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Essays on competition for customer memberships

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Content Essays on Competition for Customer Memberships Jorge Tamayo FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA Requirements for the Degree DOCTOR OF PHILOSOPHY Economics supervised by Guofu Tan May 11, 2018 Dedication This dissertation is dedicated to my loved ones. To my wife Juanita who inspires every page of this dissertation. Without your love and support, this endeavor will not be possible. To my, parents Jorge and Luz Marina, who motivate and encourage me through this journey, and always fill me with their love. I will not be pursuing a PhD in economics without my soulmates that have followed and supported me everywhere, my brother Juan Camilo and my sisters Laura Isabel and Ana Maria. 1 Acknowledgements I am extremely indebted to Guofu Tan for his continuous guidance and support. I am truly thankful to Anant Nyshadham. This dissertation would not have been possible without his guidance and unreasonable patience and support at every step of the way. My deep gratitude is also extended to Achyuta Adhvaaryu for his time and willings to help in this process. I am also grateful to Hashem Pesaran and Odilon Camara for their time and willingness to provide valuable feedback on my dissertation. I also benefited from the constant feedback and assistance from Yilmaz Kocer and Simon Wilkie. In addition, I am sincerely grateful to my dear friends, Kenneth Chuck and Michele Fioretti, who were always willing to help and provide feedback on the ideas on this disserta- tion. 2 Abstract In this collection of papers, we study competition and consumer behavior in member- ship/subscription markets. Generally, firms that implement a membership model, charge a “membership” fees that allow consumers to buy products/services at a unit price, in mul- tiple periods. There are three main questions that we attempt to answer (i) What is the optimal pricing strategy when firms use the membership model (i.e., Tariff Structure)? (ii) How the tariff structure affects competition: consumers’ behavior and firms’ profits? What are the differences between a static and a dynamic framework? In the first chapter, “Competition in Two-Part Tariffs Between Asymmetric Firms” (with Guofu Tan), we study competitive two-part tariffs (2PTs) in a general model of asymmetric duopoly firms offering (both vertically and horizontally) differentiated products. We provide a necessary and sufficient condition for marginal cost pricing to be in equilibrium, in both the Hotelling and general discrete choice approaches to horizontal differentiation. In the Hotelling setting, when the firms face symmetric demands but have asymmetric marginal costs, we show that in equilibrium the inefficient firm sets its marginal price below its own marginal cost and compensates this loss with the fixed fee, while the efficient firm sets its marginal price above its own marginal cost but below its rival’s price. The inferior firm “cross-subsidies” between the fixed fee and the marginal price. In the second chapter, “Dynamic Competition for Customer Memberships”, I study a competitive two-period membership (subscription) market, in which two symmetric firms charge a “membership” fee that allows consumers to buy products or services at a given unit price, in both periods. I explore how (i) the length of the membership (ii) the ability to price discriminate between “old” and “new” customers with the membership fee and the unit price; and (iii) the incentives to price discriminate between different consumers’ tastes, affect the competition. When firms employ long-term membership, firms have incentives to prevent their old most valuable customers from being ‘poached’ by the competitor, thus in equilibrium, firms price discriminate with their membership fee and unit price regarding customer purchased-behavior and volume of demand (second-degree). Instead, with short- term membership, firms only discriminate with their membership fee, between new and old customer. In the last chapter of my dissertation, I study how the quality of managers contributes to productivity dynamics of the teams they manage. We match two years of daily, line- level production data from six garment factories in India to rich survey data on managerial practicesoflinesupervisors. Wemeasurecontributionsof6distinctdimensionsofmanagerial quality motivated by previous literature: tenure, cognitive skills, autonomy, personality psychometrics, attention, and “relatability” to workers. We find that while tenure as a 3 supervisor contributes to all aspects of productivity dynamics, additional dimensions of managerial quality such as attention and autonomy contribute strongly as well, particularly to the rate of learning and retention of learned productivity. Personality impacts initial productivity most strongly, while cognitive skills contribute to learning. Additional results indicate that these dimensions of quality are generally undervalued in supervisor pay. 4 Contents Contents 5 1 Competition in Two-Part Tariffs Between Asymmetric Firms 7 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3 Marginal Cost Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.4 Asymmetric Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.5 Asymmetric Demands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.6 General Market Share Functions . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.6.1 Marginal Cost Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.6.2 Asymmetric Marginal Cost model with discrete types. . . . . . . . . . 33 1.7 An Extensions: Non-Linear Pricing . . . . . . . . . . . . . . . . . . . . . . . 35 1.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2 Dynamic Competition for Customer Membership 61 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.3 Long-Term Membership . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2.3.1 Subscription Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2.3.2 Restricted Membership Model. . . . . . . . . . . . . . . . . . . . . . . 78 2.4 Short-Term Membership . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 2.5 Long-Term Contracts (for the Unit Price) . . . . . . . . . . . . . . . . . . . 85 2.6 Heterogeneous Agents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 2.6.1 Membership Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 2.6.2 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3 Managerial Quality and Productivity Dynamics 143 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 3.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 3.2.1 Production Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 3.2.2 Management Survey Data . . . . . . . . . . . . . . . . . . . . . . . . 150 3.2.3 Pay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 3.3 Graphical Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 3.3.1 Dynamics of Productivity . . . . . . . . . . . . . . . . . . . . . . . . 158 5 3.3.2 Heterogeneity by Managerial Quality . . . . . . . . . . . . . . . . . . 161 3.4 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 3.4.1 Learning Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 3.4.2 Parameterization of Relationship between Learning and Managerial Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 3.5 Empirical Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 3.5.1 First Stage: Productivity Dynamics . . . . . . . . . . . . . . . . . . . 167 3.5.2 Second Stage: Latent Factors of Managerial Quality . . . . . . . . . 170 3.5.3 Third Stage: Contributions of Managerial Quality to Productivity Dy- namics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 3.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 3.6.1 First Stage: learning parameters . . . . . . . . . . . . . . . . . . . . . 174 3.6.2 Second Stage: managerial quality measures and factors . . . . . . . . 177 3.6.3 Third Stage: productivity contributions of managerial quality . . . . 179 3.6.4 Third Stage: Contributions of Managerial Quality to Pay . . . . . . . 186 3.7 Checks and Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 3.7.1 Tests for Sorting Bias: Monte Carlo Simulations . . . . . . . . . . . . 190 3.7.2 Deadline or Reference Point Effects: Robustness to Controlling for Days Left . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 3.7.3 Alternate Productivity Measure: Robustness to Using log(Quantity) in Place of log(Efficiency) . . . . . . . . . . . . . . . . . . . . . . . . 191 3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 .1 Tests for Sorting Bias: Balance Checks and Monte Carlo Simulations . . . . 193 .2 Reference Points: Robustness to Controlling for Days Left . . . . . . . . . . 201 .3 Alternate Productivity Measure: Robustness to Using log(Quantity) in Place of log(Efficiency) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 .4 Alternative Learning Measure: Robustness to Measuring Experience in Cu- mulative Quantity Produced . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 Bibliography 218 6 Chapter 1 Competition in Two-Part Tariffs Between Asymmetric Firms 1 Jorge Tamayo 2 and Guofu Tan 3 1 We are grateful to Odilon Camara, Keneth Chuk, Michele Fioretti, Yilmaz Kocer, Anthony Marino, and Simon Wilkie for their comments and thank seminar participants at the the 14th annual International Industrial Organization Conference, 91st Annual Conference - Western Economic Association International, 2016AsianMeetingoftheEconometricSociety, LACEA-LAMESAnualMeeting2016, Asia-PacificIndustrial Organisation Conference 2016, University of Southern California and Lingnan University, for their valuable comments. 2 Department of Economics, University of Southern California. jtamayo8@gmail.com. 3 Department of Economics, University of Southern California. guofutan@usc.edu. 7 Abstract We study competitive two-part tariffs in a general model of asymmetric duopoly firms of- fering (both vertically and horizontally) differentiated products. We provide a necessary and sufficient condition for marginal-cost pricing to be in equilibrium, in both the Hotelling and general discrete choice approaches to horizontal differentiation. In the Hotelling set- ting, when the firms face symmetric demands but have asymmetric marginal costs, we show that in equilibrium the less efficient firm sets its marginal price below its own marginal cost and compensates this loss with the fixed fee, while the more efficient firm sets its marginal price above its own marginal cost but below its rival’s price. When the firms have identical marginal costs but asymmetric demands, we show that in equilibrium the firm with the vertically inferior product sets its price below the marginal cost, while the superior firm sets its price above the marginal cost. The inferior firm “cross-subsidies” between fixed fee and marginal price. When the market shares are determined by Logit, even in the symmetric model we show that the marginal-cost pricing is not an equilibrium. 8 1.1 Introduction In this paper, we study competitive two-part tariffs (2PTs) offered by duopolistic firms when consumers have elastic demands and private information regarding horizontal brand prefer- ences and quality preferences over the products, and firms have asymmetric marginal costs and face asymmetric vertically differentiated demands. We show that when the necessary and sufficient condition for marginal cost pricing is not satisfied, the equilibrium strategy involves cross-subsidization between the unit price and the fixed fee for the disadvantage firm, while the advantage firm always offers a unit price above its marginal cost. The liberalization of the British electricity market at the end of the 1990s is a good example of competition in 2PTs between asymmetric firms. 4 The retail sector was separated in fourteen monopolized regional markets before it was opened to competition. In the years before and after the liberalization, firms predominantly offered single 2PTs. A stylized fact of these markets is the considerable variability of the tariffs offered to each region at each point in time, for the period following the liberalization. According to (Davies, Waddams Price, and Wilson 2014) in two-thirds of all cases, the entrant offered a lower marginal price with a higher fixed fee than the incumbent. 5 Moreover, they show that this asymmetry between the tariffs was persistent over time. Morerecently, internet-enablesubscriptionserviceshasbeengrowingexponentially. Busi- ness-to-consumer subscription services has been growing at 200% annually since 2011 (Mc- CarthyandFader2017); 6 on-demandeconomyisattractingmorethan22.4millionconsumers annually(ColbyandBell2016); Amazon hasexpandeditsmembershipservicesbeyond Ama- zon Prime, offering shipping on everyday essentials (Prime Pantry) and groceries (Amazon Fresh). 7 During 2016 and 2017 Uber tested in some cities Uber Plus which allows consumers to purchase fixed price rides for a monthly fee (Kominers 2017). In the previous examples, 4 Other traditionally examples of 2PTs include credit cards, telephone services, car rentals, club mem- berships, equipment leasing, amusement parks, TV program subscriptions and many others which charge an annual membership and usage fees; bars and nightclubs set cover fees and prices for drinks. Recently, Amazon has expanded its loyalty program, Amazon Prime, offering various benefits: access to Amazon In- stant Video, free cloud storage through Amazon Web Services and the possibility to shop “Lightning Deals” on Prime Days. According to (Morgan Stanley Research 2017) Amazon Prime penetration is expected to increase to 51% of the US households by the end of 2018. 5 In this market, there are other types of asymmetry not considered in our paper. In particular, most of the electricity suppliers were also active in the gas market. Some of the firms were vertically integrated into the generation; National Grid provides transmission, and there is a monopoly distributor in each of these regions. For a complete description of the British electricity market see (Davies, Waddams Price, and Wilson 2014). 6 Business-to-consumer subscription businesses sell a wide variety of products, from food (Hello Fresh and Blue Apron) to grooming products (Dollar Shave Club) and clothes (Stitch Fix and Trunk Club). For details please see (McCarthy and Fader 2017). 7 Prime members pay a membership of $4.99 per month for Prime Pantry and $14.99 for Amazon Fresh. 9 consumers pay a positive fixed fee that allow them to buy products and services at a given (positive or zero) unit price, i.e., firms charge 2PT. 2PTs, traditionally, were viewed as a monopoly price discrimination tool. 8 Most of the articles that have studied competition in 2PTs assume horizontally differentiated consumers with homogeneous taste for quality (or homogeneous demand), and firms with symmetric marginal costs and product demands. However, these assumptions are restrictive for many applications of interest and do not provide a complete and accurate description of the indus- tries that use 2PTs, like the British electricity market. For instance, we may be interested in studying competition in 2PTs where firms offer multiple products and consumers purchase multiple units of the same product. Typical examples where this assumption (elastic de- mand) is relevant are on-line grocery delivery services (e.g., Amazon Fresh and Instacart), membership-only retail stores (e.g., Costco and Sam’s Club), health clubs (e.g., tennis and country clubs), and the retail electricity market (e.g., British electricity market). Similarly, we may be interested in considering a model where consumers are heterogeneous and differ by unobservable tastes for quality, in addition to their brand (horizontal) preferences. For the British market, (Davies, Waddams Price, and Wilson 2014) show that the levels of con- sumption (of electricity) vary significantly across households. In fact, implementing 2PTs is often subject to uncertainty regarding consumers’ preferences, which may explain why too little price discrimination is observed compared to what theory suggests (Armstrong 2006). The most salient assumption adopted by the previous literature that have studied 2PTs and other forms of price discrimination, is that firms are symmetric. However, some of the industries in which 2PTs are widely practiced have evolved from natural monopolies before the recent worldwide liberalization of their sectors – e.g. energy and communication – and have experienced competition from more efficient firms (lower marginal costs) with new products and differentiated demands. Thus, we may want to consider competition in 2PTs between asymmetric firms with different marginal costs, and offering asymmetrically differentiated products. Finally, the recent literature on competitive price discrimination shows that when the market is fully covered, and symmetric firms offer non-linear pricing schedules, the optimal strategy in equilibrium is to offer cost-based 2PTs (Armstrong and Vickers 2001; Rochet and Stole 2002). Since the analysis with asymmetric firms is complicated, we may want to understand and characterize the equilibrium when firms use 2PTs (smaller strategy space) and infer about the general case with non-linear pricing schedules. 9 8 In fact, traditional theories viewed 2PTs as “price discrimination devices, employed exclusively by firms with market power” (Hayes 1987). A seminal contribution is the study of (Oi 1971). 9 We explain in detail the relationship between our paper and the literature on competitive price discrim- ination at the end of page 5. 10 In this paper, we assume that consumers’ types are described by a horizontal brand pref- erence parameter as well as a taste parameter for product quality, which are independently distributed. We consider two different assumptions regarding the horizontal parameter. In the first five sections we study a model in which consumers have horizontal brand preferences à la Hotelling (uniformly distributed). In Section 6, we consider a general discrete choice model of random utility maximization. We assume that there are two asymmetric firms both offering 2PTs and full market coverage, that is, all consumers buy at least from one firm and both firms sell strictly positive quantities. In Section 2, we introduce our framework and discuss the basic assumptions of the model. In Section 3, we assume that firms have asymmetric marginal costs and asymmetric demands (differentiated products). We start by considering a model where consumers have private information about their horizontal-brand preferences and homogeneous taste for quality. We show that there exists a unique equilibrium in which firms set marginal prices equal to marginal costs. This result is intimately related with (Mathewson and Winter 1997)’s proposition for a multi-product firm selling goods that are strongly complementary in de- mand. Further, we provide necessary and sufficient conditions for a marginal-cost pricing Nash equilibrium when consumers have heterogeneous tastes for quality. In Section 4, we assume that both firms have symmetric demands but asymmetric marginal costs (Asymmetric Costs model). From our results in Section 3, if consumers have heterogeneous tastes for quality, marginal-cost pricing is not a Nash equilibrium. 10 We showthattheoptimalstrategyfortheinefficientfirm(thefirmwiththehighermarginalcost) is to set its marginal price below its own marginal cost and compensate for this loss with the fixed fee. On the other hand, the optimal strategy for the efficient firm is to set its marginal price above its own marginal cost but below that of its rival. This result contrasts with the one in a model in which both firms use linear pricing (LP): as the number of tools available to firms increases (from one to two), they have incentives to establish “cross-subsidies” across the tariff instruments (fixed fee and marginal price), which, is not possible in the LP model. Section 5 presents the Asymmetric Demands model, in which both firms have symmetric marginal costs but asymmetric demands. Similar to our previous model, from Section 3 we know that if consumers are heterogeneous in their tastes for quality, marginal-cost pricing is not a Nash equilibrium. In fact, the optimal strategy for the firm whose products are verticallyinferioristosetitsownpricebelowthemarginalcost, whereastheoptimalstrategy for the firm with superior vertical goods is to set its marginal price above its rival’s price (and above the marginal cost). This result contrasts with our previous model, where the 10 Whereas if consumers are homogeneous in their tastes for quality, under the assumption of full market coverage, marginal-cost pricing is an equilibrium. 11 firm with higher marginal cost (“disadvantaged” firm) sets its price below its own marginal cost but above its rival’s marginal price. This is because in the Asymmetric Costs model, the efficient firm sets a marginal price below its rival but above its own marginal cost, while in the Asymmetric Demands model, by setting a price below the common marginal cost, the firm with the “advantage” would have to compensate this loss increasing the fixed fee, decreasing its market share. Hence, in both models, the “disadvantaged” firm uses “cross- subsidies” between the tariffs (e.g., marginal price below marginal cost and positive fixed fee). However, the equilibrium pricing strategy of the efficient firm depends on the nature of each model. Section 6 extends our analysis by considering a discrete choice model of random utility maximization. We propose a model in which consumers have private information about their horizontal brand preferences and consider homogeneous as well as heterogeneous tastes for quality. In the first case (homogeneous tastes for quality), we show that there exists a unique equilibrium in which firms set marginal prices equal to marginal costs and provide comparative statics properties of the equilibrium. In the second case, with heterogeneous tastes preferences, we provide a necessary and sufficient condition for marginal-cost pricing to be an equilibrium. We further show that the results of Section 4 hold in a simplified discrete-type model. In Section 7, we study under what conditions cost-based 2PT is an equilibrium if both firms use nonlinear tariffs instead of 2PTs, under Hotelling horizontal differentiation model. (Armstrong and Vickers 2001) and (Rochet and Stole 2002) showed that if the two firms are symmetric and under full market coverage, in equilibrium, each firm offers a cost-based 2PT. We extend this result and consider asymmetric firms and derive necessary and sufficient conditions for cost-based 2PT to be an equilibrium. Related literature: A seminal contribution to the literature on competitive price dis- crimination is (Armstrong and Vickers 2001) who study competitive non-linear pricing when consumers are differentiated á la Hotelling, have private information about their tastes for quality, and purchase all products from a single firm (one-stop shopping). They show that when the market is fully covered, and firms are symmetric, each firm offers a simple 2PTs contract with a marginal price equal to the marginal cost in equilibrium. (Rochet and Stole 2002) interpreted the quantity in (Armstrong and Vickers 2001) as quality (so consumers choose a price-quality pair) and show that if firms are symmetric and transportation cost is sufficiently low to guarantee full coverage, firms offer a cost-plus-fee pricing schedule in equilibrium. 11 However, this surprisingly “simple” result strongly depends on the assumption 11 Note that if firms are symmetric and the market is competitive (all consumers buy at least from one 12 of symmetry of the firms, excluding cases in which firms may have different marginal costs, or may offer asymmetrically differentiated products. 12 Our analysis extends (Armstrong and Vickers 2001)’s and (Rochet and Stole 2002)’s findings in two ways. First, we provide necessary and sufficient conditions under which marginal cost-based 2PTs is an equilibrium under horizontally differentiated consumers with heterogeneous quality preferences and asymmetric firms. This condition allows us to identify environments in which marginal cost-based 2PTs are not an equilibrium when firms have smaller pricing spaces like 2PTs, and hence are not an equilibrium either when they are allowed to use larger pricing spaces. Similarly, we show that even if firms are symmetric, marginal cost-based 2PTs may not be an equilibrium, like in the logit model with outside option. Second, we characterize the equilibrium outcome of the model when marginal cost- based 2PTs is not an equilibrium, and show that for the asymmetric marginal costs and asymmetric demands model, the optimal solution involves cross-subsidization between the marginal price and the fixed fee for the disadvantaged firm. (Yin 2004) considers a model of 2PTs competition with general horizontal preferences in which the transportation cost interacts with the quantity (transportation cost is a shipping cost) and consumers have homogeneous taste for quality. He shows that marginal prices are equal to marginal cost if and only if the demand of the marginal consumer (who is indifferent between buying the i-good and the j-good for i6= j in the full competition equilibrium) is equal to the average demand. For instance, if the horizontal taste parameter is additively separable from the price (transportation cost is a shopping cost), marginal price is equal to the marginal cost in equilibrium. 13 We show that this result does not hold if consumers have heterogeneous taste preferences and firms have asymmetric marginal costs or asymmetric demands. In this case, the disadvantaged firm (the one with the higher marginal cost) sets its prices below its own marginal cost. 14 firm) (Armstrong and Vickers 2001)’s and (Rochet and Stole 2002)’s result implies that there would be an efficient quantity (or quality) provision supported by the cost-based 2PTs. 12 (Armstrong and Vickers 2010) generalize (Armstrong and Vickers 2001) model assuming that consumers are allowed to multi-shop (buy from both firms or from just one) and find that in equilibrium, firms offer cost-based 2PTs. (Hoernig and Valletti 2011) consider a simple version of (Armstrong and Vickers 2010)’s model where vertical and horizontal taste parameters are correlated. They show that neither 2PTs nor full exclusivity can arise in equilibrium. 13 Note that in our model all consumers with the same taste preference purchase the same quantity of the goods, independently of their location. Thus, by construction the demand of the marginal consumer is equal to the average demand. 14 (Hoernig and Valletti 2007) consider a model where consumers are horizontally differentiated á la Hotelling and mix goods offered by two firms, and show that tariffs structure affects location decision, consumers and profits. In particular, they assume unit demands (consumers can buy from only one firm or combine products from the two firms) and quadratic transportation cost. The authors show that when both firms use 2PTs, marginal prices are equal to the marginal cost if and only if both firms are located at the same spot. (Griva and Vettas 2015) consider a duopoly model in which firms use 2PTs and offer 13 Relatedtothisarticleistheliteratureoncross-subsidization, whichiscommonlyobserved in multiproduct firms who often price some products below marginal cost and subsidize the resulting loss of the profits from other products. The literature provides different expla- nations for competitive cross-subsidization. (DeGraba 2006) shows that pricing below cost could serve as a strategy to screen the most profitable consumers in a setting in which firms face heterogeneous consumers. (Chen and Rey 2012) show that by pricing below marginal cost the products on which the large firm competes with the smaller rival and raising the price on the other products, allows the large firm to discriminate between multi-shoppers from one-stop shoppers. Note that in this context, loss leading serves as an exploitative de- vice rather than as an exclusionary instrument. (Chen and Rey 2016) study multi-product firms with different comparative advantages, competing for customers with heterogeneous transaction costs. They show that firms price strong products (on which they have a compar- ative advantage) above cost and weak products below cost. Our paper provides a different rationale for “cross-subsidization”. Here, the disadvantaged firm is the one that has incen- tives to use “cross-subsidies” between the tariffs (fixed fee and marginal price) as an optimal strategy to extract consumer surplus. 1.2 Model There are two firms, A and B, offering differentiated products to a population of hetero- geneous consumers. We assume that both firms can produce their products at constant marginal costs, denoted byc A andc B , respectively. There is a mass of consumers with types (x;) wherex is uniformly distributed on the unit interval independently of the distribution of ( 1 ;:::; n )2 ; n , which is continuously distributed with cumulative distribu- tion G (). 15 We adopt a one-stop shopping Hotelling model with heterogeneous consumers with different tastes for quality, i.e., consumers buy all products from one or the other firm, or they consume their outside option. Consumers’ preferences for the two differentiated products can be represented by the utility function u A (q A ;)tx if she buys from A and u B (q B ;) (1x)t if she buys from B, where x is the distance to firm A (and 1x the homogeneous goods to a population of vertically differentiated consumers (e.g., heterogeneous usage rate). The authors show that when one price of the components is fixed for both firms, e.g., the fixed fee or the marginal price, and the difference between these fixed components is large, the market is segmented, e.g., low usage consumers choose the low fee firm, and high usage consumers choose the low rate firm. Our analysis does not consider any of these cases e.g., interaction of the transportation cost with the quantity or location decisions. Thus, the reasons for marginal cost-based 2PTs (or not) are different from the previous models. Here, is related to the asymmetry of the firms and whether there is no correlation between each firm’s efficient quantity and the difference between the efficient consumers surpluses offered by the two firms. 15 In Section 6 we consider a general discrete choice model of random utility maximization. 14 distance to firm B), t > 0 is the consumer transportation cost per unit of distance, and represents the preference for quality. The next assumption characterizes the set of utility functions. Assumption 1. The utility function u i (q i ;) : R + ! R + is twice continuously differentiable, satisfies @u i (q i ;) @q i q i =0 > c i , @ 2 u i (q i ;) @q 2 i < 08 2 , and @ 2 u i (q i ;) @q i @ j > 0,8j 2 f1; 2;:::;ng. The firms use 2PTs, which include a marginal (unit) price, p i , and a lump-sum fee, F i , fori =A;B. To avoid expositional complications, we define the set of feasible unit prices of both firms asP. 16 Given (p i ;F i ), a consumer with vertical taste parameter2 decides to buy q i :P !R + units from firm i2fA;Bg, where q i (p i ;) = arg max q i 2R + fu i (q i ;)p i q i g; so the net utility U i (p i ;F i ;) is U i (p i ;F i ;)v i (p i ;)F i ; where v i (p i ;) is the indirect utility “offered” by firm i, defined by, v i (p i ;) max q i 2R + fu i (q i ;)p i q i g: We will focus on the case with E [v i (c i ;)] > 0, where v i (c i ;) is the maximum surplus offering a good at the marginal cost, c i , by firm i2fA;Bg for any2 . Note that the indirect utility function, v i (p i ;) satisfies q i (p i ;) = @v i () =@p i by Roy’s identity with @ 2 v i () @p i @ j < 0 for all i2fA;Bg and8j2f1; 2;:::;ng. Moreover, by continuity of the first and second derivative of v i (p i ;) and by Roy’s identity we know that v i (p i ;) is submodular in (p i ;). 17 From the properties of supermodular (submodular) functions we know thatv i (p i ;) satisfies increasing differences property. 18 That is, v i (p i ;)v i (p 0 i ;) must be monotone nondecreasing in for all p i ;p 0 i 2P and p i p 0 i 8i2fA;Bg. In order to simplify our analysis we assume full market coverage in which all consumers buy at least from one firm i2fA;Bg and both firms sell strictly positive quantities. This assumption implies a lower and an upper bound for t, which will depend on the model 16 For each model, we define the set of feasible unit prices,P. 17 Notice that, @ 2 v(p;) @p@i = @q(p;) @i > 0. From (Topkis 1978) we know thatv (p;) is supermodular in (p;). See also (Milgrom and Shannon 1994), Theorem 6. 18 See for example (Milgrom and Shannon 1994). 15 considered in each section. Moreover, (A1) implies that the buyer’s demand function and the monopoly profit func- tion, q i (p i ;) and i (p i ;), respectively, are continuously differentiable and q i (p i ;) is strictly decreasing on p i ,8i2fA;Bg. Assumption 2. @ i (p i ) @p i < 1,8i2fA;Bg where i (p i ) E[q i (p i ;)] E[q 0 i (p i ;)] and q 0 i (p i ;) @q i (p i ;) @p i . 19 Under(A2), thereisauniqueoptimalmonopolypricep m i 2P. Furthermore, theexpected value of the monopoly profit function E [ i (p i ;)] =E [q i (p i ;)] (p i c i ); is single-peaked in p i under (A2). Due to our full market coverage assumption, the share of-consumers who decide to buy from firm i2fA;Bg is 20 s i (p i ;F i ;p j ;F j ;) 1 2 + v i (p i ;)v j (p j ;)F i +F j 2t ; (1.1) and the share of firm j6= i is s j (p j ;F j ;p i ;F i ;) = 1s i (p i ;F i ;p j ;F j ;). The problem of each firm i2fA;Bg is max p i ;F i Efs i (p i ;F i ;p j ;F j ;) [ i (p i ;) +F i ]g; (1.2) for j6=i. We present conditions for marginal-cost pricing under the assumption of homogeneous and heterogeneous taste preferences for consumers. 21 We restrict our general model to study- ing the equilibrium pricing strategy of each asymmetry separately. First, we assume that 19 An identical assumption is used by (Carrillo and Tan 2015) in amodel of platform competition. Likewise, (Armstrong and Vickers 2001) have a similar assumption for a model with consumers with homogeneous tastes for quality and symmetric firms with common marginal cost, c. They assume 0 (u) 0 where (p) = q 0 (p) q(p) (pc) for u =v (p). The function (p) represents the elasticity of demand express in terms of the mark-up (pc) instead of the price p. Is straightforward to show that 0 (p) < 1 implies that 0 (u) 0. 20 The full market coverage assumption requires a lower bound for t that guarantees that both firms sell strictly positive quantities and an upper bound such that all consumers buy at least from one firm. Note for each model we need different bounds. In Section 3 we define the lower and upper bound for t. For the rest of the models, the bounds are similar, so we exclude them from the analysis. 21 Note that the term “homogeneous” and “heterogeneous” refer to the taste parameter . We will denote “homogeneous preferences” when is constant in the model and “heterogeneous preferences” when follows a distribution G () independent of x. Note that in both cases consumers are horizontally differentiated. 16 the indirect utility provided by both firms are equal, that is, v i (p;) = v j (p;) = v(p;) for all p2P and 2 where v (p;) satisfies (A1) but the marginal cost for firm A, the efficient firm, is lower than the marginal cost for firm B, the less efficient firm, that is, c A < c B . The second model, assumes that both firms have symmetric marginal costs but offer differentiated goods. In particular we assume that the products offered by firm A are vertically superior to the products offered by firm B, that is, v A (p;) > v B (p;) for all p2P and2 . 1.3 Marginal Cost Pricing In this section, we assume that firms offer differentiated products and have different marginal costs. We start by considering a model in which consumers are homogeneous in their taste for quality, whereas their horizontal brand preferences remain unknown to the firm. We show that there exists a unique equilibrium in which firms set their marginal prices equal to their marginal costs. Next, we consider a model in which consumers have heterogeneous tastes preferences and provide necessary and sufficient conditions for a marginal-cost pricing Nash equilibrium. We show that under certain conditions this equilibrium is unique. Homogeneous Preferences. The set of feasible unit prices is P = [c; p]; where c minfc A ;c B g and p = maxfp m A ;p m B g. Now consider the choice of prices and fixed fees by each firm. Due to our full market coverage assumption, the market share of consumers, s i (p i ;F i ;p j ;F j ), who decide to buy from firm i2fA;Bg is defined by the analogue of (1.1) for identical for all consumers. 22 The problem of firm i2fA;Bg is max p i ;F i i = max p i ;F i s i (p i ;F i ;p j ;F j ) i (p i ) +F i ; for j6=i. Proposition 1. Suppose the analogue of (A1) and (A2) for identical for all consumers are satisfied. Then, marginal cost-based 2PT is a unique equilibrium where F i = t + 22 The full market coverage assumption requires t2 n t2R + ; v A (c A )v B (c B ) 3 <t< v A (c A )+v B (c B ) 3 o . For the rest of the paper, we omit the conditions for t. 17 v i (c i )v j (c j ) 3 for i2fA;Bg and j6=i. 23 Proposition 1 shows that if consumers are homogeneous in their tastes for quality, under the assumption of full market coverage, the optimal strategy for each firm is to set its prices equal to their marginal costs and extract surplus through the fixed fee. Note that in this model the marginal costs of the two firms may be different, which implies that the marginal prices (and fixed fees) may also be different. 24 Proposition1iscloseto(MathewsonandWinter1997)’sresultforgoodsthatarestrongly complementary in demand. In our model of one-stop shopping and homogeneous taste preferences, consider firm i’s choices for i =A;B: we can interpret the permission to allow consumers to enter the shop as the first product (product 1) and its price to be equal to the fixed fee F i , and treat the real product offered by firm i as product 2 with price equal to p i . The demand for product 1 is the market share of firm i’s product, s i (p i ;F i ;p j ;F j ), and the demand for product 2 is the market share multiplied by the individual demand for such product, s i (p i ;F i ;p j ;F j )q i (p i ). Note that the ratio is independent of the fixed fee, F i . Hence the two “products” are strong complements. Using Proposition 2 in Mathewson and Winter we would be able to conclude that the profits are maximized for firm i at p i = c i . Hence, independently of firm j’s actions, firm i6=j always charges the marginal cost of the second product, c i . 25 Note that if we modify our model to make it compatible with (Yin 2004), Proposition 1 would be able to be derived from his Proposition 1. When the location parameter does not interact with quantity, the demand of the marginal consumer is equal to the average demand, satisfying the condition for marginal-cost pricing. However, there are three impor- tant remarks: First, he assumes a general distribution for the consumers while we assume they are uniformly distributed on [0; 1]. Second, although Yin considers the particular case in which consumers are uniformly distributed, firms have symmetric costs and demands for this case. Third, we cannot deduct uniqueness (or the conditions needed) from his result. 23 If t< v A (c A )v B (c B ) 3 then there exists an corner equilibrium in which firm B sets p B =c B and F B = 0 while firm A sets p A =c A and F A = t 2 + v A (c A )v B (c B ) 2 . For the rest of the paper we consider only interior equilibria. 24 Note that we exclude from the analysis cases in which the fixed fees offered by the two firms are equal to zero, otherwise we will end up considering LP contracts. If full market coverage assumption is satisfied, firms will have incentives to deviate and offer a 2PT scheme with positive fixed fees. 25 Note that this game (and in general the set of games presented here) satisfies strategic complementarity on rivals’ strategy, like in (Bulow, Geanakoplos, and Klemperer 1985). However, neither is a game with strategic complementarities like in (Vives 1990) nor a supermodular game like in (Milgrom and Roberts 1994). The reason is that the product under consideration and access by each firm are complements to consumers, not substitutes. So, these two “products” are substitutes across the firms but complements within each firm. Thus, we cannot use the results derived for these set of games e.g. Nash equilibria exist and have a certain order structure. 18 From Proposition 1 we know that in equilibrium both firms set their prices equal to their marginal costs, and in equilibrium, the firm that provides the highest surplus (at its own marginal cost) has the highest fixed fee, market share and total profits. Similarly, in the asymmetric marginal costs model (indirect utilities are symmetric e.g., v i (p) =v j (p) for all p2P andi6=j); ifc i <c j then in equilibriumF i >F j ,s i >s j and i > j . Likewise, note that for the asymmetric demands model (marginal cost are symmetric e.g.,c i =c j forj6=i); if v i (p) > v j (p) for all p2P, then in equilibrium F i > F j , s i > s j and i > j . Finally, note that if both marginal costs and indirect utilities are symmetric, we get the standard 2PTs symmetric result. HeterogeneousPreferences. Weshiftourattentiontothecasewhereconsumersdiffer both in their brand preferences (horizontal differentiation) and in their quality preferences (“vertical” taste parameter), and provide necessary and sufficient conditions under which marginal-cost pricing is an equilibrium. Due to our full market coverage assumption, the market share and the problem of firm i2fA;Bg is defined by (1.1) and (1.2), respectively. First order conditions for firm i2fA;Bg are [p i ] : E [q i (p i ;) i (p i ;)]E [q i (p i ;)]F i +2tE [ 0 i (p i ;)s i (p i ;F i ;p j ;F j ;)] = 0 (1.3) [F i ] : 2tE [s i (p i ;F i ;p j ;F j ;)]E [ i (p i ;)]F i = 0: (1.4) We can establish general conditions under which marginal-cost pricing is an equilibrium. From (1.3) and (1.4) we get the following condition Cov (v i (p i ;)v j (p j ;);q i (p i ;)) = 0; (1.5) forp i =c i andi2fA;Bg andi6=j. We summarize this result in the following proposition. Proposition 2. (i) For a given c i ;c j 2P, marginal cost-based 2PT is an equilibrium if and only if (1.5) holds for p i =c i for i;j2fA;Bg . (ii) If for anyp i ;p j 2P, (1.5) holds fori;j2fA;Bg andi6=j, marginal cost-based 2PT is a unique equilibrium. Note that if (1.5) holds for p i = c i , the covariance of the demand, q i (p i ;), and the market share, s i (p i ;F i ;p j ;F j ;), is also zero. The reason is that the market share is linear 19 with respect to differences of the two indirect utilities,v i (p i ;)v j (p j ;), for the uniformly distributed à la Hotelling model. Thus marginal-cost pricing is an equilibrium if and only if the covariance of the demand and the market share for both firms, evaluated at the marginal cost, is zero. Now, if the demand is independent of the market share for firm i2fA;Bg for all feasible prices, marginal-cost pricing is a unique equilibrium. We can use our previous example to explain under what conditions Mathewson and Win- ter’s result holds when consumers have heterogeneous tastes for quality in our model; re- member that we can interpret the permission to allow consumers to enter the shop as the first productandtreattherealproductofferedbyfirmiasproduct2withpricesF i andp i , respec- tively. In this case the demand for product 1 is the expected market share for firm i’s prod- uct,E [s i (p i ;F i ;p j ;F j ;)], and the demand for product 2 is the expected value of the market share multiplied by the individual demand for such product, E [s i (p i ;F i ;p j ;F j ;)q i (p i ;)]. Proposition 2(i) shows indirectly that if for i;j2fA;Bg and i6=j E [s i (c i ;F i ;c j ;F j ;)q i (c i ;)] E [s i (c i ;F i ;c j ;F j ;)] =E [q i (c i ;)]; (1.6) marginal-cost pricing is an equilibrium, that is, if (1.5) holds the ratio of the demands of the two products is independently of the fixed fee, F i , for p i =c i . Hence, from Mathewson and Winter’s result we know that marginal-cost pricing is an equilibrium. 26 Moreover, Proposi- tion 2(i) shows that for a given c i ;c j 2P this is a necessary and sufficient condition. Note that this condition is always satisfy for the symmetric case, as we discuss in the following corollary. Corollary 1. If c i = c j = c and v i (p;) = v j (p;) for all p2P, 2 and j6= i, marginal cost-based 2PT is a Nash equilibrium. Corollary 1 is related to the standard result of 2PTs (e.g., (Armstrong and Vickers 2001), and (Rochet and Stole 2002)), that is, marginal cost pricing is an equilibrium for the sym- metric case. An implication of Proposition 2 is that if is associated, 27 since q i (c i ;) is monotone 26 Note that the fact that (6) implies (7) for p i = c i is a special feature of the Hotelling’s market share. That is, this result would not be true for a model with a general market share like the model in Section 6. 27 We assume that is associated for the rest of the paper. A vector of random variables is associated if Cov[f ();g ()] 0 for all nondecreasing functions f and g for which E [f ()], E [g ()] and E [f ()g ()] exist. For a complete reference on association of random variables and its properties see (Esary, Proschan, and Walkup 1967). See also (Holmstrom and Milgrom 1994) and (Milgrom and Weber 1982) for economic applications. 20 increasing, and if v i (c i ;)v j (c j ;) is monotone increasing or decreasing (depending on marginal cost and functional forms for v i () i2fA;Bg) with respect to , marginal-cost pricingisnotanequilibrium(1.5isviolated), exceptforthecasewhenv i (c i ;)v j (c j ;) =k 82 where k is a constant. That is, note that Proposition 2 is a general result in the following sense. Corollary 2. (i) If c i 6= c j and v i (p;) = v j (p;) = v(p;) for all p2P and 2 , marginal cost-based 2PT is not a Nash equilibrium. (ii) If c i = c j and v i (p;)v j (p;) is monotonic with respect to , then marginal cost-based 2PT is not a Nash equilibrium. Corollary 2(i) shows that if marginal costs are asymmetric and the products of the two firms are symmetric then (1.5) does not hold. Likewise, if marginal costs are symmetric but v i (p;)v j (p;) is monotonic with respect to, then (1.5) does not hold from Corollary 2(ii). 28 Anillustrativeexampleofthelastcasewithsymmetricmarginalcostisthefollowing: suppose for instance that for any p2P and2 the indirect utility offered by firm i is v (p;) (satisfies A1) and the indirect utility offer by firm j, j6= i, is v (p;) for any 2 (0; 1). Then, marginal-cost pricing is not an equilibrium. However, if marginal costs are asymmetric and v i (p;),c i for i =A;B and are such that v (c i ;)v (c j ;) = 0 for all 2 we get an opposite result. 