Network externalities and downstream collusion under asymmetric costs: A note

Lee, Jen Yao; Fan, Chen-Chia; Tsai, Chien-Shu Article Network externalities and downstream collusion under asymmetric costs: A note Games Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Lee, Jen Yao; Fan, Chen-Chia; Tsai, Chien-Shu (2023) : Network externalities and downstream collusion under asymmetric costs: A note, Games, ISSN 2073-4336, MDPI, Basel, Vol. 14, Iss. 2, pp. 1-11, https://doi.org/10.3390/g14020029 This Version is available at: https://hdl.handle.net/10419/330022 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. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Citation: Lee, J.-Y.; Fan, C.-C.; Tsai, C.-S. Network Externalities and Downstream Collusion under Asymmetric Costs: A Note. Games 2023,14, 29. https://doi.org/ 10.3390/g14020029 Academic Editors: Randall Calvert, Kjell Hausken and Ulrich Berger Received: 31 December 2022 Revised: 14 February 2023 Accepted: 15 March 2023 Published: 30 March 2023 Copyright: © 2023 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 Network Externalities and Downstream Collusion under Asymmetric Costs: A Note Jen-Yao Lee 1,2 , Chen-Chia Fan 1and Chien-Shu Tsai 3,4,* 1Department of International Business, National Kaohsiung University of Science and Technology, Kaohsiung City 807618, Taiwan; [email protected] (J.-Y.L.); [email protected] (C.-C.F.) 2Center for Global Operations Research & Development, National Kaohsiung University of Science and Technology, Kaohsiung City 807618, Taiwan 3Institute of Marine Affairs and Business Management, National Kaohsiung University of Science and Technology, Kaohsiung City 811213, Taiwan 4Center for Marine Affairs Studies, National Kaohsiung University of Science and Technology, Kaohsiung City 811213, Taiwan *Correspondence: [email protected] Abstract: This paper considers the collusive stability of downstream competition in a vertical market with network externalities and cost asymmetry. A dynamic collusion game is constructed, and backward induction is employed to solve the subgame perfect Nash equilibrium. We show that larger network externalities lead to less collusive incentive for an inefficient firm, while for an efficient firm, this depends on the efficiency gap. An increase in network externalities will destabilize the downstream collusion when the cost asymmetry is large and network externalities are relatively weak. Keywords: collusion; cost asymmetry; network externalities JEL Classification: D43; L13; M21 1. Introduction The stability of collusion over time has received significant attention in the literature. It is also widely discussed under different market structures. Friedman [ 1 ] first used supergame to discuss the stability of collusion, which is dependent on the discount factor of the player. Deneckere [ 2 ] pointed out the impact of product differentiation on collusion. Cost structures also play an important role when it comes to collusion (Collie [ 3 ]). In recent years, the effects of network externalities have received extensive attention; Song and Wang [ 4 ] first took network externalities into account in the framework of collusion stability with symmetric cost. In today’s industrial structure, vertical supply chains generally exist in the consumer electronics market, which is also characterized by network externalities. Collusion and competition among manufacturers of consumer electronics products is an important research topic. To fill this literature gap, in this paper, we investigate collusion stability in a vertical structure with downstream cost asymmetries in the presence of network externalities, extending the work of Song and Wang [ 4 ] and Toshimitsu [ 5 ]. They took product substitution into consideration and found