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Essays on competition between multiproduct firms

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Content Essays on Competition between Multiproduct Firms by Yinqi Zhang A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Economics) May 2021 Copyright 2021 Yinqi Zhang To my wife, my parents and my grandparents ii Acknowledgements I am deeply indebted to Guofu Tan for his generous support and continuous guidance. He has always been an incredible advisor, and I hope I can become an economist like him. I have greatly benefited from Jonathan Libgober. This dissertation would not have been possible without his crit- ical comments and advice. I am also grateful to Micheal Leung for providing insightful feedback on my papers. His course on network economics is one of the best classes I have taken at USC. My deep gratitude is also extended to John Carlsson. He has introduced me to the field of combi- natorial optimization and helps me to develop my research ideas. I would also like to thank Simon Wilkie and Yu-Wei Hsieh for serving on my qualifying exam committee and providing insightful comments on my third-year paper. Special thanks go to Jeffrey Nugent and Eric Heikkila, who have introduced me to rigorous economic research. I sincerely thank my colleagues and friends, Kanika Aggarwal, Andreas Aristidou, Zhen Chen, Jason Choi, Weiran Deng, Bada Han, Qin Jiang, Youngmin Ju, Jeehyun Ko, Eunjee Kwon, Yinan Liu, Yiwei Qian, Lidan Tan, Yimeng Xie, Yejia Xu, Jiaqi Liu, Xiaoying Pan, Bohan Wang, Yi Yu, Xiaoshu Zeng, Chenchen Zhao, and several others, for their support and companionship. I would also like to thank the current and former administrative staff of the Department of Economics at USC. Alexander Karnazes, Young Miller, Irma Alfaro, and Morgan Ponder have solved an infinite amount of administrative problems for me. Finally, I am grateful to my family, who helps me through this whole journey. I am blessed to have my wife. She has given me the courage and wisdom to overcome so many challenges in these years. I thank my father for his great passion for knowledge. He has shown me the characteristics of a true scholar. I thank my mother for her love and optimism, which have encouraged me to be confident in my life. I miss my grandparents, who always believe in me. iii Table of Contents Dedication ii Acknowledgements iii List of Tables vi List of Figures vii Abstract ix Chapter 1: The Competitive Effects of Joint Ventures in the International Airline Indus- try 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Competitive Effects of JVs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Brief Background and Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 Descriptive Statistics and Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.6 Differences-In-Differences Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.7 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Chapter 2: Competition with Complements and Substitutes 19 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.3.1 Network of Complements and Substitutes . . . . . . . . . . . . . . . . . . 26 2.3.2 Network Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.3.3 Prices and Consumer Demand . . . . . . . . . . . . . . . . . . . . . . . . 30 2.3.4 Pricing Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.4 Equilibrium Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.4.1 Equilibrium Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.4.2 Equilibrium Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.4.3 An Example with MNL Demand . . . . . . . . . . . . . . . . . . . . . . . 37 2.5 Merger Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.5.1 Symmetric Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.5.2 Asymmetric Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 iv 2.5.2.1 One-Sided Asymmetric Network . . . . . . . . . . . . . . . . . 41 2.5.2.2 Two-Sided Asymmetric Network . . . . . . . . . . . . . . . . . 44 2.6 Policy Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Chapter 3: An Analysis of International Airline Alliances and Joint Ventures 49 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.1.1 International Airline Network . . . . . . . . . . . . . . . . . . . . . . . . 50 3.1.2 Alliance Membership and Joint Venture . . . . . . . . . . . . . . . . . . . 51 3.1.3 Current Government Regulation . . . . . . . . . . . . . . . . . . . . . . . 51 3.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.2.1 The Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.2.2 Demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.2.3 Airline Alliance Agreements . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.2.4 Important Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.3 Equilibrium Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.3.1 Fare Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.3.2 Joint Venture Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.3.3 Alliance Membership Pricing . . . . . . . . . . . . . . . . . . . . . . . . 59 3.3.4 Price Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.3.4.1 Strategic Substitute Strategies . . . . . . . . . . . . . . . . . . . 62 3.3.4.2 Strategic Complement Strategies . . . . . . . . . . . . . . . . . 63 3.3.4.3 Network Structure and Market Size . . . . . . . . . . . . . . . . 64 3.4 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.4.1 Hub Spoke Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.4.2 Impacts of Airline Alliance . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 References 70 Appendices 73 A Chapter 2 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 A.1 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 A.2 Proof of Proposition 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 A.3 Proof of Proposition 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 A.4 Proof of Proposition 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 A.5 Proof of Proposition 4 and Proposition 5 . . . . . . . . . . . . . . . . . . . 97 A.6 Proof of Proposition 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 B Chapter 3 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 B.1 Proof of Lemma 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 B.2 Proof of Lemma 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 B.3 Proof of Proposition 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 B.4 Proof of Proposition 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 B.5 Proof of Proposition 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 v List of Tables 1.1 Summary Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Comparison between AA Online Flights and AA-BA Codeshare Flights (Itinerary Level) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Comparison between AA Online Flights and AA-BA Codeshare Flights (Market Level) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 Comparison between AA Online Flights and AA-BA Codeshare Flights (Market Level) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 vi List of Figures 1.1 A Stylized Airline Network for American Airlines and British Airways . . . . . . . 3 1.2 The Flights of AA and BA in the City-pair Market between Pittsburgh and London 6 1.3 Average Airfares of AA Online Flights and AA-BA Codeshare Flights in Markets Exposed to JV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 Average Number of Passengers of AA Online Flights and AA-BA Codeshare Flights in Markets Exposed to JV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.5 The Evolution of Airfares of AA’s Online Flights in the Treatment and Control Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.6 Dynamics of the Impact on Airfares of AA’s Online Flights . . . . . . . . . . . . . 17 2.1 An Illustration of the Airline Network Offered by Delta Airlines in the Changsha- Seattle Market. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 A Network with Two Separable Components: S 1 =f1;2;3g and S 2 =f4;5g. . . . 27 2.3 An Illustration of the Three Types of Networks. . . . . . . . . . . . . . . . . . . . 29 2.4 Firm f 1 is a Dominant Firm that Produces Essential Components for All Products. . 42 2.5 The Market Structure after the Merger between f 1 and f 2 in Figure 2.4 . . . . . . . 43 2.6 A Two-sided Asymmetric Network with Three Final Products. . . . . . . . . . . . 44 2.7 The Two-sided Asymmetric Network for the Merger Analysis in Scenario (2). . . . 45 2.8 The One-sided Asymmetric Network by Adding Firm f 1 to the Final Product 3 in the Network Structure in Figure 2.7. . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.1 An Airline Network with Six Cities. Straight Lines Represent Carrier A (Air China) and Dotted Lines Represent Carrier B (United Airlines) . . . . . . . . . . . 50 3.2 Optimal Prices when a= 2:1 and c= 4 . . . . . . . . . . . . . . . . . . . . . . . 65 3.3 Optimal Prices when a= 3 and c= 4 . . . . . . . . . . . . . . . . . . . . . . . . 65 3.4 Network Structure when m= 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 vii 3.5 Network Structure when m= 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 viii Abstract This dissertation focuses on understanding competition between multiproduct firms and its antitrust implications. Multiproduct firms are frequently observed in different industries and play a crucial role in the economy. However, an important feature of competition between multiproduct firms is largely overlooked in the literature: firms may provide both complements and substitutes. In this dissertation, I systematically study this feature and its implications from different perspectives. Chapter 1 develops an empirical study about the price-increasing effect (also defined as the Edgeworth-Salinger effect in Luco and Marshall (2020)) from a merger of complements (i.e., a merger between firms producing complementary products) in the international airline industry. In this joint project with Guofu Tan, we focus on a set of city-pair markets where American Airlines offers a route operated by itself and a route jointly operated with British Airways (i.e., a codeshare flight where each carrier offers an essential component of the flight) in each market. These two major airline carriers formed a Joint Venture (JV) in 2010 to fully integrate their services in the transatlantic markets. We use the international airline data from the U.S. Department of Transporta- tion (DOT)’s Airline Origin and Destination Survey, which contains a 10 percent random sample of itineraries involving a U.S. airport and a U.S. carrier. We conduct a differences-in-differences (DID) analysis, and our results indicate that the JV increased airfares of flights operated by Amer- ican Airlines itself by three to four percent. Previous studies, which often emphasized the airfare reductions of codeshare flights in these markets, may have overestimated the benefits to consumers from JVs. Inspired by the empirical results from Chapter 1, Chapter 2 develops a network model of com- plements and substitutes like the international airline network where final products are combina- tions of intermediate products offered by (possibly) different multiproduct firms. I use the general ix demand system developed in Nocke and Schutz (2018), which nests multinomial logit (MNL) and constant elasticity of substitution (CES) demands, to model consumers’ choices of final products. Also, I generalize the aggregative games approach to characterize the equilibrium prices and solve the technical challenges in analyzing the existence and uniqueness of a pricing equilibrium. Based on this framework, I identify conditions on networks under which a merger of complements can improve or reduce consumer surplus, respectively, even though such a merger may affect prices differently. Therefore, antitrust agencies can better protect consumers in merger reviews by evalu- ating the connections between firms’ intermediate products. Chapter 3 explores another important feature of competition between multiproduct firms in the international airline industry. Specifically, each component of a connecting flight also corresponds to an independent city-pair market. I develop a theoretical model to show that intensive competition in inter-hub markets can constrain prices in other markets, which use inter-hub flights as part of their trips. As a result, depending on firms’ pricing strategies for complementary flights, a JV may not always reduce the prices of codeshare flights through the elimination of double margins. This study explains why recent JVs cannot significantly reduce the prices of codeshare flights. x Chapter 1 The Competitive Effects of Joint Ventures in the International Airline Industry 1 1.1 Introduction The rapid development of joint ventures (JVs) is a key feature of the international airline industry. Many members of the three major airline alliances (Star, SkyTeam, and One World) have further expanded cooperation by forming JVs. In 2016, 81 percent of the travelers who flew between the U.S. and Europe used an airplane operated by a JV (Bilotkach (2019)). More recently, there is a new wave of JVs in the transpacific markets. Air Canada and Air China concluded a JV agreement in 2018, and the JV between American Airlines and Qantas got approval in 2019. In addition, the formation of JVs may no longer be limited to members of the same airline alliance. Some proposed new arrangements in 2019, like the JV between Westjet and Delta, and the JV between Japan Airlines and Hawaiian Airlines, have included independent airline carriers (Westjet and Hawaiian Airlines) which do not belong to any major alliance. 1 Joint work with Guofu Tan. Yinqi Zhang gratefully acknowledges financial support from the Transportation Research Board’s ACRP Graduate Research Award, which is sponsored by the Federal Aviation Administration (FAA) and administered by the Airport Cooperative Research Program (ACRP). The authors thank Eric Amel, Frank Berardino, Xiao Fu, Nicholas Janota, Rajat Kochhar, Kevin Neels, Robert Samis, Liying Yang, Anming Zhang, and the participants in the ACRP regular meetings for their valuable comments and suggestions. The authors are particularly grateful to Robert Samis of FAA and Lawrence Goldstein of ACRP for helping them to get access to the DB1B international data. The authors also thank the Institute for Economic Policy Research at the University of Southern California for sponsoring the data subscription. 1 Except for mergers, JVs are considered the highest level of cooperation in the international air- line industry (Bilotkach (2019)). Compared to other types of cooperation (e.g., codeshare agree- ments), one unique feature of JVs is that they require partners to jointly determine airfares and operate as a single carrier in the relevant markets. Another form of cooperation that is closest to a JV is antitrust immunity (ATI). However, ATIs are essential components for the new proposed JVs and most of the ATIs have been upgraded to JVs in the recent decade. As a result, we can broadly categorize international airline cooperation into two groups. In the first group, we have standard interline agreements, codeshare agreements, and alliance membership where airline carriers can cooperate on many functions like aircraft maintenance or frequent flier programs, excluding pric- ing. Each carrier simply chooses airfares to maximize its own profits. In the second group, we have JV agreements with ATI that can fully integrate partners into a single carrier for international air travel services. Note that in this study, we do not distinguish between JV and ATI, and we will simply use the term JV to represent the second group. An additional reason that we do not distinguish is that the recent empirical paper by Brueckner and Singer (2019) shows that JV and ATI’s airfares in connecting markets which include all kinds of non-gateway-to-gateway markets are very similar. Since JV partners can jointly determine airfares in concerned markets, consumers are likely to be affected by JVs. To illustrate the impact of JVs, we look at a simplified international airline network structure with two hub-spoke components, as shown in Figure 1.1. The straight lines represent American Airlines (AA), and the dashed lines represent British Airways (BA) in this network. The two carriers, AA and BA, use New York and London as hub cities. These two carriers are key members of the Oneworld alliance, and they have had a JV since 2010. In this network, circles represent cities, and lines that connect two cities represent nonstop flights offered by the two carriers. In the gateway-to-gateway market, the carriers usually compete against each other for consumers. However, due to government regulations, air travel in the behind-to-beyond markets between a non-gateway U.S. city and a foreign non-gateway city must combine flights from both AA and BA. Existing literature often emphasizes two major effects of a JV . On the one 2 Figure 1.1: A Stylized Airline Network for American Airlines and British Airways hand, a JV removes competition in the gateway-to-gateway market and consumers might be worse off in this market. On the other hand, a JV can eliminate the problem of double marginalization and reduce airfares of codeshare flights (i.e., flights operated by more than one carrier) for consumers in the connecting markets. Without JV , each carrier adds a markup over cost for its portion of the codeshare flights to maximize its own profits without considering potential negative effects on the other carrier. This so-called double marginalization problem leads to inefficiently high airfares of codeshare flights which hurt both consumers and carriers in the connecting markets. Therefore, the government regulations on JVs seem to show a consistent pattern: “approving JVs unless carriers’ networks overlap too much” (Bilotkach (2019)). In this paper, we focus on international air travel in the behind-to-gateway markets between cities like Pittsburgh and London, where both online (i.e., flights operated by a single carrier) and codeshare flights are offered. We study how the airfares of online flights are affected by JVs and develop a new hypothesis that JVs can increase the airfares of online flights. We use the U.S. Department of Transportation (DOT)’s Airline Origin and Destination Survey (DB1B) international data from 2008 to 2013 to develop an empirical analysis of the Oneworld alliance’s transatlantic JV established in July 2010. We find that the JV increased airfares of online flights in the behind-to-gateway markets by about three to four percent. The conclusion in the literature that JVs only increase airfares of nonstop flights in the gateway-to-gateway markets and reduce 3 airfares of codeshare flights in the connecting markets fails to fully account for the impact of JVs on consumers. Therefore, DOT’s current regulation may overlook some potential costs of JVs to consumers in the connecting markets. 1.2 Literature Review The competitive effects of international airline JVs were first analyzed theoretically by Brueckner (2001). His model without product differentiation shows that air travels in the behind-to-gateway markets, where both online and codeshare flights are available, are often unaffected by a JV . Car- riers with the same constant marginal cost charge the same monopoly prices for these kinds of markets before and after forming a JV . One of the key reasons for this result is that the model does not allow product differentiation, which means that consumers only choose the flights with lower airfares. By charging monopoly prices for online flights, the carriers in these behind-to-gateway markets get the optimal profits, and they have no incentive to engage in offering codeshare flights (or codeshare flights can only have the same prices as the online flights). The JV could not stim- ulate carriers to change prices since pre-JV prices are already optimal for profit-maximization in these markets. Brueckner (2001), by assuming the presence of economies of densities, also finds that consumers in these markets generally benefit from a JV . However, our empirical evidence indicates that codeshare flights often have higher airfares than online flights in the same behind- to-gateway markets before forming a JV , and consumers might have chosen both kinds of services. This implies that modeling airline demand without product differentiation may not be realistic. More recently, Fu and Tan (2018) develop a theoretical model with product differentiation to study the impact of different forms of airline cooperation. Their model focuses on the differences be- tween ATI and JV agreements. Tan and Zhang (2020a) use the Multinomial Logit (MNL) demand system, which is widely used to model air travel (see Garrow (2010) for a comprehensive review), to evaluate the impact of JVs on different kinds of city-pair markets such as the behind-to-gateway 4 market between Pittsburgh and London. They show that the airfares of online flights increase, and the airfares of codeshare flights decrease after forming JVs in the behind-to-gateway markets. To the best of our knowledge, the outcome that the airfares of online flights in behind-to- gateway markets get higher after forming a JV has not yet been identified in previous empiri- cal studies (Bilotkach (2019); Zhang and Czerny (2012)). Papers by Calzaretta et al. (2017) and Brueckner et al. (2011) mainly show that, as airline carriers improve their cooperation, airfares for the interline or codeshare flights fall. Brueckner and Singer (2019) also show that the airfares of both ATI and JV are similar to the airfares of online flights in the connecting markets, and they emphasize the two main effects of a JV: airfares increase in the inter-hub markets, while airfares decrease in codeshare/interline flights. Our empirical study, based on a differences-in-differences (DID) methodology, identifies the new costs of JVs to consumers from online flights in the behind- to-gateway markets. Our empirical strategy is similar to the one used by Luco and Marshall (2020) who study the impact of vertical mergers between the upstream companies (e.g., the Coca-Cola Company and PepsiCo) and the downstream bottlers in the U.S. carbonated-beverage industry. 1.3 Competitive Effects of JVs Figure 1.2 decomposes the network in Figure 1.1 and illustrates the behind-to-gateway market between Pittsburgh and London, with the straight lines representing AA and the dashed line repre- senting BA. Consumers in this city-pair market can choose between the AA-BA codeshare flights and the AA online flights. Current empirical literature implies that the airfares of the codeshare flights would decrease if AA and BA form a JV due to the elimination of double marginalization. However, an interesting question is how the airfares of the AA online flights would change. Since these two kinds of flights are differentiated and competing for the same set of customers in the same market, it is reasonable to believe that JV can also affect AA’s online flights. One theory is that the decreases in the airfares of codeshare flights may be so significant that they also reduce the airfares of online flights. Fu and Tan (2018) use a linear demand model with 5 Figure 1.2: The Flights of AA and BA in the City-pair Market between Pittsburgh and London product differentiation and show that airfares of both codeshare flights and online flights under a JV are lower than the airfares under simple codeshare without ATI. However, there are other possible outcomes where a JV can raise the airfares of online flights. In the absence of a JV , AA may try to use lower airfares of online flights to incentivize BA to lower its airfares so that AA can alleviate the double marginalization problem in the codeshare flights, which hurts both consumers and suppliers (see Hendricks et al. (1997) for a similar result in a model without product differentiation). Another way to see this is that potential double marginalization in the codeshare flights generates higher costs for AA, causing AA to lower its airfares of online flights so that more consumers can take the more efficient online flights. As a JV fully integrates the two carriers, they no longer have incentives to price noncooperatively for the codeshare flights, and they choose airfares for both kinds of flights to maximize joint profits. Therefore, lower airfares of online flights are not needed anymore, and the JV increases airfares of online flights. As airfares of online flights increase, it is possible that these increases are so significant that airfares of codeshare flights also get higher (see Salinger (1991) for a detailed analysis of this problem). We tend to believe this extreme result is less likely and is generally inconsistent with existing empirical evidence which often concludes that JVs reduce airfares of codeshare flights. A more reasonable outcome is that, following a JV , airfares of online flights get higher, whereas airfares of codeshare flights get lower. These two airfares converge to the same level when the two carriers have similar product qualities and similar marginal costs. This problem is discussed systematically in a separate paper by Tan and Zhang (2020a). Based on the MNL demand system, they show that JVs reduce airfares of codeshare flights, and increase airfares of online flights in the behind-to-gateway markets. 