29 The reason for this difference -between Corollary 1 and 2- is related with the dependence of the fixed fees and marginal prices on . If both marginal costs and indirect utilities (demand of the two goods) are symmetric, the most profitable way for both firms to attract consumers and extract consumer surplus is to set its marginal prices equal to marginal cost andsetthefixedfeeequaltot. Notethatthiscost-based2PTdoesnotdependon, therefore this tariff remains an equilibrium even when is unknown for both firms (Armstrong and Vickers 2001; Armstrong 2006). However, if marginal costs or the products offered by the two firms are asymmetric, the marginal price and the fixed fee would depend on . In the next section we show that as the dispersion of increases, the “quasi”-best response function in terms of (p i ;p j ) rotate clockwise for firm i, and counterclockwise for firm j, for i6= j. 28 By “monotonic with respect to ” we refer to the following example: Let ; 0 2 such that > 0 (“high” and “low” type). Then if v i (p;)v j (p;) is monotonic with respect to, the sum of the indirect utilitiesofferedbyfirmiandj tothehighandlowtype,respectively,ishigherthanthesumofindirectutilities offered by firmi andj to the low and high type, respectively, e.g. v A (p;)+v B p; 0 >v A p; 0 +v B (p;). That is, product A is “vertically” superior to product B. 29 Marginal cost pricing is also an equilibrium if for example the indirect utilities offered by the two firms are such that v i (p;)v j (p;) is a constant for all p2P and2 . 21 This implies that firms would have incentives to deviate from marginal cost-bases 2PTs. In sum, when firms have asymmetric marginal costs or asymmetric demands, information about vertical taste preferences has a substantial effect on the equilibrium pricing strategy. Thatis, verticaluncertaintyaffectstheslopeoftheimplicitbestresponsefunctionsregarding the marginal prices. We will further investigate this issue in the next sections. 1.4 Asymmetric Costs In this section we suppose that indirect utilities offered by the two firms are symmetric but marginal costs differ. Without loss of generality we assume that the marginal cost of firm A, the efficient firm, is lower than the marginal cost of firm B, the less efficient firm, i.e., c B > c A 0. We assume that consumers are heterogeneous in their taste for quality, thus from Proposition 2 we know that marginal-cost pricing is not a Nash equilibrium. We first show in Proposition 3 that no equilibrium exists for p B c B . Indeed, we show that there exists a pure-strategy Nash equilibrium, in which (p A ;p B )2 (c A ;c B ) 2 . Next, we show that the quasi-best response functions of the two firms are increasing, 30 and that the equilibrium is unique. Finally, we provide some comparative statics with respect to. To make notation compatible with previous sections we need to redefine the set of feasible unit prices both firms can choose, ^ P = [c A ;p m B ]; where p m B corresponds to the monopoly price of firm B. We restrict the set of feasible unit prices of the efficient firm to be always above its marginal cost, c A , however, prices for firm B are allowed to be lower than its own marginal cost, c B . 31 The problem of each firm i2fA;Bg is, max p i ;F i E 1 2 + v (p i ;)v (p j ;)F i +F j 2t [ i (p i ;) +F i ] : From the first order conditions with respect to p i , for firm i, 30 By quasi-best response functions we refer to the best response functions only in terms of p A and p B (substituting both fixed fees). 31 We discuss later that this assumption is without loss of generality. 22 (p i c i )E 2tq 0 (p i ;)s i q (p i ;) 2 +E [q (p i ;) (t +v (p i ;)v (p j ;))] (1.7) +E [q (p i ;)] (F j 2F i ) = 0; where s i is the market share of firm i. When both firms use LP, the first and the second term on the left-hand side of (1.7) characterized the best response functions of each firm i. 32 Thus, there are two main differences when firms use 2PT compared to LP: first, there is a direct effect that moves the curve of “quasi-best” response function of firm i to left, in the p i ;p j plane, for j6=i. 33 This implies that firm i reacts more aggressively with its marginal price for each value ofp j . Second, there is an indirect effect, sinceF j , also makes firmj react more aggressively, decreasing p j for each value of p i , increasing F j . The intuition for these two effects is the following: when firms are allowed to use 2PT, they can extract surplus also through the fixed fee, which does not depend directly on the curvature of the demand. Thus the best strategy is to set a low price to attract consumers and extract surplus through the fixed fee. We next show that in equilibrium both firms decrease their marginal prices compare to the case when both firms use LP. Now, from the first order conditions with respect to F i for each firm i =A;B, the fixed fee in equilibrium is defined by, F i +E [ i (p i ;)] =t + TS i (p i )TS j (p j ) 3 ; (1.8) whereTS i (p i )E [v (p i ;)] +E [ i (p i ;)] is the expected total surplus for firm i2fA;Bg forj6=i. From (1.8) we know that the total profit per-consumer depends on the transporta- tion cost and the difference between the surplus offered by both firms. Similarly, from (1.7) and (1.8), p i is implicitly defined by, (p i c i )f2tE [q 0 (p i ;)s i ] Var [q (p i ;)]g+Cov (v (p i ;)v (p j ;);q (p i ;)) = 0; (1.9) where s i 1 2 + v i v j +TS i TS j 2t and v i v (p i ;)E [v (p i ;)] for i = A;B and j6= i. Note that if the second term on the left-hand side is zero, marginal cost-based 2PT is an equilibrium (Proposition 2), that is, the first term on the left-hand side is zero if and only if p i =c i . If p i >c i , the first term of the left-hand side of (1.9) is negative. This implies that the second term must be positive; p i >p j . Ifp i <c i , the first term is positive, which implies 32 If both firms use LP, the problem of firm i is, max pi E n 1 2 + v(pi;)v(pj;) 2t i (p i ;) o . 33 See Figure 1. 23 that in this case p i c B . We formalize this idea in the following proposition. Proposition 3. There exists a pure-strategy Nash equilibrium. In any equilibrium, the following hold: (i) c A <p A <p B <c B ; (ii) the expected market share, per customer profits and total revenue per consumer is greater for firm A than firm B. For existence, we first show that no equilibrium exists for values of p B c B , as we mentioned before. Thus, we show that the two curves defined by (1.9) for i = A;B cross each other at least once in the set (c A ;c B ) 2 - see Figure 1.1. 34 Note that (ii) follows from (i) and (1.8), and the fact that the expected market share is a linear function of the difference in the expected surplus. That is, from (i) we know that in any equilibrium p A < p B , then the expected total surplus and then the market share for firm A is greater than for firm B. Similarly, note that in any pure-strategy Nash equilibrium in which, c A <p A <p B <c B , we have that, E [ B (p B ;)]< 0<E [ A (p A ;)], and thus the expected revenue per consumer is greater for firm A than firm B. The intuition of this result is the following: suppose that initially both firms sell products at the marginal cost and charge a positive fixed fee. Then, the inefficient firm, B, has incentives to decrease the marginal price below its own marginal cost and compensate this loss increasing the fixed fee, keeping the market share for its products relatively constant. On the other hand, the efficient firm increases its marginal price, but keeps it below its rival’s price, and decreases slightly its fixed fee. Therefore, firm B is following the strategy suggested by the games with symmetric 2PTs: extract the largest share of the total income through the fixed fee. While firm A is using its advantage over firm B. Although firm B sets its price below its own marginal cost, in any equilibrium, firm A sets a price below its rival’s price, which guarantees it a greater market share. Thus, firm A does not need to set its price below its own marginal cost to get a higher market share than firm B. Insum, inanyequilibrium, theexpectedmarketshare, profits, totalrevenueperconsumer and total revenue is greater for firm A than for firm B. In particular, note that the marginal price is lower and the fixed fee is higher for the efficient firm than for the inefficient one. These results may explain the empirical regularities observed in the British electricity market and highlighted by (Davies, Waddams Price, and Wilson 2014); if the entrant firms are more 34 Particularly, note that asp A !c A in (1.9) fori =A, we have thatp B !c A and asp A !c B is not true that p B !c B . In fact, p B ! A >c B . Similarly, from (1.9) for i =B, as p B !c B we have that p A !c B while as p A !c A , p B ! B >c A . 24 efficient than the incumbent, we should expect lower marginal prices and higher fixed fees for the entrant than the incumbent. 35 We next show in Lemma 1 that the slope of the implicit functions defined by (1.9) for i = A;B, R i (p A ) : ^ P! ^ P, is positive, where R i (~ p A ) = ~ p B is such that ~ p A and ~ p B satisfy (1.9) for i = A;B. We show that there exists a unique equilibrium in 2PTs. Particularly, we show that in equilibrium the slope of the implicit function defined by (1.9) for i = A, @R A (p A ) @p A , is greater than the slope of the implicit function defined by (1.9) for i =B, @R B (p A ) @p A (Proposition 4). Finally, we illustrate some comparative statics properties of the equilibrium with respect to. To analyze the slope of the quasi-best response functions we need to introduce a new assumption that helps us to characterize it. 36 First we introduce the following definition. Definition 1. v (p;) : ^ P !R + is separable if there exist functions v : ^ P!R + , h : !R andm : ^ P!R + wherev () andm () are strictly decreasing and h () is strictly increasing, such that for all (p;)2 ^ P , v (p;) =v (p)h () +l () +m (p): Assumption 3. v (p;) : ^ P !R + is separable. An example of the class of indirect utilities that satisfy (A3) are the power functions (or constant elasticity demand) e.g., suppose thatu (q;) = p q thenv (p;) = 2 4p ; log function, e.g., u (q;) = logq thenv (p;) = (log 1) logp; and linear demand-type function, e.g., u (q;) =q q 2 2 then v (p;) = (p) 2 2 . Lemma 1. Suppose (A3) is satisfied. Then, the slope of the implicit functions defined by (1.9), @R i (p A ) @p A for i =A;B, is positive for (p A ;p B )2 [c A ;c B ] 2 . Note that although both firms are using fixed fees to extract surplus, the quasi-best response functions with respect to the prices of the two firms are increasing, as in the standard LP game. This result will be useful to show uniqueness of the game in the following 35 Aswementionedbefore, thereareothertypesofasymmetriesthatareimportantintheBritishelectricity market. Some of the firms were integrated upstream into generation, and some of them were active in the gas market. Although (Davies, Waddams Price, and Wilson 2014) suggest small cost asymmetries between firms, we need to assume that the other type of asymmetries (vertical integration and gas offer) can be projected into the marginal cost of the firms. This may result in firms with asymmetric marginal cost. 36 Note that (1.9) for i =A;B implicitly define quasi-best response functions for each firm in terms of the marginal prices p A and p B . 25 proposition. Proposition 4. Suppose (A3) is satisfied. Then there exists a unique equilibrium in 2PTs in which p i 2 ^ P is determined by (1.9) and F i satisfies (1.8), for i =A;B. From Proposition 3 we know that the two implicit functionsR A (p A ) andR B (p A ) derived from (1.9) for i =A;B, cross at least once (see Figure 1.1). Next, from Lemma 1 we know that the slope of the implicit functionsR A (p A ) andR B (p A ) is positive. To prove uniqueness we show that in equilibrium the slope of R A (p A ) is greater than the slope of R B (p A ). Figure 1.1: Equilibrium with Asymmetric Cost Corollary 3. In equilibrium, as c B goes to c and c A goes to c, p i converges to c and F i to t, for i =A;B. Note that Corollary 3 follows from Proposition 4 and Proposition 2. As the marginal cost for firm B and firm A converge to a common value c, both marginal prices tend to the marginal cost, and both fixed fees to the transportation cost,t; which is the symmetric 2PTs result like in (Armstrong and Vickers 2001). Finally, (A4) allows us to express (1.9) as a function of Var [h ()] Corollary 4. In equilibrium, (i) as ! 0, p A !c A and p B !c B ; (ii) as !1, p A ! p A and p B ! p B where c A < p A < p B <c B . 26 Corollary 4(i) follows from Proposition 1 and the monotonicity of the quasi-best response functions with respect to the prices for both firms. Note that when = 0, the quasi-best response function for firm A is a vertical line at p A =c A in the p A ;p B plane. Similarly, for firm B is a horizontal line atp B =c B . From different numerical simulations, we find that as increasesp A increases andp B decreases, i.e., as increases the quasi-best response function rotates to the right around (c A ;c A ). Similarly, for firm B; as increases the quasi-best response function rotates to the left (counterclockwise) around (c B ;c B ). Thus, forp i >c i , as increases firmi reacts less aggressively (sets a higher price) to eachp j , forj6=i. However, for p i <c i , as increases firm i reacts more aggressively (sets a lower price) to each p j , for j6= i. This explains why when consumers are heterogeneous in their tastes marginal-cost pricing is not a Nash equilibrium. In particular, it explains why the optimal strategy for the inefficient firm (B) is to set its marginal price below its own marginal cost and compensate for this loss with the fixed fee. On the other hand, the optimal strategy for the efficient firm (A)istosetitsmarginalpriceaboveitsownmarginalcostbutbelowthatofitsrival. Finally, from Corollary 4(ii), note that as increases, the marginal increase of p A is decreasing, and the marginal decrease of p B is also decreasing. 1.5 Asymmetric Demands This section presents the second model, which analyze the second type of asymmetry related with the goods offered (or equivalently with the demand) by the two firms. We consider a model in which both firms have the same marginal cost, c, but we assume that firms offer differentiated products. Without loss of generality, we assume that for any p2 ~ P and 2 the indirect utility offered by firm A is higher than the one offered by firm B i.e., v A (p;)v B (p;) > 0 for all 2 . In order to simplify the analysis we introduce the following assumption. Assumption 4. Let v A (p;) = v (p) and v B (p;) = v (p) for 2 (0; 1), where v (p;) satisfies (A1) and v : ^ P!R + is strictly decreasing. Intuitively, (A4i) implies that for any p2 ~ P and for two indirect utility functions that satisfies (A1) e.g. v A (p;)v B (p;)> 0 , we have that for; 0 2 such that> 0 (high and low type), the sum of the indirect utilities offered by firms A and B to the high and low type, respectively, is higher than the sum of indirect utilities offered by firm A and B to the low and high type, respectively, i.e. v A (p;) +v B (p; 0 ) > v A (p; 0 ) +v B (p;). That is, product A is “vertically” superior to product B. A second implication of (A4i) is that cases in 27 which the two indirect utilities differ by an additive constant are excluded from the analysis. Suppose, for example, that u A (q;) = p q and u B (q;) = p q then v A (p;) = 2 4p and v B (p;) = 2 4p , which satisfy (A4). The set of feasible unit prices both firms can choose is, ~ P = [~ B ; ~ p m A ] where ~ B is such that, v (c) =v (~ B ) (1.10) where ~ B is strictly less than c. We restrict our analysis to the set of indirect utilities that satisfy (A1) such that ~ B is strictly positive. This condition implies that the difference between the two indirect utilities is bounded (equivalently, we can say that the difference between the demands of the two products offered by the firms is bounded). We proceed to characterize the equilibrium of the game following a similar strategy as in the previous section. From Proposition 2 we know that marginal-cost pricing is not a Nash equilibrium. We first show that no solution exists for p B > c. Next, we show that there exists an equilibrium in 2PTs and is uniquely defined. From the first order conditions, p i is defined by, 37 Cov (v i (p i ;)v j (p j ;);q i (p i ;)) + (p i c i )f2tE [q 0 i (p i ;) ~ s i ] Var [q i (p i ;)]g = 0 (1.11) where ~ s i 1 2 + ~ v i ~ v j + ~ TS i ~ TS j 2t , ~ v i v i (p i ;)E [v j (p i ;)] and ~ TS i = E [v i (p i ;)] + E [ i (p i ;)] for i =A;B for j6=i. Proposition 5. Suppose (A4) is satisfied. Then there exist a unique equilibrium in 2PTs, in whichF i +E [ i (p i ;)] =t + 1 =3 TS i (p i )TS j p j ,p A ;p B 2 ~ P are determined by (1.11) for i =A;B, and p B <c c, which implies that if there exist any equilibrium of the game it must be in the set ~ (p A ;p B ) where ~ (p A ;p B ) 37 Note that the model is similar to the model presented in the previous section. Thus we omit the profit function and first order conditions. 28 n p A ;p B 2 ~ Pj(p A ;p B )2 [c; ~ A ] [~ B ;c] o , ~ A is such that (~ A ;c) satisfy (1.11) for i = B, and ~ B was defined in (1.10). Next we show that for (p A ;p B )2 ~ (p A ;p B ) the slope of the implicit functions defined by (1.11) fori =A;B ~ R i (p) : ~ P! ~ P where ~ R i (~ p i A ) = ~ p i B are such that ~ p i A and ~ p i B satisfy (1.11) for i =A;B, are both positives. Finally, weshowthatthereexistatleastoneNashequilibriumi.e., thetwoimplicitcurves defined by (1.11) for i =A;B always cross each other in the region ~ (p A ;p B ). Particularly, note that asp A !c in (1.11) fori =A, we have thatp B ! ~ B >c and asp A ! ~ A ,p B !c where ~ A is such that (~ A ;c) satisfy (1.11) for i = A. Similarly, from (1.11) for i = B, as p A ! c we have that p A ! ~ B while as p A ! ~ A , p B ! c, where ~ B is such that (c; ~ B ) satisfy (1.11) for i = B, and ~ B > ~ B . Similarly, we show that as p A ! ~ A in (1.11) for i =B, p B !! B <c. Thus, both curves cross each other at least once in the set ~ (p A ;p B ) (See Figure 1.2). To prove uniqueness we show that in equilibrium the slope of the implicit function ~ R A (p A ) is greater than the slope of ~ R B (p A ). Figure 1.2: Equilibrium with Asymmetric Demand In equilibrium the firm that is disadvantage in demand, B, sets its marginal price below its rival’s price, firm A. This result contrasts with Proposition 4, in which the firm that is disadvantaged in marginal cost sets its price below its own marginal cost, but above its rival’s marginal price. One reason that explains these results is that in the first model (Asymmetric Costs) the efficient firm sets a marginal price below its rival but above its own marginal cost, while in the second model (Asymmetric Products) by setting a price below the common marginal cost, firm A (advantage in demand) has to compensate this loss increasing the fixed Fee and decreasing the demand for its products. Hence, firm A has incentives to 29 deviate and sets a higher price than firm B, due to its advantage in demand. Finally note that as the difference of the indirect utilities offered by the two firms tends to 0, e.g. tends to 1, for any p i 2 ~ P fori =A;B and2 , both marginal prices tend to the marginal cost,c, and the fixed fees become independent of, equal tot (standard result of 2PT). 1.6 General Market Share Functions In this section, we extend our analysis to allow for general market share functions. We assumethatthereisamassofconsumerswithtypes (;)where ( A ; B ; 0 )isdistributed independently of the distribution of. Following Armstrong and Vickers (2001), we consider a discrete-choice model in which consumers’ preferences for the two differentiated products can be represented by the net utility function, u A (q A ;) + A if she buys from A and, u B (q B ;) + B if she buys from B, and u 0 + 0 if no purchase is made. Consumers buy all products from one or the other firm, or they take their outside option. 38 Given (p i ;F i ) the share of-consumers who choose to buy from firm A iss (v A (p A ;)F A ; v B (p B ;)F B ) and the share of consumers who choose firmB iss (v B (p B ;)F B ; v A (p A ;)F A ), where v i (p i ;) is the indirect utility “offered” by firm i, defined as before, for i =A;B. We impose the following regularity assumptions. First, s (u A ;u B ) is increasing with respect to u A and decreasing with respect to u B . Second, s (u A ;u B ) s 1 (u A ;u B ) is weakly increasing with respect to u A and weakly decreasing with respect to u B . 39 The problem of each firm is, max p i ;F i E [s (v i (p i ;)F i ;v j (p j ;)F j ) ( (p i ;) +F i )] fori;j2fA;Bg andj6=i. We first provide necessary and sufficient conditions for marginal- cost pricing being a Nash equilibrium. We show that these conditions may be violated even 38 For the-consumer the aggregate consumer utility is, V (u A ();u B ()) =E [maxfu A () + A ;u B () + B ;u 0 + 0 g]: By the envelope theorem, V 1 (u A ();u B ()) s (v A (p A ;)F A ; v B (p B ;)F B ) is the share of - consumers who choose to buy from firm A. We assume that consumers’ tastes for the two firms’ products are symmetrically distributed i.e., V (u A ();u B ()) V (u B ();u A ()). See (Armstrong and Vickers 2001) for more details. 39 A similar assumption is used by (Armstrong and Vickers 2001). 30 when firms are symmetric (e.g., same products and marginal costs), which contrasts with the Hotelling model of Section 3 in which marginal cost pricing is an equilibrium for this case. We show that a model in which consumers are homogeneous (the taste quality parameter is constantforalltypeconsumers)triviallysatisfytheseconditions, i.e., firmssettheirmarginal prices equal to their marginal costs. We further consider an asymmetric costs model with general market share with discrete types and show that the qualitatively results of section 4 hold for this general model. 40 1.6.1 Marginal Cost Pricing Here, we study necessary and sufficient conditions under which marginal-cost based 2PT is an equilibrium. Let, i () s v i ();v j () E s v i ();v j () s 1 v i ();v j () E s 1 v i ();v j () where v i ()v i (c i ;)F i and F i is jointly defined by, F i = E s v i ();v j () E s 1 v i ();v j () (1.12) forj6=iandi;j2fA;Bg. Thus, necessaryandsufficientconditionsformarginalcost-based 2PTs depend on the following condition, Cov ( i ();q i (c i ;)) = 0 (1.13) for j6=i and i;j2fA;Bg. We summarize this result in the following proposition. Proposition 6. For a givenc i ;c j 2P, marginal cost-based 2PT is an equilibrium if and only if (1.13) holds for i;j2fA;Bg. The result of Proposition 6 is quite general. The result relies upon full market coverage assumption and positive profits for both firms. For the specific case, in which i is distributed uniformly (a la Hotelling) condition (1.13) trivially depends on the covariance between each firm’s efficient quantity,q i (c i ;), and the difference between the efficient consumer surpluses offered by the two firms v i (c i ;)v j (p j ;), as we showed in Section 3. For the general model presented here, the market share may not be linear with respect to the difference of the indirect utilities offered by the two firms. This may have two implications: first, we required, generally, stronger conditions to derived necessary and sufficient conditions for 40 For this model we consider only interior equilibria with no outside option. 31 marginal-cost pricing to be an equilibrium compared to the Hotelling model, i.e., if (1.13) is satisfied then (1.5) is also satisfied, but not vice versa. Second, condition (1.13) depends on the market share, and not on the difference between the efficient consumer surpluses offered by the two firms (as in the Hotelling model). These two conditions implies that even if firms are symmetric, i.e, c i =c j andv i (p;) = v j (p;) for all p 2 P and 2 and for i 6= j, marginal-cost pricing may not be an equilibrium. In the Hotelling model presented in section 3, if firms are symmetric, marginal- cost pricing is an equilibrium. For the model presented here, even if the market share is constant with respect to in any symmetric equilibrium (e.g., a model with no outside option), the right hand side of (1.13) must be equal to zero, in order to conclude that marginal-cost pricing is an equilibrium. An example of a model in which marginal-cost pricing is not an equilibrium when firms are symmetric is the logit model with outside option. Example: Logit Model with Outside Option. In order to illustrate Proposition 6 for the case in which firms are symmetric, here we assume that i follows a type-I extremum distribution and allow for an outside option. The market share of firm i is, s (u i ();u j ()) e u i () e u i () +e u j () + 1 where u i () v i (p i ;)F i . Note that in a symmetric equilibrium, the covariance be- tween i () and firm’s efficient quantity would be positive, since s (v ();v ()) and s 1 (v ();v ()) are increasing with respect to , but the rate of increase is higher for s (v ();v ()) than s 1 (v ();v ()), thus if is associated, (1.13) is not satisfied. Proposition 7. For the logit model with outside option, in any pure-strategy Nash equilibrium, p >c. For the logit model without outside option, s (u ();u ()) and s 1 (u ();u ()) are constant, thus (1.13) is satisfied and marginal cost-based 2PT is an equilibrium. Note, however, that this result holds only if firms are symmetric; if firms have asymmetric marginal costs or offer differentiated products, marginal-cost pricing is not an equilibrium. For the case in which consumers are homogeneous, (1.13) is trivially satisfied. We show in the next corollary that for this case the equilibrium is uniquely defined. 32 Corollary 5. Suppose v i (c i ) > s(0;0) s 1 (0;0) for i2fA;Bg. Then, marginal cost-based 2PT is a unique equilibrium where F i = v i (c i )v i and v i are implicitly defined by v i (c i ) = v i + s(v i ;v j ) s 1(v i ;v j ) for i;j2fA;Bg and j6=i. If consumers are homogeneous, under the assumption of full market coverage, Corollary 5 shows that the optimal strategy for each firm is to set its prices equal to their marginal costs and extract surplus through the fixed fee. Note that v i (c i ) is the efficient (maximum) surplus offered by firm i to the consumers, and v i is the net surplus when competing firms engage in efficient surplus extraction. In the case of monopoly, the firm would setF i =v i (c i ) and hence the net consumer surplus would be v i = 0 (full extraction). In the presence of competition full extraction is not possible and hence v i > 0. Under full market coverage the equilibrium net surpluses v i ;v j are determined by the above equations, which imply that 0 < v i < v i (c i ) for each i2fA;Bg. The ratio s(v i ;v j ) s 1(v i ;v j ) represents the competitive effect that prevents firms from full extraction. Similarly, note that in equilibrium the firm that provides the highest surplus (at its own marginal cost) has the highest fixed fee, market share and total profits, similar to the model presented in section 3, i.e., if v i (c i ) > v j (c j ) then F i >F j , s i >s j and i > j for i6=j and i;j2fA;Bg. Corollary 6, extends Armstrong and Vicker’s Proposition 1 to allow for asymmetric firms. Here, firms set their marginal prices equal to the marginal costs; thus a natural framework to consideristhecompetitionin“utilityspace,” whichprovidesatractablewaytoshowexistence and uniqueness for this model with homogeneous consumers (see proof of Corollary 6). 1.6.2 Asymmetric Marginal Cost model with discrete types. In this section we suppose that the indirect utilities offered by the two firms are symmetric but marginal costs differ. We consider a discrete type model in which is drawn from the distribution on =f L ; H g, where L < H (low and high type), with probabilities and 1, respectively. 41 The next proposition generalizes the results of section 4, allowing for a general market share without outside option. Proposition 8. There exists a pure-strategy Nash equilibrium, and in any equilibrium, c A <p A <p B <c B . Note first that from Proposition 8 it follows that marginal cost-based 2PTs is not an equilibrium. The result in Proposition 8 (and the strategy used for its proof) is similar to 41 The results presented here can be generalized to the case of ntypes. 33 the one presented in Proposition 3. That is, when consumers are heterogeneous in their tastes for quality, firms have incentives to deviate from marginal-cost pricing. The efficient firm increases its marginal price, but keeps it below its rival’s price, and decreases slightly its fixed fee. On the other hand, the inefficient firm has incentives to decrease the marginal price below its own marginal cost, so that the revenue losses arising from this are more than offset the revenue gains obtained from the fixed fee. Note that Proposition 8 assumes that there is not outside option, so consumers buy either from firmA orB. A natural extension would be a general market share model with outside option. In the next example, we consider the logit model with outside option and show that in equilibrium the order for the marginal prices and the fixed fee remains constant i.e., p A F B , but marginal price are above marginal costs for both firms. We explain these results further in the following example. Example: Logit Model with outside option. Here, we present a numerical example in order to illustrate Proposition 7 for the logit model with outside option. We assume the following functional form: u (q;) = 1 q 1 1 1 1 , then we have that q (p) = p , v (p) = 1 (1) p 1 andq 0 (p) = p +1 . We use the following parameters:t = 0:3,c A = 0:2,c B = 0:25, = 0:2, = 2, L = 0:3 and H = 0:5. Suppose also for simplicity that H = L = 0:5. Figures 3A and 3B show the change on the marginal prices and the fixed fees, respectively, when varies from 0 to 1. Note that marginal prices are always above marginal costs for both firms, and lower for firm A than firm B, while fixed fees are always higher for firm A than firm B. Intuitive, when we allow for an outside option, the competition is less intense compared to a model with no outside option. In this model, firms also compete with the outsideoption, whichisneutral, i.e., variationsinthemarginalpricesorfixedfeesofthefirms do not cause changes in the outside option. Thus, firms do not have to react aggressively to its competitor pricing strategy, which in equilibrium allow them to price above the marginal costs. 34 Figure 3A: Marginal prices Figure 3B: Fixed fees 1.7 An Extensions: Non-Linear Pricing In this section, we extend our previous model and study under what conditions cost-based 2PTs is an equilibrium if both firms use nonlinear tariffs. (Armstrong and Vickers 2001) and (Rochet and Stole 2002) showed that if the two firms are symmetric and under full market coverage, each firm offering marginal cost-based 2PT is an equilibrium. Here, we keep all the assumptions of section 2; we assume that there are two firms competing on either ends of the market, offering differentiated products to a population of heterogeneous consumers, with constant but different marginal cost, c A 6= c B . We suppose that, instead of 2PTs, firm i =A;B offers a nonlinear tariff T i (q i ). A consumer with taste parameter2 has utility, u i (q i ;)T i (q i ) excluding transportation cost, if she buysq i units from firmi, in return for a paymentT i (q i ), where u i (q i ;) satisfies (A1) for i2fA;Bg. Given T i (q i ), a type2 consumer choose the price-quantity pairfq i ();T i (q i )g that maximize her utility, u i () max q i fu i (q i ;)T i (q i )g if she buys from firm i. Finally let, v i () max q i fu i (q i ;)c i q i g 35 be the type- consumers’ utility when firm i sets its prices at its own marginal cost. Using the dual approach, we can write the market share and the total expected profit of firm i, as a function of consumers maximum utility u i (). Due to our full market coverage assumption the problem of each firm i2fA;Bg is, max u i ;q i E 1 2 + u i ()u j () 2t (S i (q i ();)u i ()) where S i (q i ();) = u i (q i ;)c i q i () is the surplus from trade with firm i and q i () satisfies the first order differential equation, _ u i () =q i (). Proposition 9. Suppose both firms use non-linear tariffs. Then, marginal cost-based 2PT is an equilibrium if and only if for i;j2fA;Bg and j6= i, v i ()v j () is constant over2 . Proposition 9 extends Armstrong and Vickers’ (2001) Proposition 5 and Rochet and Stole’s (2002) Proposition 6 for the general case, in which firms offer differentiated products to a population of heterogeneous consumers, and have asymmetric marginal cost. The proof of Proposition 9 can be constructed, adapting the strategies used by (Armstrong and Vickers 2001) and (Rochet and Stole 2002), for this case with asymmetric costs. We show that cost-based 2PT is an equilibrium if and only if the difference of the utilities offered by the two firms at their marginal cost is constant (does not depend on ). Two examples that satisfy this condition are: (i) suppose thatv A (p;) =v (p;) andv B (p;) = v (p;) +k where k2R and v (p;) is derived from an utility function that satisfies (A1); (ii) suppose thatv A (p;) =v (p;) andv B (p;) =v (p;) where2 (0; 1) andv (p;) is separable. Then there exists (c A ;c B ) such that the difference is zero at (c A ;c B ). Note that if both firms offer symmetric goods and have symmetric marginal cost (as in Armstrong and Vickers and Rochet and Stole) this condition is trivially satisfied. Finally, note that if q i (p;) is increasing with respect to for i2fA;Bg, the condition needed in Proposition 10 is the same as in Proposition 3. That is, the covariance of the demand and the difference of the utilities would be equal to zero at the marginal costs, for both firms A and B. 1.8 Conclusions 2PTs are prevalent in our economy. Most of the articles that have studied competition in 2PTs assume unit demands, homogeneous consumers distributed uniformly and symmetric 36 firms (including pricing schedules, marginal cost and unit demands). However, industries that have implemented this type of tariffs are characterized by share different features from the ones established by these assumptions. Likewise, therecentliteratureoncompetitivenonlinearpricingshowsthatinasymmetric model of competition between firms offering products horizontally and vertically differenti- ated, if the market is fully covered, cost based 2PTs will arise in equilibrium. This result holds for one stop shopping models e.g. (Armstrong and Vickers 2001) and (Rochet and Stole 2002) and two stop shopping models e.g. (Armstrong and Vickers 2010). However, (Rochet and Stole 2002) show that this simple cost-based tariff result is not robust to a num- ber of changes including: asymmetric marginal cost; partial covered markets, or interactions between (correlation) the unobserved vertical and horizontal taste parameter. In this paper, we consider a general model in which the firms have asymmetric marginal cost and asymmetric demands (or offer differentiated goods). We provide necessary and sufficient conditions under which marginal-cost pricing arise in equilibrium under two infor- mation structures: horizontal differentiated consumers with homogeneous taste parameter for quality and private information about consumer’s taste for the goods (and not just about their location). Likewise, we study the equilibrium outcomes of each asymmetry separately. First, we consider a model in which both firms have symmetric demands but have asym- metric marginal cost. We show that the optimal strategy for the inefficient firm (the firm with the highest marginal cost) is to set its marginal price below its own marginal cost and compensate this loss with the fixed fee. On the other hand, the optimal strategy for the efficient firm is to set its marginal price above its own marginal cost, but below its rival. The second model assumes that both firms have symmetric marginal cost but asymmetric demands. Similar to our first model, we show that the optimal strategy for the firm that is disadvantage in demand (which products are vertical inferior than its rival) is to set its own price below the marginal cost, while in this case the optimal strategy for the superior firm is to set its marginal price above its rival. That is, the disadvantage firm has incen- tives to use “cross-subsidies” between the tariffs (fixed fee and marginal price). However, the optimal strategy of the advantage firm depends on the nature of each model. Finally, we provide conditions for uniqueness and comparative statics properties of the equilibrium of each model. 37 Appendix A Proof of Proposition 1. First order conditions for firm i =A;B are, [p i ] : q i (p i ) 2t [ i (p i ) +F i ] + 0 i (p i )s i (p i ;F i ;p j ;F j ) = 0; (1.14) [F i ] : s i (p i ;F i ;p j ;F j ) 1 2t [ i (p i ) +F i ] = 0: (1.15) From (1.15) for i =A;B, F i =t + ' i (p i ;p j ) 3 2 i (p i ) 3 j (p j ) 3 ; (1.16) where ' i (p i ;p j )v i (p i )v j (p j ). Using (1.16) in (1.14), 0 i (p i ) 2 q i (p i ) 2 t + ' i (p i ;p j ) 3 + i (p i ) 3 j (p j ) 3 = 0 (1.17) We claim that t + ' i (p i ;p j ) 3 + i (p i ) 3 j (p j ) 3 > 0: (1.18) First, note that if p i c, 0<F i =t + ' i (p i ;p j ) 3 2 i (p i ) 3 j (p j ) 3 t + ' i (p i ;p j ) 3 + i (p i ) 3 j (p j ) 3 If p i <c and both firms have strictly positive profits F i + i (p i )> 0 ) t + ' i (p i ;p j ) 3 + i (p i ) 3 j (p j ) 3 > 0: Thus we conclude that (1.18) holds. Then it follows from (1.17) that 0 i (p i )q i (p i ) = 0 which implies that, p i =c i Similarly, note that if p i 6= c i left side of (1.17) implies that i (p i ) +F i = 0 for all 38 i2fA;Bg. However, by full market coverage assumption (see footnote 19) both firms have a profitable deviation by setting marginal prices equal to marginal costs and the fixed fee equal to (1.16). Thus, marginal-cost pricing is a unique equilibrium. Sufficient conditions. The Hessian is defined by, 42 H q 0 i (c i ) 2t [F i ] q i (c i ) 2 t + 00 i (c i )s i c i ;F i ;c j ;F j q i (p i ) t q i (p i ) t 1 t ! : Then, H 11 = q i (c i ) 2 t + q 0 i (c i ) 2t 2 t + ' i (c i ;c j ) 3 < 0 and, jHj = q 0 i (c i ) 2t 2 t + ' i (c i ;c j ) 3 > 0: Thus we conclude that D 2 s i c i ;F i ;c j ;F j [ i (c i ) +F i ] is negative definite. Proof of Proposition 2. We first show the necessary and then the sufficient condition. (i) Let’s show first the “if” part. Assume marginal cost pricing is an equilibrium. Then, from (1.3) and (1.4) for firm i2fA;Bg marginal cost pricing is an equilibrium if, E [q i (c i ;)]E [' i (c i ;c j ;)] +E [q i (c i ;)' i (c i ;c j ;)] = 0; where' i (p i ;p j ;)v i (p i ;)v j (p j ;), forj6=i, which implies that (1.5) holds forp i =c i . Let’s prove the other direction. Suppose that (1.5) holds in equilibrium. Note that, if firm j6=i uses marginal-cost pricing, then from (1.3), (1.4) and (1.5) it follows that marginal-cost pricing is also an equilibria for firm i, for i;j2fA;Bg. Then it follows that marginal-cost pricing is an equilibrium. (ii) From (1.3) and (1.4) for firm i2fA;Bg F i =t + E [' i (p i ;p j ;)] 3 E [ j (p i ;)] 3 2E [ i (p i ;)] 3 ; (1.19) and 42 We exclude sufficient conditions for the rest part of the paper. 39 F i F j = 2E [' i (p i ;p j ;)] 3 + E [ j (p j ;)] 3 E [ i (p i ;)] 3 ; (1.20) fori6=j. Using (1.19) and (1.20) in (1.3) and the fact that (1.5) holds for i;j2fA;Bg and i6=j, Var [q (p i ;)] (p i c i ) + (p i c i ) (1.21) E q 0 i (p i ;) t + ' i (p i ;p j ;) 3 E [ j (p i ;)] 3 + E [ i (p i ;)] 3 = 0: Now, note that if (1.5) holds for any p i ;p j 2P then, E [q i (p i ;)]E [' i (p i ;p j ;)] +E [q i (p i ;)' i (p i ;p j ;)] = 0: (1.22) If we take derivative to both sides of (1.22) with respect to p i and multiply by p i c i 0 =E [q i (p i ;) i (p i ;)] +E [q i (p i ;)]E [ i (p i ;)] (1.23) +E [q 0 i (p i ;) (p i c i )' i (p i ;p j ;)]E [q 0 i (p i ;) (p i c i )]E [' i (p i ;p j ;)]: Using (1.23) in (1.21) E [q 0 i (p i ;) (p i c i )] t + TS i (p i ) 3 TS j (p j ) 3 = 0; (1.24) where TS i (p i )E [v i (p i ;)] +E [ i (p i ;)]. If both firms have strictly positive profits, the second term of the left side of (1.24) is strictly positive, then p i = c i for i2fA;Bg. Similarly, note that if p i 6= c i left-hand side of (1.24) implies that E [ i (p i ;)] +F i = 0 for i2fA;Bg. However, under full market coverage, both firms have a profitable deviation by setting marginal prices equal to marginal costs and the fixed fee equal to (1.19). Thus, marginal-cost pricing is a unique equilibrium. Proof of Corollary 1. Follows directly from Proposition 2, that is, ' i (c;c;) = 08. Proof of Corollary 2. Follows directly from Proposition 2. Proof of Proposition 3. (i) We first show that for every p A , p B 2 ^ P that satisfy (1.9) for i = A, it has to be true that p A < p B . Second we show that for p B c B and for every 40 p A ;p B 2 ^ P that satisfy (1.9) for i = B, we have that p B < p A . Thus we conclude that no equilibrium exists for values of p B c B . Finally we show that the two curves defined by (1.9) for i =A;B cross each other at least once in the set [c A ;c B ] 2 . Let’s first show that for every p A , p B 2 ^ P that satisfy (1.9) for i = A, it has to be true that p A p B , E [q (p A ;) (v A v B )] 2t < 0 since is associated. So, we get a contradiction. Note that we also get a contradiction if p A =p B . Thus we conclude that if p A , p B 2 ^ P satisfy (1.9) for i =A, then it has to be true thatp A <p B . Similarly, we can show that forp B c B and for everyp A ;p B 2 ^ P that satisfy (1.9) for i = B, we have that p B < p A . Thus we conclude that no equilibrium exists for values of p B c B . Now lets prove existence. Note that asp A !c A in (1.9) fori =A, we have thatp B !c A and as p A ! c B , from previous paragraph, we know that p B ! A > c B . Similarly, from (1.9) for i =B, note that as p A !c A , E [q (p B ;) (v B v A )] 2t + (p B c B ) E [q 0 (p B ;)]s B Var [q (p B ;)] 2t | {z } >0 = 0 we have thatp B ! B >c A since the first term of the right-hand side must be negative, and asp A !c B ,p B !c B . Thus, the necessary condition for existence is proved (See Figure 1). (ii) Follows from substituting F i for i2fA;Bg in the expected market share and the fact that in any pure-strategy Nash equilibrium, c A <p A <p B <c B . Proof of Lemma 1. Note that (A4) allows us to express (1.9) as a function of Var [h ()], i =h i (p i ) (p i c i ) t + TS i (p i ) 3 TS j (p j ) 3 + v (p i )v (p j ) i v (p i ) (1.25) 41 whereh i (p) = q 0 v (p) m 00 (p) i0 v (p) , i v (p) = qv (p) 2 (pc i ) i0 v (p) ,q v (p) @v(p) @p , i v (p)q v (p) (pc i ), and TS i (p i )v (p i ) + i v (p), for i2fA;Bg and j6=i. (i) @R A (p A ) @p A > 0: In Proposition 4 we show that, @ A @p A > @ B @p A , and in part (ii) of this proposition we show that @ B @p A > 0, thus, we just need to show that @ A @p B > 0, where i is equal to (1.25) for i =A;B. Note that, @ A =@p B =q v (p B ) +h A (p A ) (p A c A ) TS 0 (p B ) 3 | {z } >0 > 0 since TS 0 (p B )> 0 for p B <c B . Then from the implicit function theorem we conclude that @R A (p A ) @p A > 0 for (p A p B )2 [c A ;c B ] 2 . (ii) @R B (p A ) @p A > 0: In Proposition 4 we show that @ A @p B < @ B @p B . In part (i) we showed that @ A @p B > 0. Thus we just need to show that @ B @p A > 0 for p B <c B , @ B =@p A =q v (p A ) +h B (p B ) (p B c B ) 2 TS 0 (p A ) 3 > 0 since h B (p B ) (p B c B ) < 1, for p B < c B . Then from the implicit function theorem we conclude that @R B (p A ) @p A > 0 for (p A p B )2 [c A ;c B ] 2 . Proof of Proposition 4. Let’s show that in equilibrium @R A (p A ) @p A > @R B (p A ) @p A , i.e., @ A =@p A @ A =@p B > @ B =@p A @ B =@p B We first show that @ A =@p A > @ B =@p A . Next we show that, @ B =@p B > @ A =@p B . (i) @ A =@p A > @ B =@p A . We need to show that, q v (p A )+h B (p B ) (p B c B ) 2 TS 0 (p A ) 3 < (q v (p A ) + 0 v (p A ))+ @ @p A h A (p A ) (p A c A ) A +h A (p A ) (p A c A ) 2 TS 0 (p A ) 3 whereA n t + TS(p A ) 3 TS(p B ) 3 o . Note thath B (p B ) (p B c B )< 1 and thatTS 0 (p A ) = q 0 v (p A ) (p A c A ) thus we show that, 0<h B (p B ) (p B c B ) 2 TS 0 (p A ) 3 + @ @p A h A (p A ) (p A c A ) A + 0 v (p A ) 42 +h A (p A ) (p A c A ) 2 TS 0 (p A ) 3 Thus, since TS 0 (p A )< 0 is enough if we show that, 0< 2 q 0 v (p A ) (p A c A ) 3 + 0 v (p A )+ @ @p A h A (p A ) (p A c A ) A+h A (p A ) (p A c A ) 2 TS 0 (p A ) 3 which is equal to, 0< 2 q v (p A ) 3 q 0 v (p A ) (p A c A ) 0 (p A ) + 0 v (p A ) + @ @p A h A (p A ) (p A c A ) A Note that, we know that q 0 (p A ) (p A c A )>q (p A ), thus is enough if we show that, 2 q v (p A ) 2 3 0 v (p A ) + 0 v (p A )> 0 which is true (See Lemma A1). (ii) @ B =@p B > @ A =@p B . Here, we show that, q v (p B )+h A (p A ) (p A c A ) 2 TS 0 (p B ) 3 <q v (p B )+ 0 v (p B )+ @ @p B h B (p B ) (p B c B ) B +h B (p B ) (p B c B ) 2 TS 0 (p B ) 3 where B n t + TS(p B ) 3 TS(p A ) 3 o . Note that TS 0 (p B ) = q 0 (p B ) (p B c B ). Thus if we restrict to the set of (c A ;c B ) such that q 0 (p A )(p A c A ) 0 (p A ) < 1, for p A 2 (c A ;c B ), 0< 2 q 0 (p B ) (p B c B ) + B0 (p B ) + @ @p B h B (p B ) (p B c B ) B which is positive. Lemma A1. A0 v (p A ) A v (p A )E [q v (p A ;)] > 0 where A (p A ) E[q A (p A ;)] E[ 0A (p A ;)] . Proof. Note first that, A0 (p A )E [q A (p A ;)] (p A ) = (1.26) 43 E A0 (p A ;)q (p A ;) E A0 (p A ;) +E A (p A ;)q 0 (p A ;) E A0 (p A ;) E [ A0 (p A ;)] 2 E A (p A ;)q (p A ;) E A00 (p A ;) E [ A0 (p A ;)] 2 E [q A (p A ;)] 2 E [ A0 (p A ;)] Substituting by, E A0 (p A ;) =E [q (p A ;) +q 0 (p A ;) (p A c A )] and, E A00 (p A ;) =E [2q 0 (p A ;) +q 00 (p A ;) (p A c A )] in (1.26), A0 (p A )E [q A (p A ;)] (p A ) = 8 < : 2E [q (p A ;)q 0 (p A ;)] E [ A0 (p A ;)] 2 E [q (p A ;)] (p A c A ) E [2q 0 (p A ;)]E h q (p A ;) 2 i E [ A0 (p A ;)] 2 (p A c A ) 9 = ; | {z } 0 + 8 < : E h q (p A ;) 2 i E [ A0 (p A ;)] E [q A (p A ;)] 2 E [ A0 (p A ;)] 9 = ; | {z } >0 8 < : 2E [q (p A ;)q 0 (p A ;)] E [ A0 (p A ;)] 2 E [q 0 (p A ;)] (p A c A ) 2 E [q 00 (p A ;)]E h q (p A ;) 2 i E [ A0 (p A ;)] 2 (p A c A ) 2 9 = ; | {z } B Using (A4), B = E h h () 2 i E [h ()] E [ 0 (p A ;)] 2 (p A c A ) 2 q (p A )q 0 (p A ) 2 2 q 00 (p A )q (p A ) q 0 (p A ) > 0 which follows by (A2). Proof of Corollary 3. From Proposition 4 we know that asp A !c A in (1.9) fori =A, we have that p B !c A and as p A !c B , p B ! A >c B . Similarly, from (1.9) for i =B, as p B ! c B we have that p A ! c B while as p A ! c A , p B ! B > c A . As c A ;c B tends to c, A and B also tends toc. Thus the two best response function intersect each other only at p i =c i . Proof of Proposition 5. (i) We first show that no solution exist for p B >c; we show that for everyp A ;p B 2 ~ P that satisfy (1.11) for i =A, it must be true that ' A (p A ;p B ;)> 44 0; 82 . Similarly, we show that for every p A ;p B 2 ~ P that satisfy (1.11) for i =B, and for p B >c, ' A (p A ;p B ;)< 0;82 . Let’s first show that for everyp A ;p B 2 ~ P that satisfy (1.11) fori =A,' A (p A ;p B ;)> 0 82 . Supose not, i.e. ' A (p A ;p B ;) 0. If ' A (p A ;p B ;) < 0 for any2 , then by separability, h() (v (p A )v (p B ))< 0 82 Thus, ~ f A (p A ;p B )E h q (p A ;) ~ v A ~ v B i | {z } <0 + (1.27) (p A c A )f2tE [s A (p A ;F A ;p B ;F B ;)q 0 (p A ;)] Var [q (p A ;)]g | {z } <0 = 0 which is a contradiction. Similarly, if ' A (p A ;p B ;) = 0. Thus we conclude that for every p A ;p B 2 ~ P thatsatisfy(1.11)forA,itistruethat (p A ;p B )2fp A ;p B j ' A (p A ;p B ;)> 082 g. Now let’s show that for every p A ;p B 2 ~ P that satisfy (1.11) for i = B, it must be true that ' B (p B ;p A ;)> 082 for p B >c. Suppose not, i.e. ' B (p B ;p A ;) 0. Then from (1.11) for i =B, ~ f B (p A ;p B )E h q (p B ;) ~ v B ~ v A i | {z } <0 + (1.28) (p B c B ) 2t f2tE [s B (p B ;F B ;p A ;F A ;)q 0 (p B ;)]Var [q (p B ;)]g | {z } <0 = 0 which is a contradiction. Similarly, if ' B (p B ;p A ;) = 0 we also get a contradiction. Thus we conclude that for every p A ;p B 2 ~ P that satisfy (1.11) for i = B, for p B > c it is true that (p A ;p B )2fp A ;p B j ' B (p B ;p A ;)> 0;82 g. Thus we conclude that no equilibrium exists forp B >c, which implies that if there exist any equilibrium of the game it must be in the set ~ (p A ;p B ) where ~ (p A ;p B ) n p A ;p B 2 ~ Pj(p A ;p B )2 [c; ~ A ] [~ B ;c] o , where ~ A is such that (~ A ;c) satisfy (1.11) for i =B. (ii) Next, we that for (p A ;p B )2 ~ (p A ;p B ) the slope of the implicit functions defined by (1.11) for i = A;B, ~ R i (p) : ~ P! ~ P where ~ R i (~ p i A ) = ~ p i B are such that ~ p i A and ~ p i B satisfy (1.11) for i =A;B, are both positives. (iia) @ ~ R A (p A ) @p A > 0: In (iiib) we show that, @ ~ f A @p A > @ ~ f B @p A , and in part (iib) of this proposition we show that @ ~ f B @p A > 0, thus, we just need to show that @ ~ f A @p B > 0; taking derivative of (1.11) for i =A with respect to p B , 45 @ ~ f A @p B = ( (p A ) 1) E [q 0 (p B ;) (p B c)] 3 + E [ 0 (p A ;)q (p B ;)] E [ 0 (p A ;)] E [q (p B ;)]E [ 0 (p A ;)] E [ 0 (p A ;)] | {z } 0 > 0 The last inequality always holds for p B c andp A c. Then from the implicit function theorem we conclude that @ ~ R A (p A ) @p A is positive for for (p A ;p B )2 ~ (p A ;p B ). (iib) @ ~ R B (p A ) @p A > 0: In (iiib) we show that @ ~ f A @p B < @ ~ f B @p B . In part (iia) of this proposition we showed that @ ~ f A @p B > 0. Thus we just need to show that @ ~ f B @p A > 0; taking derivatives of (1.11) for i =B with respect to p A , @ ~ f B @p A = ( (p B ) 1) E [q 0 (p A ;) (p A c A )] 3 + E [ 0 (p B ;)q (p A ;)] E [ 0 (p B ;)] E [q (p A ;)]> 0 The last inequality always holds for p B c andp A c. Then from the implicit function theorem we conclude that @ ~ R B (p A ) @p A is positive for (p A ;p B )2 ~ (p A ;p B ). (iii) Finally, we prove existence first, and then uniqueness. (iiia) Existence: Note that as p A !c in (1.11) for i =A, ~ f A (c;p B ) =E [' (c;p B ;)] + E [q (c;)' (c;p B ;)] E [q (c;)] (1.29) is not true thatp B !c, since' (c;c;)> 0 for all2 , where' (p A ;p B ;)v A (p A ;) v B (p B ;). Thus, from part (ii), 43 we conclude that p B ! ~ B < c, as p A ! c, where ~ B is such that, E [' (c; ~ B ;)] = E [q (c;)' (c; ~ B ;)] E [q (c;)] Similarly, as p B !c in (1.11) for i =B, ~ f B (p A ;c) =E [' (p A ;c;)] E [q (c;)' (p A ;c;)] E [q (c;)] (1.30) from part (ii) we conclude that p A ! ~ A >c. 44 Likewise, as p A !c in (1.11) for i =B, we have, ~ f B (c;p B ) ( (p B ) 1) t E [' (c;p B ;)] 3 2E [ (p B ;)] 3 +E [' (c;p B ;)] +E [ (p B ;)] E [ 0 (p B ;)' (c;p B ;)] E [ 0 (p B ;)] (p B ) notice that if p B =c, 43 Note that f A (c;p B )> 0 if p B =c. 44 Note that f B (p A ;c)< 0 if p A =c. 46 ~ f B (c;c)E [' (c;c;)] E [ 0 (c;)' (c;c;)] E [ 0 (c;)] < 0 so from part (ii) we know that asp A !c in (1.11) fori =B,p B ! ~ B <c. Is easy to show that ~ B > ~ B . Suppose not, i.e. ~ B ~ B , then ~ f B (c; ~ B ) ( (~ B ) 1) t E [' (c; ~ B ;)] 3 2E [ (~ B ;)] 3 +E [' (c; ~ B ;)] +E [ (~ B ;)] E [ 0 (~ B ;)' (c; ~ B ;)] E [ 0 (~ B ;)] (~ B ) But from (1.29) ~ B is such that, E [' (c; ~ B ;)] = E [q (c;)' (c; ~ B ;)] E [q (c;)] (1.31) Thus, ~ f B (c; ~ B )> (1 (~ B )) t E [' (c; ~ B ;)] 3 2E [ (~ B ;)] 3 +E [ (~ B ;)] (~ B )> 0 since ~ B <c, (~ B )< 0 so from part (ii) we conclude that ~ B > ~ B . Finally, note that as p B !c in (1.11) for i =A p A ! ~ A >c, since, ~ f A (c;c)E [' (c;c;)] + E [ 0 (c;)' (c;c;)] E [ 0 (c;)] > 0 where ~ A is such that ~ f A (~ A ;c) = 0, i.e., ~ f A (~ A ;c) = ( (~ A ) 1) t + E [' (~ A ;c;)] 3 2E [ (~ A ;)] 3 E [' (~ A ;c;)] +E [ (~ A ;)] + E [ 0 (~ A ;)' (~ A ;c;)] E [ 0 (~ A ;)] (~ A ) = 0 (1.32) Note that as p A ! ~ A in (??) for i =B, and p B =c, ~ f B (~ A ;c) =E [' (~ A ;c;)] E [q (c;)' (~ A ;c;)] E [q (c;)] (1.33) From (1.32), E [' (~ A ;c;)] = ( (~ A ) 1) t + E [' (~ A ;c;)] 3 2E [ (~ A ;)] 3 (1.34) +E [ (~ A ;)] + E [ 0 (~ A ;)' (~ A ;c;)] E [ 0 (~ A ;)] (~ A ) substituting (1.34) in (1.33), 47 ~ f B (~ A ;c) = ( (~ A ) 1) t + E [' (~ A ;c;)] 3 2E [ (~ A ;)] 3 +E [ (~ A ;)] (~ A ) + E [ 0 (~ A ;)' (~ A ;c;)] E [ 0 (~ A ;)] E [ 0 (c;)' (~ A ;c;)] E [ 0 (c;)] < 0 From part (ii) we conclude that as p A ! ~ A in (1.11) for i = B p B ! ! B < c. Thus, both curves cross each other at least once in the set ~ (p A ;p B ). (iiib) Uniqueness: To prove uniqueness we need to show that in equilibrium the slope of the implicit function defined by (1.11) for i = A is greater than the slope defined for i =B, i.e., @ ~ R A (p A ) @p A j p A =p A > @ ~ R B (p A ) @p A j p A =p A . From (ii) we know that, @ ~ R A (p A ) @p A = N(p A ;p B ;;t) ( (p A ) 1) n E[q(p B ;)] 3 E[ 0 (p B ;)] 3 o E [q (p B ;)] + E[ 0 (p A ;)q(p B ;)] E[ 0 (p A ;)] (1.35) where, N(p A ;p B ;;t) 0 (p A ) t + E [' (p A ;p B ;)] 3 E [ (p B ;)] 3 2E [ (p A ;)] 3 + ( (p A ) 1) E [q (p A ;)] 3 2E [ 0 (p A ;)] 3 E [q (p A ;)]E [ 0 (p A ;)]+ E [ 0 (p A ;)q (p A ;)] E [ 0 (p A ;)] + 0 (p A ) Similarly, we have, @ ~ R B (p A ) @p A = ( (p B ) 1) E[q(p A ;)] 3 E[ 0 (p A ;)] 3 E [q (p A ;)] + E[ 0 (p B ;)q(p A ;)] E[ 0 (p B ;)] D(p A ;p B ;;t) (1.36) where, D(p A ;p B ;;t) 0 (p B ) t E [' (p A ;p B ;)] 3 2E [ (p B ;)] 3 E [ (p A ;)] 3 + ( (p B ) 1) E [q (p B ;)] 3 2E [ 0 (p B ;)] 3 E [q (p B ;)]E [ 0 (p B ;)] + E [ 0 (p B ;)q (p B ;)] E [ 0 (p B ;)] + (p B ) Let’s analyze first both denominators which both are positives. In particular, we can show that the denominator of (1.36), D(p A ;p B ;;t), is greater than the denominator of (1.35). That is, 48 ( (p A ) 1) E [q (p B ;)] 3 E [ 0 (p B ;)] 3 E [q (p B ;)] + E [ 0 (p A ;)q (p B ;)] E [ 0 (p A ;)] < 0 (p B ) t E [' (p A ;p B ;)] 3 2E [ (p B ;)] 3 E [ (p A ;)] 3 + ( (p B ) 1) E [q (p B ;)] 3 2E [ 0 (p B ;)] 3 E [q (p B ;)]E [ 0 (p B ;)] + E [ 0 (p B ;)q (p B ;)] E [ 0 (p B ;)] + (p B ) Using (A5ii), 0< 0 (p B ) t E [' (p A ;p B ;)] 3 2E [ (p B ;)] 3 E [ (p A ;)] 3 | {z } F B 0 + (1 (p B ))E [q (p B ;)] + 1 2 (p B ) 3 (p A ) 3 E [q 0 (p B ;)] (p B c) | {z } 0 (p B )E [ 0 (p B ;)] | {z } 0 which is true for 2 ~ Ofj (~ B )>E [ 0 (~ B ;)]g. Now lets analyze both numerators. In particular, let’s show that the numerator of (1.35) is greater than the numerator of (1.36). That is let’s show that, ( (p B ) 1) E [q (p A ;)] 3 E [ 0 (p A ;)] 3 E [q (p A ;)] + E [ 0 (p B ;)q (p A ;)] E [ 0 (p B ;)] < 0 (p A ) t + E [' (p A ;p B ;)] 3 E [ (p B ;)] 3 2E [ (p A ;)] 3 + ( (p A ) 1) E [q (p A ;)] 3 2E [ 0 (p A ;)] 3 E [q (p A ;)]E [ 0 (p A ;)]+ E [ 0 (p A ;)q (p A ;)] E [ 0 (p A ;)] + 0 (p A ) From (A5ii), 0< 0 (p A ) t + E [' (p A ;p B ;)] 3 E [ (p B ;)] 3 2E [ (p A ;)] 3 | {z } F A >0 + (p A ) 2E [q 0 (p A ;) (p A c A )] 3 E [q 0 (p A ;) (p A c A )] 3 +# (p B ) E [q 0 (p A ;) (p A c A )] 3 | {z } >0 49 + 0 (p A ) (p A )E [q (p A ;)] where# (p B ) (1 (p B ))< 1. Theprooftoshowthatf 0 (p A ) (p A )E [q (p A ;)]g 0 is very similar to the proof presented at the end of Proposition 4, so we omit it. Proof Proposition 6. We first show the sufficient condition and then the necessary condition. (i) (Sufficient condition) The first order conditions for firm i2fA;Bg are, [p i ] E s (v i (p i ;)F i ;v j (p j ;)F j ) i0 (p i ;) (1.37) s 1 (v i (p i ;)F i ;v j (p j ;)F j )q (p i ;) i (p i ;) +F i = 0 [F i ] Efs (v i (p i ;)F i ;v j (p j ;)F j ) (1.38) s 1 (v i (p i ;)F i ;v j (p j ;)F j ) i (p i ;) +F i = 0 Suppose that condition (1.13) is satisfied for p i =c i . From (1.38), marginal-cost pricing is an equilibrium if, 1 F i = v i (c i ;)F i ;v j (c j ;)F j where (u i ();u j ()) E[s 1 (u i ();u j ())] E[s(u i ();u j ())] and from (1.37), 1 F i = E s 1 v i (c i ;)F i ;v j (c j ;)F j q (c i ;) E s v i (c i ;)F i ;v j (c j ;)F j q i (c i ;) That is, E s 1 v i (c i ;)F i ;v j (c j ;)F j q (c i ;) E s v i (c i ;)F i ;v j (c j ;)F j q i (c i ;) = E s 1 v i (c i ;)F i ;v j (c j ;)F j E s v i (c i ;)F i ;v j (c j ;)F j which is true if (1.13) is satisfied. We can prove this last statement subtracting both sides by E s 1 v i (c i ;)F i ;v j (c j ;)F j E [q (c i ;)]. (Necessary conditions) Suppose that marginal-cost pricing is an equilibrium. From (1.37) and (1.38), E s v i (c i ;)F i ;v j (c j ;)F j q (c i ;) (1.39) ( 1 E s 1 v i (c i ;)F i ;v j (c j ;)F j q (c i ;) E s v i (c i ;)F i ;v j (c j ;)F j q (c i ;) 1 v i (c i ;)F i ;v j (c j ;)F j ) = 0 50 thus, the second term of the left side of (2.68) is equal to 1, which implies that condition (1.13) is satisfied for p i =c i for i2fA;Bg. Proof of Proposition 7. For completeness of the proof we first prove that marginal cost based-2PT is not an equilibrium, i.e., condition (1.13) is not satisfied. Suppose not, i.e., p i =p j =c and F i =F j =F . Then, note that in equilibrium, s (v ();v ()) e v() 2e v() + 1 ; s 1 (v ();v ())s (v ();v ()) (1s (v ();v ())) where v () =v (c;)F . Note that, ()s (v ();v ()) 1 E [s (v ();v ())] (1s (v ();v ())) E [s 1 (v ();v ())] is monotonic increasing with respect to . Thus, if is associated, (1.13) is not satisfied, since q (c;) is also monotonic increasing with respect to. We now show that in any symmetric equilibrium, p > c. Note that the first order conditions are, E [s (u i ();u j ()) 0 (p i ;)s 1 (u i ();u j ())q (p i ;) ( (p i ;) +F i )] = 0 (1.40) E [s (u i ();u j ())s 1 (u i ();u j ()) ( (p i ;) +F i )] = 0 (1.41) where u i ()v (p i ;)F i for i;j2fA;Bg and j6=i. In a symmetric equilibrium, F = E [s (u ();u ())] E [s 1 (u ();u ())] E [s 1 (u ();u ())q (p ;)] E [s 1 (u ();u ())] (p c) (1.42) where u () =v (p ;)F . Let h ()s (u ();u ()) and h 1 ()s 1 (u ();u ()), thus from (1.40) and (1.42), 8 > > < > > : E [h ()q 0 (p ;)] | {z } <0 E h 1 () q (p ;) 2 + E [h 1 ()q (p ;)] 2 E [h 1 ()] | {z } G 9 > > = > > ; (p c) +E [h ()q (p ;)]E [h 1 ()q (p ;)] E [h ()] E [h 1 ()] = 0 (1.43) Note that h 1 () = h () (1h ()), and that the last two terms of (1.43) are positive 51 i.e, E q (p ;) h () E [h ()] h () (1h ()) E [h () (1h ())] > 0 if, Cov q (p ;); h () E [h ()] h () (1h ()) E [h () (1h ())] > 0 which holds if is associated. Similarly, note that the first big bracket of (1.43) is negative i.e., G< 0 since, E s 1 (u ();u ()) q (p ;) 2 + E [s 1 (u ();u ())q (p ;)] 2 E [s 1 (u ();u ())] = E [s 1 (u ();u ())] E k () q (p ;) 2 (E [k ()q (p ;)]) 2 where k () s 1 (u ();u ()) E[s 1 (u ();u ())] . Let m () = g ()k () where g () is the density function of . Note that m () is a density function since m ()d = 1, thus, E k () q (p ;) 2 (E [k ()q (p ;)]) 2 = m () q (p ;) 2 d () m ()q (p ;)d () 2 > 0 by Lemma 2.2.1 in (Tong 1980). This last inequality was originally due to Chebyshev, know as Chebyshev’s other inequality, or Kimball’s inequality. Thus in equilibrium p >c. Proof of Corollary 5. (i) The problem of each firm is given by, max pi;Fi s (u i (p i ;F i );u j (p j ;F j )) i (p i ) +F i where the utility offered by firmi =A;B isu i (p i ;F i ) =v i (p i )F i . The first order conditions are, [p i ] q (p i ) i (p i ) +F i i0 (p i ) = s (u i (p i ;F i );u j (p j ;F j )) s 1 (u i (p i ;F i );u j ) (1.44) [F i ] i (p i ) +F i = s (u i (p i ;F i );u j (p j ;F j )) s 1 (u i (p i ;F i );u j (p j ;F j )) (1.45) Note that from (1.44) and (1.45) it follows thatp i =c i while the fixed fee,F i , in equilibrium is such that, 52 F i = s (u i ;u j ) s 1 (u i ;u j ) (1.46) To show that the equilibrium is unique, note that if p i 6=c i , from (1.44) and (1.45), q 0 (p i ) (p i c i ) = 0 which is a contradiction, given that F i > 0. Thus we conclude that p i = c i is a unique equilibrium for F i > 0. If F i = 0, there is always a profitable deviation to F i > 0. Existence. Now let’s show that there exists a unique F i that satisfy (1.46) for i2 fA;Bg. For the proof we use the dual approach, i.e., we show that there exists a unique u i that satisfy v i (c i )u i = s u i ;u j s 1 u i ;u j (1.47) for i2fA;Bg. First, note that from the first order conditions of firm j; by the implicit function theorem there exists a function R j (u i ) : ^ U! ^ U, where R i (~ u i ) = ~ u j is such that ~ u i and ~ u j satisfy (1.47) for j6=i. Moreover, note that, R j (u i ) @u i = 2 (R j (u i );u i ) 1 + 1 (R j (u i );u i ) > 0 where (u i ;u j ) s(u i ;u j ) s 1 (u i ;u j ) . Note that 2 (u i ;u j ) < 0, 1 (u i ;u j ) > 0 for u i ;u j 2U, and as u i ! 0,R j (u i )!> 0 since, foru i = 0,R j (0) is implicitly defined by the value of u j that satisfy, v j (c j )u j = u j ; 0 note that the right-hand side (RHS) is increasing w.r.t. u j , and asu j ! 0, RHS! (0; 0)> 0. Similarly, note that the left-hand side (LHS) is decreasing w.r.t. u j , and as u j ! 0, the LHS!v j (c j )> (0; 0) (by A5), and as u j !v j (c j ), the LHS! 0. Thus, as u i ! 0, there is a unique value R j (u i ) that satisfies, v j (c j )R j (u i ) = R j (u i );u i From the first order conditions of firm i, u i is defined by, ~ R (u i )s 1 u i ;R j (u i ) u i ;R j (u i ) (v i (c i )u i ) = 0 Note that as u i ! 0, ~ R (u i )< 0, since by continuity, and for u i = 0, 53 ~ R (0) =s 1 0;R j (0) 0;R j (0) v i (c i ) and, v i (c i )> (0; 0)> 0;R j (0) Similarly, note that as u i ! v i (c i ), ~ R (u i )! s (v i (c i );R j (v i (c i ))) > 0. Thus existence follows. Uniqueness (with outside option). Here we onlyproof uniqueness since existencefollows from Proposition 6. Note that, @ ~ R (u i ) @u i u i =u i =s 1 u i ;R j (u i ) 2 4 @ (u i ;R j (u i )) @u i | {z } +1 3 5 > 0 sinceR j (u i ) is increasing w.r.t. u i . Thus the solution is unique, since, asu i ! 0, ~ R (u i )< 0, and as u i !v i (c i ), ~ R (u i )!s (v i (c i );R j (v i (c i )))> 0. Sufficient conditions. Second order conditions are, evaluated at p i =c i and F i =F i , [p i ] 2 : q (c i )F i s u i (c i ;F i );u j c j ;F j 8 > > > < > > > : 0 (c i ) F 2 i + q 0 (c i ) q (c i )F i + @~ (u i ;u j ) @u i u i =u i u j =u j q (c i ) 9 > > > = > > > ; < 0 where ~ (u i ;u j ) = (u i ;u j ) 1 , [F i ] 2 : s u i (c i ;F i );u j c j ;F j F i 8 > > > < > > > : 1 F 2 i + @~ (u i ;u j ) @u i u i =u i u j =u j 9 > > > = > > > ; < 0 and, [F i ] [p i ] s u i (c i ;F i );u j c j ;F j F i 8 > > > < > > > : q (c i ) F 2 i + @ (u i ;u j ) @u i u i =u i u j =u j q (c i ) 9 > > > = > > > ; Note that, D 2 s (u i ;u j ) i (p i ) +F i u i =u i u j =u j = 54 F i s u i (c i ;F i );u j c j ;F j 2 8 > > > < > > > : q (c i ) F 2 i + @~ (u i ;u j ) @u i u i =u i u j =u j q (c i ) 9 > > > = > > > ; q 0 (c i ) q (c i ) > 0 Thus we conclude that D 2 fs (u i ;u j ) [ i (p i ) +F i ]g at p i = c i and F i = F i is negative definite. ProofofProposition8. Forcompletenessoftheproofwefirstshowthatmarginalcost- based 2PT is not an equilibrium. Next, we show that in any equilibrium,c A <p A <p B <c B . Finally, we show existence The first order conditions are, [p i ] X k2L;H k s (v (p i ; k )F i ;v (p j ; k )F j ) 0 i (p i ; k ) (1.48) X k2L;H k s 1 (v (p i ; k )F i ;v (p j ; k )F j )q (p i ; k ) ( i (p i ; k ) +F i ) = 0 [F i ] X k2L;H k s (v (p i ; k )F i ;v (p j ; k )F j ) (1.49) X k2L;H k s 1 (v (p i ; k )F i ;v (p j ; k )F j ) ( i (p i ; k ) +F i ) = 0 Let, s (u (p i ;F i );u (p j ;F j ) ;) " L s (u (p i ;F i ; L );u (p j ;F j ; L )) H s (u (p i ;F i ; H );u (p j ;F j ; H )) # where u (p i ;F i ; k )v (c i ; k )F i , and similarly for s 1 (u (p i ;F i );u (p j ;F j ) ;). Let also, q (p i ;) = " q (p i ; L ) q (p i ; H ) # and similarly for q 0 (p i ;) and q (p i ;) 2 . Finally, let, (u (c i ;F i );u (c j ;F j ) ; H ) (u (c i ;F i ; H );u (c j ;F j ; H )) (i) Here, we show that marginal cost-based 2PT is not an equilibrium. Suppose not, i.e., suppose cost-based 2PT is an equilibrium. Then, from (1.48), 55 F i = s (u (c i ;F i );u (c j ;F j ) ;) 0 q (c i ;) s 1 (u (c i ;F i );u (c j ;F j ) ;) 0 q (c i ;) (1.50) Note that from (1.49) and (1.50), (1) s 1 (u (c i ;F i ; L );u (c j ;F j ; L ))s 1 (u (c i ;F i ; H );u (c j ;F j ; H )) A (q (c i ; H )q (c i ; L )) f (u (c i ;F i );u (c j ;F j ) ; H ) (u (c i ;F i );u (c j ;F j ) ; L )g = 0 where, A s 1 (u (c i ;F i );u (c j ;F j ) ;) 0 q (c i ;) which is a contradiction since, (u (c i ;F i );u (c j ;F j ) ; H )> (u (c i ;F i );u (c j ;F j ) ; L ) for c i < c j , since s() s 1 () is increasing and, by increasing differences property v (c i ; H )F i v (c j ; H ) +F j >v (c i ; L )F i v (c j ; L ) +F j . (ii) Note that from (1.48) and (1.49), in equilibrium we have, (p i c i ) s (u i ;u j ;) 0 q 0 (p i ;) s 1 (u i ;u j ;) 0 q (p i ;) 2 (1.51) +s 1 (u i ;u j ;) 0 q (p i ;) s 1 (u i ;u j ;) 0 q (p i ;) s 1 (u i ;u j ;) 0 1 [2;1] + + s (u i ;u j ;) 0 q (p i ;) s 1 (u i ;u j ;) 0 q (p i ;) s (u i ;u j ;) 0 1 [2;1] s 1 (u i ;u j ;) 0 1 [2;1] where u i = u (p i ;F i ) and u j = u (p j ;F j ). We first show that the expression inside the first big bracket is negative. Thus, for p i >c i , the expression in the second big bracket must be positive. Now let’s analyze the first big bracket (p i c i ) 8 < : s (u i ;u j ;) 0 q 0 (p i ;) | {z } <0 s 1 (u i ;u j ;) 0 q (p i ;) 2 | {z } A 56 + s 1 (u i ;u j ;) 0 q (p i ;) | {z } B s 1 (u i ;u j ;) 0 q (p i ;) s 1 (u i ;u j ;) 0 1 [2;1] | {z } C 9 > > > = > > > ; + where, A +BC = D [q (p i ; H )q (p i ; L )] 2 < 0 where, D (1)s 1 (u (p i ;F i ; H );u (p j ;F j ; H ))s 1 (u (p i ;F i ; L );u (p j ;F j ; L )) s 1 (u i ;u j ;) 0 1 [2;1] > 0 Now we show that the second big bracket is positive only if p i <p j . That is, s (u i ;u j ;) 0 q (p i ;) s 1 (u i ;u j ;) 0 q (p i ;) s (u i ;u j ;) 0 1 [2;1] s 1 (u i ;u j ;) 0 1 [2;1] =D [q (p i ; H )q (p i ; L )] [ (u (p i ;F i );u (p j ;F j ) ; H ) (u (p i ;F i );u (p j ;F j ) ; L )] since (u i ;u j ) is increasing with respect u i and, by increasing differences property, p i c B . Moreover, in any equilibria in [c A ;c B ] [c A ;c B ], p A < p B . Using the analysis of the previous section, we know that the first order condition with respect to p i evaluated at the marginal cost of firm i, is positive for firm A and negative for B, for p A <p B . Thus, in any equilibria, c A <p A <p B <c B . (iii) By implicit function theorem, there are neighborhoods UR + and WP 2 of F i and p i ;p j on which (1.49) uniquely defines F i as a function of (p i ;p j ), for i2fA;Bg and j6=i. That is, there is a function i :W!U such that, ((p i ;p j ); i (p i ;p j )) satisfies (1.49) for i2fA;Bg. Thus (1.51) can be express as, (p i c i ) n s u c i ; i (p i ;p j ) ; j (p j ;p i ) ; 0 q 0 (p i ;)D [q (p i ; H )q (p i ; L )] 2 o (1.52) 57 +D [q (p i ; H )q (p i ; L )] u p i ; i (p i ;p j ) ;u p j ; j (p j ;p i ) ; H u p i ; i (p i ;p j ) ;u p j ; j (p j ;p i ) ; L = 0 Let, ~ D (1) s 1 (u (c i ; i (c i ;p j ); L );u (p j ; j (p j ;c i ); L ))s 1 (u (c i ; i (c i ;p j ); H );u (p j ; j (p j ;c i ); H )) L s 1 (u (c i ; i (c i ;p j ); L );u (p j ; j (p j ;c i ); L )) + H s 1 (u (c i ; i (c i ;p j ); H );u (p j ; j (p j ;c i ); H )) Note that as p i !c i , ~ D [q (p i ; H )q (p i ; L )] u c i ; i (c i ;p j ) ;u p j ; j (p j ;c i ) ; H u c i ; i (c i ;p j ) ;u p j ; j (p j ;c i ) ; L = 0 p j ! c j , due to increasing differences property and monotonicity of (u i ;u j ). Moreover, from (ii) we know that as p A !c B , in (1.48) for i =A,p B ! A >c B . Similarly, note that as p A !c A , in (1.48) for i =B, from (ii) we know that p B ! B >c A . Thus, we conclude that there exists a pure-strategy Nash equilibrium, with c A 0 for i;j2fA;Bg and j6=i. That is, we need to show that i p i ;p j > 0 for i2fA;Bg. From (1.49), F i 1 P k2L;H k s 1 v (p i ; k )F i ;v p j ; k F j (1.53) ( X k2L;H k s 1 v (p i ; k )F i ;v p j ; k F j v (p i ; k )F i ;v p j ; k F j i (p i ; k ) ) = 0 From (1.53), as F i ! 0, the left-hand side tends to, 58 1 P k2L;H k s 1 v (p i ; k );v p j ; k F j (1.54) ( X k2L;H k s 1 v (p i ; k );v p j ; k F j v (p i ; k );v p j ; k F j i (p i ; k ) ) = 0 Note that for F j = 0, LHS is negative under A7, which also implies that is negative for F j > 0 since (;) is increasing with respect to F j . As F i !v (c i ; L ), LHS tends to, X k2L;H k s 1 v (p i ; k )F i ;v p j ; k F j v (c i ; L ) + i (p i ; k ) v (p i ; k )v (c i ; L );v p j ; k F j = 0 note that for F j =v (c j ; L ), from A7, the left-hand side is positive, and is also positive for F j <v (c j ; L ), since v (c i ; L ) ( v (c i ;); v (c j ;)) + B (c j ; H )> 0. Proof of Proposition 9. (Sufficient Condition) The proof can be constructed by mod- ifying the proof of Proposition 6 in (Rochet and Stole 2002). Note that the Hamiltonian for firm i is characterized by (ignoring the monotonicity constraint), H (q i ;u i ;;) = 1 2 + u i ()u j () 2t (S i (q i ();)u i ())f () + ()q i () The necessary conditions are: () = = 0 and, 1 2 + u i ()u j () 2t (S i;qi (q i ();))f () = () 1 2 + u i ()u j () 2t f () + (S i (q i ();)u i ())f () 2t = _ () Note that the cost-based two part-tariffT i (q i ) =q (c i ;)c i +F i satisfies these constraints: in particular, note that this cost-based two part-tariff implies that () = 0 and that v i ()u i () =F i , which requires _ () = 0 for all. Thus, 1 2 + v i ()F i v j ()F j 2t = F i 2t this solution is feasible if v i ()v j () are constant with respect to. The proof can also be constructed following a similar strategy used in the proof of Propo- sition 5 in (Armstrong and Vickers 2001). Suppose firm B offers a cost-based two part-tariff (c B ;F B ). An upper bound on firm’s A profit is obtained by assuming that is observed. The optimal method to generates profits for firmA is by setting prices equal to the marginal cost and setting a fixed fee F A () such that, 59 max F A () 1 2 + v A ()F A () +v B () +F B 2t F A () Thus, from the first order conditions, 1 2 + v A ()F A () +v B () +F B 2t = F A () 2t ifv A ()v B () is constant with respect to, cost-based two part-tariff (c A ;F A ) are optimal for firm A. Note that provided that the market is fully covered F A does not depend on. (Necessary Condition) Suppose cost-based two-part tariff is an equilibrium. Then from Proposition 2, Cov (v i ()v j ();q (c i ;)) = 0. Since in this section we assumed that q () is strictly increasing with respect to, this condition implies thatv i ()v j () are constant. Note that if cost based 2PT is a Nash equilibrium in a larger pricing space (e.g. nonlinear pricing) it should also be a Nash in a smaller space (e.g. 2PTs). 60 Chapter 2 Dynamic Competition for Customer Membership 1 Jorge Tamayo 2 1 I am extremely indebted to Guofu Tan for his continuous guidance and support. I am very thankful to Odilon Camara, Keneth Chuk, Michele Fioretti, Yilmaz Kocer, Anthony Marino, and Simon Wilkie for their comments, and seminar participants at the University of Southern California and the 15th Annual International Industrial Organization Conference. I gratefully acknowledge funding from the USC Dornsife INET graduate student fellowship. All errors are mine. 2 Department of Economics, University of Southern California. jtamayo8@gmail.com. 61 Abstract A competitive two-period membership (subscription) market is analyzed. Two symmetric firms charge a “membership” fee that allows consumers to buy products or services at a given unit price, for both periods. Transactions are not anonymous, and firms price discriminate based on purchase history. Three main features of the tariff structure affect the competition. They are: (i) the length of the membership (long-term vs. short-term); (ii) the ability to price discriminate between “old” and “new” customers with the membership fee and the unit price; and (iii) the incentives to price discriminate between different consumers’ types (low- and high-demand/value customers). When firms employ long-term membership, firms have incentives to prevent their old most valuable customers from being “poached” by the competitor. In equilibrium, firms price discriminate with their membership fee and unit price regarding customer purchased-behavior and volume of demand (i.e., second-degree price discrimination). Instead, with short-term membership, they don’t discriminate with their unit price but only with their membership fee, between new and old customer, without screening on the taste (vertical) parameter. Overall, the number of consumers poached is smaller with long-term membership. Given that poaching erodes welfare and firms are better extracting surplus with short-term memberships, consumers are better off with long-term memberships. 62 2.1 Introduction In this paper, we study competition and consumer behavior in membership (subscription) markets. Generally, companies that implement a membership model, charge a “membership” fee that allows consumers to buy products/services at a given unit price. 3 A main feature of themembershipmarketsisthattransactionsarenotanonymous. Recentdevelopmentofnew information technologies has allowed firms to identify and classify consumers based on past purchase behavior, and to price discriminate according to this classification. However, the structure of the tariffs and the strategies used vary for different markets and industries. In particular, memberships may be valid for multiple periods (long-term subscriptions) or for a short period of time (short-term subscriptions). 4 Firms may offer differentiated membership fees to their current and new customers, like Amazon Fresh, which offers a discount in their subscription fee to the new customers, or may discriminate offering differentiated unit prices (or usage prices), like cable companies (e.g., Directv and Spectrum) or wireless carriers (e.g., Sprint) that offer cheaper monthly plans to new customers. 5 Finally, firms may discriminate based on the type/quality of the consumers, for example, high-value current customers may receive more benefits than low-value current customers. 6 So far, most of the literature has focused on models in which consumers have inelastic demand (i.e., buy one unit), firms use linear pricing, and consumers are homogeneous in their taste for quality. 7 These three assumptions exclude from the analysis important features shared by most of the membership (subscription) markets described in the first paragraph. First, tounderstandtheroleofthemembershipfeeandtheunitprice, weneedtoassumethat consumers have general elastic demand; otherwise, there is no information advantage about old customers. Second, firms use richer tariff structures that are generally more complex 3 Examples of markets in which membership fees are prevalent include credit-card markets, telephone ser- vices, cable companies, wireless carriers, and different on-demand economy services like grocery delivery ser- vices, online marketplaces, among others. Particularly, the digital membership economy (or Internet-enabled subscription services) has increased substantially during the last five years (e.g., Amazon has expanded its loyalty program Amazon Prime to include a variety of benefits; the grocery delivery services has expanded recently with Instacart and Amazon Fresh). See also Table 2.3. 4 That is, for long-term memberships, consumers who buy from the same firm they bought from before do not need to pay the membership fee again, whereas for short-term memberships, consumers need to renew their memberships more frequently. For example, wireless carriers used to offer plans for two-years while cable companies offered contracts for one or two years. 5 Directv offers two-year contracts that guarantees a low monthly rate for the first year (for the new customers), but does not commit to a unit price for the second year. In the recent merge between Time Warner Cable and Spectrum, the merged firm offered discounted monthly rates limited to customers who were not subscribed to applicable services within the previous thirty days. 6 For example, frequent flyer miles strategies offer extra benefits to their most valued consumers. 7 (Esteves and Reggiani 2014) and (Shin and Sudhir 2010) are two exceptions. The first, is a model in which consumers have constant elastic demands but only considers linear pricing, whereas the latter is a model in which the most valuable consumer buys more than one unit. 63 than linear pricing. Finally, consumers may have heterogeneous brand preferences and taste for quality, which, with the assumption of elastic demand, induces another layer of price discrimination between different consumer types. Here, we study a competitive membership (subscription) market and explore how differ- ent configurations of this market affect prices, consumer behavior, and the firm’s profits. In particular, three different settings are investigated: (i) long-term versus short-term mem- berships, (ii) price discrimination with the membership fee versus the unit price, and (iii) homogeneous versus heterogeneous preferences for quality. 8 In the next paragraphs, we describe three different markets that motivate our three settings discussed here. Allowing consumers to have elastic demands is important to capture certain character- istics of the market. For example, consider the grocery delivery service. 9 Two of the most important players in the industry, Instacart and Amazon Fresh, charge a membership fee of $14.99permonth. 10 Bothfirmschargeunitpricesfortheirproducts(besidesthemembership fee) and consumers generally buy more than one unit of the products. However, the tariffs offered in the subscription market may be more complex in industries other than the grocery delivery services. In particular, firms may have incentives to discrim- inate between current and new customers, not only with their membership fees, as in the grocery delivery services (see footnote 10), but also with their unit prices. 11 Cable and wire- less carriers companies are good examples of these practices. 12 Spectrum, merged with Time Warner Cable, recently offered Internet subscription services with lower monthly plans (or unit prices) to qualified customers who were not subscribed to applicable services within the previous thirty days, including “old” Time Warner Cable customers. Wireless carries compa- nies offer different monthly plan prices for new and current customers e.g.,Sprint offers lower monthly rates for new customers. The biggest carriers (e.g., Verizon, T-Mobile, and AT&T) have trade-in offers, which include cash back and, covering switching fees, among others. When should firms discriminate with their membership fees or their unit prices between new and current/old customers? Here we show that the optimal tariff is closely related to the frequency with which customers renew their memberships, and the ability firms have to 8 The assumption of horizontal differentiation for all the models considered. 9 Recent FMI-Nielsen report that online grocery spending could grow from 4.3 percent of the total food and beverage sales spending in 2016, to as much as 20 percent share by 2025, which could reach upward of $100 billion; "Put into context, that is the equivalent of nearly 3,900 grocery stores based on store volume" (Nielsen-FMI, 2017, “The Digitally Engaged Food Shopper”). 10 The former provides a free trial for fourteen days and the later for thirty days, and both firms offer every-day low prices and free delivery on orders over $35 and $40, respectively. 11 Note that in the online grocery industry, firms do not offer different unit prices to current and new customers. 12 Credit card companies also offer introductory APR, annual fees, cash back, and miles, among others, usually conditional on spending a minimum amount in a given period of time. 64 discriminate with their unit prices. Finally, note that in the previous examples, firms usually offered better deals to new customers. However, there are other markets in which the better deals may not be offered to only the new customers. 13 For example, most of the customer and loyalty programs provide differentiated tariffs to their best-value customers. Frequent-flyer miles strategies used by most of the airlines provide a good example of these practices. Airlines commonly offer extra benefits and lower prices to their best-value customers, who generally spend more and have a higher demand. (Shin and Sudhir 2010) also note that practitioner intuition suggests that previous customers should be rewarded to increase their loyalty and thus make them more profitable to the firm. Note that this is another source of price discrimination; second-degree price discrimination based on the volume of the demand, different from purchased behavior price discrimination. In this paper, we show that firms have incentives to protect their high- value customers in long-term subscription markets and avoid being poached by their rivals. However, this result does not hold if firms use short-term subscriptions; in this case, firms charge the same membership fee and unit price to current low- and high-type customers. In this article, we analyze a competitive two-period membership model with symmetric firms offering horizontal differentiated products. Consumers are forward-looking and have “general” elastic demands. We study how the tariff structure (in terms of the length of the membership, discrimination between old and new customers with the membership fee or the unit price, and discrimination between their high- and low-value customers) charged by firms with no commitment, affects prices, consumers welfare, firm profits and the ability to poach customers from rivals. Section 3.1 introduces our benchmark model in which firms use two-period membership fees and charge a unit (marginal) price for their products/services on each period. “Old” customers don’t need to pay the membership fee again in period 2, if they buy from the same firm of period 1, but they need to pay a price for each unit they buy in both periods. In the second period, firms discriminate based on prior purchase behavior and charge a single marginal price to their “old” customers and a subscription fee and a differentiated marginal price to their new customers (those who purchased from the rival in period 1). We find similarities in the way firms set up their tariffs in our two-period membership model and the previous literature on behavior price discrimination (e.g., (Fudenberg and Tirole 2000); for a review, see (Fudenberg and Villas-Boas 2006)). That is, firms charge higher unit prices to their “old” customers and charge cost-based membership fees to their new customers. 14 In 13 (Caillaud and Nijs 2014) and (Shaffer and Zhang 2000) provide many instances in which firms offer better deals to their previous customers. 14 Moreover, weshowthattheunitpricetotheoldcustomersishigherandtheshareofswitchers(customers poached) lower, compared to the case in which firms use linear pricing but are also allowed to discriminate 65 period 1, firms charge cost-based membership fees in equilibrium. In general, membership fees allow firms to extract surplus more efficiently from consumers, making the last worse off. In Section 3.2, we study a model in which firms are not allowed to price-discriminate based on purchase history (i.e., between old and new customers) with their unit price, but they are allowed to charge (subsidize) different membership fees. 15 That is, suppose that regulation does not allow firms to charge differentiated unit prices (or different prices for the same plan) to old and new customers but still allows them to charge membership fees. We show that unit prices are lower (for current and new customers) compared to the unit price offered to the old customers in our benchmark model, and the share of switchers are higher in this restricted model. Moreover, we show that firms have incentives to offer subsidies (i.e., negative membership fees) to the new customers, which are proportional to their demand, in equilibrium. The short-term membership model is examined in Section 4. Consumers must renew their memberships to buy the products in the second period, so the membership fees are paid in both periods. We show that in equilibrium, firms offer marginal-cost pricing in both periods and extract surplus through the membership fees from both new and current customers. Therefore, Firms don’t discriminate with their marginal price between old and new customers. Instead, they offer differentiated membership fees. 16 In Section 5, we introduce long-term contracts. We extend the analysis of our bench- mark two-period membership model and allow firms to offer long-term contracts with their membership fees that promise to supply products at a constant unit price in both peri- ods. In equilibrium, firms set the long-term marginal price equal to the marginal cost and extract surplus through the “long-term” contract membership fee. The long-term contract membership fee is higher compared to the standard membership fee. 17 Section6, extendsthebenchmarkmodelinanotherdirectionbyassumingthatconsumers have heterogeneous taste preferences (or heterogeneous vertical preferences). In particular, we assume that consumers differ in their taste parameter for quality, allowing for a new dimension of private information and price discrimination. Note that because consumers based on purchase history. This model is described in Appendix B 15 Most of the wireless carriers companies like Verizon, T-Mobile and At&T, offer the same monthly plan to their new and current customers, but discriminate with their membership fees. Moreover, all these companies use trade-in offers as a strategy to poach customer from their rivals. 16 Note that this equilibrium tariffs are similar to the ones used by the firms in the online grocery delivery services like Amazon Fresh, who usually set low unit prices at each period, and discriminate only with their membership fee every month between new and old customers. See also Table 2.3. 17 In Appendix C we compare our long-term membership model with a long-term contracts model in which firms use linear pricing. 66 have elastic demands, firms can infer consumers’ types in period 2. That is, high-type consumers have a larger demand than low-type consumers for a fixed unit price in period 1 (for both types). Thus, purchase history provides an additional layer of information to the firms, generating an asymmetry of information available from their old customers; each firm knows the type of their old customers but only knows type’s probability distribution of the new customers. Under this new scenario, firms compete aggressively for their highest-value customers, offeringthemlowerunitpricesthantheonesofferedtothelower-valueconsumers. In equilibrium, the share of consumers who switch to the rival firm is smaller for the high- type than the low-type consumers. Notably, unit prices offered to the new customers are below marginal cost in both periods; thus, firms compensate this loss with their membership fees. This is a remarkable result, particularly for the unit prices in period 1, as firms are symmetric. Contribution to the Literature This article contributes to the literature on several inter- related areas. Our primary contribution applies to the behavioral price discrimination and customer recognition literature. In terms of the model setup, our work comes closest to the seminal article of (Fudenberg and Tirole 2000) who study a two-period duopoly model with forward-looking firms and customers. 18 Other related literature with payoff-relevant history pertains to switching costs, pioneered by (Chen 1997). He analyzed a two-period duopoly model in which switching costs are heterogeneous, generating ex-post differentia- tion. 19 (Taylor 2003) extends Chen’s model to allow competition among more than two firms, an arbitrary number of periods, and persistent consumer heterogeneity. In general, both branches of the literature conclude that, in equilibrium, firms benefit the new customers and charge higher prices to the old customers. 20 Our membership model shares some of these results, but there are also notable differences. We assume that consumers have elastic de- mands, and firms charge a “membership” fee that allows consumers to buy multiple products at a given unit price, in both periods. Equilibrium prices and consumer welfare depend on the type of the structure of tariffs used by the firms, namely, long-term versus short-term memberships. These new features provide the best setting to study the recent increase of membership business model. The literature on behavior-based price discrimination extends in different directions. (Es- 18 (Villas-Boas 1999) extends (Fudenberg and Tirole 2000)’s model to an infinite model with overlapping generations. 19 For a comprehensive review, see (Fudenberg and Villas-Boas 2007) and, more recently, (Villas-Boas 2015). 20 As (Taylor 2003) pointed out, there are differences between both types of models: if consumers in period 2 experience a large taste shock for the firm, it may be efficient for her to switch. However, in switching cost model with homogeneous goods, changing supplier is never efficient. 67 teves and Reggiani 2014) extend the (Fudenberg and Tirole 2000) model and assume that consumers have constant elasticity demand, but they only consider linear pricing. (Shin and Sudhir 2010) and (Chen and Pearcy 2010) study a model in which brand preferences change over the two periods. The first paper shows that sufficient heterogeneity in the taste for quality of the consumer and stochastic preferences are a key ingredient to observe firms rewarding old best value customers. The latter shows that firms still offer lower prices to rival customers when commitment to future prices is infeasible; however, consumer loyalty is rewarded when firms are allowed to commit to future prices. 21 In the extension of the benchmark model with consumers’ heterogeneous taste preferences, presented in Section 6, we show that firms compete aggressively for their most valued customers, offering them lower unit prices than the ones offered to the lower-value customers, in period 2. Thus, similarly to (Shin and Sudhir 2010), in this model loyalty is rewarded only for the best-value cus- tomers. 22 Yet, this result does not hold if firms use short-term subscriptions; firms do not find profitable to discriminate between low and high-type consumers. 23 Another related literature pertains to competitive price discrimination. First, from the literatureon“static” models, itiswellknownthattheprofit-maximizingmethodofextracting surplus from the consumers when firms charge membership fees and unit prices is to set the unit price equal to the marginal cost and to extract surplus through the fixed fee. 24 However, firms in the membership business models deliver a flow of goods and services to their customers for different periods. Thus, previous predictions for the static models may not hold, given that firms extract surplus not only through the membership fee but also through the unit price in both periods. Second, when firms use long-term membership fees, in period 2, the competition for the old and new customers is asymmetric, that is, firms offer the old customers a unit price, and they offer a membership fee and a unit price to the new customers. 25 We present three novel results that contribute to this literature. First, we show that 21 Note these two articles only consider linear pricing, and the second assumes inelastic demands. 22 (Caillaud and Nijs 2014) provide a different theoretical explanation for why a firm may reward current customers; it comes from the reciprocity of the incentives of the firm to acquire information from young customers and extract surplus from their loyal customers, and to recognize their new customers. 23 Other extensions are as follows: (Chen 2008) assumes that firms are asymmetric where one firm has a stronger market position than its competitors. (Esteves 2009) extends (Fudenberg and Tirole 2000) and assumes that firms invest in advertising to generate awareness. (Pazgal and Soberman 2008) show that profits increase if only one firm uses past purchase information. (Chen 2010) studies whether firms have incentives to share customer information of past purchase behavior and shows that its effect on consumers depends on the relative magnitude of the prices in the substitute goods market and the complementary good market. 24 See, for example, (Armstrong and Vickers 2001), (Rochet and Stole 2002), and (Tamayo and Tan 2017b). 25 (Tamayo and Tan 2017a) analyze asymmetric pricing for a competitive environment with multiple (more than 2) firms. 68 when firms are not allowed to price-discriminate with their unit price based on purchase history but are allowed to charge (subsidized) membership fees to the new customers, firms have incentives to offered subsidies (i.e., negative membership fees) to the new customers, which are proportional to consumers’ demand. That is, they have incentives to offer subsi- dies to the new customers, leading to a prisoners’ dilemma that induces lower profits and higher consumer welfare. Second, in the long-term membership model with consumers with heterogeneous taste parameter for quality, unit prices offered to the new customers are below marginal cost in both periods. In period 2, firms offer differentiated unit prices to low- and high-type old customers while their rival offers them a fixed membership fee and a unit price to both types. Thus, the fact that firms infer consumers’ type in period 2 generates an information asymmetry among otherwise symmetric firms, in which the firm that is “disad- vantaged” in information sets its own unit price below the marginal cost and compensates this loss with the membership fee. 26 Third, we show the information advantage explained before over the old customers disappear, if both firms are allowed to charge a membership fee and a unit price to both new and old customers, which is our model with short-term memberships. Our work is also connected to models in competitive cross-subsidization. We provide a different theoretical explanation to use “cross-subsidization” (between the unit price and the membership fee) from the previous literature, which has tried to explain this practice within different contexts, such as, loss-leading associated with market power ((Ambrus and Weinstein 2008)); asymmetric competition by multiproduct firms ((Chen and Rey 2012)); as a mechanisms to screen consumers ((Chen and Rey 2016) and (DeGraba 2006)); or network competition and off-net usage ((Lopez and Rey 2016) and (Calzada and Valletti 2008)). In our membership model, in equilibrium, the unit price is set below the marginal cost in period 1 because the impact of first-period tariffs asymmetrically affects low- and high-type consumers in period 2, given that firms and consumers are forward-looking. Thus we provide a new explanation different from the previous literature on why firms may price below the marginal cost. 26 (Tamayo and Tan 2017b) find a similar result in a different context; they study competitive two-part tariffs in a model of asymmetric duopoly firms offering differentiated products in terms of both vertical and horizontal (à la Hotelling) differentiation. Particularly, they show that if firms have symmetric demands but asymmetric marginal costs, the firm with the higher marginal cost sets its unit price below its own marginal cost. 69 2.2 Model There are two firms, A and B, offering homogeneous products to a population of homoge- neous consumers, horizontally differentiated competing over two periods, 1 and 2. Both firms can produce their products at a constant marginal cost, c. We adopt a Hotelling-type of horizontal differentiation with consumers uniformly located on the unit interval [0; 1]. That is, consumer’s preferences for the products can be represented by u (q A )tx, if she buys from firmA andu (q B )t (1x) if she buys fromB, wherex2 [0; 1] is the distance to firm A, 1x the distance to firmB, andt is the transportation cost, at each period. We assume that consumer’s preferences remain constant for both periods. In each period, consumers either buy all products from one or the other firm, or they consume their outside option, u 0 . The next assumption characterizes the set of utility functions considered here. Assumption A1. The utility functionu :R + !R + is twice continuously differentiable, satisfiesu (0) = 0,u 00 ()< 0,u 0 (0)>c, andthereexistsauniqueq e > 0suchthatu 0 (q e ) =c. The focus of this paper is on a membership (subscription) model in which consumers pay a membership fee (lump-sum fee),F i that allows to buy products/services at a marginal price p i , for i =A;B. In the benchmark model, memberships are valid for the two periods (long-term or life-time memberships), that is, consumers who buy from the same firm in both period 1 and period 2 do not need to pay the membership fee again. They only need to pay for the units they buy in period 2 at the marginal price, p 2 i;o . If consumers decide to switch to the rival firm in period 2, they must pay the membership fee and the unit price for each good/service they purchase (new customers in period 2), p 2 i;n ;F 2 i;n . 