ABSTRACT Title of dissertation: DYNAMIC COMPETITION WITH CUSTOMER RECOGNITION AND SWITCHING COSTS: THEORY AND APPLICATION Vesela Dimitrova Grozeva, Ph.D., 2010 Dissertation directed by: Professor Daniel R. Vincent Department of Economics This dissertation aims to contribute to our understanding of dynamic interac- tion in duopoly markets. Chapter 1 motivates the study and o ers a brief overview of the results. In Chapter 2 I study the dynamic equilibrium of a market characterized by re- peat purchases. Such markets exhibit two common features: customer recognition, which allows rms to price discriminate on the basis of purchase history, and con- sumer switching costs. Both features have implications for the competitiveness of the market and consumer welfare but are rarely studied together. I employ a dynamic framework to model a market with customer recognition and switching costs. In contrast to earlier studies of dynamic competition with switching costs, these costs are explicitly incorporated in the demand functions. Two sets of market equilibria are characterized depending on the size of the switching cost. For all values of the switching cost, customer recognition gives rise to a ?bargain-then-ripo ? pattern in prices and switching costs amplify the loyalty price premium. When switching costs are low, there is incomplete customer lock-in in steady state, rm pro ts increase in the magnitude of the switching cost and introductory o ers do not fall below cost. When switching costs are high, there is complete customer lock-in in steady state, rm pro ts are independent of switching costs and introductory prices may fall below cost. Under incomplete lock-in and bilateral poaching, switching costs do not a ect the speed of convergence to steady state; under complete customer lock-in and no poaching from either rm, convergence to steady state occurs in just one period. The model also suggests that imperfect customer recognition leads to lower pro ts relative to both uniform pricing and perfect customer recognition. In Chapter 3 I use the market framework developed in Chapter 2 to exam- ine the perception that imperfect competition hinders information sharing among rivals in games of random matching. In contrast to previous studies of information sharing, I propose a new channel through which competition may deter informa- tion sharing. This approach reveals a key role for rm liquidity by showing that information sharing among rivals is more likely to arise in markets populated by more liquid rms. Employing a dynamic duopoly framework, in which competition intensity varies with the degree of product di erentiation, consumer switching costs and consumer patience, I show that more intense market competition can weaken the disincentives associated with disclosing information to a rival. I test the model?s predictions using rm-level data on the information-sharing practices of agricultural traders in Madagascar. As predicted by the model, traders operating in liquid mar- kets are shown to be more likely to share information about delinquent customers. This result is robust to the use of two alternative measures of liquidity, of which one is credibly exogenous, and two alternative ways of de ning market liquidity. Fur- thermore, traders who report more intense competition in their market are found to be signi cantly more likely to share information. DYNAMIC COMPETITION WITH CUSTOMER RECOGNITION AND SWITCHING COSTS: THEORY AND APPLICATION by Vesela Dimitrova Grozeva Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial ful llment of the requirements for the degree of Doctor of Philosophy 2010 Advisory Committee: Professor Daniel R. Vincent, Chair Professor John Ham Professor Peter Murrell Professor Ginger Jin Professor Erik Lichtenberg c Copyright by Vesela Dimitrova Grozeva 2010 Dedication To my family for their unconditional support. ii Acknowledgments I owe my gratitude to all the people who have made this thesis possible and because of whom my graduate experience has been one that I will cherish forever. First and foremost I?d like to thank my advisor, Professor Daniel Vincent, for his invaluable guidance and constant support. His insights and encouragement have been tremendously important. I would also like to thank committee members Professor John Ham, Professor Peter Murrell and Professor Ginger Jin for their guidance and detailed comments on my work. I would also like to thank Professor Rachel Kranton for encouraging my earlier ideas that led to this dissertation. I would also like to acknowledge Professor Roger Betancourt for his support and practical advice, as well as Yeon Soo Kim and David Givens for helpful discus- sions. iii Contents 1 Introduction 1 1.1 Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Introduction and Motivation of Chapter 2: . . . . . . . . . . . . . . . 4 1.3 Introduction and Motivation of Chapter 3: . . . . . . . . . . . . . . . 13 2 Dynamic Competition with Customer Recognition and Switching Costs 23 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.2.1 Demand from Newcomers . . . . . . . . . . . . . . . . . . . . 31 2.2.2 Demand from Loyal Customers . . . . . . . . . . . . . . . . . 33 2.2.3 Demand from Switchers . . . . . . . . . . . . . . . . . . . . . 37 2.2.4 Equilibrium Concept . . . . . . . . . . . . . . . . . . . . . . . 38 2.2.5 Exit Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.3 Equilibrium Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.3.1 The Equilibrium under Incomplete Lock-in and Low Switching Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.3.2 Equilibrium Results under Complete Lock-in . . . . . . . . . . 58 2.3.3 The Equilibrium under Incomplete, Asymmetric Lock-in and High Switching Costs . . . . . . . . . . . . . . . . . . . . . . . 67 2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2.4.1 Customer Recognition vs. Uniform Pricing . . . . . . . . . . . 72 2.4.2 Imperfect vs. Perfect Customer Recognition . . . . . . . . . . 73 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3 Inter-Firm Information Sharing, Competition, and Liquidity Con- straints: Theory with Evidence from Madagascar 83 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.2 Theoretical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.2.2 The market . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.2.3 The decision to share information . . . . . . . . . . . . . . . . 110 3.3 Data and Empirical Strategy . . . . . . . . . . . . . . . . . . . . . . . 131 3.3.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 3.3.2 Empirical Strategy . . . . . . . . . . . . . . . . . . . . . . . . 136 iv 3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 3.4.1 The Impact of Competition . . . . . . . . . . . . . . . . . . . 146 3.4.2 The Impact of Liquidity . . . . . . . . . . . . . . . . . . . . . 148 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 A Appendix to Chapter 2 157 A.1 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 A.1.1 Proof of Proposition 1 . . . . . . . . . . . . . . . . . . . . . . 157 A.1.2 Proof of Corollary 1 . . . . . . . . . . . . . . . . . . . . . . . 171 A.1.3 Proof of Corollary 2 . . . . . . . . . . . . . . . . . . . . . . . 174 A.1.4 Proof of Proposition 2 . . . . . . . . . . . . . . . . . . . . . . 176 A.1.5 Proof of Proposition 3 . . . . . . . . . . . . . . . . . . . . . . 181 B Appendix to Chapter 3 184 B.1 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 B.1.1 Proof of Proposition 1. . . . . . . . . . . . . . . . . . . . . . . 184 B.1.2 Proof of Corollary 1 . . . . . . . . . . . . . . . . . . . . . . . 189 B.1.3 Proof of Lemma 3. . . . . . . . . . . . . . . . . . . . . . . . . 190 B.1.4 Proof of Lemma 4. . . . . . . . . . . . . . . . . . . . . . . . . 191 B.1.5 Proof of Proposition 2 . . . . . . . . . . . . . . . . . . . . . . 192 B.1.6 Proof of Proposition 3 . . . . . . . . . . . . . . . . . . . . . . 200 B.1.7 Proof of Proposition 4 . . . . . . . . . . . . . . . . . . . . . . 204 B.1.8 Proof of Corollary 2 . . . . . . . . . . . . . . . . . . . . . . . 205 B.2 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 v List of Tables B.1 Variable De nitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 B.2 Liquidity Scores Components. . . . . . . . . . . . . . . . . . . . . . . 208 B.3 Frequency distribution of Own Liquidity Score 1. . . . . . . . . . . . 209 B.4 Frequency distribution of Own Liquidity Score 2. . . . . . . . . . . . 209 B.5 Frequency distribution of Average Liquidity Scores. . . . . . . . . . . 209 B.6 Summary Statistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 B.7 Controls. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 B.8 Determinants of Competition Intensity. . . . . . . . . . . . . . . . . . 212 B.9 Regression Results { Competition . . . . . . . . . . . . . . . . . . . . 213 B.10 Regression Results { Liquidity . . . . . . . . . . . . . . . . . . . . . . 214 B.11 Robustness Checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 B.12 Ordered Probit Results. . . . . . . . . . . . . . . . . . . . . . . . . . 216 B.13 Marginal E ects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 vi List of Figures 2.1 Steady-state pro ts. . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.1 Timeline of events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.2 The information-sharing stage game. . . . . . . . . . . . . . . . . . . 94 vii Chapter 1 Introduction 1.1 Outline of Thesis Economic agents rarely make choices independently of the choices made by others. Instead, strategic interaction underlies much of economic activity and study- ing the manifestations and outcomes of strategic interaction has opened up a vast area of research in Industrial Organization Theory and Applied Microeconomics. Strategic considerations may take many forms. Agents may condition their optimal strategies on the strategies of other agents, on the current state of the economic setting, or on their knowledge about the preceding two factors. At the same time, agents? own actions today may a ect the state of the economic environment tomor- row and in uence the information sets and future strategies of their counterparts. The heterogeneity of agents? characteristics and how these characteristics a ect pay- o s imposes ever more stringent requirements on the information that agents must have about the characteristics of their strategic partners. In this dissertation I ex- 1 plore two aspects of strategic interaction among rival rms { the determination of optimal price strategies in a dynamic duopoly market, and the decision to exchange information about the past conduct of previous contractual partners. This dissertation consists of three chapters. Chapter 1 provides the introduc- tion and motivation of the research. In Chapter 2, I present a model of dynamic competition with customer recognition and consumer switching costs and study its equilibrium properties. Customer recognition occurs when rms are able to dis- tinguish between new and repeat customers and can o er them di erent prices. I extend an earlier model of customer recognition, originally formulated by Villas-Boas (1999), and introduce consumer switching costs in the market. Consumer switching costs arise when customers incur transaction or learning costs as a result of buying from a di erent producer. In contrast to past studies of dynamic competition with switching costs, I am able to incorporate these costs explicitly in the demand func- tions and derive two sets of market equilibria depending on the size of the switching cost. I derive closed-form solutions for the equilibrium prices, which enables a com- parative statics analysis. Previous studies of dynamic competition with switching costs have limited attention to the presence of high switching costs that induce cus- tomer lock-in. I do not impose this limitation in my model. For all values of the switching cost, customer recognition gives rise to a ?bargain-then-ripo ? pattern in prices, and switching costs amplify the loyalty price premium. When switching costs are low, there is incomplete customer lock-in in steady state, rm pro ts increase in the magnitude of the switching cost and introductory o ers do not fall below cost. When switching costs are high, there is complete customer lock-in, rm pro ts are 2 independent of switching costs and introductory prices may fall below cost. Under incomplete lock-in switching costs do not a ect the speed of convergence to steady state; under complete customer lock-in, convergence to steady state occurs in just one period. The model also suggests that imperfect customer recognition leads to lower pro ts relative to both uniform pricing and perfect customer recognition. In Chapter 3, I apply the model developed in Chapter 2 in the context of rm behavior in developing countries. In developing countries, rms often cannot rely on formal institutions to enforce contracts. An alternative solution is to rely on information ows about the past performance record, or ?reputation?, of potential partners, in order to identify reliable contacts and discourage contract breach. How- ever, when rms deal with a speci c partner for the rst time, information about that partner?s contract performance is not readily observable. In a seminal paper, Kandori (1992) establishes that reputation mechanisms can limit opportunism in bilateral relationships if agents have at least some information that summarizes the past performance of their new partner. In a real-world setting, rms are often ex- posed to the risk of contract breach from customers and suppliers and the most likely source of information about the reputation of these parties are other rms in the market. However, it is commonly perceived that rms will not exchange valuable information with their market rivals. The goal of this essay is to formally exam- ine this perception and identify other key factors that may a ect rms? incentives to share information with rivals. My main nding is that rm liquidity facilitates information sharing among rivals. When rms experience breach of contract, their cash ows and inventory stock may be disrupted and their ability to compete will 3 depend on how costly it is to raise additional capital. Liquid rms will incur low costs of capital while liquidity constrained rms will face higher such costs. Firms realize that if they have liquid rivals, they cannot pro t from the rival? experience of contract breach because a liquid rival faces low cost of funds. Hence, a rm facing a liquid rival will have a weaker incentive of exposing this rival to a higher probability of contract breach by not sharing information. Therefore, information sharing will be more likely to arise in markets populated by more liquid rms rela- tive to markets populated by liquidity-constrained rms. Furthermore, I show that more intense market competition can lower the cost of disclosing information to a rival. I test the model?s predictions using rm-level data on the information sharing practices of agricultural traders in Madagascar and nd support for the proposed hypothesis that liquidity has a positive e ect on traders? propensity to share infor- mation. In addition, traders who report more intense competition in their market are found to be signi cantly more likely to share information. 1.2 Introduction and Motivation of Chapter 2: Chapter 2 builds a model of dynamic competition with imperfect customer recognition and switching costs. Customer recognition and switching costs are com- monly present in markets where rms can distinguish their repeat customers and can practice price discrimination on the basis of purchase history. However, the literature has largely reviewed the impact of these two features separately and there are no dynamic models that integrate both. In this essay, we show that the joint 4 presence of imperfect customer recognition and switching costs brings qualitative changes in the market equilibrium when compared to models that exhibit only one of these features. Furthermore, we allow for the presence of overlapping generations of consumers, which generates three groups of customers based on their purchase history { new, unattached consumers; customers who switch away from their origi- nal supplier; and customers who stay with their original suppliers. We rst present a model of ?imperfect? customer recognition { rms can distinguish between new and repeat customers but they do not know if a new customer is a switcher or a newcomer to the market. Then, we dispose of this latter assumption and show that rms? ability target all three groups of customers with a di erent price increases rm pro ts. By comparing our results of the competitive outcome under imperfect cus- tomer recognition to comparable studies of uniform pricing, it is also seen that rms would be better o in a market where repeat customers cannot be distinguished from new customers. This result holds for markets with high switching costs that induce complete customer lock-in and is due to the fact that under customer recog- nition rms compete away the gains from selling to loyal customers at a premium in the competition for market share.1 There are few models that consider the interaction of customer recognition and consumer switching costs, namely Chen (1997), Gehrig and Stenbacka (2002), and Taylor (2003). Chen (1997) and Taylor (2003) consider markets that consist of a single generation of consumers. In the initial period of the game consumers 1The comparison cannot be extended to equilibria with switching because there is no bench- mark model of uniform pricing, i.e. a dynamic model with product di erentiation, switching costs and uniform pricing that also allows switching in equilibrium. 5 enter the market and rms compete for market share. In the subsequent period(s), there are no new incoming generations. Firms recognize their previous customers and engage in price discrimination by o ering discounts to the rival?s customers (a practice commonly referred to as ?poaching?). A key aspect of this analysis is that after the initial period there are only two types of customers, { loyal customers, who stay with their original supplier, and switchers, who change suppliers. Firms can target each group with a di erent, optimally chosen price. Therefore, models that consider competition for a single generation of consumers arti cially induce a separation between unattached consumers and switchers by assuming that all consumers enter the market in some initial period while switching occurs in the subsequent periods when there are no new cohorts. I extend this line of research by considering the more realistic setting where in each period an old cohort of consumers exits the market and a new cohort enters { thus, each period rms face overlapping generations of consumers, { and explore the impact of rms? inability to distinguish between newcomers and switchers on the market equilibrium. A setting with overlapping generations of customers is particularly relevant for markets with high rates of new consumer entry and somewhat low switching costs that make the change of suppliers feasible. Examples include markets for the provision of high-speed data (e.g., cable, internet and cell phone services), credit card services, movie rentals, and others.2 In many of these markets rms are unable to distinguish between newcomers and switchers because it is easier to obtain infor- 2For example, as consumers choose to upgrade from dial-up to broad-band internet, the two main internet service providers, Comcast and Verizon, face substantial demand from new, unattached consumers. At the same time, the two rms actively engage in poaching each other?s existing customers. 6 mation on the purchase history of one?s own customers (e.g., by enticing customers to enroll in loyalty programs o ering discounts or to set up membership accounts that reduce transaction costs), rather than on the purchase history of the rival?s customers. For this reason, the main focus of this study is on the impact of imper- fect customer recognition. This analysis is presented in Sections 2.2 through 2.3.1. For the rest of the paper, the term ?customer recognition? will be used to refer to imperfect customer recognition unless speci ed otherwise. Since newcomers and switchers have di erent price elasticities, rms would be willing to set di erent prices to each group if they could separate the two mar- kets. Such practice would give rise to perfect customer recognition. Pazcal and Soberman (2007) report that Air Canada used to give promotional o ers exclu- sively to Aeroplan members. In 2006 Blockbuster ran a promotional campaign that gave free movie rentals to Net ix subscribers.3 In Section 2.4 I construct a simple two-period model that is su cient to capture the market outcome under perfect customer recognition. I show that perfect customer recognition generates higher pro ts and for su ciently low switching costs reverses the loyalty price premiums that loyal customers pay under imperfect customer recognition. For low switching costs, switchers would be o ered the lowest price and newcomers { the highest. This occurs because lower switching costs erode the pro ts from market share and relax competition for new customers. When switching costs are su ciently high, introductory o ers emerge as in the case with imperfect customer recognition This 3Net ix subscribers were required to prove membership by bringing in their Net ix envelope aps. 7 result explains the practice of inducing the rival?s customers to reveal their purchase history and o ering deeper discounts to switchers. A systematic analysis of the impact of customer recognition in a market with overlapping generations of consumers is necessary because rms face two con icting incentives in setting their price to new customers { on one hand, they want to max- imize pro ts from switchers, and, on the other, they want to invest in market share. We do not have clear intuition as to which incentive will exercise stronger downward pressure on the price to new customers and how this will a ect the competition of market share. Villas-Boas (1999) shows that customer recognition intensi es com- petition to the point that price to new customers may fall down to marginal cost. It is unclear how switching costs will a ect this nding. High switching costs may weaken the incentive to poach and thereby raise the price to new customers, but they may also increase the return to market share, so rms will bid that price down. The literature on exogenous switching costs has shown that typically switching costs give rise to loyalty premiums, i.e. loyal customers pay higher prices than new cus- tomers, and in markets for homogeneous goods rents from exploiting consumers due to switching costs are dissipated in the competition for market share. In our model we will demonstrates that the level of the loyalty premium and the degree to which rents from switching costs are competed away depend on the size of the switching cost. If switching costs are su ciently high to cause complete customer lock-in, then the incentive to poach disappears and rms would set the price to new customers with the only goal of capturing market share. As a result, any gains associated with the presence of switching costs will be competed away. In contrast, when switching 8 costs are low and rms can poach the rival?s customers, the loyalty price premium and rm pro ts will be increasing in the size of the switching cost. So far, the only model that incorporates switching costs and customer recogni- tion while considering the impact of overlapping generations of consumers appears in Gehrig and Stenbacka (2002). The authors employ a two-period model to analyze the stationary equilibrium of the dynamic game between two in nitely-lived rms. However, by limiting attention to the case where switching costs are su ciently high to prevent switching, they do not allow for the presence of switchers, which is what makes the overlapping generations setting interesting. Their two main ndings are that introductory o ers to new customers only emerge for a strictly positive level of the switching cost and the combination of price discrimination by purchase history and the presence of high switching costs reduces rm pro ts relative to a setting with zero switching costs. In the present study, which incorporates similar features but rests on an in nite-horizon dynamic model and investigates the market equilibrium for all levels of the switching costs, I nd that rm pro ts increase in the size of the switching costs. Furthermore, Villas-Boas (1999) and this study show that under imperfect customer recognition introductory o ers would appear even if switching costs were zero. The disparity between the results in Gehrig and Stenbacka (2002) and the model here as well as the original framework by Villas-Boas suggests that a dynamic model is indeed necessary to capture the complex processes in a market characterized by overlapping generations of consumers and customer recognition. The present model is also closely related to the broader strand of literature on imperfect competition with customer recognition. Customer recognition gives rise 9 to a non-traditional form of price discrimination commonly referred to as behavior- based price discrimination.4 Holmes (1989) shows that price discrimination in an imperfectly competitive market does not necessarily increase rm pro ts. Corts (1998) further demonstrates that in oligopolies with di erentiated goods best re- sponse asymmetry does not allow us to make a priori predictions about the impact of price discrimination on rm pro ts and social welfare. Chen (1997) and Fudenberg and Tirole (2000) develop two-period models to examine the impact of behavior- based price discrimination on market outcomes. Chen looks at a homogeneous-good duopoly and shows that pro ts are lower when rms engage in price discrimination. He further shows that the price to loyal customers and the discount to switchers are increasing in the cost of switching. In his model, the presence of switching costs is the only cause of introductory o ers to switchers. Fudenberg and Tirole (2000) shift the focus towards markets with product di erentiation but no switching costs and ex- amine the market equilibrium under short-term and long-term contracts. With xed consumer preferences and short-term contracts, poaching gives rise to discounts for loyal customers. This result is in contrast with the switching costs literature where discounts are geared towards new customers. Villas-Boas (1999) extends the anal- ysis of Fudenberg and Tirole to an in nite-horizon model and shows that, even in the absence of switching costs, in nitely-lived rms will optimally o er discounts to new customers. This result will persist in our model as well and is due to the fact that there is product di erentiation { customers? choice of supplier in their rst 4For an excellent survey of behavior-based price discrimination models see Fudenberg and Villas-Boas (2005). 10 purchase reveals information about their relative preferences with respect to each rm?s product. Villas-Boas shows that customer recognition intensi es competition for new customers and drives both the price to new customers and the price to loyal customers down.5 All three of these studies { Chen (1997), Fudenberg and Tirole (2000), and Villas-Boas (1999) { conclude that rms are worse o when they price discriminate on purchase history as opposed to uniform pricing. With respect to imperfect price discrimination, this result is preserved in the present study as well.6 The literature on switching costs examines the impact of these costs in the light of two distinct settings: one, based on homogeneous goods and heterogeneous switch- ing costs, and another, based on heterogeneous goods and homogeneous costs.7 The subsequent analyses of the impact of switching costs on market competitiveness have mostly relied on two-period models because of their tractability which allows for the examination of a wide variety of features and problems.8 Nevertheless, there are a few dynamic models that look at the impact of switching costs on incumbency advantages, the incentives for collusion, and the competitiveness of the market. All but one of these dynamic models are based on uniform pricing strategies. The only exception is Taylor (2003) who allows for customer recognition but his framework does not allow for overlapping generations of consumers and assumes that rms have a nite horizon. All other dynamic models feature overlapping generations of 5In Villas-Boas? model the price to loyal customers is set sequentially after the introductory prices are announced and its optimal level is increasing in the introductory price of the rival. 6The lack of closed-form solutions for the uniform-pricing models preclude a comparison be- tween pro tability under uniform pricing and perfect customer recognition. 7The lack of pure-strategy equilibrium constrains the analysis of homogeneous goods and ho- mogeneous switching costs. 8For a thorough review of the literature on switching costs see Farrell and Klemperer (2007). 11 consumers and uniform pricing strategies. Farrell and Shapiro (1988) and Padilla (1995) examine a market with homogeneous products. Farrell and Shapiro (1988) rst recognize the importance of having overlapping generations of consumers on rms? price strategies. They make the unusual assumption that rms choose a price leader in the dynamic game and show that switching costs soften competi- tion. Padilla (1995) disposes of this assumption, because of its direct impact on competition, and allows rms to set their prices simultaneously. His results con rm that switching costs relax competition and this e ect is not due to the sequential nature of the price-setting game in Farrell and Shapiro. Our own ndings are con- sistent with this result when we limit attention to the equilibrium with switching. On the other hand, Beggs and Klemperer (1992) and To (1996) analyze a dynamic market with product di erentiation and switching costs, which successfully prevent consumers from changing suppliers. The former assume that consumers have an in nite horizon as well and rea rm the results from homogeneous markets that switching costs lead to higher prices and pro ts. To (1996) modi es the model in Beggs and Klemperer by assuming that consumers have a nite horizon and, speci - cally, enter the market for two periods only. The only qualitative di erence between his results and those in Beggs and Klemperer is that in To?s model convergence to steady state is non-monotonic. In both models convergence to steady state takes a su ciently high number of periods and may be in nitely slow if rms are in nitely patient. We present a model that features customer recognition, switching costs and overlapping generations of consumers. Two sets of market equilibria are character- 12 ized depending on the size of the switching cost. For all values of the switching cost, customer recognition gives rise to a ?bargain-then-ripo ? pattern in prices and switching costs amplify the loyalty price premium. When switching costs are low, there is incomplete customer lock-in in steady state, rm pro ts increase in the magnitude of the switching cost and introductory o ers do not fall below marginal cost. When switching costs are high, there is complete customer lock-in in steady state, rm pro ts are independent of switching costs and introductory prices may fall below cost. When both rms poach in equilibrium, switching costs do not af- fect the speed of convergence to steady state; when neither rm nds it optimal to poach, convergence to steady state occurs in just one period. Furthermore, we nd that imperfect customer recognition generates lower pro ts relative to both uniform pricing and perfect customer recognition. 1.3 Introduction and Motivation of Chapter 3: In the absence of adequate legal protection against contract breach, rms can reduce their exposure to contractual risk in one-shot transactions by exchanging information about defectors. The goal of this chapter is to investigate whether competition discourages such exchange and under what conditions. Information sharing is particularly important in developing and transition economies where reliance on formal means of contract enforcement is limited. On one hand, these economies may not have adequate legal framework or e cient en- forcement institutions to provide protection against contract breach. On the other 13 hand, informal business activity, corruption and ine ciency in the legal system may discourage the use of formal contracts and rule out recourse to the court system.9 The consequences of such institutional failtures can be highly detrimental as rms may limit their transactions to long-standing partners, and forgo better economic opportunities with new partners (McMillan and Woodru , 2000). There are several theoretically and empirically established mechanisms, through which information sharing can help rms reduce their exposure to contractual risk. Information sharing can alleviate adverse selection through reputation e ects: when past performance is a signal of a player?s propensity to renege on a contract, rms can screen out defectors, conditional on receiving information about the agent?s record (Jappelli and Pagano, 1993). Information sharing can also have a discipline e ect that discourages some players from acting opportunistically because these players can foresee that information about their actions will be publicly available (Padilla and Pagano, 2000). Sharing information about players? records can also facilitate cooperation on a wide range of problems through social-norm equilibria (Kandori, 1992; Okuno-Fujiwara and Postlewaite, 1995).10 Case studies of informal coalition arrangements that sustain cooperation through social norms have given rise to vari- ations of this game (Greif, 1993; Clay, 1997) but the dissemination of information about players? past actions remains a key function of such coalitions. 9Schneider (2002) shows that the average size of the informal sector in 21 transition countries amounts to 38% of o cial GDP. Safavian and Wimpey (2007) show that the probability of an enterprise preferring to use only informal credit is inversely related to the quality of the overall quality of governance in the country. 10In fact, Kandori (1992) shows that for the cooperative equilibrium to be sustained players only need information about the ?status? of their current partner, i.e. whether the partner is to be punished in the current period for past deviations. Players do not need to know the full history of the game or the status of all players. 14 Past studies of contract enforcement based on social norms (Landa, 1981; Greif, 1989 and 1993; Clay, 1997; Bernstein, 1992 and 2001) do not endogenize the existence of information networks. Greif (1993) presents evidence of the ac- tive correspondence among Maghribi traders in the 11c. Cairo and their partners in the Mediterranean region on the performance of their overseas agents. Greif conjectures that this extensive communication network indicated the existence of an informal traders coalition that relied on reputation mechanisms to keep agents honest. However, Greif (2006) brie y acknowledges that if coalition members were rivals in a common oligopolistic market, they would be reluctant to share informa- tion that may bene t their competitors. In numerous studies Marcel Fafchamps and co-authors have recognized that competition may be responsible for the lack of information sharing networks in some African countries (Fafchamps et al., 1994; Fafchamps, 1996 and 1997; Fafchamps and Minten, 1998). However, to date there is no formal study on the subject. This paper complements the literature on so- cial norms by examining the conditions under which inter- rm information sharing networks in a competitive environment are viable. So far the literature has largely addressed the issue of information sharing on agents? contract performance from the perspective of lending institutions only (Jap- pelli and Pagano, 1993; Padilla and Pagano, 1997 and 2000; Bouckaert and Degryse, 2001; Gehrig and Stenbacka 2006; and Brown and Zehnder, 2008). Among these studies, few have focused on the impact of ex ante imperfect market competition on the endogenous emergence of information ows.11 Jappelli and Pagano (1993), 11A notable exception is Klein (1992) who models rms? decisions to pay a fee and join a credit 15 Padilla and Pagano (1997), Brown and Zehnder (2008) model banks as ex ante local monopolists and examine various aspects of information sharing on competition in- tensity, entry decisions and borrower performance. Jappelli and Pagano (1993) rst look at the trade-o s of sharing information with potential rivals. In their model banks bene t from pooling information about the borrowing histories of their local customers but also lose their monopoly power, so the threat of more intense ex post competition can deter information sharing. Brown and Zehnder (2008) present ex- perimental evidence in support of this theoretical result. Padilla and Pagano (1997) propose that information sharing may serve as a pre-commitment device that helps reduce moral hazard on behalf of borrowers { by agreeing to share information, banks pre-commit to limit their ability to extract rents from their customers, which increases borrower e ort. Bouckaert and Degryse (2001) in turn consider the incen- tives of a local monopolist to unilaterally reveal information about its customers? types to a potential entrant. They nd that when adverse selection is severe, not revealing information can deter entry, which makes information sharing suboptimal. If entry does occur, then two-way information sharing emerges when the level of adverse selection is large, consistent with the results in Jappelli and Pagano (1993). Padilla and Pagano (2000) investigate how the scope of the information shared a ects its disciplinary e ect on borrowers in the context of perfectly competitive markets.12 They nd that sharing default information only (also referred to as bureau. However, he does not consider the role of competition on rms? incentives to reveal their private information about customer performance. 