Dynamic vertical foreclosure with learning-by-doing production technologies

Kourandi, Frago; Bettas, Nikolaos Article Dynamic vertical foreclosure with learning-by-doing production technologies Games Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Kourandi, Frago; Bettas, Nikolaos (2024) : Dynamic vertical foreclosure with learning-by-doing production technologies, Games, ISSN 2073-4336, MDPI, Basel, Vol. 15, Iss. 2, pp. 1-23, https://doi.org/10.3390/g15020009 This Version is available at: https://hdl.handle.net/10419/330078 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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Games 2024,15, 9. https://doi.org/10.3390/g15020009 Academic Editors: Konstantinos Serfes, Ulrich Berger and Kaniska Dam Received: 31 December 2023 Revised: 20 February 2024 Accepted: 27 February 2024 Published: 29 February 2024 Copyright: © 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). games Article Dynamic Vertical Foreclosure with Learning-by-Doing Production Technologies Frago Kourandi 1and Nikolaos Vettas 2,3,* 1Department of Economics, National and Kapodistrian University of Athens, 1 Sofokleous Str., 10559 Athens, Greece; [email protected] 2Department of Economics, Athens University of Economics and Business, 76 Patision Str., 10434 Athens, Greece 3Centre for Economic Policy Research, London EC1V 0DX, UK *Correspondence: [email protected] Abstract: Here, we study vertical foreclosure in a dynamic setup with learning-by-doing production technologies. There is a downstream monopoly and an upstream duopoly, where manufacturers produce differentiated products and can gain proficiency through the accumulation of their production. We study the dynamic interactions in the vertical chain when the monopolist sets the prices; we find that customer foreclosure may arise in equilibrium when the products are close substitutes and be welfare-enhancing. The rate of learning is lower than the social optimal and a social planner would tend to impose exclusivity more often compared to the downstream monopolist. Keywords: dynamic interactions; learning-by-doing; exclusivity JEL Classification: L42; L13; L14; L11; L81 1. Introduction How production is organized within and across firms crucially affects the welfare of the final consumers and is, naturally, the core concern of industrial organization and related fields. One of the factors that shape the key market outcomes is the intensity of competition and the variety of available products. In this regard, different instances of exclusionary practices in vertically related industries, between upstream and downstream firms, have drawn the attention of regulators and anti-trust authorities and have been the focus of influential academic research, with the foreclosure of some producers viewed with suspicion in most of the cases. Yet, there are also other factors that play a crucial role in determining market outcomes and welfare, including efficiency in production. While exclusionary practices have been extensively studied in the literature, less attention has been given to the role and nature of exclusivity in dynamic environments. In this paper, we introduce dynamic interactions in a vertical chain through learningby-doing in production. This is performed in a simple way: there is an upstream duopoly, where the production of each firm may become more efficient according to the accumulation of production over time. The upstream firms produce either complements or substitutes, with horizontal product differentiation taking the form of “love for variety”. A downstream monopolist can receive the products of the upstream firms and has the power to set linear wholesale contract terms. Crucially, the game lasts for two periods. In each period, the retailer sets the contract terms to each of the upstream firms, which can either accept or reject their contract offers. Following that stage, the retailer sells the products to the final consumers and pays the upstream firms according to the contracts that have been agreed. When the game proceeds to the second period, the costs of the upstream firms may have been reduced based on their first period production level. Games 2024,15, 9. https://doi.org/10.3390/g15020009 https://www.mdpi.com/journal/games Games 2024,15, 9 2 of 23 In this setting, we explore the tensions between assuring access to competing varieties and lowering production costs via learning-by-doing. Exclusion arises implicitly when the dominant retailer offers very disadvantaged contract terms to an upstream firm, and, thus, forces it to stay out of the market. Thus, exclusivity does not arise by a dominant upstream firm’s denial to supply some downstream firms with the essential input that it produces. 1 Instead, the retailer is a large player, and the upstream firms can reach the final consumers only via this retailer. Our analysis can be applicable to a multiproduct firm where production is characterized by the learning hypothesis or to a strategic buyer that can manipulate the learning process through their product choice. The role of the intermediary is studied by examining whether the existence of this firm intensifies the learning process compared to the case where there is no intermediary in the market, but the upstream firms compete directly in the final market. Within such a framework, interesting questions arise: How does the final market outcome depend on the intensity of learning-by-doing and product differentiation? Does the retailer choose to carry only one product to intensify the learning process? Under what conditions do exclusivity and, thus, customer foreclosure emerge? Under what conditions may they be beneficial for the final consumers and the firms? Does the market competition favor an outcome with lower prices or with more variety in the market? Is the presence of the retailer in the market necessary to intensify the learning process? The food industry is a leading motivating example, corresponding to such vertical structure models and one that has been attracting the attention of policy makers and even the popular press. In the last couple of decades, large supermarkets have emerged and become powerful in their transactions with upstream suppliers in many industries. Other examples include large tour operators trading with airlines and hotels or general retailers such as Wal-Mart. The issue of exclusionary practices is also central in manufacturing, like in the automobile sector (see, e.g., Brenkers and