27 Similarly, memberships may be valid for one period (short-term memberships), in which case consumers need to renew their memberships also in period 2 (Short-term membership). We assume that firms observe the first-period actions of their own consumers. Thus, the price offered by firm i2fA;Bg in period 2 may depend on the purchase history. That is, firms may discriminate between “old” customers (who bought from firm i in period 1) and “new” customers (who bought from its rival in period 1) in period 2. We assume that consumers are forward-looking while firms cannot commit to a particular price level in period 2, so, only short-term contracts are allowed. 28 Finally, we assume that firms and consumers have a common discount factor 2 [0; 1]. To avoid expositional complications, we define the set of feasible unit prices for both 27 Hereafter, we use to subscript “n” (“o”) to denote the price offered to the new (old) customers in period 2. Similarly, the superscript “2” denotes the price offered in period 2. 28 In Section 5 we extend the analysis to allow for long-term contracts. 70 periods and both firms asP. 29 Given (p i ;F i ), in a given period, a consumer decides to buy q i :P!R + units from firm i2fA;Bg, where, q i (p i ) = arg max q i 2R + fu (q i )p i q i g so the aggregate utility U i (p i ;F i ) is, U i (p i ;F i )v (p i )F i where v (p i ) is the indirect utility “offered” by firm i, defined by, v (p i ) max q i 2R + fu (q i )p i q i g Note that by Roy’s identity, the indirect utility function,v (p i ) satisfiesq (p i ) = @v() =@p i . In order to simplify the analysis, we focus on the case of full market coverage in which all consumers buy at least from one firm i2fA;Bg, and both firms sell strictly positive quantities in both periods. Note that (A1) implies that the buyer’s demand function and monopoly profit function, q (p i ) and (p i ), respectively, are continuously differentiable, and q (p i ) is strictly decreasing on p i . Furthermore, the monopoly profit function, (p i ) =q (p i ) (p i c) is single-peaked in p i under the following assumption: Assumption 2. 0 (p i )< 1, where i (p i ) q(p i ) q 0 (p i ) . 30 Under (A2), there is a unique optimal monopoly price p m i 2P defined by @ i (p i ) @p i = 0. 29 For each model we defined the set of feasible unit prices,P. 30 An identical assumption is used by (Carrillo and Tan 2015) in amodel of platform competition. Likewise, (ArmstrongandVickers2001)haveasimilarassumptionforamodelwithhorizontaldifferentiatedconsumers and symmetric firms. They assume 0 (u) 0 where (p) = q 0 (p) q(p) (pc) for u =v (p). The function (p) represents the elasticity of (expected) demand express in terms of the mark-up (pc) instead of the price p. Note also that as p!c, (p)! 0 and as p!p m we have that (p)! 1. It is straightforward to show that 0 (p)< 1 implies that 0 (u) 0. 71 2.3 Long-Term Membership This section, introduces the benchmark model. Forward-looking consumers pay a member- ship fee in the first period, which allows them to buy from the same firm (without paying the membership fee again) in the second period. That is, in period 2, firms offer a unit price and a membership fee to the new customers and a single marginal price to the old customers. We show that the membership fee offered to the new customers in the second period is lower compared to the fee offered in the first period. Similarly, the marginal price offered to the old customers is higher than the price offered to the new customers in both periods, which is equal to the marginal cost. We next compare these results with a standard linear pricing (LP) model with no mem- bership fees. 31 We show that the marginal prices offered to old customers in the LP game are lower compare to the marginal prices offered in the benchmark game. Moreover, we show that the share of switchers in period 2 are smaller in the subscription model compare to the LP model, that is, poaching rival’s customers is more difficult when firms use membership fees. Next, we explore a restricted membership model, in which firms are not allowed to discriminate between old and new customers with their unit price in the second period. 32 Thus, if antitrust authorities regulate price discrimination of the unit price between old and new customers, does consumer surplus increase? Are firms harmed by this policy? How does the equilibrium change compared to our standard subscription model? We explore these questions in section 3.2. 2.3.1 Subscription Model We suppose that both firms charge a membership fee and a unit price (p i ;F i ) in the first period. The membership fee allows consumers to purchase from the same firm for the two periods, that is, consumers don’t need to pay the fee again in the second period if they decide to buy from the same firm they purchased from in period 1. In the second period, each firm charges a single marginal price to its “old” customers, p 2 i;o , and a membership fee and a marginal price to those who purchased from its rival in period 1 (or new customers), p 2 i;n ;F 2 i;n for i2fA;Bg. Thus, firms discriminate with both the marginal price and the subscription fee between new and old customers in the second period. Consumers in period 2 decide whether to buy from the same firm they bought from in 31 We present the derivation of the LP model in Appendix B. 32 That is, firms still charge a membership fee to the new customers in period 2, but offer the same marginal price to both old and new customers in period 2. 72 period 1, or if they switch to the competitor. Remember that we assume that consumers are forward-looking (i.e., are not myopic); thus, in period 1, they decide to buy from the firms that provide them the highest expected returns. The wire and cable market would be a good example of this model. Typically, customers need to pay a membership fee and a marginal price at the beginning of the contract, and in the next period (usually a year), firms offer old customers a higher marginal price compared to the marginal price and fee offered to new clients who switch from a different company. 33 There are only two periods in our model, so we need to assume that the new customers who switch were enrolled previously in a different company. Cell phone companies also offer membership fees and monthly plan rates specific to new customers who were enrolled previously with other carriers. 34 Revealed-preference argument implies that for any pair of first-period prices, there is a cut-off, x , such that all consumers with x < x buy from firm A in the first period, and all consumers with x>x buy from firm B. Called the space between 0 and x as the turf of firm A and everything to the right of x as the turf of firm B. We show that that there exists x and x such that, if x 2 [x; x], there exists an interior equilibrium with a positive share of customers poached by each firm. Second Period. For brevity, let’s start analyzing the problem on A’s turf in period 2 (i.e., the set of x2 [0; 1] such that x x ). The problem of both firms on B’s turf is equivalent, so it is omit it. The problem of firm A on its own turf is: max p 2 A s A p 2 A;o ;p 2 B;n ;F 2 B;n p 2 A;o where s A = min x ; 1 2 + v(p 2 A;o )v(p 2 B;n )+F 2 B;n 2t . Similarly, firm B charges a subscription fee and a marginal price to consumers on A’s turf (new customers), so the problem of firm B is max ~ p 2 B ; ~ F 2 B x s A p 2 A;o ;p 2 B;n ;F 2 B;n p 2 B +F 2 B BecausefirmBwillnotcapturetheentiremarketonturfA,fromthefirst-orderconditions 33 In the recent merger between Time Warner Cable and Spectrum, new offers were made exclusively to new customers. The offers available online had the following claim for old customers: “Offers are valid for a limited time only, to qualifying residential customers who have not subscribed to applicable services within the previous thirty days and have no outstanding obligation to TWC”. Similarly, DirecTV and AT&T offer a bundle of Internet plus TV, and the following claim was in the offer “New approved residential customers only.” 34 In March 2017, Sprint offered slashed prices on its unlimited plan only to new customers who were enrolled previously with carriers such as Verizon or T-Mobile. 73 the fixed fee charged by firm B in period 2 to the new customers is, F 2 B;n = t (2x 1)v (p 2 A ) +v (~ p 2 B ) (~ p 2 B ) 2 (2.1) and p 2 B;n =c . p 2 A;o is defined by, t (1 + 2x )v (c) = p 2 A (2.2) where (p) 2 (p)v (p). The next proposition summarizes the equilibrium. Proposition 1. . Given the existence of a first-period cutoff, there exists a unique interior equilibrium in period 2, in which: i. For t > 0, small, there exists x < 1 2 and x > 1 2 such that for x 2 [x; x], each firm charge a marginal price to the customers on its own turf p 2 i;o defined by (2.2), and a membership fee defined by (2.1), and a marginal price equal to the marginal cost, e.g., p 2 i;n =c, for i2fA;Bg. ii. s A p 2 A;o ;p 2 B;n ;F 2 B;n <x . When firms use membership fees, the optimal strategy in period 2 is to charge a positive membership fee (defined by 2.1) and a marginal price equal to the marginal cost to the new customers. Old customers do not need to pay the subscription fee again; thus, when price discrimination is allowed, firms set a higher marginal price than the one offered to the new customers. Note that turfs are independent markets. The game on each turf is closely related to a static asymmetric two-part tariffs game (2PTs); one firm uses a marginal price (charged to its old customers) and the other firm uses 2PTs (to its new customers). Given that consumers have homogeneous taste preferences, the firm that uses 2PTs will charge marginal cost-based two-part tariffs, independently of its rival marginal price. 35 This is a consistent result with the literature on price discrimination on a static framework. 36 Note that as x !x, F 2 B;n ! 0 and the market share for firm B decreases up to a point in which the indirect utility provided by firm B would not be enough to compensate the transportation cost. The following corollary provides some comparative statics with respect to the cutoff value, x . 35 In Section 5 we show that marginal cost-based two-part tariffs is no longer an equilibrium if consumers have heterogeneous tastes for quality. 36 See (Tamayo and Tan 2017a). 74 Corollary 1. In any interior equilibrium i. @p 2 A;o @x > 0 and @F 2 B;n @x > 0 ii. @p 2 B;o @x < 0 and @F 2 A;n @x < 0 Corollary 1 shows that as the market for new consumers for firm B increases (turf A increases, e.g., x increases), the subscription fee for the new customers increases. The analysis is similar for p 2 A;o and equivalent on B’s turf. That is, as the market share for new consumers for firm B increases (x increases), firms compete less aggressively on A’s turf, and more aggressively onB 0 s turf. Intuitively, if firm A is larger (has more consumers in the first period), then in the second period firms compete more aggressively for current firm B customers, and less aggressively for current firm A consumers First Period. Let’s consider now the first-period pricing and consumers’ decisions. Note that we assume that firms have no commitment power, so the prices and market share of the first period affect the second-period pricing strategy. Remember that consumers are forward-looking and anticipate (correctly in equilibrium) the second-period pricing strategy of the firms. Before, we guessed the existence of a cutoff,x , such that all consumers withx<x buy from firm A in the first period, and all consumers with x>x buy from firm B, in the first period. Note that consumer located at x must be indifferent between buying from firm A in period 1 (pay the subscription fee F A and the marginal price p A ) and then switch and paying the subscription fee F 2 B;n and the marginal price c to firm B in period 2 , or buying from firm B in period 1 ( pay the subscription fee F B and the marginal price p B ) and then to buying from firm A at a marginal price equal to the marginal cost and a subscription fee F 2 A;n , in period 2. 37 Thus, x () = 1 2 + v (p A )v(p B ) +F B F A + F 2 A;n (x )F 2 B;n (x ) 2t (1) (2.3) where (p A ;F A ;p B ;F B ). Note thatF 2 A;n (x )F 2 B;n (x ) is monotonically increasing with respecttox ,sothatthereexistsatmostasinglex thatsatisfies(2.3)if F 2 A;n (x )F 2 B;n (x ) is bounded. Now, the overall problem of firm A in the first period is: 37 Note that is not possible that consumerx buy from the same firm in both periods, given that we showed in Proposition 1 that the share of switchers is positive, which implies that the rival firms provides a higher utility to the new customers than the current firm. 75 max p A ;F A x () ( (p A ) +F A ) | {z } (1) +s A p 2 A;o ;c;F 2 B;n p 2 A;o | {z } (2) + s B p 2 B ;c;F 2 A;n x () F 2 A;n + p 2 A;n | {z } (3) Note that firm A’s overall objective function depends on three terms: (1) is equal to the share of consumers who buy from firm A in period 1, x , times the subscription fee and the monopoly profit function; (2) is equal to the market share of customers who buy from firm A in period 1 and buy again from firm A in period 2, times the monopoly profit function 38 ; and (3) the share of switchers, meaning those who buy from firm B in period 1 and then buy from firm A in period 2, times the subscription fee and the monopoly profit function charged to new customers. The following proposition summarizes the overall equilibrium of the game. Proposition 2. There exists a symmetric equilibrium in which firms i. in the second period, charge a marginal pricep 2 o = 1 (2tv(c)) to the old customers, and marginal cost-based membership fee to the new customers with a fixed fee equal to F 2 n = v(c)v(p 2 o) 2 , where (p) 2 (p)v (p); ii. in the first period, firms charge marginal cost-based membership fee i.e., p =c and, F =t + tq (p 2 o ) 2 0 (p 2 o ) +q (p 2 o ) p 2 o 0 (p 2 o ) +q (p 2 o ) 2 0 (p 2 o ) +q (p 2 o ) | {z } <1 + q (p 2 o ) (v (c)v (p 2 o )) 2 0 (p 2 o ) +q (p 2 o ) > 0; iii. F >F 2 n ; iv. s A = 1 2 + v(p 2 o)v(c) 4t < 1 2 and s B = 1 2 + v(c)v(p 2 o) 4t > 1 2 . From Proposition 2 we know that in the second period, firms offer a unit price to the old customers above the marginal cost, and a marginal cost-based membership fee to the new customers. Note that this extra pricing instrument allows firms to extract surplus more efficiently; they charge a membership fee to the new customers that is proportional to the 38 Remember that old customers do not have to pay the subscription fee again. 76 difference of the efficient surplus and the surplus offered by the rival firm, v (c)v (p 2 ). The membership fee is linear in the profit function of firmi, so it does not depend directly on the curvature of the demand. Moreover, the membership fee to the new customers in period 2 is lower than the fee offered in period 1. This is a similar result in the literature of behavior- based priced discrimination. Finally, in equilibrium, the share of switchers is proportional to the difference of the efficient surplus and the surplus offered to the old customers. Propositions 1 and 2 show that firms offer cost-based membership fees to the new (all) customers in period 2 and 1. As we mentioned before, in period 2, each firm’s turf can be considered as different market, in which one firm offers LP and the other firm 2PTs. Note that the firm that charges 2PT sets the marginal price equal to the marginal cost and extract surplus through the membership fee, and the other firm extracts surplus from the old customers with the marginal price. This asymmetric pricing game provides more tools to extractsurplus, increasingthepricestooldcustomers, anddecreasingthesurplus, asweshow in the next section. Note that in the first period both firms compete offering membership fees and marginal prices. From the literature on competitive price discrimination, e.g., (Tamayo and Tan 2017a), we may expect that when consumers have homogeneous taste preferences, firms charge cost-based membership fees. 39 Why do firms use cost-based membership fees in period 2? (Mathewson and Winter 1997) provide good intuition in their model with goods that are strongly complementary in demand. Let us first consider firm B’s choices onA’s turf, in period 2: we can interpret the permission to allow new consumers to enter the shop as the first product (product 1) and its price to be equal to the membership fee F 2 i;n , and treat the real product offered by firm B as product 2 with price equal to p 2 B;n . The demand for product 1 is the market share of firm B’s product, x s A p 2 A;o ;c;F 2 B;n , and the demand for product 2 is the market share multiplied by the individual demand for such product, [x s A p 2 A;o ;c;F 2 B;n ]q p 2 B;n . Note that the ratio is independent of the subscription fee,F i . Hence the two “products” are strong complements. Using Proposition 2 in Mathewson and Winter, we can conclude that the firm B’s profits are maximized by setting p 2 B;n = c, independently of p 2 A;o . A similar logic allows us to conclude why marginal price are equal to the marginal cost to the new customers for firm A. What about period 1? We need to modify our analysis in order to take into account that first-period tariffs affect second-period prices and membership fees. Notice that using envelope theorem in the first-order conditions of each firm and imposing symmetry, the ratio of the effects on the market share of p and F is equal to q(p), which is the same ratio of 39 Note, however, that the previous literature on 2PTs only consider static games; thus this result may not be obvious. 77 the marginal contribution of the monopoly profit function and the marginal contribution of the membership, at p =c. If p =c these margins cancel, which result in a cost-membership equilibrium. Of course, this intuition does not discard the existence of non-symmetric equi- libria. Comparison with the Linear Pricing Game. The membership fee is an extra pricing instrument that allows firms to extract surplus more efficiently. We may expect that this extra tool will harm consumers compared to the case in which firms use only LP. 40 Let p 2 2PT;o the price offered to old customers in equilibrium in our membership model presented before (Proposition 1 and 2), and let p 2 LP;n andp 2 LP;o be the marginal price offered to new and old customers when firms use linear pricing, respectively, in period 2. The following corollary compare these marginal prices and the share of switchers for both cases. Corollary 2. In equilibrium, i. p 2 LP;n <p 2 LP;o <p 2 2PT;o ; ii. s A LP <s A 2PT . Therefore, the membership fee allows firms to extract surplus more efficiently from the new customers, which also implies a higher marginal price to its old customers. That is, the membership fee not only helps firms extract surplus from consumers on its rival turf but also, given that firms have upward slope reaction curves, allows them to increase the marginal price charge to old customers on its turf, compared to the price offered in the LP model. Similarly, the surplus extracted from the new customers is also greater in the subscription fee model. Thus, in equilibrium, both firms extract more consumer surplus in both turfs, reducing consumers’ welfare. Note that poaching rival’s customers is more difficult when firms use subscription fees; fixed fees creates sunk costs (switching-costs) which allow firms to extract a higher share of consumers surplus, diminishing the proportion of switchers. 2.3.2 Restricted Membership Model. We now assume each firm is allowed to offer a unit price and a membership fee in the first period, (p i ;F i ), butinthesecondperiod, eachfirmoffersasinglemarginalpricetoeverybody 40 Note that a model in which firms use linear pricing will be close to the model proposed by (Fudenberg and Tirole 2000), but instead of buying a unit good, consumers have elastic demands. In Appendix B we provide the details of this model. 78 (old and new customers), p 2 i , however can also charge (subsidize) a membership fee, ~ F 2 i , to those who purchased from its rival in period 1. Intuitively, firms can charge a membership fee in order to allow consumers to buy its products in both periods, but they must offer the same marginal price to old and new customers. The cell phone industry, particularly the wireless carrier market, would be an example that fits this model. Often, firms charge an initial membership fee to consumers (e.g., for the cell phone) and a marginal price for the monthly plan. Usually, these companies offer the same monthly plan to current and new customers, although they discriminate with their subscription fee. 41 For the US market, most of the “big” carriers offer subsidies to trade- in the phone from rival firms, holding the price for the monthly plans fixed for the new and current customers. 42 Here we explore how the unit prices and membership fees change when firms are restricted to offer the same unit price (or monthly price) to new and current customers. In the second period, the problem of firm A is: max p 2 A ; ~ F 2 A s A p 2 A ;p 2 B ;F 2 B;n | {z } old p 2 A;o + s B p 2 A ;F 2 A;n ;p 2 B x | {z } New p 2 A +F 2 A;n where s A p 2 A ;p 2 B ;F 2 B;n 1 2 + v(p 2 A )v(p 2 B )+F 2 B;n 2t is the market share of consumers on its own turf (share of old customers who buy again from firm A) and s B p 2 A ;F 2 A;n ;p 2 B 1 2 + v(p 2 A )F 2 A v(p 2 B ) 2t minus x , is the market share of consumers who buy from firm A on B 0 s turf (new customers). Note that firm A needs to find a price that maximizes the profits for both turfs, and charge a membership fee to the new customers. In equilibrium, F 2 i;n =tx t 2 v p 2 j v (p 2 i ) 2 ! (p 2 i ) 2 (2.4) and p 2 A and p 2 B are defined by, t (p A ) t 2 x t 3 (p A ) 2 + (p A ) 2 (p B ) 2 (p A ) v (p A )v (p B ) 2 +v p 2 A v p 2 B = 0 (2.5) 41 From what we are aware of, during 2016-2017, only Sprint offers different monthly plans that are conditional on whether the client is a new or an old customer. 42 Verizon offers up to $650 for installment plan balance less trade-in value (or up to $350 prepaid card for early termination fees less trade in value), to customers that switch to Verizon. T-Mobile, Sprint and AT&T offer similar plans. 79 t (p B ) tx t 2 3 (p B ) 2 + (p B ) 2 (p A ) 2 (p B ) v (p B )v (p A ) 2 +v (p B )v (p A ) = 0 (2.6) where (p) = q(p) 0 (p) . Note that in this model, given that firms offered the same marginal price on both turfs, if firms set a positive membership fee, consumers will not have incentives to switch. Thus, in order to provide incentives for those consumers close to x , firms need to offer subsidies proportional to their demand. Moreover, (2.5) is not defined as p A !c; thus, marginal-cost pricing is not an equilibrium. Similarly, for (2.6) as p B ! c. The following proposition characterizes the equilibrium in period 2. Proposition 3. i. Fort> 0, small, there exists x< 1 2 and x> 1 2 such that forx 2 [x; x], there exists an interior equilibrium, in which p 2 i >c and F 2 i;n < 0; ii. ifx = 1 2 there exist a unique symmetric equilibrium, e.g., p 2 A =p 2 B andF 2 i;n = (p 2 ) =2; iii. for x < (>) 1 2 , in equilibrium, p 2 A < (>)p 2 B ; From Proposition 3(i), we conclude that when firms are not allowed to discriminate between old and new customers with their unit price (e.g., different monthly plans), marginal cost pricing is not an equilibrium. Moreover, in the proof of Proposition 3 (i) we show that the“quasi”-bestresponsefunctionswithrespecttothemarginalpricearedecreasing(contrary to a LP and a 2PT game), and that firms set a price above the marginal cost. Intuitively, astherival’sprice increases, each firmhasincentivesto decreaseitsunit price, to attract consumers with a lower subsidy, increasing the market share on both turfs. Note that from (ii) we know that when x = 1 2 , there exists a symmetric equilibrium, in which each firm offers subsidies proportional to its monopoly profits to new customers. Since firms know the consumer’s optimal demand, firms set subsidies that still allow them to extract positive surplus from the switchers. This fact explains, why “quasi”-best response functions with respect to the marginal price are decreasing. Note that if firm i sets a fixed fee equal to 0, it is optimal for its rival, firm j6= i, to set a negative fixed fee; firm i will not poach consumers from j 0 s turf in equilibrium while firm j poaches a positive share from i 0 s turf and gets a positive net revenue from them. If firm i increases its price, firm j has no incentive to increase its price; by keeping its price constant (or lower), the share of switchers increases, increasing the net revenue. This means that firms cannot extract surplus through the fixed fee from the new customers. Instead, 80 they need to offer subsidies proportional to their demand and now extract surplus through the unit price. TheresultinProposition3(i)contrastswithrelatedresultsfoundintheliteratureoncom- petitive price discrimination. Particularly, (Tamayo and Tan 2017b) show that in a standard static framework with horizontally differentiated consumers with homogeneous taste pref- erences and asymmetric firms, marginal cost-based two-part tariff is a unique equilibrium. The difference here is that now each firm sets a price for both markets, that is, both firm A’s and firm B’s turf. Thus, marginal cost-based two-part tariff is no longer an optimal strategy; otherwise, in any symmetric equilibrium e.g., x = 1 2 , it would not be possible to poach consumers. 43 Here, firms are competing asymmetrically in different markets, which explains why their optimal strategy differs compared to the static models. Comparative statics with respect to the cutoff value, x , are displayed in Corollary 3. Corollary 3. For t small, i. @p 2 A @x > 0 @p 2 B @x < 0 ; ii. @ ~ F 2 A @x < 0 @ ~ F 2 B @x > 0. Note that the intuition of Corollary 3 is similar to Corollary 1, namely, as the market share for new consumers for firm B increases (x increases), firms compete less aggressively on A’s turf and more aggressively on B 0 s turf. Working backward, we now consider the optimal prices and consumption decisions in the first period to evaluate the overall impact of the information pricing that makes poaching possible. As in the previous models, first-period prices and membership fees will influence second-period pricing; thus, firms take this into account in their first-period pricing decision. The following proposition characterize the equilibrium. Proposition 4. There exists a symmetric equilibrium in which firms i. in the second period, charge a single unit price defined by, t = 3 2 p 2 43 (Tamayo and Tan 2017b) show that when firms use 2PT and are asymmetric, and consumers have heterogeneous taste preferences, the “quasi”-best response functions with respect to the marginal price are increasing. 81 and “subsidize” new consumers to switch with a negative membership fee, F 2 = (p 2 ) 2 ; ii. in the first period, firms charge a marginal cost-based 2PT with a fixed fee equal to F =t 0 (p 2 ) 2 q (p) (pc) (q 0 (p) (pc)) 4t p 20 B (x )> 0; iii. 0<s A = 1 2 (p 2 ) 4t <x and s B = 1 2 + (p 2 ) 4t >x . In summary, in the membership fee model, when firms are not allowed to discriminate between new and old customers with their unit price (e.g., offering different monthly plans to old and new customers) in period 2, firms in equilibrium set a negative membership fee (subsidy) for the new consumers, which is proportional to the equilibrium demand, and a single unit price to the entire market (both turfs). In period 1, firms offer cost-based membership fees, that is, a unit price equal to the marginal cost and a positive membership fee. ComparisonwiththeMembershipModelandLPGame. Weshowintheprevious section that when firms are not allowed to discriminate between old and new customers with their marginal price in period 2, firms compete more aggressively for new customers, offering them subsidies (negative subscription fees) to switch. Here we compare the marginal prices offered in period 2 for the three models: subscription model, restricted subscription model, and LP. Corollary 4. Given t small, in equilibrium, p 2 LP;n <p 2 2PT;R <p 2 LP <p 2 2PT . Firms are worse off when they are not allowed to discriminate based on purchase history by means of the unit price, in period 2. Similarly, this restriction benefits consumer welfare because firms set a lower marginal price and lower (negative) membership fee to the new customers in period 2, than when are allowed to price discriminate (even using LP). When discriminationbetweenoldandnewcustomersintheLPgameisnotallowed, theequilibrium price in period 2 would be higher than the unit price offered in the restricted membership model. 44 Thus, in this case, the possibility to charge (subsidized) membership fees to new 44 Note that if firms are not allowed to discriminate between old and new customers in the LP game, prices in period 1 and 2 would be constant (i.e., the problem for both firms is the same in both periods). 82 customers in period 2 harms firms, i.e., firms would be better off if they did not have the option to charge (subsidize) membership fees in the last period. 2.4 Short-Term Membership After analyzing the effect of long-term memberships, we now study short-term ones (mem- bership is valid for one period only). This means that customers need to pay the membership fee in both the first and second periods. Two cases are studied here. The first model, as- sumes that firms are allowed to discriminate with their unit price and membership fee based on purchase history. Thus, they offer a membership fee and a unit price to both old and new customers. The second model assumes that firms offer different membership fees but are not allowed to discriminate with their marginal price based on purchased history. Compared to the previous model, here consumers need to “renew” their subscription in the second period, even if they purchase from the same firm of the first period. The assumptions of the previous section remain valid in this section. Warehouse clubs, like Costco, Sam’s Club, or Amazon Prime, are good examples of the model presented in this section. Usually, in these clubs, there is a membership fee that has to be renewed each year (period), which allows a consumer to buy products at “lower” prices compared to standard retail stores. Usually firms do not discriminate between old and new customers with their marginal prices although, some of them provide discounts in their membership fees to the new clients. 45 In the model presented here, in equilibrium, firms offer marginal-cost pricing in both periods and extract surplus through the fixed fees from both new and old customers. Thus, in equilibrium, firms don’t discriminate between current and new clients with their marginal price, although they provide a discount on the membership fee to the new customers in the second period. We start by guessing (from a standard revealed preferences argument) that at any pair of prices and fixed fees in the first period, there is a cutoff x , such that all consumers with x < x buy from firm A in the first period (A’s turf), and all consumers with x > x buy from firm B in the first period (B’s turf), similar to the previous models. In the second period, the problem of firm A on its own turf is now: max p 2 A ;F 2 A s A p 2 A;n ;F 2 A;n ;p 2 B;o ;F 2 B;o p 2 A;o +F 2 A;o where s A p 2 A;n ;F 2 A;n ;p 2 B;o ;F 2 B;o 1 2 + v(p 2 A;n )v(p 2 B;n )F 2 A;o +F 2 B;n 2t . From the first-order condi- tions, it follows that marginal cost-based 2PTs is a unique equilibrium. That is, for any 45 For example, Amazon provides a thirty-day Prime free trial to new customers. 83 marginal price and membership fee of firm B, firm A sets its marginal price equal to the marginal cost and extract surplus through the membership fee. Note that the problem is similar to a standard competitive 2PT model (see (Tamayo and Tan 2017b)). Following a similar argument, firm B uses marginal cost-based 2PT on A’s turf. Thus, in equilibrium, F 2 A;o = t (2x + 1) 3 ; F 2 B;n = t (4x 1) 3 and p 2 i;o = p 2 i;n = c. Note that there exists an interior equilibrium with positive profits on each turf for both firms for x 2 1 4 ; 3 4 . Similarly, note that F 2 i;o > F 2 i;n for i = A;B, for x < 1. Let’s consider now the first-period pricing and consumers’ decisions. Given that con- sumers are not myopic and anticipate the second-period pricing strategy of the firms, at an interior equilibrium the type-x consumer is such that: x = t 2 + 3 v (p A )v (p B ) +F B F A 2 (3 +) where we use the fact that F 2 A;o F 2 B;n = 4t(12x ) 3 and p 2 i;o = p 2 i;n = c. The following proposition characterizes the equilibrium. Proposition 5. There exists a unique symmetric equilibrium in which x = 1 2 , s A = 1 3 , F 2 = 2t 3 , F 2 n = t 3 and F = (3+) 3 . When consumers have to pay the membership fee in the first and second periods, we get a model that is close to the poaching model proposed by (Fudenberg and Tirole 2000). In the second period, firms offer efficient surplus to both types of customers, but they offer a lower membership fee to the new customers than the fee offered to its old customers. Moreover, if t< 3+ 2 , the membership fee in the first period would be higher than the membership fee offered in the second period. The opposite result holds if t> 3+ 2 . Thus, if firms offer different membership fees but are restricted to offering the same marginal price to old and new customers, marginal cost-based 2PT would be a Nash equi- librium. That is, if customers need to “renew” their memberships in the second period, and firms are not allowed to discriminate offering different marginal prices to old and new cus- tomers, in equilibrium, both firms set their prices equal to the marginal cost and extract surplus through different membership fees. Note that this result contrasts with the model presented in the previous section, in which when firms are not allowed to discriminate with their marginal price between new and old customers (restricted membership model), firms 84 are worse off, and consumers are better off. Moreover, firms would be better off if both of them were not allowed to charge a subscription fee. 2.5 Long-Term Contracts (for the Unit Price) We now explore how long-term contracts impact prices in our membership (benchmark) model. Note that in the models of the previous sections, firms do not have commitment power; thus, the size of the two first-period markets (i.e., the difference between the first- periodpricesoffirmsAandB)willinfluencepricesinthesecondperiod. Anaturalextension of the subscription model would be to allow firms to offer long-term contracts in which the marginal price is fixed for both periods. Wirelesscarriercompaniesoftenofferlong-termcontracts: two-yearcontractsthatspecify an initial membership fee (an initial one-time fee) and a monthly price for the two years. Similarly, cable companies often offer two-year contracts. Here, firms offer long-term contracts that include a “long-term” membership fee, F l i , that promises to supply the goods in both periods at a fixed marginal price, p l i . Firms also offer the standard contract analyzed in the benchmark model: a unit price and a membership fee in the first period, (p i ;F i ), and in the second period, each firm charges a single marginal price to its old customers (who did not buy the long-term contract), p 2 i;o , and a membership fee and marginal price to those who purchased from its rival in period 1, p 2 i;n ;F 2 i;n for i2fA;Bg. 46 We assume that long-term contracts are purchased by customers who mostly prefer that firm’s product, that is, the set [0;s A ] and [s B ; 1] would buy the long term contracts from firms A and B, respectively. Note that the set of consumers who buy long-term and short-term contracts from firmi are indifferent towards the two options. Consumers have homogeneous taste preferences, so the model is completely deterministic for these consumers. Second Period. Notice that here the market is shrunk by the share of consumers who prefer to buy the long-term contract in period 1. Thus, on the turf of firm A, the problem of firm A is max p A s A p 2 A;o ;p 2 B;n ;F 2 B;n s A (p A ) 46 There are different long-term contracts to be considered; we can also discuss a contract that guarantees a marginal price in period 2 distinct from the price in period 1 if consumers agree to pay the “long-term” subscription fee. 85 wheres A p 2 A;o ;p 2 B;n ;F 2 B;n 1 2 + v(p 2 A;o )v(p 2 B;n )+F 2 B;n 2t ands A p 2 A;o ;p 2 B;n ;F 2 B;n s A isthemarket share of consumers of firmA in period 2 who bought from firmA in period 1 but did not buy the long-term contract. On A’s turf, firm B offers a unit price and a membership fee to the new consumers, analogous to our membership (benchmark) model. Following the reasoning in Proposition 1, it is straightforward to show that in any interior equilibrium, firm B offers a marginal cost-based 2PT and extract surplus through the membership fee, (i.e., p 2 B;n =c) and ~ F 2 B = t (2x 1) +v (c)v (p 2 A ) 2 (2.7) where p 2 A is uniquely defined by 2t (1 2s A ) +t (2x 1)v (c) = 2 p 2 A +v p 2 A (2.8) Similarly, an interior equilibrium requires that x 2 [x; x], so that both firms poach a positive share from the rival’s turf. 47 Note that the incentive constraint for the long-term contract is such that v p l A (1 +)F l A =v (p A )F A +v p 2 A;o (2.9) Consumers are indifferent between paying the membership feeF l and paying a unit price p l A in both periods, or pay the regular membership fee and marginal price in period 1 and pay p 2 A in period 2. Using (2.7) and (2.9), and the analogous first-order conditions on B 0 s turf, the type-x consumer is such that x = 1 2 + v (p A )v(p B ) 4t + F B F A 4t (2.10) + v p l A (1 +)F l A 4t v p l B (1 +)F l B 4t which depends only on the marginal prices and membership fee of the first period. Thus, using (2.8), (2.9), and (2.10), we can determine how first-period pricing decisions would affect second-period prices and margins, e.g., s A and x . The following corollary shows the comparative statistics. Corollary 5. i. @x @F A ; @p 2 A @F A < 0 @s A @F A > 0 47 The proof of these claims follows from Proposition 3, so we omit them. 86 ii. @s A @F l A ; @x @F l A < 0; @p 2 A @F l A > 0 iii. q (p A ) h @x @F A @s A @F A @p 2 A @F A i 0 = h @x @p A @s A @p A @p 2 A @p A i 0 An increase in the membership fee charged to consumers who do not buy the long-term contracts, F A , increases the share of consumers who prefer the long-term contract (making the short-term contract/prices less attractive) and decreases the second-period prices to old customers. A similar intuition follows when the membership fee of the long-term contracts increases. Note that the effect of F A on (x ;s A ;p 2 A ) is equal to the effect of p A divided by q (p A ), suggesting that the most efficient way to extract surplus in the first period is to use marginal cost-based membership fees, that is, to set marginal prices equal to the marginal cost and extract surplus through the membership fee. First Period. The problem of firm A in the first period is max p A ;F A ;p A ;F l A s A p l A +F l A + (x ()s A ) ( (p A ) +F A ) + s A p 2 A ;c; ~ F 2 B s A (p 2 A ) + h s B p 2 B ;c; ~ F 2 A x () i ~ F 2 A s:t: (2.9) and (2.10). Note that from Corollary 5, it follows that we can use the Implicit Function Theorem and express the second-period prices, p 2 A ,F 2 A;n ,p 2 B;n and F B;n , and both margins x and s A in terms of p A ;F A p l A and F l A and find a solution to the previous problem. The following proposition characterizes the whole equilibrium of the pricing game. Proposition 6. i. There exists an interior equilibrium in which p = c, p l = c, F, and F l are defined by (2.28) and (2.29), respectively (see Appendix). ii. In any interior symmetric equilibrium, x = 1 2 , s A = 1 2 2(p 2 o)+v(c)v(p 2 o) 4t < s A , s A = 1 2 v(c)v(p 2 n) 2t <x , p 2 n =c and F 2 n = v(c)v(p 2 ) 2t ; iii. F <F l . 87 In summary, when firms use long-term contracts that guarantee a price for the first and the second periods, firms charge a marginal price equal to the marginal cost and extract surplus through the membership fee, F l . Note that in this contract, the marginal prices of period 1 and 2 are both equal to the marginal cost, which explains why the membership fee charged in the long-term contract is higher than the membership fee in the standard membership contract, i.e., F l > F. In a long-term contract, where firms offer different marginal prices in periods 1 and 2, if the marginal price of the long term contract is higher than p 2 , we would expect the opposite order of the membership fees, i.e., F l <F. 2.6 Heterogeneous Agents. We shift our attention to the case where consumers have heterogeneous taste preferences for quality (or heterogeneous demands). We adopt a Hotelling type of horizontal differentiation with heterogeneous consumers with different taste for quality (vertically differentiated). We suppose that consumers’ preferences for the products can be represented by the utility func- tion,u (q A ;)tx if a consumer buys fromA andu (q B ;) (1x)t if he or she buys from B, where x, t, and q are defined as before, and represents a taste parameter for quality. We assume that there is a mass of consumers uniformly located on the unit interval [0; 1], that is, x2 [0; 1], with drawn independently of x from the distribution on f L ; H g, where L < H (low and high type), with probabilities 1 and , respectively. Given (p i ;F i ), a consumer with “vertical” taste parameter 2 decides to buy q i : P !R + units from firm i2fA;Bg, where q i (p i ;) = arg max q i 2R + fu i (q i ;)p i q i g so the net utility U i (p i ;F i ;) when firms charge a membership fee F i is U i (p i ;F i ;)v i (p i ;)F i where v i (p i ;) is the indirect utility “offered” by firm i, defined by, v i (p i ;) max q i 2R + fu i (q i ;)p i q i g We will focus on the case with E [v i (c i ;)] > 0, where v i (c i ;) is the maximum surplus offering a good at the marginal cost, c i , by firm i2fA;Bg for any 2 . First we assume that consumers’ taste parameter for quality is constant for both periods. 88 Under this assumption, firms can infer the type of the customers who buy in the first period, allowing them to discriminate based on this characteristic. In Appendix E, we relax this assumptionandassumethatpreferenceschangebasedontheexperience,suchasthepurchase in the first period. 2.6.1 Membership Model Here, we extend the membership model of section 3.2, in which firms are allowed to price discriminate with their unit price and membership fee between new and old customers al- lowing for heterogeneity in the taste parameter for quality of the consumers. In the second period, firms infer the type (high or low) of the customers who bought from them in the first period. This fact allows the firms to offer different tariffs to each type of consumer. Note, however, that in period 2, firms cannot infer the type for the customers who bought from its rival in the first period. Thus we assume that firms offer a fixed tariff (unit price and membership fee) to both types of customers on their rival’s turf. In the next section, we compare our model with the short-term membership model of Section 4 with heterogeneous consumers. First, note that in period 2, firms offer a take-it-or-leave offer to each type of consumer who buys their products in the first period (old customers). Firms infer the type of the customersinthesecondperiod; thus,theplansoffereddonotneedtobeincentivecompatible. Second, we are assuming that firms just offer a membership fee and a marginal price to the customers who bought from their rival in the first period. This may be seen as a strong assumption, but more complex mechanisms (or tariff structures) are subject to the vertical uncertainty regarding consumers’ quality preferences, which may be too complex compared to what is observed ((Armstrong 2006)). 48 The frequent-flyer miles strategy used by most of the airlines and credit cards provides a good example of this model. Firms commonly offer extra benefits (including lower prices) to the high-type consumers, whose demand in previous periods is relatively higher compare to the rest of the buyers. 49 First we introduce the following definition. Definition 1. v (p;) : ^ P !R + is separable if there exist functions v : ^ P!R + and h : !R, where v () is strictly decreasing and h () is strictly increasing, such that 48 A natural extension could be that firms offer efficient contracts with different fixed fees that are incentive compatible. We explore this model in Appendix E. 49 Consumers that have a “high” demand in period 1 are frequently assign to a specific status (“gold” or “diamond”) that provides extra benefits in period 2. 89 for all (p;)2 ^ P , v (p;) =v (p)h () To analyze the slope of the quasi-best response functions, we need to introduce a new assumption that helps us to characterize them. Assumption 3. v (p;) : ^ P !R + is separable. An example of the class of indirect utilities that satisfy (A3) are the power functions (or constant elastic of demand) e.g., suppose that u (q;) = p q then v (p;) = 2 4p . Second Period. The standard revealed-preference argument of the previous section needs to be modified to allow for the heterogeneity in the taste parameter for quality. When consumers are heterogeneous, at any pair of first-period prices, there exists a cut-off, x k , for each type k2fL;Hg, such that all consumers with x < x k buy from firm A in the first period, and all consumers with x>x k buy from firm B. Thus the space between 0 and x k is the “turf” of firm A for a ktype consumer, and similarly, to the right of x k is the turf of firm B. The problem of firm A on its own turf is: max p A;k s A k p 2 A;k ;p 2 B;n ;F 2 B;n p 2 A;k ; k wheres A k p 2 A;k ;p 2 B;n ;F 2 B;n min x k ; 1 2 + k[v(p 2 A;k )v(p 2 B;n )+F 2 B;n ] 2t is the share of type-k con- sumers of firmA on its own turf, fork2fL;Hg and H and L 1. Similarly, the problem for firm B on A’s turf is: max ~ p 2 B ; ~ F 2 B X k2fL;Hg k x k s A p 2 A;k ;p 2 B;n ;F 2 B;n p 2 B;n +F 2 B;n The problem on B’s turf is equivalent, and it is omitted. The following proposition characterizes the equilibrium pricing strategy in period 2 on the turf of firm A. Proposition 7. For t small, and x k 2 [x; x], there exists a unique interior equilibrium: i. p 2 A;H <p 2 A;L ; ii. p 2 B;n <c<p 2 A;H <p 2 A;L :; 90 iii. p 2 B;n ;p 2 A;H ;p 2 A;L are defined by (2.34) and (2.30), and, ~ F 2 B = t + P k2fL;Hg k k v (~ p 2 B )v p 2 A;k k k (~ p 2 B ) 2 where (2x H 1) + (2x L 1) (1). The equilibrium pricing strategy in period 2, for consumers with heterogeneous prefer- ences, differs substantially compared to the homogeneous case. 50 In fact, note that on i’s rival turf, marginal cost-based membership is no longer an equilibrium strategy. This result is due to the asymmetry of the game in period 2, that is, on A 0 s turf, firm A offers a differ- entiated unit price to the high- and low-type consumers while firm B offers a membership fee and a marginal price. Firms now have extra information about the consumers’ type of their old customers that can be used to protect against their most valuable customers being poached by the rival. In equilibrium, firms set a unit price below the marginal cost and, below that, its rival price. There are two types of asymmetries on each turf. Let’s analyze A’s turf: first, firm A charges a unit price to their old customers (one for each type), whereas the rival firm uses a unit price and a membership fee. Second, firm A knows the type of their old customers whereas firm B only knows their distribution function. Thus, firm B is disadvantaged on A’s turf. This disadvantage rotates the intercept and the slope of the “quasi”-best response functions (a clockwise rotation). In equilibrium, the optimal strategy form firm B is to set the unit price below the marginal cost, and compensate the loss with the membership fee. A similar analysis follows for B’s turf. The type-k consumer, x k , who is indifferent between buying from firm A in the first period and from firm B in period 2, or vice versa, is such that x k = 1 2 + k p A ;p B ;p 2 A;n ;p 2 B;n + F A ;F B ;F 2 A;n ;F 2 B;n (2.11) for k 2fL;Hg where p A ;p B ;p 2 A;n ;p 2 B;n v(p A )v(p B )+[v(p 2 B;n )v(p 2 A;n )] 2t(1) and (F A ;F B ; F 2 A;n ;F 2 B;n ) = F B F A +(F 2 A;n F 2 B;n ) 2t(1) . Note that the functions () and () in (2.11), do not depend on k; thus, x H > x L if x L > 1 =2, and x H < x L if x H < 1 =2. Similarly, x L = x H if () = () = 0, that is, a symmetric equilibrium. 