that collusion becomes more sustainable for closer substitutes of products under relatively strong network externalities. Toshimitsu [ 5 ] demonstrated the conditions under which collusive behavior improves social welfare. In mostly related work, Choi and Lee [ 6 ] showed that if the network externality is strong (weak), the collusion of price (quantity) is more stable than quantity (price), which is different from the findings of Collie [ 3 ]. In this paper, we aim to take network externalities and cost asymmetry into account to analyze downstream collusive stability in a vertical market. We show that larger Games 2023,14, 29. https://doi.org/10.3390/g14020029 https://www.mdpi.com/journal/games Games 2023,14, 29 2 of 11 network externalities lead to less collusive incentive for an inefficient firm, while for an efficient firm, this depends on the efficiency gap. This is because the input price (cost) can be changed as efficiency changes compared to a one-tier market structure as in Pal and Scrimitore’s work [7]. The reminder of the paper is as follows. A literature review is provided in Section 2. The basic model with a linear demand curve and network externalities is presented in Section 3and solved in Section 4, with an extensive analysis of the results. Section 5presents a discussion on robustness. Section 6concludes the paper. 2. Literature Review Deneckere [ 2 ] first showed that when goods are very close substitutes, more tacit collusion is supported under Bertrand duopoly than Cournot duopoly. Rothschild [ 8 ] discussed the possibility that firms optimally choose whether to be price or quantity setters in each period is considered, pointing out that asymmetric cartels do not emerge from any of these contributions at the subgame perfect equilibria. Ross [ 9 ] used a supergame theoretic model of collusion to analyze the effects of different levels of product differentiation on cartel stability, and found that greater homogeneity can reduce cartel stability. Lambertini and Sasaki [ 10 ] derived the optimal punishments required to sustain collusion under Bertrand and Cournot duopoly with differentiated products. Collie [ 3 ] showed that collusion is more sustainable under Cournot duopoly than under Bertrand duopoly with quadratic costs for any degree of product substitutability. In another literature trend, the issue of vertical collusion is explored, e.g., Nocke and White [ 11 ], Barbot [ 12 ], Normann [ 13 ], Bian et al. [ 14 ], Biancini and Ettinger [ 15 ], Dingwei et al. [ 16 ], Gilo and Yehezkel [ 17 ], and Wang and Wang [ 18 ]. In particular, Wang and Wang [ 18 ] investigated the collusive incentive for far-sighted manufacturers selling via managerial retailers. They showed that revenue delegation can impede upstream collusion in Bertrand competition. Furthermore, the hindering result of managerial delegation is robust if it allows manufacturers to consider partial collusion. Ying et al. [ 19 ] found that consumer-oriented CSR’s effect on the stability of upstream collusion basically hinges on the downstream competition modes. In particular, for a given degree of CSR and product substitutability, upstream collusion is always less stable under downstream price competition. However, the previous studies did not take the issue of network externalities into consideration. The effect of network externalities has a non-negligible impact on industrial economic decision making, e.g., see Katz and Shapiro [ 20 – 22 ], Chou and Shy [ 23 ], Economides [ 24 ], Hoernig [ 25 ], Bhattacharjee and Pal [ 26 , 27 ], Pal and Scrimitore [ 7 ], Song and Wang [ 4 ], and Nakamura [ 28 ]. To see how network externalities impact collusive stability, Ruhmer [ 29 ] analyzed price collusion between platforms in a two-sided market model, and pointed