Verboven, 2005) [1]. The fact that learning-by-doing can be an important factor in production processes has been documented in a number of business units and sectors, such as the agriculture industry where sustainable farming relies on learning from the experience gained over multiple growing seasons. Furthermore, this is the case in manufacturing contexts, such as in the automobile industry where Toyota Production System is based on the concept of “Kaizen”, i.e., continuous improvement, through learning from experience. Other leading examples of learning-by-doing in production include the construction industry and the software development industry. In the static version of our model, the retailer always chooses to carry both products. In a one-period setting, there is no learning and exclusivity reduces the variety in the market without reducing the production costs or the prices. In the dynamic model, exclusivity arises in equilibrium when products are close substitutes. In contrast, when products are complements or not close substitutes, both are purchased in both periods. This result follows from two opposite effects. There is a trade-off between lower costs due to learning and more product varieties in the market. When product differentiation is low, the “learning” effect dominates the “product variety” effect, and the total profits of the chain and the consumer’s surplus are higher when one product is excluded from the market. It follows that exclusivity is welfare-enhancing. Furthermore, relative to the non-exclusivity case, the product prices in both periods are lower when exclusivity is imposed by the retailer. However, the rate of learning is not equal to the social optimum and the social planner would impose exclusivity more often compared to the retailer. We also maintain that the presence of the downstream intermediary is necessary in intensifying the learning-by-doing process when products are complements or when products are close substitutes, and the retailer imposes exclusivity. Our paper contributes to two different fields in the literature within industrial organization: vertical contracting and learning-by-doing and business organization. Each of these contains very important papers and it is too vast to survey here. We only refer to work that is more closely related to our analysis. First, numerous contributions highlight Games 2024,15, 9 3 of 23 the impact of vertical contracting and, in particular, the impact of exclusionary behavior on market competition. For a review and some key results on vertical foreclosure, see Rey and Tirole (2007) [ 2 ]. Krattenmaker and Salop (1986) [ 3 ] argue that contracts with input suppliers can be fertile ground for raising competitor’s cost and Aghion and Bolton (1987) [ 4 ] demonstrate that contracts between buyers and sellers will be signed for entry-prevention purposes. Mathewson and Winter (1987) [ 5 ], Besanko and Perry (1993) [ 6 ], and O’Brien and Shaffer (1993) [ 7 ] derive that exclusive arrangements can have desirable welfare properties. 2 The opportunistic behavior is faced through exclusivity according to McAfee and Schwartz (1994) [ 8 ]. We obtain that exclusivity is welfare-improving, compared to non-exclusivity, due to learning in the production. Rey and Stiglitz (1995) [ 9 ] obtain maintain that, when goods are close substitutes, producers’ profits are higher under exclusive territories. We maintain that when goods are close substitutes, retailer’s profits are higher under exclusivity since the learning effect dominates the product variety effect. Foreclosure has also recently been studied in vertical contracting frameworks with vertically integrated firms by Reisinger and Tarantino (2015) [ 10 ] and Kourandi and Pinopoulos (2023) [ 11 ]. We depart from vertical integration and study how learning affects future production efficiency and, thus, the market exclusion of upstream producers. A dynamic setting with intertemporal linkages and input foreclosure is studied by Fumagalli and Motta (2020) [ 12 ] and Sandiumenge i Boy (2023) [ 13 ]. We focus on customer foreclosure. Furthermore, there is a well-known field in the literature on common agencies that deal with exclusivity. See, for example, Martimort (1996) [ 14 ], Bernheim and Whinston (1998) [ 15 ], and Segal and Whinston (2000) [ 16 ]. Further, Marx and Shaffer (2007) [ 17 ] adopt upfront payments with bargaining power in the downstream level and find that exclusivity arises in equilibrium. In our setup, when products are close substitutes, the bottleneck retailer coordinates the purchases of the two products and takes advantage of the learning process in a more efficient way compared to the case where upstream producers directly sell their products to the final consumers. Nevertheless, relatively few papers have examined vertical chain interactions in a dynamic framework. An exemption is Chen (2005) [ 18 ], who studies vertical disintegration in a dynamic model with learning-by-doing and downstream competition. 3 In our dynamic model with learning-by-doing, we focus, instead, on product differentiation and customer foreclosure. The second related field in the literature examines strategic purchases in one-tier industries in a dynamic setting under the learning-by-doing hypothesis. Spence (1981) [ 21 ] obtains the dynamic output path of a singleand then a multi-firm model and studies the open and closed loop equilibria. Fudenberg and Tirole (1983) [ 22 ] derive the precommitment and perfect equilibria with linear demand and learning functions. A price-setting differentiated duopoly with an infinite sequence of heterogeneous buyers and uncertain demands is analyzed by Cabral and Riordan (1994) [ 23 ], where they find the Markov perfect equilibrium and study the concept of market dominance. In an enriched version of the Cabral and Riordan model, Besanko, Doraszelski, and Kryukov (2019) [ 24 ] use computational methods to analyze the Markov Perfect equilibrium behavior. Lewis and Yildirim (2002a) [ 25 ] study the trade-off between experience and competition in an industry with privately cost functions and a single buyer. Meanwhile, Lewis and Yildirim (2002b) [ 26 ] investigate how incentive regulation should be designed to encourage suppliers to develop and adopt cost-saving technologies. Recently, Sweeting et al. (2022) [ 27 ] studied dynamic price competition when sellers benefit from learning-by-doing and buyers are long-lived and strategic, while capturing, with a single parameter, the