Parallel imports and repair services∗
Jota Ishikawa†

Hodaka Morita

Hitotsubashi University and RIETI

University of New South Wales

Hiroshi Mukunoki
Gakushuin University
November 23, 2015

Abstract
This study investigates the eﬀect of parallel imports when the producer of a durable good
may refuse to provide repair and maintenance services for parallel imported units, or chargehigher prices for those services. By doing so, the producer is able to weaken intra-brandcompetition and reduce the degree of market integration, mitigating the negative eﬀect ofparallel imports on proﬁts. Although the lower degree of market integration increases the
producer’s proﬁt, it does not always mean the producer wants to improve the durability ofthe product. If the producer invests in improving the durability of the good, the service
discrimination against the parallel imported units could lower the durability of the product.
In this case, consumers in the importing country may suﬀer by permitting parallel imports,
and the negative eﬀect is ampliﬁed by trade liberalization.
∗ We wish to thank Yasuhiro Arai, Tomohiro Kuroda, Keisuke Hattori, and Keisaku Higashida and seminar par-ticipants at Kochi University and RIETI for their helpfur comments and suggestions. Jota Ishikawa acknowledges
ﬁnancial support from the Japan Society for the Promotion of Science through the Grant-in-Aid for Scientiﬁc Research (S) Grant Number 26220503, Hodaka Morita acknowledges ﬁnancial support from the Australian ResearchCouncil, and Hiroshi Mukunoki acknowledges ﬁnancial support from the the Japan Society for the Promotion of
Science through the Grant-in-Aid for Scientiﬁc Research (A) and (C) Grant Numbers 24243034 and 26380310.
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

When consumers purchase imported goods, these may be products imported in parallel throughdistributors not authorized by the original producers, who hold the property rights of those
products. These goods are called “parallel imported (PI) goods” or “gray market goods.” PI
goods are not counterfeit goods. They are genuine goods purchased from authorized distributors
in the source countries and transported to the destination countries without the consent of theproperty holders. The unauthorized distributors, or parallel traders, engage in such trade to
proﬁt from price diﬀerences of the same good between markets in diﬀerent countries. Speciﬁcally,parallel traders earn proﬁts by purchasing the good in a country where the price of the authorized
good is low, and reselling it to consumers in a country where the price of the same good is high.
The end user of a good might also be a parallel trader if he/she purchases a good in a foreigncountry and transports it to a diﬀerent country.
Although parallel trade is typically observed for pharmaceutical products (e.g., Ganslandt
and Maskus, 2004), it also occurs for many other products. NERA (1999) reported that PI
goods within the European Union (EU) include footwear and leather goods, musical recordings,
motorcars, consumer electronics, domestic appliances, cosmetics and perfumes, clothing, and soon. It is reported that the market share of parallel imported automobiles in Hong Kong andSingapore is 15%-25%, and is 15%-17% in the US and the UK (Yeung and Mok, 2013).
Parallel trade encourages international price arbitrage and promotes international market
integration. From the viewpoint of original producers, parallel trade is usually harmful because
it limits their abilities to discriminate prices among countries.1 Basically, permitting parallel
1 Some analyses have suggested that permitting parallel trade may increase the proﬁt of the producer. Ander-

son and Ginsburgh (1999) suggest that a monopolist might use consumers’ arbitrage activities through parallel
importation to undertake second-degree price discrimination, if arbitrage costs diﬀer across consumers. Knox andRichardson (2002) show that parallel imports may beneﬁt the producer because they reduce the optimal tariﬀ
of the destination country. Raﬀ and Schmitt (2007) found that, in the presence of demand uncertainty, parallel
imports lead to larger orders from retailers because they enable retailers to ship unsold inventories to countries
where demand turns out to be unexpectedly high. In a duopoly model, Matsushima and Matsumura (2010) foundthat permitting parallel imports increases proﬁts of all ﬁrms because it serves as a commitment device to soften
price competition in the destination country. Roy and Saggi (2012b) also showed a similar commitment eﬀect ofparallel imports in a diﬀerent setting.

2

imports reduces prices in destination countries and raises prices in the source country. The priceconvergence beneﬁts consumers in the countries that receive parallel imports and hurts consumers
in the country that originates them. However, the welfare eﬀects of parallel imports are much
more complicated. Malueg and Schwartz (1994) investigated how uniform pricing induced byparallel importation aﬀects global welfare. They showed that permitting parallel imports yields
lower global welfare if many markets become unserved by the monopolist. Maskus and Chen
(2004) used a model of vertical control to show that allowing parallel imports improves global
welfare if the cost of undertaking parallel trade is low, but reduces global welfare if the cost is
high.
These theoretical analyses focus mainly on the changes in prices induced by parallel imports.
Another notable subject of parallel imports is their eﬀect on producers’ innovation activities. For
example, parallel imports may lead to less cost-reducing R&D, or a deterioration of the qualityof products and/or the quality of before- and after-sales services. This is because the inability
to discriminate prices between markets may reduce the gains from producers’ investments inimproving the quality of their products and services (Grigoriadis, 2014). Li and Maskus (2006)
explore how parallel imports inhibit the producer’s cost-reducing R&D activities. Valletti (2006)shows that parallel imports reduce the quality of the product when diﬀerential pricing is demand
based but raise the quality when it is cost based. In a North-South model, Grossman and Lai(2008) show that parallel trade promotes product innovation because it induces the Southerngovernment to change their price control in favor of Northern ﬁrms.
Because there are no multilateral rules regarding the permission of parallel imports, eachcountry can choose whether to permit or prohibit them.2,3 Parallel imports are mostly permitted
in Japan. However, they are only permitted for trademarked goods in the United States as long
2 More speciﬁcally,

countries choose between adopting national exhaustion and international exhaustion of

intellectual property rights. Under national exhaustion, property rights expire only in the market where goods are
initially sold. Thus, the trademark holders can legally prevent parallel imports. Under international exhaustion,parallel trade is allowed because property rights expire globally, irrespective of where goods are originally sold by
the trademark holders.
3 A few studies have investigated countries’ choices of parallel import policies. In a monopoly model, Richardson(2002) showed that all countries permit parallel imports in a global Nash equilibrium if they can choose whether
to prohibit parallel imports. Roy and Saggi (2012a) investigated strategic interdependence between countries’ PI
policies in a North-South model of international oligopoly.

3

as the US and foreign trademarks are held by the same owner, or diﬀerent owners are in a parentsubsidiary relationship (Maskus, 2000). The EU members only allow parallel imports from otherEU members. Russia has not liberalized parallel importation.
However, even if parallel importation is allowed and parallel traders are competitive, this does
not always mean the price diﬀerence between countries is reduced to the marginal cost of parallel imports. This is because producers might use instruments to mitigate the negative eﬀect ofparallel imports. One possible way is to reduce the substitutability between the authorized units
and the PI units by providing after-sales services, such as repair and maintenance services, only
to authorized products. For instance, Nikon Inc. USA, a subsidiary of the Japanese company of
optical instruments, Nikon, does not provide support or a warranty service on parallel imported
items, including fee-based repair work.4 Samsonite, the US luggage manufacturer, employs thesame policy in Japan. World Commerce Corp, the Japanese authorized distributor of the Swiss
watch company, Franck Muller, refuses to repair parallel imported watches. Firms also discriminate prices in after-sales services and set higher maintenance and repair prices for PI units,
while they provide free repairs or charge a lower fee for authorized units. An empirical analysis
supports the view that discrimination in after-sales services weakens intra-brand competition
between authorized goods and parallel imported goods. Chen, Lai, and Yu (2013) investigatedTaiwan’s Yahoo! auctions of Nikon cameras and showed that authorized goods with a longer
duration of warranty and that are repaired by the oﬃcial dealers have higher transaction prices
than PI goods.
Given this background, this study theoretically examines the eﬀect of parallel imports whena monopolist producer may refuse to provide repair and maintenance services or may set ahigher repair price for a PI good.5 In this study, refusal of repairs and price discrimination inrepair services are together referred to as “discrimination in repair services” or simply “service
discrimination.” Speciﬁcally, we investigate a producer’s incentive to discriminate repair services,
4 Nikon Inc.