out that collusion becomes harder to sustain as indirect network externalities become stronger. Unlike the two-sided market model, Pal and Scrimitore [ 7 ] highlighted that collusion sustainability under homogenous Cournot game depends on the strength of network externalities in an infinitely repeated game with trigger strategy punishment. Song and Wang [ 4 ], in a differentiated oligopoly game, showed that collusion becomes more sustainable for closer substitutes of products under relatively strong network externalities. However, the mentioned papers did not consider the collusive stability of downstream competition in a vertical market with network externalities and cost asymmetry. 3. Basic Model Consider a market where there is one upstream firm selling an input to two downstream firms for a wholesale price w. Assume that downstream duopoly produces homogenous final products with positive consumption network externalities. For simplicity’s sake, the upstream firm’s cost is assumed to be normalized to zero and there are no other costs except the input price for downstream firms. Wang and Wang [ 18 ], in a vertical structure with many manufacturing firms, investigated the collusive incentive for far-sighted manu- Games 2023,14, 29 3 of 11 facturers selling via managerial retailers. In contrast to the existing literature, they found that revenue delegation can impede upstream collusion in Bertrand models. Firm 1 produces one unit of products with λ unit of inputs (0.5 <λ< 1), while firm 2 produces one unit of products with one unit of inputs. The more the λ , the less the cost difference. This way of modeling a firm’s cost asymmetry allows us to capture differences in firm capacity and its impact on production efficiency, i.e., firm 1 is more efficient than firm 2. The inverse demand function for product can be expressed as follows (see also Choi and Lee [30]): p=a−q1−q2+n(y1+y2)(1) where p denotes the final price charged for products, qi(i=1, 2) denotes the quantities, and yi denotes consumers’ expectations regarding firm i ’s total sales, a is the market scale, and n∈(0, 1) measures the network effects. To solve the equilibrium, we impose the “rational expectations” conditions as those set by Katz and Shapiro [ 20 ], i.e., y1=q1 , y2=q2. The profits of firms can be given by πU=λwq1+wq2(2) π1=(p−λw)q1(3) π2=(p−w)q2(4) where πU is the profit for the upstream firm, and πi(i=1, 2) is the profit for downstream firm i. Consider that the firms engage in an infinitely repeated game. We examine the effect of cost asymmetry on the stability of the collusion in a vertical structure with downstream network externalities. Along the punishment path, assume that firms use Friedman’s [ 1 ] grim trigger strategy. A two-stage dynamic production game is used to explore the equilibrium. In every following period, the upstream firm decides the input price in the first stage, and each downstream firm simultaneously chooses the outputs in the second stage. We solve the subgame perfect Nash equilibrium (SPNE) through backward induction. The game structure of the model is as follows (see Figure 1): Stage 1: upstream firm decides the input price. Stage 2: downstream firms decide to collude or deviate. Stage 3: downstream firms simultaneously choose the outputs. Games 2022, 13, x FOR PEER REVIEW 4 of 11 Time Stage 2: Downstream firms decide to collude or deviate. Stage 1: Upstream firm decides the input price. Stage 3: Downstream firms simultaneously choose the outputs. Figure 1. Flowchart of the game. 4. Market Equilibrium and Analyses 4.1. Non-Collusion Firstly, we consider that each downstream firm chooses its output to maximize its own profit (Equations (3) and (4)) independently. According to the profit maximum problems, the first-order conditions are 𝜕𝜋1 