Intellectual Property Right Protection, Parallel Imports, and

Document Sample
Intellectual Property Right Protection, Parallel Imports, and Powered By Docstoc
					Intellectual Property Right Protection, Parallel Imports,
                               and Product Piracy
                                   Yasukazu Ichino∗
                       Faculty of Economics, Konan University

                                       October 10, 2004


                                            Abstract

     We examine the welfare effect of parallel imports under the possibility of product piracy,
  by constructing a model of the monopoly firm selling its product in two countries. The
  analysis reveals that restricting parallel imports does not always make the firm better off and
  consumers worse off. Sometimes parallel imports increase both the profit of the firm and
  consumer surplus, irrespective of the existence of product piracy. However, product piracy
  makes parallel imports more preferable to consumers and less preferable to the firm. We
  also suggest that a policy regarding piracy can be internationally coordinated with a policy
  regarding parallel imports, so as to improve international welfare.

JEL Classification Numbers: F12, L12

*Address: 8-9-1 Okamoto, Higashinada-ku Kobe 658-8501, Japan.
E-mail: yichino@center.konan-u.ac.jp




                                                1
1     Introduction

When products embodying intellectual property are imported from one country to another without
authorization of a holder of intellectual property right, such imports are called parallel imports.
The products subject to parallel importing are widely ranged, from clothing to automobiles,
cigarettes to pharmaceutical products, perfumes to home appliances.
    In Japan, an example of parallel importing is music CDs from East Asian countries. Since the
1990’s, Japanese record labels have started to sell Japanese music CDs in East Asian countries
such as China, Hong Kong, and Taiwan, as popularity of Japanese pop music increasing in these
countries. Because of lower income levels in these countries, CDs are sold at a price far below that
in Japan. Although jacket design or language of the brochure would be different from “Japanese
version” CDs, since the contents are almost identical, those “Asian version” CDs are imported
to Japan, and sold to Japanese consumers. Recently, there has been a sharp increase in such
parallel imports of Japanese pop music CDs, which consequently provoked considerable public
discussion over pros and cons of allowing those imports in Japan. The main trade-off considered
in the discussion appears that restricting parallel imports would protect the profit of record labels
by sacrificing consumers’ benefit of being able to buy cheaper CDs. In this sense, parallel imports
are considered as the same as illegal copies or product piracy, since they harm the owners of
intellectual property right. However, it should be noticed that parallel imports are different from
product piracy, since, the sales of parallel imports become revenue to the intellectual property
right owners, while the sales of pirated CDs do not.
    This paper is motivated by the above discussion of parallel imports of Japanese music CDs. Of
course, however, we would like to maintain our analysis as general as possible to make our results
applicable to any kind of goods subject to parallel importing. One of the objectives of this paper
is thus to investigate whether and under what condition allowing parallel importing is beneficial
to the firms, consumers, and a country as a whole. In particular, we are interested in a situation
where the main reason of parallel importing is in the income-level differences of the countries, and
parallel imports are considered as a low-quality version of the product.
    As we explained above, the goods subject to parallel importing are, in many cases, intellectual
property right products. However, surprisingly, there is virtually no formal, analytical research of
parallel imports that explicitly takes account of the issues of intellectual property right protection.
Thus, in this paper, as a first step toward incorporating explicitly the issues of intellectual property



                                                  2
right into the analysis of parallel imports, we are going to examine the situation where the product
subject to parallel importing is also subject to product piracy. Another objective of this paper is
thus to examine how the existence of piracy changes the welfare effect of parallel imports.
    In the existing literature of analyzing parallel imports, there are three types of researches. One
type is to consider that parallel imports are to invalidate the third-degree price discrimination, as
discussed by Malueg-Schwartz (1994). The papers by Richardson (2002), and Knox-Richardson
(2002), are seen as extensions of Malueg-Schwartz. Another type is to look at parallel imports
in the context of vertical price control, first suggested by Maskus-Chen (2000). The third type is
to consider parallel imports as a device of the second-degree price discrimination, noticing that
parallel imports are often regarded as a low quality version of the product. Anderson-Ginsburgh
(1999) and Ahmadi-Yang (2000) are the papers of this type. This paper belongs to the third type,
and related to Anderson-Ginsburgh and Ahmadi-Yang, but we are going to make more discussions
than they did over the effect of allowing/banning parallel imports on profits, consumer surpluses
and welfare. A very recent paper by Cosac (2003) is a hybrid of the second type and the third type.
The basic structure of our model is similar to Cosac’s, but our motivation and interest are quite
different from hers. That is, she is interested in examining manufacturer-retailer relationships in
identical two countries, while we are interested in investigating welfare effects of parallel imports
with the possibility of product piracy, in two different counties.
    The rest of the paper is organized as follows. In the next section, we briefly explain the setting
of the model. Then, from section 3 we start our analysis of parallel imports without piracy.
Product piracy is incorporated in Section 4. Finally, Section 5 gives concluding remarks.



2     The setting

We consider two countries, the home country and the foreign country, and suppose that the home
country is a developed country and the foreign country is a developing country. The differences of
the home country and the foreign country are modeled in terms of the following three points: (1)
creation of a new product, (2) income level, and (3) the protection of intellectual property rights.
    First, we assume that it is only the firm in the home country which creates a product embodying
intellectual property. The firm creating a new product, referred to as the monopolist, is going to
sell its product both in the home county and in the foreign country. In this model, we suppose that
the monopolist sells “home version” for the home consumers, and “foreign version.” for the foreign



                                                  3
consumers. Although these two versions are intrinsically the same, they are packaged differently.
Such a difference is made in order to suit the different tastes, customs, or languages of the two
countries. For example, different versions of a music CD have different jacket pictures and different
brochure, although the contents of the CD are the same.
   Second, we model that the income level of the foreign consumers are on average lower than that
of the home consumers. Consequently, the monopolist will set the price of the home version higher
than the foreign version as long as the two markets are segmented. However, if the arbitrage cost
is low enough, the price difference between two countries give an opportunity of parallel importing,
that is, importing the foreign-versioned products and sell them to the home consumers, without
authorization of the monopolist. We assume that the home consumers think of imported foreign
version as a low-quality substitute of home version, because foreign version is made less suitable
for the home consumers. Another reason why imported foreign version is considered having a
lower quality is that it comes with no warranty, because it is imported without the authorization
of the monopolist.
   Third, we suppose that protection of intellectual property rights is weak in the foreign country,
so that the product originally created in the home country is copied, or pirated, and those pirated
versions are sold in the foreign country. The foreign consumers presume that quality of the pirated
version is lower than that of foreign version. We assume that the enforcement of intellectual
property right is strong enough in the home country, thus pirated version is not produced, nor
imported, in the home country.
   In each country, there is a continuum of consumers, who are indexed by their income levels,
θ. In the home country, θ is distributed uniformly between 0 and 1, with the population per type
equal to one. Thus, the total population of the home consumers is normalized to one. In the
foreign country, θ is distributed between 0 and η, with the population per type given by A. The
total population of the foreign consumers is thus equal to Aη. We assume that 0 < η < 1 since
the foreign consumers are supposed to have lower income. In addition, we assume that Aη > 1,
supposing that the total population of the foreign country is larger than that of the home country.
   The utility of a type-θ consumer is given as follows.1 If he buys the product of quality s at the
price p, his utility is equal to s (θ − p). For the home consumers, the quality of home version is
equal to 1, and that of imported foreign version is α, where 0 < α < 1. Similarly, for the foreign
consumers the quality of foreign version is 1, and that of pirated version is β where 0 < β < 1. If
  1 This   setting of the utility function is taken from Shaked and Sutton (1982).



                                                          4
a consumer does not buy the product at all, his utility is zero.
    Setting up the model in this way, essentially what we will do in this paper is to analyze the
monopolist’s pricing decisions and then to compare the results under different situations of parallel
imports and piracy. Specifically, we consider the following four cases:
    (The case o): No parallel imports without piracy,
    (The case i): Parallel imports without piracy,
    (The case c): No parallel imports with piracy, and
    (The case ic): Parallel imports with piracy.
    In the following section, we analyze the case o and the case i, and compare these two cases.
Then, in Section 4, we will look at the cases of piracy, i.e., the case c and the case ic.