USA declares in its website: “Nikon Inc. USA cannot provide any technical support or warranty

service on Gray Market items. Additionally Nikon Inc. USA cannot perform any fee-based repair work on Gray
Market items. Please do not contact Nikon Inc. USA for help with any Gray Market products.”
5 Ishikawa, Morita, and Mukunoki (2015) investigate the welfare eﬀects of repair services in the context ofinternational trade. However, their focus is diﬀerent to ours and does not consider parallel imports and service
discrimination.

4and how the welfare eﬀect of parallel imports is associated with such a business practice. In ourmodel, the diﬀerence in the demands for the good induces the producer to set diﬀerent prices in
diﬀerent countries. There are many parallel traders, and the competition among them reduces
the price of the PI good by an amount equal to the sum of the price of the authorized good in a
low-priced country and trade costs.
We ﬁnd that service discrimination works as a tool to weaken the intra-brand competition
and makes the domestic and foreign markets only “partially” integrated. Partial integrationmeans that the producer cannot fully discriminate prices, but can still charge a higher price in ahigh-demand country than in a low-demand country. Because of the price-discrimination eﬀect,
the producer always prefers discrimination in terms of repair services for the PI good. Comparedto the case of full market integration, in which the price diﬀerence between the two countries
reduces to the level of trade costs (tariﬀ), the price in a high-demand country becomes higher
and the price in a low-demand country becomes lower. The changes in prices beneﬁt consumers
in the source country and hurt consumers in the destination country.
We also investigate how a refusal to repair the PI good aﬀects the durability of the goodwhen the producer engages in costly activities to improve it. We show that the eﬀect of parallel
imports on the durability depends on whether the ﬁrm descriminate repair services.6 . To thebest of our knowledge, this is the ﬁrst study to explore the eﬀects of parallel imports when aproducer has an option to discriminate repair services.7
Our ﬁndings show that if the producer does not discriminate services and parallel imports
realize full market integration, permitting parallel imports has no eﬀect on the durability of thegood. In contrast, parallel imports with service discrimination worsen the durability of the good.
6 Some papers such as Fethke and Jaganatthan (2002), Goering (2007), and Goering and Pippenger (2002)

have considerd endogenous determination of product durability, but their focuses are not on the eﬀect of parallel
imports.
7 A related paper is Ahmadi and Yang (2000). They show that if consumers discount their valuations on the
PI goods because of the inconvenience due to market-speciﬁc product diﬀerentiation (e.g., right-hand drive or
left-hand drive, manuals written in diﬀerent languages) or due to service diﬀerentiation (e.g., the lack of technicalsupport and warranty), allowing parallel imports can attract consumers in importing countries who would nototherwise have bought the good. However, they do not consider the eﬀect of a change in the discounting factor
of PI goods and the endogenous determination of the product quality. Furthermore, they do not investigate theeﬀects of trade liberalization.

5

This is because a lower durability reduces the substitutability between the authorized good andthe PI good, which weakens intra-brand competition or generates a greater proﬁt opportunity
in the repair market of the PI good. Because trade liberalization reduces the cost of exporting
the authorized good, but also decreases the cost of conducting parallel trade, this may induce
the producer to invest less to improve the durability of the good. However, trade liberalization
always raises the durability if the markets are either fully segmented or fully integrated. In
this situation, permitting parallel imports with service discrimination can hurt consumers in thedestination country, and this becomes more likely as trade costs fall.
The remainder of the paper is organized as follows. Section 2 develops a monopoly modelof durable-goods producers and repair services, and derives the equilibrium of three diﬀerent
regimes. Section 3 investigates the relationship between the durability of the good and the
producer’s proﬁt. Section 4 examines the endogenous determination of the durability of the
product. Section 5 relaxes some assumptions of the basic model and discusses the robustness of
the results. Section 6 summarizes the paper and oﬀers concluding remarks. Lastly, the appendix
contains the proofs of lemmas and propositions in the paper.

2

Model

We consider a two-country model with a single ﬁrm in the domestic country (country D) that
produces good x, which it supplies locally and to the foreign country (country F ). An importtariﬀ (or a trade cost), t, is imposed on good x by the foreign country. Each consumer purchases
at most one unit of good x. Consumers in country i (i = D, F ) are identical except in their
willingness to pay for the good, which is denoted by v. If a consumer in country i purchases thegood at price pi , its utility is given by Ui = v − pi . We assume each consumer’s utility underno purchase is normalized to zero. The population in each country is identical and given by n,though the distribution of v is diﬀerent across the two countries. In country F , v is uniformly
distributed over the interval [0, b] with density fF = 1b . In country D, v is uniformly distributed
1over the interval [0, kb] with density fD = kb
, where k is positive and k < 1. This means that

consumers in country F , on average, attach a higher value to good x.
If parallel trade is allowed, consumers in each country can purchase the good from either

6authorized distributors controlled by the ﬁrm or other distributors who purchase the good in
the other country and resell it through parallel trade without authorization. We call the goodpurchased from authorized distributors “the authorized good” and that purchased from parallel
importers “the PI good.”
After consumers purchase the goods, they may break down because of imperfect quality
control. The probability of the good working correctly is q ∈ (0, 1] and it is common to both theauthorized good and the PI good. Hence, the probability of the good breaking down is (1 − q).
If the consumers utilize repair services provided by the producing ﬁrm, the broken good becomesconsumable. However, without repair services, broken goods are useless and have zero scrap
values. We assume if a consumer purchases the authorized good, the good is subject to free
repairs.8 However, the ﬁrm may refuse to repair the PI good or charge a fee for the repair. The
unit cost of production for the ﬁrm is given by c (> 0), and the unit cost of repairs is given bym, which should be lower than c.
If parallel trade is allowed, many competitive parallel traders can purchase the good in the
country where the price is low and sell it to the country where the price is high. For simplicity,
we assume parallel traders incur no additional cost other than the trade cost. We only focus on
the situation where the ﬁrm always provides the good in both countries.
We consider the following three situations: (i) Regime N in which parallel imports are bannedin country F ; (ii) Regime I in which parallel imports are allowed and the ﬁrm provides the same
repair services to the consumers purchasing the PI good as those provided to the consumersof the authorized good; and (iii) Regime R in which parallel imports are allowed and the ﬁrm
discriminates the repair services against the PI good by either refusing to repair the PI goodor charging a fee for the repair. We show that refusing repairs and charging for repairs of the
PI good have the same eﬀect on the equilibrium prices and welfare. These ﬁrms’ behaviors arereferred to collectively as service discrimination.
The model has three stages. In the ﬁrst stage, the producer sets authorized prices in country
D and country F . In the second stage, consumers in each country and parallel traders purchase
the authorized product. In the third stage, parallel traders export the purchased good to othercountries and sell it to consumers who purchase the PI good, if parallel traders earn non-negative
8 We conﬁrm that the ﬁrm is willing to oﬀer a full warranty for the authorized good. See Section 5.1.

