the service market for good F become larger. When the tariff
reaches t = tD , the further reduction of t induces entry of firm D and switches the equilibriumfrom the NR equilibrium to the RR equilibrium. The switch to the RR equilibrium causes thecollusive effect which reduces the extent of the product-market competition and discontinuously
hurts consumers (see Proposition 2).
As has been discussed in Section 2.2.5, the consumer’s loss is due to a time inconsistency
problem on the side of the consumers. Even if consumers ex ante anticipate that the repairs ofgood F by firm D raise the price of good D in the product market and eventually become harmful
for them, they can’t help but repair good F after they purchased good F . This is because, given
consumers hold the broken units of good F , it is ex post optimal for them to purchase the repair
services from firm D.
By using Propositions 1 and 3, we can also confirm that the switch discontinuously reduces
the imports, hurts and firm F , while it does not affect firm D. The loss of firm F is due to themarket-contraction effect which outweighs the collusive effect. The switch has no effect on firm
D’s net profit (i.e., the operating profits minus the fixed cost of FDI) because ∆ΠD = KD holds
at t = tD .

21

As a result, the negative effects of the equilibrium shift on consumer surplus (CS RR < CS N R ),R
RR
tariff revenues induced by the reduced imports (tD xRR< tD xN<FF ), and firm F ’s profits ΠFRΠNlead to a decline in world welfare.
F

Once the RR equilibrium is realized, further reductions of t within the RR equilibrium increases the imports of good F and benefits consumers and firm F . However, it is ambiguouswhether the trade liberalization benefits or hurts firm D. Trade liberalization decreases the salesof good D and thereby lowers the profits in the product market, while it increases the sales ofgood F and so does firm D’s profits in the after-service market. Therefore, the overall effecton firm D’s profits depends on the relative magnitudes of these two effects. Furthermore, it isalso ambiguous whether trade liberalization improves or worsens world welfare within the RR
equilibrium because it increases the sales of good F and reduces the sales of good D and thehigher cost of repairing good F (mH ≥ mL ) worsens the overall efficiency of the economy. Dueto these effects, trade liberalization may worsen world welfare within the RR equilibrium.
Table 1 summarizes the effects of trade liberalization when KF = KF0 .
[Insert Table 1 around here]
If the effect of the regime switch outweighs the effect within each regime, the overall effect oftrade liberalization from t0 ∈ (t1 , t) to t1 ∈ [0, tD ) reduces imports and hurts consumers and firm
F , and worsens world welfare. Besides that, if trade liberalization increases the profit of firm D
within the RR equilibrium, there is a case where the same tariff reduction benefits firm D.

3.2

Trade liberalization when the fixed-cost of service FDI is low

Next, suppose the fixed cost is reduced from KF0 to KF1 so that KF1 < min[ ∆ΠF |t=0 , ∆ΠF |t=tD ]holds (see Point B of Figure 4). In this case, there exists a unique threshold value of tariff,tF (∈ (tD , t)), such that ∆ΠF > KF holds if and only if t < tF . As tariff is reduced fromt0 and becomes lower than t = tF , it becomes profitable for firm F to undertake service FDI.
Because firm F offers a full-warranty on good F at the point of selling good F , firm D has noway to win the competition with firm F in the aftermarket, the RR equilibrium is no longer apossible equilibrium outcome for any t ∈ [0, tD ).25 Consequently, the trade liberalization shifts
the equilibrium from the NR to the OR equilibrium.
By using Propositions 2 and 3, we can confirm that the switch intensifies market competition
25 See the first two paragraphs of Section 2.1.

A full-warranty assumption does not affect the result as long as

mH ≥ mL holds. See Section 4.2 for details.

22and discontinuously benefits consumers, hurts firm D, and improves world welfare.26 It does
not affect firm F because ∆ΠF = KF holds at t = tF in this case. Once the service FDI isundertaken, the further trade liberalization within the OR equilibrium always has a standard
effect which increases imports, benefits consumers and firm F , hurts firm D, and improves world
welfare. Table 2 summarizes the effects of trade liberalization in this case.
[Insert Table 2 around here]
If KF is low enough so that trade liberalization induces firm F ’s service FDI while excluding
firm D’s entry, the overall effect of trade liberalization from t0 ∈ (t1 , t) to t1 ∈ [0, tD ) always
benefits consumers and hurts firm D. Regarding world welfare, although the shift from the NR
to the OR equilibrium improves world welfare, the overall effect of trade liberalization can be
negative if a tariff reduction worsens world welfare within the NR equilibrium. However, as KF1
becomes smaller, the cut-off level of tariff, tF , becomes larger and approaches t0 . In particular,
if KF1 is small enough to make t0 ≤ tF hold, the equilibrium regimes before the tariff reduction
also becomes the OR equilibrium and any tariff reductions always have a positive effect on world
welfare.

3.3

The role of liberalization in service FDI

The above two examples imply that consumer-hurting, welfare-reducing trade liberalization canbe transformed into a consumer-benefiting, welfare-improving liberalization by liberalizing FDI
in aftermarket services. This transformation is not a special case and has a general validity, as
the following proposition states.
Proposition 5 If tD > 0 and KF0 > ∆ΠF holds for some t in t ∈ [0, tD ), then a tariff reduction
from t0 ∈ (t1 , t) to t1 ∈ [0, tD ) may decrease imports, hurt consumers and firm F , benefits firm
D, and/or worsen world welfare, holding KF fixed at KF = KF0 . In this case, we can alwayse F (≤ K 0 ), such that the same tariff reduction necessarilyfind a unique cut-off level of KF , KFeF .
increases imports, benefits consumers and firm F , and improves world welfare for all KF < KThis proposition suggests that any consumer-hurting, welfare-reducing trade liberalization
turns into consumer-benefiting and welfare-improving one if KF is reduced through liberalizationof service FDI. If the fixed cost is high enough, trade liberalization induces the entry of firm D
into the aftermarket. The entry entails the collusive effect which hurts consumers and worsens
26 It is ambiguous whether the switch increases the imports of good F , though it always increases the consumptionof good F (see Proposition 1).

23

world welfare. If the fixed cost is sufficiently lowered, however, trade liberalization induces theentry of firm F into the aftermarket. The entry not only blocks the potential entry of firm Dthat causes the collusive effect, but also increases the marginal gains from selling good F in the
product market because firm F can capture the valuation effect of repairs. This makes firm
F supply good F more, given the supply of good D. As a result, trade liberalization which
induces the entry of firm F intensifies the product market competition and benefits consumers
and improves world welfare.
The result suggests that promoting FDI in aftermarket services is important to make tradeliberalization in goods consumer-benefiting and welfare-improving.

4

Discussion

We have shown that the provision of aftermarket services conducted by the other firm, withwhom the original producer competes in the product market, has a collusive effect and hurts
consumer and worsens world welfare. Even if consumers anticipate this effect and it is ex ante
beneficial for them to stop utilizing the services, it is ex post beneficial for them to purchasethe services in the aftermarket once they find the broken units of good F . Because of thetime inconsistency problem on the side of consumers, trade liberalization in goods that induces
the service provisions by the rival firm may hurt consumer and worsen world welfare. The
liberalization of service FDI, however, converts the same trade liberalization into a consumerbenefiting, welfare-improving liberalization. In this section, we explore the robustness of theseresults by relaxing some assumptions made in the basic model.

4.1

Repair services by ISOs

Up to this point, we have assumed that only firm D and firm F can provide the repair services.
In this section, we consider the case in which Independent Service Organizations (ISOs) may also
provide the repair services for good F . Under this alternative set-up, many potential ISOs, firm
F , and firm D simultaneously decide whether they provide the repair services for good F in stage
1. If an ISO enters the repair market, it must incur the fixed cost. For simplicity, we assume theISO incurs the same unit cost, mH , and the same fixed cost, KD , as those of firm D.27
27 Since the firms which produce goods have better knowledge about the goods, the unit cost and the fixed costof each ISO in the aftermarket may be higher than those of firm D. Or they may be lower if each ISO has betterknowledge and higher skills on repairing goods. Although different service costs between firm D and ISOs makeeach ISO’s entry more difficult or easiler, the qualitative nature of our analysis would remain unchanged.