12See also Verkammen (1995) and Diamond (1989) for early studies on the impact of pub- licly observable credit histories on borrowers? choice of projects and e ort. Brown and Zehnder (2007) present experimental evidence showing that the incentive e ects of information sharing are signi cant only in the absence of bilaterally repeated transactions between lender and borrower. 16 ?black? information) rather than the full borrowing history, has a stronger discipline e ect because lenders make their inferences about a player?s type on the basis of a single incident of default. In line with this result, I limit attention to the transmission of ?black? information only. More recently, Gehrig and Stenbacka (2006) consider the e ects of information sharing on the degree of market competition. They nd that information exchange has anti-competitive implications as it facilitates poaching the rival?s ?good? borrowers and reduces the returns on credit relationships, thus weakening the competition for new customers. Empirical studies of formal information sharing regimes have focused on the outcomes of these regimes in credit markets and provide strong evidence in support of the e ectiveness of the reputation mechanism. Public and private credit registries have become centralized repositories of information in credit markets.13 The opera- tion of credit bureaus is shown to reduce default rates (Jappelli and Pagano, 2002), increase the volume of lending (Jappelli and Pagano, 2002; Djankov, McLiesh and Shleifer, 2007) and reduce lenders? selection costs (Kallberg and Udell, 2003). Firm- level data shows that formal information sharing mechanisms among lenders reduce rms? cost of credit, particularly in countries with weak legal enforcement (Brown, Jappelli and Pagano, 2009) and soften rms? credit constraints (Love and Mylenko, 2003). Experimental evidence further demonstrates that sharing default informa- tion increases borrowers? incentives to repay loans and without such exchange the 13See Klein (1992) for a discussion of rms? decision to join a credit bureau when competition is not a consideration. See Jappelli and Pagano (1993) and Padilla and Pagano (2000) for a theoretical treatment of the endogenous emergence of information sharing in credit markets. Also, see Padilla and Pagano (1997) for a study of the precision of information to be shared in order to maximize borrower performance. 17 credit market can collapse (Brown and Zehnder, 2006). Evidence from microdata in developing countries stresses on the e ciency gains of having an operating credit bureau, especially when borrowers understand the implications of a traceable credit history (de Janvry, McIntosh and Sadoulet, 2006; Gin e, Goldberg and Yang, 2009). The empirical evidence on the existence of informal information sharing net- works in developing economies is somewhat limited and o ers mixed ndings. McMil- lan and Woodru (2000) nd that gossip within Vietnam?s manufacturing commu- nity serves an essential role in disseminating information about suppliers and cus- tomers who have reneged on their contracts. In contrast, Annen (2007) surveys informal textile producers in Bolivia and nds that most traders do not disclose information about dishonest agents and even if they do, such information is limited to one?s family members, rather than directed towards other traders who would bene t most from such information. Among the few studies that recognize the role of competition on the formation of information sharing networks, Fafchamps (1996) reports the following in a particularly illustrative case study of contract enforcement in Ghana: ?There seems to be no mechanism whereby information about clients? trust- worthiness is shared among rms other than direct recommendation by common acquaintances. When prompted directly, rms declare that they never bother pass- ing information about untrustworthy customers to other rms. Sharing information would provide competitors with an undue advantage, they say. In fact, several re- spondents appeared to relish the idea that their competitors have to deal with the 18 same deadbeats by whom they had been burnt.? (Fafchamps, 1996, pg. 441.) On the other hand, Fafchamps et al. (1994) point out with surprise that several competing textile producers in Kenya deliberately exchanged information about delinquent customers. Given the mixed anecdotal evidence on the existence of information sharing networks in di erent markets and the lack of a formal analysis of this issue, the current study lls an important gap in understanding how competition a ects rms? decisions to share information with rivals. The theoretical investigation in this paper di ers from past studies of infor- mation sharing in credit markets in two ways. First, all existing studies model the impact of competition through the feedback e ect of information sharing on lenders? market power. In contrast, I propose a di erent channel through which competition may hinder information sharing, based on the observation that the experience of contract breach can be particularly harmful to rms that are liquidity constrained. Firms? losses associated with contract breach include not only the value of the con- tracted goods/services, as it is assumed in studies focusing on lenders, but also the potential loss of market share and its implications for future pro tability un- der an in nite horizon. This wider impact of contract breach on rms? ability to compete, particularly in environments with imperfect capital markets, has not been addressed in the literature so far. Considering rms in a developing country setting where information sharing can act as a particularly important substitute for cred- itor protection rights further supports the thesis that credit constraints have the potential to a ect rms? strategic decisions. Second, I focus on rms rather than 19 lending institutions. This further justi es the emphasis on liquidity and allows us to make use of survey data at the rm level and empirically examine the factors that may have contributed to the presence of information sharing networks in some markets but not in others. 14 Finally, all past studies use two-period models of banking competition, through which they derive banks? net bene ts from engaging in information sharing while the model at hand uses a richer, dynamic framework to model market competition. This study is also related to the broader subject of information sharing among competing rms. A well-established strand of the literature looks into rms? in- centives to pool information about uncertain demand and cost parameters. In the presence of demand or cost uncertainty, rms? optimal strategies depend on the type of competition (Bertrand or Cournot) and the source of uncertainty, i.e. demand or cost conditions (Novshek and Sonnenschein, 1982; Clarke, 1983; Vives, 1984; Gal-Or, 1985 and 1986; Li, 1985; Raith, 1996).15 Except for models of Bertrand competition with cost uncertainty, unilaterally revealing information about inde- pendent values, private values or common values with strategic complementarity is shown to be a dominant strategy (Vives, 2006). More recent studies focus on rms? decisions to reveal information about their customers? purchase histories in order to determine if customers view their products as substitutes or complements (Kim and 14Previous studies that use rm-level data, e.g. Galindo and Miller (2001), Love and Mylenko (2003), Brown, Jappelli and Pagano (2009), have studied the impact of private credit bureaus and public credit registries on credit market performance but there is no empirical study, either based on country-level or rm-level data, that examines the determinants of the emergence of information-sharing institutions. 15See Vives (2006) for a review of the literature on pooling private signals of uncertain demand and cost conditions. 20 Choi, 2009; Liu and Serfes, 2009). Finally, this work also ts into the literature on informal risk-sharing arrange- ments (Kimball, 1988; Coate and Ravallion, 1993; Fafchamps, 1995 and 2002; Besley, 1995). Information sharing arrangements reduce rms? exposure to cost shocks trig- gered by the experience of contract breach. Thus, engaging in costly exchange of information about defectors can be viewed as an insurance mechanism against future shocks. Furthermore, rm access to low-cost, informal credit reduces the strategic cost of information sharing and facilitates information exchange. Hence, this paper suggests complementarities between credit and risk sharing institutions. The main nding of this study is that market liquidity facilitates information sharing by reducing the strategic cost of disclosing information to a rival. Firms ex- pect that withholding information about cheaters will expose the remaining rms in the market to higher risk of contract breach and the rm that withheld information may be able to pro t from the rival?s higher exposure to risk. However, access to liquid assets makes rms less vulnerable to the disruptions caused by experiencing contract breach. As a result, rms that face liquid rivals have weaker incentives to withhold information from these rivals. Assuming that rms within a market are similar in their access to liquidity, we can formulate the hypothesis that infor- mation sharing is more likely to emerge in markets populated by more liquid rms. This hypothesis is supported by empirical evidence based on the information-sharing practices of agricultural traders in Madagascar. The model also suggests that more intense competition may encourage or discourage information sharing, depending on what market features are driving the intensity of competition. The accompany- 21 ing empirical analysis shows that traders who report stronger competition in their markets are also more likely to engage in information-sharing. 22 Chapter 2 Dynamic Competition with Customer Recognition and Switching Costs 2.1 Introduction This paper studies the interaction of customer recognition and switching costs in a di erentiated duopoly with overlapping generations of consumers. Customer recognition refers to the practice of o ering di erent prices to new and repeat con- sumers. It has become a widespread market phenomenon facilitated by the the advancement of information technologies over the past two decade. Previously as- sociated predominantly with subscription markets, today this practice is feasible in a wide variety of settings as more and more consumers provide rms with unique identi ers when using non-cash methods of payment, carrying store membership 23 cards or shopping online.1 Combined with the low cost of information storage, it is easier than ever for companies to store and retrieve information about previous customers, opening the door to price discrimination based on purchase history. This paper shows that the dynamic properties of the equilibrium path of a di erentiated duopoly with overlapping generations of consumers di er substantially depending on whether customer recognition is present. At the same time, consumers in markets characterized by repeat purchases are more likely to face real or perceived costs of switching when they purchase from di erent providers over time. Such costs could be purely transactional, e.g. the cost of opening a new account with a di erent sup- plier, or they could be due to learning costs arising from the need to get accustomed to a new supplier or a new product.2 We demonstrate that there are two qualita- tively di erent equilibrium paths depending on the magnitude of the switching cost. Furthermore, since purchase history reveals information about a consumer?s relative preferences and allows rms to extract more surplus from their repeat customers, the presence of switching costs has the potential to increase the value of customer recognition. Therefore, switching costs become especially relevant in such markets. Yet, with a few notable exceptions, the literature so far has mostly considered the role of these two features separately from each other. The contribution of this paper can be best understood in light of the work of To (1996) and Villas-Boas (1999). A comparison between the present model and To (1996) allows us to understand the impact of customer recognition in the presence of 1See Taylor (2003) for a discussion of customer recognition in subscription markets. 2The classi cation of switching costs into transaction and learning costs was rst introduced by Nilssen (1992). 24 large switching costs. Similar to the model in To (1996), I examine a di erentiated duopoly with heterogeneous switching costs and consumers with two-period life spans. In addition, the current paper does not impose restrictions on the size of the switching cost; in contrast, To (1996) assumes that switching costs are su ciently high to prevent switching in equilibrium. For this reason, an analysis of the impact of customer recognition is limited to markets where switching costs are high enough to cause complete customer lock-in. Under complete lock-in, rms do not face demand from switchers. The main point of distinction between the two models is that in our model rms exercise customer recognition. As mentioned above, customer recognition and the presence of switching costs are likely to occur in the same markets and it is important to understand their interaction. I nd that customer recognition in a market with complete lock-in allows rms to set introductory prices that exclusively target newcomers, i.e. those consumers who are in their rst period in the market. As a result, the prices o ered to new customers become independent of market share and convergence to steady state occurs in just one period. In contrast, To?s model shows that under uniform pricing the rm with the larger market share charges a higher price (to all customers) and the market converges to steady-state non-monotonically and after a su ciently large number of periods.3 Given complete customer lock-in, steady-state per-cohort pro ts are lower when rms can price discriminate relative to uniform pricing. Switching costs intensify the competition for market share and rms compete away any pro ts associated with 3In a model of uniform pricing and in nitely-lived consumers Beggs and Klemperer (1992) show that convergence to steady state is monotonic but may be in nitely slow. 25 the resulting customer lock-in. Steady-state pro ts are at most equal to the pro t in the static Hotelling model with no switching costs and incumbency advantages in terms of pro ts and market share disappear within one period. Thus my results imply that under customer recognition, the presence of high switching costs does not protect the incumbency position of the dominant rm but rather encourages entry. From a practical point of view, rms? ability to price discriminate on the basis of purchase history allows us to derive the equilibrium prices in terms of the magnitude of the switching cost and perform comparative statics. Dynamic models of uniform pricing (Beggs and Klemperer, 1992; To, 1996) do not permit the identi cation of the direct impact of the switching cost on prices. This is due to the fact that in these models the switching costs do not explicitly enter the demand functions but rather justify the assumption that repeat customers cannot switch . In our setting we can derive the endogenous threshold, beyond which switching costs cause complete customer lock-in. Therefore, we can characterize the equilibrium path depending on whether switching costs are below or above this threshold level. A comparison between our model and Villas-Boas (1996) allows us to exam- ine the e ect of switching costs on prices, pro ts and convergence in the presence of customer recognition. Since the model in Villas-Boas (1996) does not feature switching costs, the equilibrium results are limited to the case where there is only partial customer lock-in. In our setting, partial lock-in is preserved as long as switch- ing costs are su ciently low. I nd that the equilibrium results from Villas-Boas? model are largely preserved except for the direct e ects of switching costs on prices, 26 on the volume of switching, and on the probability that both rms will be able to poach in equilibrium. In particular, I nd that the price to new customers is weakly decreasing in the cost of switching and as long as this cost is low enough to allow switching in equilibrium, the introductory price does not fall below marginal cost. Furthermore, switching costs raise the price to repeat customers, expand the loyal customer segment for each rm, increase pro ts and increase the likelihood that only one rm will be able to poach for a given distribution of the market. As long as both rms poach in equilibrium, switching costs do not a ect the speed of convergence to steady state. By adding switching costs to Villas-Boas? framework, we see that the size of the switching costs matters and can lead to two distinct equilibrium paths. When switching costs are high such that there is complete lock-in in steady state, the properties of the equilibrium change substantially. Convergence to steady state occurs in just one period, the price to new customers may fall below cost, and bilateral poaching is no longer possible. In addition, under the equilibrium with complete lock-in, I can characterize the rms? equilibrium strategies for all values of the consumer discount factor while under incomplete lock-in, Villas-Boas (1999) and my own treatment of the model is subject to the restriction that consumers are su ciently patient.4 Finally, I examine the e ect of imperfect customer recognition (ICR) on pro ts by examining an otherwise identical market where rms can also distinguish between 4While I derive the closed-form solutions for the coe cients that determine the optimal price policies of the two rms, the resulting expressions cannot be meaningfully analyzed unless we limit attention to a consumer discount factor close to one. 27 switchers and newcomers. I refer to the latter practice as perfect customer recogni- tion (PRC), because rms can recognize all three groups of consumers - newcomers, switchers and loyal customers. Since, the inability to price discriminate between newcomers and switchers under ICR is most relevant when switching does occur, I restrict this analysis to the case where switching costs are low enough to allow incomplete customer lock-in in steady state. I nd that under PCR loyal customers and switchers receive discounts relative to newcomers and a larger fraction of old consumers switch in equilibrium. Furthermore, rm pro ts per cohort of consumers is higher under PCR relative to ICR although this di erence is decreasing in the magnitude of the switching costs. Section 2.2 describes the model and Section 2.3 presents the equilibrium results for the case of low and high switching costs, respectively. Section 2.4 presents the comparison of imperfect versus perfect customer recognition and Section 2.5 concludes. 2.2 The Model We consider a duopoly market consisting of two in nitely-lived rms, A and B, selling a nondurable good. Consumers have uniformly distributed preferences over the products of the two rms, which gives rise to ex-ante product di erentiation. The degree of product di erentiation is exogenously determined and xed. Each rm produces the good at a constant marginal cost, c. Consumers enter the market for two periods only and demand one unit of the good in each period. They have 28 common valuation for the good given by v, which I assume to be su ciently high to induce a purchase in each period. In each period an old cohort of consumers exits the market and a new cohort enters. I assume that all cohorts are of the same size, although the analysis can be readily extended to accommodate cohorts of di erent sizes. In any given period, a rm faces two overlapping generations of consumers: old consumers in their second period in the market who have established a purchase history; and newcomers, who enter the market in the current period and have not purchased from either rm yet. If old customers purchase from the same supplier in both periods I call them ?loyal? customers, while if they purchase from two di erent suppliers over their lifetime, I call them ?switchers?. Firms recognize their own loyal customers but cannot determine if a new customer is a newcomer with no purchase history or a switcher from the rival rm. I use Hotelling?s framework to model product di erentiation. Firms are located at the opposite ends of the unit interval with rm A located at zero. Each cohort of consumers has mass normalized to one and consumers are uniformly distributed over the unit interval. For each consumer, the distance from rm A relative to rm B is a proxy for her preference towards the two rms. I assume that these preferences are time-invariant and known to the consumer ex ante.5 To model preferences, suppose that customers face a linear transportation cost of per unit of distance, such that if a consumer is located at distance x from rm A, she will have to incur transportation costs of x if she buys from A, or (1 x) if she buys from rm B. A 5The literature on experience goods considers settings where consumers are ex ante unaware of their preferences and the information they obtain through purchasing from one supplier creates an endogenous cost of switching. I do not consider such situations here but the reader is referred to Villas-Boas (2004) for a model of dynamic competition with experience goods. 29 consumer who switches suppliers in her second period also incurs a switching cost, s, which is assumed to be time-invariant, uniform across consumers and common knowledge.6 Firms compete for customers by o ering an introductory price and a regular price. All new customers, which could be either newcomers or switchers, are o ered an introductory price, pint, while loyal customers are o ered a regular price, piot. We assume that rms simultaneously announce their introductory prices at the beginning of each period. However, each rm sets its regular price only after observing the introductory price of the rival. This assumption guarantees the existence of a pure-strategy equilibrium in periods when the distribution of market share is su ciently unequal.7 It is well known that sequentially set prices are higher than the Bertrand price. However, for the purposes of our analysis an upward bias in the regular price will not a ect the results in a qualitative way. Since my setup is based on Villas-Boas (1999), I adopt the same notation, whenever possible, to facilitate comparison of the results. Let i;j =fA;Bg. Then, let qii;t indicate demand from rm i?s loyal customers in period t, qij;t { demand from old customers who switch from i to j in period t, and q1i;t { demand from newcomers who purchase from i in their rst period in the market. To simplify some of the notation that follows, without loss of generality, I normalize marginal costs to zero for both rms. In Section 2.2 I will present the equilibrium results when marginal costs are constant, symmetric and equal to c 0. 6Since customers can switch at most once, there is no need to distinguish whether switching costs arise strictly from learning costs (which are incurred only once with each supplier) or from transaction costs (which are incurred every time a buyer switches suppliers). 7This is a common assumption in models of customer recognition where customers are hetero- geneous in some characteristic (Villas-Boas, 1999; Marquez, 2002). 30 To set up the rms? problem, we rst characterize the demand functions for each of the three groups of customers: newcomers, switchers and loyal customers. I then discuss the recursive nature of the problem and set up the rms? value functions. 2.2.1 Demand from Newcomers To derive each rm?s demand from newcomers, we rst determine the location of the marginal consumer among newcomers in the market. This location also determines the distribution of the market in the current period and the market shares that rms will inherit in the following period. We indicate the location of the marginal consumer among newcomers at time t as xt+1, where xt+1 will also be used to describe rm A?s market share in period t+ 1. Indicate the consumer discount factor as c where c2 (0;1]. Assuming that consumers have perfect foresight, a newcomer located at x will purchase from rm A in her rst period in the market if this purchase yields a weakly higher surplus over the consumer?s two-period life span in the market than the one realized when purchasing from rm B: v pAnt x+ c max v pAot+1 x;v pBnt+1 s (1 x) (2.2.1) v pBnt (1 x) + c max v pBot+1 (1 x);v pAnt+1 s x The marginal newcomer, located at xt+1, will be just indi erent between the two sequences of purchases when 31 pAnt+ xt+1 + c min pAot+1 + xt+1;pBnt+1 +s+ (1 xt+1) (2.2.2) = pBnt + (1 xt+1) + c min pBot+1 + (1 xt+1);pAnt+1 +s+ xt+1 Since we assume that the regular price is set after the rival?s introductory price is known, rm A will always set pAot+1 such that its marginal loyal customer at time t + 1 is just indi erent between switching and staying after having purchased from A at time t. If rm A wants to keep all of its customers in period t + 1, it will set pAot+1 + xt+1 = pBnt+1 + s + (1 xt+1). If it wants to let some customers switch, then it must be true that for rm A?s marginal old customer pAot+1 + xt+1 > pBnt+1 +s+ (1 xt+1): In either case, we have min pAot+1 + xt+1; pBnt+1 +s+ (1 xt+1) = pBnt+1 +s+ (1 xt+1) (2.2.3) Otherwise, rm A can always increase its pro ts by raising pAot+1 without a ect- ing demand from loyal customers. Therefore, the location of the marginal newcomer at time t can be determined from: pAnt+ xt+1 + c pBnt+1 +s+ (1 xt+1) (2.2.4) = pBnt + (1 xt+1) + c pAnt+1 +s+ xt+1 32 This equality determines the location of the marginal newcomer at time t as well as the distribution of market share at the beginning of period t+ 1: xt+1 = (1 c) + c(p A nt+1 p B nt+1) +p B nt p A nt 2 (1 c) (2.2.5) Thus, demand from newcomers can be de ned as: q1A;t = xt+1 (2.2.6) q1B;t = 1 xt+1 (2.2.7) 2.2.2 Demand from Loyal Customers The marginal loyal customer for rm A at timetwill be just indi erent between switching and staying. Therefore, her location, xlt can be determined from the equality of the payo s of each alternative: pAot + xlt = pBnt +s+ (1 xlt) (2.2.8) This equality yields rm A?s demand from loyal customers: qAA;t(pAot;pBnt) = min +s+pB nt p A ot 2 ; xt (2.2.9) Similarly, we can nd rm B?s demand from loyal customers: qBB;t(pBot;pAnt) = min +s+pA nt p B ot 2 ;1 xt (2.2.10) 33 Note that the regular price does not a ect demand from newcomers or switch- ers, so rm i will choose piot independent of its own choice of pint. Thus, each rm sets piot to maximize pro ts from loyal customers taking as given the rival?s introductory price. Since we assume that rms set piot after observing pjnt, the choice of piot deter- mines the optimal mass of loyal customers that a rm would like to keep given the introductory price of its rival. If this optimal mass exceeds the rm?s actual market share, sales to loyal customers are limited to the size of the rm?s existing customer base. For c = 0 rm A?s problem with respect to pAot can be set up as follows: max pAot pAot min +s+pB nt p A ot 2 ;xt (2.2.11) Given the rival?s introductory o er, pBnt, the optimal regular price for rm A is: pAot(pBnt) = max +s+pB nt 2 ; +s+p B nt 2 xt (2.2.12) Note that the optimal regular price is increasing in the rival?s introductory o er.8 In fact, if pBnt is su ciently large, rm A would keep all of its previous customers as loyal customers so qAA;t will be constrained by rm A?s market share. On the other hand, if pBnt is low enough, rm A?s loyal customer segment will be less than xt. In the cases where sales to loyal customers are less than a rm?s market share 8For an arbitrary level of the marginal cost, the optimal regular price depends on c if sales to loyal customers are less than the rm?s market share: pAot(pBnt) = max c+ +s+pB nt 2 ; +s+p Bnt 2 xt 34 we de ne ^qii;t as rm i?s optimal sales to loyal customers in period t. We can further extend the interpretation of ^qii;t as rm i?s optimal market share in period t if we consider situations in which the acquisition of market share is costly (i.e. when the introductory price is below cost). Upon nding a deterministic optimal path for pAnt and pBnt, each rm can project how much market share it would like to capture today in order to maximize pro ts from loyal customers tomorrow. We can nd ^qAA;t by plugging pAot(pBnt) into (2.2.9) which yields rm A?s optimal market share as a function of the rival?s introductory price only: ^qAA;t(pBnt) = +s+p B nt 4 (2.2.13) Firm A?s sales to loyal customers and the resulting pro ts can be summarized as follows: qAA;t(pBnt) = min ^qAA;t(pBnt); xt (2.2.14) Aot(pBnt) = max ( +s+pB nt) 2 8 ;( +s+p B nt 2 xt)xt (2.2.15) Similarly, we nd rm B?s optimal regular price, sales to loyal customers and 35 pro ts: pBot(pAnt) = max +s+pA nt 2 ; +s+p A nt 2 (1 xt) (2.2.16) ^qBB;t(pAnt) = +s+p A nt 4 (2.2.17) qBB;t(pAnt) = min ^qBB;t(pAnt); 1 xt (2.2.18) Bot(pAnt) = max ( +s+pA nt) 2 8 ; +s+pA nt 2 (1 xt) (1 x t) (2.2.19) From (2.2.12) and (2.2.16) we can see that the optimal regular price is uniquely determined given knowledge of the rival?s introductory price. The assumption that rms set introductory and regular prices sequentially ensures that once the intro- ductory prices are announced and rms set their regular prices accordingly, neither rm has a pro table deviation in changing its regular price. Without this assump- tion, there may not be pure-strategy equilibria if the prior distribution of the market is su ciently unequal. To see this, observe from (2.2.12) and (2.2.13) that when market share is not a binding constraint on sales to loyal customers, the regular price and sales to loyal customers are increasing in the rival?s introductory price. Consider a setting where regular prices are set simultaneously with the introduc- tory price of the rival. Suppose that rm B starts the period with a relatively high market share and charges a low introductory price. Firm A?s best response would be to charge a low regular price as well in order to retain some of its clientele. But it is possible that as rm A charges a low regular price it still retains all of its old customers because of their close proximity to A (since we assumed that rm A?s 36 market share is low), so rm B?s best response may be to raise its introductory price and target newcomers only, to which rm A?s best response will be to raise its regular price as well. Realizing that it can now pro tably poach rm A?s customers, rm B?s best response would be to lower its price again, giving rise to another cycle of price cuts. So, when regular prices and introductory prices are set simultaneously an equilibrium in pure strategies may not exist. 9 2.2.3 Demand from Switchers Demand from switchers, if positive, can be represented as the di erence be- tween the rival?s market share and its optimal sales of loyal customers. For rm A, demand from switchers is given by qBA;t = max (0;(1 xt) ^qBB;t) (2.2.20) Using (2.2.13), we nd: qBA;t(xt) = max 0; 3 s p A nt 4 xt (2.2.21) Similarly, demand from switchers for rm B is given by: qAB;t = max (0;xt ^qAA;t) (2.2.22) qAB;t(xt) = max 0;xt +s+p B nt 4 (2.2.23) 9This is the same argument as applied in Villas-Boas (1999), pp. 611. At this point my model follows closely the setup in Villas-Boas and my introduction of switching costs does not preclude the need to assume sequential price setting with respect to regular prices. 37 From the demand equations in (2.2.21) and (2.2.23), it is clear that a rm?s ability to poach depends on the pre-existing distribution of market shares, sum- marized in xt. The rm that enters the period with low market share can attract the rival?s previous customers at a higher price because of the closer proximity of prospective switchers. At the same time, both newcomers and switchers are o ered the same price, pnt, so each rm chooses its optimal introductory price by balancing the incentives to gain market share and to maximize pro ts from poaching. Given the symmetry of the problem, it is clear that if the rms? only goal was to capture market share, their introductory prices would be equal. It is the incentive to poach that drives a wedge between the rms? introductory prices, unless market share is equally split at the beginning of the period. 2.2.4 Equilibrium Concept Since current market share depends on the introductory prices from the previ- ous period only, xt is the only payo -relevant state variable that a ects the optimal introductory prices in period t. We will refer to xt as the state variable in pe- riod t while pAnt and pBnt will represent the choice variables for rm A and rm B, respectively. The optimal regular price is unique for a given introductory price, so identifying the optimal pricing strategies for pint is su cient to derive the full schedule of prices in period t as well as the distribution of market share in period t+ 1. We solve the dynamic problem for each rm by looking for a Markov Perfect Equilibrium (MPE), in which rms? pricing strategies depend solely on the realized 38 distribution of the newcomers? market shares in the previous period. Based on the solution of a similar problem in Villas-Boas (1999), we look for a MPE, in which the price strategies regarding pint are piecewise a ne in xt and the value function of each rm is piecewise quadratic in xt.10 We allow for the possibility that the optimal strategies are piecewise a ne because rms may pursue di erent strategies depending on whether they are able to poach, which in turn depends on the state variable xt. 11 Therefore, we specify pAnt pBnt = ak +bkxt (2.2.24) pAnt = ek +fkxt (2.2.25) pBnt = ek ak + (fk bk)xt (2.2.26) VA(xt) = Ak + Akxt + Akx2t (2.2.27) VB(xt) = Bk + Bk (1 xt) + Bk (1 xt)2 (2.2.28) We index each of the undetermined coe cients by k to indicate that they may be di erent for di erent ranges of the state variable. Since we conjecture that the introductory prices are piecewise a ne in xt, we also write their di erence as piecewise a ne in xt. Using (2.2.5) and applying pAnt+1 pBnt+1 = ak +bkxt+1 where the subscript k refers to the region containing xt+1, we can rewrite the market distribution in period t+ 1 as a function of the current market distribution: 10Equilibria in non-a ne strategies may also exist but they are outside the scope of my study. 11In the special case when c = 1, the optimal price strategies would be identical for all xt. 39 xt+1 (xt) = (1 c) + cak +p B nt (xt) p A nt (xt) 2 (1 c) cbk (2.2.29) Finally, using f 2 (0;1] to indicate the rms? discount factor, we write out the rms? optimization problems with respect to pint: VA (xt) = max pAnt max +s+pBnt 2 8 ; +s+pB nt 2txt x t ! (2.2.30) +pAnt xt+1(pAnt;pBnt) + max 0; 3 s p A nt 4 xt + fVA xt+1(pAnt;pBnt) VB (xt) = max pBnt max +s+pAnt 2 8 ; s+pA nt + 2 xt (1 x t) ! (2.2.31) +pBnt 1 xt+1(pAnt;pBnt) + max 0;xt +s+p B nt 4 + fVB 1 xt+1(pAnt;pBnt) 2.2.5 Exit Costs Following Villas-Boas (1999) I assume that there is some minimal level of exit costs that a rm would incur at the end of the period if it does not realize sales to newcomers in that period. Similarly to the assumption that regular prices are set after introductory prices are known, the assumption on exit costs rules out the possibility that a pure-strategy equilibrium may not exist when one rm starts out 40 the period with a very small market share. For example, it may be more pro table for a rm with low or no market share to set a price that targets switchers only (because it can charge a higher price to rival?s customers who are located closer to the rm) rather than compete for newcomers. But then the rival can raise its price as well without giving up too much in demand from newcomers. Given this higher price, the rm with low market share may nd it pro table to compete for newcomers as well, so it will lower its price. This will be followed by another price cut by the rival and the rst rm may be willing to exit the newcomers? market again. The presence of some minimal level of exit costs eliminates this possibility by ensuring that a rm will always choose to sell to newcomers regardless of how small its market share is.12 It is important to note that an assumption about exit costs is necessary only for a limited range of the parameter values. Exit costs are plausible if we assume that rms enter the market with the intention to serve both new and old customers. One example of exit costs could be the erosion of goodwill a rm has if it is based on intergenerational transfer of information about the rm?s product. A rm that does not sell to newcomers in a given period may have to compensate for the dissipation of goodwill by taking costly actions to promote its product to newcomers in the next period. Alternatively, we could think of these exit costs as the cost of reentering the market when the rm?s market share is zero. The minimal required level of exit costs that would ensure 12Previous models of dynamic competition with switching costs have imposed similar conditions to guarantee the existence of a pure-strategy equilibrium. To (1996) imposes an upper bound on the consumer reservation value while Beggs and Klemperer (1992) impose restrictions on the rate of customer turnover and cost di erentials. In both models, these conditions ensure that rms will not pursue a strategy where they do not serve any newcomers. 