𝜕𝑞1=𝑎−𝑤𝜆−2𝑞1−𝑞2+𝑛(𝑦1+𝑦2)=0 (5) 𝜕𝜋2 𝜕𝑞2=𝑎−𝑤−𝑞1−2𝑞2+𝑛(𝑦1+𝑦2)=0 (6) Letting 𝑞1=𝑦1 and 𝑞2=𝑦2, from Equation (5) to Equation (6), we obtain the response functions of the output as follows: 𝑞1=𝑎+𝑤−𝑛𝑤+(−2+𝑛)𝑤𝜆 3−2𝑛 (7) 𝑞2=𝑎+𝑤−𝑛𝑤+(−2+𝑛)𝑤𝜆 3−2𝑛 (8) In the second stage, the response functions of the firms are considered. Substituting the response functions of Equations (7) and (8) into Equation (2), and deciding the input price of the upstream firm, we obtain 𝜕𝜋𝑢 𝜕𝑤 =𝑎(1+𝜆)+2𝑤(𝑛(−1+𝜆)2−2(1+(−1+𝜆)𝜆)) 3−2𝑛 =0 (9) Thus, the optimal input price can be derivable as 𝑤𝑁=(1+𝜆)𝑎 2𝐻 (10) where 𝐻≡2−𝑛(1−𝜆)2−2(1−𝜆)𝜆 and the superscript “N” denotes the non-collusion (competition) under vertical separation with homogenous goods. 𝑞1 𝑁=1 2𝑎(1−𝜆 𝐻−1 2𝑛−3) 𝑞2 𝑁=𝑎(2−𝑛−5𝜆+4𝑛𝜆+(5−3𝑛)𝜆2) 2(3−2𝑛)𝐻 𝜋1 𝑁=𝑎2(−5+𝑛(3−𝜆)(1−𝜆)+(5−2𝜆)𝜆)2 4(3−2𝑛)2𝐻2 𝜋2 𝑁=𝑎2(𝑛−2+5𝜆−4𝑛𝜆−(5−3𝑛)𝜆2)2 4(3−2𝑛)2𝐻2 Figure 1. Flowchart of the game. Games 2023,14, 29 4 of 11 4. Market Equilibrium and Analyses 4.1. Non-Collusion Firstly, we consider that each downstream firm chooses its output to maximize its own profit (Equations (3) and (4)) independently. According to the profit maximum problems, the first-order conditions are ∂π1 ∂q1 =a−wλ−2q1−q2+n(y1+y2)=0 (5) ∂π2 ∂q2 =a−w−q1−2q2+n(y1+y2)=0 (6) Letting q1=y1 and q2=y2 , from Equation (5) to Equation (6), we obtain the response functions of the output as follows: q1=a+w−nw +(−2+n)wλ 3−2n(7) q2=a+w−nw +(−2+n)wλ 3−2n(8) In the second stage, the response functions of the firms are considered. Substituting the response functions of Equations (7) and (8) into Equation (2), and deciding the input price of the upstream firm, we obtain ∂πu ∂w=a(1+λ)+2w(n(−1+λ)2−2(1+(−1+λ)λ)) 3−2n=0 (9) Thus, the optimal input price can be derivable as wN=(1+λ)a 2H(10) where H≡ 2 −n(1−λ)2− 2 (1−λ)λ and the superscript “N” denotes the non-collusion (competition) under vertical separation with homogenous goods. qN 1=1 2a1−λ H−1 2n−3 qN 2=a2−n−5λ+4nλ+(5−3n)λ2 2(3−2n)H πN 1=a2(−5+n(3−λ)(1−λ)+(5−2λ)λ)2 4(3−2n)2H2 πN 2=a2n−2+5λ−4nλ−(5−3n)λ22 4(3−2n)2H2 πN U=a2(1+λ)2 4(3−2n)H SWN=a2(7−4n(1−λ)2−(10 −7λ)λ)(17 +4n2(1−λ)2−λ(14 −17λ)−n(17 −(22 −17λ)λ)) 8(3−2n)2H2 4.2. Collusion Secondly, we discuss joint-off maximization. Of note, collusion exists only when the members produce goods with the same costs. In our paper, the strategy is providing the Games 2023,14, 29 5 of 11 advanced production tech to the inefficient firm 2 to reach the collusion. The profits of firm 2 and upstream firm will then become π0 2=(p−λw)q2and π0 U=λw(q1+q2). We assume a permanent tech transfer occurs, and then after a deviation, the game turns into a Cournot without asymmetry. Simplifying the bargaining procedure from Verboven [ 31 ], we assume the share of the joint profit from firm 1 is α , which comes from the consideration of the ad valorem side payment due to exogenous bargaining power. Hence, it is reasonable to assume 1 2<α< 1 for the tech efficiency of firm 1. The joint payoff is π1+π0 2 and it is maximized when qC 1=qC 2=a 8−4nand wC=a 2λ. The resulting payoffs of firms are πC 1=αa2 4(2−n)2, πC 2=(1−α)a2 4(2−n)2, πC U=a2 8−4n, SWC=a2(7−3n) 8(2−n)2 where the superscript “C” denotes collusion. We obtain the collusion of the two firms if the profit with collusion for each firm is superior to the Cournot–Nash equilibrium profit. That is, πC 2>πN 2 , if α<ˆ α for inefficient firm 2 and πC 1>πN 1 , if α>ˇ α for efficient firm 1, where ˆ α= 1 −(2−n)2(2−n−5λ+4nλ+(5−3n)λ2)2 (3−2n)2H2 and ˇ α=(2−n)2(5−n(3−λ)(1−λ)+(2λ−5)λ)2 (3−2n)2H2 . By checking the sensibility of the incentive to the collusion with regard to the degree of network externalities, we obtain Proposition 1. Proposition 1. With the larger degree of network externalities, the collusion incentive for the inefficient firm is always smaller, while for