7

proﬁts. There is no time discounting between the second stage and the third stage. Thus,
consumers always wait until the third stage if they anticipate that purchasing the PI good will
generate higher utility.

2.1

No parallel imports

As a benchmark, we start with Regime N , where parallel imports are prohibited. In this case, themarkets of the two countries are completely segmented (full segmentation). The monopolist isable to make independent decisions in setting the prices in each country. A consumer in country
i purchases the good if
Ui = v − pi ≥ 0
holds. Under market segmentation, all consumers purchase the authorized good if they buy. The
consumers whose willingness to pay for quality satisﬁes v ≥ pi purchase the good. The demand
for good x in each country is given byxD (pD )xF (pF )pD ,= nfD {kb − pD } = n 1 −
kb


pF
.
= nfF {b − pF } = n 1 −
b

(1)
(2)

The ﬁrm’s proﬁt is given byΠ (pD , pF ) = [pD − {c + (1 − q) m}]xD (pD ) + [pF − {c + t + (1 − q) m}]xF (pF ) .

(3)

The ﬁrm maximizes this proﬁt with respect to pD and pF . By solving the ﬁrst-order conditions,the equilibrium prices are given by
pN
D

=

pN
F

=

kb + c + (1 − q) m,2
b + c + t + (1 − q) m
.
2

(4)
(5)

 
 N
F > 0, we assume b > max[{c +To ensure positive sales in each market, xD pN
D > 0 and xF p
(1 − q) m}/k, c + t + (1 − q) m] holds.
By comparing the equilibrium prices, we have pNN
F −p
D = (1 − k) (b + t) /2 > 0. This implies
that the ﬁrm sets a higher price in the foreign country where consumers’ willingness to pay forthe good is, on average, higher than in the domestic country. The equilibrium cut-oﬀ level of v,

8above which consumers purchase the good in each country, becomes viN = pN
i . The equilibriumN
proﬁts are given by ΠN = Π pN
and the equilibrium consumer surplus in country D andD, pFcountry F are, respectively, given by
 kb
N
CSD
= nD f D(v − pN
D )dv
N
v
eD

CSFN

2


N
 kb + 
n kb − pN
vD
n 
D
N
N
kb − vD− pD =
,
=
kb
2
2kb
 b
= nfF
(v − pN
F )dv
N
v
eF

=

2.2

(6)

2


 b + vFNn b − pN
n
F
− pN
.
=
b − vFN
F
b
2
2b

(7)

Parallel imports without service discrimination

If parallel trade is allowed, the price diﬀerence of good x between the two countries creates anopportunity for parallel trade. Since there are many competitive parallel traders, the price of the
PI good in country F becomes pD + t.
The ﬁrm is able to choose whether it provides free repair services for the good sold by unauthorized, parallel traders. Suppose it does do so (Regime I). Then, the good sold directly by
the authorized distributors and the good sold by parallel importers are perfect substitutes. Thismeans that if pF > pD + t holds, all consumers purchase the PI good. By (4) and (5), N
 (1 − k)b − t.
pND + t =
F − p
2

(8)

 N

We restrict our attention to the case where (1 − k)b > t is satisﬁed so that pND + t holds.
F > p
Therefore, all consumers buy the PI good if the ﬁrm sets the same prices as those charged underfull market segmentation. We can conﬁrm that the ﬁrm has no incentive to set pF lower than N
pD + t given pND + t holds. Then, the ﬁrm maximizes the proﬁt such that pF = pD + t
F > pholds. In other words, the threat of parallel trade makes the two markets completely integrated
and the price diﬀerence is always equal to the tariﬀ level.
Consumers purchase the good if v ≥ pi holds, and the demand is given by (1) and (2). The
ﬁrm maximizes (3) such that pF = pD + t holds. Then, the respective equilibrium prices become:
pID

=

pIF

=

pN
pN
D + k(F − t),
1+kpN
pN
D + t + k
F
.
1+k
9

(9)
(10)

Note that the equilibrium prices are the weighted average of the equilibrium prices under market
segmentation. The equilibrium cut-oﬀ level of v in each country becomes 
viI = pIi . The equilibrium proﬁt becomes ΠIR = Π pID , pIF and the respective equilibrium consumer surplus in eachcountry is given by

2.3

2

kb − pID
,
nfD
I
2kb
v
eD
2

 b
n b − pIFN.
nfF
(v − pF )dv =
I
2b
v
eF
 kb

I
CSD

=

CSFI

=

n
(v − pND )dv =

(11)
(12)

Parallel imports with service discrimination

We now investigate Regime R, where the ﬁrm discriminates repair services by either (i) refusing
to repair the PI good or (ii) charging a positive fee for the repair.

2.3.1

Refusal to repair

Suppose the ﬁrm refuses to repair the PI good (Regime I). If q = 1, the good never breaks down
and the equilibrium proﬁt coincides with ΠI . However, if q < 1, the authorized good and the PI
good generate diﬀerent expected utilities, even if they are sold at the same price. Speciﬁcally, if
a consumer in country F buys an authorized good at price pF , his/her utility becomes
UFA = v − pF .

(13)

If the consumer buys a PI good at price pD + t, then his/her expected utility becomes9
UFP I = qv − (pD + t) .

(14)

Let vFA be the preference parameter of the consumer who is indiﬀerent between buying an
authorized good and a PI good (UFA = UFP I ), and vFP I be the parameter for the consumer who is
indiﬀerent between buying a PI good and no purchase (UFU = 0). We have
vFA

=

vFP I

=

pF − (pD + t),
(1 − q)pD + t.
q

9 We assume consumers are risk neutral.

10

(15)
(16)

By comparing vFA and vFP I , we havevFA − vFP I =qpF − (pD + t).
(1 − q)q(17)

This means that if pF > (pD + t) /q holds, which means that the “quality-adjusted price” of the
PI good is lower than the price of the authorized good, consumers with preference parameter v∈ [vFA , b] buy the authorized good and consumers with parameter v ∈ [vFP I , vFA ] buy the PI good.
Even if the price of the PI good is lower than that of the authorized good, the consumers withhigh v buy the authorized good because the price diﬀerence is insuﬃcient for them to compensate
for the risk of the good being broken down and not being repaired. The consumers with low v
buy the PI good because they value the good less and are more sensitive to the price of the good.
Suppose pF > (pD + t) /q holds, then the demand for the authorized good and the demand
for the PI good in country F are, respectively, given byxA
F (pF , pD )I
xPF (pF , pD )


pF − (pD + t)
= nfF b − vFA = n 1 −,(1 − q)b
qpF − (pD + t)
= nfF vFA − vFP I = n,(1 − q)qb

(18)
(19)

while the demand in country D is given by (1). An increase in pD not only raises the price of thegood in country D, but also the price of the PI good in country F . These price changes increase
the demand for the authorized good and decrease the demand for the PI good. An increase in
pF , on the other hand, has the opposite eﬀects.
The ﬁrm’s proﬁt is given byΠ (pD , pF )= [pD − {c + (1 − q) m}]xD (pD ) + [pF − {c + t + (1 − q) m}]xA
F (pF , pD )I+(pD − c)xP
F (pF , pD ).

