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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 Â i Â c Â ic and o Â c Â i Â 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 Â c Â i Â 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 Â c Â i Â 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 Â i Â ic Â c or i Â o Â c Â 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 Â i Â ic Â 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 Â i Â c Â ic and the foreign welfare ordering of ic Â i Â c Â 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

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