the efficient firm, the incentive is smaller if the efficiency gap is large or small enough. If the efficiency gap is moderate, then there is an inverse U-shaped relationship between collusive incentive and network externalities. Proof. We have ∂ˆ α ∂n< 0. If 0 <λ<1 21−r3√17 −4 or 1 21+r3√17 −4< λ< 1, then ∂ˇ α ∂n> 0. If 1 21−r3√17 −4<λ<1 21+r3√17 −4 , and n∗≡2+√2q(1−λ)λ(1+λ)4−2λ(9−λ(10−(3−λ)λ)) (1−λ)(1−λ(11(3−λ)λ)) , then (1) ∂ˇ α ∂n< 0, as 0 <n<n∗ . (2) ∂ˇ α ∂n> 0, as 1>n>n∗. This finding is consistent with Pal and Scrimitore [ 7 ], who found that in a network goods oligopoly, there is no incentive to collude unless the network externalities are sufficiently weak. It is well known that the stronger the degree of network externalities, the greater the gain from Cournot competition than that from collusion, as for any given degree of network externalities, there are higher outputs in Cournot competition, and the outward shift of the demand curve is greater than that under collusion. Nevertheless, there is a crucial difference in the case where the (relative) efficiency is moderate as there is an inverse U-shaped relationship between collusive incentive of the efficient firm and the degree of network externalities. In this case, with moderate efficiency difference and smaller network externalities, there exists a possibility that the collusive incentive for the efficient firm is positively correlated with network externalities; namely, moderate λ indicates a higher input price (When λ<e λ≡(n−2)+√2√6−7n+2n2 2−n , ∂wN ∂λ > 0; Games 2023,14, 29 6 of 11 when λ>e λ , ∂wN ∂λ < 0. Hence, when the cost asymmetry is moderate, the input price is higher.) and lower marginal profit due to network externalities, compared to the other two extreme cases. Via partial differentiation to efficiency difference, we obtain the following Proposition 2. Proposition 2. If the efficiency difference between firms is smaller, the collusive incentive of the efficient firm is larger, and the collusive incentive of the inefficient firm is larger until the critical value is reached, and then the relationship will be reversed. Furthermore, the collusion likelihood is smaller. Proof. We have ∂ˇ α ∂λ < 0 and ∂(ˆ α−ˇ α) ∂λ > 0, provided that ˆ α>ˇ α . When 0 <λ<√4−2n+n−2 n , ∂ˆ α ∂λ >0, and when λ>√4−2n+n−2 n,∂ˆ α ∂λ <0.  With smaller efficiency asymmetry, apart from underutilized network externalities due to undercut production, there is a lower implicit opportunity cost of free riding (lower market share) for firm 1 under collusion. That is, beating the rival is more important, in spite of the side payment under collusion. The overall effect of competition on the efficient firm will be positive. For firm 2, the cost-saving effect dominates the comprehensive effect of underutilized network externalities and side payment when the cost asymmetry is large. Hence, in this case, cost asymmetry has a positive effect on the collusion, while the relationship is reversed when the asymmetry is small enough. Taking both firms into account, we find that when the efficiency gap is large, the weakening effect on firm 1 is greater than the temptation for firm 2 to deviate. However, when the efficiency gap is small enough, the likelihood of collusion increases. Our finding is consistent with Ganslandt et al. [ 32 ], and Miklós-Thal’s [ 33 ] who states that collusion is sustainable under cost symmetry, while collusion may be sustainable under cost asymmetry; however, the difficulty for efficient collusion to sustain holds when costs are asymmetric. Miklós-Thal [ 33 ] built a model of price competition in a one-tier market structure. 