(20)

The third term is the proﬁt earned from selling the good to the parallel traders. By solving∂Π (pD , pF ) /∂pD = 0 and ∂Π (pD , pF ) /∂pF = 0, the equilibrium prices of this case become:
pD

=

pF

=

k[b (q − k) − {t + (1 − q) m}],
2 (q + k)(1 − q)(b + m).
pD + t +
2
pN
D +

(21)
(22)

It is apparent that pF > pD + t holds and the price diﬀerence between the two markets does not
reduce to the level of the tariﬀ. In other words, the two markets are not fully integrated, but
11only partially integrated. The extra price diﬀerence over the tariﬀ level is due to the “quality
premium” of the authorized good, because the authorized good is subject to free repair services
and its expected quality is higher than that of the PI good. As q decreases, consumers in country
F regard the authorized good and the PI good as becoming more vertically diﬀerentiated, andthe ﬁrm is able to discriminate more on pricing. By substituting pD and pF into vFA and vFP I ,the equilibrium cut-oﬀ level of v in the foreign country is given by vFA and vFP I . The cut-oﬀ levelIin country D is given by vD= vD (pID ).

These prices are consistent with the supposition that pF > (pD + t) /q holds if
pF −

Γ(q)
(pD + t)= (1 − q)>0q
2q (q + k)

is satisﬁed, where Γ(q) = (b + 2m)q 2 − {kb + c + 2t + (1 − k)m}q − k (c + t). We have the following
lemma.
Lemma 1 If b > {(1 + k) (c − m) + (2 + k) t}/(1 − k) holds, then there exists a unique value ofq, qH , such that pF > (pD + t) /q holds for q ∈ (qH , 1). Otherwise, pF ≤ (pD + t) /q holds forany q.
Proof. Γ(q) is either U-shaped or decreasing in q ∈ [0, 1]. We have Γ(0) = −k (c + t) < 0
and Γ(1) = (1 − k) b − (1 + k) (c − m) − (2 + k) t. If b > {(1 + k) (c − m) + (2 + k) t}/(1 − k)holds, then Γ(1) > 0 and Γ(q) is U-shaped in q. Therefore, there exists a unique value of q,qH (∈ (0, 1)), that satisﬁes Γ(qH ) = 0. This means that pF > (pD + t) /q holds for q ∈ (qH , 1)
and pF ≤ (pD + t) /q holds for q ∈ (0, qH ]. If b ≤ {(1 + k) (c − m) + (2 + k) t}/(1 − k) holds,
pF ≤ (pD + t) /q always holds for all q ∈ [0, 1].
Even if a decrease in q reduces the price of the good sold in the domestic country (pD ), it
increases the quality-adjusted price of the PI good ((pD + t) /q) in the foreign country. The
quality adjusted price approaches the inﬁnity as q approaches zero. If a small decrease in q fromq = 1 realizes pF > (pD + t) /q (i.e., if Γ(1) > 0 holds), xU
F (pF , pD ) is ﬁrst increased and thendecreased as q becomes smaller. This implies that if q is suﬃciently small (i.e., q ≤ qH ), consumers
do not purchase the PI good. If a small decrease in q from q = 1 results in pF ≤ (pD + t) /q (i.e.,
if Γ(1) ≤ 0 holds), no consumers purchase the PI good irrespective of the level of q if the repair
services are not available. For expositional convenience, we set qH = 1 in this case.
12The consumer surplus in country F when the producer accommodates the PI good is givenbyCSF (pD , pF )

 b
= nfF
=

A
v
eF

(v − pF )dv + veFAPIveF


{qv − (pD + t)}dv

n{b (1 − q) + (pD + t) − pF }{(b − 2pF ) (1 − q) + {pF − (pD + t)}
2b (1 − q)+n{q pF − (pD + t)}22bq (1 − q)2 d

2

.

(23)

The ﬁrst term of (23) represents the surplus from consuming the authorized good and the second
term is the surplus from consuming the PI good.
If q ≤ qH is satisﬁed, vFP I ≥ vFA holds and all consumers in country F prefer the authorized ND + t /q holds, the ﬁrm is able to block parallel
good, if they buy one. In this case, if pNF < pimports, even if it completely discriminates the prices between the two countries. Therefore, theproducer charges pNN
N and pF in equilibrium. We have the following lemma.
Lemma 2 There exists a unique value of q, qL (∈ (0, qH )) such that pF < (pD + t) /q holds forq ∈ (0, qL ).
Proof. See Appendix.
If q is suﬃciently small such that it satisﬁes q ≤ qL , the ﬁrm’s refusal to repair the PI good
blocks parallel imports, making the two markets completely segmented.
If qL < q < qH holds, the ﬁrm opts to block parallel imports by setting pF = (pD + t) /q,
rather than accommodating them. In this case, the ﬁrm maximizes (3) subject to pF = (pD + t) /q.
The optimal prices become:
pD

=

pF

=

pN
q 2 pN
D + k(qF − t)
,
2
q +k
pN
q(
pN
D + t) + kF.
q2 + k(24)(25)

Because (pD + t) /q > pD + t holds, the ﬁrm is able to make the price diﬀerential between thetwo countries more than the tariﬀ level. The quality premium of the price diﬀerence ispF − (pD + t) =(1 − q) (pD + t) > 0.
q

13

(26)

In summary, the equilibrium price in country i, pRR
Ii if q = 1, pRi , becomes pi = pi = pi ifRN
qH = 1 and q ∈ (qH , 1), pRi = pi if q ∈ (qL , qH ], and pi = pi if q ∈ (0, qL ]. We have the following

proposition.
Proposition 1 If q = 1 and the ﬁrm refuses to repair the PI good, the ﬁrm sets: (i) the prices
such that some consumers purchase the authorized good and other consumers purchase the PI
good for q ∈ (qH , 1); (ii) the prices such that all consumers purchase the authorized good forq ∈ (qL , qH ); and (iii) the same prices as those under no parallel imports and all consumers
purchase the authorized good for q ∈ [0, qL ).
The equilibrium proﬁt when the ﬁrm refuses to repair the PI good becomes
⎧
⎪
⎪
ΠI
if
q=1
⎪
⎪
⎪
⎪
⎪
⎪
⎨ Π (pD , pF ) if qH = 1 and q ∈ (qH , 1)
R
Π =
.
⎪
⎪
q ∈ (qL , qH ]⎪ Π(pD , pF ) if
⎪
⎪
⎪
⎪
⎪
⎩
ΠN
if
q ∈ (0, qL ]The equilibrium consumer surplus in each country is given by
I
CSD

CSFI

2.3.2

2

n bk − pR
D
,
=
⎧ 2kb
⎪
⎨ CS  (pD , pF )if qH = 1 and q ∈ (qH , 1)
F
=
.


2
⎪
⎩
n b − pR
/2botherwiseF

Price discrimination in repair services

The monopolist may provide repair services for the PI good, but a repair price is charged against
the PI good. Let r denote the repair price the ﬁrm charges for the PI good in country F . If
a consumer in country F buys a PI good and pays for repair services, his/her expected utility
becomes
UFI,r = v − (pD + t) − (1 − q)r.

