Competitive strategies and quality to counter parallel

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Competitive strategies and quality to counter parallel importation in global marketR Hongfu Huang a, Yong He a,∗, Jing Chen b a b

School of Economics and Management, Southeast University, Nanjing 210096, China Rowe School of Business, Dalhousie University, Halifax, NS B3H 4R2, Canada

a r t i c l e

i n f o

Article history: Received 20 December 2017 Accepted 24 July 2018 Available online xxx Keywords: Parallel importation Global distribution strategy Quality design Game theory

a b s t r a c t When selling products globally in geographically separated markets with price discrimination, manufacturers often have to compete with parallel importers (PIs) who sell “gray products,” purchasing the manufacturers’ authorized products in low-price markets and reselling them in high-price markets. In this paper, we show that manufacturers can use ex ante quality design along with distribution strategies to mitigate the negative impacts of parallel importation. We develop game-theoretic models for a supply chain with a manufacturer and a PI, in which the manufacturer needs to determine distribution structure (either selling through high-price market exclusively, or selling through two channels), product quality, and retail prices. If a dual-channel structure is selected, the manufacturer also should decide the channel opening time in the low-price market, either early or late. Our results suggest that (1) the manufacturer should strategically choose a lower quality level when designing its product to weaken the PI’s competitiveness; (2) the manufacturer’s optimal distribution strategy is determined by the gap between customers’ willingness to pay (WTP) in the two markets and the customer’s tolerance for late consumption; and (3) parallel importation hurts the customer surplus (CS) and social welfare (SW) in the low-price market, and benefits CS and SW in the high-price market when the two markets are moderately balanced. We then consider several important extensions to provide additional insights. Firstly, we show that the appropriate marketing investment in the low market can effectively counter the PI. Secondly, we examine the impact of multiple competing PIs on the manufacturer’s decisions, including distribution strategy, quality design, prices, and profit. Lastly, we find that the advertising effect may motivate the manufacturer to improve product quality and benefit both the manufacturer and PI if the effect is relatively strong. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction Imagine that a Chinese customer is planning to purchase a Nikon camera (for example, the D750 model). A search of Amazon’s websites in Japan, China, and the UK, respectively shows that the selling prices of this model are $1465.7 (or Ұ165,564), $2168.7 (or ¥14,699), and $1933.3 (or £1498), respectively. The consumer might buy the camera at a lower price by looking for a gray marketer to. Likewise, customers in some countries may buy fashion clothes at lower prices through gray markets, as the selling prices in the markets they locate are higher. What the gray marketers do is ‘parallel importation,’ and a business that engages in parallel importation is a ‘parallel importer’ (PI). Parallel importation is the unauthorized import into a country of non-counterfeit goods imR ∗

This manuscript was processed by Associate Editor G. Lai. Corresponding author. E-mail address: [email protected] (Y. He).

ported without the permission of the intellectual property owner, and generally involves high-priced branded goods such as jewellery, cameras, tablets, and watches. A competitive global market always creates a gray market [23]. When a manufacturer sells authorized products in different regions or countries, prices and availability may vary, for reasons including differences in living standards in different regions, customers’ level of acceptance of gray products, and local taxes. PIs see the price differences between different markets or regions as an opportunity to make profits by diverting manufacturers’ authorized products from low-price markets to resell in markets with higher prices [13,55]. Over the past three decades, with improvements in information and logistics technologies, the business of parallel importation has grown very fast in many product categories, such as electronic products, watches, luxury goods, textbooks, and pharmaceuticals. As Adam [3] reported, in the luxury watch market in the US, gray product sellers offer up to a 50% discount as compared to authorized sellers, and this caused a

https://doi.org/10.1016/j.omega.2018.07.009 0305-0483/© 2018 Elsevier Ltd. All rights reserved.

Please cite this article as: H. Huang et al., Competitive strategies and quality to counter parallel importation in global market, Omega (2018), https://doi.org/10.1016/j.omega.2018.07.009

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significant decline in authorized watch dealers’ sales in 2015. Antia et al. [8] claim that about 70% of cellphones in Malaysia and twothirds of personal computers in India come from gray markets. In the UK’s pharmaceutical industry, nearly 20% of the total sales are gray [32]. In 2007, about 1 million iPhones were unlocked and sold by gray marketers [62]. Brand name firms, including Nikon, Omega S.A., Coach, and Apple, are all suffering from the encroachment of gray products into their high margin markets. Annual sales of gray products in the US accounted for about ten billion dollars in the 1980s [17]. In 2007 in the IT industry alone the total sales of unauthorized products reached 58 billion dollars [35], and the market size was still growing [36]. Field [27] suggests that companies on average may lose about 4.5% of their total profit due to parallel importation. Parallel importation affects not only the mix of products in the market and the intensity of market competition, but also the status of the named manufacturers’ profits and brands. It is a concern to manufacturers, distributors, customers, and policymakers for several reasons, including, of course, the profits of the original manufacturers. A report by the New Zealand Ministry of Economic Development [45] points out that the effects on customer price can be direct or indirect. ‘Gray goods’ can be sold at lower prices and gain some market share (direct effect), but the forces of competition will also tend to drive down the prices of other products in the same market (indirect effect). Peiravian [49], focusing on the pharmaceutical industry, suggests that the presence of PIs in the market directly reduces the price of branded medicines and can be a complement to price control strategies. However, PIs may discourage the investment of original manufacturers in R&D, as PIs have no costs in R&D and marketing [43], and reduce the profitability of original manufacturers. Although some countries have restrictions on PIs (for example, Copyright Act 1968 and the Trade Marks Act 1995 in Australia that are mentioned in Smith [55]), a report by the New Zealand Ministry of Economic Development [45] suggests that the benefits of allowing parallel imports outweigh the costs. Firms strive to curb parallel importation via with various marketing strategies, including strategic pricing, service differentiation, price cutting, and supply interference [16]. With supply interference/quantity strategy, firms can close low-price sales channels to cut off the supply of gray products [4]. In the pharmaceutical industry, for example, some manufacturers take the factor of parallel importation into account in making their market entry or exit decisions [1]. This strategy can prevent PIs from entering the market, although the manufacturers suffer the cost of losing their profit gains in the closed regions. With price strategies, when firms sell through multiple channels they balance prices to defend against the attacks of PIs [59]. Pricing strategy allows firms to mitigate the negative impacts of parallel importation without losing much of the market coverage. In addition to pricing, firms can also use service differentiation strategies to compete with potential PIs [30] by providing high quality service to customers. Previous studies on the impact of parallel importation on a firm’s distribution decisions have discussed the single channel strategy (Strategy S, in which the manufacturer only sell through the high price market) and the dual channel strategy (in which the manufacturer sells though both markets) (see [4,30]). However, the timing of selecting the dual-channel structure has not been discussed in the literature; these studies assume that the manufacturer sells simultaneously through the two channels (Strategy M) if the dual-channel distribution strategy is selected. In practice, it is observed that manufacturers sometimes to sell through two different regional channels but open the two markets sequentially

(Strategy Q).1 For example, the Apple Watch was released at different times in Asia and Europe. It is expected that with Strategy Q, the manufacturer can delay the gray market’s emergency, as those customers who might prefer gray products but are impatient will have to buy the manufacturer’s authorized products in the early period. Using the timing of opening the dual-channel to counter PIs is understudied in the literature. In addition, product quality design is a major issue in positioning the product in the market. However, none of studies has discussed how the manufacturer can employ product quality in different distribution strategy to effectively counter PIs. We are interested in exploring how firms can counter PI with distribution strategy (Strategy S, or Strategy M, or Strategy Q) and product quality design. Quality has been extensively studied in a broad range of contexts. Following the studies in Moorthy [44], Desai [24], and Chen et al. [20], we define “quality” as a single attribute or a combination of attributes that has the “more is better” characteristic. For instance, a higher quality computer means that the computer may have either a larger capacity hard disk, or a higher speed CPU, or a higher frequency video card, or a longer battery duration, than a “low-quality” computer. Research shows that distribution channel strategy may significantly affect firms’ quality design (See for example [20]). In this paper, we investigate the firm’s quality design strategies when competing with gray marketers. Convention holds that a higher quality can be a competitive advantage for a firm, but in the context of parallel importation, the dynamics around a firm’s quality decision become complex. When a firm increases product quality, its rivals’ (the PIs’) product quality also increases. Since PIs compete with a firm who sells its own authorized products, and divert and sell those same products, previous studies on product quality design cannot be extended to the context of parallel importation. To the best of our knowledge, ours is the first study to examine how a manufacturer should design product quality as a marketing strategy, and implement it along with pricing and distribution strategies to counter the negative impact of parallel importation. Focusing on the research points, we develop a model for a manufacturer who sells its products in either a high-price market exclusively, or two separate and asymmetric markets, facing competition from a PI. We first derive the equilibrium for the manufacturer’s three distribution strategies: Strategy S, Strategy M, and Strategy Q. From the derivation of the manufacturer’s optimal prices, product quality and distribution strategy, several results are obtained. First, the manufacturer’s distribution strategy decision depends strongly on the gap between customers’ willingness to pay (WTP) in the two markets. Specifically, when this gap is very high, the manufacturer should adopt Strategy S to completely eliminate PI behavior; when this gap is medium, the manufacturer should open the two markets sequentially to delay the entry time of the PI and compete with it only in the late period; otherwise, when customers’ WTP in the two markets are very close, the manufacturer should adopt Strategy M (the PI’s entry is blocked). Second, when customers are less tolerant of late consumption, Strategy Q is more effective to counter the PI. Third, product quality can be an effective tool in helping to enhance the manufacturer’s profit in the presence of parallel importation. We extend our discussion by showing that the appropriate marketing investment in quality design and distribution strategy can effectively counter the PI’s behavior. Interestingly, we find that when customers’ level of tolerance of late consumption is above a certain threshold, the PI can never enter the market if the manufacturer optimally sets its marketing investment. We also extend

1 This time-linked channel strategy is studied by Choi et al. [21] and Zhang and Wang [66] in the context of offline-online channels, to mitigate channel conflict.


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our discussion to examine the impact of multiple competing PIs on the manufacturer’s decisions on distribution strategy, quality design, and prices, as well as on the manufacturer’s profit. We find that when there are more PIs in the gray market, the manufacturer will be more likely to deter the PIs’ entry. At the same time, the manufacturer’s profit will decrease as the number of PIs increases in the gray market. We further show that, when the advertising effect is relatively strong, the manufacturer can enhance its profit by setting a higher quality level. The remainder of the paper is organized as follows. Section 2 discusses the related literature. Section 3 presents the model description. Section 4 derives the equilibrium solution to each of the three channel strategies. Section 5 analyzes the optimal channel selection strategies for the manufacturer. Section 6 examines the impact of the PI on the manufacturer’s optimal decisions of distribution strategy, product quality, and prices, and on the customer surplus. In Section 7, three extensions of the base model are presented. Lastly, conclusions and implications are summarized, and future research directions are proposed, in Section 8. 2. Literature review This paper is related to the impact of parallel importation and approaches to counter parallel importation. The impacts of parallel importation on manufacturers have been examined empirically and theoretically in the literature. Some empirical studies focus on impacts on firms’ decisions and performance (for example, [28,68]), and others focus on customers’ purchasing behaviors (for example, [41]). Ganslandt and Maskus [28] investigate PI’s impacts on firms’ pricing and quantity decisions in the pharmaceutical industry during the years 1994–1999. They find that the encroachment of PIs leads to a dramatic decrease in sales quantity for authorized products and this effect increases as more PIs enter the market. Liao and Hsieh [41] study the main factors that influence customers’ willingness to pay (WTP) for gray products in China’s smart phone market. By analyzing the 34-week-period data of gray marketers and transaction records on Taobao.com, Zhao et al. [68] find that the number of online gray market sellers and the transaction quantities are closely related to product availability. Analytical models also have been developed to further examine the optimal pricing and quantity decisions in the presence of parallel importation under different settings. Some studies focus on problems with third-party PIs. Ahmadi and Yang [6] address a game problem between a manufacturer and a PI when demands in each market are price-dependent. Their analytical results show that encroachment of a PI may help the manufacturer to achieve a larger market coverage, but only increases the manufacturer’s profit under certain circumstances. Ahmadi et al. [4] and Li et al. [37] also extend the model of Ahmadi and Yang [6]. The formal study considers demand uncertainty in both markets to examine the joint impacts on the decisions of the optimal pricing and quantity. Li et al. [37] investigate the manufacturer’s quantity competition with a local differentiated firm in the presence of parallel importation. By assuming that the two manufacturers can invest to expand the market size of an emerging market, Autrey et al. [10] find that the parallel importation benefit both manufacturers when the investment spillover effect and the base market size are relatively high. Kim and Park [34] study the impacts of demand uncertainty and manufacturer’s organizational structure on the optimal price decisions and profit, when the manufacturer faces the competition from a PI. They show that the high demand variance incentivizes the manufacturer to strategically decentralize the domestic and foreign sectors to gain more profit. Some studies have focused on the strategy of the manufacturer to counter PI. Chen [19] extends the work of Ahmadi and Yang

[6] to the setting that the manufacturer sells through intermediate retailers in both markets. He identifies several factors, such as the low gray product penetration ratio, the low-price elasticity of demand, the high cross-price elasticity of demand, and the high demand convexity, that can drive the manufacturer and the retailers to block the parallel importers. Autrey et al. [9] investigate a duopolistic Cournot competition problem for two manufacturers who produce differentiated products in the presence of parallel importation. They show that strategically decentralizing the manufacturer’s production and sales sectors is an effective strategy to defend against gray marketing. Iravani et al. [30] examine a onemanufacturer-one-PI problem with competition in both prices and services. They find that offering appropriate services can also efficiently defend against the PI’s encroachment. There have also been studies on the retailer-driven gray marketing problems. Xiao et al. [59] study a firm’s pricing decisions when parallel importation is carried out by a third-party firm or an authorized dealer. They found that supply chain structure significantly affects the firm’s profit and under some conditions, the firm can benefit from parallel importation. Shao et al. [52] investigate a manufacturer’s pricing strategy when it sells through a retailer, facing a local or an international gray market. It is shown that competition between the manufacturer and the retailer acts as a main driving force leading to the PI’s emergence. Zhang [65] develops a model with gray market behavior by the retailer and shows that the gray market hurts the manufacturer’s profit. He finds that offering appropriate sales rebate to customers in the high-price market can perfectly deter the gray marketing and achieve supply chain coordination. The above literature shows that there are limited studies on multinational firms’ quality design and sales channel strategies in the context of parallel importation. Thus, this paper aims to fill this gap and extend the existing literature on parallel importation by studying product quality design and channel selection simultaneously. This paper is also related to the literature on the impact of product quality on competition. Moorthy [44] and Banker et al. [11] study duopoly models with two manufacturers who make quality decisions separately and compete with each other. Both studies find that competition will motivate both firms to invest in quality improvement. Li and Chen [39] model a Stakelberg gaming problem with two manufacturers and a single retailer, and find that when product quality is endogenous, the retailer’s backward integration induces both manufacturers to increase product quality, which helps to soften the price competition. Jing [31] studies the impacts of behavior-based price discrimination (BPD) on the price and quality decisions of two competing firms. He finds that BPD does not affect the quality decisions of the low-end firm but it does force the high-end firm to improve the quality. Karaer and Erhun [33] suggest that firms can use quality enhancement as a competitive lever to deter potential entry. The above papers demonstrate that when different entities determine qualities for their own products, high quality helps to strengthen competitiveness. However, firms may also strategically reduce product quality. Ha et al. [29] study a supplier encroachment problem for a two-level distribution chain, considering the supplier’s product quality decisions. They find that the supplier’s encroachment may result in a lower product quality when the investment cost is relatively high. They assume that product quality is the same in the two distribution channels. When the supplier sells the product through its own channel and competes with the retailer, strategic quality reduction helps to reduce the retailer’s competitiveness. Örsdemir et al. [48] incorporate the quality decision into a model in which an original equipment manufacturer (OEM) competes with an independent remanufacturer (IR). They show that choosing a lower product quality helps the OEM to deemphasize the IR’s competitive advantage, and thus protect its own


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Fig. 1. The supply chain for multinational firms facing parallel importation.