41 a pure-strategy equilibrium in my model can be found by setting the maximum discounted payo from deviating once and selling to switchers only equal to the payo from staying with the equilibrium strategy and selling to both switchers and newcomers in equilibrium. We denote exit costs by E and relegate the derivation of their minimal required level to the appendix. 2.3 Equilibrium Results We present two sets of equilibrium results depending on the magnitude of the switching cost. For each equilibrium I characterize the optimal price strategies on the equilibrium path and the resulting pattern of convergence to steady state. I present the intuition of these results in the body of the paper and relegate all technical proofs to the appendix. 2.3.1 The Equilibrium under Incomplete Lock-in and Low Switching Costs In his model of customer recognition in the absence of switching costs, Villas- Boas (1999) shows that when the current distribution of market share is not too uneven, both rms are able to attract some of the rival?s previous customers. Based on this result, I conjecture that when switching costs are not too large to preclude switching, there will be incomplete customer lock-in for xt close to the middle. I de ne this range as (xm;1 xm) and refer to it as the "poaching region". Upon identifying the optimal price strategies under the conjecture that both rms poach in 42 equilibrium, I derivexm and verify that my conjecture is correct forxt2(xm;1 xm). I suppress the subscript k for all coe cients that de ne the rms? optimal strategies and value functions for xt 2 (xm;1 xm). As long as the problem is symmetric, I also know that the coe cients determining the rms? value functions are identical across the two rms, so I write A = B = ; A = B = ; A = B = . We modify the value functions for each rm to re ect my conjecture that within the poaching region both rms attract some of the rival?s previous customers: VA (xt) = max pAnt max +s+pBnt 2 8 ; +s+pB nt 2txt x t ! (2.3.1) +pAnt xt+1(pAnt;pBnt) + 3 s p A nt 4 xt + fVA xt+1(pAnt;pBnt) VB (xt) = max pBnt max +s+pAnt 2 8 ; s+pA nt + 2 xt (1 x t) ! (2.3.2) +pBnt 1 xt+1(pAnt;pBnt) +xt +s+p B nt 4 + fVB 1 xt+1(pAnt;pBnt) After nding the best response functions and solving for the undetermined coe cients I obtain the following: 43 a = b2 (2.3.3) f = b2 (2.3.4) The equation characterizing b is given by: [4b(8 2 c cb) + 16 (2 2 c cb)][(2 2 c cb)2 fb2] + fb3(18 2 c cb) + 16 fb2(2 2 c cb) = 0 (2.3.5) In order to derive a tractable solution for b, I limit attention to the case where c! 1. All results that refer to the equilibrium with incomplete lock-in are based on this restriction. From (2.3.5) I nd that for c!1;b!0 and @b=@ c < 0. In the exposition of the rms? problem I assumed that marginal cost is zero. I now generalize the setup presented in the previous section by assuming marginal cost equals c 0. Proposition 1 below characterizes the rms? optimal price strategies on the equilibrium path within the poaching region. Proposition 1. Suppose f 0; c ! 1; xt 2 (xm;1 xm); s ;and E (3 s)2 2 f( +s)2 16 . A Markov-perfect equilibrium in a ne strategies exists and can be characterized as follows. As c!1: 44 pint!c (2.3.6) piot!c+ +s2 (2.3.7) ^qiii! +s4 (2.3.8) qij! s4 (2.3.9) xm! +s4 (2.3.10) In addition to (2.3.3), (2.3.4), and (2.3.5), the relevant coe cients governing the optimal price strategies and the rms? value functions (for c normalized to zero) are 45 given by: e = b4 + 2 2(1 c) + 4 (1 c) 3 cb 4 f 4 f 2(4 2 c cb) (2.3.11) s(2 2 c cb)2(4 2 c cb) =( +s+e a)(f b)4 (2.3.12) + e( 4 (2 2 c cb) + 4 (f b) f(6 2 c cb))4 (2 2 c cb) + f (10t 2(1 c) + 4 ca 3 c (2 2 c cb) + 4 (e a)) 4 (2 2 c cb) fe(6 2 c cb)4 (2 2 c cb) + f b2 2 c cb + f 2b(t(1 c) + ca a)(2 2 c cb)2 = b 2(2 2 c cb)(18 2 c cb) + 16 b(2 2 c cb)2 32 ((2 2 c cb)2 fb2) (2.3.13) As c ! 1; a! 0; b! 0; e! 0; ! 0 and ! 0. The limits of the poaching region, xm, are de ned as follows: 46 xm = 1 2 M +C2 pM(M +C2) e C2(3 s) 2 M +C2 pM(M +C2) f 4 C2 (2.3.14) where C = 2 2 c cb; M = 4 C 4 f In steady state the market is equally distributed, x = 1=2 and convergence is governed by xt+1 12 = b2t(1 c) cb xt 12 (2.3.15) where b=(2t(1 c) cb)2(2=3;1). Proof. See Appendix. The optimal price strategies described in Proposition 1 have a number of interesting features, which become evident when I consider the limit case where c = 1. I refer to consumers with c = 1 as very patient consumers (all consumers within the market have the same discount factor). When consumers are very patient Proposition 1 indicates that a = b = 0, which implies that pAnt = pBnt for all possible distributions of xt. That is, rms? introductory prices become independent of market share and equal to marginal cost. To provide intuition for this result I present the following lemma: Lemma 1. For c = 1, q1i;t = 0 whenever rm i deviates from the equilibrium path 47 proposed in Proposition 1 by setting pint >pjnt. Proof. First note that for c = 1, we have pAnt = pAnt+1 = pBnt+1. Designate this equilibrium price as p and note from Proposition 1 that this is true for all xt as long as c = 1. Let pAnt be the deviating price for rm A. A newcomer located at x will purchase from A if and only if: pAnt + x+ (p+ (1 x) +s) p+ x+ (p+ x+s) pAnt p It is clear that for pAnt >p, the inequality above cannot be satis ed for any x. Therefore, if A deviates to a price above the proposed equilibrium price, p, it will make no sales to newcomers. The lemma above suggests that demand from newcomers is zero whenever a rm sets its introductory price above the rival?s price. Similarly, a rm will capture the entire market of newcomers if it undercuts the rival?s price. These results suggest that on the equilibrium path demand from newcomers is perfectly elastic when c = 1. This result is also present in Villas-Boas (1999) and is preserved here even when switching costs are positive. There are a number of features present in the model that produce this result - consumers stay in the market for an even number of periods, there is cost and demand symmetry across the two rms, and there is no uncertainty about the realization of consumer preferences in the mature 48 market. As long as the presence of switching costs does not give rise to asymmetric demand, this property is preserved. In my setting switching costs are symmetric and there is no uncertainty about their realization so demand from newcomers is symmetric across the two rms. The perfect elasticity of newcomers? demand helps explain the properties of the equilibrium. Recall that we are able to derive tractable representations of the equilibrium price strategies only when we limit attention to c close to one, so under- standing the equilibrium properties in the limit, c = 1, is particularly important. The perfect elasticity of newcomers? demand explains the somewhat surprising re- sult that as c ! 1 the introductory price approaches marginal cost despite the heterogeneity of consumer preferences. In fact, as noted in Villas-Boas(1999), com- petition intensi es as consumers become more patient. To see this, note that using the equilibrium result a = b=2 we can rewrite xt+1 as follows: xt+1 = (1 c) + ca+p B nt p A nt 2 2 c cb = 12 + p B nt p A nt 2 2 c cb (2.3.16) Let w = 12 2 c cb indicate the weight of the price di erential on the location of the marginal consumer, given by xt+1. On the equilibrium path b 0 and @b=@ c < 0, so @w=@ c > 0 { as consumer patience increases, the marginal newcomer becomes more sensitive to the di erence between the introductory prices o ered today. 49 The result that consumer patience intensi es competition is in contrast with previous results in the literature on switching costs and is due to our assumption about imperfect customer recognition.13 In models of dynamic competition with switching costs and uniform pricing policies (Beggs and Klemperer, 1992; To, 1996) consumer patience softens the competition for market share. In these models rms charge a uniform price to all customers and switching costs are high enough to preclude switching. As a result, consumers recognize that the low-price rm today will charge a higher price tomorrow because it has a greater incentive to exploit its customer base today rather than invest in market share. In contrast, under customer recognition combined with the assumption that regular prices are conditioned on the introductory prices, the price that an old customer pays increases in the introductory price of the rival, regardless of the decision to switch or stay. I have shown that a!0 and b!0 as c!1, so consumers realize that on the equilibrium path the rms? introductory prices next period will be less di erentiated as c goes up. This increases the importance of the introductory o ers in the current period and makes consumers more sensitive to these o ers. Going back to the result that pAnt = pBnt = c when c = 1, we observe that this is the same price as derived in Villas-Boas? model where switching costs are zero. It may seem puzzling at rst that the introductory price does not fall below cost despite the perfect elasticity of newcomers? demand and the presence of switching costs. To provide intuition for this result, I summarize some important properties 13In subsequent results based on perfect customer recognition, we nd the opposite e ect { consumer patience relaxes competition. 50 of the equilibrium described above in the following corollary: Corollary 1. Along the equilibrium path described in Proposition 1 the following is true for c!1: 1) @pin=@s< 0 for c < 1 and @pin=@s = 0 for c = 1 (2.3.17) 2) @pio=@s> 0 (2.3.18) 3) @^qii=@s> 0 and ^qii! 12 as s! (2.3.19) 4) @xm=@s> 0 and xm! 12 as s! (2.3.20) Proof. See Appendix. The properties described by (2.3.18) and (2.3.19) show that rms? regular prices and optimal sales to loyal customers increase in the size of the switching costs. Therefore, switching costs raise pro ts from loyal customers. Furthermore, (2.3.17) indicates that the introductory price is weakly decreasing in s because market share becomes more valuable as s goes up and because the barriers to switching are higher. Interestingly, when consumers are very patient the introductory price is independent of the switching cost. This can be explained by result (2.3.19). The equilibrium I characterize in Proposition 1 is valid for s (conditional on c = 1) and I nd that rms? optimal market share, ^qii, is no greater than 1=2. When consumers are very patient, rms are unable to charge introductory prices above marginal cost (or they will not make any sales to newcomers) and they are not interested in capturing 51 more than half of the newcomers? market if this entails pricing below cost. Thus, at pin = c neither rm has an incentive to undercut the rival?s price regardless of the value of s. The fact that rms retain less than their share of old customers as loyal cus- tomers explains why the equilibrium I derive here is similar to the one in Villas-Boas? model of customer recognition, in which switching costs are zero. As long as switch- ing costs are su ciently low to allow poaching in equilibrium, we have that ^qii 12 { besides increasing the value of market share and decreasing the return to poach- ing, the presence of some low level of switching costs does not change the nature of competition in the market. The magnitude of the switching cost does a ect the equilibrium level of prices, pro ts, and the mass of consumers who switch, by varying the relative return on market share acquisition and poaching. To see how s a ects these two competing forces in the determination of the optimal introductory price, I look at the de- composition of the partial derivative of pint with respect to switching costs. Since @pAnt=@s = @e=@s+@f=@s and the latter term is zero we have: @e=@s = 2 f2 +C @ =@s C4 + 2C (2.3.21) The rst term in this expression captures the impact of the value of market share ( ) as a function of s on the introductory price. In the proof of Corollary 1 I show that @ =@s 0 and C = 2 2 c cb ! 0+ as c ! 1. As switching costs increase, market share becomes more valuable (@ =@s 0) and patient rms lower 52 their introductory o ers accordingly. The second term in @e=@s depends solely on b and captures the e ect of the incentive to poach. Recall that f = b=2 and pAnt = e+fxt, so a higher b indicates that market share becomes more important in determining pn. Thus, b can be viewed as a proxy of the magnitude of the incentive to poach. It is straightforward to show that @C=@b < 0, so the absolute value of C=(4 + 2C) increases as b goes up { a strong incentive to poach has a larger negative impact on the equilibrium pn through s because rms have to o er larger discounts to compensate switchers for the cost of switching. In fact, newcomers bene t from the ongoing competition for switchers because the rms? incentive to poach reduces introductory prices even further. Only in the limit as c!1 we have C 4 +2C ! 0 { the incentive to poach no longer a ects the equilibrium introductory price. When c < 1 the determination of the introductory o ers hinges on the trade- o between poaching and investing in market share and the size of the incoming cohort of consumers matters. If consumers are fairly impatient and there is a large cohort of newcomers (relative to the old cohort of consumers, some of which are switchers today), the equilibrium introductory price may be quite high as poaching becomes relatively less important. If there is only a small mass of newcomers in the market, rms may forgo the high pro t margin on newcomers and lower their prices to attract more switchers. When consumers are very patient, we have shown that pn converges to marginal cost, reducing the pro t margin on switchers and newcomers to zero. In this case, if the size of the newcomers cohort is relatively small, a rm may nd a pro table deviation in raising its price and targeting switchers only. This 53 will give rise to an equilibrium in mixed strategies where rms randomize between charging a high price targeted at switchers and undercutting the rival?s price to at- tract newcomers, resulting in higher introductory prices on average. Thus, a small incoming cohort of consumers may either raise or lower prices depending on the degree of consumer patience. We can also conclude that fast growing markets with large incoming cohorts of very patient consumers can more easily sustain introduc- tory o ers close to marginal cost (even in the absence of exit costs) because a larger volume of sales to newcomers counteracts the temptation to raise prices and target switchers only. From (2.3.17)and (2.3.18) it is clear that customer recognition alone gives rise to a ?bargain-then-rip-o ? price pattern that is typical of switching costs models.14 Switching costs further amplify the resulting loyalty price premium, pot pnt. Using (2.2.12) and adjusting for c> 0, we see that pot pnt = c+ +s+pnt2 pnt (2.3.22) = +s2 pnt c2 ! +s2 as c!1 Finally, for c to1 we can derive the impact of switching costs on pro ts. Firm i?s pro ts per cohort of consumers entering the market in period t are given by 14See Section 2.3.1 in Farrell & Klemperer 2007 for a thorough discussion of such models. 54 it(xt) = (pint c)(q1i +qji) + f (pot+1 c) (qii) (2.3.23) @ it @s = @pint @s (q1i +qij) + (p i nt c) @qji @s + f @pi ot+1 @s qii + (p i ot+1 c) @qii @s (2.3.24) ! f 1 2qii + +s 2 @qii @s as c!1 (2.3.25) From Proposition 1 and Corollary 1 we can see that @pin=@s!0 and pin c! 0 as c!1 while @pio=@s> 0 and @qii=@s> 0 for all c. Therefore, in the limit rm pro ts increase in the size of the switching costs and reach a steady-state maximum of f 2 as s! . Note, however, that this is the minimum level of pro ts in this market because prices are decreasing in c and I evaluate the pro t function at c!1. In the standard Hotelling model pro ts from a generation of customers who make a purchase twice would equal (1+ f) 2 . Compared to that level, pro ts in a market with customer recognition, low switching costs and very patient consumers are always lower. Based on the same result for s = 0 and = 1, Villas-Boas (1999) concludes that rms would be better o if they did not recognize their previous customers. Chen (1997) reaches the same conclusion for a homogeneous market with heterogeneous switching costs and customer recognition. In the current setting, we see that switching costs can alleviate the disparity in pro ts. Unfortunately, the literature does not provide us with a basis of comparison to determine if customer recognition alone leads to lower pro ts relative to uniform pricing when switching 55 occurs in equilibrium. Corollary 1 also states that the presence of switching costs shrinks the ?poach- ing? region given by (xm;1 xm). I identify this region by looking for the range of xt such that neither rm has a pro table deviation in a price strategy that does not attract switchers. I nd that rm A will not deviate from the equilibrium strategy described in Proposition 1 if xt 1 xm and, similarly, rm B will not deviate if 1 xt 1 xm. Intuitively, these restrictions follow from the fact a rm with large market share has to charge a lower introductory price in order to poach. If the rm?s market share is su ciently large and c < 1, the desire to poach will lower pn to a level, at which the rm is better o raising its price and selling to newcomers only. Since higher switching costs lower the payo from poaching, a deviation to a no-poaching strategy becomes pro table at lower levels of market share. As the poaching region contracts, bilateral switching becomes less likely to be observed in equilibrium. As (2.3.20) shows, the poaching region shrinks to a mass of zero as switching costs approach and c!1 and poaching becomes unfeasible. When the market is very unevenly distributed, i.e. xt is outside (xm;1 xm), only the rm with the smaller market share engages in poaching while its rival pursues a no-poaching strategy. Villas-Boas (1999) investigates in detail the market dynamics outside the poaching region for c < 1 and nds that it takes no more than two periods for the market to enter the poaching region. If market share falls in the ?very small? region, xt2[0;xs], only rm A poaches; next period the market enters the ?small? region, given by (xs;xm), where A poaches and B does not, and in the next period market share is such that xt 2 (xm;1 xm) and both rms 56 poach. When switching costs are low, the market will follow a similar path before entering the poaching region. Switching costs will a ect the boundary xs, which will a ect the probability that xt falls within the ?very small? region and that it will take an additional period before entering the poaching region. Since convergence to steady state may be rather slow in this region, a change in the probability that the market takes an additional period to enter the poaching region does not seem to be signi cant enough to merit a detailed investigation of how s a ects xs. For this reason, I do not characterize the equilibrium price strategies in the ?small? and ?very small? regions. I note, however, that when c = 1 the price strategies outlined in Proposition 1 constitute an equilibrium for all xt and convergence to steady-state occurs in just one period from all possible realization of market share. This occurs because the perfect elasticity of newcomers? demand implies that both rms would set their introductory prices equal to marginal cost regardless of xt. The market dynamics for c < 1 are governed by the equation de ning b in (2.3.5). Note that s does not enter (2.3.5) and, hence, within the no-poaching region switching costs do not impact the speed of convergence to steady state. Villas-Boas (1999) shows that convergence is monotonic and may take a large number of pe- riods. Thus, the incumbent rm preserves its incumbency advantage in terms of market share and this result is una ected by the presence of switching costs as long as the latter are not too high to prevent switching. However, the ultimate impact of the incumbency advantage is unclear. Entering the period with larger market share bene ts the incumbent because the rival is setting a higher introductory price allowing the incumbent to set a higher regular price and retain more regular cus- 57 tomers. However, large market share also hurts pro tability because it implies a lower introductory price { the incumbent has a lower pro t margin on new cus- tomers. When c = 1 incumbency advantages in terms of market share disappear within one period, while incumbency advantages in terms of pro ts are zero. 2.3.2 Equilibrium Results under Complete Lock-in When s > c and c ! 1, the price strategies described in Proposition 1 no longer constitute an equilibrium because bilateral poaching is not feasible. I now conjecture that when switching costs exceed the threshold c , there will be a middle region for xt, within which each rm retains its entire previous clientele. When switching costs are very high, this region will extend to the entire market - all customers will be locked-in to their original supplier for all xt. Such high switching costs are normally assumed to exist in the dynamic models of uniform pricing (Beggs and Klemperer, 1992; To, 1996). Under complete customer lock-in, both rms will pursue price strategies tar- geting newcomers and loyal customers only. Assuming (and later verifying) that next period the distribution of the market falls within the same no-poaching region, we can modify the rms? value functions as follows: 58 VA (xt) = max pAnt ( +s+pBnt 2txt)xt (2.3.26) +pAnt xt+1(pAnt;pBnt) + fVA xt+1(pAnt;pBnt) VB (xt) = max pBnt s+pA nt + 2 xt (1 x t) (2.3.27) +pBnt (1 xt+1(pAnt;pBnt) + fVB 1 xt+1(pAnt;pBnt) Solving for the rms? best response functions and checking for deviations, we can characterize the equilibrium price strategies under complete lock-in in Proposi- tion 2: Proposition 2. Suppose c 2 (0;1); f 2 (0;1); xt 2 (exm;1 exm); and s > c . A Markov-perfect equilibrium in a ne strategies exists and can be characterized as 59 follows: pAnt = pBnt = c fs (1 c + f) 1 + f ; (2.3.28) pAot = pBot = c+ + s c 1 + f ; (2.3.29) qAA = xt qBB = 1 xt; qAB = qBA = 0 (2.3.30) xt+1 = x = 12 (2.3.31) (2.3.32) The limits of the no-poaching region, (exm;1 exm) are de ned as follows: exm = max (0; 1 ^qBB; ^xm) (2.3.33) where ^qBB = +s+p A nt 4 (2.3.34) ^xm = 3 s4 2( pM(M +C2) M)pA nt 4tC2 (2.3.35) C = 2 (1 c) (2.3.36) M = 4 C 4 f (2.3.37) The relevant coe cients governing the optimal price strategies and the rms? value 60 functions (for c normalized to zero) are given by: a = b = f = 0 (2.3.38) e = (1 c + f) fs1 + f (2.3.39) = (2 + f c) +s1 + f (2.3.40) = 2 (2.3.41) Proof. See Appendix. Proposition 2 characterizes the equilibrium in the no-poaching region, (exm;1 exm). First, note that the equilibrium price strategies are independent of market share (f = b = 0) for all values of c2(0;1). This is in contrast to previous models of dynamic competition with switching costs. Beggs and Klemperer (1992), Padilla (1992) and To (1995) establish that in the presence of switching costs prices are increasing in market share because the rm with higher market share has a stronger incentive to exploit its customer base and forgo investment in future market share. All three of these models, however, consider rms that charge uniform prices to all customer segments. In a setting where rms can price discriminate on the basis of purchase history, there is no tradeo between exploiting one?s clientele and investing in future market share. The main factor that drives this result is the fact that within (exm;1 exm) neither rm can poach. In our setting poaching is the only channel that establishes a relationship between the introductory price and market share. When rms realize that they cannot successfully poach in equilibrium, their only 61 objective in selecting pnt is to compete for newcomers and this renders market share irrelevant. Hence, customer recognition in the presence of switching costs, which are su ciently high to induce complete customer lock-in, breaks up the relationship between introductory prices and market share. For the same reason, we see that a = b = 0 for allxt within the no-poaching region. This is also true in the equilibrium with low switching costs and incomplete lock-in but only when c = 1 since in the latter case poaching is pro t-neutral. While the inability to poach explains f = b = 0, the symmetry of the problem explains the fact that a = 0 - rms? introductory prices will always be identical. From this it is clear that within the no-poaching region, convergence to steady state will occur in just one period. In addition, when switching costs are su ciently high, the no-poaching region encompasses the entire market so Proposition 2 and the convergence result extend to all values of xt. I present this result in the following corollary: Corollary 2. If s max((2 + c + 2 f) ;min(s1;s2;s3)), the no-poaching region extends to the entire unit interval and the market converges to steady state in just one period from all xt in [0,1]. Proof. See Appendix. The condition on s outlined here is su cient to ensure that exm = 0, so that there is complete customer lock-in for all possible distributions of the market. This guarantees that the introductory prices will be independent of xt and convergence will take place within one period for all xt. This result has important implications 62 for the incumbency advantages in the market. Despite the fact that a single rm can lock in all old customers due to switching costs, it loses any incumbency advantages from possessing larger market share in just one period upon entry by a rival rm. Furthermore, from Proposition 2 we can see that the value function of each rm increases in s through , the value of market share. Thus, switching costs facilitate market entry not only because the incumbent loses her dominant position in just one period, but also because the value of entering the market is higher. Similar to the equilibrium with incomplete lock-in and c away from one, the introductory prices we derive under complete lock-in decrease in the level of switch- ing costs and fall as consumers become more patient and competition intensi es. Furthermore, the equilibrium pn falls below cost when switching costs are su - ciently high: pint (1 c + f) = f. Recall that under complete lock-in, rms are competing for more than half of the newcomers? market because at the proposed equilibrium prices the optimal sales to loyal customers next period exceed one-half. The optimal regular price is increasing in s, so switching costs increase the value of market share. Similar to the equilibrium with low switching costs, the loyalty price premium is increasing in s but it is also decreasing in the rm?s market share because the regular price under complete lock-in depends on xt: pAot pAnt = s+ 2 (1=2 xt) (2.3.42) pBot pBnt = s+ 2 (xt 1=2) (2.3.43) From (2.3.28) and (2.3.29) we see that rms compete away all rents associated 63 with the presence of switching costs { the introductory price o ers a discount of fs (1+ f), which is extracted next period in the form of a price premium conditional on s, i.e. s(1+ f) . As a result, rm pro ts per cohort of consumers are independent of the magnitude of the switching costs: (x) = pint(1=2) + f[piot(1=2)] (2.3.44) = 1 c + f2 It should also be noted that there is no pure-strategy equilibrium for c = 1. This is due to the market property that newcomers demand becomes perfectly elastic as consumers become very patient. When newcomers demand is perfectly elastic and rms compete for more than half of the market as is the case when s > c , each rm has an incentive to undercut the rival by !0 in order to capture the entire market and realize a discrete gain in pro ts. In contrast, under incomplete lock-in rms anticipated that they will keep less than half of the market as loyal customers and did not have a pro table deviation in undercutting when demand was perfectly elastic. Per-cohort pro ts are increasing in the degree of product di erentiation, de- crease in consumer patience and are at most equal to . This pro t level is equivalent to the level of pro ts in a Hotelling model with no switching costs and no customer recognition. Since switching costs do not a ect pro ts and only act as a barrier to switching, this result is largely driven by the presence of customer recognition. Similar to the result in Chen (1997), we nd that in steady state rms are worse 64 o under customer recognition than under uniform pricing since in the latter case the equilibrium price is above the standard Hotelling price, as shown in Beggs and Klemperer (1992) and To(1996). Figure 2.1 shows the relationship between steady-state pro ts and switching costs under complete and incomplete lock-in conditional on c ! 1. I choose to set c = :99 because the equilibrium with incomplete lock-in is only analyzed for c!1 and we need to compare rm pro ts under the equilibrium paths conditional on the same parameter values. The positively-sloped section of the pro t function in Figure 2.1 illustrates that pro ts under incomplete lock-in are increasing in the size of the switching cost. When switching costs are low and both rms nd it optimal to poach, rms anticipate that they will keep less than their market share as loyal customers next period and competition for newcomers is not as intense as under complete lock-in. As a result, the introductory price does not fall below cost (despite the presence of switching costs) and pro ts increase in s. When switching costs approach the threshold c , per-cohort pro ts approach their peak level. As soon as switching costs exceed the threshold c , rms nd it optimal to keep all of their attached customers and complete customer lock-in intensi es completion. Firms undercut until they dissipate all rents from the presence of switching costs and, pro ts become independent of s. Nevertheless, steady-state pro ts per cohort remain positive because of the underlying product di erentiation. Firms undercut until they dissipate all rents from the presence of switching costs and, pro ts be- come independent of s. Nevertheless, steady-state pro ts per cohort remain positive because of the underlying product di erentiation. Finally, pro ts are decreasing in 65 0.5 1.0 1.5 2.0s 0.1 0.2 0.3 0.4 0.5 0.6 Profit Figure 2.1: Steady-state pro ts per cohort ( c = :99; f = 1; = 1) consumer patience because newcomers? demand becomes more elastic as c goes up and this puts downward pressure on the introductory price (regular prices are also increasing in the introductory price). Therefore, the pro t levels in Figure 2.1 should be interpreted as showing the minimum level of steady-state per-cohort pro ts in a market with switching costs and customer recognition. Finally, I draw attention to the continuity of the equilibrium results with respect to s by assuming c ! 1, in which case the threshold level for s, { the level that determines whether we have an equilibrium with switching or not, - is approaching . We have shown in Proposition 1 that for s! and c ! 1, the optimal introductory price converges to marginal cost from above. From Proposition 2 it can be seen that for c ! 1, pn ! c f(s )=(1 + f) { as s ! +, the equilibrium introductory price approaches marginal cost from below. Also, when c ! 1, ^qii ! ( + s)=(4 ) (from (2.3.8)) when s c and ^qii ! 1=2 as s! . Similarly, we can show that under complete lock-in, ^qii! 1=2 as s! +. We can nd ^qii under complete lock-in by plugging in the equilibrium pnt into (2.2.13).15 15Note that in nding (2.2.13) I assumed that marginal cost is normalized to zero so I adopt the same assumption in deriving ^qii here 66 ^qAA = +s f(s )1+ f 4 (2.3.45) =(1 + 2 f) +s4 (1 + f) As switching costs approach the threshold value, the introductory price approaches marginal cost and rms? optimal market share approaches one-half. Once switching costs exceed that threshold, pn falls below cost because rms now compete for more than half of the market of newcomers. Steady-state pro ts peak as switching costs reach c and level o for higher levels of s. 2.3.3 The Equilibrium under Incomplete, Asymmetric Lock- in and High Switching Costs I now present the equilibrium price strategies when the no-poaching region does not encompass the entire market. Suppose that rm B starts period t with low market share such that xt2 (1 exm;1). In this case, rm B will not play the strategy outlined in Proposition 2 because it has a pro table deviation in choosing a poaching strategy. Therefore, I conjecture that in period t rm B is able to poach some of A?s previous customers while retaining its entire customer base. I further suppose that next period the market moves into the no-poaching region characterized by Proposition 2, which is true for a wide range of parameter values. Under these assumptions, I designate the value function of a rm outside the no- 67 poaching region as Vi1 where the subscript 1 indicates the relevant range of xt such that unilateral poaching occurs for one period only (since I assume that next period the market distribution falls within the no-poaching region). Thus, the a ne functions governing the equilibrium price strategies in this region for period t are given by: pAnt pBnt = a1 +b1xt (2.3.46) pAnt = e1 +f1xt (2.3.47) pBnt = e1 a1 + (f1 b1)xt (2.3.48) VA(xt) = 1 + 1xt + 1x2t (2.3.49) VB(xt) = 1 + 1 (1 xt) + 1 (1 xt)2 (2.3.50) Since I suppose that xt+1 2(exm;1 exm), next period prices will be governed by the coe cients valid for the no-poaching region (pAnt+1 pBnt+1 = a + bxt+1 and so on). I modify the value functions accordingly: 68 VA1 (xt) = max pAnt +s+pB nt 2 8 (2.3.51) +pAnt xt+1(pAnt;pBnt) + f + xt+1 + x2t+1 VB1 (xt) = max pBnt s+pA nt + 2 xt (1 x t) (2.3.52) +pBnt 1 xt+1(pAnt;pBnt) +xt +s+p B nt 4 + f + (1 xt+1) + (1 xt+1)2 Solving for the best response functions and checking for deviations, I nd an equilibrium in pure strategies for some, though not all, parameter values. I characterize the MPE in pure strategies in Proposition 3: Proposition 3. Suppose c2(0;1); f 2(0;1); xt2(max(1 exm;1 exs);1); s2 ( c ; (2 + c + 2 f) ) and xt+1 2(exm;1 exm). A Markov-perfect equilibrium in pure 69 strategies exists and can be characterized as follows: ^pAnt =(7 2 3 c + 13 f + 6 2 f + 2 c(11 + f) + 2 c( 8 7 f + 2 f)) 2(1 + f)(5 + 2 2c + 6 f c(7 + 2 f) (2.3.53) (1 + 11 f + 10 2 f + 2 c(1 + 3 f) 2 c(1 + 7 f + 2 f))s 2(1 + f)(5 + 2 2c + 6 f c(7 + 2 f)) + 2(1 c)(1 c + 2 f) 5 + 2 2 c + 6 f c(7 + 2 f) xt ^pBnt = (c4 + 4 f 2f) 2 2c(2 f) 5 f 3 2f (1 + f)(5 + 2 2c + 6 f c(7 + 2 f)) (2.3.54) + 1 + 5 f + 5 2f + 2c(2 + 6 f + 2f) s (1 + f)(5 + 2 2c + 6 f c(7 + 2 f) + 4(1 c)(1 c + 2 f) 5 + 2 2 c + 6 f c(7 + 2 f) xt The boundary 1 exs is de ned by: 1 exs = +s4 + pM(M +C2) M 4tC2pM(M +C2) p B nt (2.3.55) where C and M are de ned by (2.3.36) and (2.3.37), respectively. A similar argument for the existence of an equilibrium when xt 2 (0;min(exs;exm)) applies. Proof. See Appendix. The equilibrium described above is valid for a wide range of parameter values, suggesting that for these parameter values next-period market share falls into the no-poaching region. Since the equilibrium price strategies are not straightforward 70 to interpret, I employ a numerical analysis. It can be shown that when xt is close to 1, rm B?s introductory price is higher than rm A?s price and xt+1 approaches the midpoint on the unit interval from above. On the other hand, whenxt is farther away from 1, rm B?s price is lower than the rival?s price, so xt+1 approaches the midpoint of the market from below. This suggests that for xt outside the no-poaching region convergence may be non-monotonic. The lower bound of the region, where B poaches for one period only, is de ned as max(1 exm;1 exs). Recall that 1 exm is the upper limit of the no-poaching region: when xt > 1 exm rm B has a pro table deviation in poaching while A plays a no-poaching strategy. In addition, when xt 1 exs rm B has a pro table deviation in not poaching conditional on A not poaching either. If 1 exs 1 exm, then rm B has no pro table deviation away from ^pBnt and Proposition 3 applies to the entire region (1 exm;1). By symmetry, similar arguments can be applied towards nding an equilibrium when xt2(0;exm) and A has a pro table deviation in poaching. If 1 exs > 1 exm and xt 1 exs, then Proposition 3 still applies. If 1 exs > 1 exm but xt 2 (1 exm;1 exs) there will be no equilibrium in pure strategies. If neither rm poaches, rm B will deviate to a poaching strategy since xt > (1 exm). As B adopts the poaching strategy, ^pBnt, and A responds by setting ^pAnt, B now has a pro table deviation in not poaching since xt < (1 exs). A numerical analysis shows that (1 exm;1 exs) is the empty set for a wide range of parameters. For this reason, I do not explore the equilibrium when xt2(1 exm;1 exs). Recall that the price strategies in Proposition 3 are optimal conditional on 71 the conjecture that next period the distribution of the market falls within the no- poaching region. For a limited range of the parameter values this conjecture is not correct. In this case we have to solve for the equilibrium strategies by conjecturing that next period the market stays in region 1 - the region where rm B poaches and A does not. If xt+1 falls within (1 exm;1 exs), then the problem is further complicated by the lack of pure-strategy equilibria for this region. I believe that investigating these scenarios will not contribute substantially to our present discussion so I limit the equilibrium results to the cases described in Propositions 2 and 3. 