Impacts of gray market selling on the supply chain under product upgrade and pricing flexibility decisions

Gray markets are infamous for unauthorized diversion of authentic products at lower prices to harm the manufacturer's authorized distribution channel. Hence, manufacturers are becoming more proactive in shaping strategies that can counter gray markets. By considering the risk of unauthorized selling through gray markets, we analyze the manufacturer's strategic channel choice and product upgrade decision. We analytically explore the provision of granting flexibility to the legitimate retailer to adjust price during the selling season as a strategy to cope with gray market. We find that unauthorized selling through gray markets has a severe negative impact on the manufacturer's profitability depending on the degree of channel differentiation and product brand equity. We counter‐intuitively reveal that the overall decentralized supply chain can be better‐off, owing to higher product leakage, especially because the legitimate retailer gains an increased sales volume through product diversion to the gray market. However, the manufacturer's loss is more severe in such a decentralized supply chain. We show that the manufacturer can eliminate unauthorized sales through gray market by offering price adjustment flexibility to the retailer. Another compelling finding exhibits that the manufacturer's decision to launch an upgraded product can intensify the diversion of an existing product. However, unfair competition from the unauthorized channel cannibalizes the sales of an upgraded product, thereby making the manufacturer worse‐off. Interestingly, despite the cannibalization of upgraded product sales, the manufacturer can be better‐off in terms of overall profitability via price adjustment flexibility if both the degree of upgrade and brand equity are higher. Overall, in addition to deploying monitoring mechanisms for supervising legitimate retailers, price adjustment flexibility can reduce product diversion if the degree of upgrade is moderate and channel differentiation is higher.

and popular in various industries, such as pharmaceutical, apparel, consumer electronics, IT/computers, automobile, cosmetics, and copyrighted works (Altug & Sahin, 2019).The extent of gray market's popularity can be understood given that large e-commerce players, including Amazon, e-Bay, and Alibaba, have often been responsible for facilitating gray marketers (Drucker & Gregg, 2019).Furthermore, the gray market's omnipresence can be easily gauged with industry-level sales within diverse industries and across geographies.For instance, the European Federation of Pharmaceutical Industries and Associations (EFPIA) estimated that the volume of trade "transpiring" through the gray market in the European Union in 2017 exceeded €5400 million (EFPIA, 2019).Furthermore, in 2008, the IT industry in the United States was ascertained to have diverted products worth $58 billion into gray markets (KPMG, 2016).Similarly, gray market sales have accounted for approximately 70% of personal computers as well as cell phone sales in the Southeast Asian countries, such as India and Malaysia (Antia et al., 2004).Strikingly evident during the COVID-19 pandemic, gray markets have been observed to flourish in a variety of industries.The healthcare industry has experienced a severe blow in terms of "artificial shortages" and illegal diversion1 .For example, 3M, a US-based multinational enterprise, has taken legal action against unauthorized distributors, such as Performance Supply (a New Jersey-based company) and Rx2Live (an American healthcare company based in Utah), who have been found selling 3M-branded N95 masks (Maynard, 2020).Generally, the pandemic has propelled e-commerce and triggered brands to have greater presence on third-party marketplaces, thus providing an impetus vis-à-vis accessibility to unauthorized intermediaries.
Gray market diversion can rely on diverse mechanisms.Parallel imports, being a significant one, tend to exploit the price discrimination strategy of manufacturers.A manufacturer can adopt differential pricing schemes based on either geography or consumer's willingness to pay.Notorious gray marketers through parallel imports can source a product from the market with a lower price and place it for resale in another market that sells the same product at a higher price, however, at a price relatively lower than that of the authorized channel (Ahmadi et al., 2017;Altug & Sahin, 2019;Li et al., 2022).The case of Nestle's Perugina (NP) chocolate could explain parallel imports whereby gray marketers imported NP chocolates from Venezuela and subsequently diverted them to Puerto Rico at lower prices without Nestle's authorization (Societe des Produits Nestle, S. A. v. Casa Helvetia, 1992).
Apart from parallel imports, gray markets can flourish domestically (without imports).Such gray marketers are usually in search of opportunistic retailers who desire additional sales and the retailers holding unsold inventory, and seek prospects of luring such retailers to clear their inventory at lower prices (Altug, 2017;Dasu et al., 2012).Several electronic equipments manufactured by Pioneer and Harman Kardon were recently reported to have been acquired by gray marketers from the authorized channel and were subsequently diverted within the same market for low-price sales.
While the domestic bootlegging-based gray market activity evidently signifies a moral hazard on the part of an authorized retailer, it may point to a failure of supply chain governance as well as malign from the focal manufacturer's perspective.To explain these aspects, the extant literature recognizes that authorized distributors/retailers may unduly escalate their order quantities to acquire manufacturer's channel incentives, such as trade promotions, quantity discounts, and restrictive buying (Su & Mukhopadhyay, 2012).Capital City Micro, an authorized distributor for Hewlett-Packard (HP) was once accused of ordering laptops in bulk to exploit quantity discount and later diverting the excess inventory to gray markets (Hu et al., 2013).Similarly, the healthcare industry has documented cases whereby downstream firms dealing in pharmaceuticals or medical devices have been found to misuse quantity discounts via gray markets (Grant, 2018).
For the harms, without good measures, gray markets create negative externalities for the manufacturer (focal manufacturer or brand) in terms of reduced profit margins, inducement of tension in channel relationships, and erosion of brand image owing to the availability of products to unintended consumer segments (Antia et al., 2006).The damage imposed on the manufacturer can be further gauged through gray markets' free ride on the sales promotion efforts of authorized members (Ahmadi et al., 2015).Consequently, sensitive manufacturers, such as "Burberry" in the United Kingdom, chose to retaliate by destroying their unsold product inventory (Table 1) at the end of their selling season to counter gray markets and protect brand image (Paton, 2018).Thus, devoid of auditing, monitoring, and control mechanisms, manufacturers are now becoming watchful to counter unintended product diversion to gray markets (KPMG, 2016).In the short run, brand dilution owing to gray market sales largely affects the manufacturer with a negligible direct consequence on the retailer.However, in the long run, it can reduce sale quantity of the retailer due to negative word-of-mouth about the brand because of its visibility in the unauthorized gray market channel.Given the difficulties such diversions are posing to truthful authorized dealers, intended customer segment, and public/private sector policymakers, the need to devise safeguards against gray markets, is a latent concern within the sphere of practice.
To combat the operations of gray markets through either parallel importation or domestic bootlegging, market-level as well as supply chain level interventions have been proposed in prior studies.On the one hand, market-level interventions toward controlling such gray markets can target parallel importation by focusing on price gap reduction between two

Brand(s)
Actions to protect their brands from unauthorized selling Burberry Burned unsold inventory worth £28.6 million in year 2018.The total value of goods destroyed in the last 5 years is more than £90 million (Paton, 2018).
Richemont (Cartier, Piaget, and Baume & Mercier) Destroyed about $563 million worth of watches over the past two years to prevent unauthorized selling (Lieber, 2018).
Louis Vuitton Burned excess merchandise to avoid damage to their brand reputation owing to product diversion to the unauthorized channel (i) .
markets, so that gray marketers find it difficult or unprofitable to divert a product (Ahmadi et al., 2017;Autrey et al., 2015).
To achieve this, the manufacturer would require a better understanding of the taxation policy prevailing in its operative market to rationalize price discrimination, and can also focus on altering product functionality to make the product of one market inoperative in the other.Supply chain interventions, on the other hand, target domestic bootlegging via strategies based on contract design, pricing mechanisms, offering service benefits and technological sales monitoring mechanisms to minimize the incentive for any entity in the supply chain to promote gray market diversion (Srivastava & Mateen, 2020;Altug, 2017).However, assumptions of extant supply chain mechanisms deliberated are quite static in terms of their consideration of dynamic aspects, such as short product life cycle, launch of an upgraded version of the existing product, and the retailer's flexibility to adjust the price of the existing product during the selling season.Most importantly, regarding pricing, the extant research has assigned the authorized channel's price as unchanged during the selling season to curtail a realistic view (Smith & Shulman, 2022;Srivastava & Mateen, 2020).Therefore, given this insufficiency in the literature, we examine the viability of capitalizing on empowering the retailer with significant flexibility over pricing decisions to control gray market diversion based on domestic bootlegging.More specifically, we explicate the possibility of deploying a price adjustment flexibility mechanism under a two-period scheme.Within this mechanism, a manufacturer can empower its authorized retailers with the autonomy to markdown the existing product's price in the second selling period.Markdown pricing is a strategy that enables the retailers to not only increase sales and revenue but also aids in clearing inventory.Thereby, apart from limiting gray markets, the supply chain can draw additional surplus by capturing the demand of price-sensitive consumers otherwise satiated through gray markets.However, in practice, most manufacturers do not grant such price adjustment flexibility to the retailers.Instead, the manufacturers aim at neutralizing pricing variability and ensure standard product prices across retailers by communicating the "manufacturer suggested retail price (MSRP)".For example, the automakers in the US automobile industry display the MSRP on the wind shield to ensure transparency in prices2 .Another classic example is of Apple Inc. which employs the control pricing strategy to minimize price variability by using minimum advertised price to keep prices relatively comparable to its own stores and offers minimal wholesale discounts to prevent retailers from passing discounts onto consumers3 .The two-period scheme further includes a critical consideration from a product life cycle (PLC) perspective that extends the view to product upgrades.Therefore, manufacturers tend to incrementally upgrade their products by augmenting them with additional attributes and features every year until the PLC declines.Manufacturers usually design the next generation of an existing product to generate more revenue by revitalizing the demand (Mehra et al., 2014).However, in the ongoing selling season, it is important to note that despite the availability of a newly upgraded product, the previous generation product usually remains available for sale for a while within the authorized distribution channels of the manufacturer.This configuration therefore risks the onset of domesticated gray market bootlegging by an opportunistic retailer, wherein the retailer can specifically divert an original product from the previous generation to gray marketers.
In addition to the opportunity to earn additional revenue, the necessity to serve continuously changing consumer needs is yet another reason for brands to launch upgraded versions of their products.For example, Samsung launched the upgraded version of Galaxy S21 Ultra after almost a year keeping in view the consumers' preferences with respect to display, aperture, camera resolution, and internal storage 4 .Therefore, in order to survive in the competitive market amidst ongoing technological progress it becomes essential for brands to upgrade their products to keep pace with dynamic consumer preferences.In this pursuit of continuous product upgrade, the authorized retailers feel pressurized to clear existing product inventory due to its reduced shelf-life.Therefore, retailer's risk of salvaging the product at a lower price and opportunity to increase the sales volume are the two key reasons for product diversion to unauthorized gray markets.
From the above discussion, an interesting question arises on whether a greater degree of price adjustment flexibility granted to the retailer can effectively restrict gray market diversion.Through a greater control over the pricing of an existing product in its second selling season, a product from the previous generation can be sold at a markdown price by the authorized retailer.This price adjustment flexibility can possibly discourage the retailers who resort to gray market diversion.Extending further, the present analysis considers the manufacturer's decision to launch the next generation of a product within the ongoing selling season.Thus, we attempt to understand whether the manufacturer's strategy of granting price adjustment flexibility to the authorized retailer and the careful launch of an upgraded version in the second period is capable of shifting consumers' preference from the gray market to the authorized channel.Furthermore, how it impacts consumer surplus (CS) and social welfare (SW) is important.The following research questions are put forward to encapsulate the preceding discussion: RQ.1 What is the impact of internal channel structure on the profitability of supply chain members in the presence of unauthorized gray market selling?How does the manufacturer's product upgrade decision in the presence of unauthorized gray market selling affect the performance of supply chain members?