extent to which each buyer internalizes future buyer surplus. They find that even moderate degrees of forwardlooking buyer behavior may eliminate the multiplicity of equilibria and that equilibria with competition are more likely to survive. In these models, learning-by-doing can be understood either as cost-reducing innovations or as the result of economies of scale across multiple market segments, where the different time periods play the role of different market segments. Our setup adds an extra Games 2024,15, 9 4 of 23 tier in the vertical supply chain. Products are reached by the consumers via a retailer that may decide to reduce product availability. We explore the conditions under which the learning effect dominates the product variety effect in a vertical chain and find, consistently with the learning-by-doing literature, that equilibria with either more (non-exclusivity, i.e., competing suppliers) or less (exclusivity, i.e., monopoly supplier) variety can emerge depending on the level of product differentiation. Yet, our model is kept simple, especially in the sense that the power to set wholesale prices is given to the retailer. Thus, while the analysis captures the main tension between product variety and dynamic cost reduction, the richer model is necessary to address more complex issues that would arise if bargaining power was more evenly distributed across upstream and downstream firms. The remainder of the paper is presented as follows. Section 2characterizes the equilibrium when the game lasts only one period. The dynamic model is analyzed in Section 3. Consumer’s surplus and total welfare are calculated in Section 4. In Section 5, we derive the social planning solution and compare it to the equilibrium. Section 6studies the role of the intermediary in the learning process and Section 7concludes. 2. The Static Model We begin our analysis by studying the potential for exclusivity to arise in a simple setting where firms compete for a single period. We consider two manufacturers, A and B, who each produce a single product with these products to be either substitutes or complements. Product differentiation is horizontal and takes the form of “love for variety”. Manufacturers supply the downstream market and have the same constant marginal production cost, c , since we focus on the possibility of exclusivity to emerge as a result of strategic interaction and not due to cost asymmetries in the upstream industry. The downstream market is monopolized by a retailer, R, that has the power to set the contract terms which take the form of linear wholesale prices. 4 One unit of the manufacturers’ product becomes one unit of final good at zero marginal cost. The inverse final demand functions take the linear form pi=a−qi−bqj , where pi is the retail price and qi is the quantity of product i , i=A , B , a≥c and b∈(− 1, 1 ) is the product differentiation parameter. 5 When b equals zero, the two products are independent. As b approaches the unity ( b→ 1) the two goods become closer substitutes and as b approaches minus one ( b→ − 1) the two goods become closer complements. When b is equal to one, the goods are homogeneous and when b is equal to minus one, the two goods are perfect complements. There is no uncertainty and all contracts are observable. The static framework is presented in Figure 1. 1 c c A B R wAwB pApB final consumers pA=α-qA-bqB pB=α-qB-bqA b delta -10 1.0 0.5 1.0 0.5 0 0.0 10 0.0 Figure 1. The static model. Games 2024,15, 9 5 of 23 We consider a three-stage game. First, the retailer sets the contract terms. Then, the manufacturers either accept or reject the contract offers and finally the retailer sells the products to the final consumers and pays the manufacturers according to their contracts. The game is solved backwards, using subgame perfection as the equilibrium concept. In the first stage of the game, the retailer can set very disadvantaged terms to one of the two suppliers (say B) to force this supplier to reject this offer and stay out of the market. The retailer implicitly denies access of the final consumers to that firm’s product by offering a wholesale price lower than the production cost of that manufacturer. 6 First, we consider the case where both products are carried by the retailer (non-exclusivity) and then we study the corner case where the retailer chooses to carry only one product (exclusivity). Non-exclusivity: Solving backwards, in the final stage of the game, the retailer maximizes its profits, ΠR= (pA−wA)qA+ (pB−wB)qBs . t . pi=a−qi−bqj , in the final goods market. The optimal quantities are derived by the first-order conditions. Then, the retailer sets the wholesale prices wi , i=A , B , given the individual rational constraints of the manufacturers. The manufacturers accept the offer if they obtain non-negative profits, since at the downstream level there is no alternative retailer, which leads to zero outside option for the manufacturers. Thus, the downstream monopolist sets the wholesale prices equal to the minimum possible level, that is, wi=c for both manufacturers.7 Lemma 1. In the static model with both products carried by the retailer, we obtain: qi=a−c 2(1+b),pi=a+c 2,wi=c,Πi=0, ΠR=(a−c)2 2(1+b). The final prices are equal to the monopoly prices. The chain profits are maximized and captured by the retailer. Exclusivity: Assume now that, in the first stage of the game, the retailer chooses to distribute only one product, say A. Now, it maximizes its profits, ΠR= (pA−wA)qAs . t . pA=a−qA, and obtains the optimal quantity by the first-order condition. The retailer then determines the wholesale prices wi for the manufacturer A to accept the offer and for B to reject the offer. The retailer charges a wholesale price equal to the unit production cost to manufacturer A, wA=c , and a wholesale price lower than the production cost to manufacturer B, wB<c , to exclude this product from the market. The monopoly results are reached with the retailer to extract the vertical chain’s profits. Lemma 2. In the static model, with one product in the market, we obtain: qA=a−c 2,pA=a+c 2,wA=c,wB<c,ΠA=0, ΠR=(a−c)2 4. When products are perfect substitutes ( b= 1), the results from the two subgames coincide, apart from the fact that the total quantity is equally split to the two manufacturers when they are both selling. Under imperfect substitutability ( b∈( 0, 1 ) ), the final prices are equal in the two subgames. However, the total quantity and the profit obtained by the retailer increase as b decreases under non-exclusivity, since consumers prefer to have both varieties available. When products are complements ( b∈(− 1, 0 ) ), final quantities and profits under non-exclusivity are further increased. The consumer’s surplus when both products are available is equal to (a−c)2 4(1+b) , and when only one product is distributed it is equal to (a−c)2 88 Since profits and consumer’s surplus under non-exclusivity are higher, total welfare is also higher. In the static model, exclusivity harms both consumers and firms. Games 2024,15, 9 6 of 23 Proposition 1. In the static model, exclusivity does not arise in equilibrium. The total profits of the chain, the consumer’s surplus, and the total welfare when both products are purchased are higher than when one product is excluded from the market. 