3     A model without piracy

3.1    No parallel imports without piracy: the case o

First of all, we consider the case where there is no piracy of the product in the foreign country,
and there is no parallel imports in the home country, due to, say, the prohibitively high arbitrage
cost or the government policy. Here, the case of no parallel imports without piracy, to which we
give a shorthand name “the case o”, can be analyzed as a standard model of monopoly. Let ph
denote the price of home version in the home country, and pf the price of foreign version in the
foreign country. In the home country, the home consumer whose θ is equal to ph is indifferent
between buying home version and not buying. Therefore the demand for the home version in the
home country is given by
                                                  Z    1
                                      o
                                     Dh (ph ) =            dθ = 1 − ph .
                                                      ph

Similarly, the demand for the foreign version in the foreign country is


                                        o
                                       Df (pf ) = A (η − pf ) .


Now, look at the price choice of the monopolist. To simplify the algebra, we assume that the
marginal cost of production is constant and equal to zero. Then, we have the following standard
profit-maximization problem of the monopolist:


                                                o             o
                               max Πo (p) = ph Dh (ph ) + pf Df (pf ),


                                                      5
where p = (ph , pf ). The first-order conditions

                                     ∂Πo
                                             = 1 − 2ph = 0                                      (1a)
                                     ∂ph
                                     ∂Πo
                                             = Aη − 2Apf = 0                                    (1b)
                                     ∂pf

gives the solutions
                                                            µ         ¶
                                                                1 η
                                     po = (po , po ) =
                                            h f                  ,        .
                                                                2 2

Then, we can calculate the profit of the monopolist:

                                                                              1 Aη 2
                                          o             o
                           Πo (po ) = po Dh (po ) + po Df (po ) =
                                       h      h      f      f                   +    .
                                                                              4   4

                                                                                            o
The consumer surplus of the home country when there are no parallel imports is denoted by CSh ,
                                                                                        o
and the consumer surplus of the foreign country when there is no piracy is denoted by CSf . Then,

                                             Z
                              o
                                                  1
                                                                       (1 − ph )2
                            CSh   (p) =               (θ − ph ) dθ =              ,
                                                 ph                        2
                              o              A (η − pf )2
                            CSf (p) =                     .
                                                  2

When these are evaluated at po , we have

                                             1                   Aη 2
                                 o
                               CSh (po ) =             o
                                               , and CSf (po ) =      .
                                             8                    8


3.2    Parallel imports without foreign demand

Since the home consumers think of parallel imports as a low-quality substitute of home version,
for the monopolist, parallel imported products work as a device to price discriminate the home
consumers thorough self selection (that is, the monopolist is able to make the second-degree price
discrimination). To isolate this price-discriminating effect of parallel imports, here we consider
a hypothetical situation where there is no demand for the product in the foreign country, but
the monopolist still sells the foreign version, solely in order to let the foreign-versioned products
imported back to the home country. We assume that the parallel importing is perfectly competi-
tive, and that the marginal cost of transportation is constant and equal to t. Then, the price of
imported foreign version in the home country is equal to pf + t.



                                                        6
     When foreign version as well as home version becomes available to the home consumers, there
are two marginal consumers. One marginal consumer is indifferent between buying home version
and buying imported foreign version. This consumer is found from θ − ph = α (θ − pf − t). Then,
the demand for home version, when foreign version is available through parallel importing, is

                                     Z   1
                        i                                         (1 − α) + α (pf + t) − ph
                       Dh (p)    =                         dθ =                             ,
                                         ph −α(pf +t)
                                             1−α
                                                                           1−α

where the superscript i (stands for “imports”) is to denote the existence of parallel imports. The
other marginal consumer is indifferent between buying the foreign version and not buying. This
consumer is simply θ = pf + t. Thus, the demand for imported foreign version, denoted by Di (p),
is
                                               Z    ph −α(pf +t)
                                                        1−α               ph − pf − t
                                 Di (p) =                          dθ =               .
                                                   pf +t                     1−α

Given these demand functions, the monopolist’s profit-maximization problem, when there is no
demand in the foreign county, is


                                         ˜           i
                                     max Π (p) = ph Dh (p) + pf Di (p).


The first-order conditions are

                           ∂Π˜               (1 − α) + (1 + α) pf + αt − 2ph
                                     =                                       =0                   (2a)
                           ∂ph                            1−α
                           ∂Π˜               (1 + α) ph − 2pf − t
                                     =                            =0                              (2b)
                           ∂pf                     (1 − α)

which gives the solution

                                                    µ                                     ¶
                                                        2 − t (1 + α) − (2 + α) t
                           ˜ p ˜
                           p =(˜h , pf ) =                   ,                                .
                                                        3+α         3+α

Now, we can calculate that
                                                                   1 − α − 2t
                                             ph − pf − t =
                                             ˜    ˜                           .
                                                                     3+α

Thus, in order to have Di (˜) > 0, it must be that 1 − α − 2t > 0. Otherwise, parallel importing
                           p
will not occur, and the monopolist simply chooses p = po . So, we restrict our attention to the set
of parameters such that 1 − α − 2t > 0 is satisfied.
     Now, let us compare the results here with the results of the case o. First, it is shown that,


                                                              7
when parallel imports work solely as a device of the second-degree price discrimination, the price
of the home version is above, and the price of the imported foreign version is below, the standard
monopoly price of the home market:

                                                         1 − α − 2t
                                        ph − po
                                        ˜     h        =             > 0,                                          (3a)
                                                          2 (3 + α)
                                                           1 − α − 2t
                                     pf + t − po
                                     ˜         h       = −            < 0.                                         (3b)
                                                            2 (3 + α)

Second, calculate the profit of the monopolist


                          ˜ p                             (1 − α) (1 − t) + t2
                                  ˜ i p       ˜
                          Π (˜) = ph Dh (˜) + pf Di (˜) =
                                                     p                         ,
                                                            (1 − α) (3 + α)

                          o
and compare this with po Dh (po ) to show that the monopolist’s profit from the home market rises
                       h      h

when parallel imports serves as a device of the second-degree price discrimination. That is,


                                 ˜ p                   1 (1 − α − 2t)2
                                             o
                                 Π (˜) − po Dh (po ) =
                                          h      h                       > 0.
                                                       4 (1 − α) (3 + α)

                                                                        i
Finally, let us look at the consumer surplus of the home country. Let CSh (p) denote the consumer
surplus of the home country when there are parallel imports.
                                                                                                                
                 Z   1                       Z   ph −α(pf +t)
                                                     1−α
                                                                                    Z   ph −α(pf +t)
                                                                                            1−α
     i
   CSh (p) =             (θ − ph ) dθ + α                      (θ − pf − t) dθ −                      (θ − ph ) dθ
                  ph                           pf +t                                ph

                             2                         2
                 (1 − ph )    (ph − pf − t)
             =             +α               .
                     2          2 (1 − α)

                     i            o
Then, by comparing CSh (˜) with CSh (po ), we can see that the home consumers are worse off
                        p             h

when imported foreign version is available:

                                            (1 − α − 2t) (2 (1 + 3α) t + (1 − α) (5 − α))
              i         o
            CSh (˜) − CSh (po ) = −
                 p          h                                                                          < 0.
                                                               8 (3 + α)2 (1 − α)

We summarize these results below.

Proposition 1 Suppose that there is no demand for the product in the foreign country.
   (1) When there is parallel importing, the price of home version is above, and the price of
imported foreign version is below the standard monopoly price.
   (2)Parallel imports make the monopolist better off and the home consumers worse off.