RQ.2 Can the provision of price adjustment flexibility deter unauthorized product diversion toward gray markets? RQ.3 How do factors such as internal channel structure, product upgrade decision, and price adjustment flexibility affect CS and SW in the presence of gray markets?
Addressing these research questions is important for several reasons.First, it expands the understanding of gray market selling by considering dynamic forces that impact the decision of a retailer to resort to the gray market based on domestic bootlegging.Second, in doing so, diverse scenarios created to explicate profitability implications can unearth strategies to control gray market diversion.For instance, it can determine whether profitability and brand erosion restrict the manufacturer's freedom of channel choice between centralized and decentralized supply chain structures in the presence of gray markets.Third, regarding product upgrade, we uncover the critical aspects of gray markets' existence that a manufacturer needs to consider.Fourth, we find the conditions (and hence factors) wherein offering price adjustment flexibility to the retailer can be an optimal strategy for the manufacturer to deter product diversion toward the gray market.
Overall, our key findings include the following: (i) It is counterintuitive to note that in the presence of a gray market based unauthorized selling, the total supply chain profit can be higher under a decentralized supply chain when the product possesses higher brand equity.However, the manufacturer remains worse-off.(ii) The manufacturer can eliminate gray market selling by offering price adjustment flexibility to the retailer.This makes the manufacturer better-off irrespective of the reduced total channel sales due to gains from the elimination of unauthorized gray market sales.(iii) A manufacturer's myopic product launch decision can intensify product diversion to gray markets.Thus, a manufacturer in this scenario becomes worse-off due to the associated product upgrade costs and demand cannibalization from gray market selling.(iv) In the presence of upgraded product launch, offering price adjustment flexibility can be a helpful mechanism to reduce gray market diversion as it encourages the retailer to adhere to the authorized channel.Therefore, the manufacturer and the retailer can benefit by reinforcing the prominence of the authorized channel.
The remaining paper is organized as follows: we present a literature review of pertinent research in Section 2. In section 3, we conceptualize the modeling framework under domestic gray market setting.Subsequently, comparative analysis is presented to draw implications in Section 4. The models under the parallel import setting and the comparative analysis is entailed in Section 5. Several important extended models are presented in Section 6.The last section concludes to encapsulate key implications and recommendations for theory and practice.To enhance the presentation, detailed sensitivity analysis is included in the Online Appendix.All proofs are provided in the Online Supplementary Appendix.

Gray markets: Parallel import
The literature on gray markets primarily revolves around the agenda of preventing unintended inventory diversion to gray markets.Parallel imports have been a traditional anchor to examine the emergence of gray markets by focusing on the unauthorized flow of products across countries (Altug & Sahin, 2019;Autrey et al., 2014).The literature herein primarily focuses on examining how the presence of parallel importation can curtail a manufacturer's degree of freedom with regard to product launch decision across countries, brand erosion, competition in terms of price setting, and profitability (Antia et al., 2004).Control strategies for parallel importation often rely on shaping legal mechanisms that prevent imports, achieve careful channel management of global value chains, involve re-thinking market segmentation, and make use of market intelligence (Altug & Sahin, 2019;Antia et al., 2004).For instance, Ahmadi et al. (2015) show that within parallel importation, minimizing the price difference between high-and low-priced markets proves to be more beneficial in curbing gray markets than reducing the product supply to a low-priced market.

Domestic gray markets: Reasons for emergence and potential business solutions
Gray markets are more than parallel importation in the form of domestic bootlegging.For instance, an authorized retailer can divert its leftovers to unauthorized retailers in the local market (Ahmadi et al., 2017;Su & Mukhopadhyay, 2012).The motivation to divert here mainly involves the abuse of channel incentives as well as mitigating the salvage risk at the end of the selling season (Altug, 2017;Shao et al., 2016).Products available through gray markets can be available at cheaper prices than the authorized channel.While such domestic diversions can deteriorate the manufacturer's reputation and brand image, the intended target audience of the product can also perceive loss after buying the same product at its original price (Liao & Hsieh, 2013).
Contracts, being the fundamental realm of interventions within domestic bootlegging, have often been focused upon to examine the performance of different contractual options under the risk of gray market-oriented product diversion (Su & Mukhopadhyay, 2012).For instance, the extant research herein largely suggests that apart from strengthening its sales monitoring capability, the manufacturer must be especially careful with its offering of quantity discounts and revenue sharing contracts to avoid its exploitation by untrustworthy partners downstream (Srivastava & Mateen, 2020;Ahmadi et al., 2017).Furthermore, a wholesale price contract which traditionally results in profit loss owing to double marginalization, in the case of gray markets, has been reported to perform better by regulating and lowering the risk of channel incentive abuse by an opportunistic supply chain entity (Altug, 2017).Srivastava and Mateen (2020) find that the wholesale price contract proves to be most beneficial when product diversion to gray market is highly detrimental to the manufacturer.Similarly, the replenishment policy of the manufacturer is another frontier within interventions that can better share risks between the manufacturer and its retailers, while preventing gray market infiltration.Dasu et al. (2012), for instance, note that a manufacturer operating on a multiple replenishment policy can better control product diversion to the gray market by regulating the supply quantity for individual replenishments.
Apart from channel incentives, demand uncertainty is another important reason for the emergence of gray markets.Smith and Shulman (2022) reveal that an honest authorized retailer may suffer in terms of shrinking profit margins because of inventory diversion by a dishonest authorized retailer to gray marketers who can in turn compete with these retailers.Given the unstoppable prevalence of domestic bootlegging, scholars have also attempted to evaluate whether gray markets can be utilized to a manufacturer's benefit.The answers typically suggest that while the manufacturer can gain from the bulk clearance of unsold stocks, brand erosion is the critical loss which can still prevent the manufacturer to be tolerant of gray markets (Li et al., 2020).Dasu et al. (2012), for instance, discuss potential strategies that could be embraced by the manufacturer to control gray market activities or exploit them.Apart from brand erosion, they further analyze the impact on supply chain agents due to inventory diversion.Hence, recent literature on domestic bootlegging has considerably addressed supply chain coordination in the presence of gray markets from diverse facets.

Product upgrade decision
In the literature, product upgrade decisions have been examined in various contexts (e.g., Wu & Deng, 2021;Xiong et al., 2016).For instance, Xiong et al. (2016) theoretically analyze the impact of a manufacturer's product upgrade strategy on a third-party entrant's decision to sell the remanufactured version of an existing brand in the same market.They find that the entrant's decision to participate in the same market is highly dependent on the degree of the innovation (e.g., major, moderate, or minor).To our knowledge, our study is the first to analyze the impact of product launch decision in the presence of an unauthorized gray market channel.We believe that the launch of an upgraded product becomes very crucial for the manufacturer, when the retailer can divert the product inventory in an unauthorized gray market channel which can have severe negative impacts on the manufacturer's brand reputation.Rapid changes in technology and consumer preferences are raising a need for updating products' features continuously.This may reduce the shelf life of existing products and increase the inventory risk for the retailer.Hence, we explore the product upgrade challenge herein.
Against this backdrop, our study seeks to employ a more realistic two-period supply chain model to analyze consumer choice and potential manufacturers' strategies.In the first period, the product is sold only through the authorized channel.Price sensitive consumers walk away from the market in the first period expecting product availability at a lower price in the gray market in the second period.The product becomes available in the gray market in the second period as a result of inventory diversion by the authorized retailer.Such a supply chain model has been previously addressed in the literature (Srivastava & Mateen, 2020;Ahmadi et al., 2017).However, we extend these prior studies by introducing another choice to the price-sensitive consumers in the second period, that is, to buy an upgraded version of the same product with additional functionalities from the authorized channel or purchase the initial product version from the gray market.It can be seen in practice that companies launch upgraded product versions occasionally.For instance, Apple launched iPhone 11 Pro in September, 2019 and its upgraded version, iPhone 12 Pro, in October, 2020.Major functional upgrades involved larger display, 5G support, compatibility with MagSafe accessories, ceramic shield front, and A14 bionic chip, which were attractive features the consumers wanted5 .
This study is unique as it considers a real-life scenario wherein the manufacturer may decide to launch an upgraded version of the same product in the second period.Moreover, we have introduced the possibility of the manufacturer granting price adjustment flexibility to the retailer in the second period, thereby enabling the latter to markdown the price to lure price-sensitive consumers who are otherwise attracted to the gray market.It is important to note that a manufacturer's decision to launch an upgraded product version and employ price adjustment flexibility as a mechanism to deter gray markets are under-explored in the gray market operations literature.Another highlight of the study is the consideration of strategic consumer behavior (Cachon & Swinney, 2009;Lee et al., 2021;Su & Zhang, 2009).Strategic consumers aim at utility maximization by interpreting market dynamics and exercising a rational choice between the authorized and gray market channel.Such consumers are prepared to postpone consumption in the anticipation of lower future prices (Ahmadi et al., 2017).In light of these considerations, we aim to study the consumer choice and potential strategies that may prove beneficial for the manufacturer.

