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University of Southern California Dissertations and Theses
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Price competition among firms with a secondary source of revenue
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Price competition among firms with a secondary source of revenue
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PRICE COMPETITION AMONG FIRMS WITH A SECONDARY SOURCE OF REVENUE by Yejia Xu A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ECONOMICS) May 2022 Copyright 2022 Yejia Xu I dedicate this dissertation to my beloved wife Wenting for her constant love, support and companion, and our cats Mochi, Yitong, and Yoko for all the happiness and joy they bring us. ii Acknowledgements First of all, I would like to express my sincere gratitude to my advisor, Professor Guofu Tan, who has been a mentor to me in academia, in life, and in many other aspects, and has been providing me with invaluable support and guidance for the past six years. I would like to thank my committee members, Professor Jonathan Libgober and Professor Anthony Dukes, for their great help and advice during my preparation of the dissertation and other research projects. And I would like to thank my co-authors, Professor Feng Zhu from Harvard Business School and Professor Jin Li from Hong Kong University, for the valuable and inspiring experience working with them throughout an excellent project, which broadens my horizon and shows more possibilities for my future research. I am also grateful to the Department of Economics at University of Southern California, for both the financial support and our top-notch faculty, students and staffs that have helped and sup- ported me throughout my study for the Ph.D. program. It is not only where I met Zhen Chen and Yinqi Zhang, my best friends and comrades in exploring the fields of microeconomic theory and industrial organization, but also my wife Wenting Jiang, who has been providing me with constant love, support and companion, and has completed my life as my better half (along with our cats, Mochi, Yitong and Yoko, who are the cutest in the world). Finally, I would like to thank my parents and my grandparents. My grandparents are farmers living in the countryside. Unlike many people of their times, they insisted on sending my parents to the city for better education. My parents managed to finish high school but they never had the chance to go to college. I am the first in my family to receive a bachelor’s degree, a master’s degree, and soon a doctoral degree, but it would never have been possible if not for my parents’ and my grandparents’ emphasis on the importance of education and all their best wishes for me along the way. I love you all. iii Table of Contents Dedication ii Acknowledgements iii List of Tables vii List of Figures viii Abstract ix Chapter 1: Delegated Bundling 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Model Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Equilibrium Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4.1 No Bundling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4.2 Delegated Bundling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4.3 Product Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.4.4 Royalty Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.4.5 Equilibrium Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.5 Extensions and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.5.1 Royalty Rate Determination . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.5.2 Price Discrimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.5.3 Early Subscription . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.5.4 Flat-Rate Payment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Chapter 2: Competition in Freemium Services and the Impact of Ad-Avoidance Technologies 31 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.3 Model Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.3.1 Firms with Freemium Services . . . . . . . . . . . . . . . . . . . . . . . . 38 2.3.2 Consumers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.3.3 Ad-Avoidance Technologies . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.3.4 Timeline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.4 Monopoly Pricing Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 iv 2.4.1 Consumer Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.4.2 Optimal Pricing Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.4.3 Comparative Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.5 Oligopoly Pricing Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.5.1 Consumer Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.5.2 Freemium Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.5.3 Comparative Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.5.4 Welfare Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.6 Impact of Ad-Avoidance Technologies . . . . . . . . . . . . . . . . . . . . . . . . 62 2.6.1 Impact on Equilibrium Outcome . . . . . . . . . . . . . . . . . . . . . . . 63 2.6.2 Service Provision Game . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.6.3 Service Quality Investment . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.7 Calibration and Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.7.1 Parametric Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.7.2 Industry Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.7.3 Parameter Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2.7.4 Impact on Equilibrium Outcomes . . . . . . . . . . . . . . . . . . . . . . 69 2.7.5 Impact on Service Provisions . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.7.6 Impact on Quality Investment . . . . . . . . . . . . . . . . . . . . . . . . 77 2.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Chapter 3: Equilibrium Existence of Price Competition among Moonlighting Firms 80 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.3 Model and Equilibrium Existence . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.4.1 Per-Unit vs. Ad Valorem Royalty Licensing . . . . . . . . . . . . . . . . . 91 3.4.2 Profitable Outside Option . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.4.3 Equilibrium Uniqueness under Cross Holding . . . . . . . . . . . . . . . . 98 3.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 References 101 Appendices 106 A Proofs of Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 A.1 Technical Notes on Log-Concave (Density) Functions . . . . . . . . . . . 106 A.2 Proof of Lemma 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 A.3 Proof of Proposition 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 A.4 Proof of Proposition 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 A.5 Proof of Corollary 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 A.6 Proof of Lemma 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 A.7 Proof of Proposition 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 A.8 Proof of Proposition 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 A.9 Proof of Proposition 1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 A.10 Proof of Proposition 1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 v A.11 Proof of Proposition 1.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 A.12 Proof of Corollary 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 A.13 Proof of Proposition 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 A.14 Proof of Corollary 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 A.15 Proof of Proposition 1.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 B Proofs of Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 B.1 Proof of Proposition 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 B.2 Proof of Lemma 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 B.3 Proof of Corollary 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 B.4 Proof of Corollary 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 B.5 Proof of Corollary 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 B.6 Proof of Porposition 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 B.7 Proof of Corollary 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 B.8 Proof of Corollary 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 B.9 Proof of Corollary 2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 B.10 Proof of Corollary 2.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 B.11 Proof of Proposition 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 B.12 Proof of Proposition 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 B.13 Proof of Proposition 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 B.14 Proof of Proposition 2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 C Proofs of Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 C.1 Proof of Lemma 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 C.2 Proof of Lemma 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 C.3 Proof of Lemma 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 C.4 Proof of Lemma 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 C.5 Proof of Proposition 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 C.6 Proof of Lemma 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 C.7 Proof of Lemma 3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 C.8 Proof of Lemma 3.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 C.9 Proof of Corollary 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 C.10 Proof of Corollary 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 C.11 Proof of Proposition 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 C.12 Proof of Lemma 3.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 C.13 Proof of Proposition 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 C.14 Proof of Lemma 3.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 C.15 Proof of Proposition 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 C.16 Proof of Corollary 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 vi List of Tables 2.1 Market Performances under Alteredd,e = 5 . . . . . . . . . . . . . . . . . . . . 69 2.2 Market Performances under Alteredd,e = 1 . . . . . . . . . . . . . . . . . . . . 69 2.3 Market Performances under Alteredd,e = 0:5 . . . . . . . . . . . . . . . . . . . 69 2.4 Market Performances under Altered n,e = 1 . . . . . . . . . . . . . . . . . . . . . 71 2.5 Payoff Matrix of the Triopoly Game . . . . . . . . . . . . . . . . . . . . . . . . . 71 vii List of Figures 1.1 Consumer Segmentation for the Third-Party Firm (Equilibrium) . . . . . . . . . . 11 1.2 Consumer Segmentation for Product Firm 1 (p 1 > p ) . . . . . . . . . . . . . . . . 13 1.3 Consumer Segmentation for Product Firm 1 (p 1 < p ) . . . . . . . . . . . . . . . . 14 1.4 Consumer Segmentation for Product Firm 1 (Equilibrium) . . . . . . . . . . . . . 14 1.5 Royalty Rate Thresholda (t U[0;1]) . . . . . . . . . . . . . . . . . . . . . . . 22 1.6 Royalty Rate Determineda (t U[0;1], b 1 = b 2 ) . . . . . . . . . . . . . . . . . 25 2.1 Monopoly Consumer Segmentation (Premium Only) . . . . . . . . . . . . . . . . 43 2.2 Monopoly Consumer Segmentation (Freemium) . . . . . . . . . . . . . . . . . . . 44 2.3 Monopoly Consumer Segmentation (Free Only) . . . . . . . . . . . . . . . . . . . 44 2.4 Optimal Service Provision (v U[0;10];a = 0:5) . . . . . . . . . . . . . . . . . . 47 2.5 Optimal Service Provision (v U[0;10];a! 1) . . . . . . . . . . . . . . . . . . . 49 2.6 Oligopoly Consumer Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.7 Oligopoly Consumer Segmentation (Grouped) . . . . . . . . . . . . . . . . . . . . 53 2.8 Oligopoly Consumer Segmentation (p> p ) . . . . . . . . . . . . . . . . . . . . . 55 2.9 Oligopoly Consumer Segmentation (v 0 < p< p ) . . . . . . . . . . . . . . . . . . 56 2.10 Oligopoly Consumer Segmentation (p v 0 ) . . . . . . . . . . . . . . . . . . . . . 57 2.11 Optimal Service Provision (Comparison) . . . . . . . . . . . . . . . . . . . . . . . 59 2.12 Payoff Comparison (Triopoly) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 2.13 Consumer Surplus Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 2.14 Total Surplus Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 2.15 Quality Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 viii Abstract Firms may earn a secondary source of revenue in addition to their primary operations, and they usually have only partial or indirect control over their secondary revenues. In this dissertation, I study the price competition among these firms. The first two chapters discuss in detail two business models emerging in recent years. Chapter 1 studies delegated bundling, where inter-firm bundles are operated by a third-party firm and in competition with the single products. While product firms control the prices of their own products, they earn profits from both selling their own products and royalty fees paid by the third-party firm. I show that product firms are better off with than without delegated bundling if and only if products are less differentiated and the royalty rate is sufficiently high, and consumer and social welfare always improve. Chapter 2 (which is a joint work with Guofu Tan) studies freemium, a business strategy where firms provide both a free ad-sponsored service and a paid premium service. Firms control the subscription fee of the premium service and also earn an advertising revenue from the free service. Consumers may use ad-avoidance technologies (AATs) to avoid watching ads, which jeopardizes the freemium business model. We show that an increase in the penetration rate of AATs would reduce equilibrium prices and profits but benefit consumer welfare in the short run. However, if the firms respond by dropping free services or reducing investment in service qualities, both consumer and social welfare would suffer a loss. Chapter 3 summarizes the features of these firms and refers to them as “moonlighting firms”, which also nests patent and IP licensing, public firms and cross holding. I provide conditions for the quasi-concavity of moonlighting firms’ profit functions in their own prices and establish the equilibrium existence for the price competition. I show further three simple applications where I can apply these results to the literature and improve our understandings on some traditional topics of interest including royalty licensing, profitable outside option and cross holding. ix Chapter 1 Delegated Bundling 1.1 Introduction Many inter-firm bundles are operated by an independent third-party firm other than the product firms that produce the single products included in the bundle. I refer to this phenomenon as “Del- egated Bundling”. For a typical delegated bundle, take a look at Apple News Plus. Apple News Plus is a service provided by Apple that charges $9.99 per month and provides its users with access to over 200 magazines and newspapers owned by several major publishers in the United States and Canada, including Cond´ e Nast, Hearst Magazines, Meredith Corporation, News Corp, Rogers Media, and Time Inc. Here, the publishers are the product firms, and they delegate the operation of an inter- firm bundle of their magazines and newspapers to a third-party firm, Apple. Some other examples of delegated bundling include: • Many music and video streaming platforms, such as Spotify and Hulu, can be viewed as del- egated bundling, where the content providers serve as the product firms while the streaming platform is the third-party firm. • Tourists may purchase sightseeing passes from firms such as Go City, who bundles the attrac- tion tickets together and sells at a relatively lower price. Here, the agencies who manage the 1 attractions serve as the product firms and the sightseeing pass issuing firm is the third-party firm. To further differentiate from other business models, I focus on markets of delegated bundling with the following features: • The single products included in the bundle are in general differentiated substitutes. This differs my model from those where the single products are complements and the incentive to bundle comes from complementary synergies. • The single products are not only included in the bundle but also sold independently by the product firms, while the third-party firm cannot sell the single products individually but only in the form of a bundle. This differs my model from many traditional manufacturer-retailer models. Product firms turn to delegated bundling because they want to benefit from the introduction of an inter-firm bundle and the expansion of market, but they cannot run the bundle by themselves, due to either high negotiation costs or possible antitrust concerns. Delegating the operation of the bundle to a third-party firm can partially resolve these concerns, but it introduces new competition between product firms and the third-party firm. The overall effects of delegated bundling on prod- uct firms’ profits as well as social and consumer welfare are thus ambiguous and require further examination. To better understand the behavior of firms and consumers and the possible economic conse- quences of delegated bundling, I propose a modified Perloff-Salop model with discrete choice demand as the general framework to study delegated bundling, which can be described as follows. On the consumer side, each consumer is endowed with a random valuation for each single product. For the bundle, she can guarantee the maximum valuation among single products, and in addition enjoy an extra random utility from mix-and-match. Each consumer has unit demand and there is no outside option, and thus she chooses one single product or the bundle that maximizes her net utility. 2 On the firm side, product firms and the third-party firm are involved in a simultaneous pricing game. All firms incur zero marginal cost, but the third-party firm pays a royalty fee that is equally divided between the product firms, where the royalty rate is exogenously given. I show first that, when consumers are unlikely to earn an extremely high extra utility from mix- and-match, delegated bundling arises in equilibrium. The equilibrium price for single products is lower with than without delegated bundling due to the intensified competition; the equilibrium price for the bundle is higher than that for single products since consumers gain extra utility from mix-and-match, but cannot be too high as such extra utility is unlikely to be extreme as assumed. The equilibrium outcome under delegated bundling depends largely on the level of product dif- ferentiation and the royalty rate. As products become more differentiated, the competition within product firms softens, and thus both equilibrium prices and equilibrium profits increase, which follows the traditional intuition from the standard Perloff-Salop model. An increase in the roy- alty rate softens the competition between product firms and the third-party firm and increases the equilibrium prices, but has three effects on equilibrium profits and welfare: Profit redistribution, price increase, and bundle favoring. Comparing the relative magnitudes, I show that an increase in the royalty rate benefits product firms and the total surplus, but harms the third-party firm and consumer surplus. Compared to the no-bundling case, delegated bundling is always social welfare enhancing as it realizes the extra utility from mix-and-match for consumers, and it is also consumer welfare enhancing as it lowers the equilibrium price for single products and provides consumers with an extra option, i.e., the bundle. In terms of firm profits, the third-party firm is always better off, but product firms are better off if and only if the single products are less differentiated and the royalty rate is sufficiently high, i.e., product firms benefit from the introduction of an inter-firm bundle through a high royalty fee, and meanwhile they do not hurt much from the new competition against the third-party firm as the current competition within themselves is sufficiently intense already. 3 I also provide discussions on some other salient features in delegated bundling and how I can accommodate them in this model. If product firms negotiate their respective royalty rates with the third-party firm via Nash bargaining, there is a unique Nash bargaining solution where the royalty rates are symmetric and product firms are always better off in the delegated bundling implemented. If product firms are allowed to fully price discriminate the consumers under delegated bundling, they will propose a threshold pricing scheme, where they charge a higher price to consumers with higher valuations than without price discrimination, but they don’t necessarily earn higher profits. Product firms are always better off with than without bundling in equilibrium if the bundle has to be subscribed in advance, but such an equilibrium does not always exist. On the contrary, product firms are always worse off if the third-party firm pays product firms a discounted price per usage instead of a royalty fee. The rest of the paper is organized as follows: Section 1.2 reviews some related literature. Section 1.3 presents the model setup. Section 1.4 examines and compares the pricing equilibrium with and without delegated bundling. Section 1.5 discusses some extensions to the main model. Section 1.6 concludes the paper and provides some directions for future work. All proofs are presented in Appendix A. 1.2 Literature Review This paper is related to the massive literature on bundling, which starts with the idea that multi- product monopolists use bundling as a price discrimination device to extract more surplus from consumers (Adams & Yellen, 1976; McAfee, McMillan, & Whinston, 1989; Schmalensee, 1982, 1984; Stigler, 1963). This channel is not available in this model as the bundle is not owned directly by the product firms, so product firms cannot perform price discrimination based on the purchasing behavior. As an extension, however, I consider the possibility that, through the usage data collected by the third-party firm, product firms are still capable of learning the preferences of the consumers and, in the extreme case, perform full price discrimination on them. 4 The study of bundles is then extended to the oligopoly market and can be further categorized based on the overall effects of bundling. In the most common case, bundling is expected to deter entry, foreclose, or reduce the profits of opponents, possibly through the leverage of market powers (Nalebuff, 2004; Peitz, 2008; Whinston, 1990). Bundling may, however, intensify competition and harm all firms, which is more likely to happen when mixed bundling is used (Ahn & Yoon, 2012; Anderson & Leruth, 1993; Innocenti, Menicucci, et al., 2019; J. Zhou, 2021) or when the firms are rather symmetric (Armstrong, 2006b; Hurkens, Jeon, & Menicucci, 2019; Matutes & Regibeau, 1988; Reisinger, 2004; Shuai, Yang, & Zhang, 2022). Conversely, bundling can also relax competition and benefit all firms, when the number of firms is above some threshold (S.-H. Kim & Choi, 2015; J. Zhou, 2017) or the firms are quite asymmetric (Hurkens, Jeon, & Menicucci, 2019; Prasad, Venkatesh, & Mahajan, 2010; S. Zhou, Song, & Gavirneni, 2020), possibly through increased product differentiation (Carbajo, De Meza, & Seidmann, 1990; Chen, 1997). Alternatively, pure bundling can eliminate the self- cannibalization of products within the same firm (Giri, Mondal, & Maiti, 2020; Jeon & Menicucci, 2012). Among the different ways of bundling, this paper is most closely related to the mixed bundling case, where both the bundle and every single product included in the bundle are available in the market. The results thus follow from the literature, that the equilibrium prices and profits are likely to be lower than those without bundling due to intensified competition. So far I have been looking at the literature on intra-firm bundles, while in this paper I consider inter-firm bundles. Despite a large amount of work done on the former topic, there are only a few papers that look into the theory of the latter. A major difference of this paper with the literature of inter-firm bundles lies in how the bundle is operated. In the literature, the inter-firm bundle is either provided by a strategic alliance, where the consumer is forced to buy both or neither (S.-H. Kim & Hahn, 2021; Mialon, 2014), or through a bundle discount, where the consumer pays a lower price for purchasing the bundle than individual prices combined (Armstrong, 2013; Brito & Vasconcelos, 2015; Gans & King, 2006; Hahn & 5 Kim, 2016; Huang, Nagarajan, & Soˇ si´ c, 2013; Jeitschko, Jung, & Kim, 2017). Either way, the final effective price of the bundle is determined by product firms. Instead, in this model, the bundle is operated and priced independently by the delegated third-party firm, while the product firms can only affect the bundle price through the delegation choice as well as the prices of their own single products in the pricing game. In terms of product differentiation, these papers fall either under the two-dimensional Hotelling framework introduced in Matutes and Regibeau (1988) for a market of two products and four single-product firms, or under a more general random utility framework for a market of two prod- ucts and two single-product firms. The only exception is Huang, Nagarajan, and Soˇ si´ c (2013), which skips the micro-structure and makes assumptions directly on demand curves. I also adopt the random utility framework but consider n 2 single-product firms. For tractability, I focus primarily on products that are substitutes, while Armstrong (2013) and Jeitschko, Jung, and Kim (2017) allow arbitrary patterns, and the other papers consider independent products or comple- ments. 1.3 Model Setup Consider a market of n 2 product firms, a third-party firm, and a continuum of consumers. The measure of consumers is normalized to 1. Each product firm produces and sells a single product while the third-party firm sells a bundle of all n single products. The unit production cost of any single product is normalized to zero. The third-party firm incurs no extra cost in producing and selling the bundle but has to pay the product firms a royalty fee equal to a share a of the profits from bundle sales, which is equally divided among product firms. The single products are horizontally differentiated across the product firms. To model such product differentiation, I adopt the random utility framework in Perloff and Salop (1985). Let q i denote the random valuation of single product i for each consumer. I assume that q i is IID across consumers and firms i2 N where N =f1;2;:::;ng. Let F q (x) be the common cumulative 6 distribution function (CDF) for allq i with supportR and a corresponding density function f q (x). I assume that f q (x) is log-concave in x. The log-concavity of the density function is often used to guarantee the existence of a pricing equilibrium and that the equilibrium price can be determined from first-order conditions (see for example J. Zhou (2017)). Alternatively, the consumer may purchase the bundle. Since the bundle provides the consumer with access to all n products, the consumer can guarantee the maximum among allq i by consuming only the single product with the highest valuation. In addition, the consumer enjoys a random extra utilityt 0 from mix-and-match between single products. I assume thatt is IID across consumers and is independent of the random valuationsq =(q i ) i2N . Let F t (x) be the CDF fort with support R + and a corresponding density function f t (x). I also assume that f t (x) is log-concave in x. I consider a discrete-choice framework where each consumer purchases either a single product or the bundle. As often assumed in the literature, the market is fully covered, which will be the case if consumers do not have outside options, or if on top of the random valuations, consumers have a sufficiently high basic valuation for each single product as well as the bundle. For a consumer with random valuations q =(q i ) i2N , if she consumes single product i priced at p i , she obtains a net utility u i =q i p i : (1.1) If she consumes the bundle priced at p b , she obtains a net utility u b = max i2N q i +t p b : (1.2) As a rational agent, each consumer will purchase a single product or the bundle which gives her the highest net utility. Product firms and the third-party firm compete in prices. The third-party firm chooses p b to maximize P b (p; p b )=(1a)p b (p; p b )=(1a)p b D b (p; p b ); (1.3) 7 where p=(p i ) i2N denotes the price vector of all single products,p b the profits from bundle sales and D b the demand for the bundle. Each product firm i chooses p i to maximize P i (p; p b )=p i (p; p b )+ a n p b (p; p b )= p i D i (p; p b )+ a n p b D b (p; p b ); (1.4) wherep i denotes profits from sales of single product i and D i the demand for single product i. The timing of this model is as follows: Product firms and the third-party firm choose the prices for single products and the bundle simultaneously, and then consumers make their purchase decisions after observing all the prices and realized valuations q and t. The equilibrium concept used in this paper is sub-game perfect equilibrium. 