3. The Dynamic Model Here, we depart from the static model by assuming that the game lasts for two periods. We study the dynamic interactions that emerge in the vertical framework by introducing the learning-by-doing hypothesis. Over time, upstream producers gain proficiency through the repetition of their production. The unit production cost decreases as the producer gains more experience; that is, the unit cost function is a decreasing function of past accumulated production. Interesting issues arise in this dynamic environment: Does exclusivity emerge in equilibrium and under what conditions does it emerge? Does market competition favor an outcome with lower prices or with more varieties in the market? In our model, production is characterized by the linear learning-by-doing hypothesis. The unit production cost of the manufacturers in the second period reduces proportionally with the production of the first period. Firms learn from their own production. Specifically, the unit cost functions in the second period are given by: ci2=c−λqi1if qi1<c λ 0 if qi1≥c λfor i=A,B. (1) The first subscript refers to the manufacturer and the second to the time period ( t= 1, 2), while λis the learning parameter. The timing of the game is the same as in the static game, with the difference being that firms now interact for two periods. In each period, the retailer first sets the contract terms; then, the upstream firms either accept or reject the offers and, subsequently, the retailer sells the products to the final consumers and pays the manufacturers. Two cases are examined in each period: both products are purchased by the retailer or only one product is purchased either in period one or in period two. Thus, in terms of product availability, four alternative cases may arise in the two-period model. Crucially, there is interdependence between the two periods due to the learning-bydoing process. The unit costs in the second period are affected by the quantities produced in the first one. Therefore, the retailer maximizes the present value of its profits in the first period: ΠR1+δΠR2 , where δ∈( 0, 1 ) is the discount factor. We proceed backwards to solve for the subgame’s perfect equilibrium. 3.1. Period Two Product availability and quantities purchased in the first period shape the production costs in the second period. As a matter of notation, if the retailer chooses to carry one product in the first period, this will be product A. So, in the second period, the more cost-efficient firm (if any) is firm A with cA2≤cB2 . Taking as given the production costs cA2 , cB2 , we consider first the case where the retailer does not strategically exclude an upstream supplier by offering disadvantaged contract terms and then the case where the retailer purchases only one product in period two. Non-exclusivity: We begin from stage three. The retailer maximizes its profits: max qA2,qB2 ΠR2= (pA2−wA2)qA2+ (pB2−wB2)qB2 s.t.pi2=a−qi2−bqj2with i=A,Band i=j. Similarly to the static model, the upstream firms accept the offers and the retailer sets the wholesale prices equal to the unit production costs. wi2=ci2. We obtain Games 2024,15, 9 7 of 23 qi2=a 2(1+b)+bcj2−ci2 2(1−b2),pi2=a+ci2 2,wi2=ci2,Πi2=0, ΠR2=a(2a−cA2−cB2) 4(1+b)+(a−cA2)(bcB2−cA2) + (a−cB2)(bcA2−cB2) 4(1−b2). Note that qA2≥ 0 always holds since b≤ 1 ≤a−cA2 a−cB2 and cA2≤cB2 , but qB2≥ 0 holds only when b≤a−cB2 a−cA2≤ 1, where the cost asymmetry in the second period is not high enough. Even if the retailer does not strategically exclude one supplier by offering a wholesale price lower than the production cost, product B may be excluded from the market in period two when firm B has a production cost too high compared to the rival’s cost, i.e., when b>a−cB2 a−cA2. Taking into account this possibility of a corner solution, we obtain:9 qA2=   a 2(1+b)+bcB2−cA2 2(1−b2)i f b ≤a−cB2 a−cA2 a−cA2 2i f b >a−cB2 a−cA2 ,qB2=   a 2(1+b)+bcA2−cB2 2(1−b2)i f b ≤a−cB2 a−cA2 0i f b >a−cB2 a−cA2 , pA2=a+cA2 2,pB2=(a+cB2 2i f b ≤a−cB2 a−cA2 −i f b >a−cB2 a−cA2 ,(2) wA2=cA2,wB2=(cB2i f b ≤a−cB2 a−cA2 −i f b >a−cB2 a−cA2 , ΠA2=0, ΠB2=(0i f b ≤a−cB2 a−cA2 −i f b >a−cB2 a−cA2 , ΠR2=   a(2a−cA2−cB2) 4(1+b)+(a−cA2)(bcB2−cA2)+(a−cB2)(bcA2−cB2) 4(1−b2)i f b ≤a−cB2 a−cA2 (a−cA2)2 4i f b >a−cB2 a−cA2. Exclusivity: Now, assume that the retailer purchases only one product in period two (product A) by offering to supplier B very disadvantaged contract terms. The retailer maximizes: max qA2 ΠR2= (pA2−wA2)qA2s.t.pA2=a−qA2. The upstream firm A accepts the offer and firm B rejects the offer where wA2=cA2 and wB2<cB2. Therefore:10 qA2=a−cA2 2,qB2=0 (3) pA2=a+cA2 2,wA2=cA2,wB2<cB2 Πi2=0, ΠR2=(a−cA2)2 4. Outcome in period two: Given the possible values of the two production costs in period two, the retailer decides if it will impose exclusivity in period two or not, by comparing its profits in this period for the two alternative cases above (by Expressions (2) and (3)). When − 1 <b≤a−cB2 a−cA2 , we find that it is profitable for the retailer to purchase both products in the final period of the game. 11 The second period is the last period of the game and excluding one supplier from the market by offering disadvantaged terms only reduces the variety in this period without opting for future cost reduction. When 1 >b>a−cB2 a−cA2 , the retailer’s profits in (2) or (3) are identical with product B not available in period two. In (2), product B cannot survive in the market due to high cost asymmetry; meanwhile, in (3), product B is strategically excluded by the retailer via disadvantaged wholesale pricing. Summing up, the equilibrium outcome in period two is given by Expression (2).12 Games 2024,15, 9 8 of 23 3.2. Period