24

We can confirm that, even if ISOs enter the repair market, all broken units of good F are
repaired in equilibrium.28 Then, how does the presence of ISOs affect the equilibrium of the
product market? Given that firm F does not undertake service FDI, if more than two ISOs
or an ISO and firm D enter the repair markets for good F , then the price competition in theaftermarket leads to the marginal-cost pricing in equilibrium: r = mH . As long as KD > 0, it
is unprofitable for each ISO to enter the repair market if other ISOs or firm D enter the repair
market. This means that at most a single ISO enters the aftermarket in equilibrium. We call theequilibrium where a single ISO monopolizes the aftermarket the ISO equilibrium.
Given that a single ISO monopolizes the repair services for good F , it sets r = VF (xD , xF )to maximize its profit and the inverse demand for good F are given by pF = qVF (xD , xF ). The
equilibrium operating profit of the ISO is given by ΠISO = {VF (xD , xF ) − mH } (1 − q) xF . Instage 2, firm D sets xD such that it maximizes ΠD = [VD (xD , xF ) − {c + (1 − q) mL }]xD andfirm F sets xF such that it maximizes ΠF = [qVF (xD , xF ) − (c + t)]xF .
Because firm F cannot capture the rents associated with the repairs, its maximization problem
becomes the same as the RR subgame. Meanwhile, firm D also cannot capture any rents fromthe repair services for good F , and so its maximization problem is the same as that in the OR
subgame. Hence, firm F ’s reaction curve becomes the f f line in Figures 2 and 3, while firm D’sreaction curve becomes the dd line in these figures. The equilibrium sales of the two goods under, are obtained at the intersection ofand xISO
the ISO equilibrium, which are denoted as xISO
F
D
R
and
= xN
the f f line and the dd line. It is obvious that the equilibrium sales satisfy xISO
D
D
R
xISO
= qxN
F
F . The equilibrium consumer surplus and the firm’s profits are respectively denotedISO.
by CS ISO , ΠISO
D , and ΠF

Because the valuation effect is captured by the ISO and the collusive effect is absent in thiscase, holding t fixed, the equilibrium workable units of both goods become the same between the
NR equilibrium and the ISO equilibrium. This means that consumer surplus and firm D’s profits
RISO
also remain unchanged (i.e., ΠISO
= ΠN
= CS N R given t). Because of the marketD
D and CS
R
R
contraction effect, however, the switch reduces the volume of imports (i.e., xISO
= qxN
< xN
F
F
F )
R
and hurts firm F (ΠISO< ΠN
F
F ). Note that the lack of the collusive effect means that the entryof the ISO into the aftermarket for good F hurts firm F more than the entry of firm D does
(ΠISO
< ΠRR
F
F ). The loss of firm F and the decline in tariff revenue mean that world welfare
28 Suppose a single ISO monopolizes the repair services for good F . The ISO’s profit-maximization problem instage 3 is the same as that of firm D in the RR case. The ISO sets r such that RF = (1 − q) xF holds. Besides
that, the repair price becomes lower if more than two ISOs or both an ISO and firm D enter the repair market.
This means that all broken units will be repaired in equilibrium if at least one ISO enters the repair market. See
Appendix for details.

25

in the ISO equilibrium is lower than that in the NR equilibrium if the net profit of the ISO,ΠISO − KD ≥ 0, is small.
We have compared the NR equilibrium and the ISO equilibrium given the tariff level. Now
we examine the effect of trade liberalization in the presence of ISOs. We can confirm that a tariff
reduction increases an ISO’s gains from providing the repair services (i.e, ∂{∆ΠISO }/∂t < 0).
The following proposition suggests that the ISO equilibrium can be the equilibrium of the entire
game.
Proposition 6 If ∆ΠF ≤ KF and ΠISO |t=0 > KD hold, then there exists a threshold value oftariff, tISO ∈ (0, t), such that the equilibrium of the entry game becomes: (i) the NR equilibrium if
max[tISO , tD ] ≤ t holds, (ii) the RR equilibrium if tISO ≤ t < tD holds, (iii) the ISO equilibrium
if tD ≤ t < tISO holds, and (iii) either the ISO equilibrium or the RR equilibrium if 0 ≤ t <min[tISO , tD ] holds.
When the tariff level is high, the import of good F is small and the ISO’s operating profit
from providing repair services does not exceed the fixed cost, KD . If the tariff is sufficiently
reduced, however, the market size for repair services become sufficiently large and it becomes
profitable for an ISO to enter the aftermarket if both firms or other ISOs do not enter.
Proposition 6 implies that, given that the fixed cost of service FDI and the import tariff
initially satisfy ∆ΠF < KF0 and t0 > max[tISO , tD ], a tariff reduction from t0 to t1 ∈ [0, tISO )may switch the equilibrium from the NR to the ISO equilibrium. Since a tariff reduction ineach equilibrium benefits consumers and hurts firm D while the switch from the NR to the ISO
equilibrium does not affect consumer surplus nor firm D given t, the trade liberalization always
benefits consumers and hurts firm D. Although a tariff reduction always benefits firm F withineach regime, the switch from the NR to the ISO equilibrium hurts firm F given t, and if this effectdominates the effects within each regime, the trade liberalization from t0 ∈ (max[tISO , tD ], t) to
t1 ∈ [0, tISO ) hurts firm F . Furthermore, if the difference between t1 and tISO is small enough,the profit loss of firm F dominates the profit gain of the ISO from the trade liberalization.29
Furthermore, trade liberalization may worsen world welfare within the ISO equilibrium by thesame reason as it does in the RR equilibrium. Consequently, trade liberalization from t0 to t1
may hurt firm F and worsens world welfare, while it always benefits consumers and hurts firm
D.
Under this situation, if the fixed cost of service FDI is sufficiently reduced, the same tariff
reduction induces service FDI by firm F and becomes consumer-benefiting and welfare-improving.
These results suggest that even if the presence of ISOs prevent the repairs by the rival firm,
29 Since ΠISO = KD holds at t = tISO , ΠISO − KD becomes smaller if tISO approaches t1 .

26

trade liberalization could still worsen world welfare and the liberalization of service FDI is still
important to guarantee welfare-improving trade liberalization.

4.2

Full-warranty assumption

We have assumed that firm D and firm F provide a full warranty if they provide the repair
services for their own products. Here, we show that the main results of our model do not dependon the assumption.
Suppose firm i (i ∈ {D, F }) charges consumers a fee for repairing good i. Let si denote theprice that firm i sets for repairing good i in Stage 3. Suppose only firm F establishes the facilities
for repairing good F . In this case, each consumer anticipates that she will pay sF in Stage 3 perunit of repairs if the purchased unit is defective. Since firm F always set sF so that all broken
units are repaired in equilibrium, the consumer maximizes V (xD , xF ) + Z with respect to xD and
xF , subject to pD xD + pF xF ≤ I − (1 − q)sF xF . The demand for good D and for good F arerespectively determined by pD = VD (xD , qxF ) and pF = VF (xD , xF ) − (1 − q)sF . Then, firm F ’s
maximization problem in stage 2 is to maximize ΠF = {pF − (c + t)}xF + (sF − mL ) (1 − q)xF =[VF (xD , xF ) − {c + t + (1 − q) mL }]xF , which is independent of sF and exactly the same as firm
F ’s maximization problem in the OR subgame (see Eq.(5)). Therefore, irrespective of the levelof sF , the equilibrium coincides the OR equilibrium.
Intuitively, if firm F sets sF in the aftermarket, each consumer’s willingness to pay for good
F in the product market decreases by (1 − q)sF . This means that the optimal price of goodF also decreases by (1 − q)sF . Hence, the ”full price” of good F that each consumer pays andfirm F receives, pF + (1 − q)sF , becomes the same as the price of good F under full-warranty
assumption. This also means that, even if both firm F and firm D enters the aftermarket forgood F , the equilibrium coincides with the OR equilibrium.
If both firms enter, they engage in Bertrand competition in the aftermarket for good F . If mH> mL holds, the price competition leads to limit-pricing by firm F , and the equilibrium prices
satisfy sF = sD = mH . In this case, firm F takes all demands and earns (mH − mL ) (1 − q)xF
in the aftermarket. If mH = mL holds, then the “race to the bottom” in repair prices results
in sF = sD = mH and both firms earn zero profits in the aftermarket for good F . In either
case, by the above discussion, the two firms’ optimization problems coincide with those in the
OR subgame. This means that, even if firm F does not provide a full warranty, firm D does notenter into the aftermarket for good F whenever firm F does it. As long as mH ≥ mL holds, theassumption of a full warranty is not essential to obtain the main results.
If mL > mH holds, firm D wins the price competition and earn (mL − mH )(1 − q)xF in the27