(27)

By comparing (13) and (27), consumers prefer to buy the authorized good, rather than purchasing
the PI good and paying for the repair services, if the overall price of the PI good is less thanthe price of the authorized good, pF > pD + t + (1 − q)r. Because the expected quality of theauthorized good and the quality of the PI good become identical, the consumers purchase the14good with the lower overall price. If pF = pD + t + (1 − q)r holds, consumers are indiﬀerent
between the authorized good and the PI good. If pF < pD + t + (1 − q)r holds, all consumers
prefer the authorized good because the repair price is too high. This case can be regarded as therefusal of repairs.
Given that pF ≥ pD + t + (1 − q)r holds, let r∗ denote the equilibrium repair price for the PI
good, and p∗D and p∗F denote the equilibrium price of the authorized good in countries D and F ,respectively. We assume the ﬁrm continues to provide a full warranty for the authorized good.
We have the following proposition.
Proposition 2 In equilibrium, the overall price of the PI good under the fee-based repair service
is the same as the price of the authorized good under the refusal of repairs.
Proof. See Appendix.
The intuitive explanation is as follows. Under the refusal of repairs, the ﬁrm diﬀerentiates
the quality of the good between the authorized good and the parallel imported good. If the ﬁrm
accommodates the PI good, (22) suggests pF − (pD + t) = (1 − q)(b + m)/2 holds in equilibrium,
where the price diﬀerence between the authorized good and the PI good, (1 − q)(b + m)/2, isthe quality premium of the authorized good. Here, instead of refusing the repair services anddiﬀerentiating the quality, the ﬁrm closes the quality gap by providing the repair services andcharging a fee. The amount of the fee is set to extract the quality premium that would have been
generated had the repair services for the PI good not been provided: (1 − q)r∗ = (1 − q)(b + m)/2.
Because the expected repair price is equal to the quality premium generated by the refusalof repairs, the price gap between the unrepaired good and the repaired good remain unchanged.
This implies that the ﬁrm sets the same price in the domestic country and p∗D = pD and p∗D +t + (1 − q)r∗ = pF . Then, any price of the authorized good that satisﬁes p∗F ≥ p∗D + t + (1 − q)r∗
becomes the equilibrium price of the authorized good. Consequently, we have p∗D = pD andp∗F ≥ pF = pD + t + (1 − q)r∗ .
If the ﬁrm blocks the PI good by setting pF = (pD + t) /q, the quality premium in equilibrium

becomes pF − (pD + t) = (1 − q) (pD + t)/q (see (26)). Again, the repair price is set suchthat (1 − q)r∗ is equal to the quality premium. Hence, we have p∗D = pD and p∗F ≥ pF =pD + t + (1 − q)r∗ .
15

Therefore, the refusal of repairs and the fee-based repair services for the PI good are qualitatively indiﬀerent for the ﬁrm and consumers, and their eﬀects on the durability of the productand the quality of repair services are also indiﬀerent. In other words, the equilibrium in the caseof the fee-based repairs of the PI good practically coincides with the equilibrium under the refusalof repairs.
The equilibrium pricing of the ﬁrm is summarized as follows: the ﬁrm sets (i) the prices suchthat some consumers of the PI good are unwilling to repair the broken units for q ∈ (qH , 1); (ii)the prices such that all consumers of the authorized good and the PI good repair the broken
units for q ∈ (qL , qH ); and (iii) the same overall prices as those under no parallel imports and all
consumers repair the broken units for q ∈ [0, qL ).

2.4

Comparison

Now, we compare the equilibrium prices under the three regimes. We have the following proposition.
Proposition 3 Given the level of q, allowing parallel imports decreases the price of the good in
the foreign country and increases the price in the domestic country under no service discrimination. Under service discrimination, allowing parallel imports leads to the same direction of pricechanges if q > qL holds, but has no eﬀect if q ≤ qL holds. Service discrimination reduces theextent of the price changes.
Proof. See Appendix.

By comparing ΠR and ΠI , we can conﬁrm that ΠR ≥ ΠI always holds given the level of q.
Proposition 4 The producer always prefers to discriminate repair services against the parallel
imported good.
Proof. See Appendix
Service discrimination makes the quality of the authorized good higher than that of the PI
good when the ﬁrm refuses to repair the PI good, or it enables the ﬁrm to earn additional proﬁts
16

in the repair market for the PI good when the ﬁrm charges more to repair the PI good. As aresult, the ﬁrm is able to sustain the price diﬀerence between the two countries above the tariﬀ
level. Note that the ﬁrm always has an option to block the PI good by manipulating the levelof prices. This means that if the ﬁrm sets the prices such that some consumers purchase the PI
good (or some consumers of the PI good are refrain from paying the repair price), the ﬁrm earns
a higher proﬁt than if it blocks the PI. More speciﬁcally, the ﬁrm accommodates parallel trade
because it can earn an additional proﬁt from selling the good to parallel traders and the (overall)
price in each country becomes closer to the price under market segmentation. However, if q issmall, the prices that accommodate parallel imports substantially reduce the proﬁts from sellingthe authorized good. Thus, the ﬁrm chooses to block them.

3

Parallel imports and the durability of the good

Here, we explore the eﬀect of a change in q on the ﬁrm’s proﬁt. If parallel imports are banned,or the ﬁrm cannot discriminate the repair services, a decline in q always reduces the ﬁrm’s proﬁt.
This is because the presence of the repair cost means the increased probability of the good
breaking down increases the ﬁrm’s expected cost of undertaking the repairs. We call this eﬀect
the cost-increasing eﬀect of a decline in durability. By diﬀerentiating ΠN and ΠI with respect
to q, we have ∂ΠN /∂q = ∂ΠI /∂q = N [(1 + k) [bk − {c + (1 − q)m}]+{(1−k)b−t}k]m/{2bk} > 0,
because b > {c+(1−q)m}/k and (1−k)b > t hold. Therefore, a higher durability always increases
the ﬁrm’s proﬁt, and the magnitude of the eﬀect is unaﬀected by parallel imports.
However, if parallel imports are allowed and the ﬁrm refuses to repair the PI good (or provides
fee-based repairs), the eﬀect of a decline in q becomes more complicated. On top of the costincreasing eﬀect, a decline in q has two additional eﬀects on the ﬁrm’s proﬁt.
Firstly, it increases the importance of the repair services and the ﬁrm is able to charge ahigher price for the authorized good or charge a higher repair price for the PI good. Hence,service discrimination makes the price diﬀerence between countries D and F much closer to theprice diﬀerence under full market segmentation. This eﬀect is called the market-segmentationeﬀect. Secondly, if the market for the authorized good and the market for the PI good coexist
in country F (q ∈ (qH , 1)), this aﬀects the proﬁt from selling the good to parallel traders since a17

decrease in q changes the relative demand for the PI good. We call this eﬀect the PI-market
eﬀect.
The cost-increasing eﬀect has a negative eﬀect on the ﬁrm’s proﬁt, while the market-segmentationeﬀect has a positive eﬀect. The PI-market eﬀect is either positive or negative because the demand
for the PI good is an inverse-U shaped in q. Whether a decrease in q increases or decreases theﬁrm’s proﬁt depends on the relative magnitudes of these eﬀects.
Therefore, in the presence of parallel imports, a lower durability of the good may generatehigher proﬁts. This is because, by discriminating the repair services against the PI good, the ﬁrm
can mitigate the pressure of price arbitrage. We show a numerical example of the eﬀect of q on
the ﬁrm’s proﬁt. In what follows, the parameters other than q are set as follows: n = 100, b = 40,
k = 1/3, c = 5, m = 1, and t = 3. In this case, the cut-oﬀ levels of q become qH = 0.68758 and
qL = 0.51191. Figure 1 shows that ΠR is decreasing in q for q ∈ (qH , 1] and an inverse-U shapedin q for q ∈ (qL , qH ].
[Insert Figure 1 around here]
In this case, ΠR is an inverse-U shaped in q, and it is maximized at q = q ∗ = 0.52611(∈ (qL , qH )), where the ﬁrm blocks the PI good by refusing to provide repair services, or sets therepair price for the PI good so that all consumers of the PI good pay the repair price for repairing
the good.
Proposition 5 If the ﬁrm discriminates repair services against the parallel imported good, thena decrease in q may increase the proﬁt of the ﬁrm. Under no parallel imports or no service
discrimination, a decrease in q always decreases the proﬁt of the ﬁrm.
This proposition suggests that service discrimination substantially changes the relationshipbetween the durability of the good and the proﬁt of the ﬁrm.