                                                           8
3.3       Parallel imports without piracy: the case i

Now we are ready to analyze the price decision of the monopolist when there is parallel importing
and when there is demand for the product in the foreign country. This case is named “the case
i.” The profit maximization of the monopolist in this case is given by


                                               i                       o
                              max Πi (p) = ph Dh (p) + pf Di (p) + pf Df (pf )


The first-order conditions are

                           ∂Πi          (1 − α) + (1 + α) pf + αt − 2ph
                                   =                                    = 0 and                                 (4a)
                           ∂ph                       1−α
                           ∂Πi          (1 + α) ph − t − 2pf
                                   =                         + A (η − 2pf ) = 0.                               (4b)
                           ∂pf                (1 − α)

Let pi = (pi , pi ) denote the solution to equation (4). For pi to be actually chosen by the
           h f

monopolist, it must be that Di (pi ) > 0 (otherwise parallel importing will not occur), as in the
case of no foreign demand. Furthermore, even when pi satisfies Di (pi ) > 0, parallel importing
can be naturally blockaded if po − po − t ≤ 0: that is, when the monopolist choosing the prices as
                               h    f

if there is no parallel importing, it may actually result in no parallel importing. Since our interest
is to investigate how the introduction of parallel imports will change the price decision, we want
to confine the parameter space such that the parallel importing will occur when it is allowed. To
guarantee this, we impose the condition 1 − η − 2t > 0, on top of the condition 1 − α − 2t > 0.
Notice that 1 − η − 2t > 0 is equivalent to po − po − t > 0.2
                                             h    f

       Now, we are going to compare pi with po . To do this, we first show that pi is between pf and
                                                                                f            ˜
po .
 f


Lemma 1 If po < pf , then po < pi < pi . If po > pf ,then po > pi > pf . Finally, if po = pf ,
            f   ˜          f    f   ˜f       f   ˜         f    f   ˜                 f   ˜
then po = pi = pf .
      f    f   ˜

                                                  ˜                                ˜ p
Proof. Equation (4b) is rewritten as ∂Πi /∂pf = ∂ Π/∂pf +A (η − 2pf ) = 0. Since ∂ Π (˜) /∂pf = 0
      ˜
and ∂ Π (p) /∂pf is deceasing in pf , we see that pi < pf if and only if η −2pi < 0. Namely, pi < pf
                                                       ˜                                          ˜
                                                   f                          f               f

if and only if po < pi . Similarly, we see that pi = pf if and only if po = pi .
                f    f                           f   ˜                  f    f


       The intuition is straightforward. When the foreign-versioned products are sold only to the
                                                                                                              ˜ p
   2 Technically, we impose the following additional condition that η is not too low, so that Πi (pi ) > Π(˜ ) is

satisfied. If this inequality is violated, the monopolist will ignore the foreign consumers entirely, and sell foreign
version only for the purpose of making price discrimination of the home market. We want to avoid such absurdity.


                                                         9
foreign buyers, the optimal price is equal to po . On the other hand, when they are sold only to
                                               f

the home buyers as a low-quality substitute of home version, the monopolist should set pf = pf .
                                                                                            ˜
Therefore, in the present case where parallel imports are allowed and there is nonzero foreign
demand, the monopolist has to balance the marginal profit from the home market and that from
the foreign market. The result is that pi is set between the two extremes, pf and po .
                                        f                                  ˜       f

   Although we are interested in to which direction the prices are changed, Lemma 1 alone does
not determine whether the prices rise or fall as parallel imports are allowed. However, at least, it
is certain that when one of these prices falls, the other does not fall.

Proposition 2 When parallel imports are allowed, if one price falls, then the other price does
not fall.

Proof. Suppose that both prices fall: pi < po and pi < po . From Lemma 1, pi < po implies
                                       h    h      f    f                  f    f

pf < pi < po . Since both p and pi satisfy equation (4a), pf < pi implies ph < pi . Therefore, we
˜     f    f              ˜                               ˜     f         ˜     h

have ph < po . But this contradicts ph − po > 0 (see equation (3a)).
     ˜     h                        ˜     h



   To derive more specific results, we have to rely on the algebraic solution to the first order
conditions (4), which is

                               4 (1 + A (1 − α)) − (2 − 4Aα) t + 2 (1 + α) Aη
                     pi
                      h    =                                                  ,
                                                2 (3 + α + 4A)
                               2 (1 + α) − 2 (2 + α) t + 4Aη
                     pi
                      f    =                                 .
                                      2 (3 + α + 4A)

Using this explicit solution, we are now able to give the following proposition about the effect of
parallel imports on the prices.

Proposition 3 If η ≤ α, then pi < po and pi > po .
                              h    h      f    f


Proof. A straightforward calculation gives that

                                  (1 − α − 2t) (2Aα − 1) + 2 (1 + α) A (α − η)
                  pi − po = −
                   h    h                                                      < 0,             (5)
                                                 2 (3 + α + 4A)

and
                                    (2 + α) (1 − α − 2t) + (3 + α) (α − η)
                       pi − po =
                        f    f                                             > 0.                 (6)
                                                2 (3 + α + 4A)




                                                   10
   The intuition of the proposition is seen as follows. Recall that α represents the quality level
of imported foreign version relative to that of home version, and that η can be thought of as the
income level of the foreign country relative to that of the home country. The condition η ≤ α
essentially means that the income level of the foreign country is relatively far from that of the
home country, while the quality of imported foreign version is relatively close to that of home
version. In such a case, po , which reflects the relative income level of the foreign country, turns
                          f

out to be too far apart from po for price discrimination to work well, provided that two versions
                              h

are only narrowly differentiated. Hence, when parallel importing is allowed, the price gap between
po and po is narrowed, with po decreasing to pi , and po increasing to pi .
 h      f                    h                h        f                f

   We now turn to examine the profit of the monopolist. The proposition below tells us that the
monopolist prefers parallel imports if income level of the foreign country is close enough to the
level of the home country, and if the quality of imported foreign version is not very close to that
of home version.

Proposition 4 If η ≥ α, Then Πi (pi ) > Πo (po ) unless pi = po .

                                i               o
Proof. Recall that Πi (p) = ph Dh + pf Di + pf Df . Then, note that the first two terms are
rewritten as follows:


                     i                                 α (ph − (pf + t))      ph − (pf + t)
                 ph Dh + pf Di   = ph (1 − ph ) − ph                     + pf
                                                             1−α                 1−α
                                       o
                                 = ph Dh + (pf − αph ) Di .                                     (7)


             η                                                  η−α
Since po =
       f     2   and po = 1 , η ≥ α implies that po − αpo =
                      h   2                       f     h        2    ≥ 0. Then,


                                       o             o
                        Πo (po ) = po Dh (po ) + po Df (po )
                                    h      h      f      f

                                        o
                                  ≤ po Dh (po ) + (po − αpo )Di (po ) + po Df (po )
                                     h      h       f     h              f
                                                                            o
                                                                                f

                                  = Πi (po ) ≤ Πi (pi ).


The last inequality is from the definition of pi . Since pi is unique, the strict inequality holds
unless pi = po .


   The intuition is as follows. When the income level of the foreign country is close to the level
of the home country, the monopolist is going to set pf close to ph . At the same time, however,
as foreign version is not a very good substitute (i.e., α is small), the optimal price gap between

                                                    11
home version and foreign version will be large. So, this allows the monopolist to set pi above po ,
                                                                                       h        h

extracting more from high-income consumers while not losing low-income consumers, making price
discrimination successful. Notice that the best situation for the monopolist is where it achieves
the most effective price discrimination in the home market, without sacrificing at all the profit
from the foreign market. In this best situation, Πi (pi ) is surely larger than Πo (po ). In fact, this
best situation happens when η > α, as the example below demonstrates.

Example 1 When pf = po , the monopolist is deriving the maximum profit from the home market,
               ˜     f

without sacrificing at all from the foreign market. Using the expressions for pf and po , pf = po
                                                                             ˜       f ˜       f

is rewritten as η = 2 (1+α)−(2+α)t . Then, it is straightforward to see that 2 (1+α)−(2+α)t − α =
                          3+α                                                      3+α
(2+α)(1−α−2t)
     3+α        > 0. Therefore, pf = po happens when η > α.
                                ˜     f




   Since η ≥ α is just a sufficient condition for Πi (pi ) > Πo (po ), one may suspect that Πi (pi ) >
Πo (po ) holds for any parameter values. The example below, however, suggests that Πi (pi ) <
Πo (po ) can actually happen when η is small enough relative to α.

Example 2 First, we show that Πi (pi ) < Πo (po ) when pi < αpi .
                                                        f     h


                                               ¡         ¢
                                      o
                       Πi (pi ) = pi Dh (pi ) + pi − αpi Di (pi ) + pi Df (pi )
                                   h      h      f     h             f
                                                                        o
                                                                            f

                                         o             o
                                   < pi Dh (pi ) + pi Df (pi )
                                      h      h      f      f

                                   = Πo (pi ) ≤ Πo (po ).