2.4 Discussion 2.4.1 Customer Recognition vs. Uniform Pricing The results regarding the equilibrium with complete lock-in allow us to isolate the impact of customer recognition on pro ts by comparing my results to those in Beggs and Klemperer (1992) and To (1996). Both papers consider markets charac- terized by product di erentiation, overlapping generations of consumers, in nitely- lived rms, complete customer lock-in due to switching costs and uniform pricing. Beggs and Klemperer assume that consumers have in nite life spans and show that in a symmetric steady state rms charge prices above c+ and generate per-cohort pro ts exceeding (1+ f) 2 . To (1996) modi es this analysis by considering consumers with nite lifespans and shows that this assumption does not qualitatively alter the level of prices. 72 My own results from Section 2.3.2 show that per-cohort pro ts remain below (1+ f) =2 for alls, which leads us to conclude that steady-state rm pro ts are lower when sellers can imperfectly price discriminate based on customers purchase history. This result is similar to the one derived in Chen (1997), despite the dissimilar settings (Chen?s model is based on homogeneous markets with low switching costs and incomplete lock-in). Unfortunately, we cannot compare the outcomes under incomplete lock-in with their equivalent under uniform prices since there are no models of uniform pricing, product di erentiation and low switching costs in the literature yet. 2.4.2 Imperfect vs. Perfect Customer Recognition We can also show that perfect customer recognition (PCR) yields higher pro ts relative to imperfect customer recognition (ICR). The next section presents the modi ed model that captures the e ect of PCR on prices and pro ts. Perfect Customer Recognition Let pnt indicate the price to newcomers, pst - the price to switchers, and pot - the price to loyal customers. Since imperfect customer recognition hampers rms? ability to price discriminate between switchers and newcomers only when switching occurs in equilibrium, I suppose that switching costs are low enough to prevent complete lock-in. The upper bound on s that allows switching in equilibrium will be determined below. When rms can separate switchers from newcomers, current market share has no impact on pnt. Therefore, even in the presence of overlapping 73 generations the rms? problem can be represented through a two-period model where each period rms choose pst and pot on the basis of their current market shares while pnt is chosen by taking into account expected future pro ts from the customer base installed today. Let t =f1;2g, where t = 1 indicates the period when customers are new and t = 2 is the period when customers are in their second period on the market. Let x stand for rm A?s market share captured in period 1. I rst consider rm A?s problem with respect to pAo and pAs in period 2 where I maintain the assumption that regular prices are set after the rival?s o er to switchers is known. Demand from loyal customers can be expressed as qAA(pAo;pBs ) = min(xt;s+ +p B s p A o 2 ) (2.4.1) so rm A?s optimal regular price is pAo = arg maxpAo min(xt;p B s +s p A o + 2 ) (2.4.2) pAo = max(p B s +s+ 2 ;p B s +s+ 2 ( 1 2 x)) (2.4.3) In turn, rm A?s sales to loyal customers are given by: qAA = +s+P B s 4 (2.4.4) 74 Similarly, pBo = max(p A s +s+ 2 ;p A s +s 2 ( 1 2 x)) qBB = +s+PAs 4 (2.4.5) Firm A?s demand from switchers in period 2 is given by: qBA = max(0;(1 x) (1 p B o +s p A s + 2 )) (2.4.6) where (1 x) is rm B?s market share in period 2 and 1 pBo +s pAs + 2 is rm B?s mass of loyal customers, conditional on pAs . Assuming qBA > 0 and using pBo = pAs +s+ 2 , rm A?s optimal price to switchers is pAs = arg maxpAs 3 pA s s 4 x (2.4.7) pAs = 3 s 4 x2 (2.4.8) from which we nd that qBA = 3 s 4 x8 . Similarly, for rm B: pBs = 4 x s 2 (2.4.9) qAB = 4 x s 8 (2.4.10) Having found the optimal price to switchers, we can substitute it into the expressions 75 for pio and qii: pAo = +s+p B s 2 (2.4.11) = +s+ 4 x s 2 2 = +s+ 4 x4 qAA = +s+P B s 4 (2.4.12) = +s+ 4 x s 2 4 = +s+ 4 x8 pBo = +s+ 3 s 4 x 2 2 (2.4.13) = 5 +s 4 x4 qBB = +s+ 3 s 4 x 2 4 (2.4.14) = 5 +s 4 x8 We can now nd the rms? total pro ts, i2, from their old customers in period 2 as a function of the beginning-of-the-period market share, x: A2 (x) = Aswitchers + Aloyal (2.4.15) = (3 s 4 x) 2 16 + ( +s+ 4 x)2 32 B2 (x) = Bswitchers + Bloyal (2.4.16) = (4 x s) 2 16 + (5 +s 4 x)2 32 In period 1 rm A sets pAn in order to maximize the present value of pro ts 76 per cohort entering in period 1. The marginal newcomer will purchase from A if pAn + x+ c min pAo + x;pBs + (1 x) +s (2.4.17) pBn + (1 x) + c min pBo + (1 x);pAs + x+s Using (2.4.8) and (2.4.9), rm A?s demand from newcomers can be expressed as q1A = x = p B n p A n + (1 + c) 2 (1 + c) (2.4.18) Note that under PCR demand from newcomers becomes less elastic as consumers become more patient. The rms? optimal introductory price to newcomers are de- termined simultaneously and are given by: pAn = arg maxpAn p B n p A n + (1 + c) 2 (1 + c) + f A 2 (x) (2.4.19) pBn = arg maxpBn 1 p B n p A n + (1 + c) 2 (1 + c) + f B2 (x) (2.4.20) 77 This produces the following solution: pAn = pBn = 1 + c f4 34 fs (2.4.21) x = 1=2 (2.4.22) pAs = pBs = s2 (2.4.23) qBA = qAB = s8 > 0 8s< (2.4.24) pAo = pBo = 3 +s4 (2.4.25) qAA = qBB = 3 +s8 (2.4.26) (2.4.27) First, the results above show that under PPD the price to new customers again does not fall below cost (here normalized to zero) for the admissible range of s, i.e. s< . Second, pn decreases in the rm discount factor (because rms place higher value on pro ts from market share tomorrow) and increases in consumer patience (because we showed consumers become less price sensitive as they grow more patient). The newcomers? price also decreases in the switching costs and @pn=@s only depends on the rm discount factor { from x(pAn;pBn) we saw that consumers will incur s as a cost either by switching or by paying at least that much more if they stay, so the consumer discount factor is irrelevant; on the other hand, rms anticipate that their pro ts from loyal customers are increasing in s and the more they value these pro ts the more they are willing to lower the price to newcomers today in order to capture market share. Note also that switching costs 78 must be below for switching to occur in equilibrium. Naturally, the optimal price to switchers is below the price to loyal customers, i.e. ps 4 c+ 1 f1 + 3 f (2.4.28) We can further verify that the above condition will be satis ed for some range of values in s2[0; ) if c < f, because the latter condition ensures that 4 c+1 f1+3 f is less than , the upper bound on switching costs in the model. On the other hand, if c f or s 4 c+1 f1+3 f , we nd that loyal customers are o ered a loyalty discount: po pn. This occurs because low s makes switching more attractive and rms are induced to charge lower prices to loyal customers. This, in turn, reduces the payo from market share and relaxes competition for new customers, resulting in higher introductory prices. Sincepo >ps, having low switching costs results inpn >po, which leads topn > po >ps { switchers pay the lowest price, followed by loyal customers who are o ered a discount, and nally new customers pay the highest price. Hence, low switching costs and perfect customer recognition lead to endogenous loyalty discounts. If switching costs are high (s> 4 c+1 f1+3 f ) and rms are su ciently patient ( f > c), then new customers still receive a discount relative to loyal customers as in the ICR case. Furthermore, if s > 4 c+2 f2+3 f (a condition stronger than s > 4 c+1 f1+3 f ) we 79 also obtain that pn < ps { i.e., for very high levels of s, newcomers are charged a lower price than switchers because market share becomes more attractive while sales to switchers become less pro table. The equilibrium pro ts per cohort can also be shown to be higher under PCR relative to ICR. Under PCR, the pro t per cohort equals PCR = 12 (1 + c f4 ) 34 fs + f (3 s 4 (1=2))2 16 + ( +s+ 4 (1=2))2 32 (2.4.29) = 1 + c f2 + 38 f( s) + f ( s)2 16 + (3 +s)2 32 We can write: PCR ICR = 1 + c f2 + 38 f( s) + f ( s)2 16 + (3 +s)2 32 f ( +s) 2 8 (2.4.30) = 2(16 + 16 c + 2 f) + 6 fs2 + f( s)2 > 0 The comparison above suggests that rms would be better o if they could distin- guish switchers from newcomers. This result may explain why some stores carry- ing product lines from multiple suppliers issue coupons for the rival?s product at the point of sale. The widespread implementation of bonus cards in grocery and convenience stores could be motivated by the stores? willingness to uniquely iden- tify customers and trace their purchase patterns as this facilitates perfect customer 80 recognition at the supplier level and increases industry pro ts. 2.5 Conclusion In this paper I present an analysis that integrates imperfect customer recog- nition and consumer switching costs in the context of dynamic competition in a di erentiated-goods duopoly. The model presented here builds upon Villas-Boas (1999) and complements the study of dynamic competition with switching costs by introducing customer recognition. This allows us to incorporate the switching cost explicitly in the demand functions and derive closed-form solutions for the equilib- rium prices, which enables a comparative statics analysis. There are two sets of market equilibria depending on the level of the switch- ing cost. For all values of the switching cost, customer recognition gives rise to a ?bargain-then-ripo ? pattern in prices { this feature would be present even if switch- ing costs were zero and is due to rms? ability to price discriminate between new and repeat customers. When switching costs are low enough to allow customer switching in equilibrium, they only amplify the loyalty price premium and increase rms pro ts. The price to new customers does not fall below cost because rms keep only a fraction of their equilibrium market share as loyal customers. Switching costs do not a ect the speed of convergence to steady state. When consumers are very patient, demand from newcomers is perfectly elastic and convergence to steady state occurs in just one period. When switching costs are high, there can be complete customer lock-in, such 81 that neither rm is able to ?poach? the rival?s customers. Firm pro ts are indepen- dent of switching costs because any discounts that are conditional on the size of the switching costs are extracted from the captured consumers in the form of loy- alty premiums when in their second period in the market. Because rms compete for more than half of the market, introductory prices may fall below cost. Under complete customer lock-in, convergence to steady state occurs in just one period when the current distribution of the market is such that both rms nd it optimal to retain all of their previous customers. The model also suggests that imperfect customer recognition leads to lower pro ts relative to both uniform pricing and perfect customer recognition. If rms can distinguish new, unattached consumers from switchers, they can increase their pro ts by price discriminating between these two types. Under such perfect customer recognition, loyalty discounts would emerge if switching costs are su ciently low; otherwise, newcomers will be o ered introductory o ers, which are below the price paid by loyal customers. 82 Chapter 3 Inter-Firm Information Sharing, Competition, and Liquidity Constraints: Theory with Evidence from Madagascar 3.1 Introduction In the absence of adequate legal protection against contract breach, rms can reduce their exposure to contractual risk in one-shot transactions by exchanging in- formation about defectors. The goal of this paper is to investigate whether compe- tition discourages such exchange. Previous studies that analyze information sharing in a competitive environment have focused exclusively on lending institutions and on 83 the impact that pooling information on borrowers? histories has on lenders? market power over their customers. In contrast, I look at this issue from the perspective of rms and consider rms? exposure to risk from trade partners in both the upstream and downstream markets. Furthermore, I propose a new channel through which competition may deter information exchange among rivals. Foreseeing that sharing information reduces the rival?s exposure to such risk, a rm holding private informa- tion about a defector may have an incentive not to reveal this information depending on how much it can bene t from exposing its rival to higher risk of default. Hence, information sharing agreements may not be sustained among rivals. This approach gives rise to two novel theoretical insights: i) in imperfect credit markets liquidity plays a key role in facilitating the exchange of information between rivals; ii) in- formation sharing may be easier to sustain in ex-ante more competitive markets. I test the model?s predictions using a unique rm-level dataset on the information sharing practices of agricultural traders in Madagascar.1 I nd strong support for the predicted positive impact of liquidity on information sharing and establish that traders who report stronger competition in their markets are more likely to share information. The main premise of my model is that su ering contract breach can trans- late into an unanticipated cash out ow that weakens the rm?s ability to compete depending on its liquidity position. When a customer defaults on a payment or a supplier does not deliver goods on time, the rm has to employ additional resources 1The dataset comes from a survey conducted by the International Food Policy Research In- stitute and the Malagasy Ministry of Scienti c Research. I am grateful to Marcel Fafchamps and Bart Minten for sharing this dataset with me. 84 in order to meet its own payment or delivery obligations. Some rms may antici- pate an average rate of default in their operations and hold precautionary capital or inventory.2 However, when a higher than the expected rate of default occurs and the rm does not have su cient resources, it has to borrow money to maintain the rm?s operations - the cost of securing short-term capital will temporarily raise the rm?s marginal cost. Hence, an episode of contract breach can be viewed as a tran- sitory adverse cost shock to the rm that was cheated. Depending on how much the rival rm can bene t from the resulting cost advantage, it may have an incentive to expose the other rm to a higher risk of default and pro t from its vulnerability.3 I refer to rms with low cost of funds as ?liquid? { for these rms the experience of contract breach will have minimal impact on their marginal cost and, hence, on their rival?s pro tability. As a result, the rival has a weaker incentive to withhold information and expose the rm to higher default risk. Alternatively, ?liquidity constrained? rms face a high cost of funds and upon experiencing above average rates of default, the shock to their marginal cost is larger. As a result, liquidity constrained rms are more vulnerable to the risk of default and their rivals can realize higher pro ts by withholding information. The key insight here is that the detrimental impact of competition on information sharing depends on the liquidity positions of the rms in a given market. I model information sharing as the Pareto-optimal non-cooperative equilibrium 2See Fafchamps et al. (2000) for empirical evidence on Zimbabwean rms? holdings of precau- tionary inventory stocks in the face of high contract risk. 3This reasoning is somewhat similar to the ?deep pockets? concept in the predatory pricing literature (McGee, 1958; Telser, 1966; Bolton and Scharfstein, 1990) although in our case, it is only the deep pockets of the rival that matter. 85 of the information-sharing supergame between two rival rms (Friedman, 1971). Firms use grim trigger strategies and punish rivals, who do not reciprocate in the information exchange, by not revealing information in the future. The strategic cost of information sharing determines the short-term incentive to deviate from the cooperative strategy of sharing information and is given by the additional pro ts a rm can realize when it withholds information and exposes the rival to higher risk of default. On the other hand, playing the cooperative strategy gives rise to long-term bene ts arising from lower exposure to contractual risk for both rms. The main nding of the model is that information sharing can be sustained at lower rm discount factors when rms are liquid, because the temptation to deviate is lower when a rm?s rival faces a low cost of funds. Furthermore, if we assume that rms within a market are either both liquid, or are both liquidity constrained, then we can formulate the testable prediction that information sharing is more likely to be observed among liquid rms. I next examine the intuitive claim that more intense competition would pro- vide stronger disincentives to information sharing. I employ a market framework, in which competition intensity varies along three dimensions - product di erentiation, switching costs, and consumer patience. For tractability, I limit attention to markets in which consumers are very patient (consumers live two periods only) and charac- terize the pro ts that a rm will forgo when it reveals information about a defector to its rival - these pro ts comprise the strategic cost of information sharing. I show that this cost is zero when the rival?s cost of funds is su ciently low - i.e. when it is below an endogenously determined liquidity threshold. This threshold depends 86 on the the level of product di erentiation and the magnitude of the switching costs, which allows for a comparative statics exercise. The e ect of competition intensity is ambiguous and depends on the under- lying market features. When driven by lower switching costs or lower degree of product di erentiation, more intense competition lowers the liquidity threshold and on average raises the strategic cost of information sharing. However, consumer pa- tience is also shown to play a role. The model exhibits the feature that the market becomes more competitive as consumers become more patient. I show that the strategic cost of information sharing is zero when consumers are in nitely patient. I also provide an intuitive discussion for the case where consumers are not in nitely patient, which illustrates that the strategic cost of information sharing is likely to be positive in less competitive markets when the degree of competition is varied along the degree of consumer patience. Previous studies of competition and information sharing have placed little emphasis on the features of the competitive environment.4 Furthermore, the approach in these studies is to analyze how information sharing a ects competition and the the desirability of an information-sharing regime then depends on the change in the competitive environment and its implications for the banks? pro tability. This study contributes to the literature by presenting a more detailed picture of the various market parameters through which competition a ects information sharing behavior rather than vice versa. I use survey data from Madagascar to test the model?s predictions regarding 4In Jappelli and Pagano (1993) the exogenous level of competition is proxied by the cost advantage of the incumbent bank relative to a potential entrant, while in Gehrig and Stenbacka (2006) it is proxied by the dispersion of switching costs that borrowers incur when changing lenders. 87 the impacts of liquidity and competition intensity on information sharing practices. The data does not allow the identi cation of a trader?s rivals. However, assuming that traders within a market have similar liquidity positions such that a trader?s liquidity position is a proxy for the position of its rival, the model suggests that more liquid rms would be more likely to share information because they are also facing more liquid rivals. Consistent with this hypothesis, I nd that rms operating in liquid markets are signi cantly more likely to share information about delinquent customers. The results are qualitatively equivalent under two alternative measures of liquidity { access to informal credit and availability of own liquid funds, { and robust to the inclusion of a rich set of controls. One of the liquidity measures is credibly exogenous, so its coe cient can be interpreted as a causal estimate of the positive e ect of liquidity on information sharing. Next, I nd a positive and statistically signi cant relationship between the intensity of competition, as reported by the trader, and information sharing. Unfor- tunately, a lack of suitable instruments prevents me from establishing a statistically signi cant causal relationship. However, to the best of my knowledge, this study is the rst to present empirical evidence on the correlation between competition intensity and information sharing practices based on observational data.5 The rest of the paper is organized as follows. Section 2 presents the theoretical model that links the strategic cost of information sharing to the rival?s cost of funds and discusses rms? incentives to share information. Section 3 describes the data 5Brown and Zehnder (2008) present experimental evidence on this relationship in a credit mar- ket environment and nd that stronger competition reduces information sharing. Their experimen- tal design follows Pagano and Jappelli (1993) who theoretically explore the e ects of competition and adverse selection on the incentives of competing lenders to pool information. 88 and the empirical strategy. Section 4 presents the results and Section 5 concludes. 3.2 Theoretical Model 3.2.1 Preliminaries I consider the sustainability of a self-enforcing information-sharing agreement between two rival rms. The rms may agree to share timely information about defectors but their agreement cannot be enforced in court. In the one-shot non- cooperative game, sharing information about defectors will be shown to be a weakly dominated strategy. However, information sharing can be sustained as the Pareto- optimal, non-cooperative equilibrium of the in nitely-repeated stage game. Information- sharing practices in credit markets are often based on reciprocity. For example, members in credit bureaus have an obligation to report information and, in turn, can access credit reports at a much lower cost than non-members (Klein, 1992). Case study evidence on informal information sharing arrangements suggests that rms report defectors to other rms in the market in expectation that the favor will be returned (Vinogradova, 2006).6 Firms have an incentive to participate in information-sharing networks because they reduce their exposure to contractual risk. First, information sharing networks within a market allow rms to screen out defectors if history of past default is any indication of future propensity to cheat. Second, the existence of such networks 6In a detailed account of the information networks among small business owners in Russia, Vinogradova (2006) reports that one business owner explicitly states that other rms share infor- mation with him because they expect this favor to be reciprocated in the future. 89 may have a disciplinary e ect that reduces the fraction of cheaters in the population (Padilla and Pagano, 2000; Jappelli and Pagano, 2002). On the other hand, while rms may ex ante agree to share information, such commitment is credible ex post only if the costs of disclosing information today do not exceed the net bene ts of reduced exposure to risk in the future. The main contribution of this paper is to illustrate how liquidity can a ect the cost of disclosing information to a rival and, hence, on the viability of information-sharing practices. The sequence of events is summarized in Figure 3.1. In some initial period t0 rms declare their intent to share information; the ensuing obligation is not enforce- able by courts or other third parties, so even though we use the term ?agreement? it should be noted that this agreement is non-binding. At the beginning of each subse- quent period rms simultaneously choose whether to disclose their private informa- tion or not; afterwards they sign contracts with agents (i.e. customers, suppliers or other trade partners), a fraction of whom have unilateral incentive to breach their contracts with the rm. Firms do not sign contracts with agents who are known to have defaulted in the past (defectors). Not revealing private information about the identity of defectors is equivalent to deviating from the information sharing agree- ment. Deviations in the current period are discovered at the end of that period { rms are assumed to be able to verify if the information transmitted by their rival at the beginning of the period re ected the full scope of the information the rival possessed at the time. Also at the end of the period, rms discover the aggregate default rate for the period. Naturally, a discussion of the emergence of an information sharing regime is 90 Figure 3.1: Timeline of events. warranted to the extent that it bene ts rms by reducing their exposure to con- tractual risk.7 Hence, I presume that i) in the absence of competition rms have incentives to share information ex-post (e.g. the bene ts from information sharing outweigh the physical cost of information transmission) and that there are no other feasible and possibly cheaper means of enforcing contracts (e.g. engaging in long- term bilateral relations or resorting to third parties to enforce contracts); and ii) there is a time lag between two cheating incidents by the same agent so that the exchange of information can have any value. Sharing information is assumed to be a game of complete information in the sense that deviations from the cooperative strategy are detected perfectly and with no lag. Failing to report on the identity of a defector or falsely reporting honest 7The magnitude of these bene ts may depend on the number of rms pooling information, the share of dishonest agents in the population, the share of agents who are deterred from cheating by the knowledge that their actions will be public knowledge, and the magnitude of the contractual losses being avoided. I assume that these factors are held constant throughout the analysis. 91 agents as defectors is considered to be a deviation from the information-sharing strategy. These assumptions are consistent with the theoretical setting in Kan- dori (1992) and Okuno-Fujiwara and Postlewaite (1995), whose insights we use to motivate information sharing as a feasible device that facilitates the emergence of reputation mechanisms in games of random matching. They are needed to avoid the plethora of issues related to truth-telling in repeated games where the actions of agents are observed by only a subset of the population.8 Within this context, An- nen (2007) tackles truthful information sharing and shows that truth-telling can be obtained as a unique dominant strategy equilibrium.9 He also points out that this equilibrium is harder to sustain among competing players who have an incentive to slander each other because slandering triggers a punishment on the opponent. This result suggests that truth-telling may be a serious issue in information sharing if rms can bene t from misreporting the performance on honest agents. At this point, it is assumed in our model that rms report information truthfully. Firms start each period with the expectation that some fraction of their trade partners will cheat. Contract breach can take many forms - for example, customers may cheat by not repaying their credit or by pre-ordering goods that they do not purchase later; similarly, suppliers may cheat by not delivering the contracted goods on time or by delivering goods of lower quality. Such incidents can cause a disruption to the cheated rm?s expected stream of cash and inventory ows. To cope with this disruption, the rm can either seek additional funds, internally through retained 8See Ben-Porath and Kahneman (1996 and 2003), and Anderlini and Laguno (2005). 9This equilibrium does not satisfy the assumption in Kandori (1992) and Okuno-Fujiwara and Postlewaite (1995) that players only need to communicate ?simple? information in the form of labels. 92 earnings or externally through borrowing, or it can rely on excess inventory whenever available. I do not distinguish between a rm borrowing capital or negotiating the delivery of goods on credit as they both serve the same purpose of supporting the rm?s operations.10 Either alternative is costly and the additional costs of securing additional inventory or cash holdings add up to the marginal cost of operation in the next period. Originally, rms are assumed to be symmetric in their average default rate but informational asymmetries, arising from one rm sharing information while the other one does not, give rise to asymmetric exposure to risk and asymmetric marginal costs of operation in the next period. The rm that is exposed to a higher average rate of default (e.g. because its rival deliberately did not disclose information) has to incur higher costs of maintaining its operations in the next period. Hence, a higher risk of default for one of the rms in the market can be interpreted as causing an adverse cost shock to that rm in the following period. All else equal, the magnitude of the cost shock will be increasing in the rm?s cost of funds. Naturally, the cost shock will be smaller for a rm that relies on retained earnings or has cheap access to trade credit relative to a rm that borrows from a moneylender. Hence, I refer to rms with a low cost of securing liquid assets (such as cash and inventory) as liquid; rms for which these costs are high will be referred to as liquidity-constrained. While I recognize that a rm may be unable to borrow at times of need and face capacity constraints, I do not explicitly model 10Fafchamps et al (2000) propose that excess inventory holdings may be motivated by the desire to insure against contractual risk and nd evidence of this motive in their data on Zimbabwean export rms who are particularly prone to contract risk from overseas partners. 93 this possibility as it entails analyzing a Bertrand-Edgeworth version of the dynamic model described below with potentially no pure-strategy equilibria.11 Therefore, the case where a rm in reality experience binding capacity constraints due to contract breach can be modeled as a very high cost of funds. A rm?s decision to share information has a direct e ect on the probability that its rival experiences a higher than anticipated rate of cheating. If a rm can derive su ciently high bene ts from the rival?s higher exposure to risk, then it will have incentives to deviate from the information sharing agreement by withholding information. To x ideas, consider the stage game in Figure 3.2. Firms can take two actions: they can reveal information (?Share?) or they can withhold information (?Do not share?). Figure 3.2: The information-sharing stage game. Firm B Do not share Share Firm A Do not share k; k m; l Share l; m n; n The parameters, k;l;m and n, that can describe the one-shot information sharing game are as follows. It is assumed that rms bene t from receiving infor- mation about defectors because it reduces their exposure to contractual risk: n>k and m>k.12 However, unilaterally revealing information is costly not only because the rm does not lower its exposure to risk (since it does not receive information 11For characterization of the equilibria in a one-period model of Bertrand-Edgeworth competi- tion with product di erentiation, see Boccard and Wauthy (2005). 12Note that n, the rms? payo when they both share information, can be adjusted to absorb the physical cost of information transmission and it is assume that this cost is su ciently small to make information sharing desirable in the absence of competition. 94 from its rival) but also because its risk of default is now higher relative to that of its rival; hence, l n { rms are better o when they receive information from their rival but do not return the favor. One of our main goals is to identify conditions under which m = n. Note that if m > n, the only Nash equilibrium of the stage game is (Do Not Share, Do Not Share) information sharing can only be sustained if rms are su ciently patient. However, if m = n, there are two Nash Equilibria { (Do Not Share, Do Not Share) and the Pareto-optimal (Share, Share), { and information sharing becomes a question of coordinating on the latter or it can be sustained under minimal requirements on the rms? discount factors. Based on the asymmetry in the rms? exposure to risk when one of them withholds information, the model will explore to what extent that rm can bene t from this deviation. Such bene ts will arise from the competitive advantage that the deviating rm obtains when it exposes its rival to a higher risk of default. The key nding of the model will show that for a given rm m = n when the rm?s rival is su ciently liquid and m>n if the rival is liquidity constrained. The natural corollary of this result is that an information sharing agreement is more easily sustained among liquid rms than it is among rms that are liquidity-constrained as the latter have a strictly positive incentive to deviate; the liquid rms have no incentive to deviate at all. I illustrate rms? strategic motives to withhold information by using a dynamic model of imperfect competition featuring in nitely-lived rms and overlapping gen- erations of consumers. While a simpler static model would be su cient to illustrate 95 how liquidity can a ect rms? incentives to share information, the dynamic model I develop can additionally shed light on the complex ways in which competition inten- sity can a ect information sharing incentives. I begin by characterizing the steady- state equilibrium of the market when the rms have symmetric cost structures. This implies that they also have symmetric average probabilities of experiencing contract breach from agents. Next, I re-examine the market equilibrium when one rm is exposed to relatively higher probability of contract breach during a single period { this is equivalent to the experience of a transitory cost shock that a ects a single rm only. If a rm realizes higher pro ts when its rival is hit by such shock, then the additional pro ts would constitute the rm?s strategic cost of information shar- ing as revealing information about defectors reduces the probability that the rm will realize these pro ts. In terms of the notation in Figure 3.2 the strategic cost of information sharing will is captured by the di erence m n. Finally, I discuss how the cost of funds a ects rms? incentives to share information, perform com- parative statics with respect to the competition parameters and formulate testable hypotheses. 3.2.2 The market This section characterizes the market and its equilibrium properties when there is no uncertainty regarding the rm?s next period costs. I will discuss how relaxing this assumption a ects the market equilibrium in Section 3.2.3. The model of mar- ket competition that I introduce below possesses a number of features that make 96 it particularly suitable for our analysis and justify the level of sophistication and the limitations that come with it. First, I use a duopoly model because the po- tential bene ts from a rm?s distress accrue to a single rival and the incentive to withhold information, if any, is strongest.13 Second, the model has features, such as product di erentiation, consumer switching costs and consumer patience, that provide exogenous variation in the intensity of competition and allow us to inves- tigate how ex ante more intense competition a ects information sharing, based on di erent competition parameters. Third, I introduce imperfect customer recogni- tion { rms distinguish between new customers and repeat customers but cannot distinguish a newcomer from a switcher. Customer recognition is exogenous and further intensi es the competition for market share. It also makes the model more tractable as it allows us to derive intuitive closed-form solutions for the equilibrium prices.14 It is also important to note that aside from their impact on competition, customer recognition and consumer switching costs are two features that add more realism to the model as they are both commonly found in markets characterized by relational contracting and weak rule of law.15 Fourth, by assuming that rms are in nitely-lived, face overlapping generations of customers and consumers incur switching costs, I allow rm pro ts to be path dependent. This captures the possi- 13Greif (2006, pg. 446) notes that in ?thick? markets the cost of providing information could be negligible but this would not be the case in ?thin? markets where rms may be unwilling to help their rivals. 14For a discussion of the role of customer recognition on prices in dynamic models of product di erentiation and switching costs, see Grozeva (2009). For a comparable model without customer recognition and assuming complete customer lock-in, see Beggs and Klemperer (1992) and To (1996). 15Numerous case studies document that in relation-based market interaction, customer recog- nition and switching costs arise as rms prefer to deal with their established partners even if this entails forgoing better deals from new partners (McMillan and Woodru , 1999; Vinogradova, 2006). 