6 In summary, although JVs can reduce or eliminate the double marginalization problem in the codeshare flights, its impact on online flights is uncertain. Different theoretical models illustrate that JVs can lead to higher or lower airfares of online flights in the behind-to-gateway markets. In the next two sections, we will introduce the Oneworld Transatlantic JV and the data we use for the empirical analysis and develop a DID analysis to evaluate the impact of the Oneworld Transatlantic JV on online flights in the behind-to-gateway markets. 1.4 Brief Background and Data In July 2010, AA, Finnair (AY), BA, Iberia (IB), and Royal Jordanian (RJ), five major carriers from the Oneworld alliance, received approval for their ATI (U.S. Department of Transportation (2020a)). In October 2010, BA, IB, and AA officially launched their JV . Note that we consider 2010Q3 as the quarter when the JV started following our previous discussions about the equiva- lence between JV and ATI in recent years. Since AY already entered into an ATI agreement with AA in 2002, we try to evaluate the impact of new members, BA and IB, on the U.S.-U.K.-Spain- France markets (RJ’s service is negligible in this region). Due to a large amount of empirical research about the impact of JVs on codeshare flights, we focus on evaluating the impact of the Oneworld Transatlantic JV on online flights in the behind-to-gateway markets. To simplify the notation in the following analysis, we also combine BA and IB, which were officially merged in January 2011. Flight information in the connecting markets is from the Airline Data Inc’s “GatewaySup” survey from 2008 to 2013. It is drawn from the DOT’s DB1B international data which contain a 10 percent random sample of itineraries involving a U.S. airport. Each observation in the database contains information on airfare, the number of passengers (original record times 10), operating and marketing carriers, the number of coupons, fare class, the route (which includes the origin airport, the connecting points, and the destination airport) and the total distance at the itinerary level by quarter. The database provided by Airline Data Inc is often used in academic research 7 (e.g., papers by Calzaretta et al. (2017) and Ito and Lee (2007)). Like many other studies using the DOT’s DB1B international data, trips without any U.S. carrier segment are unobservable. This does not affect our study because the existing data have sufficient information about codeshare flights, and we only study the impact of the JV on AA’s online flights. Since BA’s online flights are unobservable in this survey, we cannot examine whether its online flights are affected by the JV in a similar manner. However, as discussed in Brueckner and Singer (2019), foreign carriers and U.S. carriers have similar airfares, and using only U.S. carriers’ airfares of online flights is a reasonable approach in studying connecting markets. Therefore, we expect that the JV has a similar impact on BA’s online flights. We process the data following standard approaches in the literature. We only include round trips with no more than eight segments, and each travel leg cannot have more than four segments. We exclude itineraries with fares less than $200 since these tickets may be purchased using frequent flier programs. We also only include economy class itineraries. Additionally, AA’s regional carriers are recoded with AA’s carrier code, since regional carriers are not involved in determining airfares. We group airports into cities and define markets as city pairs following DOT’s Master Coordinate table (U.S. Department of Transportation (2020b)). Following the results from Ito and Lee (2007), we identify three types of flights in the behind- to-gateway markets based on the operating carriers of the segments of the itineraries. Each city-pair market in the sample must have online flights offered by AA (type 1 flight), and we identify two other categories of flights: codeshare flights between AA and BA (type 2 flight) and codeshare flights between AA and non-BA carriers (type 3 flight). Based on our previous discussion, we assume that the carriers start their joint pricing after receiving ATI in 2010Q3. Therefore, 2008Q1 to 2010Q2 is the pre-JV period, and 2010Q4 to 2013Q4 is the post-JV period. We drop data from 2010Q3 in our regression analysis since we cannot determine whether the itineraries were before or after the JV was approved. We anticipate that carriers’ pricing will change from noncooperative to fully integrated in these two periods. 8 Table 1.1: Summary Statistics Note: the total amount of observations is 14,138. 1.5 Descriptive Statistics and Variables We identify 687 city-pair markets in the U.S.-U.K.-Spain-France region, where AA offers online connecting flights. These markets are all between U.S. cities and five major European cities (Lon- don, Manchester, Madrid, Barcelona, and Paris). We also make sure that these markets do not have AA nonstop flights. Based on the three types of flights, we aggregate itineraries up to the type-market-quarter level by computing passenger-weighted average variables (airfare, coupons, total itinerary distance) and the total amount of observations (i.e., aggregate itineraries) is 14,138. Summary statistics for the key variables are shown in Table 1.1. Note that, following the paper by Gillespie and Richard (2012), our regression analysis also reports results using individual itineraries. Itineraries that are different from the three types of flights are dropped, but we still use information from these itineraries to calculate the total number of passengers and share of carriers other than AA and BA in the city-pair markets. Table 1.1 shows that, for each aggregate itinerary, the average airfare is $1,114.65 and the aver- age number of coupons is 4.38. On average, each aggregate itinerary has 113.81 passengers and a total distance of 10,178.39 miles. We use the total number of roundtrip passengers to measure the 9 Table 1.2: Comparison between AA Online Flights and AA-BA Codeshare Flights (Itinerary Level) Notes: This table reports the average of airfares, coupons, passengers, and total itinerary distance of type 1 and type 2 itineraries. Standard deviations of the variables are in brackets. The total amounts of observations for AA online flights and AA-BA codeshare flights are 8,611 and 4,983, respectively. size of the market, and we use the market share of carriers other than AA and BA to measure the level of competition from other carriers in the markets. We also note that these two variables may not be accurate for some city-pair markets because the DB1B international data do not have infor- mation about flights operated only by foreign carriers. We observe foreign carriers only when they offer connecting flights with some U.S. carriers. However, the two variables do not significantly affect our results and the main conclusion still holds if we drop them in the regression analysis. Additional information about the city population and income will be incorporated into our analysis in the future. Lastly, we use three dummy variables, type 1, type 2, and type 3, to identify whether the aggre- gate itinerary is an AA online flight, or an AA-BA codeshare flight, or an AA-non-BA codeshare flight. Since we only have a small amount of AA-non-BA flights in this kind of market, to find a suitable control group for AA-BA flight is quite difficult with our current sample. The variable treatment group identifies whether an aggregate itinerary is from a city-pair market affected by the JV or not. If a market has both type 1 and type 2 services before and after forming a JV , then each observation in such a market has the treatment group equal to one. Our current sample includes large amounts of observations that are affected and not affected by the JV . We also compare AA online flights and AA-BA codeshare flights in Tables 1.2 and 1.3. Table 1.2 reports the itinerary-level characteristics of AA online flights and AA-BA codeshare flights. Ta- ble 1.3 reports the market-level characteristics of AA online flights and AA-BA codeshare flights. 10 Table 1.3: Comparison between AA Online Flights and AA-BA Codeshare Flights (Market Level) Notes: This table reports the average of total market passengers and the average share of other car- riers in the city-pair markets for type 1 and type 2 itineraries. Standard deviations of the variables are in brackets. We can see that codeshare flights are, on average, more expensive than online flights with more coupons and longer distances. Also, more passengers choose online flights. In terms of the mar- ket level comparison, we see that codeshare flights are offered in the markets with higher total passengers and more competition from other carriers. In Figure 1.3, where the dashed vertical line indicates the quarter when the JV started (2010Q3), we show the evolution of the airfares of AA’s online flights (type 1) versus the airfares of AA-BA codeshare flights (type 2) in markets that are affected by the JV (treatment group). Before 2010Q3, we can see a pattern that online flights are cheaper than codeshare flights. It is consistent with our theoretical model with MNL demand which shows that when carriers have similar marginal costs and qualities of services, online flights have airfares lower than codeshare flights. After 2010Q3, we see that airfares of these two types of flights were converging, and gradually online flights were even slightly more expensive than codeshare flights. It provides some preliminary evidence that a JV can decrease the airfares of codeshare flights and increase the airfares of online flights. However, we note that the global financial crisis in 2008 might also reduce airfares before 2010. A simple comparison of airfares before and after the formation of the JV in the treatment group cannot accurately measure the impact of the JV . In Figure 1.4, where the dashed vertical line indicates the quarter when the JV started (2010Q3), we illustrate the trends of the average number of passengers of AA online flights and AA-BA code- share flights. The average number of passengers using AA online flights slightly decreased after forming the JV , while AA-BA codeshare flights attracted a lot more consumers after 2010Q3. It 11 Figure 1.3: Average Airfares of AA Online Flights and AA-BA Codeshare Flights in Markets Exposed to JV 12 Figure 1.4: Average Number of Passengers of AA Online Flights and AA-BA Codeshare Flights in Markets Exposed to JV also supports our discussion that theoretical models in the airline industry should incorporate the feature of product differentiation. Airfare is not the only factor in determining the consumers’ choices. Even though codeshare flights were more expensive than online flights without JV , con- sumers still chose codeshare flights due to other considerations like schedules or brands. Figure 4 also illustrates the importance of understanding the impact of JVs on online flights. Since a higher number of consumers were using online flights before 2010Q3, a small increase in the airfares of online flights, following the theoretical model with MNL demand, can lead to a high impact on consumer welfare. 1.6 Differences-In-Differences Analysis The DID analysis attempts to simulate an experimental research design. Our treatment group includes city-pair markets that were exposed to the JV , and the control group includes city-pair 13 markets that were unaffected by the JV . The control group helps to provide a counterfactual for what would have happened to the treatment group in the absence of the JV . By comparing airfare changes of the online flights in the treatment group to the airfare changes of the online flights in the control group from 2008 to 2013, we can more accurately measure the impact of the JV on online flights. We define P j;m;t as the airfare of AA’s online aggregate itinerary j paid by consumers in the market m at quarter t. Our main empirical model is the following: log(P j;m;t )=b 1 JV j;m;t + x j;t + y j;m +g 0 z j;m;t +q 0 u m;t +e j;m;t : In this model, dummy variable JV j;m;t equals 1 if the itinerary j’s city-pair market m has AA-BA codeshare flights before and after 2010Q3 and the quarter t is after 2010Q3. Here b 1 is the main coefficient of interest and represents the impact of the JV on airfares of online flights. Note that our dependent variable is the logarithm of airfare andb 1 measures the average percentage change of airfare caused by the AA-BA JV . We incorporate quarter fixed effects and city-pair market fixed effects where dummy variable x j;t is equal to 1 if itinerary j is observed in quarter t and dummy variable y j;m is equal to 1 if itinerary j is from city-pair market m. We also control for variables that measure the itinerary j’s characteristics in the city-pair market m in quarter t by the vector z j;m;t which includes the itineraries’ total amount of coupons and total distances. This vector helps to control for potential adjustments of the flights by the carriers in this period. The vector of market- quarter level characteristics, denoted by u m;t , is also included in the estimation. The vector u m;t includes the total amount of passengers and the share of other carriers in the city-pair market m in quarter t. It helps to control for additional market changes in this period that may affect the airfares. The error term,e j;m;t , is clustered at the city-pair market level. The key assumption in the DID analysis is the parallel trend, which is a critical requirement for identifying a valid control group. The airfares in the treatment group should have the same trend as the airfares in the control group before the formation of the JV . Figure 1.5, where the dashed vertical line indicates the quarter when the JV started (2010Q3), illustrates that the average 14 Figure 1.5: The Evolution of Airfares of AA’s Online Flights in the Treatment and Control Group airfares of the online flights, in both the treatment and the control groups, followed a similar trend before 2010Q3. We also estimate the regression model with leads and lags and find no evidence of differential trends before forming the JV (we provide more details of these findings in the next section). All these results suggest that the treatment group and the control group follow a similar trend before the JV . We have, therefore, identified a reliable control group. Moreover, we have found that the carriers continued to offer both codeshare flights and online flights in most of the city-pair markets after forming the JV in the treatment group. Only about 6 percent of the markets no longer observe AA-BA codeshare flights or AA online flights after the JV , and most of these city-pair markets have relatively small market sizes with scarce observations of AA online flights or AA-BA codeshare flights before 2010Q3. We may fail to observe these two kinds of flights after 2010Q3 due to other demand shocks unrelated to the JV or the fact that the DB1B data only contain a 10 percent random sample of itineraries. Therefore, AA did not select specific city-pair 15 Table 1.4: Comparison between AA Online Flights and AA-BA Codeshare Flights (Market Level) Notes: *p< 0:1, **0:01< p< 0:05, ***p< 0:01. Standard errors in parentheses are clustered at the city-pair market level. markets for the JV , and we believe that the potential selection issue is not a major concern in our DID analysis. 1.7 Results and Discussion Table 1.4 shows our empirical results, where Model 1 uses the aggregate itineraries for AA’s online flight, Model 2 uses the aggregate itineraries and excludes all control variables, Model 3 uses indi- vidual itineraries. Model 1 shows our main result that the Oneworld Transatlantic JV has increased the airfares of AA’s online flights in behind-to-gateway markets by 3.56 percent. Other empirical studies (e.g., papers by Calzaretta et al. (2017) and Brueckner and Singer (2019)) repeatedly show that JVs can reduce airfares of codeshare flights in connecting markets by about 7 percent. Since the number of passengers who chose online flights was larger than the number of passengers who chose codeshare flights without the JV , the costs from higher airfares of online flights might have offset much of the benefits from lower airfares of codeshare flights for consumers. 16 Figure 1.6: Dynamics of the Impact on Airfares of AA’s Online Flights Notes: The coefficient for 1 year prior to the JV is normalized to zero. Vertical bands show1:96 times the standard error of each point estimate. We further explore the robustness of our analysis with different specifications. Model 2 and Model 3 show that the Oneworld Transatlantic JV has increased airfares of AA’s online flights by 3.44 percent and 4.49 percent, respectively. Like the results from Gillespie and Richard (2012), models with aggregate itineraries and models with individual itineraries can lead to different esti- mations of the parameters. Model 3 shows that the JV has a slightly higher impact on the airfares of online flights, though its p-value is 0.109. Expanding our sample size to all European countries and including better control variables (like population and average income) may lead to a more accurate estimation of the impact of the JV . Lastly, we estimate the baseline empirical model with leads and lags to study whether there were differential trends before forming the JV and when the airfares of AA’s online flights in markets exposed to the JV started to change. Figure 1.6 shows no statistical evidence that online 17 flights in the treatment and control groups followed different trends before the JV . It also shows that the airfares of AA’s online flights did not immediately change after forming the JV . The airfares of online flights in markets exposed to the JV took more than a year to increase by about 5 percent. Many time-consuming institutional changes caused by the JV (e.g., building a revenue-sharing system) may delay the impact on airfares of online flights. 1.8 Concluding Remarks The rapid development of JVs in the international airline industry has helped more and more airline carriers to coordinate their airfares. Existing literature indicates that JVs can reduce airfares for consumers in connecting markets and increase airfares for consumers in inter-hub markets. We focus on one set of behind-to-gateway markets with both online flights and codeshare flights which is largely overlooked in the literature. In these markets, the competition between online flights and codeshare flights can lead to different results regarding the impact of JVs. Specifically, JVs may reduce airfares of codeshare flights and increase airfares of online flights. We use the DOT’s DB1B international data and develop a DID analysis to show that the Oneworld alliance’s transatlantic JV increased airfares for online flights in the behind-to-gateway markets. Therefore, DOT can better protect consumers by carefully evaluating this possibility of airfare increase in the behind- to-gateway markets when reviewing new JV applications. There are several directions for future research. We will expand our sample size to include all European countries and evaluate the impact of JVs on codeshare flights to see if our DID analysis would lead to consistent and robust conclusions. In addition, since JVs can help carriers to coordinate their schedules and provide better services, we need to incorporate potential quality changes of the flights into the analysis and develop a structural model to accurately measure the impact of JVs on consumer welfare in the connecting markets. Moreover, more theoretical studies that compare different models with product differentiation can help us to understand the impact of JVs in different markets. 18 Chapter 2 Competition with Complements and Substitutes 2.1 Introduction The impact of mergers on consumer surplus and market efficiency is a long-standing question in antitrust economics. Classical models, like Bertrand or Cournot, which feature either substitutes or complements, provide basic intuitions for the effects of price competition. These models show that mergers lead to higher prices if firms’ products are substitutes and lower prices if firms’ prod- ucts are complements. Since we can consider horizontal mergers as mergers of firms providing substitutes and vertical mergers as mergers of firms providing complements, we often believe that horizontal mergers raise prices and vertical mergers reduce prices if other factors like marginal costs or entries of new competitors remain unchanged. 1 However, in real-world markets, firms of- ten provide multiple products, and the members of a merger may produce complementary products and competing substitutes in the same market. In the Vertical Merger Guidelines, the government also observes similar patterns and indicates that ”mergers often present both horizontal and verti- cal elements”(U.S. Department of Justice and Federal Trade Commission (2020)). As a result, we cannot directly apply these classical models to understand the impact of mergers on markets where multiproduct firms may provide both complements and substitutes. 1 We note that standard vertical mergers and mergers of complements mainly differ when the pricing is sequential or simultaneous. In this paper, we try to focus on the nature of firms’ pricing strategies and do not incorporate sequential pricing into the model. Therefore, we can consider mergers of complements and vertical mergers as equivalent, which is consistent with the analyses in Quint (2014a,b). 19 Figure 2.1: An Illustration of the Airline Network Offered by Delta Airlines in the Changsha- Seattle Market. We can illustrate this more complex market structure with both complements and substitutes in an example from the international airline industry. Figure 2.1 shows the flight services offered by Delta Airlines (henceforth, DL) between Changsha and Seattle, two major cities of China and the US. Specifically, DL offers three routes by working with three different airline carriers: flying to Seoul with Korean Air (henceforth, KE) and then switching to DL (Route 1); flying to Beijing with China Southern Airlines (henceforth, CZ) and then switching to DL (Route 2); flying to Shanghai with China Eastern Airlines (henceforth, MU) and then switching to DL (Route 3). In this market, each route is a combination of different carriers’ flights, which are perfect complements, and the total price consumers paid for each route is a combination of prices of the flights used for that route. Moreover, the three routes, which have different characteristics, are competing for consumers who have heterogeneous preferences. This airline network of complementary and substitute products can be easily generalized to understand competition in other similar types of industries. 2 We can 2 Quint (2014b) analyzes the pricing of patents, which are often purchased in combinations. We can similarly interpret links to be patents owned by different firms and routes from s to t to be potential combinations of the patents. 20 simply consider each route connecting the starting point s and the end point t as a final product that is composed of intermediate products supplied by different firms. Even in this simple market structure illustrated in Figure 2.1, it is unclear whether the merger between DL and MU, which are members of the Skyteam alliance, can only reduce prices and improve consumer surplus. 3 The flights offered by DL and the flight offered by MU share features of both complements and substitutes. The flight offered by MU is essential for Route 3, but it also competes against DL’s Route 1 and Route 2. Following the terminology used in the Vertical Merger Guidelines, we call mergers, like the one between MU and DL, mergers of complements (or vertical mergers), where the partnered firms have some intermediate products that are comple- mentary to each other. 4 We can conjecture two major effects from the merger between DL and MU based on the existing empirical studies. On the one hand, Route 3 becomes cheaper to sell because of the elimination of double marginalization. This is the efficiency effect of the merger. On the other hand, as Route 3 becomes more profitable to sell, DL has incentives to increase the prices of its flights in Route 1 and Route 2 so that more consumers choose Route 3. As a result, the total prices of Route 1 and Route 2 may get higher due to the merger. The increases in the total prices of Route 1 and Route 2 are called the Edgeworth-Salinger effect, which leads to major concerns on whether vertical mergers can still improve consumer surplus (Luco and Marshall (2020)). 5 In this case, we should evaluate these simple intuitions more carefully. Even though DL may raise the prices of its flights in Route 1 and Route 2, KE and CZ may also choose to reduce the prices of their flights in Route 1 and Route 2 to better compete against Route 3. Then the total prices of Route 1 and Route 2 may still be lower after the merger. Moreover, the outside option in this market (e.g., flights offered by other airline alliance members) also constrains DL’s incentives to raise prices for Route 1 and Route 2. Therefore, the existence of the Edgeworth-Salinger effect 3 Due to government regulations, it is more likely that these two carriers will form a joint venture to fully integrate their operations as a single carrier for international air travel (Bilotkach (2019); Tan and Zhang (2020b)). 4 All other kinds of mergers where partnered firms do not provide any complementary products are defined as mergers of substitutes (or horizontal mergers). 5 We generalize the notion of the Edgeworth-Salinger effect since we consider more arbitrary combinations of firms’ intermediate products. Any increases in the total prices of final products caused by a merger of complements are considered as the Edgeworth-Salinger effect in this paper. 