Then, what we need to do next is to demonstrate that pi < αpi can happen. Using the expressions
                                                      f     h

for pi and pi , we can calculate that
     h      f

                                   ¡       ¢
                                    1 + Aα2 (1 − α − 2t) + A (2 − α (1 + α)) (η − α)
                  pi
                   f   −   αpi
                             h   =                                                   .
                                                    3 + α + 4A

When 1−α−2t is very close to zero while η −α is negative enough, this can be negative. Therefore,
Πi (pi ) < Πo (po ) can happen when η is small enough relative to α.



   Intuitively, when the foreign version is a close substitute (i.e., α is large), the monopolist has
to choose a narrow price difference between home version and foreign version in order to make
price discrimination successful. However, as the income level of the foreign firm is very low (i.e.,


                                                     12
η is low), the price of foreign version should be kept low enough in order not to lose the foreign
customers, which in turn hampers successful price discrimination. Therefore, instead of increasing
the price of foreign version very much, the monopolist has to somewhat lower the price of the home
version to narrow the price difference, which deteriorates the profit of the monopolist. Hence the
profit falls as parallel importing is allowed in this case.
   On the other hand, for the consumers, such a case that η ≤ α is preferable, since parallel
imports in this case give them an opportunity to buy a close substitute of home version at a
reasonable price, and at the same time, the price of home version is decreased. This argument is
formalized in the proposition below.

                               i           o
Proposition 5 If η ≤ α, then CSh (pi ) > CSh (po ).

                    i           o
Proof. We rewrite CSh (pi ) − CSh (po ) as follows:


             i           o             i            o         o            o
           CSh (pi ) − CSh (po ) = CSh (pi ) − CSh (pi ) + CSh (pi ) − CSh (po )
                                     ³        ³       ´´2 Ã                            !
                                                            ¡         ¢2
                                       pi − pi + t
                                         h      f            1 − pi        (1 − po )
                                                                                     2
                                                                    h            h
                                 = α                     +               −               .      (8)
                                           2 (1 − α)             2             2


The first term is positive. The second term is positive if pi < po , which is implied by η ≤ α (see
                                                           h    h

Proposition 3).

                                                                                i
   Although the proposition does not say anything about whether the opposite, CSh (pi ) <
  o
CSh (po ), can happen, it is straightforward to demonstrate that it can. Recall that when p = p,
                                                                                              ˜
parallel imports decrease the home consumer surplus as we saw in Section 3.2. So, when pi
happens to equal p, the home consumers are made worse off.
                 ˜
   Regarding the profit and the home consumer surplus, there seems a casual perception that
parallel imports are beneficial to the consumers but harmful to the monopolist. This view is
supported by Maskus-Chen’s model (2000) where the home version and the foreign version are
identical. However, as the results of our analysis suggest, the casual perception is not necessarily
true. In Cosac (2003) where home version and foreign version are differentiated but two countries
are identical, it is shown that the monopolist is better off and the consumers are worse off by
parallel importing. On the other hand, we found that whether parallel importing is beneficial
or harmful to the monopolist and to the home consumers is in general ambiguous. Moreover, in
contrast to the results of previous papers, our model showed that the interests of the monopolist


                                                  13
and the consumers are not always in conflict. Proposition 4 and 5 combined tells us that both
the consumers and the monopolist prefer allowing parallel imports when η = α. Then, from
this result, we can immediately infer that welfare of the home country, which is the sum of the
monopolist’s profit and the home consumer surplus, is increased by parallel imports when η and
α are close enough.
    In order to have more complete results about the welfare effect of the parallel imports, we now
proceed to a graphical representation of the results. Let us consider η-t plane, and draw a curve
of Πi (pi ) − Πo (po ) = 0 and that of CSh (pi ) − CSh (po ) = 0 on this plane.3 (See Figure 1. In this
                                         i           o


figure, the small arrows indicate the direction to which the parallel importing becomes preferred
to no parallel importing).



               [Figure 1 here]



                                  i                              o
    We define W i (p) = Πi (p) + CSh (p) and W o (p) = Πo (p) + CSh (p). Then, using the graph,
we fix the location of the W i (pi ) − W o (po ) = 0 curve, and derive the following proposition.

                   i           o
Proposition 6 If CSh (pi ) ≥ CSh (po ), then W i (pi ) ≥ W o (po ).

Proof. See appendix B.


    This proposition shows that even when the monopolist loses by parallel imports, the consumers’
gain is large enough to offset the loss of the monopolist. A policy implication of this proposition
would be as follows. For the government pursuing welfare maximization, it should ban parallel
imports only if the consumers do not like it, which occurs when η > α, in words, when the foreign
country is not very poor in relative to the home country, while foreign imported version is not a
very good substitute of home version.
    Before closing this section, we mention the effect of allowing parallel imports on the foreign
consumer surplus. Since parallel importing does not change the functional form of the foreign
consumer surplus, we can simply calculate

                                                       ³       ´2          ³       ´2
                                                      A η − pi
                                                             f            A η − po
                                                                                 f
                             o           o
                           CSf (pi ) − CSf (po ) =                    −                 .
                                                             2                 2
  3 In appendix A, we formally prove that there is only one Πi (pi ) − Πo (po ) = 0 curve, and only one CS i (pi ) −
                                                                                                          h
  o
CSh (po ) = 0 curve.



                                                        14
This equation shows that the foreign consumers prefer the home country allowing parallel imports
if and only if parallel importing lowers the price of the foreign version, that is, pi < po . As we
                                                                                     f    f

saw in Proposition 3, this happens only when η > α.



4     A model with piracy

4.1    No parallel imports with piracy: the case c

In this section we are going to look at the case where the product of the monopolist is pirated in
the foreign country. First, as a benchmark, consider the product piracy when parallel imports are
not allowed. This case is called “the case c.”
    In the foreign country, now two versions of the product is available. One is the foreign version
sold by the monopolist (which is authentic one), and the other is “pirated version” sold by pirates.
Recall that in our model the foreign consumers can distinguish pirate version from authentic foreign
version, and pirate version is considered as a lower quality version of foreign version. Let pc denote
the price of pirated version. One marginal consumer who is indifferent between buying foreign
version and buying pirate version is found by θ − pf = β (θ − pc ). The foreign demand for foreign
version, when there is pirated version, is thus given by

                                     Z   η
                            c                                (1 − β) η + βpc − pf
                           Df   =A                 dθ = A                         .
                                         pf −βpc
                                           1−β
                                                                   (1 − β)

where the superscript c (stands for “copies”) is to denote the existence of pirates. The other
marginal consumer who is indifferent between buying pirate version and not buying is simply
θ = pc . So the foreign demand for pirated version is

                                              Z    pf −βpc
                                                     1−β              pf − pc
                                 Dc = A                      dθ = A           .
                                               pc                      1−β

    As we have assumed that parallel importers are perfectly competitive, here we suppose that
pirates are perfectly competitive. Assuming the constant marginal cost of making a pirated copy
equal to c, we have the price of the pirated version just equal to c.
    The profit maximization problem of the monopolist is


                                              o             c
                                 max Πc = ph Dh (ph ) + pf Df (pf ).



                                                        15
The first-order conditions

                                  ∂Πc
                                         = 1 − 2ph = 0
                                  ∂ph
                                  ∂Πc          (1 − β) η + βc − 2pf
                                         = A                        =0
                                  ∂pf                (1 − β)

give us
                                                     µ                         ¶
                                     ¡       ¢           1 η − β (η − c)
                                  p = pc , pc =
                                   c
                                       h f                 ,                       .4
                                                         2       2

   Here, to see the effect of piracy briefly, we compare the results here with the results of no
parallel imports without piracy. When there are no parallel imports, piracy does not have any
impact on the home market, so that the price of home version in the case c is the same as the
one in the case o. The effect of piracy here is simply to lower the demand for foreign version, and
hence to lower the price of foreign version. Consequently, compared to the case of no piracy, the
profit of the monopolist is smaller in this case:

                                                             1    (η − β (η − c))2
                             o             c
              Πc (pc ) = pc Dh (pc ) + pc Df (pc ) =
                          h      h      f      f               +A                  < Πo (po ) .
                                                             4       4 (1 − β)

The home consumer surplus here is the same as that of no piracy case:

                                                      1
                                          o
                                        CSh (pc ) =         o
                                                        = CSh (po ) .
                                                      8

On the other hand, the foreign consumer surplus has now the different functional form. Let
  c
CSf (p) denote the foreign consumer surplus when there is piracy.