50 Compare Proposition 7 with Proposition 1. 91 First Period. Working backwards, we now consider equilibrium first-period pricing and consumption decisions. As firms cannot infer consumers’ type in the first period, they offer a standard unit price and membership fee to both types of customers. Thus, the problem of firm A is max p A ;F A X k2fL;Hg k x k ( (p A ;) +F A ) | {z } h1i + k s A k p 2 A;k ; ~ p 2 B ; ~ F 2 B p 2 A;k ; k | {z } h2i + s B k ~ p 2 A ; ~ F 2 A ;p 2 B;k x k ~ p 2 A ; k + ~ F 2 A | {z } h3i The following proposition characterizes the equilibrium pricing of the game. Proposition 8. There exists an interior symmetric equilibrium, that is, x k = 1 2 , such that i. c<p 2 n <p 2 H <p 2 L , s A L <s A H < 1 2 , p 2 B;n ;p 2 A;H ;p 2 A;L are defined by (2.40)-(2.42), and F 2 B = P k2fL;Hg k k [v (p 2 n )v (p 2 k )] k k (p 2 n ) 2 ; ii. p<c and F is defined by (2.37). In a symmetric equilibrium, firms charge a unit (marginal) price below the marginal cost and a positive membership fee in the first period. In period 2, firms can infer consumers’ type, that is, high-type consumers have a larger demand than low-type consumers in period 1, for a fixed membership fee and unit price price. 51 In period 2, we have an asymmetric competition on each firm’s turf; each firm knows the type of its old customers and offers a unit price to each type, whereas its rival doesn’t know their type and will offer them a fixed membership fee and marginal price. Firms compete aggressively for their old high-type consumers, offering them a lower unit price than the one offered to low-type consumers. Their rival offer them a fixed marginal price and membership fee that attracts more high- type than low-type consumers. In equilibrium, the share of high-type consumers who switch to the competing firm in period 2 is smaller than the share of low-type consumers. When consumers have private information about their quality (e.g., firms don’t know the consumers’ type on their rivals’ turf), higher types have a higher marginal utility over all 51 In period 1, firms only know the distribution function of the consumers. 92 units, thus a lower average price per unit, for a fixed marginal price and membership fee. Given that firms compete more aggressive for their old high-type consumers, the inability of the rival to price discriminate between the high and low type, prevents the rival firm from retaining a larger share of high-type consumers in period 2. Note that although firms are symmetric, marginal prices are below marginal cost in both periods for the new customers. As was mentioned, in period 2, each turf is an independent market. Firm i offers a different unit price to its old high- and low-type consumers whereas firm j6= i offers a fixed membership fee and a unit price to the new customers. We can use our previous example to explain under what conditions Mathewson and Winter’s (1997) result does not hold when consumers have heterogeneous taste preferences. Remember that for period 2, we can interpret the permission to allow consumers to enter the shop as the first product and treat the real product offered as product 2 with pricesF 2 j;n andp 2 j;n , respectively. In this case, the demand for product 1 is the expected market share for firm j’s product, E x k s i p 2 i;k ;p 2 j;n ;F 2 j;n and the demand for product 2 is the expected value of the market share multiplied by the individual demand for such product: E x k s i p 2 i;k ;p 2 j;n ;F 2 j;n q (p j ; k ) : When consumers are horizontally and vertically differentiated, these two “goods” are no longer strong complements. 52 Finally, note that in a symmetric equilibrium, the marginal price offered in period 1 is below the marginal cost. This is a surprising result, given that firms are symmetric. The fact that consumers are forward looking implies that the impact of first-period tariffs (membership fee and marginal cost) affect asymmetrically high and low-type consumers in period 2. Thus, the margins of both types differ, which explains why marginal cost pricing is no longer an equilibrium. 53 In Proposition 8 we show that marginal cost-based membership is an equilibrium if the marginal effect of the first-period tariffs are constant for the high- and low-type consumers in period 2, that is, if the marginal effect of F andp onh2i andh3i is the same for high- and low-type consumers (see the proof of Proposition 8 in Appendix A). This asymmetry incentivizes firms to extract as much surplus as possible in the first period, pricing below marginal cost and extracting surplus with the membership fee, i.e., there is a rotation of the inter-temporal best response functions. Corollary 6. In equilibrium, 52 For a complete discussion, see (Tamayo and Tan 2017b) and (Tamayo and Tan 2017a). 53 See (Tamayo and Tan 2017b), (Rochet and Stole 2002), and (Armstrong and Vickers 2001) for a related discussion on this topic. 93 i. as ! 0, p!c, p 2 n !c and p!c ; ii. as ! 1, p!c and p 2 n !c; iii. as ! 0, p!c. Corollary 6 shows that as the share of low-type (or high-type) consumers tends to zero, in equilibrium, the marginal price of the goods tends to the marginal cost for all consumers in period 1 and for the new customers in period 2. Note also that as tends to 0, the marginal price in period 1 tends to c, as in the 2PT static symmetric game with horizontally and vertically differentiated consumers. 54 ComparisonwithShort-TermMembership. Ifmembershipsonlylastforoneperiod -consumersneedtorenewtheirmembershipeachperiod-inperiod2, firmsofferdifferentiated membership fees and unit prices to their high- and low-type customers. Since firms do not know the type of the new customers, they charge them a fixed membership fee and a unit price, similar to the model presented in Section 4. Proposition 9. There exists an interior symmetric equilibrium, that is, x k = 1 2 , in which i. c =p 2 H =p 2 L =p 2 n , F 2 H =F 2 L = 4t 6 , s A L =s A H < 1 2 , and, F 2 B = t 3 ; ii. p =c and F are defined by (2.50). There are substantial differences with respect to the previous case in which consumers do not have to renew their membership in period 2. First, note that firms offer the same unit price (equal to the marginal cost) to their old (high- and low-type) and new customers in period 2, similar to the model in Section 4. Thus, high-type consumers get a higher marginal utility over all units compared to low-type consumers. Moreover,firms charge the same membership fee to both low- and high-type old consumers. On the other hand, the rival firm charges (to the new customers) a lower membership fee and a marginal price equal to the marginal cost. Note that the share of consumers who switch to the rival firm in period 2 is the same for both types of consumers. This contrasts with our previous model, in which firms charge long-term membership fees, in which the share of high-type consumers who switch to the competing firm in period 2, is smaller than the share of low-type consumers. 54 See (Tamayo and Tan 2017b). 94 In this model firms have three powerful tools to extract surplus from consumers: (i) they can discriminate between new and old customers; (ii) given that consumers have elastic demand, firms can infer and may discriminate among high- and low-type consumers; and (iii) firms charge a membership fees again in period 2 and a unit price. Compared to the previous model, in which firms offer long-term membership fees, firms are allowed to extract surplus more efficiently with the membership fee in period 2 from their old customers, that is, firms set marginal prices equal to the marginal cost and extract surplus through the membership fee. Surprisingly, inequilibrium, firmsdonotfindprofitabletodiscriminatebetweenlow-and high-type consumers and offer them the same subscription fee and marginal price. Moreover, note that the tariffs (both the marginal price and the membership fee) do not depend on consumers’ type. This fact suggests that this equilibrium, may be the same equilibrium of a game with a larger set of strategies. 2.6.2 Simulation Here we provide a numerical example to illustrate and compare consumers’ utility, firm’s profits and the share of customers poached across the two tariff structures analyzed in this paper (long- versus short term membership). We the following utility function: u (p; k ) = ( k ) 1 q 1 1 1 1 , that results in a constant elasticity demand curve (with elasticity equal to ), i.e., q (p; k ) = k p fork2fL;Hg. Inthiscase,v (p; k ) = k (1) p 1 . When = 2, (p) = pc p(2cp) . We assume the following parameters: = 1, H = 1:5, H = 1, c = 0:3, = 0:1, and = 2. Under this assumptions, when firms use long-term contracts, in equilibrium, p 2 L = 0:373, p 2 H = 0:329, F 2 = 0:3802, and p 2 = 0:293, in period 2, and p = 0:2995 and F = 1:87, in period 1. Note that in equilibrium F + (p; H ) = 1:859 is positive. When firms use short- term memberships, from Proposition 9, F 2 H =F 2 L = 4 3 andF 2 = 2 3 andF = 2:139. Table 2.1 summarizes the results for the aggregate utility without including the transportation cost in period 2. First, when firms use short-term memberships, they are allowed to extract surplus more efficiently with the subscription fee in period 2 from their old customers. This also incen- tivizestherivaltoincreasesthemembershipfeeforthenewcustomers,reducingtheconsumer welfare in general. As we mentioned in Proposition 8, the share of high-type switchers is smaller than the share of low-type consumers when firms use long-term memberships. Sec- ond, when firms use short-term memberships, the share of switchers is the same for low- and high-type consumers and substantially larger than the case in which firms use long-term memberships. Table 2.2 summarizes the results for the aggregate utility without including the transportation cost in period 1. 95 Table 2.1: Aggregate Utility in Period 2 Table 2.2: Aggregate Utility in Period 1 Table 2.2 shows that there are not substantial differences between the two types of mem- bership models for the aggregate utility in period 1. 2.7 Conclusions In this paper, we study competition and consumer behavior in membership (subscription) markets. Firms that implement a membership model, charge a “membership” fee that al- lows consumers to buy products/services at a given unit price. Here, we study a competitive two-period membership (subscription) market, in which two symmetric firms charge a “mem- bership” fee that allows consumers to buy products or services at a given unit price, in both periods. I explore how (i) the length of the membership (ii) the ability to price discriminate between “old” and “new” customers with the membership fee and the unit price; and (iii) the incentives to price discriminate between different consumers’ tastes affect the competition. Inourbenchmarkmodel,firmsusetwo-periodmembershipfeesandchargeaunit(marginal) price for their products/services on each period. “Old” customers don’t need to pay the mem- bership fee again in period 2, if they buy from the same firm of period 1, but they need to pay a price for each unit they buy in both periods. In the second period, firms discriminate based on prior purchase behavior and charge a single marginal price to their “old” customers and a subscription fee and a differentiated marginal price to their new customers (those who 96 purchased from the rival in period 1). In equilibrium, firms charge higher unit prices to their “old” customers and charge cost-based membership fees to their new customers. In period 1, firms charge cost-based membership fees in equilibrium. Instead, with short-term membership, (e.g., consumers must renew their memberships to buy the products in the second period, so the membership fees are paid in both periods) we show that in equilibrium, firms offer marginal-cost pricing in both periods and extract surplus through the membership fees from both new and current customers. Therefore, Firms don’t discriminate with their marginal price between old and new customers. Instead, they offer differentiated membership fees. Weextendouranalysisfurtherbyassumingthatconsumershave heterogeneous tastepref- erences. When employing long-term membership, firms have incentives to prevent their old most valuable customers from being ‘poached’ by the competitor, thus in equilibrium, firms price discriminate with their membership fee and unit price regarding customer purchased- behavior and volume of demand (second-degree). Instead, with short-term membership, firms don’t discriminate with their unit price but only with their membership fee, between new and old customers, without screening on the taste (vertical) parameter. 97 Table 2.3: Examples Industries with Memberships 98 Appendix A Proof of Proposition 1. (i) The first-order condition of firm A is, p 2 A : s A p 2 A ; ~ p 2 B ; ~ F 2 B 0 (p A ) + @s A p 2 A ; ~ p 2 B ; ~ F 2 B @p 2 A (p A ) = 0 Similarly, the first-order conditions of firm B on A’s turf are ~ p 2 B : x 1 2 v (p 2 A )v (~ p 2 B ) + ~ F 2 B 2t ! 0 ~ p 2 B = q (~ p 2 B ) 2t ~ p 2 B + ~ F 2 B (2.12) h ~ F 2 B i : x 1 2 v (p 2 A )v (~ p 2 B ) + ~ F 2 B 2t ! = 1 2t ~ p 2 B + ~ F 2 B (2.13) We know that firm B will not capture the entire market on turf A then, from (2.13) ~ F 2 B = t (2 1)v (p 2 A ) +v (~ p 2 B ) (~ p 2 B ) 2 (2.14) and from the first-order conditions (2.12) and (2.14), x 1 2 v (p 2 A )v (~ p 2 B ) + ~ F 2 B 2t ! 0 ~ p 2 B q ~ p 2 B = 0 (2.15) Claim: Marginal-cost pricing is a unique equilibrium. Suppose not. Note that in any interior equilibrium, x 1 2 v (p 2 A )v (~ p 2 B ) + ~ F 2 B 2t ! > 0 thus, from (2.15), we conclude that ~ p B =c is a unique equilibrium. Thus, ~ p B =c ~ F 2 B = 2tx tv (p 2 A ) +v (c) 2 and p 2 A is defined by tv (c) + 2tx = p 2 A where (p) 2 (p)v (p). Notethat 0 ()> 0sop 2 A isuniquelydefinedby 1 (tv (c) + 2t ). Note that t < v (c) and from the definitions of p 2 A and ~ F 2 B there exists x < 1 2 such that for 99 x x, there is an unique interior equilibrium. The problem on B’s turf is symmetric, that is, there exists x> 1 2 such that for x x there is a unique interior equilibrium on B’s turf. (ii) Follows directly from the fact that t<v (c) and the definition of p 2 A and ~ F 2 B . Proof Corollary 1. (i) From the problem on firm A’s turf, 2tx +tv(c) 2 (p A ) +v p 2 A = 0 Thus, @p 2 A @x = 2t 2 0 (p 2 A ) +q (p 2 A ) > 0 Similarly, for firm B on A’s turf, @ ~ F 2 B @x =t + q (p 2 A ) 2 @p 2 A @x =t + tq (p 2 A ) 2 0 (p 2 A ) +q (p 2 A ) Similarly on firm B’s turf, @p 2 B @x = 2t 2 0 (p 2 B ) +q (p 2 B ) @ ~ F 2 A @x =t tq (p 2 B ) 2 0 (p 2 B ) +q (p 2 B ) Proof Proposition 2. (i) The problem of the first-period of firm A is max p A ;F A x () ( (p A ) +F A ) +s A p 2 A ; ~ p 2 B ; ~ F 2 B p 2 A + h s B p 2 B ; ~ p 2 A ; ~ F 2 A x () i ~ F 2 A + ~ p 2 A The first-order conditions after using the envelope theorem are [p A ] : @x () @p A ( (p A ) +F A )+x () 0 (p A )+ 8 < : @s A p 2 A ;c; ~ F 2 B @ ~ F 2 B p 2 A @ ~ F 2 B @ 9 = ; @x @p A (2.16) 100 + h s B p 2 B ;c; ~ F 2 A x () i @ ~ F 2 A @x ! @x @p A + 2 4 @s B p 2 B ;c; ~ F 2 A @p 2 B @p 2 B @x + @s B p 2 B ;c; ~ F 2 A @ ~ F 2 A @ ~ F 2 A @x 1 3 5 ~ F 2 A @x @p A [F A ] : ( (p A ) +F A ) +x () @x () @F A 1 + 8 < : @s A p 2 A ;c; ~ F 2 B @ ~ F 2 B p 2 A @ ~ F 2 B @x 9 = ; (2.17) + h s B p 2 B ;c; ~ F 2 A x () i @ ~ F 2 A @x ! + 2 4 @s B p 2 B ;c; ~ F 2 A @p 2 B @p 2 B @ + @s B p 2 B ;c; ~ F 2 A @ ~ F 2 A @ ~ F 2 A @x 1 3 5 ~ F 2 A Note that from (2.17) [F A ] : F A =x () @x () @F A 1 + 8 < : @ A p 2 A ;c; ~ F 2 B @ ~ F 2 B p 2 A @ ~ F 2 B @x 9 = ; + h s B p 2 B ;c; ~ F 2 A x () i @ ~ F 2 A @x ! + 2 4 @s B p 2 B ;c; ~ F 2 A @p 2 B @p 2 B @x + @s B p 2 B ;c; ~ F 2 A @ ~ F 2 A @ ~ F 2 A @x 1 3 5 ~ F 2 A and x () @x () @F A 1 =x () 0 (p A ) @x () @p A 1 But we know that @x @p A = q (p A ) 2t (1) 2 ~ F 20 A (x ) Similarly, @x @F A = 1 2t (1) 2 ~ F 20 A (x ) Thus we conclude that marginal-cost pricing is an equilibrium. 101 (ii) In any symmetric equilibrium, x = 1 2 , p A =c, and F A =t (1) ~ F 20 A (x ) ( 1 2t p 2 A @ ~ F 2 B @x ) v (c)v (p 2 B ) 4t @ ~ F 2 A @x ! " q (p 2 B ) 2t @p 2 B @x + 1 2t @ ~ F 2 A @x 1 # ~ F 2 A Thus we have F A =t (1) ~ F 20 A (x ) 1 2t p 2 A t + tq (p 2 A ) 2 0 (p 2 A ) +q (p 2 A ) v (c)v (p 2 B ) 4t t tq (p 2 B ) 2 0 (p 2 B ) +q (p 2 B ) q (p 2 B ) 2t 2t 2 0 (p 2 B ) +q (p 2 B ) + 1 2t t tq (p 2 B ) 2 0 (p 2 B ) +q (p 2 B ) 1 ~ F 2 A and F =t + tq (p 2 ) 2 0 (p 2 ) +q (p 2 ) p 2 0 (p 2 ) +q (p 2 ) 2 0 (p 2 ) +q (p 2 ) | {z } <1 + q (p 2 ) (v (c)v (p 2 )) 2 0 (p 2 ) +q (p 2 ) > 0 (iii) Note that in a symmetric equilibrium t = p 2 + v (c)v (p 2 ) 2 thus it follows that F > ~ F 2 . Proof of Corollary 2. (i) From Proposition 2 we know that ~ p 2 LP ;p 2 L are such that, tv ~ p 2 LP = p 2 LP v p 2 LP (2.18) v p 2 LP = ~ p 2 LP v ~ p 2 LP (2.19) From Corollary B1 we know that ~ p 2 LP c. Similarly, from Proposition 1 we know that, 102 t [v (c) +v (p 2 2PT )] 2 = p 2 2PT v p 2 2PT (2.20) First note that as t! 0, p 2 2PT ; ~ p 2 LP ;p 2 L !c. Note that, for the symmetric case we have, " @p 2 LP @t @ ~ p 2 LP @t # = 1 0 (p 2 LP ) 0 (~ p 2 LP )q (~ p 2 LP )q (p 2 LP ) " 0 (~ p 2 LP ) q (p 2 LP ) # and, @p 2 2PT @t = 1 0 (p 2 2PT ) q(p 2 2PT ) 2 If t = 0, " @p 2 LP @t @ ~ p 2 LP @t # = " 2 3q(c) 1 3q(c) # and, @p 2 2PT @t = 2 3q (c) If t> 0, " @p 2 LP @t @ ~ p 2 LP @t # = 1 0 (p 2 LP ) (~ p 2 LP ) + 0 (p 2 LP )q (~ p 2 LP ) + (~ p 2 LP )q (p 2 LP ) " (~ p 2 LP ) +q (~ p 2 LP ) q (p 2 LP ) # Note that, 1 0 (p 2 LP )( ~ p 2 LP )+ 0 (p 2 LP )q( ~ p 2 LP )+( ~ p 2 LP )q(p 2 LP ) 0( ~ p 2 LP )+q( ~ p 2 LP ) = 1 0 (p 2 LP ) + 0 ( ~ p 2 LP )q(p 2 LP ) 0( ~ p 2 LP )+q( ~ p 2 LP ) < 1 0 (p 2 LP ) + q(p 2 LP ) 2 and, @p 2 2PT @t = 1 0 (p 2 2PT ) + q(p 2 2PT ) 2 Thus, @p 2 2PT @t > @p 2 LP @t 103 In equilibrium, we have that ~ p 2 p 2 LP , thus, s A;LP <s A;2PT . Proof of Proposition 3. (i) From the first order conditions with respect to F i for firms i2fA;Bg, note that, F B F A = 2tx t (v (p A )v (p B )) + (p A ) 2 (p B ) 2 (2.21) Using (2.21) in the first order conditions with respect to p i for i2fA;Bg, [p A ] : 0 (p A ) t +v p 2 A v p 2 B + (p A ) 2 (p B ) 2 (2.22) q (p A ) (p A )q (p A ) (p A ) 2 + t 2 x t + v (p A )v (p B ) 2 = 0 [p B ] 0 (p B ) t +v (p B )v (p A ) + (p B ) 2 (p A ) 2 (2.23) q (p B ) (p B )q (p B ) (p B ) 2 +tx t 2 + v (p B )v (p A ) 2 = 0 Claim: Marginal cost pricing is not an equilibrium. Suppose that marginal cost pricing is a Nash equilibrium for both firms, then, q (c) t 2 +x t = 0 which is a contradiction. Proof of Proposition 4 (i) Follows from Proposition 5 (ii). 104 (ii) The Problem of the first period: let = (p A ;F A ;p B ;F B ) x () ( (p A ) +F A ) +s A p 2 A ;p 2 B ;F B p 2 A + s B p 2 A ;p 2 B ;F 2 A x () p 2 A +F 2 A The first order conditions are, [p A ] x () @p A ( (p A ) +F A ) +x () 0 (p A ) + 8 > > < > > : s A p 2 A ;p 2 B ;F 2 B 0 ^ p 2 A @ ^ p 2 A @x + @ A (p 2 A ;p 2 B ;F 2 B ) @p 2 A ^ p 2 A @ ^ p 2 A @x | {z } a + @ A (p 2 A ;p 2 B ;F 2 B ) @p 2 B ^ p 2 A @ ^ p 2 B @x + @ A (p 2 A ;p 2 B ;F 2 B ) @F 2 B p 2 A @F 2 B @x x () @p A + 8 > < > : s B p 2 A ;p 2 B ;F 2 A x () 0 B @ 0 ^ p 2 A @p 2 A @x | {z } b + @F 2 A @x |{z} d 1 C A 9 > = > ; x () @p A + x () @p A 2 6 6 4 @s B (p 2 A ;p 2 B ;F 2 A ) @p 2 A @p 2 A @x | {z } c + @s B (p 2 A ;p 2 B ;F 2 A ) @p 2 B @p 2 B @ + @s B (p 2 A ;p 2 B ;F 2 A ) @F 2 A @F 2 A @x | {z } e 1 3 7 7 5 ^ p 2 A +F 2 A and, [F A ] : @x () @F A ( (p A ) +F A ) +x () + x () @F A 8 > > < > > : s A p 2 A ;p 2 B ;F 2 B 0 ^ p 2 A @ ^ p 2 A @x + @s A (p 2 A ;p 2 B ;F 2 B ) @p 2 A ^ p 2 A @ ^ p 2 A @x | {z } a + @s A (p 2 A ;p 2 B ;F 2 B ) @p 2 B ^ p 2 A @ ^ p 2 B @x + @s A (p 2 A ;p 2 B ;F 2 B ) @F 2 B p 2 A @F 2 B @x 105 + 8 > < > : s B p 2 A ;p 2 B ;F 2 A x () 0 B @ 0 ^ p 2 A @p 2 A @x | {z } b + @F 2 A @x |{z} d 1 C A 9 > = > ; x () @F A + x () @F A 2 6 6 4 @s B (p 2 A ;p 2 B ;F 2 A ) @p 2 A @p 2 A @x | {z } c + @s B (p 2 A ;p 2 B ;F 2 A ) @p 2 B @p 2 B @x + @s B (p 2 A ;p 2 B ;F 2 A ) @F 2 A @F 2 A @x | {z } e 1 3 7 7 5 ^ p 2 A +F 2 A Using envelope theorem (e.g. a +b +c = 0 and d +e = 0) for [p A ], (p A ) +F A +x () 0 (p A ) x () @p A 1 + @s A (p 2 A ;p 2 B ;F 2 B ) @p 2 B ^ p 2 A @ ^ p 2 B @x + @s A (p 2 A ;p 2 B ;F 2 B ) @F 2 B p 2 A @F 2 B @x (2.24) + @s B (p 2 A ;p 2 B ;F 2 B ) @p 2 B @p 2 B @x 1 ^ p 2 A +F 2 A = 0 similarly for [F A ], (p A ) +F A +x () @x () @F A 1 (2.25) + @s A (p 2 A ;p 2 B ;F 2 B ) @p 2 B ^ p 2 A @ ^ p 2 B @x + @s A (p 2 A ;p 2 B ;F 2 B ) @F 2 B p 2 A @F 2 B @x + @s B (p 2 A ;p 2 B ;F 2 A ) @p 2 B @p 2 B @x 1 ^ p 2 A +F 2 A = 0 using symmetry, we have that in equilibrium, x () @p A = x () @F A 0 (p A ) which implies that p =c is the unique symmetric equilibrium with, F A = 1 2 x () @F A 1 + (p 2 ) (q 0 (p 2 ) (p 2 c))p 0 B ( ) 4t which is equal to, 106 F A =t 0 (p 2 ) 2 q (p) (p 2 c) (q 0 (p 2 ) (p 2 c)) 4t p 20 B (x )> 0 Similarly, note that using symmetry, F 2 = (p 2 ) 2 t = 3 2 p 2 thus, F A =t + 2 6 6 4 2q 0 (p 2 )q (p 2 ) (p 2 c) 6q (p 2 ) | {z } >0 q (p 2 ) 2 2q (p 2 ) + q 0 (p 2 ) 2 (p 2 c) 2 6q (p 2 ) | {z } >0 3 7 7 5 p 20 B ( )> 0 Proof of Corollary 4 (i) From Corollary 3 we know that p 2 LP < p 2 2PT : Let’s show first that p 2 2PT;NE < p 2 LP . From Corollary 3 we know that (~ p 2 LP ;p 2 LP ) are defined by, tv ~ p 2 LP = p 2 LP v p 2 LP (2.26) v p 2 LP = ~ p 2 LP v ~ p 2 LP (2.27) Moreover, we know that, as t! 0, p 2 2PT;NE ;p 2 2PT ; ~ p 2 LP ;p 2 L !c. From Corollary 3 we also know that, @p 2 LP @t = 1 0 (p 2 LP ) + 0 ( ~ p 2 LP )q(p 2 LP ) 0( ~ p 2 LP )+q( ~ p 2 LP ) > 1 0 (p 2 LP ) + 0 (p 2 LP ) 2 since 0 (~ p 2 LP )q (p 2 LP ) 0 (~ p 2 LP ) +q (~ p 2 LP ) < 0 (~ p 2 )q (p 2 ) 2q (~ p 2 ) > 0 (~ p 2 ) 2 note that as t! 0, @p 2 LP @t ! 2 3q(c) . Similarly, from Proposition 5 we know that p 2PT;NE is defined by, 2t 3 = p 2 2PT;NE 107 Note that, @p 2PTe @t = 2 3 0 (p 2PTe ) and that as t! 0, @p 2PTe @t ! 2 3q(c) ProofofProposition5. Fromthefirstorderconditionswecanfollowasimilarstrategy as Proposition 6 and 4 to show that marginal cost based 2PT is an equilibrium in period 1. The first order conditions with respect to F i is, @x () @F A ( (p A ) +F A ) +x () + @s A c;F 2 A ;c; ~ F 2 B @ ~ F 2 B @ ~ F 2 B @x x () @F A F 2 A + 2 4 B c;F 2 B ;c; ~ F 2 A @F 2 B @ ~ F 2 B @x 1 3 5 x () @F A ~ F 2 A = 0 which follows after using envelope condition. In a symmetric equilibrium, @F 2 A @x = 2t 3 ; @ ~ F 2 B @x = 4t 3 ; @x @F A = 3 2 (3 +) ; @F 2 B @x = 2t 3 ; F 2 A = 2t 3 ; ~ F 2 A = t 3 ; Thus we have, F A = (3 +) 3 Proof of Corollary 5. Let (p 2 A ) = 2 (p 2 A ) +v (p 2 A ), thus, f 1 = 2t (1 2 A ) +t (2x 1)v (c) p 2 A f 2 =4tx + 2t +v (p A )v(p B ) +F B F A +v p l A (1 +)F l A v p l B (1 +)F l B f 3 =v p l A (1 +)F l A v (p A ) +F A v p 2 A 108 Thus using implicit function theorem, @x @p A @s A @p A @p 2 A @p A =q (p A ) 2 6 6 6 4 1 4t 1 8t 0 (p 2 A ) 4tq(p 2 A ) 1 q(p 2 A ) 3 7 7 7 5 note that, 2 6 4 @x @F A @s A @F A @p 2 A @F A 3 7 5 q (p A ) = 2 6 6 4 @ @p A @ A @p A @p 2 A @p A 3 7 7 5 Similarly, @x @F l A @s A @F l A @p 2 A @F l A = 2 6 6 6 4 1 4t 1 8t + 0 (p 2 A ) 4tq(p 2 A ) 1 q(p 2 A ) 3 7 7 7 5 Proof Proposition 6. Lets show first that marginal-cost pricing is an equilibrium. The first order conditions after using the envelope theorem are, [p A ] : @s A @p A p l A +F l A + @x () @p A @s A @p A ( (p A ) +F A ) + (x ()s A ) 0 (p A ) + 8 < : @s A @p A p 2 A + @s A p 2 A ;c; ~ F 2 B @ ~ F 2 B p 2 A @ ~ F 2 B @p A 9 = ; + 2 6 6 4 @s B p 2 B ;c; ~ F 2 A @p 2 B dp 2 B dp A | {z } =0 @s @p A 3 7 7 5 ~ F 2 A = 0 and for F A , [F A ] : @s A @F A p l A +F l A + @x () @F A @s A @F A ( (p A ) +F A ) + (x ()s A ) 109 + 8 < : ds A dF A p 2 A + @s A p 2 A ;c; ~ F 2 B @ ~ F 2 B p 2 A @ ~ F 2 B @F A 9 = ; + 2 6 6 4 @s B p 2 B ;c; ~ F 2 A @p 2 B @p 2 B @F A | {z } =0 @x @F A 3 7 7 5 ~ F 2 A = 0 Similarly for p l A , p l A : @s A @p l A p l A +F l A +s A 0 p l A + @x () @F l A @s A @F l A ( (p A ) +F A ) + 8 < : ds A dF A p 2 A + @s A p 2 A ; ~ p 2 B ; ~ F 2 B @ ~ F 2 B p 2 A @ ~ F 2 B @F l A 9 = ; + 2 6 6 4 @s B p 2 B ;c; ~ F 2 A @p 2 B dp 2 B dp l A | {z } =0 @x @F l A 3 7 7 5 ~ F A = 0 and for F l A , F l A : @s A @F l A p l A +F l A +s A + @x () @F l A @s A @F l A ( (p A ) +F A ) + 8 < : ds A dF A p 2 A + @s A p 2 A ; ~ p 2 B ; ~ F 2 B @ ~ F 2 B p 2 A @ ~ F 2 B @F l A 9 = ; + 2 6 6 4 @s B p 2 B ;c; ~ F 2 A @p 2 B dp 2 B dF l A | {z } =0 @ @F l A 3 7 7 5 ~ F A = 0 Claim. Marginal cost pricing is a symmetric equilibrium, e.g. p A =p l A =c. Proof. Suppose that marginal cost pricing is an equilibrium, e.g. p A = p l A = c. Using Corollary 6, and dividing by q (p A ), [p A ] : @s A @F A F l A + @x () @F A @s A @F A F A + 1 2 s A 110 + 8 < : @s A @F A p 2 A + @s A p 2 A ;c; ~ F 2 B @ ~ F 2 B p 2 A @ ~ F 2 B @F A 9 = ; @x @F A ~ F 2 A = 0 and for F A [F A ] : @s A @F A F l A + @x () @F A @s A @F A F A + 1 2 s A + 8 < : ds A dF A p 2 A + @s A p 2 A ;c; ~ F 2 B @ ~ F 2 B p 2 A @ ~ F 2 B @F A 9 = ; @x @F A ~ F 2 A = 0 Following a similar strategy for the first order conditions with respect to p l A and F l A we conclude that marginal cost pricing is an equilibrium. Note that in a symmetric equilibirum, from Corollary 6, @x () @F A @s A @F A = 1 8t 0 (p 2 A ) 4tq (p 2 A ) < 0 and, @ ~ F 2 B @F A = 4 2 4 = (2 +) 4 Similarly from (2.8), s A = 1 2 (p 2 A ) 2t v (c)v (p 2 A ) 4t Thus we have the following system of equations 1 8t + 0 (p 2 A ) 4tq (p 2 A ) F A = 1 8t + 0 (p 2 ) 4tq (p 2 A ) F l A + (p 2 ) 2t + v (c)v (p 2 ) 4t (2.28) p 2 1 4t p 2 0 (p 2 ) 4tq (p 2 A ) + (v (c)v (p 2 )) 8t 1 8t 0 (p 2 A ) 4tq (p 2 A ) F l A = 1 2 (p 2 ) 2t v (c)v (p 2 ) 4t + 1 8t + 0 (p 2 ) 4tq (p 2 ) F A (2.29) 0 (p 2 A ) 4tq (p 2 A ) p 2 A p 2 A (1 +) 4t + (v (c)v (p 2 )) 8t = 0 111 which can be express as, AF A =BF l A +C DF l A =EF A +G where A = 1 8t + 0 (p 2 A ) 4tq(p 2 A ) , B = 1 8t + 0 (p 2 ) 4tq(p 2 A ) , C = (p 2 ) 2t + v(c)v(p 2 ) 4t (p 2 ) 1 4t (p 2 ) 0 (p 2 ) 4tq(p 2 A ) + (v(c)v(p 2 )) 8t , D = 1 8t 0 (p 2 A ) 4tq(p 2 A ) , E 1 8t + 0 (p 2 ) 4tq(p 2 ) G = 1 2 (p 2 ) 2t v(c)v(p 2 ) 4t 0 (p 2 A ) 4tq(p 2 A ) (p 2 A ) (p 2 A ) (1+) 4t + (v(c)v(p 2 )) 8t , thus, F l D EB A = EC A +F Note that, D EB A = 1 8t 0 (p 2 ) 4tq(p 2 ) 1 8t + 0 (p 2 A ) 4tq(p 2 A ) > 0 Similarly, we show that, EC A +F > 0 If we replace back F l in (2.28), we conclude that F > 0. Proof Proposition 7. (i) Note that from the first order conditions of firm A (maximization on its own turf) it follows that, t + k v p 2 A;k p 2 A;k v ~ p 2 B + ~ F B = 0 (2.30) for k =L;H. Note that, t H v (~ p 2 B ) ~ F B H < t L v (~ p 2 B ) ~ F B H < t L v (~ p 2 B ) ~ F B L p 2 A;H < p 2 A;L which implies that p 2 A;H <p 2 A;L . 112 (ii) Similarly from the first order condition for firm B (maximization on A’s turf), [p B ] : ( x H 1 2 + H v (~ p 2 B )v p 2 A;H ~ F 2 B 2t ) H q (c) H q (c) 2t h ~ F 2 B + ~ p 2 B i (2.31) + (1) ( x L 1 2 + L v (~ p 2 B )v p 2 A;L F B 2t ) H q (c)(1) L q (c) 2t h L p 2 B + ~ F 2 B i = 0 and with respect to ~ F 2 B , h ~ F 2 B i : ( x H 1 2 + H v (~ p 2 B )v p 2 A;H ~ F 2 B 2t ) 1 2t h H ~ p 2 B + ~ F 2 B i (2.32) + (1) ( x L 1 2 + L v (~ p 2 B )v p 2 A;L ~ F 2 B 2t ) (1) 1 2t h L ~ p 2 B + ~ F 2 B i = 0 which implies that, ~ F 2 B = t + P k2fL;Hg k k v (~ p 2 B )v p 2 A;k k k (~ p 2 B ) 2 (2.33) where t (2x H 1) +t (2x L 1) (1). Lemma A1. Marginal cost-based two part tariffs is not a Nash equilibrium. Proof A1. Suppose not, e.g., p B =c B . Note that from (2.33), F B = + H v (c)v p 2 A;H + (1) L v (c)v p 2 A;L 2 then from [p B ], t (1) ( H L ) [2x H 2x L ] + H (1) ( H L ) v p 2 A;L v p 2 A;H = 0 which is a contradiction. Thus, it follows that ifjx H x L j is sufficiently small, p B <c. 113 (iii) We now show existence. Note that substituting (2.33) in (2.30), t + t 2 k v ~ p 2 B + 1 2 X k2fL;Hg k k v ~ p 2 B 1 2 X k2fL;Hg k k v p 2 A;k 1 2 X k2fL;Hg k k ~ p 2 B = k v p 2 A;k + p 2 A;k where = P k2fL;Hg k k . Letp 2 A = p 2 A;H ;p 2 A;L and D p 2 A 2 4 (p H ) (1) L q(p 2 A;L ) 2 H q(p 2 A;H ) 2 (p L ) 3 5 where (p k ) = k q(p 2 A;k ) 2 + (1 k ) k q(p 2 A;k ) 2 + 0 p 2 A;k thus, D 1 p 2 A = 1 2 4 (p L ) (1) L q(p 2 A;L ) 2 H q(p 2 A;H ) 2 (p H ) 3 5 where (p H ) (p L ) H q(p 2 A;H ) 2 (1) L q(p 2 A;L ) 2 > 0. Thus, since the inverse exists, from the Implicit Function Theorem, there exists (~ p 2 B ) = H (~ p 2 B ); L (~ p 2 B ) such that ( (~ p 2 B ); ~ p 2 B ) satisfy (2.30), for k2fL;Hg. Note that from (2.31), = (p B c) ( q 0 (p B ) s H B H + (1)s L B H 2 H q (p 2 B ) 2 2t (1) 2 L q (p 2 B ) 2 2t ) (p B c) ( 2 2 H q (p 2 B ) 2 4t + (1) H L q (p 2 B ) 2 4t + (1) H L q (p 2 B ) 2 4t + (1) 2 2 L q (p 2 B ) 2 4t ) +s H B H q (p B ) + (1)s L B H q (p B ) H q (p 2 B ) 2t + H v (p 2 B )v p 2 A;H + (1) L v (p 2 B )v p 2 A;L 2 ! 114 (1) L q (p 2 B ) 2t + H v (p 2 B )v p 2 A;H + (1) L v (p 2 B )v p 2 A;L 2 ! = 0 remember that s k B = x H 1 2 + H[v(c)v(p 2 A;H )]F B 2t , thus we have, = (p B c) ( q 0 (p B ) s H B H + (1)s L B L q (p 2 B ) 2 2t (1) ( H L ) ( H L ) ) | {z } >0 (2.34) q (p B ) (1) ( H L ) ( x H x L + H v (p 2 B )v H (~ p 2 B ) 2t L v (p 2 B )v L (~ p 2 B ) 2t ) | {z } <0 if x L e x it follows that c < ~ p 2 B < p 2 A;H < p 2 A;L . Existence follows from the following fact: note that, as ~ p 2 B !c RHS of (2.34) is negative 55 , and as ~ p 2 B ! p<c RHS is positive. Thus existence follows. Proof Proposition 8. We first show that the solution exists. Note that, @x k @p A = q (p A ) k h 2t (1) + k [q (~ p 2 B ) ~ p 20 B (x k )q (~ p 2 A ) ~ p 20 A (x k )] ~ F 20 A (x k ) ~ F 20 B (x k ) i (2.35) and, @x k @F A = 1 h 2t (1) + k [q (~ p 2 B ) ~ p 20 B (x k )q (~ p 2 A ) ~ p 20 A (x k )] ~ F 20 A (x k ) ~ F 20 B (x k ) i (2.36) Thus, @x k @p A = @x k @F A q (p A ) k 55 This follows from Lemma A1. 115 max p A X k2fL;Hg k x k ( (p A ;) +F A ) + k s A k p 2 A;k ; ~ p 2 B p 2 A;k ; k + k s B k ~ p 2 A ; ~ F 2 A ;p 2 B;k x k ~ p 2 A ; k + ~ F 2 A From the first order conditions and envelope theorem, [p A ] X k2fL;Hg k @x k p A ( (p A ;) +F A ) + X k2fL;Hg k x k 0 (p A ; k ) + 2 4 k @s A k p 2 A;k ; ~ p 2 B ; ~ F 2 B @p B;k @p B;k @x k p 2 A;k ; k + k @s A k p 2 A;k ; ~ p 2 B ; ~ F 2 B @ ~ F 2 B @ ~ F 2 B @x k p 2 A;k ; k 3 5 @x k @p A + 0 @ @s B k ~ p 2 A ; ~ F 2 A ;p 2 B;k @p 2 B;k 1 1 A @x k @p A ~ p 2 A ; k + ~ F 2 A and, [F A ] X k2fL;Hg k @x k F A ( (p A ;) +F A ) + X k2fL;Hg k x k + 2 4 k @s A k p 2 A;k ; ~ p 2 B ; ~ F 2 B @p B;k @p B;k @x k p 2 A;k ; k + k @s A k p 2 A;k ; ~ p 2 B ; ~ F 2 B @ ~ F 2 B @ ~ F 2 B @x k p 2 A;k ; k 3 5 @x k @F A + 0 @ @s B k ~ p 2 A ; ~ F 2 A ;p 2 B;k @p 2 B;k 1 1 A ~ p 2 A ; k + ~ F 2 A @x k @F A Using symmetry, i.e., x k = 1 2 for k2fL;Hg, [p A ] X k2fL;Hg k @x k p A ( (p A ; k ) +F A ) + X k2fL;Hg k 1 2 0 (p A ; k ) 116 + X k2fL;Hg k 2 4 @s A k p 2 A;k ; ~ p 2 B ; ~ F 2 B @p B;k @p B;k @x k p 2 A;k ; k + @s A k p 2 A;k ; ~ p 2 B ; ~ F 2 B @ ~ F 2 B @ ~ F 2 B @x k p 2 A;k ; k 3 5 @x k @p A + X k2fL;Hg k 0 @ @s B k ~ p 2 A ; ~ F 2 A ;p 2 B;k @p 2 B;k 1 1 A @x k @p A ~ p 2 A ; k + ~ F 2 A and, [F A ] X k2fL;Hg k @x k F A ( (p A ; k ) +F A ) + 1 2 + X k2fL;Hg k 2 4 @s A k p 2 A;k ; ~ p 2 B ; ~ F 2 B @p B;k @p B;k @x k p 2 A;k ; k + @s A k p 2 A;k ; ~ p 2 B ; ~ F 2 B @ ~ F 2 B @ ~ F 2 B @x k p 2 A;k ; k 3 5 @x k @F A + X k2fL;Hg k 0 @ @s B k ~ p 2 A ; ~ F 2 A ;p 2 B;k @p 2 B;k 1 1 A ~ p 2 A ; k + ~ F 2 A @x k @F A Let C k 2 4 @s A k p 2 A;k ; ~ p 2 B ; ~ F 2 B @p B;k @p B;k @x k p 2 A;k ; k + @s A k p 2 A;k ; ~ p 2 B ; ~ F 2 B @ ~ F 2 B @ ~ F 2 B @x k p 2 A;k ; k 3 5 ; and, D k 0 @ @s B k ~ p 2 A ; ~ F 2 A ;p 2 B;k @p 2 B;k 1 1 A ~ p 2 A ; k + ~ F 2 A Thus in equilibrium p and F are defined by, [p A ] X k2fL;Hg k @x k p A ( (p A ;) +F A ) + X k2fL;Hg k 1 2 0 (p A ; k ) + X k2fL;Hg k C k q (p A ) k @x k @F A 117 + X k2fL;Hg k D k q (p A ) k @x k @F A and, [F A ] X k2fL;Hg k @x k F A ( (p A ; k ) +F A ) + 1 2 + X k2fL;Hg C k @x k @F A + X k2fL;Hg D k @x k @F A Thus marginal-cost pricing is an equilibrium if C H = C L and D H = D L , which is not true. In equilibrium, F A = 0 @ X k2fL;Hg k @x k @F A 1 A 1 8 < : 1 2 + X k2fL;Hg k @x k @F A (p A ; k ) + X k2fL;Hg C k @x k @F A + X k2fL;Hg D k @x k @F A 9 = ; (2.37) and p is defined by, X k2fL;Hg k q (p) @x k @F A (p; k ) ( k )+ q (p) 2 0 @ X k2fL;Hg k k 1 A +q 0 (p) (pc) X k2fL;Hg k k 2 + + X k2fL;Hg k C k q (p) @x k @F A ( k ) + X k2fL;Hg k D k q (p) @x k @F A ( k ) = 0 where P k2fL;Hg k k @x k @F A P k2fL;Hg k @x k @F A : Note that H P k2fL;Hg k k @x k @F A P k2fL;Hg k @x k @F A = ( H L ) (1) @x L @F A P k2fL;Hg k @x k @F A and, 118 L P k2fL;Hg k k @x k @F A P k2fL;Hg k @x k @F A = ( L H ) @x H @F A P k2fL;Hg k @x k @F A and, H + (1) L P k2fL;Hg k k @x k @F A P k2fL;Hg k @x k @F A = ( H L ) (1) P k2fL;Hg k @x k @F A @x L @F A @x H @F A > 0 Thus, p is such that, X k2fL;Hg k q (p) @x k @F A (p; k ) ( k ) + (2.38) q (p) 2 (1) ( H L ) P k2fL;Hg k @x k @F A @x L @F A @x H @F A +q 0 (p) (pc) X k2fL;Hg k k 2 + +q (p) (1) @x H @F A @x L @F A ( H L ) P k2fL;Hg k @x k @F A 0 @ C H C L +D H D L | {z } >0 1 A = 0 Let, (1) @x H @F A @x L @F A ( H L ) P k2fL;Hg k @x k @F A 0 @ C H C L +D H D L | {z } >0 1 A < 0 Note that as p!c, q (c) 2 (1) ( H L ) P k2fL;Hg k @x k @F A @x L @F A @x H @F A | {z } <0 + q (c) 2 < 0 since that @x L @F A @x H @F A > 0. 56 Thus in equilibrium p < c. To show existence, note that as p!<c X k2fL;Hg k q () @x k @F A (; k ) ( k ) + q () 2 (1) ( H L ) P k2fL;Hg k @x k @F A @x L @F A @x H @F A 56 This follows from (2.36) after imposing symmetry. 119 +q 0 () (pc) X k2fL;Hg k k 2 + +q () (1) @x H @F A @x L @F A ( H L ) P k2fL;Hg k @x k @F A 0 @ C H C L +D H D L | {z } >0 1 A > 0 thus existence follows. (i) In a symmetric equilibrium x k = 1 2 for k2fL;Hg, F 2 n = P k2fL;Hg k k [v (~ p 2 n )v (p 2 k )] k k (~ p 2 n ) 2 (2.39) and (p 2 H ;p 2 L ;p 2 n ) are defined by the following system of equations, p 2 n c ( q 0 p 2 n s H B H + (1)s L B L q (p 2 n ) 2 2t (1) ( H L ) ( H L ) ) (2.40) +q p 2 n (1) ( H L ) H [v (p 2 n )v (p 2 H )] 2t L [v (p 2 n )v (p 2 L )] 2t = 0 t + H v p 2 o;H p 2 o;H v ~ p 2 n + P k2fL;Hg k k v (p 2 n )v p 2 o;k k k (p 2 n ) 2 = 0 (2.41) t + L v p 2 o;L p 2 o;L v p 2 n + P k2fL;Hg k k v (p 2 n )v p 2 o;k k k (p 2 n ) 2 = 0 (2.42) where p 2 H H v p 2 A;H v p 2 B;n > L v p 2 A;L v p 2 B;n , thus, s A H >s A L : Proof of Proposition 9. In the second period, the problem of firm A on its own turf is, 120 max p A;k ;F A;k s A k p 2 A;k ;F A;k ;p 2 B;n ;F 2 B;n p 2 A;k ; k +F A;k wheres A k p 2 A;k ;p 2 B;n ;F 2 B;n min x k ; 1 2 + k[v(p 2 A;k )v(p 2 B;n )]F 2 A;k +F 2 B;n 2t . The first order con- ditions for firm A are, p 2 A;k : 1 2 + k v p 2 A;k v p 2 B;n F 2 A;k +F 2 B;n 2t ! 0 p 2 A;k ; k (2.43) k q p 2 A;k 2t p 2 A;k ; k +F A;k = 0 and F 2 A;k : t + k v p 2 A;k v p 2 B;n +F 2 B;n p 2 A;k ; k = 2F 2 A;k (2.44) Note that from (2.43) and (2.44) its straightforward to show thatp 2 A;k =c fork2fL;Hg. Thus, F 2 A;k = t + k v p 2 A;k v p 2 B;n +F 2 B;n 2 (2.45) Similarly, the problem for firm B is, max p 2 B;n ;F 2 B;n X k2fL;Hg k ( x k 1 2 k v p 2 A;k v p 2 B;n F 2 A;k +F 2 B;n 2t ) p 2 B;n +F 2 B;n The first order conditions are, p 2 B;n : X k2fL;Hg k ( x k 1 2 k v p 2 A;k v p 2 B;n F 2 A;k +F 2 B;n 2t ) 0 p 2 B;n ; H (2.46) k q p 2 B;n ; k 2t p 2 B;n ; k +F 2 B;n and, F 2 B;n : X k2fL;Hg k ( x k 1 2 k v p 2 A;k v p 2 B;n F 2 A;H +F 2 B;n 2t ) (2.47) 121 k 1 2t p 2 B;n ; k +F 2 B;n We first show that marginal-cost pricing is not an optimal strategy for firm B in the next lemma. Lemma A3. If x H 6=x L , marginal cost pricing is not an equilibrium for firm B. Proof Lemma A3. Suppose marginal cost-pricing is an equilibrium, i.e., p 2 B;n =c, then from (2.46) and (2.47), it follows that t (2x H 1) + (1)t (2x L 1) +F 2 A;H + (1)F 2 A;L = 2F 2 B;n where from (2.43) and (2.44) it follows that t +F 2 B;n 2 =F 2 A;k for k2fL;Hg. Thus it follows that (1) (x H x L ) ( H L ) = 0 Thus if x H <x L , p 2 B;n <c, and x H >x L , p 2 B;n >c. Thus, for x H 6=x L , F 2 B;n = 2 3 X k2fL;Hg k ( 2tx k t 2 + H v p 2 B;n v (c) 2 p 2 B;n ; k ) and p 2 B;n is uniquely defined by X k2fL;Hg k k ( 2tx k t 2 k v (c)v p 2 B;n 2 F 2 B;n 2 ) k k p 2 B;n p 2 B;n ; H +F 2 B;n = 0 Note that @F 2 B;n @x k = 4t k 3 2 3 X k2fL;Hg k k k q p 2 B;n 2 + 0 p 2 B;n ; k ! Using the Implicit Function Theorem, 122 @A @p 2 B;n = X k2fLHg k 2 k q p 2 B k k 0 ~ p 2 B k p 2 B +F B k k ~ p 2 B k 0 p 2 B @A @x k = 2t k k Thus, @p 2 B;n @x k = 2t k k P k2fLHg k 2 k q (p 2 B ) + k k 0 (~ p 2 B ) [ k (p 2 B ) +F B ] + k k (~ p 2 B ) k 0 (p 2 B ) The consumer who is indifferent between buying from firmA in the first period and then from firm B in period 2, and buying from firm B in the first period and then from firm A in period 2, is such that x = 1 2 + v (p A ; k )v (p B ; k ) +F B F A + v p 2 B;n ; k F 2 B;n v p 2 A;n ; k +F 2 A;n 2 (1)t Thus, the problem of firm A in period 1 is max p A ;F A X k2fL;Hg k x k ( (p A ;) +F A ) | {z } h1i + k s A k p 2 A;k ;F 2 A;k ; ~ p 2 B ; ~ F 2 B p 2 A;k ; k +F 2 A;k | {z } h2i + s B k ~ p 2 A ; ~ F 2 A ;p 2 B;k ;F B;k x k ~ p 2 A ; k + ~ F 2 A | {z } h3i Using the first-order conditions, the envelope theorem and symmetry, F A = 0 @ X k2fL;Hg k @x k @F A 1 A 1 8 < : 1 2 + X k2fL;Hg k @x k @F A (p; k ) + (2.48) X k2fL;Hg k C k @x k @F A + X k2fL;Hg k D k @x k @F A 9 = ; and p is such that 123 X k2fL;Hg k @x k @p A ( (p; k ) +F ) + k x k 0 (p; k ) 2 + (2.49) + k C k @x k @p A +D k @x k @p A = 0 where C k @s A k p 2 A;k ;F 2 A;k ; ~ p 2 B ; ~ F 2 B @ ~ F 2 B @ ~ F 2 B @x k F 2 A;k and D k k 0 @ @s B k ~ p 2 A ; ~ F 2 A ;p 2 B;k ;F 2 B;k @F 2 B;k @F 2 B;k @x k 1 1 A ~ F 2 A Note that in a symmetric equilibrium @F 2 B;k @x k = 2t k 3 , @F 2 B;n @x k = 4t k 3 , @s B k ( ~ p 2 A ; ~ F 2 A ;p 2 B;k ;F 2 B;k ) @F 2 B;k = @s A k (p 2 A;k ;F 2 A;k ;~ p 2 B ; ~ F 2 B ) @ ~ F 2 B = 1 2t , and, F 2 B;n = t 3 F 2 A;k = 4t 3 . Note that C k and D k are constant for k2fL;Hg. Thus it follows that marginal cost-based membership is an equilibrium, and F A = 0 @ X k2fL;Hg k @x k @F A 1 A 1 8 < : 1 2 + X k2fL;Hg k C k @x k @F A + X k2fL;Hg k D k @x k @F A 9 = ; (2.50) 124 Appendix B: Long-Term Membership with Linear Pricing In this section, we suppose that both firms use LP in both periods and are allowed to discriminate based on customers’ purchase history, similar to (Fudenberg and Tirole 2000). That is, each firm offers a price in the first period, p i , and in the second period, each firm offers a price to its own past customers, p 2 i , as well as a price to those who purchased from its rival, ~ p 2 i for i2fA;Bg . A standard revealed-preference argument implies that at any pair of first-period prices there is a cutoff, x , such that all consumers withx<x buy from firm A in the first period, and all consumers with x > x buy from firm B, in period one. We named the space between 0 and x as the “turf” of firm A, and we consider the space to the right of x as the turf of firm B. Second Period. On the turf of firm A, the problem of firm A is max p 2 A s A p 2 A ; ~ p 2 B p 2 A where s A (p A ; ~ p B ) min x ; 1 2 + v(p 2 A )v( ~ p 2 B ) 2t is the share of consumers on A’s turf who buy from firmA. Similarly, given that firmB will not capture the entire market onA’s turf, the problem of firm B on A’s turf is max p B x s A p 2 A ; ~ p 2 B ~ p 2 B In an interior equilibrium, the two equations that defined (p 2 A ; ~ p 2 B ) are, tv ~ p 2 B = p 2 A (2.51) 2tx tv p 2 A = ~ p 2 B (2.52) where (p) (p)v (p) and (p) (p)q(p) 0 (p) . The following proposition characterizes the equilibrium in period 2. Proposition B1. Suppose (A1) and (A2) are satisfied. (i) In any equilibrium, if x < 1, ~ p 2 B 0, small, there exists x < 1 2 and x > 1 2 such that for x 2 [x; x], there exists a unique interior equilibrium, in which (p 2 A ; ~ p 2 B ) are defined by (2.51) and (2.52), and s A (p 2 A ; ~ p 2 B )x . 125 If x = 1, then from (Armstrong and Vickers 2001), we know that there exists a unique symmetric equilibrium with p 2 i = ~ p 2 j for j6= i and i;j2fA;Bg. That is, the turf of firm A in the second period would be the entire market, and the problem would be symmetric regarding the market share of both firms. Similarly, note that as x ! x, there is no p2P such that the market share for firmB would be positive, that is, at ~ p 2 B =c the indirect utility provided by firm B would not be enough to compensate the transportation cost. Note that the analysis is symmetric for firm B 0 s turf, thus for an interior equilibrium we need that x 2 [x; x]. 57 This interior equilibrium is unique with ~ p 2 j <p 2 i for j6=i and i;j2fA;Bg. That is, the poacher’s price (and demand for the switchers) is strictly lower (higher) than the incumbents’ price. 58 The following corollary characterizes the interior equilibrium in the second period. Corollary B1. For an interior equilibrium, (i) ~ p 20 B (x );p 20 A (x )> 0 and p 20 B (x ); ~ p 20 A (x )< 0 (ii) There exists ~ x B such that @ ~ p 2 B @t < (>)0 if x > (<) ~ t B . (iii:) If 0 (p 2 )< 2q (p 2 ), there exists ~ x A such that @ ~ p 2 A @t > (<)0 if x > (<)~ x A . FirstPeriod. Let’sconsidernowthefirst-periodpricingandconsumers’decisions. Note that we assume that firms have no commitment power, so the prices and market share of the first period affect the second-period pricing strategy. Similarly, as we mentioned before, we assume that consumers are not myopic and they do anticipate the second-period pricing strategy of the firms. Thus, in any interior equilibrium, first-period prices imply that there is a consumer located at x who is indifferent between the two options, such as buying from firm A in period 1 and from firm B in period 2, and buying from firm B in period 1 and from firm A in the second period: v (p A )tx + v ~ p 2 B (1x )t =v(p B ) (1x )t + v ~ p 2 A tx which defines x as x = 1 2 + v (p A )v(p B ) + [v (~ p 2 B )v (~ p 2 A )] 2t (1) 57 See the proof of Proposition B1 for the analysis on B’s turf. 58 This result is similar to FT and consistent with Esteves and Reggiani (2014), who also study a model, in which firms face a demand that can vary with the price level, using a different framework compared to the one used in this paper. 126 Lemma B1. In equilibrium, @x @p A < 0 and @x @p B > 0. Note that v (~ p 2 B )v (~ p 2 A ) is decreasing with respect to x . 59 Moreover, x is decreasing with respect to p A and increasing with respect to p B . Thus, for any p A ;p B , x is uniquely defined. Let = (p A ;p B ; ~ p 2 A ; ~ p 2 B ); then the problem of the firm A in the first period is max p A x () (p A ) +s A p 2 A ; ~ p 2 B p 2 A + s B p 2 B ; ~ p 2 A x () ~ p 2 A Remember that x is the share of consumers who buy from firm A in period 1; s A is the share of consumers who buy in period 2 from firmA and who also buy form firmA in period 1, and s B (p 2 B ; ~ p 2 A )x () is the share of consumers who buy from firm B in period 1, and buy from firm A in period 2 (the switchers). The problem for firm B is symmetric, and we excluded it here. The next proposition characterizes the equilibrium of the game. Proposition B2. (i) There exists an interior symmetric equilibrium in which p>p 2 is such that, (p) + 1 2 0 (p) @x () @p A 1 + (p 2 )q (~ p 2 ) 2t @ ~ p 2 B @x + (~ p 2 )q (p 2 ) 2t @p 2 B @x ~ p 2 = 0 where @x @p A = q (p) 2ft (1) +q (~ p 2 ) ~ p 20 B ( )g (ii) In any interior symmetric equilibrium, x = 1 2 , and, (~ p 2 ;p 2 ) are uniquely defined by tv ~ p 2 = p 2 (2.53) v p 2 = ~ p 2 (2.54) (iii) s A = 1 2 + v(p 2 )v( ~ p 2 ) 2t < 1 2 =x and s B = 1 2 + v( ~ p 2 )v(p 2 ) 2t > 1 2 . 59 Note that p 2 i and ~ p 2 i for i2fA;Bg depends on x , which depends on p A ;p B , i.e., p 2 i =p 2 i (p A ;p B ). To simplify notation, we omit this for the rest of the appendix. 127 Note that in equilibrium, the prices in the first period are higher than the prices in the second period. Intuitively, there are two effects: first, the market in period 2 is divided into two turfs, which makes firms compete more aggressively. Moreover, in order to attract new consumers firms need to offer attractive prices that compensate for the higher transportation cost of the switchers. Similarly, note that in this general model the share of switcher, v( ~ p 2 )v(p 2 ) t , depends on t and is different from 1 =3, which contrasts with previous literature (e.g., (Fudenberg and Tirole 2000)) Appendix B: Proofs Proof of Proposition B1. (i) Note that from the first-order conditions, tv ~ p 2 B = p 2 A (2.55) 2tx tv p 2 A = ~ p 2 B (2.56) Let’s first assume that x = 1; then tv ~ p 2 B = p 2 A tv p 2 A = ~ p 2 B from(ArmstrongandVickers2001), weknowthatthereexistauniquesymmetricequilibrium with p 2 i = ~ p 2 j for i;j2fA;Bg. Similarly, s A (p 2 A ; ~ p 2 B ) = 1 2 <x . Note that in any equilibrium p 2 A > ~ p 2 B for x < 1. Suppose not. That is, suppose p 2 A < ~ p 2 B . Then, from (2.55) and (2.56), 2tx tv p 2 A = ~ p 2 B > p 2 A =tv ~ p 2 B thus, 2t (x 1)>v p 2 A v ~ p 2 B > 0 which is a contradiction. 128 (ii) For x = 1, from (Armstrong and Vickers 2001), we know that there exists a unique symmetric equilibrium, in which ~ p 2 i is such that ~ p 2 i =t. Note that forx < 1, asx ! 0 the intercept of B (p A ) tends to 1 (tv (c)), where B (p A ) is such that p A ; B (p A ) satisfies (2.56), but the slope remains constant. Similarly, note that the intercept and the slop of A (p A ) remain constant, where A (p A ) is such that p A ; A (p A ) satisfies (2.55). Thus, there exists a x such that for > x there exists a unique interior equilibrium. Formally, using (2.55) in (2.56), (2x 1) p 2 A v ~ p 2 B v p 2 A = ~ p 2 B note that as x ! 