21 is uncertain. We may overestimate the potential negative effects of mergers of complements on consumers in this case. However, if the merger leads to the Edgeworth-Salinger effect, the potential negative impact on consumers can be significant. In this simple case, the merger may raise two routes’ total prices and reduce only one route’s total price. 6 More importantly, if we incorporate the fact there are additional competitors offering substitutable routes in the same market (e.g., United Airlines), price increases in Route 1 and Route 2 can potentially generate additional price increases in many other competing substitutes. Therefore, the merger of complements can lead to various changes in the total prices of the products, and we need to develop a theoretical model that can incorporate different firms’ strategic responses, to characterize equilibrium prices and understand the impact of a merger of complements on consumers. Following our discussions above, the theoretical model must incorporate one important feature: multiproduct firms. With single-product firms, a merger of complements simply reduces the total marginal cost of the final product and eliminates the possibility of having the Edgeworth-Salinger effect. Therefore, mergers of complements do not lead to any negative effects on consumers (Quint (2014a)). However, multiproduct pricing in this more general environment with complements and substitutes significantly complicates our theoretical analysis due to many factors, including the problem that discrete/continuous choice models with multiproduct firms often have difficulty en- suring the existence and uniqueness of a pricing equilibrium (Konovalov and S´ andor (2010), Nocke and Schutz (2018), Armstrong and Vickers (2018)). We first show that the general demand system with multiproduct firms developed by Nocke and Schutz (2018) can be extended to include arbi- trary complementary products even though their aggregative games approach needs to be further generalized. We define a new (i-)markup that satisfies the ”common markup property,” where each firm chooses the same markup for all of its products to maximize its profit. Even though markups for firms providing complements are interrelated, we can identify a one-to-one relationship be- tween equilibrium prices and an aggregator using systems of equations. This aggregator measures 6 Another extreme result is that the Edgeworth-Salinger effect can be so significant that it dominates the efficiency effect and leads to price increases in all three routes offered by DL (Salinger (1991)). 22 consumer surplus, and we can simplify the equilibrium analysis to finding an aggregator that sat- isfies an aggregate fitting-in function. With this method, we can analyze price competition with arbitrary complements and substitutes. We also show that multinomial logit (MNL) and constant elasticity of substitution (CES) demands can lead to a unique pricing equilibrium that helps us develop robust results in our merger analysis. After characterizing the pricing equilibrium, we analyze price competition under different net- works of complements and substitutes and study the impact of mergers of complements. Similar to Rey and Tirole (2019), we concentrate on the impact of mergers on consumer surplus, which is consistent with antitrust agencies’ objectives. We first show that allowing multiproduct firms is a necessary but insufficient condition for the potential anticompetitive effects from mergers of complements. By focusing on symmetric network structures, where any two firms with comple- ments are producing intermediate products for the same final products, we show that the insights from models with single-product firms continue to hold and mergers of complements are gener- ally beneficial to consumers. We then extend the model to analyze asymmetric networks, where firms offering complements to each other may produce intermediate products for different final products. The market structure illustrated in Figure 2.1 gives an example of asymmetric networks where firms offering complements, like DL and MU, produce intermediate products for different final products. We also distinguish two important types of asymmetric networks: one-sided asym- metric networks and two-sided asymmetric networks. Figure 2:1 is also an example of a one-sided asymmetric network where for any two firms with complements, like DL and MU, one firm only produces complementary products for a subset of the other firms’ intermediate products. Under a two-sided asymmetric network, the two firms, producing complements, may both offer additional competing substitutes (possibly with other firms) in the same market. If we further incorporate the fact CZ also offers an additional competing route in the Changsha-Seattle market, the network in Figure 2:1 becomes a two-sided asymmetric network. Our main finding is that when the network is one-sided asymmetric, any merger of complements can improve the overall consumer surplus even though there may exist the Edgeworth-Salinger effect. Competition from firms outside the 23 merger constrains the size of the Edgeworth-Salinger effect. Only when the network is two-sided asymmetric may we get the results that the Edgeworth-Salinger effect dominates the efficiency effect and consumers may be worse off after a merger of complements. This type of network can effectively enhance firms’ incentives to raise prices after a merger. The rest of the paper proceeds as follows. Section 2 discusses the related literature. In Section 3, we define the network structure, the demand system, and the pricing game. In Section 4, we conduct equilibrium analysis and show the existence of a pricing equilibrium under Marshall’s second law of demand and the uniqueness of a pricing equilibrium with MNL or CES demands. In Section 5, we study mergers of complements with MNL and CES demands. Section 6 discusses policy implications from our theoretical analysis. Section 7 concludes. All proofs are presented in Appendix A. 2.2 Related Literature Quint (2014a) develops a similar discrete choice model to analyze price competition in a set- ting with both complements and substitutes. His merger analysis shows results that are generally consistent with the classical models, like Bertrand or Cournot, with either complements or sub- stitutes. His model focuses on single-product firms, and he indicates that the supermodular prop- erties, which play a crucial role in his analysis, are no longer satisfied after extending the model to multiproduct firms. Several other theoretical and technical challenges, including the lack of quasi-concavity in a multiproduct firm’s profit function, make the analysis of the existence and uniqueness of a pricing equilibrium even more complicated with multiproduct firms (Hanson and Martin (1996)). Not surprisingly, there is only small literature focusing on competition between single-product firms with complements and substitutes (Tan and Yuan (2003), Quint (2014b)). Therefore, Quint (2014a) indicates that extending the discrete/continuous choice model to multi- product firms in this type of market with complements and substitutes is a ”significant challenge for future work.” The recent empirical studies by Luco and Marshall (2020) and Tan and Zhang 24 (2020b) provide more evidence that a merger of complements can affect prices differently in in- dustries with multiproduct firms and developing a comprehensive theoretical analysis in this more general setting is very important for antitrust agencies to better protect consumers. Also, unlike papers by Hart and Tirole (1990) and Ordover et al. (1990), which allow for more general post- merger responses, our analysis in this paper concentrates on the impact of a merger of complements on prices and consumer surplus, which is more consistent with the studies by Quint (2014a) and Salinger (1991). In recent years, a series of papers have successfully characterized pricing equilibrium with multiproduct firms producing pure substitutes under (nested) MNL and (nested) CES demands (Konovalov and S´ andor (2010), Li and Huh (2011), Gallego and Wang (2014)). These contri- butions are further generalized in the papers by Nocke and Schutz (2018, 2019a), which build a demand system that nests both MNL and CES demands. Our model attempts to develop the de- mand system in Nocke and Schutz (2018) by incorporating arbitrary amounts of complementary products, which reflect the fact that the final products’ prices are often determined by multiple pro- ducers. We generalize their aggregative games approach to solve the technical challenges, show the existence and uniqueness of a pricing equilibrium, and provide new insights regarding the impact of mergers of complements on prices and consumer surplus. Our analysis also contributes to the small literature about Edgeworth’s paradox of taxation (Edgeworth (1925), Hotelling (1925), Salinger (1991)). Under our framework, a merger of com- plements between two firms that jointly offer one final product can be similarly considered as a reduction in the final product’s total marginal cost through a tax reduction. Classical literature fo- cuses on the case where one monopolist offers multiple final products and receives a tax reduction in one of the final products. It mainly tries to show the possibility that such a tax reduction may lead to price increases in all of the monopolist’s final products. Our model extends the analysis to a more general environment with multiple multiproduct firms that may provide complements and substitutes. Instead of focusing on the case where all final products get higher total prices, we provide a more detailed analysis of the impact on consumer surplus when the cost reductions from 25 a merger of complements raise the total prices of some final products (i.e., the Edgeworth-Salinger effect) and also decrease the total prices of some other final products (i.e., the efficiency effect). 2.3 The Model 2.3.1 Network of Complements and Substitutes We consider an economy with a network of complementary and substitute products. There is a set of nodes V =fs;t;1;2;:::;Ig, where s is the source node and t is the destination node. The links connecting two nodes represent firms that are producing different intermediate products. We use F =f f 1 ; f 2 ;:::; f K g to denote the set of links in this network, where we assume that the same link can be used multiple times (e.g., f k =fsi 1 ;i 2 i 3 g, where i 1 ;i 2 ;i 3 2 V , means that the link f k is used to connect node s and node i 1 and connect node i 2 and node i 3 ). A path from s to t is a sequence of distinct nodesfi 1 ;i 2 ;:::;i l g from Vnfs;tg and links from F such that g Si 1 = g i 1 i 2 = :::= g i l t = 1. We also use N =f1;2;:::;ng to denote the set of paths between s and t in this network. We can consider each node as a sector in the production chain and each path as a final product that combines intermediate products produced by different firms and competes with other final products. Therefore, a combination of V , F, and N defines a network of complements and substitutes. 7 We also assume that there exists a path 0, omitted in all figures in this paper, representing the outside option for consumers. For each path i2 N, we use the set M i F to denote the links of path i, which represent the firms that are involved in producing the final product i. Additionally, the set G k N for any firm f k 2 F represents the final products (or paths) such that, for any i2 G k , f k 2 M i . 8 This indicates that firm f k produces intermediate products for the final products G k N. This network of complements and substitutes can be used to illustrate market competition in different industries. We can consider nodes to be connecting cities and paths between s and t to be 7 Such network is also similarly defined in Acemoglu et al. (2012) and Choi et al. (2017). 8 To simplify the notation, we omit the letter ’ f ’ for the set G k . In general, we use superscripts of the sets to denote firms (i.e. links) and subscripts of the sets to denote final products (i.e. paths). 26 Figure 2.2: A Network with Two Separable Components: S 1 =f1;2;3g and S 2 =f4;5g. different routes for air travel between city s and t. Similarly, in the personal computer market, we can consider the computers offered by Dell or Apple as paths, and essential intermediate products of the computers (e.g., the operating systems or the central processing units), which are often produced by multiple hardware and software companies, as links in the network. Moreover, we introduce a useful notation of the network that is important for our generalization of the aggregative games approach in Section 4. Definition 1 A separable component of the network, S a N with a2Z + , is a minimal set of paths such that for any path i = 2 S a , there does not exist f k 2 F, where i2 G k and G k \ S a 6=?. Figure 2.2 gives an illustration of a network with two separable components where we order the paths N=f1;2;3;4;5g from top to bottom. These separable components are S 1 =f1;2;3g and S 2 =f4;5g. We can partition the paths in any network into separable components, and an impor- tant interpretation is that firms from different separable components are producing pure substitutes, whereas firms inside the same separable components may produce complements for different firms. Potentially, an indirect complementary relationship can exist within the same separable compo- nent. In S 1 , f 2 produces complements for f 1 , f 1 produces complements for f 3 , and f 3 produces complements for f 4 . This concept of separable components is also connected to the international 27 airline industry. We can interpret each separable component as an airline alliance where carriers within the same alliance may provide complements to each other and flights across alliances are substitutes. Lastly, we note that our network structure is a generalization of products in Quint (2014a) and Nocke and Schutz (2018). WhenjG k j = 1 for any f k 2 F andjM i j 1 for any i2 N, it corresponds to the case of Quint (2014a), where single-product firms may produce complements to other firms. In this case, each final product is a separable component. WhenjG k j 1 for any f k 2 F andjM i j= 1 for any i2 N, it illustrates the market structure in Nocke and Schutz (2018), where multiproduct firms can only produce substitutes. In this case, each G k N, where f k 2 F, is a separable component. In this paper, we generalize the settings in these two papers by allowing firms to produce intermediate products for multiple final products, and every final product in this economy can have multiple intermediate products. 2.3.2 Network Topology We classify the network of complements and substitutes into two major groups: (S) Symmetric network: For any two firms, f a and f b , where G a \ G b 6=?, then G a = G b . (A) Asymmetric network: There exists two firms, f a and f b , where G a \ G b 6=?, such that G a 6= G b . Under a symmetric network, any two firms that are producing intermediate products for the same final product are ”loyal” to each other in the sense that neither of the two firms tries to work with other firms (or by themselves) to introduce an additional substitutable final product. The network of complements and substitutes in Quint (2014a) can be considered as a special case of the symmetric network where each firm only produces one intermediate product. However, as discussed in Salinger (1991) and Luco and Marshall (2020), we frequently observe firms produc- ing multiple intermediate products for several firms in markets, and we introduce the asymmetric network to identify the more general relationships between different firms’ intermediate products. Under an asymmetric network, we may have firms that produce only complements for each other, 28 Figure 2.3: An Illustration of the Three Types of Networks. (a) Symmetric Network (b) One-sided Asymmetric Network (c) Two-sided Asymmetric Network but we must also have firms that produce both complements and substitutes for each other. More- over, we also distinguish two types of asymmetric networks: A1 and A2. (A1) One-sided asymmetric network: For any two firms, f a and f b , where G a \ G b 6=?, then either G a G b or G b G a . (A2) Two-sided asymmetric network: There exists two firms, f a and f b , such that G a \G b 6=?, G a 6 G b , and G b 6 G a . The relationships between firms’ products in Salinger (1991) and Luco and Marshall (2020), where one firm produces intermediate products for other firms as illustrated in Figure 2.1, can be considered as special cases of the A1 network. We generalize these cases by relaxing restrictions onjM i j and allowing additional competitors (i.e., other separable components) in the same market. However, we notice that the asymmetric network A1 could not fully reveal the relationship between firms’ products. Therefore, we introduce asymmetric network A2 to incorporate the fact that when two firms are producing complements to each other, both of them may work with other firms (or by themselves) to introduce additional competing final products. Figure 2.3 provides a basic graphical illustration of the network topology in this paper. 29 2.3.3 Prices and Consumer Demand With a given network of complements and substitutes, denoted by (V;F;N), we assume that firms determine prices for their intermediate products in the network, and the set(s k i ) f k 2M i 2R jM i j ++ con- tains the intermediate product prices charged by firms M i F for the final product i2 N. 9 We assume that the total price for any i2 N, denoted by P i , is determined by the following equation: P i å f k 2M i s k i : This means that the total price of any final product i2 N in this network is a summation of i’s intermediate products’ prices. We assume that there is a population of consumers located at the source node s. Each consumer first decides which final product to buy and then the quantity of the purchase. It can be interpreted as choosing the optimal path and the optimal frequency for consumers to go from source node s to the destination node t. As we discussed before, we also assume that consumers have an outside option denoted by 0. Each final product i2 N has an indirect utility of logh i (P i )+e i , where e i represents the taste shock of product i for each consumer, and h i (P i ) represents the intrinsic value of the final product i. Note that the outside option 0 has an indirect utility of logH 0 +e 0 . After choosing the optimal final product, each consumer consumes(logh i (P i )) 0 units of the final product i following Roy’s identity. We assume that consumers’ taste shocks(e i ) i2N[f0g are drawn i.i.d. from a type-I extreme value distribution. We can determine the probability of choosing product i to be: Prob i (P i )= h i (P i ) å j2N h j (P j )+ H 0 : We can then derive the average demand per consumer for the final product i2 N[f0g as D i (P)= h i (P i ) å j2N h j (P j )+ H 0 h 0 i (P i ) h i (P i ) = h 0 i (P i ) å j2N h j (P j )+ H 0 = ˆ D i (P i ;H(P)); (2.1) 9 To simplify the notation, we omit the letter f for the intermediate price, s k i , charged by the firm f k 2 M i for the final product i. 30 where P (P i ) i2N is the set of all final product prices. Additionally, we assume that for any i2 N, h i (P i ) is a C 3 and positive-valued function such that h 0 i < 0, h 00 i > 0 and h i is log-convex. Also, H 0 is a constant greater than 0. 10 The consumer surplus is given by logH(P), where H(P)å j2N h j (P j )+ H 0 . This demand system, which depends on the early research about the discrete/continuous choice model by Anderson et al. (1987), is formally introduced in the paper by Nocke and Schutz (2018), where they also show that this demand system can be derived from quasi-linear utility maximization. The CES demand (with h i (P i )= a i P 1s i , where a i > 0 ands > 1 are parameters) and MNL demand (with h i (P i )= e a i P i l , where a i 2R and l > 0 are parameters) are both special cases of the demand system. Also, note that the MNL demand withl = 1 is the unique case, where each consumer only purchases one unit of the final product after choosing the optimal final product. 2.3.4 Pricing Game Based on a given network of complements and substitutes,(V;F;N), and our demand system, the profit of firm f k 2 F, which produces intermediate products for final products G k N, is defined as follows: P k (s)= å j2G k (s k j c k j ) h 0 j (s k j +å l2(M j nk) s l j ) H(s) å j2G k (s k j c k j ) ˆ D j (s k ;s k 0 ;H(s)); s2(0;¥] Q ; (2.2) where s(s k i ) f k 2F;i2G k is a vector of prices charged by all firms in F and c(c k i ) f k 2F;i2G k is a vector of marginal costs where c k i 2R ++ represents the marginal cost for firm f k in producing the intermediate products for the final product i. We define s k =(s k j ) j2G k as prices charged by firm f k . As we discussed before, we can partition any network of complements and substitutes into a finite amount of separable components. Suppose firm f k is in S a where a2Z + . We can then use f k 0 to denote firms in S a nf f k g (i.e., firms other than f k in the separable component S a ) 10 Unlike in Nocke and Schutz (2018), we do not consider the case where H 0 = 0. We need to study networks like in Figure 2.1, where H 0 = 0 can lead to inconsistent interpretation of the infinite prices. 31 and s k 0 (s k 0 i ) i2G k 0 to denote the vector of prices charged by firms other than f k in the separable component S a . Q=å i2N jM i j is the total number of prices for intermediate products charged by different firms. Note that we focus on the case where we allow firms to sell the same intermediate product, which may be used for different final products, with different prices. Many industries share this feature. Hendricks et al. (1997) show that airline carriers often charge consumers on the same plane different prices contingent on the flights offered by other carriers for the remaining parts of the trips. Similarly, Luco and Marshall (2020) indicates that bottlers in the US carbonated- beverage industry can charge different prices for the bottling services they offered to different upstream firms. In practice one firm may have to charge the same price for its products that are used in several final products, as illustrated in Quint (2014b). We will study this problem in a separate paper, and we allow firms to price discriminate effectively in this paper. We focus on the pricing game in which each firm sets prices for its intermediate products simul- taneously to maximize its profit, which is given by equation (2.2). We define a pricing equilibrium as a pure-strategy Nash equilibrium of that pricing game. 2.4 Equilibrium Analysis In this section, we first discuss the existence of a pricing equilibrium and characterize the set of pricing equilibria with a given network (V;F;N). We illustrate that we can generalize the aggrega- tive games approach to this more general setting. Then we study the uniqueness of a pricing equilibrium under MNL and CES demands. 2.4.1 Equilibrium Existence As discussed in Nocke and Schutz (2018) and Quint (2014a), we have similar difficulties in the problem of the existence of a pricing equilibrium. 11 To solve this problem, we define a new version 11 If f k is a multiproduct firm (i.e.,jG k j > 1), then P k is neither supermodular nor log-supermodular in (s k j ) j2G k . In the case without complementary products, Gallego and Wang (2014) show that the pricing game is supermodular in markups. However, we no longer have such property since the markups for complementary products are strategic substitutes. 