                                                                                                              
                        Z   η                        Z       pf −βc
                                                              1−β
                                                                                        Z   pf −βc
                                                                                             1−β
            c
          CSf (p) = A           (θ − pf ) dθ + Aβ                    (θ − c) dθ − A                 (θ − pf ) dθ
                         pf                              c                              pf

                      A (η − pf )2 Aβ (pf − c)2
                 =                +             .                                                                    (9)
                           2        2 (1 − β)

            c              o
Comparing CSf (pc ) with CSf (po ), we can show that the foreign consumers are, of course, better




                                                      16
off by the pirates.


              c           o            c          o            o          o
            CSf (pc ) − CSf (po ) = CSf (pc ) − CSf (pc ) + CSf (pc ) − CSf (po )
                                        ³      ´2  ³              ´2    ³         ´2 
                                    Aβ pc − c
                                          f           A η − pc  f      A η − po 
                                                                                 f
                                  =               +                  −               .         (10)
                                      2 (1 − β)              2               2


This is positive since pc < po .
                        f    f



4.2       Parallel imports with piracy: the case ic

The last case we are going to analyze is the one where parallel imports are coming from the foreign
country to the home country, and the pirates are active in the foreign country. The shorthand
name of this case is “the case ic.” The monopolist’s problem is


                                              i                       c
                                max Πic = ph Dh (p) + pf Di (p) + pf Df (pf ).


The first-order conditions are

                     ∂Πic          (1 − α) + (1 + α) pf + αt − 2ph
                              =                                     =0                          (11a)
                     ∂ph                        1−α
                     ∂Πic          (1 + α) ph − t − 2pf     (1 − β) η + βc − 2pf
                              =                         +A                       = 0.           (11b)
                     ∂pf                 (1 − α)                   (1 − β)

Let pic = (pic , pic ) denote the solution to (11).5 Comparing the first-order condition (11) with
            h     f

(4), we see that ∂Πic /∂ph is identical to ∂Πi /∂ph , while ∂Πic /∂pf < ∂Πi /∂pf . As a result, we
have pic < pi and pic < pi . This means that, in the case where parallel importing is allowed,
      h     h      f     f

as the pirates in the foreign country become active, not only the foreign consumers but also the
home consumers are made better off. On the other hand, of course, the monopolist gets worse off
by the pirates. We summarize this discussion in the proposition below.

                i           i           o           c
Proposition 7 CSh (pi ) < CSh (pic ), CSf (pi ) < CSf (pic ), and Πic (pic ) < Πi (pi ).

               i                                                                  i
Proof. Since CSh (p) is decreasing in ph and pf , pic < pi and pic < pi implies CSh (pi ) <
                                                   h     h      f     f
  i
CSh (pic ). For the foreign consumer surplus, we can use the same argument as used in equation
                                                                                            i
(10). Finally, for the profit, we have the following chains of inequality: Πic (pic ) = pic Dh (pic ) +
                                                                                        h
  5 The   algebraic expression for pic is shown in Appendix C.




                                                        17
pic Di (pic ) + pic Df (pic ) < pic Dh (pic ) + pic Di (pic ) + pic Df (pic ) = Πi (pic ) ≤ Πi (pi ), where the
 f               f
                     c
                                 h
                                     i
                                                 f               f
                                                                     o

                            c    o          o
first inequality comes from Df = Df − βDc < Df .



4.3     Welfare comparison and welfare ordering

When we compare the results of the case c with the results of the case ic, it turns out that the
comparison will look very similar to the comparison we have made between the case o and the
case i. Namely, whichever there is piracy or not, the effects of allowing parallel imports on the
prices, profits, and consumer surplus are qualitatively similar. Specifically, when examining the
case o and case i, the key condition was whether η ≥ α or η ≤ α, as seen in Proposition 3 to 6. On
the other hand, for comparing the case c with the case ic, the key condition is the one in which η
is replaced by η − β (η − c). That is, the similar propositions to Proposition 3 to 6 are derived by
comparing η − β (η − c) with α. Because of the similarity, we just present those results without
proofs and discussions.

Proposition 8 (1) If η − β (η − c) ≤ α, then pic < pc and pic > pc . (2) If η − β (η − c) ≥ α,
                                                 h     h        f    f
        ¡ ic ¢
then Πic p                                                                   i            o
               > Πc (pc ) unless pic = pc . (3) If η − β (η − c) ≤ α, then CSh (pic ) > CSh (pc ).
         i            o                                                                   i
(4) If CSh (pic ) ≥ CSh (pc ), then W ic (pic ) > W c (pc ), where W ic (p) = Πic (p) + CSh (p), and
                     o
W c (p) = Πc (p) + CSh (p).

   In addition, for the foreign consumers, parallel importing is preferable if and only if pic ≤ pc .
                                                                                            f     f

This is because the functional form of the foreign consumer surplus when there is piracy is the
same whether there are parallel imports or not, and the foreign consumer surplus is a decreasing
function of pf (see equation (9)).


   Let us move on to the graphical representation of the results on η-t plane. We are going to draw
                                       i            o
the Πic (pic ) − Πc (pc ) = 0 curve, CSh (pic ) − CSh (pc ) = 0 curve, and the W ic (pic ) − W c (pc ) = 0
curve on this plane, and compare the location of these curves with the corresponding curves
comparing the case i with the case o. Before, when comparing the case i with the case o, the focal
                                                  ¡      ¢
point which all curves pass through was (η, t) = α, 1−α . Now, in comparing the case ic and the
                                                       2
                                   ³           ´
                                     α−βc                   α−βc
case c, the focal point is (η, t) = (1−β) , 1−α . Since α < (1−β) , the focal point is shifted to the
                                             2

right. Therefore, the curves comparing the case ic with the case c are located to the right of the
corresponding curves comparing the case i with the case o. The following propositions formally
prove this (see Figure 2 to 5 also).

                                                      18
             [Figure 2 to 5 here]




Proposition 9 Πic (pic ) > Πc (pc ), then Πi (pi ) > Πo (po ).

Proof. See Appendix D.


                    i           o               i            o
Proposition 10 If CSh (pi ) > CSh (po ), then CSh (pic ) > CSh (pc ).

               o           o                                 i            i
Proof. Since CSh (po ) = CSh (pc ), it suffices to show that CSh (pic ) > CSh (pi ), which is shown
in Proposition 7.


                    c            c               o           o
Proposition 11 If CSf (pic ) > CSf (pc ), then CSf (pi ) > CSf (po )

Proof. See Appendix E.


   In terms of the graph, these propositions fix the location of the ic-c curves in relative to
the corresponding i-o curves, and show that these curves do not cross each other. For welfare,
however, we are not able to derive the similar proposition. We cannot exclude the possibility that
the W i (pi ) − W o (po ) = 0 curve crosses the W ic (pic ) − W c (pc ) = 0 curve.
   The graphical representation suggests that piracy of the product in the foreign country makes
parallel importing less preferable for the monopolist, in the sense that there is a set of the parameter
values at which the monopolist prefers parallel imports if there is no piracy, but it does not prefer
parallel imports if there is piracy. In a similar sense, it can be seen that piracy makes the parallel
imports more preferable for the home consumers, and less preferable for the foreign consumers.
   As a summary of the discussions so far, we consider the possible orderings of the four cases,
o, i, c, and ic, in terms of the profit, the consumer surpluses, and welfare. See Figure 2 to 5.
In these graphs, for each relevant region on the η-t plane, all possible orderings of the four cases
are indicated, with pairwise orderings that must be satisfied in each region given in parentheses.
For example, in Figure 2, it is shown that the orderings o  i  c  ic and o  c  i  ic are
possible in the region left to the Πi (pi ) − Πo (po ) = 0 curve. Thus, we can see that in this region
the monopolist most prefers the case of no parallel imports without piracy, and least prefers the
case of parallel imports with piracy, but it is ambiguous if the monopolist prefers the case i to the
case c.


                                                    19
   Now, by looking at the graph, we want to comment on some interesting orderings. First, as
just mentioned above, the monopolist may have a preference ordering such that o  c  i  ic,
which actually occurs in the following example.

Example 3 Let A = 3, α = 0.8, η = 0.4, t = 0.05, β = 0.4, and c = 0.05. Then, Πo (po ) = 0.37,
                      ¡ ¢                    ¡ ¢
Πc (pc ) = 0.334 5, Πi pi = 0.332 82, and Πic pic = 0.262 63.