97 bility that a transitory adverse cost shock may a ect the future stream of pro ts of both rms, thus amplifying the shock?s e ect. I consider a duopoly market consisting of two in nitely-lived rms, i =fA;Bg, selling a nondurable good. Consumers have uniformly distributed preferences over the products of the two rms, which gives rise to ex-ante product di erentiation. The degree of product di erentiation is exogenously determined and xed. Each rm produces the good at a constant marginal cost, c. Consumers enter the market for two periods only and demand one unit of the good in each period. They have common valuation for the good given by v, which I assume to be su ciently high to induce a purchase in every period. Each period an old cohort of consumers exits the market and a new cohort of equal size enters. In any given period, a rm faces two overlapping generations of consumers: old consumers in their second period in the market who have established a purchase history; and newcomers, who enter the market in the current period and have not purchased from either rm yet. If customers purchase from the same supplier in both periods they are referred to as ?loyal? customers, while if they purchase from two di erent suppliers over their lifetime, they are referred to as ?switchers?. Firms recognize their own loyal customers but cannot determine if a new customer is a newcomer with no purchase history or a switcher from the rival rm.16 Using Hotelling?s framework to model product di erentiation, suppose that rms are located at the opposite ends of the unit interval with rm A located at 16This setup is an extension of the model of dynamic competition developed in Villas-Boas (1999). The main di erence lies in the fact that I include switching costs in the analysis, which enables us to extend the analysis to markets with complete customer lock-in. See Grozeva (2009) for a detailed exposition of the equilibria under incomplete and complete customer lock-in. 98 0 and rm B { at 1. Each cohort of customers has mass normalized to one and consumers are uniformly distributed over the unit interval. Consumer preferences, as proxied by location on the unit interval, are time-invariant and known to the consumer ex ante. I stipulate that customers face a linear transportation cost of per unit of distance, so a consumer located at x will incur transportation costs of x if she buys from A, or (1 x) if she buys from rm B. A consumer who switches suppliers in her second period also incurs a switching cost, s, which is assumed to be time-invariant, uniform across consumers and common knowledge. All new customers (i.e. newcomers and switchers) are o ered an introductory price, pint, where the superscript i indicates the rm, the subscript t indicates the time period and the subscript n indicates that this is the price o ered to new customers. Loyal customers are o ered a regular price, piot, where the notation is similar except that the subscriptoindicates that this is the price to old customers. Firms simultaneously announce their introductory prices at the beginning of each period but each rm sets its regular price only after observing the introductory price of the rival. This assumption guarantees the existence of a pure-strategy equilibrium in periods when the distribution of market share is very unequal.17 Letqii;t indicate demand from rmi?s loyal customers in periodt, qij;t { demand from old customers who switch from i to j, j = fA;Bg, in period t, and q1i;t { demand from newcomers at time t who purchase from i in their rst period in the market. Initially, I suppose that marginal costs are constant, symmetric and equal 17This is a common assumption in models of customer recognition where customers are hetero- geneous in some characteristic: Villas-Boas (1999) applies it to rms, and Marquez (2002) applies it to banks. 99 to c 0. I start by characterizing the demand functions for each of the three groups of customers: newcomers, switchers and loyal customers. Then, I present the steady-state equilibrium of the market and discuss the market dynamics. Demand from newcomers To derive each rm?s demand from newcomers, I rst determine the location of the marginal consumer among newcomers in the market. This location also determines the distribution of the market in the current period and the market shares that rms will inherit in the following period. I indicate the location of the marginal consumer among newcomers at time t as xt+1, where xt+1 will also stand for rm A?s market share in period t + 1. Indicate the consumer discount factor as c where c 2 (0;1]. Assuming that consumers have perfect foresight, a newcomer located at x will purchase from rm A in her rst period in the market if this purchase renders a weakly higher surplus than purchasing from rm B over the consumer?s two-period life span in the market: v pAnt x+ c max v pAot+1 x; v pBnt+1 s (1 x) (3.2.1) v pBnt (1 x) + c max v pBot+1 (1 x); v pAnt+1 s x The marginal newcomer, located at xt+1, will be just indi erent between the 100 two sequences of purchases when pAnt+ xt+1 + c min pAot+1 + xt+1; pBnt+1 +s+ (1 xt+1) (3.2.2) = pBnt + (1 xt+1) + c min pBot+1 + (1 xt+1); pAnt+1 +s+ xt+1 Since it is assumed that the regular price is set after the rival?s introductory price is known, rm A will always set pAot+1 such that its marginal loyal customer at time t + 1 is just indi erent between switching and staying after having purchased from A at time t. If rm A wants to keep all of its customers in period t + 1, it will set pAot+1 + xt+1 = pBnt+1 + s + (1 xt+1). If it wants to let some customers switch, then for rm A?s marginal customer located at xt+1 it must be true that pAot+1 + xt+1 >pBnt+1 +s+ (1 xt+1). In either case, we have min pAot+1 + xt+1; pBnt+1 +s+ (1 xt+1) = pBnt+1 +s+ (1 xt+1) (3.2.3) Otherwise, rm A can always increase its pro ts by raising pAot+1 without a ect- ing demand from loyal customers. Therefore, the location of the marginal newcomer at time t can be determined from: pAnt+ xt+1 + c pBnt+1 +s+ (1 xt+1) (3.2.4) = pBnt + (1 xt+1) + c pAnt+1 +s+ xt+1 101 This equality also determines the distribution of market share at the beginning of period t+ 1: xt+1 = (1 c) + c(p A nt+1 p B nt+1) +p B nt p A nt 2 (1 c) (3.2.5) Demand from newcomers can be de ned as: q1A;t = xt+1; and q1B;t = 1 xt+1 (3.2.6) Demand from loyal customers The marginal loyal customer for rm A at timetwill be just indi erent between switching and staying. Therefore, her location, xlt can be determined from the equality of the payo s of each alternative: pAot + xlt = pBnt +s+ (1 xlt) (3.2.7) This equality yields rm A?s demand from loyal customers: qAA;t(pAot;pBnt) = min +s+pB nt p A ot 2 ; xt (3.2.8) Similarly, qBB;t(pBot;pAnt) = min +s+pA nt p B ot 2 ; 1 xt (3.2.9) Note that the regular price does not a ect demand from newcomers or switch- 102 ers, so rm i will choose piot independent of its own choice of pint. Thus, each rm sets piot to maximize pro ts from loyal customers taking as given the rival?s introductory price. The choice of piot also determines the optimal mass of loyal customers that a rm would like to keep, given the introductory price of its rival. If this optimal mass exceeds the rm?s actual market share, sales to loyal customers are limited to the size of the rm?s existing customer base - xt for rm A, and 1 xt for rm B. For c> 0 rm A?s regular price can be found as follows: max pAot (pAot c) min +s+pB nt p A ot 2 ;xt (3.2.10) pAot(pBnt) = max c+ +s+pB nt 2 ; +s+p B nt 2 xt (3.2.11) Similarly, pBot(pAnt) = max c+ +s+pA nt 2 ;s+p A nt + 2 xt (3.2.12) Note that the optimal regular price is increasing in the rival?s introductory o er. For example, if pBnt is su ciently large, rm A would keep all of its previous customers as loyal customers so qAA;t will be constrained by rm A?s market share. On the other hand, if pBnt is low enough, rm A?s loyal customer segment will be less than xt. In the cases where sales to loyal customers are less than a rm?s market share 103 I de ne ^qii;t as rm i?s optimal sales to loyal customers in period t. One can further extend the interpretation of ^qii;t as rm i?s optimal market share in period t if we consider situations in which the acquisition of market share is costly. Upon nding a deterministic optimal path for pAnt and pBnt, each rm can project what is the optimal market share to invest in today in order to maximize pro ts from loyal customers tomorrow. We can nd ^qAA;t by plugging pAot(pBnt) into (3.2.8), which yields ^qAA;t as a function of pBnt only: ^qAA;t(pBnt) = +s+p B nt c 4 (3.2.13) Also, ^qBB;t(pAnt) = +s+p A nt c 4 (3.2.14) Sales to loyal customers and the resulting pro ts can be summarized as follows: qAA;t(pBnt) = min ^qAA;t(pBnt); xt (3.2.15) Aot(pBnt) = max ( +s+pB nt c) 2 8 ;( +s+p B nt 2 xt)xt (3.2.16) qBB;t(pAnt) = min ^qBB;t(pAnt); 1 xt (3.2.17) Bot(pAnt) = max ( +s+pA nt c) 2 8 ;(s+p A nt + 2 xt )(1 xt) (3.2.18) From (3.2.11) and (3.2.12) we can see that the optimal regular price is uniquely determined, given knowledge of the rival?s introductory price. The assumption that rms set introductory and regular prices sequentially ensures that once the introduc- tory prices are announced and rms set their regular prices accordingly, neither rm 104 has a pro table deviation in changing its regular price. Without this assumption, there may not be pure-strategy equilibria when the distribution of market shares is very unequal. Demand from switchers Demand from switchers, if positive, can be represented as the di erence be- tween the rival?s market share and its optimal sales of loyal customers. For rm A, demand from switchers is given by qBA;t = max (0;(1 xt) ^qBB;t) (3.2.19) Using (3.2.13), we nd: qBA;t(xt) = max 0; 3 s p A nt +c 4 xt (3.2.20) Similarly, demand from switchers for rm B is given by: qAB;t(xt) = max 0;xt +s+p B nt c 4 (3.2.21) From the demand equations in (3.2.20) and (3.2.21), it is clear that a rm?s ability to poach depends on the pre-existing distribution of market shares, sum- marized in xt { the rm that enters the period with low market share can attract the rival?s previous customers at a higher price because of the closer proximity of prospective switchers. At the same time, both newcomers and switchers are o ered 105 the same price, pnt, so each rm chooses its optimal introductory price by balancing the incentives to gain market share and to maximize pro ts from poaching. Given the symmetry of the problem, if the rms? only goal was to capture market share, their introductory prices would be equal. However, the incentive to poach causes the introductory price to be dependent on market share, which leads to path depen- dence of current period pro ts. In fact, xt is the only payo -relevant state variable in period t that a ects the choice variables pAnt and pBnt. The optimal regular price is unique for a given introductory price, so identifying the optimal pricing strategies for pnt is su cient to derive the full schedule of prices in period t as well as the distribution of the market at the beginning of period t+ 1. The symmetric market equilibrium I solve the dynamic problem for each rm by looking for a Markov Perfect Equilibrium (MPE), in which rms? pricing strategies depend solely on the realized distribution of the newcomers? market shares in the previous period. Speci cally, based on the solution of a similar problem in Villas-Boas (1999), I look for a MPE, in which the price strategies regarding pint are piecewise a ne in xt and the value function of each rm is piecewise quadratic in xt.18 Again following Villas-Boas (1999) I assume that there is some minimal level of exit costs, E, that a rm would incur at the end of the period if it does not realize sales to newcomers in that period.19 This assumption rules out the possibility that a pure-strategy equilibrium 18Equilibria in non-a ne strategies may also exist but they are outside the scope of our study. The equilibrium presented here is robust to deviations in a ne strategies only. 19One can think of exit costs as arising from the loss of goodwill when the rm does not invest in market share. For example, if newcomers can obtain information about a rm?s product only 106 may not exist when one rm starts out the period with a very small market share. It is necessary for some parameter ranges only and does not have a qualitative impact on the results that follow. I characterize the equilibrium of the market for c = 1 and s . There is switching in equilibrium, except when s = in which case the equilibrium mass of switchers reaches zero.20 Restricting attention to the limit of the consumer discount factor is justi ed for two reasons. First, letting c = 1 allows us to derive intuitive closed-form solutions of the equilibrium price strategies. Second, as I will show below, competition intensi es as consumers become more patient. Thus, assuming c = 1 allows us to focus on the most competitive version of this market, which is in line with our interest in analyzing whether competition hinders information sharing. I discuss how lower values of the consumer discount factor a ect the equilibrium results in Section 3.2.3. Proposition 1. Let f 2(0;1); c = 1; s ;and E (3 s)2 2 f( +s)216 . A unique Markov-perfect equilibrium in a ne strategies exists and can be characterized as through existing loyal customers, a rm that made no sales to newcomers in the past period may have to incur expenditures on promoting its product. 20For a discussion of the equilibrium when s> see Grozeva (2009). 107 follows: pAnt = pBnt = c (3.2.22) pAot = max c+ +s2 ; +s+c 2 xt (3.2.23) pBot = max c+ +s2 ; +s+c 2 (1 xt) (3.2.24) ^qAA = ^qBB = +s4 (3.2.25) qAB = max(0;xt ^qAA) (3.2.26) qBA = max(0;1 xt ^qBB) (3.2.27) xt+1 = 1=2 8xt2[0;1] (3.2.28) Proof. All proofs are contained in the Appendix. In equilibrium, the price to new customers equals marginal cost and is inde- pendent of market share, loyal customers are charged a premium, and the market is equally split. To provide intuition for these results I present certain features of the market equilibrium in the following lemmas. I rst draw attention to the property that demand from newcomers becomes more elastic as c increases.21 Lemma 1. Competition for newcomers intensi es as consumers become more pa- tient. As a result, pin and pio fall as c goes up. The next property is closely related to Lemma 1. I nd that as consumers become very patient, i.e. c = 1, newcomers? demand becomes perfectly elastic. 21This property is also present in Villas-Boas? model of dynamic competition where switching costs are zero. (Villas-Boas, 1999) 108 Lemma 2. When c = 1, demand from newcomers is perfectly elastic. Lemmas 1 and 2 help explain why the introductory prices do not fall below marginal cost despite the positive return on market share and the perfect elasticity of demand. Lemma 2 shows that when c = 1 competition for market share can be characterized by Bertrand price competition with homogeneous goods. Unlike most models with switching costs and product homogeneity where all future pro ts from customer lock-in are dissipated in the competition for market share, in this setting rms do not have an incentive to undercut when the price falls down to marginal cost. First, when pn = c additional sales to newcomers do not increase pro ts. Sec- ond, since demand from newcomers is perfectly elastic, consider a market sharing rule such that the market is split anywhere within the (^qAA;1 ^qBB) range whenever consumers are indi erent between the two rms. Pro ts from new customers are zero, so a sharing rule that allows rms to capture at least their future loyal cus- tomers will be robust to unilateral deviations. For example, if rm A can capture customers in (0; ^qAA), i.e. its future loyal customers, selling to newcomers located outside this range does not raise rm A?s pro ts neither in the present period, nor in the next period. At this point, I assume that the market is equally split in case of a tie and consumers in each half of the market buy from the rm that is closest to them. In the next sections, we will see that the sharing rule has to be modi ed to guarantee existence of a pure-strategy equilibrium. Proposition 1 reveals that the equilibrium introductory price is independent of the current distribution of the market, given by xt. This is due to the fact that 109 sales to switchers are pro t-neutral and xt becomes irrelevant in the determination of pnt.22 It also follows that convergence to steady state, x = 1=2 under the equal sharing rule, occurs in just one period. This result is summarized in the next corollary: Corollary 1. Under the conditions outlined in Proposition 1 and given xt 2 [0;1] the market converges to steady state, x = 1=2, in just one period. From the equilibrium price expressions we can see that the introductory price is independent of s while the regular price is increasing in s. Naturally, sales to loyal customers increase in s while sales to switchers decrease in s. Therefore, rm pro ts unambiguously increase in the magnitude of the switching cost and competition becomes more relaxed as switching costs increase. Similarly, rm pro ts are increasing in the degree of product di erentiation, as higher transportation costs further relax competition. Positive steady-state per-period rm pro ts arise solely from pro ts in the loyal customer segment of the market and are given by: = ( +s) 2 8 ! 2 as s! (3.2.29) 3.2.3 The decision to share information I now examine how the market equilibrium changes when one rm experiences a higher incidence of contract breach relative to its rival as this will illustrate how 22It should be noted that for some parameter values, it is the presence of exit costs that guar- antees that a rm with no market share would choose to compete for newcomers instead of raising its price and targeting switchers exclusively. However, the presence of exit costs will not have a qualitative e ect on the rest of the results. 110 the latter can bene t from withholding information. I rst characterize the equilib- rium path following a period of cost asymmetry. Corollary 1 facilitates the analysis because the impact of a transitory, one-period cost shock will be limited to this period?s strategies only { as soon as cost symmetry is restored next period, rms? strategies will depend on the current distribution of the market only and the market reaches steady state in the subsequent period. The impact of an asymmetric cost shock In the absence of a shock, in period t the equilibrium prices are given by: p n = c; p o = c+ +s2 (3.2.30) and the corresponding sales to newcomers, loyal customers and switchers as q 1i = 1=2; q ii = +s4 ; q ij = s4 (3.2.31) while steady-state pro ts are given by (3.2.29). As long as the rms are facing symmetric marginal costs and the market is on the equilibrium path, these values are identical across the two rms, so I suppress the rm-speci c notation for the steady-state values from now on. Consider the general case where in period t 1 rm j?s default rate is higher than of the rival so in period t it operates with marginal cost ~c > c, re ecting the additional cost of securing funds. The shock?s magnitude is given by c = ~c c, 111 where c> 0. The shock lasts one period only and its duration and magnitude are common knowledge. Firm i?s cost is unchanged and equal to c. At this point, I seek to characterize the equilibrium outcome in period t and assume that the shock is unanticipated by both rms. This assumption will be relaxed in Section 3.2.3 In order to derive the equilibrium in the period in which the shock occurs, I present two lemmas that help us analyze rm behavior when the market goes through a period of cost asymmetry. I will discuss rms? best responses in terms of their introductory prices only, because the regular prices are set sequentially and are conditional on the rival?s introductory price. Recall from Corollary 1 that as soon as symmetry is restored, the market converges to steady state in just one period. Thus, both rms correctly anticipate that after a shock in period t, in the equilibrium introductory price in period t+ 1 will be equal to its steady-state level, p n = c , and pro ts from loyal customers will depend on market share gained in period t. The following two lemmas hold regardless of the current distribution of the market. Lemma 3. Let pi nt indicate rm i?s best response to the rival?s price, pjnt. Then, (a) pi nt ci ( rm i undercuts); (b) pi nt pjnt when pjnt ci ( rm i matches or exceeds the rival?s price). Part (a) of Lemma 3 states that whenever the rival?s introductory price is above rm i?s marginal cost, rm i has an incentive to undercut because it can capture the entire market of newcomers by lowering its price just below the rival?s price, pjnt. Since pjnt > ci rm i?s sales to newcomers are pro table and a discrete 112 increase in the demand from newcomers justi es a price decrease. Furthermore, for some parameter values it is possible that rm i?s optimal response is to lower its price well below pjnt in order to optimize pro ts from switchers and newcomers. In either case, rm i?s best response is to sell below the rival?s price. The second part of Lemma 3 states that if the rival?s price is below rm i?s marginal cost, rm i can take on two actions. On one hand, it can match the rival?s price, e ectively selling at a price below cost. Unless the rival?s price is too low, this is a best response for any market sharing rule that allows the rm to capture at least its loyal customer segment when prices are equal. On the other hand, rm i can set a price above the rival?s and target switchers only. In that case, Lemma 2 points out that demand from newcomers will be zero so rm i will forgo the payo from establishing market share. To establish which one of these two actions would be a best response for rm i, we need to consider the corresponding stream of pro ts from either strategy. Since the market reverts to its steady-state equilibrium next period, the decision to invest in market share today or target switchers exclusively will only a ect pro ts from loyal customers in period t+1 but it will have no impact on the rm?s stream of pro ts in periods t + 2 and onwards. Note that the future value of market share and the presence of exit costs when the rm makes no sales to newcomers motivate rms to sell at an introductory price below cost when necessary. However, if this price is too low, rm i may be better o raising its introductory price above cost and selling to switchers only. 23 23The minimum level of exit costs is not su cient to deter rm i from taking this action because the discounted payo from market share is lower when the introductory price is below cost. 113 Let pi indicate rm i?s ?break-even? introductory price, i.e. the introductory price that renders rm i indi erent between investing in market share (by lowering its price to the rival?s level) and targeting switchers only (by raising its price above cost). If the rival?s price is below rm i?s break-even price, rm i?s payo from targeting switchers exceeds the payo from investing in market share so rm i?s best response is to raise its price above cost. Otherwise, rm i is better o matching the rival?s price and capturing market share. This result is summarized in the next lemma, which characterizes rm i?s best response in terms of pi, conditional on the rival?s price being equal to or below rm i?s cost. Lemma 4. If pjnt ci there exists a break-even price pi(ci) pjnt when pjnt p o; pBot = p o; (3.2.33) qAA;t c: pAnt = pA >c; p n p o; pBot >p o; (3.2.39) qAA;t q BB; (3.2.40) xt+1 = 0 (3.2.41) A;t < ; B;t > (3.2.42) A;t+1 < ; B;t+1 = (3.2.43) These strategies describe the unique pure-strategy equilibrium when c c. When c2( c; c), there is no equilibrium in pure strategies. For all values of c the market reverts to the equilibrium characterized in Proposition 1 in period t+ 1. From part (a) of Proposition 2 we see that a su ciently small cost shock ( c 115 c) does not disturb the market away from its symmetric-cost equilibrium aside from rm A?s adjustment of the price to loyal customers to re ect its higher marginal cost. Consequently, rm B?s pro ts remain unchanged despite its temporary cost advantage. This important result illustrates that the low-cost rm does not realize any bene ts from the distress of its rival when the cost shock to the latter is not too big. The intuition for this result is based on the observation that for c = 1 demand from newcomers is perfectly elastic. Suppose that rm B sets pBnt = c. Lemma 3 states that undercutting is not a best response for rm A when the rival?s price is below rm A?s marginal cost. Therefore, A can match B?s price and capture half of the newcomers market or it can set a higher price and sell to switchers only. Matching the rival?s price is costly for rm A because the rival?s price is below rm A?s cost. Hence, rm A would incur a loss of ~c c, equivalent to c, for each unit sold to a new customer. The payo from doing so is equal to the pro t realized on the loyal customers next period and the exit costs that are avoided by the acquisition of market share. By nding the highest value of c such that rm A is willing to invest in market share, we identify an upper limit on rm A?s cost shock, c. As long as c c rm A?s best response is to match the rival?s price because this allows it to capture market share. Note also that rm A?s break-even price, pA, can be identi ed by pA = ~c c. Part (b) of Proposition 2 describes the impact of a ?large? shock, i.e. c> c. When A?s cost shock is large, it cannot compete successfully for new customers in period t. Upon setting pAnt = pA, rm A is outbid by rm B because pA is greater 116 than B?s marginal cost (Lemma 3).24 Firm B captures the full market of newcomers today at a price above its own cost. The main implication of this result is that given the realization of a large shock to rm A, rm B realizes higher pro ts from all three customer segments in period t. In particular, it is able to maintain an introductory price above cost and sell to all newcomers while its regular price and the size of the loyal customer segment increase as well because the rival?s introductory price is above p n.25 Thus, it realizes higher pro ts from both segments of the market in period t. In period t + 1, cost symmetry is restored, rm B realizes its normal level of pro ts, while rm A makes zero pro t because it had failed to build a loyal customer base. The market goes back to steady state at the end of period t+ 1. Note also that for intermediate values of c there is no equilibrium in pure strategies. The condition c> c is stronger than c> c { the latter condition implies that A cannot compete for newcomers while the former also ensures that rm A faces no demand from switchers at all. Hence, when c > c A does not have a pro table deviation in raising its price away from pA once it has been undercut. A pro table deviation would exist if rm A faced demand from switchers at pA; therefore, c > c is needed to ensure that rm A?s cost is su ciently high to induce no demand from switchers. I do not characterize the equilibrium with mixed strategies but note that as s! , the range of c, for which pure-strategy equilibria do not exist, approaches the empty set, ( c! c).26 Furthermore, it will be shown 24Alternatively, rm B may set pBnt = pA and this strategy may still be a candidate for an equilibrium if the market sharing rule is modi ed to split the market xt+1 = 0. For clarity, I suppose that rm B outbids A by setting a slightly lower price. 25Recall from (3.2.13) that @^qii;t=@pjnt > 0. 26See the proof to Proposition 2 for more details. 117 that c c when the shock is anticipated since the market distribution from the preceding period will be such that qBA;t will be zero. As demonstrated in the proof to Proposition 2, in the presence of a small shock the equilibrium is not unique. Any pair of prices such that pAnt = pBnt = p and p 2 [pA;c] would constitute a Nash equilibrium in period t and a subgame perfect equilibrium of the subgame from period t onwards. Part (a) of Proposition 2 presents the Pareto-optimal equilibrium, assuming that this is the equilibrium that rms will coordinate on. Intuitively, pAnt = pBnt = c is not the unique equilibrium ( despite the framework?s similarity to the model of Bertrand competition with homogeneous market demand), because sales today also represent an investment in market share. Firms are willing to sustain small losses today and sell below cost in order to gain market share that will bring in pro ts tomorrow. The perfect elasticity of demand makes deviations above the rival?s price unpro table and if for any reason one rm sets a price slightly below cost, the other rm is forced to match this price in order to capture any market share at all. The lower bound on the price range within which all identical prices constitute an equilibrium is determined by max(pA;pB). When rms are symmetric, rms? break-even prices are identical so pA = pB is an equilibrium that exhausts pro ts from future market for both rms. In the asymmetric cost case here, this lower bound is given by the price that exhausts all future pro ts for the high-cost rm { this is pA in Proposition 2. If rm A sets pAnt = pA and, rm B has no pro table deviation in undercutting this price (the shock is small, pA c, rm B realizes larger gains on its loyal customers segment and its optimal market share in period 2 is larger than its steady-state level (^qBB;2 > q BB). At the same time, A?s optimal market share shrinks because it realizes smaller (or zero) pro ts from its loyal customers in period 2. If ^qAA;2 + ^qBB;2 1, the rms can continue selling at marginal cost in period 1 and, again, under a sharing rule that allows each rm to keep its respective loyal customer segment, neither rm will deviate from the proposed equilibrium, pAn;1 = pBn;1 = c. In the proof of this proposition it is shown that ^qAA;2 + ^qBB;2 is 122 always less than one. Similar to the period-2 equilibrium when rm A is hit by a ?small? shock, the period-1 equilibrium described in Proposition 3 is not unique. Under the sharing rule x2 = 1 ^qBB;2 in case of a tie, any price pair (pAn;pBn) where pAn = pBn = p and p2[max(pA1 ;pB1 );c] would be an equilibrium in period 1.29 We see that regardless of the magnitude of the anticipated shock, neither rm changes its optimal price strategy in period 1 - both rms continue to set pAn = pBn = c. This implies that rms will not change their equilibrium price strategies in period 0 either. Recall that rms agree to share information in period 0 so the anticipation that one rm may deviate could have a ected their period-0 price strategies. The results in Proposition 3 show that despite the anticipation of one rm deviating (in this case, rm B), rms optimal price strategies in periods 1 and 2 remain the same, where the price rigidity in the market is largely due to the interaction of the perfectly elastic newcomers? demand and the pro tability of the mature market. Finally, note that there is only one relevant threshold level in period 2 as c c. By splitting the market at x = 1 ^qBB;2, rm A will face no demand from switchers in period 2 as rm B will retain all of its customers { in period 1 rm B captures only its future loyal customers. Under this distribution of the market, by applying Proposition 2 we see that there is a pure strategy equilibrium in period 2 for all values of c. Recall that in the case where rm B successfully outbids 29pA 1 and p B 1 are the period-1 break-even prices for rm A and rm B, respectively. Their derivation is similar to the derivation of pA in the proof of Proposition 2. 123 A a pure-strategy equilibrium exists as long as rm A has no pro table deviation in raising its price, e.g. when it faces no demand from switchers when its price is below cost. In the derivation of the equilibrium during a period of cost asymmetry in Proposition 2, it was assumed that rms begin the period with equal market shares and rm A faces demand from switchers unless its break-even price is su ciently high. In contrast, when the market in period 2 is distributed such that qBA;2 = 0, the condition that rm A?s break-even price is su ciently high is becomes obsolete: even when rm A is outbid (i.e. when c > c), it has no pro table deviation in raising its price above pA. As a result, c c. Proposition 3 shows that rm B?s potential gains from its cost advantage in period 2 are not dissipated through the competition for market share in period 1. Recall that rm B pro ts from its cost advantage only if A?s shock is su ciently large. We see that anticipating a large shock in period 2 also does not change the equilibrium prices in period 1. The role of liquidity All else equal, when one rm experiences a higher risk of default relative to its rival, the size of the cost disparity is determined by this rm?s cost of funds. Therefore, I will refer to c as the liquidity threshold that determines whether a rm is ?liquid?, i.e. c c or ?liquidity-constrained?, i.e. c> c. Proposition 4. In a market with customer recognition, heterogeneous goods, ho- mogeneous switching costs, and in nitely patient consumers, a rm?s strategic cost of information sharing is zero when its rival is liquid ( crival crival) and strictly 124 positive when its rival is liquidity constrained ( crival > crival). Proposition 4 states our main result: rms who have cheap access to liquid as- sets endure a small cost shock when their rival deviates by withholding information. As a result, the rival does not derive any bene ts from its deviation. In contrast, when a rm is liquidity-constrained it is vulnerable to information asymmetries and the rival pro ts from withholding information. Hence, a rm?s temptation to devi- ate from the information sharing agreement is strictly positive only when the rm?s rival is liquidity-constrained. This proposition makes two important contributions. First, it pins down the key role of liquidity on the cost of sharing information with a rival { rms realize that they can only bene t from the distress of their rivals if the latter are liquidity constrained. Second, in light of the market setup so far Proposition 4 identi es con- ditions, under which imperfect competition does not discourage information sharing, i.e. the strategic cost of disclosing information can be as low as zero. A simpler di erentiated-goods duopoly model would not produce this result, which empha- sizes the need to extend the analysis of competition and information sharing to a richer market framework. In this example, it is the interaction of features such as customer recognition, overlapping generations of in nitely patient consumers and product di erentiation that produce this nding. We can use Proposition 4 to make predictions about the market characteristics that foster information ows between rivals. If the rms within a market are very dissimilar in their cost of funds, such that one rm is liquid while the other is liquidity 125 constrained, then the liquid rm has an incentive to deviate and information sharing can be sustained only if the that rm is su ciently patient. Now suppose that rms who share a market have similar liquidity positions so that they are either both liquid or both are liquidity constrained { we will refer to such markets as homogeneous. In a market populated by liquid rms, the strategic cost of sharing information is zero for both rms, so neither of them has any incentive to deviate from the information sharing agreement. In the context of the exposition in Section 3.2.1 and Figure 3.2, when rms are liquid we have that m = n. This equality implies that sharing information is a Nash equilibrium of the one-shot game in Figure 3.2. Since it is also the Pareto-optimal equilibrium, information sharing is a matter of coordinating on the desirable equilibrium outcome. In contrast, in a market with liquidity constrained rms the strategic cost of information sharing is positive: m> n. The only Nash equilibrium in the one-shot game is (Share;DoNotShare) but the payo s (k;k) are not Pareto optimal. By a straight-forward application of the Folk theorem (Friedman, 1971), liquidity-constrained rms will play the cooperative strategy of sharing information if the discounted long-term bene ts of doing so outweigh the short-term gains from deviating. Based on this discussion we can formulate the following testable hypothesis: Hypothesis 1. If the net bene ts of information sharing are strictly positive and rms within a market are homogeneous in the cost of securing liquid funds, all else equal, information sharing is more likely to be sustained in markets populated by liquid rms relative to markets populated by liquidity constrained rms. 126 This hypothesis will be tested in Section 3.3.1. The role of competition One of our main questions of interest is whether the strategic cost of informa- tion sharing is higher under more intense competition. Therefore, we can examine how the liquidity threshold c varies with competition. If c falls, then the require- ment that the shock is small become more stringent { there is a higher probability that some rms will face a positive strategic cost. Thus, more intense competition would decreases the likelihood that information sharing occurs. It is our goal to establish how c varies with changes in the competition parameters. As mentioned earlier, the degree of competition in the market varies along three dimensions { consumer patience ( c), the degree of product di erentiation in the market (proxied by ), and the level of switching costs (s). Since we limit attention to c = 1, the endogenously determined liquidity threshold c depends explicitly on s and and we can perform comparative statics. The analysis of how c varies with c is not so straight-forward because the model is not tractable for c away from one. For this reason, I present some intuition for the market dynamics when c is away from one and then discuss how a temporary cost asymmetry may a ect the rms? price strategies. This discussion will illustrate how more intense competition can reduce the strategic cost of information sharing to zero and facilitate information sharing. As shown in Villas-Boas (1999), greater consumer patience intensi es com- petition for newcomers and lowers both the introductory and regular prices. This result is con rmed in Grozeva (2009) for the augmented model of dynamic com- 127 petition with switching costs, which is used here. As competition becomes more intense, newcomers? demand becomes more elastic and the high-cost rm has less exibility in adjusting its introductory price in response to the cost shock. In the limit, when c ! 