32 of thei-markup to accommodate complementary products, introduce a different way to character- ize the fitting-in function, and show that Marshall’s second law of demand is still sufficient for the existence of a pricing equilibrium. We provide a brief discussion of the method, and interested readers can refer to the Appendix for details. We know that firm f k ’s profit is given by P k (s)= å j2G k (s k j c k j ) ˆ D j (s k ;s k 0 ;H(s)): Suppose first-order conditions are both necessary and sufficient for optimality of each firm’s profit maximization problem. We can show that equilibrium intermediate prices must satisfy the following first-order condition for every f k 2 F and i2 G k : ¶P k ¶s k i = ˆ D i +(s k i c k i ) ¶ ˆ D i ¶s k i + ¶H ¶s k i ( å j2G k (s k j c k j ) ¶ ˆ D j ¶H ) = ˆ D i (1 s k i c k i P i j ¶ log ˆ D i ¶ logP i j+ ¶H ¶s k i ˆ D i ( å j2G k (s k j c k j ) ¶ ˆ D j ¶H ))= 0: We can rewrite the first-order condition as s k i c k i P i j ¶ log ˆ D i ¶ logP i j= 1+ å j2G k (s k j c k j ) ¶ ˆ D j ¶H ¶H ¶s k i ˆ D i : This is equivalent to s k i c k i P i i i (P i )= 1+P k (s); (2.3) wherei i (P i ) P i h 00 i (P i ) h 0 i (P i ) =j ¶ log ˆ D i ¶ logP i j. Condition (2.3) implies that the right-hand side is the same for every i2 G k . However, unlike in Nocke and Schutz (2018), the left-hand side of equation (2.3) also depends on prices charged by other firms. In any pricing equilibrium, for any f k 2 F and for any i;g2 G k , we have s k i c k i P i i i (P i )= s k g c k g P g i g (P g ): 33 We define firm f k ’s i-markup, m k , where s k i c k i P i i i (P i )=m k for every i2 G k , and we say that firm f k ’s profile of prices,(s k i ) i2G k, satisfies the commoni-markup property. Note that for any final product i2 N, å l2M i m l = å l2M i s l i c l i P i i i (P i )= P i C i P i i i (P i )Y i ; (2.4) where C i å l2M i c l i is the summation of all firms’ marginal costs in providing final product i2 N. We define Y i =å l2M i m l to be the aggregate markup for the final product i. Suppose that the function P i 7! P i C i P i i i (P i ) is one-to-one for any i2 N. We can denote its inverse function by t i (). Then we can rewrite equation (2.3) to be m k = 1+m k å j2G kg j (t j (Y j )) H(Y) = 1+ ˆ P k (m k ;(t j (Y j )) j2G k;H(Y)); (2.5) where we define function g i (P i )= h 0 2 i (P i ) h 00 i (P i ) for any final product i2 N. Equation (2.5) implies that the aggregative games approach in Nocke and Schutz (2018) may not be directly applied to this more general setting, since the right-hand side of equation (2.5) depends on both the aggregator H and thei-markups of firms providing complements for f k in the separable component, S a , where G k S a . We cannot use only equation (2.5) to determine the relationship between the aggregator H and thei-markup of f k . Instead, we rewrite equation (2.5) for firm f k to be m k = H Hå j2G kg j (t j (Y j )) : (2.6) Then we can rewrite the aggregate markup for each final product i2 N to be Y i = å l2M i m l = å l2M i H Hå j2G lg j (t j (Y j )) : (2.7) 34 For the separable component of the network, S a N, a system of equations following (2.7) can ef- fectively determine a unique(Y i ) i2S a for any given aggregator H > 0, and this system of equations implicitly defines continuous functions m i (H)=Y i for any i2 S a . We call m i (H) final product i’s fitting-in function. Similarly, we can also determine the function m g (H) for any other final product g = 2 S a based on the equations in product g’s separable component. Therefore, the equilibrium existence problem becomes finding an aggregator H such that H = H 0 + å i2N h i (t i (m i (H)))G(H); whereG(H) is the aggregate fitting-in function. To complete the proof of the existence of pricing equilibrium, we just need a revision of the assumption used in Nocke and Schutz (2018): Assumption 1 For every i2 N and P i > 0,i 0 i (P i ) 0 wheneveri i (P i )> 1 and lim P i !¥ i i (P i )=¥. 12 Theorem 1 Suppose that the demand system((h i ) i2N ;H 0 ) satisfies Assumption 1. Then the pricing game has a pricing equilibrium for every N and F. The set of equilibrium aggregators H is determined by the fixed points of the equationG. 13 Suppose H is an equilibrium aggregator. Then we have the following: (a) The equilibrium price for each final product i2 N is: P i = t i (m i (H )). (b) The equilibrium i-markup for firm f k 2 F, m k , is given by equation (2.6), and firm f k ’s profit is m k 1. The equilibrium price of the intermediate product offered by f k for final product j2 G k can be determined following the definition of thei-markup. (c) Consumer surplus is given by: log(H ). 12 Similar to Nocke and Schutz (2019b), we use the differential approach based on the Poincare-Hopf index the- orem to identify the function m i (H) for any i2 N, which cannot deal with the points where the demand is not differentiable. These are the points where Y i = lim P i !¥ i i (P i ). By imposing the condition that lim P i !¥ i i (P i )=¥, ˆ P k (m k ;(t i (Y i )) i2G k;H(Y)) is differentiable for anym k 2[1;¥). 13 Note that we exclude the ”no-trade”-type equilibria, where all firms are choosing infinite prices for their interme- diate products in one final product (Quint (2014a)). We impose a tie-breaking rule that when f k ’s complements in final product i have infinite prices, f k will choose a finite price for its product in i. This leads to equivalent results with the assumption in Nocke and Schutz (2019b), where the firms’ action sets are not compactified (i.e., s k i 2(0;¥) for any i2 G k and f k 2 F). Another potential method is to introduce a notion similar to the pairwise stability in Jackson and Wolinsky (1996). 35 Theorem 1 is flexible enough to characterize pricing equilibrium for any network of comple- ments and substitutes. Assumption 1, which is sometimes called Marshall’s second law of demand, assumes that the price elasticity of the monopolistic competition demand for each final product i2 N should be nondecreasing in its own price. Similar assumptions that each final product’s own- price elasticity of demand is nondecreasing are often used in standard oligopoly models (Hendricks et al. (1997), Vives (2000), Quint (2014a)). Assumption 1 holds with MNL demand, which means that this frequently used demand continues to have a pure-strategy pricing equilibrium in this more general setting. However, CES demand cannot satisfy Assumption 1, which is more restrictive than the assumption used in Nocke and Schutz (2018). In the next subsection, we impose addi- tional conditions so that we can also apply this generalized aggregative games approach to analyze the properties of the pricing equilibrium under CES demand. 2.4.2 Equilibrium Uniqueness With an aggregative games approach, the uniqueness of the pricing equilibrium can be shown by evaluating whether the aggregate fitting-in function, G(H), has a unique fixed point or not. Proposition 1 indicates that in a network with arbitrary complementary and substitute products, the pricing equilibrium is unique under MNL demand. Proposition 1 In any given network of (V;F;N), the pricing equilibrium is unique under MNL demand. With CES demand, by assuming that s is sufficiently large for any given a i > 0, we can still use the aggregative games approach and show that in the pricing game there always exists a pricing equilibrium. If the network is not A2, the pricing equilibrium is unique. Proposition 2 In any given symmetric or one-sided asymmetric network with a i > 0 for any i2 N, there exists a unique pricing equilibrium under CES demand whens is sufficiently large. 14 14 With CES demand,Y i = P i C i P i s <s for any i2 N. Unlike the case with MNL, CES demand may lead to a result thatå l2M i m l =Y i jM i j,Y i s and the inverse function t i (Y i ) may not be differentiable. Therefore, there may not exist a pricing equilibrium whereY i <s whens is small. 36 Based on MNL and CES demands, we can effectively evaluate the impact of mergers of com- plements on prices and consumer surplus in the next section. Even though we cannot formally show that the pricing equilibrium is always unique with arbitrary networks under CES demand, it does not affect the main findings in the merger analysis in Section 5. 2.4.3 An Example with MNL Demand In this subsection, we use MNL demand with h i (P i )= exp(v i P i ) to illustrate our aggregative games approach in characterizing the pricing equilibrium of the market structure in Figure 2.1 with final products N =f1;2;3g. 15 Also, DL, KE, CZ, and MU are denoted as ( f k ) k2f1;2;3;4g , respectively. Based on all firms’ first-order conditions, we know that the equilibrium prices are determined by the following system of equations. m 1 = 1+m 1 ( 3 å j=1 h j (Y j +C j ) H(Y) ); m 2 = 1+m 2 ( h 1 (Y 1 +C 1 ) H(Y) ); m 3 = 1+m 3 ( h 2 (Y 2 +C 2 ) H(Y) ); m 4 = 1+m 4 ( h 3 (Y 3 +C 3 ) H(Y) ): (2.8) In this case with MNL demand, t i (Y i )=Y i +C i and m k = s k i c k i for any i2 N and i2 G k . Also, any f k ’s first-order condition depends on both the aggregator H and some other firm’s markup. Applying the Poincare-Hopf index theorem, we know that for any given H > 0, there exists a unique(m k ) k2f1;2;3;4g , wherem k 2[1;¥) for any f k , that satisfies all firms’ first-order conditions in (2.8). We can further simplify the equations in (2.8) to be Y i ( H Hå 3 j=1 exp(v j (Y j +C j )) + H H exp(v i (Y i +C i )) )= 0; 8i2 N; (2.9) 15 We assumel = 1 to simplify the presentation. 37 whereY i 2[2;¥) for any i2 N. We define the left-hand side of equations in (2.9) to be T(Y;H) and apply the implicit function theorem to T(Y;H)= 0, which shows that det(DT(Y;H))=(det 2 6 6 6 6 4 1 h 1 + Z 1 + Z 2 Z 1 Z 1 Z 1 1 h 2 + Z 1 + Z 3 Z 1 Z 1 Z 1 1 h 3 + Z 1 + Z 4 3 7 7 7 7 5 | {z } We denote this matrix to be ¯ A )h 1 h 2 h 3 ; where DT(Y;H) is the Jacobian matrix of T(Y;H) with respect to Y and Z k = (m k ) 2 H > 0. An interesting observation is that the matrix ¯ A is positive definite since we can decompose the matrix ¯ A into four positive semi-definite matrices,(B k ) k2f1;2;3;4g , and a diagonal positive matrix C. B 1 = 2 6 6 6 6 4 Z 1 Z 1 Z 1 Z 1 Z 1 Z 1 Z 1 Z 1 Z 1 3 7 7 7 7 5 ;B 2 = 2 6 6 6 6 4 Z 2 0 0 0 0 0 0 0 0 3 7 7 7 7 5 ;B 3 = 2 6 6 6 6 4 0 0 0 0 Z 3 0 0 0 0 3 7 7 7 7 5 ; and B 4 = 2 6 6 6 6 4 0 0 0 0 0 0 0 0 Z 4 3 7 7 7 7 5 identify the final products that have intermediate products offered by f 1 , f 2 , f 3 , and f 4 , and the matrix C is given by C= 2 6 6 6 6 4 1 h 1 0 0 0 1 h 2 0 0 0 1 h 3 3 7 7 7 7 5 : Therefore, ¯ A=å 4 l=1 B l +C, which means that the matrix ¯ A is positive definite and det(DT(Y;H))> 0 for any H > 0. Also, the equation T(Y;H) = 0 implicitly defines differentiable functions m i (H)=Y i for any i2 N. Applying Cramer’s rule and the matrix determinant lemma, we can prove that ¶å 3 j=1 h i (t i (m i (H))) H ¶H = å 3 j=1 R j ( ¯ A 1 ) H 2 < 0 38 where the function R j (X) gives the sum of the entries in the row j of matrix X. This result implies that the function G(H) H is strictly decreasing, and G(H) H = 1 has a unique solution H > 0. Therefore, the network structure in Figure 2.1 has a unique pricing equilibrium with MNL demand. 2.5 Merger Analysis One major objective of this paper is to analyze the impact of a merger of complements on consumer surplus and prices within a given network of complements and substitutes. This is consistent with the practice of antitrust agencies in many countries where they mainly focus on promoting the interests of consumers. We will systematically illustrate how connections among firms’ products can affect a merger’s impact on consumers. 2.5.1 Symmetric Network In an arbitrary symmetric network, we can further simplify the analysis we did in Section 4 and characterize the equilibrium prices more clearly. In this type of network, we can combine all i- markups of the firms producing the final product i2 N and rewrite equation (2.5) for product i2 N to be Y i =jM i j+Y i ˆ P(t i (Y i ) i2G k;H): (2.10) Equation (2.10) has a unique solution inY i , which is again denoted by m i (H). Also,Y i =Y j if i; j2 G k . The assumption of a symmetric network, which implies that any G k N is a separable component (G k = S a ), leads to the case where we can use one single equation to characterize each aggregate markupY i for any i2 S a . In this case, a merger of complements between two firms is equivalent to the reduction ofjM i j by one. The following proposition shows that such a merger can enhance consumer surplus and reduce all prices of the final products under both CES and MNL demands. This result is also consistent with the findings in the single-product model in Quint (2014a), which is a special case of the symmetric network. 39 Proposition 3 A merger between two firms, f k and f l , which are suppliers of components of the same final products (i.e., G k = G l ) leads to lower prices for all final products, and higher consumer surplus. Moreover, profits of all firms in M i nf f k ; f l g for all i2 G k are higher, and profits of all firms in producing intermediate products for Nn G k get lower after the merger. 16 The result from Proposition 3 is consistent with our traditional understanding about mergers of complements. Under symmetric networks, the two parties in a merger of complements combine all their products together, and such a merger cannot have the Edgeworth-Salinger effect we discussed before, since both firms do not offer additional products. The elimination of double marginalization reduces the price of the merged products and also intensifies competition with other final products. Consumers are clearly better off in this case. We also have the results that other firms, M i nf f k ; f l g, where i2 G k , which provide complements for firms, f k and f l , benefit from the merger since such a merger is equivalent to a reduction in marginal costs for all firms in M i nf f k ; f l g. 2.5.2 Asymmetric Network Under an asymmetric network, as illustrated in Figure 2.1, a firm may provide several final products by working with different firms. Unlike the symmetric network, a merger between DL and MU can affect the merged product’s price and stimulate DL to change its prices for the other two routes jointly supplied with CZ and KE. Since the merged product becomes more efficient to sell, which is called the efficiency effect, DL may have incentives to raise the total prices for the other two routes to encourage more consumers to use the merged route. This potential price increase, which is called the Edgeworth-Salinger effect, makes the overall impact of a merger of complements on consumer surplus uncertain. In this section, we focus on studying the existence of the Edgeworth- Salinger effect and evaluate the impact of a merger of complements on consumer surplus under asymmetric networks. 16 We omit a discussion regarding the incentives for firms k and l to merge here. As discussed in Tan and Yuan (2003) and Quint (2014b), such a merger may not be beneficial to f k and f l ’s joint profits. One sufficient condition for the profitability of such a merger is thatjM i j must be sufficiently large so that the benefit from the elimination of double marginalization is larger than the cost of more intensive competition from final products that are not in G k . 40 2.5.2.1 One-Sided Asymmetric Network We start from a one-sided asymmetric network structure illustrated in Figure 2.4. In this case, we have one dominant firm (firm f 1 ), which produces intermediate products for all the final products, N =f1;2g. Firms f 2 and f 3 offer essential components for the final productsf1;2g. This market structure is the focus of the empirical analysis in Luco and Marshall (2020), where f 1 is the bot- tler that provides essential services for US carbonated-beverage companies like Coca-Cola or Dr. Pepper. Suppose f 1 merges with f 2 . We find that there always exists the elimination of double marginal- ization, which makes the price of the final product 1 lower. However, the merger’s impact on the price of the final product 2 is unclear. In this case, two forces make the price change in the final product 2 uncertain. On one hand, the merger makes the final product 1 more efficient to sell and it stimulates f 1 to raise prices for the final product 2. On the other hand, f 3 ’s price change is uncer- tain. The final product 1 becomes cheaper after the merger, which stimulates f 3 to decrease prices. However, a more challenging issue is that the prices of the intermediate products offered by f 1 and f 3 for the final product 2 can have either strategic complement or strategic substitute relationships under different demand systems. If the merger stimulates f 3 to reduce its prices, the overall impact of the merger on the price of the final product 2 is uncertain. If we also consider the existence of the outside option, the incentives for f 1 to raise prices of its intermediate products are restricted, which makes the total price change in the final product 2 even more unclear. Our first proposition illustrates the price changes in Figure 2.4 due to the merger and provides a simple condition for the existence of the Edgeworth-Salinger effect under MNL demand: the value of the outside option, H 0 , is sufficiently low. Note that this is consistent with observations from Luco and Marshall (2020), where they find that ”the Edgeworth-Salinger effect was larger in stores belonging to small and local chains.” In local areas, the outside option for consumers may be very limited compared to large cities. As a result, firm f 1 has less concern about driving consumers away to the outside option and has more incentive to increase the total price of the final product 2. 41 Figure 2.4: Firm f 1 is a Dominant Firm that Produces Essential Components for All Products. Proposition 4 In the market illustrated in Figure 4, the merger between f 1 and f 2 has the following effects: (a) Under both MNL and CES demands, the merger stimulates f 1 to increase itsi-markup and reduces the total price of the final product 1. The price change in the final product 2 is uncertain. (b) Under MNL demand, when the outside option, H 0 , is sufficiently small, the merger raises the total price of the final product 2. After the merger between f 1 and f 2 , the market structure, illustrated in Figure 2.5, is still a one-sided asymmetric network. This kind of network is common in the international airline industry, which is the focus of the empirical study in Tan and Zhang (2020b). In this case, suppose f 1 merges with f 3 . Based on MNL and CES demands, we continue to find a price reduction in the final product 2 due to the elimination of double marginalization, and we identify a clear Edgeworth-Salinger effect in the final product 1. This result is discussed more generally in the next proposition. Our previous discussions show that under one-sided asymmetric networks, a merger of com- plements can lead to both the efficiency effect and the Edgeworth-Salinger effect. One important question is about how such mergers can affect the overall consumer surplus. Proposition 4 seems to imply that the Edgeworth-Salinger effect may not be so significant, since many additional factors 42 Figure 2.5: The Market Structure after the Merger between f 1 and f 2 in Figure 2.4 may constrain the price increases. However, even though the Edgeworth-Salinger effect may be small for each individual final product, we may conjecture that when firm f 1 has more final prod- ucts as we expand the network in Figure 2.4, a merger of complements between f 1 and f 2 could lead to price increases in multiple final products and reduce the total consumer surplus. Also, the differences in the qualities of the final products and firms’ costs in producing the intermediate products may also make the overall impact of a merger of complements on consumer surplus un- certain. A small increase in a high-quality product’s price may lead to a large drop in consumer surplus. Moreover, adding additional separable components into the network, which may increase their prices due to the Edgeworth-Salinger effect, further complicates the problem. However, our following proposition shows a general outcome. In any arbitrary one-sided asymmetric network, a merger of complements between two firms always has the result that the efficiency effect dominates the Edgeworth-Salinger effect, and consumers are better off after the merger. Proposition 5 In any one-sided asymmetric network under CES and MNL demands, a merger between two firms, f k and f l , where G l G k , and G k ;G l S a , always leads to: (a) lower total prices of the final products in G k \ G l , (b) higher total prices of the final products withjM i j= 1, and i2 S a , (c) lower total price of any product i2 Nn S a , and 43 Figure 2.6: A Two-sided Asymmetric Network with Three Final Products. (d) higher consumer surplus. Proposition 5, which allows for arbitrary one-sided asymmetric networks, causes us to raise an interesting conjecture: Even though a merger between multiproduct firms may not always lead to price reductions, consumers are still generally better off. Our previous discussions rule out the possibility of having a consumer-surplus-declining merger of complements in an economy with symmetric or one-sided asymmetric networks under MNL and CES demands. Therefore, we focus on the two-sided asymmetric network in the next section and try to understand whether a merger of complements will affect prices and consumer surplus in a different way. 2.5.2.2 Two-Sided Asymmetric Network In real-world markets, we observe that f 2 or f 3 in Figure 2.4 may also provide their own products to compete with firm f 1 , which can lead to a network structure illustrated in Figure 2.6. For example, in the Changsha-Seattle market, CZ not only works with DL to provide Route 2 but also offers its own routes. In Luco and Marshall (2020), the major carbonated-beverage companies, like Coca-Cola or Dr. Pepper, may work with other bottlers to produce different products in the same 44 Figure 2.7: The Two-sided Asymmetric Network for the Merger Analysis in Scenario (2). markets. To reflect this more complicated market structure, we study the two-sided asymmetric network, illustrated in Figure 2.3c, where f 1 and f 2 offer final product 2 jointly, and each of the two firms also provides a competing final product in the same market. To concentrate on mergers of complements, which reduce one firm’s prices to its marginal costs, we study the following problem. 17 Suppose we have three firms, F =f f 1 ; f 2 ; f 3 g, and three final products, N =f1;2;3g, in the economy. Intermediate products from firms f 1 and f 2 form a two-sided asymmetric network, which is shown in Figure 2.3c. Also, f 3 may provide an intermediate product for one of the three final products. We consider three different scenarios: Scenario (1): Firm f 3 provides an essential component for final product 1, and there is a merger between f 3 and f 1 . Scenario (2): Firm f 3 provides an essential component for final product 2, and there is a merger between f 3 and f 2 . Scenario (3): Firm f 3 provides an essential component for final product 3, and there is a merger between f 3 and f 2 . 17 Under symmetric and one-sided asymmetric networks, a merger of complements always reduces one firm’s prices to its marginal costs. However, we can see that, from Figure 2.6, a merger of complements between f 1 and f 3 includes an additional change where the competition between final product 1 and final product 3 is eliminated. We do not study this kind of merger, which is introduced into the model as we expand our network structures, and focus instead on the mergers of complements that are consistent with our previous analysis (e.g., a merger between firms f 1 and f 2 ). 45 We illustrate Scenario (2) in Figure 2.7 and Scenario (1) and (3) are similar to Figure 2.6. Our first result about this merger analysis indicates that only in Scenario (2) will the merger reduce consumer surplus. Proposition 6 The mergers of complements in these three scenarios have the following effects: (a) Consumer surplus is always improved after the mergers in Scenario (1) and Scenario (3) under both MNL and CES demands. (b) When h 1 (P 1 )= h 2 (P 2 )= h 3 (P 3 ), C 1 = C 2 = C 3 , and H 0 is sufficiently small, consumer surplus is reduced after the merger in Scenario (2) with MNL demand. In Scenario (2), the final product 2 becomes more efficient to sell for both f 1 and f 2 after the merger. f 1 and f 2 have the same incentives to raise the prices of the final product 1 and final product 3 so that more consumers choose the more efficient final product 2. These incentives are also mutually enhancing; f 1 ’s price increase the in final product 1 relieves f 2 ’s concerns that its price increases may drive its consumers to choose the final product 1. The additional assumption that the outside option is sufficiently weak also encourages f 1 and f 2 to raise prices without worrying about driving consumers to the outside option. In contrast, mergers in Scenario (1) or Scenario (3) do not have the mutually enhancing incentives to raise prices. Similar to the one-sided asymmetric network in Figure 2.4, the merger in Scenario (1) or Scenario (3) makes the merged firm more competitive and may stimulate the other firm to reduce its prices. An interesting observation is that the network structure in this merger analysis is not signifi- cantly different from a one-sided asymmetric network. If final product 3 is jointly offered by f 1 and f 2 , which is a minor revision of the network structure, we can change this network to a one-sided asymmetric network, and eliminate the possibility of having a consumer-surplus-declining merger of complements. In this one-sided asymmetric network, if f 3 provides an essential intermediate product for the final product 2, which is illustrated in Figure 2.8, the merger between f 2 and f 3 cannot generate the mutually-enhancing incentives for f 1 and f 2 to raise prices. The merger stim- ulates f 2 to raise the price for its intermediate product in the final product 3, which can also hurt 46 Figure 2.8: The One-sided Asymmetric Network by Adding Firm f 1 to the Final Product 3 in the Network Structure in Figure 2.7. f 1 ; f 1 ’s attempt to control f 2 ’s response to the merger limits the overall price increases in different final products. 2.6 Policy Implications In many industries (e.g., airline, carbonated-beverage, patent), double marginalization is consid- ered a major problem that reduces the efficiencies of the markets (Brueckner (2001), Luco and Marshall (2020), Quint (2014b)). Antitrust agencies often believe that a vertical merger (or a merger of complements) is an efficient tool in solving the double marginalization problem and improving market efficiency. However, the newly issued Vertical Merger Guidelines raised sim- ilar concerns about the potential Edgeworth-Salinger effect when members of a merger might be multiproduct firms and indicated that ”these new Vertical Merger Guidelines reaffirm our commit- ment to challenge vertical mergers that are anticompetitive and would harm American consumers” (U.S. Department of Justice and Federal Trade Commission (2020)). Our theoretical analysis shows that even when we allow for multiproduct firms, vertical mergers continue to benefit con- sumers in symmetric or one-sided asymmetric networks. Therefore, we illustrate an important set of market structures where the government can still be more optimistic about the impact of vertical mergers on consumers. Also, we point out a major source of market inefficiency caused by a ver- tical merger: the existence of two-sided asymmetric networks. These show that antitrust agencies 47 should carefully evaluate the connections between different firms’ intermediate products in merger reviews. The government should pay more attention while regulating vertical mergers in the mod- ern high-tech companies (e.g., computer manufacturing industry), where the final products often involve many different firms’ intermediate products and the existence of two-sided asymmetric networks is very likely. 