1.4 Equilibrium Analysis 1.4.1 No Bundling Let us start with the benchmark case of no bundling. Without bundling, the model reduces to the standard Perloff-Salop model with n product firms and no outside options. Since the product firms are ex-ante symmetric, I focus on the symmetric equilibrium where all product firms charge the same price p c , which I refer to as the “no-bundling equilibrium”. To facilitate the analysis, I introduce the following notation as the consumer’s excessive valua- tion of single product i compared to other single products: c i =q i max j6=i q j : (1.5) Due to the symmetry ofq i ’s, all c i ’s follow a common CDF denoted as F c (x) with supportR and a corresponding density function f c (x), and f c (x) inherits the log-concavity from f q (x). 1 1 Note thatc i ’s are not independent across firms. 8 To determine the equilibrium price p c , suppose product firm 1 deviates to charge p 1 unilater- ally. The demand for product 1 becomes D 1 (p 1 ; p c )= Pr(q 1 p 1 > max i6=1 fq i p c g) = Pr(c 1 > p 1 p c )= 1 F c (p 1 p c ) and product firm 1 earns p 1 (p 1 ; p c )= p 1 D 1 (p 1 ; p c )= p 1 [1 F c (p 1 p c )]: In equilibrium, product firm 1 should maximize its profit at p 1 = p c , which yields the following first-order condition: p c = 1 F c (0) f c (0) : (1.6) To interpret the equilibrium condition, rewrite it as 1 F c (0)= p c f c (0): In equilibrium, 1 F c (0)= 1=n on the left-hand side denotes the demand for each single product, and f c (0) on the right-hand side denotes the density of marginal consumers who are indifferent between this single product and the best among its competitors. For any product firm, the marginal benefit of a price increase is proportional to its demand, and the marginal cost is losing all the marginal consumers, which should be equal in equilibrium. Lemma 1.1. There exists a unique no-bundling equilibrium. The existence of a no-bundling equilibrium is established on the log-concavity of f c (x) and the uniqueness follows directly from the unique expression of p c . In the no-bundling equilibrium, product firms split the market equally and each earns P c = 1 n p c : (1.7) 9 Since the product firms and the third-party firm incur no cost in production, the total surplus for this economy equals the consumers’ realized valuations. In the no-bundling equilibrium, each consumer realizes her highest valuation among single products and thus T S c =E[max i2N q i ]: (1.8) On the other hand, the consumer surplus for this economy equals the difference between total surplus and the firms’ total profits, and thus CS c =E[max i2N q i ] p c : (1.9) 1.4.2 Delegated Bundling Now consider the case with delegated bundling. Similarly, I focus on the symmetric equilibrium where all product firms charge the same price p , while the third-party firm charges a possibly different price p b , which I refer to as the “delegated-bundling equilibrium”. Let us start with examining the third-party firm’s pricing decisions. When the single products are priced at p , the demand for the bundle priced at p b is D b (p b ; p )= Pr(max i2N q i +t p b > max i2N fq i p g) = Pr(t > p b p )= 1 F t (p b p ) and the third-party firm earns P b (p b ; p )=(1a)p b D b (p b ; p )=(1a)p b [1 F t (p b p )]: In equilibrium, the third-party firm should choose p b to maximize its profits, which yields the following first-order condition: 1 p b = f t (p b p ) 1 F t (p b p ) : (1.10) 10 Figure 1.1: Consumer Segmentation for the Third-Party Firm (Equilibrium) To interpret this equilibrium condition, rewrite it as 1 F t (p b p )= p b f t (p b p ): In equilibrium, 1 F t (p b p ) on the left-hand side denotes the demand for the bundle, and f t (p b p ) on the right-hand side denotes the density of marginal consumers who are indifferent between the bundle and the best single product. For the third-party firm, the marginal benefit of a price increase is proportional to its demand, and the marginal cost is losing all the marginal consumers, which should be equal in equilibrium. Also note that the third-party firm earns a fixed proportion of the profits from bundle sales, and thus the royalty rate a does not enter the equilibrium condition. Now turn to the product firms. Suppose product firm 1 deviates to charge p 1 unilaterally. Since product firm 1 not only earns profits from the sales of single product 1 but also shares the royalty fee from the third-party firm, it cares about the demand for both single product 1 and the bundle. 11 Under this deviation, consumers will purchase product 1 if and only if 8 > > < > > : q 1 p 1 > max i6=1 fq i p g q 1 p 1 > max i2N q i +t p b , 8 > > > > > < > > > > > : c 1 > p 1 p t <c 1 p 1 + p b t > < > > : max i2N q i +t p b > max i6=1 fq i p g max i2N q i +t p b >q 1 p 1 , 8 > < > : maxfc 1 ;0g+t > p b p maxf0;c 1 g+t > p b p 1 : The demand for both single product 1 and the bundle thus depends on the relations of p 1 and p . When p 1 p , the demand for product 1 is D 1 (p 1 ; p ; p b )=[1 F c (p 1 p )]F t (p b p 1 ); and the demand for the bundle is D b (p 1 ; p ; p b )= Z 0 ¥ f c (x)[1 F t (p b p )]dx + Z p 1 p 0 f c (x)[1 F t (x+ p b p )]dx + Z ¥ p 1 p f c (x)[1 F t (p b p 1 )]dx: When p 1 < p , the demand for product 1 is D 1 (p 1 ; p ; p b )= Z 0 p 1 p f c (x)F t (x+ p b p 1 )dx+[1 F c (0)]F t (p b p 1 ); 12 Figure 1.2: Consumer Segmentation for Product Firm 1 (p 1 > p ) and the demand for the bundle is D b (p 1 ; p ; p b )= Z p 1 p ¥ f c (x)[1 F t (p b p )]dx + Z 0 p 1 p f c (x)[1 F t (x+ p b p 1 )]dx + Z ¥ 0 f c (x)[1 F t (p b p 1 )]dx: Overall, product firm 1 earns P 1 (p 1 ; p ; p b )= p 1 D 1 (p 1 ; p ; p b )+ a n p b D b (p 1 ; p ; p b ): In equilibrium, product firm 1 should maximize its profits at p 1 = p . Although the profit function changes its functional form before and after p 1 = p , examining the left and right derivatives respectively yields the same first-order condition for the equilibrium price p : 1 p = f c (0) 1 F c (0) +(1 a p b np ) f t (p b p ) F t (p b p ) : (1.11) 13 Figure 1.3: Consumer Segmentation for Product Firm 1 (p 1 < p ) Figure 1.4: Consumer Segmentation for Product Firm 1 (Equilibrium) 14 To interpret this equilibrium condition, rewrite it as [1 F c (0)]F t (p b p )= p f c (0)F t (p b p )+(p a n p b )[1 F c (0)] f t (p b p ): In equilibrium, [1 F c (0)]F t (p b p ) on the left-hand side denotes the demand for any single product. As for the right-hand side, f c (0)F t (p b p ) in the first term denotes the density of marginal consumers who are indifferent between this single product and the best among its com- petitors, and[1 F c (0)] f t (p b p ) in the second term denotes the density of marginal consumers who are indifferent between this single product and the bundle. For any product firm, the marginal benefit of a price increase is proportional to its demand, and the marginal cost is two-fold: it loses the entire profits p from marginal consumers who turn to other single products, but loses only part of the profits(p a n p b ) from marginal consumers who turn to the bundle due to the compensation from the royalty fee. Again, the marginal benefit and marginal cost should be equal in equilibrium. To sum up, the equilibrium prices p and p b should solve 8 > > > < > > > : 1 p b = f t (p b p ) 1 F t (p b p ) 1 p = f c (0) 1 F c (0) +(1 a p b np ) f t (p b p ) F t (p b p ) : (1.12) It requires more than log-concavity to ensure the existence and uniqueness of the delegated- bundling equilibrium. To present the regularity condition I need, define the generalized virtual valuation function as J t;a (v)= v(1 a) 1 F t (v) f t (v) : (1.13) The log-concavity of f t (x) ensures that J t;a (v) is strictly increasing in v for any a2[0;1] and thus its inverse function J 1 t;a (v) is well-defined. Assumption 1.1. 1 F c (0) f c (0) > J 1 t;1=n (0) n 1 : (1.14) 15 Proposition 1.1. There exists a unique delegated-bundling equilibrium. Assumption 1.1 links the distributions of random valuationsq and the extra utilityt. The left- hand side of the inequality is simply the no-bundling equilibrium price p c , and it depends only on F q (x), while the right-hand side of the inequality depends only on the distribution F t (x). Fixing the distribution F q (x) and thus the no-bundling equilibrium price p c , Assumption 1.1 requires that, when consumers purchase the bundle and mix-and-match between the single prod- ucts, they are unlikely to earn an extremely high extra utilityt. To illustrate such an interpretation, consider the following examples with specific distributions: Example 1: Ift follows an exponential distribution with parameterl, i.e., for any x 0, F t (x)= 1 e 1 l x ; f t (x)= 1 l e 1 l x ; then the assumption reduces to l < np c : Assumption 1.1 places an upper bound on the parameterl, which implies that the corresponding exponential distribution has a larger mass on lower values. Example 2: If t follows a uniform distribution on the interval [0;K] where K > p c , i.e., for any 0 x K, F t (x)= x K ; f t (x)= 1 K ; then the assumption reduces to K <(2n 1)p c : Similarly, Assumption 1.1 places an upper bound on K, which implies that the corresponding uniform distribution has all its mass on lower values. 16 Corollary 1.1. In the delegated-bundling equilibrium, p < p c ; p < p b < n a p : (1.15) To see why Corollary 1.1 holds, note first that the introduction of the delegated bundle further intensifies competition for consumers, leading to a lower equilibrium price p < p c for the single products. Meanwhile, since consumers who purchase the bundle not only guarantee the highest valuation among all single products but also earn an extra utility from mix-and-match, the bundle must be priced higher than the single products, but not too high as restricted by Assumption 1.1. In the delegated-bundling equilibrium, each product firm earns P = 1 n p F t (p b p )+ a n p b [1 F t (p b p )] (1.16) and the third-party firm earns P b =(1a)p b [1 F t (p b p )]: (1.17) For the total surplus, consumers are guaranteed to realize their highest valuations for single prod- ucts, and consumers who purchase the bundle further realize their extra utilityt. Thus, T S b =E[max i2N q i ]+E[tjt > p b p ][1 F t (p b p )]: (1.18) For the consumer surplus, the firms earn a total profit of nP +P b = p F t (p b p )+ p b [1 F t (p b p )] and thus CS b = T S b p F t (p b p ) p b [1 F t (p b p )]: (1.19) 17 1.4.3 Product Differentiation In this subsection, I examine how product differentiation, as reflected in the distribution F q (x), affects the equilibrium outcome. To measure the product differentiation, I introduce first the idea of dispersive order: Consider two random variables X F and X G . Let F and G be their CDFs and supposeE[X F ]=E[X G ]. The random variable X G is said to be less dispersed than X F in the dispersive order if G 1 (t)G 1 (t 0 ) F 1 (t)F 1 (t 0 ) for any 0< t 0 t< 1. For simplicity, I refer to this relation as “G is less dispersed than F”. Further, when the consumers’ valuations IID follow a more dispersed distribution, I interpret it as that the single products are more differentiated. Lemma 1.2 (Lemma 2(i), J. Zhou (2017)). The no-bundling equilibrium price p c increases as F q (x) becomes more dispersed. Lemma 1.2 shows that a higher level of product differentiation softens the competition within the product firms and leads to a higher equilibrium price in the no-bundling equilibrium. In the delegated-bundling equilibrium, note first that the distribution F q (x) and its implied distribution F c (x) enter the equilibrium only through the term p c = 1 F c (0) f c (0) : In other words, the no-bundling equilibrium price p c is a sufficient statistic for the effects of any changes in F q (x) on the delegated-bundling equilibrium. Therefore, combined with Lemma 1.2, p c can be regarded as an exogenous parameter that measures product differentiation. Proposition 1.2. Both equilibrium prices p and p b and equilibrium profitsP andP b increase as F q (x) becomes more dispersed. Proposition 1.2 suggests that the intuition in the no-bundling equilibrium from the standard Perloff-Salop model goes through in the delegated-bundling equilibrium. When the single products become more differentiated, as reflected in a higher p c , it also softens the competition within the 18 product firms in the delegated-bundling equilibrium, leading to higher equilibrium prices for both single products and the bundle, and both product firms and the third-party firm are better off. As a side note, if I fix the distribution F q (x) but increase the number of product firms n, such increased competition can be regarded as if the single products become less differentiated. Further- more, each product firm earns a smaller share of the royalty fee and is even less willing to increase its price. Corollary 1.2 summarizes the combined effects on the equilibrium outcome. Corollary 1.2. Both equilibrium prices p and p b and equilibrium profits nP andP b decrease as the number of product firms n increases. 1.4.4 Royalty Rate As for the effects of the royalty rate a on the equilibrium outcome, I start with the following comparative statics on equilibrium prices. Lemma 1.3. Both equilibrium prices p and p b increase in the royalty rate a, and p increases faster than p b . Recall that product firms divide equally the royalty fee from the third-party firm, creating an overlap in their interests. As a increases, product firms have more interests in common, which softens the competition within themselves and raises the equilibrium prices. Between the two equilibrium prices, the single product price p is affected more by such competition than the bundle price p b , and thus the former increases faster than the latter. Following Lemma 1.3, an increase in the royalty rate a has three effects on the equilibrium outcome: • Profit redistribution: More profits from bundle sales are redistributed from the third-party firm to product firms. • Price increase: Equilibrium prices of both single products and the bundle increase. 19 • Bundle favoring: Since the equilibrium price of single products increases faster than that of the bundle, more consumers are attracted to the bundle. Proposition 1.3. Product firms’ equilibrium profitsP and social welfare T S b increase ina, while the third-party firm’s equilibrium profitsP b and consumer welfare CS b decrease ina. To understand the reasons behind Proposition 1.3, examine the three effects above in terms of profits and welfare. For the firms’ profits: • Profit redistribution benefits product firms but harms the third-party firm. • Price increase benefits both product firms and the third-party firm. • Bundle favoring benefits the third-party firm but harms the product firms. It turns out that the profit redistribution effect dominates the other two, and overall an increase in a benefits product firms and hurts the third-party firm. Similarly, for social and consumer welfare: • Profit redistribution is neutral to both social and consumer welfare as it deals with transfers between firms. • Price increase is neutral to social welfare as it deals with transfers between firms and con- sumers, but harms consumer welfare. • Bundle favoring benefits both social and consumer welfare as it attracts more consumers to the bundle and realizes more extra utilityt. Social welfare is only affected by the bundle favoring effect and thus it always benefits from an increase in a. Consumer welfare, however, is affected more by the price increase effect and thus suffers a loss. 20 1.4.5 Equilibrium Comparison Finally, I provide a comparison between the no-bundling and delegated-bundling equilibrium. Proposition 1.4. Delegated bundling is social and consumer welfare enhancing. The introduction of delegated bundling allows consumers to realize the extra utility t from mix-and-match. Sincet is always non-negative, social welfare must be improved under delegated bundling. Consumer welfare is affected by delegated bundling in two ways. Delegated bundling creates new competition between the single products and the bundle, which lowers the price of single products and benefits the consumers. Besides, since consumers are rational agents, they move away from single products to the bundle if and only if they get a higher consumer surplus from the latter. Both effects increase consumer welfare. In terms of profits, the third-party firm is always better off under delegated bundling, as it earns zero profit in the no-bundling equilibrium. Product firms, however, face two effects in opposite directions: They suffer a loss from the price reduction due to competition with the third-party firm, and meanwhile, they are compensated by the royalty fee paid by the third-party firm. The overall effect can be summarized as follows: Proposition 1.5. There exists a threshold ¯ p c and an increasing threshold functiona(p c )< 1 such that product firms are better off with than without delegated bundling if and only if p c < ¯ p c and a >a(p c ). Proposition 1.5 provides a sufficient and necessary condition for product firms to be better off with than without delegated bundling. Recall that the no-bundling equilibrium price p c measures the level of product differentiation. When single products are less differentiated, so that p c is sufficiently low, the competition within product firms is already intense, and the new competition with the third-party firm will do little harm to them. If further the royalty rate a is sufficiently high, the loss due to this new competition can be covered by the royalty fee from the third-party firm, and eventually product firms are better off with delegated bundling. 21 As I have mentioned, sometimes product firms turn to delegated bundling because they cannot run the bundle by themselves. Proposition 1.5 thus suggests that, when the single products are quite differentiated, product firms already earn a high profit from the sales of single products, and delegated bundling cannot be implemented. On the contrary, when the single products are less differentiated, a third-party firm can propose to run the bundle for the product firms, and product firms will accept the offer provided that the proposed royalty rate is sufficiently high. When delegated bundling is implemented in the latter case, it will be a Pareto improvement to the no-bundling equilibrium, as product firms, the third-party firm, and the consumers will all be better off. Figure 1.5: Royalty Rate Thresholda (t U[0;1]) Figure 1.5 illustrates the thresholda(p c ) in the duopoly (n= 2) case wheret follows a uniform distribution over[0;1]. The vertical line on the left denotes the lower bound of p c for equilibrium 22 existence as required by Assumption 1.1, while the vertical line on the right denotes the upper bound ¯ p c stated in Proposition 1.5. 1.5 Extensions and Discussions In this section, I provide discussions on some other salient features in the practice of delegated bundling and how to accommodate them in this model. In section 1.5.1, I study how the royalty rates are determined by Nash bargaining between product firms and the third-party firm. Sec- tion 1.5.2 considers the possibility that product firms can price discriminate the consumers under delegated bundling. Sometimes consumers have to subscribe to the bundle before the random val- uations are realized, and I study this timing issue in section 1.5.3. Finally, I consider in section 1.5.4 if the third-party firm pays product firms a discounted flat-rate price per usage instead of a royalty fee. 1.5.1 Royalty Rate Determination In the main model, I assume that the royalty rate a is exogenously given and the royalty fee is equally divided among product firms. In practice, however, the royalty rate is usually determined by negotiations between product firms and the third-party firm and does not have to be equal across product firms. To address this possibility, consider a duopoly version of the model, where two product firms negotiate with a third-party firm for their respective royalty ratesa 1 anda 2 in a three-way Nash bargaining. To set up the Nash bargaining, I assume first that, for anya 1 anda 2 satisfying 0a 1 1, 0 a 2 1 anda 1 +a 2 1, there exists a unique pricing equilibrium. DenoteP e 1 (a 1 ;a 2 ),P e 2 (a 1 ;a 2 ) andP e b (a 1 ;a 2 ) as the equilibrium profits for product firm 1, product firm 2 and the third-party firm respectively. As in the main model, when the negotiation breaks down, both product firms earnP c and the third-party firm earns 0 as in the no-bundling equilibrium. Note thatP c does not vary with a 1 anda 2 . 23 Suppose both product firms have the same bargaining power b 1 and the third-party firm has a bargaining power b 2 . Therefore, in the Nash bargaining, the firms choosea 1 anda 2 to maximize the following Nash product (P e 1 (a 1 ;a 2 )P c ) b 1 (P e 2 (a 1 ;a 2 )P c ) b 1 (P e b (a 1 ;a 2 )) b 2 ; (1.20) which yields the following first-order conditions 8 > > > < > > > : b 1 ¶P e 1 (a 1 ;a 2 )=¶a 1 P e 1 (a 1 ;a 2 )P c + b 1 ¶P e 2 (a 1 ;a 2 )=¶a 1 P e 2 (a 1 ;a 2 )P c + b 2 ¶P e b (a 1 ;a 2 )=¶a 1 P e b (a 1 ;a 2 ) = 0 b 1 ¶P e 1 (a 1 ;a 2 )=¶a 2 P e 1 (a 1 ;a 2 )P c + b 1 ¶P e 2 (a 1 ;a 2 )=¶a 2 P e 2 (a 1 ;a 2 )P c + b 2 ¶P e b (a 1 ;a 2 )=¶a 2 P e b (a 1 ;a 2 ) = 0 : (1.21) To solve this Nash bargaining solution in general requires us to fully characterize the asymmetric pricing equilibrium for any valid pair of (a 1 ;a 2 ) which is a challenging task. For the ease of exposition, I further assume that, if there exists a solution to equations (1.21), it must be a unique solution to the Nash bargaining problem. Proposition 1.6. For any p c < ¯ p c , there exists a uniquea 2(a(p c );1) such that(a =2;a =2) is the unique solution to the Nash bargaining problem, and P P c P b = 2b 1 b 2 dP =da dP b =da (1.22) evaluated ata =a . Note thatP andP b denote the equilibrium profits in the symmetric case wherea 1 =a 2 =a=2 as in the main model. Proposition 1.6 thus shows that, due to the ex-ante symmetry in product firms, the unique solution to the Nash bargaining problem must also be symmetric, and to deter- mine this royalty rate it is sufficient to solve the pricing equilibrium for only the symmetric case wherea 1 =a 2 . Moreover, when the single products are less differentiated and thus the no-bundling equilibrium price p c falls below the threshold ¯ p c , the Nash bargaining always results in a royalty ratea above 24 the threshold a(p c ) so that the delegated bundling can be implemented where product firms are better off than the no-bundling case. Figure 1.6 illustrates the royalty ratea determined by Nash Figure 1.6: Royalty Rate Determineda (t U[0;1], b 1 = b 2 ) bargaining whent follows a uniform distribution over[0;1] and product firms and the third-party firm have the same bargaining power b 1 = b 2 . As suggested in Proposition 1.6, for any p c < ¯ p c , the determined royalty rate a always falls above a(p c ) so that the delegated bundling can be implemented and product firms are better off. 1.5.2 Price Discrimination Product firms may delegate the inter-firm bundle to a third-party firm if the third-party firm has an advantage in data collection and analysis. If the third-party firm can exploit more from the bundle usage data and share their findings with product firms, product firms may be able to perform price discrimination over the consumers and gain extra profits. 25 To address such a possibility, consider the following modification to the main model: Under delegated bundling, product firms can perform full price discrimination on the consumers and offer a personalized pricing scheme p i (q) contingent on the consumers’ valuations q. Note that the consumers still havet as their private information, since it is sunk and cannot be learned from bundle usage. I also assume that the third-party firm cannot price discriminate and charges a fixed price p b . The model remains the same otherwise. Proposition 1.7. With price discrimination, there exists a pricing equilibrium under delegated bundling where the third-party firm charges ˜ p b and each product firm i uses a threshold pricing scheme p i (q)= minf ˜ p;c i (q)g: (1.23) The intuition behind Proposition 1.7 is that each product firm is involved in two competitions: A complete information Bertrand competition within the product firms, where they race to the bottom and the winning product firm i must charge a price lower than the excessive valuationc i (q), and a competition between the winning product firm and the third-party firm, where consumers make their decisions based only on the extra utility t as the max i2N q i term cancels out, and thus the optimal price ˜ p is non-contingent onq. To compare the equilibrium with and without price discrimination, I consider the specific case wheret follows the Exponential distribution. Corollary 1.3. Ift follows the Exponential distribution with parameterl, then ˜ p b = p b =l; ˜ p> p : (1.24) With price discrimination, the product firms’ maximum price ˜ p exceeds the delegated-bundling equilibrium price p . Nevertheless, since product firms also charge a lower price than p to con- sumers with a worse realization ofq, the changes to equilibrium profits are ambiguous. 26 1.5.3 Early Subscription The timing might be another issue involved with delegated bundling, as some bundles take the form of monthly subscriptions. Consumers need to decide ahead of time whether to subscribe to the bundle before their random valuationsq are realized, and their decisions rely only ont, which is more consistent over time. Suppose that the bundle is priced at p b and each product firm i chooses a price p i for their products. Consumers will purchase the bundle in advance if and only if E[max i2N q i ]+t p b max i2N fE[q i ] p i g,tk+ p b min i2N p i ; where k D =E[max i2N q i ]E[q i ]> 0: (1.25) Since it is less likely for all single products to realize an unfavorable valuation than one of them does, the bundle becomes more attractive in this case. Consumers who decide not to purchase the bundle stay on the market and wait for the realiza- tion of theirq, after whicht becomes sunk and they purchase the single product with the highest net utility. Proposition 1.8. With early subscription, if there exists a symmetric pricing equilibrium where all product firms charge ¯ p and the third-party firm charges ¯ p b , then ¯ p b >a ¯ p b ¯ p p c ; (1.26) and product firms are always better off with than without delegated bundling. Note that when consumers decide whether to purchase the bundle, only the minimum sin- gle product price matters. Therefore, in the symmetric pricing equilibrium, when a product firm increases its price unilaterally, it does not affect consumers’ bundle purchase decision and is essen- tially competing with only the other product firms, similar to the no-bundling case. Thus, to avoid 27 such deviation, the equilibrium price ¯ p must be higher than p c . The high equilibrium price for single products ¯ p thus allows a higher equilibrium bundle price ¯ p b as well, and eventually product firms can earn a higher profit with than without bundling. However, such an equilibrium does not always exist: Corollary 1.4. If such an equilibrium exists, then p c a 1a [k+ J 1 t;a (k)]: (1.27) And for any given p c , there exists ¯ a2(0;1] such that the condition holds if and only ifa2[ ¯ a;1]. Similar to Proposition 1.5, for product firms to be better off with than without delegated bundling, it is necessary that the single products are less differentiated, reflected in a sufficiently low no-bundling equilibrium price p c , and that the royalty ratea is sufficiently high. 1.5.4 Flat-Rate Payment Since the third-party firm is usually capable of tracking the bundle usage, product firms may require the third-party firm to pay a flat-rate price for every actual use of each product in the bundle instead of a royalty fee. The price is usually discounted compared to that when purchased individually. Suppose the third-party firm now pays every product firm a discounted price for every usage of its single product. Denote the discount rate as b. For simplicity, the third-party firm only counts one single product per consumer, and without loss of generality, assume that it is the single product with the highest valuation. Proposition 1.9. With flat-rate payment, if there exists a symmetric equilibrium where all product firms charge ˆ p and the third-party firm charges ˆ p b , then ˆ p p c ; ˆ p b b ˆ p; (1.28) and product firms are always worse off with than without delegated bundling. 