profit. Pun and Deyong [50] study a manufacturer’s quality decision when it competes with a copycat and deals with strategic customers. Since the copycat will freeride the manufacturer’s brand image, its valuation is always tied to the manufacturer’s efforts on quality improvement. They show that in some conditions, the manufacturer will choose to decrease its quality level to defend against copycats. The above papers have discussed the interactions between the competition and product quality development. Only a few papers, however, have considered the issue of parallel importation. Therefore, in this paper, we aim to fill this gap in the literature by studying quality design problem of international firms when parallel importers act as competitors. This paper is also relevant to studies on channel selection under competition. Some studies focus on online-offline competition considering factors, such as value adding service [38], delivery lead time [64], asymmetry information [42], and customer behavior [46]. Other papers consider the interaction between the downstream firm’s outsourcing decisions and the upstream supplier’s own brand development (for example, [47,57]). The abovementioned studies focus on channel selection in a single market without parallel importation. A few other studies (such as, [9,10]) have considered the impact of parallel importation on the manufacturer’s time-linked channel strategy, but in a single period. Here, we study a multinational manufacturer’s channel selection in two separate markets when dealing with strategic customers. 3. Model setup We consider that a Stackelberg manufacturer may sell an authorized product either through two separate markets, a high market (Market H) and a low market (Market L), or through Market H exclusively. If it sells through two markets, following the common assumption (See [12,26,63]), we assume that the customers in one market cannot purchase product from the other market directly (for example, the two markets are in two different countries). For simplicity, we assume that the product sold in two markets is identical. The PI might divert product from Market L to Market H (as shown in Fig. 1). Following studies in the marketing literature (for example, [18,51]), we assume that customers are heterogeneous in their marginal WTP on the product in both markets (Vj ), and WTP is uniformly distributed in the interval [0, vj ], where j = H, L. Customers in Market H have an average higher WTP than those in Market L, which suggests vH ≥ vL . vj represents a proxy for the customers’ purchasing power in market j [6]. Without loss of generality, to simplify the analysis, we assume that vH = 1 and vL ≤ 1. In practice, since a manufacturer sets different selling prices for its two markets (pH , pL ) to tailor customers with differentiated WTP to its authorized products, a PI can potentially gain a profit by purchasing product in Market L and reselling them at a higher price (pG ) as gray products in Market H. PI cannibalizes the manufacturer’s market share. In this study, the manufacturer also should decide the quality level (β ) for its authorized product sell-

ing in both markets. As in studies of Shi et al. [53] and Ha et al. [29], the manufacturer incurs a quadratic investment cost kβ 2 to achieve a quality level of β . Following the study of Chen et al. [20], k is normalized to 1 in this study. Since the PI diverts authorized products in Market L to Market H (as gray products), the two products (authorized and gray products) are identical in quality. However, gray products may not be eligible for warranty coverage, technical support, and other services (see websites [7] and [25] such as www.usa.canon.com). Following previous studies (for example, [5]), we assume customers’ valuation on gray products with a discounted ratio θ ∈ (0, 1), relative to the authorized products. θ = 1 means that customers perceive no difference between the two products, and θ = 0 means that they will never buy gray products. Following the studies of Moorthy [44] and Desai [24], we model the customers’ gross valuation on the manufacturer’s and the PI’s products as β Vj and θ β VH , respectively. All notations are summarized in Table 1. 3.1. Manufacturer’s distribution strategy options and customer utilities Manufacturer can combat the PI through one of the three distribution strategies, single channel strategy by selling in Market H (Strategy S), dual channel simultaneous open strategy (Strategy M), and dual channel sequential open strategy (Strategy Q). For Strategy Q, we model the problem in two periods, an early period and a late period. To investigate the manufacturer’s distribution strategy effectiveness, we assume that all customers whose net expected utility buying gray products is higher than buying authorized products in Market H will buy gray products (if gray products are available), and the manufacturer sells through Market H only in the early period and sells through Market L exclusively in late period. The manufacturer’s distribution strategy Y = {S, M, Q } and decision tree for customers are illustrated in Fig. 2. Strategy S can perfectly eliminate the effects of PI at the expense of losing low-value customers in Market L. Buying in Market H, a customer has a net expected utility β VH − pH . In Strategy M, the manufacturer opens sales channels in both markets once the product is launched and the PI may immediately enter the market selling gray products to compete with the manufacturer’s authorized products in Market H. The net expected utilities of customers buying authorized products in Market j and buying gay products in Market H are β V j − p j and θ β VH − pG , respectively, where j = H, L. In Strategy Q, the manufacturer first opens the channel by selling in Market H at the time of product launch, and then opens the Market L channel in the late period. Customers buying authorized products in Market H have the net expected utility β VH − pH . The gray market emerges only when the manufacturer sells its products in Market L in the late period. We assume that the customers are strategic. Since the second period price for gray products is unknown in the first period, customers in Market H will anticipate the price ( peG ) of gray products to make their strategic decisions on either buying an authorized product in the early pe-


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Table 1 Notations. Indexes X Y j Decision variables pYj , pYG

βY Parameters Vj

θ γ ρ

N

λ

Other notation qYj , qYG

Description Subscript, index of manufacturer and PI respectively, where X = {m, g}. Superscript, index of manufacturer’s distribution strategy: Strategy S (single channel), Strategy M (simultaneous dual-channel), Strategy Q (sequential dual-channel), respectively, where Y = {S, M, Q }. Index of market, where j = {H, L}. Selling prices of authorized products in Market j and of gray products in Market H. Quality of the authorized product. Marginal WTP of Market j’s customer for products, uniformly distributed in the interval [0, vj ]. Discount rate on customers’ WTP of gray products in Market H, where 0 ≤ θ ≤ 1. Customer’s tolerance level of late consumption, where 0 ≤ γ ≤ 1. Manufacturer’s unit marketing cost in Market L. Number of PIs in the gray market. Level of advertising effect. Sales quantity of authorized products in Market j and of gray products in Market H.

πXY

Profit of member X in the market.

, C SNj C SPI j

Customer’s surplus in Market j with and without PI, respectively.

Fig. 2. The manufacturer’s distribution strategy Y = {S, M, Q } and decision tree for customers.

riod or waiting for the gray product in the late period. The utility of customers who buy products (either authorized or gray) in the late period will be discounted to a proportion γ (0 < γ < 1) as they can only experience the product with delay. γ reflects an aggregate level of discount on the perceived value of product that customers can experience in the later period. We refer to γ , for simplicity, as the level of tolerance of late consumption. There are two extreme cases: when γ → 1, the customers perceive that the product bought in the late period is no different from the product bought in early period; and when γ → 0, customers perceive no value for products purchased in the late period and they therefore will not buy a product in the late period. Thus, in the late period, customers’ net expected utility of buying authorized products in Market L and buying gray products in Market H are γ (β VL − pL ) and γ (θ β VH − peG ), respectively. 3.2. The manufacturer and customer decisions To focus on our main research issues and for simplicity, we normalize the manufacturer’s production cost and selling cost, and the PI’s transportation cost to zero, as these costs can be easily included and do not affect the main results in our paper. In addition, the sales quantity of gray products cannot exceed the total sales quantity of authorized products in Market L, i.e., qL ≥ qG . Following the assumption in Ahmadi et al. [4], the PI can buy prod-

ucts before normal customers and the manufacturer cannot distinguish between purchases by the PI and normal customers. This is because the PI can hire agents to promptly purchase authorized products, and it is very difficult and costly for most sellers to distinguish these buyers [8]. The manufacturer’s profit is as follows:

πmY = pYH qYH + m − (β Y )2 ,

(1)

0Y =S pYL qYL Y = M, Q The PI’s profit is:

where m =

πgY = g ,

(2)

s.t.

qG ≤ qL where g =

pYG

pYL

0Y =S

− qYG Y = M, Q We study the manufacturer’s decision on pricing, product quality, and channel strategy in the presence of PI. Since the manufacturer’s decisions on channel strategy and quality level have longer term implications than the pricing decisions, following studies on channel selection (for example, [22]) and on quality level design (for example, [60]), these decisions are made before prices decision. In this paper, we focus on “quality” design for the manufacturer’s single attribute or a combination of attributes that


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has the “more is better” characteristic. As compared to channel strategy decision, it is a relatively short-term decision. Under the manufacturer’s strategy Y = {S, M, Q }, a customer makes a purchasing decision to maximize utility. The game proceeds in three or four steps, depending on the manufacturer’s channel strategy. Stage 1: As a Stackelberg leader, the manufacturer determines its distribution strategy (either S, or M, or Q). Stage 2: The manufacturer determines its product quality β Y . Stage 3: The manufacturer announces the selling prices ( pYH , pYL ) in both channels simultaneously. Stage 4: The PI sets the selling price of gray products ( pYG ) in Market H. Stage 5: Each customer decides whether to buy authorized products or gray products. 4. Optimal decisions on prices and product quality In this section, we study the decisions on retail prices and product quality for the cases in which the manufacturer sells its product through Market H only (Strategy S) or through a dual-channel structure (Strategy M or Strategy Q). 4.1. Single channel With Strategy S, as the manufacturer only sells through Market, customers in Market H with non-negative net expected utilities (presented in Section 3) on authorized products will buy; otherwise, they will not. Therefore, the demand is qSH = 1 − pSH /β S . With the manufacturer’s function in (1), for Y = S, we can obtain its optimal decisions on price and product quality, as well as demand and profit: S∗ pS∗ H = 1/16, qH = 1/2,

β S∗ = 1/8, and πmS∗ = 1/64

(3)

4.2. Dual-channel We first discuss the case when manufacturer sell its product through both markets simultaneously after product launch. The manufacturer may face the competition from gray products sold by the PI (Strategy M). 4.2.1. Strategy M With the customers’ net expected utility (presented in pM

L Section 3), the demand in Market L is qM L = 1 − vL β M . For customers in Market H, given product quality and prices for the authorized and gray products, customers will buy either an authorized or a gray product, depending on which yields a higher net expected utility. With the utility functions presented in Section 3, the demands for the authorized and gray products in Market H are

pM −pM

pM −pM

pM

H G H G qM = 1 − ( 1− and qM = ( 1− − θ βGM respectively. Then only H G θ )β M θ )β M M M if pG ≤ θ pH , the demand of the PI is positive. Obviously, if the manufacturer sets a low price for authorized products such that M pM G ≥ θ pH , the demand of gray products drops to zero and the demand for authorized products becomes qM = 1 − pM /β M . Only H H when the PI gains a positive profit margin and has a positive demand will it be motivated to divert Market L products to Market H and sell them as gray products. With qM ≥ qM and profits of manL G ufacturer and PI in (1) and (2), we have the following results.

Lemma 1. There exists unique optimal quality (β M ∗ ) and retail prices M M∗ ( pM H , pL ) for the manufacturer, and optimal retail price for PI ( pG ), which are summarized in Table B1 in Appendix B. )(1−θ ) 1−θ ) Defining vM1 (θ ) = 2θ (2θ5−1 and vM2 (θ ) = 4θ5(−3 L L −4θ θ , where

vM2 (θ ) > vM1 (θ ), with Lemma 1, the manufacturer’s optimal deL L

cision for Strategy M (illustrated in Fig. 3(a)) is summarized in Proposition 1.

Proposition 1. With the manufacturer’s Strategy M, a) the PI diverts all authorized products sold in Market L to Market H (Region I); b) the PI diverts a portion of authorized products sold in Market L to Market H (Region II); c) the manufacturer deters the entry of the PI (Region III); and d) the manufacturer ignores the existence of PI (Region IV). Proposition 1 implies that when the customer’s maximum valuation in Market L (vL ) is very low while the perceived value on gray products is high (Region I, 0 ≤ vL ≤ vM1 L (θ ) ), the PI enters the market and is motivated to buy and divert all the products sold in Market L to Market H. The intuition is that when vL is very low, the manufacturer has to set a low retail price in Market L, leading to a big retail price gap between the two markets for the authorized product (refer to extremely unbalanced markets). Knowing about the big price gap, strategic customers in Market H are more willing to buy gray products at a low price. Buying and diverting as many products as possible makes the PI more profit. The total number of gray products, however, is constrained by the sales of authorized products in Market L. The PI diverts all authorized products in Market L’s to Market H. As vL increases but is still low (Region II, Max(vM1 (θ ), 0) ≤ vL ≤ vM2 ( θ ) ), L L the price gap between authorized products in the two markets decreases, but the price of gray products is still attractive to those strategic customers in Market H who seek the low price. The PI is still competitive in the market and the manufacturer must compete with it. The total demand of gray products drops, and the PI only has the incentive to divert a portion of authorized products from Market L to Market H. As vL increases to the moderate range (Region III, vM2 (θ ) ≤ vL ≤ θ ), the manufacturer drives the PI out L of Market H by reducing the price in Market H while raising the price in Market L such that the PI cannot be profitable. Although there are no sales for gray products, the manufacturer loses the profit due to the threat of the PI. As vL continues to increase to a large range (Region IV, θ ≤ vL ≤ 1), the PI has no incentive to enter the market even if the manufacturer does not block it. It is interesting that there is 50% of chance that the manufacturer can ignore the existence of PI. The implication is that when the manufacturer is aware that (θ ≤ vL ), it will not worry about the threat of PI. Proposition 1 suggests that the PI can enter Market H and gain positive profit (vL ≤ vM2 (θ )) only when vL is under a certain L threshold. Notice that when θ approaches 1, both Regions I and II will converge to zero (see Fig. 3(a)). The intuition is that under Strategy M, when θ is very high, the customer perceives almost no difference between the manufacturer’s authorized products and gray products, which results in a perfect competition between the manufacturer and the PI in Market H. The fierce competition will significantly reduce the manufacturer’s profit. To mitigate the negative impact of PI on its profit, the manufacturer behaves more aggressively in deterring the PI’s entry. 4.2.2. Strategy Q With Strategy Q, the manufacturer delays selling the product in Market L, which changes the customers’ purchasing behavior. The pQ

demand in Market L is qQ = 1 − v βL Q . Customers in Market H will L L Q find the selling price ( pH ) in the early period and anticipate the selling price of gray products ( peG ) in the late period, and those pQ −γ pe

G H with high WTP (VH ≥ (1− ) will buy authorized products in γ θ )β Q

pQ

pQ −γ pe

G H the early period. Those with moderate WTP ( θ βGQ ≤ VH ≤ (1− ) γ θ )β Q will wait strategically and purchase gray products in the late pe-

pQ

riod. Those with low WTP (VH ≤ θ βGQ ) will buy nothing. Demands are

pQ −γ pe

G H qQ = 1 − ( 1− H γ θ )β Q

and

pQ −γ pe

pQ

G H qQ = ( 1− − θ βGQ , G γ θ )β Q

respectively.


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Fig. 3. the manufacturer’s optimal decision for (a) Strategy M (b) Strategy Q.

Following the studies of Cachon et al. [14], Cachon and Swinney [15], and Su and Zhang [56], we will derive the Rational Expectation equilibrium for the parallel importer ( pQ∗ , qQ∗ ) by defining: G G Definition 1. A Rational Expectation equilibrium, ( pQ∗ , qQ∗ ), in the G G game between the parallel importer and strategic customers satisfies: a) the customers play a best response given beliefs about the parallel pQ −γ pe

pQ

importer’s behavior: qQ∗ = qQ ( peG ) = (1−H γ θ )βGQ − θ βGQ ; b) the parallel G G importer plays a best response given beliefs about the customer’s behavior: pQ∗ ∈ arg max pQ ≥0 πg ( pQG , peG ); c) beliefs are consistent with G G

the equilibrium outcome: pQ∗ = peG and qQ∗ = qQ | . G pe = pQ∗ G G G

G

With PI’s profit in (2), we can derive the parallel importer’s optimal price as pQ∗ = G

Q e 1 θ ( pH −γ pG ) 2 1−γ θ

+ pQL . With Definition 1, the Ra-

tional Expectations equilibrium is then obtained by equating pQ∗ = G peG , which gives pQ∗ = G θ pQ −pQ

θ pQH +(1−γ θ ) pQL . The total sales of gray prod2−γ θ

H L ucts are qQ∗ = θ ( 2− G γ θ )β . Comparing Strategy M to Strategy Q, for Q Q M any pH = pH and pL = pM , we always have: L

Lemma 2. For Y = {M, Q } and any pQH = pM and pQL = pM , if pYH ≥ H L Q Q M pYL /θ , we have qM G ≥ qG and ∂ (qG − qG )/∂ γ ≤ 0.