3.1

Endogenous durability of the productAs explained in the Introduction, one concern over allowing parallel imports is that the reduced
proﬁts of ﬁrms owing to market integration will decrease the gains from ﬁrms’ innovation activities
and will lead to a deterioration of product quality. Here, we investigate how parallel imports aﬀect
18

the producer’s incentive to provide greater product durability that increases the “quality” of thegood in the sense that consumers worry less about the good breaking down. To this end, we add
a new stage before the ﬁrst-stage (i.e., stage zero) in which the producer chooses the durability ofthe product. Existing studies that have examined the quality eﬀect of parallel imports have notexplicitly considered the possibility that the good breaks down and the durability of the good.
We examine endogenous determinations of the product durability.
We suppose the producer must incur the ﬁxed cost, C(q), to produce the good with the
durability, q. The ﬁxed cost reﬂects the costs of conducting a more intensive durability inspection(e.g., increasing the percentage of units to be tested), using more durable parts and components
in the manufacturing process, and so on.
It is natural to suppose that this ﬁxed cost is increasing in q. Hence, it is assumed that
C(0) = 0, C  (q) > 0 and C  (q) ≥ 0. Without loss of generality, we suppose the ﬁxed cost isquadratic in q and given by C(q) = βq 2 /2, where β is positive and constant.
Diﬀerent eﬀects of a change in q on ΠN , ΠI , and ΠR imply that the ﬁrm will choose diﬀerent
levels. The ﬁrm chooses the optimal level of q so that it maximizes the proﬁt net of the ﬁxed cost,
C(q). Let qN , qI , and qR respectively represent the proﬁt-maximizing levels of q under Regime
N , IR, and I. By Proposition 5, qN = qI always holds. However, qR may be lower than them.
For instance, if β = 100, then we have qN = qI = 0.69737 > qR = 0.51538. In this case, service
discrimination induces the ﬁrm to choose lower durability.
Proposition 6 In the presence of service discrimination against the parallel imported good, parallel imports may reduce the product durability. If there is no service discrimination, parallel
imports have no eﬀect on the durability.
Note that, if qN > qR holds, it becomes ambiguous whether parallel imports beneﬁt consumers
in country F . This is because, while the direct eﬀect of parallel trade reduces pF , it decreases
the durability of the good that, in turn, increases pF . If the latter eﬀect is relatively signiﬁcant,then parallel imports hurt consumers in the importing country. In fact, under this numerical




I 
N
example, CSD− CSD= −4. 093 6 and CSFI q=eqR − CSFN q=eqN = −0.70619, and
q=e
qR
q=e
qN
parallel imports hurt consumers in the importing country.
Corollary 1 When the ﬁrm can change the durability of the good and implement service dis19

crimination, parallel imports may hurt consumers in both countries.
When the producer optimally manipulates q, the level of country F ’s import tariﬀ, t, aﬀects
the level of q. This implies that trade liberalization may have a negative eﬀect on the consumer
surplus in country F . On the one hand, trade liberalization enhances the ﬁrm’s incentive toincrease q by increasing the proﬁt in the foreign country and, thereby, the gains from increasing
the durability. On the other hand, trade liberalization decreases the durability because it increases
the producer’s damage from parallel imports and, thereby, diminishes its gains from providing
higher q. We have ∂ 2 ΠN / (∂t∂q) = ∂ 2 ΠI / (∂t∂q) = −N m/2b < 0. Thus, without service
discrimination or without parallel imports, the former eﬀect always dominates the latter, andtrade liberalization leads to a higher level of q.
However, with service discrimination, trade liberalization may induce the producer to reduceq. This is because, by reducing q, the producer can partly oﬀset the damage from the increased
arbitrage pressure.
For instance, if β = 120, trade liberalization from t = 4 to t = 0 increases the optimal level
of durability under no parallel imports from qN t=4 = 0.56522 to qN t=0 = 0.60870. However, itdecreases the optimal level of durability under service discrimination from qR t=4 = 0.54224 toqN t=0 = 0.41919.
Proposition 7 Trade liberalization increases the durability of the good if parallel imports arebanned or there is no service discrimination against the parallel imported good. In the presenceof service discrimination, there is a case where trade liberalization decreases the durability of thegood.
Thus, trade liberalization may widen the gap between qN and qR and worsen the consumers’ losses from allowing parallel imports in both countries. With the numerical calculations





above, changes in consumer surplus become:CSFI q=eqR − CSFN q=eqN = 0.004874 and
t=4





CSFI q=eqR − CSFN q=eqN = −0.66338.
t=0Corollary 2 When the ﬁrm can change the durability of the good, there is a case where allowing
parallel imports beneﬁts consumers in the importing country when the import tariﬀ is high andhurts consumers when the import tariﬀ is low.
20

4Discussion

In this section, we explore the robustness of these results by relaxing some assumptions made in
the basic model.

4.1

Full warranty assumption for the authorized good

We have assumed that the ﬁrm oﬀers a full warranty to the buyers of the authorized good. Here,
we show that this assumption does not aﬀect the qualitative results of our basic model and theﬁrm actually prefers the full warranty in some cases. Suppose parallel importation is prohibitedand let si denote the repair price that the ﬁrm sets for repairing the authorized unit in countryi. Consumers repair the broken units of the authorized good if vi ≥ si holds. That is, the utility
gain from the repair exceeds the repair price. Consumers primarily purchase the authorized good
if qvi ≥ pi . These inequalities suggest that there exist consumers who buy the authorized good,
but never purchase the repair services if and only if si > pi /q holds.
Given that si > pi /q holds, the proﬁt of the ﬁrm in country i is given byqsi − pisπ i (pi , si ) = [pi + (1 − q)si − {c + (1 − q) m}]xi (pi ) + nfi (pi − c)
,
δq

(28)

where xsD (pD ) = nfD (kb−sD ) and xsF (pF ) = nfF (b−sF ). By maximizing π i (pi , si ) with respect
to pi and si , the optimal prices become ṗD = (c + bkq)/2, ṗF = (c + bq)/2, ṡD = (bk + m)/2, andṡF = (b + m)/2. However, these prices fail to satisfy the supposition that si > pi /q holds.
Therefore, the ﬁrm sets the repair prices such that every consumer of the authorized good
prefers to pay for the repair services. In this case, consumers whose willingness to pay satisﬁes
vi ≥ pi +(1−q)si buy the good and all of them purchase the repair services. Let Pi = pi +(1−q)si
denote the overall price of the authorized good. Then, the ﬁrm’s overall proﬁt becomesΠ̇(PD , PF ) =