1, we saw that the high-cost rm cannot raise its introductory price without losing demand from all newcomers. This produced the equilibrium result that for a small shock, price remains unchanged and the low-cost rm real- izes no additional bene ts from its cost advantage. This ensured that the strategic cost of information sharing can be as low as zero. If newcomers? demand was not perfectly elastic, the high-cost rm would be able to raise its price and still capture market share. The rival?s best response then would be to also raise its price. The di erence between the two prices will depend on the discount factors of rms and consumers. This is driven by the fact that convergence to steady state is monotonic and becomes slower as rms become more patient { rms realize that larger market share hampers their ability to compete for switchers next period so they compete less aggressively for new customers today. Hence, the low-cost rm will raise its price in response to the price increase by the rm experiencing the shock and the more patient rms are, the closer the two prices will be to each other. All of this implies that when c < 1 a temporary cost asymmetry will always generate addi- tional bene ts to the low-cost rm because it allows it to charge higher prices to all customer segments and increases its sales to new and loyal customers. At the same time, the losses of the high-cost rm are mitigated by the rm?s ability to raise its price without entirely forgoing sales to newcomers. In summary, when c < 1 the strategic cost of information sharing will be strictly positive (m>n) for all values 128 of the cost shock. A lower consumer discount factor makes competition less intense, thereby increasing the strategic cost of information sharing. This suggests that in a less competitive market, where the degree of competition is only varied along the consumer discount factor, a self-enforcing information-sharing agreement will be harder to sustain because it is more costly. This argument provides a counterpoint to the perception that more intense competition is necessarily more detrimental to information-sharing practices. What happens when competition varies with the degree of product di erentia- tion or switching costs? Limiting attention to c = 1 allows us to perform compara- tive statics with respect to the liquidity threshold, c. The next corollary describes how the liquidity threshold varies with the degree of product di erentiation and switching costs: Corollary 2. Under the conditions presented in Propositions 2 and 3, the liquidity threshold, c, increases in the magnitude of switching costs and the degree of product di erentiation. Not surprisingly, the liquidity threshold increases in s and because they increase the value of market share. As market share becomes more valuable the break-even price of the a ected rm falls even lower so the liquidity threshold goes up. It should be noted, however, that the result with respect to can be ambiguous if the distribution of the market was such that rm A faced demand from switchers during the period of the shock. When c = 1, increases in switching and transportation costs unambiguously 129 relax competition. The equilibrium introductory price is independent of s and while the regular price increases in both parameters. 30 Sales to loyal customers, the pro table customer segment in the market, also increase as switching and trans- portation costs go up. Holding the cost of liquid funds constant, more intense competition through lower s and lowers the liquidity threshold, making the liq- uidity requirement more stringent. This increases the strategic cost of information sharing for those rms, whose cost of funds was below the original liquidity thresh- old and is now above the new threshold. The proof of Corollary 2 also shows that the cross-partial derivatives, @2 c@s@ and @2 c@ @s, are negative. This indicates that the rate at which higher switching costs raise the liquidity threshold is lower when there is already a large degree of product di erentiation (and vice versa). This nding is intuitive because higher switching and transportation costs raise the return on market share (given by ( + s)=2 per unit sold) and, therefore, raise the liquidity threshold. If the return on market share is already high due to large product di er- entiation, the relative e ect of the switching costs on the return to market share is smaller. The result in this section is important because it provides an analysis of how parameters de ning the ex-ante level of competition a ect information sharing. Past studies of information sharing and competition have looked at the interaction of the two as in their models information sharing relaxes or intensi es competition and the desirability of an information-sharing regime depends on the change in the compet- 30For c < 1 and complete customer lock-in (the case where s > ), switching costs intensify the competition for market share as they increase the return from loyal customers. 130 itive environment.31 In contrast, in our model, we are able to determine how the ex ante competition intensity a ects information sharing. If competition intensity is driven by variation in switching or transportation costs, then the strategic cost of information sharing is on average higher in more competitive markets. In ad- dition, our dynamic model also identi es a role for consumer patience as a ecting competition. Our heuristic discussion above suggests that more intense competition through more patient consumers can lower the strategic cost of information sharing and encourage such practices. Proposition 4 illustrate this result in the limit { when c = 1 the market is at its most competitive level, holding s and constant, and the strategic cost of information sharing can be as low as zero, ensuring that sharing information is an equilibrium in the one-shot information-sharing game. 3.3 Data and Empirical Strategy 3.3.1 Data I use data from a cross-sectional survey of agricultural traders in Madagas- car, conducted by the International Food Policy Research Institute (IFPRI) and the Malagasy Ministry of Scienti c Research (FOFIFA), and previously used by Fafchamps and Minten (1999, 2000). The survey was designed to be representa- tive of traders along the entire food marketing chain { wholesalers, retailers, and 31Of these, only Jappelli and Pagano (1993) and Gehrig and Stenbacka (2006) explicitly look at changes in the competition parameters to determine how they will a ect information sharing. Jappelli and Pagano (1993) use a permanent cost disparity between the local monopolist and a potential entrant to provide exogenous variation in the incumbent?s market power. Gehrig and Stenbacka (2006) use the distribution of borrowers? switching costs to model the degree of competition in a duopoly market. 131 assemblers, { whose main product traded was a local staple food.32 Traders from three main agricultural areas (Fianarantsoa, Majunga, Antananarivo) were sam- pled and particular attention was paid to obtaining observations from both urban and rural communities. The survey was administered in two rounds - 850 traders were surveyed in the rst round in May-August 1997 and 738 of those respondents were traced for a follow-up survey in September-November of the same year.33 The dataset is unique in providing a rich set of measures on traders? information shar- ing practices, liquidity position, access to credit, reliance on formal institutions, contractual risk, and con ict resolution. Table B.1 presents de nitions of relevant variables to be used in the estimation and Table B.6 presents summary statistics. The business environment captured in the survey is representative of the type of settings that motivate our study. The data reveals that traders are exposed to contractual risk from both suppliers and customers { 31% of the second-round sample report recieving late or no payment from customers in the past twelve months and 21% report late or no delivery from suppliers for the same period. Only 2 of the 738 traders in the sample have resorted to formal means of contract enforcement such as the police or the courts.34 Yet, the con ict resolution rate is 77% for the traders reporting issues with customers and 84% for those reporting issues with 32For a more detailed discussion of the survey design and the sample composition, see Fafchamps and Minten (1999). 33The questionnaire from the rst round gathered information on trader and rm characteristics while the questionnaire from the second round focused on the traders? relationships with customers, suppliers and other traders. The data used in this investigation is drawn from both rounds of the survey. 34The survey does not ask if traders resort to private enforcers, hence it cannot be determined if private enforcement is a feasible alternative that may be preferred to the use of information networks. 132 suppliers, suggesting that informal contract-enforcement mechanisms may be at work.35 There is also evidence that traders do screen customers on the basis of past performance: among the 285 traders who were asked if delinquent customers will be refused credit from their other suppliers as well, 74:7% respond that at least some suppliers will stop extending credit and 24:9% report that most or all suppliers will do so; less than 1% of the traders state that they would sell on credit to a new customer. Furthermore, in a developing country context liquidity is likely to represent a sizeable hurdle to information sharing practices, which facilitates a test of the model?s liquidity hypothesis. I obtain data on the traders? information-sharing practices from a question that asks respondents about the frequency with which they talk to other traders about delinquent customers.36 There is also information about the frequency of discussing the product quality of di erent suppliers. I choose to use only the question on discussions about customers because it is more likely to capture the exchange of information regarding contract performance. Speci cally, the data indicate that there is less room for opportunistic breach of contract with respect to product quality { 84:5% of the traders report that they always check the supplier?s product quality before purchase and another 12:5% of the traders report they do so often. Those traders who state that product quality varies a lot were asked to indicate the reason for this variation and none of the respondents attribute it to malfeasance on behalf 35The con ict resolution rate is computed as the share of traders who resolved their contractual disputes with customers/suppliers. 36Fafchamps and Minten (1999) point out that by ?other traders? respondents understood ?other traders who operate in a manner similar to yours?. See Footnote 11 in Fafchamps and Minten (1999). 133 of suppliers.37 The question about traders? discussions of delinquent customers does not make it clear whether traders disclose information about their own customers or receive information from other rms. Case study evidence from other markets indicates that information sharing is based on reciprocity, so I assume that discussions about delinquent customers point to the existence of information sharing networks with two-way information ows. The question is directed only at those respondents who report having regular customers. This reduces our sample size to 344 observations. While it is not clear if traders communicate directly with their rivals, it is reasonable to assume that traders who want to withhold information from their rivals will not disclose it to anyone in that line of business. The possible answers to the question are ?daily?, ?weekly?, ?monthly?, ?occasionally/less frequently than once a month?, and ?never?. Only 3:2% of the traders report having such discussions at least weekly compared to 13:3% of traders who engage in such discussions at least monthly; 24% of the traders report that they never discuss customers with other traders. Traders who report discussing delinquent customers at least once a month are coded as respondents who share information, giving rise to a binary dependent variable. This de nition increases the likelihood that the transmission of information is timely and deliberate. Admittedly, the frequency of communication also depends on the frequency with which rms experience customer issues. In the sample 35:1% 37Among the full sample of 738 traders, 147 report having received products of lower quality from their suppliers in the past 12 months. Among those, 50 respondents state that the quality of products they purchase varies at least somewhat, but none of those 50 respondents identi es cheating on behalf of the supplier as a cause for this variability. However, when asked why they think product quality varies a lot, only 3 out of the 738 respondents attribute this variation to manipulation on behalf of the suppliers. 134 of the rms who extend sales credit have not experienced late or no payment from customers in the past year. Under a binary speci cation of the dependent variables traders who face a signi cantly lower default rate may appear to be less likely to share information. To use the full information contained in the traders? reponses I also report results from coding the dependent variable as categorical according to the reported frequency of information exchange. Since information sharing is driven by exposure to contractual risk, I limit the sample to traders who extend sales credit. 38 I exclude traders who are non- Christian, a total of ten observations, because of the concerns that they may be part of closely-knit ethnic networks.39. Traders from the regions Majunga Plaines and Majunga Hauts Plateaux are also dropped during the estimation procedure because of their limited representation in the sample, which causes collinearity issues. Finally, among the traders who state the number of competitors they have over their regular customers, I keep only those who have at least one competitor in order to exclude monopolies. I also keep those traders who respond that they do not know how many competitors they have. The nal sample used in the estimation consists of 279 observations. Table B.1 presents de nitions of all variables relevant to the discussion that follows and Table B.6 shows presents summary statistics. 38Of the traders who are asked about their information sharing practices, 96:5% are selling on credit. Only 1 trader in the sample reports that he discusses customers with other traders but does not extend sales credit himself. 39Fafchamps and Minten (1999) note that the Malagasy society tends to be fairly homogeneous in terms of ethnicity and religion and respondents who identify as non-Christian could represent the few ethnic minorities in the country 135 3.3.2 Empirical Strategy The theoretical model yields the unequivocal result that rms are more likely to share information if their rivals are liquid, but it is ambiguous about the impact of operating in a market with more intense competition. In principle, one would like to use the liquidity prediction to test the model, and the correlation between information sharing and competition to gain insight of the relationship between the two, based on rm-level observational data. This suggests one would estimate a model of the form y i = 0 Liquidityi + 1 Competitioni + 2Xi + i (3.3.1) where Yi = 1 if y i > 0 Yi = 0 if y i 0 and y i > 0 indicates information sharing while y i 0 { no information sharing; i ~ N(0; 2 ) and i = 1;2;:::K indexes observations; Liquidity is a measure of the rival?s liquidity position, Competition measures the intensity of competition in the market, and X is a vector of controls. 136 Liquidity measures Consistent estimation of 0 requires that E[Liquidityi ijXi] = 0. Again, in my model causality runs directly from liquidity to information sharing so this as- sumption is satis ed in the model. Furthermore, I would argue that for at least one of the liquidity measures discussed below, it is reasonable to assume that it is not correlated with i. Thus, I should be able to test the implication of the model that 0 > 0. The dataset does not allow us to identify a trader?s local competitors. Hence, I have no explicit information about their number or their liquidity position. Assum- ing that the model?s predictions hold for more than one competitor and assuming that markets are homogeneous, I focus on identifying a proxy for the liquidity posi- tion of traders who operate in the respondent?s market. I construct two such proxies { one set of liquidity measures (to be discussed below) uses own liquidity to proxy for market liquidity, and a second set of liquidity measures uses the average liquidity of traders in the same market. A market is de ned as the intersection of main prod- uct traded and distribution category. With six products traded and 7 distribution categories, there are 42 possible markets although not all of them are represented in the sample. Restricting attention to markets with at least two observations, we are left with 27 markets in the sample. The average value of liquidity within a market is based on the liquidity measures of all traders in this market after excluding the observation at hand since we are mainly interested in obtaining information about the rivals? liquidity position. I will refer to this set of liquidity proxies as average 137 liquidity and to the former set of proxies { own liquidity. The use of either proxy rests on the assumption that rms within a market are not too dissimilar in their liquidity positions. This assumption is reasonable if traders? liquidity positions are related to the characteristics of agricultural markets in Madagascar. For example, being able to rely on family members for credit requires that the trader?s family is not employed in the same enterprise. The data shows that among the three broad trade categories (assemblers, wholesalers and retailers) assemblers are the most likely to have non-family members as employees and also the most likely to have more than three family members with jobs. The availability of own liquid funds and a credit network can also lead traders to self-select into markets with high or low volatility of earnings, or markets with high or low entry costs. For these reasons, I believe the assumption about traders? similar cost of capital is likely to be satis ed. In case this assumption is not satis ed for all traders, then the estimate of 0 would provide a lower bound on the true value of 0 as the model suggests that in non-homogeneous markets liquid rms are less likely to share information with their liquidity-constrained rivals. To measure the liquidity position of each observation and create the proxies for the market liquidity, I construct two di erent liquidity scores, referred to as LS 1 and LS 2. The rst liquidity score, LS 1, uses those variables that give us most con dence in their exogeneity { these are the dummy variable for being able to borrow from friends/family at the present if nancial hardship arises and the categorical variables for number of family members or friends the trader can borrow from and number of family members with jobs. A more detailed description of 138 these variables is provided in Table B.2. The questions that produce these variables exclude the respondent?s spouse from the de nition of family and friends. Since we have one dummy variable and two categorical variables that range from 0 to 2, the maximum liquidity score is equal to 5 and its frequency distribution is presented in the rst column of Table B.3. Using LS 1, liquidity is exogenous if having a network of friends or family who are able to extend credit is uncorrelated with unobservable trader/ rm char- acteristics that may also drive information sharing behavior. Ideally, this liquidity measure would be entirely driven by exogenous variation in family size, geographic proximity to family and friends, or area-speci c characteristics that a ect income levels and diversi cation of economic activities among the local population. How- ever, we cannot rule out that the availability of a network of friends or family may be cultivated over time and is likely based on reciprocal relationships, which raises the concern that traders who have such networks may be more patient or may have an unobservable taste for cooperation.40 Hence, I include a dummy variable equal to one if the trader reports discussing input or output prices with other traders and a dummy variable equal to one if the trader reports discussing the quality of suppliers? products with other traders.41 Both variables may capture unobserved characteristics that a ect the trader?s propensity to cooperate with other traders. For the second speci cation of the liquidity score, LS 2, I use variables that re- ect the availability of personal funds - these are the dummy variables for possessing 40For example, traders who expect to stay in the market/area for a longer period may be more likely to engage in reciprocal relationships with friends/family as well as local business rivals. 41Similar to the de nition of the dependent variable, only traders reporting at least monthly discussions are coded as actively discussing prices or product quality. 139 a bank account, having another source of income (own or through spouse), reporting formal savings, reporting informal savings, and having access to overdraft facility (see Table B.4 for the resulting frequency distribution).42 This liquidity score also ranges from 0 to 5 although none of the traders possesses a score above 3. It should be noted that the dataset also provides information on variables that could proxy for the traders? ability to negotiate favorable payment terms with customers and suppliers in case of a liquidity crunch. The results from using these measures of liquidity are qualitatively the same as those reported for LS 1 and LS 2. I do not report them here because they are ambiguous in terms of their implications for the trader?s liquidity position. For example, receiving supplier credit on a regular basis may indicate ability to obtain credit during a liquidity crunch but it may also indicate that the trader is liquidity constrained even in the absence of shocks. Under LS 2, liquidity is exogenous if own liquid assets are uncorrelated with unobservables that a ect information sharing behavior. This assumption is more contestable because traders? personal assets can be a by-product of the market in which they operate and unobservable market characteristics can a ect both infor- mation sharing behavior and liquidity. Furthermore, information sharing may also a ect traders? liquidity. As discussed below I include a set of controls capturing a number of market characteristics that should minimize the possible bias in 0. I report results using both LS1 and LS 2 as alternative measures of liquidity. When using own liquidity as a proxy for the rival?s liquidity, the liquidity score of the 42While the dataset also has information whether the trader has received formal bank credit in the past, I do not include this variable in the liquidity score because bank credit likely cannot be obtained quickly enough to cover unexpected cash out ows. Fewer than 10% of the traders in the nal sample had applied for bank loan in the past. 140 current observation is used. While LS 1 is more likely to be exogenous and provide causal estimates, LS 2 is likely to capture market liquidity with less noise because it is a direct measure of the availability of personal funds. Table B.5 summarizes the distribution of the liquidity variables based on average market liquidity (for each of the two measures, LS 1 and LS 2). Control variables Table B.7 provides a summary of all control variables included in the esti- mation of (3.3.1). To capture a large set of market characteristics that may a ect market power and exposure to risk, I include controls for rm size (based on total sales and coded by the data collectors), rm age, trade category (e.g. wholesalers, assemblers or retailers), geographic region (or city, as a robustness check), capital location, main product traded, and access to a telephone. Market-level controls mit- igate the potential omitted variables bias when using LS 2 and can also proxy for unobservable characteristics correlated with the level of competition. I also control for demographic characteristics such as sex, age, and education level. Information sharing incentives are stronger when the rm is exposed to higher default risk. To control for exposure to risk, I rst include categorical variables for the number of traders the respondent knows personally.43 Traders with larger trade networks can disseminate and receive information about delinquent customers in a more timely manner and they can also anticipate that customers would be aware of the stronger reputation e ects if they default to a well-connected trader. 43Since Fafchamps and Minten report that the data is subject to considerable measurement error I recode some continuous variables as categorical. 141 Hence, traders with larger networks may face signi cantly lower default risk and communicate information about customers less frequently. Second, I control for the trader?s percent of sales on credit. Credit sales reduce a potential omitted variable bias in the estimate of the liquidity measures { since liquidity may be positively correlated with credit sales, we can erroneously interpret a positive coe cient of the liquidity measure as a validation of the model, while in e ect it may be driven by higher exposure to risk (since the latter provides a stronger incentive to engage in information sharing). As a robustness check, I also estimate (3.3.1) by restricting the sample to traders with credit sales between 15% and 50% and compare the results to those for the full sample. I include discussing prices and discussing suppliers? product quality as proxies for unobserved market characteristics that may facilitate cooperation and infor- mation exchange in the market. As it will be evident from the discussion of the competition controls below, discussing prices does not suggest price collusion. Fi- nally, I do not include number of competitors in the set of controls because the variable appears to be very noisy and does not have any meaningful impact on the regression results. Competition measures I use three measures of competition to establish a correlation between in- formation sharing and competition intensity. The most straight-forward measure, Competition, is a self-reported measure of the strength of competition the trader faces: Competition is equal to one if the trader responds that the level of compe- 142 tition in their market is high, and zero { if competition is reported to be low or moderate. To consistently estimate 1 we need E[Competitioni ijXi] = 0. This is satis ed in my model since causality runs directly from competition to information sharing. As already mentioned, a major distinction between my model and the pre- vious literature is that I can identify a role for competition that does not generate a feedback e ect of information sharing on competition. Nevertheless, the feedback e ects pointed out by previous studies can also have a role in a real-world setting and can lead to a simultaneity bias in our estimates. For example, traders who share information may anticipate lower default rates and extend more sales credit, leading to more intense competition for customers. This potential endogeneity can be overcome if there is a suitable instrumental variable. Such an instrument must be i) correlated with Competition, ii) uncorrelated with and iii) not included in X. Table B.8 shows the results of regressing Competition on a number of rm characteristics. It shows that variables such as having completed high school edu- cation, being more liquid as measured by LS 1, engaging in product processing as a secondary activity and having a large share of sales to regular customers are all associated with a lower propensity of reporting strong competition, so they satisfy i). None of the remaining rm characteristics, including the dummy for discussing prices, have signi cant coe cients. Number of competitors also does not have a sig- ni cant e ect on reported competition possibly because it is a very noisy measure of the true number of direct competitors. How do we interpret the signi cant negative coe cients of product processing and share of regular customers? Product processing can add more variation in the 143 quality of the nal product so it can be interpreted as indicating greater product di erentiation. A larger share of regular customers could be associated with higher switching costs and/or a higher degree of product di erentiation. Both variables may satisfy iii) because they are likely to a ect information sharing only through competition. Furthermore, one can argue that they are uncorrelated with , so ii) is satis ed too. However, they do not pass the tests for weak IV by Staiger and Stock (1997) and Stock and Yugo (2005). In the absence of stronger instruments, I estimate (3.3.1) to obtain the OLS (partial) correlation between self-reported competition intensity, given by Compet and information sharing. As indicated by the results in Table B.8, the coe cients of sales to regular customers, and in column (6) those of product processing, are precisely estimated and consistent with the interpretation that these two variables capture to some extent the degree of switching costs and product di erentiation in the market. For this reason, I include them as proxies for competition intensity. They are also less likely to be endogenous and subject to simultaneity bias from the feedback e ect of information sharing on competition.44 This allows us to test the implications of Corollary 2 { speci cally, the corollary predicts that information sharing is more likely among traders who engage in product processing or have a high share of sales to regular customers because the latter are indicative of high switching and/or transportation costs, which raise the liquidity threshold. Because of concerns that self-reported competition may be endogenous and bias all coe cient estimates, I 44This statement would not be valid if information sharing a ects consumer switching costs. In the model switching costs are assumed to be exogenous and driven by factors, other than the choice of information-sharing regime. 144 add it to the regression separately from the preceding two competition proxies. The results from Table B.8 also help us rule out variables that may be sus- pect of indicating collusive practices in the market. For example, discussing prices with other traders is shown to have no impact on the perception of competition intensity. This is consistent with the ndings in Fafchamps and Minten (2002) who use this dataset to examine the impact of social capital on rm productivity and nd no evidence that social capital is associated with collusion on prices.45 In fact, the question that asks traders if they discuss prices with other traders does not distinguish between input and output prices. Since discussing suppliers? product quality is highly correlated with discussing prices, it is more plausible that traders discuss input prices. Therefore, the price discussion dummy cannot be interpreted as indicative of collusion. Interestingly, rms with liquidity score LS 1 greater than 1 are signi cantly less likely to report strong competition (column 4), while our second and more explicit measure of liquidity, LS2, does not have a signi cant e ect (column 5). However, when I use a liquidity dummy equal to one when LS1 > 2 (not reported), this e ect disappears { LS 1 is no longer signi cant (p-value = 0:265). Hence, it is possible that the signi cance of the LS 1 scores is due to the small number of observations in the omitted category, i.e. observations with LS 1 equal to or less than 1. Finally, note that controlling for credit sales in (3.3.1) helps us reduce a poten- tial omitted variable bias in the estimate of the impact of self-reported competition. 45Fafchamps and Minten (2002) measure social capital in terms of number of traders known, number of family members in agricultural business and number of people the trader can borrow from and nd that it has no signi cant impact on traders? pro t margin. 145 Evidence from developing countries shows that market power and credit sales can be positively correlated due to enforcement concerns (McMillan and Woodru , 1999) or negatively correlated if sales credit is used as a competitiveness tool (Fisman and Raturi, 2004; Van Horen, 2007; Fabbri and Klapper, 2008). Hence, competition may a ect the percent of credit sales that a rm extends and, therefore, its exposure to risk and likelihood to share information. Since we are interested in the impact of competition ex-post, it is important to control for exposure to risk and the share of sales on credit is an adequate proxy. 3.4 Results Unless speci ed otherwise, all regression results below report probit coe cients with robust standard errors and include controls for trader and market characteris- tics as listed in Table B.7. 3.4.1 The Impact of Competition The raw correlation between the perceived intensity of competition and infor- mation sharing is 0:078 based on the responses of 377 traders. The correlation is positive, but not statistically signi cant. Table B.9 shows the partial correlation between the two after estimating 3.3.1. Column (1) shows the results of including only Competition and the main market and demographic controls listed in Table B.7. Column (2) adds the additional set of controls: % of sales on credit, competi- tion proxies, and discussing prices and suppliers. Columns (3) - (6) each include a 146 di erent liquidity measure based on the two liquidity speci cations, LS 1 and LS 2, and the two alternative proxies, own liquidity and average market liquidity. Across all columns the dummy variable for reporting strong competition has a positive and sign cant coe cient, indicating that traders who perceive their markets as more competitive are more likely to share information about delinquent customers. As mentioned before, this result has to be interpreted with caution due to possible si- multaneity bias. If we believe that there is a feedback e ect of information sharing on competition and Competition captures the ex-post level of competition, then this result would be consistent with the ndings in Jappelli and Pagano (1993), Padilla and Pagano (1997), and Gehrig and Stenbacka (2006) that information sharing inten- si es competition. Otherwise, the positive coe cient on Competition is consistent with the conjecture in our model that information sharing is more likely in ex-ante more competitive markets (if competition is driven by consumer patience). The two objective competition measures, product processing and the percent of sales to regular customers, are included in columns (2) - (6). Product processing has the expected positive coe cient but is imprecisely estimated. The coe cient on sales to regular customers is negative and signi cant in columns (4) - (6), suggesting that traders with a higher share of sales to regular customers are less likely to share information. In line with the interpretation of this variable as indicating higher switching costs or greater product di erentiation, this result is inconsistent with the model?s predictions. Among the coe cients for the trader and market characteristics, we obtain that semi-wholesalers and retailers with a xed point of sale exhibit a higher propen- 147 sity to share information relative to wholesalers. Large and medium-sized rms as well as rms who have been in operation between 5 and 10 years and rms whose main product is beans or peanuts, relative to traders whose main product is rice (re- sults for these variables are not reported for brevity), all exhibit a higher propensity to share information. The estimates also indicate that traders who share informa- tion about prices are signi cantly more likely to share information about delinquent customers. The coe cient on discussing prices is particularly large and signi cant at the 1% level. These results largely persist for the rest of the reported regressions. 3.4.2 The Impact of Liquidity In this subsection the emphasis is on the e ect of the liquidity measures on information sharing. Table B.10 presents the estimates of the liquidity coe cients, based on several speci cations of the liquidity variable. Column (1) lists the re- sults from using only the main market and demographic characteristics. Columns (2) through (5) add alternative sets of controls: percent of credit sales, self-reported competition intensity, competition proxies, and discussing prices and suppliers. Col- umn (6) presents the results when all above-mentioned controls are included. Fi- nally, in column (7) I also control for the trader?s beliefs that if other suppliers knew about the delinquency of a customer, they would not extend credit to this customer. This question is asked of only a subsample of the traders, which reduces the sample size to 198 observations. Hence, we report the results from including this control separately. 