As compared to Strategy M, Lemma 2 suggests that Strategy Q leads to fewer customers purchasing gray products if the PI enters the market (when pYH ≥ pYL /θ ). In addition, as customers’ level of tolerance of late consumption (γ ) decreases, the difference in sales of gray products between the two strategies becomes large, implying that Strategy Q is more efficient in mitigating the PI’s competitiveness. With qQ ≥ qQ and profits of the manufacturer and the PI in L G (1) and (2), we have the following result. Lemma 3. There exists unique optimal quality (β Q ∗ ) and prices ( pQ∗ , H

pQ∗ ) for the manufacturer, and optimal price for the PI ( pQ∗ ), which L G are summarized in Table B2 in Appendix B. We define vQ (θ , γ ) = L

2θ (2−θ γ ) . 4+(1−θ )γ

With Lemma 3, we summarize

the manufacturer’s optimal decision for Strategy Q in Proposition 2, as illustrated in Fig. 3(b) (for γ = 0.5). Proposition 2. With the manufacturer’s Strategy Q,

a) the PI diverts authorized products sold in Market L to Market H (Region I); b) the manufacturer deters the entry of the PI (Region II); and c) the manufacturer ignores the existence of PI (Region III). Proposition 2 shows that when the maximum customers’ WTP in Market L (vL ) is relatively low (0 ≤ vL ≤ vQ (θ , γ )), the PI enters L Market H and competes with the manufacturer (Region I). When vL has a moderate value (vQ (θ , γ ) ≤ vL ≤ θ ), the two markets are L more balanced, and the manufacturer can prevent the entry of the PI with the proper pricing policy (Region II). When vL is sufficiently high (θ ≤ vL ≤ 1), the PI has no chance to enter Market H, and the manufacturer can ignore the existence of the PI (Region III). As compared to Strategy M, we can obtain the following results for Strategy Q: (1) the ‘all-diversion’ scenario never happens; (2) the manufacturer will not worry about the threat of PI for both strategies if vL > θ ; (3) the PI can still survive to be profitable in the market (for 0 ≤ vL ≤ vQ (θ , γ )) even if θ = 1. For StratL egy M, when θ = 1, competition between the manufacturer and the PI is perfect such that the manufacturer will always prevent the entry of the PI into the market. However, for Strategy Q, customers’ utilities are affected by two factors: θ and γ , which suggests that the gross value of gray products in the late period is lower than that of authorized products in the early period. Thus, the PI can survive due to the less intensive competition with the manufacturer; (4) the deterrence region (Region II, in Fig. 3(b)) is smaller than that in Strategy M (Region III, in Fig. 3(a)). In region vM2 (θ ) ≤ vL ≤ vQL (θ , γ ), as compared to Strategy M, the manufacL turer’s pricing strategy switches from deterrence to competition in Strategy Q. That is, the PI is less competitive entering the market in the late period, due to lower customer’s utility on gray products. The PI must set a lower selling price for gray products if it enters the market. In order to deter the entry of the PI, the manufacturer must also set a lower selling price in Market H, which leads to lower sales margin and lower profit. When the manufacturer competes with the PI, however, it can still maintain a relatively high selling price with a low cannibalization effect. Consequently, the manufacturer prefers to compete rather than deter the PI. 4.3. Product quality change under Strategy M, Q and S We now analyze the quality decisions of the manufacturer 4θ γ (2−θ γ ) under three strategies. By defining vSL (θ , γ ) = 16+8(1−θ )γ +(1−4θ )γ 2 and comparing the manufacturer’s optimal quality under three


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Fig. 5. The manufacturer’s optimal distribution strategy for a given γ . Fig. 4. Comparison of optimal quality under the three strategies.

channel strategies (In Eq. (3), Table B1, and Table B2 in Appendix B), we obtain the results and summarize in Lemma 4. Lemma 4. The manufacturer provides the highest quality level: a) under Strategy S if and only if 0 ≤ vL ≤ vSL (θ , γ );

b) under Strategy Q if and only if vSL (θ , γ ) ≤ vL ≤ vQ ( θ , γ ); L c) under Strategy M or Strategy Q if and only if

vQL (θ , γ

) ≤ vL ≤ 1.

Lemma 4 and Fig. 4 show that when the gap between the customers’ WTP in the two markets is very large (vL is small), the manufacturer will set the highest product quality if it sells through Market H exclusively. The intuition is that selling the product at high quality allows the manufacturer to charge a high selling price in its exclusive market H without competing with the PI. When the gap shrinks (vL becomes moderate), Strategy Q leads to the highest product quality. The reason is that the manufacturer can be more profitable in a dual-channel structure when vL is moderate. Setting a high product quality allows the manufacturer to set a higher selling price in Market L for its authorized product, resulting in a higher cost for the PI’s gray products; as a result, the PI will raise its selling price for gray products. More impatient customers see less value in waiting for gray products in the late period (as the manufacturer delays the sale in Market L), due to the higher selling price of gray products. They buy authorized product in market H. This implies that product quality can be a useful mechanism to counter the PI, if the manufacturer can control the timing of selling products in Market L. When the gap is much smaller (vL is large), Strategies M and Q result in the same quality level, which is higher than that of Strategy S. This suggests that with a small gap between the two markets, the PI cannot gain a high profit by diverting the manufacturer’s products from Market L to Market H. The threat of the PI becomes weak. The dual-channel strategy, either Strategy M or Strategy Q, can prevent the PI from entering the market (Propositions 1 and 2). 5. The manufacturer’s optimal sales strategy We now discuss the manufacturer’s optimal sales strategy. Comparing the optimal profits under three strategies (In Eq. (3), Tables B1 and B2 in Appendix B), the manufacturer’s optimal distribution strategy can be derived and summarized in Proposition 3 (illustrated in Fig. 5). Proposition 3. The manufacturer’s optimal distribution strategy is: a) Strategy S (Region S) if and only if 0 ≤ vL ≤ vSL (θ , γ );

b) Strategy Q (Region QC) if and only if vSL (θ , γ ) ≤ vL ≤ vQ ( θ , γ ); L the manufacturer competes with the PI; c) Strategy M if and only if vQ (θ , γ ) ≤ vL ≤ 1; the manufacL turer deters (Region MD, vQ (θ , γ ) ≤ vL ≤ θ )/ignores (Region MI, L θ < vL ≤ 1) the entry of the PI.

Proposition 3 presents the manufacturer’s optimal distribution strategy in the presence of PI. The decision zone is shown in Fig. 5. Firstly, the manufacturer will choose Strategy S when the customers’ WTP on the authorized products in Market L is very low (0 < vL ≤ vSL (θ , γ )). The intuition is that the manufacturer will give up Market L and only sell in Market H to avoid competition from the PI, as it cannot set a high price in Market L and the loss due to the PI’s negative effect cannot be offset by gain in Market L. Ahmadi et al. [4] have the same observation and pointed out that the ‘Block using Quantity’ strategy (equivalently to Strategy S in this paper) is optimal for multinational firms when customers’ WTP for gray products is high and the prices in the two markets are very unbalanced. The strategy is adopted by firms in the pharmaceutical industry, who aim to mitigate the threat of parallel importation in order to protect their profitability in a global environment [1]. Secondly, when vL is at a moderate level (vSL (θ , γ ) ≤ vL ≤ vQ (θ , γ )), L the manufacturer is more profitable with Strategy Q if the timing of selling the product through Market L is delayed. As compared to Strategy M, the manufacturer can achieve higher market share and profit in the early period in Market H because there is no competition between authorized and gray products. The result shows that, for some multinational firms who are facing moderately balanced markets, the sequential dual-channel strategy (Strategy QC) can serve as a useful tool to mitigate the threat of parallel importation activities. In practice, it is common for companies to release their new version of products in different countries at different times. For example, the Apple Watch’s releasing time in China and Europe is different. Thirdly, when vL is high (vQ (θ , γ ) ≤ vL ≤ θ ), with L Strategy M, and a small gap in customers’ WTP in the two markets, the PI has no chance to enter the market. Notice that in this region, the equilibriums of prices, demands and profits are identical for Strategies M and Q, and this suggests that the two strategies are equivalent, from the perspective of the manufacturer. Although selecting Strategy Q does not affect the manufacturer’s profit, the customers’ utility is significantly decreased in Market L. Therefore, the manufacturer should select Strategy M in this region in favor of the customers. The result implies that when the customers’ WTP in different markets is very close, multinational firms can deter the entry of PIs by adjusting prices in the two markets only. Similar strategies have been discussed in previous research, such as


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customers in Market H choose to wait for and buy gray products in the late period if γ is high under Strategy Q since the cannibalization effect is stronger. Therefore, to protect its market share, the manufacturer will more aggressively work to deter the PI’s entry. 6. The impact of PI 6.1. Without PI To examine the impact of PI, we consider a benchmark case where PI’s behavior is not allowed (denoted with a superscript N). Since the two markets are separated, the manufacturer will always choose to sell though both markets simultaneously. The optimal prices and quantities for the two markets, and product quality and the manufacturer’s profit are:

pNH = Fig. 6. The manufacturer’s strategy switches with θ and vL if Strategy Q is an option.

Ahmadi et al. [4] and Iravani et al. [30]. Lastly, when vL is very high (θ ≤ vL ≤ 1), the two markets are such balanced that the PIs can never benefit from the price gap between the two markets. Consequently, the manufacturers can decide the optimal prices and quality decisions without considering the issue of parallel importation. Proposition 3 also suggests that the PI can enter the market to be profitable only when the customers’ WTP in the two markets are moderately balanced; otherwise, the manufacturer will always prevent PI from entering the market. The implication of Proposition 3 is that Strategy Q is more effective for the manufacturer in competing with the PI who sells gray products. In addition, the size of region (QC) increases with θ , suggesting that the higher the customers’ acceptance for gray products, the easier it is for the PI to enter the market and the more difficult it is for the manufacturer to deter the PI. Comparing the profits under Strategy M (in Table B1) and Strategy S (in Eq. (3)), we can obtain that, when vL ≥ vˆ L (θ ), Strategy M dominates Strategy S, and vice versa, where vˆ L (θ ) = 8θ (1−θ ) , θ ≤ 0.75 8θ 2 −32θ +25 . We further show the impacts of Strategy θ , θ ≥ 0.75 θ +2 Q on the change of channel strategy in Corollary 1. Corollary 1. The manufacturer’s equilibrium strategy switches from a) S to QC if vSL (θ , γ ) ≤ vL ≤ vˆ L (θ ); M2 b) MC (when vL ≤ vM2 L (θ )) or MD (when vL ≥ vL (θ )) to QC if

vˆ L (θ ) ≤ vL ≤ vQL (θ , γ ).

Fig. 6 numerically illustrates the switch of the manufacturer’s strategy if Strategy Q is an option. In the shadow area, Strategy QC dominates the other two strategies. Specifically, when vL and θ are located above a threshold line (vL ≥ vˆ L (θ )), the manufacturer’s optimal strategy switches from MD (or MC) to QC; however, when vL and θ are located under the threshold line (vL ≤ vˆ L (θ )), the optimal strategy switches from S to QC. Lemma 5 shows the impact of the customer’s tolerance of late consumption of the product on the manufacturer’s selection of distribution strategy. Lemma 5. The size of region of QC decreases, while the sizes of regions of MD and S increase with γ . Lemma 5 indicates that, when the customer’s level of tolerance of late purchase (γ ) is high, the manufacturer acts more aggressively to deter the PI from entering the market. It becomes more difficult for the PI to enter the market. The intuition is that more

1 + vL (1 + vL )vL N 1 N 1 , pNL = , qH = , qL = , 16 16 2 2 2 1 + v ( ) L and πmN = . 64

βN =

1 + vL , 8 (4)

Eq. (4) suggests that the manufacturer should set prices and product quality for the two markets such that it can have the same sales in the two markets. 6.2. The impact of PI on the manufacturer’s optimal decisions Since the PI has no influences on the manufacturer’s optimal decisions for vL > θ Proposition 3), our discussion in this section focuses on the scenario when vL ≤ θ . With Eqs. (3) and ((4), and Tables B1 and B2 in Appendix B, the impacts of PI on the manufacturer’s profit, optimal product quality, prices and demands are summarized in Proposition 4 and illustrated in Fig. 7 (the superscript PI denotes the optimal decisions with the PI). Proposition 4. For vL ≤ θ , in the presence of the PI, the manufacturer a) reduces selling price in Market H and raises selling price in Market L; b) reduces product quality; c) sells more in Market H and sells less in Market L, resulting in a decrease in total sales; d) incurs a profit loss. Proposition 4 shows that the presence of the PI results in a reduction in the selling price in Market H and an increase in the selling price in Market L (see Fig. 7(a)). In the meantime, the manufacturer should choose a lower product quality (See Fig. 7(b)). The intuition is that a larger price gap between the two markets gives the PI the opportunity to gain more profit. To discourage the PI from diverting authorized products to Market H as gray products, the manufacturer should reduce the price gap between the two markets by raising the price in Market L and cutting the price in Market H (the price-balancing strategy). The increased selling price in Market L increases the PI’s purchasing cost, while the decreased selling price of authorized products in Market H forces the PI to lower the selling price of gray products, and this weakens the PI’s competitiveness. The manufacturer’s price-balancing strategy leads to increased demand in Market H and decreased demand in Market L. Although the PI’s competitiveness is weakened by the manufacturer’s price-balancing strategy, the manufacturer’s total demand in the two markets drops. The decreasing sales in Market L cannot be offset by the increasing sales in Market H (See Fig. 7(c)). The decrease in the total sales quantity discourages the manufacturer’s incentives for quality investment and pushes the manufacturer to set a lower product quality to attract more low-value customers. This result also provides practical managerial insights to firms, in that, when the rival’s product quality is tied to their own product


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Fig. 7. The impacts of the PI on the manufacturer’s prices, product quality, sales, and profit (for γ = θ = 0.5).

quality, setting a lower quality level may be profitable under certain conditions. For example, when a laptop manufacturer knows that the PI will divert its new style laptops from its low price market, it is a feasible and reasonable management strategy to use normal performance components (such as the Intel Core i5-6300HQ) instead of high performance components (such as the Intel Core i7-7700HQ) to lower quality. Our result is also consistent with the practice in computer and automobile industries. In 2007, Lenovo noticed the evidence that many laptops from gray markets were diverted from New Zealand to China with lower price (as compared to the price of authorized products), which badly hurts its benefit in China. Lenovo thus reduced the quality of some products, such as ‘ThinkPad X60’ in response to the competition with gray market products. Lowering product quality results in a low selling price and the attraction of more customers to buy Lenovo’s authorized products [69]. In addition, quality reduction to deal with parallel importation issue is also witnessed in the Chinese automobile industry [40]. The evidence further confirms our finding that reducing the product quality can be a useful tool to combat parallel importation. The impacts of the PI on the selling prices, quality, and demands all contribute to loss of profit for the manufacturer (See Fig. 7(d)). Fig. 7 also shows the impact of the PI as vL increases; the prices in both markets, product quality, and total profit increase. In

the absence of PI, the demand in both markets remains constant (=1/2). However, the PI results in an increasing demand first and then a decrease to 1/2 in Market H. Fig. 7(c) demonstrates that the gap of total sales due to the impact of PI shrinks when the two markets are more balanced. In addition, the impact of PI on the manufacturer’s profit is diluted as vL increases. The impact of the PI on the manufacturer’s profit with changes of customer’s level of tolerance of late consumption (γ ) and maximum value in Market L (vL ) is illustrated in Fig. 8(a). We define PI and its profit change rate the manufacturer’s profit with PI as πm PI − π N )/π N × 100%, respectively. Interestingly, we find that as (πm m m when customers are more tolerant of late consumption (γ = 0.9), the PI’s detrimental effect on the profit is more severe, reaching 25%. When customers are less tolerant of late consumption (γ = 0.1), however, the profit loss is very small, as low as 5%. The reason is that when customers are more tolerant of late consumption, they are less willing to buy and experience the products in the late period. Therefore, Strategy Q becomes a very efficient strategy to counter the impact of the PI. The impact of parameters (vL , γ , θ ) on the parallel importer’s profit is illustrated in Fig. 8(b). It conforms the intuition that the parallel importer’s profit always increases in θ (the PI can charge a high price when the customer’s perceived value on gray products is high), while it decreases in vL (the PI cannot divert as many


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Fig. 8. The manufacturer’s and the parallel importer’s profit change.

for a low vL , then drops gradually until to zero as vL further increases. Moreover, Fig. 9 also illustrates that, when customers become more tolerant for the late consumption (higher γ ) or more willing to buy gray products (higher θ ), endogenous-quality design also serves as an attractive tool for the manufacturer to combat parallel importation and gain more profit.