BASIC MODELS
Under a two-period model (Srivastava & Mateen, 2020;Ahmadi et al., 2017;Altug, 2017), we structure our analysis based on domestic gray market as follows: first, we analyze the centralized and the decentralized supply chains in the presence of unauthorized selling through gray markets.
Second, under the decentralized supply chain structure, we analyze three models.In the first model, the manufacturer extends price adjustment flexibility to the retailer, however, there is an absence of the decision to launch an upgraded version product.In the second model, the manufacturer decides to launch the upgraded version of a current product in the second period; however, the retailer does not have price adjustment flexibility.In the third model, the manufacturer decides to launch the upgraded version of a product in the second period, and the retailer has price adjustment flexibility.
We use the superscript C to show the equilibrium solution of the centralized supply chain model, D for the decentralized supply chain model, DF for the decentralized supply chain model featuring price adjustment flexibility in the absence of product upgrade decision, N for the model characterized by product upgrade decision with no price adjustment flexibility offered to the retailer, and F for the model featuring product upgrade decision along with price adjustment flexibility.The Models DF and N are special cases of Model F. Here, please note that all the afore-mentioned models are based on domestic gray market.Additionally, in order to derive interesting insights we have considered two models under the parallel import setting, Model FI featuring product upgrade decision with price adjustment flexibility (see Section 5) and Model NI featuring product upgrade decision without price adjustment flexibility (see Appendix A8).We use the subscript m for the manufacturer, r for the retailer, g for the gray market, a for the authorized channel, sc for the supply chain, and u for the upgraded version product.We describe notations used throughout the study in Table 2.

Centralized supply chain in the presence of gray market (Model C)
The presence of gray markets can prove to be beneficial for the manufacturers'/brands due to various reasons: (i) For brands which have relatively lower brand equity can use the gray markets to increase their market share by selling products using the alternative selling channel (Sugden, 2009).(ii) Gray markets can help manufacturers to overcome international regulations (Antia et al., 2004).For example, in China multinational companies (MNCs) need to have local partner firm in order to establish their facilities.In such scenarios, many MNCs can use gray markets as a mechanism to avoid such market entry barriers to ensure availability of their product.(iii) Many-a-times manufacturers and distributors use gray markets to clear their excess inventory in order to make a room for the upcoming new product (Ahmadi et al., 2015).In our case, the first argument supports the logic for considering a gray market in the centralized supply chain.
Under Model C, the manufacturer announces its authorized channel selling price p a at the beginning of period 1.Consumers form a belief of gray market expected price p e g .At the beginning of period 2, the manufacturer decides to capture additional sales by diverting a part of inventory to the gray market at a wholesale price w g .Then, the gray marketer decides the selling price p g and sells the product in the same market.The consumer decides through which channel (either authorized or unauthorized) the product can be bought as well as the time of buying (either the first or second period).This is based on the utility that the consumer receives from each purchasing option.Let  be the consumer's valuation for the product, which is uniformly distributed in a unit interval 0 to 1. Consumers often perceive that the product being sold in the unauthorized gray market channel may be inferior in terms of after-sales services or warranty.Thus, consumer valuation decreases for the product in the unauthorized gray market channel.Let  be a consumer's decreased valuation when the consumer decides to purchase the product from the unauthorized channel.Here, the parameter  ∈ [0, 1] captures the consumer's perceived channel differentiation.The higher the value of , the lower the channel differentiation is.However, in terms of the product,  = 1 shows that the consumer perceives that a product bears the same attribute irrespective of the channel (authorized or gray market) from where it is purchased.Therefore, a consumer will prefer to purchase from  Manufacturer's total profit Profit earned by the manufacturer in the second period Retailer's total profit under the model j

𝜋 j rg
Retailer's profit from the gray market channel Profit earned by the retailer in the second period Total supply chain profit the authorized channel in the first period if  ≥ p a .However, a consumer would delay its purchase decision if  < p a and decide to purchase from the unauthorized gray market channel if  ≥ p e g .Here,  ∈ (0, 1) is the time discount factor for delaying purchase (Srivastava & Mateen, 2020).Consumers who are characterized by  < p e g will leave the market without purchasing.
Let U a (, p a ) be the utility received by a consumer who decides to purchase from the authorized channel in the first period and continues to use the authorized product in the next period as well, then U a (, p a ) =  − p a + .Although the product will deteriorate with time yet the consumer who is using the first period product in the second period as well will still receive some increased utility due to the continued usage of the product bought in the first period.Therefore, the factor "" in the utility expression captures the decreased valuation but still some addition in the utility of such segment of consumers.Similarly, let U g (, p e g ) be the utility received by a consumer who decides to delay the purchase decision and waits for the second period to purchase from the unauthorized gray market channel, then U g (, p e g ) =  − p e g .Let  1 (p a , p e g ) be the threshold valuation at which a consumer is indifferent between buying from the manufacturer in period 1 and delaying its purchase decision to buy from the unauthorized gray market seller in period 2. Therefore, at such an indifferent point the utility received by the consumer from the authorized seller would be equal to the utility received from the unauthorized gray market.Thus, U a (, ( p e g ) be the threshold valuation of a consumer who is indifferent between buying from the gray market seller in period 2 and not buying the product at all when the unauthorized gray marketer announces its price p g .Thus, U g (, The demand for the authorized d, while that for the gray market d, and the total channel demand is d.Next, we solve the second period equilibrium game between the unauthorized gray market seller (Stackelberg follower) and the manufacturer (Stackelberg leader).The unauthorized seller maximizes its profit by deciding p g to maximize . Following the rational expectations (RE) equilibrium (Su & Zhang, 2009), we establish p e g = p g to solve the game.The RE equilibrium framework requires p e g = p g , thus the unauthorized seller's best response to the manufacturer's decision is: . Given the best response of the gray marketer, the manufacturer decides w g to maximize Π sc2 where, ]

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟
Profit from gray market channel . The function Π sc2 is strictly concave in w g , and solving the first order condition (FOC) results in the optimal wholesale price w * g = p a  2(+1) .Following the period 2 problem, we solve period 1 problem of the manufacturer to derive the unique equilibrium price p a .We have max In this model, we assume the per unit penalty cost incurred by the manufacturer due to product diversion in the unauthorized gray market channel as 0 (i.e.,  = 0).This is because the manufacturer has complete control over the supply chain under a centralized structure and will not voluntarily indulge in gray market product diversion at the cost of brand dilution, especially when brand equity is high.Checking the concavity condition of the manufacturer profit function confirms the existence of a unique optimal authorized channel retail price at which its profit is the maximum.It is to be noted that for model tractability, we normalize the manufacturer's unit production cost to be zero for all the models.This assumption is widely used in relevant studies such as Xiong et al. (2016) and it is commonly known that having a zero or non-zero production cost does not affect the core qualitative insights.We solve the game using backward induction and equilibrium results are presented in Table 3.We use the superscript C to denote the solution under Model C.

Decentralized supply chain in the presence of gray market (Model D)
In the decentralized supply chain, the manufacturer sells the product through a retailer at a wholesale price w m .The retailer sets the market retail price p a .However, the retailer has access to the unauthorized gray market channel where it can divert product inventory for maximizing sales at the beginning of the second period at a wholesale price w g .Here, it is important to note that the retailer's incentive to maximize sales may involve additional costs in terms of shipping and warehousing.However, we have not taken into account such costs as it will make the analysis complex without having any significant effect on our results.Next, the gray marketer decides the selling price p g and sells the product in the same market.The consumer decides through which channel (either authorized or unauthorized) the product can be bought as well as the time of buying (either the first or second period).This is based on the utility the consumer receives from each purchasing option.The consumer forms a belief about the product price in an unauthorized gray market channel as p e g .The decision sequence and decision variables of supply chain members are shown in the Online Appendix (Figure A2.1).Let U a (, p a ) be the utility received by a consumer who decides to purchase from the authorized channel in the first period and continues to use the authorized product in the next period as well, then U a (, p a ) =  − p a + .Similarly, let U g (, p e g ) be the utility received by a consumer who decides to delay the purchase decision and waits for the second period to purchase from the unauthorized gray market channel, then U g (, p e g ) = −p e g .Let,  1 (p a , p e g ) be the threshold valuation at which a consumer is indifferent between buying from the authorized seller in period 1 and delaying its purchase decision to buy from the unauthorized gray market seller in period 2. Therefore, at such an indifferent point the utility received by the consumer from the authorized seller would be equal to the utility received from the unauthorized gray market.Thus, ( p e g ) be the threshold valuation of a consumer who is indifferent between buying from the gray market seller in period 2 and not buying the product at all when the unauthorized gray marketer announces its price p g .Thus, U g (,  2009), we assume that 0 <  2 (p e g ) <  1 (p a , p e g ) ≤ 1.This assumption shows that the valuation of consumers who are willing to purchase only from the gray market is smaller than the valuation of consumers who are willing to purchase either from the authorized channel in period 1 or from the gray market in period 2. Following these conditions, we derive various constraints on price: p a ( p g  , p g + (1 − )) and p g (0, p a ).The demand for the authorized channel is d, while that for the gray d, and the total channel Equilibrium solutions Next, we solve the second period equilibrium game followed by the first period equilibrium game through backward induction.The second period equilibrium game is between the unauthorized gray market seller (Stackelberg follower) and the retailer (Stackelberg leader).The unauthorized seller maximizes its profit by deciding p g as: max .
Here, the profit function, Π g , represents the profit earned by the gray marketer by selling Q g quantity to the consumers.Following the RE equilibrium (Su & Zhang, 2009), we establish p e g = p g to solve the game.The first order condition of the profit function provides the following solution: . The RE equilibrium framework requires p g = p e g .Thus, the unauthorized seller's best response to the retailer's decision is: . Hence, p * g is a function of p a and w g , in fact p * g will change in accordance with the retailer's choice of p a and w g .The retailer's second period profit is: .
In the profit function,  r 2 shows the profit earned by the retailer in period 2. Please note that in the second period, the retailer earns profit by diverting Q g quantity to the gray marketer.After solving the second period equilibrium game, we perform backward substitution of the obtained equilibrium results to solve the first period game.The retailer's total profit function can be written as: ]