148 Panel A presents the coe cient estimates of own liquidity, as measured by liquidity score 1, as a categorical variable. Recall that LS 1 captures traders? ability to borrow from friends and family and own liquidity is assumed to proxy for the liquidity of the trader?s rival. Observations with liquidity scores of 0 are grouped together with those who have liquidity scores of 1 and constitute the omitted cate- gory. Relative to this group, the coe cient on observations with liquidity scores of 2 is negative and signi cant. The coe cients for all other categories are insigni cant. Overall, the results in Panel A do not present a clear picture of the role of liquidity on information sharing. This may be due to the fact that the omitted category, traders with liquidity scores of 1 or less, is too small (only 21 observations) to pro- duce informative results. Therefore, in Panel C we replace the category dummies with a single dummy variable that seeks to more clearly distinguish liquid from illiquid rms based on LS 1. Panel B again presents the results from using own liquidity as a proxy for the rival?s liquidity position, but uses liquidity score 2 to measure liquidity. In columns (1) - (4) traders with liquidity scores of 2 or 3 are signi cantly more likely to share information relative to traders with liquidity scores equal to 0. This result persists upon the inclusion of all control variables (column 6) and is consistent with the model?s prediction. When we include the control for whether the trader believes a delinquent customer will be refused credit by most traders, the coe cients on the liquidity score dummies change in a nontrivial way: traders with LS 2 = 1 are now signi cantly less likely to share information relative to the more liquidity constrained group of traders with LS 2 = 0 while the coe cients for having a score of 2 or 3 149 remain positive but insigni cant. In Panels C and D, we replace the category dummies for own liquidity with one aggregate indicator. In Panel C, we use a dummy equal to 1 if LS 1 is greater than two. The chosen cuto point treats the bottom one-third (33.55%) of the traders as operating in liquidity-constrained markets. Admittedly, the cuto point is arbitrary chosen but a cuto of 3 or 4 leads to qualitatively identical results, except that in some speci cations a cuto of 3 leads to very imprecise, albeit positive, estimates of the liquidity dummy coe cient. In Panel D we use a dummy equal to 1 if LS 2 > 1 (see Table B.10). This procedure again treats roughly the bottom one-third (26:56%) of the traders as facing liquidity constraints in their markets. Again, the model predicts a positive coe cient for the so-constructed liquidity indicator variables. In both panels C and D, the liquidity estimates are consistently positive and signi cant throughout the inclusion of all control variables (columns 1 - 7), as predicted by the model. When controlling for discussing prices and suppliers, the liquidity coe cients in both panels increase in magnitude and become more precisely estimated, possibly due the fact that discussing prices explains a large part of the variation that was not picked up by liquidity. In column (7) I again add a dummy variable equal to one if the trader believes a delinquent customer will be refused credit by most traders. With this additional control, the liquidity coe cients almost do not change their value and standard error, giving us more con dence in the stability of the results. In results that are suppressed in this table for conciseness, it can be seen that in column (7) the coe cient on product processing becomes large, positive and signi cant at the 1% level { this estimate is consistent 150 with the interpretation that product processing as a secondary activity re ects a larger degree of product di erentiation, which was shown to relax competition and facilitate information sharing. In Panels E and F, I use average market liquidity as a proxy for the rival?s liquidity. In Panel E, the liquidity variable is based on liquidity score 1, while in Panel F { on liquidity score 2. Higher average liquidity scores indicate higher average liquidity in the trader?s market. In Panel E the coe cient on liquidity is positive and signi cant upon the inclusion of all controls (columns 6 and 7). In Panel F, the coe cient on liquidity is positive and signi cant across all columns. Hence, the results using average market liquidity as a proxy are consistent with the results that rely on own liquidity (Panels A - D). Together, these results present consistent evidence that rivals? liquidity has a positive e ect on traders? propensity to share information. Results, based on liquidity score 1 are arguably providing causal estimates, but they are also less precisely estimated, which is consistent with our conjecture that they are a more noisy measure of the rival?s liquidity position. Robustness checks Table B.11 presents several robustness checks for the full set of liquidity mea- sures used in Table B.10. Column (1) replicates column (6) from Table B.10. Col- umn (2) replaces the region xed e ects with city xed e ects. The resulting changes in the estimates are negligible across all liquidity measures. I use region xed ef- fects in the main speci cation because fewer observations are dropped during the estimation process. In column (3) I restrict the sample to include only rms with 151 credit sales between 15% and 50% of total sales.46 This reduces the variation in exposure to risk through credit sales and still includes about 60% of the sample. Again, there are no qualitative changes in the results when compared to column (1). Column (4) limits the sample to only those traders who have experienced late or no payment from customers in the past twelve months, which I would refer to as customer default. This reduces the sample to 152 observations but ensures that we are only looking at markets where cheating is known to occur { hence, some of the unobserved variation in default risk is reduced. Th set of measures using own liquid- ity as a proxy (panels A - D) preserve their signs and signi cance levels compared to the baseline results. The measures using average market liquidity (panels E and F) lose signi cance but remain positive. Overall, these robustness checks con rm that our results are stable across di erent speci cations, with the exception of average liquidity measures which become insigni cant when restricting the sample to traders who have experienced customer default. Table B.12 presents the results from estimating an ordered probit model where the dependent variable is categorical and re ects the frequency with which traders exchange information about customers. The dependent variable takes on discrete values from 1 to 5 where a value of 1 indicates the trader never discusses customers and 5 indicates that the trader discusses customers daily. Columns (1) - (2) report results using the own liquidity proxies and columns (3) - (4) show results for the av- erage market liquidity proxies. This speci cation of the dependent variable is much less informative. The only signi cant predictors of the frequency of the information 46During the estimation, certain controls were dropped given the smaller sample size. 152 exchange are the dummies for discussing prices and being a large rm { their signs are consistent with the probit results. The liquidity coe cient is positive but not signi cant for the own liquidity proxy using LS 1 and for the average market liquid- ity proxy based on LS 2. Again, this is consistent with the prediction of the model and with the wider set of results presented in Tables B.10 and B.11. In summary, the results in Tables B.10 and B.11 suggest that rms operating in liquid markets are more likely to share information about delinquent customers, as proposed by the model. This result is conditional on the validity of the two sets of proxies of rival?s liquidity { own liquidity and average market liquidity. The estimates are robust to the inclusion of various controls and to the use of various estimation speci cations and liquidity measures. Since the rst speci cation of the liquidity score can be credibly viewed as exogenous, the positive estimate of 0 implies that liquidity has a positive causal e ect on information sharing. Marginal e ects To get a better idea about the magnitude of the e ect of liquidity, I compute the marginal e ects of the two liquidity measures. The baseline speci cation is based on column (6) of Table B.10. For ease of interpretation, the liquidity proxy based on own liquidity uses liquidity dummies as de ned in Table B.10 instead of indicator variables for the underlying categorical liquidity variable. Because most of the explanatory variables are binary estimating marginal e ects at the mean values is not very informative. Therefore, I compute the average marginal e ects 153 (see Table B.13).47 The probability of information sharing is on average 9:7 points higher for liquid rms, if liquidity is de ned in terms of ability to borrow from friends and family (i.e., based on the liquidity dummy using LS 1), and 13 points higher if liquidity is measured by personal funds (i.e., using LS 2). Using average market liquidity as a proxy, we nd that the probability of information sharing is on average 8:6 points higher for each 1 point increase in the liquidity score of the market as measured by LS 1 and 17:3 points higher for each 1 point increase in the liquidity score of the market as measured by LS 2. All reported marginal e ects are signi cant at the 10% level. These estimates indicate non-trivial changes in the probability that a trader would share information about delinquent customers if that trader operates in a liquid market. 3.5 Conclusion Inter- rm information sharing is of key importance in developing and tran- sition economies where reliance on formal contracting institutions is limited. The transmission of information about defectors helps rms screen out bad risks and reduce their exposure to contractual risk. At the same time, liquidity constraints are a common characteristics of economies with weak contract enforcement insti- tutions and imperfect capital markets. In this study I derive a causal relationship between liquidity and information sharing and o er a systematic investigation into why inter- rm information sharing practices emerge in some markets and not in 47The marginal e ect for each observation is computed using the user-written Stata command -marge -. 154 others. Competition is commonly perceived as the main impediment to voluntary information exchange. I have shown that the presence of competition does not necessarily create barriers to information sharing. Second, I have identi ed con- sumer patience as a market characteristic, which intensi es market competition and encourages information sharing behavior. Third, this study sheds light on the importance of liquidity in the decision of rival rms to share information. The model suggests that markets populated by liquid rms are more likely to exhibit information-sharing networks. Using a relatively unexploited dataset on the infor- mation sharing practices of agricultural traders in Madagascar, I nd support for this hypothesis, based on two alternative measures of liquidity. An important pol- icy implication of this result is that improved access to low-cost credit can foster the formation of information-sharing networks and mitigate problems of contract breach. I have limited attention to information sharing agreements between two rival rms in order to focus on the determinants of the strategic cost of information sharing. A natural extension of the research question would be to consider larger coalitions where each rm has one or more local rivals but does not face competition from the rest of the coalition members. In those cases the timing of information revelation becomes relevant because the informed rms may reveal information only after their rival has been exposed to the risk of cheating. Furthermore, as originally pointed out by Pagano and Jappelli (1993), an information sharing coalition can be a ?natural monopoly? { the returns from participating in the coalition increase as 155 more rms join in. These are two important features that are outside the scope of this study but merit further investigation within the context of the model at hand. Another important component of this line of research is to perform a more rigorous test of the causal e ect of competition intensity on information sharing. The lack of suitable instruments limits my ability to establish causality using obser- vational data. The developments in the literature on information sharing in credit markets clearly show that empirical work in this area lags behind its theoretical counterpart. Brown and Zehnder (2008) make a signi cant contribution to this area by using an experimental setting that allows them to distinguish the impact of competition from the impact of default risk on lenders? incentives to pool infor- mation. The study of information sharing among rms can bene t tremendously from a similar experimental approach, given the paucity of observational data on information-sharing practices. 156 Appendix A Appendix to Chapter 2 A.1 Proofs A.1.1 Proof of Proposition 1 Proof. Using the speci cation of the price strategies and value functions in terms of the state variable, xt, we nd the optimal price strategies, pnt, and the resulting distribution of the market. We then show that when consumers are very patient prices are independent of rms? market shares and the market converges to the proposed equilibrium for all values of xt in just one period. We proceed to nd pot and the resulting mass of customers who switch in equilibrium. Finally, we determine the minimum value of exit costs that ensure that the proposed prices constitute an equilibrium in pure-strategies and show that within xm;1 xm neither rm has a pro table deviation given the proposed equilibrium prices. Without loss of generality, in the exposition that follows we normalize marginal costs to zero. 157 First we specify the rms? problem in period t by assuming that xt is in some middle range, (xm;1 xm), such that both rms poach: qij > 0. The relevant value functions are given by: VA (xt) = max pAnt max +s+pBnt 2 8 ; +s+pB nt 2txt x t ! (A.1.1) +pAnt (1 c) + ca+pBnt pAnt 2 2 c cb + 3 s pAnt 4 xt + f + (1 c) + ca+p B nt p A nt 2 2 c cb + (1 c) + ca+pBnt pAnt 2 2 c cb 2! VB (xt) = max pBnt max +s+pAnt 2 8 ; s+pA nt + 2 xt (1 x t) ! (A.1.2) +pBnt (1 c) ca cb+pAnt pBnt 2 2 c cb xt +s+pBnt 4 + f + (1 c) ca cb+p A nt p B nt 2 2 c cb + (1 c) ca cb+pAnt pBnt 2 2 c cb 2! The corresponding best response functions are: 158 pAnt(pBnt) = 8 (2 2 c cb) + 2(2 2 c cb)2 8 f 1 (A.1.3) h (4 (2 2 c cb) 8 f ) (1 c) + ca+pBnt + (2 2 c cb)2(3 s 4 xt) 4 (2 2 c cb) f i pBnt(pAnt) = 8 (2 2 c cb) + 2(2 2 c cb)2 8 f 1 (A.1.4) h (4 (2 2 c cb) 8 f ) (1 c) ca cb+pAnt + (2 2 c cb)2(4 xt s) 4 (2 2 c cb) f i The optimal price strategies are given by: 159 pAnt = 1D " 2(6 2 c cb)(2 2 c cb) 8 f (A.1.5) (2 2 c cb) h 10 2(1 c) + 4 ca 3 cb 4 (2 2 c cb)xt i 4 f (2 2 c cb) (2 2 c cb)2s 8 f ( (1 c) + ca) ! + 4 (2 2 c cb) 8 f (2 2 c cb) h 2 2(1 c) 4 ca 3 cb + 4 (2 2 c cb)xt i 4 f (2 2 c cb) (2 2 c cb)2s 8 f ( (1 c) ca cb) !# 160 pBnt = 1D " 2(6 2 c cb)(2 2 c cb) 8 f (A.1.6) (2 2 c cb)[2 2(1 c) 4 ca 3 cb + 4 (2 2 c cb)xt] 4 f (2 2 c cb) (2 2 c cb)2s 8 f ( (1 c) ca cb) ! + 4 (2 2 c cb) 8 f (2 2 c cb) h 10 2(1 c) + 4 ca 3 cb 4 (2 2 c cb)xt i 4 f (2 2 c cb) (2 2 c cb)2s 8 f ( (1 c) + ca) !# where D = ((16 4 c 2 cb)(2 2 c cb) 16 f ) (2(4 2 c cb)(2 2 c cb)) Using pAnt pBnt = a + bxt, we can derive a and b based on the optimal price strategies shown above. We nd that b = 8 F (2 2 c cb)2 (A.1.7) a = 8 F (( (1 c) + ca)(2 2 c cb) f (2 ca+ cb)) (A.1.8) 161 where F = ((16 4 c 2 cb)(2 2 c cb) 16 f ). From (A.1.7) and (A.1.8) we can establish that a = b=2 (A.1.9) Using pAnt = e+fxt we can also derive e and f as functions of b: f = 2 (2 2 c cb) 2 (8 2 c cb)(2 2 c cb) 8 f = b 2 (A.1.10) e = 2 2(1 c) + 4 (1 c) 3 cb 4 f 4 f (2 2 c cb)s 2(4 2 c cb) b 4 (A.1.11) To identify ; ; we use the prespeci ed quadratic functions for VA(xt) and VB(xt) and the a ne functions for pAnt and pBnt: + xt + x2t (A.1.12) = ( +s+e a+ (f b)xt) 2 8 + e+fxt 4 (2 2 c cb) 10 2(1 c) + 4 ca 3 cb (2 2 c cb)s 4 (2 2 c cb)xt + 4 (e a+ (f b)xt) (6 2 c cb)(e+fxt) ! + f + (1 c) + ca a bxt2 2 c cb + (1 c) + ca a bxt 2 2 c cb 2! Rearranging and matching the terms, we obtain the expressions for and : 162 = b 2(2 2 c cb)(18 2 c cb) + 16 b(2 2 c cb)2 32 ((2 2 c cb)2 fb2) (A.1.13) =( +s+e a)(f b)4 (A.1.14) + e( 4 (2 2 c cb) + 4 (f b) f(6 2 c cb))4 (2 2 c cb) + f (10 2(1 c) + 4 ca 3 cb (2 2 c cb)s) 4 (2 2 c cb) + f (4 (e a) e(6 2 c cb))4 (2 2 c cb) + fb 2 2 c cb + 2b( (1 c) + ca a)(2 2 c cb)2 Plugging in (A.1.7) we can characterize b: [4b(8 2 c cb) + 16 (2 2 c cb)][(2 2 c cb)2 fb2] + fb3(18 2 c cb) + 16 fb2(2 2 c cb) = 0 (A.1.15) This equation is independent of s and if we normalize = 1, it is equivalent to the one in Villas-Boas (1999). 1 Following the approach in Villas-Boas?s paper, we let y = (2 2 c cb)=b and rewrite (A.1.15) in terms of y. For c! 1 this equation reduces to 2y3 + 3y2 f = 0 (A.1.16) For f 2(0;1) the equation above has three roots in the intervals ( 3=2;1); ( 1;0) and (0;1=2). The appropriate solution must also satisfy the second-order conditions 1Equation A15 on pg. 626 163 for each rm, which are: 8 (2 2 c cb) 2(2 2 c cb)2 + 8 f < 0 (A.1.17) Rewriting this expression in terms of C = (2 2 c cb), we note that the second- order conditions are satis ed when < (4 C + C2)=(4 f). Rewriting (A.1.13) in terms of C, we get = (b2C(16 +C) + 16 C2b)=(32 (C2 fb2)). Then, we can show that the condition < (4 C + C2)=(4 f) converges to 2y2 + fy f > 0 as c! 1. Among the three possible ranges for y, the second-order conditions are satis ed only when y2( 3=2; 1). Also, y increases in f and y! 1 as f !1. From y 2 ( 3=2; 1) we can see that for c ! 1 the coe cient b must be negative. Furthermore, @b=@ c < 0, and b!0 as c!1. Using the fact that b ! 0 as c ! 1, it can also be seen that a; e; f; and all converge to zero as well. Therefore, pnt converges to zero or marginal cost whenever c > 0. Obtaining the results for pot; ^qii and qij is straightforward after substituting pint in the appropriate expressions. We just note that when c> 0, piot(pjnt) = (c+ +s+pjnt)=2 and ^qii(pjnt) = (c+ +s+pjnt)=(4 ). From a = b=2 and xt+1 = ( (1 c) + ca+pBnt pAnt)=(2 2 c cb) we can see that convergence to steady state occurs according to xt+1 12 = b2 2 c cb xt 12 (A.1.18) Since b=(2 2 c cb) = y 1 and y2 ( 3=2; 1), it is clear that convergence 164 is monotonic and becomes in nitely slow when f !1. We now nd the limits of the poaching region (xm;1 xm) by looking for the range of xt, within which neither rm has a pro table deviation in a price strategy that does not attract the rival?s previous customers. Normalizing marginal cost to zero again, consider deviations for rm A such that it does not poach (qBA = 0). Therefore, the introductory price is given by pAnt = arg maxpAnt (1 c) + ca+p B nt p A nt 2 2 c cb + f (A.1.19) + f (1 c) + ca+p B nt p A nt 2 2 c cb + f (1 c) + ca+pBnt pAnt 2 2 c cb 2 The rst-order condition is (2 2 c cb)( (1 c) + ca+pBnt 2pAnt) 2 2 c cb (A.1.20) f 2 2 c cb 2 f (1 c) + ca+pBnt pAnt 2 2 c cb = 0 Firm A?s best response function is pAnt(pBnt) = 2 2 c cb 2 f 2 (2 2 c cb f ) (1 c) + ca+pBnt (A.1.21) 2 2 c cb2 (2 2 c cb f ) f 165 Therefore, rm A?s deviation payo is given by VA(pAnt) =pAnt (1 c) + ca+p B nt p A nt 2 2 c cb + f (A.1.22) + f (1 c) + ca+p B nt p A nt 2 2 c cb + f (1 c) + ca+pBnt pAnt 2 2 c cb 2 The payo from poaching is VA(pAnt) =pAnt (1 c) + ca+pBnt pAnt 2 2 c cb + 3 s pAnt 4 xt (A.1.23) + f + f (1 c) + ca+p B nt p A nt 2 2 c cb + f (1 c) + ca+pBnt pAnt 2 2 c cb 2 Firm A will not deviate if xt is such that the payo from poaching is weakly greater than the deviation payo , i.e. VA(pAnt) VA(pAnt). After some regrouping and canceling of common terms this inequality reduces to (pAnt pAnt) (1 c) + ca+p B nt 2 2 c cb (pAnt)2 2 2 c cb (A.1.24) +pAnt 3 s 4 xt4 (p A nt) 2 4 f pAnt 2 2 c cb 2 f ( (1 c) + ca+p B nt)(p A nt p A nt) (2 2 c cb)2 + f (pAnt)2 (2 2 c cb)2 (p A nt) 2 2 2 c cb f pAnt 2 2 c cb + f (pAnt)2 (2 2 c cb)2 166 Using C = 2 2 c cb and plugging in the best response functions pAnt(pBnt) = (4 C 8 f )( (1 c) + ca+p B nt) 4 C f +C 2(3 s 4 xt) 2 (4 C +C2 4 f ) (A.1.25) and pAnt(pBnt) = (C 2 f )( (1 c) + ca+p B nt) C f 2(C 2 f ) (A.1.26) the inequality VA(pAnt) VA(pAnt) simpli es to (4 C +C2 4 f )(pAnt)2 (4 C 4 f )(pAnt)2 (A.1.27) From the second-order conditions in (A.1.17) we can see that 4 C + C2 4 f > 0 and, therefore, 4 C 4 f > 0 as well. Thus, the inequality above can be written as (pAnt)p4 C +C2 4 f pAntp4 C 4 f . Let M = 4 C 4 f . We can rewrite pAnt in terms of pAnt: pAnt = M +C 2 M p A nt C2(3 s 4 xt 2M (A.1.28) 167 Using pAnt = e+fxt, (A.1.27) can be stated as pM +C2(e+fx t) pM M +C2 M (e+fxt) C2(3 s 4 xt 2M (A.1.29) from which we obtain the critical value for xt such that rm A is strictly better o engaging in poaching: xt 1 xm = 2 M +C2 pM(M +C2) e C2(3 s) 2 M +C2 pM(M +C2) f 4 C2 (A.1.30) Thus, when xt 1 xm poaching is an equilibrium strategy for rm A and, by symmetry, when xt xm poaching is an equilibrium strategy for rm B. As c!1, e!0 and f!0, so 1 xm!(3 s)=(4 ). Note that xm!( +s)=(4 ) which is equal to the equilibrium level of ^qAA. We also derive an alternative expression for 1 xm because it will allow us to sign its derivative with respect to s, which is one of the results stated in Corollary 1. First, using pBnt = e a+ (f b)xt = (2e+b bxt)=b and rewriting pAnt in terms of C we obtain pAnt = C(C 2 f ) + (C 2 f )(2e+b bxt) 2 f C4C 4 f (A.1.31) Rewriting again in terms by using M = 4 C 4 f pAnt = C(C 2 f 2 f ) + (C 2 f )(2e+b bxt)M= (A.1.32) 168 Then, (4 C +C2 4 f )(pAnt)2 (4 C 4 f )(pAnt)2 becomes pM +C2 e+ b 2xt (A.1.33) pM C(C 2 f 2 f ) + (C 2 f )(2e+b bxt) M= which is satis ed for xt 1 xm = C(C 2 f 2 f ) + (C 2 f )(2e+b) 2Web[W + (C 2 f )] (A.1.34) where W = pM(M +C2)=2. There are two implicit assumptions used in the derivation of xm. First, in setting up VA(pAnt) VA(pAnt) we assume that qBA(pAnt) > 0, which is true for 3 s (e+fxt) 4 xt 4 > 0. This condition is satis ed for xt pBn^t while pBn^t!0. From Lemma 1 we know that for c ! 1 a rm that sets a price above the rival?s price does not make any sales to newcomers so x^t+1 = 0. Therefore, rm A will set pAn^t so as to maximize pro ts from switchers in period ^t: pAn^t = arg maxpAn^t 3 s p A n^t 4 (A.1.37) =3 s2 (A.1.38) At this price, sales to switchers equal qBA;^t = (3 s)=(8 ) and the maximum pro t from poaching in period ^t equals ^ A^t = (3 s)2 16 (A.1.39) The net present value of rm A?s deviation is (3 s)2 16 E + 1X =2 f (A.1.40) whereE stands for exit costs and indicates the per-period level of pro ts within the 170 poaching region, which is equal to ( +s)2=(8 ). We also use to designate pro ts in the period when the market transitions from (xs;xm) to (xm;1 xm) because we do not have a straightforward expression for pro ts when xt2(xs;xm) while provides an upper bound on these pro ts, which slightly strengthens the minimum required value of exit costs. Comparing the payo from a one-time deviation to the payo from staying on the equilibrium path we can obtain the minimum level of exit costs that guarantees that rm A has no pro table deviation in not selling to newcomers: (3 s)2 16 E + 1X =2 f 1X =1 f (A.1.41) E E = (3 s) 2 16 f ( +s)2 8 (A.1.42) A.1.2 Proof of Corollary 1 Proof. 1. @pin=@s< 0 for c < 1 and @pin=@s = 0 for c = 1 We will show the proof for pAnt. Recall that pAnt = e + fxt and f = b=2. From (A.1.15) we can see that b is independent of s and, therefore, f is independent of s. Therefore, to sign @pAnt=@s we need to nd @e=@s. Note that e is also a function of and the latter depends on s as well. In the proof of Proposition 1 we show that (A.1.14) and (A.1.11) jointly characterize and e. We now explicitly solve for and e and di erentiate the resulting expression for e with 171 respect to s.2 We obtain @e=@s = 4C cb 4C fb+ 3 f cb2 + 8 C 8 cCb 8 C c (A.1.43) 22 fb+ 14 f cb+ 16 2 16 2 c 16 2 f + 16 2 c f 1 2( cb+C fb f cb2 2 C + 2 cC + 2 fb 2 c fb) and @e=@s! (C fb)4 (1 f) (A.1.44) as c!1. Note that C fb> 0 since b< 0 and, therefore,@e=@s< 0. In the limit @e=@s!0, since b!0 and C!0. 2. @pio=@s > 0 Within the poaching region, rm A?s optimal regular price is de ned by pAot(pBnt) = +s+pBnt2 . Di erentiating with respect to s, we obtain @pAot=@s =1 +@e=@s2 (A.1.45) Since @e=@s!0 , we conclude that for c!1; @pAot=@s> 0. 3. @^qii=@s > 0 and ^qii ! 12 as s ! The rst part of this statement is a straightforward derivation of @^qii=@s from (2.2.13) and using the fact that @e=@s!0. 2The explicit solutions for and e were found using Mathematica 7.0. We do not include these solutions here because of their length. Files containing the exact solutions for and e are available upon request. 172 To show that ^qii ! 12 as s ! , note that as pn ! 0(or marginal cost), ^qii!( +s)=4 which clearly converges to 1=2 as s! . 4. @xm=@s> 0 and xm! 12 as s! To sign the derivate @xm=@s we use (A.1.34). @(1 xm) @s = 2 C f(@ @s) + 2 (C 2 f )(@e@s) 2W(@e@s) b[W + (C 2 f )] (A.1.46) The denominator of this expression is negative because b < 0 while W > 0 and C 2 f > 0. We will now show that for c!1 the numerator of @(1 xm)=@s is positive. From (A.1.11) we can derive @e @s = 8 f(@ @s) 2C 4(2 +C) (A.1.47) which produces @ @s = 2(C + 2 ) 4 f @e @s C 4 f (A.1.48) Plugging this expression into the numerator and using @e=@s ! (C fb)=(4 (1 f)), the numerator of @(1 xm)=@s above can be expressed as 173 1 2 (2C2 + 8 C 8 f 4W) (C fb) 4 (1 f) +C2 (A.1.49) = 14 (1 f) C2(C fb) (M 2W)(C fb) + 2 C2(1 f) ! 14 (1 f) (M 2W)(C fb) + 2 C2(1 f) This expression is positive since M 2W < 0 and C fb > 0. Therefore, the numerator of @(1 xm)=@s is positive. Combined with the fact that the denominator is negative, we obtain that @(1 xm)=@s< 0, or @xm=@s> 0. Showing that xm! 12 as s! is a straightforward application of the result that xm!( +s)=(4 ) as c!1. A.1.3 Proof of Corollary 2 Proof. Corollary 2 states that if switching costs are su ciently high, the no-poaching region extends to the entire market, so the result from Proposition 2 regarding convergence to steady state applies automatically to all xt2[0;1]. First, note that for (exm;1 exm) to extend to [0;1], it must be that ^qii(pjnt) = 1 and no deviations are pro table within (1 ^qBB; ^qAA). The rst condition holds when s (2 + c + 2 f) (A.1.50) The second condition is satis ed if x = 0, or pAnt 0, or pAnt 0. First, if 174 x = 0, then from (A.1.69) poaching is not feasible. In terms of s, x = 0 whenever s s1 = (11 + 8 f)(1 + f) 10 c(1 + f) 2 c(1 3 f) (5 + 2c)(1 + f) 2 c(3 + f) (A.1.51) Second, pAnt 0 implies that poaching is not pro table so rm A will not deviate. This condition is satis ed when s s2 = 7(1 c) 2 + f(1 c)(11 3 c) + 4 2 f 4(1 + f)(1 c) 2xt (1 + f)(1 + 4 f + 2c) 2 c(1 + 3 f) (A.1.52) Finally, pAnt 0 ensures that the deviation price must be below zero in order for rm A to attract a positive mass of switchers and this condition is satis ed when s s3 = 1 + f c f (A.1.53) Therefore, combining (A.1.50), (A.1.51), (A.1.52) and (A.1.53) we obtain suf- cient conditions for (exm;1 exm) to cover the whole market: s max((2 + c + 2 f) ;min(s1;s2;s3)) (A.1.54) 175 A.1.4 Proof of Proposition 2 Proof. The rms? value functions within the no-poaching region were presented in (2.3.26) and (2.3.27). Taking the rst-order conditions, we obtain the following best response functions: pAnt(pBnt) = 2 2 c cb 2 f 2(2 2 c cb f ) (1 c) + ca+pBnt (A.1.55) 2 2 c cb2(2 2 c cb f ) f pBnt(pAnt) = 2 2 c cb 2 f 2(2 2 c cb f ) (1 c) ca cb+pAnt (A.1.56) 2 2 c cb2(2 2 c cb f ) f Let A = 2 2 c cb 2 f ; B = 2 2 c cb f and C = 2 2 c cb. We solve for the optimal price strategies and express them in terms of these A;B and C: pAnt = 1 A 2 (2B)2 1 A2 (2B)2 ( (1 c) ca cb) A C (2B)2 f (A.1.57) + A2B( (1 c) + ca) C2B f ! pBnt = 1 A 2 (2B)2 1 A2 (2B)2 ( (1 c) + ca) A C (2B)2 f (A.1.58) + A2B( (1 c) ca cb) C2B f ! 176 Note that the optimal price strategies are independent of xt. Applying pAnt pBnt = a+bxt we see that b = 0 for all c, which also leads to a = 0: pAnt pBnt = 1 A 2 (2B)2 1 A 2B 1 A 2B (2 ca+ cb) ! + 0 xt (A.1.59) = a+bxt (A.1.60) From a = 2A c2B+Aa, we obtain that a = 0. Therefore, in equilibrium pAnt = pBnt. Applying a = b = 0, we can match the expression for the optimal pAnt with e+fxt and identify e: pAnt =[2B A] 1 (A (1 c) C f ) (A.1.61) = e + 0 xt where now A = 2 (1 c) + 4 f ; B = 2 (1 c) + 2 f and C = 2 (1 c). Since a = b = 0, we also have pBnt = e. To nd e, we match the coe cients in the value functions VA(xt) = + xt + x2t (A.1.62) =( +s+e 2 xt)xt +e12 + f + 12 + 1 2 2! 177 from which we obtain = +s+e and = 2 . Substituting for in e, e =(2 (1 c) + 4 f ) (1 c) 2 (1 c) f( +s+e)2 (1 c) e =(1 c) f(s )1 + f (A.1.63) Therefore, pAnt = pBnt = ((1 c) f(s ))=(1 + f) and = ((2 + f c) + s)=(1 + f). From the equality of the rms? prices, we obtain that xt+1 = 1=2, which is also the steady-state distribution of the market since it falls within the no-poaching region where rms always set pAnt = pBnt. Having found the optimal pnt, it is straightforward to show that indeed ^qii > 1=2 when s> c and ^qii = 1 when s (2 + 2 f + c) since ^qii = +s+p i nt 4 (A.1.64) =(2(1 + f) c) +s4 (1 + f) (A.1.65) Hence, for s> c our conjecture that there is complete customer lock-in for some xt close to the middle is correct. We now nd the limits of the no-poaching region, (exm;1 exm), which are determined by the values of xt that guarantee that neither rm has a pro table deviation in a strategy that involves poaching. Suppose that rm A starts period t with a relatively low market share and considers deviating in period t by selecting a price pAnt > 0 such that qBA(pAnt) > 0. From the proof of Proposition 1 we know that when rm A intends to poach its best response function is given by (A.1.3). 178 Applying a = b = 0 and using C = 2 (1 c) , we can write the optimal deviating price as pAnt(pBnt) =[8 C + 2C2 8 f ] 1 4 (C 2 f )( (1 c) +pBnt) (A.1.66) C2(4 xt +s 3 ) 4 C f The proposed equilibrium price for rm A is given by pAnt(pBnt) = [2(C f )] 1 (C f )( (1 c) +pBnt) C f (A.1.67) and we can express pAnt in terms of pAnt: pAnt = 8 C 8 f 8 C + 2C2 8 f pAnt + C 2 8 C + 2C2 8 f (3 s 4 xt) (A.1.68) Note that as c!1, C!0 and, therefore, pAnt!pAnt - the optimal deviation price converges to the equilibrium price and we have shown that qBA(pAnt) = 0. However, when consumers are not very patient, a deviation to a poaching strategy is feasible when qBA(pAnt) > 0. De ning M = 4 C 4 f we can restate this requirement in terms of (A.1.68) and identify the highest level of rm A?s market share that would render poaching feasible: xt pAnt and since qBA is decreasing in pAnt, qBA(pAnt) = 0. Therefore, it is reasonable to consider deviations to poaching only when pAnt 0. Note that if ^xm 1 ^qBB, rm A will have no pro table deviations within the no-poaching region. On the other hand if ^xm > 1 ^qBB, then (^xm;1 ^xm) will de ne the boundaries of the no-poaching region. For this reason, we state that exm = max(0; 1 ^qBB; ^xm). 180 Finally, we check that the second-order conditions are satis ed for the coe - cients found above: 2(2 2 c cb f )2 2 c cb < 0 (A.1.72) 2(2 2 c f( 2 ))2 2 c < 0 2(1 c + f) 1 c < 0 8 f > 0; c > 0 A.1.5 Proof of Proposition 3 Proof. Using (2.3.51) and (2.3.52) we derive the best response functions for each rm: pAnt(pBnt) = 2 2 c cb 2 f 2(2 2 c cb f ) (1 c) + ca+pBnt (A.1.73) 2 2 c cb2(2 2 c cb f ) f pBnt(pAnt) = 8 (2 2 c cb) + 2(2 2 c cb)2 8 f 1 (A.1.74) h (4 (2 2 c cb) 8 f ) (1 c) ca cb+pAnt + (2 2 c cb)2(4 xt s) 4 (2 2 c cb) f i Note that because we assume that xt+1 2(exm;1 exm), next period prices are formed according to the a ne functions that we speci ed for the no-poaching region. 181 Therefore, we can apply a = b = 0; = ((2 + f c) +s)=(1 + f); = 2 , in which case the best response functions above fully determine the optimal values of pAnt and pBnt. In particular, pAnt =(7 2 3 c + 13 f + 6 2 f + 2 c(11 + f) + 2 c( 8 7 f + 2 f)) 2(1 + f)(5 + 2 2c + 6 f c(7 + 2 f) (A.1.75) (1 + 11 f + 10 2 f + 2 c(1 + 3 f) 2 c(1 + 7 f + 2 f))s 2(1 + f)(5 + 2 2c + 6 f c(7 + 2 f)) + 2(1 c)(1 c + 2 f) 5 + 2 2 c + 6 f c(7 + 2 f) xt pBnt = (c4 + 4 f 2f) 2 2c(2 f) 5 f 3 2f (1 + f)(5 + 2 2c + 6 f c(7 + 2 f)) (A.1.76) + 1 + 5 f + 5 2f + 2c(2 + 6 f + 2f) s (1 + f)(5 + 2 2c + 6 f c(7 + 2 f) + 4(1 c)(1 c + 2 f) 5 + 2 2 c + 6 f c(7 + 2 f) xt We now nd the range of xt such that, conditional on pAnt as described in (A.1.76), rm B does not have a pro table deviation in choosing an introductory price such that it does not attract any switchers. Let pBnt stand for the optimal deviation price, which is given by: pBnt = 2 2 c cb 2 f 2(2 2 c cb f ) (1 c) ca cb+pAnt (A.1.77) 2 2 c cb2(2 2 c cb f ) f 182 Checking that VB(pBnt) VB(pBnt) reduces to (M +C2)(pBnt)2 M(pBnt)2 (A.1.78) which implies that rm B will not deviate to a no-poaching strategy if xt 1 exs = +s4 + pM(M +C2) M 4 C2pM(M +C2)p B nt (A.1.79) Note that if 1 exs < 1 exm, then the region, in which B poaches in equilibrium while A does not is de ned by (1 exm;1). On the other hand, if 1 exs > 1 exm, there are no pure-strategy equilibria in (1 exs;1 exm) { if the rms follow the strategies prescribed in (A.1.76) and (A.1.77), B has a pro table deviation in selecting a price such that it does not poach. If B does not poach, then A?s best response is described by Proposition 2. However, as soon as A chooses to follow this price strategy, B has a pro table deviation in poaching. By similar arguments we can identify an equilibrium in the case xt2(0;exm), where rm A has a pro table deviation in poaching. 183 Appendix B Appendix to Chapter 3 B.1 Proofs B.1.1 Proof of Proposition 1. Proof. The set-up of the problem is identical to the one discussed in Section 2.3.1 of Chapter 2. Recall that the rms? optimal price strategies are expected to depend on the state variable, xt, and we de ned the rms? price strategies and value functions as a ne and quadratic functions in xt, respectively. In the Proof of Proposition 1 in Appendix A, where we have shown that the undetermined coe cients characterizing these functions, a; e; f; and converge to zero as c!1 and the rms? optimal price strategies become independent of the market share. In the limit, c = 1, we have that a = b = 0, which immediately implies that pAnt = pBnt regardless of the current distribution of the market. From e = f = 0, we also see that pAnt = pBnt = 0, or marginal cost if c> 0. Throughout the rest of the proof I assume that marginal 184 cost is positive so that pAn = pBn = c. At this point we assume that in case of a tie the market is evenly split. Since prices are independent of market share, the equilibrium where pAn = pBn = c and xt = 1=2 is a steady state. In addition, from the Proof of Proposition 1 in Appendix A we have that in steady state: piot(pjnt = c) = (c+ +s+c)=2 = c+ +s2 (B.1.1) ^qii(pjnt = c) = ( +s+c c)=(4 ) = +s4 (B.1.2) qAB(pBnt = c) = 1=2 +s4 = s4 (B.1.3) qBA(pAnt = c) = 1=2 +s4 = s4 (B.1.4) (B.1.5) To show that neither rm has a pro table deviation from the proposed equi- librium price, pnt = c, we need the two lemmas stated in the text to describe market dynamics when c = 1. Lemma 1: Competition for newcomers intensi es as consumers become more patient. As a result, pin and pio fall as c goes up. Proof. Using the equilibrium result a = b=2 (see (A.1.9)) we can rewrite xt+1 185 as follows: xt+1 = (1 c) + ca+p B nt p A nt 2 2 c cb = 12 + p B nt p A nt 2 2 c cb (B.1.6) Let w = 1=(2 2 c cb) indicate the weight of the price di erential on the location of the marginal consumers, xt+1. On the equilibrium path b 0 and @b=@ c < 0, so @w=@ c > 0 - as consumer patience increases, the marginal newcomer becomes more sensitive to the di erence between the introductory prices o ered today. As a result, pn goes down. Since po is increasing in the rival?s pn, as introductory prices fall, regular prices fall as well. Lemma 2: When c = 1, demand from newcomers is perfectly elastic. Proof. First note that for c = 1, we have pAnt = pAnt+1 = pBnt+1. Designate this equilibrium price as p and consider a deviation price pAnt for rm A. The marginal newcomer will either switch next period or will stay with the current supplier. In both cases her consumption expenditure in the second period will be at least as high as the expenditure from switching. For example, if she purchases from rm A in her rst period she will switch if p+ (1 x) +s< p + x. If the consumer stays, she will be the marginal stayer and will be o ered a price such that she is just indi erent between switching and staying: p + (1 x) + s = p + x. In either case, in her second period the marginal consumer will spend at least p + (1 x) + s if she rst purchases from A 186 and p + x + s if she purchases from B. Hence, a newcomer located at x will purchase from A if: pAnt + x+ (p+ (1 x) +s) p+ x+ (p+ x+s) pAnt p It is clear that for pAnt > p, the inequality above cannot be satis ed for any x. Therefore, if A deviates to a price above the proposed equilibrium price, p, it will make no sales to newcomers. Similarly, if rm A o ers a price below p it will capture the entire market of newcomers. Hence, newcomers? demand becomes perfectly elastic when consumers are in nitely patient. Exit Costs Lemma 2 shows that if one rm deviates by raising its price above marginal cost, it forgoes sales to newcomers and starts next period with no market share. The only incentive for a rm to raise its price is to maximize pro ts from switchers. Pro ts from switchers would be greatest when the rm starts the period with no market share because of the higher mass of potential switchers. Suppose that in period ^t rm A starts with x^t = 0 and deviates from the proposed equilibrium by setting pAn^t > pBn^t while pBn^t ! 