6.3. The impact of PI on customer surplus and social welfare

Fig. 9. Profit incremental rate with changes of (vL , γ , θ ) due to endogenous quality.

of the manufacturer’s authorized products from Market L to Market H as gray products as it desires) when it can enter the market successfully. In addition, the parallel importer’s profit declines with γ . This is because, when customers become more tolerant of the delay in purchasing, their perceived valuations on purchasing gray products will increase, leading to a more intense competition between the manufacturer and the parallel importer. The manufacturer becomes more aggressive to fight the PI; consequently, the profits of both the manufacturer and the retailer decrease. We now compare the manufacturer’s equilibrium profits between the cases when the quality is endogenous and when the quality is exogenous. We can show that the quality level is β N = (1 + vL )/8 when the quality is exogenous, which is the same as FQ the quality level in the absence of PI. We denote πm as the manufacturer’s profit with PI in the exogenous-quality model and PI −π F Q ) (πm m × 100% as the profit incremental rate when quality is πmF Q

endogenous. As shown in Fig. 9, when the two markets are relatively unbalanced (vL is relatively small), if the quality is endogenous, the manufacturer enjoys a higher profit, as compared to the exogenous-quality case. Specifically, the incremental rate increases

In this subsection, we investigate the impacts of PI on customer surplus (CS) and social welfare (SW). Without PI, CSN H = (1 + vL )/48 and CSN L = vL (1 + vL )/48. Then, in the presence of PI, the CS in each market is presented in Table B4 in Appendix B. The derivation is presented in Appendix A. SW in each market can be obtained by N adding CS to firms’ profits in that market, i.e., SWN H = CSH + πg + N pH qH − β 2 and SWN = CS + p q . The impacts of PI on CS and SW L L L L (when γ = 0.5 and θ = 0.5) are illustrated in Fig. 10(a) and (b), respectively. Fig. 10(a) demonstrates that the presence of PI hurts CS in both markets in Regions S. In this region, a very small vL pushes the manufacturer to leave Market L and sell authorized products only in Market H. Since customers in Market L have no access to the manufacturer’s products sold in Market H, their surplus remains zero, which is strictly lower than that without parallel importation. Interestingly, CS in Market H also declines due to the presence of parallel importation. This because when vL is sufficiently small, the manufacturer has to abandon Market L to avoid fierce competition with the PI; the total demand decreases from 1 (without parallel importation) to 1/2 (with parallel importation) (see Fig. 7(c)). Therefore, the manufacturer will be less willing to invest in product quality improvement. Although the sales quantity in Market H is not affected by parallel importation, CS declines as the quality of the products decreases. Fig. 10(a) also illustrates that parallel importation will benefit customers in Market H, while hurting those in Market L in Regions QC and MD. In these two regions, to combat the PI, the manufacturer will set a lower product quality and a relatively higher price in Market L, leading to a decrease in the total sales; consequently, CS decreases in Market L. Conversely, the manufacturer will lower the selling price in Market H to combat parallel importation. Therefore, CS increases since the decrease in the price can


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Fig. 10. The impact of PI on customer surplus and social welfare in both markets (when γ = θ = 0.5).

offset the negative impact on CS in Market H due to the decrease in product quality. Fig. 10(a) shows that the increase in customers’ WTP in Market L always contributes to the improvement of CS in Market L. Commonly, when customers are more willing to buy authorized products in Market L, their total surplus increases. It is also interesting to see that in Market L, customers’ WTP can significantly affect CS in Market H. Specifically, when Strategy QC is adopted, CS in Market H increases, while it decreases in vL if Strategy MD is adopted. This is because, in Region QC, as vL increases, customers are willing to pay a high price, which motivates the manufacturer to invest more in improving product quality. When the effect of product quality improvement dominates that of the increase in price, the customer’s utility increases, leading to an increase in CS. On the other hand, in Region MD (where the PI is blocked), the CS decreases with vL as the PI cannot enter the market to cannibalize the manufacturer’s market share. The total market coverage of both products shrinks in Market H. When vL becomes higher, the manufacturer will set a higher selling price. Although the manufacturer improves product quality with a higher vL , the decrease in the total market coverage and the increased selling price still lead to a decrease in CS in Market H. We also show the impact of PI on SW in Fig. 10(b). Firstly, the existence of a parallel importer always hurts SW in both markets when vL is located in Region S. SW will drops to 0 in the presence of PI in Market L, as the manufacturer chooses to close this sales channel. As the total potential market size shrinks, the manufacturer has less incentive to invest in product quality. A higher total profit in Market H in the presence of PI (see Fig. A.1 in Appendix A) cannot offset the decrease of CS (see Fig. 10(a)). Secondly, in the presence of PI, SW in Market H increases in Regions QC and MD. Although the total profit in market H decreases due to PI (see Fig. A.1), it can be offset by the increase in CS, which leads to an increase in SW. Thirdly, as compared to SW in Market L (always increases in vL ), SW in Market H may decrease in both Regions MD and MI. Fig. 10(b) also shows that when vL is sufficiently high (vL ≥ 0.5), the total SW in Market L is higher than that in Market H. The reduction in SW is due to the rapid increase in the total investment cost. As Fig. A.1 shows, the total profit in Market H reduces gradually while that in Market L increases rapidly with vL . It is easy to verify that the changing rate of the revenue to the cost in Market H is negative (

2 ∂ ( pH qH ) v − ∂ ∂(βv ) = − 32L ) in Region MI. This ∂ vL L

implies that the investment cost increases much faster than the revenue gained in Market H. When the effect of investment cost is very strong (i.e., when vL is very high), SW in Market H will be badly hurt and it can be even lower than that in Market L. The above discussion suggests that parallel importation can harm both the importing and exporting countries when the two

countries are sufficiently unbalanced, as it may harm customers, firms, and society. However, when the importing and exporting countries are moderately balanced, parallel importation can be positive for some importing countries, as the parallel importation can benefit both customers and society. Our study suggests that policy makers should consider whether or not exporting and importing countries are balanced, when policies, regulations, or laws on parallel importation are proposed. 7. Extensions In this section, we discuss the impact of PI in three extensions to our main model by considering the manufacturer’s marketing investment, multiple competing PIs, and the advertising effect. 7.1. The manufacturer’s marketing investment The above discussions assume that the customers’ WTP in Market L vL is exogenously determined. In practice, customers’ WTP can be positively impacted by firms’ marketing efforts, such as advertising the products, building more stores, or providing high quality service [10]. However, these marketing activities are costly and controllable. In this subsection, we extend our main model by endogenizing vL , which can be enhanced by the manufacturer’s investment. We assume that the manufacturer needs to decide on its marketing investment to influence vL after its decision on distribution strategy but before its decision on product quality. We also assume that vL0 is the initial value of vL . The manufacturer’s ρ (v −v )2

L L0 investment cost function is , which is increasing and con2 cave in vL , where ρ denotes the unit investment cost. Without loss of generality, vL0 is normalized to 0, implying that the customers’ WTP remains zero without marketing investment (as introducing a none-zero vL0 into our model will not change the main conclusions). Based on the results presented in Proposition 3 and Table B3 in Appendix B, the manufacturer’s profit functions considering marketing investment can be formulated as:

πm ( v L ) =

⎧ v2 1 S ⎪ ⎪πm (vL ) = 64 − ρ 2L , ⎪ 2 ⎪ v2 ⎪ (2−γ θ )2 [2−γ θ +(γ +2 )vL ] ⎪ − ρ 2L , ⎨πmQC (vL ) = 2 4 [ 8 ( 2 −γ θ ) −γ 2 v L ] v2L (1+θ )4 v2 L − 2 2 , 64(θ 2 +vL ) 2 2 v 1+vL − 2L , 64

⎪ ⎪ πmMD (vL ) = ⎪ ⎪ ⎪ ⎪ ⎩ πmMI (vL ) = (

ρ

)

ρ

vL ≤ vSL (θ , γ ), vSL (θ , γ ) ≤ vL ≤ vQL (θ , γ ), vQL (θ , γ ) ≤ vL ≤ θ , θ ≤ vL ≤ 1.

(5) We have shown in Proposition 4 that the manufacturer’s profit (without marketing investment) weakly increases with vL . However, when considering investment cost, a higher vL incurs higher


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is because, when ρ is not very large, the manufacturer will invest more to enhance the customer WTP in Market L. The enhanced WTP results in more customers purchasing authorized products in the two markets. Thus, the manufacturer has an incentive to set a higher product quality in the presence of the PI. Last, when ρ is relatively high, the manufacturer reduces product quality as the investment of improving vL becomes costly. At the same time, due to the low customer WTP in Market L, the manufacturer chooses to sell through only Market H. 7.2. Oligopolistic PIs in the gray market

Fig. 11. Impact of γ and ρ on the manufacturer’s strategy choice and optimal investment (for θ = 0.5).

investment cost. Therefore, the manufacturer needs to balance the revenue gain and the investment cost to maximize the total profit by choosing an appropriate level of vL and distribution strategy. The optimal decision on investment and distribution strategy is presented in Proposition 5 and illustrated in Fig. 11. Proposition 5. In the presence of PI, the manufacturer a) sets v∗L = min{vMI∗ , 1} and chooses Strategy MI if 0 < ρ ≤ ρ 1 ; L ∗ and chooses Strategy MD if (γ ≤ γ b) sets v∗L = vMD ˆ and L

ρ 1 ≤ ρ ≤ ρ 2 ) or (γ ≥ γˆ and ρ1 ≤ ρ ≤ ρ ); c) sets v∗L = vQC∗ and chooses Strategy QC if (γ ≤ γˆ and ρ2 ≤ ρ ≤ L ρˆ ); d) sets v∗L = vS∗ = 0 and chooses Strategy S if (γ ≤ γˆ and ρ ≥ ρˆ ) or L

∗ , and vQC∗ (γ ≥ γˆ and ρ ≥ ρ ), where γˆ , ρ 1 , ρ 2 , ρˆ , ρ , vMI∗ , vMD L L L are given in Table B5 in Appendix B.

With Proposition 5, Fig. 11 illustrates some interesting results. If customers’ level of tolerance of late consumption is low (γ ≤ γˆ ), for a given γ , the manufacturer’s distribution strategy depends on the unit investment cost (ρ ) to improve vL : choosing Strategy MI if ρ is very low (0 < ρ ≤ ρ 1 ), choosing Strategy MD if ρ is low (ρ 1 ≤ ρ ≤ ρ 2 ), choosing Strategy QC if ρ is high (ρ2 ≤ ρ ≤ ρˆ ), and choosing Strategy S (selling only in Market H) by not investing if ρ is very high (ρ ≥ ρˆ ). However, when customers are very tolerant of late consumption (γ ≥ γˆ ), Strategy Q is never an optimal strategy. It implies that once the manufacturer is aware that γ is sufficiently high (γ ≥ γˆ ), Strategy Q should not be considered, as the PI never has a chance to enter the market. Fig. 11 suggests that even if the unit investment cost (ρ ) is sufficiently high, as long as the customer’s level of tolerance of late consumption is sufficiently low, Strategy Q is an effective strategy to counter PI behavior. Fig. 12 shows that v∗L , product quality, and the manufacturer’s profit always decrease with ρ (when θ = 0.5) for two cases: γ = 0.5 and γ = 0.95. First, it illustrates that a higher customer’s level of tolerance of late consumption (γ ) hurts the manufacturer’s profit and PI behavior has a negative impact on the manufacturer’s profit in both cases. Second, a relatively low investment cost (i.e., ρ ≤ ρ 1 ), enables the manufacturer to perfectly deter the entry of the PI because it can achieve a higher vL with a low investment. We have shown that the PI cannot enter the market if vL is relatively high (Proposition 5). Thus, the manufacturer always chooses to drive the PI out of the market. Third, the encroachment of the PI pushes the manufacturer to raise product quality when ρ is not very large, to enhance the customer WTP in Market L. This result is the converse of the case without marketing investment. It

We now extend our discussion to examine the impact of the presence of multiple competing PIs (N symmetric PIs) in the gray market on the manufacturer’s distribution strategy, as well as on the associated decisions on quality and price. The distribution structure is illustrated in Fig. 13. PIi can divert a quantity qGi (i = 1, 2, . . . , N ) of authorized products from Market L to Market H by setting the retail price (pG ). We first derive the equilibrium of the manufacturer’s three distribution strategies. For Strategy S, the optimal decisions are unchanged (in Eq. (3)) as there is no PI effect. For Strategy M, the demand functions for authorized products in Markets L and H are unchanged from the main model (presented in Section 3). The total quantity of gray products in Market H becomes: N

qGi =

i=1

pH − pG

(1 − θ )β

−

pG

θβ

.

(6)

The competition between N symmetric PIs is assumed as a Cournot competition and each PI chooses the optimal sales quantity qGi to maximize its individual profit. With simple algebra, the inverse function of the gray product’s selling price is:

pG = θ pH − (1 − θ )θ β

N

qGi .

(7)

i=1

Then, the ith PI’s problem becomes:

Max qGi

πgi (qGi |qG,−i ) = ( pG − pL )qGi

=

θ pH − pL − (1 − θ )θ β

N

qGi qGi .

(8)

i=1

With qM L ≥

N i=1

qGi and profit functions of the manufacturer and

the ith PI in (1) and (8), we have the following result. Lemma 6. There exists unique optimal quality (β M ∗ ) and prices ( pM∗ , H M∗ ), which pM∗ ) for the manufacturer, and optimal price for the PIs ( p L G are summarized in Table B6 in Appendix B. For strategy Q, considering an anticipated selling price of gray products peG , the expected demands of authorized products in the first period in Market H and in the late period in Market L are unchanged from the main model presented in Section 3. The total demand of gray products in the second period becomes: N

qGi =

i=1

pH − γ peG pG − . (1 − γ θ )β θ β

(9)

The inverse function of gray products market price is:

pG =

N pH − γ peG θ − θβ qGi . (1 − γ θ )

(10)

i=1


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Fig. 12. The changes in optimal v∗L , product quality, and the manufacturer’s profit with ρ for θ = 0.5.

Fig. 13. The diagram for a manufacturer with multiple competing PIs in gray market.

Then, PI i’s problem is:

πgi (qGi |qG,−i ) =

Max qGi

N pH − γ peG θ − pL − θ β qGi qGi . (1 − γ θ )

(11)

i=1

Following the logic of definition of the Rational Expectation equilibrium for a single PI, we define: Definition 2. A Rational Expectation equilibrium, ( pQ∗ , qQ∗ ), in G Gi the game between the parallel importer i and the strategic consumers should satisfy: a) given customers’ belief about the PI’s acN p −γ pe tions, the market clearance price is: pG = (H1−γ θ )G θ − θ β qGi (see i=1

Eq. (10)); b) given beliefs about the consumer’s behavior qQ∗ ∈ Gi arg max πgi (qGi |qG,−i ), PIs simultaneously play a best response; c) qGi

beliefs are consistent with the equilibrium outcome: pQ∗ = peG , qQ∗ = G Gi qQ | . Gi pe = pQ∗ G

G

With qQ ≥ L

N i=1

qGi and profit functions of the manufacturer and

Fig. 14. Thresholds of change of manufacturer’s distribution strategy for N = 1, 5, 20, +∞ (for γ = 0.5).

the PIs in (1) and (11), we have the following result. Lemma 7. There exists unique optimal quality (β Q ∗ ) and prices ( pQ∗ , H

pQ∗ ) for the manufacturer, and optimal price for the PIs ( pQ∗ ), which L G are summarized in Table B7 in Appendix B. Let

vˆ SL (θ , γ , N ) =

vˆ QL (θ , γ , N ) =

4γ θ N (1+N −γ θ N )

4[1+(2+γ −γ θ )N]+[ (γ +2 ) 2[θ +θ (1−γ θ )N] . Comparing the manufacturer’s 2+(2+γ −γ θ )N 2

−4γ θ (1+γ )]N 2

and profit

in three strategies in Eq. (3), and Tables B6 and B7 in Appendix B, the manufacturer’s distribution strategy facing multiple identical PIs can be summarized in Proposition 6 and illustrated in Fig. 14.

Proposition 6. In the presence of N identical PIs, the manufacturer a) selects Strategy S if and only if 0 ≤ vL ≤ vˆ SL (θ , γ , N );

b) selects Strategy Q if and only if vˆ SL (θ , γ , N ) ≤ vL ≤ vˆ Q ( θ , γ , N ), L and competes with the PIs (Strategy QC); c) selects Strategy M if and only if vˆ Q (θ , γ , N ) ≤ vL ≤ 1; it can deL ter (Strategy MD if vˆ Q (θ , γ , N ) ≤ vL ≤ θ ) /ignore (Strategy MI if L θ ≤ vL ≤ 1) the entry of the PI.


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Fig. 15. The changes of the optimal product quality and the profits of the manufacturer and PIs with N (for γ = 0.5, θ = 0.6, and vL = 0.25).

Fig. 16. The manufacturer’s optimal channel strategy for combinations of (a) θ and vL ; and (b) λ and vL .

Proposition 6 presents the manufacturer’s optimal distribution strategy when it faces multiple competing PIs. The strategy switching thresholds (vˆ SL (θ , γ , N ) and vˆ Q (θ , γ , N )) change with the numL

ber of PIs (N), where vˆ Q (θ , γ , N ) decreases while vˆ SL (θ , γ , N ) inL creases in N (See Fig. 14), suggesting that with more PIs in the gray market, the space for PIs’ market entry (Region QC) shrinks. This implies that when the competition is intensified in the gray market, the manufacturer is more willing to try to prevent the entry of the PIs, either through Strategy M using the price-balancing strategy or through Strategy S. The intuition is that when competition is fierce (reflecting a large N), each PI can enter the market even with a very low sales margin, leading to more products being diverted to Market H. The competition between the authorized and gray products in Market H is thus intense, pushing the manufacturer to reduce the selling prices. Therefore, the manufacturer is more willing to deter the PIs’ entry. Fig. 15 shows that as more PIs enter the market to compete with the manufacturer, the manufacturer reduces product quality (a). Profits of both the manufacturer (b) and individual PI (c) decrease due to the intensified competition, which is consistent with the intuition. However, the intense competition contributes to the growth of profit in the total gray market (c). In addition, the impact of number of competing PIs on product quality and profits of both the manufacturer and PIs is significant when N is smaller but becomes insignificant when N is sufficiently large. This implies that the manufacturer should be careful in setting product quality when there are fewer PIs in the market.