28 As in the main model, the introduction of delegated bundling creates competition between product firms and the third-party firm, and thus the equilibrium price ˆ p is lower than the no- bundling equilibrium price p c . However, for each single product, the group of consumers who consume it, either from individual purchases or from the bundle, remains unchanged. Hence, with flat-rate payment, product firms effectively charge a lower price to the same group of consumers and thus are always worse off. 1.6 Concluding Remarks In this paper, I propose a general framework to study “Delegated Bundling”, the business strategy where product firms delegate the operation of an inter-firm bundle to a third-party firm. Under certain regularity conditions, I show the existence and uniqueness of a delegated-bundling equi- librium, which allows us to examine and compare the economic outcomes before and after the introduction of delegated bundling. Among other things, I show that delegated bundling is always social and consumer welfare enhancing, and it also benefits the product firms if the products are less differentiated and the royalty rate is high. These results thus provide an explanation to the prevalence of delegated bundling as well as business implications on its adoption and regulation. For tractability and better illustration, I focus on a specific market structure with some critical assumptions, including single-product firms, unit-demand consumers, and the independence be- tween random valuations q and the extra utility t. It would be a challenging yet interesting task to examine if these results continue to hold if these assumptions are relaxed but I shall expect the intuitions behind these results to go through. I am also interested in extending the framework to include other types of bundles, such as intra-firm bundles and inter-firm but non-delegated bundles, and to study the competition between them. Product firms may decide to join or leave a certain bundle or even launch a new bundle by 29 themselves, which suggests the consideration of a bigger game of bundle formation. The contract- ing for delegated bundling is also quite specifically modeled and in practice it can be much more complex than that in the main model and two related extensions. 30 Chapter 2 Competition in Freemium Services and the Impact of Ad-Avoidance Technologies (Joint Work with Guofu Tan) 2.1 Introduction “Freemium” refers to the business model where a firm provides both a free version, usually ad- sponsored and of lower quality, and a paid premium version of its services or products. The business model has been a common practice in the computer software industry since the 1980s, where developers offer basic programs for free and charge users a price for the full package. In fact, any firm that can vertically differentiate its services or products can migrate to freemium, and no wonder such a business model has been widely adopted in the Internet era: On the one hand, firms with paid services or products may provide a free version with limited functions to attract users (e.g., newspaper and magazine publishers may allow readers to read a limited number of articles on their websites, but readers must subscribe to access full contents). On the other hand, firms with free services or products can also join freemium by providing users with an upgrade option (e.g., Quora, which has always been free-to-use, curates some of its best contents into Quora Plus as a paid subscription; Twitter also announces Twitter Blue which includes many useful features such as undoing tweets that are not available in the basic free service). 31 For many digital service providers who use the freemium model, the free version of their ser- vices is usually ad-sponsored, mostly for two reasons: it helps the firm to profit from free service users, and also provides an incentive for free service users to upgrade to the premium service. Therefore, advertising plays an important role in these freemium models. However, consumers may turn to ad-avoidance technologies (AATs) to avoid watching ads. Broadly speaking, AATs can refer to anything that allows users to not watch the ads. For example, to avoid watching ads on TV , people would switch channels, go to the restroom, or simply turn off the TV . But AATs for digital services are much more complicated, and are often specifically designed to target a certain type of services. Among all the industries which adopt freemium, the video streaming market might have been the one suffers most from AATs. As an important form of entertainment, the market size of the video streaming industry has been increasing dramatically. The more mature U.S. video steaming market earned a total revenue of $10.443 billion U.S. dollars in 2018 and is expected to have an annual growth rate of 3.1% in coming years. 1 Meanwhile, the less-developed video streaming market in China had witnessed an annual growth of 54% from 2013 to 2017, and the estimated total revenue in 2018 reached $110.46 billion CNY (approximately $15.98 billion U.S. dollars). 2 Many major video streaming platforms adopt a freemium business model: users choose be- tween the free service, to watch the videos with bundled ads for free, and the premium service at a subscription fee. Therefore, the platforms have two major sources of revenues: subscription fee from premium service users, and advertising fees from advertisers which are proportional to the number of free service users. Although there exist platforms that rely on one of the sources only (e.g., Netflix), most platforms rely on both. The latter source is especially important for video streaming platforms in China, where the advertising revenue accounts for almost half of the total revenue in the industry. 1 Statista. Available at https://www.statista.com/outlook/206/109/video-streaming--svod-/united-states# market-revenue. Accessed on May 25th, 2019. 2 BigData Research. Available at http://www.bigdata-research.cn/content/201901/810.html. Accessed on May 25th, 2019. 32 AATs targeting at the video streaming platforms emerge along with the prospering of the in- dustry. They are usually in the form of third-party software or browser plug-ins, and some new browsers even have ad-blocking as a built-in feature to attract more users. The overall penetration rate was reported to be 18% for U.S. and 13% for China, and the annual global growth rate stood at 30% in 2016. 3 A recent lawsuit in China, Tencent Video vs. The World Browser, provides a handy real-life example. Tencent Video is one of the three major video streaming platforms in China, and it provides both a free membership and a premium membership. The World Browser, a browser software developer, provides its users with a built-in function of blocking ads on major video streaming platforms, including Tencent Videos, so that the users with a free membership can skip the ads and watch the videos right away. The World Browser argued that users could benefit from not watching the ads, which was supported in the trial of first instance on January 26th, 2018. In the court decision, the judge regarded AATs as technological advancements which the users have the right to enjoy, and claimed that such practices should be encouraged as long as it is beneficial to the society as a whole. However, such claim obscured the difference between consumer welfare and social welfare, neglecting the possible harm to the platforms’ profits and thus the social welfare. Moreover, in the long run, if the platforms decide to drop the free services or lower their investment due to the decline in their revenues, it would backfire on the users as well. Therefore, in the trial of second instance on December 28th, 2018, the appeal court took this stand, and supported Tencent Video’s claim that the built-in ad-blocking feature of The World Browser is a business foul play. To thoroughly examine the impact of AATs, in this paper, we propose a game-theoretic frame- work of competition between firms providing freemium services, where each firm provides both a free service and a premium service, but takes the advertising rate as given and decides only the price for the premium service. Consumers with random valuations over the services choose the one that provides them with the highest net utility. We assume that a certain share (i.e., the penetration 3 PageFair. Available at https://pagefair.com/downloads/2017/01/PageFair-2017-Adblock-Report.pdf. Accessed on May 25th, 2019. 33 rate of AATs) of consumers have access to AATs which could eliminate the disutility from watch- ing ads, independently from their valuations over the services. Therefore, it is weakly dominant for these users to choose from only the free services provided by the firms. The exogenous advertising rate is assumed to be negatively correlated with the penetration rate, as advertisers would be less interested in advertising in this market if fewer people are watching ads here. We start with providing conditions under which freemium is supported in equilibrium. On the basis of that, we show that both the equilibrium price and profits increase in the advertising rate, and thus decrease in the penetration rate. The decrease in the equilibrium price is due to that, as the advertising rate decreases, firms earn less from free service users, and thus would lower the price for the premium service to attract free service users to upgrade. The decrease in equilibrium profits thus come from two sources: users who newly gain access to AATs create no revenues for the firms any more, and users without access to AATs create fewer revenues due the lower advertising rate as well as the lower equilibrium price for premium services. In terms of welfare, we show that, in the short run, there will be an increase in consumer welfare if the penetration rate increases, but the change in social welfare is ambiguous. However, in the long run, firms may respond by dropping the free services and providing only premium services, or reducing the investment in service qualities and lowering the consumers’ overall valuation for services, and thus an increase in the penetration rate may end up hurting consumers as well as the society. To illustrate our findings, we use actual industry data from the Chinese video streaming market to calibrate the parameters and perform counterfactual simulations under various levels of penetrate rates of AATs. The simulation results confirm our main findings that, an increase in the penetration rate would lower the equilibrium price and profits for the firms, and the firms may drop free services or reduce their investment in service quality which result in a lower total surplus in the long run. The rest of this paper is organized as follows: Section 2.2 reviews the related literature. Section 2.3 provides the basic model setup for firms, consumers and AATs. We provide a full characteriza- tion of the monopoly pricing rule in Section 2.4, which provides us with some helpful tools for the 34 derivation of the oligopoly pricing rule in Section 2.5. Section 2.6 then considers the impact of an increase in the penetration rate, both in the short run and in the long run. Section 2.7 presents our calibration results with actual data from the Chinese video streaming market and counterfactual simulations under various levels of penetration rates. Section 2.8 concludes our work. All proofs are provided in Appendix B. 2.2 Literature Review The literature on freemium markets can be divided into three categories depending on their per- spectives. The first strand of the literature, mostly from the field of marketing, considers freemium as a business strategy with the intention to sell the premium products or services, and thus firms may not need to profit from the free products. For example, Niculescu and Wu (2014) compared two business models in the software industry, namely the featured-limited freemium and uniform seeding, in terms of their performance in commercializing a software product with two parts. For featured-limited freemium, the firm would give away part A for free and make money only on part B. Due to such a starting point, papers in this strand have a specific focus on the interaction with network effects. For instance, Boudreau, Jeppesen, and Miric (2021) empirically investigated firms in Apple’s App Store and found that stronger network effects would not amplify the advantage of leaders over followers on its own but only in the settings where freemium strategies are used. The second strand deals with the freemium model as product line design problems. Shi, Zhang, and Srinivasan (2019) considered the pricing problem of market dominant firms on their low-end and high-end products. They showed that freemium, i.e., setting a zero-price for the low-end product, emerge as an optimal pricing choice only if the high-end product provided larger utility gain from an expansion of the firm’s user base. Sato (2019) solved the optimal menu pricing problem for advertising firms on two-sided market and argued that the optimal menu consists of only two services: ad-supported basic service and ad-free premium service. Further, if the 35 willingness to pay of advertisers is sufficiently high, the basic service would be zero-priced and thus freemium. The last strand, where this paper falls in, takes the freemium arrangement as given and studies the competition between freemium firms. Li, Nan, and Li (2018) studied the competition between two social network platforms where they choose from either an advertising strategy (free service with ads) or a freemium strategy (free and premium services without ads). Lambrecht and Misra (2017) allowed online content providers to change their offering between free and paid contents, and found that firms should use “counter-cyclical offering”, or offering more free contents in pe- riods of high demand, provided that consumers’ heterogeneity varies over time. The two papers closest to ours are Carroni and Paolini (2019) and Zennyo (2020). Carroni and Paolini (2019) stud- ied the business model of a streaming platform which we also pay a special focus on. The main difference is that they studied a monopolistic platform which serves three sides, including content providers, advertisers and users, while we consider the competition between such platforms who serve mostly the users. Although we also consider the possible change in advertising rate and the firms investment in service quality (for streaming platforms, it means they have to purchase more contents from upstream content providers), but for the ease of exposition we only consider them as a mechanical process and do not model those two sides explicitly. Zennyo (2020) considered freemium competition among ad-sponsored platforms. Like Carroni and Paolini (2019), the plat- forms are two-sided serving both users and advertisers, but the author assumed that platforms start as purely ad-sponsored platform with the intention to introduce a premium service. We acknowl- edge that this reflects the reality of some industries that we mention earlier, including Quora (with Quora Plus) and Twitter (with Twitter Blue), while our model applies better to other industries such as music and video streaming markets, where firms have experienced the stage of introducing a premium service (for instance, the major video streaming platforms in China all started with only free services, and introduced the premium service later as they expanded), and are in the stage where they are trying to benefit more from subscriptions rather than advertising (the video streaming platforms in China are all actively seeking to increase the share of subscription revenues 36 in their total revenues). Our model has yet another advantage that we allow more flexible choice of consumer utility representations, compared to the uniform distribution in Carroni and Paolini (2019) and the Hotelling model in Zennyo (2020). Our paper also contributes to the literature on advertising bundled with valuable contents and AATs targeting at these ads. As early as in 1950, economist Kaldor has already pointed out that ads are often jointly supplied with entertaining content over media platforms (Kaldor, 1950). The media platforms cross-subsidize the audience by providing the bundle of media contents or services with ads at a price lower than the marginal cost, and get compensated by the advertising revenue. Such a cross-subsidy is a typical feature of a two-sided market (Armstrong, 2006a; Tan & Zhou, 2021). Anderson and Coate (2005) provided a standard two-sided market framework for the commer- cial TV market. The platforms have to decide the optimal bundles of programs and a certain level of ads for audience with heterogeneous tastes. The framework enabled the authors to perform a welfare analysis to the market performances. On the basis of this framework, Anderson and Gans (2011) provided a theoretical analysis on AATs. The authors pointed out that AATs in the digital era, such as TiV o and ad-blocking softwares, are completely different from the traditional ways of avoiding ads, in that the marginal cost of using AATs is very low or essentially zero. Moreover, the audience using AATs would be those who dislike ads the most, and as the penetration rate continues to rise, the remaining audience would be more tolerant, which allows the platform to increase the advertising intensity, driving more audience to use AATs. Using the US commercial TV market data, Wilbur (2008) estimated that a decrease of 10% in ads would result in an increase of 25% in the number of audience. Moreover, he estimated with simulation that as the penetration rate of AATs increases, the TV network would increase the level of ads, but the effective views of these ads decrease, as well as the advertising revenue of the TV network. Despite the fact that we are not modeling the advertising intensity in our model, we come to a similar result that fewer consumers would watch ads and thus the advertising revenues decline as the penetration rate increases. 37 Shiller, Waldfogel, and Ryan (2018) performed empirical studies on ad-supported websites. The regression analysis shows that, for each additional percentage point of site visitors blocking ads, the website traffic reduces by 0.67% over 35 months, along with a decline in content qualities. The authors argue that the usage of AATs has a compound effect on reducing the website revenues both through reduced actual views of ads directly and reduced visits due to quality decline indi- rectly. We follow the same idea and consider the possibility in the long run welfare analysis that the firms may respond by reducing their investment in service qualities, which may eventually lead to a loss in consumer and social welfare. 2.3 Model Setup 2.3.1 Firms with Freemium Services Consider a market with n firms and a continuum of consumers. The measure of consumers is normalized to 1. Each firm provides two types of services: a free service where the consumer has to watch ads before enjoying the service, and a premium service where the consumer pays a fixed price and enjoys the service ad-free. 4 The marginal costs for providing the services are normalized to zero. Each firm i has thus two sources of revenue: For each consumer who uses its free service and watches ads, the advertiser pays f to the firm, and for each consumer who uses its premium service, the consumer pays a fixed price p i . We assume that the advertising market is competitive and thus the advertising ratef is exogenously determined and taken as given by the firms. Therefore, in the pricing game, firm i chooses p i to maximize p i (p)= p i D iP (p)+fD iF (p): (2.1) 4 We relax the setting later in Section 2.6 to that the firms can choose to provide only the premium service. 38 p=(p i ) i2N denotes the price vector of all premium services where N =f1;2;:::;ng. D iP (p) and D iF (p) denote the demand for the premium and free service provided by firm i respectively, whose specific expressions will be derived in the next sections. 2.3.2 Consumers Each consumer is endowed with a vector v=(v i ) i2N , where v i denotes the consumer’s basic val- uation of the service provided by firm i, and all v i ’s are independent across firms and consumers. For each firm i, the distribution formed by the v i ’s of all consumers shares a common cumulative distribution function (CDF), denoted as F(x). The corresponding density function is denoted as f(x) with support[v; ¯ v), where 0 v< ¯ v+¥. We focus on distributions whose density function f(x) is log-concave on [v; ¯ v). The log- concavity of density functions is often used to guarantee the existence of a pricing equilibrium and that the equilibrium price can be determined from first-order conditions (see for example J. Zhou (2017)). We also assume that lim x! ¯ v x[1 F(x)]= 0, which automatically holds if ¯ v is finite. If instead ¯ v is infinite, this condition rules out the distributions with fat tails, so that firms cannot earn a significant profit by pricing arbitrarily high and targeting only consumers with extremely high valuations. Each consumer can choose at most one service from all 2n services provided by the n firms. If the consumer chooses the premium service provided by firm i, her net utility is v i p i , where p i is the fixed price chosen by firm i. If the consumer chooses the free service provided by firm i, her net utility isa(v i v 0 ), wherea2[0;1) denotes the advertising intensity and v 0 > 0 is the reserve price for watching ads. We assume that the advertising intensity is exogenous and symmetric across firms. Besides, each consumer has an outside option with a normalized net utility of 0. Consumers are rational and always choose the option with the highest net utility. We shall note that, following our setup, the services in the freemium market are both horizon- tally differentiated in terms of the random valuations v and vertically differentiated in terms of the 39 choice between the outside option and the free and premium service from the same firm. Moreover, the market differs from the traditional vertically-differentiated multi-product firm models in two ways: In the usual models with vertical differentiation, firms control the pricing of all their own products, and they earn the price as posted. In the freemium market, the firms do not control the pricing of the free services, and although the free service has a reserved price v 0 , instead the firms receive an advertising ratef as the effective price from each free service user. 2.3.3 Ad-Avoidance Technologies We now add ad-avoidance technologies (AATs) into our model. Suppose that, independent from their valuations v, consumers of mass d2[0;1) have access to AATs, which allow them to com- pletely skip the ads and thus eliminate all the disutilities from watching ads. Since botha and v 0 are due to ads, consumers with access to AATs get a final utility of v i when using the free service from firm i, which always dominates the outside option or the premium service from the same firm. Therefore, these consumers would always choose the free service from the firm for which they have the highest basic valuation. Furthermore, since they never watch the ads or pay for the premium service, they generate no revenue to the firms. Meanwhile, for the rest of the consumers without access to AATs, the service choice problem remains unchanged. As the penetration rate of AATs increases, the advertisers would be less willing to advertise in this market, and the advertising rate decreases. Without referring to a specific mechanism on how the advertising rate is determined, we address such economic force by assuming thatf takes the form of an exogenous, monotonically decreasing function of the penetration rate d, with the boundary casef(1)= 0. We make two remarks here regarding our discussion of optimal pricing rules in the next two sections: First, in the main model we always assume that the free services are provided. Even if the firms price their premium services at sufficiently low prices such that the premium services dominate the free services for all consumers without access to AATs, those with access to AATs are still allowed 40 to use the free services. Only in the later discussion of the service provision game we allow firms to not provide free services at all, and in the case where all firms drop the free services simultaneously, consumers with access to AATs will then have no choice but to choose from premium services or the outside option. Second, due to the independence assumption as well as our first remark, firms only need to compete for consumers without access to AATs, and thus their optimal pricing rule would remain unchanged for a given advertising ratef. In other words, the change ind only affects the optimal pricing rule indirectly through its negative correlation with the advertising rate f. This allows us to consider the optimal pricing rule underd = 0 in the next two sections and consider the impact of AATs later in Section 2.6. 2.3.4 Timeline The timeline of our model is as follows: 1. Given the penetration rated and thus the advertising ratef, each firm chooses the fixed price of its premium service p i simultaneously. 2. Each consumer chooses one service or the outside option with the highest net utility accord- ing to her realized valuations v. Our equilibrium concept is sub-game perfect equilibrium. 2.4 Monopoly Pricing Rules We start with n= 1 or the monopoly market. Each consumer has a basic valuation v for the services provided by the monopolist firm and the monopolist firm charges a fixed price p for the premium service. As we have mentioned, we consider the optimal monopoly pricing rule underd = 0 and takingf as given. Note that our discussion here only applies to consumers without access to AATs, 41 as those with access to AATs would always use the free service and create no revenues for the firm and thus are irrelevant. 2.4.1 Consumer Segmentation Depending on the valuation v, each consumer chooses from the following options: • Premium service: The net utility is v p. • Free service: The net utility isa(v v 0 ). • Outside option: The net utility is 0. Clearly, consumers with valuation v close to 0 would prefer the outside option, while consumers with valuation v sufficiently high would prefer the premium service. We are thus interested in consumers choosing the free service whose valuation v must satisfy: 8 > < > : a(v v 0 ) 0; a(v v 0 ) v p; , 8 > < > : v v 0 ; v pav 0 1a D =m(p): For there to be a positive mass of consumers using the free service, we must have m(p)> v 0 , p> v 0 : Therefore, when p v 0 , the firm is essentially providing the premium service only and con- sumers are divided into two segments: • Consumers with valuation v p choose the outside option. • Consumers with valuation v> p choose the premium service. In this case, the monopolist firm’s profit function is denoted as p p (p)= p[1 F(p)]: (2.2) 42 Figure 2.1: Monopoly Consumer Segmentation (Premium Only) When p> v 0 and m(p)< ¯ v, p<av 0 +(1a) ¯ v; the firm is in freemium and consumers are divided into three segments: • Consumers with valuation v v 0 choose the outside option. • Consumers with valuation v 0 < vm(p) choose the free service. • Consumers with valuation v>m(p) choose the premium service. In this case, the monopolist firm’s profit function is denoted as p b (p)=f[F(m(p)) F(v 0 )]+ p[1 F(m(p))]: (2.3) Finally, when pav 0 +(1a) ¯ v and thusm(p) ¯ v, the firm is essentially providing the free service only and consumers are divided into two segments: • Consumers with valuation v v 0 choose the outside option. • Consumers with valuation v> v 0 choose the free service. 43 Figure 2.2: Monopoly Consumer Segmentation (Freemium) In this case, the monopolist firm’s profits are fixed at p f =f[1 F(v 0 )]: (2.4) Figure 2.3: Monopoly Consumer Segmentation (Free Only) 2.4.2 Optimal Pricing Rules To derive the optimal pricing rules for the monopolist firm, we need to compare the firm’s profits in all three cases discussed in the previous subsection. 44 To facilitate the comparison, we note first that the log-concavity of f(x) implies that both profit functions p p (p) and p b (p) are quasi-concave in p and thus the optimal price in each case can be pinned down by the first-order conditions and the boundary constraints. The global optimal price is then determined by comparing the local maximum profits. Specifically, for anya2[0;1), define the general virtual valuation function as J a (x)= x(1a) 1 F(x) f(x) ; and the virtual valuation function is simply J 0 (x)= x 1 F(x) f(x) : The log-concavity of f(x) ensures that J a (x) is strictly increasing in x for anya2[0;1] and thus the inverse function J 1 a (z) is well-defined. Further denote ¯ J 0 = lim x! ¯ v J 0 (v); and let ¯ v 0 solves [a ¯ v 0 +(1a) ¯ J 0 ][1 F( ¯ v 0 )]= max p p[1 F(p)]: Note that ¯ J 0 cannot be negative, as otherwise x[1 F(x)] must be strictly increasing in x and does not converge to 0. In addition to non-negative values, ¯ J 0 may also take the value of¥ if ¯ v=¥ or the distribution has an infinite support. The optimal pricing rule for the monopolist firm can thus be summarized as follows: 5 Proposition 2.1. For any pair of(a;v 0 ), there exist two thresholdsf and ¯ f with 0f ¯ f such that: 5 Proposition 2.1 holds under a weaker assumption than log-concave f(x) that the virtual valuation function J 0 (x) is strictly increasing in x. However, the log-concavity of f(x) is critical for the following comparative statics results as well as the equilibrium existence under oligopoly, and thus we focus on log-concave f(x) for consistency throughout the paper. 