0, and the fact that in any equilibria p 2 A > ~ p 2 B , we conclude that there exists x such that for x< x, there is not an interior equilibrium. Now let’s show that there exists a unique interior equilibrium for x> x. From the first-order condition of firm A @ (p 2 A ) @p 2 A = 0 (p 2 A ) 2t t +v p 2 A v ~ p 2 B (p A ) (2.57) and in equilibrium we have that @ 2 (p 2 A ) (@p 2 A ) 2 = 00 (p 2 A ) 2t t +v p 2 A v ~ p 2 B (p A ) + 0 (p 2 A ) 2t q p 2 A 0 (p A ) < 0 Similarly the problem of firm B is, @ (p 2 B ) @p 2 B = x 1 2 v (p 2 A )v (~ p 2 B ) 2t 0 p 2 B (p 2 B )q (p 2 B ) 2t In equilibrium, t (2x 1) v p 2 A v ~ p 2 B ~ p 2 B = 0 Note that @ B (p A ) @p A = q (p 2 A ) q (~ p 2 B ) + 0 (~ p 2 B ) Thus, (2.57) can be expressed in terms of p A : @ (p 2 A ) @p 2 A = 0 (p 2 A ) 2t t +v p 2 A v B (p A ) (p A ) (2.58) Note that asp A !c, the right-hand side of (2.58) tends to a positive value, as B (c)>c. 129 Similarly, asp A !p m A the right-hand side of (2.58) tends to a negative value. Thus a solution exists. Moreover, note that @ 2 (p 2 A ) (@p 2 A ) 2 = 00 (p 2 A ) 2t | {z } <0 t +v p 2 A v ( (p A )) (p A ) | {z } >0 if p A <p A + 0 (p 2 A ) 2t q ( (p A )) 0 (p A )q p 2 A (p A ) | {z } <0 @ 2 (p 2 A ) (@p 2 A ) 2 p A =p A = 0 (p 2 A ) 2t q ( (p A )) q (p 2 A ) q (~ p 2 B ) + 0 (~ p 2 B ) q p 2 A (p A ) < 0 Thus, second-order conditions are satisfied and p 2 A is uniquely defined. Proof of Corollary B1. (i) From the Implicit Function Theorem, @p 2 A @x @ ~ p 2 B @x = " @f 1 @p A @f 1 @p B @f 2 @p A @f 2 @p B # 1 " @f 1 @x @f 2 @x # = " 0 (p 2 A ) q (~ p 2 B ) q (p 2 A ) 0 (~ p 2 B ) # 1 " 0 2t # Note that " 0 (p 2 A ) q (~ p 2 B ) q (p 2 A ) 0 (~ p 2 B ) # 1 = 1 0 (p 2 A ) 0 (~ p 2 B )q (p 2 A )q (~ p 2 B ) " 0 (~ p 2 B ) q (~ p 2 B ) q (p 2 A ) 0 (p 2 A ) # thus we have " @p 2 A @x @ ~ p 2 B @x # = 2 6 4 2tq( ~ p 2 B ) 0 (p 2 A ) 0 ( ~ p 2 B )q(p 2 A )q( ~ p 2 B ) 2t 0 (p 2 A ) 0 (p 2 A ) 0 ( ~ p 2 B )q(p 2 A )q( ~ p 2 B ) 3 7 5 Note that 0 (p 2 A ) 0 (~ p 2 B )q (p 2 A )q (~ p 2 B )> 0, thus ~ p 20 B ( )> 0 and p 20 A ( )> 0 130 (ii) From the Implicit Function Theorem, @p 2 A @t @ ~ p 2 B @t = " @f 1 @p A @f 1 @p B @f 2 @p A @f 2 @p B # 1 " @f 1 @t @f 2 @t # which is equal to = 1 0 (p 2 A ) 0 (~ p 2 B )q (p 2 A )q (~ p 2 B ) " 0 (~ p 2 B ) + 2x q (~ p 2 B ) (2x 1) 0 (p 2 A ) 2x q (p 2 A ) # Thus we conclude that there exists ~ x B such that @ ~ p 2 B @t > (<)0 if x > (<)~ x B . Similarly, if 0 (p 2 )< 2q (p 2 ), there exists ~ x A such that @ ~ p 2 A @t > (<)0 if x < (>)~ x A . Proof of Lemma B1. We know that x must be indifferent between the two options, thus, x = 1 2 + v (p A )v(p B ) + [v (~ p 2 B )v (~ p 2 A )] 2t (1) thus we have @x @p A = q (p A ) f2t (1) +q (~ p 2 B ) ~ p 20 B (x )q (~ p 2 A ) ~ p 20 A (x )g Similarly, @x @p B = q (p B ) f2t (1) +q (~ p 2 B ) ~ p 20 B (x )q (~ p 2 A ) ~ p 20 A (x )g Proof of Proposition B2. (i) From the first-order condition, [p A ] : @x () @p A (p A ) +x () 0 (p A ) + 0 p 2 A s A p 2 A ; ~ p 2 B @p 2 A @x + p 2 A @ A (p 2 A ; ~ p 2 B ) p 2 A @p 2 A @x @x () @p A + p 2 A @s A (p 2 A ; ~ p 2 B ) ~ p 2 B @ ~ p 2 B @x + B p 2 B ; ~ p 2 A x () 0 ~ p 2 A @ ~ p 2 A @x @x () @p A + @s B (p 2 B ; ~ p 2 A ) p 2 B @p 2 B @x + @s B (p 2 B ; ~ p 2 A ) p 2 B @ ~ p 2 A @x 1 ~ p 2 A @x () @p A = 0 131 Using the envelope theorem, [p A ] : @x () @p A (p A ) +x () 0 (p A ) + p 2 A @s A (p 2 A ; ~ p 2 B ) ~ p 2 B @ ~ p 2 B @x @x () @p A + ~ p 2 A @s B (p 2 B ; ~ p 2 A ) p 2 B @p 2 B @x ~ p 2 A @x () @p A = 0 Using symmetry e.g. p 2 A =p 2 B , ~ p 2 A = ~ p 2 B and x = 1 2 , [p A ] : (p) + 1 2 0 (p) @x () @p A 1 + (p 2 )q (~ p 2 ) 2t @ ~ p 2 B @x + (~ p 2 )q (p 2 ) 2t @p 2 B @x ~ p 2 = 0 note that in any symmetric equilibria ~ p 20 A ( ) =~ p 20 B ( ), andj~ p 20 B (x )j>jp 20 B (x )j. [p A ] : 2t (p) +t 0 (p) @x () @p A 1 (2.59) + p 2 q ~ p 2 @ ~ p 2 B @x + ~ p 2 q p 2 @p 2 B @x 2t ~ p 2 = 0 Existence follows by the fact that asp!p 2 the right-hand side of (2.59) is negative, and as p!p m the right-hand side is positive. Thus, in equilibrium p>p 2 . (ii) In any interior symmetric equilibrium, x = 1 2 , and (~ p 2 ;p 2 ) are defined by, tv ~ p 2 = p 2 (2.60) v p 2 = ~ p 2 (2.61) 132 Appendix C: Long-Term Contracts with Linear Pricing Suppose now that firm i offers a spot price p i in the first period and a long-term contract that promises to supply the good in both periods at prices p l i , for i2fA;Bg. The second period is similar to previous sections; each firm offers a price to its past customers, p 2 i , as well as a price to those who purchased from its rival, ~ p 2 i , for i2fA;Bg. As we mentioned before we assume that long-term contracts are purchased by customers who mostly prefer that firm’s product, namely, the set [0;s A ] and [s B ; 1] would buy the long term contracts from firm A and B, respectively. Second Period. On the turf of firm A, the problem of firm A is max pa s A p 2 A ; ~ p 2 B s A p 2 A where s A (p 2 A ; ~ p 2 B ) = 1 2 + v(p 2 A )v( ~ p 2 B ) 2t . Similarly, the problem for firm B on A’s turf is, max p B x s A p 2 A ; ~ p 2 B ~ p 2 B In an interior equilibrium, the optimal prices on A’s turf are defined by t (1 2s A )v ~ p 2 B = p 2 A (2.62) t (2x 1)v p 2 A = ~ p 2 B (2.63) We have an equivalent problem on B’s turf. Proposition C1. Suppose (A1) and (A2) are satisfied. (i) In any equilibria, if x < 1 A , ~ p 2 B 0, small, there exists x < 1 2 and x > 1 2 such that for x maxfx;s A g and x < minfx;s B g, there exists a unique interior equilibrium, in which (p 2 A ; ~ p 2 B ) are defined by (2.62) and (2.63), and equivalently on B’s turf. Note that Proposition 8 follows from Proposition 1 with the difference that now we need to consider the margin of the consumers who buy the long-term contract in period 1. Note that this margin is unknown for the firm on the rival’s turf; thus, conditions for an interior equilibrium are tighter. As we mentioned before, these contracts make consumers indifferent between long-term and short-term contracts, i.e., the indirect utility provided by these two 133 contracts for the two periods must be equal, which implies that p l i and p i are such that v p l i (1 +) =v(p i ) +v p 2 i (2.64) fori2fA;Bg. Similarly, the consumer which will be indifferent between A and B is defined by v (p A )tx + v ~ p 2 B (1x )t =v(p B ) (1x )t + v ~ p 2 A tx where ~ p 2 B = ~ p 2 B (s A ;x ) and ~ p 2 A = ~ p 2 A (s B ;x ); then x = 1 2 + v (p A )v(p B ) + (v (~ p 2 B )v (~ p 2 A )) 2t (1) (2.65) Note that prices in the second period depend on x and s i for i = A;B, which depend on p i and p l i . Thus, in order to understand the impact of first-period pricing decision on second-period prices we need to consider the whole system of equations, i.e., (2.62), (2.63), and the equivalent on B’s turf, (2.64) for i2fA;Bg, and (2.65). The following corollary shows this relationship. Corollary C1. For any interior equilibrium, (i) @x @p A , @p 2 A @p A 0 and @ ~ p 2 A @p A ; @s A @p A 0 (i) @x @p l A ; @s A @p l A 0 and @ ~ p 2 A @p A ; @p 2 A @p l A 0 First Period. Let’s consider now the first-period pricing and consumers’ decisions. Let = (p A ;p B ; ~ p 2 A ; ~ p 2 B ), then the problem of firm A in the first period is max p A ;p l A s A p l A (1 +) + (x ()s A ) (p A ) + (s A (p 2 A ; ~ p 2 B )s A ) (p 2 A ) + [s B (p 2 B ; ~ p 2 A )x ()] (~ p 2 A ) s:t: (2.64) and (2.65) Proposition C2. (i) There exists an interior equilibrium, in whichp andp l are defined by (2.72) and (2.73), respectively. 134 (ii) In any interior symmetric equilibrium, x = 1 2 , s A = 1 2 v( ~ p 2 )+(p 2 ) 2t < s A = 1 2 v( ~ p 2 )v(p 2 ) 2t ; (iii) ~ p 2 <p 2 <p l <p. Appendix C: Proofs Proof of Proposition C1 (i) Note that from the first order conditions, t (1 2s A )v ~ p 2 B = p 2 A (2.66) t (2x 1)v p 2 A = ~ p 2 B (2.67) Note that if x = 1s A , then from Armstrong and Vickers (2001) we know that there exist a unique symmetric equilibrium with p 2 i = ~ p 2 j for i;j 2fA;Bg. Note that from Proposition 1, in any equilibrium p 2 A > ~ p 2 B for x < 1s A . And, p 2 A < ~ p 2 B for x < 1s A . (ii) Forx = 1s A , we know that there exist a unique symmetric equilibrium, in which ~ p 2 i is such that ~ p 2 i =t. The rest of the proof follows from Proposition 2. Proof of Corollary C1. Consider the following equations, f 1 =t (1 2s A )v ~ p 2 B p 2 A = 0 (2.68) f 2 =t (2x 1)v p 2 A ~ p 2 B = 0 (2.69) and, p 2 B ; ~ p 2 A are such that, f 3 =t (1 2x )v p 2 B ~ p 2 A = 0 (2.70) f 4 =t (2s B 1)v ~ p 2 A p 2 B = 0 (2.71) f 5 =2t (1)x +t (1) +v (p A )v(p B ) +v ~ p 2 B v ~ p 2 A 135 f 6 =v p l A (1 +)v(p A )v p 2 A f 7 =v p l B (1 +)v(p B )v p 2 B Note that we have 7 unknowns, s A , s B , p 2 A , p 2 B ,~ p 2 A , ~ p 2 B and x , that can be expressed in terms of p A ,p B ;p l A and p l B using the implicit function theorem. 2 6 6 6 6 6 6 6 6 6 6 6 6 4 @s A @p A @s B @p A @x @p A @p 2 A @p A @p 2 B @p A @ ~ p 2 A @p A @ ~ p 2 B @p A 3 7 7 7 7 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 4 @f 1 @s A @f 1 @s B @f 1 @x @f 1 @p 2 A @f 1 @p 2 B @f 1 @ ~ p 2 A @f 1 @ ~ p 2 B @f 2 @s A @f 2 @s B @f 2 @x @f 2 @p 2 A @f 2 @p 2 B @f 2 @ ~ p 2 A @f 2 @p 2 B @f 3 @s A @f 3 @s B @f 3 @x @f 3 @p 2 A @f 3 @p 2 B @f 3 @ ~ p 2 A @f 3 @p 2 B @f 4 @s A @f 4 @s B @f 4 @x @f 4 @p 2 A @f 4 @p 2 B @f 4 @ ~ p 2 A @f 4 @p 2 B @f 5 @s A @f 5 @s B @f 5 @x @f 5 @p 2 A @f 5 @p 2 B @f 5 @ ~ p 2 A @f 5 @p 2 B @f 6 @s A @f 6 @s B @f 6 @x @f 6 @p 2 A @f 6 @p 2 B @f 6 @ ~ p 2 A @f 6 @p 2 B @f 7 @s A @f 7 @s B @f 7 @x @f 7 @p 2 A @f 7 @p 2 B @f 7 @ ~ p 2 A @f 7 @p 2 B 3 7 7 7 7 7 7 7 7 7 7 7 7 7 5 1 | {z } A 2 6 6 6 6 6 6 6 6 6 6 6 6 4 @f 1 @p A @f 2 @p A @f 3 @p A @f 4 @p A @f 5 @p A @f 6 @p A @f 7 @p A 3 7 7 7 7 7 7 7 7 7 7 7 7 5 Note that, 2 6 6 6 6 6 6 6 6 6 6 6 6 4 @f 1 @p A @f 2 @p A @f 3 @p A @f 4 @p A @f 5 @p A @f 6 @p A @f 7 @p A 3 7 7 7 7 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 6 6 6 4 0 0 0 0 q (p A ) q (p A ) 0 3 7 7 7 7 7 7 7 7 7 7 7 5 Similalrly, A = 2 6 6 6 6 6 6 6 6 6 6 6 4 2t 0 0 0 (p 2 A ) 0 0 q (~ p 2 B ) 0 0 2t q (p 2 A ) 0 0 0 (~ p 2 B ) 0 0 2t 0 q (p 2 B ) 0 (~ p 2 A ) 0 0 2t 0 0 0 (p 2 B ) q (~ p 2 A ) 0 0 0 2t (1) 0 0 q (~ p 2 A ) q (~ p 2 B ) 0 0 0 q (p 2 A ) 0 0 0 0 0 0 0 q (p 2 B ) 0 0 3 7 7 7 7 7 7 7 7 7 7 7 5 Then, 136 2 6 6 6 6 6 6 6 6 6 6 6 6 4 @s A @p A @s B @p A @x @p A @p 2 A @p A @p 2 B @p A @ ~ p 2 A @p A @ ~ p 2 B @p A 3 7 7 7 7 7 7 7 7 7 7 7 7 5 =q (p A ) 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 q(p 2 A )q( ~ p 2 B )( 0 ( ~ p 2 A )+q( ~ p 2 A )) 2tq(p 2 A )(~ p A ;~ p B ) 0 (p 2 A ) 2tq(p 2 A ) q( ~ p 2 A )( 0 ( ~ p 2 B )q( ~ p 2 B )) 2t(~ p A ;~ p B ) 0 ( ~ p 2 A )( 0 ( ~ p 2 B )q( ~ p 2 B )) 2t(~ p A ;~ p B ) 1 q(p 2 A ) 0 0 ( ~ p 2 B )q( ~ p 2 B ) (~ p A ;~ p B ) 0 ( ~ p 2 A )+q( ~ p 2 A ) (~ p A ;~ p B ) 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 where (~ p A ; ~ p B ) [ 0 (~ p 2 A ) 0 (~ p 2 B ) (1) +q (~ p 2 A ) 0 (~ p 2 B ) +q (~ p 2 B ) 0 (~ p 2 A )] > 0. Similarly, with respect to p l A 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 @s A @p l A @s B @p l A @x @p l A @p 2 A @p l A @p 2 B @p l A @ ~ p 2 A @p l A @ ~ p 2 B @p l A 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 =q (p A ) (1 +) 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 0 ( ~ p 2 A )q(p 2 A )q( ~ p 2 B )(1)+q( ~ p 2 A )q(p 2 A )q( ~ p 2 B ) 2tq(p 2 A )(~ p A ;~ p B ) 0 (p 2 A ) 2tq(p 2 A ) q( ~ p 2 A )q( ~ p 2 B ) 2t(~ p A ;~ p B ) q( ~ p 2 B ) 0 ( ~ p 2 A ) 2t(~ p A ;~ p B ) 1 q(p 2 A ) 0 q( ~ p 2 B ) (~ p A ;~ p B ) 0 ( ~ p 2 A )(1)+q( ~ p 2 A ) (~ p A ;~ p B ) 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 Proof of Proposition C2. The first order conditions are [p A ] : @s A @p A p l A (1 +) + @x () @p A @s A @p A (p A ) + (x ()s A ) 0 (p A ) + 0 p 2 A s A p 2 A ; ~ p 2 B s A + p 2 A @s A (p 2 A ; ~ p 2 B ) p 2 A p 2 A @ A p 2 A @p 2 A @p A + p 2 A @s A (p 2 A ; ~ p 2 B ) ~ p 2 B d~ p 2 B dp A + s B p 2 B ; ~ p 2 A x () 0 ~ p 2 A d~ p 2 A dp A + 0 B B @ @s B (p 2 B ; ~ p 2 A ) p 2 B dp 2 B dp A |{z} =0 + @s B (p 2 B ; ~ p 2 A ) @ ~ p 2 A d~ p 2 A dp A @s () @p A 1 C C A ~ p 2 A = 0 137 and, p l A : @s A @p l A p l A (1 +) +s A 0 p l A (1 +) + @x () @p l A @s A @p l A (p A ) + s A p 2 A ; ~ p 2 B s A 0 p 2 A dp 2 A dp A + @s A (p 2 A ; ~ p 2 B ) @p 2 A dp 2 A dp A + @s A (p 2 A ; ~ p 2 B ) @ ~ p 2 B d~ p 2 B dp A @s A @p A p 2 A + s B p 2 B ; ~ p 2 A () 0 ~ p 2 A d~ p 2 A dp A + @s B (p 2 B ; ~ p 2 A ) @p 2 B dp 2 B dp A + @s B (p 2 B ; ~ p 2 A ) @ ~ p 2 A d~ p 2 A dp A @x () @p A ~ p 2 A from Corollary C1 we know dp 2 B dp A = 0 and dp 2 B dp l A = 0. Using envelope condition, [p A ] : @s A @p A p l A (1 +)+ @x () @p A @s A @p A (p A )+(x ()s A ) 0 (p A ) p 2 A @s A p 2 A @p 2 A @p A + p 2 A @ A (p 2 A ; ~ p 2 B ) @ ~ p 2 B d~ p 2 B dp A @x () @p A ~ p 2 A = 0 and p l A : @s A @p l A p l A (1 +) +s A 0 p l A (1 +) + @x () @p l A @s A @p l A (p A ) + @s A (p 2 A ; ~ p 2 B ) @ ~ p 2 B d~ p 2 B dp l A @s A @p A p 2 A @x () @p l A ~ p 2 A = 0 In a symmetric equilibrium we have that the following conditions are satisfied, x = 1 2 s A = t (p 2 ) +v (p 2 )v (~ p 2 ) 2t t (1 2s A )v ~ p 2 = p 2 v p 2 = ~ p 2 138 v p l (1 +) =v(p) +v p 2 and, 2 6 6 6 6 6 6 6 6 6 6 6 6 4 @s A @p A @s B @p A @x @p A @p 2 A @p A @p 2 B @p A @ ~ p 2 A @p A @ ~ p 2 B @p A 3 7 7 7 7 7 7 7 7 7 7 7 7 5 =q (p A ) 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 q(p 2 )q( ~ p 2 )( 0 ( ~ p 2 )+q( ~ p 2 )) 2tq(p 2 A )(~ p A ;~ p B ) 0 (p 2 ) 2tq(p 2 A ) q( ~ p 2 )( 0 ( ~ p 2 )q( ~ p 2 )) 2t(~ p A ;~ p B ) 0 ( ~ p 2 )( 0 ( ~ p 2 )q( ~ p 2 )) 2t(~ p A ;~ p B ) 1 q(p 2 ) 0 0 ( ~ p 2 )q( ~ p 2 ) (~ p A ;~ p B ) 0 ( ~ p 2 )+q( ~ p 2 ) (~ p A ;~ p B ) 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 where (~ p A ; ~ p B ) = [ 0 (~ p 2 ) 0 (~ p 2 ) (1) + 2q (~ p 2 ) 0 (~ p 2 )]. Similarly, note that, @x () @p A @s A @p A =q (p A ) " 0 (~ p 2 ) 2 0 (~ p 2 )q (~ p 2 ) (1 +) 2t (~ p A ; ~ p B ) q (p 2 )q (~ p 2 ) 2 2tq (p 2 A ) (~ p A ; ~ p B ) 0 (p 2 ) 2tq (p 2 A ) # Thus we have, [p A ] : @s A @p A p l A (1 +) | {z } >0 + @x () @p A @s A @p A (p A ) | {z } <0 + v (~ p 2 ) + (p 2 ) 2t 0 (p A ) | {z } >0 (2.72) q (p A ) p 2 A 0 (p 2 ) 2tq (p 2 A ) +q (p A ) 0 (~ p 2 ) 2 0 (~ p 2 )q (~ p 2 ) 2t (~ p) ~ p 2 = 0 and for p l A , p l A : @s A @p l A p l A (1 +) +s A 0 p l A (1 +) + @s () @p l A @s A @p l A (p A ) (2.73) + q (~ p 2 B ) 2t d~ p 2 B dp l A @s A @p l A p 2 A @x () @p l A ~ p 2 A = 0 Thus, existence follows as in Proposition B2. 139 AppendixD:LinearPricingwithHeterogeneousConsumers Here we assume that firms use LP in both periods. As we mentioned before, in the second period firms infer the type (high or low) of the customers who bought from them in the first period. This allows the firms to offer differentiated tariffs to each type of consumer. Note, however, that firms cannot infer the type for the customers who bought from their rival in the first period. Thus we assume that firms offer a single price for both types of customers on rival turf. In the model considered in this section, firms offer a take-it-or-leave offer to each type of consumer who bought their products in the first period. That is, firms know the type of the customers in the second period; thus, the plans offered do not need to be incentive compatible. Second, we are assuming that firms simply offer a single price to the set of customers who bought from their rival in the first period. Second Period. The standard revealed-preference argument of section B needs to be modified; when consumers are heterogeneous, at any pair of first-period prices, there is a cutoff, x k , for each type k2fL;Hg, such that all consumers with x<x k buy from firm A in the first period, and all consumers withx>x k buy from firmB. Thus the space between 0 and x k is the “turf” of firm A for a ktype consumer, and similarly to the right of x k as the turf of firm B. The problem of firm A on its own turf is max p A;k X k2fL;Hg k s A k p 2 A;k ; ~ p 2 B p 2 A;k ; k where s A k p 2 A;k ; ~ p 2 B min x k ; 1 2 + k[v(p 2 A;k )v( ~ p 2 B )] 2t is the share of type-k consumers who buy from firmA that lie onA’s turf fork2fL;Hg , H and L 1. The first-order conditions are s A k p 2 A;k ; ~ p 2 B 0 p 2 A;k ; k + @s A k (p 2 A ; ~ p 2 B ) @p 2 A p 2 A;k ; k = 0 (2.74) for k2fL;Hg. Similarly, the problem for firm B on turf A is max p B X k2fL;Hg k x k s A k p 2 A;k ; ~ p 2 B ~ p 2 B ; k and, from the first-order conditions of firm B, 140 X k2fL;Hg k t (2x k 1) k + k 2 k v ~ p 2 B v p 2 A;k k 2 k ~ p 2 B = 0 (2.75) The problem for each firm on B’s turf is analogous to (2.74) and (2.75). Proposition D1. For t small, for x k 2 [x; x], there exists a unique interior equilibrium, in which p A;H <p A;L . The intuition of the proof of Proposition D1 is similar to Proposition B1. There is a lower bound x under which the market share of firm B is 0 at ~ p B = c for a ktype consumer, and it is similar for firm A on B’s turf. As we mentioned before, a priori, given any pair of first-period prices the share of customers who buy from firm A in the first period (and firm B), is not the same for the high- and low-type consumers. The type-k consumer, x k , who is indifferent between buying from firm A in period 1 and then buying from B in period 2, or vice versa, is such that, x k = 1 2 + k p A ;p B ; ~ p 2 A ; ~ p 2 B for k2fL;Hg where (p A ;p B ; ~ p 2 A ; ~ p 2 B ) v(p A )v(p B )+[v( ~ p 2 B )v( ~ p 2 A )] 2t(1) . Note that x H > x L if x L > 1 =2, and x H < x L if x H < 1 =2. Similarly, note that x L = x H if () = 0, that is, a symmetric equilibrium. Using this fact, we characterized the second-period prices in the following corollary. Corollary D1. There exists a ~ x such that if 1 2 < ~ x < x H p 2 A;H < ~ p 2 B < p 2 A;L , and if 1 2 <x H < ~ x , ~ p 2 B <p 2 A;H <p 2 A;L . An analogous result forp 2 B;H ,p 2 B;L and ~ p 2 A follows from Corollary D1. Intuitively, note that when x H = 1, the turf of firm A for the high-type consumer is the entire market; thus, firm B, does not need to set a very low marginal price to attract high- and low-type consumers. On the other hand, if x k = 1 2 , the competition for firm B is more aggressive on A’s turf, thus, in order to attract high-type consumers firm B needs to lower its price below its rival prices. First Period. Remember that firms in the first period cannot infer consumers’ type, so we assume that firms offer a single price in the first period. The problem of firm A in the first period is 141 max p A X k2fL;Hg k x k (p A ; ) + k s A k p 2 A;k ; ~ p 2 B p 2 A;k ; k + s B k ~ p 2 A ;p 2 B;k x k ~ p 2 A ; k Proposition D2. There exists an interior symmetric equilibrium. In this symmetric equilibrium, x k = 1 2 , ~ p 2 < ~ p 2 H < ~ p 2 L and s A k = 1 2 + k[v(p 2 k )v( ~ p 2 )] 2t < 1 2 . Proofs Appedinx D Proof of Proposition D1. Note that from the first order conditions of firm A with respect to p A;k we have, t k v p 2 B = k p 2 A;L k v p 2 A;L thus using implicit function theorem, @ A;k (p B ) @p B = q (p 2 A ) + 0 (p 2 A ) q (p 2 B ) > 0 Then, from the first order conditions with respect to p B on A’s turf, H 0 (p B ) 2t ( 2t (x H 1) + H v (p 2 B )v A;H (p B ) 2t p 2 B ) + (1) L 0 (p B ) 2t ( 2t (x L 1) + L v (p 2 B )v A;L (p B ) 2t p 2 B ) = 0 Note that as p B !c, H 0 (p B ) 2t ( 2t (x H 1) + H v (c)v A;H (c) 2t ) + (1) L 0 (p B ) 2t ( 2t (x L 1) + L v (c)v A;L (c) 2t ) = 0 LHS > 0. Similarly, as p B !p m , LHS is less than zero. Thus existence follows. 142 Chapter 3 Managerial Quality and Productivity Dynamics 1 AchyutaAdhvaryu 2 ,AnantNyshadham 3 andJorgeTamayo 4 1 Thanks to Nick Bloom, Hashem Pesaran, John Van Reenen, Raffaella Sadun, and seminar participants at Clark, Harvard, McGill, Michigan, USC, Columbia, HEC Paris, and the MIT/Harvard/Stanford Meeting on Empirical Management for helpful comments. This research has benefited from support from the Pri- vate Enterprise Development in Low-Income Countries (PEDL) initiative. Tamayo gratefully acknowledges funding from the USC Dornsife INET graduate student fellowship. All errors are our own. 2 University of Michigan & NBER; adhvaryu@umich.edu 3 Boston College & NBER; nyshadha@bc.edu 4 University of Southern California; jtamayo8@gmail.com 143 Abstract What combination of skills and traits makes a good manager? We study this question by matching two years of daily, line-level production data from six garment factories in India to rich survey data on the managerial practices of line supervisors. We structurally esti- mate a non-linear latent factor model that: 1) addresses the common issues of noise and redundancy in comprehensive management survey data and 2) flexibly identifies the con- tributions of different aspects of managerial quality to the various productivity dynamics observed over line-product runs. We measure the contributions of 7 distinct dimensions of managerial quality motivated by previous literature: Tenure, Cognitive skills, Autonomy, Personality, Control, Attention, and Relatability to workers. We find that while Tenure of a supervisor contributes to all aspects of productivity dynamics, additional dimensions of managerial quality such as Attention and Autonomy contribute strongly as well. Cognitive skills and Control impact initial productivity more strongly than the rate of learning, while Autonomy impacts learning and retention most strongly. Other dimensions of Personality and Relatability to workers do not contribute much to productivity outside of their corre- lation with other factors. Additional results indicate that these dimensions of quality are generallyundervaluedinsupervisorpay, withlesseasilyobservedorlessobviouslyproductive dimensions such as Attention and Control particularly underweighted in pay. The different dimensions of quality are not substitutable, implying that firms with shorter tenured or less cognitively skilled supervisors can still increase productivity most cost-effectively by screen- ing on and/or training in Attention and Control. 144 3.1 Introduction Management matters for firm productivity. Both across and within countries, large pro- ductivity gaps exist and are tightly linked to variation in management practices (Bloom and Van Reenen 2007; Bloom and Van Reenen 2011; McKenzie and Woodruff 2016). Re- cent studies from around the world verify that this linkage is at least in part causal, by demonstrating the impacts of general management consulting interventions (Bloom, Eifert, Mahajan, McKenzie, and Roberts 2013; McKenzie and Woodruff 2013; Karlan, Knight, and Udry 2015) on productivity. These studies are useful in that they demonstrate a role for managers in determining firm productivity, and show that intervening to improve the overall quality of management prac- tices can generate meaningful impacts in many contexts. But what dimensions of managerial quality matter the most for productivity? That is, what makes a “good” manager good? And does the market appropriately price these dimensions (i.e., are managers compensated for the features that matter)? These are largely unanswered questions, likely due to two main empirical challenges. First, to study managerial quality, one must extract signal from noisy measures of quality across many dimensions, and relate these underlying signals to produc- tivity in a flexible way. Second, in contexts where productivity dynamics are salient – e.g., learning by doing is critically important in production in many manufacturing sectors (Ar- row 1962; Lucas 1988; Jovanovic and Nyarko 1995) – it is necessary to estimate the effects of managerial practices on the parameters governing these dynamics. Overcoming these challenges is the scope of inquiry of the present study. We study the way in which managerial quality interacts with the learning by doing process, focusing on the case of ready-made garments production in India. We match granular production data from several garment factories in India to rich data from a management survey conducted on all line supervisors to answer the following research questions: Do production teams supervised by better managers start at higher productivity levels? Do they learn faster? Do they retain more learning from previous productions runs? Do they forget previous stock of learning at a lower rate? What managerial characteristics matter most for each dimension of learning (i.e., the intercept or initial productivity, slope of the learning curve, retention of past learning and rate of forgetting)? We begin by documenting the presence and scope of learning in our empirical context. Productivity, as measured by the proportion of target production realized by a line per unit time (denoted “efficiency”), is strongly increasing in experience. Efficiency rises by roughly 50% or more over the life of a production run. 5 This pattern is identical irrespective of 5 Efficiency rises from roughly 40 points when a line first starts production of a garment style to around 145 whether experience is measured as days the line has been producing the current product or cumulative quantity produced to date. 6 Learning curves show strong concavity: learning slows markedly after roughly the first 10 days of an order’s production cycle. We also document the presence of retained learning from previous runs of the same style on a line and the depreciation of this retained stock of learning over the intervening time elapsed between runs. Experience from a previous run contributes roughly 50% of the productivity gains of an equivalent unit of experience from the current run on average, with each log day of intervening time between runs eroding gains by roughly 15-20% (i.e., retained learning is depreciated by roughly 50% after three and a half production weeks away from a style). Next, we analyze the relative contribution of various dimensions of managerial quality to the aspects of productivity dynamics seen in the data. Our structural estimation procedure isolates each quality dimension’s contribution, as well as allows for interactions between di- mensions. We also address the common issues of measurement error and redundancy likely in a large set of survey measures of quality. (That is, many survey measures likely proxy for the same underlying dimension of managerial quality, but it is difficult to know which measure does so with the strongest signal.) Accordingly, to leverage the full breadth of the managerial survey data collected in this context and to explore agnostically the degree to which different managerial characteristics impact these dimensions of the learning curve, we propose a structural estimation of a learning function using a non-linear latent factor measurement system to obtain the inputs of managerial quality, similar to the one used in recent studies of the cognitive and noncognitive components of the skill production func- tion (Cunha, Heckman, and Schennach 2010; Attanasio, Meghir, and Nix 2015; Attanasio, Cattan, Fitzsimons, Meghir, and Rubio-Codina 2015). Ourempiricalanalysisproceedsinthreesteps. Weestimateacanonicallearningfunction, which takes a very similar form to the functions estimated in, e.g.,(Levitt, List, and Syverson 2013), (Benkard2000), and(Kellogg2011), exceptthatweallowfortheparametersgoverning the shape of the learning curve to vary by managers. Second, in the spirit of (Cunha, Heckman, and Schennach 2010), we estimate a nonlinear latent factor model using the data 60 points by the end of the production run. 6 Previous studies have addressed possible endogeneity in the dynamics of production decisions and there- fore the sequence of productivity shocks or innovations by instrumenting for differences in quantity produced each period with demand shifters or the contemporaneous productivity of other production teams (Benkard 2000; Thompson 2001; Levitt, List, and Syverson 2013). By conducting our analysis using a time-based measure of accrued experience (and documenting qualitatively identical patterns as those obtained using quantity based measures), we circumvent this issue. That is, if production is mean 0 conditional on past productivity and determinants of learning and i.i.d. from a stationary distribution each day of the production run, then this sort of endogeneity is not an issue. The similarity in patterns when using time- and quantity- based experience results, as well as robustness of main results to controlling for days left to complete the order, strongly supports this assumption. 146 from our managerial survey to recover information about the joint distribution of k latent factors of managerial quality, and the learning parameters estimated in the first stage using maximum likelihood and minimum distance. In our exploratory analysis, we identify seven distinct factors related to well-studied dimensions of managerial quality: Tenure, Cognitive skills, Autonomy, Personality, Control, Attention, and Relatability to workers. We then draw a synthetic dataset from this joint distribution and estimate, using nonlinear least squares, a CES function for each learning parameter with the 7 factors of managerial quality as arguments. We find that Tenure, Attention, and Autonomy impact all elements of the learning curve strongly. Control and Cognitive skills contributes most substantially to initial productivity, while the contributions of Autonomy and Tenure are strongest for learning and retention. Relatability and Personality do not contribute incrementally to productivity dynamics be- yond their correlations with other factors. Additional results indicate that these dimensions of quality are not substitutable. That is, irrespective of their tenure or cognitive skills, managers can achieve higher productivity by exhibiting enough autonomy, control, and at- tentiveness. This implies that screening on or training in these skills may be quite effective in raising productivity. Complementary analysis on manager pay indicates that some dimensions of managerial quality are also more cost-effective in raising productivity than others. More easily measured dimensions of quality like Tenure, though still undervalued, contribute to wages in closer pro- portions to their impacts on productivity. Less easily observed or less obviously productive dimensions such as Attention and Control are less rewarded. Estimates of pass-through of productivity increases as a result of simulated managerial quality increases to managers’ pay are quite small, ranging from 5% for Control to 48% for Autonomy (20 to 32%, re- spectively when accounting for correlations among factors). In sum, firms could employ more productive line supervisors and more quickly and consistently achieve peak production by better measuring, screening for, training in, and rewarding a broad array of dimensions of managerial quality, particularly less traditionally valued dimensions like Attention and Control. Ourstudycontributestoafast-growingliteratureineconomicsontheimportanceofgood management in organizations across the world (Bandiera, Hansen, Prat, and Sadun 2017; Bloom and Van Reenen 2007; Bloom, Eifert, Mahajan, McKenzie, and Roberts 2013; Ad- hvaryu, Kala, and Nyshadham 2016; Aghion, Bloom, Lucking, Sadun, and Van Reenen 2017; Bloom, Mahajan, McKenzie, and Roberts 2017; Bloom, Brynjolfsson, Foster, Jarmin, Pat- naik, Saporta-Eksten, and Van Reenen 2017; Macchiavello, Menzel, Rabbani, and Woodruff 2015; Schoar 2011; McKenzie and Woodruff 2016). We add to this work by evaluating the 147 relative importance of many (error-ridden) measures of managerial quality simultaneously in one holistic structural analysis, and by calculating the pass-through of managerial quality to pay. Our study is also related to the rich body of work on the role of learning by doing in determining firm productivity dynamics (Levitt, List, and Syverson 2013; Benkard 2000; Kellogg 2011; Atkin, Khandelwal, and Osman 2016). In particular, we answer a pointed call madein(Levitt,List,andSyverson2013)toconduct“moreresearchonthecomplementarities betweenthelearningprocessandmanagerialpractices.” Thecrucialheterogeneityinlearning along the distribution of managerial quality is implicit in much of the earlier work on learning but until the present study has not been directly estimated. The rest of the paper is organized as follows. Section 3.2 explains the garment production process, our data sources, and the construction of key variables. Section 3.3 presents prelimi- nary graphical evidence of productivity dynamics and heterogeneity by various dimensions of managerial quality. Section 3.4 develops a structural model to formalize these relationships. Section 3.5 describes our strategy for estimating the model in three stages and section 3.6 describes the results. Section 3.7 discusses checks and robustness, and section 3.8 concludes. 3.2 Data We use data from two main sources for this study. The first source is line-daily data on productivity and specific style (product being produced by each line each day), and the second is survey data on managerial characteristics and practices at the supervisor level that we match to the production lines they manage. 3.2.1 Production Data We use line productivity data at the daily level for two years, from July 2013 to June 2015, from six garment factories in Bengaluru, India. The data include the style or product the line is working on, the number of garments the line assembles and the target quantity for each day. Target quantities are lower for more complex garments (since lines can produce fewer complex garments in a given day), and therefore are an appropriate way to normalize productivity across lines producing garments of varying complexity. Our primary measure of productivity is efficiency, which equals garments produced divided by the target quantity of that particular garment per day. Efficiency is the global industry standard measure of productivity in garments. The target quantity for a given garment is calculated using a measure of garment com- 148 plexity called the standard allowable minute (SAM). SAM is taken from a standardized global database of garment industrial engineering that includes information on the universe of garment styles. It measures the number of minutes that a particular garment should take to produce. For instance, a line producing a style with SAM of 30 is expected to produce 2 garments per hour per worker on the line. Accordingly, a line of 60 workers producing a style with SAM of 30 for 8 hours in a day will have a daily target of 960 units. 7 If the line produces 600 garments by the end of the day its efficiency would be 600/960 = .625 for that day. We use daily line-level efficiency as the key dependent variable of interest. 8 Fromtheproductivitydata, wecancalculatehowlongaproductionlinehasbeenproduc- ing a particular garment style. We can measure learning-by-doing in 2 ways: as a function of the consecutive number of days that a line has been working on a particular style, or as a function of the cumulative quantity the line has produced of that style to date. By conducting our analysis of learning using a time-based measure of accrued experience (while documenting qualitatively identical patterns using a quantity-based measure of experience), we circumvent the issue of endogenous productivity innovations across unit time. That is, serial correlation in production innovations are less concerning when the unit of experience is deterministic like time rather than stochastic like quantity produced to date. 9 We show graphical evidence using quantity-based experience, but use time-based experience as our preferred measure in the structural estimation as it is more robust to endogeneity concerns. 10 We can also see in the data whether a line is producing a style that it has produced in the past, and how that changes current learning-by-doing. In particular, we define three variablesthatmeasureretainedpriorlearningandforgetting: 1)thenumberofdayssincethe production line last produced the style it is currently producing, 2) the total number of days that the line produced the same style over prior production runs, and 3) the total quantity that the line produced of a particular style prior to the start of the current production run. Of course, these three variables are positive only when lines have produced a particular style more than once and are all 0 when a line is running a style for the first time. Table 3.1 presents summary statistics of key variables of interest. We use data from 120 7 That is, the line has 60 minutes 8 hours 60 workers = 28,800 minutes to make garments that take 30 minutes each, so 28,800/30 = 960 garments by the end of the day. 8 We run all the same analysis with log quantity as the outcome instead of log efficiency and find qual- itatively identical results (see Section 7.3 ). We keep log efficiency as our preferred outcome as this most closely corresponds to outcomes used in related studies like defect rates in (Levitt, List, and Syverson 2013) and labor per unit produced (Benkard 2000) and (Thompson 2012). 9 This issue is discussed and investigated in detail in previous studies. See, e.g., (Thompson 2001). 10 In additional robustness results, we also include days left to the end of each order to control for any reference point effect (i.e., productivity increasing as the end of the order approaches). These results are presented in Appendix .2 and discussed in section 7.3. They appear nearly identical to the main results. 149 production lines with a total of 153 supervisors. 11 Our sample comprises roughly 50,000 production line-date observations, and we observe nearly 2,740 line-style pairings with 88% of lines producing the same style more than once. Mean efficiency is about 0.51 overall, but less than 0.41 on the first day of a new production run. Production runs last for an average of around 15 days and produce on average 6,200 total pieces. Prior experience values are slightly more than the length of time and total quantity of an average order, consistent with lines having on average more than one previous run of experience. On average, the intervening time between runs of the same style on a line is similar in magnitude to the length of a single run. 3.2.2 Management Survey Data Each line is managed by 1 to 3 supervisors who assign workers to tasks and are charged with motivating workers and diagnosing and solving production problems (such as machine mis- alignment or productivity imbalances across the line) to prevent and relieve bottlenecks and keep production on schedule. To measure managerial quality, we conducted a survey of all line supervisors. We drew from several sources to construct the management questionnaire, in particular borrowing heavily from (Lazear, Shaw, Stanton, et al. 2015), (Schoar 2014), (Bloom and Van Reenen 2011) and (Bloom and Van Reenen 2010). The survey consisted of several different modules intended to measure both traditional dimensions of managerial skill like job and industry-specific tenure and cognitive skills as well as leadership style and specific managerial practices that have been emphasized in the literature. Additional mod- ules on personality and risk and time preferences were also administered. Overall the survey covered work history, leadership style, management practices, personality psychometrics, cognitive skills, demographic characteristics and discriminatory attitudes. We comprehensively utilize the entirety of the survey in constructing measures to include in the non-linear factor system. 12 We allocate this full set of measures to factors by first conducting exploratory factor analyses within each module of the survey to determine if measures within a module appeared to inform a single factor or multiple factors. We then pool measures across related modules (e.g., leadership style and managerial practices) and perform the exploratory factor analysis again on this pooled set to check that measures 11 We restrict our analysis to the largest connected set of styles-lines, which includes 120 of the 130 lines for which we have data available. We use the bgl toolbox in matlab to extract the largest connected set. Finally, we use an iterative conjugate gradient algorithm suggested by (Abowd, Creecy, and Kramarz 2002) to solve for the standard normal equations. 12 In the end, we include all measures from the survey except for a few additional demographic (e.g., mode of transportation to work) and work history (e.g., second sources of income and agricultural experience) variables that were irrelevant to the research questions in this study. 150 Table 3.1: Summary Statistics (a) Note: We keep the largest connected set between lines and styles, which corresponds to 96 lines and 1003 styles. Efficiency is equal to the garments produced divided by the target quantity of that particular garment. The target quantity is calculated using a measure of garment complexity called the standard allowable minute (SAM), which is equal to the number of minutes that a particular garment should take to produce. are being correctly mapped to the factor for which they are most informative. 13 We follow (Cunha, Heckman, and Schennach 2010),(Attanasio, Meghir, and Nix 2015),(Attanasio, Cat- tan, Fitzsimons, Meghir, and Rubio-Codina 2015) in conducting this exploratory analysis to define factors and determine the mapping of measures to factors. Like them, we perform orthogonal rotations of the factor loadings to confirm that measures are mapped to the factor they most strongly inform. 13 Note that the measurement system we implement allows for the recovered factors to be correlated with each other, so it is permissible for measures to load incidentally onto other factors. However, we ultimately want to identify each factor from the set of measures which load primarily onto that factor. Accordingly, we check for each mapping that the measure most strongly informs the factor to which it is mapped above all other factors. 151 We first construct factors that capture the traditional dimensions of skill emphasized in the literature. We construct a Tenure factor to measure the importance of on-the-job human capital accumulation as emphasized in the long-standing literature on wage growth andproductivity. WealsoconstructaCognitiveSkillsfactorfromdirectmeasuresofmemory and arithmetic. To inform the Tenure factor, we use 4 measures: total years working, years working in the garment industry, years working as a garment line supervisor, and years supervising the current line. In exploratory factor analysis, these four measures load onto a single eigenvector with an eigenvalue greater than 1 indicating that a single factor summarizes their contribution. In additional pooled analyses with other demographic characteristics, cognitive skills, and managerial measures discussed below, this factor persistently appears as distinct from the other factors and all of these four measures consistently inform this factor more strongly than any other. The literature on productivity contributions industry, firm, and job-specific accrued human capital, is large and well-established (Mincer et al. 1974; Jovanovic 1979; Mincer and Ofek 1982; Topel 1991; Neal 1995; Gibbons and Waldman 2004). Any contribution of additional dimensions of managerial quality described below should be measured after accounting for this long-studied dimension. To inform the Cognitive Skills factor, we use a measure of short-term memory and two measures of arithmetic skill. Digit span recall captures the largest number of digits in an expanding sequence the respondent was able to successfully recall. We use both the number of correct responses on a timed arithmetic test we administered as well as the percent of the attempted problems that had correct responses. Exploratory factor analysis of these three measures yields only 1 factor with a positive eigenvalue. Pooled factor analyses once again show that this factor is distinct from the others and that these three measures inform this factor above all others. 14 The literature on returns to cognitive skills in productivity and earnings is nearly as long-standing and well-established as that for tenure (Boissiere, Knight, and Sabot 1985; Bowles, Gintis, and Osborne 2001). Once again, as has been emphasized in recent studies of the returns to cognitive and non-cognitive skills (Heckman, Stixrud, and Urzua 2006), we must account for, and even benchmark against, these traditional dimen- sions of ability when studying additional dimensions of managerial quality like Autonomy, Personality, and Attention. We next construct two factors meant to capture non-cognitive skills or personality di- mensions and attitudes not readily captured by traditional measures of cognitive skills and 14 The preliminary analyses show that these cognitive skills measures are positively correlated with mea- sures of Autonomy, Attention, Control and Personality discussed below, but an orthogonal varimax rotation confirms that these three measures load more strongly onto a separate factor than those primarily informed by these other measures. 152 tenure. The survey included a standard module for conscientiousness meant to capture commonly measured personality psychometrics. 15 In addition, we collected measures of per- severance, self-esteem, and internal locus of control as well as risk aversion, patience, and Kessler’s psychological distress scale. 16 We started by checking if the two measures of risk and time preferences informed distinct factors. Exploratory factor analysis showed that risk aversion and patience loaded onto the same factor. Analogous factor analysis on the four measures from the personality psycho- metrics module (i.e., conscientiousness, perseverance, psychological distress, self-esteem, and internal locus of control) revealed two distinct factors. Conscientiousness, perseverance, self- esteem, and psychological distress are highly correlated and load onto a single factor, while internal locus of control loads onto a distinct factor. Factor analysis on the pooled set of measures across these two modules yields two distinct factors with internal locus of control loading clearly onto the same factor as risk aversion and patience. Once again additional factor analyses alternately pooling these measures with other modules of the survey confirm that these two factors are distinct and that these measures load more strongly onto these factors than any others. Recent empirical studies have begun to document the importance of personality psychometrics for earnings and productivity (Borghans, Duckworth, Heckman, and Ter Weel 2008; Heckman and Kautz 2012). Next, we pool measures from the two management related modules to construct factors. These two modules measured leadership behaviors with respect to “initiating structure” and “consideration” (Stogdill and Coons 1957) and specific management practices such as production monitoring frequency, problem identification and solving, efforts to meet targets, communication with subordinates and upper level management, and personnel management activities. Additional self-reported measures of issues overcoming worker resistance and motivating workers as well as a self-assessment measure of managerial quality relative to peer supervisors were also collected. We pooled these measures from the two modules together for the exploratory factor analysis to be most agnostic about which dimensions of management styles and practices are being measured by these survey modules. The factor analysis yields two eigenvectors with eigenvalues above 1. Both measures of leadership style (“initiating structure” and “consideration”) load onto the same factor with initiating structure having the higher loading. “Initiating structure” is said to capture the degree to which a manager plays a more active role in directing group 15 Piloting showed that the other Big 5 modules produced measures that were highly correlated with conscientiousness. This is consistent with what other recent studies have found among blue-collar workers in developing countries (Bassi and Nansamba 2017). Accordingly, we did not administer the other Big 5 modules and rely on conscientiousness alone. 16 ModulesforriskandtimepreferenceswereadaptedfromthoseusedintheIndonesianFamilyLifeSurvey. 