7.3. The advertising effect We now extend the base model by considering the advertising effect of gray products on the manufacturer’s authorized products. In recent years, more and more parallel importers (also commonly called as “Daigou” in China) sell gray products through ITenabled marketplaces, such as Taobao, AmazonGlobal, and WeChat [58,67]. Many manufacturers, however, are still selling through offline channels with limited market coverage. When detailed information on the products is available online through PIs, due to the advertising effect, more customers will know more about the manufacturer’s authorized products. As a result, some high-value customers who were previously not familiar with the products may be willing to buy authorized products from the manufacturer. That is, although manufacturers often suffer from the intense competition with PIs, they can also gain more sales by freeriding PIs’ advertising efforts. Following Abhishek et al. [2] (who consider the spillover effect of online channel to offline channel), we assume that with the sales quantity of gray products (qYG ), the sales of authorized products in market H will increase by λqYG , where the level of advertising effect (λ) satisfies λ ∈ [0, 1]. The manufacturer’s profit function in Eq. (1) then can be rewritten as:

πmY = pYH qYH + λqYG + m − (β Y )2 , where m =

(12)

0Y =S pYL qYL Y = M, Q


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We now use a numerical example to illustrate the equilibrium channel strategy under different values of λ, θ , and vL . We set λ = 0.3 and γ = 0.5 to examine the impact of θ and vL on the manufacturer’s equilibrium channel strategy (Fig. 16(a)). We also set θ = 0.5 and γ = 0.5 to examine the impact of λ and vL on the manufacturer’s equilibrium channel strategies (Fig. 16(b)). Comparing Fig. 16(a) with Fig. 6, we have several observations. Firstly, we find that Strategy QC is a more dominant strategy when considering advertising effect (λ = 0.3). With the advertising effect, the total sales of authorized products may increase, and the manufacturer becomes less likely to block parallel importations. Secondly, different from the results presented in Proposition 4, when both θ and vL are small, the manufacturer will set a higher quality level in the presence of parallel importation. The reason is that with low θ and vL , the competition from the parallel importer is relatively weak, and the advertising effect on the manufacturer’s profit dominates the cannibalization effect due to the competition of PI. This result, however, will change when θ is relatively high (θ ≥ 0.6 in this example), reflecting the fact that the cannibalization effect comes to dominate the advertising effect. The impacts of the advertising effect on the manufacturer’s optimal channel strategy are illustrated in Fig. 16(b), as λ and vL change. The figure shows that as λ increases, Strategy QC becomes more dominant, but the feasible areas of all Strategies MI, MD, and S shrink. Moreover, when λ is above a threshold (λ = 0.25 in this example), neither Strategy MD nor Strategy S can be a dominant strategy. In addition, it is observed that the manufacturer may set a higher quality level in the presence of parallel importation when λ is above the threshold (λ = 0.25) and vL is relatively small. The results in Fig. 16 suggest that the advertising effect can sometimes benefit both the manufacturer and the parallel importer. This may explain why some large companies are not striving to eliminate the gray market. 8. Conclusions This paper investigates a manufacturer’s distribution strategy considering product quality and pricing decisions in the presence of PI, when the manufacturer has three strategy options available: Strategy S, Strategy M, and Strategy Q, to compete with a PI. We identify the manufacturer’s distribution strategy and discuss the impacts of parallel importation and some critical systems parameters on the manufacturer’s optimal decisions, to obtain several managerial insights. We show that distribution strategy can be a competitive advantage for the manufacturer to counter the PI. The optimal distribution decision depends strongly on the customers’ WTP gap between the two markets. Specifically, when the gap is sufficiently small, the manufacturer can prevent the entry of the PI and choose Strategy M; when the gap is moderate, Strategy Q is chosen and the manufacturer competes with the PI; when the gap is very large, the manufacturer chooses Strategy S and sells the product in Market H only. We find that product quality design plays an important role to counter the PI. As compared to the case without parallel importation, the manufacturer should strategically lower product quality to mitigate the competiveness of PIs. In addition, a low customer’s level of tolerance of late consumption eases the competition between the manufacturer and the PI, thus both firms’ profits increase. To customers, the existence of parallel importation hurts the customer surplus in the low-price market, but benefits the customer surplus in the high-price market (when the customers’ WTP gap is moderate). Here we show how the manufacturer can effectively counter the PI through selection of distribution market, quality design, and price setting. The presence of PI results in loss of profit for the manufacturer, and the manufacturer should reduce product quality when it faces parallel importation.

Our first extension shows that an appropriate marketing investment with quality design and distribution strategy can effectively counter the PI’s behavior. Interestingly, when customers’ level of tolerance of late consumption is above a certain threshold, and the manufacturer can optimally set the marketing investment to deter the PI’s entry, the PI can never enter the market; this is different in the base model. In addition, the manufacturer will never use Strategy Q when customers’ tolerance level is high, as under this condition, delaying the timing of selling products in Market L is less efficient to suppress parallel importation activities. In the second extension, we consider multiple identical competing PIs in the gray market. We show that the number of PIs has strong impacts on the manufacturer’s and PIs’ profit. With more PIs, the manufacturer’s profit and the single PI’s profit drop significantly, while the total profit gain in the gray market increases. In addition, with more potential PIs, the manufacturer is more aggressive in blocking their entry. The third extension, which examines the impact of the advertising effect of gray products on the manufacturer’s optimal channel strategy suggests that the manufacturer may have motivations to improve its product quality in the presence of parallel importation if the advertising effect is strong. The advertising effect may benefit both the manufacturer and the PI because it softens the competition between the manufacturer and the PI in Market H and expands the total market coverage. Our research could be further extended in several directions. In the present paper, we consider a single manufacturer with a single PI. In practice, manufacturers may sell through common retailers. One extension for future work would be to study the impacts of vertical competition on manufactures’ product quality and distribution decisions. In addition, in this paper, we assume that all the products in different markets share the same quality level, but in fact the quality of products sold in different regions can be different. Another extension would be to consider the manufacturer’s product line decisions. Using product line design (producing products with differentiated quality) as a strategy to deal with parallel importation would be interesting. Furthermore, we assume that the manufacturer has unlimited production capacity. In industrial practice, however, firms often face capacity constraint, especially for those producing multiple products or servicing multiple groups of customers. Thus, it would be interesting to examine the manufacturer’s capacity allocation problems across the two markets under capacity constraint in the presence of parallel importation. Finally, governments’ tax regulations significantly affect international trade (See, for example, [54,61]), but they are not considered in the models discussed in this paper. Examining the impacts of tax policies on firms’ quality design, pricing decisions, and channel strategies in the context of parallel importation would be an interesting direction for future research. It is expected that high tax rates would hinder the entry of PIs, and hurt CS and SW in the import market. Setting of appropriate tax rates, from the perspectives of the government, would merit significant worth study.

Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant Nos. 71771053, 71628101, 71671081, 71331004 and 71371003), the Scientific Research Foundation of the Graduate School of Southeast University (No. YBJJ1526), the Fundamental Research Funds for the Central Universities (No. 2242017K41036), Research and Innovation Program of Postgraduates in Jiangsu Province (No. KYLX_0140), and the Natural Sciences and Engineering Research Council of Canada (Grant No. 372400).


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Declarations of interest None. Appendix A. Proofs Proof of Lemma 1 and Proposition 1. We first solve the PI’s maximization problem, which is

πg pMG = pMG − pML

Max pM G

M pM pM H − pG − GM M θβ (1 − θ )β

pM −pM

pM

,

(A.1)

pM

H G s.t. pM ≤ θ pM and (1− − θ βGM ≤ 1 − v βL M . G H θ )β M L M∗ M M The profit function in (A.1) is concave in pM G . The unique optimal price pG = (θ pH + pL )/2 can be obtained by equating the first order derivative to zero, which is increasing in selling prices of both markets. To satisfy the condition pM∗ ≤ θ pM ⇔ pM /θ ≤ pM . In addition, to H L H G M ⇔ pM ≤ pM /θ + 2 (1 − θ )(β − pM /v ). Conversely, when pM /θ ≥ pM , the demand and profit of the PI will be satisfy the constraint qM ≥ q L L G H L L L H zero and it cannot enter the market. We summarize the PI’s optimal price as:

pM∗ G

M i f pM H ≤ pL /θ ,

N/A,

=

θ

M pM H + pL

2

β − pML /vL .

M M i f pM L /θ ≤ pH ≤ pL /θ + 2 (1 − θ )

,

(A.2)

Substituting the response function of pM∗ into the demand function for authorized products, the sales quantity in Market H can be G obtained as

M

qM H , qG

⎧ pM ⎪ ⎪ ⎨ 1 − βH , 0 , = ⎪ (2−θ ) pM −pM θ pM −pM ⎪ L ⎩ 1 − 2(1−Hθ )β L , 2(1H−θ )θβ ,

pM

i f pM ≤ θL , H pM

pM

pM

i f θL ≤ pM ≤ θL + 2 (1 − θ ) β − vL H L

(A.3) .

Substituting demand functions of (qM , qM ) into the manufacturer’s profit function, we obtain the maximization problem as: H G

Max

pM ,pM ,β H L

πm

⎧ ⎨ pMH 1 − M pM , p , β = H L ⎩ pM 1 −

H

pM H

β

pM

L + pM L 1 − vL β

(2−θ ) pM H −pL 2 ( 1 − θ )β

pM

− β 2, pM

L + pM L 1 − vL β

L i f pM H ≤ θ ,

pM

− β 2,

pM

L i f θL ≤ pM H ≤ θ + 2 (1 − θ )

M

β−

pM L

vL

M

(A.4) .

M

M

pH pL pL pH ∂L M 2 M The KKT simultaneous equations for the first scenario are given as L = pM H (1 − β ) + pL (1 − vL β ) − β + λ ( θ − pH ). ∂ pM = 1 − 2 β − H

λ,

M M2 M2 M ∂ L = 1 − 2 pL + λ , ∂ L = −2β + pH + pL , ∂ L = pL − pM . Complementary slackness conditions: vLβ H θ ∂β θ β2 vL β 2 ∂λ ∂ pM L

pM L

λ(

pM L

M θ − pH ) = 0. From complemen-

pM L

M tary slackness conditions, we get (1) λ = 0, θ > pM H or (2) λ > 0, θ = pH . Using the above conditions, we have the optimal pricing decisions for a given β in this scenario:

vL β β ∗ M∗ 1) When θ ≤ vL ≤ 1, the PI has no impact on the manufacturer’s decisions. The optimal decisions are pM H (β ) = 2 , pL (β ) = 2 , and

πmM∗ (β ) = ∗ (β ) = pM L

(1+vL )β 4

v (1+θ )β

∗ L − β 2 . 2) When 0 ≤ vL ≤ θ , the PI is deterred by the manufacturer. The optimal decisions are pM H ( β ) = 2 ( θ 2 +vL ) ,

vL (1+θ )θ β , and 2 ( θ 2 +vL )

πmM∗ (β ) =

vL (1+θ )2 β − β 2. 4 ( θ 2 +vL )

After obtaining the profit functions with respect to β , we determine the optimal quality. When θ ≤ vL ≤ 1, then β M∗ = 0 ≤ vL ≤ θ , then

β M∗

2

(1+θ ) v = 8(θ 2 +v L) . L

1+vL 8 ;

when

Substituting the optimal quality level into the price and profit functions, we obtain: ∗ 1) θ ≤ vL ≤ 1, pM H =

2) 0 ≤ vL ≤ θ ,

∗ pM H

=

1+vL M∗ 16 , pL v2L (1+θ )3 2

16 (θ 2 +vL )

= ,

vL (1+vL )

∗ pM L

16

=

, β M∗ =

v2L (1+θ )3 θ

2

16 (θ 2 +vL )

,

1+vL 8 ,

β M∗

M∗ = and πm

(1+vL )2 64

(1+θ )2 v = 8(θ 2 +v L) , and L

.

πmM∗

=

(1+θ )4 v2L

2

64 (θ 2 +vL )

.

The KKT simultaneous equations for the second scenario are given as

M pM (2 − θ ) pM M H − pL L L= 1− + pL 1 − 2(1 − θ )β vL β M M pL pM pL 2 M M L −β + λ1 pH − + λ2 + 2 (1 − θ ) β − − pH . θ θ vL pM H

M M ∂ L = 1 − 2(2−θ ) pH −pL + λ − λ , 1 2 2(1−θ )β ∂ pM H M

M

M pM λ λ 2λ (1−θ ) ∂ L = 1 − 2 pL + H − θ1 + θ2 − 2 v , vL β 2(1−θ )β ∂ pM L L

pM ( (2−θ ) pM −pM ) pM2 ∂L H H L + v Lβ 2 , ∂β = −2β + 2(1−θ )β 2 L M

M

M

pL ∂L M ∂ λ1 = θ − p H , M

p p p p p and ∂∂λL = θL + 2(1 − θ )(β − vL ) − pM . Complementary slackness conditions: λ1 ( pM − θL ) = 0 and λ2 ( θL + 2(1 − θ )(β − vL ) − pM ) = 0. H H H L L 2 pM

pM

pM

pM

From complementary slackness conditions, we get (1) λ1 = 0, pM > θL or (2) λ1 > 0, pM = θL and λ2 = 0, θL + 2(1 − θ )(β − vL ) > pM or H H H L pM

pM

(3) λ2 > 0, θL + 2(1 − θ )(β − vL ) = pM . H L Please cite this article as: H. Huang et al., Competitive strategies and quality to counter parallel importation in global market, Omega (2018), https://doi.org/10.1016/j.omega.2018.07.009

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Using the combination of the above conditions, we have the optimal pricing decisions for the manufacturer under the scenario: θ 2 −2) 1) When 0 ≤ vL ≤ θ (6θ5−4 , PI competes with the manufacturer and all the Market L products will be diverted to Mar−4θ [4θ 2 (1−θ )2 −4θ (1−θ )2 vL +v2L ]β

∗ ket H. The optimal decisions are pM H (β ) = 4θ 2 (1−θ )2 −4θ (1−2θ )(1−θ )vL +(8θ 2 −8θ +1 )v2L β 4[2θ 2 (2−θ )(1−θ )−4θ (1−θ )vL +v2L ]

2[2θ 2 (2−θ )(1−θ )−4θ (1−θ )vL +v2L ]

∗ , pM L (β ) =

[2θ 2 (3−2θ )(1−θ )vL +θ (4θ −3 )v2L ]β

2[2θ 2 (2−θ )(1−θ )−4θ (1−θ )vL +v2L ]

M∗ (β ) = , and πm

θ 2 −2) 1−θ ) − β 2 . 2) When Max( θ (6θ5−4 , 0 ) ≤ vL ≤ 4θ5(−3 −4θ θ , PI competes with the manufacturer and 2(1−θ )(vL −4θ +4 )β ∗ (β ) = , pM L 8(2−θ )(1−θ )−vL vL (1+θ )β M ∗ we have: pH (β ) = 2(θ 2 +v ) , L

∗ (β ) = part of the Market L products will be diverted to Market H. The optimal decisions are pM H 2(1−θ )(5−2θ )vL β , 8(2−θ )(1−θ )−vL

∗ (β ) = pM L

M∗ (β ) = and πm

vL (1+θ )θ β , and 2 ( θ 2 +vL )

2(1−θ )[2−2θ +(3−θ )vL ]β 8(2−θ )(1−θ )−vL

πmM∗ (β ) =

1−θ ) − β 2 . 3) When 4θ5(−3 θ ≤ vL ≤ 1, PI is deterred. Then

vL (1+θ )2 β − β 2. 4 ( θ 2 +vL )

θ 2 −2) After obtaining the profit functions with respect to β , we determine the optimal quality. When 0 ≤ vL ≤ θ (6θ5−4 , then β M∗ = −4θ

4θ 2 (1−θ )2 −4θ (1−2θ )(1−θ )vL +(8θ 2 −8θ +1)v2L 8[2θ 2 (2−θ )(1−θ )−4θ (1−θ )vL +v2L ]

(1−θ )[2−2θ +(3−θ )v ] θ 2 −2) ; when 0 ≤ vL ≤ θ (6θ5−4 , then β M∗ = 8(2−θ )(1−θ )−v L ; when −4θ L

4θ (1−θ ) 5−3θ

≤ vL ≤ 1, then β M∗ =

(1+θ )2 vL . 8 ( θ 2 +vL )

Substituting β M ∗ into the price and profit functions, we obtain:

θ 2 −2) 1) when 0 ≤ vL ≤ θ (6θ5−4 , −4θ

[4θ 2 (1−θ )2 −4θ (1−θ )2 vL +v2L ][4θ 2 (1−θ )2 −4θ (1−2θ )(1−θ )vL +(8θ 2 −8θ +1 )v2L ]

∗ = pM H

16[2θ 2 (2−θ )(1−θ )−4θ (1−θ )vL +v2L ]2

∗ = pM L

16[2θ 2 (2−θ )(1−θ )−4θ (1−θ )vL +v2L ]2

4θ 2 (1−θ )2 −4θ (1−2θ )(1−θ )vL +(8θ 2 −8θ +1 )v2L

β M∗ =

πmM∗ =

8[2θ 2 (2−θ )(1−θ )−4θ (1−θ )vL +v2L ]

2 2 L + (8θ −8θ +1 )vL 8[2θ 2 (2−θ )(1−θ )−4θ (1−θ )vL +v2L ]

2

θ 2 −2) 1−θ ) Max( θ (6θ5−4 , 0 ) ≤ vL ≤ 4θ5(−3 −4θ θ ,

πmM∗ =

4θ (1−θ ) 5−3θ

16 (θ 2 +vL )

∗ = ≤ vL ≤ 1, pM H

∗ pM H =

(1−θ )2 [2−2θ +(3−θ )vL ]2 . [8(2−θ )(1−θ )−vL ]2 2 (1+θ )3 θ v2L (1+θ )3 v M∗ L

(1−θ )[2−2θ +(3−θ )vL ] , and 8(2−θ )(1−θ )−vL

2

, pL

=

,

, and

4θ 2 (1−θ )2 −4θ (1−2θ )(1−θ )v

2) when

3)

,

[2θ 2 (3−2θ )(1−θ )vL +θ (4θ −3 )v2L ][4θ 2 (1−θ )2 −4θ (1−2θ )(1−θ )vL +(8θ 2 −8θ +1)v2L ]

2

16 (θ 2 +vL )

2 (1−θ )2 (vL −4θ +4 )[2−2θ +(3−θ )vL ] [8(2−θ )(1−θ )−vL ]2

(1+θ )2 v

M∗ = , β M∗ = 8(θ 2 +v L) , and πm L

∗ , pM L =

(1+θ )4 v2L

2

64 (θ 2 +vL )

2 (1−θ )2 (5−2θ )[2−2θ +(3−θ )vL ]vL [8(2−θ )(1−θ )−vL ]2

,

β M∗ =

.