One The first question now is whether the retailer has an incentive to exclude one upstream supplier in the first period to intensify the learning process by purchasing higher quantity by only producer A. The second question is, given that the retailer chooses to carry only product A in the first period, whether it will purchase high enough quantity by producer A to make this producer efficient enough in period two as well (which will lead to high cost asymmetry in the next period). Without exclusion in the first period, producers are equally cost-efficient in the future, in contrast to the case where exclusivity is imposed in the first period. There are two alternative cases: one where the retailer purchases both products and one where the retailer purchases only one product in period one by offering disadvantaged contract terms to producer B. Non-exclusivity: Given that the suppliers initially have equal costs, the equilibrium in period one will be symmetric and the suppliers equal efficient in the second period. The retailer maximizes the present value of its profits ΠR1+δΠR2 , where ΠR2 is given by Expression (2). The quantities purchased in period one affect the production costs in period two and, subsequently, the prices and the profits obtained are also affected. The retailer solves: max qA1,qB1 ΠR1+δΠR2= (pA1−wA1)qA1+ (pB1−wB1)qB1 +δa(2a−cA2−cB2) 4(1+b)+(a−cA2)(bcB2−cA2) + (a−cB2)(bcA2−cB2) 4(1−b2) s.t.ci2=c−λqi1for i=A,B. By the fact that the upstream firms accept the offers and the wholesale prices are set to the marginal cost wi1=c, we have:13 Lemma 3. In the dynamic model without exclusivity in period one, we have:14 ΠNE R1+δΠNE R2=2(a−c)2((δ+1)(b+1)+λδ) 8b−λ2δ+4b2+4. (4) Exclusivity: Assume now that, in the first period, the retailer chooses to distribute only product A. Then, manufacturer A will be more cost-efficient in period two ( cA2<cB2 ) since it has benefited from the learning-by-doing ( qA1> 0 and qB1= 0). This cost reduction determines whether product B will be produced in period two, depending also on the product differentiation parameter. The retailer maximizes: max qA1 ΠR1+δΠR2=      (pA1−wA1)qA1+δa(2a−cA2−cB2) 4(1+b)+(a−cA2)(bcB2−cA2)+(a−cB2)(bcA2−cB2) 4(1−b2)i f b ≤a−cB2 a−cA2 (pA1−wA1)qA1+δ(a−cA2)2 4i f b >a−cB2 a−cA2 s.t.cA2=c−λqA1and cB2=c. The wholesale price for manufacturer A is set to be equal to the production cost in period one and for manufacturer B lower than the production cost, making this firm not operate in the first period. So, we have wA1=c and wB1<c . Choosing the optimum level of qA1 now affects the subsequent cost asymmetry and, thus, the subsequent product availability. When products are either complements or not so close substitutes, − 1 <b≤a−cB2 a−cA2 , or equivalently when the quantity of firm A in period one is not high enough, qA1≤(a−c)(1−b) λb , both products are purchased in period two. The retailer’s maximand is Games 2024,15, 9 15 of 23 WE−S 1+δWE−S 2=     (a−c)2(λ2δ2(2b−2λ+2bλ+2λ2δ+3b2−5)+(b+1)(b−1)2(b+4δ+4λδ+1)) 2((1−b2)−δλ2)2i f b ≤1−δλ2 1+λ (a−c)2(−2λ3δ2−λ2δ2+4λδ+2δ+1) 2(1−δλ2)2i f b >1−δλ2 1+λ. (9) Again, here the social planner intensifies the learning process by setting prices lower than the production costs in the first period and, thus, incurring some losses. In the second period, prices are equal to the production costs. We find that exclusivity is imposed in the first period of the game which also leads to exclusivity in the second period of the game when b>1−λ2δ 1+λ ; however, when b≤1−λ2δ 1+λ , the quantity of product A purchased in the first period of the game is not high enough to exclude product B in the second period. Characterization of the social optimum: To fully characterize the social planner’s solution we have to check when the social planner excludes an upstream supplier in the first period of the game to manipulate the learning process. We compare the present value of the total welfare using Expressions (8) and (9) for every product differentiation parameter. ∆W=WE−S 1+δWE−S 2−WNE−S 1+δWNE−S 2=      (a−c)2(λ2δ2(2b−2λ+2bλ+2λ2δ+3b2−5)+(b+1)(b−1)2(b+4δ+4λδ+1)) 2((1−b2)−δλ2)2−(a−c)2(b+δ+bδ+2λδ+1) (1+b)2−λ2δi f b ≤1−δλ2 1+λ (a−c)2(−2λ3δ2−λ2δ2+4λδ+2δ+1) 2(1−δλ2)2−(a−c)2(b+δ+bδ+2λδ+1) (1+b)2−λ2δi f b >1−δλ2 1+λ. When ∆W is positive and b≤1−δλ2 1+λ , exclusivity is imposed only in period one and in period two both products are purchased. However, when ∆W is positive and b>1−δλ2 1+λ , exclusivity is imposed in both periods. Numerically, we find that: Lemma 6. For low values of the product differentiation parameter, the social planner chooses to have both products in the market in both periods. For intermediate values of the product differentiation parameter product B is excluded in the first period but not in the second period. Finally, for high values of the product differentiation parameter, product B is excluded in both periods. Furthermore, we have: Lemma 7. The social planner tends to impose exclusivity more often than the retailer. To better understand these results, we present a numerical example. For given learning and discount factor parameters, we plot in Figure 4(for each b ): the difference in the present value of the welfare ( ∆W=WE 1+δWE 2−WNE 1+δWNE 2 , the objective function of the social planner) and the difference in the present value of the profits (∆Π =ΠE R1+δΠE R2−ΠNE R1+δΠNE R2 , the objective function of the retailer from Section 3) . We observe that, when the products are close complements, the social planner never imposes exclusivity and this also holds for the retailer ( ∆W< 0 and ∆Π < 0 for low and negative b ). However, when the products are not close complements ( b is negative but not very low), the social planner excludes product B only in the first period, which is never an equilibrium choice for the retailer. The retailer never excludes product B in the first period when the products are complements (close or not, ∆Π < 0 for b< 0). Furthermore, the social planner excludes product B from the market only in the first period when products are not close substitutes and excludes product B from the market in both periods when products are close substitutes. In contrast, the retailer never excludes product B only in the first period, it excludes product B in both periods for high and positive values of b. Games 2024,15, 9 16 of 23 To summarize, as b increases (from − 1 to 1), the social planner first carries both products in both periods, then, carries only product A in the first period and both products in the