   The ordering o  c  i  ic means that the monopolist prefers its product to be pirated
rather than to be parallel imported. This seems counterintuitive, given that the monopolist
always prefers not to be pirated over to be pirated, while it often prefers parallel importing over
no parallel importing. However, note that this happens when α is well above η: in such a case,
   ¡ ¢
Πi pi can be far below Πo (po ), thus the case c can be preferred to the case i.
   Second, in terms of welfare of the home country, ordering such that o  i  ic  c or i  o Â
c  ic may happen. When the home country has this kind of welfare orderings, the government
should be aware of whether there is piracy in the foreign country in making a decision of allowing
or banning parallel imports. Below, we give an example in which the ordering o  i  ic  c can
happen.

Example 4 Let A = 1.5, α = 0.5, η = 0.95, t = 0.02, β = 0.1, c = 0.05. Then, W o (po ) = 0.713 44,
                         ¡ ¢                     ¡ ¢
W c (pc ) = 0.683 17, W i pi = 0.706 28, and W ic pic = 0.687 53.

   When this is the case, parallel imports improve welfare if there is piracy, but worsens welfare
if there is no piracy.


4.4    Policy game

In this section, we briefly discuss an issue of policy choices by the home government and the
foreign government. Suppose that the home government and the foreign government are welfare
maximizer. Here, it is reasonable to assume that the policy choice of the home government is
to allow or ban parallel imports, and that of the foreign government is to allow or ban piracy.
Since welfare of the home country and that of the foreign country depend on parallel importing
and piracy, when allowing or banning those activities becomes a policy choice, the welfare of each
country becomes interdependent on each-other’s policy choice. Thus, we consider the following
policy-choice game, where the home government and the foreign government move simultaneously.




                                                20
                                               Foreign Government
                                               Allow piracy              Ban piracy
     Home           Allow parallel imports                    c
                                               W ic (pic ), CSf (pic )                o
                                                                         W i (pi ), CSf (pi )
     Government     Ban parallel imports                    c
                                               W c (pc ), CSf (pc )                   o
                                                                         W o (po ), CSf (po )

    From the analysis so far, we know that allowing piracy is the dominant strategy of the foreign
government. Therefore, the equilibrium of the game is either “allowing parallel imports, allowing
piracy” or “banning parallel imports, allowing piracy,” namely, the outcome is either the case ic
or the case c. The case ic is going to be the outcome if η ≤ α, while the case c is going to be
the outcome when η is very large. Now, it is interesting to point out that there is a possibility
of prisoners’ dilemma situation. Suppose that η is very large so that the home government has a
dominant strategy of not allowing parallel imports. The equilibrium of the policy game is then
“banning parallel imports, allowing piracy”, and the the equilibrium outcome of the game is the
case c. From Figure 4 and Figure 5, we see that a possible welfare orderings giving this equilibrium
outcome is that the home welfare ordering of o  i  c  ic and the foreign welfare ordering of
ic  i  c  o. The following example demonstrate that these orderings can actually happen.

Example 5 Let A = 1.5, α = 0.5, η = 0.95, t = 0.02, β = 0.03, and c = 0.05. Then, W o =
                                                                 o                     c
0.713 44, W c = 0.704 35, W i = 0.706 28, and W ic = 0.700 75; CSf (po ) = 0.169 22, CSf (pc ) =
           o                         c
0.182 9, CSf (pi ) = 0.185 53, and CSf (pic ) = 0.195 3.

    If this is going to be the case, the home government and the foreign government could be
better off through the following policy coordination: by the home government switching its policy
from banning parallel imports to allowing parallel imports, and the foreign government switching
its policy from allowing piracy to banning piracy, they can move together from the case c to the
case i. In other words, if each government is able to commit to its policy choice, it might be
beneficial for both countries to negotiate over the issues of piracy and parallel imports together.
The example above suggests that such a negotiation is successful when the transportation cost is
very low, and the quality of pirated version is very low.



5     Concluding remarks

In this paper, we have examined the welfare effect of parallel imports under the possibility of
product piracy. The key feature of our model is that the monopoly firm sells its product in two


                                                 21
countries of different income levels, and that parallel imports are considered having a low quality.
In the first half of the paper where we focused purely on parallel imports, we found that there
is no clear-cut answer whether parallel imports are beneficial or harmful to the monopolist or
to the consumers. Our model suggests that it crucially depends on the relative income size of
the two countries and the relative quality of parallel imports. In the second half of the paper,
where we incorporated product piracy, we found that product piracy makes parallel imports more
likely to be beneficial to the consumers and less likely to beneficial to the monopoly firm. We
also pointed out that a policy regarding piracy can be internationally coordinated with a policy
regarding parallel imports, so as to improve the welfare of international economy.
   As a study of intellectual property right product, a limitation of the analysis in this paper
is that we paid attention only to short-run, static effects of parallel imports and product piracy
on national welfare. Namely, we have not taken account of a long-run, dynamic effect of parallel
imports and product piracy on, say, the incentives of the firms to keep creating new products.
In this sense, our results might be biased against protecting the holders of intellectual property
rights. Incorporating dynamic effects of parallel imports can be an interesting extension of the
research.




                                                22
   Appendix
                                                                              i
   Appendix A. There is only one Πi (pi ) − Πo (po ) = 0 curve and only one CSh (pi ) −
  o
CSh (po ) = 0 curve.
                                                                                   i
  First, in Lemma 2 we show that there are a Πi (pi ) − Πo (po ) = 0 curve and a CSh (pi ) −
                                                                 ¡ 1−α ¢
  o
CSh (po ) = 0 curve both of which pass through the point (η, t) = α, 2 .

                   ¡      ¢
Lemma 2 If (η, t) = α, 1−α , then Πi (pi ) = Πo (po ) and CSh (pi ) = CSh (po ).
                        2
                                                            i           o


                              ¡       ¢
    Proof. Given that (η, t) = α, 1−α , from equation (5) and equation (6) it is seen that pi = po .
                                   2
                     ¡      ¢
In addition, (η, t) = α, 1−α implies Di (pi ) = 0. Then, from equation (7), pi Dh (pi )+pi Di (pi ) =
                          2                                                  h
                                                                                i
                                                                                         f
    o
po Dh . Therefore, Πi (pi ) = Πo (po ). For the consumer surplus, equation (8) tells that po −po −t = 0
 h                                                                                         h   f
                    i           o
and pi = po imply CSh (pi ) = CSh (po ).¤


   From Proposition 4 and 5, we know that the Πi (pi ) − Πo (po ) = 0 curve is located in the area
                                i           o
left of η = α line, while the CSh (pi ) − CSh (po ) = 0 curve is located in the area right of η = α
line. Then, on the Πi (pi ) − Πo (po ) = 0 curve, the derivative of Πi (pi ) − Πo (po ) is

                                ¡                   ¢
                               ∂ Πi (pi ) − Πo (po )   ¡        ¢
                                                      = pi − po A > 0
                                                         f    f
                                        ∂η

since pi > po if η ≤ α. Therefore, there is no other Πi (pi ) − Πo (po ) = 0 curve. Similarly, on the
       f    f
  i           o
CSh (pi ) − CSh (po ) = 0 curve, the derivative of CS i (pi ) − CS o (po ) is

                     ¡                        ¢
                    ∂ CS i (pi ) − CS o (po )              ∂pi            ∂pi
                                                                            f
                                                    i             i
                                                = −Dh (pi ) h − αDf (pi )     <0
                               ∂η                          ∂η             ∂η

since
                       ∂pi
                         h     (1 + α) A          ∂pi
                                                    f        2A
                           =              > 0 and     =              > 0.
                       ∂η    (3 + α + 4A)         dη    (3 + α + 4A)

So, there is no other Πi (pi ) − Πo (po ) = 0 curve. ¥



   Appendix B. Proof of proposition 6.
    Lemma 2 implies that there is a W i (pi ) − W o (po ) = 0 curve which go through the point
        ¡      ¢
(η, t) = α, 1−α . From Proposition 4 and 5, we know W i (pi ) − W o (po ) = 0 curve does not cross
             2
                                                                i           o
the area bounded by the Πi (pi ) − Πo (po ) = 0 curve and the CSh (pi ) − CSh (po ) = 0 curve.