Comparing the profits in the two scenarios, we finally get the optimal solutions for Strategy M, which are presented in Table B1 in Appendix B. The results in Proposition 1 can also be obtained. Proof of Lemma 2. We first solve the PI’s maximization problem, which is

Max pQG

π

Q g pG

=

pQG

− pL

pQ pQH − γ peG − G ( 1 − γ θ )β θ β

The profit is maximized at pQ∗ = G

.

(A.5)

Q e 1 θ ( pH −γ pG ) 2( 1−γ θ

+ pQL ). Similar to Cachon et al. [14], Cachon and Swinney [15], and Su and Zhang [56], we

obtain the Rational Expectations (RE) equilibrium by equating pQ∗ = peG . Then, the optimal responsive price for gray products can be solved G as pQ∗ = G

θ pQH +(1−γ θ ) pQL . When the condition pQH ≥ pQL /θ is satisfied, the sales margin (i.e., pQH − pQL ) of PI is positive and it can enter Market 2−γ θ

H successfully. To satisfy the constraint qQ ≥ qQ ⇔ pQH ≤ pQL /θ + (2 − γ θ )(β − pQL /vL ). However, when pQH ≤ pQL /θ , the PI cannot achieve a L G positive sales margin and it is prevented from entering Market H. We summarize the PI’s optimal response price as

pQ∗ G

=

pQ

N/A, θ

( 1 −γ θ ) 2 −γ θ

pQH +

pQL

,

i f θL ≥ pQH , pQ pQ pQ i f θL ≤ pQH ≤ θL + (2 − γ θ ) β − vLL .

(A.6)

Substituting the response function p∗G into the demand functions of the manufacturer and the PI, we obtain the demand for the authorized and gray products in Market H as

⎧ Q Q ⎨ 1 − qH , qG = ⎩ 1−

pQH

pQ

β ,0 ,

2 pQH −γ pQL ( 2 − γ θ )β

θ pQ −pQ

, θ (2−H γ θ L)β ,

i f θL ≥ pQH , pQ

pQ

i f θL ≤ pQH ≤ θL + (2 − γ θ )

β−

pQL

vL

(A.7) . θ pM −pM

θ pQ −pQ

H L Comparing the results in (A.3) and (A.7), for the same price, i.e., pQH = pM and pQL = pM , then qM − qQ = 2(1H−θ )θLβ − θ (2− H L G G γ θ )β =

∂ ( qM −qQ ) (θ pM −pM ) θ (θ pM −pM ) (2−γ )θ G G H L > 0 and = 2(1−Hθ )θ Lβ 2(θ −1)2 ≤ 0. θβ 2(1−θ )(2−γ θ ) ∂γ (2−γ θ )


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19

Proof of Lemma 3 and Proposition 2. Substituting the responsive demand functions of qQ and qQ in (A.7) into the manufacturer’s profit H L function, we obtain its maximization problem.

⎧ Q Q Q ⎨ pH 1 − Max πm pH , pL , β = ⎩ pQ 1 − pQH ,pQL ,β

pQH

β

2 pQH −γ pQL ( 2 − γ θ )β

H

pQ

+ pQL 1 − v Lβ L

pQ

− β 2, pQ

+ pQL 1 − v Lβ L

i f θL ≥ pQH , pQ

− β 2,

pQ

i f θL ≤ pQH ≤ θL + (2 − γ θ )

pQL

β−

vL

(A.8) .

The KKT simultaneous equations for the first scenario are given as: pQ

pQ

pQ

pQ

2 pQ

pQ2

pQ2

pQ

∂ L = −2β + H + L , and ∂ L = L − pQ . L = pQH (1 − βH ) + pQL (1 − v Lβ ) − β 2 + λ( θL − pQH ). ∂ LQ = 1 − 2 βH − λ, ∂ LQ = 1 − v L + λθ , ∂β H ∂λ θ β2 vL β 2 Lβ L ∂ pH ∂ pL

pQ

Complementary slackness conditions: λ θL − pQH = 0.

pQ

pQ

From complementary slackness conditions, we get (1) λ = 0, θL > pQH or (2) λ > 0, θL = pQH . Using the above conditions, we have the optimal pricing decisions for the manufacturer under this scenario: vβ Q∗ 1) When θ ≤ vL ≤ 1, the PI has no impact on the manufacturer’s decisions. Then pQH∗ (β ) = β2 , pQL ∗ (β ) = L2 , and πm (β ) = (1+4vL )β − β 2 . v (1+θ )β

2) When 0 ≤ vL ≤ θ , the PI is deterred by the manufacturer. Then pQH∗ (β ) = 2L(θ 2 +v ) , pQL ∗ (β ) = L 1+vL 8 ;

Q∗ With πm (β ), we can obtain: when θ ≤ vL ≤ 1, then β Q ∗ =

and profit functions, we can obtain: 1+vL Q∗ 16 , pL v2L (1+θ )3

1) When θ ≤ vL ≤ 1, pQH∗ = 2) When 0 ≤ vL ≤ θ , pQH∗ =

16 (θ 2 +v

=

vL (1+vL ) 16

Q∗ 2 , pL =

L)

, βQ∗ = 3

v2L (1+θ ) θ

1+vL 8 ,

2 pQ −γ pQ

pQ

(1+vL )2

Q∗ and πm =

64

. (1+θ )4 v2L

(1+θ )2 v

pQ

2 pQ L

2

64 (θ 2 +vL )

Q

.

Q

Q

Q

p p 4 p −γ p + λ2 θL + (2 − γ θ )(β − vL ) − pQH . Then ∂ LQ = 1 − (2H−γ θ )βL + λ1 − λ2 , ∂ LQ = 1 − L ∂ pH ∂ pL

γ pQH pQ (2 pQ −γ pQ ) pQ2 pQ pQ pQ λ1 λ2 λ2 (2−γ θ ) ∂ L , ∂β = −2β + H(2−γHθ )β 2L + v Lβ 2 , ∂∂λL = θL − pQH , and ∂∂λL = θL + (2 − γ θ )(β − vL vL vL β + (2−γ θ )β − θ + θ − L 1 2 L

pQ

Complementary slackness conditions: λ1 pQH − θL

vL (1+θ )2 β − β 2. 4 ( θ 2 +vL )

(1+θ )2 v

Q∗ Q∗ = L , and πm = 2, β 8 ( θ 2 +vL ) 16 (θ 2 +vL )

+ pQL 1 − v Lβ − β 2 + λ1 pQH − θL L

πmQ∗ (β ) =

when 0 ≤ vL ≤ θ , then β Q ∗ = 8(θ 2 +v L) . Substituting β Q ∗ into the price L

The KKT simultaneous equations for the second scenario are given as L = pQH 1 − (2H−γ θ )βL

vL (1+θ )θ β , and 2 ( θ 2 +vL )

pQ

) − pQH .

pQ

= 0, λ2 ( θL + (2 − γ θ )(β − vL ) − pQH ) = 0. L pQ

pQ

pQ

pQ

From the complementary slackness conditions, we get (1) λ1 = 0, pQH > θL or (2) λ1 > 0, pQH = θL and λ2 = 0, θL + (2 − γ θ )(β − vL ) > L pQ

pQ

pQH or (3) λ2 > 0, θL + (2 − γ θ )(β − vL ) = pQH . L Using the combination of the above conditions, we have the optimal pricing decisions for the manufacturer under this scenario: 4θ −2θ 2 γ , PI competes with the manufacturer and part of Market L’s products will be diverted to Market H. Then 4+(1−θ )γ (2−γ θ )[2(2−γ θ )+γ vL ]β γ θ )β )vL ]β Q∗ decisions are pQH∗ (β ) = , pQ∗ (β ) = (2−γ8θ()(2−4+γ γθ ))−vLγ(22 − , and πm (β ) = (2−γ θ8)([22−−γγθθ)+−(γγ2+2 − β 2. L 8(2−γ θ )−γ 2 vL vL vL

1) When 0 ≤ vL ≤ optimal 2) When

4θ −2θ 2 γ 4+(1−θ )γ

πmQ∗ (β ) =

v (1+θ )β

≤ vL ≤ 1, PI is deterred by the manufacturer. The optimal decisions are pQH∗ (β ) = 2L(θ 2 +v ) , pQL ∗ (β ) = L 2

vL (1+θ ) β − β 2. 4 ( θ 2 +vL )

the

vL (1+θ )θ β , and 2 ( θ 2 +vL )

3) When the condition θL + (2 − γ θ )(β − v L ) = pH is satisfied, then λ2 < 0. Thus, the solution for this condition is omitted. L p

p

4θ −2θ 2 γ 4+(1−θ )γ

Q∗ With πm (β ), we can obtain: when 0 ≤ vL ≤

, then β Q ∗ =

(2−γ θ )[2−γ θ +(γ +2 )vL ] 4θ −2θ 2 γ ; when 4+(1−θ )γ ≤ vL ≤ 1, then 16(2−γ θ )−2γ 2 vL

Substituting β Q ∗ into the price and profit functions, we can obtain: 1) when 0 ≤ vL ≤

4θ −2θ 2 γ 4+(1−θ )γ

, then pQH∗ =

(2−γ θ )2 [2(2−γ θ )+γ vL ][2−γ θ +(γ +2 )vL ] (2−γ θ )2 (4+γ )vL [2−γ θ +(γ +2 )vL ] , pQL ∗ = , 2 2 2[8(2−γ θ )−γ 2 vL ] 2[8(2−γ θ )−γ 2 vL ]

(2−γ θ )2 [2−γ θ +(γ +2 )vL ]2 . 2 4[8(2−γ θ )−γ 2 vL ] 2 v2 (1+θ )3 v2L (1+θ )3 θ 4θ −2θ γ Q∗ 2) when 4+(1−θ )γ ≤ vL ≤ 1, then pQH∗ = L 2 , pL = 2, 16 (θ 2 +vL ) 16 (θ 2 +vL )

βQ∗ =

βQ∗ =

(1+θ )2 vL . 8 ( θ 2 +vL )

(2−γ θ )[2−γ θ +(γ +2 )vL ] , 16(2−γ θ )−2γ 2 vL

Q∗ and πm =

βQ∗ =

(1+θ )2 vL , 8 ( θ 2 +vL )

Q∗ and πm =

(1+θ )4 v2L

2

64 (θ 2 +vL )

.

Comparing the profit for the manufacturer under the two scenarios, we get the optimal solutions for Strategy Q, which is presented in Table B2 in Appendix B. The results in Proposition 2 can also be obtained. Proof of Lemma 4. The results are obtained by comparing the manufacturer’s optimal quality under Strategy S, Strategy M (shown in Table B1), and Strategy Q (shown in Table B2). Solving the two inequalities β S ∗ ≥ β Q ∗ and β S ∗ ≥ β M ∗ , we obtain that the manufacturer will set the highest quality in Strategy S if 0 < vL ≤ 4θ −2θ 2 γ 4+(1−θ )γ

, then

βQ∗

> β M∗

and

β Q∗

> β S∗;

when

8θ γ −4θ 2 γ 2 . Following 16+8(1−θ )γ +(1−4θ )γ 2 4θ −2θ 2 γ vL ≥ 4+(1−θ )γ , then β Q∗ = β M∗

the same logic, we have when >

β S∗ .

8θ γ −4θ 2 γ 2 16+8(1−θ )γ +(1−4θ )γ 2

≤ vL ≤

Proof of Proposition 3. The results are obtained by comparing the manufacturer’s profit under Strategy S, Strategy M (shown in S ≥ π Q and π S ≥ π M , we can obtain that when 0 < v ≤ Table B1), and Strategy Q (shown in Table B2). Solving the two inequalities, πm L m m m 8θ γ −4θ 2 γ 2 , 16+8(1−θ )γ +(1−4θ )γ 2

Strategy S dominates the other two strategies. Following the same logic, we can also obtain the dominant regions of

Strategy Q and Strategy M Appendix B.

8θ γ −4θ 2 γ 2 16+8(1−θ )γ +(1−4θ )γ 2

≤ vL ≤

4θ −2θ 2 γ 4+(1−θ )γ

and vL ≥

4θ −2θ 2 γ 4+(1−θ )γ

, respectively. The results are presented in Table B3 in


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Proof of Lemma 5.

∂vQL (θ ,γ ) ∂vSL (θ ,γ ) 8θ (4 (2−θ γ )2 −γ 2 ) = = −4θ (1+θ ) 2 < 0. Thus, the size of region QC (vQ (θ , γ ) − vSL (θ , γ )) 2 > 0, L ∂γ ∂γ (4+(1−θ )γ ) (16+8(1−θ )γ +(1−4θ )γ 2 )

decreases in γ . Besides, the size of region MD (θ − vQ (θ , γ )) and S (vSL (θ , γ )) increases in γ . L

Proof of Proposition 4. It is clearly shown that in the MI region (vL ≥ θ ), the two profits, prices and demands are equal. Then we compare the optimal results with and without PI in regions MD, QC and S. 1) We compare the manufacturer’s profit with and without PI MD∗ − π N ∗ = In MD region, πm m

πmQC∗ − πmN∗ =

−(vL−θ )2 ( (1+θ )2 vL+(1+vL )(θ 2+vL ) ) 2

64 (θ 2+vL ) {4(2−γ θ )[2−γ θ+(γ +2 )vL ]+(1+vL )(8(2−γ θ )−γ 2 vL )}M (vL ) 2

4[8(2−γ θ )−γ 2 vL ]

≤ 0. In the QC region, ,

where

M (vL ) = 4γ 2 θ 2 − 8γ θ + (8γ − 4γ 2 θ + γ 2 )vL + γ 2 v2L ,

vL ∈

[vSL (γ , θ ), vQ (γ , θ )], which also satisfies vL ∈ [0, θ ]. The first and second order derivatives are M (vL ) = (8γ − 4γ 2 θ + γ 2 ) + 2γ 2 vL L and M (vL ) = 2γ 2 > 0. If vL ∈ [vSL (γ , θ ), vQ (γ , θ )], then M (vL ) > 0 always holds. Thu, M(vL ) increases in vL . Since M (vL ) ≤ M (vL )|max = L 2 2γ θ (2−γ θ )(1+θ )(4+γ ) QC ∗ N ∗ < 0. − < 0, 16 − 8γ θ − γ 2 vL > 0, and 4(2 − γ θ )[2 − γ θ + (γ + 2 )vL ] > 0, we finally can obtain πm − πm (γ −γ θ +4 )2 2 S∗ − π N∗ = 1 − (1+vL ) < 0. Therefore, the manufacturer’s profit in the presence of PI is always lower than that In the S region, πm m 64 64 without PI.

2) We compare the optimal product quality for the manufacturer with and without PI.

β MD∗ − β N∗ =

−(vL −θ )2 8 ( θ 2 +vL )

M (v )

< 0 and β QC ∗ − β N∗ = 8( (16−8γ θL)−γ 2 v ) , where M(vL ) = γ 2 v2L + (8 − 4γ θ + γ )vL − 4γ θ (2 − γ θ ). The first order L θ γ 2 (θ + 1 )

derivative is M (vL ) = 2γ 2 vL + (8 − 4γ θ + γ ) > 0. In addition, M(vL ) ≤ M(vL = vQ ( θ , γ ) ) = − 8 ( θ 2 γ 2 − θ γ 2 − 8θ γ L vL QC ∗ N ∗ S ∗ N ∗ β − β < 0 and β − β = − 8 < 0.

+ 4γ + 16)

< 0. Therefore

3) We compare the prices in the two markets with and without PI.

∗ − pN ∗ = pMD H H

(4−2γ θ −γ )γ θ

v2L (1+θ )3 16 (θ 2 +v

2

L)

−

1+vL 16

0 and pQC∗ − pN∗ > 0. In a similar logic, we obtain that pMD L L L L Thus, the price in Market H decreases, and the price in Market L increases in the presence of PI.

4) We compare the demands in the two markets with and without PI. (θ −v )θ

(4−γ )γ v

v −θ

γ (−4+2γ θ −γ v )

QC ∗ ∗ − qN ∗ = N∗ = N ∗ = 0; qMD∗ − qN ∗ = L L L qMD > 0, qH − qH > 0, and qSH∗ − qH < 0, qLQC ∗ − qLN∗ = 2(16−8γ θ −γ 2 vL ) < 0, H H L L 2θ 2 +2vL 2(16−8γ θ −γ 2 vL ) 2 ( θ 2 +vL ) L and qSL∗ − qLN∗ = − 12 < 0. Thus, the demand in Market H increases while in Market L it decreases under the impacts of PI.