second period and, finally, it carries only product A in both periods. However, the retailer, first, carries both products in both periods and then carries only product A in both periods. Note also in the figure that exclusivity occurs more often in the social optimum setting compared to the equilibrium outcome where the retailer imposes exclusivity. The area of the parameter bwhere ∆Wis positive is greater than the one where ∆Π is positive. -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.1 0.2 0.3 0.4 0.5 0.6 -3 -2 -1 1 2 3 b To better understand these results, we present a numerical example. For given learning and discount factor parameters, we plot in Figure 4 (for each b) the di¤erence in the present value of the welfare (W=WE 1+WE 2WNE 1+WNE 2, the objective function of the social planner) and the di¤erence in the present value of the pro…ts ( = E R1+E R2 NE R1+NE R2, the objective function of the retailer from Section 3). We observe that, when the products are close complements, the social planner never imposes exclusivity and this also holds for the retailer (W < 0and  <0for low and negative b). However, when the products are not close complements (bis negative but not very low) the social planner excludes product B only in the …rst period which is never an equilibrium choice for the retailer. The retailer never excludes product B in the …rst period when the products are complements (close or not,  <0for b < 0). Also, the social planner excludes product B from the market only in the …rst period when products are not close substitutes and excludes product B from the market in both periods when products are close substitutes. In contrast, the retailer never excludes product B only in the …rst period, it excludes product B in both periods for high and positive values of b. To summarize, as bincreases (from -1 to 1) the social planner …rst carries both products in both periods, then, carries only product A in the …rst period and both products in the second period and, …nally, it carries only product A in both periods. However, the retailer, …rst, carries both products in both periods and then carries only product A in both periods. Note also in the …gure that exclusivity occurs more often in the social optimum setting compared to the equilibrium outcome where the retailer imposes exclusivity. The area of the parameter bwhere Wis positive is greater than the one where  is positive. By direct comparison of the quantities in the …rst period under the planning solution and 23 ΔΠ ΔW Figure 4. Welfare and profit comparisons for λ=1, δ=0.5. By direct comparison of the quantities in the first period under the planning solution and the equilibrium outcome of Section 3, we conclude to the following proposition. Proposition 5. The rate of learning in equilibrium is lower than the social optimum. The production costs in the second period are lower under the social optimal due to the higher quantity produced in the first period, compared to the equilibrium outcome without a social planner. 6. The Role of the Intermediary Finally, we study the role of the retailer in the learning process. Is the presence of such a retailer indeed necessary to intensify this process? Or can firms take advantage of the learning process equally efficiently without the downstream monopolist? First, we analyze the role of the retailer in the static model where there is no learning and then we study the dynamic model. We compare the equilibrium outcome when the upstream firms sell their products to the final market indirectly, via the retailer, to the equilibrium outcome when the upstream firms distribute their products directly to the final consumers. When there is no intermediary, the outcome is the Cournot duopoly outcome with differentiated products and when a retailer exists, the outcome is the monopoly outcome (given in Section 2). The key is the product differentiation parameter. When the products are substitutes ( b∈( 0, 1 ) ) and there is no intermediary, the final prices and total profits are lower and the consumer’s surplus and total welfare are higher compared to the case where a retailer exists. Remark 4. In the static model, when the products are substitutes, the presence of the intermediary hurts the market, since the retailer gains the monopoly profits by increasing the final prices and, thus, lowering the consumer’s surplus and total welfare. In contrast, when the products are complements, the role of the retailer in the market is positive, since it internalizes the externalities from the two complement goods. The prices are lower and the consumer’s surplus, profits, and total welfare are higher when there exists a retailer in the downstream level. Games 2024,15, 9 17 of 23 Then, we solve for the dynamic Cournot differentiated duopoly model with learningby-doing technology and no intermediary. Both products are purchased in both periods and the equilibrium quantities for the first period are given by26 qC i1=(a−c)(2−b)(2+b)2+4δλ (2−b)(2+b)3−4δλ2for i=A,B. We compare these quantities to the quantities in our equilibrium model with the presence of the retailer:27 q∗ A1=   (a−c)(2(1+b)+δλ) 4(1+b)2−δλ2NE in both periods i f b ∈−1, b (a−c)(2+δλ) 4−δλ2E in both periods i f b ∈b, 1 and obtain that: Remark 5. For b∈(−1, 0) we have qC A1<q∗ A1 , for b∈0, b we have qC A1>q∗ A1 and for b∈b, 1we have qC A1<q∗ A1. The presence of the intermediary intensifies the learning process when the two products are complements or when the products are close substitutes and exclusivity is imposed by the intermediary (i.e., b∈(−1, 0)∪b, 1 ). The retailer coordinates the purchases of the two products and takes advantage of the learning process in a more efficient way compared to the case where upstream producers sell their products directly to the final consumers. The rate of learning is closer to the social rate of learning (for these parameter values of b ) when there is an intermediary in the market. 