45 • When 0ff, the monopolist firm charges a price p v 0 and provides only the premium service. Specifically, p = minfJ 1 0 (0);v 0 g: • Whenf <f < ¯ f, the monopolist firm charges a price v 0 < p <av 0 +(1a) ¯ v and provides both services. Specifically, p solves p (1a) 1 F(m(p )) f(m(p )) =f: • Whenf ¯ f, the monopolist firm charges any price p av 0 +(1a) ¯ v and provides only the free service. The intuition for Proposition 2.1 is that, as the advertising ratef increases, the monopolist firm has a higher incentive to provide the free service as well as to drop the premium service. When the advertising rate is sufficiently low, the monopolist firm provides only the premium service, and it is a well known result that the firm should set the price where the virtual valuation equals zero. When the advertising rate is sufficiently high, the monopolist firm provides only the free service by pricing extremely high. Only when the advertising rate is intermediate will the monopolist firm provide both services and thus be in freemium. To interpret the pricing rule under freemium, rewrite it as 1 F(m(p ))= p f 1a f(m(p )): When the monopolist firm increases the price for the premium service, its marginal revenue from existing consumers of its premium service is shown on the left-hand side. Meanwhile, the mo- nopolist firm loses some marginal consumers from its premium service to its free service. Each of these consumers costs p for not paying for the premium service any more, but also generates f for using the free service, and the marginal cost is shown on the right-hand side. The optimal price under freemium is thus achieved when marginal benefit equals marginal cost. Since the marginal 46 consumers are comparing only the free service and the premium service, the outside option is irrelevant in this condition. Figure 2.4: Optimal Service Provision (v U[0;10];a = 0:5) Note that the latter two cases described in Proposition 2.1 may vanish under certain settings: When v 0 ¯ v 0 ,f and ¯ f converge and it is never optimal for the monopolist firm to provide both services at any advertising ratef, as the profit from doing so is dominated by that from providing the premium service only. When ¯ J 0 takes an infinity value, which happens if and only if ¯ v also takes an infinity value, it is never optimal for the monopolist firm to provide the free service only. An infinite ¯ J 0 implies that, for any finite f, there exists a sufficiently high price p such that the marginal users account for a significant share of existing users of the premium service, and thus the firm is always willing to lower the price and attract them to use the premium service. Lemma 2.1. (i) Bothf and ¯ f increase in v 0 . (ii)f increases ina while ¯ f decreases ina. To interpret Lemma 2.1, recall that at f the monopolist firm is indifferent between premium only and freemium, and at ¯ f the firm is indifferent between free only and freemium. As a or v 0 changes, the thresholds need to be adjusted to re-balance the maximum profits under different provision. 47 We start with examining the effect of an increase inf. Due to the respective share of consumers using the free service, it leads to no changes in the firm’s maximum profits under premium only, a small increase under freemium and a large increase under free only. As v 0 increases, the firm’s maximum profits increase under premium only as the constraint p v 0 relaxes, and decrease under free only as fewer consumers use the free service. The effect under freemium falls in-between. Therefore, to re-balance the maximum profits, we need to raise bothf and ¯ f. As a increases, the firm’s maximum profits do not change under either premium only or free only, but decrease under freemium as consumers switch from the premium service to the free service. Therefore, to re-balance the maximum profits, we need to raisef and reduce ¯ f. 2.4.3 Comparative Statics For the discussion below, suppose the initial parameters are such that it is optimal for the monop- olist firm to choose freemium and provide both services. Therefore, the optimal price p satisfies p (1a) 1 F(m(p )) f(m(p )) =f; (2.5) and the maximum profitsp are p =f[F(m(p )) F(v 0 )]+ p [1 F(m(p ))]: (2.6) Corollary 2.1. (i) Asf increases(decreases), eventually the firm may drop the premium(free) ser- vice. (ii) If the firm remains in freemium, both the optimal price p and the maximum profits p increase inf. Part (i) of Corollary 2.1 follows directly thatf ¯ f. As for part (ii), an increase inf reduces the marginal cost for the firm to increase p, and thus the optimal price p is higher. The maximum profitsp are also higher since the firm now earns more from users of both the free service and the premium service due to the higherf and p . 48 Consider the limiting case wherea approaches 1 and the free service provides a net utility of v v 0 . 6 Depending on the price p for the premium service, either the free service is dominated when p< v 0 or the premium service is dominated when p> v 0 . The monopolist firm is thus never optimal to provide both services and the thresholdsf and ¯ f converge to a single threshold f = 8 > > < > > : v 0 ; if v v 0 J 1 0 (0) max p p[1 F(p)] 1 F(v 0 ) ; if J 1 0 (0)< v 0 ¯ v : Figure 2.5: Optimal Service Provision (v U[0;10];a! 1) Corollary 2.2. (i) As a decreases, the firm always remains in freemium. As a increases, if f > f (f <f ), eventually the firm may drop the premium(free) service. (ii) If the firm remains in freemium, both the optimal price p and the maximum profitsp decrease ina. Part (i) of Corollary 2.2 follows directly Lemma 2.1. As for part (ii), an increase in a leads to consumers switching from the premium service to the free service and reduces the marginal benefit for the firm to increase p, and thus the optimal price p is lower. The maximum profits p are also lower since consumers who used to choose the premium service either stick to the 6 We specifically requiresa2[0;1) in our main model to avoid the discussion on possible dis-continuity ata = 1. 49 premium service and pay a lower price p or switch to the free service and the advertising rate is even lower asf < p . Corollary 2.3. (i) As v 0 increases(decreases), eventually the firm may drop the free(premium) service. (ii) If the firm remains in freemium, the optimal price p always increases in v 0 , while for the maximum profits p , there exists a threshold f such that p increase in v 0 if and only if f <f . Similarly, part (i) of Corollary 2.3 follows directly Lemma 2.1. As for part (ii), an increase in v 0 leads to consumers switching from the free service to the premium service and increases the marginal benefit for the firm to increase its price, and thus the optimal price p is higher. However, the effect on the maximum profits p is two-fold: the firm benefits from consumers who switch from the free service to the premium service and consumers who stick to the premium service and pay a higher p , but also hurts from consumers who switch from the free service to the outside option as they no longer generate the advertising revenues. Hence, the maximum profitsp increase if and only if the advertising rate is sufficiently low such that the benefit would dominate the loss. 2.5 Oligopoly Pricing Rules We now turn to the oligopoly market where n 2. Again we discuss the pricing competition under d = 0 and take f as given. Since we are mostly interested in studying freemium, we focus on a symmetric equilibrium where all firms charge the same price p > v 0 such that in equilibrium, all firms are providing both the free service and the premium service. We refer to it as the “freemium equilibrium”. 50 2.5.1 Consumer Segmentation We start with examining the consumer segmentation in equilibrium, where all premium services are priced at p > v 0 . For each consumer with valuation vector v, the highest net utility from all 2n possible services and the outside option can be expressed as maxf0;max i2N fa(v i v 0 )g;max i2N fv i p i gg= maxf0;a(max i2N v i v 0 );max i2N v i p i g: Therefore, the consumer’s choice of service depends entirely on u= max i2N v i : • If u< v 0 , the consumer chooses the outside option. • If v 0 u < m(p ), the consumer uses the free service provided by firm i with the highest basic valuation v i . • If u m(p ), the consumer uses the premium service provided by firm i with the highest basic valuation v i . From a single firm’s point of view, we can further decompose u= max i2N v i = maxfv 1 ;max i6=1 v i g: Denote u 1 = max i6=1 v i (2.7) as the maximum basic valuation from services provided by all other firms except firm 1, and denote its CDF as G(x)= F n1 (x) with a density function g(x)=(n 1)F n2 (x) f(x): 51 For firm 1, a symmetric equilibrium in the symmetric oligopoly market is thus equivalent to an equilibrium in an asymmetric duopoly market, where firm 1 with a basic valuation v 1 competes against a derived firm (i.e., the aggregation of all other firms) with a basic valuation u 1 and a premium service price p > v 0 . Our analysis above suggests that, in equilibrium, when firm 1 also charges p > v 0 , the con- sumer segmentation can be illustrated as follows: Figure 2.6: Oligopoly Consumer Segmentation There is yet another way to derive the consumer segmentation. Instead of asking each consumer to choose among all the services and the outside option at once, each consumer can choose her favorite service (or the outside option) for each firm depending on her basic valuations first, and then compare across firms. This way of derivation is especially useful when we re-frame the symmetric oligopoly market into the asymmetric duopoly market. Using this new way of thinking, for firm 1: • The consumer prefers the outside option when v 1 < v 0 . 52 • The consumer prefers the free service when v 0 v 1 <m(p ). • The consumer prefers the premium service when v 1 m(p ). For the derived firm (the aggregation of all other firms): • The consumer prefers the outside option when u 1 < v 0 . • The consumer prefers the free service when v 0 u 1 <m(p ). • The consumer prefers the premium service when u 1 m(p ). Based on their favorite services for firm 1 and the derived firm, consumers can be divided into 3 3= 9 groups. Within each group, we only need to compare the two net utilities of the two favorite services to further differentiate the consumers. Figure 2.7: Oligopoly Consumer Segmentation (Grouped) 53 For example, for the group of consumers with v 0 u 1 <m(p ) and v 1 m(p ), we need to compare the net utility of the free service provided by the derived firm and that of the premium service provided by firm 1. Since v 1 p m(p ) p =a(m(p ) v 0 )>a(u 1 v 0 ); all consumers within this group end up using the premium service provided by firm 1. To determine the freemium equilibrium price p , we need also the consumer segmentation when any firm deviates from the equilibrium unilaterally. Without loss of generality, suppose again firm 1 is deviating to a different price p for its premium service. When p> p , the consumers’ favorite service of firm 1 becomes: • The consumer prefers the outside option when v 1 < v 0 . • The consumer prefers the free service when v 0 v 1 <m(p). • The consumer prefers the premium service when v 1 m(p). Compared to the consumer segmentation in equilibrium, two groups of consumers are essentially affected: • For consumers who prefer the free service of firm 1 and the premium service of the derived firm, they would end up using the former if and only if a(v 1 v 0 )> u 1 p , u 1 u 1 p , u 1 < v 1 p+ p : 54 Figure 2.8: Oligopoly Consumer Segmentation (p> p ) Firm 1’s profit function is thus p(p; p )=f[ Z m(p ) v 0 G(v) f(v)dv+ Z m(p) m(p ) G(a(v v 0 )+ p ) f(v)dv] + p Z ¯ v m(p) G(v p+ p ) f(v)dv: When v 0 < p < p , although consumers’ favorite service of firm 1 remains unchanged, the comparison between services from firm 1 and the derived firm is slightly modified. Compared to the consumer segmentation in equilibrium, again two groups of consumers are essentially affected: • For consumers who prefer the premium service of firm 1 and the free service of the derived firm, they would end up using the former if and only if v 1 p>a(u 1 v 0 ), u 1 < v 1 p a + v 0 : 55 Figure 2.9: Oligopoly Consumer Segmentation (v 0 < p< p ) • For consumers who prefer the premium service of both firm 1 and the derived firm, they would end up using the former if and only if v 1 p> u 1 p , u 1 < v 1 p+ p : Firm 1’s profit function is thus p(p; p )=f Z m(p) v 0 G(v) f(v)dv + p[ Z m(p )p +p m(p) G( v p a + v 0 ) f(v)dv+ Z ¯ v m(p )p +p G(v p+ p ) f(v)dv]: When p v 0 , the consumers’ favorite service of firm 1 becomes: • The consumer prefers the outside option when v 1 < p. • The consumer prefers the premium service when v 1 p. 56 Figure 2.10: Oligopoly Consumer Segmentation (p v 0 ) Consumers are thus separated into 2 3= 6 groups. Compared to the consumer segmentation in equilibrium, three groups are no longer available (where consumers prefer the free service of firm 1). Among the remaining six groups, two groups of consumers are essentially affected: • For consumers who prefer the premium service of firm 1 and the free service of the derived firm, they would end up using the former if and only if v 1 p>a(u 1 v 0 ), u 1 < v 1 p a + v 0 : • For consumers who prefer the premium service of both firm 1 and the derived firm, they would end up using the former if and only if v 1 p> u 1 p , u 1 < v 1 p+ p : 57 Firm 1’s profit function is thus p(p; p )= p[ Z m(p )p +p p G( v p a + v 0 ) f(v)dv+ Z ¯ v m(p )p +p G(v p+ p ) f(v)dv]: 2.5.2 Freemium Equilibrium Recall that the “freemium equilibrium” refers to a symmetric equilibrium where all firms charge the same price p > v 0 . By examining the first-order derivative of firm 1’s profit function at p 1 = p , we can show that any freemium equilibrium price p must satisfy: 1 n [1 F n (m(p ))] p f 1a F n1 (m(p )) f(m(p )) p Z ¯ v m(p ) f(v)dF n1 (v)= 0: To ensure the existence and uniqueness of the freemium equilibrium, we further make the following assumptions (denoteb =(n 1)=a): Assumption 2.1. v 0 1F(v 0 ) f(v 0 ) +bv 0 1F(v 0 ) F(v 0 ) 0: Assumption 2.2. f a (1a) 1F(m( f a )) f(m( f a )) +b f a (1a) 1F(m( f a )) F(m( f a )) 0: Assumption 2.1 and 2.2 require that v 0 and f cannot be too high. If v 0 is sufficiently high, pricing the premium service above v 0 would attract very few consumers and thus it would be better for the firm to price below v 0 and provide only the premium service. If instead the advertising ratef is sufficiently high such that it exceeds the equilibrium price p , firms may have incentives to unilaterally increase the price of their premium services so that consumers can switch from the premium service to the free service and generate higher profits. As a comparison, set n= 1 and thusb = 0, and the conditions become J 0 (v 0 )= v 0 1 F(v 0 ) f(v 0 ) 0 and f a (1a) 1 F(m( f a )) f(m( f a )) 0: 58 Since n= 1 corresponds to the monopoly market, we can show that the two conditions restrict us to the shaded area illustrated in the following graph of the monopolist firm’s optimal service provision. Figure 2.11: Optimal Service Provision (Comparison) Proposition 2.2. There exists a unique freemium equilibrium. To interpret the equilibrium condition, rewrite it as 1 n [1 F n (m(p ))]= p f 1a F n1 (m(p )) f(m(p ))+ p Z ¯ v m(p ) f(v)dF n1 (v): We note that the condition resembles that from Proposition 2.1 when the monopolist firm prefers freemium. The left-hand side denotes the marginal benefit when firm 1 unilaterally increases its price for the premium service and gains extra profit from the existing consumers. Meanwhile, firm 1 also loses some marginal consumers. Marginal consumers with higher u 1 switch to the premium service of other firms and costs p each for firm 1, while marginal consumers with lower u 1 switch to the free service of firm 1, and thus costs only p f as they still generate the advertising revenue. The equilibrium price is thus determined by equating marginal benefit and marginal cost. Again, since the marginal users do not care about the outside option, the outside option is also irrelevant in the oligopoly pricing rule. 59 2.5.3 Comparative Statics In this section we examine how the equilibrium price p and the equilibrium profitsp are affected by the model parameters. Corollary 2.4. The equilibrium price p increases inf and v 0 , and decreases ina and n. The intuition behind Corollary 2.4 is essentially the same as in the monopoly case: If a firm unilaterally increases the price for its premium service, its marginal cost decreases in f and its marginal benefit decreases ina and increases in v 0 . Therefore, the equilibrium price p increases in f and v 0 where the marginal benefit exceeds the marginal cost, and decreases in a where the marginal cost exceeds the marginal benefit. Meanwhile, as the number of firms n increases, the competition is intensified and the equilibrium price p decreases. Given the equilibrium price p , each firm earns in equilibrium p = 1 n [f[F n (m(p )) F n (v 0 )]+ p [1 F n (m(p ))]]: Corollary 2.5. The equilibrium profitsp increase inf, and decrease ina and n. Again, the intuition behind Corollary 2.5 follows that from the monopoly case. When f in- creases, each firm earns more from both the free service and the premium service. When a in- creases, each firm earns less as consumers who used to choose the premium service either stick to the premium service and pay a lower price p or switch to the free service and generate the lower advertising rate f < p . Finally, as the number of firms n increases, each firm suffers from the shrinking consumer base as well as a lower equilibrium price for the premium services. We note that the effect of a change in v 0 on the equilibrium profits is ambiguous and more complicated than that in the monopoly case. Despite that an increase in v 0 leads to a higher equilibrium price p as well as marginal consumers switching from the free service to the premium service, it also results in another group of marginal consumers switching from the free service to the outside option. The overall effect thus depend on the relative magnitudes of all these effects. 60 We also note that, the total profits earned by all the firms np may not decrease in the number of firms. As more firms enter the market, the consumers have a better draw for the maximum basic valuations, and thus more consumers will switch from the outside option to the free service or from the free service to the premium service, which benefits the firms. However, the firms still suffer a loss from the lower equilibrium price due to intensified competition. The overall effect is thus still ambiguous. 2.5.4 Welfare Analysis We now move on to examine the social and consumer welfare in the freemium equilibrium. We have shown that, in equilibrium, consumers always go to the firm with the highest basic valuation u= max i2N v i , and then choose among the outside option, the free service and the premium service depending on u. The consumer surplus in the freemium equilibrium is thus CS= Z m(p ) v 0 a(u v 0 )dF n (u)+ Z ¯ v m(p ) (u p )dF n (u); (2.8) where F n (u) is the distribution function of u. Corollary 2.6. The consumer surplus CS decreases inf and v 0 , and increases ina and n. To interpret Corollary 2.6, rewrite the consumer surplus as CS= Z ¯ v 0 maxf0;a(u v 0 );u p gdF n (u): An increase inf raises the equilibrium price p and reduces the net utility for the premium service. Similarly, an increase in v 0 raises the equilibrium price p and reduces the net utility for both the free service and the premium service. The consumer welfare suffers a loss in both cases. Conversely, an increase ina increases the net utility for the free service and benefits the con- sumer welfare. Finally, when there are more firms in the market, the equilibrium price is lower due to intensified competition and the net utility for the premium service increases, and meanwhile 61 consumers get a better draw for the maximum basic valuation u, both leading to a higher consumer surplus. The total surplus is the sum of the consumer surplus and the firms’ total profits 7 T S= Z m(p ) v 0 [a(u v 0 )+f]dF n (u)+ Z ¯ v m(p ) udF n (u): (2.9) Since Corollary 2.6 goes in the exact opposite direction to Corollary 2.5, suggesting a sharp conflict in interests between consumers and firms, the effects of parameter changes on total surplus are in general ambiguous. The only result we have here is when the number of firms increases: Corollary 2.7. The total surplus T S increases in n. To see why Corollary 2.7 holds: On the one hand, since p >f > 0, the total surplus created by each consumer’s transaction with the firms is increasing in the realized maximum basic valuation u, and thus when consumers have a better draw of u due to having more firms on the market, the overall total surplus increases. On the other hand, although the lower equilibrium price due to intensified competition does not enter the total surplus directly as it serves only as a transfer between the consumers and firms, it still attracts more consumers to use the premium service and indirectly increases the total surplus. 2.6 Impact of Ad-Avoidance Technologies We are now ready to take into consideration the impact of AATs. Denote p (f) andp (f) as the equilibrium price and profits we determine in the last section, where f is taken as an exogenous parameter andd is essentially set at 0. Thus, under a non-zero penetration rated, the equilibrium price becomes p(d)= p (f(d)); 7 The advertisers receive no profits as the advertising market is assumed to be competitive. 62 and the equilibrium profits are p(d)=(1d)p (f(d)): 2.6.1 Impact on Equilibrium Outcome Proposition 2.3. Both p(d) andp(d) decrease ind. Note that the penetration rate d affects the equilibrium price p only indirectly through f. As the penetration rate increases, the advertising rate decreases, which in turn forces the firms to lower the price of premium services. The firms thus suffer a revenue loss from three sources: they lose all the revenues from consumers who newly gain the access to AATs, receive lower advertising revenue from current free service users, and have to charge current premium service users a lower price. The overall profitsp thus decrease ind. In terms of welfare, consumers are always better off with an increase in the penetration rate: Proposition 2.4. The consumer surplus CS(d) increases ind. Note that, without AATs, consumers with maximum basic valuation u= max i2N v i used to earn the maximum among 0 (from the outside option),a(uv 0 )(from the free service) and u p (from the premium service). With AATs, consumers with access to AATs are guaranteed to earn u, which dominates all three options, while consumers without access to AATs are also better off since the premium service price p is lower. As in the previous section, Proposition 2.3 and Proposition 2.4 go in exact opposite directions, and thus the effect of an increase in the penetration rate on the total surplus remains ambiguous. 2.6.2 Service Provision Game In the long run, in addition to passively changing their prices for the premium services, firms may have other options to cope with an increase in the penetration rate of AATs, such as changing the provision of services. For example, firms may decide to stop providing the free services to all consumers. In the extreme case where all firms provide only premium services, consumers with 63 access to AATs have no choice but to use either the outside option or one of the premium services. As a real-life example, the video streaming platform Hulu used to have both a free service and a premium service, but later in August 2016 it dropped the free service and started to offer two tiers of paid services where the cheaper one comes with ads. Formally, consider a service provision game, where all firms have two options: provide both services (BOTH) or provide only the premium service (PO). To simplify our calculations, we assume that the prices for the premium services are fixed at the current level, and thus we can isolate the effect from only the change in service provision. As firms turn away from providing both services to only the premium services, consumers with access to AATs are restricted in their choices. For instance, a consumer with access to AATs may have a high realization of the basic valuations for firms not providing free services, but low realizations for firms that do, and end up choosing the premium services from the former, which creates extra profits for these firms. Meanwhile, consumers without access to AATs who have an intermediate realization of the basic valuations for firms dropping free services may turn away to other firms. Thus, firms face a trade-off between which type of consumers to attract. If there are no consumers with access to AATs ord = 0, provided that the advertising ratef(0) is sufficiently high, firms who drop the free services would suffer a significant loss due to losing all the free service users, and thus all firms providing both services constitutes an equilibrium; while if all consumers have access to AATs ord = 1, all firms providing only premium services should constitute an equilibrium as any firm deviating to providing both services would result in losing all the revenues, as consumers who favor this firm the most would always use its free service and generate no revenue. Proposition 2.5. (1) Whend = 0, all firms choosing BOTH is an equilibrium of the service provi- sion game iff(0) is sufficiently large. (2) Whend = 1, all firms choosing PO is an equilibrium of the service provision game. 64 For more general cases ofd2(0;1), we may observe other equilibria where some firms choose BOTH while the others choose PO. We will examine these possibilities later through simulations in Section 2.7. 2.6.3 Service Quality Investment Alternatively, firms may reduce their investment in service qualities to cope with the decline in revenues due to an increase in the penetration rate of AATs. For instance, Shiller, Waldfogel, and Ryan (2018) shows empirically that ad-supported websites observe a decline in content qualities along with the increase in usage of AATs. Similarly, video streaming platforms have to purchase new contents (movies, TV series and TV shows) to maintain their attractiveness to users so that they will stay on these platforms, and with a decline in revenues the platforms will have to spend less on new contents and consumers’ overall valuations should drop. To provide a measurement of the service quality, we consider a specific form of distribution for the basic valuations v i . Suppose all v i follows the exponential distribution with parameterl, i.e., F(x)= 1 e 1 l x ; f(x)= 1 l e 1 l x : The expectation of v i is l which can be regarded as a measurement of the overall quality for the services provided by firm i. As in the previous subsection, to simplify our calculations, we assume that the prices of pre- mium services are fixed at the current level. We also assume that the firms change their service qualities simultaneously and thus we do not have to consider asymmetricl’s. Under such settings, the equilibrium profits for the firms are np(d)=(1d)[p[1 F n (m(p))]+f(d)[F n (m(p)) F n (v 0 )]]; 65 the consumer surplus is CS(d)=d Z ¯ v v udF n (u)+(1d) Z ¯ v v maxf0;a(u v 0 );u pgdF n (u); and the total surplus is T S(d)= CS(d)+ np(d): Proposition 2.6. The total profits np(d), the consumer surplus CS(d) and the total surplus T S(d) all increase inl. Proposition 2.6 is not surprising: as l increases, consumers realize higher valuations, which improves consumer surplus; meanwhile, more consumers upgrade from the outside option to free services or from free services to premium services, generating higher profits for the firms. Since both consumer surplus and total profits increase inl, so does the total surplus. As it is a challenging task to determine the exact decision process for firms to choose their service quality levels, in the simulation exercise, we instead consider the “quality equivalence” for different levels of penetration rates. Suppose that the penetration rate increases from the currentd 0 to somed. We then calculate a new quality level, such that the consumer surplus or the total surplus is restored to the initial level, and denote the two quality levels asl CS (d) andl T S (d) respectively. Due to Proposition 2.6, the two quality levels serve as handy thresholds that, if the firms turn out to choose a service quality level lower than them in response to the change in the penetration rate of AATs, the consumer surplus or total surplus will end up lower than the initial level, and we can conclude that the consumers are harmed by AATs. 2.7 Calibration and Simulation To further evaluate the impact of AATs, we now move on to a numerical exercise where we con- struct counter-factual results based on the model from previous sections. As a starting point, we use the actual data from the Chinese video streaming market to calibrate the parameters. Combining 66 these parameters with various penetration ratesd would allow us to simulate the possible outcome of consumer segmentation, equilibrium prices and profits, and consumer and social welfare if the altered d is realized. This calibration and simulation exercise not only checks our theoretical re- sults from the model, but also allows us to have a better understanding of the interests involved in the Chinese lawsuit case between Tencent Videos and The World Browser. 2.7.1 Parametric Setting For the purpose of calibration, we need a more specific parametric setting. For the negative corre- lation between the penetration rated and the advertising ratef, we use the functional form f(d)=h(1d) e : The parameter e allows some flexibility in the magnitude of advertisers’ reactions to a decline in the number of viewers of their ads. While for the distribution F(v), we use the exponential distribution Exp(l) as mentioned earlier. The exponential distribution has a constant hazard rate ofl, and is also left-skewed, which corresponds to the fact that the density of consumers decreases in valuation. 