2.7 Conclusion In this paper, we generalize the aggregative games approach developed in Nocke and Schutz (2018) to analyze price competition among firms providing both complements and substitutes. We show that Marshall’s second law of demand is still sufficient in ensuring the existence of a pricing equi- librium in a more general setting, and we illustrate that the classical demands, MNL and CES, continue to have a unique pricing equilibrium under different conditions. Based on these demands, we concentrate on analyzing mergers of complements. When the network is symmetric, we show that the results are generally consistent with our traditional understanding of mergers of comple- ments, which is also illustrated in Quint (2014a). When the network is asymmetric, the elimination of double marginalization and the Edgeworth-Salinger effect lead to new findings. We distinguish two types of asymmetric networks: one-sided and two-sided. Our results show that mergers of complements are still generally beneficial to consumers under one-sided asymmetric networks, and antitrust agencies should review mergers more carefully when the network is two-sided asym- metric, which can make mergers of complements harmful to consumer surplus. 48 Chapter 3 An Analysis of International Airline Alliances and Joint Ventures 3.1 Introduction Airline carriers in the U.S. got the freedom to design their own network structure and prices af- ter the Airline Deregulation Act of 1978. After that, we saw many changes in the industry: the emergence of hub-spoke networks, failure of entrants, complicated pricing system, and interlining. More freedom to the airline market also stimulated international cooperation. The international airline alliance, which mainly started in the late 1990s, developed very fast in the past twenty years. In 2017, the three major alliances (Star, SkyTeam, and Oneworld) accounted for 57 per- cent of total scheduled traffic in the world (International Air Transport Association (2018)). In the meantime, members of the alliances also received increasingly more freedom in coordinating different operations. Joint ventures which are currently implemented within SkyTeam and Star Alliance are ”operationally very close to mergers” (Bilotkach (2019); Bilotkach and H¨ uschelrath (2011)). In this paper, we analyze the growing cooperation among airline carriers, and we dis- cuss how policymakers can improve their decisions in granting antitrust immunities to form joint ventures. 49 Figure 3.1: An Airline Network with Six Cities. Straight Lines Represent Carrier A (Air China) and Dotted Lines Represent Carrier B (United Airlines) 3.1.1 International Airline Network International airline carriers often form hub-spoke networks because of the economies of density (Brueckner (2001); Hendricks et al. (1995)). Hub-spoke networks can easily pool consumers from different origins to reduce marginal costs and save fixed investments on airplanes and ground equipment. Therefore, our analysis relies on the network structure illustrated in Figure 3.1, where two airline carriers operate on two hub-spoke components with an overlapping spoke to connect six cities. Between any two cities, consumers need flight services to travel. Also, each edge of the network represents the direct/nonstop flights offered by an airline carrier to travel between the two cities. This network structure is similar to the international airline network operated by Air China (denoted by CA) and United Airlines (denoted by UA). We can consider City 1 and City 2 to be CA and UA’s international gateway cities: Beijing and San Francisco. Based on Beijing, CA offers direct flights to and from local cities 3 and 4 (e.g., Dalian and Wuhan) and a global destination, San Francisco. Similarly, UA operates direct flights from San Francisco to local cities 5 and 6 (e.g., Seattle and Las Vegas) and a global destination, Beijing. Due to government regulations, foreign airline carriers often cannot offer domestic flight services. Therefore, in our network, CA and UA do not have any overlapping services in their local spoke markets. 1 1 Carriers may face competition from other local carriers in their spoke markets. Their effects are not the focus of our paper. 50 3.1.2 Alliance Membership and Joint Venture Alliance members are allowed to cooperate in many activities. For example, CA and UA, which are members of the Star Alliance, focus on four major issues in their cooperation: expanding con- necting flights, creating a more comfortable experience to travel at the gateway cities, enhancing partnership in frequent-flier programs, and coordinating marketing. As a result, alliance members often provide more and more similar quality of service. However, basic alliance membership does not allow members to coordinate in the price setting due to government regulations (Bilotkach 2018). For any interlining market where both carriers are involved in providing the services (e.g., market between City 3 and City 5), each carrier will set prices for its segments of the interlining trips to maximize its own profits with basic alliance membership. The sum of the resulting fares determines the total prices of the interlining trips. To improve cooperation between alliance members, many members got approval from the gov- ernment and successfully formed joint ventures, which allow them to coordinate their prices. In 2016, 81 percent of the travelers who flew between America and Europe were on airplanes op- erated by different joint venture member carriers (Bilotkach (2019)). With a joint venture, each carrier can set prices for all city-pair markets to maximize joint profits, and in our network, the two carriers are merged to become a monopolist after forming a joint venture. 3.1.3 Current Government Regulation In our network structure, flight services are complementary products for many city-pair markets where direct/nonstop flights are not available. As we discussed before, with simple alliance mem- bership, each carrier will noncooperatively set prices for its segments. Standard economic theory tells us that government should encourage airline carriers to form a joint venture because it can eliminate the problem of double marginalization (International Air Transport Association (2012)). However, we also have direct competition in the inter-hub market where consumers can either go with CA or UA, and we can expect that price will increase after eliminating the competition. Therefore, to determine whether carriers can be allowed to form joint ventures, current government 51 regulation shows a consistent pattern: approving antitrust immunities unless carriers’ networks overlap too much. In our simple network where we only have one overlapping inter-hub market, it is very likely the two carriers can gain antitrust immunity from the government. Since international airline carriers often have similar network structures, we are not surprised that most of the flights between American and Europe are operated by joint ventures. In this paper, we build a network flow model to analyze price changes when the two alliance members form a joint venture. We show that inter-hub competition can affect the equilibrium prices of local spoke markets due to carriers’ different pricing strategies and consumers’ fare arbitrage. As a result, current government interventions may overestimate the benefits of eliminating double margins, and a joint venture may lead to higher prices for the interlining markets. The rest of the paper proceeds as follows. In the second section, we will formally introduce the environment of our model and key assumptions. In the third section, we will discuss consumers’ fare arbitrage on the network and characterize equilibrium prices under the two different cases: alliance membership and joint venture. Then we can show the price changes which depend on the properties of the demand function. We will also connect our results with different network structures and market sizes. In the fourth section, we will relate our results to the existing literature. Section 5 concludes. All proofs are in Appendix B. 3.2 The Model 3.2.1 The Environment In this economy, there are six cities, and each city is represented by a node in the airline network. We use N =f1;2;:::;6g to denote the set of cities. If two cities are connected by an edge, then some airline carrier provides direct (or nonstop) flight services for customers to travel between these two cities. There are two airline carriers in this economy, and we denote them A and B. We assume that the network structure is exogenously determined with two hub-spoke components and an overlapping spoke. Figure 3.1 illustrates this network structure. We can see that airline carrier 52 A provides direct flights to connect hub-city 1 to cities 2,3,4 while carrier B provides direct flights to connect hub-city 2 to cities 1,5,6. 3.2.2 Demand We assume that there are consumers who want to travel between any two cities in the network. Also, we assume that consumers who need a round trip to travel from city g to h do not want to go anywhere else. Consumers only care about reaching their destinations at the lowest cost and they only choose the shortest route (i.e., the smallest amount of edges) between any two cities. Formally, we assume that the flow from one city to another is determined by a given demand function, D(p), which is the same for every city pair. This function reflects consumers’ willingness to pay, and if p is the price of the cheapest return air ticket from city g to h, then the amount of g h consumers is D(p). We assume that this demand function satisfies the standard assumptions: strictly positive, downward sloping, and differentiable. In this economy, even though the demand function is the same for every city-pair market, airline carriers can charge different prices so that the actual demand may be different in each city-pair market. 3.2.3 Airline Alliance Agreements We focus on two major airline cooperation agreements: alliance membership and joint venture. With simple alliance membership, each carrier will noncooperatively set prices for its segments, and the sum of both carriers’ prices determines the price of each interlining trip where both carriers are involved in providing the service. If alliance members form a joint venture, carriers will set city-pair market prices to maximize joint profits. 53 3.2.4 Important Assumptions Based on Hendricks et al. (1997), we impose several assumptions on the demand function and costs. We assume that carrier A and B have the same per traveler cost of a round trip on a direct flight (i.e., c A = c B = c). Assumption 1 e(p)=pD 0 (p)=D(p) is nondecreasing in p. If e(p) equals a constant a, then a > 1. Ife(p) is an increasing function, then there exists a ˆ p> 0 such that if p> ˆ p,e(p)> 1. Assumption 2 D 0 (p)=D(p) is strictly monotonic in p. Assumption 1 can be similarly considered as Marshall’s second law of demand (Nocke and Schutz (2018)). To understand the implications of these two assumptions, we consider a city-pair market composed of two direct flights. Each carrier operates one flight independently, and carrier i will choose price p, given the other carrier charges price s for its segment, to maximize its profits: (p c)D(p+ s): (3.1) We use f i (s) to denote the solution of the optimal p to maximize (3.1). When s= 0, f i (0) is equal to the monopoly price of a city pair market with marginal cost c, which is also denoted by p M (c). We also usep(c) to denote the corresponding monopoly profit with marginal cost c. With Assumption 1 and Assumption 2, we can derive some properties of the best reply function,f i (s). Lemma 1 (i) f i (s) is single valued. (ii) If D 0 (p)=D(p) is strictly decreasing (increasing), then f i (s) is strictly decreasing (increasing) in s for i= A;B, which means that the flights of the two carriers are strategic substitutes (complements). (Hendricks et al. (1997)) The key thing about this lemma is that we can focus our analysis on two cases. In one case, the best reply functions are strategic substitutes (i.e., f i (s) is decreasing in s). It means that the best response to an increased price by the other carrier is price decreasing. In the other case, the best replies are increasing so that the pricing strategies are strategic complements. These two 54 cases are critical in market competition as illustrated in Bulow et al. (1985). A simple analysis of Assumption 2 will also show that whether the two carriers’ pricing strategies in interlining are strategic substitutes or strategic complements depends on the concavity of the demand function. Specifically, prices are strategic substitutes if D 00 (p) is less than D 0 (p) 2 =D(p). It means that if the demand function is concave then interlining prices are strategic substitutes. If the prices are strategic complements, then the demand function must be strictly convex. Rearranging the first order condition of the profit in (3.1), we can get that: e(p+ s) p c p+ s = 1 (3.2) Based on this equation, we can easily see that strategic substitute strategies require that e(p) must be a strictly increasing function of p while for strategic complement strategies e(p) can be a constant. As a result, iso-elastic demand function, D(p)= p b where b> 1, leads to strategic complement pricing strategies. We can also verify that standard linear demand function, D(p)= a p where a> c, leads to strategic substitute pricing strategies. 3.3 Equilibrium Prices In this section, we will introduce consumers’ fare arbitrage into our model, characterize equilib- rium prices and then analyze the price changes after forming a joint venture. Since all alliance members can monopolize certain parts of their segment, we always start by discussing the joint venture and then extend to the alliance membership scenario. 3.3.1 Fare Arbitrage One key feature of the airline market is the complicated pricing system. Here we allow airline carriers to charge consumers different prices even though they may travel on the same flight. How- ever, there are several pricing constraints to guarantee the effectiveness of discriminatory pricing and make consumers willing to reveal their private information about their destinations. 55 In the joint venture, the price for the market from city g to h, denoted by p J gh , where g;h2 N, which is composed of two or three direct flights, must satisfy the following condition: p J g j + p J jh p J gh ; i f h6= g6= j; h;g; j2 N: (3.3) It rules out the possibility that g h customers buy a g j ticket and a j h ticket instead of buying a g h ticket. Clearly, if this constraint is violated, the pricing on the g h market is not effective. Note that if the g j market in the above condition contains two direct flights, then we can get: p J gk + p J k j + p J jh p J g j + p J jh p J gk ; i f h6= g6= j6= k; h;g; j;k2 N: Now if airline carriers are only alliance members without joint venture, and we allow carriers to charge prices contingent on a traveler’s purchase of a ticket from the other carrier, we need to impose additional constraints on prices. For example, if we look at the pricing for the 3 1 edge, carrier A can potentially charge different prices for 3 1 travelers, 3 2 travelers, 3 5 travelers, 3 6 travelers, and 3 4 travelers who all need to use the 3 1 flight for their trips. For travelers buying interlining flights in the 3 2 market, we use s A 31 and s B 12 to denote the prices of buying tickets from the two carriers (we use p i jh to denote the price for traveling in the city-pair market j h with carrier i2fA;Bg only.). Comparing with the travelers who just need to go from 3 to 1, interlining travelers in the 3 2 market need to buy two tickets and they will only reveal that information if they can get lower prices. In other words, interlining travelers in the 3 2 market can pretend to be 3 1 travelers but 3 1 travelers do not have the information to distinguish themselves. Therefore, interlining prices must satisfy this condition: p A 31 s A 31 and p B 12 s B 12 : For the 3 5 city-pair market, consumers have to buy tickets from both carriers. We assume that 35 consumers can hide their private information and airline carriers cannot distinguish them 56 from all the other travelers in the intermediate city-pair markets (i.e., 3 1, 3 2, 1 5, 2 5, and 1 2 markets which comprise 3 5 market) unless 3 5 travelers receive lower prices when they purchase any one of these intermediate tickets. We use w A 32 and v A 31 to denote prices charged on the 35 consumers depending on whether carrier A chooses to monopolize the 32 segment or share with carrier B for the 3 2 segment. Therefore, we must have the following pricing constraints: 8 > > < > > : p A 31 s A 31 v A 31 ; p B 12 s B 12 v B 12 if carrier A shares with carrier B for the 3-2 segment p A 31 + minfp A 12 ; p B 12 g p A 32 w A 32 if carrier A monopolize the 3-2 segment: If carrier A decides to share with B in the 3 2 segment, it only chooses a price of the 3 1 segment and lets 3 2 and 3 5 consumers use the other carrier for the 1 2 segment. The above inequality ensures that 3 5 consumers can only be charged a lower (or the same) price for using the 3 1 flight. If carrier A wants to monopolize the 3 2 segment, then the monopoly price of the 3 2 segment must be lower than buying two components (i.e., 3 1 and 1 2) separately, and 3 5 travelers should be able to get cheaper price than the 3 2 consumers. In summary, we can generalize previous examples and impose these pricing conditions for alliance members (i2fA;Bg): 8 > > < > > : p i gh s i gh v i gh if g;h2 N;g6= h;and g-h market contains one edge p i gh w i gh if g;h2 N;g6= h;and g-h market contains two edges: (3.4) As we discussed before, we use v i and w i to denote the prices for customers who use interlining services to travel between two rim cities (e.g., 3 5 and 36). s i is the price for the interlining trips between one rim city and one hub city (i.e., j 2 and g 1, where j2f3;4g and g2f5;6g). For p i gh where g h market contains two edges, constraints from (3.3) also need to be satisfied so that p i gh is effective. Also, we add superscripts to identify city-pair markets that use the same flight but can be distinguished from each other. s i b 12 where b2f3;4;5;6g are used to distinguish interlining prices charged on 3 2, 4 2, 1 5 and 1 6 travelers. Similarly, v A g j1 , w A g j2 (where 57 j2f3;4g, g2f5;6g) and v B q g2 , w B q g1 (where g2f5;6g, q2f3;4g) are used to denote different prices for interlining tickets between rim cities. 2 3.3.2 Joint Venture Pricing If the two alliance members A and B form a joint venture, the joint venture chooses prices for all city-pair markets to maximize total profit, and the total profit is: P J = 2 4 å j=2 (p J j1 c)D(p J j1 )+ 2 6 å g=5 (p J g2 c)D(p J g2 ) + 2 4 å j=3 (p J j2 2c)D(p J j2 )+ 2 6 å g=5 (p J g1 2c)D(p J g1 )+ 2(p J 34 2c)D(p J 34 )+ 2(p J 56 2c)D(p J 56 ) + 2 4 å j=3 6 å g=5 (p J g j 3c)D(p J g j ): (3.5) Therefore, the joint venture will choose prices p J gh (g6= h;g;h2 N) to maximize (3.5) subject to constraints (3.3). With assumption 1 we can characterize joint venture pricing in the following lemma: Lemma 2 Suppose the two airline carriers form a joint venture. The equilibrium prices are that p J gh = p M (kc) where k is the number of edges in the gh market, and p M (2c) 2p M (c), p M (3c) 3p M (c), p M (3c) p M (2c)+ p M (c). Therefore, the joint venture’s total profits are: P J = 10p(c)+ 12p(2c)+ 8p(3c): (3.6) This lemma shows that under the joint venture, airline carriers choose monopoly prices to maximize total profits, and fare arbitrage constraints are not binding. This result is not affected by 2 Consumers in the intermediate markets can potentially choose to buy longer distance tickets and give up part of their trip if the price is lower. This practice is often not possible in the international airline industry since consumers will not be able to take the return flight. Airline carriers often require consumers to board their return flights at the city designated on the ticket. Also, we can easily show that in equilibrium prices are higher for longer distance flights. 58 assumption 2, which means that both strategic complement and strategic substitute strategies have the same result. 3.3.3 Alliance Membership Pricing When the two carriers are alliance members without a joint venture, each of them chooses prices for its city-pair markets to maximize its own profits. For the inter-hub market 1 2, the two carriers engage in Bertrand type of competition which means that the carrier offers a lower price wins the entire market. Each airline carrier is also able to monopolize some part of the economy. For example, customers in the 3 4 market can only travel with carrier A. For city-pair markets j2 ( j2f3;4g) or g1 (g2f5;6g) which involve one rim city and one oversea hub city, carriers can choose between monopolize the market and share them with the other carrier depending on whether p A j2 (p B g1 ) is lower or higher than s A j1 + s B 12 (s B g2 + s A 12 ). Similarly, for markets between cities g and j, each carrier can also choose to monopolize or share the j 2 and g 1 components. We first define a functiond(x;y): d(x;y)= 8 > > > > > > < > > > > > > : 1 if x< y 1 2 if x= y 0 if x> y (3.7) 59 We assume that when two carrier’s prices are the same, then the market is equally shared between the two carriers. The airline carrier A’s total profit is: P A = 2d(p A 12 ; p B 12 )(p A 12 c)D(p A 12 )+ 2 4 å j=3 d(s A j1 + s B j 12 ; p A j2 )(s A j1 c)D(s A j1 + s B j 12 ) + 2 4 å j=3 (1d(s A j1 + s B j 12 ; p A j2 ))(p A j2 2c)D(p A j2 )+ 2 6 å g=5 d(s B 2g + s A g 12 ; p B 1g )(s A g 12 c)D(s A g 12 + s B 2g ) + 2 6 å g=5 4 å j=3 d(v A g j1 + w B j 1g ;w A g j2 + v B j 2g )(v A g j1 c)D(v A g j1 + w B j 1g ) + 2 6 å g=5 4 å j=3 (1d(v A g j1 + w B j 1g ;w A g j2 + v B j 2g ))(w A g j2 2c)D(w A g j2 + v B j 2g ) + 2 4 å j=3 (p A j1 c)D(p A j1 )+ 2(p A 34 2c)D(p A 34 ) (3.8) Carrier A chooses its prices to maximize equation (3.8) subject to constraints (3.3) and (3.4) depending on whether carrier A chooses to monopolize the market or not. Here carrier B’s profit maximization problem is symmetric to carrier A. We can analyze carriers’ profit maximization problem and show that both airline carriers com- pete intensively in the inter-hub market. They focus on choosing prices of its local spokes contin- gent on different destinations of the consumers. Lemma 3 Suppose two airline carriers only form an alliance without joint venture. Then p i 12 = s i 12 = c, where i= A;B, and without loss of generality, we can assume that carrier A monopolizes all the j2 segments while carrier B shares in all the 1g segments where j2f3;4g and g2f5;6g, then each carrier’s optimal prices are as follow: (p A j1 ; p A j2 ;w A 5 j2 ;w A 6 j2 )=(p;s+ c;v 1 + c;v 2 + c) (p B 2g ;s B 2g ;v B 3 2g ;v B 4 2g )=(p;s;v 1 ;v 2 ); 60 where (p;s;v 1 ;v 2 )= argmax p;s;v 1 ;v 2 f(pc)D(p)+(sc)D(s+c)+(v 1 c)D(v 1 +c+v 0 1 )+(v 2 c)D(v 2 +c+v 0 2 )g subject to p s v 1 p s v 2 : (3.9) Here we use v 0 1 and v 0 2 to denote the price charged by the other carrier for its component in the interlining trips between rim cities. Intensive competition makes the inter-hub market unprofitable. This result shows that each carrier will instead try to charge different prices on its local spoke edges contingent on consumers’ destinations in order to maximize profits since the local spoke edges are essential complements for many city-pair markets. With alliance membership, both carriers in fact divest their inter-hub services into independent divisions which compete against each other. We can also easily show that each carrier does not have the incentive to completely quit the inter-hub market. If carrier A completely quits the inter-hub market, carrier B will charge higher prices for the inter-hub edge and the total profits of carrier A will be reduced. This analysis is also related to the findings of Tan and Yuan (2003) which shows that a firm that produces several complements can obtain higher profits through divesting complementary products when these products have substitutes produced by another company. 3.3.4 Price Comparison We now compare the equilibrium prices between alliance membership and joint venture. We will show that the results are different depending on whether the pricing strategies are strategic sub- stitutes or strategic complements. In the former case, all constraints in Lemma 3 are not binding because the best reply functions are downward sloping. In the latter case, all constraints (3.9) in Lemma 3 are binding because of the upward sloping best reply functions. As a result, competition 61 in the inter-hub market generates externalities on other spoke markets. Carrier A (B) has to charge same price for j 1 (2 g), j 2 (1 g) and j g ( j g) travelers who use the same j 1 (2 g) flight service. 