⏟⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏟
Purchase cost (2) In the retailer's total profit function, Π r , the first term represents the retailer's revenue from the authorized channel sales in period 1, the second term shows the cost of total quantity Q tot purchased from the manufacturer, and the third term represents the retailer's period 2 profit.The manufacturer receives the best response function of the retailer with respect to p a and maximizes its profit Π m by deciding w m as follows: max ]

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟
Loss due to brand dilution Here,  is the per unit penalty cost incurred by the manufacturer due to product diversion in the unauthorized gray market channel.A higher value of  means that the product has a higher brand equity.In the manufacturer's total profit function, Π m , the first term shows the profit earned by the manufacturer from selling the total quantity Q tot to the retailer and the second term represents the total penalty cost incurred by the manufacturer due to gray market product diversion.We use the superscript D to denote the solution under Model D.

Proposition 1. In the presence of the GM: (i)
The supply chain is better-off in the decentralized case, that is, Proposition 1 reveals that under some scenarios, the overall decentralized supply chain can be better off (i.e., the total supply chain profit is higher) as compared to the centralized supply chain in the presence of gray market.We find that the manufacturer loses its control over the pricing and the selling decisions under the decentralized supply chain.As a result, the retailer attempts to maximize its profit by diverting a greater quantity to the gray market.Therefore, if the product diversion to the unauthorized gray market channel has a higher impact on the brand or manufacturer's brand reputation, then the manufacturer becomes worse-off under the decentralized supply chain.However, the total channel sales (i.e., authorized sales as well as unauthorized sales) are higher under the decentralized supply chain, leading to a higher total supply chain profit.In contrast, under the centralized case, if the brand erosion due to product diversion in the gray market is higher, the manufacturer tries to curb gray market diversion to avoid loss due to brand erosion.This leads to decline in total channel sales as well as the total supply chain profit.This finding challenges the conventional wisdom in the literature whereby the total supply chain profit is always lower under the decentralized supply chain as compared to the centralized supply chain due to double marginalization effect (Cachon, 2003).Existing studies on gray market, such as Ahmadi et al. (2017), mainly focus on the performance of various contracts (e.g., quantity discount and wholesale price contract) in the presence of gray market.They reveal that the wholesale price contract can outperform the quantity discount contract due to the absence of channel incentives (e.g., discount).However, our focus is mainly on the channel choice decision in the presence of product upgrade and price adjustment flexibility.
Proposition 1 suggests that the manufacturers selling a high brand equity product should carefully monitor their channel partners' sales under Model D. In practice, luxury brands Yves Saint Laurent, Calvin Klein, and Omega once filed a lawsuit against Costco for product leakage and sale to the unauthorized channel (Bryant, 2017).Similarly, HP also faced the problem of product diversion to the gray market by its distributor Maxicom PC Inc. and finally recovered $4 million from Maxicom PC6 .Next, we analyze a decentralized supply chain model, wherein the manufacturer grants price adjustment flexibility to the retailer in the absence of product upgrade decision (Model DF).The details and analysis of Model DF (a special cases of Model F) is entailed in Appendix A6.Proposition 2 shows an interesting result that the manufacturer can eliminate the gray market selling by offering price adjustment flexibility to the retailer.Curbing the unauthorized sales becomes more important when the product has a high brand equity.In such a scenario, selling a discounted product through the authorized channel in the second period is a better strategy for the manufacturer in comparison to the cases when the product is being sold through the unauthorized gray market channel.Proposition 2 (ii) indicates that the authorized channel sales are greater under Model DF because the retailer is able to sell more through the authorized channel due to price adjustment flexibility (i.e., at prices p a and p 2 in the first and the second period, respectively).Interestingly, the manufacturer is better-off under Model DF irrespective of reduced total channel sales (Proposition 2 (iii) and (iv)).This could be attributed to the fact that the manufacturer's gain due to elimination of gray market outweighs the loss from the reduced total channel sales.However, the retailer can lose its profit margin due to reduced sales through the gray market channel under Model DF.Next, we analyze a decentralized supply chain model wherein, the manufacturer decides to launch an upgraded version product but does not grant price adjustment flexibility to the retailer (Model N).The details and analysis of Model N (a special case of Model F) is given in Appendix A7.The manufacturer's product upgrade decision in the presence of the price adjustment flexibility and domestic gray market (Model F) is analyzed in the following subsection.

Product upgrade decision: With pricing flexibility to the retailer (Model F)
In practice, manufacturers standardize prices across retailers and selling channels to reduce price variability by implementing the MSRP scheme.However, the MSRP may create conflicts between the manufacturer and opportunistic retailers as the latter aim at sales maximization by selling at discounted prices as well as through inventory diversion to unauthorized channels.Therefore, in order to mitigate the diversion of inventory to the gray market by opportunistic retailers, the brand should allow price adjustment flexibility to the authorized retailers.In this model, we analyze a scenario where, in addition to offering flexibility to the retailer to markdown an existing product's price, the manufacturer launches an upgraded version of the product in the second period.The sequence of decisions as well as decision variables of each player are depicted in Online Appendix (Figure A2.2).
The consumer-type space can be divided into five segments.Segment 1: A consumer who buys from the authorized channel in the first period at a price p a (receives a surplus equivalent to  − p a ), resells the used product at a return price r at the end of period 1 or in the beginning of period 2, and decides to purchase the upgraded version product in period 2 (receives a surplus equivalent to (1 + ) − p e u ).Here, the parameter 'r' is the product reverse cost, that is, the price that the manufacturer pays to the consumers who are willing to sell their used product.The utility received will be U 1 (p a , p e u ) =  − p a + r + (1 + ) − p e u , where  is the degree of product upgrade.Segment 2: A consumer who buys the existing version product from the authorized channel in the first period and continues to use the same product in the second period.The utility received will be U 2 (p a ) = −p a +.Segment 3: A consumer who does not buy in the first period and decides to buy an upgraded version of the product in the second period.
Here, the utility received will be U 3 (p e u ) = (1+)−p e u .Segment 4: A consumer who buys from the authorized channel in the second period at a markdown price p 2 .The utility received will be U 4 (p 2 ) = −p 2 .Segment 5: A consumer who decides to delay the purchase in order to buy from the unauthorized gray market channel in the second period receives utility equivalent to U 5 (p e g ) =  − p e g .Please note that we consider consumers' salvage behaviour in the models where upgraded version product is available in period 2. However, in the models where the upgraded version product is not available in the second period, we have assumed that the consumers continue to use the previous version (period 1 product) in period 2 as well.As the upgraded version of the product is not available in the market, consumers have no motivation to salvage the product and also manufacturer would not participate in taking back the product or trade-in.Similar to previous models, we derive threshold valuations.Figure 1depicts the market segmentation and demand distribution for each consumer segment under Model F.
Next, we solve the equilibrium game similar to the earlier cases.The analysis consists of a two-stage equilibrium game in the second period.The first stage of the game is between the retailer and the gray marketer.The second stage game is between the retailer (who decides to markdown the existing product's price p 2 and sets the upgraded version product price p u ) and the manufacturer (who decides the wholesale price for the upgraded version product w u ).In the first stage, the gray marketer's as well as the retailer's decision problems can be formulated as: max

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟
Profit from gray market channel )

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟
Profit from selling markdown product . (5) Here, the profit function, Π g , represents the total profit earned by the gray marketer by selling Q 5 (p 2 , p e g ) units of the product to the consumers.On the other hand, the profit function,  rc , is the sum of retailer's profit received from unauthorized gray market selling ( rg ) and the profit received on account of the sales of original version product at a markdown price.Next, in the second stage game, the retailer's and the manufacturer's decision problems can be formulated respectively as: max The profit function,  r2 , represents the profit earned by the retailer in period 2. It is the sum of profit received by the retailer from the gray market and markdown product sales ( rc ), and the profit earned by selling upgraded version product.The profit function,  m2 , represents the profit earned by the manufacturer in period 2. The first term shows the profit earned by the manufacturer by selling the upgraded version product to the retailer and the second term represents the product upgrade cost incurred by the manufacturer.
After solving the second period game, we solve the first period game using backward induction.The retailer's and the manufacturer's total profit functions can now be written as follows: max

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟
Profit from existing version product max ]

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟
Loss due to brand dilution The retailer's total profit function, Π r , is the total profit received from selling the original version product in the authorized channel in period 1 and the retailer's period 2 profit ( r 2 ).In the manufacturer's total profit function, Π m , the first term shows the profit received from selling the original version product to the retailer, the second term represents the total penalty cost incurred by the manufacturer due to gray market diversion, and the third term is the profit earned by the manufacturer in period 2 ( m2 ).We use the superscript F to denote the solution under Model F.