2.7.2 Industry Data The following industry data of video streaming platforms in China are used for calibration: n= 3; p= 15;d = 13%; D P + D F = 75%; D P D P + D F = 22:5%; p P p F = 24:8% 48:6% ; 67 where D P , D F ,p P andp F denote the mass of premium service users, the mass of free service users, subscription revenue from premium service users and advertising revenue from free service users respectively, and their specific expressions are D P =(1d)[1 F n (m(p)]; D F =d+(1d)[F n (m(p) F n (v 0 )]; p P = 1d n p[1 F n (m(p)]; p F = 1d n f[F n (m(p) F n (v 0 )]: Although there are more than three video streaming platforms in China, the three platforms in the top tier (iQiyi, YoukuTudou and Tencent Videos) cover most of the markets while the rest have little say in the industry, and hence we set n= 3. All three platforms have set the monthly price of their premium services at p= 15 Chinese Yuan. The AATs penetration rated can be found in “2017 Global Adblock Report” by PageFair. 8 The rest can be found in “The Report of China’s Business Situation in Online Video 2018” by iReserach. 9 2.7.3 Parameter Calibration Since all three platforms provide both free and premium services in reality, the model calibration will be based on the symmetric freemium equilibrium. There are 5 parameters that need to be calibrated: a, v 0 ,d,l,f (or equivalentlyh provided the choice ofe). The calibration results are as follows: a 0:4227;v 0 8:103;d = 0:13;l 7:513; 8 Available at https://pagefair.com/downloads/2017/01/PageFair-2017-Adblock-Report.pdf. Accessed on May 25th, 2019. 9 Available at http://report.iresearch.cn/report pdf.aspx?id=3216. Accessed on May 25th, 2019. 68 f 10:99) 8 > > > > > < > > > > > : e =5; h 22:06; e =1; h 12:64; e =0:5; h 11:79: 2.7.4 Impact on Equilibrium Outcomes Using the calibrated parameters we can now construct counter-factual results with altered d. We have three choices of parameter e: e = 5(convex f), e = 1(linear f) and e = 0:5(concave f). For each e, we investigate the firms’ performances at 5 different values of d: 13%(status quo), 8%(status quo5%), 18%(status quo+5%), 23%(status quo+10%), 0(completely forbidden). Table 2.1: Market Performances under Alteredd,e = 5 d p D P D F np P np F np 0% 25.00 2.06% 69.20% 0.52 7.61 8.12 8% 18.67 7.96% 65.60% 1.49 6.33 7.82 13% 15.00 16.88% 58.13% 2.53 4.96 7.49 18% 12.08 29.12% 47.32% 3.52 3.22 6.74 23% 9.87 41.34% 36.54% 4.08 1.49 5.57 Table 2.2: Market Performances under Alteredd,e = 1 d p D P D F np P np F np 0% 16.71 13.39% 57.88% 2.24 6.36 8.60 8% 15.66 15.49% 58.08% 2.43 5.50 7.93 13% 15.00 16.88% 58.13% 2.53 4.96 7.49 18% 14.34 18.30% 58.14% 2.62 4.41 7.04 23% 13.68 19.72% 58.15% 2.70 3.86 6.56 Table 2.3: Market Performances under Alteredd,e = 0:5 d p D P D F np P np F np 0% 15.83 16.23% 55.03% 2.57 6.05 8.62 8% 15.32 16.64% 56.92% 2.55 5.38 7.93 13% 15.00 16.88% 58.13% 2.53 4.96 7.49 18% 14.67 17.08% 59.36% 2.51 4.55 7.05 23% 14.32 17.25% 60.62% 2.47 4.14 6.61 69 These results conform to Proposition 2.3 that the equilibrium price p and profits p always decrease ind. We can further tell from the tables that, as the penetration rated increases: • The mass of premium service users D P increases. • The mass of free service users D F increases for smalle while decreases for largee. • The mass of free service users who actually watch the ads which equals D F d decreases. • The overall subscription revenue from premium service users np P increases for largee while decreases for smalle. • The overall advertising revenue from free service users np F decreases. For the mass of premium service users D P , as we have discussed in the monopoly case, the firms have incentive to attract more free service users to subscribe when d increases. Note that a largere makesf drop faster around status quo, and thus D P fluctuates more violently at the same time. For the mass of free service users D F , it is obvious that D F d would always decrease in d: more free service users gain access to AATs, and some of the rest are attracted to upgrade to the premium service due to the lower price p. Similar to D P , the fluctuation of D F d is more violent with largee and thus the decreasing trend dominates; with smalle, D F d is rather stable and the increasing trend ofd makes D F increase as well. For the overall advertising revenue from free service users np F , note that it equals to the product of D F d andf. Since both decrease ind, np F must be decreasing ind as well. Finally, for the overall revenue np, since np F usually accounts for a larger part in revenue than np P , the decreasing trend of np F dominates and passes along to np. A by-product of our simulation is to examine the impact of changes in the number of firms. As is predicted in Corollary 2.4, the equilibrium price p decreases in n. There are two effects playing crucial roles in determining the firms’ other performances as n goes up: On the one hand, since the valuations are IID across firms, the distribution of the maximal valuation u skews to the right as n 70 Table 2.4: Market Performances under Altered n,e = 1 n p D P D F np P np F np p P p F p 1 15.33 5.59% 37.00% 0.86 2.64 3.50 0.86 2.64 3.50 2 15.17 11.21% 50.90% 1.70 4.17 5.87 0.85 2.08 2.93 3 15.00 16.88% 58.13% 2.53 4.96 7.49 0.84 1.65 2.50 4 14.81 22.60% 60.90% 3.35 5.27 8.61 0.84 1.32 2.15 5 14.60 28.43% 60.69% 4.15 5.24 9.39 0.83 1.05 1.88 6 14.36 34.39% 58.43% 4.94 4.99 9.93 0.82 0.83 1.66 7 14.08 40.55% 54.71% 5.71 4.59 10.29 0.82 0.66 1.47 8 13.75 47.01% 49.86% 6.46 4.05 10.51 0.81 0.51 1.31 9 13.32 54.00% 43.94% 7.19 3.40 10.59 0.80 0.38 1.18 10 12.72 62.01% 36.63% 7.89 2.60 10.49 0.79 0.26 1.05 increases. In other words, the consumers are more likely to get a better draw of u. The fact that the probability of getting a high u is always increasing but the probability of getting a medium u would increase first and decrease later is reflected in the increasing trend of D P and np P , and the inverted-U shape of D F and np F , which in turn leads to the drop of np for sufficiently large n. On the other hand, the more firms there are, the fiercer competition would be, and thus the revenue for each firmp P ,p F andp always decrease in n. 2.7.5 Impact on Service Provisions We now consider the service provision game as mentioned in Section 2.6. Since there are n= 3 firms in the market, consider the following triopoly game: BOTH BOTH PO BOTH p p 21b PO p 21p p 12p PO BOTH PO BOTH p 21b p 12b PO p 12p p c Table 2.5: Payoff Matrix of the Triopoly Game Firm 1 is the row player, firm 2 is the column player, and the strategy chosen by firm 3 is shown in the top-left corner of the tables. Due to the symmetry of our settings, we only list the payoffs for firm 1 in the table, which should all be regarded as functions of d. For this part, we assume that the price of premium services p is fixed at the current level, and for the advertising rate, we sete = 1 and thush 12:64. 71 Case 1: When all firms provide both services, the payoffp follows our main model p(d)= 1d 3 [p[1 F 3 (m(p))]+f(d)[F 3 (m(p)) F 3 (v 0 )]]: Case 2: When all firms provide only the premium services, in which case all consumers, with or without access to AATs, choose from either the outside option or the premium services, the payoff is denoted as p c . We can tell that those who choose the outside option account for F 3 (p) and thus the firms equally split the rest of the consumers: p c (d)= 1 3 p[1 F 3 (p)]: Case 3: When two firms choose to provide both services while the other chooses to provide only the premium service, we denote the payoff for the former asp 21b and the latterp 21p . Without loss of generality, suppose firm 1 chooses PO. For consumers with access to AATs, they choose between the premium service of firm 1 with net utility v 1 p, the free service of firm 2 with net utility v 2 and the free service of firm 3 with net utility v 3 (note that the outside option is dominated). They will only generate profits for firm 1 if they use the premium service of firm 1 given that 8 > < > : v 1 p> v 2 ; v 1 p> v 3 : For consumers without access to AATs, they choose from the outside option, all three premium services and the two free services from firms 2 and 3. For firm 1, it only earns subscription revenue from consumers who use its premium service v 1 p> maxf0;a(v 2 v 0 );v 2 p;a(v 3 v 0 );v 3 pg: 72 Combining with its revenue from consumers with access to AATs, firm 1’s payoff becomes p 21p (d)=d p Z ¯ v p f(v)F 2 (v p)dv+(1d)p[ Z m(p) p f(v)F 2 ( v p a + v 0 )dv+ Z ¯ v m(p) f(v)F 2 (v)dv] For firm 2 (and similarly firm 3), it earns both subscription revenues and advertising revenues from only consumers without access to AATs. Consumers would use the free service of firm 2 if and only if 8 > > > > > < > > > > > : v 0 < v 2 <m(p); v 2 > v 3 ; a(v 2 v 0 )> v 1 p; and the premium service of firm 2 if and only if 8 > > > > > < > > > > > : v 2 >m(p); v 2 > v 3 ; v 2 p> v 1 p: Therefore, firm 2’s payoff is p 21b (d)=(1d)[p Z ¯ v m(p) f(v)F 2 (v)dv+f(d) Z m(p) v 0 f(v)F(v)F(a(v v 0 )+ p)dv]: Case 4: When two firms choose to provide only the premium services and one firm chooses to provide both services, we denote the payoff for the former asp 12p and the latterp 12b . Without loss of generality, suppose firm 1 chooses BOTH. For consumers with access to AATs, they choose from the free service of firm 1 with net utility v 1 , the premium service of firm 2 with net utility v 2 p and the premium service of firm 3 with net 73 utility v 3 p (note again that the outside option is dominated). They will generate profits for firm 2 (similarly firm 3) when they use the premium service given that 8 > < > : v 2 p> v 3 p; v 2 p> v 1 : For consumers without access to AATs, they choose from the outside option, all three premium services and the free service from firm 1. For firm 2, it only earns subscription revenue from consumers who use its premium service v 2 p> maxf0;v 3 p;a(v 1 v 0 );v 1 pg: Combining with its revenue from consumers who have access to AATs, firm 2’s payoff becomes p 12p (d)=d p Z ¯ v p f(v)F(v)F(v p)dv +(1d)p[ Z m(p) p f(v)F(v)F( v p a + v 0 )dv+ Z ¯ v m(p) f(v)F 2 (v)dv]: For firm 1, it earns both subscription revenues and advertising revenues from only consumers without access to AATs. Consumers would use the free service of firm 1 if and only if 8 > > > > > < > > > > > : v 0 < v 1 <m(p); a(v 1 v 0 )> v 2 p; a(v 1 v 0 )> v 3 p; and the premium service of firm 1 if and only if 8 > > > > > < > > > > > : v 1 >m(p); v 1 p> v 2 p; v 1 p> v 3 p: 74 Therefore, firm 1’s payoff is p 12b (d)=(1d)[p Z ¯ v m(p) f(v)F 2 (v)dv+f(d) Z m(p) v 0 f(v)F 2 (a(v v 0 )+ p)dv]: Using the simulated parameters, we can numerically calculate the payoffs under different levels of penetration rates. To determine the Nash equilibrium of the triopoly service provision game, we are especially interested in the differencesp(d)p 21p (d),p 21b (d)p 12p (d) andp 12b (d)p c (d), which we show in the following graph: Figure 2.12: Payoff Comparison (Triopoly) As shown in the graph, there are 4 possible scenarios for the triopoly service provision game: • Ford <d 1 0:35, it is a dominant strategy equilibrium for all three firms to choose BOTH. • Ford 1 <d <d 2 0:42, we have p(d)>p 21p (d);p 21b (d)>p 12p (d);p 12b (d)p 21p (d);p 21b (d)d 3 , it is a dominant strategy equilibrium for all three firms to choose PO. To summarize the implications from the discussion above: The current penetration rate d = 0:13 falls in the first scenario where it is a dominant strategy equilibrium for all firms to provide both services. As the penetration rate d increases, all firms providing only the premium services emerges as a new equilibrium parallel to the existing one. And for sufficiently larged, it becomes a dominant strategy equilibrium for all firms to provide only the premium services. In terms of welfare, we only need to compare the consumer surplus or total surplus between the two equilibria: Figure 2.13: Consumer Surplus Comparison 76 As shown in the diagram, consumer surplus in the all BOTH equilibrium CS is always higher than that in the all PO equilibrium CS c , and thus as the penetration rate increases, consumers are worse off when firms switch from the former to the latter. Figure 2.14: Total Surplus Comparison Similarly, the total surplus in the all BOTH equilibrium T S is also always higher than that in the all PO equilibrium T S c . Despite that firms are better off when they switch to the all PO equi- librium by forcing the consumers with access to AATs to use the premium services and generate subscription revenues when d is close to 1, the magnitude of such effect is too small compared to the difference in consumer surplus. We conclude that, if firms respond to the increase in the penetration rate of AATs by switching to the equilibrium where all firms simultaneously provide only the premium services, then both the consumers and the society as a whole would suffer a loss in welfare. 2.7.6 Impact on Quality Investment The two thresholdsl CS (d) andl T S (d) are calculated and shown in the following graph: 77 Figure 2.15: Quality Equivalence Since the consumer surplus increases in both d and l, the threshold l CS (d) decreases in d, which means that, asd increases, consumers will be harmed if and only if the firms have an even stronger incentive to reduce their investment in service qualities and decreasel significantly. On the other hand, we note thatl T S (d) falls above the calibratedl for intermediated, which means for thesed, the total surplus is already below the initial level even if the firms do not reduce their investment in service qualities. We also note thatl T S (d) is rather flat ind overall, suggesting that it requires much smaller reduction in service quality investment for the total surplus to drop. To summarize, the graph provides certain evidences to support our argument that, if firms respond to an increase in the penetration rate of AATs by reducing their investment for service qualities, the society as a whole would likely suffer a loss in total surplus, even if the magnitude of the reduction is rather small and the consumers may still be gaining in consumer surplus. 2.8 Concluding Remarks In this paper we propose a game-theoretic framework to study the competition of freemium ser- vices. We show that, while firms can essentially provide only the free service or only the premium service through different pricing, providing both services and thus freemium can be supported in 78 equilibrium under certain regularity conditions. Based on the existence of a freemium equilibrium, we consider the impact of an increase in the penetration rate of AATs, and show that in the short run it would lower the equilibrium price and profits for the firms and benefit the consumers. In the long run, however, if the firms respond by dropping the free services or reducing their investment in service qualities, consumers and the society as a whole may end up suffering a loss in welfare. To simplify our analysis, throughout the paper we assume a mechanical process of how the advertising rate is determined. We also assume a fixed level of advertising intensity and disutility from watching ads across firms and consumers. As a further extension, we would like to consider an endogenous determination of the advertising rate f and allow more flexibility in the firms’ choice of advertising intensities. Meanwhile, the access to AATs is assumed to be independent from the consumer’s basic valu- ations for services provided by firms. This may be the case if AATs are provided to the consumers unexpectedly (e.g., consumers download a browser which happens to have a built-in ad-blocking function), but may not work well if consumers actively seek AATs (e.g., consumers spend some time looking for ad-blocking software and plug-ins such as AdBlock to get rid of the ads). To bet- ter model the latter case, we may include a consumer search model for AATs so that their access to AATs would actually be correlated with their valuations, in which case our intuitions are that consumers with intermediate realization of service valuations are more likely to get AATs. Finally, we are interested in exploring more possibilities in the vertical differentiation structure. As we have mentioned, Hulu dropped its free service in 2016 and modified it into a paid service with ads which is cheaper than its original premium service. In other words, in addition to the usual “quality-price” pair representation from the traditional vertical differentiation models, we may add advertising intensity as a third parameter which works like a quality parameter while it also affects the pricing. 79 Chapter 3 Equilibrium Existence of Price Competition among Moonlighting Firms 3.1 Introduction The term “moonlighting” means holding a second job outside of normal working hours. For exam- ple, a programmer who works 9-to-5 at Google may freelance on WeWork at night. Similarly, firms may have a secondary source of revenue in addition to their primary operations, and analogously I call them “moonlighting firms”. Some examples of moonlighting firms include: • Licensing: Firms with patents and intellectual properties can use these intangible assets in their own productions, while licensing them to other firms for licensing fees. • Public Firms: Public firms are firms that maximize total surplus instead of own profits. Equivalently, they can be regarded as moonlighting firms whose secondary source of revenue is the sum of profits of other firms and consumer surplus. • Delegated Bundling: Product firms may sell their products both individually or in a bundle. Instead of running the bundle by themselves, firms can delegate the operation of the bundle to a third-party firm and receive a royalty fee in return. 80 • Freemium: Firms may provide a free version of their products or services, which is usually of lower quality and ad-sponsored. Advertisers then pay advertising fees to the firms based on the number of users. • Cross Holding: Firms may hold shares of other firms in the same industry, and thus the manager who runs the firm has to take into consideration the equity values of other firms when deciding the optimal action in the competition. I shall note two prominent features of moonlighting firms from these examples: On the one hand, the firms’ secondary source of revenue is usually related to their primary operations, due to either that they share the same set of assets or skills during the production process (e.g., licensing, delegated bundling, freemium), or that they originate from the same market (e.g., public firms and cross holding), or both. On the other hand, while the firms have full control over their primary operations, they have only partial or indirect control over the secondary source of revenue. For example, in the freemium model, firms can arbitrarily set the price for their premium services, but since there is no price for free services, they can only alter the number of free service users indirectly. This contrasts my model with the usual multi-product firms where firms have full control over all products (and thus all sources of revenue). The presence of a secondary source of revenue leads to a challenge in proving equilibrium existence for price competitions among moonlighting firms. Even under a mild assumption that both profits from the primary and secondary sources are log-concave in the firms’ pricing decisions, the sum of log-concave functions are not necessarily log-concave. Even worse, it is not guaranteed to be quasi-concave, which is rather a common sufficient condition for the existence of a pure- strategy Nash equilibrium. Therefore, in this paper, I propose first several conditions under which each moonlighting firm’s profit function is quasi-concave in its own pricing decisions. Utilizing these conditions, I show that an equilibrium may exist for price competition among moonlighting firms, and further 81 the equilibrium can be characterized by first-order conditions, which provides a solid foundation for further analysis of the equilibrium outcomes. I then move on to illustrate how these results can be applied to economic activities involv- ing moonlighting firms. The first application compares per-unit and ad valorem royalty licensing when firms are involved in a price competition, which complements an earlier paper Fan, Jun, and Wolfstetter (2018) who discussed the same issue under quantity (Cournot) competition. I show that, if the incumbent patent holder is equally efficient in utilizing the cost-reduction patent as the licensee, it always prefer per-unit to ad valorem royalty licensing under price competition, which is in contrast to the findings of Fan, Jun, and Wolfstetter (2018) that the incumbent patent holder should be indifferent. The second application considers the horizontal merger in certain industries where firms can profit from the outside option. The outside option can be viewed as a basic service provided at a fixed price, possibly due to government regulations. Meanwhile, firms compete in providing a premium service. An example would be the vehicle insurance market in China, where all vehicle owners have to purchase a mandatory insurance of fixed quality and price from any insurance company, and in addition may purchase some commercial insurance with better coverage. I show that the profitable outside option always results in a higher increment in product prices after the merger, and thus to offset such effects requires larger synergies in terms of cost-reduction Finally, I consider conditions for the equilibrium existence and uniqueness under arbitrary cross-holding structures. Suppose the manager of firm i who controls the price p i for product i also owns shares of other firms. Thus, when deciding the optimal price for product i, the manager does not maximize the profits of product i, but instead the equity value of her own asset portfolio. I show that, there exists an equilibrium where all managers price their products above the corresponding marginal costs. Further, such an equilibrium is unique if the magnitudes of all cross holdings are relatively small. The rest of this paper is organized as follows: Section 3.2 reviews some related literature. Sec- tion 3.3 sets up the model and provides conditions for equilibrium existence. Section 3.4 considers 82 three applications on related topics of interest. Section 3.5 concludes the paper. All proofs are presented in Appendix C. 3.2 Literature Review In their seminal work, Caplin and Nalebuff (1991) considered the equilibrium existence problem for price competitions among single-product firms. The demand for each product is determined through a discrete choice framework and the uncertainty of consumer types or product attributes enters the utility function linearly. The authors showed that, if the joint distribution of uncertainty has a density function that is [1=(n+ 1)]-concave, then each firm’s demand function is (1)- concave and its profit function is quasi-concave in its price, and thus there exists a pure-strategy Nash equilibrium for the price competition. However, their method does not apply to the moonlighting firms. The concavity assumption impose on density functions can only guarantee that each of the firm’s source of revenue (primary or secondary) is quasi-concave in its pricing decision, but not necessarily the summation. Thus, I need to find some other conditions in addition to the general concavity. A recent paper Nocke and Schutz (2018) considered the price competition of multi-product oligopoly under discrete-continuous choice demand. In other words, consumers choose in a discrete- choice way which brand to buy from, and then decide a continuous amount of purchase. The au- thors utilized the aggregative game approach to reduce the multi-product firms’ decision to a single decision of the i-markup, and further show that any equilibrium can be summarized by a single aggregator. Unlike multi-product firms, however, moonlighting firms have only partial or indirect control over their secondary source of revenue, and thus even if I consider the discrete-continuous choice demand such as multinomial logit (MNL) demand or constant elasticity of substitution (CES) demand, there are no pricing decisions to be aggregated in the first place. 83 This paper also contributes to the literature on licensing, public firms, cross holdings and other types of economic activities that can be regarded as moonlighting firms. I note that, many pioneer- ing works in the literature deal with these problems under quantity (Cournot) competition rather than price competition, even if the latter is equally, if not more commonly, observed in reality. (For example, Fan, Jun, and Wolfstetter (2018), Kamien and Tauman (1986), Katz and Shapiro (1985), San Mart´ ın and Saracho (2010), Wang (1998) in the licensing literature, and Clayton and Jorgensen (2005), Farrell and Shapiro (1990a), Flath (1991), Reynolds and Snapp (1986) in the cross holding literature.) The choice of modeling these economic activities with quantity competitions over price competitions is likely due to the relative difficulty in establishing a pricing equilibrium, and thus the discussion on the equilibrium existence of pricing competition among moonlighting firms sets an important first step for further analysis. 3.3 Model and Equilibrium Existence Consider a market consists of n firms competing in price. Firm i’s profit function can be written as: p i (p)=(p i c i )D i (p)+ r i (p); (3.1) where p i , c i and D i (p) are respectively the price, marginal cost and demand for product i, and thus (p i c i )D i (p) denotes firms i’s profits from its primary operations. In addition, firm i receives a secondary source of revenue r i (p) which is related to its primary operation and pricing decision (as well as the pricing decisions of other firms). Finally, each firm i chooses the price p i to maximize its total profitsp i (p). For ease of exposition, I assume that all functions mentioned in this paper is sufficiently smooth unless otherwise noted. Further, I assume throughout the paper that: Assumption 3.1. D i (p) is strictly decreasing and log-concave in p i . The log-concavity assumption is satisfied by many demand functions, such as linear demands or multinomial logit (MNL) demands. If the demand functions are derived from a discrete choice 84 framework, according to Caplin and Nalebuff (1991), the log-concavity of these demand func- tions follows immediately from the log-concavity of the joint density function of the underlying uncertainties entering linearly into the consumers’ utility. For now, I impose no conditions on the secondary source of revenue r i (p), but will unravel as I move on to examine the equilibrium existence. As is well known, a sufficient condition for the existence of a pure-strategy Bertrand-Nash equilibrium for the price competition among firms is that each firm’s profit function is quasi- concave in its pricing decision, given the prices of other firms. Without loss of generality, consider firm 1 and examine its profit function: 1 p 1 (p 1 )=(p 1 c 1 )D 1 (p 1 )+ r 1 (p 1 ): (3.2) The first-order derivative of this function is thus p 0 1 (p 1 )= D 1 (p 1 )+(p 1 c 1 )D 0 1 (p 1 )+ r 0 1 (p 1 ): (3.3) Before I examine the quasi-concavity ofp 1 (p 1 ) in p 1 , I state the following lemma which pro- vides an important sufficient condition for quasi-concavity. Lemma 3.1. If there exists a function g(x) > 0 such that f 0 (x)g(x) decreases in x, then f(x) is quasi-concave in x. Following Lemma 1, consider first h 1 (p 1 )= p 0 1 (p 1 ) D 0 1 (p 1 ) = D 1 (p 1 ) D 0 1 (p 1 ) p 1 + c 1 r 0 1 (p 1 ) D 0 1 (p 1 ) : (3.4) Lemma 3.2. If r 0 1 (p 1 ) D 0 1 (p 1 ) decreases in p 1 , thenp 1 (p 1 ) is quasi-concave in p 1 . 1 When the other prices are given, I can regard the profit function as a function of p 1 only. 85 For illustration purposes, consider a specific setting where firm 1 earns the secondary revenue from (and thus proportional to) the demand of another product (or the outside option) on the same market, say product 2. Thus, firm 1’s secondary source of revenue can be written as r 1 (p 1 )= rD 2 (p 1 ); (3.5) where r can be viewed as an effective price. Under this specification, I can rewrite r 0 1 (p 1 ) D 0 1 (p 1 ) =r D 0 2 (p 1 ) D 0 1 (p 1 ) = rd 12 (p 1 ); (3.6) where d 12 (p 1 ) denotes the diversion ratio from product 1 to product 2 when the price of product 1 increases. Therefore, the required condition in Lemma 3.2 translates into that the diversion ratio d 12 (p) is decreasing in p 1 . Example 1: In a duopoly market with linear demands, suppose 8 > < > : D 1 (p)=ab p 1 +g p 2 ; D 2 (p)=ab p 2 +g p 1 : Then the diversion ratio from product 1 to product 2 d 12 (p)= ¶D 2 (p)=¶ p 1 ¶D 1 (p)=¶ p 1 = g b is constant. Thus, if firm 1 earns its secondary revenue from product 2, then p 1 (p 1 ) is always quasi-concave in p 1 . 