π i (Pi ) = (Pi − {c + (1 − q) δm}) xi (Pi ),

(29)

i=D,F

where xi (·) is deﬁned in (1) and (2). Because (29) has the same functional form as (3), theproﬁt-maximizing level of each price also coincides (Pi = pi ), as does the equilibrium proﬁt:Π̇(PD , PF ) = Π pN
N
D, p
F .
Because the ﬁrm’s proﬁt is larger when all consumers utilize the repair services, the combination of the good’s price and the repair prices does not matter to the ﬁrm, as long as the overall
21

price is equals to pi and they are consistent with si ≤ pi /q. If the ﬁrm set a higher repair price,
the good’s price becomes lower. If the ﬁrm provides a full warranty, it means that the ﬁrm setsa higher price to cover the cost of the repairs.
In the presence of parallel imports, the ﬁrm becomes more willing to provide the full warranty
in country D to keep the price of the authorized good high and to mitigate the arbitrage pressure.
If the ﬁrm does not provide a full warranty in country F , the authorized good and the PI good
becomes perfectly substitutable and the ﬁrm captures the quality premium of the authorized
good by setting a high repair price. After all, the overall price of the authorized good in country
F becomes the same as the price of the authorized good with the full warranty.

4.2

Repair services by ISOs

Until now, we have restricted our attention to the case where only the original producer can
provide the repair services. Here, we consider the case in which independent service organizations(ISOs) are also able to provide the repair services for the PI good. We assume there are many
ISOs and the competition among them reduces the repair prices to their marginal cost.
The original producer of the good should have more technical information about the goodthan the ISOs, and will stock more parts and components that are necessary to repair the good.
Hence, it is natural to assume that the quality of the repair services performed by the ﬁrm is not
lower than the quality performed by the ISOs. Let μ ∈ (0, 1] represents how well a repair recovers
the original quality of good x. As μ approaches one, the quality gap between the unbroken unit
and the repaired unit narrows, and the quality is fully recovered when μ = 1. The level of μ
reﬂects the extent to which the repair services performed by ISOs recover the “physical” qualityof the good.
Let μw denote the unit cost of undertaking repair services when performed by ISOs. The
unit cost of repairs is increasing in μ because it should be more costly for the ﬁrm to returnthe quality to the original quality. The ISOs may also incur a higher unit cost of repairs, but
their less eﬀective repair services may come with a lower cost. Therefore, we leave w to be either
higher or lower than m.
Firstly, suppose μ = 1. In other words, ISOs provide the same quality of repair services as the22original producer. In this case, even if the ﬁrm refuses to repair the PI good, consumers can repair
the PI good by paying w to an ISO. The overall price of the PI good becomes pD + t + (1 − q)w.
Then, the ﬁrm needs to set pF ≤ pD + t + (1 − q)w to sell the authorized good. If (1 − q)w is N
larger than pND + t given by (8), the ﬁrm can set the same prices as those under market
F − psegmentation. Otherwise, the equilibrium international price diﬀerence becomes t + (1 − q)w,
and becomes larger as q becomes smaller. Note that, unless w = 0, the entry of competitive ISOs
realizes a higher proﬁt for the ﬁrm than the proﬁt that is realized if the ﬁrm cannot discriminate
repair services.
Secondly, suppose μ < 1. To make clear the analysis, we set w = 0 so that ISOs provides
virtually free repairs. However, the free repairs cannot realize full market integration because theauthorized unit and the PI unit repaired by ISOs are imperfect substitutes. If the ﬁrm refuses toprovide the repair services for the PI good, it can attract consumers to buy the authorized goodby setting pF and pD , such thatpF =pD + tμholds. Because μ < 1, we have pF − (pD + t) = (1 − μ) (pD + t) /μ > 0. If this price diﬀerence is N
larger than pND + t , the full market segmentation is realized by the service discrimination.
F − p
If μ < 1 and w > 0: the equilibrium price diﬀerence reﬂects the above two eﬀects. Accordingly,
if the ISOs’ quality of repair services is lower than the ﬁrm’s quality or the ISOs incur a positive
cost of repair services, the presence of competitive ISOs does not realize full market integration.
The original producer’s service discrimination becomes less eﬀective, but it is still eﬀective, anda decrease in q may expand the international price diﬀerence and increase the producer’s proﬁt.

5

Conclusion

In this study, we have re-examined the eﬀect of parallel imports when the producer (trademark
holder) of a durable good might discriminate after-market services against a parallel imported
good. We have shown that the producer is willing to diﬀerentiate the authorized good and theparallel imported good by not providing repair and maintenance services for the parallel imported
units. A refusal to repair parallel imported goods realizes full market segmentation between
countries if the durability of the good is suﬃciently low. In the middle range of durability,23the producer successfully blocks parallel importation even if the price of the authorized good ishigher than the price of the parallel imported good. If the durability is suﬃciently high, theproducer accommodates the parallel imported good and consumers in the importing country
buy either the authorized good or the parallel imported good, depending on their willingness
to pay for the quality. In the latter two cases, the two markets are only partially integrated.
Instead of refusing the repair services, the ﬁrm can also mitigate the pressure of price arbitrage
by charging a relatively high repair price for the parallel imported good. The entry of many
competitive independent service organizations helps to narrow the international price gap, but it
is not suﬃcient to eliminate the ﬁrm’s service discrimination.
Although service discrimination decreases the extent of market integration and realizes ahigher proﬁt for the ﬁrm, it does not always increase the producer’s incentive to improve thedurability of the good. This is because the durability of the product is itself a strategic tool tomanipulate the degree of vertical diﬀerentiation between the authorized good and the parallel imported good. Speciﬁcally, the service discrimination may reduce the durability of the good below
the level under full market integration and under full market segmentation. When the durability
decreases, consumers in the importing country may suﬀer by permitting parallel imports, and thenegative eﬀect of parallel importation will be ampliﬁed as trade liberalization proceeds. Thereis a case where allowing parallel imports beneﬁts consumers in the importing country when an
import tariﬀ is high, but hurts consumers under free trade.
There is room for further research. Introducing other producers that provide the same good,but of diﬀerent quality will enable us to investigate how competition among producers is inﬂuenced by service discrimination. Then, it would be interesting, though cumbersome, to consider
the endogenous determination of the tariﬀ level.

Appendix
Proof of Lemma 2
 N
2
We have pND + t /q = −Ω(q)/(2q) where Ω(q) = (1 − q) (kb + m) + (1 − q) {(2k − 1) qb +F − p
c + 2t} + q{t − (1 − k) qb}. Since Ω(q) is a quadratic function in q, Ω(0) = kb + c + 2t + m > 0,24

and Ω(1) = −{(1 − k)b − t} < 0, there exists a unique value of q, qL (∈ (0, 1)), that satisﬁes
 N

 N

D + t /q holds for q ∈ [qL , 1) and pND + t /q holds
Ω(qL ) = 0. This means that pN
F ≥ p
F < p
for q ∈ (0, qL ].