COMPARISONS AND ANALYSIS
In this section, we present two sets of comparative analysis toward the optimal sale quantity, optimal price, and optimal profit under the domestic gray market setting.We begin by comparing the "decentralized supply chain model" (Model D) and the "no price adjustment flexibility model" (Model N), followed by a comparison between the "no price adjustment flexibility model" (Model N) and the "price adjustment flexibility model" (Model F).Proposition 3 implies that when the degree of product upgrade is higher than a threshold (i.e., a sufficiently large ), a consumer is more likely to prefer the upgraded product featuring additional functionalities over the earlier version in period 2. Therefore, a manufacturer's decision to launch an upgraded product and the possible subsequent shift in consumer preference in favor of the upgraded product can accentuate the retailer's fear of being left with the unsold inventory of the original version.To reduce this risk, the retailer diverts a large quantity of the original version product at a lower wholesale price to the gray marketer during period 1 which in turn allows the gray marketer to charge a lower retail price from the consumers in period 2 under Model N (P.S. Proposition 3 (i), (iv), (v)).Notably, the higher the gray market diversion, the greater the total channel sales quantity (sum of sales through the authorized channel and the gray market) of the original product becomes (Proposition 3 (ii)).Here, it is noteworthy that under Model D, the retailer is not required to cushion against risks caused by the upgraded product launch; hence, it is not forced to clear the inventory in period 1. Whereas, under Model N, the retailer is largely focused on diverting original version product to the gray market as soon as possible to safeguard against potential threats posed by the upgraded product in period 2, thereby leading to a lower authorized channel sales quantity in period 1 (Proposition 3 (iii)).Although the total channel sales of the original version product are higher under Model N, as attributed mainly to the greater sale of the original version of the product in the gray market but the authorized channel sales are lower in period 1.

Comparisons of Model D and
Proposition 4 (i) indicates that when the degree of upgrade () is lower and the penalty cost for the loss of brand reputation due to product diversion () is higher, launching an upgraded version of the product in the second period is not an optimal strategy for the manufacturer.This can be attributed to the fact that under Model N, the retailer diverts a greater quantity of the earlier version of the product to the gray market in the first period which reduces the earlier version's sale through the authorized channel (see Proposition 3).Instead of controlling product diversion to the gray market, the decision of upgraded product launch aggravates gray market diversion.Thus, increased gray market product diversion by the retailer adversely affects brand equity and tarnishes the reputation of the manufacturer.Thus, the manufacturer not only suffers in terms of loss of reputation and tarnished brand image but also incurs additional cost on account of the investment involved in upgraded product development, thus leaving it worse-off under Model N.However, when  is above a threshold level, the manufacturer becomes better-off under Model N.
Proposition 4 (ii) uncovers that the retailer is better-off under Model N, wherein it not only is able to divert more quantity to the gray market but also gauges an opportunity to sell the upgraded version of the product at a premium price without bearing any investment cost related to upgraded product development.Proposition 4 (iii) suggests that when the degree of upgrade and channel differentiation is lower, upgraded product sales make the gray marketer better-off.This is because the retailer offers the original version inventory to the gray marketer at a lower price as compared to Model D. For example, unauthorized intermediaries, such as https://Jomashop.comand https://IconicWatches.co.uk,tactfully purchase unsold inventory of watches from the authorized retailers.Then, they sell these watches at a competitively lower price compared to the authorized channel to lure the price-conscious consumers (Shannon, 2017) Proposition 5 (i) and (iii) imply that when the degree of upgrade is moderate and channel differentiation is higher, the manufacturer grants greater flexibility to the retailer in terms of adjusting the price of the original version product in period 2. This arrangement not only raises the quantity of the original version product sold through the authorized channel but also shrinks the retailer's incentive to indulge in gray market diversion.Consequently, the retailer diverts a smaller quantity to the gray market and the authorized channel sales of the earlier version product are higher under Model F. Price adjustment flexibility eliminates the risk that a retailer otherwise faces under the model in terms of unsold inventory.The retailer may be burdened by unsold or surplus inventory due to quantity discounts (AGMA, 2020) and pressure by the manufacturer to stock inventory that exceeds the market demand (Shannon, 2017).Owing to this threat, the retailer resorts to gray market diversion to ensure the timely clearance of the original version inventory under Model N.However, under Model F, the retailer has incentives to adhere to the authorized channel to sell the earlier version of the product thus inducing the retailer to charge a higher price from the gray marketer which consequently forces the latter to charge a higher price from the consumers.
Under Model F, the original version and the upgraded version of the product are sold simultaneously through the authorized channel in period 2 due to the presence of price adjustment flexibility and the opportunity to gauge a premium price, respectively.Such a setting culminates in the cannibalization of the sales of the upgraded product mainly when (a) consumers are price sensitive; (b) additional functionalities associated with the upgraded product are non-appealing to consumers and/or; (c) brand-conscious consumers have a strong desire to get associated with the brand but are indifferent to product versions.Owing to these reasons, consumers tend to choose the original version available at a marked-down price in the authorized channel instead of the premium-priced upgraded version, thus resulting in the cannibalization of upgraded version sales.This explains Proposition 5 (ii) in which the price adjustment flexibility mechanism is found to be the cause of cannibalization of the upgraded product's sales.

Proposition 6. Comparative analysis of the optimal profits under Model F and Model N gives the following results:
(i) Price adjustment flexibility can be an optimal strategy for the manufacturer (i.e., Proposition 6 (i) proposes that when both the degree of upgrade as well as the brand equity are higher, the strategy of permitting price adjustment flexibility to the retailer in the second period proves to be an optimal strategy for the manufacturer.This can be attributed to the fact that offering price adjustment flexibility dissuades the retailer from exercising gray market diversion which is beneficial to the manufacturer in terms of superior brand image and favorable reputation.Here, it is intriguing to observe that in spite of the cannibalization of the upgraded product's sale owing to price adjustment flexibility, this strategy proves to be beneficial for the manufacturer under certain conditions.This is because the positive effect of reduction in availability of the earlier version product in the gray market outweighs the negative impact of demand cannibalization of the upgraded version product.Contrary to prior literature wherein demand cannibalization is shown to be harmful to the manufacturer (Desai, 2001), Proposition 6 (i) subtly implies that despite the cannibalization of sales, the manufacturer can be better-off in terms of overall profitability by adopting the price adjustment flexibility strategy only if certain conditions are fulfilled.Here, it is important to note that when the degree of upgrade () is above a threshold level, the manufacturer's profitability is the highest under Model F followed by Model N and Model D (see Propositions 4 (i) and 6 (i)).Proposition 6 (ii) indicates that if the degree of price adjustment flexibility is higher, the retailer is empowered to sell the earlier version product through the authorized channel in the second period under Model F.Even if the retailer wishes to sell the product to the gray marketer in the second period, the former is in a position to charge a higher price from the latter.However, under Model N, the threat of low salvage value of the leftover inventory makes the retailer hastily opt for the clearance of the earlier version inventory through gray market diversion in the first period.Such a threat is caused by consumer preference for the upgraded version product over the earlier version in the second period, especially given that the upgrade degree is higher.Thus, the retailer is better-off under Model F. Proposition 6 (iii) suggests that the gray marketer is worse-off under Model F, given that both the price adjustment flexibility and upgrade degrees are higher.This is attributable to two reasons: first, under Model F, a smaller quantity is diverted to the gray market and at a higher price compared to Model N. Second, as the earlier version product is available at a markdown price in the authorized channel accompanied by after-sale benefits, consumers are hesitant to tap into the gray market where such benefits are not guaranteed.
To provide additional insights of Propositions 4 and 6, we conduct the following numerical analysis with respect to parameters  and .Following the model assumptions and physical meanings of parameters, we set  = 0.5,  = 0.5, and vary  and  from 0.1 to 0.9.It can be observed from Figure 2 that for lower to moderate values of , the manufacturer's optimal profit is highest under Model C.However, for larger (> 0.712), the manufacturer's optimal profit becomes highest under Model F due to considerably reduced product diversion.Moreover, launching the upgraded version of the product enables the manufacturer to capture greater demand.At a larger degree of upgrade ( > 0.33), the manufacturer's profit remains the highest under Model F (see Figure 2).
As the demand for the upgraded product decreases due to higher price, the manufacturer is able to sell the earlier version product in large quantities through the authorized channel, the retailer's profit is, thus, always the highest under Model F as it is able to increase the sale by offering the existing product at a markdown price as well as the upgraded version product.Moreover, under Model F, the retailer does not bear the risk of low salvage value of the existing version product at the end of the selling season due to the price adjustment capability.Thus, it can be concluded that selling an upgraded version of the product along with price adjustment flexibility is the optimal strategy for the retailer.
We derive CS by integrating consumer utilities with respect to valuation parameter  over the segments.Thus, for example, CS for the model where the manufacturer offers price adjustment flexibility to the retailer can be derived as follows: We observe that Zhang and Choi (2021) follow a similar approach to derive CS function from consumer utilities.Similar to Hassin and Roet-Green (2017), the SW function is defined as the sum of total supply chain profit and CS, which is given as follows: SW = Π j sc + CS j .Proposition 7. The comparative analysis of CS gives results as follows: CS F > CS N > CS D > CS C for  >  6 .
Proposition 7 reveals that CS is largest under Model F as a greater variety of product versions is being offered to the consumers (i.e., original version product, upgraded version product, gray market product, and the markdown product).To provide supplementary insights, we conduct sensitivity FIGURE 3 The optimal CS and SW with respect to  and .
analysis with respect to parameters  and  for comparing CS function across models.Regarding SW, it is important to mention that the mathematical expressions for the SW function are very complex; hence, we could not derive meaningful closed form conditions for the comparative analysis across models.Therefore, we have conducted a detailed sensitivity analysis for comparing the SW function across models.
It can be seen from Figure 3 that CS and SW are consistently decreasing with an increase in  across all the models.This can be attributed to the negative consequences of brand erosion on the manufacturer and total supply chain profit.Figure 3 depicts the trend of CS and SW with respect to  and .It can be observed that under Model N, CS continuously increases with a rise in ; however, SW increases only up to a certain level (i.e.,  < 0.5).Under Model F, CS and SW are increasing with  only up to a certain level (i.e.,  < 0.7).Additionally, when the degree of upgrade is low to moderate (i.e.,  < 0.7), CS is highest under Model F. However, CS becomes highest under Model N when the degree of upgrade is higher (refer to the Online Appendix for the detailed sensitivity analysis).On the other hand, SW always remains the largest under Model F. Next, in order to derive interesting insights we have considered two models under the parallel import setting, Model NI featuring product upgrade decision without price adjustment flexibility (for details and analysis please refer Appendix A8) and Model FI featuring product upgrade decision with price adjustment flexibility.