aftermarket. The repairs of good F by firm D reduces pF in the product market. Therefore,
compared to the OR equilibrium, the profit of firm D becomes higher and that of firm F becomes
lower. Even if firm F chooses to enter, Firm D also enters if KD is not so large, and the gap
between mL and mH is large. In this case, firm F cannot earn profits in the aftermarket if firm D
enters. Nevertheless, firm F may still establish its own facility in order to make the repair pricekept at sF = mH and increase its profit in the product market.30 Therefore, even if ∆ΠF > KF
holds, firm D may monopolize the repair market or both firms may establish service facilities inequilibrium if full warranty is not attached to good F . In this case, however, firm F has a strong
incentive to offer a full-warranty to exclude firm F from the aftermarket for good F if she canchoose it. Therefore, the results would not change even if firm F can choose whether or not it
offers full warranty.

4.3

Repurchase of good F

Here, we consider the case where each consumer is able to repurchase good F from firm F if
she found the unit of good F she purchased is broken down and the repair services for good F
are not provided. We can show that the equilibrium outcomes in this situation coincides withthose under the NR equilibrium (see Appendix for details). If consumers can repurchase good
F under no repair services, ceteris paribus, they reduce the amount of the original purchase ofgood F because they no longer need to make a precautionary purchase. However, the reductionin the original purchase of good F coincides with the amount of good F consumers repurchasein equilibrium. In other words, the precautionary purchase of good F in the NR subgame fully
supplements the repurchase of good F .
Therefore, the qualitative nature of the results would remain unchanged even if we allow the
repurchase of good F .31

5Conclusion

Using a duopoly model with horizontal product differentiation, this paper examines the interaction between trade liberalization in good and the liberalization of FDI in services. Our focus ison the post-production services as represented by repair services and maintenance services, which
can be demanded by consumers after they purchased the goods. The aftermarket services for
30 If firm F

does not enter, the equilibrium repair price becomes rb = VF (xD , xF ) (> mL ). Since higher repair

price decreases the profit of selling good F , firm F may enter to keep the repair price low.
31 Furthermore, if the imported good is sold as a limited version with special specifications, the repurchase of
the good is impossible.

28

imports can be provided either by the original producer or by the rival producer in the domestic
country.
When the fixed cost of FDI in repair services is sufficiently high, trade liberalization in goods
enhances the domestic firm’s entry into the repair services for imports. The repair services forimports induce the domestic firm to reduce the supply of her own product in the product market
because it has an effect to increase the domestic firm’s profit from the repair market. As aresult of this collusive effect, the repair services increase the price of the domestic product andconsumers become worse off compared to the case without any repairs for imports. It may also
hurt the foreign firm since the repairs eliminate the extra purchase for the precautionary motive
and reduce the sales of good F in the product market.
However, if the foreign firm, who is the original producer of the imported goods, undertakes
an FDI in aftermarket services, the services encourage the competition in the product market
and benefit consumers and the foreign firm. Trade liberalization in goods, however, does notnecessarily encourage the service FDI unless the fixed cost of the FDI is sufficiently low.
Furthermore, even if the repair services are provided by ISOs, it neither helps nor hurts
consumers if a single ISO monopolizes the repair market. It still reduces the profit of the foreign
firm and the loss is bigger than the case where the domestic firm provides the repair services.
The entry of ISOs benefits consumers and the foreign firm only if there is no fixed costs for entry
so that innumerable ISOs can enter the repair market.
These results suggest the importance of liberalization in service sectors, especially the liberalization of FDI in services. The liberalization not only reduces the cost of providing services,
but also diminishes the possibility that trade liberalization in good leads to an anti-competitive
behavior by the domestic firm in the product market. In other words, countries should promote
FDIs in aftermarket services to guarantee welfare-improving trade liberalization which enhances
the product market competition.
There remain some directions to extend our analysis. Our results indicate that a tariff-jumping
FDI in production may make consumers worse off by inviting the domestic firm’s entry into theaftermarket. To guarantee consumer’s benefits of a tariff-jumping FDI, it should be accompanied
by an FDI in repair services. Considering the problem of parallel imports in our framework would
be an interesting extension since the original producers sometimes refuse to repair the broken
units sold by unauthorized distributors.

29

Appendix
Proof of Lemma 1
At stage 3, the firm D’s maximization problem is to choose r that maximizes (r − mH ) RF subject
to RF ≤ (1 − q) xF . Let the Lagrangian function as L = (r − mH ) RF + λ{(1 − q) xF − RF }where λ is the Lagrangian multiplier. The first-order conditions are given by
VF (xD , qxF + RF ) + VF F (xD , qxF + RF )RF = mH + λ;(A1)

(1 − q) xF − RF ≥ 0; λ ≥ 0; λ [(1 − q) xF − RF ] = 0.
bF = (1 − q) xF and rb = VF (xD , xF ) hold at stage(i) Suppose λ > 0. This implies that R
3. At stage 2, the representative consumer anticipates that all broken units will be repaired
and its maximization problem is given by maxxD ,xF V (xD , qxF + (1 − q) xF ) +Z subject topD xD + pF xF ≤ I − rb (1 − q) xF . The first-order conditions yield pD = VD (xD , xF ), pF +(1 − q) rb = VF (xD , xF ). The profit-maximization problems of firm D and firm F are respectively
given bymax ΠD = (pD − c − (1 − q) mL ) xD + (1 − q) (br − m H ) xF
xD

= {VD (xD , xF ) − c − (1 − q) mL }xD + (1 − q) (VF (xD , xF ) − mH ) xF
max ΠF = {pF − (c + t)}xF = {qVF (xD , xF ) − (c + t)}xF
xF

RR
By solving the first-order conditions, the optimal sales of the two firms, (xRR
D , xF ), must satisfy
RR
RR
RR RR
RR
RR RR
VD (xRR
D , xF ) + VDD (xD , xF )xD + (1 − q) VF D (xD , xF )xF = c + (1 − q) mL ,
RR
RR
RR RR
VF (xRR
D , xF ) + VF F (xD , xF )xF =

c+t
.
q

(A2)
(A3)

By (A1), (A3), and c ≥ mH ,
RR
RR
RR RR
λ = VF (xRR
D , xF ) + VF F (xD , xF )xF − mH =c+t
− mH > 0.
q

RR
RR
RR
Therefore, (xRR
actually constitute an equilibrium.
D , xF ) and RF = (1 − q) xF(ii) Suppose λ = 0. This means that firm D sets RF so that only a part of the broken units
is repaired (i.e., RF < (1 − q) xF ). By (A1),
VF (xD , qxF + RF ) + VF F (xD , qxF + RF )RF = mH

(A4)

holds. Since we have assumed that VF F (dD , dF ) < 0 and 2VF F (dD , dF )+(∂VF F (dD , dF )/∂dF )dF <0 hold, 2VF F (dD , dF ) + (∂VF F (dD , dF )/∂dF )D < 0 holds for any D ∈ (0, dF ]. Combined this30

property with (A3) and c ≥ mH , we haveVF (xD , qxF + RF ) + VF F (xD , qxF + RF )RF>VF (xD , xF ) + (1 − q) VF F (xD , xF )xF>

VF (xD , xF ) + VF F (xD , xF )xF
c+t=> mH .
q

This inequality contradicts (A4). Therefore, λ = 0 cannot hold in equilibrium.