( θ , γ ). Proof for Proposition 5. We first verify that the profit function is continuous and differentiable at the threshold points θ and vQ L

S (v ), π MI (v ), π MD (v ), and π QC (v ) can be respectively expressed as The first and second order derivatives for πm L L L L m m m

∂πmS (vL ) ∂2π S v = −ρvL < 0 and ∂vm 2( L ) = −ρ < 0; ∂ vL L MI ∂πm (vL ) ∂ 2 πmMI (vL ) +1 1 = vL32 − ρvL and = 32 − ρ; ∂ vL ∂v2L θ 2 (1+θ )4 (2vL −θ 2 ) ∂πmMD (vL ) ∂ 2 πmMD (vL ) vL θ 2 (1+θ )4 = =− − ρ < 0; 3 − ρvL and 2 4 ∂ vL ∂v 2 32 (θ +vL ) 32 (θ 2 +vL ) L 3 2 QC ∂πm (vL ) = (2−θ γ ) (4+γ ) 2(2vL −θ γ +3 vL γ +2) − ρvL and ∂ vL 2 (16−γ vL −8θ γ ) (2−θ γ )3 (4+γ )2 (16γ −16θ γ −8γ 2 θ −3γ 3 θ +4γ 2 vL +2γ 3 vL +6γ 2 +32) ∂ 2 πmQC (vL ) =− − ρ < 0. 4 ∂v2L 2 (16−γ 2 vL −8θ γ )

At

the

threshold

point

vL = θ , ∂πm∂ v (vL ) |vL =θ = MI

L

∂πmMD (vL ) θ |vL =θ = 1+ 32 − ρθ . ∂ vL

At

point

vL = vQL (θ , γ ) =

4θ −2θ 2 γ 4+(1−θ )γ

,

∂πmQC (vL ) ∂ vL |vL =vQL =

2 ∂πmMD (vL ) |v =vQ = (2−θ γ )(1+θ )(4+2γ −θ γ ) − ρθ . This means that the function is continuous and differentiable at the two threshold points in ∂ vL 16 (4−θ γ ) L L

interval [vSL , 1]. Because the total profit function is decreasing in interval [0, vSL ], the optimum is v∗L = 0 and maximum profit is 1/64. To derive the MD∗ , optimal decisions, we also need to compare the maximum profit in interval [0, vSL ] with that in [vSL , 1]. In the following analysis, vMI∗ L , vL and vQC∗ are obtained by solving L

∂πmQC∗ ∂πmMI∗ ∂πmMD∗ ∂ vL = 0, ∂ vL = 0, and ∂ vL = 0, respectively.

We analyze the optimal decisions in interval [vSL , 1]. When ρ ≤

∂πmMI (vL ) 1 32 , ∂ vL |vL =θ

=

QC MD 2 MI (v ) ∂πmMD (vL ) L m |vL =θ > 0, ∂πm∂ v (vL ) |v =vQ = ∂πm∂ v (vL ) |v =vQ ≥ 0, and ∂ π∂v ≥ 0. Thus, the optimal decision is 2 ∂ vL L L L L L L L

v∗L = 1. In addition, πmMI (v∗L ) ≥ 1/64. Thus, the optimal decision is v∗L = min{vMI∗ , 1}. L ∂ 2 πmMI (vL ) 1 When ρ ≥ 32 , we have < 0, which means the function is concave in vL . ∂v2 L


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⎧ ⎨

∂πmMI (vL ) ∂πmMD (vL ) |vL =θ ≥ 0 ∂ vL |vL =θ = ∂ vL 1 If ⇒ 32 ≤ QC MD ⎩ ∂πm∂ v (vL ) | Q = ∂πm∂ v (vL ) | Q ≥ 0 vL =vL vL =vL L L

v∗L = min{vMI∗ L , 1}.

⎧ ⎨

1+θ 32θ

, v∗L = min{vMI∗ , 1}. We find that the profit is always higher than 1/64. Thus, L

∂πmMI (vL ) ∂πmMD (vL ) vL =θ = vL =θ ≤ 0 ∂ vL ∂ vL

⎩ ∂π∂ v(vL ) v =vQ = L L L ⎧ MI ⎨ ∂πm∂ v(vL ) vL =θ = L If ∂π QC v ⎩ m∂ v( L ) v =vQ = L L L If

ρ≤

21

QC m

1+θ ( 1 + θ ) ( 4 + γ − θ γ )3 ⇒ ≤ ρ ≤ , v∗L = vLMD∗ . 3 ∂π (vL ) 32θ 32 θ 4 − γ θ ( ) Q ≥ 0 vL =vL ∂ vL MD m

∂πmMD (vL ) vL =θ ≤ 0 ∂ vL

( 1 + θ ) ( 4 + γ − θ γ )3 ∗ ⇒ ρ ≥ , vL = vQC∗ . L 3 ∂π (vL ) 32 θ 4 − γ θ ( ) Q ≤ 0 vL =vL ∂ vL MD m

QC ∗ Comparing the maximum profit with 1/64, we find that πm (vL ) ≥ 1/64 and πmMD (v∗L ) ≥ 1/64 cannot always be satisfied, which depends QC ∗ on the values of θ and γ . For a certain ρ , if the maximum point is located at the threshold of v∗L = vQ and πm (vL = vQL ) = L

(2−θ γ )3 (1+θ ) 1 ≤ 64 ⇒ 8 (4−θ γ )3

√ 3

(2−θ γ )3 (1+θ ) ≤ 8 (4−θ γ )3

θ +1−1 ) , 1}, depending √ θ (2 3 θ +1−1 ) θ, on the values of ρ for the case that three strategies (Strategy S, Strategy MD, and Strategy MI) are feasible. Specifically, (1) if ρ ≤ 1+ 32θ

1+θ ∗ MI∗ ∗ MD ∗ ∗ MD ∗ then vL = min{vL , 1}; (2) if 32θ ≤ ρ ≤ ρ , then vL = vL ; (3) if ρ ≥ ρ , then vL = 0. (ρ is obtained by solving πm (vL (ρ ) ) = 1/64.) 1 64 ,

then the optimal point can never be located in QC region. Therefore, when

γ ≥ γˆ = min{

4(

However, when γ ≤ γˆ , the optimal point can be located in the QC region, and there will be four strategies to choose from. Specifically, θ , then v∗ = min{vMI∗ , 1}; (2) if 1+θ ≤ ρ ≤ ρ , then v∗ = vQC∗ ; (3) if ρ ≤ ρ ≤ ρˆ , then v∗ = vMD∗ ; (3) if ρ ≥ ρˆ , then v∗ = 0, (1) if ρ ≤ 1+ 2 2 L L L L L L L 32θ 32θ where ρ2 =

∂πmMD (vL ) (1+θ )(4+γ −θ γ )3 is obtained by solving |v =vQ = 0 and ρˆ is obtained by solving ∂ vL 32θ (4−γ θ )3 L L

πmQC (v∗L (ρ )) = 1/64.

∂π

Proof for Lemma 6. The first and second order derivatives of ith PI’s profit function are ∂ q gi = θ pH − pL − (1 − θ )θ β (qGi + Gi

N i=1

qGi ) and

∂ 2 πgi ∂π = −2(1 − θ )θ β < 0, respectively. With the symmetry of the PIs, ∂ q gi = θ pH − pL − (1 − θ )θ β (N + 1 )qGi = 0. The optimal diverting ∂ q2Gi Gi θ p −pL θ pH +N pL p quantity is then qGi = (1−θ )θHβ (N+1 . Substituting q into Eq. (7) , we obtain pG = N+1 . The positive sales quantity requires θL ≤ pH , Gi ) (N+1−θ ) pH −N pL pL which is not dependent on N. We further derive qH = 1 − (1−θ )β (N+1 ) . Conversely, when θ ≥ pH , PIs cannot enter the market. Substi-

tuting qH and pG into the profit functions, we formulate the manufacturer’s maximization problem as:

Max

pH , pL ,β

πm ( p H , p L , β ) =

pH 1 − pβH + pL 1 − vpβL − β 2 , L −θ ) pH −N pL pH 1 − (N+1 + pL 1 − vpβL − β 2 , (1−θ )β (N+1 ) L

i f pH ≤ pθL , i f pθL ≤ pH ≤ pθL + (N+1)N(1−θ ) β − vpLL .

(A.9)

This optimization problem can be solved using KKT conditions, which is similar to the proof of problem (A.4). We summarize the optimal decisions in Table B6 in Appendix B. ∂π

p −γ pe

Proof of Lemma 7. The first and second order derivatives of ith PI’s profit function is ∂ q gi = (H1−γ θ )G θ − pL − θ β (qGi + Gi

N

i=1

qGi ) and

∂ 2 πgi ∂π p −γ pe = −2θ β < 0, respectively. With symmetric PIs, the first order condition becomes ∂ q gi = (H1−γ θ )G θ − pL − θ β (N + 1 )qGi = 0. Then ∂ q2Gi Gi e pH −γ peG N pL 1 θ pH −γ pG qGi = θ β (N+1 ) ( (1−γ θ ) θ − pL ). Substituting qGi into Eq. (10), we obtain pG = N+1 (1−γ θ ) + N+1 . The Rational Expectations (RE) equilibrium θ p +(1−γ θ )N p is obtained by equating pG = peG , which yields pG = HN+1−Nγ θ L . Substituting pG into the manufacturer’s and PI’s demand functions, we (N+1 ) p −γ N p θ pH −pL have qH = 1 − (N+1−Hγ θ N )β L and qGi = θ β (N+1 . When pH ≥ pL /θ , the demand of each PI is positive and all PIs can enter Market H −N γ θ )

successfully. However, when pH ≤ pL /θ , all the PIs are deterred out of Market H. Then qH = 1 − βH . Substituting qH and pG into the profit functions, we formulate the manufacturer’s maximization problem for Strategy Q as p

Max

pH , pL ,β

πm ( p H , p L , β ) =

pH 1 − pβH + pL 1 − vpβL − β 2 , L ) p H −γ N p L pH 1 − (N+1 + pL 1 − vpβL − β 2 , (N+1−γ θ N )β L

i f pH ≤ pθL , i f pθL ≤ pH ≤ pθL +

N+1−γ θ N N

β−

pL

.

(A.10)

vL

This optimization problem of (A.10) can be solved using KKT conditions, which is similar to the proof of problem (A.8). We summarize the optimal decisions in Table B7 in Appendix B. Proof of Proposition 6. The results are obtained by comparing the manufacturer’s profit under Strategy S, Strategy M (shown in Table B6) S ≥ πQ, and Strategy Q (shown in Table B7). For a profit maximizing manufacturer, Strategy S dominates the other two strategies when πm m

πmS ≥ πmM . Solving the two inequalities, we obtain that 0 < vL ≤

4γ θ N (1+N −γ θ N )

4[1+(2+γ −γ θ )N]+[ (γ +2 )2 −4γ θ (1+γ )]N 2

the manufacturer’s best choice is Strategy S. Following the same fashion, we can also obtain the dominant region of Strategy Q 2[θ +θ (1−γ θ )N] 2+(2+γ −γ θ )N

and the Strategy M dominant region vL ≥

2[θ +θ (1−γ θ )N] . 2+(2+γ −γ θ )N

. This implies that under this condition, 4γ θ N (1+N −γ θ N )

4[1+(2+γ −γ θ )N]+[ (γ +2 )2 −4γ θ (1+γ )]N 2

≤ vL ≤

The results are presented in Table B8 in Appendix B. Calculations of CS in both markets There are four decision regions, i.e., MI, MD, QC, and S. The customers of Market H, in region MI, MD, and S, will not buy gray products, pi

1

i = ∫ (β V − pi )dV = and the indifference point is VH1 = βH . Then CSH H H H V H1

2

(β −piH ) , i = {MI, MD, S )}. However, in region QC, customers with 2β


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Fig. A.1. The total profit change in market H and L with and without PI (when γ = θ = 0.5). Table B1 Equilibrium optimal solutions and profits for Strategy M. MC1 : 0 ≤ vL ≤ vM1 L (θ ) pM∗ H pM∗ L

βM∗ pM∗ G qM∗ H qM∗ L qM∗ G

πmM∗ πgM∗

[4θ 2 (1−θ ) −4θ (1−θ ) vL +v2L ][4θ 2 (1−θ ) −4θ (1−2θ )(1−θ )vL +(8θ 2 −8θ +1)v2L ] 2

2

2

2

[2θ

2

16[2θ 2 (2−θ )(1−θ )−4θ (1−θ )vL +v2L ] (3−2θ )(1−θ )vL +θ (4θ −3)v2L ][4θ 2 (1−θ )2 −4θ (1−2θ )(1−θ )vL +(8θ 2 −8θ +1)v2L ] 2

16[2θ 2 (2−θ )(1−θ )−4θ (1−θ )vL +v2L ] 2 4θ 2 (1−θ ) −4θ (1−2θ )(1−θ )vL +(8θ 2 −8θ +1)v2L 2 8[2θ (2−θ )(1−θ )−4θ (1−θ )vL +v2L ] θ [2θ 2 (1−θ )2 +(1−θ )θ vL +(2θ −1)v2L ][4θ 2 (1−θ )2 −4θ (1−2θ )(1−θ )vL +(8θ 2 −8θ +1)v2L ] 2

16[2θ 2 (2−θ )(1−θ )−4θ (1−θ )vL +v2L ] 2θ 2 (2−θ )(1−θ )+θ (5θ −4 )vL +(−2θ +1 )v2L 2[2θ 2 (2−θ )(1−θ )−4θ (1−θ )vL +v2L ] 2θ 2 (1−θ )+θ (4θ −5 )vL +2v2L 2[2θ 2 (2−θ )(1−θ )−4θ (1−θ )vL +v2L ] 2θ 2 (1−θ )+θ (4θ −5 )vL +2v2L 2[2θ 2 (2−θ )(1−θ )−4θ (1−θ )vL +v2L ] 2 2 4θ 2 (1−θ ) −4θ (1−2θ )(1−θ )vL +(8θ 2 −8θ +1)vL2 8[2θ 2 (2−θ )(1−θ )−4θ (1−θ )vL +v2L ]

θ (1−θ )(−2θ 3 +2θ 2 +(4θ 2 −5θ )vL +2v2L )2 [4θ 2 (1−θ )2 −4θ (1−2θ )(1−θ )vL +(8θ 2 −8θ +1)v2L ] 3 32[2θ 2 (2−θ )(1−θ )−4θ (1−θ )vL +v2L ] M2 MC2 : Max(vM1 L ( θ ), 0 ) ≤ vL ≤ vL ( θ )

MD : vM2 L ( θ ) ≤ vL ≤ θ

MI: θ ≤ vL ≤ 1

pM∗ H

2(1−θ ) (vL −4θ +4 )[2−2θ +(3−θ )vL ] 2 [8(2−θ )(1−θ )−vL ]

v2L (1+θ )3 2 16(θ 2 +vL )

1+vL 16

pM∗ L

2(1−θ ) (5−2θ )[2−2θ +(3−θ )vL ]vL 2 [8(2−θ )(1−θ )−vL ] (1−θ )[2−2θ +(3−θ )vL ] 8(2−θ )(1−θ )−vL (1−θ )2 [(5−θ )vL +4θ (1−θ )](2−2θ +(3−θ )vL ) 2 [8(2−θ )(1−θ )−vL ] (2−θ )(vL +4−4θ ) 8(2−θ )(1−θ )−vL −2θ (5−2θ )+6−vL 8(2−θ )(1−θ )−vL 4θ (1−θ )+(−5+3θ )vL θ (8(2−θ )(1−θ )−vL ) (1−θ )2 [2−2θ +(3−θ )vL ]2 2 [8(2−θ )(1−θ )−vL ] 2 (1−θ )2 [(5−θ )vL −4θ 2 +4θ ] [2−2θ +(3−θ )vL ] θ [8(2−θ )(1−θ )−vL ]3

v (1+θ ) θ

vL (1+vL )

βM∗ pM∗ G qM∗ H qM∗ L qM∗ G

πmM∗ πgM∗

2

2

2 L

3

16(θ 2 +vL ) (1+θ )2 vL 8 ( θ 2 +vL )

2

16

1+vL 8

N/A

N/A

2 θ 2 + ( 1 −θ ) v L 2 ( θ 2 +vL ) θ 2 −θ +2vL 2 ( θ 2 +vL )

1 2 1 2

0

0

(1+θ )4 v2L 2 64(θ 2 +vL )

(1+vL )2

0

0

64

Note: For Strategy M, regions of competing with full and partial diversion, deterrence, and ignorance are denoted as MC1 , MC2 , MD, and MI, respectively. pQC −γ pQC

high valuations will buy the authorized products (with indifference point VH2 = (H1−γ θ )Gβ ), and those with lower valuations will buy gray 2

2

VH2 1 pQC γ ( pQC ) ( pQC −γ pQC ) QC products (with indifference point VH3 = θGβ ). Then CSH = ∫ (β VH − pQC )dVH + ∫ γ (θ βVH − pQC )dVH = β2 − pQC + 2θGβ + 2H(1−γ θG)β . H G H V V H2