3.3.4.1 Strategic Substitute Strategies Our first proposition indicates that if we have strategic substitute strategies, equilibrium prices with only alliance membership are higher than joint venture for all interlining markets between any rim cities. We denote the equilibrium prices under alliance membership for interlining market between rim cities j and g to be p jg , and clearly we have p jg = w A g j2 + v B j 2g . Proposition 1 With strategic substitute pricing strategies, p jg > p M (3c) for any j2f3;4g and g2f5;6g. For the inter-hub market, p 12 < p M (c). All other markets have same prices in alliance membership and joint venture. Note that this result is not affected by the number of spokes in the network. Following this result, we can see that a joint venture can reduce the prices of all interlining markets between rim cities and increase the price of the inter-hub market. Therefore, customers in the interlining markets will benefit from the joint venture while customers in the inter-hub market will be worse off in the joint venture. This finding is similar to the results of Brueckner (2001) and we can draw the same policy recommendation that government should allow airline carriers to form joint ventures because we normally have much more interlining markets between rim cities than inter- hub markets. 3 As long as we have a sufficiently large amount of spokes, then the benefits of joint ventures dominate their costs. The reason for this result is also consistent with our traditional understanding of double marginal- ization. For any interlining market between two rim cities, both carriers’ flight services are perfect complements under alliance membership. The unbinding constraints (3.9) in Lemma 3 due to strategic substitute strategies mean that each carrier can freely charge prices for its segments of the 3 If the number of spokes in each hub-spoke segment is m, then the interlining markets between rim cities will be (m 1) 2 . 62 interlining trips without considering its negative externalities on other carrier’s profits. Therefore, both carriers overcharge prices for their segments under alliance membership and as a result, the joint venture can eliminate the double marginalization problem and reduce prices because it is a merger of the producers of complementary goods. 3.3.4.2 Strategic Complement Strategies Now we move to the case of strategic complement strategies. For any interlining market between rim cities, each carrier still tries to charge optimal prices to maximize its profits without consider- ing potential negative externalities on the other carrier. However, due to travelers’ fare arbitrage conditions, each carrier needs to consider both the pricing of local spoke markets and its impact on other related markets. Therefore, unlike in the previous case, we are uncertain about the price changes if we allow alliance members to form the joint venture. In this subsection, we will derive a sufficient condition under which we can determine whether the joint venture can lead to lower or higher prices for interlining markets between rim cities. Proposition 2 When e(p) is constant a (a > 2), then there exists an ˆ a such that when a > ˆ a, p jg ˆ a. For the inter-hub market, p 12 p M (c). All other markets have same prices in alliance membership and joint venture. In this proposition, we attempt to understand the connection between elasticity of demand and the price changes after forming the joint venture. Due to strategic complement strategies, each carrier’s pricing of any spoke market j1 or 2g is no longer independent of other markets. They may want to increase the price, p A j1 (p B 2g ), so that they can charge higher prices on j2 (1g) and j g ( j g) travelers. The following graphs illustrate the carrier’s optimal choice of spoke market price. Comparing Figure 3.2 and Figure 3.3, we can see that when elasticity is small, an increase of price in the spoke market will lead to much higher profits from the other two markets (e.g., 4 Assuming a > 2 can guarantee that interlining market between rim cities has a positive maximum point in the unconstrained problem. Gayle (2013) shows that own-price elasticity of demand is 4.72 in the U.S. 63 j 2 and j g) than the profit reduction in the spoke market. As a result, the equilibrium price under simple alliance membership will be much further away from the optimal price of the spoke market when fare arbitrage constraints are not binding. Also, the equilibrium price is higher than p= ac+ c 2 a1 which means that the interlining market between rim cities will be: p jg > p M (3c)= 3ac a1 . As elasticity gets higher, we can see from Figure 3.3 that the spoke market, which has the highest profit level, becomes much more sensitive to price changes, and the equilibrium price moves closer to the spoke market’s optimal price without fare arbitrage constraints and this price makes p jg less than p M (3c). 3.3.4.3 Network Structure and Market Size So far, we focused on the exogenously given network structure as defined in Figure 3.1. In the international airline market, each hub city may connect to much more local cities. Under strategic substitute pricing strategies, the previous results still hold. However, if more spokes are added to each carrier, then under strategic complement strategies, alliance members will have higher incentives to raise their spoke market prices. It can lead to higher prices and profits in the interlining trips between rim cities, and the equilibrium prices with the alliance membership may become higher than the prices under the joint venture. Another factor we did not consider in the previous section is that city-pair markets between a rim city and a hub city often have much bigger market sizes than the city-pair markets between rim cities. There may be a large number of city-pair markets between rim cities, but they may not have a significant impact on the whole economy due to their small sizes. To see how these two factors will affect our previous results, we first assume that the amount of spokes that each carrier has is m, and m 1. Let n be the total amount of cities in the network and we must have m= n 2 . Figure 3.4 and Figure 3.5 illustrate the cases when we have m= 2 and m= 4. We also revise our previous assumption on the demand between any two cities. If j and g are two rim cities, then D jg (p)= D(p), while if j or g is a hub city, then D jg (p)= lD(p) where 64 Figure 3.2: Optimal Prices when a= 2:1 and c= 4 Figure 3.3: Optimal Prices when a= 3 and c= 4 65 Figure 3.4: Network Structure when m= 2 Figure 3.5: Network Structure when m= 4 l is a constant greater than 1. We can easily extend our previous model to incorporate these two factors and derive the following result. Proposition 3 When e(p) is constant a (a > 2), there exists d > 0, such that for any rim cities j and g which belong to carrier A and B respectively, p jg < p M (3c) if and only if m1 l <d. If m1 l <d, we also have p j2 = p 1g < p M (2c). Other markets have same result in Proposition 2. This proposition shows that even though the airline network is large with lots of spokes, we cannot easily conclude that forming the joint venture between the two carriers can reduce the prices of interlining tickets between two rim cities. Markets between a rim city and a hub city may be so large that airline carriers are unwilling to increase spoke markets prices to increase the profits from interlining trips between rim cities. 66 3.4 Related Literature 3.4.1 Hub Spoke Network The theoretical foundation of hub-spoke network structure comes from the seminal papers by Ken Hendricks, Michele Piccione, and Guofu Tan. Hendricks et al. (1995) first study the network for- mation in a monopoly environment and show that monopolist will choose to build a hub-spoke network when there exists economies of density. Hendricks et al. (1999) focus on the competition between two large airline carriers, which are allowed to form different airline networks. They show the conditions under which airline carriers choose to form hub-spoke networks. Our network with two hub-spoke segments may seem to contradict their conclusion that when carriers compete in- tensively for customers, there is no equilibrium in which both carriers choose hub-spoke networks. However, our paper is very different from the assumptions in Hendricks et al. (1999) where air- line carriers can connect any two cities they want. As we discussed before, the international airline market has the feature that carriers are not allowed to enter a different country’s market. Therefore, our network structure is more like Hendricks et al. (1995) where each carrier is a monopoly of its local cities, and they build one connection with international markets to increase market coverage. Therefore, our exogenous network structure does not contradict the existing literature, and it is also a common assumption in several papers (Bilotkach (2019)). 3.4.2 Impacts of Airline Alliance Our finding is mostly related to the paper by Brueckner (2001) which considers airline competition and alliance in the same network structure shown in Figure 1.1. It analyzes the price difference before and after forming a codesharing alliance between the two competing airline carriers. The pre-alliance agreement is similar to our alliance membership agreement, and the modeling of the alliance is also equivalent to our joint venture. It concludes that airline alliance can reduce the prices of interlining flights between two rim cities due to the elimination of double marginalization, while in the inter-hub market where carriers provide competing services, market price increases 67 because alliance reduces competition. Therefore, these results lead to the conclusion that airline al- liances are often beneficial to consumers as a whole because we only have a few inter-hub markets between countries, and the interlining markets are much more frequent. However, these results underestimate the impact of the inter-hub competition due to some of the assumptions in Brueckner (2001). This paper assumes that the inter-hub market has a Cournot type of competition and it imposes the assumption that each carrier is a monopolist in its own hub- spoke segment so that consumers travel on its home airline as much as possible (this simplifying assumption is also used in Brueckner and Proost (2010) and Bilotkach (2005)). 5 As a result, carriers cannot charge a lower inter-hub market price to attract customers from the other carrier’s rim cities. Its inter-hub market is separated from the rest of the markets in the model. Therefore, each carrier can easily monopolize its component, and noncooperative pricing leads to the standard problem of double marginalization. In contrast, our model shows that once carriers compete more intensively in the inter-hub market, monopoly (or joint venture) may not always lead to lower prices in the interlining markets. Many theoretical papers about this problem heavily rely on the assumption of linear demand and they often notice that fare arbitrage constraints are not binding. As a result, different mar- kets are independent of each other (Brueckner (2001); Brueckner and Proost (2010); Lin (2008)). Our model provides a more general result on this issue and shows that the theoretical airline mar- ket model needs to focus on the two critical cases: strategic substitute and strategic complement strategies. Lastly, we notice that many papers also show that airline cooperation may not always benefit consumers. For example, Chen and Gayle (2007) show that when one carrier offers competing services for a city pair market (one direct flight and one indirect flight), then more cooperation between carriers (codesharing with price coordination) can lead to higher prices. Lin (2008) shows that airline alliance can be used as a credible threat to deter entrants, which do not have significant cost advantages. Many empirical papers find that airline cooperation leads to significant price 5 It may be a more reasonable assumption for the modeling of pre-alliance competition. 68 reductions, but recent research also shows that the price reduction effect is less and less significant (Brueckner and Whalen (2000); Bilotkach (2019)). Combining with our results, we believe the government needs to be much more careful in granting antitrust immunities to airline carriers, and more empirical research about carriers’ competition strategies will be very useful. 3.5 Conclusion In this paper, we build a network flow model to analyze the international airline alliance and the joint venture. We rely on a critical distinction: strategic complement strategies and strategic substi- tute strategies. We show that intensive competition in the inter-hub market can generate externali- ties to local markets due to consumers’ fare arbitrage and carriers’ strategic complement strategies. As a result, joint ventures may not always lead to lower interlining prices comparing with simple alliance membership. The traditional viewpoint that inter-hub markets are separated from the rest of the markets is mainly a result of strategic substitute strategies. 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Assuming also that h satisfies Assumption 1, and we define g(p) = h 0 2 (p)=h 00 (p), r(p) = h(p)=g(p), andi(p)= ph 00 (p)=(h 0 (p)). Then: (b) There exists a unique scalar p such that for every p > 0, i(p) > 1 if and only if p > p. Moreover,i 0 (p) 0 for all p> p. (c) ¯ m lim p!¥ i(p)> 1. (d) For every p> p,g 0 (p)< 0. (e) lim p!¥ g(p)= 0. (f) If lim p!¥ h(p)= 0 and ¯ m <¥, then lim p!¥ r(p)= ¯ m ¯ m1 . Proof: See Supplemental Material of Nocke and Schutz (2018). Next, from Lemma 5 to Lemma 10, we provide a detailed proof that the generalized first-order conditions are still necessary and sufficient for global optimality. By redefining the i-markup for each firm, we can follow the similar methods developed in Nocke and Schutz (2018) to show these results. The newi-markup is also naturally connected to the aggregate markup for each final product defined in Lemma 11. 74 We fix a firm f k 2 F and a price vector for all firm f k ’s rivals(s l j ) l2Fnfkg; j2G l . Also, for some i2 G k ,å l2M i nfkg s l i 6=¥. 1 We evaluate the following maximization problem: max s k 2[0;¥] jG k j P k (s k ;s k 0 ;H(s)); (A.1) where we have P k (s k ;s k 0 ;H(s))= å i2G k (s k i c k i ) h 0 i (s k i +å l2M i nfkg s l i ) H(s) ; = å i2G k (s k i c k i ) h 0 i (s k i +å l2M i nfkg s l i ) (å j2G k h j (s k j +å l2M j nfkg s l j )+ H 0 0 , and H 0 0 å l2NnG k h l (P l )+ H 0 . This maximization problem has the following property: Lemma 5 Maximization problem (A.1) has a solution. Moreover, if (s k i ) i2G k solves (A.1), then s k i c k i for any i2 G k , and s k i <¥ for some i2 G k where P i 6=¥. We omit the proof for this lemma, which is very similar to Lemma C in Nocke and Schutz (2018). The idea of the proof is that by compactifying the space of the prices we can conclude that the continuous function P k (s k ;s k 0 ;H(s)) on the compact set attains its supremum and infimum (we focus on the case with H 0 > 0 so that the continuity at infinity is maintained.). Also, changing any price s k i where i2 G k from s k i < c k i to s k i c k i can stop making losses in product i and improve demand for the final product j2 G k where s k j c k j , since i and j are substitutes. We use(s k j ;(s k i ) i2G k nf jg ) to denote firm f k ’s price vector with intermediate product price s k j and other intermediate prices(s k i ) i2G k nf jg . The generalized first-order conditions of the maximization problem (A.1) is the following: 1 The case where for any i2 G k ,å l2M i nfkg s l i =¥ leads to the result thatP k is always zero for any s k > 0. Similar to our discussions later, we assume that f k will choose s k i = c k i for any i, whereå l2M i nfkg s l i =¥. (The results are the same if firm f k chooses some other finite s k i .) We impose this tie-breaking rule to exclude the ”no-trade”-type equilibria where all intermediate products of the final product provided by multiple firms have infinite prices. 75 Definition 1 We say that the generalized first-order conditions of maximization problem (A.1) hold at price vector( ˜ s k i ) i2G k2(0;¥] jG k j ) if, for every j2 G k , (a) ¶P k ¶s k j (( ˜ s k i ) i2G k;s k 0 ;H(s))= 0 whenever ˜ s k j <¥, and (b)P k (( ˜ s k i ) i2G k;s k 0 ;H(s))P k ((s k j ;( ˜ s k i ) i2G k nf jg );s k 0 ;H(s)) for every s k j 2R ++ whenever ˜ s k j = ¥. The generalized first-order conditions are clearly necessary for optimality. Lemma 6 If (s k i ) i2G k2(0;¥] jG k j solves profit maximization problem (A.1), then the generalized first-order conditions are satisfied at price vector(s k i ) i2G k. We attempt to show that generalized first-order condtions are also sufficient for global opti- mality. To see this, we first illustrate that if the generalized first-order conditions are satisfied at a price vector, then this price vector satisfies the commoni-markup property discussed in Section 2.4. We look at function v i : s k i ! s k i c k i å l2M i s l i i i (å l2M i s l i )= s k i c k i P i i(P i ) for i2 G k whereå l2M i nfkg s l i 6=¥. Let s k mc i denote the unique solution of the equation v i (s k i )= 1, which corresponds to the case of monopolistic competition. Compared to Nocke and Schutz (2018), the function v i for i2 G k is different because it includes additional terms for prices of firm f k ’s complementary products in the denominator. However, several key properties of the v i functions are maintained with Assumption 1, and we can prove the following lemma: Lemma 7 For any i2 G k withå l2M i nfkg s l i 6=¥ and m k 2(1; ¯ m i ), the equation v i (s k i )=m k has a unique solution in (0;¥), denoted by r i (m k ). The function r i () is strictly increasing and C 1 on (1; ¯ m i ), which satisfies lim m k !1 r i (m k )= s k mc i , and lim m k ! ¯ m i r i (m k )=¥, and r 0 i (m k )= g i (r i (m k )+å l2M i nfkg s l i ) (m k 1)h 0 i (r i (m k )+å l2M i nfkg s l i )m k g 0 i (r i (m k )+å l2M i nfkg s l i ) > 0: (A.2) Proof: Notice that since we fix all firm f k ’s rivals’ prices, the new i-markup defined in this paper simply includes a few additional constants and the function v i is still strictly increasing with Assumption 1. 76 First, s k mc i is well-defined. By lemma 1, v i (s k i )< 1 for every s k i < max(p i å l2M i nfkg s l i ;c k i ) and lim s k i !¥ v i (s k i )= ¯ m i > 1. With Assumption 1, we can see that the continuous function v i is strictly increasing on(max(p i å l2M i nfkg s l i ;c k i );¥). Therefore, the equation that v i (s k mc i )= 1 has a unique solution on(max(p i å l2M i nfkg s l i ;c k i );¥), which means that s k mc i is well-defined. Similarly, we can conclude that for any 1<m k < ¯ m i , the equation v i (s k i )=m k has a unique solution in the interval (max(p i å l2M i nfkg s k i ;c k i );¥). Clearly, no solution exists ifm k ¯ m i . Second, as we know that s k mc i >(max(p i å l2M i nfkg s l i ;c k i ), this implies thati i (s k i +å l2M i nfkg s l i ) is a non-decreasing function on (s k mc i ;¥), and v 0 i (s k i ) > 0 for every s k i > s k mc i . We can apply the inverse function theorem to get that r 0 i (m k )= 1 v 0 i (r i (m k )) . Since we know that v 0 i (s k i )=( (s k i c k i )h 0 i (P i ) g i (P i ) ) 0 , = h 0 i (s k i c k i )h 00 i +g 0 i (s k i c k i ) h 0 i g i g i , = (v i 1)h 0 i v i g 0 i g i . This leads to the equation (A.2). Since v i is continuous and strictly increasing on (s k mc i ;¥), lim s k i !s k mc i = 1 < v i (s k i ) < ¯ m i = lim s k i !¥ v i (s k i ). Q.E.D. We can extend the function r i by continuity: r i (1)= s k mc i and r i (m k )=¥ for any m k ¯ m i , and generalize the commoni-markup property to price vectors with infinite components. We also set r j (m k )= c k j for any final product j2 G k whereå l2M j nfkg s l j =¥ which is consistent with our assumption that f k chooses its marginal cost as the price for its component in final product j when other firms choose infinite prices for their components in j (Note that whenå l2M j nfkg s l j =¥, firm f k can only earn a zero profit from the final product j2 G k since all prices for the intermediate products are non-negative). Definition 2 The price vector(s k i ) i2G k2(0;¥] jG k j satisfies the commoni-markup property if there exists a scalarm k 1 such that s k i = r i (m k ) for any i2 G k . 77 We define ¯ m k max i2G k ¯ m i and extend g i by continuity at infinity such that g i (¥)= 0 for any i2 N. We can further simplify the first-order conditions. Lemma 8 Suppose that the generalized first-order conditions for problem (A.1) hold at price vec- tor (s k i ) i2G k2 (0;¥] jG k j . Then (s k i ) i2G k satisfies the common i-markup property. The common i-markup,m k , solves the following equation on(1; ¯ m k ): m k = 1+m k å j2G kg j (r j (m k )+å l2M j nfkg s l j ) å j2G k h j (r j (m k )+å l2M j nfkg s l j )+ H 0 0 : (A.3) Also,P k (s k ;s k 0 ;H(s))=m k 1. Proof: Without loss of generality, we assume that G k =f1;2;:::;n k g N and G ˆ k =f j2 G k : s k j <¥;å l2M j nfkg s l j 6=¥g=f1;2;:::;Kg, where 1 K n k . Clearly, this set is nonempty since firms can obtain positive profits from G ˆ k by setting s k j > c k j . For productsfK+ 1;:::;Tg, where K < T n k , we assume that for any j2fK+ 1;:::;Tg,å l2M j nfkg s l j =¥ which implies thatg j = 0 and h j (¥)= lim p j !¥ h j (p j ) 0. The first-order condition for product i2 G ˆ k implies the following ¶P k ¶s k i = D i (1 v i (s k i )+P k (s k ;s k 0 ))= 0: (A.4) It shows that v i (s k i )= v j (s k j )=m k > 1 for any 1 i; j K, and s k i = r i (m k ) wherem k < ¯ m i for any 1 i K. The profitP k (s k ;s k 0 ;H(s))=m k 1. Next, for any i T+ 1, we show that ¯ m i m k , which means that r i (m k )=¥. To see this, we assume that ¯ m i >m k for i= T+ 1. Then r i (m k )= x<¥. Let ˜ P k (x) and ˜ D i (x) be the profit of firm f k and the demand for product i= T+ 1 at price vector(s k 1 ;:::;s k T ;x;¥;:::;¥). Then the derivative of the profit function with respect to x implies d ˜ P k dx = ˜ D T+1 (1 v T+1 (x)+ ˜ P k (x)): 78 Since lim x!¥ v T+1 (x)= ¯ m T+1 and lim x!¥ ˜ P k (x)=m k 1, it means that d ˜ P k dx < 0 when x is sufficiently large. Therefore, there must exist s k i <¥ such that ˜ P k (s k i )> ˜ P k (¥). This contradicts with the assumption that the generalized first-order conditions hold at price vector (s k i ) i2G k. We can conclude that r i (m k )=¥ for any i> T . Based on equation (A.4) m k = 1+P k (s k ;s k 0 ;H(s)); = 1+ å K j=1 (s k j c k j )(h 0 i (s k j +å l2M j nfkg s l j )) å j2G k h j (s k j +å l2M j nfkg s l j )+ H 0 0 ; = 1+ å K j=1 (s k j c k j ) P j i j (P j )g j (P j ) å j2G k h j (P j )+ H 0 0 ; = 1+m k å j2G kg j (r j (m k )+å l2M j nfkg s l j ) å j2G k h j (r j (m k )+å l2M j nfkg s l j )+ H 0 0 : Q.E.D. Next, we show that equation (A.3) has a unique solution inm k for firm f k . Lemma 9 These exists a uniquem k 2(1; ¯ m k ) that solves equation(A:3). Proof: Previous results show that there must exist such a solution of equation (A.3). Lemma 2 indicates that the maximization problem (A.1) has a solution s k =(s k i ) i2G k. With lemma 3, the generalized first-order conditions are satisfied at s k . Lemma 5 further implies that s k satisfies the commoni-markup property and the correspondingi-markupm k 2(1; ¯ m k ) is a solution. To show the uniqueness of the solution, we define a function f(m k ) where m k 2(1; ¯ m k ) as follows: f(m k )=(m k 1)( å i2G k h i (r i (m k )+ å l2M i nfkg s l i )+ H 0 0 )m k å i2G k g i (r i (m k )+ å l2M i nfkg s l i ): 79 Therefore, m k is a solution of equation (A.3) if and only if f(m k )= 0. Since f is continuous on (1; ¯ m k ) and C 1 on (1; ¯ m k )nf ¯ m i g i2G k by lemma 5, we only need to show that f 0 (m k ) > 0 for m k 2(1; ¯ m k )nf ¯ m i g i2G k. For anym k 2(1; ¯ m k )nf ¯ m i g i2G k, we let G ˆ k be the set of products in G k with finite prices. f 0 (m k )= H 0 0 + å i2G k nG ˆ k h i (¥)+ å i2G ˆ k (h i g i )+(m k 1)( å i2G ˆ k h 0 i r 0 i )m k ( å i2G ˆ k r 0 i g 0 i ); = H 0 0 + å i2G k nG ˆ k h i (¥)+ å i2G ˆ k (h i g i )+ å i2G ˆ k g i ; = H 0 0 + å i2G k h i > 0: We can conclude that equation (A.3) has a unique solution on(1; ¯ m k ). Q.E.D. With all these results, we can conclude that the generalized first-order conditions are necessary and sufficient for global optimality in solving maximization problem (A.1). Lemma 10 Profit maximization problem (A.1) has a unique solution. The generalized first-order conditions of this problem are both necessary and sufficient for global optimality. The optimal price vector satisfies the commoni-markup property, and the correspondingi-markup,m k , is the unique solution of equation (A.3). The optimal profit ism k 1. Proof: Suppose(s k i ) i2G k is a solution of maximization problem (A.1). By Lemma 2 and 3, such (s k i ) i2G k exists and s k i <¥ for some i2 G k , and the generalized first-order conditions hold at (s k i ) i2G k. Lemma 5 then indicates that(s k i ) i2G k satisfies the commoni-markup property and the correspondingi-markup, m k , solves equation (A.3). Lemma 6 shows that the solution of equation (A.3), denoted by m k , is unique. Therefore, the solution of the maximization problem (A.1) is unique and(s k i ) i2G k = r i (m k ) i2G k andm k 1 is the optimal profit. Suppose that the generalized first-order conditions hold at price vector(s k i ) i2G k. Then Lemma 5 shows that(s k i ) i2G k satisfies the commoni-markup property, and we can find the uniquei-markup, m k , by Lemma 6. Therefore, (s k i ) i2G k =(r i (m k )) i2G k =(s k i ) i2G k and the generalized first-order conditions are sufficient for global optimality. Q.E.D. 80 Now we can discuss the existence of pricing equilibrium. Based on Lemma 10, we know that each firm f k 2 F chooses the optimal i-markup, m k . For each final product i2 N with a finite total price, we can define C i =å l2M i c l i and sum alli-markups of product i to derive the aggregate markup,Y i , where å k2M i m k = P i C i P i i i (P i )Y i : Similar to Lemma 7, we can define a function w i : P i ! P i C i P i i i (P i ), and P mc i is the unique solution for w i (P i )= 1. With Assumption 1, we know that: Lemma 11 For every i2 N with a finite price P i andY i 2(1; ¯ m i ), the equation w i (P i )=Y i has a unique solution in the interval (0;¥), denoted t i (Y i ). If Y i ¯ m i , then that equation does not have a solution. Moreover, t i () is strictly increasing and C 1 on(1; ¯ m i ) and has the properties that lim Y i !1 t i (Y i )= P mc i and lim Y i ! ¯ m i t i (Y i )=¥. Proof: We omit the proof of this lemma which is very similar to the proof of Lemma 7. Q.E.D. Similarly, we extend this function t i by continuity: t i (1)= P mc i and t i (Y i )=¥ for anyY i ¯ m i . Based on Lemma 10, Lemma 11 and Assumption 1, we can summarize the problem about the existence of a pricing equilibrium in the following way. Lemma 12 Under Assumption 1, for any given network of complements and substitutes, (V;F;N), there exists a pricing equilibrium if and only if there exists a profile of i-markups(m k ) k2F where m k <¥ for any f k 2 F, such that m k = 1+m k å i2G kg i (t i (Y i )) å j2N h j (t j (Y j ))+ H 0 = 1+m k å i2G kg i (t i (Y i )) H(Y) ; 8 f k 2 F: (A.5) Proof: Assumption 1 implies that for any i2 N, lim P i !