Proof of Proposition 1
By solving (2) and (3), we have the equilibrium sales in the RR case as
2{a − c − (1 − q) mL }q − ab 2 − q 2 + (2 − q) b (c + t)
RR
xD =
,
{4 − (2 − q) b2 }q(2 − b) qa + bq{c + (1 − q) mL } − 2 (c + t).
{4 − (2 − q) b2 }q

xRR
F =

(A5)

(A6)

RR
To guarantee xRR
D > 0 and xF > 0, we assume a > a := [2 (c + t) − bq{c + (1 − q) mL }]/(2 − b)q

are satisfied.
By solving (4) and (5), we have the equilibrium sales in the OR case asxOR
D =(2 − b) {a − c − (1 − q) mL } + bt,
4 − b2

(A7)

xOR
=
F

(2 − b) {a − c − (1 − q) mL } − 2t.
4 − b2

(A8)

R
NR
RR
We can easily confirm that xN> 0 hold as long as xRR
D > 0 and xF
D > 0 and xF > 0 hold.

By solving (4) and (5), and using (A5) and (A6), the equilibrium sales in the NR case are
given by
R
RRxN
D = xD +2b (1 − q) RRx ,(4 − b2 ) q F

R
xN
=
F

{4 − (2 − q) b2 } RRxF .
(4 − b2 ) q

(A9)

NR
RR
NR
By (A9), it is obvious that xRR
hold. By (A5) and (A6), and (A9), we
D < xD and xF < xF

RR
NR
2
NR
R
have xD −xD = − (1 − q) b (c − qmL + t) /{ 4 − b q} < 0, xRR= (1 − q) qb2 xN
F −qxFF /{4−
NR(2 − q) b2 } > 0, and xRR= 2 (1 − q) (c − qmL + t)/{ 4 − b2 q} > 0.
F − qxF

Proof of Proposition 2
The equilibrium consumer surplus under the quadratic utility function in the k (k ∈ {RR, OR, N R})
case is given by
CS k =(dkD )2 + (dkF )2+ b(dkD )(dkF ).
2

31

Since all broken units of good F are repaired both in the RR case and in the OR case, dRR
= xRR
i
i
R
R
hold for i ∈ {D, F }. In the NR case, on the other hand, dN
= xN
and
= xRR
and dRR
i
i
D
D
R
R
dN
= qxN
hold because the broken units of good F remain unrepaired. We haveFF

(1 − q) bB2 xRR
D

CS RR − CS N R = −2

2 (4 − b2 ) {4 − (2 − q) b2 }qwhere B2 = q (2 − b) {16 + 4b − 16b2 − b3 + 4b4 + qb 4 + 4b − b2 − 2b3 }a + 2b{4 − 7b2 + 2b4 −
4 − 3b2 + b4 q}t − q 4 − 3b2 (1 − q) {8 − b2 (3 − q)}mL + {2b 4 − 7b2 + 2b4 − b2 4 − 3b2 q 2 −
q (2 + b) 16 − 4b − 16b2 + 5b3 + 2b4 }c. We can verify that ∂B3 /∂a > 0 holds. Hence, we have
B3 > B3 |a=a = 2 (2 + b) 2 − b2 {4 − b2 (2 − q)}{(1 − q) (c − qmL ) + t} > 0 and so CS RR <CS N R holds. Similarly, we have
CS OR − CS N R =(1 − q) (t + c − qmH ) B3
2

2 (4 − b2 ) q 2

2
where B3 = 2q (b + 1) (2 − b) a − { 4 − 3b2 (1 + q) + 2b3 q}c − 4 − 3b2 (1 + q) t − (4 − 3b2 +2b3 ) (1 − q) qmL .

Since B4 is decreasing in a, we have B3 > B3 |a=a = (1 − q) (4 + 4b −b2 ) (c − qmL ) + {4 + 4b − b2 − q 4 − 3b2 }t > 0. Consequently, we have CS RR < CS N R < CS OR .

Proof of Proposition 3
Under the quadratic utility function, the operation profit of firm D in each equilibrium is cal2
2RR
culated as follows: ΠRR= xRR+ (1 − q) [ xRR+ {(c + t)/q − mH }xRR
+ bxRR
DDF
F
D xF ],

2
R 2
R
= xN
. Similarly, the operation profit of firm F is given by:= xOR, and ΠN
ΠOR
D
D
D
D



R 2
NR
RR 2
OR
OR 2
RR
. In the RR case, in addition to the profit, ΠF = qxN, Π F = xF
ΠF = q x F
Ffrom selling good D presented in the first term, firm D can grab a part of the profits generated
from the consumption of good F by providing the repairs services for good F . This is reflectedin the second term of the first equation.

2
2
NR
(i) We have ΠRR= − (1 − q) {16 1 − b2 + (4 − q) b4 } xRR/ 4 − b2 < 0 andF − ΠFFNR
ΠOR= 4 (1 − q) (c + t − qmL ) [{a(2−b)+bc}q−q (1 − q) (1 − b) mL −(q + 1) (c + t)]/{(4−
F −ΠF
2

2

b2 )2 q 2 } > 4 (1 − q) (c + t − qmL ) /{(4 − b2 )2 q 2 } > 0 where the inequalities are due to a >a.
NR
Hence, ΠRR< ΠOR
is satisfied.
F < ΠF
F


2
NR
RR
2 2
(ii) We have ΠRR{4−(2 − q) b2 }q where B1 = a (2 − b) {4 (1 − b) (2 + b) +D −ΠD = {(1 − q) B1 xF }/ 4 − b(3 + 2b) b4 + 4 − 2b2 − b3 b2 q}q +2{16−20b2 +5b4 +2(2−b2 )b2 q}t+bq (1 − q) {16−4 (1 − q) b2 −2b4 }mL − 4 − b2 {4 − (2 − q) b2 }qmH + {2 16 − 20b2 + 5b4 + (2 + b) (8 − 2b2 − b3 )bq + 4b3 q 2 }c.
By using a >a and c > mH , we can confirm that B1 > (2+b){4−(2 − q) b2 }{(8−4b−2b2 +b3 q)c−
2(2 + b) (2 − b) qmH +2 4 − 2b − b2 t+b3 (1 − q) qmL } > (2+b){4−(2 − q) b2 }{2c 4 − 2b − b2 ((1 − q) c+
32

NR
OR
NR
t)+b3 (1 − q) qmL } > 0. The inequality means that ΠRR
D > ΠD holds. Besides that, xD < xD
NR
OR
NR
RR
(see Proposition 1) implies that ΠOR
D < ΠD holds. Consequently, we have ΠD < ΠD < ΠD .

Proof of Lemma 2
NR2
We have ∂{ΠRR
D −ΠD }/∂t = −2 (1 − q) B4 /[ 4 − b2{4−(2 − q) b2 }2 q 2 ] where B4 = (2 − b) {8−
2 (2 + b) b+(2 − q) b3 }b2 qa+4{16−20b2 +5b4 +2b2 2 − b2 q}t−(4−b2 )2 {2 2 − b2 +b2 q}qmH +
2b3 {8 − (3 − q) b2 }q (1 − q) mL + 4 (1 + b) 2 − b2 b2 q + b5 (1 + q) q}c. Since ∂B5 /∂a > 0 holds,
B5 > B5 |a=a = (2 + b) {4−b2 (2 − q)}[ 8 − 4b − 2b2 + b3 q (c − mH )+2 4 − 2b − b2 {t+(1 − q) mH }+NR(1 − q) qb3 mL ] > 0. Hence, ∂{ΠRR
D − ΠD }/∂t < 0 is satisfied.