PARALLEL IMPORTS: PRODUCT UPGRADE DECISION WITH PRICE ADJUSTMENT FLEXIBILITY TO THE RETAILER (MODEL FI)
We consider that the manufacturer introduces its product in the low-priced market (market 1) at a wholesale price w m1 .The retailer observes w m1 and offers the product to the consumers at a retail price p a1 in market 1.Now, the manufacturer decides to launch its product in the high-priced market (market 2) in another country at a wholesale price w m2 and the retailer charges a price p a2 from the consumers in period 1.Such a decision may induce threat of parallel import from an unauthorized intermediary.Thus, in the second period, the parallel importer purchases the branded product from the authorized retailer in market 1 at a lower price w g and resells it in market 2 at a higher price p g to earn profit.In the process of buying a product from one market and selling it in another market, the parallel importer incurs per unit transportation cost c t .Usually, parallel imports are accentuated by opportunistic retailers who willingly divert a part of the inventory to the parallel importer to maximize sales and profit.Furthermore, the manufacturer launches an upgraded version of the product and grants price adjustment flexibility to the retailer in the high-priced market in period 2. Therefore, in addition to the parallel import product, an upgraded version product and a markdown version is available in the high-priced market in period 2, whereas in period 1 only the original version product is available in both the markets.Figure 4 depicts the key players and decision variables in the parallel import setting.It is important to note that the consumers' valuation in the high-priced market is relatively higher compared to the low-priced market (Altug & Sahin, 2019).Thus, consumers valuation in the high-priced market is denoted by θ, whereas consumers' valuation in the low-priced market is represented by .The consumer-type space in the high-priced market can be divided into five segments: Segment 1: One who buys from the authorized channel in period 1 and sells the used product to buy an upgraded version product in period 2 to receive a utility equivalent to U 1 (p a2 , p e u ) = θ − p a2 + r + (1 + ) θ − p e u .Segment 2: One who does not buy in period 1, but purchases the upgraded product in period 2 to receive a utility equivalent to U 2 (p e u ) = (1 + ) θ − p e u .Segment 3: One who buys from the authorized channel in period 1 and continues to use the same product in the second period to receive a utility equivalent to U 3 (p a2 ) = θ − p a2 +  θ.Segment 4: One who buys from the authorized channel in period 2 at a markdown price p m and obtains a utility equivalent to U 4 (p m ) =  θ − p m .Segment 5: One who delays the purchase decision and waits for the second period to buy from the parallel importer receives utility equivalent to U 5 (p e g ) =  θ − p e g .Similar to previous models, we derive threshold valuations and demand.
Unlike the previous models based on domestic gray market, we have two separate markets under the parallel import scenario which makes the analysis significantly different.Under the parallel import scenario, we solve the first and second period equilibrium games distinctly for the high-priced and the low-priced market.We begin by solving the equilibrium game in the high-priced market followed by the equilibrium game in the low-priced market.

High-priced market (Market 2)
Under the high-priced market, the analysis entails a two-stage equilibrium game in the second period.The first stage game is between the retailer (who decides to markdown the existing product's price p m ) and the parallel importer.The second stage game is between the retailer (who sets the upgraded version product price p u ) and the manufacturer (who decides the wholesale price for the upgraded version product w u ).
Please note that we use the symbol ∼ to represent functions and expressions under the high-priced market.In the first stage, the parallel importer's as well as the retailer's decision problems can be formulated as: Here, the profit function, Πg , represents the total profit earned by the parallel importer by selling Qg units of the product to the consumers.On the other hand, the profit function, πrc , is the retailer's profit received on account of the sales of original version product at a markdown price.Next, in the second stage game, the retailer's and the manufacturer's decision problems can be formulated respectively as: max )

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟
Profit from selling upgraded product . The profit function, πr2 , represents the profit earned by the retailer in period 2. It is the profit earned by the retailer from selling the upgraded version product and markdown product sales ( πrc ).The profit function, πm2 , represents the profit earned by the manufacturer in period 2. The first term shows the profit earned by the manufacturer by selling the upgraded version product to the retailer and the second term represents the product upgrade cost incurred by the manufacturer.After solving the second period game, we solve the first period game using backward induction.The first period equilibrium game is between the retailer and the manufacturer.Thus, the retailer's and the manufacturer's decision problems can be formulated respectively as:

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⏟
Profit from existing version product max ]

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟
Loss due to brand dilution The retailer's total profit function, Πr , is the total profit received from selling the original version product in the authorized channel in period 1 and the retailer's period 2 profit ( πr2 ).In the manufacturer's total profit function, Πm , the first term shows the profit received from selling the original version product to the retailer, the second term represents the total penalty cost incurred by the manufacturer due to gray market diversion, and the third term is the profit earned by the manufacturer in period 2 ( πm2 ).

Low-priced market
Under the low priced market, consumers either buy the product from the authorized channel or leave the market without purchasing.Therefore, utility is U a =  − p a1 and U O = 0, respectively.We derive threshold valuation similar to the previous case.The retailer's and the manufacturer's profit maximizing problem is: max max The retailer's total profit function, Π r , is the total profit received from selling the original version product in the authorized channel in period 1 and the retailer's profit from diverting inventory to the parallel importer in period 2 ( rg ).Here, please note that the retailer aims to maximize gray market sales as: max w g  rg = w g × ( 4 −  5 ).
The manufacturer's total profit function, Π m , shows the profit received from selling the original version product to the retailer.We use the superscript FI to denote the solution under Model FI.Additionally, we solve the parallel import model characterized by product upgrade decision without price adjustment flexibility (Model NI) (for details refer Appendix A8).Further, to gain additional insights we compare Model NI and Model FI under the parallel import setting to derive the following Proposition.In the high-priced market under Model FI, price-sensitive consumers who have a strong desire to be associated with the brand prefer to buy the original version product available at a markdown price instead of a costlier upgraded version product.Thus, offering price adjustment flexibility cannibalizes manufacturer's upgraded product sales (Proposition 8 (i)).When the manufacturer extends price adjustment facility to the retailer in the high-priced market, the latter adjusts the price of the original version product in period 2 to maximize sales.As a result, the consumers tend to switch their purchase choice from unauthorized gray market product to a genuine product available at a markdown price (markdown version product).Consequently, the sales of the parallel importer shrink only if the per unit transportation cost is greater than a threshold value (Proposition 8 (ii)).Furthermore, when both the brand reputation loss due to gray market diversion and the cost of transportation are higher, the parallel importer's profit decreases due to decline in sales (Proposition 8 (iv)).In the domestic setting also, the gray marketer's sales and profit declines under the price adjustment flexibility model (Model F).
Owing to reduced sales of the parallel importer, it buys lesser inventory from the retailer operating in low-priced market, thereby reducing manufacturer's sales in the low-priced market.However, it is important to note that increase in the manufacturer's sales in the high-priced market outweighs the decline in the manufacturer's sales in the low-priced market leading to improvement in the manufacturer's global sales (sum of market 1 and market 2 sales).This explains why the manufacturer's profit in the low-priced (high-priced) market decreases (increases) and the global profit increases under the price adjustment flexibility model (Proposition 8 (iii) and (v)).Also, it is interesting to note that inspite of cannibalization of upgraded version sales, the manufacturer's profit increases under the high-priced market under Model FI.This can be attributed to the fact that consumers change their purchase preference from the unauthorized gray market product in favour of the markdown version product, thus increasing the authorized channel sales in the high-priced market under price adjustment flexibility.
Proposition 8 (vi) states that given both the per unit transportation cost and brand reputation loss due to gray market diversion are higher than a threshold value, the retailer in the high-priced market can be worse-off in the presence of price adjustment flexibility.The retailer suffers not only because of sales cannibalization by the parallel importer but also due to cannibalization of upgraded product sales through price adjustment flexibility.This is different from our finding in the domestic gray market setting wherein the retailer becomes better-off in the presence of price adjustment flexibility as the same authorized retailer is involved in the authorized channel sales as well as gray market diversion.However, under the parallel import setting, the authorized retailer operating in the high-priced and the low-priced market are different.Next, we compare Model N and Model F when upgrade degree is an endogenous decision for the manufacturer.Proposition 9. When the upgrade degree is an endogenous decision for the manufacturer, the comparative analysis of Model N and Model F results: (i)  The explanation for Proposition 9 (i), (ii), and (iv) is similar to Propositions 5 and 6 that deal with comparative analysis of Model N and Model F when upgrade degree is not an endogenous decision for the manufacturer.However, the only finding that is different is with respect to the retailer.It can be observed that the retailer is better-off with price adjustment flexibility (i.e., Π F r ≥ Π N r ) when the upgrade degree is not an endogenous decision for the manufacturer (see Proposition 6 (ii)).However, the retailer becomes worse-off under Model F when the upgrade degree is an endogenous decision for the manufacturer (i.e., Π F r ≤ Π N r ).This can be attributed to the fact that under Model F not only the retailer's sales volume through gray market diversion shrinks but also the upgraded product sales is cannibalized.Furthermore, the upgraded product sales is adversely affected when the upgrade degree is less than a threshold value as the consumers are less likely to buy a premium priced upgraded product having minimal upgraded functionalities.Hence, the retailer mainly relies on the markdown version sales in period 2, lowering the retailers product.
Corollary 1.In the presence of product upgrade decision without price adjustment flexibility, the optimal level of upgrade degree is  * = 1  + 8+ 16 .The proof for Corollary 1 is given in Appendix A4.On similar lines, we can derive the  * for Model F as well.