7. Conclusions and Further Research Learning-by-doing technologies can play a significant role in vertical chains. In this paper, we have examined how the learning-by-doing process affects the final market outcome in a vertical framework. Upstream firms produce differentiated products, either substitutes or complements, and the downstream monopolist sets the linear contract terms. The unit production cost of the upstream firms reduces with the accumulated production. We study how the dynamic interactions between upstream and downstream firms affect the exclusive dealing decisions in a two-period game. The existing literature has either examined vertical contracting without dynamic interactions due to learning effects or the learning-by-doing process in an oligopolistic industry without vertical considerations. Our paper is the first that studies a dynamic vertical foreclosure with learning-by-doing production technologies. In the static model, both products are carried by the retailer. Exclusivity never arises in equilibrium, since final consumers like product variety. When we introduce dynamic considerations, the decision on imposing exclusivity depends on the product differentiation parameter. Close substitutability leads to exclusivity in the dynamic model, since the “learning” effect (leading to lower prices) dominates the “product variety” effect. In contrast, complementarity or a lack of close substitutability leads to an equilibrium where the retailer purchases both products in both periods. More varieties in the market are preferred to lower final prices. Consumer’s surplus and total welfare are also higher under exclusivity when it is imposed by the retailer compared to the case where exclusivity is not imposed (for example, due to regulatory restrictions). Therefore, exclusivity is welfare-improving. Nevertheless, the equilibrium rate of learning is not equal to the social optimum. A social planner would more often impose exclusivity and would further reduce the production costs in the second period compared to the retailer. Finally, when products are complements or close substitutes the presence of the retailer is necessary to coordinate the learning process. Games 2024,15, 9 18 of 23 There is often intense criticism for various exclusionary practices, which underlies economic policy decisions. Often, these practices are anti-competitive, but sometimes can be pro-competitive. In this paper, we characterize conditions under which exclusivity is beneficial for firms and consumers in a simple dynamic vertical framework with learningby-doing production technologies. Our analysis enriches, thus, the set of results concerning exclusionary practices by highlighting the role of possible efficiencies through cost reduction. Further research can, of course, shed light on additional aspects of the issue. In particular, our model is kept very simple regarding the allocation of bargaining power and endows the downstream firm with the ability to set linear wholesale prices. In a more complex setting, where the bargaining power may be more evenly distributed across the upstream and downstream firms, additional tensions will likely arise, partly reflecting results from earlier research on downstream ‘bottleneck’ models. If the upstream firms actively participate in the pricing decisions, the implicit coordinating role of the retailer would tend to be diminished, perhaps also leading to a reduced ability to motivate and exploit learning-by-doing. In parallel, competition between the upstream firms may tend to be more intense in the first stage of the game, so that they strategically improve their efficiency in the second stage, and partly benefit from this cost reduction. How the cost reduction versus product variety tension will play out in such a rich setting, and how profits will be allocated among the firms, would of course largely depend on the type of contracts that can be employed. Finally, it would be interesting to extend the present model by considering alternative demand specifications and learning-by-doing processes. Author Contributions: Both authors contributed to the study conception, design and writing equally. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Data Availability Statement: No new data were created or analyzed in this study. Conflicts of Interest: The authors declare no conflict of interest. Appendix A All expressions for Lemma 3: qNE i1=(a−c)(2(1+b) + δλ) 4(1+b)2−δλ2,qNE i2=a−c 2(1+b)+λ(a−c)(2(1+b) + δλ) 2(1+b)(4(1+b)2−δλ2), pNE i1=2(b+1)2(a+c)−aλ2δ−λδ(a−c)(b+1) 4(1+b)2−δλ2, pNE i2=2(b+1)2(a+c)−aλ2δ−λ(a−c)(b+1) 4(1+b)2−δλ2, wNE i1=c,wNE i2=4c(1+b)2−λ2δ−2λ(a−c)(1+b) 4(1+b)2−δλ2, ΠNE i1=ΠNE i2=0, ΠNE R1=2(a−c)22(1+b)2−λ2δ−λδ(1+b)(2b+λδ +2) (4(1+b)2−δλ2)2, ΠNE R2=2(a−c)2(2b+λ+2)2(1+b) (4(1+b)2−δλ2)2. All expressions for Lemma 4: Games 2024,15, 9 19 of 23 qE A1=   (1−b)(a−c)(2b+λδ+2) 4(1−b2)−δλ2i f b ≤4−λ2δ 2(λ+2)both in t = 2 (a−c)(2+δλ) 4−δλ2i f b >4−λ2δ 2(λ+2)only A in t = 2 , qE A2=   (a−c)(2(1−b)+λ) 4(1−b2)−δλ2i f b ≤4−λ2δ 2(λ+2) (a−c)(2+λ) 4−δλ2i f b >4−λ2δ 2(λ+2) ,qE B2=   (a−c)(4−δλ2−2b(λ+2)) 2(4(1−b2)−δλ2)i f b ≤4−λ2δ 2(λ+2) 0i f b >4−λ2δ 2(λ+2) , pE A1=   2(1−b)(b+1)(a+c)−δλ((1−b)(a−c)+aλ) 4(1−b2)−δλ2i f b ≤4−λ2δ 2(λ+2) c(λδ+2)+a(2−δλ(λ+1)) 4−δλ2i f b >4−λ2δ 2(λ+2) , pE A2=   2(1−b)(b+1)(2(a+c)−λ(a−c))−δλ2(2a−b(a−c)) 2(4(1−b2)−δλ2)i f b ≤4−λ2δ 2(λ+2) c(λ+2)+a(2−λ(λδ+1)) 4−δλ2i f b >4−λ2δ 2(λ+2) ,pE B2=   a+c 2i f b ≤4−λ2δ 2(λ+2) −i f b >4−λ2δ 2(λ+2) , wE A1=c,wE B1<c,wE A2=   2(1−b)(b+1)(2c−aλ+cλ)−δλ2(a−b(a−c)) 4(1−b2)−δλ2i f b ≤4−λ2δ 2(λ+2) c(2λ+4)−aλ(λδ+2) 4−δλ2i f b >4−λ2δ 2(λ+2) ,wE A2=c, ΠE A1=ΠE A2=ΠE B2=0, ΠE R1=   (a−c)2(1−b)(2(1−b)(b+1)−δλ(1−b+λ))(2(1+b)+λδ) (4(1−b2)−δλ2)2i f b ≤4−λ2δ 2(λ+2) (a−c)2(2+δλ)(2−δλ−δλ2) (4−δλ2)2i f b >4−λ2δ 2(λ+2) , ΠE R2=     (a−c)2(λ4δ2−8λ2δ(1−b)(b+1)+4(1−b)(b+1)(λ(λ+4(1−b))+8(1−b))) 4(4(1−b2)−δλ2)2i f b ≤4−λ2δ 2(λ+2) (a−c)2(2+λ)2 (4−δλ2)2i f b >4−λ2δ 2(λ+2) . All expressions for Proposition 19: q∗ A1=   (a−c)(2(1+b)+δλ) 4(1+b)2−δλ2NE in both periods i f b ∈−1, b (a−c)(2+δλ) 4−δλ2E in both periods i f b ∈b, 1, q∗ B1=   (a−c)(2(1+b)+δλ) 4(1+b)2−δλ2i f b ∈−1, b −i f b ∈b, 1, q∗ A2=   a−c 2(1+b)+λ(a−c)(2(1+b)+δλ) 2(1+b)(4(1+b)2−δλ2)i f b ∈−1, b (a−c)(2+λ) 4−δλ2i f b ∈b, 1, q∗ B2=   a−c 2(1+b)+λ(a−c)(2(1+b)+δλ) 2(1+b)(4(1+b)2−δλ2)i f b ∈−1, b −i f b ∈b, 1, p∗ A1=   2(b+1)2(a+c)−aλ2δ−λδ(a−c)(b+1) 4(1+b)2−δλ2i f b ∈−1, b c(λδ+2)+a(2−δλ(λ+1)) 4−δλ2i f b ∈b, 1, p∗ B1=   2(b+1)2(a+c)−aλ2δ−λδ(a−c)(b+1) 4(1+b)2−δλ2i f b ∈−1, b −i f b ∈b, 1, Games 2024,15, 9 20 of 23 p∗ A2=   2(b+1)2(a+c)−aλ2δ−λ(a−c)(b+1) 4(1+b)2−δλ2i f b ∈−1, b c(λ+2)+a(2−λ(λδ+1)) 4−δλ2i f b ∈b, 1, p∗ B2=   2(b+1)2(a+c)−aλ2δ−λ(a−c)(b+1) 4(1+b)2−δλ2i f b ∈−1, b −i f b ∈b, 1, w∗ A1=c,w∗ B1=   c i f b ∈−1, b <c i f b ∈b, 1, w∗ A2=   4c(1+b)2−λ2δ−2λ(a−c)(1+b) 4(1+b)2−δλ2i f b ∈−1, b c(2λ+4)−aλ(λδ+2) 4−δλ2i f b ∈b, 1, w∗ B2=   4c(1+b)2−λ2δ−2λ(a−c)(1+b) 4(1+b)2−δλ2i f b ∈−1, b −i f b ∈b, 1, Π∗ A1=Π∗ B1=Π∗ A2=Π∗ B2=0, Π∗ R1=   2(a−c)2(2(1+b)2−λ2δ−λδ(1+b))(2b+λδ+2) (4(1+b)2−δλ2)2i f b ∈−1, b (a−c)2(2+δλ)(2−δλ−δλ2) (4−δλ2)2i f b ∈b, 