                                                   23
                                                                                              i
   To fix the location of the W i (pi )−W o (po ) = 0 curve, we take the total derivative of CSh (pi )−
    o
CSh (po ) = 0 and W i (pi ) − W o (po ) = 0, and we find that the slopes of these curves are equal at
        ¡       ¢
(η, t) = α, 1−α :
             2

   ¯                             ¯
dt ¯
   ¯                          dt ¯
                                 ¯
                            =
dη ¯CS i (pi )=CS o (po )     dη ¯W i (pi )=W o (po )
       h          h
                                Ã                                      ! ,Ã                                 !
                                   ∂CSh (pi ) ∂pi
                                         i
                                                    h   ∂CSh (pi ) ∂pi
                                                           i
                                                                     f
                                                                               i
                                                                            ∂CSh (pi ) ∂pi
                                                                                         h   ∂CSh (pi ) ∂pi
                                                                                                i
                                                                                                          f
                            = −                       +                                    +
                                       ∂pi h     ∂η       ∂pif     ∂η         ∂pih      ∂t     ∂pif      ∂t

                            ¡      ¢
This is because, at (η, t) = α, 1−α , the following equalities hold:
                                 2


                                ∂Πi (pi )                  ∂Πo (po )
                                             = pi A = po A =
                                                f      f             , and
                                  ∂η                          ∂η
                                ∂Πi (pi )        pi − αpi
                                                  f     h    po − αpo
                                                              f        h
                                             = −          =−             = 0.
                                  ∂t               1−α         1−α

                                                            i           o
Since W i (pi ) > W o (po ) holds in the area left of the CSh (pi ) − CSh (po ) = 0 curve, the W i (pi ) −
                                                         ¡      ¢
W o (po ) = 0 curve that passes through (η, t) = α, 1−α comes the right of the CSh (pi ) −
                                                              2
                                                                                                i

  o
CSh (po ) = 0 curve, as drawn in Figure 1.
   Before moving on the next step, we need to show the following lemma.

Lemma 3 When η is large enough, W i (pi ) − W o (po ) is decreasing in η.

   Proof. Consider the derivative

         ¡                      ¢
        ∂ W i (pi ) − W o (po )        ∂Πi (pi ) ∂Πo (po ) ∂CSh (pi ) ∂pi
                                                                  i
                                                                         h   ∂CSh (pi ) ∂pi
                                                                                    i
                                                                                            f
                                     =           −        +         i      +
                  ∂η                       ∂η      ∂η            ∂ph   ∂η        ∂pi  f   ∂η
                                       ¡ i       ¢           (1 + α) A                  2A
                                                      i
                                     = pf − po A − Dh (pi )
                                               f                       − αDi (pi )
                                                            3 + α + 4A             3 + α + 4A

This is negative if pi < po . Recall equation (6)
                     f    f


                                            (2 + α) (1 − α − 2t) + (3 + α) (α − η)
                              pi − po =
                               f    f
                                                        2 (3 + α + 4A)

to consider the pi − po = 0 line on η-t plane. It is easily seen that this line go through (η, t) =
                 f    f
¡ 1−α ¢
 α, 2 . The slope of the line is

                                            ¯
                                         dt ¯
                                            ¯            3+α        1
                                                    =−           <−
                                         dη ¯pi =po    2 (2 + α)    2
                                               f   f




                                                        24
Thus, pi − po = 0 line is left of the 1 − η − 2t line. Therefore, when η large enough, pi < po can
       f    f                                                                           f    f

happen.¤


   Next, we examine on which side of the W i (pi ) − W o (po ) = 0 curve we will have W i (pi ) >
W o (po ). Suppose that W i (pi ) > W o (po ) were to hold on the right side of this W i (pi )−W o (po ) =
0 curve. This would imply that W i (pi ) − W o (po ) were increasing in η on this curve. Since it
                                                        i           o
must be W i (pi ) > W o (po ) in the area left of the CSh (pi ) − CSh (po ) = 0 curve, we should have
                                                      i           o
another W i (pi ) − W o (po ) = 0 curve between the CSh (pi ) − CSh (po ) = 0 curve and the first
W i (pi ) − W o (po ) = 0 curve. Then, because W i (pi ) − W o (po ) is quadratic, there would be no
other W i (pi )−W o (po ) = 0 curve on the right side of the first W i (pi )−W o (po ) = 0 curve, meaning
that W i (pi ) − W o (po ) would be increasing in η for any η left of the first W i (pi ) − W o (po ) = 0
curve. However, this could not be the case, since W i (pi ) − W o (po ) is decreasing in η for large
enough η. Therefore, W i (pi ) > W o (po ) holds on the left side of the W i (pi ) − W o (po ) = 0 that
                     ¡        ¢
pass through (η, t) = α, 1−α .
                          2

   Finally, we show that there is no W i (pi ) − W o (po ) = 0 curve in the area of η ≤ α. Since
W i (pi ) − W o (po ) is quadratic in η, and W i (pi ) > W o (po ) when η = α, it suffices to show that
W i (pi ) > W o (po ) for the minimum possible η. From the assumption of Aη ≥ 1, for any given A,
the minimum possible η is 1/A. Substitute η = 1/A into the expression of W i (pi ) − W o (po ), we
have


           W i (pi ) − W o (po )
                                                  ³            ´2
                                                                    ¡        ¢2 µ             ¶
                                           ¡ i¢    pi − pi − t
                                                    h     f          1 − pi       1 + Aη 2 1
          i   i  i      i     i      i   o                                 h
       = ph Dh (p ) + pf Di (p ) + pf Df p + α                    +            −          +
                                                     2 (1 − α)           2           4      8
                      1               £     ¡                           ¢ 3
       =                  2            16α 1 − α − 2t + 4t2 + 2tα − t2 α A
         8A (3 + α + 4A) (1 − α)
            ¡                                                                             ¢
         + 16 + 8t2 α3 − 16tα3 + 32t2 + 48t2 α + 8α3 − 80t + 80tα + 8t2 α2 + 16tα2 − 24α A2
            ¡                                                                   ¢
         + 24tα + 40tα2 − 3 − 9α2 + 17α + 20t2 α + 8tα3 − 72t − 5α3 + 28t2 A
                                   ¤
         +6α − 18 + 10α2 + 2α3                                                                (12)


We want to show that the numerator, i.e., the bracketed term of expression above, is positive.
Notice that the minimum possible A is 1/α since 1 < Aη ≤ Aα. Substitute A = 1/α into the




                                                   25
numerator of the expression above, we have

                            ¡¡ 2                 ¢                        ¡           ¢¢
            (1 − α − 2t)      7α + 15α + 2α3 + 24 (1 − α − 2t) + 2 (1 − α) α2 + 3α + 4
                                                  α2

This is positive for 1 − α − 2t > 0. Next, we take the first, second, and third derivative of the
numerator of equation (12) with respect to A, and show that all of the derivatives evaluated at
A = 1/α are positive. The first derivative evaluated at A = 1/α is


             (64+4α3 +19α+9α2 )(1−α−2t)2 +2(1−α)(2α4 +4tα3 +5α3 +4α2 +10tα2 +13α+2tα+8)
                                                  α                                       >0


the second derivative evaluated at A = 1/α is

  ¡               ¢                         ¡                                       ¢
   112 + 4α3 + 4α2 (1 − α − 2t)2 + 4 (1 − α) α4 + 4tα3 + 12tα2 + 24α − 5α2 + 4 + 24t > 0,


and the third derivative evaluated at A = 1/α is

                                     ¡                             ¢
                                  96α 1 − α − 2t + 2tα − t2 α + 4t2 > 0


since 1 − α − 2t > 0. Therefore, the numerator of (12) is positive for any A > 1/α. Therefore,
W i (pi ) > W o (po ) at the minimum possible η. ¥



   Appendix C. Algebraic expression of pic .
   By solving the first-order conditions