153 activities; while “consideration” is meant to capture a good rapport with subordinates (Kor- man 1966). These two behaviors are often hypothesized to be somewhat distinct from each other, but the factor analysis shows that in our context initiating structure and considera- tion are highlycorrelated. Nevertheless, both havebeenconsistently validatedas informative measures of successful leadership (Judge, Piccolo, and Ilies 2004). Our two measures of the degree to which the supervisor takes the lead in and responsibility for identifying and solving production problems also load onto this same factor, along with the self-assessment measure of managerial quality relative to peers. Given the higher loading of “initiating structure” and the contributions of our measures of problem identification and solving, we interpret this factor as capturing Autonomy on the part of the supervisor, both in terms of leader- ship style and management practices. The empirical literature on the value of autonomy among lower level managers is small, but a few recent papers on decentralization of man- agement have emphasized the importance of this dimension. (Aghion, Bloom, Lucking, Sadun, and Van Reenen 2017) find that more empowered lower-level management allows for stronger resilience during economic slowdowns. Similarly, (Bresnahan, Brynjolfsson, and Hitt 2002) find that the productivity returns to information technology are highest when management is decentralized. Indeed, (Bloom and Van Reenen 2011) emphasize managerial autonomy/decentralization as an important dimension of managerial quality, drawing from earlier evidence of the value of autonomy at higher levels of organizational hierarchy (Groves, Hong, McMillan, and Naughton 1994). The second factor from these management modules reflects contributions from five man- agerial practice measures: efforts to achieve production targets, production monitoring fre- quency, active personnel management, communication, and issues motivating workers and overcoming resistance. Each of these is meant to measure effort and attention on the part of the supervisor in accomplishing managerial tasks. The first measures the number of dif- ferent practices the supervisor engages in to ensure production targets are met. The second records the number of times in a day the supervisor makes rounds of the production line to identify any production problems. The third measures the number of different practices the supervisor engages in to retain workers, motivate low performing workers, and encourage high performing workers. The fourth measures the frequency of communication regarding production with both workers and upper level managers, with a higher value representing less communication. The fifth measures the frequency with which the supervisor reports issues motivating workers and overcoming resistance to initiatives and change. Accordingly, we interpret this factor as capturing managerial attention. The literature on managerial at- tention is long-standing in theory and has added some recent empirical evidence (Reis 2006; Ellison and Snyder 2014). For example, (Adhvaryu, Kala, and Nyshadham 2016) find that 154 more attentive managers are better able to diagnose and relieve bottlenecks that arise from shocks to worker productivity. The last two measures we analyze are meant to capture demographic similarity between the supervisor and workers on the line they manage and any discriminatory attitudes the supervisor might have regarding demographic characteristics of their workers. The first is a simple count of the number of similarities between supervisor and majority of workers on the line in the following dimensions: age, gender, religion/caste, migrant status, and native language. The second measure is a count of the number of demographic dimensions (total of 9) over which the supervisor expressed no discriminatory preference. These measures load onto the same factor in the exploratory analysis and do not load more strongly onto any other factors in additional pooled factor analyses. In pooled factor analyses this factor appears distinct but weak with a positive eigenvector smaller than one. Nevertheless, we include this additional factor as dimensions of ethnic and other demographic similarity and discrimination have been emphasized in the literature (Hjort 2014). Summary statistics for these measures across all 153 supervisors are presented in Table 3.2. As discussed above, lines have between 1 and 3 permanent supervisors. While we have management characteristics for each manager, productivity data is common across managers of the same line. Co-supervisors generally share all production responsibilities, so it is only appropriate to match the productivity of a given line equally to each of the supervisors responsible. 3.2.3 Pay In additional analysis, we explore the degree to which the contributions of various managerial quality measures to productivity dynamics translate into supervisor pay. Given the difficulty in accurately measuring dimensions of managerial quality, as outlined in our approach below, andthecomplexityandnuanceintherelationshipsbetweendimensionsofqualityandvarious aspects of productivity, we might expect that the firm struggles to appropriately identify and reward supervisor quality. To investigate this, we obtained pay data for each supervisor from the month in which the survey was completed (November 2014). These data include both monthly salary as well as any production bonus earned by the supervisor when the production line exceeds targets. Summary statistics for these pay vari- ables are reported in the bottom rows of Table 3.1. Note that there appears only a negligible difference between the monthly salary alone and complete pay inclusive of production bonus. That is, while supervisors can in theory be rewarded for their productivity by way of pro- duction bonuses, these bonuses make up only a small fraction of supervisor compensation. 155 Accordingly, in order to appropriately reward supervisor quality in practice, the firm must adjust monthly salary to reflect quality. We explore the degree to which we observe this occurring below. 156 Table 3.2: Managerial Quality Measures Note: Tenure variables are measure in years. Digit span recall measures the number of correct digits a manager remember from a list of 12 numbers; arithmetic (% correct of attempted) is the ratio of the number of correct answers in a math test with 16 questions to the number of questions attempted; arithmetic (number correct) counts the number of correct answers in a math test with 16 questions; initiating structure capture the degree to which a manager plays a more active role in directing group activities (range 30 to 50) and consideration capture a good rapport with subordinates (range 32 to 55); autonomous problem solving (range -3 to 2) and identifying production problem (range 1 to 7) measure the ability of the managers to identify and solve production problems alone; locus of controls is an index from -15 to 1; risk averse and patience are index from 0 to 4; monitoring frequency is the number of rounds of the line to monitor production (range 2 to 5); effort to achieve targets is a composite index of dummy variables that measure the activities the supervisors reports engaging in to ensure that production targets are met (range 0 to 5); active personnel management is constructed analogously for activities related to reinforcing high level performance from star and under-performer workers (range 3 to 13); lack of communication measures the frequency of communication regarding production with both workers and upper level managers (range 3 to 18); issues motivating workers, resistance measures the frequency with which the supervisor reports issues motivating workers and overcoming resistance to initiatives and change (range 5 to 18) demographic similarity measures the similarities between the managers and the workers (range 0 to 9) and egalitarianism measures the preferences of the managers about the workers of the line (range 0 to 3). 157 3.3 Graphical Motivation Before adapting the canonical function shared by most recent empirical studies of learning- by-doing to allow for heterogeneity across managers, we present graphical evidence that illustrates the learning patterns in our empirical context. 3.3.1 Dynamics of Productivity We first present figures that depict how efficiency evolves as a function of the number of days that a production line has been producing a particular style consecutively. As an al- ternative to the number of days that the line has been producing a style, we also present efficiency as a function of the cumulative quantity that the line has produced to date. 17 As noted above, quantity-based experience measures may be subject to endogenous produc- tion decisions and serial correlation in production volume. That is, if factory management ramps up production for a series of consecutive days, then higher quantity produced one day (and therefore a larger experience increment) would look like it increased productivity on subsequent days through learning erroneously. On the other hand, when the increment of experience is fixed and deterministic like in time-based experience measures, this concern is less salient. Accordingly, we conduct this preliminary analysis using both a quantity-based measure of experience to conform with the convention set by previous studies and a time- based measure to demonstrate robustness to these endogeneity concerns. 18 We demonstrate the robustness of the empirical patterns across both experience measures here; however, in the main estimation, we present results using the experience defined in days producing a style as our preferred measure. Figures 1A and 1B show the learning curve for our two measures of experience of the current run: days line has been producing the current style and cumulative quantity of the current style produced to date, respectively. Both figures reflect that productivity, as measured by efficiency, is increasing and concave in the line’s current experience. Lines start the production of a new style at around 40% efficiency and approach a maximum of around 60% efficiency. The majority of this roughly 50% rise in productivity over the course of a production run occurs over the first 10 production days or first 3000 units produced of a given style. 19 17 The two are highly correlated, with a correlation of over 0.9, but either may plausibly be considered as the appropriate unit of learning. 18 We also control for days left to complete production in the current order as an additional check of reference point type dynamics in productivity. The results are presented Appendix .2. The additional control does not impact the results and so is not included in the preferred specification. 19 We also show the full set of results using log(quantity) instead of log(efficiency) as our measure of productivity. We present these results in Appendix .3, but find that results are qualitatively identical. 158 Figure 1A: Efficiency by Days Running .4 .45 .5 .55 .6 Efficiency 0 3 6 9 12 15 18 21 24 27 30 33 Days Producing Style in Current Run Nonparametric Fit 95% CIs Figure 1B: Efficiency by Quantity Produced .2 .3 .4 .5 .6 .7 Efficiency 0 3000 6000 9000 12000 15000 Cumulative Quantity Produced in Current Run Nonparametric Fit 95% CIs Note: Figures 1A and 1B depict learning curves of efficiency by experience with experience defined by consecutive number of days a style has been running on the production line and cumulative quantity produced to date, respectively. The raw mean of efficiency by bin of experience is depicted in the scatter plot in both figures and the fitted curve (solid line) is the result of a lowess smoothed non-parametric estimation. Dashed lines represent 95% confidence intervals. Experience is trimmed at the 90th percentile in this graphical depiction to ignore outliers, but not from any regression analysis below. Next, we explore the degree to which learning is retained from the past. That is, if a line has produced a style in the past, are the productivity gains accrued during that production run retained when the line starts producing that style again? Does the line start at higher initial levels of productivity in subsequent runs of the same style? Does it have less to learn to achieve peak productivity? Figures 2A and 2B show learning curves analogous to those depicted in Figures 1A and 1B, respectively, but with the data split into first runs of a style on a line and subsequent runs. Figures 2A and 2B show clearly that productivity gains accrued during first runs of a style are indeed retained, with lines starting at higher initial productivity levels and leaving less scope for additional learning. The next pressing question, then, is whether this previous retained learning depreciates with the time elapsed between runs of the same style. That is, if a line accrues productivity gains through experience on a first run of a style, does the effect of these gains on subsequent production runs of the same style vary by how much time has elapsed between runs of the same style. We explore this in Figures 3A and 3B by repeating the exercise depicted in Figures2Aand2B,respectively,butwiththesampleofsubsequentrunsofthesamestyleona Accordingly, we keep log(efficiency) as our preferred measure of productivity as it relates closely to the measures of productivity used in previous studies (e.g., defect rate in (Levitt, List, and Syverson 2013) and labor cost per unit in (Thompson 2012)). 159 Figure 2A: Retention (Prior Days) .35 .4 .45 .5 .55 .6 Efficiency 0 3 6 9 12 15 18 21 24 27 30 33 Days Producing Style in Current Run First Run Produced Before Figure 2B: Retention (Prior Quantity) .3 .4 .5 .6 .7 Efficiency 0 3000 6000 9000 12000 15000 Cumulative Quantity Produced in Current Run First Run Produced Before Note: Figures 2A and 2B depict the results of repeating the exercise from Figures 1A and 1B, respectively, but separately by whether the line has every produced the same style before. Dotted lines represent 83% confidence intervals to emphasize significant differences between the two curves. Experience is trimmed at the 90th percentile in this graphical depiction to ignore outliers, but not from any regression analysis below. line further split by days elapsed since last run. Figures 3A and 3B show clearly that retained productivity gains from prior learning depreciates over the time elapsed before the line produces the same style again. It appears that roughly a third to a half of the productivity value of retained prior learning is depreciated after 12 days (or two full production weeks) of elapsed time between runs of the same style. In summary, the graphical evidence of the productivity dynamics in line-style production run data closely matches the patterns of learning and forgetting presented in previous studies (Benkard 2000; Thompson 2012; Levitt, List, and Syverson 2013). Accordingly, we start in section 3.4 with a model nearly identical to those used in these previous studies, differing mainly by allowing production dynamics to be heterogeneous in the characteristics of the line supervisor. As empirical evidence of this heterogeneity is novel to the literature and a main contribution of this study, we present preliminary evidence of heterogeneity in production dynamics by several supervisor characteristics in the next subsection before formalizing the relationships we find in section 3.4. 160 Figure 3A: Forgetting (Prior Days) .3 .4 .5 .6 .7 Efficiency 0 3 6 9 12 15 18 21 24 27 30 33 Days Producing Style in Current Run First Run 2-11 Days Since Last Run 12+ Days Since Last Run Figure 3B: Forgetting (Prior Quantity) .3 .4 .5 .6 .7 Efficiency 0 3000 6000 9000 12000 15000 Cumulative Quantity Produced in Current Run First Run 2-11 Days Since Last Run 12+ Days Since Last Run Note: Figures 3A and 3B depict the results of repeating the exercise from Figures 2A and 2B, respectively, but further splitting previous runs by the number of days that have elapsed since the style was last produced. Dotted lines represent 83% confidence intervals to emphasize significant differences between the two curves. Experience is trimmed at the 90th percentile in this graphical depiction to ignore outliers, but not from any regression analysis below. 3.3.2 Heterogeneity by Managerial Quality Havingestablishedaclearpatternoflearningdynamicsinourempiricalsetting, wenextturn to heterogeneity by supervisor quality. As discussed above, we focus on seven dimensions of supervisor characteristics: Tenure, Autonomy, Cognition, Personality, Control, Attention and Relatability. These 7 dimensions of managerial quality have been emphasized in pre- vious literature, as mentioned in section 2.2, and are therefore well-motivated as important aspects on which to focus. Here we provide preliminary evidence that suggests how these characteristics relate to the productivity dynamics shown in the figures above. Figures 4A and 4B repeat the exercise from Figures 1A, but splitting the sample into lines managed by supervisors with above and below median tenure and cognitive skills, respectively. 20 For this exercise, we use tenure supervising current line as our measure of tenure (Figure 4A) and digit span recall as our measure of cognitive skills (Figure 4B). Figure 4A shows clearly that lines managed by longer tenured supervisors have higher efficiency at the start of a production run and also appear to learn faster over the life of the product run. The pattern is different in Figure 4B with initial levels of productivity appearing higher 20 For the rest of this section we the use number of days that a production line has been producing a particular style consecutively as our measure of current experience. The time-based experience measure is preferred given the endogeneity concerns discussed in section 2.1 above. 161 for lines managed by supervisors with higher cognitive skills, but no apparent difference in productivity later in the product run. Figure 4A: Tenure Supervising Current Line .3 .4 .5 .6 .7 Efficiency 0 3 6 9 12 15 18 21 24 27 30 33 Days Producing Style in Current Run Low Tenure High Tenure Figure 4B: Digit Span Recall .35 .4 .45 .5 .55 .6 Efficiency 0 3 6 9 12 15 18 21 24 27 30 33 Days Producing Style in Current Run Low Recall High Recall Note: Figures 4A and 4B depict learning curves of efficiency by current-style experience defined by consecutive number of days a style has been running on the production line. We split the sample into lines managed by supervisors with above and below median tenure defined by years supervising current line (4A); and above and below median cognitive skills defined by digit span recall (4B).The fitted curves (solid and dashed lines) are the result of a lowess smoothed non-parametric estimation. Dotted lines represent 83% confidence intervals to emphasize where the curves are significantly different from each other. The number of days a style has been running is trimmed at the 90th percentile in this graphical depiction to ignore outliers, but not from any regression analysis below. Figures 5A and 5B depict analogous comparisons across lines managed by supervisors with above and below median autonomy and attention, respectively. In Figure 5A, we use an index of autonomous problem-solving measuring the degree to which managers identify and solve production problems on their own. In Figure 5B, we use the manager’s reported number of rounds of the line made to monitor production per day as a measure of attention. These figures show a different pattern compared to the two previous graphs. Productivity at the start of a new production run appears indistinguishable across lines managed by more and less autonomous (attentive) supervisors, but subsequent learning appears faster for lines with more autonomous (attentive) supervisors. We next repeat the exercise using two measures of supervisor personality: internal locus of control (Figure 6A) and psychological distress (Figure 6B). Figure 6A shows a higher initial productivity at the start of new production runs for lines managed by supervisors with higher internal locus of control, but subsequent learning appears indistinguishable. Figure 6B shows lines supervised by more psychologically distressed managers start at lower levels of initial productivity, but productivities converge later in the order. 162 Figure 5A: Autonomous Problem-Solving .4 .45 .5 .55 .6 Efficiency 0 3 6 9 12 15 18 21 24 27 30 33 Days Producing Style in Current Run Low Autonomy High Autonomy Figure 5B: Monitoring Frequency .4 .45 .5 .55 .6 Efficiency 0 3 6 9 12 15 18 21 24 27 30 33 Days Producing Style in Current Run Low Monitoring Freq High Monitoring Freq Note: Figures 5A and 5B depict the results of repeating the exercise from Figure 4A, but splitting the sample by supervisors with above and below median managerial autonomy and attention skills, respectively. In Figures 5A we use an index of autonomous problem-solving related to the ability of the managers to identify and solve production problems alone. In figure 5B, we use a monitoring frequency index. The fitted curves (solid and dashed lines) are the result of a lowess smoothed non-parametric estimation. Dotted lines represent 83% confidence intervals to emphasize where the curves are significantly different from each other. The number of days a style has been running is trimmed at the 90th percentile in this graphical depiction to ignore outliers, but not from any regression analysis below. In summary, this preliminary graphical evidence confirms that indeed productivity dy- namics of the production lines vary by our measures of managerial quality. Furthermore, the figures discussed above suggest that the relationship between managerial quality and productivity dynamics of the line differs by dimension of quality. Some dimensions appear to impact both the initial productivity and the rate of learning (e.g., tenure); others seem to contribute mainly to the initial productivity (e.g., cognition and control) or rate of learning (e.g., autonomy and attention). However, this preliminary evidence falls short of a formal investigation of these relation- ships. That is, ultimately we are interested in investigating the simultaneous, incremental contributions of each of these dimensions of quality to each of the aspects of productivity dynamics present in the line-style production run data (i.e., initial level of productivity, rate of learning, degree of retention, and rate of forgetting). Such an exercise requires a more formal modeling of the learning function that both allows for each quality dimension to flex- ibly contribute to the various aspects of productivity dynamics and acknowledges the noise and redundancy inherent in survey measures of managerial quality. 163 Figure 6A: Internal Locus of Control .4 .45 .5 .55 .6 Efficiency 0 3 6 9 12 15 18 21 24 27 30 33 Days Producing Style in Current Run Low LOC High LOC Figure 6B: Psychological Distress .4 .45 .5 .55 .6 Efficiency 0 3 6 9 12 15 18 21 24 27 30 33 Days Producing Style in Current Run Low Conscientiousness High Conscientiousness Note: Figures 6A and 6B depict the results of repeating the exercise from Figure 4A, but splitting the sample by supervisor with high and low internal locus of control and psychological distress, respectively. The fitted curves (solid and dashed lines) are the result of a lowess smoothed non-parametric estimation. Dotted lines represent 83% confidence intervals to emphasize where the curves are significantly different from each other. The number of days a style has been running is trimmed at the 90th percentile in this graphical depiction to ignore outliers, but not from any regression analysis below. 3.4 Model 3.4.1 Learning Function In the previous section, we provided evidence of the learning-by-doing process in our garment factory data and showed preliminary results on how managerial quality impacts productivity dynamics. In this section, we build a theoretical framework that formalizes the relationships implied by the preliminary results presented in the previous section. We start with a learning function with similar intuition and structure to that employed in (Levitt, List, and Syverson 2013), log (S ijt ) = i + i log (E ijt ) + i log (P ij ) [1 + i log (D ij )] +" ijt (3.1) where S ijt is the efficiency of line i2f1;:::;Ng, producing style j2f1;:::;Jg at period t2f1;:::;Tg. 21 E ijt is the experience that line i has in producing style j at date t in the 21 In Appendix .3, we present the results of this estimation using log(quantity produced) on the left-hand side instead of log(efficiency). Given that the results are qualitatively identical but with a smaller R-squared, we continue the rest of the estimation using log(efficiency) on the left-hand side. Given that efficiency is measured as the actual quantity produced exceeding minimum quality standards per worker-hour, it is also a closer analogue to the the defect rates and labor cost per unit used in previous studies (Levitt, List, and 164 current production run, as measured by the number of consecutive days spent producing that style. i measures the initial level of productivity and i the rate of learning of the line i. P ij is line i’s experience with style j in the previous production runs (i.e., the number of total days in the prior production run). D ij is the measure of forgetting, which is defined as the number of days since line i last produced style j. i measures the contribution of previous stock learning (retention) and i is the depreciation rate of previous stock learning (rate of forgetting) of linei. t is a time trend that is included in all specifications. 22 Finally, " ijt , is an idiosyncratic error term. 23 Note that the learning function in equation (3.1) differs primarily from those considered by previous literature (Levitt, List, and Syverson 2013; Benkard 2000; Thompson 2001) in that we allow for the parameters governing the shape of the learning curve ( i , i , i and i ) to vary across lines. This is done to reflect the graphical evidence presented in section 3.3.2 showing that learning curves differ across lines supervised by managers with varying skills and characteristics. However, we cannot tell from the simple exploratory graphs in section 3.3.2 the functional form these relationships take. Accordingly, we next describe the flexible functional form we use to relate each parameter ( i , i , i and i ) to underlying dimensions of managerial quality and to arrive at an estimable model. 3.4.2 ParameterizationofRelationshipbetweenLearningandMan- agerial Quality Here we impose a structural form to understand how managerial quality affects each of the learning parameters. We assume that there are k latent factors that describe managerial quality. We assume that each of the learning parameters depends nonlinearly on these k factors, i.e., i =f ( 1;i ; 2;i ;:::; k;i ) (3.2) where 2 f;; ;g for line i 2 f1;:::;Ng, and k;i is the k-th quality factor. Note we assume that the functions for initial level of productivity (f ), rate of learning (f ), degree of retention (f ) and rate of forgetting (f ) take the same set of underlying factors Syverson 2013; Thompson 2012). 22 The time trend is to account for any incidental serial correlation in productivity which may not reflect actual learning. We also show robustness to the inclusion of an additional control for days left to complete the order as a further check against this type confounding of incidental serial correlation with true learning, perhaps through “reference point” mechanisms. This robustness check is presented in Appendix .2 and does not appear to impact the results. 23 Note that this function also matches closely to that used in and (Benkard 2000) and (Thompson 2001) with the factor allocations of capital ignored, given the fixed man-to-machine ratio in garment factories. 165 as arguments, but want to allow for the contributions of the factors to differ across these functions. We assume that f for 2f;; ;g can be approximated by a Constant Elasticity of Substitution (CES) function. The CES form considered here allows us to explore the degree of complementarity or substitutability between the factors included in the function for each learning parameter. That is, we assume that f takes the following functional form, i =A [ ;1 1;i + ;2 2;i + + ;k k;i ] 1 exp( ;i ) (3.3) where ;k 0 and P k ;k = 1 for 2f;; ;g and line i2f1;:::;Ng. Note that any of the factors can be irrelevant in any of these functions when ;k = 0. determines the elasticity of substitution between the latent factors, which is defined by 1 1 , and A is a factor-neutral productivity parameter. Under this technology, 2 [1; 1]; as approaches 1, thelatentfactorsbecomeperfectsubstitutes, andas approaches1, thefactorsbecome perfect complements. In summary, we assume a common functional form across the learning parameters 2 f;; ;g, but we allow the loadings for each latent factor k ( ;k ) and the degree of com- plementarity ( ) to differ across learning parameters. 3.5 Empirical Strategy Having adapted the canonical learning function to allow different dimensions of managerial quality to flexibly determine the shape of the learning curve, we next develop our strategy for estimating these relationships in the presence of measurement error. Remember that our goal is to be able to estimate equation (3.3) for 2f;; ;g. However, to do so, we must first recover i , i , i and i for the LHS of equation (3.3) by estimating equation (3.1) in our production data, and also extract thek latent factors k;i for the supervisors of each line i from the management survey data. Accordingly, our empirical strategy consists of three steps. First, we estimate equation (3.1) line by line to recover i , i , i , and i for each line i2f1;:::;Ng using ordinary least squares. Second, we follow (Cunha, Heckman, and Schennach 2010; Attanasio, Meghir, and Nix 2015; Attanasio, Cattan, Fitzsimons, Meghir, and Rubio-Codina 2015) in estimating a nonlinear latent factor measurement system using the data from our managerial survey. This step allows us to recover information about the joint distribution (approximated as a mixture oftwonormals)ofk latentfactors( k )underlyingthemultitudeofnoisysurveymeasuresand the learning parameters estimated in the first stage ( i , i , i , i ) using maximum likelihood 166 and minimum distance. We finally draw a synthetic dataset from this joint distribution and estimate equation (3.3) for 2f;; ;g using nonlinear least squares and bootstrapping to obtain the error distribution. 3.5.1 First Stage: Productivity Dynamics Homogenous Learning Function We start by estimating the conventional model of learning-by-doing assuming homogeneous learning parameters across lines. This model matches the specification used in previous studies on learning-by-doing (Levitt, List, and Syverson 2013; Benkard 2000; Thompson 2001) and is represented by equation (3.1) with homogenous parameters for , , , and . We perform this estimation by ordinary least squares using different sets of cross-sectional andtemporalfixedeffects. Inparticular, weincludestylefixedeffectstoaccountforvariation inproductivityduetocomplexityofthestyleandsizeoftheorder, aswellasyear, monthand day of the week fixed effects, to account for common seasonality and growth in productivity across lines. These estimations serve to validate that the patterns observed in Figures 1A through 3B indeed persist in a more formal regression framework and that the functional form in equation (3.1) fits the patterns well. We also use these estimations to demonstrate that the patterns of learning and forgetting are robust to varying sets of controls. These controls include time-varying worker characteristics to account for any compositional changes in the workforce of lines and days left to complete the order throughout the run to account for any reference point effects. Heterogeneous Learning Functions Next, we estimate the learning function from equation (3.1) as it is written, allowing for initial levels of productivity, rate of learning, degree of retention and rate of forgetting to vary across lines. That is, we estimate i , i , i , and i for each line i2f1;:::;Ng in a preferred specification including controls for worker characteristics (age, gender, language, tenure, skill grade, and salary) and fixed effects for style and time (year, month, and day of theweek). Thecontrolsforworkercharacteristicsaremeanttoaccountforanycompositional differences in the workforce across lines and even within line over the production run or across styles. As we discuss below, balance checks across lines managed by supervisors with differing managerial quality show no systematic compositional differences in the work forces across lines. The style fixed effect in addition to the line-specific learning parameters being estimatedamountstoatwo-sidedfixedeffectmodeloflinesmatchedtostyles. Thistwo-sided 167 fixed effect model is analogous to the worker-firm sorting model studied (Abowd, Kramarz, and Margolis 1999) (also known as AKM). 24 Accordingly, we must address, as they do, the potential obstacles to identification of the parameters of interest due to any possible sorting in the match between lines and styles in the data. First, note that to be able to the identify the line and style fixed effects separately, lines must be observed producing different styles for multiple production runs during the sample period, and each style should be observed being produced by multiple lines (not necessarily contemporaneously). Second, identification is possible only within a group of lines and styles that are connected. A group of lines and styles are connected when the group comprises all the styles that have ever matched with any of the lines in the group, and all of the lines at which any of the styles have been matched during the sample period. Third, we assume that the probability of a style being produced by a certain line is conditionally mean independent of contemporaneous, past, or future shocks to the line. Fourth, we assume that there is no complementarity between lines and styles. The third and fourth assumptions are quite strong. For example, if the firm is aware of the heterogeneous productivity dynamics depicted in the figures in section 3.3, it stands to reason that the firm would consider these differences in productivity levels and dynamics when allocating styles so as to optimize overall productivity. This type of sorting on the basis of learning dynamics (and, implicitly, any underlying managerial characteristics) would be a violation of the assumptions inherent in the two-sided fixed effect (AKM) model we have proposed. However, if either the firm does not actively measure and analyze these differences in dynamics or the underlying managerial characteristics, or the firm is incapable of practicing this type of optimal allocation of styles to lines due to difficulty in forecasting the arrival of future orders and/or a high cost of leaving lines vacant to await optimally matched orders in the future, then we might expect that assumptions 3 and 4 might actually hold in the data. It is difficult to know which might the be the case, so choose to simply test using Monte Carlo simulation whether the additively separable representation of line and style effects in equation (3.1) is sufficient to capture any line-style sorting. We also test empirically whether managers of differing quality tend to produce styles of different complexity or orders of differing size on average. Tests for Sorting Bias: Balance Checks and Monte Carlo Simulations To establish the validity of this first stage of our strategy, we check for two types of sorting: workers to managers and styles to managers. A priori, we may expect the workforce compo- 24 We have a two-sided FE model in which the lines and styles map to the firms and workers, respectively, in the context of the AKM model. 168 sitions of lines to be relatively homogeneous; lines are comprised of around 70-80 workers, and line assignments are not determined by the line supervisor. Rather, line supervisors log demand for more workers centrally with the firm’s Human Resources (which is above the the factory level) and these demands queue and get filled on a first come first serve basis. To check that indeed this quasi-random line assignment leads to homogenous work-forces across lines on average, we perform balance checks for worker characteristics by managerial characteristics used in our latent factor measurement system. Tables A1-A5 compare differ- ent characteristics of the workers (efficiency, skill grade, salary, age, tenure, gender, language, and migrant status) for high and low-type managers defined by the 26 different measures included in the measurement system (summarized in Table 3.2). The comparisons in Tables A1-A5 show that the groups are quite balanced across high and low-type managers. Only 29 out of 234 differences are statically significant with significant differences spread across various manager characteristics. Tests of joint significance cannot reject balance overall. 25 We perform similar balance checks for style to manager sorting, checking that the complex- ity of the style being assigned (measured by the target quantity) and the size of the order (schedule quantity) are balanced across these same managerial characteristics. The compar- isons presented in Table A6, once again, show very few (7 of the 52) significant differences, and joint tests fail to reject balance overall. Nevertheless, to further assess if there is any bias due to endogenous sorting of styles to lines in our estimation of the two-sided FE model proposed in equation (3.1), we use a Monte Carlo experiment (following (Abowd, Lengermann, and Pérez-Duarte 2004)) which relies on thein-samplepatternoftheobservedrelationshipsbetweenlinesandstyles. Wefirstestimate themodelinequation(3.1)andkeepalltheobservedcharacteristics, lineandstyleidentifiers, the autocorrelation structure of the residuals, and the estimated coefficients. We generate for each style a style effect, and for each line an initial productivity, rate of learning, retention and forgetting (our proposed decomposition of the line effect) from a normal distribution which resembles the distribution of the line and style effects as estimated in the first step. 26 Finally, we draw idiosyncratic error terms and construct a simulated outcome based on the simulated fixed effects, the observed characteristics and the simulated error terms, and estimate the model using the simulated data. 27 We repeat the procedure 10,000 times, and 25 The incidental individual differences do not appear to systematically match to the pattern of findings presented and discussed below. 26 That is, we compute the mean and standard deviation of the line effect paramters (e.g., initial pro- ductivity, rate of learning, retention and forgetting) and style effects. We simulate the new lines and styles effects using these moments. Note that by construction, each line effect (initial productivity, rate of learning, retention and forgetting rate) and each style effect is endowed with independent effects. 27 We first assume that the errors are i.i.d. across lines and time, and then relax this assumption by using the autocorrelation structure estimated for the residuals. 169 compute the percentage mean bias in absolute value for the coefficients of interest ( i , i , i and i ). If we find minimal bias, we can conclude that the full set of assumptions imposed in this first stage estimation including those related to sorting are valid in the data and proceed to the next stage of our empirical strategy. As discussed in section 3.7 below, we find little evidence of bias in the results of the Monte Carlo experiment. That is, it appears in the data that the firm is not sorting styles to lines on the basis of the relationships between managerial quality and productivity dynamics we find in this study. This is surprising given the clear benefits to the firm from doing so, but seems plausiblegiventhemeasurementandcomputationalcomplexitiesinvolvedinextractingthese insights. That is, the firm was not even storing these granular productivity data prior to our intervention,letaloneanalyzingthem,andthemeasurementofthemanagerialcharacteristics was completed first hand by our research team. Nevertheless, we might imagine that some coarse insights might be gleaned from less rigorous measurement and analysis which might allow the firm to optimize the allocation of styles to lines. Such dynamic optimal assignment would, however, require both predictability of future orders and a willingness to delay the start of an order and leave some lines vacant for some periods of time to achieve a more optimal match of style to line. We find no evidence that lines are left vacant or that lines supervised by managers with differing quality show different patterns of order start and completion. Furthermore, the number of lines completing an order or starting a new order on any given day is rarely more than 1 indicating a limited scope for optimizing the style to line assignment. This evidence is all consistent with a limited predictability of future orders and a high cost of slackness as communicated by factory management. 3.5.2 Second Stage: Latent Factors of Managerial Quality We do not directly observe i . Instead, we observe a set of measurements that can be thought of as imperfect proxies of each factor with an error. We adapt from (Cunha, Heck- man, and Schennach 2010) a non-linear latent factor framework that explicitly recognizes the difference between the available measurements and the theoretical concept used in the production function. We set the number of the latent factors to k = 7, comprised of the following: Tenure, Autonomy, Cognition, Personality, Control, Attention, and Relatability. As discussed in section 3.2.2, we use the original survey module delineations and exploratory factor analyses, following (Cunha, Heckman, and Schennach 2010; Attanasio, Meghir, and Nix 2015; Attanasio, Cattan, Fitzsimons, Meghir, and Rubio-Codina 2015), to map the full set of survey measures to these 7 factors, each corresponding to dimensions of managerial 170 qualitypreviouslyproposedandstudiedintheliterature. Thatis, weletboththeintuitionof the modules and the data itself determine which are the distinct factors and which measures map to each factor. Let m l;k denote the lth available measurement relating to latent factor k. Following (Cunha, Heckman, and Schennach 2010) and (Attanasio, Meghir, and Nix 2015), we assume a semi-log relationship between measurements and factors such that m l;k =a l;k + l;k ln k +" l;k (3.4) where l;k is the factor loading,a l;k is the intercept and" l;k is a measurement error for factor k2KfT;Aut;Cog;P;Ctrl;Att;Rg (Tenure, Autonomy, Cognition, Personality, Control, Attention, and Relatability) and measure l2f1; 2;:::;M k g. Thus, for eachk we construct a set of M k measures. For identification purposes, we normalize the factor loading of the the first measure to be equal to 1 (i.e., 1;k = 1 fork2K). Similarly, log-factors are normalized to have mean zero, so a lk is equal to the mean of the measurement. Finally, " l;k are zero mean measurement errors, which capture the fact that them lk are imperfect proxies. Three assumptions regard- ing the measurements and factors are required for identification. First, we assume that the latent factor and the respective measurement error are independent. Second, we assume that measurement errors are independent of each other. Finally, we assume that each measure is affected by only one factor. 28 Note that the estimation of (3.3) requires the construction of a synthetic dataset from the joint distribution of management factors and estimated learning parameters. We follow (Attanasio, Meghir, and Nix 2015) and augment the set of latent factors with ^ i , ^ i , ^ i and ^ i , estimated in the first stage, and the average of the log of supervisor pay, w i , for each line i. 29 As we explain later in Section 6, we are able to recover i and i for 120 lines, which is the largest connected set, but we are only able to recover i and i for 28 This assumption can be relaxed to allow some subset of measures to inform more than one factor; how- ever, in our setting, these cross-factor loadings are not well-motivated, as factors come from distinct modules of the survey which were designed to capture different aspects of managerial quality. For identification of the system, we need at least two dedicated measures per factor and at least one measure for each factor conditionally independent of the other measures. See (Cunha, Heckman, and Schennach 2010) and (Attana- sio, Meghir, and Nix 2015). Note as discussed in 3.2.2 that in exploratory analyses across pooled sets of measures across modules we find some correlations; however, we always assign the measure to the factor for which its loading is strongest. Note that the factors obtained can be correlated with each other and indeed do appear to be in the final results as shown in the Appendix. Accordingly, this assumption preserves the interpretation of each factor while not restricting that measures assigned to different factors be unrelated. 29 We use total compensation of the supervisor for the month which includes the monthly salary for November 2014, the month in which the management survey was completed, and any production bonus associated with the productivity of the line. 171 99 lines. The 21 lines for which we cannot recover i and i are those that we do not observe producing more than one style multiple times in the observation period. We restrict the sample in the second stage to the number of managers that are in these 99 lines (129 managers) for which we can estimate the full model. 30 Finally, we assume that the learning parameters from the first stage and the log of supervisor pay are measured with no error. 31 Let 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; ^ i ; ^ i ; ^ i ; ^ i ;w i , thus we can express the extended demeaned measurement system in vector notation as, ~ M =MA = ln () + " " (3.5) where is the matrix of factor loadings, " is a vector of measurement errors and " is a diagonal matrix with the standard deviation of the measurement error defined before. 32 In order to capture complementarities in the learning parameter functions, we follow (Cunha, Heckman, and Schennach 2010) and (Attanasio, Meghir, and Nix 2015) in assuming that the joint distribution of the log latent factors, f (), follows a mixture of two normals, f (ln) =f A (ln) + (1)f B (ln) (3.6) where f i () is the joint CDF of a normal distribution with mean vector, i , and variance covariance matrix, i , and mixture weight, 2 [0; 1], for i2fA;Bg. 33 Finally, we assume that the log-factors have mean zero, i.e., A + (1) B = 0 (3.7) Note that if " is normally distributed, the distribution of the observed measurements is 30 We use all 120 lines (153 managers) in the first stage. As a robustness check, we estimate the full results in the second and third stage using only the ^ i and ^ i for all 153 managers lines and omitting the ^ i and ^ i from the model. The insights regarding the and are nearly identical to those in the main results reported below, confirming that restricting attention in the main estimation to the 129 managers of the 99 lines for which we can recover the full set of learning parameters does not meaningfully impact the conclusions we draw. 31 This assumption with respect to the pay measure is similar to that imposed by (Attanasio, Meghir, and Nix 2015) in their extended measurement system. With respect to the learning parameters, we are including constructed variables in our second stage. From the validity of the identification in the first stage, we regard the error remaining in the constructed variables ( ^ i , ^ i , ^ i and ^ i ) to be near 0 asTN!1. In our data, TN = 37; 192. Finally, relaxing this assumption would require multiple measures for each of the learning parameters which we do not have. 32 As we mentioned before we assume that learning parameters and the log of pay are measured with no error. This implies that the corresponding factor loadings are set equal to one in , and the corresponding standard deviations of the error in equal to zero. 33 The departure from the joint normality assumption is important, otherwise the log of the production function would be linear and additively separable in logs (i.e., Cobb-Douglas, as discussed in (Attanasio, Meghir, and Nix 2015)). 172 F (m) = ( m A ; m A ) + (1) ( m B ; m B ) (3.8) where, m A = A (3.9) m B = B (3.10) m A = 0 A + " (3.11) m B = 0 B + " (3.12) Estimation in this second stage proceeds in three steps. First, we construct the set of measures for each latent factor by matching the appropriate survey modules to each of the seven dimensions of quality previously studied in the literature, as discussed in section 3.2.2. Second, we use maximum likelihood to estimate an unconstrained mixture of normals for the distribution of measurements. 