EXTENSIONS
To establish the robustness of the derived findings from the basic models, we provide extensions on three aspects, namely, reselling of used products, dual marketing strategy, and trade-in rebates and non-zero salvage value.All the extensions discussed below carry good physical meanings.

Reselling of used products
In our models, under the product upgrade decision, the customer segment buying from the authorized channel in the first period and selling its used product to buy an upgraded version of the product in the second period receives a utility U 1 =  − p a + r + (1 + ) − p u .In this part, we explore the effect of refurbishing of the used product collected back from the consumers at a reversed cost r.For analysis, we consider that the manufacturer handles the refurbishing activity and sells the remanufactured product in the same market.For instance, in developing countries, most manufacturers, such as in the automobile or electronic equipment segment, facilitate the exchange of used vehicles or gadgets with newer ones.The used vehicle/gadget is then channeled in the second hand selling platform of the manufacturer or sent back for refurbishing 7 .Let  be the unit cost of refurbishing/ maintenance cost and p R r be the remanufactured product's selling price.The introduction of a refurbished product in the market adds a new segment of consumers, that is, those who are willing to buy refurbished products at a lower price.The manufacturer will have an additional variable to decide in terms of p R r .In the second stage game, the retailer's and the manufacturer's decision problem can be formulated as: ]

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⏟
Profit from upgrade product Here,  R rg is the retailer's profit received due to unauthorized selling alongside the sale of the earlier version of the product at a markdown price.After solving the second period game, we solve the first period game using backward induction.The retailer's and manufacturer's total profit functions can now be written as:

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟
Profit from selling existing version product (iv) Consumer surplus is higher under the remanufacturing case if and only if This implies that the manufacturer can sell the remanufactured product at a lower price when the cost of remanufacturing is comparatively low.However, setting a lower price for the remanufactured product would cannibalize the demand for the existing product that is being sold by the retailer at a markdown price in the second period.Consequently, the retailer's risk of salvaging the product would increase, leading to greater product diversion to the unauthorized gray market channel.On the one hand, the manufacturer can earn additional revenue by selling the remanufactured product.However, on the other hand, the manufacturer might suffer due to increased product diversion to the unauthorized channel.Therefore, the manufacturer can benefit through remanufacturing if and only if the price of the remanufactured product is neither too low (demand cannibalization of the existing product in the second period) nor too high (consumer would not be incentivized to buy the remanufactured product).

Dual marketing strategy
Although remanufacturing can serve as a mechanism for the manufacturer to capture additional revenue, the cannibalization problem that occurs between the remanufactured product and the existing product sold by the retailer at a markdown price needs to be resolved.Consequently, entering into a separate market for selling the remanufactured product appears to be a solution to this problem.Figure 5 portrays the manufacturer's dual marketing strategy.Fortunately, the emergence of online platforms for selling remanufactured products has made it easy for firms to enter into a different market by incurring minimal additional market entry cost.For example, Toyota sells and leases its used/remanufactured automobiles to an international secondary market (Lacourbe, 2016).Other automobile companies, such as BMW8 , Ford9 , and Mercedes-Benz10 , often collect their used products through trade-in programs and then sell them to a secondary market after refurbishment and remanufacturing.
In the secondary market, consumers' willingness to pay for the remanufactured product will be lower.Therefore, following Aflaki and Mazahir (2015), we assume that the valuation for the remanufactured product in secondary market is  ∈ [0, b] and b ≤ 1.Let p r be the remanufactured product's price in the secondary market; then, the demand for the remanufactured product will be Q ) .Thus, the manufacturer's second period profit function can be expressed as: max In the second period profit function, the term (p r − ) ) represents the revenue received from selling the remanufactured product in the secondary market.The retailer's and gray marketer's profit maximization problems would be similar to the pricing flexibility case.
Proposition 11.Selling the remanufactured product through a dual marketing strategy is always an optimal strategy for the manufacturer.
To avoid the existing product's demand cannibalization in the second period and increase productivity, the manufacturer should sell the remanufactured product through a separate secondary market.By doing so, the manufacturer can explore additional profit avenues by selling a greater quantity of the remanufactured product and the existing product at the same time.

Trade-in rebates and non-zero salvage value
In the basic models, we have assumed the salvage value as zero.Thus, we explore the effect of a non-zero salvage value on firms' decisions.This not only serves the purpose of checking robustness, but also investigates how a non-zero salvage value and trade-in rebates together bring some new findings.
Thus, we also analyze the effect of the manufacturer's trade-in rebate as a competitive strategy to cope with unauthorized gray market selling.Trade-in programs offer redeemable trade-in rebates to consumers in exchange for their used products (Cao & Choi, 2022).Furthermore, such programs can be a mechanism to expand product sales and stimulate market demand (Xiao et al., 2020).In reality, many companies offer trade-in programs; for instance, Apple itself offers a trade-in program wherein trade-in can be executed both online and offline: mail-back trade-in and in-store trade-in.The trade-in program provides credit for consumer's next purchase, or an Apple gift card that the consumer can avail anytime and get the certified refurbished product11 .Ikea, the world's giant furniture company, started a buyback program in 2020.Consumers can sell used furniture and in turn receive an Ikea refund card for an agreed amount with no expiry date.Refund cards may be used for an indefinite period after they are issued12 .Let s be the salvage value and p t be the rebate offered by the retailer.Then, the segment that participates in a trade-in program will have a utility equivalent to U 1 =  − p a + p t + (1+)−p u and the segment of consumers' willing to buy the refurbished product from the secondary market will receive a utility equivalent to U r =  − p r .However, other segments' utilities will remain same.The manufacturer's profit maximization functions in the second and first periods will be: max In the second period profit function, the first term represents revenue received from selling the upgraded version of the product.The second term represents revenue received from selling the remanufactured product in the secondary market.The third term refers to the salvage from used products that are collected through trade-in but not remanufactured.We find that offering a trade-in rebate under a non-zero salvage can be an optimal strategy for the manufacturer when (i) the product belongs to a higher brand equity category and (ii) the salvage value is not very low.Offering a rebate would result in increased sales for the upgraded version of the product.Offering trade-in rebates can be an indirect strategy for price discrimination as the consumers who opt for trade-in rebates pay less for the upgraded product.Therefore, having a non-zero salvage value and increased sales from trade-in rebates for the upgraded product makes the retailer better-off and reduces the risk of product diversion to the unauthorized gray market.Additionally, following a dual-market strategy with a trade-in rebate would not only result in reduced risk of demand cannibalization for the retailer but also increases remanufactured product sales.

CONCLUSION
In this paper, we analyze strategic channel choices by comparing profitability as well as product diversion risk between a decentralized supply chain and a centralized supply chain in the presence of gray market.Further, we proceed to investigate the capability of price adjustment flexibility mechanism in limiting gray market product diversion under a decentralized supply chain.Additionally, we explore plausible consequences of a manufacturer's decision to launch an upgraded product in the second period under unauthorized selling via the gray market.We also traverse the provision of granting price adjustment flexibility to the legitimate retailer upon the launch of an upgraded product as a strategy to deter gray market selling.Our findings reveal that total supply chain profit is higher under the decentralized supply chain due to greater product leakage which contradicts with the argument usually made in the existing literature.Notwithstanding the better performance of the supply chain in terms of greater profitability, the manufacturer becomes worse-off under such a setting.Another interesting finding is that offering price adjustment flexibility to the retailer can eliminate gray market product diversion and makes the manufacturer better-off in spite of reduced total channel sales.Furthermore, we find that a manufacturer's decision to launch an upgraded product can aggravate the problem of gray market product diversion due to the reduced product selling cycle of an existing product.Lastly, we conclude that offering price adjustment flexibility in addition to launching an upgraded product in the presence of gray market selling makes the manufacturer better-off in spite of the demand cannibalization of the upgraded product.