The effect of tariff on ∆ΠFR
Firm F ’s gains from entry is given by ∆ΠF = ΠOR− ΠN
if ∆ΠF ≤ KF and ∆ΠF =
F
F
2

R
− ΠN
otherwise. We have ∂ 2 (ΠOR
− ΠRR
ΠOR
F )/ (∂a∂t) = 4 (1 − q) /{q (2 − b) (b + 2) } > 0
F
F
F
2

2
2
2
2
RR
and ∂ 2 (ΠORF − ΠF )/ (∂a∂t) = 4 (1 − q) {8 − b (3 − q)}b /[(2 − b) (2 + b) {4 − b (2 − q)} ] > 0.
NR
OR
RR
Besides that, we have ∂(ΠORF − ΠF )/∂t a=a = ∂(ΠF − ΠF )/∂t a=a = −8 (1 − q) {c + t −2qmL }/{q 4 − b2 } < 0. Hence, we can derive the unique cutoff level of a, e
aN = [c{2 +R− ΠN(2 − b) q} + 2 (1 + q) t − {2q + (1 − q) b}qmL ]/{(2 − b) q}, such that ∂(ΠORF )/∂t > 0
F
R
R
− ΠN
aN , and ∂(ΠOR− ΠN
holds for a > e
aN , ∂(ΠORF )/∂t < 0FF )/∂t = 0 holds for a = eF2holds for a ∈ (a, eaN ). Similarly, we can derive eaE = [2{ 4 − b2 + b2 q(8 − b2 (3 − q))}t +2(2 − b) {2 (2 − b) (2 + b) + b2 q(8 − b2 (3 − q))}c − (2 − b) {16 + 8b − 12b2 − 2b3 + 3b4 + b2 q(8 −

RR
aE ,
b2 (4 − q))}qmL ]/{(2 − b) 8 − b2 (3 − q) b2 q} such that ∂(ΠORF − ΠF )/∂t > 0 holds for a > e
RR
RR
∂(ΠOR
aE , and ∂(ΠORaE ).
F − ΠF )/∂t = 0 holds for a = e
F − ΠF )/∂t < 0 holds for a ∈ (a, e

We can easily confirm that ∂eaN /∂c > 0, ∂eaE /∂c > 0, ∂eaN /∂t > 0, ∂eaE /∂t > 0, ∂eaN /∂mL <0, and ∂eaE /∂mL < 0. Hence, ∂(∆ΠF )/∂t > 0 (resp. ∂(∆ΠF )/∂t < 0) is more likely to hold as cand t become smaller (resp. large) and mL becomes larger (resp. small).

Proof of Proposition 4
Let σ i ∈ {E, N } denote firm i’s (i ∈ {D, F }) action and ∆Πi (σ −i , t) denote firm i’s gains inoperating profits from providing the repair services for good F given the action of the otherNR
firm, σ −i , and the tariff level. The firm D’s gains are given by ∆ΠD (N, t) = ΠRRandD − ΠD∆ΠD (E, t) = 0. We have ∆ΠD (E, t) = 0 because firm D cannot earn positive profits from therepair services if firm F chooses σ F = E. Regarding firm F ’s gains from the entry, we haveRNR
∆ΠF (N, t) = ΠOR− ΠNand ∆ΠF (E, t) = ΠOR
− ΠRR
> ΠRR
holds given t,
FFF
F . Since ΠFF

33∆ΠF (E, t) > ∆ΠF (N, t) holds which means that firm F ’s gains from the entry are larger whenfirm D also chooses the entry.
First, we consider firm D’s best response to firm F ’s action. Because ∆ΠD (E, t) = 0 < KDholds, firm D’s best response is σ D = N if firm F chooses σ F = E. Firm D enters the service
market only if firm F chooses σ F = N . When ∆ΠD (N, 0) > KD is satisfied, there exists a unique
cut-off level of t, denoted by tD , such that∆ΠD (N, t) > KD∆ΠD (N, t) = KD ∆Π (N, t) < K
D

D

for t ∈ [0, tD )for t = tD
for t ∈ (tD , t)holds. For tractability, we set tD = 0 if ∆ΠD (N, 0) ≤ KD holds. Hence, firm D’s best response isσ D = E if firm F chooses σ F = N and the tariff level is less than tD , and it is σ D = N otherwise.
Given firm D’s action, firm F ’s gains from entry is expressed as ∆Π (E, t) for t ∈ [0, t )
F
D
∆ΠF =
.
 ∆Π (N, t) for t ∈ [t , t)FD(i) Suppose ∆ΠF > KF holds. In this case, choosing σ F = E becomes the firm F ’s dominant
strategy. Since ∆ΠD (E, t) = 0 ≤ KD is always satisfied, the firm D’s best response to firm F ’s
entry is to choose σ D = N . As a result, the OR case become the unique equilibrium outcome.
(ii) Suppose t < tD holds. In this case, ∆ΠD (N, t) > KD is satisfied. Since ∆ΠF =∆ΠF (E, t) < KF is also satisfied, choosing σ F = N becomes the firm F ’s dominant strategy
and firm D’s best response is to choose σ F = E. As a result, the RR case becomes the uniqueequilibrium outcome.
(iii) Suppose t ≥ tD holds. In this case, ∆ΠD (N, t) ≤ KD is satisfied. Since ∆ΠF =∆ΠF (N, t) < KF holds, choosing σ F = N becomes the firm F ’s dominant strategy and firm D’s
best response is to choose σ F = N . As a result, the OR case becomes the unique equilibriumoutcome.

The effects of trade liberalization within each regime(i) Consumer surplus: By (A6), (A8), and (A9), we can easily verify that ∂xRR
F /∂t < 0,NR
∂xOR
F /∂t < 0, and ∂xF /∂t < 0 hold. Hence, given the structure of the repair market, tradeliberalization in goods always increases the imports of good F . Regarding the profit of firm F ,RR
RR
RR
we have ∂ΠRR
F /∂t = 2qxF (∂xF /∂t) < 0 because ∂xF /∂t < 0 holds in the RR case. Regard
ing the consumer surplus, we have ∂CS RR /∂t = [q{2 1 − b2 (2 − b) + b 2 − b2 q + b2 q 2 }a −c (1 + b) {2 (1 − b) (2 − qb) + b (2 − b) q 2 } + bq (1 − q) {2 1 − b2 − q 2 − b2 }mL − {4 1 − b2 +34

b2 q 2 }t]/[2{4−(2 − q) b2 }2 q 2 ] > ∂CS RR /∂t a=a = −b{(1 − q) (c − qmL )+t}/[(2 − b) q{4−(2 − q) b2 }] <0. Hence, ∂CS RR /∂t < 0 holds.
OR
OROR
In the OR case, since ∂xOR
F /∂t < 0 holds, we have ∂ΠF /∂t = 2xF (∂xF /∂t) < 0. Be
2sides that, we have ∂CS OR /∂t = −[(1 + b) (2 − b) {a − c − (1 − q) mL } − 4 − 3b2 t]/(4 −b2 )2 < ∂CS OR /∂t a=a = −[2 (1 + b) (2 − b) (1 − q) (c − qmL ) + {4 (1 − q) 1 − b2 + (2 + (2 −q)b)b}t]/{(4 − b2 )2 q} < 0. Hence, ∂CS OR /∂t < 0 holds.
R
NR
2 NR
NR
Lastly, in the NR case, since ∂xNF /∂t < 0 holds, we have ∂ΠF /∂t = 2q xF (∂xF /∂t) <0. In addition, ∂CS N R /∂t = −[q(1 + b)(2 − b)2 a − {4 − b2 (3 − bq)}c − b3 q(1 − q)mL − (4 −3b2 )t]/{(4 − b2 )q}2 < ∂CS N R /∂t a=a = −b{(1 − q) (c − qmL ) + t} (2 + b) /{(4 − b2 )q}2 < 0.
Hence, ∂CS N R /∂t < 0 holds.
(ii) Firms’ profits: In the RR case, we have ∂ 2 ΠRR
D /(∂a∂t) = 2b{2 − b (2 − q)}/[q{4 −(2 − q) b2 }2 ] > 0 and∂(ΠRR
2[(1 − q) {2 (1 − b) (c − mH ) + bqmL } + γ{t + (1 − q)mH })]D )=∂tq 2 (2 − b) {4 − (2 − q) b2 }a=awhere γ := 2 (1 − b) − (2 − b) q. Suppose 2 (1 − b) / (2 − b) ≥ q holds so that γ ≥ 0 holds. In
RR
this case, ∂(ΠRR
D )/∂t a=a > 0 and so ∂(ΠD )/∂t > 0 holds irrespective of the other parametervalues. Alternatively suppose 2 (1 − b) / (2 − b) < q holds so that γ < 0 holds. In this case,∂(ΠRR
D )/∂t a=a < 0 holds if c and mL are sufficiently small and t and mH are sufficiently large.
This means that ∂(ΠRR
D )/∂t < 0 can hold if a, q, c, and mL are small and t and mH are large.
NR
In the OR case and in the NR case, since ∂xOR
D /∂t > 0 and ∂xD /∂t > 0 hold, we have
NR
NR
NR
OR
OR
∂ΠOR
D /∂t = 2xD (∂xD /∂t) > 0 and ∂ΠD /∂t = 2xD (∂xD /∂t) > 0.