¥ P i C i P i i i (P i )= ¯ m i =¥. Since we as- sume that every firm f k chooses its marginal cost c k j for j2 G k when å l2M j nfkg s l j =¥, then the common i-markup property under Lemma 10 implies that no firm will choose infinite prices for its components in equilibrium. 81 Suppose we have a pricing equilibrium s =(s k i ) i2G k ;k2F , which means that the generalized first-order conditions and the commoni-markup property are satisfied for every firm f k 2 F with s . Then we can get a profile of i-markups (m k ) k2F which satisfies the equation (A.3) for any f k 2 F following Lemma 10 andm k <¥ for any f k 2 F with m k = 1+m k å i2G kg i (å l2M i r l i (m l )) å j2N h j (å l2M j r l j (m l ))+ H 0 ; 8 f k 2 F; where r l i (m l ) is the r i function defined in Lemma 7 for firm f l 2 F. Since å l2M i r l i (m l ) = å l2M i s l i = P i = t i (Y i ) for any i2 N, the equation above can be rewritten as m k = 1+m k å i2G kg i (t i (Y i )) å j2N h j (t j (Y j ))+ H 0 ; 8 f k 2 F: Suppose there exists a profile of i-markups (m k ) k2F where m k <¥ for any f k 2 F such that equations (A.5) are satisfied. Then we can get the correspondingY i for any i2 N, and derive the corresponding price for each final product i, P i = t i (Y i ). Based on the definition of thei-markup, we can calculate a unique price vector s =(s k ) i2G k ;k2F . Since P i = t i (Y i )=å l2M i r l i (m l ), we know that the generalized first-order conditions are satisfied for every firm f k 2 F and s is a pricing equilibrium. Q.E.D. As we discussed in the Section 4, we cannot directly apply the aggregative games approach developed in Nocke and Schutz (2018) to show the existence of a pricing equilibrium. And only using equation (A.5) for firm f k 2 F is not enough to show that there is a one-to-one relationship between the aggregator H and a unique m k . m k also depends on thei-markups of other firms that provide complementary products for firm f k . Instead, we first rewrite equation (A.5) to be m k = H Hå j2G k g j (t j (Y j )) : 82 For any final product i2 N, we can combine all i-markups of the firms that are involved in producing final product i, and derive an equation for each final product’s aggregate markup Y i = å l2M i m l = å l2M i H Hå j2G l g j (t j (Y j )) : (A.6) All final products’ aggregatei-markups form a system of equations, and based on the definition of the separable component, when an aggregator H > 0 is given, aggregatei-markups from different separable components are independent from each other. We can show that each aggregatei-markup can be uniquely determined by the aggregator H based on a system of equations from (A:6) for products within the same separable component under Assumption 1. Lemma 13 For any i2 N and H > 0, a system of equations given by (A.6) for all final products N have a unique solution inY=(Y i ) i2N , whereY i 2(jM i j; ¯ m i ). We denote the solution for each final product i to be m i (H), and the function m i (:) is continuous. Proof. For a given network of (V;F) with a total of n final products, suppose we can partition the network into y 1 separable components, denoted by S=fS 1 ;S 2 ;:::;S y g. Clearly, changes in the system of functions (A.5) in one separable component do not affect aggregate markups in other separable components when an aggregator H > 0 is given. Without loss of generality, we study equations (A.5) within S 1 =f1;2;:::;qg N, which includes a set F 1 =f f 1 ; f 2 ;:::; f m g F of firms. We first show that the right-hand side of equations (A.5) for firms in F 1 must be bounded and greater than 1 when m k 1 for any f k 2 F 1 . Since for any f k 2 F 1 , m k å i2G k g i (t i (Y i )) H(Y) can be considered as f k ’s profit, and our analysis in Lemma 7 shows that by assigning all f k ’s complementary products’ prices to be zero and any firm f j ’s prices to be infinity where G k \ G j = ?, there exists an optimal 1 < m k <¥ that maximizes f k ’s profits. Therefore, m k å i2G k g i (t i (Y i )) H(Y) is less than infinity for any f k 2 F 1 . It implies that if we fix any H > 0, m k å i2G k g i (t i (Y i )) H is also less than infinity for 1m k < ¯ m i . Let ˆ m be a large number which is greater than the maximum of 1+m k å i2G k g i (t i (Y i )) H for any f k 2 F 1 . Following Brouwer’s fixed-point theorem, we can conclude that for a given H > 0, there exists a solutionm2[1; ˆ m] m that satisfies equations (A.5) for firms in 83 F 1 . Also, we can rewrite equations (A.5) with a given H > 0 for firms in S 1 and define a function t(m) on[1; ˆ m] m . t(m)= 8 > > > > > > > > > > < > > > > > > > > > > : 1 å j2G 1 g j (t j (Y j )) H 1 m 1 1 å j2G 2 g j (t j (Y j )) H 1 m 2 . . . 1 å j2G mg j (t j (Y j )) H 1 m m : To determine whether there exists a unique solutionm2[1; ˆ m] m that satisfies equations (A.5) with a given H > 0, it is equivalent to evaluate whether t(m)= 0 has a unique solution in m2[1; ˆ m] m or not. We can apply the Poincare-Hopf index theorem (Simsek et al. (2007)) to show this result. First, it is clear that the set [1; ˆ m] m satisfies the boundary conditions of the Poincare-Hopf index theorem. Second, for anym2[1; ˆ m] m , we can derive the Jacobian of the functiont(m), denoted by Dt(m), to be Dt(m)= 2 6 6 6 6 6 6 6 4 å j2G 1 g 0 j t 0 j H +( 1 m 1 ) 2 å j2(G 1 \G 2 ) g 0 j t 0 j H å j2(G 1 \G m ) g 0 j t 0 j H å j2(G 2 \G 1 ) g 0 j t 0 j H å j2G 2 g 0 j t 0 j H +( 1 m 2 ) 2 å j2(G 2 \G m ) g 0 j t 0 j H . . . . . . . . . . . . å j2(G m \G 1 ) g 0 j t 0 j H å j2(G m \G 2 ) g 0 j t 0 j H å j2G mg 0 j t 0 j H +( 1 m m ) 2 3 7 7 7 7 7 7 7 5 : Clearly, the matrix Dt(m) is symmetric. Next, we define matrix E i , where i2 S 1 , to be E i = 2 6 6 6 6 6 6 6 4 c G 1(i) c G 1(i)c G 2(i) c G 1(i)c G m(i) c G 2(i)c G 1(i) c G 2(i) c G 2(i)c G m(i) . . . . . . . . . . . . c G m(i)c G 1(i) c G m(i)c G 2(i) c G m(i) 3 7 7 7 7 7 7 7 5 g 0 i t 0 i H ; 84 where c G k(i) is an indicator function which equals to one when final product i2 G k , and we can easily see that the matrix E i is positive semi-definite. E i identifies the firms that are involved in the production of the final product i. Then we can see that Dt(m)= å i2S 1 E i + 2 6 6 6 6 6 6 6 4 ( 1 m 1 ) 2 0 0 0 ( 1 m 2 ) 2 0 . . . . . . . . . . . . 0 0 ( 1 m m ) 2 3 7 7 7 7 7 7 7 5 = å i2S 1 E i + D: Since the matrix Dt(m) is a summation of positive semi-definite matrix E i and a positive definite matrix D, we can conclude that det(Dt(m))> 0 for any m2[1; ˆ m] m . By the Poincare-Hopf index theorem, we know thatt(m)= 0 has a unique solution m2[1; ˆ m] m , which means that there exists a unique solution m2[1; ˆ m] m that satisfies equations (A.5) with a given H > 0. Therefore, we also know that, with any H > 0, the system of equations (A.6) for S 1 have a unique solution Y2[jM i j; ˆ mjM i j] q , wherem k = H Hå j2G k g j (t j (Y j )) > 1 for any f k 2 F 1 . Then we define a function T(Y) on [jM i j; ˆ mjM i j] jqj following equation (A.6) for each final product in S 1 such that T(Y)= 8 > > > > > > > > > > < > > > > > > > > > > : Y 1 å l2M 1 H Hå j2G l g j (t j (Y j )) Y 2 å l2M 2 H Hå j2G l g j (t j (Y j )) . . . Y q å l2M q H Hå j2G l g j (t j (Y j )) : 85 Next, we show that with given H > 0, the function T(Y) = 0 defines an implicit function m i (H)=Y i for any i2 S 1 . To simplify the notation, we define a function Z k for every f k 2 F such that Z k = H (Hå j2G k g j (t j (Y j ))) 2 > 0. We can derive the Jacobian of the function T(Y) to be DT(Y)= 2 6 6 6 6 6 6 6 4 1(å l2M 1 Z l )g 0 1 t 0 1 (å l2(M 1 \M 2 ) Z l )g 0 2 t 0 2 (å l2(M 1 \M q ) Z l )g 0 q t 0 q (å l2(M 2 \M 1 ) Z l )g 0 1 t 0 1 1(å l2M 2 Z l )g 0 2 t 0 2 (å l2(M 2 \M q ) Z l )g 0 q t 0 q . . . . . . . . . . . . (å l2(M q \M 1 ) Z l )g 0 1 t 0 1 (å l2(M q \M 2 ) Z l )g 0 2 t 0 2 1(å l2M q Z l )g 0 q t 0 q 3 7 7 7 7 7 7 7 5 A: The determinant of the Jacobian can be further simplified to be det(DT(Y))= det( 2 6 6 6 6 6 6 6 4 1 g 0 1 t 0 1 +(å l2M 1 Z l ) å l2(M 1 \M 2 ) Z l å l2(M 1 \M q ) Z l å l2(M 2 \M 1 ) Z l 1 g 0 2 t 0 2 +(å l2M 2 Z l ) å l2(M 2 \M q ) Z l . . . . . . . . . . . . å l2(M q \M 1 ) Z l å l2(M q \M 2 ) Z l 1 g 0 q t 0 q +å l2M q Z l 3 7 7 7 7 7 7 7 5 | {z } We denote this matrix as ¯ A )( q Õ i=1 g 0 i t 0 i )(1) q ; = det( ¯ A)( q Õ i=1 g 0 i t 0 i )(1) q : Note that following Lemma 1 (d), the term (Õ q i=1 g 0 i t 0 i )(1) q is always positive, and the sign of det(DT(Y)) is determined by det( ¯ A). Next, we define matrix B k for any f k 2 S 1 to be B k = 2 6 6 6 6 6 6 6 4 c G k(1)Z k (c G k(1)c G k(2))Z k (c G k(1)c G k(q))Z k (c G k(2)c G k(1))Z k c G k(2)Z k (c G k(2)c G k(q))Z k . . . . . . . . . . . . (c G k(q)c G k(1))Z k (c G k(q)c G k(2))Z k c G k(q)Z k 3 7 7 7 7 7 7 7 5 ; 86 where function c G k(i) is an indicator function that equals to one when final product i2 G k . All entries in B k are positive, and B k is also positive semi-definite. 2 Therefore, we can decompose the matrix ¯ A into the following equation. ¯ A= å l2F 1 B l + 2 6 6 6 6 6 6 6 4 1 g 0 1 t 0 1 0 0 0 1 g 0 2 t 0 2 0 . . . . . . . . . . . . 0 0 1 g 0 q t 0 q 3 7 7 7 7 7 7 7 5 | {z } We denote this matrix as C = å l2F 1 B l +C: Since matrix C is a strictly diagonally dominated symmetric matrix, matrix C is positive defi- nite, and we can conclude that det( ¯ A)= det(( å l2F 1 B l )+C) å l2F 1 det(B l )+ det(C)> 0: Therefore, we can apply the implicit function theorem to conclude that there exists a unique contin- uously differentiable function, m i (H) on(0;¥), for each final product i2 S 1 , such that m i (H)=Y i . Similarly, we can apply all procedures above to other separable components in S and determine a unique continuously differentiable function m j (H)=Y j for any j = 2 S 1 . Q.E.D. To show the existence of a pricing equilibrium, we just need to show that there exists an H such that: G(H )= H 0 + å i2N h i (t i (m i (H )))= H Lemma 14 There exists a H 2(0;¥) such thatG(H )= H . Proof: Based on Lemma 13, the functionG(H) is a continuous function on(0;¥). Since we assume that H 0 > 0 and h i > 0,G(H)> H 0 . Also, since for any f k 2 F and H > 0, m k 1 which implies that s k i > c k i . Since we know that h 0 i < 0, we can conclude thatG(H)= H 0 +å i2N h i (t i (m i (H)))< 2 To see this, note that any matrix of ones is positive semi-definite, and any B k matrix is derived from a matrix of ones by adding rows of zeros and columns of zeros and times Z k > 0. Clearly, such matrices are always positive semi-definite. 87 H 0 +å i2N h i (å l2M i c l i ). Therefore, the codomain of the continuous function G(H) is [H 0 ;H 0 + å i2N h i (å l2M i c l i )], and as we know that G(H) is continuous on [H 0 ;H 0 +å i2N h i (å l2M i c l i )], by applying the Brouwer fixed-point theorem, we know that there exists a H 2 (0;¥) such that G(H )= H . Q.E.D. For any H , we can use functions m i (H)=Y i and t i (Y)= P i and firms’ equilibrium conditions in (A.5) to determine all relevant variables in (a), (b) and (c) of Theorem 1. This series of lemmas completes a proof of Theorem 1. A.2 Proof of Proposition 1 Based on results from Lemma 10 and Lemma 11, the uniqueness of the pricing equilibrium with MNL and CES demands can be proved by showing that the functionG(H) has a unique fixed point. Under MNL demand, we know that g i = h 02 i (P i ) h 00 i (P i ) = h i (P i ), where h i (P i )= exp( a i P i l ), a i 2R, and l = 1. (l = 1 is the standard MNL demand. The result is the same when l > 1.) Since Y i = P i C i P i i i (P i )= P i C i under MNL demand, we can derive the inverse function t i (Y i )=Y i +C i . Suppose we can decompose the given network structure(V;F;N) into a finite amount of separable components denoted by S=fS 1 ;S 2 ;:::;S J g. Within one of the separable components S 1 , which 88 has final productsf1;2;:::;Kg N, we can totally differentiate equation (A.6) for each product in S 1 to derive the following equation: 2 6 6 6 6 6 6 6 4 1+(å l2M 1 Z l )h 1 (å l2(M 1 \M 2 ) Z l )h 2 (å l2(M 1 \M n ) Z l )h n (å l2(M 2 \M 1 ) Z l )h 1 1+(å l2M 2 Z l )h 2 (å l2(M 2 \M n ) Z l )h n . . . . . . . . . . . . (å l2(M K \M 1 ) Z l )h 1 (å l2(M K \M 2 ) Z l )h 2 1+(å l2M K Z l )h K 3 7 7 7 7 7 7 7 5 | {z } We denote this matrix as A 2 6 6 6 6 6 6 6 4 ¶m 1 ¶H ¶m 2 ¶H . . . ¶m K ¶H 3 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 4 (å l2M 1 Z l (å j2G l h j ) H ) (å l2M 2 Z l (å j2G l h j ) H ) . . . (å l2M K Z l (å j2G l h j ) H ) 3 7 7 7 7 7 7 7 5 | {z } We denote this vector as b : The determinant of the matrix A, as we discussed before in the proof of Lemma 13, is always positive. We can simplify the determinant of matrix A to be det(A)= det( 2 6 6 6 6 6 6 6 4 1+(å l2M 1 Z l )h 1 (å l2(M 1 \M 2 ) Z l )h 2 (å l2(M 1 \M K ) Z l )h K (å l2(M 2 \M 1 ) Z l )h 1 1+(å l2M 2 Z l )h 2 (å l2(M 2 \M K ) Z l )h K . . . . . . . . . . . . (å l2(M K \M 1 ) Z l )h 1 (å l2(M K \M 2 ) Z l )h 2 1+(å l2M K Z l )h K 3 7 7 7 7 7 7 7 5 ); = det( 2 6 6 6 6 6 6 6 4 1 h 1 +å l2M 1 Z l å l2(M 1 \M 2 ) Z l å l2(M 1 \M K ) Z l å l2(M 2 \M 1 ) Z l 1 h 2 +å l2M 2 Z l å l2(M 2 \M K ) Z l . . . . . . . . . . . . å l2(M K \M 1 ) Z l å l2(M K \M 2 ) Z l 1 h K +å l2M K Z l 3 7 7 7 7 7 7 7 5 | {z } We denote this matrix as ¯ A )( K Õ ( j=1) h j ); = det( ¯ A)( K Õ ( j=1) h j ): 89 Then we apply Cramer’s rule to determine ¶m 1 ¶H . First, we calculate the determinant of A 1 , which is formed by replacing the first column of the matrix A by the vector b. det(A 1 )= det( 2 6 6 6 6 6 6 6 4 (å l2M 1 Z l (å j2G l h j ) H ) (å l2(M 1 \M 2 ) Z l )h 2 (å l2(M 1 \M n ) Z l )h n (å l2M 2 Z l (å j2G l h j ) H ) 1+(å l2M 2 Z l )h 2 (å l2(M 2 \M n ) Z l )h n . . . . . . . . . . . . (å l2M K Z l (å j2G l h j ) H ) (å l2(M K \M 2 ) Z l )h 2 1+(å l2M K Z l )h K 3 7 7 7 7 7 7 7 5 ); = det( 2 6 6 6 6 6 6 6 4 (å l2M 1 Z l (å j2G l h j )) (å l2(M 1 \M 2 ) Z l )h 2 (å l2(M 1 \M K ) Z l )h K (å l2M 2 Z l (å j2G l h j )) 1+(å l2M 2 Z l )h 2 (å l2(M 2 \M K ) Z l )h K . . . . . . . . . . . . (å l2M K Z l (å j2G l h j )) (å l2(M K \M 2 ) Z l )h 2 1+(å l2M K Z l )h K 3 7 7 7 7 7 7 7 5 ) | {z } We can subtract all other columns from the first column. ( 1 H ); = det( 2 6 6 6 6 6 6 6 4 (å l2M 1 Z l )h 1 (å l2(M 1 \M 2 ) Z l )h 2 (å l2(M 1 \M K ) Z l )h K (å l2(M 2 \M 1 ) Z l )h 1 1 1+(å l2M 2 Z l )h 2 (å l2(M 2 \M K ) Z l )h K . . . . . . . . . . . . (å l2(M K \M 1 ) Z l )h 1 1 (å l2(M K \M 2 ) Z l )h 2 1+(å l2M K Z l )h K 3 7 7 7 7 7 7 7 5 )( 1 H ); = det( 2 6 6 6 6 6 6 6 4 (å l2M 1 Z l ) (å l2(M 1 \M 2 ) Z l ) (å l2(M 1 \M K ) Z l ) (å l2(M 2 \M 1 ) Z l ) 1 h 1 1+(å l2M 2 Z l ) (å l2(M 2 \M K ) Z l ) . . . . . . . . . . . . (å l2(M K \M 1 ) Z l ) 1 h 1 (å l2(M K \M 2 ) Z l ) 1+(å l2M K Z l ) 3 7 7 7 7 7 7 7 5 )( 1 H K Õ j=1 h j ); = det( ¯ A+ uv T )( 1 H K Õ j=1 h j ): where we assume u=( 1 h 1 ) 2 6 6 6 6 6 6 6 4 1 1 . . . 1 3 7 7 7 7 7 7 7 5 ; v= 2 6 6 6 6 6 6 6 4 1 0 . . . 0 3 7 7 7 7 7 7 7 5 : 90 We apply the matrix determinant lemma and get the following result: det(A 1 )=(1+ v T ( ¯ A) 1 u)det( ¯ A)( 1 H K Õ j=1 h j )=(1 1 h 1 R 1 ( ¯ A 1 ))det( ¯ A)( 1 H K Õ j=1 h j ): where the function R i ( ¯ A 1 ) denotes the sum of the entries in row i of matrix ¯ A 1 . Therefore, we have ¶m 1 ¶H = det(A 1 ) det(A) = 1 1 h 1 R 1 ( ¯ A 1 ) H : Similarly, we can derive that for any final product i2 S 1 , ¶m i ¶H = det(A i ) det(A) = 1 1 h i R i ( ¯ A 1 ) H : Within the first separable component S 1 , we know that ¶(å i2S 1 h i (t i (m i (H))) H ) ¶H = (å i2S 1 (h i ¶m i ¶H H))å i2S 1 h i H 2 ; = å i2S 1 R i ( ¯ A 1 ) H 2 : The numerator in the above equation, å i2S 1 R i ( ¯ A 1 ), is the summation of all entries in the matrix ¯ A 1 . The fact that ¯ A is a positive definite matrix implies that ¯ A 1 is also a positive definite matrix and as a result, we can determine that ¶(å i2S 1 h i (t i (m i (H))) H ) ¶H < 0: We then derive similar results for other separable components of the network, and conclude that the ratio ofG(H) and H, G(H) H = å J a=1 å i2S a h i (t i (m i (H)))+ H 0 H ; 91 is strictly decreasing. Therefore, under the MNL demand, the pricing equilibrium is unique. Q.E.D. A.3 Proof of Proposition 2 We first prove an important property. Lemma 15 Suppose we have an arbitrary positive diagonal n n matrix A and a set of positive matrices B=(B (i; j) ) i; j2N =(b i j u i j u T i j ) i; j2N , where i< j and b i j 2(0;¥) and u i j is a positive n 1 column vector with ones for the entries from row i to row j and the remaining entries are all zero. Also, assume that for any two matrices B (s;t) 2 B and B (u;v) 2 B, either[s;t][u;v],[u;v][s;t], or [s;t]\[u;v]=?. Then the matrix C=(A+å i< j B (i; j) ) is a positive definite matrix and every row m of the matrix C 1 has the property that 0< R m (C 1 ) 1 A mm . Proof. We prove this lemma by induction. Without loss of generality, suppose the set B only has one element that B=(B (1; j) ). The matrix C is a combination of the diagonal matrix A, which has diagonal entries denoted byfa 1 ;a 2 ;:::;a n g, and the fact that matrix B (1; j) is a positive semi-definite matrix implies that the matrix C is positive definite. The matrix C becomes a block matrix such that C= 2 6 6 6 6 6 6 6 6 6 6 4 C 1 0 0 0 0 A j+1 j+1 0 0 0 0 A j+2 j+2 0 . . . . . . . . . . . . 0 0 0 0 A nn 3 7 7 7 7 7 7 7 7 7 7 5 ; 92 where C 1 = ˆ A 1; j + ˆ B 1; j , ˆ A 1; j is a j j diagonal matrix with diagonal entries equal tofa 1 ;a 2 ;:::;a j g, and ˆ B 1; j is a j j matrix with the same entry of b 1 j . Therefore, we know that C 1 = 2 6 6 6 6 6 6 6 6 6 6 4 C 1 1 0 0 0 0 1 A j+1 j+1 0 0 0 0 1 A j+2 j+2 0 . . . . . . . . . . . . 0 0 0 0 1 A nn 3 7 7 7 7 7 7 7 7 7 7 5 ; and by applying the Sherman-Morrison formula, we can determine that C 1 1 = ˆ A 1 1; j b 1 j 1+ b 1 j (å j i=1 a i ) ( ˆ A 1 1; j J j ˆ A 1 1; j ); where J j is a j j matrix of ones. Then we know that for any 1 m j, R m (C 1 )= 1 1+b 1 j (å j i=1 a i ) 1 a m < 1 a m . For any other j< m n, we have R m (C 1 )= 1 A mm . Now suppose for any matrix B with n elements and the matrix C n =(A+å i< j B (i; j) ) satisfies the claims in Lemma 15. We then know that the matrix C is a block diagonal matrix with blocks denoted by C 1 ;C 2 ;:::;C k where 1 k n. Without loss of generality, we assume C 1 is a k 1 k 1 (k 1 > 1) matrix and C 2 is a k 2 k 2 (k 2 1) matrix. Suppose we add another matrix B (1;t) to C, we concentrate on the case with t =(k 1 + k 2 ) since if t < k 1 + k 2 the addition of the matrix B (1;t) is either trivial (it may be combined with one of the matrices in B we have) or not compatible since we may find another matrix B (k 1 ;s) with the property that k 1 > 1 and s> t. Then the matrix B (1;t) combines the blocks C 1 and C 2 , and we can denote the matrix G to be G=( ˆ C+ ˆ B 1;t )=( 2 6 4 C 1 0 0 C 2 3 7 5 + ˆ B 1;t ): 93 Similarly applying the Sherman-Morrison formula, we know that G 1 = ˆ C 1 b 1t 1+ b 1t å t i=1 R i ( ˆ C 1 ) ( ˆ C 1 J t ˆ C 1 ); which leads to the result that 0< R j (G 1 )= 1 1+b 1tå t i=1 R i ( ˆ C 1 ) R j ( ˆ C 1 )< 1 a j where 1 j t. We can conduct similar analysis for the case with t >(k 1 + k 2 ), and generate a new block diagonal matrix G. Therefore, for a matrix B with n+ 1 elements, the statement is still correct and we complete the proof. Q.E.D. CES demand does not satisfy Assumption 1, and here we show that for any givens > max(jM i j) i2N , whens is sufficiently large, there exists a unique pricing equilibrium under CES demand. 3 Due to the tie-breaking rule we imposed to exclude the ”no-trade” equilibrium, we cannot haveY i s for any i2 N in any pricing equilibrium. In the case with a symmetric network, we can combine firms’ i-markups following our discussions in subsection 2.5.1 and show that when s > max(jM i j) i2N , the pricing equilibrium is unique. For one-sided asymmetric networks, the maximum of the right- hand side of equation (A.5) is decreasing ins, and we can easily conclude that there exists a ¯ s <¥ such that when s > ¯ s, there exists a solution (m k ) k2F which satisfies the equation (A.5) for any given 0< H 0 < H and t i (Y i )<¥ for any i2 N (i.e.,Y i <s). We can then follow the similar procedure we used in Theorem 1 to show the existence of a pricing equilibrium with H 0 < H . Also, just like the case with MNL demand, we derive the result that ¶m 1 ¶H = det(A 1 ) det(A) = 1+ v T ( ¯ A) 1 u Ht 1 ; 3 A similar argument can be developed with H 0 to be sufficiently large. 94 wheret i = 1s sY i < 0, d= s1 s , and v T = 2 6 6 6 6 6 6 6 4 1 0 . . . 0 3 7 7 7 7 7 7 7 5 ; u= 2 6 6 6 6 6 6 6 4 1 dt 1 h 1 1 dt 2 h 1 . . . 1 dt n h 1 3 7 7 7 7 7 7 7 5 : We also know that, within the first separable component S 1 , ¶(å i2S 1 h i (t i (m i (H))) H ) ¶H = (å i2S 1 (h 0 i t 0 i ¶m i ¶H )H)(å i2S 1 h i ) H 2 = å i2S i 1 dt i R i ( ¯ A 1 ) H 2 : In the case with any one-sided asymmetric network, Lemma 15 implies that every sum of the row of the matrix ¯ A 1 is positive and we can conclude that ¶(å i2S 1 h i (t i (m i (H))) H ) ¶H < 0: Therefore, we can get same results for other separable components and conclude that the pric- ing equilibrium is unique. Q.E.D. A.4 Proof of Proposition 3 Without loss of generality, we assume that firm f 1 and firm f 2 are two suppliers of the final products G 1 = G 2 =f1;2;:::; jg N and we have S 1 = G 1 = G 2 . Each final product i2 N hasjM i j firms producing essential components. We only show the case with MNL demand as the case with CES demand is very similar (g i = h i for MNL whileg i = s1 s h i for CES). The equilibrium prices within S 1 are characterized by the following equations: Y i (1 å i2G 1 h i (t i (Y i )) H )=jM i j; 8i2 G 1 : (A.7) 95 If firm f 1 and firm f 2 are merged, then the equilibrium conditions become ˆ Y i (1 å i2G 1 h i (t i ( ˆ Y i )) H )=jM i j 1; 8i2 G 1 : The left-hand side of the equation (A.7) is strictly increasing inY i , and therefore, given the same aggregator H, the function ˆ m i (H) = ˆ Y i after the merger is lower than m i (H) =Y i before the merger. For all other products, we still have the same function m j (H) ( j2 Nn G 1 ). With the same H, the merger leads to a new ˆ G(H) H > 1. Since we have illustrated that ˆ G(H) H function is strictly decreasing in H with CES and MNL demands under any symmetric network. We can conclude that the equilibrium aggregator ˆ H after the merger is higher than the equilibrium aggregator H before the merger. Therefore, after a vertical merger,jM i j drops and the equilibrium aggregator H increases. By evaluating the equations above, we can conclude that ˆ Y i <Y i . Similarly, for any other final product j2 Nn G 1 , which includes firm f k , we have Y j (1 å l2G k h l (t l (Y l )) H )=jM j j: (A.8) An increase in the equilibrium aggregator H means thatY j must get lower after the merger. We know that each final product price i2 N is determined by t i (Y i ) which is a strictly increasing function. We can conclude that all prices of the final products are lower, and the increase of the equilibrium aggregator H means that the overall consumer surplus is improved in this economy after the merger. Note that the equation (A.8) for any j2 Nn G 1 , which includes firm f k can also be rewritten as Y j jM j j Y j = å l2G k h l (t l (Y l )) H : Since we know that the post-merger ˆ Y j is lower than the pre-mergerY j , the equilibrium å l2G k h l (t l (Y l )) H becomes lower after the merger, which implies that all firms producing intermediate products 96 for Nn G 1 get lower profits after the merger. Note that å i2N h i (t i (Y i )) H = 1, and H 0 H and å j2(NnG 1 ) h j H get lower, then å i2G 1 h i H becomes higher after the merger. Therefore, for an arbitrary firm f g 2 M i nf f 1 ; f 2 g, itsi-markup is determined by the following equation m g 1 m g = å i2G 1 h i (t i (Y i )) H ; which implies that f g ’s profit gets higher after the merger. Q.E.D. A.5 Proof of Proposition 4 and Proposition 5 We combine the proof for Proposition 4 and Proposition 5. We start from proving Proposition 5, which also shows some of the results in Proposition 4. Without loss of generality, suppose firm f 1 and firm f 2 from S 1 =f1;2;:::;Kg N, where G 2 G 1 , are merged together. To see the impact of the merger on consumers, we prove a stronger result that under one-sided asymmetric network any exogenous increases in final products’ aggregate markups reduce consumer surplus. For simplicity, we assume G 2 =f1g, and we replace thei-markup of firm f 2 with a parameter a. We study how a change in a will affect the equilibrium aggregator H given the MNL demand. (This analysis can be similarly extended to the case with CES demand.) We follow the same 97 procedure in Proposition 1 where we totally differentiate the equation (A.6) within S 1 to get the following result 2 6 6 6 6 6 6 6 4 1+(å l2M 1 Z l )h 1 (å l2(M 1 \M 2 ) Z l )h 2 (å l2(M 1 \M n ) Z l )h n (å l2(M 2 \M 1 ) Z l )h 1 1+(å l2M 2 Z l )h 2 (å l2(M 2 \M n ) Z l )h n . . . . . . . . . . . . (å l2(M K \M 1 ) Z l )h 1 (å l2(M K \M 2 ) Z l )h 2 1+(å l2M K Z l )h K 3 7 7 7 7 7 7 7 5 | {z } We denote this matrix as A 2 6 6 6 6 6 6 6 4 ¶m 1 ¶a ¶m 2 ¶a . . . ¶m K ¶a 3 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 4 1 0 . . . 