4.3. Deviation and Tacit Collusion Thirdly, we discuss deviations from tacit collusion. Given the collusive output of the rival, firm 2 (or 1) maximized its profit. If the deviant firm is firm 1, it is reasonable to assume that the deviated firm still holds the efficient technology for the deviant firm and would not let the competitor detect the deviation. If the deviant firm is firm 2, it makes sense that it still has the tech since deviated firm 1 is unaware of the betrayal, so the deviation case is actually the same for the two firms. The upstream firm still makes the same price in the collusion case for the unknown betrayal. The outputs are obtained as follows: qD i=2aλ−a 8λ−4nλ,qD j=a(1−n+2λ) 4(n−2)2λ(11) where the superscript “D” denotes the deviation under vertical separation and the subscript idenotes the deviated firm and jdenotes the deviant firm. The profits of firm iand jand social welfare are as listed below: πD i=a2(1−n+2λ)(1−2λ) 16(n−2)3λ2,πD j=a2(1−n+2λ)2 16(n−2)4λ2,πD U=a2(2(3−n)λ−1) 8(n−2)2λ, (12) SWD=a2(2(3−n)λ−1)(1+26λ−n(1+6(4−n)λ)) 32(2−n)4λ2(13) Comparing the social welfare under the three regimes, we see that in the static analysis, SWN>SWD>SWC , which is in line with the work of Ciarreta and Gutiérrez-Hita [ 34 ]. The latter inequality can be explained for the same shared efficiency tech (the total marginal Games 2023,14, 29 7 of 11 cost of the industry), but firm 1 has greater production. The former inequality can be explained for the production extension effect dominating the social efficiency effect. By letting δ denote the discount factor between periods, based on the above analysis, the tacit collusion is sustainable if and only if these two conditions are correct for the case of firm 1 or 2 deviation, respectively: πC 1 1−δ∗ 1≥πD 1+δ∗ 1πN 1 1−δ∗ 1 ,πC 2 1−δ∗ 2≥πD 2+δ∗ 2πN 2 1−δ∗ 2 (14) Let δ∗ i be the δ satisfying Equation (14) with equality. Given the cost function proposed by Friedman [ 1 ], this critical discount factor for the cartel is the lowest one that maintains collusion between the two firms. This means that we should have δ∗=max(πD 1−πC 1 πD 1−πN 1 ,πD 2−πC 2 πD 2−πN 2)=max{δ∗ 1,δ∗ 2}=δ∗ 2 As in Rothschild [ 9 ], we can find that firm 2 has the stronger incentive to deviate from the collusion, which is in contrast with Brandão et al. [ 35 ]. The explanation is intuitive: with the more efficient production skill, firm 2 gains the most from deviation and loses the least from the punishment, while firm 1 holds on to its shared technology on the other hand. Considering the infinitely repeated game, again, we make the comparison of social welfare in three cases. With simple calculation, we have SWC 1−δ∗≥SWD+δ∗SWN 1−δ∗ , as long as this collusion is sustainable. That is, collusion benefits social welfare, which is in contrast to the conventional literature. We relax the assumption that 1 2<α< 1, and derive the profit ratio when the critical discount factor is minimized, δ∗ 1=δ∗ 2=δ∗∗(α,a,λ,n), πD 1−πC 1 πD 1−πN 1 =πD 2−πC 2 πD 2−πN 2 (15) Solving Equation (15), we have δ∗∗ =δ(a,λ,n) (Due to the complexity of the function, only its implicit function is written here. If the calculation process is required, it can be obtained from the author.) Of note, whether the collusion strategy is via ad valorem or fixed fee (side payment), the discount factor is the same. Comparative static analyses for the influences of network externalities and profit ratio are shown as follows in Propositions 3 and 4. Proposition 3. Collusion sustainability is more stable as cost differences decrease. Proof. We have ∂δ∗∗ ∂λ >0.  This result is in line with the finding of Miklós-Thal [ 33 ]. There is larger deviation motivation for the sunk cost (firm 1) and a smaller punishment effect for larger Cournot profit (firm 1) and inefficiency (firm 2). Hence, cheating becomes less favorable as λincreases. Proposition 4. An increase in network externalities will destabilize the collusion when they are small and will stabilize it when they are large. Proof. As shown in Figure 2, if n>n∗∗,∂δ∗∗ ∂n>0, and if n<n∗∗,∂δ∗∗ ∂n<0.  