(iii) World welfare: Regarding the effects on world welfare, we have ∂(W W OR )/∂t =2
2−[(2 − b) {a−c−(1 − q) mL }+ 4 − 3b2 t]/ 4 − b2 < 0. In the NR case, we have ∂(W W N R )/∂t =22−[aq (b − 2) −{4 (1 − bq)+b2 }c+(4−3b2 )t+4bq (1 − q) mL ]/{q 2 4 − b2 }. Hence, ∂(W W N R )/∂t ≥
20 holds if a ≤ b
a := [{4 (1 − bq) + b2 }c − 4 − 3b2 t − 4b (1 − q) qmL ]/{q (2 − b) } holds. Since2ba−a= (2 + b) [b(1 − q)(c − qml ) − (4 − 3b) t]/{q (2 − b) } holds, b
a >a is satisfied if c is large andmL and t are small. Putting it altogether, ∂(W W N R )/∂t ≥ 0 holds if c is large enough and a,
mL , and t are small enough. Otherwise, ∂(W W N R )/∂t < 0 holds.
2

In the RR case, we have ∂(W W RR )/∂t = −B5 /[q 2 {4−b2 (2 − q)}2 ] where B5 := aq{2 (2 + b) (1 − b) +

2 + 2b − 3b2 bq−(1 − b) b2 q 2 }+{4 1 − b2 −2 (1 − b) 4 + b − b2 q−(1 − b) (3b + 2) bq 2 −b3 q 3 }c
+ 4 − 4b2 + b2 q 2 t − 2 (1 − q) (4 − 2b2 + b2 q)qmH + (1 − q) {2 3 − b2 − 2 − 3b2 q − b2 q 2 }bqmL .
Hence, B5 ≤ 0 holds if a < b
a0 := −[{4 1 − b2 − 2 (1 − b) 4 + b − b2 q − (1 − b) (3b + 2) bq 2 −
b3 q 3 }c + 4 − 4b2 + b2 q 2 t − 2(1 − q)(4 − 2b2 + b2 q)qmH + (1 − q) {2 3 − b2 − 2 − 3b2 q −2

b2 q 2 }bqmL ]/[q{2 (2 + b) (1 − b) + (2 + 2b − 3b2 )bq − (1 − b) b2 q 2 }] is satisfied. Since we haveb
a0 −a= {4−b2 (2 − q)}[2 (1 − q) (2 − b) qmH −{4 (1 − b)+bq}{(1 − q) c+t}−b (1 − q) (2 − q) qmL ]35

2/[q (2 − b) {2 (2 + b) (1 − b) + (2 + 2b − 3b2 )bq − (1 − b) b2 q 2 }] and (ba0 − a)|mH =c,t=0,mL =0 =[q (4 − 3b) − 4 (1 − b)](1 − q){4 − b2 (2 − q)}c/[q(2 − b){2(2 + b)(1 − b)2 + (2 + 2b − 3b2 )bq −(1 − b) b2 q 2 }], ba0 > a holds if t and mL are sufficiently small, mH is sufficiently large, and q islarge enough to satisfy q > 4 (1 − b) / (4 − 3b). In sum, ∂(W W RR )/∂t ≥ 0 holds if a, t, and mL
are small enough and mH and q are large enough. Otherwise, ∂(W W RR )/∂t < 0 holds.

Proof of Proposition 5(i) To prove the proposition, we provide a numerical example in which trade liberalization reduces
the imports of good F , hurts consumers and firm F , and worsens world welfare. Parameters are set
at a = 20, c = 5, mH = 2, mL = 1, q = 0.5, b = 0.5, and KD = 18. Under the parameterization,
we have tD = 0.35634 and ∆ΠF (E, 0) = 22.08 < ∆ΠF (E, tD ) = 22.401.
(a) The shift from the NR case to the RR case by a tariff reduction. Consider a tariff
reduction from t0 = 0.4 to t1 = 0 and suppose KF0 > min[∆ΠF (E, 0), ∆ΠF (E, tD )] holds. Sincet0 > tD holds, ∆ΠF (N, t0 ) < ∆ΠF (E, tD ) is satisfied. Because ∆ΠF (E, 0) < ∆ΠF (E, tD ) < KF0
holds under the parameterization, the equilibrium service regime becomes the NR case at t = t0
and the RR case at t = t1 . The changes in the amount of imports, consumer surplus, the profitRof each firm, and world welfare are respectively given by xRR− xN= −2.429 4 <
F
F
t=t1
t=t0
NR
0, CS RR t=t − CS N R t=t = −1.057 4 < 0, ( ΠRR
D t=t − KD ) − ΠD t=t = 0.31014 > 0,
1

0

1

0

R
= −2. 6552 < 0, and W W RR t=t1 − W W N R t=t0 = −5.7812 < 0.
− ΠN
ΠRR
F
F
t=t0
t=t1

(b) The shift from the OR case to the RR case by a tariff reduction. Suppose KF0 =22.2 and a tariff reduction from t0 = 0.2 to t1 = 0. Since t0 < tD and ∆ΠF (E, 0) = 22.08 < KF0 <∆ΠF (E, t1 ) = 22. 259 hold, the equilibrium regime under t = t1 and under t = t0 respectively
becomes the OR case and the RR case. The changes in the amount of imports, consumer
surplus, the profit of each firm, and world welfare are respectively given by xRR
− xOR
=
F
F
t=t1
t=t0
OR
−2.429 4 < 0, CS RR t=t1 − CS OR t=t0 = −15. 564 < 0, ( ΠRR
D t=t1 −KD )− ΠD t=t0 = 8.696 8 >0, ΠRR− ΠOR− KF = −4.0286 < 0, and W W RR t=t − W W OR t=t = −12.035 < 0.
F
F
t=t
t=t
1

0

1

0

As these numerical examples show, there exists a case where the tariff reduction reduces theimports, decreases consumer surplus and the profits of the foreign firm, increases the profits ofthe domestic firm, and worsens world welfare.
(ii) If a tariff reduction from t0 ∈ (t1 , t) to t1 ∈ [0, tD ) given KF = KF0 increases the imports,
consumer surplus, the profits of firm F , and improve world welfare, we have the same effects fore F = K 0 holds.
all KF ∈ (KD , KF0 ]. In this case, KF
e F = K 0 does not hold. Suppose the case where the tariff
Next consider the case where K
F
reduction improves world welfare at KF = KF0 . Note that if KF satisfies KF < ∆ΠF (E, t1 ), the36

post-liberalization regime is the OR case. By combining Propositions 2 and 4, KF < ∆ΠF (E, t1 )is necessary and sufficient so that the tariff reduction always increases the imports, consumer
surplus, and the profits of firm F irrespective of the pre-liberalization service regime. Hence, we
e F = ∆ΠF (E, t1 ) in this case.
have K
Alternatively, suppose the tariff reduction worsens world welfare at KF = KF0 . In this case,
KF < ∆ΠF (E, t1 ) is necessary but may not be sufficient for a welfare-improving tariff reduction.
eF =If ∂(W W N R )/∂t ≤ 0 holds, KF < ∆ΠF (E, t1 ) becomes a sufficient condition and so K
∆ΠF (E, t1 ) holds. If ∂(W W N R )/∂t > 0 holds, on the other hand, we need to derive KF0 suchthat W W OR t=t1 − W W N R t=t0 = 0 holds at KF = KF0 . Naturally, we have W W OR t=t1 >W W N R t=t for all KF < KF0 . Furthermore if t0 ≥ tD and KF < ∆ΠF (N, t0 ) hold or t0 < tD
0

and KF < ∆ΠF (E, t0 ) hold, the pre-liberalization regime is also the OR case so that the tariff
reduction necessarily increases world welfare given that KF < ∆ΠF (E, t1 ) holds.
In summary, when the tariff reduction worsens world welfare at KF = KF0 , it is transformed
e F = max[K 0 , ∆ΠF (E, t1 ), ∆ΠF (N, t0 )] when t0 ≥ tD
to be welfare-improving (a) for all KF < K
F
e F = max[K 0 , ∆ΠF (E, t1 ), ∆ΠF (E, t0 )] when t0 < tD holds. As
holds, and (b) for all KF < K
Flong as KD is small enough to satisfy KD < ∆ΠF (E, t) for all t, we can always find a uniquee F in KF ∈ (KD , K 0 ].
level of KF