86 Example 2: In a duopoly market with MNL demand and an outside option: 8 > > > > > > < > > > > > > : D 0 (p)= 1 e v 1 p 1 + e v 2 p 2 + 1 ; D 1 (p)= e v 1 p 1 e v 1 p 1 + e v 2 p 2 + 1 ; D 2 (p)= e v 2 p 2 e v 1 p 1 + e v 2 p 2 + 1 : If firm 1 earns its secondary revenue from product 2, the diversion ratio d 12 (p)= ¶D 2 (p)=¶ p 1 ¶D 1 (p)=¶ p 1 = D 1 (p)D 2 (p) D 1 (p)[1 D 1 (p)] = D 2 (p) 1 D 1 (p) = e v 2 p 2 e v 2 p 2 + 1 is constant in p 1 . Instead, if firm 1 earns its secondary revenue from the outside option, the diver- sion ratio d 10 (p)= ¶D 0 (p)=¶ p 1 ¶D 1 (p)=¶ p 1 = D 0 (p) 1 D 1 (p) = 1 e v 2 p 2 + 1 is also constant in p 1 . Either way, firm 1’s profit functionp 1 (p 1 ) is always quasi-concave in p 1 . Example 3: Consider a random utility framework, where a continuum of consumers of mass 1 have random valuations v 1 and v 2 for product 1 and 2. Respectively, v 1 and v 2 follow distribution functions F 1 (v) and F 2 (v) with support onR + and log-concave density functions f 1 (v) and f 2 (v). Given the prices p 1 and p 2 , consumers choose from product 1 (with net utility v 1 p 1 ), product 2 (with net utility v 2 p 2 ) and an outside option (with normalized net utility 0). The demand functions can thus be derived as 8 > > > > > < > > > > > : D 0 (p)= F 1 (p 1 )F 2 (p 2 ); D 1 (p)= Z ¥ 0 F 2 (v+ p 2 ) f 1 (v+ p 1 )dv; D 2 (p)= Z ¥ 0 F 1 (v+ p 1 ) f 2 (v+ p 2 )dv: 87 As mentioned earlier, the demand functions are all log-concave in the prices following the results in Caplin and Nalebuff (1991). The diversion ratios are d 12 (p)= R ¥ 0 f 1 (v+ p 1 ) f 2 (v+ p 2 )dv F 2 (p 2 ) f 1 (p 1 )+ R ¥ 0 f 1 (v+ p 1 ) f 2 (v+ p 2 )dv ; d 10 (p)= F 2 (p 2 ) f 1 (p 1 ) F 2 (p 2 ) f 1 (p 1 )+ R ¥ 0 f 1 (v+ p 1 ) f 2 (v+ p 2 )dv : Since d 12 (p)+ d 10 (p)= 1 and neither function is constant, only one of them can be decreasing in p 1 . Since f 1 (v) is log-concave, I have ¶ ¶ p 1 [lnd 12 (p) lnd 10 (p)]= R ¥ 0 f 0 1 (v+ p 1 ) f 2 (v+ p 2 )dv R ¥ 0 f 1 (v+ p 1 ) f 2 (v+ p 2 )dv f 0 1 (p 1 ) f 1 (p 1 ) 0; and thus d 12 (p) decreases in p 1 while d 10 (p) increases in p 1 . Therefore, if firm 1 earns its sec- ondary revenue from product 2, its profit function is guaranteed to be quasi-concave by Lemma 3.2. However, if firm 1 earns its secondary revenue from the outside option, I need some other conditions to guarantee the quasi-concavity. Alternatively, I can manipulate the first-order derivativep 0 1 (p 1 ) in a different way: h 2 (p 1 )= p 0 (p 1 ) D 1 (p 1 ) = 1(p 1 c 1 ) D 0 1 (p 1 ) D 1 (p 1 ) + r 0 1 (p 1 ) D 1 (p 1 ) : (3.7) Lemma 3.3. Suppose p 1 solves the first-order condition p 0 1 (p 1 )= 0. If p 1 c 1 and r 0 1 (p 1 ) D 1 (p 1 ) de- creases in p 1 , thenp 1 (p 1 ) is quasi-concave in p 1 . Under the specification that firm 1 earns its secondary revenue from another product, I can rewrite r 0 1 (p 1 ) D 1 (p 1 ) = r D 0 2 (p 1 ) D 1 (p 1 ) : If further the demand system admits a smooth indirect utility function V(p) and thus ¶ 2 V(p) ¶ p 1 ¶ p 2 = ¶D 1 (p) ¶ p 2 = ¶D 2 (p) ¶ p 1 ; 88 then the required condition in Lemma 3.3 becomes ¶ 2 lnD 1 (p) ¶ p 1 ¶ p 2 0: (3.8) Example 3 (Continued): Suppose again that firm 1 earns its secondary revenue from the outside option, and r 0 1 (p 1 ) D 1 (p 1 ) = r D 0 0 (p 1 ) D 1 (p 1 ) = r F 2 (p 2 ) f 1 (p 1 ) R ¥ 0 F 2 (v+ p 2 ) f 1 (v+ p 1 )dv : Since ¶ ¶ p 1 [lnD 0 0 (p 1 ) lnD 1 (p 1 )]= f 0 1 (p 1 ) f 1 (p 1 ) R ¥ 0 F 2 (v+ p 2 ) f 0 1 (v+ p 1 )dv R ¥ 0 F 2 (v+ p 2 ) f 1 (v+ p 1 )dv 0; the condition in Lemma 3.3 does not hold and firm 1’s profit function is still not guaranteed to be quasi-concave in p 1 . Before I move on to the last Lemma for quasi-concavity, let us ponder for a second on the possible causes for firm 1 to not have a quasi-concave profit function in Example 3. Due to the log-concavity of demands, firm 1’s profits from its primary operation is log-concave and thus quasi- concave in its pricing decisions p 1 . Furthermore, as p 1 increases, the demand for the other product (either product 2 or the outside option) increases, and thus firm 1’s secondary revenue is always increasing in p 1 . From that perspective, as firm 1’s effective price r for the secondary revenue increases, it has a larger incentive to price its own product 1 higher and get compensated from the increased demand for the other product. Intuitively, to ensure that firm 1 will not deviate to setting a higher price for its own product 1, the price p 1 for its own product should be comparatively higher than the effective price r for the secondary source. I summarize the result in the following lemma: 89 Lemma 3.4. Suppose p 1 satisfies the-first-order conditionp 0 1 (p 1 )= 0. If there exists m 1 > 0 such that p 1 m 1 + c 1 , and both r 0 1 (p 1 ) D 1 (p 1 ) and D 0 1 (p 1 )r 0 1 (p 1 )=m 1 D 1 (p) are positive and increase in p 1 , then p 1 (p 1 ) is quasi-concave in p 1 . Example 3 (Continued): I have shown that, if firm 1 earns its secondary revenue from the outside option, then r 0 1 (p 1 ) D 1 (p 1 ) is positive and increases in p 1 . Further let m 1 = r, then D 0 1 (p 1 ) r 0 1 (p 1 )=m 1 D 1 (p 1 ) = D 0 1 (p 1 ) D 0 0 (p 1 ) D 1 (p 1 ) = D 0 2 (p 1 ) D 1 (p 1 ) = D 0 2 (p 1 ) D 0 1 (p 1 ) D 0 1 (p 1 ) D 1 (p 1 ) which is also positive and increases in p 1 . Therefore, if the first-order conditionp 1 (p 1 )= 0 solves for some p 1 > c 1 + r, then firm 1’s profit functionp 1 (p 1 ) is indeed quasi-concave in p 1 . Recall that, a sufficient condition for the existence of a pure-strategy Nash equilibrium is that all firms’ profit functions are quasi-concave in their own pricing decisions. I am now ready to present the following Proposition on equilibrium existence utilizing the previous Lemmas: Proposition 3.1. Suppose a vector of prices p solve all the first-order conditions, and for each firm i, either of the following holds: •r 0 i (p)=D 0 i (p) decreases in p i , or • r 0 i (p)=D i (p) decreases in p i and p i c i , or • r 0 i (p)=D i (p) is positive and increases in p i , and there exists some m i > 0 such that, p i m i + c i and(D 0 i (p) r 0 i (p)=m i )=D i (p) is positive and increases in p i . Then p constitutes a pure-strategy Nash equilibrium. 90 3.4 Applications 3.4.1 Per-Unit vs. Ad Valorem Royalty Licensing Patent and intellectual property owners can license these intangible assets to other firms and earn royalty fees in addition to using them directly in their own productions, and thus can be viewed as moonlighting firms. In an earlier paper, Fan, Jun, and Wolfstetter (2018) discussed the comparison between per- unit and ad valorem royalty licensing in a market where the licensor and licensee compete in Cournot competition. The authors concluded that, the licensor prefers per-unit to ad valorem royalty licensing if and only if the licensor is more efficient than the licensee in utilizing the patent. However, I shall note that price competitions are equally, if not more commonly observed in reality. Therefore, in this subsection, complementing their discussions, I set up a similar licensing game under a price competition to examine how the results will change. Consider the same dynamic licensing game as described in Fan, Jun, and Wolfstetter (2018). The duopoly product market consists of an incumbent patent holder (firm 0) and a competitor (firm 1). The incumbent offers a license contract in the form of a two-part tariff that prescribes either a per-unit royalty rate, r, or an ad valorem royalty rate s, together with a fixed fee f . If the license contract is rejected, in the subsequent price competition, firm 0 has a constant marginal cost d and firm 1 has a constant marginal cost c > d, reflecting the advantage of firm 0 due to the patent. Conditional on accepting the license contract, the constant marginal cost for firm 0 remains unchanged at d, and the constant marginal cost for firm 1 reduces to x. Following Fan, Jun, and Wolfstetter (2018), I call firm 0 “more efficient” if x> d and firm 1 “more efficient” 91 if x < d. Thus, the firms’ payoff functions under per-unit and ad valorem royalty licensing are respectively: p r 0 (p 0 ; p 1 )=(p 0 d)D 0 (p 0 ; p 1 )+ rD 1 (p 0 ; p 1 )+ f r ; p r 1 (p 0 ; p 1 )=(p 1 x r)D 1 (p 0 ; p 1 ) f r : p s 0 (p 0 ; p 1 )=(p 0 d)D 0 (p 0 ; p 1 )+ sp 1 D 1 (p 0 ; p 1 )+ f s ; p s 1 (p 0 ; p 1 )=((1 s)p 1 x)D 1 (p 0 ; p 1 ) f s : (3.9) For ease of exposition, I impose the following regularity conditions on the demand functions D 0 (p) and D 1 (p). Assumption 3.2. The demand functions D 0 (p) and D 1 (p) satisfy: (1) Full market coverage: D 0 (p)+ D 1 (p)= 1. (2) Roy’s identity:9V(p) such that for i2f0;1g, D i (p)= ¶V(p) ¶ p i : (3) Substitutes: ¶D 0 (p) ¶ p 1 = ¶D 1 (p) ¶ p 0 > 0: (4) Log-concavity: ¶ 2 lnD i ¶ p 2 i < 0: Following Assumption 3.2, I can show that, forfi; jg=f0;1g: ¶ 2 lnD i ¶ p 0 ¶ p 1 = ¶ ¶ p i ¶D i =¶ p j D i = ¶ ¶ p i ¶D j =¶ p i D i = ¶ ¶ p i ¶D i =¶ p i D i = ¶ 2 lnD i ¶ p 2 i > 0: Lemma 3.5. There exists a unique pricing equilibrium for any r 0 or 0 s< 1. The equilibrium existence follows from Proposition 3.1. For the equilibrium uniqueness, I refer to the Gale-Nikaido Theorem proposed in Gale and Nikaido (1965). Provided the existence and 92 uniqueness of a pricing equilibrium, I can show further that the equilibrium prices increase in both r and s, but at different speeds. Lemma 3.6. (1) For per-unit royalty licensing, both p r 0 and p r 1 increase in r, and p r 0 = p r 1 if x= d. (2) For ad valorem royalty licensing, both p s 0 and p s 1 increase in s, and p s 0 > p s 1 if x= d. Note that as either royalty rate r or s increases, firm 0 cares more about the demand for product 1 and the two firms have more common interests, which relaxes competition and raises the equilib- rium prices. If further the firms are equally efficient, under per-unit royalty licensing, the increase in r always has the same level of effects on both firms, and thus p r 0 = p r 1 ; while under ad valorem royalty licensing, the increase in s affects firm 0 more than firm 1, and eventually p s 0 > p s 1 . Intuitively, firm 0 should always prefer relaxing the competition and pushing up the prices, for at least two reasons: On the one hand, due to the full market coverage assumption, higher prices will not drive consumers out of the market. On the other hand, since firm 0 can charge firm 1 a fixed fee f through the licensing contract, it is always to the firm 0’s interest to fully exploit firm 1’s gain from licensing, and leave firm 1 only its pre-licensing profits, denoted as p n 1 . In other words, when firm 0 is designing the contract in the first stage, if firm 0 chooses per-unit royalty licensing, it sets r to maximize P r =(p r 0 d)D 0 (p r 0 ; p r 1 )+(p r 1 x)D 1 (p r 0 ; p r 1 )p n 1 ; (3.10) and if firm 0 chooses ad valorem royalty licensing, it sets s to maximize P s =(p s 0 d)D 0 (p s 0 ; p s 1 )+(p s 1 x)D 1 (p s 0 ; p s 1 )p n 1 : (3.11) Both profits should be increasing in the corresponding royalty rates r or s. Lemma 3.7. P r increases with r andP s increases with s. Nonetheless, it is neither desirable nor realistic for firm 0 to choose an infinite r or s arbitrarily close to 1. In Fan, Jun, and Wolfstetter (2018), the authors considered an antitrust constraint 93 that, the post-licensing equilibrium prices cannot be higher than the pre-licensing prices. Here, I impose the same constraint that, whether firm 0 chooses per-unit or ad valorem royalty licensing, the resulting equilibrium prices should not exceed the pre-licensing prices, denoted as p n 0 and p n 1 . Since the marginal cost for firm 1 before licensing is c, which is always greater than the marginal cost d for firm 0, I shall expect that firm 0 sets a lower price than firm 1 before licensing. Corollary 3.1. The pre-licensing prices satisfy p n 0 < p n 1 . Combining Lemma 3.6, Lemma 3.7 and Corollary 3.1, when designing the contract, firm 0 would keep increasing r or s until either equilibrium price hits the antitrust constraint, and an immediate result is as follows: Corollary 3.2. Suppose x= d. When firm 0 optimally chooses r and s subject to the antitrust constraint, p s 1 < p s 0 = p r 0 = p r 1 = p n 0 < p n 1 : The main results in Fan, Jun, and Wolfstetter (2018) state that, under Cournot competition and subject to the antitrust constraint, the incumbent patent holder (firm 0) prefers per-unit royalty licensing over ad valorem royalty licensing if and only if firm 0 is more efficient, i.e., P r (r )RP s (s ), xR d: (3.12) Although I am unable to perform a comprehensive examination of all cases, I present the fol- lowing result for the indifferent case x= d: Proposition 3.2. Suppose x= d, then the incumbent patent holder (firm 0) always prefers per-unit over ad valorem royalty licensing. Intuitively, due to the existence of an antitrust constraint, firm 0 would prefer to raise the price of product 1 more than product 0, since product 1 observes a larger price drop due to the lower marginal cost before and after licensing. However, the best firm 0 can do is to use the per-unit royalty licensing, where the prices are raised equally as the royalty rate r increases, while under ad 94 valorem royalty licensing, as the royalty rate s increases, it raises the price of product 0 more than product 1, which is undesirable for firm 0. 3.4.2 Profitable Outside Option In discrete choice frameworks, the outside option is often modeled in a way that it creates no profit for the firms in the market. This may not be the case in certain industries. For example, vehicle owners in China have to purchase a mandatory vehicle insurance from any insurance company, and both the quality and price of such insurance are regulated by the government. In addition, vehicle owners can purchase commercial insurance from these companies for better coverage. In other words, insurance companies compete in the provision of commercial insurances, while profiting from the outside option which is the mandatory insurance at a fixed price. The existence of a profitable outside option would affect the firms’ pricing behavior, and further would possibly result in changes when a horizontal merger happens in the industry. In this section, following the idea of Farrell and Shapiro (1990b), I examine how the presence of a profitable outside option affects the price changes before and after a horizontal merger in the market, given the possibility of post-merger synergies. Consider a symmetric duopoly market consisting of two firms 1 and 2. There are three types of services available on the market: The basic service, which is like the outside option, is of a constant valuation v 0 for consumers and its price is fixed at p 0 . In addition, each firm i provides a premium service i at a constant marginal cost c, which adds v i to the valuation of the basic service, and is sold at a total price of p i . Thus, a continuum of consumers of mass 1 choose one service from the following: • Basic service with net utility v 0 p 0 ; • Premium service 1 with net utility v 1 + v 0 p 1 ; • Premium service 2 with net utility v 2 + v 0 p 2 . 95 Assume that v 1 and v 2 IID follow the same distribution function F(v) with support onR + and a log-concave density function f(v). The demands for each service are: 8 > > > > > < > > > > > : D 0 (p)= F(p 1 p 0 )F(p 2 p 0 ); D 1 (p)= Z ¥ 0 f(v+ p 1 p 0 )F(v+ p 2 p 0 )dv; D 2 (p)= Z ¥ 0 f(v+ p 2 p 0 )F(v+ p 1 p 0 )dv: (3.13) Each firm earns the profits from the sales of its own premium service. In addition, the firms equally split the demand for the outside option. Therefore, their profit functions are respectively: p 1 (p 1 ; p 2 )=(p 1 c)D 1 (p)+ 1 2 p 0 D 0 (p); p 2 (p 1 ; p 2 )=(p 2 c)D 2 (p)+ 1 2 p 0 D 0 (p): (3.14) For the ease of exposition, I impose first the following regularity conditions: Assumption 3.3. For i2f1;2g: j ¶ 2 lnD i (p) ¶ p 0 ¶ p i j> 1 2 j ¶ 2 lnD i (p) ¶ p 2 0 j: (3.15) Assumption 3.3 serves as a technical condition for equilibrium uniqueness. Assumption 3.4. p 0 < 1 F 2 (c) F(c) f(c)+ 2 R ¥ c f 2 (v)dv : (3.16) Following the idea of Lemma 3.4, Assumption 3.4 ensures that the firms’ effective price from their secondary source of revenue is relatively low. Example I: If F(v) admits a uniform distribution U[0;k] with k> 0, then the condition in Assump- tion 3.4 becomes p 0 < k 2 c 2 2k c : 96 Example II: If F(v) admits an exponential distribution Exp(l), then the condition in Assumption 3.4 becomes p 0 <l(2 e c l ): As usual, I start with establishing the equilibrium existence and uniqueness for the pre-merger price competition. Suppose p solves the symmetric equilibrium condition: 1 2 [1 F 2 (p p 0 )]+ 1 2 p 0 F(p p 0 ) f(p p 0 )= (p c)[F(p p 0 ) f(p p 0 )+ Z ¥ p p 0 f 2 (v)dv]: (3.17) Lemma 3.8. There exists a unique symmetric equilibrium for the pre-merger price competition where both firms charge p > p 0 + c. Now I consider a horizontal merger of firm 1 and firm 2 in this market. Suppose that the merger results in synergies reflected as a reduction in marginal cost from c to c. Therefore, the merged firm’s combined profits are simply p m (p 1 ; p 2 )=(p 1 c)D 1 (p)+(p 2 c)D 2 (p)+ p 0 D 0 (p) =(p 1 p 0 c)D 1 (p)+(p 2 p 0 c)D 2 (p)+ p 0 : (3.18) Without loss of generality, assume that the merged firm maximizes its profits when it prices both premium services at p m . The existence of a profitable outside option thus worsens the price and welfare effect of a horizontal merger and requires a larger remedy in terms of cost-reduction synergies: Proposition 3.3. Compared to a non-profitable outside option, a profitable outside option at fixed price p 0 raises the optimal monopoly price p m by exactly p 0 while raises the pre-merger equilib- rium price p by less than p 0 . 97 As shown in the combined profit function, the merged firm cares only about the markup p 1 p 0 and p 2 p 0 , and thus an increase in p 0 would be completely transferred to the consumers. Meanwhile, firms involved in the pre-merger competition will not raise their prices for premium services by exactly p 0 due to the competition pressure. Therefore, given a fixed level of synergies, compared to a non-profitable outside option, a profitable outside option would increase the difference between the post-merger monopoly price and the pre-merger equilibrium price, and to offset such impacts a higher level of cost reduction from post-merger synergies is required. 3.4.3 Equilibrium Uniqueness under Cross Holding Consider a market with n single-product firms, where the manager of firm i controls the price p i for product i. The managers’ cross holding portfolio is denoted by a, where a i j 0 means the manager of firm i owns a i j shares of firm j. In general, a ii should be much larger than a i j for j6= i, meaning that the manager of firm i controls most of the shares of her own firm, but I make no explicit assumptions on that for now, except for that a ii > 0 for the purpose of normalization later. When choosing the optimal price for product i, the manager of firm i maximizes the total valuation of her portfolio instead of the profits from product i, and thus the pricing equilibrium would be different from the benchmark case where all managers care only about their own firms (which is equivalent to the special case where a ii = 1 for all i and a i j = 0 for all i6= j). That is, the manager of firm i solves max p i å j a i j (p j c j )D j (p): (3.19) Further assume that product i is produced at constant marginal cost c i , and its demand follows the MNL demand system: D i (p)= e v i p i å k e v k p k + H 0 : (3.20) Lemma 3.9. There exists an equilibrium where all products are priced above their marginal costs. 98 The equilibrium existence follows the discussion of MNL demand after Lemma 3.2. Moreover, given that the other managers price their products above the marginal costs, the manager of firm i is always better off by pricing at c i compared to pricing below c i , so that she can avoid the loss from selling product i below marginal cost while also increase the equity value for her shares of all other firms. Consider the normalized portfolio matrix as follows A= 2 6 6 6 6 6 6 6 4 1 a 12 a 11 a 1n a 11 a 21 a 22 1 a 2n a 22 . . . . . . . . . . . . a n1 a nn a n2 a nn ::: 1 3 7 7 7 7 7 7 7 5 (3.21) and denote a 1 i j as the entries for its inverse matrix A 1 . Proposition 3.4. If for all i, å j a 1 i j > e v i c i e v i c i + H 0 ; (3.22) then such equilibrium mentioned in Lemma 3.9 is unique. The phrase “unique” here means that there is no other equilibrium where all products are priced above their maginal costs. There still might be equilibrium where some products are priced below their marginal costs and there could be multiple of them. In general, the condition in Proposition 3.4 requires that the manager of firm i cannot hold a significant share of a different firm j. In the benchmark case when the managers care only about the profits from their own firms, A= A 1 = I andå j a 1 i j = 1, and the condition always holds. To better illustrate the intuition, consider the symmetric case where the cross holding of any manager of firm i over a different firm j is constant at t2[0; 1 n1 ), and thus every manager is left with 1(n 1)t share of her own firm. 99 Corollary 3.3. Suppose all managers hold the same amount of t shares of all other firms, then Proposition 3.4 holds if t < 1 n 1 H 0 e max i (v i c i ) + H 0 : 3.5 Concluding Remarks In this paper, I introduce the concept of “moonlighting firms”, where firms have a secondary source of revenue with indirect or partial control in addition to their primary operations, and show that the concept nests many economic phenomena including licensing, public firms, delegated bundling, freemium and cross holding. I note that a challenge in solving the equilibrium for a price compe- tition among these firms is the quasi-concavity of profit functions in their pricing decisions due to that sum of quasi-concave functions are not necessarily quasi-concave. To deal with this challenge, I present several conditions under which each firm’s profit function is indeed quasi-concave in its own price, and thus the equilibrium existence is established. 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Therefore, if the density function f(x) is log-concave, so are the CDF F(x) and the survival function 1 F(x). • If the function f(x) is log-concave, so is f(x). Therefore, if a random variable X F has a log-concave density function, so doesX F . • If the functions f(x) and g(x) are log-concave, so is the product f(x)g(x). Therefore, if two random variables X F and X G are independent and both have log-concave density functions, then their maximum maxfX F ;X G g also has a log-concave density function. • If the functions f(x) and g(x) are log-concave, so is the convolution: h(x)= Z f(t)g(xt)dt: 106 Therefore, if two random variables X F and X G are independent and both have log-concave density functions, then the sum X F + X G also has a log-concave density function. • If the function f(x) is log-concave and the function g(x) is monotone and concave, then the composite function f(g(x)) is log-concave. • If the joint density function f(x) is log-concave, and S(p) is a mapping from p to a convex set specified by a set of fixed linear inequalities involving only x and p, then the cumulative integral D(p)= Z S(p) f(x)dx is log-concave in p. This is a special case of the Pr´ ekopa-Borell Theorem introduced in Caplin and Nalebuff (1991). • The log-concavity of the CDF F(x) implies that f(x) F(x) decreases in x and F(x) f(x) increases in x, and thus x+ F(x) f(x) strictly increases in x. • The log-concavity of the survival function 1 F(x) implies that f(x) 1F(x) increases in x and 1F(x) f(x) decreases in x, and thus x 1F(x) f(x) strictly increases in x. A.2 Proof of Lemma 1.1 To show the existence of a symmetric pricing equilibrium, I am left to show that p 1 (p 1 ; p c ) is quasi-concave in p 1 . The first-order derivative can be derived as dp 1 (p 1 ; p c ) d p 1 = 1 F c (p 1 p c ) p 1 f c (p 1 p c ): Dividing by p 1 [1 F c (p 1 p c )]> 0, I have f(p 1 )= 1 p 1 f c (p 1 p c ) 1 F c (p 1 p c ) ; 107 which strictly decreases in p 1 , and thus p 1 (p 1 ; p c ) is quasi-concave. The uniqueness of such equilibrium thus follows that p c is uniquely determined. A.3 Proof of Proposition 1.1 I start with showing that, under Assumption 1.1, the equilibrium prices p and p b are unique and satisfy p < p c ; a n p b < p < p b : Denoted = p b p , and note first that I must haved 0 in equilibrium. If otherwised < 0 or p b < p , the bundle strictly dominates all single products, and all consumers would purchase the bundle, in which case it is strictly profitable for the third-party firm to raise the bundle price p b . Rewrite the equilibrium conditions as 8 > > < > > : p =d + 1 F t (d ) f t (d ) (1 a n )p a n d =(1 p p c ) F t (d ) f t (d ) and regard them as two curves on the d p plane. Any pair of equilibrium prices thus corre- sponds to an intersecting point of the two curves. For the first equilibrium condition, p is decreasing ind , and whend = 0, the corresponding p = 1 f t (0) > 0. For the second equilibrium condition, whenever p p c , the implicit function theorem sug- gests that p is increasing in d . Moreover, when d = 0, the corresponding p = 0, and when p = p c , the correspondingd =( n a 1)p c . Putting together these two properties, I can show that p is increasing ind ford 2[0;( n a 1)p c ] and p > p c for anyd >( n a 1)p c . 108 Therefore, if the point(( n a 1)p c ; p c ) falls above the curve representing the first equilibrium condition, then there exists a unique intersection point of the two curves. A sufficient condition for this to happen regardless of the choice ofa is thus (by replacinga with 1) p c >(n 1)p c + 1 F t ((n 1)p c ) f t ((n 1)p c ) , np c > 1 F t ((n 1)p c ) f t ((n 1)p c ) ; ,(n 1)p c n 1 n 1 F t ((n 1)p c ) f t ((n 1)p c ) = J t;1=n ((n 1)p c )> 0,(n 1)p c > J 1 t;1=n (0) which is exactly Assumption 1.1. At the intersection point, I must have d > 0 or p b > p , as well as p < p c . To show that p > a n p b , simply rewrite the second equilibrium condition as p a n p b =(1 p p c ) F t (d ) f t (d ) : The right-hand side is positive when p < p c , so the left-hand side is also positive and p > a n p b . I now turn to show the existence of a symmetric equilibrium. For the third-party firm, I am left to show thatP b (p b ; p ) is quasi-concave in p b , which immediately follows from that both p b and 1 F t (p b p ) are log-concave in p b . For product firm 1, I need to consider the two cases p 1 > p and p 1 < p separately. When p 1 > p , the first order derivative ofP 1 (p 1 ; p ; p b ) is dP 1 d p 1 =[1 F c (p 1 p )]F t (p b p 1 ) p 1 f c (p 1 p )F t (p b p 1 ) p 1 [1 F c (p 1 p )] f t (p b p 1 )+ a n p b [1 F c (p 1 p )] f t (p b p 1 ): Dividing by p 1 [1 F c (p 1 p )]F t (p b p 1 )> 0, I have f(p 1 )= 1 p 1 f c (p 1 p ) 1 F c (p 1 p ) (1 a p b np 1 ) f t (p b p 1 ) F t (p b p 1 ) ; 109 which strictly decreases in p 1 . Since f(p )= 0, for any p 1 > p , I have f(p 1 ) < 0 and hence dP 1 d p 1 < 0. When a n p b < p 1 < p , I can follow the same steps to get f(p 1 )= 1 p 1 f c (p 1 p )F t (p b p ) D 1 (p 1 ; p ; p b ) (1 a p b np 1 ) R 0 p 1 p f c (x) f t (x+ p b p 1 )dx+[1 F c (0)] f t (p b p 1 ) D 1 (p 1 ; p ; p b ) : I show first that f c (p 1 p )F t (p b p ) D 1 (p 1 ; p ; p b ) < f c (p 1 p )F t (p b p ) [1 F c (p 1 p )]F t (p b p ) f c (0)F t (p b p ) [1 F c (0)]F t (p b p ) = f c (0)F t (p b p ) D 1 (p ; p ; p b ) ; and thus 1 p 1 a p b np 1 f c (p 1 p )F t (p b p ) D 1 (p 1 ; p ; p b ) > 1 p 1 f c (p 1 p )F t (p b p ) D 1 (p 1 ; p ; p b ) > 1 p f c (0)F t (p b p ) D 1 (p ; p ; p b ) >f(p )= 0: Now rewritef(p 1 ) as f(p 1 )= 1 p 1 a p b np 1 f c (p 1 p )F t (p b p ) D 1 (p 1 ; p ; p b ) (1 a p b np 1 ) dD 1 (p 1 ;p ;p b ) d p 1 D 1 (p 1 ; p ; p b ) = 1 p 1 (1 a p b n f c (p 1 p )F t (p b p ) D 1 (p 1 ; p ; p b ) )(1 a p b np 1 ) dD 1 (p 1 ;p ;p b ) d p 1 D 1 (p 1 ; p ; p b ) : Since D 1 (p 1 ; p ; p b ) can be written as a cumulative integral on the convex set S 1 (p 1 )=f(c;t)jc > p 1 p ;c+t > p 1 p b ;t > p 1 p b g; 110 it must be log-concave in p 1 , which implies 0< dD 1 (p 1 ;p ;p b ) d p 1 D 1 (p 1 ; p ; p b ) < dD 1 (p ;p ;p b ) d p 1 D 1 (p ; p ; p b ) : Therefore, for any a n p b < p 1 < p , f(p 1 )> 1 p (1 a p b n f c (0)F t (p b p ) D 1 (p ; p ; p b ) )(1 a p b np ) dD 1 (p ;p ;p b ) d p 1 D 1 (p ; p ; p b ) =f(p )= 0; and hence dP 1 d p 1 > 0. When p 1 a n p b , I derive the samef 1 (p) as in the last case: f(p 1 )= 1 p 1 f c (p 1 p )F t (p b p ) D 1 (p 1 ; p ; p b ) (1 a p b np 1 ) R 0 p 1 p f c (x) f t (x+ p b p 1 )dx+[1 F c (0)] f t (p b p 1 ) D 1 (p 1 ; p ; p b ) : Note that the third term must be non-negative. Using the inequality derived in the last case, I have f(p 1 ) 1 p 1 f c (p 1 p )F t (p b p ) D 1 (p 1 ; p ; p b ) > 1 p f c (0)F t (p b p ) D 1 (p ; p ; p b ) >f(p )= 0; and hence dP 1 d p 1 > 0. The uniqueness of such equilibrium thus follows the uniqueness of equilibrium prices p and p b as I have shown above. A.4 Proof of Proposition 1.2 Lemma 1.2 shows that p c increases as F q (x) becomes more dispersed. I am left to show that both equilibrium prices p and p b and equilibrium profitsP andP b increase in p c . 