34 Using equations (3.7) through (3.12) as restrictions, we perform minimum distance estimation to recover A ; A ; B ; B . Finally, we draw a synthetic dataset from the joint distribution of the learning parameters (and log pay) and factors of managerial quality to produce data for both the LHS and RHS of equation (3.3). 3.5.3 Third Stage: Contributions of Managerial Quality to Produc- tivity Dynamics Remember that our goal is to estimate equation (3.3) for 2f;; ;g. We first recover the learning parameters (initial level of productivity, rate of learning, retention rate and forgetting rate) for the LHS of equation (3.3) for each line by estimating the line-specific learning function in equation (3.1) using ordinary least squares. Second, we estimate a latent factor model similar to (Cunha, Heckman, and Schennach 2010) and (Attanasio, Meghir, and Nix 2015) and recover the joint distribution of the latent factors and the learning parameters obtained in the first stage. That is, from the full set of error-ridden survey measures we observe, we recover the RHS of (3.3). This procedure allows us to construct a synthetic dataset of the factors (RHS) and the learning parameters (LHS). Finally, in the third stage, 34 We use EM algorithm and k-means clustering to select the initial values with uniform initial proportions. We replicate the procedure 10,000 times and select the model with largest loglikelihood. 173 we estimate equations (3.3) for2f i ; i ; i ; i g using nonlinear least squares. We bootstrap this third stage 100 times to construct the standard errors of the estimated coefficients. 3.6 Results In this section, we formally test for the patterns depicted in Section 3. We first report and discuss the results of estimating equation (3.1) assuming homogeneous learning parameters across lines (i.e., ;; ;) to verify that the patterns observed in Figures 1A through 3B persist and are statistically significant in a more formal regression analysis. We then move on to present the results of the regression analysis of the learning function with heteroge- neous parameters, and recover i , i , i and i for each production line. Next, we discuss the measures used in the latent factor model to recover the underlying dimensions of man- agerial quality and the informative content of each. Then, we present the results of the estimation of equation (3.3) for2f i ; i ; i ; i g and perform simulations to investigate how productivity dynamics change with increases in each of the dimensions of managerial quality (i.e., Tenure, Cognitive Skills, Autonomy, Personality, Control, Attention, and Relatability). We perform two types of simulations: independent shocks to each factor analogous to what a focused training might accomplish and correlated shocks using the covariance structure between factors to mirror what focused screening among candidates in the hiring process might accomplish. Finally, we use our procedure to investigate the relationship between the latent factors for managerial quality and the observed pay of supervisors, and perform anal- ogous simulations to recover pass through of productivity contributions of each dimension of managerial quality to pay. 3.6.1 First Stage: learning parameters Table3.3presentstheresultsofthelearningfunctionwithhomogeneouslearningparameters. Column 1 of Table 3.3 includes experience from the current run of a style, measured by the number of consecutive days spent producing that style, retained learning from previous runs and its interaction with days since the style was last produced on the line along with style fixed effects and time varying characteristics of the workers on the line (average skill grade, share of the highest skill, average gross salary, average age, share of females, share of workers speaking Kannada, and average tenure) as baseline controls. Column 2 adds additional fixed effects for year, month, and day of week to account for any seasonality in productivity and buyer demand. Column 3 adds the number of days left to the end of the order to control for any reference point effect related to the end of the order. 174 Table 3.3: Learning (Experience in Days) Note: robust standard errors in parentheses ( p < 0:01, p < 0:05, p < 0:1). Standard errors are clustered at the line level. Table3.3showsthattheestimatedlearningrateisbetween0.143and0.146. Thislearning rate implies that productivity will increase on average 50% over roughly 30 days of producing the same style, which is very close to what we inferred from the graphical evidence in Figure 1A. The productivity contribution of retained learning from previous runs is around 0.075, which is just over 50% of contemporaneous learning magnitudes. Every unit of log days since the last run erodes roughly 16-17% of the impact of retained learning such that, after 20 intervening days, 50% of the productive value of retained learning has depreciated. These results are quite robust to alternative specifications and measures of productivity and experience. Note that the coefficients are very similar across the three specifications when we control for time fixed effects and days left of the order. In Appendix .3 we present the analogous results to those in Table 3.3 using log(quantity produced) on the left-hand side and controlling for the target quantity on the right-hand side. Table C1 shows nearly identical results to Table 3.3. Note that the coefficient for target quantity is close to 1, which suggeststhatthereisnoscaleeffectsontheefficiencyduetothecomplexityofdifferentstyles. For the rest of the paper, we only present and discuss the results using log efficiency on the left hand side and use the specification in column 2 of Table 3.3 as our preferred specification in the main results that follow. Full estimation results from these alternative specifications are presented in the Appendix sections .2 through .3 175 Next, we estimate model (3.1) with heterogeneous learning parameters using ordinary least squares line by line. 35 Figures 7A, 7B, 7C, and 7D show the distribution of the es- timated initial productivity ( ^ i ), rate of learning ( ^ i ), degree of retention ( ^ i ) and rate of forgetting ( ^ i ), respectively. Figures 7A through 7D depict a large degree of variation in each of the parameters governing the shape of the learning function which corresponds well to heterogeneity depicted in Figures 4A through 6B. Figure 7A: Initial Productivity ( ^ i ) 0 .5 1 1.5 Density 3 3.5 4 4.5 5 α Estimated Distribution of the estimated α in the first stage, mean=4.05. Figure 7B: Learning ( ^ i ) 0 2 4 6 Density 0 .1 .2 .3 .4 .5 β Estimated Distribution of the estimated β in the first stage, mean=.0.16 Note: Figures 7A and 7B show the distribution of the estimates of the initial productivity (line-specific intercepts) and the rate of learning (line-specific slopes) for the 120 lines, which is the largest connected set. 35 For the estimation, we use the largest connected set, which represents 98.5% of the available data 176 Figure 7C: Retention ( ^ i ) 0 2 4 6 8 Density -.2 0 .2 .4 .6 γ Estimated Distribution of the estimated γ in the first stage, mean=0.06. Figure 7D: Forgetting ( ^ i ) 0 .5 1 Density -2 -1 0 1 2 δ Estimated Distribution of the estimated δ in the first stage, mean = -0.21. Note: Figures 7C and 7D show the distribution of the estimates of the retention rate and forgetting rate for the 99 lines for which we are able to recover these parameters. 3.6.2 Second Stage: managerial quality measures and factors In this section, we report and discuss the results of the measurement system. Remember from the discussion in section 3.2.2 that we map the complete set of measures from the different modules of the survey using exploratory factor analysis into the following seven dimensions of managerial quality: Tenure, Cognitive Skills, Autonomy, Personality, Control, Attention, and Relatability. Table 3.4 presents the set of measures used to proxy each latent factor and the estimated loading for each. To establish the informativeness of each measure, we compute the signal content in each measure (i.e., the variance of the contribution to the latent factor over the residual variance of the measure). Remember that for each factor we normalized the highest loading measure to a loading of 1 such that the loadings of all other measures are relative to that highest loading measure. Table 3.4 shows that the most informative measures for Tenure are years supervising current line and years as supervisor with signals of 59% and 20% and loadings 1 and 0.5, respectively. Tenure in the garment industry is also informative with a loading of .36, but total years working is less informative than the more job and industry-specific measures. For Cognitive Skills, Table 3.4 shows that digit span recall, arithmetic (number correct) and arithmetic correct (%) are all quite informative, although the signal is higher for the memory measure (63%) than for the other two arithmetic measures (23% and 37%). For Autonomy, the two leadership behavior measures, initiating structure and consideration, are 177 Table 3.4: Loadings and Signals Note: The first loading of each factor is normalized to 1. Signal of measure j of factor k is s k j = ( j;k) 2 Var(ln k ) ( j;k) 2 Var(ln k )+Var(" j;k) . The measures were standardized across all supervisors who were surveyed. Learning parameters (, , , and ) and the mean of log wage (including both monthly salary and production bonus) from November 2014 across supervisors of a line are all included in the extended system but measured with no error, i.e., the corresponding factor loadings are set equal to 1 but omitted from this table. highly informative with loadings of 1 and .86 and signals of 83% and 77%, respectively. Autonomous Problem-Solving, Problem Identification, and Self-Assessment contribute less with loadings of .05, .17, and .11, and are much noisier with signals of only 0.2%, 3.4% and 1.7%, respectively. Note that the sign of the loadings for all measures in these first three factors are positive as would be expected. With respect to Personality, conscientiousness, perseverance, and self-esteem are all highly informative. The three measures present signal above 73%, 75%, and 69%, respec- tively, and all have loadings near 1. Psychological Distress is less informative than the other three with a loading of -0.24 and a signal of 2.6%. Note that a higher score on the Kessler 178 scale corresponds to more distress, so a negative loading is what we would expect. With respect to Control, internal locus of control has the highest loading and a signal of 53% jus- tifying our naming this factor after this measure. Risk aversion and patience also contribute with loadings of .13 and .22, but both contain much more noise with signals of only 0.7% and 1.5%, repsectively. For Attention, monitoring frequency and active personnel management are the strongest contributors, both with loadings of roughly 1, and both with strong signals (52% and 48%, respectively). Efforts to meet targets also contributes strongly with a loading of .57, but is less precise with a signal of 21%. Lack of communication and issues motivating workers both contribute with loadings of -.44 and -.13, but appear quite noisy with signals of 13% and 0.8%, respectively. Note that we would expect less communication with workers and uppermanagementregardingproductionandmoreissuesmotivatingworkersandovercoming resistance to initiatives to both indicate less managerial attention or effort, so negative loadings for these measures is what we would expect. Finally, for Relatability, the loading is largest for demographic similarity with signal of 32%; while the contribution of egalitarianism is negative with a loading of -0.18, but less informative (7.1% signal). Once again, a negative loading on egalitarianism is as expected, as the factor is informed by demographic similarity and more egalitarianism on the part of the supervisorwoulderodetheproductivevalueofanydemographicsimilarity. Itisimportantto note in summary the heterogeneity in the amount of information contained in each measure for each factor. This demonstrates the importance of allowing for measurement error in the system. Note also that even measures with low loading and high degree of noise are valuable to the system in efforts to purge informative measures of error. 3.6.3 ThirdStage: productivitycontributionsofmanagerialquality Table 3.5 reports the estimates of the CES functions for the initial level of productivity, the rate of learning, retained learning, and rate of forgetting. We see in column 1 that the initial level of productivity is most strongly impacted by Attention and Control, followed by Tenure, Autonomy, and Cognitive Skills. The estimated coefficients for Personality and Relatability are not significantly different from zero. 179 Table 3.5: Contributions of Managerial Quality to Productivity Dynamics Note: p< 0:01, p< 0:05, p< 0:1. Standard errors in parentheses based on 100 bootstrap replications. For the rate of learning, we find that Attention and Tenure still contribute strongly along with Autonomy which contributes more to the rate of learning than to initial productivity. Control, on the other hand, contributes nearly half as strongly to the rate of learning as compared to its contribution to initial productivity. Similarly, the Cognitive Skills contri- bution to the rate of learning is smaller than its contribution to initial productivity. Once again, Personality and Relatability exhibit no discernible contribution. Table 3.5 shows that the pattern of contributions to retention are quite similar to those for learning. That is, Tenure and Attention contribute most strongly and Autonomy con- tributes more strongly to retention than to initial productivity. Cognitive skills contribute more strongly to retention than the rate of learning, consistent with the memory-based measure digit span recall being the most informative measure underling this factor. Con- trol contributes less to retention than to learning and initial productivity. Personality and Relatability continue to be insignificant. 180 With respect to forgetting, we find that Tenure contributes most strongly. Autonomy and Attentioncontributelessstronglytoforgettingthanotherlearningparameters, whileControl and Cognitive Skills do not contribute to forgetting. We find a positive and significant contribution of Personality to the rate of forgetting. This is consistent with the Personality factor being most informed by perseverance and conscientiousness. We also see in column 4 that the contribution of Relatability is marginally significant though small in magnitude. For all the CES functions across the learning parameters, we find that the complementar- ity parameter is close to zero and not generally statistically significant, except for the rate of learning which is positive and weakly significant. This indicates that the different dimensions of managerial quality are not strongly substitutable in their contributions to productivity. That is, the factors appear only weakly complementary in initial productivity and weakly substitutable in learning, indicating that deficiencies in one dimension of managerial quality are not easily compensated with other dimensions. For example, a long-tenured and cogni- tively skilled supervisor can benefit greatly from training in Autonomy, Attention, and/or Control. Overall, given the complex relationships between the factors and productivity at different points along the learning curve, it is difficult to evaluate the composite impacts of higher stocks of different dimensions of managerial quality on productivity from the estimates in Table 3.5. Additionally, the relative value of screening on or training in these different dimensions is also hard to evaluate without considering how variable is each factor. In order toperformthistypeofcomparison, simulationsofproductivityundersupervisorswithhigher values of different factors would be most informative. Simulated Learning Curves with Higher Quality Managers In this section, we simulate the contribution of a one standard deviation (SD) increase in each of the seven factors to productivity. Specifically, we substitute the estimated function of each learning parameter presented in Table 3.5 into the first stage (equation 3.1) and compute the impact of an increase of one standard deviation of each factor (as estimated in the second stage) on productivity at all points along the learning curve. We first evaluate evaluateproductivitywitheachfactorineachlearningparameterfixedtoitsmean(baseline), and then increase sequentially each factor by one standard deviation. We consider two candidate shocks to each factor. We first assume that shocks to the factors are independent. That is, we assume that a potential intervention on different di- mensions of managerial quality affect only the treated factor, as might be the case under a focused training intervention. Second, we use the covariance structure and compute the im- pact of an increase of factori by i , i.e.,E (lnj ln i = i ) where i = p ii and ii =var ( i ). 181 The computation ofE (lnj ln i = i ) depends on the nature of the multivariate distribution assumed for ln, thus E (lnj ln i = i ) = ( 1i = ii ; ; Ki = ii ) 0 i where ij =var ( i ; j ). This procedure is similar to the generalized impulse response func- tions proposed in the time series context by (Pesaran and Shin 1998). 36 This type of correlated shock is more analogous to what might result from a screening intervention in which supervisors with a SD more of a given factor than the average candidate would come along with more or less of the other correlated factors as well. Figure 8A: Tenure Simulation Figure 8B: Cognitive Skills Simulation Note: Figures 8A and 8B show the contribution of Tenure and Cognitive Skills to the learning curve (log efficiency), respectively. We fix the learning parameters to their mean and increase sequentially each factor by one standard deviation. Figures 8A through 8G show the contribution to the learning curve for Tenure, Cognitive Skills, Autonomy, Personality, Control, Attention, andRelatability, respectively. Weconduct the simulations assuming alternately independent and correlated shocks to each factor of managerial quality as compared to the baseline learning curve evaluated with each factor at its mean value. In the simulations, we evaluate the learning curves with previous experience and days since last run of the same style at average levels observed in the data to reflect contributions to all parameters of the learning curve. From Figures 8A-8G, we observe that Attention has the largest impact on productivity when we assume that the intervention can 36 See also (Pesaran 2015). 182 Figure 8C: Autonomy Simulation Figure 8D: Personality Simulation Note: Figures 8C and 8D show the contribution of Autonomy and Personality to the learning curve (log efficiency), respectively. We fix the learning parameters to their mean and increase sequentially each factor by one standard deviation. Figure 8E: Control Simulation Figure 8F: Attention Simulation Note: Figures 8E and 8F show the contribution of Control and Attention to the learning curve (log efficiency), respectively. We fix the learning parameters to their mean and increase sequentially each factor by one standard deviation. 183 Figure 8G: Relatability Simulation Note: Figure 8G shows the contribution of Relatability to the learning curve (log efficiency). We fix the learning parameters to their mean and increase the factor Relatability by one standard deviation. independently increase each factor, followed by Tenure and Control. However, Control has thelargestimpact, followedbyAttentionandCognitiveSkills, whentheinterventionimpacts correlated factors along with the primary factor being targeted. For example, if we compare productivity on day 15 (the mean length) of the order, an increase of one SD of Control increases productivity from roughly .5 to more than 1 if we assume independent interventions and 2.2 for the correlated scenario. A one SD increase in Attention raises productivity on day 15 to roughly 1.5 in both simulations; while the anal- ogous exercise for Tenure shows an increase in productivity on day 15 to nearly 1.1 in both simulations. A one SD increase in Autonomy yields an increase in productivity to roughly 0.9 for the independent simulation and 1.1 for the correlated one. The day 15 comparisons for Personality and Relatability depict increases from .5 to 1.3 and 1.5, respectively, for the correlated simulation, but we find negligible differences for the independent simulation. Similarly, the Cognitive Skills simulations yield a small increase in productivity from .5 to .6 for the independent simulation, but a large increase to 1.4 for the correlated simulation. Table 3.6 summarizes the results of the simulations of both independent and correlated shocks to each factor in turn, evaluated on average across the learning curve. That is, we simply evaluate the mean difference between the simulated curves for each of the indepen- dent and correlated shocks and the baselines in Figures 8A through 8G. Note that Tenure has nearly identical impacts under both types of simulations, reflecting the limited corre- lation between Tenure and other factors. Autonomy and Attention also show only slightly 184 larger impacts under the correlated shock simulation, while Cognitive Skills, Personality and Relatability all exhibit much stronger impacts on productivity under correlated shocks as compared to independent shocks. Table 3.6: Simulated Contributions to Productivity Note: Table 3.6 shows the impact on productivity of an increase of each factor by one standard deviation. The second column, Independent, assumes that the intervention only affects the specific dimension of managerial quality considered, while in the third column Correlated, we use the covariance structure of the factors to compute the impact on productivity. We present the correlation structure between factors in Table A9 in the Appendix. The Cognitive Skills factor is positively correlated with all other factors, most strongly with Control (.335) and Personality (.326). Personality is strongly positively correlated with Autonomy (.852), as well as moderately correlated with Relatability (.383) and Control (.268). Relatability is correlated with all factors except for Tenure, most strongly with Control (.476), Autonomy (.383), and Attention (.308). The comparison between the two simulations sheds some light on whether screening on some dimensions of quality in the hiring process will be more effective in raising productivity than would a focused training program that increases the stock of some dimension indepen- dent of others. That is, the results indicates that screening on Cognitive Skills, Personality, and Control would yield larger increases in productivity than would a focused training in any of these skills because of the correlations with other productive factors. On the other hand, a focused training in Attention or Autonomy would be nearly as impactful as would selection in the hiring process. It is not clear whether any focused training might be able to raise the Tenure dimension of skill, but screening on Tenure would deliver roughly the same impact on productivity. This is interesting as one might suspect that with greater Tenure other dimensions of skill might also rise, but simulations do not support this hypothesis. 185 3.6.4 Third Stage: Contributions of Managerial Quality to Pay Having estimated the contributions of the seven latent factors to the learning parameters and simulated impacts of skill increases on composite productivity, we next test if there exists a relationship between these seven factors and supervisor pay. If pay reflects the marginal productivity of labor, as a standard model of a perfectly competitive labor market would predict, we may expect similar results to the ones presented in Table 3.5. However, imperfect information on the part of the employer (or competing employers) regarding quality of the managers, particularly less easily measured or observed dimensions of quality, may lead the firm to rely just on the observable characteristics, like Tenure and maybe Cognitive Skills to determine the pay scheme (or only force the firm to reward these observable dimensions). Furthermore, if the firm’s market power approaches a monopsony, the firm may not have incentives to adjust the wages fully in response to productivity. To test the link between the seven latent factors and supervisor pay, we follow the same approach as we did for productivity. We use data on salary paid by the firm to each of the managers during the moth of the survey, November 2014, and include the monetary bonuses that are associated with the productivity of the lines. Remember that we included the log of thispaymeasureinthemeasurementsysteminstage2ofourempiricalstrategy. Accordingly, we can draw synthetic datasets from the joint distribution of factors and supervisor pay just as we did for the learning parameter analysis above. Finally, we estimate equation (3.3) with log of supervisor pay as the outcome. Table 3.7 presents the results of this analysis of supervisor pay. Attention and Tenure are reflected most strongly in supervisor pay, followed by Autonomy. Control and Cognitive Skills are not strongly reflected in pay; neither estimate is statistically significant. Perhaps unsurprisingly Personality and Relatability are not reflected in pay at all with estimates of 0, consistent with the lack of contributions to productivity. Note, however, that overall this pattern is not entirely consistent with the rank of factors’ contributions to productivity. For example, Control showed fairly larg impacts on productivity in the simulations (one of the largest in the correlated shock simulation) but is not reflected in pay. To best assess the relative pass-through of productivity contributions of factors to pay, we should perform analogous simulations for pay to the productivity simulations summarized in Table 3.6 in which we increase each dimension of quality by one SD from the mean in turn and note impacts on pay. We can then compare these simulated impacts on pay to the simulated impacts on productivity presented in Table 3.6. 186 Table 3.7: Contributions of Managerial Quality to Pay Note: p< 0:01, p< 0:05, p< 0:1. Standard errors in parentheses based on 100 bootstrap replications. Simulation: Pass-through of Productivity Contributions of Managerial Quality to Pay In this section we compare the contribution of a simulated 1 SD increase in each of the 7 factors to productivity vs. supervisor pay. For productivity, we simply plot the coefficients from Table 3.6 along with corresponding bootstrapped errors. For analogous pay simula- tions, we substitute the estimated coefficients of factors presented in Table 3.7 back into the estimating equation (3.3) using the mean value of each factor at baseline and an increase of one standard deviation of each factor sequentially to simulate pay for the higher skilled supervisors. We once again perform this pay simulation for both independent and correlated shocks. Finally, we compute the pass-through of productivity to pay by dividing the simu- lated change in pay by the simulated change in productivity for the one SD increase in each 187 factor. 37 We compute the results assuming both independent and correlated shocks to each dimension of managerial quality. Figure 3.9: Contribution to Productivity and Pay of Each (Independent) Factor .646 .174 .512 .007 .588 .972 .087 .28 .021 .248 .006 .029 .262 0 0 .2 .4 .6 .8 1 Cognitive Skills Autonomy Personality Control Attention Relatability Tenure Note: thesquares arethecontribution(percentagechange)ofanincreaseofonestandarddeviationofeachfactortoproductivity and the triangles to pay. The vertical lines are the 95% confidence intervals for each mean. Figure3.9comparesthemeansimulatedproductivitygainstothesimulatedpayincreases assuming that interventions (shocks) to each factor are independent of other dimensions (factors) of managerial quality. The squares in Figure 3.9 are the mean of the percentage increase in productivity across days of an order on first and subsequent runs and the triangles are the percentage increases in pay, both due to an increase of one standard deviation of each factor. The vertical lines are the 95% confidence intervals. Figure 3.9 shows that the increase in productivity from an increase of one standard deviation in Tenure is 64%, while the analogous increase in pay is only 28%. A similar gap appears for all factors except for 37 To compute the standard errors of the percentage increase we follow a similar procedure as in the previous section. From stage 2, we draw a synthetic dataset for the learning parameters, factors and log of pay that allow us to estimate a CES function for each learning parameter and log pay. We compute the impact (difference) on the log efficiency and log pay due to an increase of 1 standard deviation of each factor. Finally, we replicate this procedure 100 times, and compute the standard deviation of the percentage increase of productivity and pay and the ratio of the two, each divided by the square root of N. 188 Personality, for which the impact on both pay and productivity is 0. The gap appears largest for Control, Attention, and Autonomy. Figure 3.10: Contribution to Productivity and Pay of Each (Correlated) Factor .684 .91 .603 .783 1.379 1.123 .943 .211 .202 .197 .215 .276 .251 .184 0 .5 1 1.5 Tenure Cognitive Skills Autonomy Personality Control Attention Relatability Note: thesquares arethecontribution(percentagechange)ofanincreaseofonestandarddeviationofeachfactortoproductivity and the triangles to pay. The vertical lines are the 95% confidence intervals for each mean. Figure 3.10 compares the mean simulated productivity gains to the simulated pay in- creases assuming that shocks to each factor are correlated to other dimensions (factors) of managerial quality. Once again we see that there is a large gap between the impacts on productivity and pay of each factor. The impacts on pay appear much more balanced across eachfactornow, thoughstillsmallrelativetothelargeimpactsofeachfactoronproductivity. Table 3.8 summarizes the pass-through of impacts on productivity to pay as the ratio of the percent change in pay to the percent change in productivity as a result of a one SD increase in each factor. We see that the pass-through is in general quite low with a maximum of 48% when evaluating independent shocks (ignoring Personality which has effectively no impact on both pay and productivity) and 33% when evaluating correlated shocks. This is consistent with the firm paying almost entirely fixed salaries with limited role for performance-contingent bonuses as indicated by the summary statistics on pay. This is also consistent with the executives of each factory being unable to effectively measure 189 dimensions of managerial quality and evaluate which dimensions to reward. Table 3.8: Pass-through of Productivity to Pay Note: we compute the pass-through of productivity to the wages, dividing the contribution to productivity by the contribution to the wages, of an increase of one standard deviation of each factor, i.e., the coefficients in Figure 3.9 Additionally, we see that some factors produce larger pass-through (e.g., Autonomy, and Tenure) than do others (e.g., Control and Attention). We interpret these results as consistent with differences in the observability of these skills on the part of the firm and awareness of their importance for productivity. Tenure is a traditional dimension of ability that is often reflectedinapplicationsandinterviews. Autonomythoughlikelylessimmediatelyobservable in the hiring process reflects a style of leadership perhaps more obviously productive in this high pressure manufacturing environment. On the other hand, whether a manager will take control of the production environment and avoid unnecessary risks or how much attention and effort the manager will put forth in daily personnel and production activities are likely difficult to assess in the hiring process. The limited impact on pay of these productive but hard to measure dimensions of quality are consistent with information frictions in the hiring and wage-setting process. 3.7 Checks and Robustness 3.7.1 Tests for Sorting Bias: Monte Carlo Simulations We present the result of the Monte Carlo experiment discussed in section 3.5.1 for the initial productivity, i , the rate of learning, i , retention, i and rate of forgetting, i . We compute the percentage mean bias for the estimated coefficients for the 120 lines for which we recover i and i and the 99 lines for which we recover i and i , and then we compute the average of the absolute value of the mean bias for each line. We conduct this simulation twice: 190 first assuming i.i.d. errors and then assuming the errors are AR(1). The results of this experiment show that the bias is small (less than 0.7%) for both the initial productivity and the learning rate under both error structures. For the retention rate and the forgetting rate, the average of the absolute value of the mean bias for each line is slightly higher but still only 8% or less under both error structures. We interpret these results as strong evidence that the identifying assumptions underlying the first stage estimation, including the absence of sorting of styles to lines, are valid. 3.7.2 Deadline or Reference Point Effects: Robustness to Control- ling for Days Left We repeat our full three step estimation controlling for days left to complete the order in the first stage (equation 3.1), to account for any reference point effect (e.g., productivity rising as the deadline draws near). Table B1 reports the estimated measurement system (analogous to Table 3.4). Tables B2 and B3 report the estimates of the CES production functions for the learning parameters (analogous to Table 3.5) and pay (analogous to Table 3.7), respectively. Figures B1 and B3 present the results of the simulations for both productivity and pay under independent and correlated shocks, respectively, (analogous to Figures 3.9 and 3.10). Note that the loadings and the signals of each measure are very similar to our previous results in Table 3.4, and the coefficients of the CES function for the learning parameters and pay are almost identical to the previous results. Finally, note that the pattern of contributions of each factor productivity and pay are nearly identical to our main results. 3.7.3 AlternateProductivityMeasure: RobustnesstoUsinglog(Quantity) in Place of log(Efficiency) Similarly, we repeat our three-step estimation procedure using log quantity produced instead of log efficiency as the outcome in the first stage and control for log of target quantity. Table C2 reports the results of the estimated measurement system, and Tables C3 and C4 report the estimates of the CES production functions for the learning parameters pay, respectively. Finally,FiguresC1andC3showthecontributionofanincreaseofeachfactorbyonestandard deviation to both productivity and pay for independent and correlated shocks, respectively. Again, the results show a qualitatively similar to the main results in Tables 3.4, 3.5 and 3.7, and Figures 3.9 and 3.10. 191 3.8 Conclusion We match granular production data from several garment factories in India to rich data from a management survey conducted on all line supervisors to answer the following research questions: Do production teams supervised by better managers start at higher productivity levels? Do they learn faster? What managerial characteristics matter most? Estimating a non-linear latent factor model in 3 stages, we identify 5 distinct dimensions of managerial quality: vocation-specific experience, managerial autonomy, cognitive skills, personality (e.g., risk and time preferences and psychometrics), and demographic relatability to workers. We find that some dimensions of quality (e.g., cognitive skills and personality) of the supervisor contribute to initial productivity of the line, but do not significantly im- pact the rate of learning. On the other hand, production lines with supervisors exercising greater managerial autonomy learn at significantly faster rates, but do not start product runs at higher initial productivity levels. Vocation-specific experience of the supervisor impacts both initial productivity and the rate of learning of the line. Additional results indicate that these dimensions of quality are imperfectly substitutable in the production function and generally undervalued in existing wage contracts. More easily observed dimensions of quality like experience and cognitive skills, though still undervalued, contribute to wages in closer proportions to their impacts on productivity; while less easily measured or less obvi- ously productive dimensions such as managerial autonomy and demographic relatability are negatively rewarded. Firms could employ more productive line supervisors and more quickly and consistently achieve peak production by better measuring and rewarding dimensions of managerial quality beyond traditional measures like experience. 192 APPENDIX .1 Tests for Sorting Bias: Balance Checks and Monte Carlo Simulations 193 Table A1: Sorting of Workers’ and Managers Characteristics 194 Table A2: Sorting of Workers’ and Managers Characteristics 195 Table A3: Sorting of Workers’ and Managers Characteristics 196 Table A4: Sorting of Workers’ and Managers Characteristics 197 Table A5: Sorting of Workers’ and Managers Characteristics 198 Table A6: Sorting of Workers’ and Managers Characteristics 199 Table A7: Correlation Learning Parameters Table A8: Bias Learning Parameters Table A9: Correlation of the factors 200 .2 Reference Points: Robustness to Controlling for Days Left Table B1: Loadings and Signals Note: The first loading of each factor is normalized to 1. Signal of measure j of factor k is s k j = ( j;k) 2 Var(ln k ) ( j;k) 2 Var(ln k )+Var(" j;k) . The measures were standardized across all supervisors who were surveyed. Learning parameters (, , , and ) and the mean of log wage (including both monthly salary and production bonus) from November 2014 across supervisors of a line are all included in the extended system but measured with no error, i.e., the corresponding factor loadings are set equal to 1 but omitted from this table. 201 Table B2: CES Production of the Learning Parameters Note: p< 0:01, p< 0:05, p< 0:1. Standard errors in parentheses based on 100 bootstrap replications. 202 Table B3: CES Function Wages Note: p< 0:01, p< 0:05, p< 0:1. Standard errors in parentheses based on 100 bootstrap replications. 203 Figure B1: Contribution to Efficiency and Pay of Each Factor, Independent (%) .599 .119 .528 .001 .69 .85 .013 .276 .034 .252 0 .044 .238 0 0 .2 .4 .6 .8 1 Tenure Cognitive Skills Autonomy Personality Control Attention Relatability Factor Note: the squares are the contribution (percentage change) of an increase of one standard deviation of each factor to the efficiency and the triangles to the wages. The vertical lines are the 95% confidence intervals for each mean. 204 Figure B2: Contribution to Efficiency and Pay of Each Factor, Correlated (%) .628 .543 .49 .658 1.209 .826 .544 .202 .134 .169 .181 .25 .199 .106 0 .5 1 1.5 Tenure Cognitive Skills Autonomy Personality Control Attention Relatability Factor Note: the squares are the contribution (percentage change) of an increase of one standard deviation of each factor to the efficiency and the triangles to the wages. The vertical lines are the 95% confidence intervals for each mean. 205 .3 Alternate Productivity Measure: Robustness to Using log(Quantity) in Place of log(Efficiency) Table C1: log(Units Produced) 206 Table C2: Loadings and Signals 207 Table C3: CES Production of the Learning Parameters Note: p< 0:01, p< 0:05, p< 0:1. Standard errors in parentheses based on 100 bootstrap replications. 208 Table C4: CES Function Wages Note: p< 0:01, p< 0:05, p< 0:1. Standard errors in parentheses based on 100 bootstrap replications. 209 Figure C1: Contribution to Efficiency and Pay of Each Factor, Independent (%) .456 .123 .335 0 .896 .899 .014 .266 .007 .232 0 .04 .252 0 0 .2 .4 .6 .8 1 Tenure Cognitive Skills Autonomy Personality Control Attention Relatability Factor Note: the squares are the contribution (percentage change) of an increase of one standard deviation of each factor to the efficiency and the triangles to the wages. The vertical lines are the 95% confidence intervals for each mean. 210 Figure C2: Contribution to Efficiency and Pay of Each Factor, Correlated (%) .558 .771 .43 .656 1.24 .825 .528 .185 .155 .142 .165 .21 .194 .087 0 .5 1 1.5 Tenure Cognitive Skills Autonomy Personality Control Attention Relatability Factor Note: the squares are the contribution (percentage change) of an increase of one standard deviation of each factor to the efficiency and the triangles to the wages. The vertical lines are the 95% confidence intervals for each mean. 211 .4 Alternative Learning Measure: Robustness to Mea- suring Experience in Cumulative Quantity Produced In this section, we test for the patterns depicted in Section 3 using cumulative quantity as a measure of experience of current run, in the first stage, instead of days running of the order. We use log efficiency in the left hand side. We first report and discuss the results assuming homogeneous learning parameters across the lines, and then move on to present the results of the regression analysis of the production function with heterogeneous learning parameters. Table D1 presents the results of the production function with homogeneous learning parameters. As before, Column 1 of Table 3.3 includes experience from the current run of a style, measured by the number of consecutive days spent producing that style, retained learning from previous runs and its interaction with days since the style was last produced on the line along with style fixed effects as baseline controls. Column 2 adds additional fixed effects for year, month, and day of week to account for any seasonality in productivity and buyer demand. Column 3 adds the number of days left to the end of the order, to control for any reference point effect related to the end of the order. Table D1 shows a nearly identical pattern to Table 3.3; learning rates are around 0.066, which implies that the average productivity will increase 50% roughly after 465 units produced. Table D1: Learning (Experience in Cumulative Quantity Produced) Table D2 describes the set of measures used to proxy each latent factor and the respec- 212 tively estimated loading, which are exactly the same measures used before. We also compute the signal (ratio of the variance of the latent factor to the variance of each measurement) for each measurement. Table D2 presents the results. Table D2: Loadings and Signal-to-Noise Ratios Table D3 reports the estimates of the CES productions function for the initial level of productivity the rate of learning, previous experience (retention) and forgetting rate, for the model with cumulative units produced as a measure of experience of current run. Figure D1 presents the contribution (percentage change) of one standard deviation of each factor to the efficiency and the wages for the average number of units produced of an order. Thesquares in FigureD1 arethemean ofthe percentageincrease onthe efficiencyand the triangles on the wages. Figure D1 shows that an increase of one standard deviation of Experience increase efficiency by 71%, Autonomy 16.4%, Personality 7.1%, Cognitive Skills, 22.18 %, and Relatability 4.9%. Note that the contribution to the wages are lower, i.e., for Experience is 17%, Autonomy 3.5%, Personality 0.475% and Cognitive Skills, 6.24%. 213 Table D3: CES Production Function Note: p< 0:01, p< 0:05, p< 0:1. Standard errors in parentheses based on 100 bootstrap replications. 214 Table D4: CES function Wages Note: p< 0:01, p< 0:05, p< 0:1. Standard errors in parentheses based on 100 bootstrap replications. 215 Figure D1: Contribution to Efficiency and Wages of Each Factor (%) .246 .473 .009 .116 .955 .221 .149 .049 .019 .065 .001 .216 .141 .001 0 .2 .4 .6 .8 1 Tenure Cognitive Skills Autonomy Personality Control Attention Relatability Factor Note: the squares are the contribution (percentage change) of an increase of one standard deviation of each factor to the efficiency and the triangles to the wages. The vertical lines are the 95% confidence intervals for each mean. 216 Figure D3: Contribution to Efficiency and Wages of Each Factor (%) .56 .974 .484 .629 1.394 .571 .458 .104 .141 .105 .138 .286 .199 .062 0 .5 1 1.5 Tenure Cognitive Skills Autonomy Personality Control Attention Relatability Factor Note: the squares are the contribution (percentage change) of an increase of one standard deviation of each factor to the efficiency and the triangles to the wages. 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Abstract (if available)

Abstract In this collection of papers, we study competition and consumer behavior in membership/subscription markets. Generally, firms that implement a membership model, charge a “membership” fees that allow consumers to buy products/services at a unit price, in multiple periods. There are three main questions that we attempt to answer (i) What is the optimal pricing strategy when firms use the membership model (i.e., Tariff Structure)? (ii) How the tariff structure affects competition: consumers' behavior and firms' profits? What are the differences between a static and a dynamic framework? ❧ In the first chapter, ""Competition in Two-Part Tariffs Between Asymmetric Firms"" (with Guofu Tan), we study competitive two-part tariffs (2PTs) in a general model of asymmetric duopoly firms offering (both vertically and horizontally) differentiated products. We provide a necessary and sufficient condition for marginal cost pricing to be in equilibrium, in both the Hotelling and general discrete choice approaches to horizontal differentiation. In the Hotelling setting, when the firms face symmetric demands but have asymmetric marginal costs, we show that in equilibrium the inefficient firm sets its marginal price below its own marginal cost and compensates this loss with the fixed fee, while the efficient firm sets its marginal price above its own marginal cost but below its rival's price. The inferior firm “cross-subsidies” between the fixed fee and the marginal price. ❧ In the second chapter, ``Dynamic Competition for Customer Memberships'', I study a competitive two-period membership (subscription) market, in which two symmetric firms charge a “membership” fee that allows consumers to buy products or services at a given unit price, in both periods. I explore how (i) the length of the membership (ii) the ability to price discriminate between “old” and “new” customers with the membership fee and the unit price

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Creator Tamayo Castano, Jorge Andres (author)

Core Title Essays on competition for customer memberships

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Economics

Publication Date 05/06/2018

Defense Date 03/20/2018

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

Tag competition,cross-subsidies,marginal-cost pricing,OAI-PMH Harvest,product differentiation,two-part tariffs

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Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Tan, Guofu (committee chair), Camara, Odilon (committee member), Kocer, Yilmaz (committee member)

Creator Email jtamayo8@gmail.com,tamayoca@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c40-501046

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