                     ∂Πic         (1 − α) + (1 + α) pf + αt − 2ph
                              =                                    =0
                     ∂ph                       1−α
                     ∂Πic         (1 + α) ph − t − 2pf     (1 − β) η + βc − 2pf
                              =                        +A                       = 0,
                     ∂pf                (1 − α)                   (1 − β)

we obtain

                            4 ((1 − β) + A (1 − α)) − (2 (1 − β) − 4Aα) t + 2 (1 + α) Aˆ
                                                                                       η
               pic
                h     =
                                               2 ((1 − β) (3 + α) + 4A)
                            2 (1 + α) (1 − β) − 2 (2 + α) (1 − β) t + 4Aˆ
                                                                        η
               pic
                f     =
                                       2 ((1 − β) (3 + α) + 4A)

where η = (1 − β) η + βc.
      ˆ


                                                      26
   Appendix D. Proof of proposition 9.
                                 α−βc              α−βc
Proof. First, consider η ≥       (1−β) .   Since   (1−β)   > α, in this case both Πic (pic ) − Πc (pc ) > 0
                                                                                 α−βc
and Πi (pi ) − Πo (po ) > 0. Second, consider the case of α ≤ η <                (1−β) ,   and suppose that
Πic (pic ) − Πc (pc ) > 0. In this case, too, Πi (pi ) − Πo (po ) > 0. Finally, consider η < α and suppose
that Πic (pic ) − Πc (pc ) > 0. In this case, consider the following expressions:


                               −A(α+3)(1−α)(η2 −α2 )+4A(1−α)(1−2t+α−αt)(η−α)+(α+2t−1)2 (Aα2 +1)
     Πi (pi ) − Πo (po ) =                              4(1−α)(3+α+4A)
                               −A(α+3)(1−α)(η2 −α2 )+4A(1−α)(1−2t+α−αt)(ˆ−α)+(α+2t−1)2 (Aα2 +1−β )
                                            ˆ                            η
    Πic (pic ) − Πc (pc ) =                           4(1−α)((1−β)(3+α)+4A)



where η = (1 − β) η + βc. Subtracting the numerator of Πic (pic ) − Πc (pc ) from the numerator of
      ˆ
Πi (pi ) − Πo (po ), we get

                            ¡         ¢                                                      2
          −A (α + 3) (1 − α) η 2 − η 2 + 4A (1 − α) (1 − 2t + α − αt) (η − η ) + (α + 2t − 1) β
                                   ˆ                                       ˆ
                                                                                           2
      = (η − η ) A (1 − α) [4 (1 − 2t + α − αt) − (α + 3) (η + η )] + (α + 2t − 1) β
             ˆ                                                 ˆ


This is positive when the bracketed term is positive. So, now consider the bracketed term:


             [4 (1 − 2t + α − αt) − (α + 3) (η + η )]
                                                 ˆ

         = 4 (1 − η − 2t) + 4 (α − η) + 4β (η − c) − 4αt + η (1 − α) + η (1 − α)
                                                                       ˆ

         = 4 (1 − η − 2t) + (4 − 2 (1 − α)) (α − η) + (3 + α) β (η − c) + 2α (1 − α − 2t) > 0


Therefore, if Πic (pic ) − Πc (pc ) > 0, then Πi (pi ) − Πo (po ) > 0.



   Appendix E. Proof of proposition 11.
               c            c                                       o           o
Proof. Since CSf (pic ) > CSh (pc ) if and only if pic < pc , and CSf (pi ) > CSh (po ) if and only
                                                    f     f

if pi < po , what we need to show is that pic < pc implies pi < po . Note that the discussion of
    f    f                                 f     f          f    f

Lemma 1 is applied to the comparison of pc , pic , and pf . So, if pic < pc , then pf < pc . Combined
                                         f    f        ˜            f     f        ˜     f

with pc < po , this implies pf < po . Then, by Lemma 1, pi < po .
      f    f                ˜     f                      f    f




                                                      27
References

[1] Ahmadi, Raza and Yang, B. Rachel (2000), “Parallel Imports: Challenges from Unauthorized
   Distribution Channels,” Marketing Science, 19(3), 279-294.

[2] Anderson, Simon P. and Ginsburgh, Victor A. (1999), “International Pricing with Costly
   Consumer Arbitrage,” Review of International Economics, 7(1), 126-139.

[3] Cosac, Teodora (2003), “Vertical Restraints and Parallel Imports with Differentiated Products,
   ” mimeo.

[4] Knox, Daniel and Richardson, Martin (2002), “Trade Policy and Parallel Imports,” European
   Journal of Political Economy 19, 133-151.

[5] Malueg, David A. and Schwartz, Marius (1994), “Parallel Imports, Demand Dispersion, and
   International Price Discrimination,” Journal of International Economics 37, 167-195.

[6] Maskus, Keith E. and Chen, Yongmin (2000), “Vertical Price Control and Parallel Imports:
   Theory and Evidence,” mimeo.

[7] Richardson, Martin (2002), “An Elementary Proposition Concerning Parallel Imports,” Jour-
   nal of International Economics 56, 233-245.

[8] Shaked, Avner and Sutton, John (1982), “Relaxing Price Competition Through Product Dif-
   ferentiation,” Review of Economic Studies 49, 3-13.




                                               28
t
              1–η–2t = 0
                                   1−α 
                                  α ,   
                                      2 

                                                         1–α–2t = 0

    Π i (p i ) − Π o (p o ) = 0                    Wi(pi)–Wo(po) = 0


                                                         CSfo(pi)–CSfo(po) = 0




                           CShi(pi)–CSho(po) = 0
                                                                           η
                                     α                                 1

                                     Figure 1:




                                        29
                   The monopolist’s preference orderings
t
                 1–η–2t = 0
                                      1−α           α − β cc 1 − α 
                                     α ,            1− β , 2 
                                                                     
                                         2                         

                                                                        1–α–2t = 0
    Π i (p i ) − Π o (p o ) = 0
                                                                       Π ic (p ic ) − Π c (p c ) = 0
        o f i f c f ic

        o f c f i f ic            i f o f c f ic     i f o f ic f c
                                                     i f ic f o f c
     o f i , c f ic,      i f o, c f ic, 
     o f c, i f ic 
                                                              i f o, ic f c, 
    
                      
                           o f c, i f ic                    o f c, i f ic 
                                                                               
                                                                             
                                                                                          1     η


                                            Figure 2:




                                                30
                  The home consumers’ preference orderings
t
                1–η–2t = 0
                                 1−α       α − β cc 1 − α 
                                α ,       
                                             1− β , 2      
                                    2                     

                                                                  1–α–2t = 0
    CShi(pi)–CSho(po) = 0                                    CShi(pic)–CSho(pc) = 0




                                       ic f o ~ c f i
               ic f i f o ~ c                                   o f i , c f ic , 
                                                                                 
                                            o f i , ic f c,   o ~ c, ic f i 
     i f o, ic f c,                                                           
                                           o ~ c, ic f i  
     o ~ c, ic f i                                       
                                                             o ~ c f ic f i
                                                                                      1   η


                                    Figure 3:




                                       31
            The welfare orderings of the home country

                                                         o f i f ic f c
t                                                                          o f i, 
                                                         o f ic f c f i            
              1–η–2t = 0                                                   ic f c, 
                                                         o f ic f i f c   o f c 
                                                                                   
                                                          ic f o f i f c
                                                          ic f o f c f i
                                                                    1–α–2t = 0
    Wi(pi)–Wo(po) = 0
                                                                     Wic(pic)–Wc(pc) = 0
         i f o f ic f c

         i f ic f o f c                                                    o f i, 
                                i f o,                                           
                                                      o f i f c f ic     ic f c, 
         ic f c f i f o         ic f c,                                          
                               o f c                     o f c f i f ic  o f c 
         ic f i f o f c                 
                                                                  o f c f ic f i
                                                                                    1   η
               (i f o, c f ic, o f c ) i f o f c f ic

                                        Figure 4:




                                             32
                   The foreign consumers’ preference orderings
t
                 1–η–2t = 0
                                 1−α         α − βcc 1 − α 
                                α ,          1− β , 2 
                                                             
                                    2                      

                                                                  1–α–2t = 0
    CSfo(pi)–CSfo(po) = 0                                         CSfc(pic)–CSfc(pc) = 0
                                                                           i f o, ic f c, 
        c f ic f o f i                                                     c f o, ic f i 
                                                                                          
                                                                                          
           c f o f ic f i                                                  ic f c f i f o
                                                   i f o, c f ic, 
                                                  
                                                   c f o, ic f i        ic f i f c f o
     o f i , c f ic,                                            
     c f o, ic f i 
                     
                                                       c f ic f i f o
                                                                                    1     η


                                      Figure 5:




                                         33