The equilibrium repairs in the presence of ISOs
In the monopoly-ISO case, the ISO’s maximization problem at Stage 3 coincides with that of firm
D in the RR case. Hence, the first-order condition is given by (A1). Suppose λ > 0. This implies
bF = (1 − q) xF and r = VF (xD , xF ) at stage 3 where r is the service price set by the ISO. At
R
stage 2, by the consumer’s utility maximization as to xD and xF , the inverse demand functions
are given by pD = VD (xD , xF ) and pF = VF (xD , xF ) − (1 − q) r = qVF (xD , xF ). Each firm’s
maximization problems are respectively given by maxxD ΠD = {VD (xD , xF ) − c − (1 − q) mL }xD
and maxxF ΠF = {qVF (xD , xF ) − (c + t)}xF . By the first-order conditions, the optimal sales ofISO
the two firms, (xISO), must satisfy
D , xFISOISO ISO
VD (xISO) + VDD (xISO)xD = c + (1 − q) mL ,D , xFD , xFISOISO ISOVF (xISO) + VF F (xISO)xF =D , xFD , xF(c + t)
.
q

(A11)

ISO
ISO ISO
By the above equations and c ≥ mH , λ = VF (xISO) + VF F (xISO)xF − mH =D , xFD , xFISO(c + t)/q − mH > 0 holds. Therefore, (xISO) and RFISO = (1 − q) xISO
actually constituteD , xFF

an equilibrium.

37

Suppose λ = 0. This means RF < (1 − q) xF and VF (xD , qxF + RF ) + VF F (xD , qxF +RF )RF = mH hold.

Since we have assumed that VF F (dD , dF ) < 0 and 2VF F (dD , dF ) +(∂VF F (dD , dF )/∂dF )dF < 0 hold, 2VF F (dD , dF ) + (∂VF F (dD , dF )/∂dF )D < 0 holds for any
D ∈ (0, dF ]. With this property, by equation (A5), and c ≥ mH , we haveVF (xD , qxF + RF ) + VF F (xD , qxF + RF )RFVF (xD , xF ) + (1 − q) VF F (xD , xF )xF
c+t> VF (xD , xF ) + VF F (xD , xF )xF => mH .
q>

This inequality contradicts VF (xD , qxF + RF ) + VF F (xD , qxF + RF )RF = mH . Hence, λ = 0
cannot hold in equilibrium.

Proof of Proposition 6Since ∂ (ΠISO ) /∂a∂t = − (1 − q) b2 /{q 4 − b2 (2 + b)} < 0 holds, we have ∂ (ΠISO ) /∂t <
∂ (ΠISO ) /∂t|a=a = −2 (1 − q) (c + t − mH q)/{q 2 4 − b2 } < 0. When ΠISO |t=0 > KD is
satisfied, there exists a unique cut-off level of t, denoted by tISO , such that ΠISO > KD holds
for t ∈ [0, tISO ), ΠISO = KD holds at t = tD , and ΠISO < KD holds for t ∈ (tISO , t). BecauseRholds, possible entry of ISOs does not affect ∆ΠD and so the level of tD . Given
= ΠN
ΠISO
D
D

that ∆ΠF ≤ KF and ΠISO |t=0 > KD hold, the equilibrium of the enter game becomes (i) theNR equilibrium if max[tISO , tD ] < t holds, because max[ΠISO , ∆ΠD ] ≤ KD holds, (ii) the RR
equilibrium if tISO ≤ t < tD holds, because ΠISO ≤ KD < ∆ΠD holds, (iii) the ISO equilibrium
if tD ≤ t < tISO holds because ∆ΠD ≤ KD < ΠISO holds, and (iv) either the ISO equilibriumor the RR equilibrium if 0 ≤ t < min[tISO , tD ] holds, because min[ΠISO , ∆ΠD ] > KD meansthat both firm D and ISOs have an incentive to enter the aftermarket for good F , and at mosta single firm or a single ISO enters in equilibrium due to the price competition in the market.

The repurchase of good F
Let x0F and x00F respectively denote the amount of good F that is originally purchased and thatof good F that is repurchased. Likewise, let p0F and p00F respectively denote the original price ofgood F and the repurchase price of good F .
Suppose firm F cannot differentiate between the price of good F that is originally purchased0

and that of good F that is repurchased: p00F = pF . This case corresponds to the situation inwhich firm F cannot re-export good F immediately after consumers find the broken units. In
stage 3, each consumer maximizes V (xD , qx0F + qx00F ) − p0F x00F with respect to x00F . The demand forthe repurchase of good F is determined by p0F = qVF (xD , qx0F + qx00F ). In stage 2, the consumer
maximizes V (xD , qx0F +qe
x00F )+Z with respect to xD and x0F , subject to pD xD +p0F x0F ≤ I −p0F x00F .
38

The demand for good D and that for good F are respectively given by pD = VD (xD , qx0F + qx00F )and p0F = qVF (xD , qx0F + qx00F ). Given the demand functions, firm F determines the supply of
good F , xF = x0F + x00F . The maximization problems of the two firms in stage 2 are written as
max ΠD = [pD − {c + (1 − q) mL }]xD = {VD (xD , q(xF + x0F )) − c − (1 − q) mL }xD ,xD

max ΠF = {p0F − (c + t)} (x0F + x00F ) = {qVF (xD , qxF ) − (c + t)}xF .
xF

These maximization problems coincide with those in the NR subgame. Analytically, the repurchase of good F in this case and the extra-purchase of good F in the NR subgame are identical.
Therefore, the equilibrium outcomes in the two cases becomes the same.
Alternatively, suppose firm F can set a different price for the repurchase of good F . In stage 3,
each consumer maximizes V (xD , qx0F +qx00F ))−p00F x00F with respect to x00F subject to x00F ≤ (1−q)x0F .
We have two cases: (i) all broken units are repurchased if qVF (xD , qx0F + qx00F ) ≥ p00F holds at
x00F = (1 − q) x0F , which means that VF (xD , q(2 − q)x0F ) ≥ p00F holds, and (ii) only a fraction of
the broken units is repurchased if qVF (xD , qx0F ) ≥ p00F > qVF (xD , qx0F + qx00F ) holds.
Firstly, let us consider the case where all broken units are repurchased. The demand for the
repurchase is given by x00F = (1 − q) x0F , which is inelastic in p00F . In this case, the equilibrium
price becomes pe00F = qVF (xD , q(2 − q)x0F ). In Stage 2, the consumer anticipates x00F = (1 − q) x0F
holds in stage 3 and she maximizes V (xD , q(2 − q)x0F ) + Z with respect to xD and x0F , subject to
p00F x0F . The demand for good D and that for good F are respectively
pD xD + p0F x0F ≤ I − (1 − q)egiven by pD = VD (xD , q(2 − q)x0F ) and p0F = qVF (xD , q(2 − q)x0F ). Note that p0F = pe00F holds inthis case. Then, the maximization problems of the two firms in stage 2 are given by:max ΠD = [pD − {c + (1 − q) mL }]xD = {VD (xD , q(2 − q)x0F ) − c − (1 − q) mL }xD ,xDmax
ΠF = {p0F − (c + t)}x0F + {p00F − (c + t)}(1 − q)x0F = {qVF (xD , q(2 − q)x0F ) − (c + t)}(2 − q)x0F .
0xF

The first-order conditions become:VD (xD , q(2 − q)x0F ) + VDD (xD , q(2 − q)x0F )xD
q[VF (xD , q(2 − q)x0F ) + q(2 − q)VF F (xD , q(2 − q)x0F )x0F ]= c + (1 − q) mL ,= c + t.