0 3 7 7 7 7 7 7 7 5 |{z} We denote this vector as b : We apply Cramer’s rule first and get the same result in Proposition 1 that det(A)= det( ¯ A)(Õ K j=1 h j ). By applying the matrix determinant lemma, we know that ¶m i ¶a = ¯ A 1 i1 h i : Therefore, we can determine that ¶ G(a;H) H ¶a = å i2S 1 h i ¶m i ¶a H = R 1 ( ¯ A 1 ) H : Based on Lemma 15 and the assumption that we have a one-sided asymmetric network, the matrix ¯ A is positive definite and has the property that R i ( ¯ A 1 )> 0 for any i2 S 1 . We can imply that as a decreases, with any given aggregator H, the functionG(a;H) is higher. Since we can effectively consider the merger between f 1 and f 2 as reducing a from some positive number to zero, it means that such a merger of complements always raisesG(a;H) for any given H, and using the aggregate fitting-in function, we get a higher equilibrium aggregator H , which means that the merger of 98 complements raises consumer surplus. This method can be easily modified to analyze any other mergers of complements in any one-sided asymmetric network, and we can conclude that in this case, any merger of complements always improves consumer surplus as claimed in (d). Also, under any one-sided asymmetric network, we can follow the analysis in the proof of Proposition 1 and use Lemma 15 to show that ¶m i (H) ¶H < 0 for any i2 N. This means that higher consumer surplus leads to lower aggregate markup and price for each final product. Therefore, for products outside S 1 which are affected by the merger only through changes in H, the total prices become lower after the merger which gives the result in (c). Since we know that ¯ A 1 11 > 0, which follows from the property that ¯ A is a positive definite matrix, the reduction in a due to the merger and the increase in the equilibrium aggregator H leads to the result that m 1 (a;H) is lower after the merger which means that the merger always reduces the price of final product 1. This shows the result in (a). Within S 1 , the assumption that we have a one-sided asymmetric network implies that there exists f g such that G g = S 1 . Based on (c), we can determine thatm g increases after the merger, and sincem g =Y j for any j2 S 1 withjM j j= 1, we prove the result in (b). Now we focus on the market structure in Figure 1.5 to prove the results in Proposition 4. Under both CES and MNL demands, we know that m 1 1 m 1 = g 1 +g 2 H : Since the merger between f 1 and f 2 raises H and following the equation that H 0 H + h 1 + h 2 H = 1; we know that the merger leads a higher h 1 +h 2 H , which implies thatm 1 in equilibrium is higher after the merger. Since the decrease in a raises m 2 (a;H) and the increase in the equilibrium aggregator H reduces m 2 (a;H), the overall impact of the merger on final product 2 is uncertain. These give the results in (a) of Proposition 4. 99 Under MNL demand, we get the following equations which characterize the pricing equilib- rium when thei-markup of firm f 2 is given by a m 1 = 1+m 1 h 1 (m 1 + a)+ h 2 (m 1 +m 3 ) h 1 (m 1 + a)+ h 2 (m 1 +m 3 )+ H 0 ; m 3 = 1+m 3 h 2 (m 1 +m 3 ) h 1 (m 1 + a)+ h 2 (m 1 +m 3 )+ H 0 : Using the Lambert W function, we can rewrite the above equations. P 1 W( h 1 (1+ a)+ h 2 (2+P 3 ) H 0 )=P 1 W 1 = 0; P 3 W( h 2 (P 1 + 2) h 1 (P 1 + a+ 1)+ H 0 )=P 3 W 3 = 0; (A.9) whereP 1 =m 1 1 andP 3 =m 3 1. Similarly, we can totally differentiate the functions above, and derive the strategic responses inP 1 andP 3 as the variable a changes. 2 6 4 1 W 1 1+W 1 h 2 (2+P 3 ) h 1 (1+a)+h 2 (2+P 3 ) W 3 1+W 3 H 0 h 1 (P 1 +a+1)+H 0 1 3 7 5 | {z } ˆ A 2 6 4 dP 1 da dP 3 da 3 7 5 = 2 6 4 W 1 1+W 1 h 1 (1+a) h 1 (1+a)+h 2 (2+P 3 ) W 3 1+W 3 h 1 (P 1 +a+1) h 1 (P 1 +a+1)+H 0 3 7 5 (A.10) To apply the Cramer’s rule, we first get det( ˆ A)= 1( W 1 1+W 1 W 3 1+W 3 h 2 (2+P 3 ) h 1 (1+ a)+ h 2 (2+P 3 ) H 0 h 1 (P 1 + a+ 1)+ H 0 )> 0: Therefore, we have dP 1 da = ( W 1 1+W 1 h 1 (1+a) h 1 (1+a)+h 2 (2+P 3 ) + W 1 1+W 1 W 3 1+W 3 h 1 (P 1 +a+1) h 1 (P 1 +a+1)+H 0 h 2 (2+P 3 ) h 1 (1+a)+h 2 (2+P 3 ) ) det( ˆ A) < 0; dP 3 da = W 3 1+W 3 h 1 (P 1 +a+1) h 1 (P 1 +a+1)+H 0 + W 1 1+W 1 W 3 1+W 3 h 1 (1+a) h 1 (1+a)+h 2 (2+P 3 ) H 0 h 1 (P 1 +a+1)+H 0 det( ˆ A) > 0: 100 It shows that as f 1 and f 2 are merged, f 1 increases its i-markup and f 3 decreases its i-markup. When H 0 approaches zero, we show that as the merger reduces a from the pre-merger a > 0 to the post-merger a = 0, the total price of final product 2 becomes higher. We first rewrite dP 1 da and ignore its denominator to get dP 1 da = W 1 1+W 1 ( h 1 (1+ a) h 1 (1+ a)+ h 2 (2+P 3 ) + W 3 1+W 3 h 1 (1+ a) h 1 (1+ a)+ H 0 expP 1 h 2 (2+P 3 ) h 1 (1+ a)+ h 2 (2+P 3 ) ): Based on equation (A.10), we know that dP 1 dH 0 = W 1 1+W 1 1 H 0 + W 1 1+W 1 h 2 (2+P 3 ) h 1 (1+a)+h 2 (2+P 3 ) ( W 3 (1+W 3 )(h 1 (P 1 +a+1)+H 0 ) ) 1 W 1 1+W 1 h 2 (2+P 3 ) h 1 (1+a)+h 2 (2+P 3 ) W 3 1+W 3 H 0 h 1 +H 0 < 0: Also, equations (A.9) show that lim H 0 !0 P 1 =¥. Since we also know that m 1 = h 1 (m 1 + a)+ h 2 (m 1 +m 2 ) H 0 = h 1 (a)+ h 2 (m 2 ) H 0 exp(m 1 ) ; it implies that lim H 0 !0 H 0 exp(P 1 )= 0. Similarly, we get lim H 0 !0 H 0 exp(P 1 + a )= 0 based on the firm f 2 ’s FOC. Therefore, we have lim H 0 !0 h 1 (P 1 + a + 1) h 1 (P 1 + a + 1)+ H 0 = 1; lim H 0 !0 H 0 h 1 (P 1 + a + 1)+ H 0 = 0: This also implies that at a= a , lim H 0 !0 dP 3 da = W 3 1+W 3 : Since as H 0 ! 0, we also know that j dP 1 da j! h 1 (1+ a ) h 1 (1+ a )+ h 2 (2+P 3 ) + W 3 1+W 3 h 2 (2+P 3 ) h 1 (1+ a )+ h 2 (2+P 3 ) > W 3 1+W 3 101 Therefore, when H 0 is sufficiently small,j dP 1 da j> dP 3 da . Also, as we reduce a from a > 0 to a = 0, the inequality thatj dP 1 da j> dP 3 da always holds, which means that the merger increases the price of the final product 2. It completes the proof of (b) in Proposition 4. Q.E.D. A.6 Proof of Proposition 6 We start from analyzing scenario (2) under MNL demand. Before the merger, the prices with a given aggregator H are characterized by the following equations: Y 1 = H H(h 1 (Y 1 )+ h 2 (Y 2 )) , Y 2 = H H(h 1 (Y 1 )+ h 2 (Y 2 )) + H H(h 2 (Y 2 )+ h 3 (Y 3 )) + H H h 2 (Y 2 ) , Y 3 = H H(h 2 (Y 2 )+ h 3 (Y 3 )) : (A.11) We denote H Hh 2 (Y 2 ) = a > 0, and we can see that a merger of complements between f 2 and f 3 simply makes a= 0. First, we try to understand how a change in a can affect Y i with a given H > 0 in the system of equations above. We totally differentiate the system of equations (A.11), and derive 2 6 6 6 6 4 1+ Z 1 h 1 Z 1 h 2 0 Z 1 h 1 1+(Z 1 + Z 2 )h 2 Z 2 h 3 0 Z 2 h 2 1+ Z 2 h 3 3 7 7 7 7 5 2 6 6 6 6 4 dY 1 da dY 2 da dY 3 da 3 7 7 7 7 5 = 2 6 6 6 6 4 0 1 0 3 7 7 7 7 5 : Applying Cramer’s rule and the matrix determinant lemma, we can show that dY 1 da = ¯ A 1 12 h 1 < 0; dY 2 da = ¯ A 1 22 h 2 > 0; dY 3 da = ¯ A 1 32 h 3 < 0; where the matrix ¯ A is 2 6 6 6 6 4 1 h 1 + Z 1 Z 1 0 Z 1 1 h 2 +(Z 1 + Z 2 ) Z 2 0 Z 2 1 h 3 + Z 2 3 7 7 7 7 5 : 102 Therefore, fixing an aggregator H > 0, we know that ¶ G(H) H ¶a = H 0 H + ( ¯ A 1 12 + ¯ A 1 22 + ¯ A 1 32 )) H : By applying the Sherman-Morrison formula, we find that the sign of ( ¯ A 1 12 + ¯ A 1 22 + ¯ A 1 32 ) is determined by( 1 h 1 1 h 3 Z 1 Z 2 ). Then we focus on analyzing how the function G(H) H is changed at H= H (where H is the premerger equilibrium) when we change a from the pre-merger equilibrium level a > 0 to the post-merger level of zero. We assume that h 1 (P 1 ) = h 2 (P 2 ) = h 3 (P 3 ) and C 1 = C 2 = C 3 . Also, at the pre-merger aggregator H , we know that 1 h 1 1 h 3 Z 1 Z 2 = 1 h 1 h 3 m 1 m 2 (H 0 + h 3 )(H 0 + h 1 ) : Therefore, at the pre-merger equilibrium aggregator H (and a ), when H 0 is sufficiently small, we know that 1 h 1 h 3 Z 1 Z 2 < 0, which means that ¶ G(H) H ¶a > 0. Also, we know that at the pre-merger H m 1 1 m 1 = h 1 + h 2 h 1 + h 2 + h 3 + H 0 ; m 2 1 m 2 = h 2 + h 3 h 1 + h 2 + h 3 + H 0 : Therefore, when H 0 is sufficiently small, we know thatm 1 > 2 andm 2 > 2. Then at this H , as we choose any 0 a< a , the correspondingm 1 andm 2 get even higher, and we know that ¶(m 1 m 1 )h 1 ¶a = 2m 1 h 1 ¶Y 1 ¶a +m 1 m 1 (h 1 ) ¶m 1 ¶a > 0; ¶(m 2 m 2 )h 3 ¶a = 2m 2 h 3 ¶Y 3 ¶a +m 2 m 2 (h 3 ) ¶m 2 ¶a > 0: Then we can show that for any 0< a< a at H 1 (H ) 2 > 1 m 1 m 1 h 1 m 2 m 2 h 3 ; 103 which implies that 1 h 1 h 3 Z 1 Z 2 < 0: Therefore, at this pre-merger aggregator H , as we set a= 0, we have the result that G(H ) H < 1: Then since we know that the function G(H) H is strictly decreasing in H under MNL demand. We can conclude that the post-merger aggregator H is lower than the pre-merger aggregator H . We can easily follow the same procedure to analyze Scenario (1) and Scenario (3) to show that under these two scenarios with any H > 0, we always have ¶ G(H) H ¶a < 0: As a result, as a is reduced to 0 following the merger, we have the new equilibrium aggregator H > H . To see the case with CES demand, we first show that the pricing equilibrium is unique in this case. Following the proof of Proposition 2 with a sufficiently larges, we need to show that ¶(å 3 i=1 h i (t i (m i (H))) H ) ¶H = å 3 i=1 1 dt i R i ( ¯ A 1 ) H 2 < 0: Applying the Sherman-Morrison formula, we can see that in this two-sided asymmetric network structure we always have R 1 ( ¯ A 1 ) and R 3 ( ¯ A 1 ) to be positive and R 2 ( ¯ A 1 ) can be negative. Also, the property that the matrix ¯ A is positive definite implies thatå 3 i=1 R i ( ¯ A 1 )> 0. Sincet i = 1s sY i < 0, we know thatjt 2 j>jt 1 j andjt 2 j>jt 3 j. Therefore, even when R 2 ( ¯ A 1 ) is negative, we always have å 3 i=1 1 dt i R i ( ¯ A 1 ) < 0, which implies that the pricing equilibrium is still unique with CES demand in this two-sided asymmetric network. We can then follow the same procedure with MNL demand to show that the mergers under Scenario (1) and Scenario (3) always improve consumer surplus. Q.E.D. 104 B Chapter 3 Appendix B.1 Proof of Lemma 2 In the unconstrained problem, the first-order condition shows that: (p M 31 c)D 0 (p M 31 )+ D(p M 31 )= 0 (p M 34 2c)D 0 (p M 34 )+ D(p M 34 )= 0 (p M 35 2c)D 0 (p M 35 )+ D(p M 35 )= 0: Then we can rearrange these equations to get: p M 31 (1 1 e(p M 31 ) )= c p M 34 (1 1 e(p M 34 ) )= 2c p M 35 (1 1 e(p M 35 ) )= 3c: We clearly have: p M 31 = p M 41 = p M 52 = p M 62 = p M 12 = p M (c), p M 34 = p M 56 = p M 32 = p M 42 = p M 51 = p M 61 = p M (2c) and p M 35 = p M 36 = p M 45 = p M 46 = p M (3c). Since in Assumption 1 we assume that e(p) is nondecreasing, then we must have p M (c) p M (2c) p M (3c). Also, we can get from the previous equations that: 2p M 31 (1 1 e(p M 31 ) )= p M 34 (1 1 e(p M 34 ) ): It implies that p M (2c) 2p M (c) with assumption 1. Similarly, we can obtain: 3p M 31 (1 1 e(p M 31 ) )= p M 35 (1 1 e(p M 35 ) ): It shows that p M (3c) 3p M (c) with Assumption 1. 105 Lastly, we can obtain 3p M 34 (1 1 e(p M 34 ) )= 2p M 35 (1 1 e(p M 35 ) ): It implies that: 2p M (3c) 3p M (2c) with Assumption 1. Therefore, we can also get: p M (3c)= 1 3 p M (3c)+ 2 3 p M (3c) p M (c)+ p M (2c): Q.E.D. B.2 Proof of Lemma 3 We first show that for any market j g, where j2f3;4g and g2f5;6g, one of the carriers will choose to share in the j 2 or g 1 segments. Suppose both carriers monopolize the j 2 and g 1 segments for any j g market, travelers can go from city j to 2 with carrier A and from 2 to g with carrier B (denoted by trip A) or they can go from j to 1 with carrier A and from 1 to g with carrier B (denoted by trip B). Then consumers choose trip A if and only if u A j2 + v B 2g is less than or equal to v A j1 + u B 1g . Since both carriers want to monopolize the j 2 and 1 g segments, price competition will lead to set u A j2 = u B 1g = 2c and v A j1 = v B 2g =¥ and it means that both carriers earn zero profit for any j g market. Similarly, we can conclude that it is impossible to have both carriers share in the j 2 and g 1 segments for any j g market. Therefore, we can assume without loss of generality that carrier A monopolizes all the j 2 segments while carrier B shares in all the 1 g segments since both carriers can earn positive profits. In the 12 market, two airline carriers offer competing direct flights. Suppose we have p B 12 = c, then we show that carrier A’s best reply is p A 12 = c. If carrier A raises the price, then it will lose the 1 2 market since consumers will shift to carrier B, and other markets will not be affected. If career A reduces this price, then it generates negative profit in the 12 market, and profits in other markets will not be higher since constraints (3.3) and (3.4) get tighter. Similarly, it is easy to see 106 that s i 12 = c for i= A;B since it cannot be higher than c and carriers will earn negative profits if s i 12 < c. In all other connecting markets, carrier A chooses p A j1 , p A j2 , w A 5 j2 and w A 6 j2 for any j2f3;4g to maximize (p A j1 c)D(p A j1 )+(p A j2 2c)D(p A j2 )+(w A 5 j2 2c)D(w A 5 j2 + v B j 25 )+(w A 6 j2 2c)D(w A 6 j2 + v B j 26 ) subject to: p A j1 + c p A j2 w A 5 j2 p A j1 + c p A j2 w A 6 j2 : If we define p A j2 c to be s A j2 and define w A g j2 c to be v A g j2 , then we can equivalently transfer carrier A’s profit maximization problem to be: (p A j1 c)D(p A j1 )+(s A j2 c)D(s A j2 + c)+(v A 5 j2 c)D(v A 5 j2 + c+ v B j 25 )+(v A 6 j2 c)D(v A 6 j2 + c+ v B j 26 ) (B.1) subject to: p A j1 s A j2 v A 5 j2 p A j1 s A j2 v A 6 j2 : Despite that carrier A monopolizes all the j 2 segments, its optimal pricing is similar to sharing in the j2 segments due to the Bertrand competition in the inter-hub market. Equivalently, carrier A only focuses on the pricing of the j 1 segments contingent on consumers’ destinations. From carrier B’s perspective, it chooses p B 2g , s B 2g , v B 3 2g and v B 4 2g for any g2f5;6g to maximize: (p B 2g c)D(p B 2g )+(s B 2g c)D(s B 2g + c)+(v B 3 2g c)D(v B 3 2g + w A g 32 )+(v B 4 2g c)D(v B 4 2g + w A g 42 ) 107 subject to: p B 2g s B 2g v B 3 2g p B 2g s B 2g v B 4 2g : As we have redefined w A g 32 and w A g 42 in carrier A’s problem, we can easily rewrite carrier B’s problem to be: (p B 2g c)D(p B 2g )+(s B 2g c)D(s B 2g + c)+(v B 3 2g c)D(v B 3 2g + c+ v A g 32 )+(v B 4 2g c)D(v B 4 2g + c+ v A g 42 ) (B.2) subject to: p B 2g s B 2g v B 3 2g p B 2g s B 2g v B 4 2g : Therefore, based on (B.1) and (B.2) we get the results in Lemma 3. Clearly, for city pair mar- kets like 34 or 56, each carrier is an monopoly of the market and its pricing is not constrained by fare arbitrage conditions as we illustrated in Lemma 2 and price competition can only increase spoke market prices. 1 Q.E.D. B.3 Proof of Proposition 1 With strategic substitute strategies, we can easily see that all constraints in Lemma 4 (3.9) are not binding by using Lemma 1. The first-order conditions of the unconstrained problem shows that (v 1 c)( D 0 (v 1 + c+ v 0 1 ) D(v 1 + c+ v 0 1 ) )= 1: 1 We can easily verify this result in proposition 1 and 2. 108 Due to the symmetric assumption, v 1 = v 0 1 . We define p = 2v 1 +c. Then we can rewrite above equation to be (p 3c)( D 0 (p ) D(p ) )= 2: (B.3) Here p is the equilibrium price for interlining trips between rim cities. If two carriers form a joint venture, the first-order condition shows that the prices of interlining trips between rim cities, p= p M (3c), must satisfy the equation: (p 3c)( D 0 (p) D(p) )= 1: (B.4) Suppose p p , then if D 0 (p) D(p) is strictly decreasing in p, we must have: (p 3c)( D 0 (p ) D(p ) )(p 3c)( D 0 (p) D(p) ): Then it contradicts with equations B.3 and B.4. Therefore, p< p . Joint venture leads to lower prices for interlining tickets between rim cities than alliance membership alone. For inter-hub market, we can easily see that p 12 c. City pair markets like 3 4 and 5 6 are still monopolized by carrier A and carrier B and prices will be the same in the joint venture and the alliance membership. Lastly, since constraints in Lemma 3 are not binding, markets between city j and 2 or between city 1 and g have the same prices under the alliance membership or the joint venture. We can easily see that the profit maximization problem under the alliance membership is max s (s c)D(s+ c): We can denote s+ c to be p and rewrite above maximization problem: max p (p 2c)D(p): 109 This is equivalent to the profit maximization problem under the joint venture for markets be- tween j and 2 and between 1 and g. Q.E.D. B.4 Proof of Proposition 2 First, under the joint venture, the prices of interlining trips between rim cities can be determined from the profit maximization problem: max p (p 3c)D(p): Due to constant elasticity of demand, we can derive that p M (3c)= 3ac a1 . Therefore, to show p jg 2 so that the interlining market between rim cities has a meaningful equilibrium price. 2 Also, with constant elasticity, carriers’ pricing strategies are strategic complement and we can see that constraints (3.9) are binding in equilibrium: Let p , s and v be optimal prices to maximize the unconstrained functions: g 1 (p)=(p c)D(p), g 2 (s)=(s c)D(s+ c) and g 3 (v)=(v c)D(v+ c+ v 0 ) respectively. With strategic com- plement strategies, p < s < v and all these functions only have one unique global maximum. 3 To maximize the sum of g 1 (p), g 2 (s) and g 3 (v) subject to the constraint that v s p from Lemma 3, clearly we must have p p v . If p p s , then carrier will choose s= v= p as g 2 (s) and g 3 (v) are increasing function when s< s and v< v . If s < p v and constraint is unbinding, we must have v= s and s< p since g 3 (v) increases when v< v . Then this case cannot be optimal and carrier will reduce p to s since g 1 (p) decreases when p < p. Therefore, the constraint should be binding when s < p v . In equilibrium, we must have p= s= v. 2 This guarantees that each interlining market has a positive maximum point in the unconstrained problem. Other- wise, its optimal p goes to infinity 3 Our assumption that a > 2 ensures that all these functions are increasing if price is below optimum point and decreasing if price is higer than optimum point. 110 Since we havee(p)= pD 0 (p) D(p) = a, we can derive the general form of demand function to be D(p)= K p (a) ; (B.5) where K is a positive constant. Therefore, the profit maximization problem in Lemma 3 under the alliance membership be- comes max p f(p)= K(p c)p (a) + K(p c)(p+ c) (a) + 2K(p c)(p+ c+ p 0 ) (a) : (B.6) Note that in symmetric equilibrium p= p 0 , the derivative of the objective function, f 0 (p), is f 0 (p)=aK(p (a+1) )(p(1 1 a ) c) aK(p+ c) (a+1) ((p+ c)(1 1 a ) 2c) 2aK(2p+ c) (a+1) ((2p+ c)(1 1 a )(p+ 2c)): (B.7) When p > c(1+ 1 a ) 1 2 a , then f 0 (p) < 0; when p= c, then f 0 (p) > 0. Therefore, by intermediate value theorem, there must exist a p (c 0, and for any p> p , f 0 (p)< 0. Therefore, there exists a unique global maximum of (B.6) at p . For ˆ p= ac+ c 2 a1 , we plug it into the function f 0 (p) and we can get f 0 ( ˆ p)=aK(p (a+1) ) c 2a + aK(p+ c) (a+1) ( c 2a )+ 2aK(2p+ c) (a+1) ( 3 2 c a 1 ): The sufficient and necessary condition for ˆ p> p is that f 0 ( ˆ p)< 0 which is equivalent to have: ( 1 3 + 1 6a ) (a+1) ( 2 3 1 6a ) (a+1) > 6 a a 1 : 111 The left hand side of the above inequality is an increasing function of a> 0 and approaches infinity as a increases. The right hand side of the above inequality approaches 6 as a increases. As a result, there must exist an ˆ a> 0 such that when a> ˆ a the above inequality is satisfied, and thus, p jg = 2p + c< 2 ˆ p+ c= 3ac a1 = p M (3c). For city-pair markets j 2 and 1 g, if under the joint venture, the price is simply p M (2c)= 2ac a1 with constant elasticity. Since we showed that when a is big enough, we have p < ˆ p= ac+ c 2 a1 , and we can conclude that p j2 = p 1g = p + c < ˆ p+ c < p M (2c). The inter-hub market has a Bertrand competition under alliance membership and like before we always have p 12 < p M (c). All other markets are not changed after the alliance members form the joint venture. 4 Q.E.D. B.5 Proof of Proposition 3 We can follow the proof of Proposition 3 and show that the sufficient and necessary condition for ˆ p> p is that: (( 1 3 + 1 6a ) (a+1) ( 2 3 1 6a ) (a+1) ) a 1 3a > m 1 l (B.8) For any given a> 2, we can get that: d =(( 1 3 + 1 6a ) (a+1) ( 2 3 1 6a ) (a+1) ) a 1 3a (B.9) If m1 l <d, then ˆ p> p which means that p jg = 2p + c< 2 ˆ p+ c= 3ac a1 = p M (3c). All other results can be proved in the same way in Proposition 2. 4 Strategic complement pricing can only make spoke market prices increase and monopoly prices for all other markets are still not constrained by fare arbitrage conditions. 112

Abstract (if available)

Abstract This dissertation focuses on understanding competition between multiproduct firms and its antitrust implications. Multiproduct firms are frequently observed in different industries and play a crucial role in the economy. However, an important feature of competition between multiproduct firms is largely overlooked in the literature: firms may provide both complements and substitutes. In this dissertation, I systematically study this feature and its implications from different perspectives. ❧ Chapter 1 develops an empirical study about the price-increasing effect (also defined as the Edgeworth-Salinger effect in Luco and Marshall (2020)) from a merger of complements (i.e., a merger between firms producing complementary products) in the international airline industry. In this joint project with Guofu Tan, we focus on a set of city-pair markets where American Airlines offers a route operated by itself and a route jointly operated with British Airways (i.e., a codeshare flight where each carrier offers an essential component of the flight) in each market. These two major airline carriers formed a Joint Venture (JV) in 2010 to fully integrate their services in the transatlantic markets. We use the international airline data from the U.S. Department of Transportation (DOT)'s Airline Origin and Destination Survey, which contains a 10 percent random sample of itineraries involving a U.S. airport and a U.S. carrier. We conduct a differences-in-differences (DID) analysis, and our results indicate that the JV increased airfares of flights operated by American Airlines itself by three to four percent. Previous studies, which often emphasized the airfare reductions of codeshare flights in these markets, may have overestimated the benefits to consumers from JVs. ❧ Inspired by the empirical results from Chapter 1, Chapter 2 develops a network model of complements and substitutes like the international airline network where final products are combinations of intermediate products offered by (possibly) different multiproduct firms. I use the general demand system developed in Nocke and Schutz (2018), which nests multinomial logit (MNL) and constant elasticity of substitution (CES) demands, to model consumers' choices of final products. Also, I generalize the aggregative games approach to characterize the equilibrium prices and solve the technical challenges in analyzing the existence and uniqueness of a pricing equilibrium. Based on this framework, I identify conditions on networks under which a merger of complements can improve or reduce consumer surplus, respectively, even though such a merger may affect prices differently. Therefore, antitrust agencies can better protect consumers in merger reviews by evaluating the connections between firms' intermediate products. ❧ Chapter 3 explores another important feature of competition between multiproduct firms in the international airline industry. Specifically, each component of a connecting flight also corresponds to an independent city-pair market. I develop a theoretical model to show that intensive competition in inter-hub markets can constrain prices in other markets, which use inter-hub flights as part of their trips. As a result, depending on firms' pricing strategies for complementary flights, a JV may not always reduce the prices of codeshare flights through the elimination of double margins. This study explains why recent JVs cannot significantly reduce the prices of codeshare flights.

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Creator Zhang, Yinqi (author)

Core Title Essays on competition between multiproduct firms

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Economics

Publication Date 04/29/2021

Defense Date 03/18/2021

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag industrial organization,joint ventures,mergers of complements,multiproduct firms,OAI-PMH Harvest

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Tan, Guofu (committee chair), Carlsson, John Gunnar (committee member), Leung, Michael (committee member), Libgober, Jonathan (committee member)

Creator Email yinqizha@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-458023

Unique identifier UC11669042

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Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

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