H3

By substituting the equilibrium results in Proposition 3 into the surplus functions, we obtain the customer surplus in Table B4 in Appendix B. The customers of Market L can only buy authorized products. In region S, CSLS = 0 because no products will be sold in Market L. In regions MI and MD, they will buy authorized products in the first period without PI. Customers with high valuations will buy authorized p

2 vL (βV −p j ) (βv −pi ) L L dVL = 2Lv βL , j = {MI, MD}. In region QC, customer will buy authovL L

j

j products (with indifference point VL1 = βL ). Then CSL = ∫ V

L1

pQC L

2 vL γ (βV −pQC ) (qQC −qQC ) −qQC ) γ (βvL −pQC ) (qQC L L G L L ∗ L QC G ]dVL = . vL 2vL β q qQC

rized products in the second period when VL ≥ VL2 = β and CSLQC = ∫ [ V L2

L

L

By substituting the equilibrium results in Proposition 3 into the surplus functions, we obtain the customer surplus of Market L in Table B4 in Appendix B. Appendix B. Tables


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23

Table B2 Equilibrium optimal solutions and profits for Strategy Q. QC : 0 ≤ vL ≤ vQL (θ , γ ) pQ∗ H pQ∗ L

β

Q∗

pQ∗ G qQ∗ H qQ∗ L qQ∗ G

πmQ∗ πgQ∗

QD : vQL (θ , γ ) ≤ vL ≤ θ

(2−γ θ ) [2(2−γ θ )+γ vL ][2−γ θ +(γ +2 )vL ] 2 2 [ 8 ( 2 −γ θ ) −γ 2 v L ] (2−γ θ )2 (4+γ )vL [2−γ θ +(γ +2 )vL ] 2 2 2 [ 8 ( 2 −γ θ ) −γ v L ] (2−γ θ )[2−γ θ +(γ +2 )vL ] 16(2−γ θ )−2γ 2 vL (2−γ θ )[2θ (2−γ θ )+(−3γ θ −γ 2 θ +γ +4)vL ][2−γ θ +(γ +2 )vL ] 2 2 [ 8 ( 2 −γ θ ) −γ 2 v L ] 2(4−2γ θ +γ vL ) 8 ( 2 −γ θ ) −γ 2 v L (4−γ )(2−γ θ )−γ 2 vL 8 ( 2 −γ θ ) −γ 2 v L 2θ (2−γ θ )+(γ θ −γ −4 )vL θ [ 8 ( 2 −γ θ ) −γ 2 v L ] (2−γ θ )2 [2−γ θ +(γ +2 )vL ]2 2 4 [ 8 ( 2 −γ θ ) −γ 2 v L ] 2 (2−γ θ )[−4θ +2γ θ 2 +(γ +4−γ θ )vL ] [2−γ θ +(γ +2 )vL ] 3 2θ (16−8γ θ −γ 2 vL ) 2

QI: θ ≤ vL ≤ 1

v (1+θ ) 2 L

3

1+vL 16

16(θ 2 +vL ) v2L (1+θ )3 θ

2

vL (1+vL )

16(θ 2 +vL ) (1+θ )2 vL 8 ( θ 2 +vL )

2

16

1+vL 8

N/A

N/A

2 θ 2 + ( 1 −θ ) v L 2 ( θ 2 +vL ) θ 2 −θ +2vL 2 ( θ 2 +vL )

1 2 1 2

0

0

(1+θ )4 v2L

(1+vL )2

64(θ 2 +vL )

2

64

0

0

Note: For Strategy Q, regions of competition, deterrence, and ignorance are denoted as QC, QD, and QI respectively.

Table B3 Manufacturer’s distribution choice and corresponding optimal decisions. S : 0 ≤ vL ≤ vSL (θ , γ )

QC : vSL (θ , γ ) ≤ vL ≤ vQL (θ , γ )

MD : vQL (θ , γ ) ≤ vL ≤ θ

MI: θ ≤ vL ≤ 1

p∗H

1 16

(2−γ θ )2 [2(2−γ θ )+γ vL ][2−γ θ +(γ +2 )vL ] 2 2 [ 8 ( 2 −γ θ ) −γ 2 v L ] (2−γ θ )2 (4+γ )vL [2−γ θ +(γ +2 )vL ] 2 2 [ 8 ( 2 −γ θ ) −γ 2 v L ] (2−γ θ )[2−γ θ +(γ +2 )vL ] 2[8(2−γ θ )−γ 2 vL ] (2−γ θ )[2θ (2−γ θ )+(−3γ θ −γ 2 θ +γ +4)vL ][2−γ θ +(γ +2 )vL ] 2 2 [ 8 ( 2 −γ θ ) −γ 2 v L ] 2(4−2γ θ +γ vL ) 2 8 ( 2 −γ θ ) −γ v L (4−γ )(2−γ θ )−γ 2 vL 8 ( 2 −γ θ ) −γ 2 v L 2θ (2−γ θ )+(γ θ −γ −4 )vL θ [ 8 ( 2 −γ θ ) −γ 2 v L ] (2−γ θ )2 [2−γ θ +(γ +2 )vL ]2 2 4 [ 8 ( 2 −γ θ ) −γ 2 v L ] 2 (2−γ θ )[−4θ +2γ θ 2 +(γ +4−γ θ )vL ] [2−γ θ +(γ +2 )vL ] 3 2θ (16−8γ θ −γ 2 vL )

v2L (1+θ )3 2 16(θ 2 +vL ) v2L (1+θ )3 θ 2 16(θ 2 +vL ) (1+θ )2 vL 8 ( θ 2 +vL )

1+vL 16

N/A

N/A

2 θ 2 + ( 1 −θ ) v L 2 ( θ 2 +vL ) θ 2 −θ +2vL 2 ( θ 2 +vL )

1 2 1 2

0

0

(1+θ )4 v2L 2 64(θ 2 +vL )

(1+vL )2

0

0

p∗L

N/A

β∗

1 8

p∗G

N/A

q∗H

1 2

q∗L q∗G

0

πm∗

1 64

πg∗

N/A

0

vL (1+vL ) 16

1+vL 8

64

Table B4 Customers’ surplus (CS) in markets H and L in different regions. Ranges

In market H

Region MI

vL +1

In market L (vL +1 )vL

64

2

vL (θ + 1)2 (2θ 2 − vL θ +vL ) 3 64(θ 2 + vL ) 2 2 γ ( pQC∗ ) ( pQC∗ −γ pQC∗ ) β QC∗ QC∗ G − p + + 2(H1−γ θ )βG QC∗ 2 H 2θ β QC∗

Region MD Region QC

1 16

Region S

64

2

vL 2 (θ + 1)2 (θ 2 − θ + 2vL ) 3 64(θ 2 + vL ) 2 −qQC∗ ) γ (β QC∗ vL −pQC∗ ) (qQC∗ L G L QC∗ 2vL β qQC∗ L

0

Note: In Region QC, pQC∗ , pQC∗ , pQC∗ , β QC ∗ , qQC∗ , and qQC∗ are given in Table B2. H G L L G

Table B5 ∗ MD∗ and vLQC ∗ . Definitions of γˆ , ρ 1 , ρ 2 , ρˆ , ρˇ , vMI L , vL

√

3 4( θ +1−1 ) √ ,1 θ (2 3 θ +1−1)

γˆ

min

ρ1 ρ2 ρˆ

1+θ 32θ (1+θ )(4+γ −θ γ )3 3 32θ (4−γ θ ) Root of mQC ∗L

π (v (ρ ) ) =

1 64

∂π MI∗

vMI∗ L

Root of ∂ vm = 0 L

vLMD∗ vQC∗ L ρˇ

Root of

∂πmMD∗ ∂ vL = 0 QC∗ Root of ∂π∂ vm = 0 L

Root of πmMD (v∗L (ρ ) ) =

1 64


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Table B6 Equilibrium optimal solutions and profits for Strategy M when N ≥ 2. MC1 : 0 ≤ vL ≤ vˆ M1 L (θ , γ , N )

1 −θ ) + (N+1)( βM∗ C1 − N

p∗L|MC 1

p∗H

θ

p∗L|MC 1

vL

ˆ M2 MC2 : Max(vˆ M1 L ( θ , γ , N ), 0 ) ≤ vL ≤ vL ( θ , γ , N )

MD : vˆ M2 L ( θ , γ , N ) ≤ vL ≤ θ

MI: θ ≤ vL ≤ 1

p∗H |MC 2

v2L (1+θ )3 2 16(θ 2 +vL ) v2L (1+θ )3 θ 2 16(θ 2 +vL ) (1+θ )2 vL 8 ( θ 2 +vL )

1+vL 16

N/A

N/A

2 θ 2 + ( 1 −θ ) v L 2 ( θ 2 +vL )

1 2 1 2

θ p∗H|MC −p∗L|MC 2 2 θ (1−θ )(N+1 )βM∗ C

θ 2 −θ +2vL 2 ( θ 2 +vL )

0

0

(β

(1+θ )4 v2L 2 64(θ 2 +vL )

(1+vL )2

0

0

p∗L

p∗L|MC

p∗L|MC

β∗

βM∗ C1

βM∗ C2

p∗G

θ p∗H|MC1

1

q∗H

1−

q∗L

2

∗ − (1 − θ )θ βM C1 −

p∗H|MC 1

1

vL

θ p∗H|MC +N p∗L|MC

1

2

L

1−

1

π

(β ) 1 θ p∗H|MC1 − p∗L|MC1 − (1 − θ )θ × βM∗ C1 − N

πgi∗

2

1− v β

1 N

1− v β

(N+1−θ ) p∗H|MC −N p∗L|MC 2 2 (1−θ )(N+1 )βM∗ C p∗L|MC 2 ∗ L MC 2

q∗Gi ∗ m

2

1+N

βM∗ C

p∗L|MC 1 ∗ L MC 1 p∗L|MC 1 ∗ L MC 1

1− v β

p∗L|MC

p∗ L|MC1 ∗ + θ βM C − v

2

∗ 2 MC2

p∗L|MC

vL

1

)

∗ 2 MC2

(θ p∗H|MC −p∗L|MC )

2

p∗L|MC

× (1 − v β ∗ 1 ) L MC 1

2

2 2 ∗ MC2

θ (1−θ )(N+1 ) β

vL (1+vL ) 16

1+vL 8

64

(θ −θ 2 )N2 +(2θ 3 −3θ 2 +θ )N+2θ 3 −4θ 2 +2θ 2θ (1−θ )(1+N ) where vˆ M1 , and vˆ M2 L (θ , γ , N ) = L (θ , γ , N ) = 2−2θ +(3−θ )N . 2(1−θ )N +(3−2θ )N 2 (N+1 )(θ −1 )

N−2θ +2

( 2− ( θ + ) N )βMC ( (θ 4 −4θ 3 vL −2θ 3 +4θ 2 v2L +6θ 2 vL +θ 2 −4θ v2L −2θ vL +v2L )N2 +(2θ 4 −4θ 3 vL −4θ 3 +4θ 2 v2L +6θ 2 vL +2θ 2 −4θ v2L −2θ vL )N+θ 4 −2θ 3 +θ 2 ) ∗ N vL ∗ 1 , βM pH |MC C1 = 8( (−θ 3 +2θ 2 vL +θ 2 −2θ vL +v2L )N2 +(θ 4 −3θ 3 +2θ 2 vL +2θ 2 −2θ vL )N+θ 4 −2θ 3 +θ 2 ) 2 ( v2 +( θ2 + 2(N +1N)(v θ − 1) )(N−( θ1 + (N + 1N)(v θ − 1) )(N−θ +1 )) (N+1)(1 θ −1) ) L L L ∗ (N+1 )(1−θ )(2−2θ +(2−2θ +vL )N )βM∗ C v L (N+1 )(1−θ )(2−2θ +3N )βMC (N+1 )(1−θ )( (1−θ )(1+vL )+(1−θ +2vL )N ) ∗ ∗ 2 2 , p , β = = 2 2 2 MC2 L|MC2 4(1−θ ) +(8−12θ +4θ 2 )N +(4−4θ −vL )N2 4(1−θ ) +(8−12θ +4θ 2 )N +(4−4θ −vL )N2 8(1−θ ) +8(2−3θ +1θ 2 )N +2(4−4θ −vL )N2

p∗L|MC =

1

∗

1

=

Table B7 Equilibrium optimal solutions and profits for Strategy Q when N > 2. QC : 0 ≤ vL ≤ vˆ QL (θ , γ , N )

QD : vˆ QL (θ , γ , N ) ≤ vL ≤ θ

QI: θ ≤ vL ≤ 1

p∗H

2+4N +(γ vL −4γ θ )N +(1−γ θ )(2−2γ θ +γ vL )N 2 4+4(2−γ θ )N +(4−γ 2 vL −4γ θ )N 2

v2L (1+θ )3 2 16(θ 2 +vL )

1+vL 16

p∗L

(2N+γ N+2)(N+1−γ θ )vL 4+4(2−γ θ )N +(4−γ 2 vL −4γ θ )N 2

v2L (1+θ )3 θ

vL (1+vL )

β∗

(N−γ θ N+1)[1+vL +(1+vL −γ θ +γ vL )N] 2[4+4(2−γ θ )N +(4−γ 2 vL −4γ θ )N 2 ]

p∗G

2θ +(2vL +2θ −γ θ vL −2γ θ 2 )N+(2vL +γ vL −2γ θ vL −γ 2 θ vL )N 2 4+4(2−γ θ )N +(4−γ 2 vL −4γ θ )N 2

q∗H

(N+1 )(2+2N−2γ θ N+γ vL N ) 4+4(2−γ θ )N +(4−γ 2 vL −4γ θ )N 2

q∗L q∗Gi

πm∗ π

∗ gi

β∗

β∗

16(θ 2 +vL )

2

16

(1+θ ) vL 8 ( θ 2 +vL )

1+vL 8

N/A

N/A

2 θ 2 + ( 1 −θ ) v L 2 ( θ 2 +vL )

1 2

2+(4−γ −2γ θ )N +(2−γ +γ 2 θ −γ 2 vL −2γ θ )N 2 4+4(2−γ θ )N +(4−γ 2 vL −4γ θ )N 2

θ 2 −θ +2vL 2 ( θ 2 +vL )

1 2

[2θ −2vL +(2θ −2vL −γ vL −2γ θ +γ θ vL )N] θ [4+4(2−γ θ )N+(4−γ 2 vL −4γ θ )N2 ]

0

0

2

β∗

2


2

(1+vL )2

64(θ 2 +vL )

2

2

N [2vL −2θ +(2vL −2θ +γ vL +2γ θ 2 −γ θ vL )N]

θ [4+4(2−γ θ )N+(4−γ 2 vL −4γ θ )N2 ]2

(1+θ ) v

4 2 L

β

∗

64

0

0

Table B8 Manufacturer’s distribution choice and corresponding optimal decisions for N > 2.

p∗H

S : 0 ≤ vL ≤ vˆ SL (θ , γ , N )

QC : vˆ SL (θ , γ , N ) ≤ vL ≤ vˆ QL (θ , γ , N )

MD : vˆ QL (θ , γ , N ) ≤ vL ≤ θ

MI: θ ≤ vL ≤ 1

1 16

2+(4−4γ θ +γ vL )N+(1−γ θ )(2−2γ θ +γ vL )N 2 4+4(2−γ θ )N +(4−γ 2 vL −4γ θ )N 2

v2L (1+θ )3 2 16(θ 2 +vL )

1+vL 16

v2L (1+θ ) θ

vL (1+vL )

β

∗

3

N/A

(2N+γ N+2)(N+1−γ θ )vL 4+4(2−γ θ )N +(4−γ 2 vL −4γ θ )N 2

1 8


p∗G

N/A

2θ +(2vL +2θ −γ θ vL −2γ θ 2 )N +(2+γ )(1−γ θ ))vL N 2 4+4(2−γ θ )N +(4−γ 2 vL −4γ θ )N 2

q∗H

1 2

(N+1 )(2+2N−2γ θ N+γ vL N ) 4+4(2−γ θ )N +(4−γ 2 vL −4γ θ )N 2

p∗L

β

∗

q∗L q∗Gi

πm∗ π

∗ gi

β∗

16(θ 2 +vL )

2

16

(1+θ ) vL 8 ( θ 2 +vL )

1+vL 8

N/A

N/A

2 θ 2 + ( 1 −θ ) v L 2 ( θ 2 +vL )

1 2

0

2+(4−γ −2γ θ )N +(2−γ +γ 2 θ −γ 2 vL −2γ θ )N 2 4+4(2−γ θ )N +(4−γ 2 vL −4γ θ )N 2

θ 2 −θ +2vL 2 ( θ 2 +vL )

1 2

0

[2θ −2vL +(2θ −2vL −γ vL −2γ θ +γ θ vL )N] θ [4+4(2−γ θ )N+(4−γ 2 vL −4γ θ )N2 ]

0

0

1 64

2

2


2

2

N [2vL −2θ +(2vL −2θ +γ vL +2γ θ 2 −γ θ vL )N]

θ [4+4(2−γ θ )N+(4−γ 2 vL −4γ θ )N2 ]2

(1+θ ) v

4 2 L

64(θ 2 +vL ) 2

N/A

β∗

β

∗

0

(1+vL )2 64

0


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Competitive strategies and quality to counter parallel

Competitive strategies and quality to counter parallel

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