1, Π∗ R2=   2(a−c)2(2b+λ+2)2(1+b) (4(1+b)2−δλ2)2i f b ∈−1, b (a−c)2(2+λ)2 (4−δλ2)2i f b ∈b, 1. The relevant expressions for non-exclusivity under the social optimum: qNE−S i1=(a−c)(b+λδ +1) (1+b)2−λ2δ,qNE−S i2=(a−c)(b+λδ +1) (1+b)2−λ2δ, pNE−S i1=c(b+1)(b+λδ +1)−aλδ(b+λ+1) (1+b)2−λ2δ <c, pNE−S i2=c−λ(a−c)(b+λδ +1) (1+b)2−λ2δ=ci2, ΠNE−total−S 1=−2(a−c)2(b+λδ +1)λδ(b+λ+1) (1+b)2−λ2δ2<0, ΠNE−total−S 2=0, CSNE−S 1=(a−c)2(b+1)(b+λδ +1)2 (1+b)2−λ2δ2≥0, CSNE−S 2=(a−c)2(b+1)(b+λ+1)2 (1+b)2−λ2δ2≥0, WNE−S 1=(a−c)2(b+λδ +1)2b−λδ −2λ2δ+b2−bλδ +1 (1+b)2−λ2δ2, WNE−S 2=(a−c)2(b+1)(b+λ+1)2 (1+b)2−λ2δ2≥0. The relevant expressions for exclusivity under the social optimum: Games 2024,15, 9 21 of 23 qE−S A1=   (a−c)(1−b)(b+λδ+1) (1−b2)−δλ2i f b ≤1−δλ2 1+λboth in t = 2 (a−c)(1+λδ) 1−δλ2i f b >1−δλ2 1+λonly A in t = 2 , qE−S A2=   (a−c)(1−b+λ) (1−b2)−δλ2i f b ≤1−δλ2 1+λ (a−c)(1+λ) 1−δλ2i f b >1−δλ2 1+λ , qE−S B2=   (a−c)(1−b−bλ−λ2δ) (1−b2)−δλ2i f b ≤1−δλ2 1+λ 0i f b >1−δλ2 1+λ , pE−S A1=(c−b2c−aλδ+cλδ−aλ2δ+abλδ−bcλδ (1−b2)−δλ2<c i f b ≤1−δλ2 1+λ c−aλδ+cλδ−aλ2δ 1−δλ2<c i f b >1−δλ2 1+λ , pE−S A2=(c−aλ+cλ−b2c+ab2λ−b2cλ−aλ2δ+abλ2δ−bcλ2δ (1−b2)−δλ2=cA2i f b ≤1−δλ2 1+λ c−aλ+cλ−aλ2δ 1−δλ2=cA2i f b >1−δλ2 1+λ , pE−S B2=(c i f b ≤1−δλ2 1+λ −i f b >1−δλ2 1+λ , ΠE−total−S 1=     (a−c)2(b−1)λδ(1−b+λ)(b+λδ+1) ((1−b2)−δλ2)2<0i f b ≤1−δλ2 1+λ −(a−c)2(λδ+1)λδ(λ+1) (1−δλ2)2<0i f b >1−δλ2 1+λ , ΠE−total−S 2=(0i f b ≤1−δλ2 1+λ 0i f b >1−δλ2 1+λ , CSE−S 1=     (a−c)2(b−1)2(b+λδ+1)2 2((1−b2)−δλ2)2i f b ≤1−δλ2 1+λ (a−c)2(λδ+1)2 2(1−δλ2)2i f b >1−δλ2 1+λ , CSE−S 2=     (a−c)2(λ4δ2+(1−b2)(2λ−2b−2bλ+λ2−2λ2δ+2)) 2((1−b2)−δλ2)2i f b ≤1−δλ2 1+λ (a−c)2(λ+1)2 2(1−δλ2)2i f b >1−δλ2 1+λ , WE−S 1=   (a−c)2((1−b)(b+2δ+2λδ+1)−λ2δ2) 2((1−b2)−δλ2)i f b ≤1−λ2δ 1+λ (a−c)2(δ+2λδ+1) 2(1−λ2δ)≥0i f b >1−λ2δ 1+λ , WE−S 2=     (a−c)2(λ4δ2+(1−b2)((2λ−2b−2bλ+λ2+2λ2δ+2)−4λ2δ)) 2((1−b2)−δλ2)2i f b ≤1−δλ2 1+λ (a−c)2(λ+1)2 2(1−δλ2)2i f b >1−δλ2 1+λ . Notes 1 Exclusivity contracts or refusal to deal are often viewed with suspicion by the Antitrust Authorities. Note that exclusion may arise in a market through vertical integration too. 2 Mathewson and Winter (1987) [ 5 ] argue that manufacturers under exclusive dealing arrangements compete on the basis of wholesale prices for the right to be selected by the retailer. Besanko and Perry (1993) [ 6 ], in a differentiated products oligopoly, support that exclusive dealing can eliminate inter-brand externalities due to increased promotional investments. O’Brien and Shaffer (1993) [7] examine corner solutions in an oligopolistic vertical setting. 3 There is also work on vertical contracting with inventories and dynamic considerations or renegotiating contracts, e.g., Jong-Say Jong (1999) [19] or Taylor and Plambeck (2006) [20]. 4 If the bargaining power was more evenly distributed across the upstream and downstream firms, additional tensions would arise, partly reflecting results from earlier research on downstream ‘bottleneck’ models. A short discussion on such further research is delegated to the Conclusion. Games 2024,15, 9 22 of 23 5 The representative consumer is characterized by the quadratic and strictly concave utility function U(qA , qB) = a(qA+qB)− q2 A+q2 B+2bqAqB 2(see Singh and Vives (1984)) [28]. 6 An alternative way to foreclosure an input supplier is the retailer to explicitly sign an exclusive dealing contract with the rival manufacturer. 7 We assume that if the manufacturer is indifferent between obtaining zero profits from production and not producing at all, they will choose to produce. 8A more detailed discussion about the consumers’ surplus and the total welfare is deferred for Section 4. 9 The analysis here contains the subcase (i) non-exclusivity in the first period, i.e., no subsequent cost asymmetry, and nonexclusivity in the second period, and (ii) exclusivity in the first period, i.e., subsequent cost asymmetry, and non-exclusivity in the second period. Nevertheless, a corner solution may arise in the latter case. 10 The analysis here contains the subcase (i) non-exclusivity in the first period and exclusivity in the second period, and (ii) exclusivity in the first period and exclusivity in the second period. Production cost cA2is a function of qA1. 11 Since a(2a−cA2−cB2) 4(1+b)+(a−cA2)(bcB2−cA2)+(a−cB2)(bcA2−cB2) 4(1−b2)≥(a−cA2)2 4. 12 The subcase non-exclusivity in period one, i.e., subsequent cost symmetry, and exclusivity in period two, cannot be an equilibrium outcome. 13 We obtain positive quantities and the second-order conditions are satisfied when 4 ( 1 +b)2−λ2δ> 0, 4 (1−b)2−λ2δ> 0, 41−b2−λ2δ>0. 14 All relevant expressions obtained under non-exclusivity are in the Appendix A. This subcase corresponds to non-exclusivity in both periods. 15 This subcase corresponds to exclusivity in period one and non-exclusivity in period two. 16 This subcase corresponds to exclusivity in both periods. 17 We need 4 1−b2−δλ2> 0 for the second-order conditions to be satisfied and to obtain positive quantity in period one. Otherwise, no production occurs. All remaining expressions for this Lemma are in the Appendix A. 18 (N)E stands for (non-)exclusivity. We have already proved that NE in period one and E in period two, cannot be an equilibrium outcome. 19 We have 1 >b=√λ4δ4−2λ2δ3(λ2+2)+δ2(λ(24λ+λ3+32)+16)+δ(4(λ(8−λ)+8))+16−λδ(λ(1+δ)+4) 4(δ+λδ+1)≥4−λ2δ 2(λ+2)>0 20 For b≤4−δλ2 2(λ+2), we obtain ∆Π =ΠE R1+δΠE R2−ΠNE R1+δΠNE R2<0. 21 For b∈(4−λ2δ 2(λ+2) , b) we obtain ∆Π < 0 while for b∈(b , 1 ) we obtain ∆Π > 0. Therefore, it is never an equilibrium outcome to purchase product A in the first period and both products in the second period. 22 See Singh and Vives, 1984 [28] 23 We prove that d(CSE t−CSNE t) db =qNE it −qNE it −2(1+b)dqNE it db  > 0, with dqNE it db < 0, t= 1, 2, that is, ∆CS is increasing in b . Furthermore, we obtain that ∆CS(b=0)<0 and ∆CS(b=1)>0. 24 To obtain positive quantities and to satisfy the second-order conditions, we should have (1+b)2−δλ2≥ 0, (1−b)2−δλ2> 0 and 1−b2−δλ2>0.All relevant expressions obtained in this case are in the Appendix A. 25 We need 1−b2−δλ2> 0 for the second-order conditions to be satisfied and to obtain positive quantity in period one. All relevant expressions obtained in this case are in the Appendix A. 26 For the second period, we have qC i2=a−c 2+b+λ(a−c)((2−b)(2+b)2+4δλ) (2+b)((2−b)(2+b)3−4δλ2). 27 With the presence of the retailer, we also obtain q∗ B1=   (a−c)(2(1+b)+δλ) 4(1+b)2−δλ2i f b ∈−1, b −i f b ∈b, 1. References 1. 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