 NBy comparing pND + t /q and pF − (pD + t) /q, we have
F − p


 
  2
q + k {(b + m)q − (kb + t + m)}pD + tpD + tN.
pF −− pF −=qq
2q (q + k)

The above equation tells us that pN
pD + t) /q − [pF − (pD + t) /q] is positive if q > q1 ≡F − (
(kb + t + m) /(b + m) is satisﬁed. At q = q1 , we have



pN
N
D +t 
pF −

q
q=q1

=



pD + t 
pF −

q
q=q1
−

{(1 − k)b − t}q12 [b (c − km) + m (c − m) + bt] < 0.
2 (kb + t + m)
 ND + t /q = 0 and pF − (pD + t) /q < 0 is satisﬁed at q = qL .
This means that qL > q1 and pN
F − p

=

Therefore, qH at which pF − (pD + t) /q = 0 holds is higher than qL.

Proof of Proposition 2
For the time being, let us suppose pF > (pD + t) + (1 − q)r holds and all consumers in country
F purchase the PI good. If a consumer buys a PI good but does not utilize the repair services
provided by the ﬁrm, then her expected utility is given by (14). Therefore, consumers purchasethe repair services if their willingness to pay for quality satisﬁes v ≥ r. By comparing r and vFP I ,we haver − vFP I =qr − (pD + t)q

If r > (pD + t) /q holds, which means that the repair price for the PI good is suﬃciently high,consumers with parameter v ∈ [vFP I , r] buy the PI good but do not purchase the repair services.
Suppose this inequality holds, the demand for the authorized good and the demand for the PI
good in country F are respectively given by
I,r
(r)
xP
F

=

I,nr
xP
(pD , r)
F

=


r
,
nfF [b − r] = n 1 −b
qr − (pD + t)
nfF r − vFP I = n
,
qb

25

while the demand in country D is given by (1). The ﬁrm’ proﬁt is given byΠ (pD , r)=I,r[pD − {c + (1 − q) m}]xD (pD ) + [pD + (1 − q)r − {c + (1 − q) m}]xP(r)
F(pD , r).
+(pD − c)xIP,nrF
By maximizing Π (pD , rI ) with respect to pD and r, we have the optimal prices as
p∗D

=

r∗

=

(2qb − t) k + mq (1 − q) + c (k + q),
2 (k + q)(b + m).
2

We conﬁrm that p∗D = pD holds. By using (22), pF = pD + t + (1 − q) (b + m) /2 = p∗D + t +(1 − q) r∗ holds. Therefore, the overall price for the PI good in country F coincides with theprice of the authorized good under the refusal to repair. The condition of xIP,nr(p∗D , r∗ ) > 0,Fr∗ > (p∗D + t) /q, can be transformed into pF > (pD + t) /q, the condition under which the ﬁrm
accommodates the PI good in the case of the refusal of repairs. Therefore, some consumers whopurchased the PI good leave the PI good unrepaired if q ∈ (qH , 1) holds.
I,nr ∗
If r∗ ≤ (p∗D + t) /q holds, which means that q ≤ qH holds, we have xP(pD , r∗ ) ≤ 0 withFI,nr
(pD , r) ≥ 0 becomes binding constraint and the ﬁrm cannot setp∗D and r∗ . In this case, xPF

pD and r separately. In this case, the ﬁrm sets the prices such that all consumers who purchase
I,nr(pD , r) = 0). The ﬁrm maximizes
the PI good also purchase the repair services (i.e., xPFΠ (pD , rI ) such that r = (pD + t) /q holds, and we conﬁrm the equilibrium prices satisfy pF =∗p∗
D + t + (1 − q) rI .

N
Under full market segmentation, by (8), pN
F = p
D + t + {(1 − k)b − t}/2 holds. This implies
that if the ﬁrm sets (1 − q)r∗ = {(1 − k)b − t}/2 and r∗ < (pN
D + t)/q is satisﬁed, the ﬁrm canpractically set the same prices as those under the full price discrimination. The inequality can betransformed into pF < (pD + t) /q, the same condition under which the refusal of repairs realizes
the full market segmentation. The full market segmentation occurs if q ∈ (0, qL ) holds.
In sum, the overall price of the good in each country and the cut-oﬀ values of q that are
obtained with the price discrimination in repair services becomes the same as those obtainedwith the refusal of repairs.

26

Proof of Proposition 3
By comparing pNIi , we have pID − pN
pN
pN
IF − pN
pN
i and pD = k{F −(D +t)}/(1+k) > 0 and p
F = −{
F −
N
N
(
pN
i and pi , we have p
D + t)}/(1 + k) < 0. By comparing p
D − pD = k(q − q1 )/2[(b + m) (q + k)]
and pNF − pF = −q(q − q1 )/2 (q + k) where q1 is deﬁned in the proof of Lemma 2. Since q > q1
holds whenever q > qL holds, pNN
Ni
D > pD and p
F < pF are always satisﬁed. With regard to p
2and pi , pN
pN
pN
N
pN
pN
D − pD = −k{q
F − (
D + t)} < 0 and p
F − pF = q{q
F − (
D + t)}/(q + k) > 0 hold.

Therefore, parallel imports always raises the price in country D and reduces the price in country
F.
Next, we compare the prices under Regime I and those under Regime R. We have pD − pID =−k (1 − q) {t+(1 + k) m+2kb}/{2 (k + 1) (q + k)} < 0 and pF − pIF = (1 − q) [ k + q 2 (pD + t)+2
k (1 + q) {qpN(k + 1)} > 0 for q ∈ (qL , 1) because qpN
pN
pN
F − (
D + t)}]/{q k + q
F − (
D + t) > 0
holds. Besides that, for q ∈ (qH , 1), pD − pD = −{q pF − (pD + t)}kq/ k + q 2 < 0 and pF − pF ={q pF − (pD + t)}q 2 /(k + q 2 ) > 0 hold because q pF − (pD + t) > 0 holds in this range of q.
Putting it all together, pID > pNN
IF hold for q ∈ [0, qL ), pID > pD > pN
D and p
F > p
D and

IF hold for q ∈ (qL , qH ], and pID > pD > pNN
IF hold for q ∈ (qH , 1].
pN
F > pF > p
D and p
F > pF > p

This means that parallel imports raise the price in country D and reduce the price in country F ,and the degrees of the changes are moderate when the ﬁrm refuses to repair the PI good.

Proof of Proposition 4
For q ∈ (0, qL ], ΠR = ΠN holds, and we have ΠN −ΠI = n{(1 − k) b−t}2 /{4 (1 + k) b} > 0. Whenthe ﬁrm provides the repair services for the PI good, the ﬁrm’s proﬁt function becomes the same asthat under no parallel imports and it is given by (3). For q ∈ (qL , 1], if the ﬁrm eliminates parallel
imports by setting pD = pD and pF = pF , the ﬁrm’s proﬁt function is also given by (3). Under noparallel imports, the ﬁrm sets the price that maximizes the proﬁt it can earn in each market. Thismeans that the proﬁt earned in each country is higher as the price becomes closer to the optimal

price under no parallel imports. By Proposition 2, pID > pD > pNN
IF and so
D and p
F > pF > p

Π(p , pF ) > ΠI holds for q ∈ (qL , 1]. If qH = 1, Π (pD , pF ) > Π(pD , pF ) holds for q ∈ (qH , 1)
because the ﬁrm chooses pD and pF that satisﬁes q pF > pD + t and accommodates the importsof the PI good even if it can choose pF and pD that satisﬁes q pF = pD + t to block the imports27

of the PI good. Therefore, ΠR > ΠI holds for all q ∈ [0, 1).

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29

Figure 1: The product durability and the firm’s profit

Profit 760
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0.1

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