111 Denote y(d )=d +(1 a n ) 1 F t (d ) f t (d ) [1 1 p c (d + 1 F t (d ) f t (d ) )] F t (d ) f t (d ) = 0; and recall from the proof of Proposition 1.1 that y(d ) decreases in d for d 2(0;(n 1)p c ). Since ¶y(d ) ¶ p c = 1 (p c ) 2 (d + 1 F t (d ) f t (d ) ) F t (d ) f t (d ) < 0; the implicit function theorem suggests that dd d p c = ¶y(d )=¶ p c ¶y(d )=¶d < 0: Hence, d p d p c = d dd (d + 1 F t (d ) f t (d ) ) dd d p c > dd d p c > 0 and 0< d p b d p c = d p d p c + dd d p c < d p d p c ; i.e., both equilibrium prices p and p b increase in p c . For the product firms’ equilibrium profitsP , I want to show that d d p c (nP )= d p d p c F t (d )+a d p b d p c [1 F t (d )]+(p a p b ) f t (d ) dd d p c > 0: Dividing by f t (d ) and plugging in the equilibrium conditions, the inequality can be simplified as d p d p c [ F t (d ) f t (d ) +a p b ]+ dd d p c p > 0: Since d p d p c > dd d p c > 0; 112 it is sufficient to show that p a p b F t (d ) f t (d ) < 0, p a p b < p a n p b 1 p p c ; which automatically holds in the delegated-bundling equilibrium. For the third-party firm’s equilibrium profitsP b , I want to show that dP b d p c =(1a) d p b d p c [1 F t (d )](1a)p b f t (d ) dd d p c > 0 which automatically holds since d p b d p c > 0; dd d p c < 0: A.5 Proof of Corollary 1.2 Definey(d ) as in the proof of Proposition 1.2. Note that the number of product firms n not only entersy(d ) directly, but also indirectly by changing p c . Lemma A.1 (Lemma 1(1), J. Zhou (2017)). The no-bundling equilibrium price p c decreases in n. Therefore, ¶y(d ) ¶n = a n 2 1 F t (d ) f t (d ) + d p c dn ¶y(d ) ¶ p c > 0; and the implicit function theorem suggests that dd dn = ¶y(d )=¶n ¶y(d )=¶d > 0 and d p dn = d dd (d + 1 F t (d ) f t (d ) ) dd dn < dd dn < 0 and 0> d p b dn = d p dn + dd dn > d p dn : 113 For the product firms’ total equilibrium profits nP , I want to show that d dn (nP )= d p dn F t (d )+a d p b dn [1 F t (d )]+(p a p b ) f t (d ) dd dn < 0: As in the proof of Proposition 1.2, it reduces to p a p b F t (d ) f t (d ) < 0 which automatically holds in the delegated-bundling equilibrium. For the third-party firm’s equilibrium profitsP b , I want to show that dP b dn =(1a) d p b dn [1 F t (d )](1a)p b f t (d ) dd dn < 0 which automatically holds since d p b dn < 0; dd dn > 0: A.6 Proof of Lemma 1.3 Definey(d ) as in the proof of Proposition 1.2. Since ¶y(d ) ¶a = 1 n 1 F t (d ) f t (d ) < 0; the implicit function theorem suggests that dd da = ¶y(d )=¶a ¶y(d )=¶d < 0: Hence, d p da = d dd (d + 1 F t (d ) f t (d ) ) dd da > dd da > 0 114 and 0< d p b da = d p da + dd da < d p da ; i.e., both equilibrium prices p and p b increase in the royalty ratea, and p increases faster than p b . A.7 Proof of Proposition 1.3 For the third-party firm d da P b =p b [1 F t (p b p )]+(1a)p b f t (p b p ) d p da =[(1a) d p da p b ][1 F t (p b p )]: I am left to show that (1a) d p da p b < 0 for anya2[0;1]. Recall the equilibrium condition (1 a n )p a n d =(1 p p c ) F t (d ) f t (d ) : Take derivatives of both sides with respect toa 1 n p b +(1 a n ) d p da a n dd da = 1 p c d p da F t (d ) f t (d ) +(1 p p c ) d dd ( F t (d ) f t (d ) ) dd da ; and rearrange the terms to get (1a) d p da p b = n p c d p da F t (d ) f t (d ) + n(1 p p c ) d dd ( F t (d ) f t (d ) ) dd da +a dd da (n 1) d p da < 0: 115 For product firms d da (nP )= d p da F t (d )+(p b +a d p b da )[1 F t (d )]+(p a p b ) f t (d ) dd da : Rearrange the terms, and I am left to show that for anya2[0;1], d p da [ F t (d ) 1 F t (d ) +a]+ p b + p p b dd da > 0: Since d p da > dd da > 0; it is sufficient to show that p p b a F t (d ) 1 F t (d ) < 0: Derive from the equilibrium conditions that F t (d ) 1 F t (d ) = p c p c p p a n p b p b and thus p p b a = p a p b p b < p a n p b p b < p c p c p p a n p b p b = F t (d ) 1 F t (d ) as desired. For the total surplus, rewrite it as T S b =E[max i2N q i ]+ Z ¥ d x f t (x)dx: Therefore, dT S b da =d f t (d ) dd da > 0: 116 For the consumer surplus, rewrite it as CS b =E[max i2N q i ] p + Z ¥ d (xd ) f t (x)dx: Therefore, dCS b da = d p da [1 F t (d )] dd da < F t (d ) dd da < 0: A.8 Proof of Proposition 1.4 For total surplus, T S b T S c =E[tjt > p b p ][1 F t (p b p )]> 0: For consumer surplus, rewrite CS b =E[max i2N q i ] p +E[t p b + p jt > p b p ][1 F t (p b p )] and thus CS b CS c =(p c p )+E[t p b + p jt > p b p ][1 F t (p b p )]> 0: A.9 Proof of Proposition 1.5 I start with proving the following two lemmas: Lemma A.2. d p d p c < p p c 2 : Proof. Following the proof of Proposition 1.1, rewrite the second equilibrium condition as p = 1+ a n d f t (d ) F t (d ) (1 a n ) f t (d ) F t (d ) + 1 p c : 117 Holdingd fixed, ¶ p ¶ p c = 1+ a n d f t (d ) F t (d ) [(1 a n ) f t (d ) F t (d ) + 1 p c ] 2 1 (p c ) 2 <( p p c ) 2 : In other words, a marginal increase in p c shifts the curve representing the second equilibrium condition up by less than ( p p c ) 2 , and in turn it shifts the intersection point up by less than ( p p c ) 2 , i.e., d p d p c < p p c 2 : Lemma A.3. p p b increases in p c and p b p c decreases in p c . Proof. To show that p p b increases in p c , take derivative of ln p p b with respect to p c to get d d p c (ln p p b )= 1 p d p d p c 1 p b d p b d p c > 0 since p < p b and d p d p c > d p b d p c > 0: To show that p b p c decreases in p c , note first that d p b d p c <( p p c ) 2 < p b p c (p c ) 2 = p b p c : Thus, take derivative of ln p b p c with respect to p c to get: d d p c (ln p b p c )= 1 p b d p b d p c 1 p c < 0: 118 Now I am ready to compare the equilibrium profits. If the product firms are better off with than without delegated bundling, I have P >P c , p F t (p b p )+a p b [1 F t (p b p )]> p c : Rearrange the terms to get (a p b p c )[1 F t (p b p )]>(p c p )F t (p b p ): Use the equilibrium conditions to replace 1 F t (p b p ) and F t (p b p ): (a p b p c )p b >(p c p ) p a n p b 1 p p c =(p a n p b )p c : Divide both side by p b p c to get p p b < a p b p c 1+ a n : I have shown that p p b increases in p c and p b p c decreases in p c . Further, for any given a, when p c approaches 0, the left-hand side approaches 0 while the right-hand side approaches infinity; and when p c approaches infinity, the left-hand side is positive and the right hand side is negative. Therefore, for any given a, there exists a threshold for p c such that product firms are better off if and only if p c falls below that threshold. Specifically, denote ¯ p c as the threshold for p c when a = 1. Now, for any p c < ¯ p c , product firms are better off with delegated bundling if a = 1. On the other hand, ifa = 0, it must be that P = 1 n p F t (p b p )< 1 n p c =P c ; 119 i.e., product firms are worse off with delegated bundling. Since product firms’ equilibrium profit increase ina according to Proposition 1.3, there exists a thresholda(p c )2(0;1) such that product firms are better off with delegated bundling if and only ifa >a(p c ). I am left to show thata(p c ) increases in p c . Suppose instead that there exist p 1 < p 2 such that a(p 1 )>a(p 2 ). It implies that for anya2(a(p 2 );a(p 1 )), product firms are better off if p c = p 2 but worse off if p c = p 1 < p 2 , which leads to contradiction. A.10 Proof of Proposition 1.6 Suppose first thata 1 =a 2 = a 2 . As I have shown, the unique pricing equilibrium in this case is the symmetric delegated-bundling equilibrium P e 1 ( a 2 ; a 2 )=P e 2 ( a 2 ; a 2 )=P ; P e b ( a 2 ; a 2 )=P b : Due to the symmetric role ofa 1 anda 2 , I must have ¶P e 1 ( a 2 ; a 2 ) ¶a 1 = ¶P e 2 ( a 2 ; a 2 ) ¶a 2 ; ¶P e 1 ( a 2 ; a 2 ) ¶a 2 = ¶P e 2 ( a 2 ; a 2 ) ¶a 1 ; and ¶P e b ( a 2 ; a 2 ) ¶a 1 = ¶P e b ( a 2 ; a 2 ) ¶a 2 ; which implies d da (2P )= d da [P e 1 ( a 2 ; a 2 )+P e 2 ( a 2 ; a 2 )] = 1 2 ¶P e 1 ( a 2 ; a 2 ) ¶a 1 + 1 2 ¶P e 1 ( a 2 ; a 2 ) ¶a 2 + 1 2 ¶P e 2 ( a 2 ; a 2 ) ¶a 1 + 1 2 ¶P e 2 ( a 2 ; a 2 ) ¶a 2 = ¶P e 1 ( a 2 ; a 2 ) ¶a 1 + ¶P e 2 ( a 2 ; a 2 ) ¶a 1 = ¶P e 1 ( a 2 ; a 2 ) ¶a 2 + ¶P e 2 ( a 2 ; a 2 ) ¶a 2 120 and dP b da = 1 2 ¶P e b ( a 2 ; a 2 ) ¶a 1 + 1 2 ¶P e b ( a 2 ; a 2 ) ¶a 2 = ¶P e b ( a 2 ; a 2 ) ¶a 1 = ¶P e 2 ( a 2 ; a 2 ) ¶a 1 : Therefore, if there exists somea satisfying b 1 d(2P )=da P P c + b 2 dP b =da P b = 0, P P c P b = 2b 1 b 2 dP =da dP b =da ; then the pair ( a 2 ; a 2 ) must be a solution to equations (1.21) and thus is the unique solution to the Nash bargaining problem. Examining the last equation, for any p c < ¯ p c , the right-hand side is always positive, while the left-hand side is zero ifa =a(p c ) and approaches positive infinity ifa approaches 1. Hence, there must exist ana 2(a(p c );1) such that the equation holds. A.11 Proof of Proposition 1.7 Since all product firms can fully price discriminate, they are essentially involved in a continuum of smaller price competitions for every slice of consumers with known valuationsq and an unknown t. Each of these smaller competitions is a Bertrand competition with symmetric (zero) marginal costs and asymmetric valuations. The product firms will race to the bottom, and the product firm whose product has the highest valuation wins in the end. Suppose without loss of generality that the consumer values product 1 the most, which happens with probability 1 n : q 1 max i6=1 q i ,c 1 0: Product firm 1 cannot win by pricing strictly abovec 1 : if the winning price p 1 >c 1 , the losing product firm i whose product has the second highest valuation can price slightly below p i =q i q 1 + p 1 =c 1 + p 1 > 0 to win over the consumers while maintaining the edge against the bundle. 121 Meanwhile, if product firm 1 is winning by pricing belowc 1 , it is not profitable for any losing product firm i to deviate to pricing strictly below p i =q i q 1 + p 1 c 1 + p 1 0; as it is always better off for the product firm i to marginally increase the price p i , which not only reduces the loss from the sales of single product i but also increases its share of royalty from the third-party firm. Recall that product firm 1 is still in competition with the third-party firm. When product firm 1 sets a price p 1 c 1 and the third-party firm chooses the equilibrium price ˜ p b , the consumer will purchase product 1 if and only if q 1 p 1 q 1 +t ˜ p b ,t ˜ p b p 1 and will purchase the bundle otherwise. Product firm 1 thus earns P 1 (p 1 ; ˜ p b )= p 1 F t ( ˜ p b p 1 )+ a n ˜ p b [1 F t ( ˜ p b p 1 )]: In equilibrium, product firm 1 should choose p 1 to maximize the profit, subject to the constraint that p 1 c 1 . Suppose ˜ p solves the first-order condition 1 ˜ p =(1 a ˜ p b n ˜ p ) f t ( ˜ p b ˜ p) F t ( ˜ p b ˜ p) : Since ˜ p does not depend onc 1 , if ˜ pc 1 , product firm 1 would price at p 1 = ˜ p. If ˜ p>c 1 and the profit function is quasi-concave, product firm 1 would price at p 1 =c 1 . Instead, if product firm 1 falls in the losing group and c 1 < 0, it is still weakly dominant for product firm 1 to price at c 1 and not winning. The optimal pricing scheme for product firm 1 is thus p 1 (q)= minf ˜ p;c 1 (q)g: 122 Now let us turn to the third-party firm. From the discussion of product firms, we note that the third-party firm only competes with the product firm whose product has the highest valuation. Suppose it is product firm 1, which happens with probability 1 n . When the bundle is priced at p b and product firm 1 prices according to the optimal rule derived above, the demand for the bundle is 1 n D b (p b ; ˜ p)= Z ¥ 0 f c (x)[1 F t (p b p 1 (x))dx = Z ˜ p 0 f c (x)[1 F t (p b x)]dx+ Z ¥ ˜ p f c (x)[1 F t (p b ˜ p)]dx: The third-party firm earns in total P b (p b ; ˜ p)=(1a)np b [ Z ˜ p 0 f c (x)[1 F t (p b x)]dx+ Z ¥ ˜ p f c (x)[1 F t (p b ˜ p)]dx]: In equilibrium, the third-party firm should choose p b to maximize the profit, which yields the following first-order condition: 1 ˜ p b = R ˜ p 0 f c (x) f t ( ˜ p b x)dx+ R ¥ ˜ p f c (x) f t ( ˜ p b ˜ p)dx R ˜ p 0 f c (x)[1 F t ( ˜ p b x)]dx+ R ¥ ˜ p f c (x)[1 F t ( ˜ p b ˜ p)]dx : For the existence of such an equilibrium, I need to show first the quasi-concavity of the profit functions. ForP 1 (p 1 ; ˜ p b ), the first-order derivative is dP 1 (p 1 ; ˜ p b ) d p 1 = F t ( ˜ p b p 1 ) p 1 f t ( ˜ p b p 1 )+ a n ˜ p b f t ( ˜ p b p 1 ): Dividing by p 1 F t ( ˜ p b p 1 ), I have f(p 1 )= 1 p 1 (1 a ˜ p b np 1 ) f t ( ˜ p b p 1 ) F t ( ˜ p b p 1 ) : When p 1 a n ˜ p b ,f(p 1 )> 0. When p 1 > a n ˜ p b ,f(p 1 ) decreases in p 1 . Hence,P 1 (p 1 ; ˜ p b ) is quasi- concave in p 1 . 123 ForP b (p b ; ˜ p), since the demand for the bundle D b (p b ; ˜ p)= Z ˜ p 0 f c (x)[1 F t (p b x)]dx+ Z ¥ ˜ p f c (x)[1 F t (p b ˜ p)]dx can be written as a cumulative integral on the convex set S b (p b )=f(c;t)jc > 0;t > p b ˜ p;c+t > p b g; it must be log-concave in p b , which implies thatP b (p b ; ˜ p) is log-concave in p b . The existence of such an equilibrium further rely on the existence and uniqueness of the prices ˜ p and ˜ p b , or equivalently the uniqueness of ˜ p and ˜ d = ˜ p ˜ p b . For equation (1 a n )p= a n d+ F t (d) f t (d) ; the left-hand side is strictly increasing in p and the right-hand side is increasing in d, and thus there exists a unique p + (d) that solves the equation for a givend, and p + (d) increases ind. For equation p=d+ R p 0 f c (x)[1 F t (p+d x)]dx+ R ¥ p f c (x)[1 F t (d)]dx R p 0 f c (x) f t (p+d x)dx+ R ¥ p f c (x) f t (d)dx ; the left-hand side is strictly increasing in p, and the right-hand side is decreasing in p and strictly decreasing ind. Thus there exists a unique p (d) that solves the equation for a givend, and p (d) strictly decreases ind. Any pair of ˜ p and ˜ d thus constitutes an intersection point of curves representing the functions p + (d) and p (d). Since p + (0)= 0 p (0) and lim d!+¥ p + (d)> 0; lim d!+¥ p (d)=¥; 124 the equilibrium prices ˜ p and ˜ p b exist and are unique, and thus the equilibrium exists. A.12 Proof of Corollary 1.3 Whent follows the exponential distribution with a parameterl, I have 1 F t (x)=l f t (x) which immediately implies p b = ˜ p b =l: Meanwhile, p = a n l+(1 p p c ) F t (l p ) f t (l p ) < a n l+ F t (l p ) f t (l p ) and ˜ p= a n l+ F t (l ˜ p) f t (l ˜ p) ; which implies p < ˜ p: A.13 Proof of Proposition 1.8 Let us start with the third-party firm. When all product firms choose the same price ¯ p and the third-party firm chooses p b , then the consumer purchases the bundle if and only if tk+ p b ¯ p; and the third-party firm earns P b (p b ; ¯ p)=(1a)p b [1 F t (k+ p b ¯ p)]: 125 In equilibrium, the third-party firm should choose p b to maximize the profit, which yields the following first-order condition: ¯ p b = 1 F t ( ¯ p b ¯ pk) f t ( ¯ p b ¯ pk) : Now let us turn to the product firms. Suppose product firm 1 deviates to price p 1 unilaterally. If p 1 > ¯ p, since the minimum single product price is not affected, then consumers who purchase the bundle remain unaffected, but consumers who don’t purchase the bundle will be redistributed, and these consumers will purchase product 1 if and only if q 1 p 1 > max i6=1 fq i ¯ pg,c 1 > ¯ p p 1 : The demand for product firm 1 becomes D 1 (p 1 ; ¯ p; ¯ p b )= F t ( ¯ p b ¯ pk)[1 F c ( ¯ p p 1 )] and product firm 1 earns P 1 (p 1 ; ¯ p; ¯ p b )=p 1 F t ( ¯ p b ¯ pk)[1 F c ( ¯ p p 1 )] + a n ¯ p b [1 F t ( ¯ p b ¯ pk)]: In equilibrium, product firm 1 should maximize the profit at p 1 = ¯ p, and the necessary first-order condition is 1 F c (0) ¯ p f c (0) 0, ¯ p p c : If p 1 < ¯ p, then p 1 becomes the new minimum price, and thus it changes the demand for the bundle: D b (p 1 ; ¯ p; ¯ p b )= 1 F t ( ¯ p b p 1 k): 126 Product firm 1 now earns P 1 (p 1 ; ¯ p; ¯ p b )=p 1 F t ( ¯ p b p 1 k)[1 F c ( ¯ p p 1 )] + a n p b [1 F t ( ¯ p b p 1 k)]: In equilibrium, product firm 1 should maximize the profit at p 1 = ¯ p, and the necessary first-order condition is 1 ¯ p 1 p c +(1 a ¯ p b ¯ p ) f t ( ¯ p b ¯ pk) F t ( ¯ p b ¯ pk) : Since ¯ p p c , 1 ¯ p 1 p c 0; it follows that 1 a ¯ p b ¯ p 0, ¯ pa ¯ p b < ¯ p b : Finally, if such an equilibrium exists, each product firm earns ¯ P= 1 n ¯ p[1 F t (k+ ¯ p ¯ p b )]+ a n ¯ p b F t (k+ ¯ p ¯ p b ) 1 n p c [1 F t (k+ ¯ p ¯ p b )]+ 1 n p c F t (k+ ¯ p ¯ p b ) = p c n =P c ; i.e., product firms are better off with than without bundling. A.14 Proof of Corollary 1.4 Since ¯ p b >a ¯ p b ¯ p p c ; we have ¯ p b 1 a p c 127 and ¯ p b ¯ p(1a) ¯ p b 1a a p c : Therefore, 1 a p c ¯ p b = 1 F t ( ¯ p b ¯ pk) f t ( ¯ p b ¯ pk) 1 F t ( 1a a p c k) f t ( 1a a p c k) ; which can be rewritten as p c a 1a [k+ J 1 t;a (k)]: For the inequality 1 a p c 1 F t ( 1a a p c k) f t ( 1a a p c k) ; the left-hand side is decreasing ina while the right-hand side is increasing ina. Whena= 0, the left-hand side approaches positive infinity while the right-hand side is bounded by 1 f t (0) and the left-hand side is larger than the right-hand side. Whena= 1, the left-hand side equals p c while the right-hand side approaches positive infinity and the left-hand side is smaller than the right-hand side. Hence, given p c , there must be a unique ¯ a such that the inequality holds for anya2[ ¯ a;1]. A.15 Proof of Proposition 1.9 The switch of payment between the third-party firm and product firms does not change the demand system. Therefore, the demand for the bundle remains D b (p b ; ˆ p)= 1 F t (p b ˆ p); but the third-party firm now earns P b (p b ; ˆ p)=(p b b ˆ p)[1 F t (p b ˆ p)]: 128 In equilibrium, the third-party firm should choose p b to maximize the profit, which yields the following first-order condition: ˆ p b =b ˆ p+ 1 F t ( ˆ p b ˆ p) f t ( ˆ p b ˆ p) ; which implies ˆ p b b ˆ p: Now let us turn to the product firms and suppose product firm 1 deviates to price p 1 unilaterally. The demand for product 1 remains the same as in the main model: When p 1 ˆ p, the demand for product 1 is D 1 (p 1 ; ˆ p; ˆ p b )=[1 F c (p 1 ˆ p)]F t ( ˆ p b p 1 ): When p 1 < ˆ p, the demand for product 1 is D 1 (p 1 ; ˆ p; ˆ p b )= Z 0 p 1 ˆ p f c (x)F t (x+ ˆ p b p 1 )dx +[1 F c (0)]F t ( ˆ p b p 1 ): While the demand for the bundle also remains the same, product firm 1 cares only about those consumers who purchase the bundle and end up using product 1: 8 > > > > > < > > > > > : t >maxfc 1 ;0g+ ˆ p b ˆ p t >maxf0;c 1 g+ ˆ p b p 1 c 1 0 , 8 > > > > > < > > > > > : t >c 1 + ˆ p b ˆ p t > ˆ p b p 1 c 1 0 : Therefore, when p 1 ˆ p, these consumers account for D b1 (p 1 ; ˆ p; ˆ p b )= Z p 1 ˆ p 0 f c (x)[1 F t (x+ ˆ p b ˆ p)]dx +[1 F c (p 1 ˆ p)][1 F t ( ˆ p b p 1 )]: 129 When p 1 < ˆ p, these consumers account for D b1 (p 1 ; ˆ p; ˆ p b )=[1 F c (0)][1 F t ( ˆ p b p 1 )]: Overall, product firm 1 earns P 1 (p 1 ; ˆ p; ˆ p b )= p 1 D 1 (p 1 ; ˆ p; ˆ p b )+b p 1 D b1 (p 1 ; ˆ p; ˆ p b ): In equilibrium, product firm 1 should maximize the profit at p 1 = ˆ p. Examining the left and right derivative respectively yields the same first-order condition for the equilibrium price ˆ p: 1 ˆ p = 1 p c +[1b+ b ˆ p 1 F t ( ˆ p b ˆ p) f t ( ˆ p b ˆ p) ] f t ( ˆ p b ˆ p) F t ( ˆ p b ˆ p) ; which implies 1 ˆ p 1 p c , ˆ p p c : Therefore, in equilibrium, each product firm earns ˆ P= ˆ p[1 F c (0)]F t ( ˆ p b ˆ p)+b ˆ p[1 F c (0)][1 F t ( ˆ p b ˆ p)] = 1 n ˆ p[b+(1b)F t ( ˆ p b ˆ p)] 1 n p c ; i.e., product firms are always worse off with than without delegated bundling. B Proofs of Chapter 2 B.1 Proof of Proposition 2.1 When the monopolist firm prices p v 0 , its profit function is denoted as p p (p). Consider its first-order derivative dp p (p) d p = 1 F(p) p f(p); 130 and divide by f(p) to get y p (p)= 1 F(p) f(p) p=J 0 (p): Since J 0 (p) strictly increases in p, p p (p) is strictly quasi-concave in p, and the unconstrained optimal price is determined by the first-order condition 1 F(p) p f(p)= 0, J 0 (p)= 0, p= J 1 0 (0): Taking into consideration the constraint p v 0 , we know that, conditional on the monopolist firm pricing below v 0 , the profit is maximized at v 0 if and only if v 0 J 1 0 (0); and the profit is maximized at the unconstrained optimal price J 1 0 (0) otherwise. When the monopolist prices p v 0 , its profit function is denoted asp b (p). Consider its first- order derivative dp b (p) d p = 1 F(m(p))+ f p 1a f(m(p)); and divide it by f(m(p))=(1a) to get y b (p)=f p+(1a) 1 F(m(p)) f(m(p)) : Since y 0 b (p)=1+(1a) d dm(p) 1 F(m(p)) f(m(p)) 1 1a 1< 0; p b (p) is also strictly quasi-concave in p, and the unconstrained optimal price is determined by the first-order condition p(1a) 1 F(m(p)) f(m(p)) =f: 131 Taking into consideration the constraint p v 0 , we know that, conditional on the monopolist firm pricing above v 0 , the profit is maximized at v 0 if and only if v 0 (1a) 1 F(v 0 ) f(v 0 ) f, J a (v 0 )f; the profit is maximized by pricing at some p such thatm(p)= ¯ v if and only if fav 0 +(1a) lim m! ¯ v J 0 (m)=av 0 +(1a) ¯ J 0 ; and the profit is maximized at the unconstrained optimal price otherwise. To find the global optimal price, we need to combine our discussion of the two scenarios above: (a) When v 0 J 1 0 (0),p p (p) is strictly increasing in p for p v 0 , and the global maximum is achieved where p v 0 . Thus, when v 0 J 1 0 (0), we have two thresholds ¯ f =av 0 +(1a) ¯ J 0 and f = maxf0;J a (v 0 )g which pin down the optimal pricing rule for different f: the monopolist firm provides only the premium service ifff, both services iff <f < ¯ f, and only the free service iff ¯ f. We requiref 0 becausef cannot take a negative value. The thresholds of v 0 above whichf takes a positive value is 0= J a (v 0 ), v 0 = J 1 a (0): (b) When v 0 > J 1 0 (0) and f J a (v 0 ), p b (p) is strictly decreasing in p for p v 0 and the global maximum is achieved where p v 0 , and more specifically, the unconstrained optimal price J 1 0 (0). The monopolist firm thus always provides the premium service only under this parameter setting. 132 (c) When v 0 > J 1 0 (0) and J a (v 0 )<f <av 0 +(1a) ¯ J 0 , we need to compare the monopolist firm’s maximum profit for p v 0 , which is max p p p (p)= max p p[1 F(p)]; and its maximum profit for p v 0 , which is max p p b (p)= max p f[F(m(p)) F(v 0 )]+ p[1 F(m(p))]: The envelope theorem suggests that the latter is strictly increasing in f, and thus the threshold f between providing only the premium service and providing both services must solve max p f[F(m(p)) F(v 0 )]+ p[1 F(m(p))]= max p p[1 F(p)]: For any v 0 , clearly we havef > J a (v 0 ), butf may intersect with the upper bound ¯ f =av 0 +(1 a) ¯ J 0 . Suppose v 0 = ¯ v 0 at the intersection, and then ¯ v 0 must solve f[1 F( ¯ v 0 )]= max p p[1 F(p)] ,[a ¯ v 0 +(1a) ¯ J 0 ][1 F( ¯ v 0 )]= max p p[1 F(p)]: (d) When v 0 > J 1 0 (0) andfav 0 +(1a) ¯ J 0 , the monopolist firm either provides only the premium service and earns max p p p (p)= max p p[1 F(p)]; or provides only the free service and earns p f =f[1 F(v 0 )]: 133 The thresholdsf and ¯ f thus converges to f = ¯ f = max p p[1 F(p)] 1 F(v 0 ) ; and due to continuity we know that it must intersect with the curveav 0 +(1a) ¯ J 0 at v 0 = ¯ v 0 . B.2 Proof of Lemma 2.1 Note first that both f and ¯ f are continuous in v 0 and a. Therefore, we only need to examine the piece-wise monotonicity. (a) Whenf = 0, it remains constant. (b) Whenf = J a (v 0 ), we have ¶f ¶v 0 = J 0 a (v 0 )> 0 and ¶f ¶a = 1 F(v 0 ) f(v 0 ) 0: (c) Whenf solves max p f[F(m(p)) F(v 0 )]+ p[1 F(m(p))]= max p p[1 F(p)]; let H(f;a;v 0 )= max p f[F(m(p)) F(v 0 )]+ p[1 F(m(p))] and denote the optimal price as p. Using the envelope theorem, we have ¶H ¶f = F(m(p)) F(v 0 ) 0; ¶H ¶v 0 =(f p) f(m(p)) a 1a f f(v 0 ); ¶H ¶a =(f p) f(m(p)) p v 0 (1a) 2 : 134 Sincef J a (v 0 ), we have p(1a) 1 F(m(p)) f(m(p)) =f p and thus¶H=¶a 0 which leads to¶f=¶a 0. Besides, we can rewrite ¶H ¶v 0 =a[1 F(m(p))]f f(v 0 ) using the first-order condition. Take derivative with respect tof again: ¶ 2 H ¶v 0 ¶f =a f(m(p)) 1 1a d p df f(v 0 ): Sincep b (p) is super-modular in p andf, we have d p=df 0 and thus¶ 2 H=¶v 0 ¶f < 0. Further- more, whenf = J a (v 0 ) and p= v 0 , we have ¶H ¶v 0 j f=J a (v 0 ) =a[1 F(v 0 )] J a (v 0 ) f(v 0 )=J 0 (v 0 ) f(v 0 ) 0: Therefore we always have¶H=¶v 0 0 which leads to¶f=¶v 0 0. (d) When ¯ f =av 0 +(1a) ¯ J 0 , we show first that v 0 ¯ J 0 as long as v 0 ¯ v 0 . If ¯ J 0 approaches infinity, ¯ v 0 approaches ¯ v, and thus v 0 ¯ v 0 = ¯ v ¯ J 0 =¥: If ¯ J 0 is finite, then ¯ v 0 < ¯ v and [a ¯ v 0 +(1a) ¯ J 0 ][1 F( ¯ v 0 )]= max p p[1 F(p)] ¯ v 0 [1 F( ¯ v 0 )]; which leads to v 0 ¯ v 0 ¯ J 0 : 135 We can now conclude that ¶ ¯ f ¶v 0 =a 0 and ¶ ¯ f ¶a = v 0 ¯ J 0 0: (e) When f = ¯ f = max p p[1 F(p)] 1 F(v 0 ) ; both increase in v 0 and remain constant fora. B.3 Proof of Corollary 2.1 Part (i) follows directly from thatf ¯ f. For part (ii), recall that the optimal price p satisfies y b (p )=f p +(1a) 1 F(m(p )) f(m(p )) = 0: Sincey 0 b (p)< 0 and ¶y b (p) ¶f = 1> 0; the optimal price p increases inf. Meanwhile, the maximum profits are p = max p p b (p)= max p f[F(m(p)) F(v 0 )]+ p[1 F(m(p))]: By envelope theorem we have dp df = F(m(p )) F(v 0 ) 0 and thus the maximum profitsp also increase inf. 136 B.4 Proof of Corollary 2.2 Part (i) follows directly from thatf increases ina, ¯ f decreases ina and both converges tof as a approaches 1. For part (ii), since ¶y b (p) ¶a = 1 F(m(p)) f(m(p)) +(1a) d dm(p) 1 F(m(p)) f(m(p)) 1 1a 0; the optimal price p decreases ina. Meanwhile, by envelope theorem we have dp da =(f p ) f(m(p )) p v 0 (1a) 2 0 and thus the maximum profitsp also decrease ina. B.5 Proof of Corollary 2.3 Part (i) follows directly from that bothf and ¯ f increase in v 0 . For part (ii), since ¶y b (p) ¶v 0 =(1a) d dm(p) 1 F(m(p)) f(m(p)) a 1a 0; the optimal price p increases in v 0 . Meanwhile, by envelope theorem we have dp dv 0 =(f p ) f(m(p )) a 1a f f(v 0 )=a[1 F(m(p ))]f f(v 0 ): We know from the proof of Lemma 2.1 that d 2 p dv 0 df =a f(m(p )) d p df f(v 0 )< 0 and thus there exists a threshold denoted asf such that the maximum profitsp increase in v 0 if and only iff <f . 