The equilibrium sales, (exD , xe0F ), are obtained by solving these equations. By comparing the aboveR
first-order conditions with the first-order conditions in the NR subgame, we have x
eD = xN
D andR
RR(2 − q)ex0F = xN
eD = pN
e0F = pe00F = pN
F . This means that the equilibrium prices satisfy p
D and pF .

The opportunity to repurchase good F reduces the amount of the initial sales of good F comparedRto the case without the repurchase, that is, x
e0F < xN
F . However, if the amount of the repurchaseRof good F is taken into account, the equilibrium total sales of firm F satisfy x
e0F +(1−q)ex0F = xNF .

39

Since both the equilibrium sales and the equilibrium prices of the two goods become identical across the two cases, the equilibrium firms’ profits, the equilibrium consumer surplus, theequilibrium welfare in the presence of the repurchase coincide with those in the NR equilibrium.
Secondly, let us move on to the case where only a fraction of the broken units is replacedby the repurchased good. In this case, the inverse demand for the repurchase is given by p00F =qVF (xD , q(x0F + x00F )). Since VF F < 0 holds, the demand for the repurchase is decreasing in p00F .
Firm F ’s maximization problem is written by
max{p00F − (c + t)}x00F = {qVF (xD , q(x0F + x00F )) − (c + t)}x00F ,00
xF

and the first-order condition is given byqVF (xD , q(x0F + x
e00F )) + q 2 VF F (xD , q(x0F + x
e00F ))ex00F = c + t.

(8)e00F )) + Z with respect to xD and x0F , subject to
In stage 2, the consumer maximizes V (xD , q(x0F + x
e00F . The inverse demand for good D and that for good F are respectively
pD xD + p0F x0F ≤ I − pe00F x
given by pD = VD (xD , q(x0F + x
e00F )) and p0F = qVF (xD , q(x0F + x
e00F )). Note that p0F = pe00F holds.
Given the demand functions, the maximization problems of the two firms at stage 2 are writtenasmax ΠD = [pD − {c + (1 − q) mL }]xD
xD

= {VD (xD , q(x0F + x
e00F )) − c − (1 − q) mL }xD ,maxΠF = {p0F − (c + t)}xF + {ep00F − (c + t)}e
x00F
0
xF

= {qVF (xD , q(x0F + x
e00F )) − (c + t)}(x0F + x
e00F )

By differentiating ΠF with respect to x0F and using (8), we have
∂ΠF
∂x0F

=
=


∂e
x00F 
) qVF (xD , q(x0F + x
e00F )) + q 2 VF F (xD , q(x0F + x
e00F ))(x0F + x
e00F ) − (c + t)
0∂xF
∂ex00F 2(1 +)q VF F (xD , q(x0F + x
e00F ))x0F < 0.
∂x0F

(1 +

The last inequality is due to the properties that VF F < 0 and ∂ex00F /∂x0F > −1 hold. Thismeans that the initial sales of good F must satisfy x
e0F = 0, which contradicts the condition that
x00F ≤ (1 − q)x0F must hold. Therefore, this case cannot be the equilibrium outcome.
In sum, even if consumers have an option to replace the broken units of good F with the newunits by repurchasing them from firm F who can re-sale the good with a different price in theaftermarket, the equilibrium sales of the two goods, the equilibrium prices of them, and otherequilibrium outcomes coincide with those in the NR equilibrium.
40References[1] Ausubel, Lawrence M., and Raymond J. Deneckere (1989) “Reputation in bargaining anddurable goods monopoly”, Econometrica 57(3), 511-531.
[2] Borenstein, Severin, Jeffrey K. MacKie-Mason, and Janet S. Netz (1994) “Antitrust policy
in aftermarkets”, Antitrust Law Journal 63(2), 455-482.
[3] Bulow, Jeremy I. (1982) “Durable-goods monopolists ”, The Journal of Political Economy
90(2), 314-332.
[4] Chen, Zhiqi and Thomas W. Ross (1994) “Refusals to deal, price discrimination, and independent service organizations”, Journal of Economics and Management Strategy 2(4),593-614.
[5] Chen, Zhiqi and Thomas W. Ross (1998) “Orders to supply as substitutes for commitments
to aftermarkets”, Canadian Journal of Economics 31(5), 1204-1224.
[6] Chen, Zhiqi and Thomas W. Ross (1999) “Refusals to deal and orders to supply in competitive markets”, International Journal of Industrial Organization 7(3), 399-417.
[7] Chen, Zhigi, Thomas W. Ross, and W. T. Stanbury (1998) “Refusal to deal and aftermarkets”, Review of Industrial Organization 13(1), 131-151.
[8] Coase, Ronald H. (1972) “Durability and monopoly ”, Journal of Law and Econnomics 15,
143-149.
[9] Djajić, Slobodan and Henryk Kierzkowski (1989) “Goods, services and trade”, Economica
56, 83-95.
[10] Eschenbach, Felix and Bernard Hoekman (2005) “Services Policy Reform and Economic
Growth in Transition Economies, 1990-2004”, CEPR Discussion Papers, No.5625.
[11] Francois, Joseph F. (1990) “Trade in producer services and returns due to specialization
under monopolistic competition”, Canadian Journal of Economics 23(1), 109-124.
[12] Francois, Joseph F. and Ian Wooton (2010) “Market structure and market access”, The
World Economy 33(7), 873–893.
[13] Gul, Faruk, Hugo Sonnenschein, and Robert Wilson (1986) “Foundations of dynamic
monopoly and the Coase conjecture”, Journal of Economic Theory 39(1), 155-190.

41

[14] Ishikawa, Jota, Hodaka Morita, and Hiroshi Mukunoki (2010) “FDI in post-production services and product market competition”, Journal of International Economics 82(1), 73–84.
[15] Mann, Duncan P. (1992) “Durable goods monopoly and maintenance”, International Journalof Industrial Organization 10(1), 65-79.
[16] Markusen, James R. (1989) “Trade in producer services and in other specialized intermediate
inputs”, American Economic Review 79(1), 85-95.
[17] Markusen, James R., Thomas F. Rutherford, and David Tarr (2005), “Trade and direct
investment in producer services and the domestic market for expertise”, Canadian Journalof Economics 38(3), 758–777.
[18] Shapiro, Carl (1995) “Aftermarkets and consumer welfare: Making sense of Kodak”, Antitrust Law Journal 63, 483-511.
[19] Stokey, Nancy L. (1981) “Rational expectations and durable goods pricing”, The Bell Journal of Economics 12(1), 112-128.
[20] Waldman, Michael (2003) “Durable goods theory for real world markets”, Journal of Economic Perspectives 17(1), 131-154.
[21] Wong, Clement Y.P., Jinhui Wu, and Anming Zhang (2006) “A model of trade liberalizationin services”, Review of International Economics 14(1), 148-168

42

Tables and Figures
Table 1: The welfare eﬀects of trade liberalization when KF is highImports

Consumers

Firm D

Firm F

World Welfare

Within the NR equilibrum

+

+

−

+

+ or −The switch form the NR to the RR−−

no eﬀect−−

Within the RR equilibrum+++ or −++ or −

Table 2: The welfare eﬀects of trade liberalization when KF is low
Within the NR equilibrum
The switch form the NR to the OR
Within the OR equilibrum

Imports

Consumers

Firm D

Firm F

World Welfare

+

+

−

+

+ or −+ or −+−

no eﬀect

+

+

+

−

+

+

Figure 1: The valuatoin eﬀect and the market-contraction eﬀectpF
OVF (xD , d˜F )qVF (xD , d˜F )

C
R

B

A

d˜F

d˜F
q

N

R

Figure 2: The comparision of the NR and the RR equilibrium
xF

d

D
f
xRR
F
R
qxN
F
d
xRR
D

R
xN
D

f

xD

xF

Figure 3: The comparison of the NR and the OR equilibrium
xF

d

F
f
xOR
F
R
qxN
F

F
f

d
xOR
D

xD

R
xN
D

Figure 4: The equilibrium regimes
KF

RR

NR

KF0

A
ΔΠF

OR

KF1

B

ΔΠD
KD

0

tD

tF t0

t̄

t