Games 2023,14, 29 8 of 11 Games 2022, 13, x FOR PEER REVIEW 8 of 11 This result is in line with the finding of Miklós-Thal [33]. There is larger deviation motivation for the sunk cost (firm 1) and a smaller punishment effect for larger Cournot profit (firm 1) and inefficiency (firm 2). Hence, cheating becomes less favorable as 𝜆 increases. Proposition 4. An increase in network externalities will destabilize the collusion when they are small and will stabilize it when they are large. Proof. As shown in Figure 2, if 𝑛>𝑛∗∗, 𝜕𝛿∗∗ 𝜕𝑛 >0, and if 𝑛<𝑛∗∗, 𝜕𝛿∗∗ 𝜕𝑛 <0. □ In Figure 2, if 𝑛<𝑛∗∗ (the superscript “**” denotes the critical value of network externalities), we have 𝜕𝛿∗∗ 𝜕𝑛 <0, and so, when the network externalities are relatively weak, the critical discount factor decreases in the degree of network externalities. When the cost asymmetry is large and network externalities are relatively weak (strong), 𝑛<(>)𝑛∗∗, collusion becomes less (more) difficult to sustain. There are two opposite effects. With larger network externalities, the gains from output expanding will be unilaterally larger and the deviating motivation is stronger. Additionally, the losses from the punishment will be smaller for a smaller input price (With simple calculation, we find 𝜕𝑤 𝜕𝑛 >0.). For firm 2 especially, there is an amplified output reduction effect for the loss of efficient tech. As a result, the deviation effect is dominated by (dominates) the punishment effect for weak (strong) network externalities. 𝜆 𝜕𝛿∗∗ 𝜕𝑛 >0 𝑛∗∗ 𝜕𝛿∗∗ 𝜕𝑛 <0 𝑛 Figure 2. Cost asymmetry and network effects. (**: critical value of variable) 5. Robustness of Our Claims 5.1. Collusion without Technology Transfer If there is no technology transfer, cost asymmetry always exists in the market. Firm 2′s profit will remain at 𝜋2=(𝑝−𝑤)𝑞2. The joint payoff is 𝜋1+𝜋2, and the first-order conditions are the following: 𝜕(𝜋1+𝜋2) 𝜕𝑞1=𝑎−𝑤𝜆−2𝑞1−2𝑞2+𝑛(𝑦1+𝑦2)=0 Figure 2. Cost asymmetry and network effects. (**: critical value of variable). In Figure 2, if n<n∗∗ (the superscript “**” denotes the critical value of network externalities), we have ∂δ∗∗ ∂n< 0, and so, when the network externalities are relatively weak, the critical discount factor decreases in the degree of network externalities. When the cost asymmetry is large and network externalities are relatively weak (strong), n<(>)n∗∗ , collusion becomes less (more) difficult to sustain. There are two opposite effects. With larger network externalities, the gains from output expanding will be unilaterally larger and the deviating motivation is stronger. Additionally, the losses from the punishment will be smaller for a smaller input price (With simple calculation, we find ∂w ∂n> 0.). For firm 2 especially, there is an amplified output reduction effect for the loss of efficient tech. As a result, the deviation effect is dominated by (dominates) the punishment effect for weak (strong) network externalities. 5. Robustness of Our Claims 5.1. Collusion without Technology Transfer If there is no technology transfer, cost asymmetry always exists in the market. Firm 2 0 s profit will remain at π2=(p−w)q2 . The joint payoff is π1+π2 , and the first-order conditions are the following: ∂(π1+π2) ∂q1=a−wλ−2q1−2q2+n(y1+y2)=0 ∂(π1+π2) ∂q2=a−w−2q1−2q2+n(y1+y2)<0 It is maximized when qCN 1=2a 8−4n , qCN 2= 0 and wCN =a 2λ . The resulting payoffs of firms are πCN 1=αa2 4(2−n)2,πCN 2=(1−α)a2 4(2−n)2,πCN U=a2 (8−4n)λ. (16) where the superscript “CN” denotes collusion without technology transfer. As in Section 4, we examine deviations from tacit collusion, and obtain outputs as follows: qDN i=2aλ−a 8λ−4nλ,qDN j=a(3−n)(2λ−1) 4(n−2)2λ(17)