Research findings and theoretical insights
The findings of this study are presented in line with the research questions discussed earlier.Exploring the impact of the internal channel structure on the profitability of supply chain members in the presence of gray market selling, it is counterintuitive to find that the total supply chain profit is higher under the decentralized supply chain compared to a centralized internal structure given low channel differentiation and high brand equity.However, the existing literature has shown that the decentralized supply chain results in suboptimal supply chain profit due to the double marginalization effect (Cachon, 2003).Nonetheless, the manufacturer is worse-off under a decentralized supply chain featuring possibilities of gray market diversion because of tarnished reputation and brand dilution owing to gray market product diversion by the retailer.Here, it is important to understand that the retailer is not directly affected by brand reputation loss, hence promptly diverts inventory to the unauthorized channel just to maximize sales volumes and profits even when there are severe consequences of brand dilution for the manufacturer.
Though a few studies have examined the launch of the upgraded version of products in the durable goods industry (e.g., Xiong et al., 2016), the impact of the manufacturer's product upgrade decision on supply chain members in the presence unauthorized gray market selling is completely unexplored.It is observed that such a decision increases the risk of product diversion to the gray market (Smith & Shulman, 2022).This is because the decision of launching an upgraded product pressurizes the retailer to clear the unsold stock of the earlier version of the product to avoid the risk of low salvage value at the end of season.The increased risk and fear experienced by the retailer would further facilitate markdown sales in the unauthorized channel, thereby leading to the demand cannibalization of the upgraded product.Additionally, it is noteworthy that if the manufacturer deals with a high brand equity product, the decision to launch an upgraded product leads to a greater loss in profitability mainly owing to two reasons: (i) increased gray market diversion causes a severe damage to brand reputation; (ii) burden of cost associated with the development of upgraded product and its demand cannibalization.Surprisingly, the existing literature on new product launch decisions and demand cannibalization has only considered the supply chain structure in which the manufacturer directly sells to consumers (e.g., Xiong et al., 2016), and not through the retailer.Therefore, prior studies have not considered the retailer's risk of salvage and impacts on the manufacturer's brand reputation, leading to less severe impacts on the manufacturer's profitability.Additionally, their context of product upgrade decision is related to the remanufacturing product competition, which is mainly derived from demand cannibalization.
Corresponding to the third research question regarding on the capability of price adjustment flexibility in deterring gray market product diversion, our results suggest that the manufacturer can eliminate gray market sales by extending price adjustment capability to the authorized retailer.When the product is characterized by having a high brand equity, it is beneficial for the manufacturer to limit gray market sales and promote authorized channel sales by selling the original version product at a discounted price in the second period.Under such a scenario, the manufacturer becomes better-off in spite of the reduced total channel sales because the manufacturer's gain from eliminating gray market diversion outweighs the loss owing to reduced total channel sales.However, the retailer's profitability is negatively affected by reduced sales through the unauthorized gray market channel.Furthermore, it is observed that the manufacturer's decision to launch an upgraded product and offer price adjustment flexibility simultaneously expands the sales of the earlier version product through the authorized channel, thereby improving the profitability of the retailer.Additionally, such an arrangement results in the demand cannibalization of the upgraded version of the product.It is intriguing to find that the manufacturer dealing with luxury products enjoys higher profitability under the simultaneous practice of price adjustment flexibility and product upgrade, irrespective of the losses incurred due to the demand cannibalization of the upgraded version of the product.These findings partially explain why some companies (e.g., Apple) apply self-cannibalization as a competitive strategy by launching an upgraded product in the presence of the older version of the product in the market 13 .
The analysis of how internal channel structure, product upgrade decision, and price adjustment flexibility affect CS and SW in the presence of gray markets leads to four important observations: (i) CS and SW are always greater under the decentralized supply chain structure compared to the centralized supply chain; (ii) CS continuously increases with upgrade degree under the no pricing flexibility model; (iii) SW increases with upgrade degree but only up to a threshold level followed by a subsequent decline; (iv) SW always remains the maximum under the pricing flexibility model.In a study on contracting and supply chain coordination, Ahmadi et al. (2017) found that supply chain coordination through their proposed contract increases the overall consumer welfare in the presence of gray markets.On similar lines, Srivastava & Mateen (2020) compared the performance of different contracts for a supply chain with gray markets 13 https://corporatefinanceinstitute.com/resources/knowledge/strategy/market-cannibalization/ [accessed on 12 January 2022] and revealed that the consumers derive maximum surplus under the centralized supply chain.However, these studies mainly focus on supply chain coordination and do not consider product upgrade decision, price adjustment flexibility, and competition among various versions of the same product.

7.2
Managerial implications (i) Internal channel structure: Companies are advised to choose the relevant channel structure through careful deliberation.In this regard, companies that face higher brand erosion penalty should be extremely careful while using different channel structures.Furthermore, companies that choose the decentralized supply chain must remember that the total supply chain profit can be higher in the presence of gray market, albeit at the cost of manufacturer's reputation and profitability.In case of inadequate trust over channel partners, managers are advised to use centralized control over distribution channels.For example, Capital City Micro, Inc., an authorized distributor for HP, is reported to have abused channel incentives by obtaining large discounts on laptop computers (worth $8.6 million) and diverted it to unauthorized gray markets (Mendelsohn & Stanton, 2010).In a decentralized structure, authorized retailers, such as Pioneer and Harman Kardon electronics, have been found to have diverted products to unauthorized sellers, leading to severe demand cannibalization.Similarly, some luxury brands (e.g., Yves Saint Laurent, Calvin Klein, and Omega) have filed a lawsuit against Costco for product leakage and sale to the unauthorized channel (Bryant, 2017).
(ii) Upgraded product launch: Companies should cautiously strategize the decision regarding the launch of upgraded product when under severe risk of brand dilution due to product diversion in the unauthorized channel.For retailers, the launch of an upgraded version of a product may reduce the shelf life of the existing product, thereby increasing inventory risk.Therefore, building a connected inventory concept using various analytical tools can help lower the retailers' risk.For instance, in 2015, Zalando and Adidas experimented a novel inventory risk sharing model by linking up Adidas' distribution center to Zalando's inventory system.Consequently, Zalando's customers could access a larger assortment offering of Adidas products.Adidas could also fulfil orders for products that Zalando no longer had in stock 14 .Moreover, it is of utmost significance for the companies to strike a trade-off between the investment cost associated with upgraded product development and the loss in terms of its demand cannibalization due to gray market competition, alongside the benefits associated with product upgrade decision.Companies are also advised to diligently review and monitor their channel partners, as well as set risks and rewards to control product leakage from the authorized channel.
(iii) Price adjustment flexibility: Implementing the MSRP scheme can lead to the standardization of product prices across multiple retailers and selling channels, neutralization of price variability, and elimination of the need to analyze consumer data and competitor behavior while pricing the product.However, the MSRP often creates direct conflicts between the brand and retailers.For example, many retailers sell products below the MSRP to reduce inventory, attract more consumers, or when facing the risk of having a low salvage value of the product 15 .Therefore, to cope with risk faced by authorized intermediaries in terms low salvage value, companies should offer flexibility for price adjustment to incentivize their distributors to sell only in authorized channels.In addition to offering price adjustment flexibility, companies should consider the adoption of technological solutions, such as unique serial numbers and maintenance of active records, to deter diversion to unauthorized gray market channels.

Policy implication(s)
For policy makers, it is paramount to understand the trade-off between tolerating and limiting gray markets.The presence of gray markets can prove to be beneficial for a large consumer segment comprising of price-sensitive consumers; thus, enabling them to purchase the same product at a lower price from the gray market.Moreover, the unauthorized gray market channel facilitates wider product accessibility and caters to a larger section of the society, thereby resulting in increased SW.Conversely, brand-sensitive consumers (e.g., those who invest in luxury fashion brands) may experience decline in utility owing to a loss of exclusivity because of availability of the same product in the unauthorized channel for which they paid a premium price to the authorized retailer.However, when the product belongs to a luxury brand category characterized by high brand equity or long-term investment in research and development (R&D), gray market diversion can result in huge losses for companies, causing disincentive for investments in R&D.Therefore, under such scenarios, policymakers should exercise legal mechanisms to deter gray market activities to safeguard the interest of companies.

Limitations and future scope of research
Our research has a few limitations.First, it is limited to investigating the consequences of manufacturers' decision to launch an upgraded version of the product in the second 15 https://www.investopedia.com/terms/m/manufacturers-suggested-retailprice-msrp.asp[accessed on 12 January 2022] period on supply chain members.However, it would be meaningful to analyze the impact of product launch timing (i.e., early or delayed launch) on product diversion risk.Second, our model is bound by the context of domestic gray markets.In future studies, relaxing this restriction may prove to be a feasible avenue to explore the position of supply chain members in a setting whereby the manufacturer exercises price discrimination between two different markets of operations (high-vs.low-priced markets).Furthermore, future research may analyze the impact of gray markets on sustainability and design a suitable mechanism that not only strengthens the positive impact but also controls the negative influence of gray market operations on sustainability.
Per unit retail price of the original version product under the model j, where j = C, D, DF, per unit wholesale price charged to retailer for the upgraded version product Parameters  Consumers' valuation of the product  Perceived differentiation between the authorized and the unauthorized channel's product  Degree of product upgrade  Time discount factor for delaying purchase  Degree of brand erosion due to product diversion in the unauthorized gray market Profits Π j m

F
in the presence of gray market (Model C) D Decentralized supply chain in the presence of gray market (Model D) DF Decentralized supply chain with price adjustment flexibility: in the absence of product upgrade decision (Model DF) N Product upgrade decision: No price adjustment flexibility to the retailer (Model N) Product upgrade decision: With price adjustment flexibility to the retailer (Model F) NI Product upgrade decision: No price adjustment flexibility to the retailer in the presence of parallel import (Model NI) FI Product upgrade decision: With price adjustment flexibility to the retailer in the presence of parallel import (Model FI) p a −p e g 1+− .Similarly, let  2 ) .Similarly, let  2 Desai et al. (2004),Nie et al. (2022),  and Rao et al. (

)) 2 0
the unauthorized gray market (i.e., Q DF g = 0 and Π DF g = 0); (ii) Authorized channel sales are higher under Model DF (i.e., Q DF a ≥ Q D a ); (iii) Total channel sales are lower under Model DF (i.e., Q DF tot ≤ Q D tot ); (iv) The manufacturer is better-off under Model DF (i.e., Π DF m ≥ Π D m ) when  ≥  2 ; (v) The retailer can be worse-off under Model DF (Π DF r ≤ Π D r ) when  ≥ .

FIGURE 1
FIGURE 1 Market segmentation under Model F.

FIGURE 2
FIGURE 2Manufacturer's optimal profits with respect to  and .

FIGURE 4
FIGURE 4 Parallel import setting.

FIGURE 5
FIGURE 5 Manufacturer's dual market strategy.

TABLE 1
Some notable instances of inventory destruction.
Table 3 entails the equilibrium solutions under Model C, Model D, and Model DF.The next proposition compares the equilibrium results under Model D and Model DF.

TABLE 3
Equilibrium results for Model C, Model D, and Model DF.