137 B.6 Proof of Porposition 2.2 The proof is divided into five parts: (a) Show the existence and uniqueness of p . (b) Show that no firm would unilaterally deviate to p> p . (c) Show that no firm would unilaterally deviate tof=a < p< p . (d) Show that no firm would unilaterally deviate to v 0 < pf=a. (e) Show that no firm would unilaterally deviate to p v 0 . (a) Show the existence and uniqueness of p : We want to show that there exists a unique p solving the equilibrium condition and that p >f=a. Rewrite the equilibrium condition as Z ¯ v m(p) G(v) f(v)dv pf 1a G(m(p)) f(m(p)) p Z ¯ v m(p) g(v) f(v)dv= 0; and divide both side by p R ¯ v m(p) G(v) f(v)dv to get 1 p 1 1a (1 f p ) G(m(p)) f(m(p)) R ¯ v m(p) G(v) f(v)dv R ¯ v m(p) g(v) f(v)dv R ¯ v m(p) G(v) f(v)dv = 0: Decompose the last term using integration by part R ¯ v m(p) g(v) f(v)dv R ¯ v m(p) G(v) f(v)dv = R ¯ v m(p) f(v)dG(v) R ¯ v m(p) G(v) f(v)dv = G(v) f(v)j ¯ v m(p) R ¥ m(p ) G(v) f 0 (v)dv R ¯ v m(p) G(v) f(v)dv = G(m(p)) f(m(p)) R ¥ m(p) G(v) f 0 (v)dv R ¯ v m(p) G(v) f(v)dv ; and the equilibrium condition becomes y(p)= 1 p 1 1a (a f p ) G(m(p)) f(m(p)) R ¯ v m(p) G(v) f(v)dv + R ¯ v m(p) G(v) f 0 (v)dv R ¯ v m(p) G(v) f(v)dv = 0: 138 Note that we specifically require the support of the distribution function F(x) to be open at the right bound x= ¯ v, which means f( ¯ v)= 0 and thus the integration by part is valid. Therefore, when talking about distributions where the exact support is closed on the right bound (such as uniform distributions), we need to extend the right bound by an arbitrarily small amount e > 0. Such practice may jeopardize the differentiability of f(x), in which case the integral R ¯ v m(p) G(v) f 0 (v)dv should be regarded as a shorthand for the Stieltjes integral R ¯ v m(p) G(v)d f(v). We show next thaty(p) is decreasing in p for p>f=a, by examining each term respectively: The first term is clearly decreasing in p. For the second term, when p>f=a, bothaf=p and m(m(p))= G(m(p)) f(m(p)) R ¯ v m(p) G(v) f(v)dv are positive. Further, af=p is increasing in p; with f(v) being log-concave, so do F(v), G(v), G(v) f(v) and the integral R ¯ v v G(s) f(s)ds, which implies that m(m(p)) is increasing in m(p) and thus in p. Hence, the entire second term (along with the negative sign at front) is decreasing in p. For the third term, take the derivative with respect to m(p), and we want to show that (denote m =m(p) for short) G(m) f 0 (m) Z ¯ v m G(v) f(v)dv+ G(m) f(m) Z ¯ v m G(v) f 0 (v)dv 0 , Z ¯ v m G(v) f 0 (v)dv f 0 (m) f(m) Z ¯ v m G(v) f(v)dv= Z ¯ v m G(v) f(v) f 0 (m) f(m) dv: Since f(v) is log-concave, f 0 (v)= f(v) is decreasing in v, and thus Z ¯ v m G(v) f(v) f 0 (m) f(m) dv Z ¯ v m G(v) f(v) f 0 (v) f(v) dv= Z ¯ v m G(v) f 0 (v)dv; i.e., the third term is decreasing inm(p) and thus in p. Therefore, when p>f=a,y(p) is decreas- ing in p. 139 We are left to consider the case when pf=a. If p=f=a, Assumption 2.2 guarantees that y( f a )= a f G(m( f a )) f(m( f a )) R ¯ v m( f a ) G(v) f(v)dv R ¯ v m( f a ) g(v) f(v)dv R ¯ v m( f a ) G(v) f(v)dv > 0: When p<f=a, since the first and third term is decreasing in p and the second term is positive, we know that y(p)>y( f a )> 0: To sum up, there exists a unique p > f=a satisfying y(p )= 0 and thus the equilibrium condition. (b) Show that no firm would unilaterally deviate to p> p : Suppose all firms but firm 1 charge p and firm 1 deviates to p> p . firm 1 thus earns p(p; p )=f[ Z m(p ) v 0 G(v) f(v)dv+ Z m(p) m(p ) G(a(v v 0 )+ p ) f(v)dv] + p Z ¯ v m(p) G(v p+ p ) f(v)dv: Take derivative with respect to p ¶p(p; p ) ¶ p = Z ¯ v m(p) G(v p+ p ) f(v)dv pf 1a G(m(p) p+ p ) f(m(p)) p Z ¯ v m(p) g(v p+ p ) f(v)dv: Denote D 1P (p)= Z ¯ v m(p) G(v p+ p ) f(v)dv as the demand for platform 1’s premium service. Also denote h(p)= 1 D 1P (p) ¶p(p; p ) ¶ p ; x(p)= h(p) p ; 140 A(p)= G(m(p) p+ p ) f(m(p)) R ¯ v m(p) G(v p+ p ) f(v)dv ; B(p)= R ¯ v m(p) g(v p+ p ) f(v)dv R ¯ v m(p) G(v p+ p ) f(v)dv : Therefore we can rewrite h(p)= 1 pf 1a A(p) pB(p); x(p)= 1 p 1 1a (1 f p )A(p) B(p); and we know thath(p )=x(p )= 0. Following Caplin and Nalebuff (1991), we know that D 1P (p) must be log-concave in p, and thus 1 D 1P (p) ¶D 1P (p) ¶ p = 1 1a A(p)+ B(p) is increasing in p. Moreover, use again integration by part A(p) B(p)= R ¯ v m(p) G(v p+ p ) f 0 (v)dv R ¯ v m(p) G(v p+ p ) f(v)dv : We want to show that this is decreasing in p. Take derivative with respect to p and examine the numerator [ 1 1a G(m(p) p+ p ) f 0 (m(p)) Z ¯ v m(p) g(v p+ p ) f 0 (v)dv] Z ¯ v m(p) G(v p+ p ) f(v)dv [ 1 1a G(m(p) p+ p ) f(m(p)) Z ¯ v m(p) g(v p+ p ) f(v)dv] Z ¯ v m(p) G(v p+ p ) f 0 (v)dv 0 , R ¯ v m(p) G(v p+ p ) f 0 (v)dv R ¯ v m(p) G(v p+ p ) f(v)dv 1 1a G(m(p) p+ p ) f 0 (m(p))+ R ¯ v m(p) g(v p+ p ) f 0 (v)dv 1 1a G(m(p) p+ p ) f(m(p))+ R ¯ v m(p) g(v p+ p ) f(v)dv : 141 A sufficient condition would be that both R ¯ v m(p) G(v p+ p ) f 0 (v)dv R ¯ v m(p) G(v p+ p ) f(v)dv f 0 (m(p)) f(m(p)) and R ¯ v m(p) G(v p+ p ) f 0 (v)dv R ¯ v m(p) G(v p+ p ) f(v)dv R ¯ v m(p) g(v p+ p ) f 0 (v)dv R ¯ v m(p) g(v p+ p ) f(v)dv hold. The former condition follows immediately that f 0 (v)= f(v) is decreasing in v. Denote H(v)= G(v p+ p ); h(v)= g(v p+ p ); and consider Z ¯ v m H(v) f 0 (v)dv Z ¯ v m h(v) f(v)dv Z ¯ v m H(v) f(v)dv Z ¯ v m h(v) f 0 (v)dv = Z [m; ¯ v) 2 [H(x) f 0 (x)h(y) f(y) H(x) f(x)h(y) f 0 (y)]dxdy = Z [m; ¯ v) 2 H(x)h(y) f(x) f(y)[ f 0 (x) f(x) f 0 (y) f(y) ]dxdy = Z myx ¯ v [H(x)h(y) H(y)h(x)] f(x) f(y)[ f 0 (x) f(x) f 0 (y) f(y) ]dxdy = Z myx ¯ v [ h(y) H(y) h(x) H(x) ]H(x)H(y) f(x) f(y)[ f 0 (x) f(x) f 0 (y) f(y) ]dxdy 0 The last inequality is due to that both f 0 (v)= f(v) and h(v)=H(v) are decreasing in v, and we can now conclude that A(p)+ B(p) is increasing in p. We now go back to firm 1’s pricing decision. We show first that, at any p> p >f=a, either h 0 (p)< 0 orx 0 (p)< 0. Rewrite h(p)= 1(pf)[ 1 1a A(p)+ B(p)]fB(p); x(p)= 1 p 1 1a (a f p )A(p)[A(p)+ B(p)]: 142 If A(p) is weakly increasing at p, then clearly x(p) is strictly decreasing at p. If instead A(p) is strictly decreasing at p, and since A(p)+B(p) is increasing at p, we must have that B(p) is strictly increasing at p, and thush(p) is strictly decreasing at p. We show next that, for any p> p , we must have x(p) 0. Suppose instead x(p)> 0 (and recall thatx(p )= 0), in which case we can always find some p 0 2(p ; p) such thatx(p 0 )> 0 and x 0 (p 0 )> 0, and thus h(p 0 )=x(p 0 )+ p 0 x 0 (p 0 )> 0 which leads to contradiction. Sincex(p) 0 implies that¶p(p; p )=¶ p 0, firm 1 would never unilaterally deviate to p> p . (c) Show that no firm would unilaterally deviate to f=a < p< p : Suppose all firms but firm 1 charge p and firm 1 deviates tof=a < p< p . Firm 1 thus earns p(p; p )=f Z m(p) v 0 G(v) f(v)dv + p[ Z m(p )p +p m(p) G( v p a + v 0 ) f(v)dv+ Z ¯ v m(p )p +p G(v p+ p ) f(v)dv]: Take derivative with respect to p ¶p(p; p ) ¶ p = Z p+a(m(p )v 0 ) m(p) G( v p a + v 0 ) f(v)dv+ Z ¯ v p+a(m(p )v 0 ) G(v p+ p ) f(v)dv pf 1a G(m(p)) f(m(p)) p Z p+a(m(p )v 0 ) m(p) 1 a g( v p a + v 0 ) f(v)dv p Z ¯ v p+a(m(p )v 0 ) g(v p+ p ) f(v)dv: Denote again D 1P (p)= Z p+a(m(p )v 0 ) m(p) G( v p a + v 0 ) f(v)dv+ Z ¯ v p+a(m(p )v 0 ) G(v p+ p ) f(v)dv 143 as the demand for firm 1’s premium service, and also h(p)= 1 D 1P (p) ¶p(p; p ) ¶ p ; x(p)= h(p) p ; A(p)= G(m(p)) f(m(p)) R p+a(m(p )v 0 ) m(p) G( vp a + v 0 ) f(v)dv+ R ¯ v p+a(m(p )v 0 ) G(v p+ p ) f(v)dv ; B(p)= R p+a(m(p )v 0 ) m(p) 1 a g( vp a + v 0 ) f(v)dv+ R ¯ v p+a(m(p )v 0 ) g(v p+ p ) f(v)dv R p+a(m(p )v 0 ) m(p) G( vp a + v 0 ) f(v)dv+ R ¯ v p+a(m(p )v 0 ) G(v p+ p ) f(v)dv : Therefore, as in (b), we have h(p)= 1 pf 1a A(p) pB(p) x(p)= 1 p 1 1a (1 f p )A(p) B(p) andh(p )=x(p )= 0. We now turn to show that A(p) B(p)= R p+a(m(p )v 0 ) m(p) G( vp a + v 0 ) f 0 (v)dv+ R ¯ v p+a(m(p )v 0 ) G(v p+ p ) f 0 (v)dv R p+a(m(p )v 0 ) m(p) G( vp a + v 0 ) f(v)dv+ R ¯ v p+a(m(p )v 0 ) G(v p+ p ) f(v)dv is again decreasing in p. Denote H(v)= 8 > < > : G( v p a + v 0 ); ifm(p) v< p+a(m(p ) v 0 ) G(v p+ p ); if v p+a(m(p ) v 0 ) h(v)= 8 > < > : 1 a g( v p a + v 0 ); ifm(p) v< p+a(m(p ) v 0 ) G(v p+ p ); if v p+a(m(p ) v 0 ) Note that, although the function is defined piece-wise, overall we still have h(v)=H(v) decreasing in v, and follow the exactly same steps as in (b) we conclude that A(p)+ B(p) is increasing in p. Going back to platform 1’s pricing decision, we can show once again that, at anyf=a< p< p , eitherh 0 (p)< 0 orx 0 (p)< 0, and hence, for anyf=a < p< p , we must haveh(p) 0. Suppose 144 insteadh(p)< 0 (and recall thath(p )= 0), in which case we can always find some p 0 2(p; p ) such thath(p 0 )< 0 andh 0 (p 0 )> 0, and thus x 0 (p 0 )= 1 p 0 2 h(p 0 )+ 1 p 0 h 0 (p 0 )> 0 which leads to contradiction. Sinceh(p) 0 implies that¶p(p; p )=¶ p 0, firm 1 would never unilaterally deviate tof=a < p< p . (d) Show that no firm would unilaterally deviate to v 0 < pf=a: Suppose all firms but firm 1 charge p and firm 1 deviates to v 0 < pf=a. Firm 1’s profit function is the same as that in (c), and we note that its profits from free service users increase in p, while its profits from premium service users are log-concave in p (following again Caplin and Nalebuff (1991)). Therefore, a sufficient condition for platform 1 to never charge a price v 0 < pf=a is that (with notations following those in (c)): D 1P (p)+ p ¶D 1P (p) ¶ p j p= f a 0, 1 p 1 1a A(p) B(p)j p= f a 0: Since A(p)< f(m(p)) 1 F(m(p)) ; B(p)< 1 a g(m(p)) G(m(p)) = n 1 a f(m(p)) F(m(p)) ; the desired condition is guaranteed by Assumption 2.2. (e) Show that no firm would unilaterally deviate to p v 0 : Suppose all firms but firm 1 charge p and firm 1 deviates to p v 0 . Firm 1 thus earns p(p; p )= p[ Z p+m(p )p p G( v p a + v 0 ) f(v)dv+ Z ¯ v p+m(p )p G(v p+ p ) f(v)dv]; 145 which is log-concave in p. A sufficient condition for platform 1 to never charge a price p v 0 is thus ¶p(p; p ) ¶ p j p=v 0 0 , 1 p G(v 0 ) f(p) R p+m(p )p p G( vp a + v 0 ) f(v)dv+ R ¯ v p+m(p )p G(v p+ p ) f(v)dv R p+m(p )p p 1 a g( vp a + v 0 ) f(v)dv+ R ¯ v p+m(p )p g(v p+ p ) f(v)dv R p+m(p )p p G( vp a + v 0 ) f(v)dv+ R ¯ v p+m(p )p G(v p+ p ) f(v)dv j p=v 0 0 which is guaranteed by Assumption 2.1. B.7 Proof of Corollary 2.4 Recall from the proof of Proposition 2.2 part (a) that y(p)= 1 p 1 1a (a f p ) G(m(p)) f(m(p)) R ¯ v m(p) G(v) f(v)dv + R ¯ v m(p) G(v) f 0 (v)dv R ¯ v m(p) G(v) f(v)dv is decreasing in p for p>f=a, and there exists a unique price p satisfyingy(p )= 0. Denote y 1 (m)= G(m) f(m) R ¯ v m G(v) f(v)dv and y 2 (m)= R ¯ v m G(v) f 0 (v)dv R ¯ v m G(v) f(v)dv : We have also shown that y 1 (m) is increasing in m and y 2 (m) is decreasing in m in the proof of Proposition 2.2 part (a). Since for p>f=a > v 0 : dy(p) df = 1 (1a)p y 1 (m(p)) 0; dy(p) da = 1 f p (1a) 2 y 1 (m(p)) a f p 1a y 0 1 (m(p)) p v 0 (1a) 2 +y 0 2 (m(p)) p v 0 (1a) 2 0; 146 dy(p) dv 0 = a f p 1a y 0 1 (m(p)) a 1a y 0 2 (m(p)) a 1a 0; the implicit function theorem suggests that p increases inf and v 0 while decreases ina. To show that p decreases in n, we follow the definition in Zhong (2015) to rewrite: y 1 (m)= F n1 (m) f(m) R ¯ v m F n1 (v) f(v)dv = f(m) 1 F(m) F (n1) (mjX (n) >m); y 1 (m)+y 2 (m)= R ¯ v m f(v)dF n1 (v) R ¯ v m F n1 (v) f(v)dv = Z ¯ v m f(x) 1 F(x) dF (n1) (xjX (n) >m); where F (n1) (xjX (n) > p) denotes the distribution of a statistic for the second largest value X (n1) conditional on the largest value X (n) to be larger than a certain level p. Lemma B.1 (Lemma 3.1 (Zhong, 2015)). X (n1) (xjX (n) p) is increasing in n in the sense of first order stochastic dominance, i.e. F (n1) (xjX (n) p) is decreasing in n. Following the lemma, we can show that y 1 (m) is increasing in n and y 1 (m)+y 2 (m) is de- creasing in n, and thus dy(p) dn = 1 f p 1a dy 1 (m(p)) dn + d dn [y 1 (m(p))+y 2 (m(p))] 0: It further leads to that p is decreasing in n. B.8 Proof of Corollary 2.5 Taking the derivatives ofp with respect tof anda gives us dp df = 1 n [F n (m(p )) F n (v 0 )]+ 1 n d p df [1 F n (m(p ))]+ 1 n f p 1a nF n1 (m(p )) f(m(p )) d p df = 1 n [F n (m(p )) F n (v 0 )]+ d p df p Z ¯ v m(p ) f(v)dF n1 (v) 0; 147 dp da = 1 n (f p )nF n1 (m(p )) f(m(p ))( 1 1a d p da + p v 0 (1a) 2 )+ 1 n d p da [1 F n (m(p ))] =(f p )F n1 (m(p )) f(m(p )) p v 0 (1a) 2 + d p da p Z ¯ v m(p ) f(v)dF n1 (v) 0: To examine the effect of n, we show first that, for any 0 b< a 1, the function z(n)= a n b n n decreases in n. Taking derivative with respect to n, z 0 (n)= (a n lna n b n lnb n )(a n b n ) n 2 = (a n lna n a n )(b n lnb n b n ) n 2 < 0 since d dx (xlnx x)= lnx< 0 for x< 1. Hence dp dn =f ¶ ¶n F n (m(p )) F n (v 0 ) n + p ¶ ¶n 1 F n (m(p )) n + d p dn p Z ¯ v m(p ) f(v)dF n1 (v)< 0: B.9 Proof of Corollary 2.6 Rewrite the consumer surplus as CS= Z ¯ v 0 maxf0;a(u v 0 );u p gdF n (u): Recall that p increases inf and v 0 and decreases ina and n. Therefore, the integrand maxf0;a(u v 0 );u p g decreases inf and v 0 , and increases ina and n. It immediately follows that the consumer surplus CS decreases inf and v 0 and increases ina. 148 As for n, denote p n and p n+1 as the equilibrium prices when there are n and n+1 firms respec- tively, and we know that p n > p n+1 . Therefore, the consumer surplus with n firms CS n = Z ¯ v 0 maxf0;a(u v 0 );u p n gdF n (u) Z ¯ v 0 maxf0;a(u v 0 );u p n+1 gdF n (u): Meanwhile, note that F n+1 (x) represents a random variable which first-order stochastically domi- nates the one represented by F n (x) and the integrand maxf0;a(u v 0 );u p n+1 g is an increasing function in u. It follows that CS n+1 = Z ¯ v 0 maxf0;a(u v 0 );u p n+1 gdF n+1 (u) Z ¯ v 0 maxf0;a(u v 0 );u p n+1 gdF n (u): The two inequalities suggest that CS n CS n+1 , i.e., the consumer surplus CS increases in n. B.10 Proof of Corollary 2.7 Similarly, denote p n and p n+1 as the equilibrium prices when there are n and n+ 1 firms respec- tively. Rewrite the total surplus as T S n = Z ¯ v 0 m n (u)dF n (u) where m n (u) is a piecewise function m n (u)= 8 > > > > > < > > > > > : 0, if u< v 0 a(u v 0 )+f, if v 0 u<m(p n ) u, if um(p n ) : 149 We show next that for any u, m n+1 (u) m n (u). It is sufficient to examine that, for any consumers withm(p n+1 )< u<m(p n ), a(u v 0 )+f < u,m(f)< u; which is guaranteed byf < p n+1 < p n . Hence, we have T S n = Z ¯ v 0 m n (u)dF n (u) Z ¯ v 0 m n+1 (u)dF n (u): Similarly, since m n+1 (u) increases in u, following first-order stochastic dominance, we have T S n+1 = Z ¯ v 0 m n+1 (u)dF n+1 (u) Z ¯ v 0 m n+1 (u)dF n (u); and eventually T S n T S n+1 , i.e., the total surplus T S increases in n. B.11 Proof of Proposition 2.3 For the equilibrium prices, using the chain rule we have d p dd = d p df df dd 0: For the equilibrium profits, similarly we have dp df =p +(1d) dp df df dd 0: B.12 Proof of Proposition 2.4 Under a non-zero penetration rated, the consumer surplus becomes CS(d)=d Z ¯ v v udF n (u)+(1d) Z ¯ v v maxf0;a(u v 0 );u p(d)gdF n (u): 150 Therefore dCS dd = Z ¯ v v udF n (u) Z ¯ v v maxf0;a(u v 0 );u p(d)gdF n (u) (1d)[1 F n (m(p ))] d p dd 0: B.13 Proof of Proposition 2.5 Whend = 0, suppose all firms are now choosing BOTH and firm 1 is considering deviating to PO. Since the price of premium services is fixed, consumers who prefer to use the premium service of firm 1 will not be affected, but firm 1 loses all the consumers using its free service and the loss is f(0) Z m(p) v 0 F n1 (v)dF(v): Among these consumers, those who have a valuation for firm 1 higher than the price p would consider switching to its premium service, provided that such a service gives them a higher net utility than the free services of other firms, or v 1 p> max i6=1 a(v i v 0 ): Therefore, firm 1 gains the subscription fees paid by these users which is p Z m(p) p F n1 ( v p a + v 0 )dF(v): To compare the loss and the gain, we compare first the mass of consumers involved. Since v> v p a + v 0 , v<m(p) holds on v2(p;m(p)), we know that Z m(p) p F n1 ( v p a + v 0 )dF(v)< Z m(p) v 0 F n1 (v)dF(v): 151 Further iff(0) is sufficiently large, sayf(0)> p, then firm 1’s loss from deviating to PO is greater than its gain, and thus all firms choosing BOTH constitutes an equilibrium for the service provision game. When d = 1, suppose all firms are now choosing PO and firm 1 is considering deviating to BOTH. When all firms choose PO, consumers with access to AATs have no choice but to choose from the outside option and the premium services, and thus each firm earns a positive subscription revenue. If firm 1 deviates to BOTH and provides a free service, its free service will dominate its premium service since all consumers have access to AATs and none of them generate any revenue for firm 1. Therefore, firm 1 would not deviate to BOTH and all firms choosing PO constitutes an equilibrium for the service provision game. B.14 Proof of Proposition 2.6 Asl increases, since dF(v) dl =e v l v l 2 < 0; dF n (v) dl = nF n1 (v) dF(v) dl < 0; we have dnp(d) dl =(1d)[(pf(d)) dF n (m(p)) dl f(d) dF n (v 0 ) dl ]> 0: Meanwhile, dF n (u)=dl < 0 suggests that, as l increases, the maximum valuation u improves in the first-order stochastic dominance sense, and thus CS(d) increases inl. Finally, since the total surplus T S(d)= np(d)+CS(d), it also increases inl. 152 C Proofs of Chapter 3 C.1 Proof of Lemma 3.1 Suppose instead that f(x) is not quasi-concave, and thus there exists x 1 < x 2 < x 3 such that f(x 2 )< f(x 1 ) and f(x 2 )< f(x 3 ). Hence, there exists x 0 1 2[x 1 ;x 2 ] such that f 0 (x 0 1 )= f(x 2 ) f(x 1 ) x 2 x 1 < 0; and x 0 2 2[x 2 ;x 3 ] such that f 0 (x 0 2 )= f(x 3 ) f(x 2 ) x 3 x 2 > 0: Meanwhile, since f 0 (x)g(x) decreases in x and x 0 1 x 2 x 0 2 , we have f 0 (x 0 1 )g(x 0 1 ) f 0 (x 0 2 )g(x 0 2 )> 0> f 0 (x 0 1 )g(x 0 1 ); which leads to contradiction. C.2 Proof of Lemma 3.2 This Lemma follows Lemma 3.1 immediately by letting f(x)=p 1 (x) and g(x)=1=D 0 1 (x)> 0. C.3 Proof of Lemma 3.3 Under the required conditions, when p 1 c 1 , h 2 (p 1 ) is decreasing in p 1 and thusp 1 (p 1 ) is quasi- concave in p 1 for p 1 2[c 1 ;¥). While when p 1 < c 1 , we have h 2 (p 1 ) h 2 (c 1 ) h 2 (p 1 )= 0: Overall,p 1 (p 1 ) is quasi-concave in p 1 . 153 C.4 Proof of Lemma 3.4 Rewrite h 2 (p 1 )= 1(p 1 c 1 ) D 0 1 (p 1 ) r 0 1 (p 1 )=m 1 D 1 (p 1 ) +(1 p 1 c 1 m 1 ) r 0 1 (p 1 ) D 1 (p 1 ) = 1(p 1 c 1 ) D 0 1 (p 1 ) r 0 1 (p 1 )=m 1 D 1 (p 1 ) 1 m 1 (p 1 c 1 m 1 ) r 0 1 (p 1 ) D 1 (p 1 ) : Under the required conditions, when p 1 c 1 m 1 , h 2 (p 1 ) is decreasing in p 1 and thus p 1 (p 1 ) is quasi-concave in p 1 for p 1 2[m 1 + c 1 ;¥). When p 1 c 1 < m 1 , we have h 2 (p 1 ) h 2 (m 1 + c 1 ) h 2 (p )= 0: Overall,p 1 (p 1 ) is quasi-concave in p 1 . C.5 Proof of Proposition 3.1 For each firm i, either of the conditions ensures that the profit function is quasi-concave in p i given any p i . Thus, when p i solves the first-order condition for firm i given p i , it must be the best-response to p i , and thus p constitutes a pure-strategy Nash equilibrium. C.6 Proof of Lemma 3.5 Consider first per-unit royalty licensing. Since D 0 (p)+ D 1 (p)= 1, we can rewrite firm 0’s payoff function as p r 0 (p)=(p 0 d r)D 0 (p)+ r: Therefore, the first-order conditions are respectively 8 > > < > > : ¶p r 0 ¶ p 0 = D 0 (p)+(p 0 d r) ¶D 0 (p) ¶ p 0 = 0; ¶p r 1 ¶ p 1 = D 1 (p)+(p 1 x r) ¶D 1 (p) ¶ p 1 = 0: 154 If firm 0 prices p 0 below d+r, it earns negative profit, and thus any p 0 that solves firm 0’s first- order condition must satisfy p 0 d+ r. Similarly, any p 1 that solves firm 1’s first-order condition must satisfy p 1 x+ r. The equilibrium existence thus follows immediately Proposition 3.1. To show the equilibrium uniqueness, we refer to the Gale-Nikaido Theorem proposed in Gale and Nikaido (1965). We start with rewriting the first-order conditions as 8 > > < > > : 1(p 0 d r) ¶ lnD 0 (p) ¶ p 0 = 0; 1(p 1 x r) ¶ lnD 1 (p) ¶ p 1 = 0: The corresponding Jacobian matrix is J r (p)= 2 6 4 ¶ lnD 0 ¶ p 0 (p 0 d r) ¶ 2 lnD 0 ¶ p 2 0 (p 0 d r) ¶ 2 lnD 0 ¶ p 0 ¶ p 1 (p 1 x r) ¶ 2 lnD 1 ¶ p 1 ¶ p 0 ¶ lnD 1 ¶ p 1 (p 1 x r) ¶ 2 lnD 1 ¶ p 2 1 3 7 5 : Both diagonal entries of J r (p) are positive, and both off-diagonal entries are negative. Moreover, since ¶ lnD 0 ¶ p 0 (p 0 d r) ¶ 2 lnD 0 ¶ p 2 0 (p 0 d r) ¶ 2 lnD 0 ¶ p 0 ¶ p 1 = ¶ lnD 0 ¶ p 0 > 0; ¶ lnD 1 ¶ p 1 (p 1 x r) ¶ 2 lnD 1 ¶ p 2 1 (p 1 x r) ¶ 2 lnD 1 ¶ p 1 ¶ p 0 = ¶ lnD 1 ¶ p 1 > 0; the Jacobian matrix J r (p) is a diagonally dominant matrix with positive diagonal entries, and thus is a P-matrix as defined in Gale and Nikaido (1965). Hence, we can conclude that there is a unique pair of (p r 0 ; p r 1 ) in the rectangular region [d+ r;¥)[x+ r;¥) that satisfies the first-order conditions, i.e., the equilibrium is unique. Now we move on to ad valorem royalty licensing. Similarly, rewrite firm 0’s profit as p s 0 (p)=(p 0 d sp 1 )D 0 (p)+ sp 1 ; 155 and the first-order conditions are respectively 8 > > < > > : ¶p s 0 ¶ p 0 = D 0 (p)+(p 0 d sp 1 ) ¶D 0 (p) ¶ p 0 = 0; ¶p s 1 ¶ p 1 =(1 s)D 1 (p)+((1 s)p 1 x) ¶D 1 (p) ¶ p 1 = 0: As in the previous case, in equilibrium we always have p 1 > x 1s and p 0 > d+sp 1 > d+ s 1s x, and the equilibrium existence follows from Proposition 3.1. In terms of equilibrium uniqueness, rewrite the first-order conditions as 8 > > < > > : 1(p 0 d sp 1 ) ¶ lnD 0 (p) ¶ p 0 = 0; 1(p 1 x 1 s ) ¶ lnD 1 (p) ¶ p 1 = 0; and the corresponding Jacobian matrix is J s (p)= 2 6 4 ¶ lnD 0 ¶ p 0 (p 0 d sp 1 ) ¶ 2 lnD 0 ¶ p 2 0 (p 0 d sp 1 ) ¶ 2 lnD 0 ¶ p 0 ¶ p 1 + s ¶ lnD 0 ¶ p 0 (p 1 x 1s ) ¶ 2 lnD 1 ¶ p 1 ¶ p 0 ¶ lnD 1 ¶ p 1 (p 1 x 1s ) ¶ 2 lnD 1 ¶ p 2 1 3 7 5 : Similarly, the Jacobian matrix J s (p) is a diagonally dominant matrix with positive diagonal entries and thus is a P-matrix, which in turn guarantees the equilibrium uniqueness. C.7 Proof of Lemma 3.6 Following the proof of Lemma 3.5, under per-unit royalty licensing, we have J r (p) 2 6 4 d p r 0 dr d p r 1 dr 3 7 5 = 2 6 4 ¶ lnD 0 (p) ¶ p 0 ¶ lnD 1 (p) ¶ p 1 3 7 5 , 2 6 4 d p r 0 dr d p r 1 dr 3 7 5 =[J r (p)] 1 2 6 4 ¶ lnD 0 (p) ¶ p 0 ¶ lnD 1 (p) ¶ p 1 3 7 5 : Since J r (p) is a P-matrix of Leontief type, all entries of[J r (p)] 1 are non-negative, and thus both p r 0 and p r 1 increase in r. 156 Meanwhile, if x= d, due to the symmetry in demand functions D 0 (p) and D 1 (p), the unique equilibrium must be p r 0 = p r 1 = p r , where p r solves 1 2 +(p r d r) ¶D 0 (p r ; p r ) ¶ p 0 = 0: Under ad valorem royalty licensing, we have J s (p) 2 6 4 d p s 0 ds d p s 1 ds 3 7 5 = 2 6 4 p 1 ¶ lnD 0 (p) ¶ p 0 x (1s) 2 ¶ lnD 1 (p) ¶ p 1 3 7 5 , 2 6 4 d p s 0 ds d p s 1 ds 3 7 5 =[J s (p)] 1 2 6 4 p 1 ¶ lnD 0 (p) ¶ p 0 x (1s) 2 ¶ lnD 1 (p) ¶ p 1 3 7 5 : Again J s (p) is a P-matrix of Leontief type, and thus all entries of[J s (p)] 1 are non-negative, and thus both p s 0 and p s 0 increase in s. Meanwhile, if x= d, suppose instead p s 0 p s 1 . From the first-order conditions we know that (p s 0 d sp s 1 ) ¶ lnD 0 (p s ) ¶ p 0 =(p s 1 d 1 s ) ¶ lnD 1 (p s ) ¶ p 1 : Since p s 1 > d 1s , we have 0< p s 0 d sp s 1 < p s 1 d s d 1 s = p s 1 d 1 s : On the other hand, following the log-concavity conditions, we have 0> ¶ lnD 0 (p s 0 ; p s 1 ) ¶ p 0 ¶ lnD 0 (p s 1 ; p s 1 ) ¶ p 0 = ¶ lnD 1 (p s 1 ; p s 1 ) ¶ p 1 ¶ lnD 1 (p s 0 ; p s 1 ) ¶ p 1 : Hence j(p s 0 d sp s 1 ) ¶ lnD 0 (p s ) ¶ p 0 j<j(p s 1 d 1 s ) ¶ lnD 1 (p s ) ¶ p 1 j; which leads to contradiction, and we conclude that, if x= d, we always have p s 0 > p s 1 . 157 C.8 Proof of Lemma 3.7 Under per-unit royalty licensing, dP r dr =(p r 0 d r) ¶D 0 (p r ) ¶ p 1 d p r 1 dr +(p r 1 x r) ¶D 1 (p r ) ¶ p 0 d p r 0 dr > 0: Under ad valorem royalty licensing, dP s ds =[(p s 0 d sp s 1 ) ¶D 0 (p s ) ¶ p 1 + sD 1 (p s )] d p s 1 ds +[(1 s)p s 1 x] ¶D 1 (p s ) ¶ p 0 d p s 0 ds > 0: C.9 Proof of Corollary 3.1 The firms’ profit functions before licensing are respectively p n 0 (p)=(p 0 d)D 0 (p); p n 1 (p)=(p 1 c)D 1 (p): The equilibrium existence and uniqueness follows Lemma 3.5. Moreover, following the same logic as Lemma 3.6, we can show that assuming p n 0 p n 1 would always lead to contradiction, and we must have p n 0 < p n 1 in the pre-licensing equilibrium. C.10 Proof of Corollary 3.2 Lemma 3.7 suggests that firm 0 will keep increasing r and s until it hits the antitrust constraint. Given the price comparison in Lemma 3.6, in both cases p r 0 and p s 0 will hit p n 0 first, and thus at the optimal royalty rate we have p r 1 = p r 0 = p n 0 < p n 1 ; and p s 1 < p s 0 = p n 0 < p n 1 : 158 C.11 Proof of Proposition 3.2 Consider a monopoly firm controlling both products whose profit function is P m (p)=(p 0 d)D 0 (p)+(p 1 d)D 1 (p) Whenever p 0 > p 1 , the monopoly firm would prefer increasing p 1 as ¶P m (p) ¶ p 1 = D 1 (p)+(p 0 d) ¶D 0 (p) ¶ p 1 +(p 1 d) ¶D 1 (p) ¶ p 1 = D 1 (p)+(p 0 p 1 ) ¶D 0 (p) ¶ p 1 > 0 We know from Corollary 3.2 that, with optimal choice, the prices are p n 0 = p r 0 = p r 1 under per-unit royalty licensing and p n 0 = p s 0 > p s 1 under ad valorem royalty licensing, and thus firm 0 who shares the same profit function as the monopoly firm prefers the former over the latter. C.12 Proof of Lemma 3.8 Rewrite the first-order conditions as 8 > > < > > : 1(p 1 c) ¶ lnD 1 (p) ¶ p 1 1 2 p 0 ¶ lnD 1 (p) ¶ p 0 = 0; 1(p 2 c) ¶ lnD 2 (p) ¶ p 2 1 2 p 0 ¶ lnD 2 (p) ¶ p 0 = 0: The corresponding Jacobian matrix is J(p)= 2 6 4 ¶ lnD 1 ¶ p 1 (p 1 c) ¶ 2 lnD 1 ¶ p 2 1 1 2 p 0 ¶ 2 lnD 1 ¶ p 0 ¶ p 1 (p 1 c) ¶ 2 lnD 1 ¶ p 1 ¶ p 2 1 2 p 0 ¶ 2 lnD 1 ¶ p 0 ¶ p 2 (p 2 c) ¶ 2 lnD 2 ¶ p 1 ¶ p 2 1 2 p 0 ¶ 2 lnD 2 ¶ p 0 ¶ p 1 ¶ lnD 2 ¶ p 2 (p 2 c) ¶ 2 lnD 2 ¶ p 2 2 1 2 p 0 ¶ 2 lnD 2 ¶ p 0 ¶ p 2 3 7 5 : Under Assumption 3.3, we can show that, J(p) is a diagonally dominant matrix with positive diagonal entries, and thus a P-matrix on the rectangular region [p 0 + c;¥) 2 . Thus, there exists a unique pair of(p 1 ; p 2 )2[p 0 + c;¥) 2 that constitutes a pricing equilibrium. 159 We now move on to show that p 1 = p 2 = p as defined in the main text. Consider the following function H(D)= 1 2 [1 F 2 (D)](D+ p 0 c)[F(D) f(D)+ Z ¥ D f 2 (v)dv]+ 1 2 p 0 F(D) f(D): Assumption 3.4 shows that H(c)> 0. Meanwhile lim D!¥ H(D) 1 2 [1 F 2 (D)](D+ p 0 c) = lim D!¥ F(D) f(D) 1 2 [1 F 2 (D)] < F(c) f(c) [1 F 2 (c)] < 0: Due to continuity, there existsD 2(c;¥) such that H(D )= 0, and we note that p =D + p 0 > c+ p 0 . Since (p ; p ) solves the first-order conditions and p > p 0 + c, it constitutes the unique equilibrium of the pre-merger pricing competition. C.13 Proof of Proposition 3.3 For the pre-merger price competition, note that dH(D) d p 0 = 1 2 F(D) f(D) Z ¥ D f 2 (v)dv< 0; and thus when p 0 increases,D decreases, which implies d p d p 0 = 1+ dD d p 0 < 1: For the merged firm, the prices p 1 and p 2 affect its profits only through the markups p 1 p 0 and p 2 p 0 . As p 0 increases, the merged firm can raise p m by exactly the same amount and continue to earn the optimal monopoly profits. 160 C.14 Proof of Lemma 3.9 The first-order condition faced by the manager of firm i is a ii D i (p)+ å j a i j (p j c j ) ¶D j (p) ¶ p i = 0; which can be rewritten as a ii D i (p) ¶D i (p)=¶ p i å j a i j (p j c j ) ¶D j (p)=¶ p i ¶D i (p)=¶ p i = 0 Since ¶D j (p)=¶ p i ¶D i (p)=¶ p i = D j (p)D i (p) D i (p)[1 D i (p)] = D j (p) 1 D i (p) = e v j p j å k6=i e v k p k + H 0 is a constant not involving p i , we know that the objective function for the manager of firm i is quasi-concave in p i . Consider a restricted game where the managers cannot price strictly below the marginal cost. The quasi-concavity of their objective functions thus ensures that there exists a pricing equilibrium p where all products are priced above their marginal costs. We are left to show that p still constitutes a pricing equilibrium if we lift the restrictions. For the manager of firm i, given that the other managers follow the equilibrium p , pricing at p i < c i is strictly worse off than pricing at c i : a ii (p i c i )D i (p i ; p i )+ å j6=i a i j (p j c j )D j (p i ; p i )< 0+ å j6=i a i j (p j c j )D j (c i ; p i ): Meanwhile, the first-order derivative evaluated at p i = c i is a ii D i (c i ; p i )+ å j6=i a i j (p j c j ) ¶D j (c i ; p i ) ¶ p i > 0: Hence, the manager would not deviate from p i to any p i < c i , and further we know that p i must be strictly greater than c i in equilibrium, and p must solve all the first-order conditions as it is in the interior. 161 C.15 Proof of Proposition 3.4 Since ¶D i (p) ¶ p i =D i (p)[1 D i (p)]; ¶D j (p) ¶ p i = D i (p)D j (p); we can rewrite the first-order condition for the manager of firm i as p i c i 1 å j a i j a ii (p j c j )D j (p)= 0: Denote A as the normalized portfolio matrix as defined in the main text. Further denote a vector m where m i = p i c i 1; and a vectorp where p i =(p i c i )D i (p): Then we can stack the first-order conditions as m Ap = 0, A 1 mp = 0 To examine the equilibrium uniqueness, consider the Jacobian matrix J(p)= d d p [A 1 mp]= A 1 dp d p : A sufficient condition for J(p) to be a P-matrix is that, for every row of J(p), the sum of all entries is always positive, i.e., å j a 1 i j D i (p)(p i c i ) ¶D i (p) ¶ p i å j6=i (p i c i ) ¶D i (p) ¶ p j = å j a 1 i j D i (p)> 0: 162 As we are only interested in equilibrium where all products are priced above marginal costs, the maximum demand for product i is D i (¥;:::;¥;c i ;¥;:::;¥)= e v i c i e v i c i + H 0 ; and thus the required condition becomes å j6=i a 1 i j > e v i c i e v i c i + H 0 which must hold for all i. C.16 Proof of Corollary 3.3 If a matrix A has all diagonal entries equal to a ii = x and all off-diagonal entries equal to a i j = y, then it’s inverse shares the same property, with all diagonal entries equal to a 1 ii = 1 x y 1 c(x y) 2 ; and all off-diagonal entries equal to a 1 i j = 1 c(x y) 2 ; where c= 1 y + n x y : Plug in x= 1 and y= t=[1(n 1)t] and c= 1(n 1)t t + n 1 t 1(n1)t = 1(n 1)t t(1 nt) ; 163 the diagonal entries are a 1 ii = [1(n 1)t](1t) 1 nt ; and the off-diagonal entries are a 1 i j = [1(n 1)t]t 1 nt : The required condition in Proposition 3.4 becomes [1(n 1)t](1t) 1 nt (n 1) [1(n 1)t]t 1 nt > max i e v i c i e v i c i + H 0 ,1(n 1)t > max i e v i c i e v i c i + H 0 ,t < 1 n 1 H 0 e max i (v i c i ) + H 0 : 164
Abstract (if available)
Abstract
Firms may earn a secondary source of revenue in addition to their primary operations, and they usually have only partial or indirect control over their secondary revenues. In this dissertation, I study the price competition among these firms. The first two chapters discuss in detail two business models emerging in recent years. Chapter 1 studies delegated bundling, where inter-firm bundles are operated by a third-party firm and in competition with the single products. While product firms control the prices of their own products, they earn profits from both selling their own products and royalty fees paid by the third-party firm. I show that product firms are better off with than without delegated bundling if and only if products are less differentiated and the royalty rate is sufficiently high, and consumer and social welfare always improve. Chapter 2 (which is a joint work with Guofu Tan) studies freemium, a business strategy where firms provide both a free ad-sponsored service and a paid premium service. Firms control the subscription fee of the premium service and also earn an advertising revenue from the free service. Consumers may use ad-avoidance technologies (AATs) to avoid watching ads, which jeopardizes the freemium business model. We show that an increase in the penetration rate of AATs would reduce equilibrium prices and profits but benefit consumer welfare in the short run. However, if the firms respond by dropping free services or reducing investment in service qualities, both consumer and social welfare would suffer a loss. Chapter 3 summarizes the features of these firms and refers to them as "moonlighting firms", which also nests patent and IP licensing, public firms and cross holding. I provide conditions for the quasi-concavity of moonlighting firms' profit functions in their own prices and establish the equilibrium existence for the price competition. I show further three simple applications where I can apply these results to the literature and improve our understandings on some traditional topics of interest including royalty licensing, profitable outside option and cross holding.
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Creator
Xu, Yejia
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Core Title
Price competition among firms with a secondary source of revenue
School
College of Letters, Arts and Sciences
Degree
Doctor of Philosophy
Degree Program
Economics
Degree Conferral Date
2022-05
Publication Date
04/08/2022
Defense Date
03/09/2022
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University of Southern California
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delegated bundling,equilibrium existence,freemium,moonlighting firms,OAI-PMH Harvest,price competition
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English
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Dukes, Anthony (
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), Tan, Guofu (
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), Libgober, Jonathan (
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wssx349@gmail.com,yejiaxu@usc.edu
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Tags
delegated bundling
equilibrium existence
freemium
moonlighting firms
price competition