0. From Lemma 2 we see that at this price q1A = 0, so x^t+1 = 0. Therefore, rm A will set pAn^t so as to 187 maximize pro ts from switchers in period ^t: pAn^t = arg maxpAn^t 3 s p A n^t 4 (B.1.7) =3 s2 (B.1.8) At this price, sales to switchers equal qBA;^t = (3 s)=(8 ) and the maximum pro t from poaching in period ^t equals ^ A^t = (3 s)2 16 (B.1.9) The net present value of rm A?s deviation is (3 s)2 16 E + 1X =2 f (B.1.10) where E stands for exit costs (to be incurred at the end of the period in which the rm makes no sales to newcomers) and indicates the per-period level of pro ts within the poaching region, which is equal to ( + s)2=(8 ). Comparing the payo from a one-time deviation to the payo from staying on the equilibrium path we can obtain the minimum level of exit costs that guarantees that rm A has no pro table deviation in not selling to newcomers: (3 s)2 16 E + 1X =2 f 1X =1 f (B.1.11) E E = (3 s) 2 16 f ( +s)2 8 (B.1.12) 188 As long as E E rm A does not have a pro table deviation in raising its price above marginal cost. The same argument goes for rm B. Next, consider a deviation such that rm A undercuts the rival by setting a price just below marginal cost. Note from (B.1.2) that at pn = c, the optimal mass of loyal customers for each rm, ^qii(pjnt = c) = +s4 , does not exceed one half since we speci ed that s . Therefore, capturing the full market of newcomers in the preceding period brings no additional gains since neither rm keeps more than half of the market. In addition, sales to switchers are pro t-neutral as well (price equals marginal cost), implying that neither rm can derive additional gains from undercutting the rival. Hence, the rms do not have a pro table deviation in lowering pn below cost. From the arguments above it also becomes clear that any sharing rule can ensure a pure strategy equilibrium at marginal cost as long as the rms capture their respective segments of loyal customers. B.1.2 Proof of Corollary 1 Proof. From a = b=2 and xt+1 = ( (1 c) + ca + pBnt pAnt)=(2 2 c cb) we can see that convergence to steady state occurs according to xt+1 12 = b2 2 c cb xt 12 (B.1.13) Since b=(2 2 c cb) = y 1 and y 2 ( 3=2; 1), convergence is monotonic and becomes in nitely slow when f !1. 189 For c = 1, b = f = 0, which implies that rms? introductory prices are inde- pendent of market share and equal to each other (a = b = 0) { hence, convergence to xt+1 = 1=2 occurs in just one period starting from any initial distribution of the market, xt. B.1.3 Proof of Lemma 3. Proof. (a) From Lemma 2 we see that the demand from newcomers is perfectly elastic. Therefore, rm i can capture the entire market of newcomers by un- dercutting, i.e. setting pi nt = pjnt where ! 0. When pjnt > ci the payo from undercutting is strictly higher than the payo from matching the rival?s price because sales to newcomers double, the pro t margin on newcomers is pos- itive and the corresponding loss of revenue from sales to switchers is negligible as ! 0. The payo from matching the rival?s price is also higher than the payo from exceeding it because the latter strategy results in sales to switchers only and the presence of exit costs as outlined in Proposition 1 ensures that this strategy is strictly dominated. Finally, note that rm i may also choose to undercut by more than if pjnt exceeds the price that maximizes i?s joint pro ts from newcomers and switchers. (b) First note that when pjnt ci undercutting is costly for rm i and capturing mar- ket share beyond rm i?s loyal customer segment has no future value. Therefore, undercutting will yield a strictly lower payo than matching the rival?s price. Furthermore, when pjnt is su ciently low, the payo from investing in market 190 share falls below the payo from targeting switchers only at a price above cost. Therefore, rm i will raise its price above the rival?s when pjnt is su ciently low. B.1.4 Proof of Lemma 4. Proof. We de ne pi as the break-even price that yields rm i indi erent between competing in the newcomers market, in which case it sells to both newcomers and switchers, and targeting switchers only. The decision not to sell to newcomers takes into account the forgone pro ts from loyal customers next period, the exit costs to be incurred at the end of the current period and the possibly lower per-generation pro ts next period due to starting with no market share. If the market price falls below pi, rm i is better o targeting switchers only because the investment in market share would exceed the returns. Thus, pi represents the minimum price at which rm i would be willing to sell to newcomers. (a) By the de nition of pi, when pi pjnt rm i will compete for market share. Thus, it will either match or undercut the rival?s price. By Lemma 3 undercutting is a dominated strategy when pjnt ci so rm i?s maximizes its payo by setting pi nt = pjnt. (b) Similarly, when pi > pjnt rm i realizes a higher payo if it targets switchers only, which entails setting pint > ci (for sales to switchers to be pro table). Together with ci pjnt, we obtain that pi nt > pjnt. Note that even if ci is high enough to raise pi such as to suppress demand from switchers (qji = 0 when 191 pint cj + s), rm i is still strictly better o setting pint >pjnt, in which case it makes no sales to new customers. B.1.5 Proof of Proposition 2 Proof. (a) To show that pAnt = pBnt = p n, we will rst derive conditions, under which rm A does not have a pro table deviation from the proposed equilibrium. Let pBnt = p n. Since p n = c, which is less than ~c, rm A has no pro table deviation in undercutting rm B?s price as established in part (b) of Lemma 3. The only possible pro table deviation for rm A would be to raise its price above cost, which would imply that it makes no sales to newcomers (Lemma 2). Therefore, it will set pAnt with the objective of maximizing pro ts from switchers in the current period. Note that at this point the minimum level of exit costs, E is not su cient to deter A from targeting switchers only because the payo from investing in market share is lower than the one used in (B.1.11) to derive E. Under cost asymmetry, rm A?s demand from switchers is given by qBA(pAnt) = ( s pAnt + c)=(4 ). Since xt = 1=2, rm A cannot induce all of rm B?s previous customers to switch unless it lowers its price below c s, which is clearly not pro table. Therefore, the price that would maximize pro ts 192 from switchers, is given by max (pAnt ~c) s p A nt +c 4 (B.1.14) pAnt = s+c+ ~c2 (B.1.15) from which it can be seen that rm A?s optimal price that targets switchers only is greater than the rival?s current price, p n. If rm A sets pAnt = p n, it will capture half of the newcomers market because pAnt = pBnt, but it will also sell at a price below cost to both newcomers and switchers. However, if c is not too high, rm A will recuperate some of these losses next period when it sells to the loyal customers attracted today. On the other hand, if rm A sets pAnt > pBnt, it will make no sales to newcomers, will forgo pro ts from loyal customers next period and will incur exit costs at the end of period t. Since the loss per unit sold to new customers and the maximum pro t from targeting switchers only depend on c, we can nd the highest cost di erential, within which rm A?s stream of pro ts from matching B?s price and investing in market share is higher than its stream of pro ts if selling to switchers only. Since we assume the shock lasts one period only and this is common knowledge, both rms know that they will start next period with equal marginal cost. Therefore, pAnt+1 = pBnt+1 = c and rm A?s pro t from investing in market share today equals its steady-state level of pro t: = ( +s)2=(8 ). In period t+ 1 the market goes back to equilibrium so we only need to analyze rm A?s discounted payo over periods t and t+1. Firm A?s payo from setting 193 pAnt = pBnt when pBnt = c weakly dominates the payo from targeting switchers. (pAnt = pBnt) (pAnt >pBnt) (B.1.16) (c ~c) (q1A +qBA) + f ( +s) 2 8 + 1X =t+1 f (B.1.17) E + s+c+ ~c 2 ~c max s+c ~c 8 ; 1 2 + f 0 + 1X =t+1 f where ( s+c ~c)=(8 ) is the pro t-maximizing level of sales to switchers. At pAnt = pBnt = c rm A?s sales to switchers are given by qBA = ( s)=(4 ) as stated in Proposition 1. At pAnt = ( s + c + ~c)=2, demand from switchers is positive as long as ~c c< s. Therefore, if ~c c< s, equation (B.1.16) becomes (c ~c) 1 2 + s 4 + f ( +s) 2 8 (B.1.18) E + s+c ~c 2 s+c ~c 8 and can be rewritten as (~c c)2 + 2(5 s)(~c c) 2 f( +s)2 16 E + ( s)2 0 (B.1.19) Using c = ~c c, the positive root of (B.1.19) is given by c1 = q (5 s)2 + 2 f( +s)2 + 16 E ( s)2 (5 s) (B.1.20) 194 and the inequality from (B.1.18) is satis ed for c2 [0; c1]. Therefore, for c c1, the payo from maintaining pAnt = pBnt = c weakly dominates the payo from targeting switchers only by setting pAnt >pBnt. Alternatively, if ~c c s, then the condition in B.1.16 becomes (c ~c) 1 2 + s 4 + f ( +s)2 8 E (B.1.21) and is satis ed for c c2 = 4 3 s E + f( +s) 2 8 (B.1.22) Therefore, we have established that for c c where c = c1 if c< s and c = c2 if c s, rm A does not have an incentive to deviate to a higher introductory price despite its higher marginal cost. Proposition 1 has established that rm B does not have a pro table deviation when pAnt = c, either, so pAnt = pBnt = c is indeed an equilibrium in pure strategies. The rest of the statements in part (a) follow immediately since pAnt = pBnt = p n = c. I only note that rm A?s price to loyal customers adjusts upwards to re ect the rm?s higher marginal cost; as a result, its loyal customers segment 195 in period t shrinks (qAA;t(pBnt) p o qAA;t(pBnt) = +s+p B nt ~c 4 (B.1.24) = +s+c ~c4 ~c and suppose that rm A deviates from the steady- state equilibrium pAnt = pBnt = c by raising its price to ^pA. Then, rm B has an incentive to raise its price to some level above its own marginal cost but below ^pA, because it can still capture the entire market of newcomers and can sell to new customers at a price above cost. Let rm B?s best response to pAnt = ^pA be 197 given by pB nt (^pA) = ^pA , where 2(0; ^pA c). If pB nt (^pA) > ~c, by part (a) of Lemma 3 rm A has an incentive to undercut B?s price. If pB nt (^pA) ~c by part (a) of 4 rm A would match B?s price. Similarly, since A?s price is above B?s marginal cost, B has an incentive to undercut as well, and as a result the introductory price drops to pA. At this point, rm A sets pA and does not have an incentive to undercut B?s price, while B settles at pBnt < pA and captures the entire market of newcomers. However, at pBnt < pA, rmAmay have an incentive to raise its price to ^pA again, which triggers another round of undercutting. Therefore, without imposing additional conditions on pA, there will be no equilibrium in pure strategies when ~c c> c. Suppose that (pAnt;pBnt) = (pA; pA ) and let ~c> s+c. Therefore, qBA(pAnt = ~c) = 0 { rm A cannot sell to switchers at a pro t and therefore targeting switchers with any price above ~c is not a pro table deviation. Note, however, that qBA(pA) may still be positive, so rm A would be selling to switchers at a price below cost if it maintains pAnt = pA. Therefore, it would have a pro table deviation in raising its price to at least ~c to avoid costly sales to switchers. Then, rm B would also raise its price to just below A?s price and a round of undercutting will follow again. However, rm A will have no pro table deviation if qBA(pA) = 0, which is true for pA s+c (B.1.26) 198 Using (B.1.25), we can rewrite the above condition as ~c c c = s+c (B.1.27) which produces ~c c c+ s (B.1.28) Therefore, we have derived a su cient condition to ensure that at (pAnt;pBnt) = (pA; pA ), rm A has no pro table deviation. So far we have shown that when c > c, rm A?s best response to a price at or below pA is to set pA itself. On the other hand, rm B?s best response to pAnt = pA is given by min(^pB;pA ), where ^pB is the price that would optimize rm B?s pro ts from new customers, conditional on being below A?s price.1 Hence, for c> c there is an equilibrium in pure strategies, given by (pAnt;pBnt) = (pA;min(^pB;pA )), where !0. Pro ts: The result that B;t > follows immediately by noting that pA > p n, pBot > p o, qBB;t > ^q BB and q1B = 1: rm B realizes higher pro ts on both loyal and new customers. Speci cally, B?s pro t from loyal customers is given by ( +s+pA c)2=(8 ) for !0, which is greater than its steady-state pro t of 1^pBnt+1 = arg max(^pBnt c) 1+xt+1 3 s ^pBnt 4 s.t. ^p Bnt c. Next period, the market goes back to steady state so pAnt = pBnt = c, implying that B;t+1 = Firm A?s pro t from loyal customers in period t is given by A;t = ( +s+pA ~c)2=(8 ) for !0, which is less than its steady-state pro t from loyal customers since pA < ~c. Also, A;t+1 < follows from the fact that in period t rm A makes no sales to newcomers and does not have loyal customers in period t+ 1, hence A;t+1 = 0. B.1.6 Proof of Proposition 3 Proof. (a) When c c we see from Proposition 2 that pAn;2 = pBn;2 = c where (pAn;2;pBn;2) are the two rms? prices to new customers in period 2. Firm B?s optimal market share in period 2 is unchanged, ^qBB;2 = q BB, so any distribution of the market at time 1 that allows rm B to capture newcomers in the range (^qBB;2;1), i.e. its future loyal customers, will bring in the same pro t in period 2. Let (pAn;1;pn;1B) indicate the two rms? introductory prices in period 1. Suppose that pAn;1 = pBn;1 = c and the market sharing rule is such that B captures all newcomers in (1 qBB;2). Firm B does not have a pro table deviation away from pBn;1 = c because undercutting is costly without bringing in additional revenues, and setting a higher price forgoes next-period pro ts altogether. Now consider rm A?s motivation to deviate from pAn;1 = pBn;1 when pBn;1 = c. For the same reasons as rm B, rm A does not have a strictly pro table deviation 200 in raising its price. Also, rm A does not have a pro table deviation in lowering its price if it can capture all newcomers in the range (0;1 qBB;2). Under a price tie, a market sharing rule that allows rm A to sell to newcomers located within (0;1 ^qBB;1) is crucial to nding a pure-strategy equilibrium because it ensures that rm A will face no demand from switchers in period 2 when it sells below cost. If, instead, the current price strategy led to a distribution of the market such that x2 < ^qBB;2, then rm A would have a pro table deviation in undercutting in order to capture the full market and avoid costly sales to switchers next period. Hence, under a market sharing rule that splits the market at x2 = ^qBB;2 under a tie, pAn;1 = pBn;1 = c constitutes a Nash equilibrium in period 1. By the same argument as in the proof of Proposition 2, any price strategy pair such that pAn;1 = pBn;1 = p 0) and also a larger loyal customer segment, ^qBB;2 > ^q BB. On the other hand, rm A?s pro ts from loyal customers will be lower, and its optimal market share, ^qAA;2, will be lower as well (and possibly zero if the cost shock is very large). 201 Consider again pAn;1 = pBn;1 = c and suppose that ^qAA;2 + ^qBB;2 1. Under the same sharing rule and arguments used in part (a), pAn;1 = pBn;1 = c is a Nash equilibrium in period 1 as long as the rms split the market such that x2 = 1 ^qBB;2. As long as A and B capture their future loyal customer segments, ^qAA;2 and ^qBB;2 respectively, while rm A also captures all customers in the range (^qAA;2;1 ^qBB;2), then neither rm has an incentive to undercut or to raise its price above the rival?s price. The key condition here is that the rms? loyal customer segments do not overlap, i.e. it is necessary that ^qAA;2 + ^qBB;2 1. We can now show that this condition is always satis ed. Recall from Proposition 2 that rm B?s equilibrium price in period 2 is given by p n c. Hence, we can state that c c and conclude that in period 1, there exists a pure strategy equilibrium for all values of the cost shock. B.1.7 Proof of Proposition 4 Proof. The proof is a straightforward result of Propositions 2 and 3. De ne the strategic cost of information sharing for rm B as the net present value of the bene ts that it will forgo as a result of revealing information to its rival when the latter has revealed information as well. Conditional on rm A?s cost shock being small, i.e. c c, Propositions 2 and 3 demonstrate that rm B does not realize higher pro ts in period 1 (pBn;1 = p n) or period 2 (pBn;2 = p n) upon exposing rm A to a higher risk of default by withholding information. On the other hand, rm B realizes strictly higher pro ts when rm A?s cost of funds is su ciently high, c > c, since rm B sells at a price above cost to all new customers in period 204 2 and also generates higher pro ts from loyal customers. These pro ts are not competed away in period 1 as demonstrated by Proposition 3. B.1.8 Proof of Corollary 2 Proof. To sign the derivative of c with respect to s, we need to modify the expres- sion for c from (B.1.18) to re ect the fact that in period 2 rm A faces no demand from switchers. Therefore, we rewrite (B.1.18) as: F( c) = ( c)(12 + 0) + f ( +s) 2 8 +E s c 2 max s c 8 ;0 = 0 (B.1.36) Since qBA(pA) = 0, it follows that qBA = 0 for any price above pA. Therefore, max s c 8 ;0 = 0 and (B.1.36) becomes: F( c) = ( c)12 + f ( +s) 2 8 +E = 0 (B.1.37) I apply the Implicit Function Theorem with respect to (B.1.36) to obtain @ c=@s and @ c=@ : 205 @ c=@s = @F=@s@F=@ c (B.1.38) = 2 f( +s) 8 1=2 (B.1.39) = f( +s)2 (B.1.40) > 0 (B.1.41) @ c=@ = @F=@ @F=@ c (B.1.42) = f(2( +s)8 8( +s)2) (8 )2 1=2 (B.1.43) = f(16 2+16 s 8 2 16 s 8s2) (64 2 1=2 (B.1.44) = f 8( 2 s2) 32 2 (B.1.45) 0 since s (B.1.46) We can also obtain the cross-partial derivatives: Fs = F s = f 2s4 2 (B.1.47) 206 B.2 Tables 207 Table B.1: Variable De nitions. Variable Name De nition age traders?s age capital =1 if the trader operates in the capital city, 0 otherwise categ rm category; =1 if wholesaler, =2 if semi-wholesaler; =3 if retailer with a xed point of sale; =4 if retailer without a xed points of sale; =5 if assembler for the manufacturing sector; =6 if assembler for the own purposes discuss prices =1 if the trader shares information about (input or output) prices at least once a month, 0 otherwise discuss suppliers =1 if the trader shares information about supplier quality at least once a month, 0 otherwise HS education educational level of the trader; =1 if trader has at least high-school education, 0 otherwise family members =0 if the respondent has no family members with salary jobs; =1 if 1 2 with jobs family members with jobs; = 2 if > 2 family members with jobs rm age age of the rm: =1 if < 5 years, =2 if 5 10 years, =3 if > 10 years main product main product traded: 1-rice, 2-tapioca, 3-corn, 4-beans, 5-sweet potatoes, 6-peanuts product processing =1 if the trader processes the product as a secondary activity, 0 otherwise region geographic region: 1-Tana Hauts Plateaux, 2-Vakinankaratra, 3-Fianar Hauts Plateaux, 4-Fianar C^ote et falaise, 5-Majunga Plaines, 6-Majunga Hauts Plateaux sex =1 if trader is male, 0 otherwise sizecat rm size category: =1 if small, =2 if medium, =3 if large shares info =1 if the trader shares information about delinquent customers at least once a month, 0 otherwise strong competition =1 if the trader perceives the level of competition as strong, 0 otherwise telephone access =1 if the trader has access to a telephone, 0 otherwise traders known # other traders the respondent knows personally; =1 if the trader knows < 4 traders, =2 if 4 9 traders, and =3 if 10 or more traders Table B.2: Liquidity Scores Components. Liquidity Dummy Variables Liquidity Speci cation Used Score (LS) Range LS 1 can borrow from friends/family 0 to 5 has 1 3 family members/friends to borrow from has > 3 family members/friends to borrow from has 1 2 family members with jobs has > 2 family members with jobs LS 2 has formal savings 0 to 5 has informal savings has bank account has another source of income has overdraft facility 208 Table B.3: Frequency distribution of Own Liquidity Score 1. LS 1 Freq. Percent Cum. 0 5 1.63 1.63 1 16 5.21 6.84 2 82 26.71 33.55 3 90 29.32 62.87 4 82 26.71 89.58 5 32 10.42 100.00 Total 307 100 Table B.4: Frequency distribution of Own Liquidity Score 2. LS 2 Freq. Percent Cum. 0 81 26.56 26.56 1 114 37.38 63.93 2 83 27.21 91.15 3 27 8.85 100.00 Total 305 100 Table B.5: Frequency distribution of Average Liquidity Scores. Variable Obs Mean Std. Dev. Min Max Median Avg. Liq. Score 1 299 3.031 .352 1.5 4 3.018 Avg. Liq. Score 2 299 1.156 .320 0 3 1.222 209 Table B.6: Summary Statistics. IS stands for information sharing. Signi cance level of di erences in means across the two samples are indicated as follows: *** indicate signi cance at the 1% level, ** { signi cance at the 5% level, and * { signi cance at the 10% level. All No IS IS Variable Mean Obs. Mean Obs. Mean Obs. Information sharing (IS) 0.132 304 0 264 1 40 Age 38.96 300 38.96 260 39.86 40 Sex 0.587 300 0.585 260 0.6 40 HS education 0.046 304 0.042 264 0.075 40 Wholesaler 0.267 303 0.274 263 0.225 40 Semi-wholesaler 0.125 303 0.125 263 0.125 40 Retailer w/ xed selling point 0.436 303 0.418 263 0.55 40 Retailer w/o xed selling point 0.036 303 0.042 263 0 40 Assembler manufacturing 0.026 303 0.027 263 0.025 40 Assembler private use 0.106 303 0.11 263 0.075 40 Assembler hired 0.003 303 0.004 263 0 40 Firm age: < 5 yrs 0.322 304 0.333 264 0.25 40 Firm age: 5 10 yrs 0.497 304 0.492 264 0.525 40 Firm age: > 10 yrs 0.181 304 0.174 264 0.225 40 Main Product: Rice 0.75 304 0.784 264 0.525 40 *** Main Product: Tapioca 0.046 304 0.045 264 0.05 40 Main Product: Corn 0.016 304 0.015 264 0.025 40 Main Product: Beans 0.118 304 0.102 264 0.225 40 * Main Product: Potatoes 0.026 304 0.023 264 0.05 40 Main Product: Peanuts 0.043 304 0.03 264 0.125 40 ** Traders known: < 4 0.08 303 0.072 263 0.1 40 Traders known: 4 9 0.396 303 0.384 263 0.475 40 Traders known: > 9 0.528 303 0.544 263 0.425 40 Small rm 0.15 301 0.169 261 0.025 40 * Medium-sized rm 0.342 301 0.326 261 0.45 40 Large rm 0.508 301 0.506 261 0.525 40 Capital region 0.214 304 0.2 264 0.325 40 Region: Tana Hauts Plateaux 0.248 303 0.232 263 0.35 40 Vakinankaratra 0.274 303 0.289 263 0.175 40 Fianar Hauts Plateaux 0.281 303 0.278 263 0.3 40 Fianar Cte et falaise 0.149 303 0.144 263 0.175 40 Majunga Plaines 0.013 303 0.015 263 0 40 Majunga Hauts Plateaux 0.036 303 0.042 263 0 40 Telephone access 0.508 303 0.513 263 0.475 40 Discusses suppliers 0.244 303 0.205 263 0.5 40 *** Discusses prices 0.322 304 0.25 264 0.8 40 *** Processing products 0.083 303 0.087 263 0.05 40 % sales to reg. customers 36.753 304 37.682 264 30.625 40 % credit sales 30.987 304 31.64 264 26.675 40 Strong competition 0.759 303 0.745 263 0.85 40 Liquidity score 1 3.049 304 3.011 264 3.3 40 Liquidity score 2 1.185 302 1.156 262 1.375 40 210 Table B.7: Controls. Group Control variables Market and Demographic Characteristics trader?s age education sex rm age rm size trade category product region number of traders known capital telephone Competition strong competition log(% of sales to regular clients) product processing Additional controls log(% of credit sales) discusses suppliers discusses prices 211 Table B.8: Determinants of Competition Intensity. Probit estimates; the dependent variable is a dummy equal to one if the trader reports strong market competition. All regressions control for the market and demographic characteristics listed in Table B.7. Coe cients reported, robust standard errors in parentheses. *** indicate signi cance at the 1% level, ** { signi cance at the 5% level, and * { signi cance at the 10% level. Dep. Variable: (1) (2) (3) (4) (5) (6) (7) Strong Competition HS education -1.285*** -1.309*** -1.299*** -1.278*** -1.249*** -1.339*** -1.450*** (0.434) (0.430) (0.437) (0.417) (0.451) (0.431) (0.433) Processing -0.629 -0.629 -0.633 -0.619 -0.552 -0.723* -0.693 (0.422) (0.423) (0.426) (0.421) (0.418) (0.438) (0.437) Log(sales to -0.409** -0.373** -0.408** -0.421** -0.396** -0.441*** -0.435** regular customers) (0.165) (0.184) (0.165) (0.167) (0.162) (0.169) (0.194) Log(% credit sales) -0.071 -0.048 (0.175) (0.179) Discuss prices -0.027 0.016 (0.244) (0.254) Discuss suppliers 0.018 -0.099 (0.253) (0.279) Liquidity Score 1 = 2 -0.933* -0.926* (0.531) (0.476) Liquidity Score 1 = 3 -1.111** -1.194*** (0.510) (0.445) Liquidity Score 1 = 4 -0.964* -1.013** (0.536) (0.491) Liquidity Score 1 = 5 -0.821 -0.794 (0.565) (0.520) Liquidity Score 2 = 1 -0.216 (0.275) Liquidity Score 2 = 2 0.141 (0.332) Liquidity Score 2 = 3 -0.314 (0.385) 6 10 competitors -0.358 -0.605 (0.405) (0.423) 11 15 competitors 0.390 0.029 (0.979) (0.868) > 15 competitors 0.455 0.322 (0.687) (0.669) Unknown # competitors -0.127 -0.190 (0.246) (0.253) Observations 273 273 271 273 272 273 271 2 69.14 71.02 71.61 77.80 73.33 71.74 83.83 212 Table B.9: Probit estimates. All regressions control for the market and demographic characteristics listed in Table B.7. Coe cients reported, robust standard errors in parentheses. *** indicate signi cance at the 1% level, ** { signi cance at the 5% level, and * { signi cance at the 10% level. Dep. Var.: IS (1) (2) (3) (4) (5) (6) Strong Competition 0.478* 0.794** 0.789* 0.767* 0.925** 0.804** (0.280) (0.388) (0.428) (0.415) (0.385) (0.400) Semi-wholesaler 0.507 1.160** 1.401** 1.304** 1.338** 0.723 (0.375) (0.545) (0.601) (0.604) (0.591) (0.575) Retailer with 0.810*** 1.105** 1.392*** 1.361** 1.565*** 0.596 xed selling point (0.306) (0.520) (0.539) (0.617) (0.588) (0.536) Assembler manufacturing 0.674 -0.094 -0.326 0.207 0.232 -0.171 (0.589) (0.881) (0.888) (0.990) (0.891) (0.936) Assembler individual 0.414 0.797 1.185* 1.178 1.398* 0.396 (0.419) (0.663) (0.646) (0.764) (0.747) (0.705) Firm Age: 5 10 yrs 0.469 0.745* 0.682 0.825* 0.654 0.694 (0.295) (0.425) (0.443) (0.447) (0.421) (0.434) Firm Age: > 10 yrs 0.304 0.538 0.404 0.352 0.684 0.640 (0.334) (0.446) (0.452) (0.430) (0.463) (0.451) Medium-sized rm 1.456*** 2.232*** 2.381*** 2.732*** 2.396*** 2.309*** (0.533) (0.656) (0.633) (0.655) (0.615) (0.675) Large rm 1.073** 1.793*** 2.056*** 2.149*** 1.852*** 1.769** (0.520) (0.674) (0.672) (0.632) (0.627) (0.690) 4 9 traders known -0.338 -0.198 -0.123 -0.168 -0.068 -0.275 (0.447) (0.479) (0.444) (0.587) (0.527) (0.490) > 9 traders known -0.392 -0.863 -0.798* -0.738 -0.842 -0.979* (0.448) (0.530) (0.481) (0.616) (0.609) (0.563) Capital region 0.048 1.329 1.648* 2.097* 1.313 1.527 (0.553) (0.900) (0.979) (1.113) (0.966) (0.931) Telephone access -0.322 -0.366 -0.423 -0.379 -0.345 -0.381 (0.273) (0.304) (0.303) (0.356) (0.324) (0.313) Processing 0.599 0.897 0.974 0.683 0.504 (0.717) (0.689) (0.828) (0.750) (0.788) Log(% sales to -0.417 -0.281 -0.522* -0.697** -0.540* regular customers) (0.276) (0.290) (0.306) (0.306) (0.288) Discuss prices 2.050*** 2.168*** 2.346*** 2.291*** 2.090*** (0.369) (0.394) (0.406) (0.388) (0.382) Discuss suppliers 0.223 0.375 0.320 0.174 0.251 (0.338) (0.331) (0.364) (0.346) (0.335) Log(% credit sales) 0.226 0.038 0.208 0.465 0.378 (0.262) (0.260) (0.259) (0.301) (0.279) Liquidity Controls: Own Liq. Score 1 Y Own Liq. Score 2 Y Avg. Liq. Score 1 Y Avg. Liq. Score 2 Y Observations 277 253 253 253 253 253 2 39.75 99.76 125.46 111.20 93.84 98.91 213 Table B.10: Probit estimates of the probability of sharing information using Liquid- ity Score 1. All regressions control for the market and demographic characteristics listed in Table B.7. Coe cients reported, robust standard errors in parentheses. *** indicate signi cance at the 1% level, ** { signi cance at the 5% level, and * { signi cance at the 10% level. Dep. Var.: IS (1) (2) (3) (4) (5) (6) (7) Panel A Own Liq. Score 1 = 2 -0.911** -0.953** -0.853* -1.152** -1.001** -1.075** -1.026* (0.437) (0.427) (0.439) (0.458) (0.490) (0.526) (0.573) Own Liq. Score 1 = 3 -0.007 -0.011 0.074 -0.281 0.028 0.078 -0.041 (0.425) (0.421) (0.427) (0.457) (0.474) (0.514) (0.612) Own Liq. Score 1 = 4 -0.491 -0.489 -0.425 -0.764 -0.214 -0.162 0.124 (0.465) (0.457) (0.462) (0.502) (0.538) (0.569) (0.623) Own Liq. Score 1 = 5 0.082 0.101 0.123 -0.165 0.757 0.853 0.785 (0.495) (0.494) (0.495) (0.512) (0.523) (0.590) (0.664) Observations 279 279 279 254 279 254 198 2 59.70 59.76 61.45 65.58 106.38 125.90 103.28 Panel B Own Liq. Score 2 = 1 0.163 0.181 0.192 0.264 -0.553 -0.378 -1.016** (0.285) (0.277) (0.281) (0.281) (0.415) (0.416) (0.472) Own Liq. Score 2 = 2 0.813** 0.825** 0.796** 0.907*** 0.719 0.798* 0.499 (0.339) (0.331) (0.339) (0.329) (0.440) (0.434) (0.548) Own Liq. Score 2 = 3 0.912** 0.925** 0.967** 1.092** 0.826 1.122* 0.548 (0.463) (0.458) (0.464) (0.486) (0.564) (0.597) (0.725) Observations 278 278 278 253 278 253 198 2 48.54 47.59 48.83 59.15 93.93 111.20 100.37 Panel C Own Liq. Score 1 > 2 0.553** 0.575** 0.576** 0.478* 0.812*** 0.887*** 0.887*** (0.256) (0.258) (0.254) (0.276) (0.306) (0.319) (0.331) Observations 279 279 279 254 279 254 198 2 44.73 43.84 47.90 51.41 96.89 112.51 89.70 Panel D Own Liq. Score 2 > 1 0.725*** 0.725*** 0.706*** 0.764*** 1.039*** 1.105*** 1.103*** (0.250) (0.249) (0.251) (0.258) (0.345) (0.331) (0.418) Observations 279 279 279 254 279 254 198 2 47.88 47.42 48.45 60.41 93.82 101.90 87.48 Panel E Avg Liq. Score 1 -0.026 -0.053 0.404 0.244 0.017 1.135** 1.484*** (0.438) (0.440) (0.427) (0.438) (0.440) (0.485) (0.560) Observations 277 277 277 253 277 253 198 Chi2 39.45 38.73 93.26 47.77 39.82 93.84 85.71 Panel F Avg Liq. Score 2 1.279* 1.273* 1.137** 1.053* 1.163* 1.100* 1.219** (0.663) (0.679) (0.574) (0.633) (0.619) (0.588) (0.584) Observations 277 277 277 253 277 253 198 Chi2 41.41 40.73 95.17 48.26 42.42 98.91 101.90 Controls: Market/ Y Y Y Y Y Y Y Demographics Log(% credit sales) Y Y Y Strong Competition Y Y Y Processing / Log(% sales Y Y Y to regular customers) Discuss prices / Y Y Y Discuss suppliers Suppliers will not extend credit Y to delinquent customers 214 Table B.11: Robustness Checks { probit estimates of the probability of sharing information. All regressions control for the market and demographic characteristics listed in Table B.7, except that in column (2) we replace the regional FE with city FE. Coe cients reported, robust standard errors in parentheses. *** indicate signi cance at the 1% level, ** { signi cance at the 5% level, and * { signi cance at the 10% level. Dependent Variable: IS Baseline City FE 15 50% Credit Sales Past Default (1) (2) (3) (4) Panel A Own Liq. Score 1 = 2 -1.075** -7.153*** -1.382** -1.596** (0.526) (2.180) (0.573) (0.720) Own Liq. Score 1 = 3 0.078 -1.744 -0.136 -0.107 (0.514) (1.146) (0.582) (0.513) Own Liq. Score 1 = 4 -0.162 -0.673 -0.412 -1.056 (0.569) (1.125) (0.671) (0.733) Own Liq. Score 1 = 5 0.853 1.516 0.797 -0.495 (0.590) (1.411) (0.662) (0.771) Observations 254 158 233 152 2 125.90 58.27 116.09 59.55 Panel B Own Liq. Score 2 = 1 -0.378 -0.133 -0.378 -0.642 (0.416) (0.680) (0.455) (0.692) Own Liq. Score 2 = 2 0.798* 2.266** 1.073** 1.593** (0.434) (0.882) (0.491) (0.694) Own Liq. Score 2 = 3 1.122* 1.475 1.314** 2.206** (0.597) (0.903) (0.573) (0.911) Observations 253 157 232 152 2 111.20 90.75 97.87 63.13 Panel C Own Liq. Score 1 > 2 0.887*** 3.457*** 0.867** 0.700* (0.319) (1.113) (0.378) (0.376) Observations 253 157 232 152 2 111.85 57.82 103.48 67.52 Panel D Own Liq. Score 2 > 1 1.105*** 2.129*** 1.377*** 2.007*** (0.331) (0.560) (0.368) (0.572) Observations 253 157 232 152 2 100.71 86.82 84.11 60.32 Panel E Avg Liq. Score 1 1.135** 1.426 1.201** 0.914 (0.485) (0.922) (0.512) (0.612) Observations 253 157 232 152 Chi2 93.84 88.29 90.82 67.64 Panel F Avg Liq. Score 2 1.100* 2.810*** 1.330** 0.984 (0.588) (0.846) (0.602) (0.848) Observations 253 157 232 152 Chi2 98.91 81.87 97.85 70.81 215 Table B.12: Ordered probit estimates. Columns report estimates for the own and average market liquidity. All regressions control for the market and demographic characteristics listed in Table B.7. Coe cients reported, robust standard errors in parentheses. *** indicate signi cance at the 1% level, ** { signi cance at the 5% level, and * { signi cance at the 10% level. Dep. Var.: IS Liquidity Variables (1) (2) (3) (4) Own Liq. Score 1 = 2 0.190 (0.327) Own Liq. Score 1 = 3 0.435 (0.322) Own Liq. Score 1 = 4 -0.058 (0.361) Own Liq. Score 1 = 5 0.890** (0.453) Own Liq. Score 2 = 1 -0.162 (0.213) Own Liq. Score 2 = 2 0.076 (0.256) Own Liq. Score 2 = 3 0.227 (0.347) Avg Liq. Score 1 0.113 (0.306) Avg Liq. Score 2 0.778** (0.348) Log(% credit sales) 0.065 0.125 0.128 0.141 (0.137) (0.130) (0.129) (0.126) Processing 0.323 0.275 0.328 0.304 (0.306) (0.316) (0.306) (0.315) Log(% sales to regular customers) 0.140 0.068 0.052 0.066 (0.158) (0.153) (0.152) (0.151) Discuss prices 1.552*** 1.430*** 1.455*** 1.457*** (0.247) (0.224) (0.225) (0.229) Discuss suppliers 0.044 0.274 0.200 0.221 (0.226) (0.205) (0.202) (0.201) Strong Competition 0.020 -0.035 -0.016 -0.044 (0.176) (0.178) (0.174) (0.176) Semi-wholesaler 0.193 0.123 0.112 -0.115 (0.280) (0.261) (0.264) (0.278) Retailer with xed selling point 0.290 0.215 0.239 -0.134 (0.224) (0.221) (0.236) (0.261) Assembler manufacturing -0.603 -0.578 -0.675 -0.682 (0.545) (0.533) (0.508) (0.518) Assembler individual 0.104 0.026 0.064 -0.251 (0.287) (0.278) (0.304) (0.305) Firm Age: 5 10 yrs 0.059 0.015 0.014 0.008 (0.229) (0.216) (0.213) (0.212) Firm Age: > 10 yrs -0.332 -0.247 -0.206 -0.182 (0.253) (0.242) (0.239) (0.244) Medium-sized rm 0.110 0.109 0.128 0.169 (0.231) (0.234) (0.227) (0.229) Large rm 0.432** 0.397* 0.422** 0.431** (0.208) (0.216) (0.209) (0.211) 4 9 traders known -0.193 -0.161 -0.177 -0.202 (0.325) (0.313) (0.306) (0.306) > 9 traders known -0.277 -0.290 -0.310 -0.341 (0.307) (0.289) (0.278) (0.277) Observations 270 270 270 270 Chi2 132.52 146.62 124.80 145.02 216 Table B.13: Average marginal e ects of liquidity. Estimates are based on column (6) of Table B.10 Avg. Marginal E ect St. Error Z-stat P-value Own Liq. Score 1 > 2 0.097 0.032 3.00 0.003 Own Liq. Score 2 > 1 0.130 0 .038 3.43 0.001 Avg. Liq. Score 1 0.086 0.052 1.66 0.096 Avg. Liq. Score 2 0.173 0.065 2.66 0.008 217 Bibliography [1] Anderlini, L., and Lagunoff, R. Communication in dynastic repeated games: ?whitewashes? and ?coverups?. Economic Theory 26, 2 (08 2005), 265{ 299. [2] Annen, K. Lies and Slander: Truth-Telling in Repeated Matching Games with Private Monitoring. 2007. [3] Annen, K. Small Firms and Cooperative Competition in Developing Coun- tries. May 2007. [4] Beggs, A. W., and Klemperer, P. Multi-period Competition with Switch- ing Costs. Econometrica 60, 3 (May 1992), 651{66. [5] Ben-Porath, E., and Kahneman, M. Communication in repeated games with private monitoring. Journal of Economic Theory 70, 2 (1996), 281 { 297. [6] Ben-Porath, E., and Kahneman, M. Communication in repeated games with costly monitoring. Games and Economic Behavior 44, 2 (2003), 227 { 250. [7] Bernstein, L. Opting out of the Legal System: Extralegal Contractual Re- lations in the Diamond Industry. The Journal of Legal Studies 21, 1 (1992), 115{157. [8] Bernstein, L. Private Commercial Law in the Cotton Industry: Creating Cooperation through Rules, Norms, and Institutions. Michigan Law Review 99, 7 (2001), 1724{1790. [9] Besley, T. Nonmarket Institutions for Credit and Risk Sharing in Low-Income Countries. The Journal of Economic Perspectives 9, 3 (1995), 115{127. [10] Biggs, T., Conning, J., Fafchamps, M., and Srivastava, P. Private Order under Dysfunctional Public Order. Regional Program on Enterprise De- velopment (1994). [11] Boccard, N., and Wauthy, X. Equilibrium Payo s in a Bertrand- Edgeworth Model with Product Di erentiation. Economics Bulletin 12, 11 (2005), 1{8. 218 [12] Bolton, P., and Scharfstein, D. A theory of predation based on agency problems in nancial contracting. American Economic Review 80, 1 (1990), 93{106. [13] Brown, M., Jappelli, T., and Pagano, M. Information Sharing and Credit: Firm-level Evidence From Transition Countries. Journal of Financial Intermediation 18, 2 (2009), 151 { 172. [14] Brown, M., and Zehnder, C. Credit reporting, relationship banking, and loan repayment. Journal of Money, Credit and Banking 39, 8 (December 2007), 1883{1918. [15] Brown, M., and Zehnder, C. The Emergence of Information Sharing in Credit Markets. Working Papers 2008-1, Swiss National Bank, Apr. 2008. [16] Carr, Jack L. and Landa, Janet T. The economics of symbols, clan names, and religion. The Journal of Legal Studies 12, 1 (1983), 135. [17] Chen, Y. Paying customers to switch. Journal of Economics & Management Strategy 6, 4 (December 1997), 877{897. [18] Clarke, R. N. Collusion and the Incentives for Information Sharing. Bell Journal of Economics 14, 2 (Autumn 1983), 383{394. [19] Clay, K. Trade without Law: Private-Order Institutions in Mexican Califor- nia. Journal of Law, Economics, & Organization 13, 1 (1997), 202{231. [20] Coate, S., and Ravallion, M. Reciprocity without Commitment : Char- acterization and Performance of Informal Insurance Arrangements. Journal of Development Economics 40, 1 (February 1993), 1{24. [21] Corts, K. S. Third-degree price discrimination in oligopoly: All-out compe- tition and strategic commitment. RAND Journal of Economics 29, 2 (Summer 1998), 306{323. [22] de Janvry, A., McIntosh, C., and Sadoulet, E. The Supply and De- mand Side Impacts of Credit Market Information. Proceedings, Nov (2006). [23] Diamond, D. W. Monitoring and reputation: The choice between bank loans and directly placed debt. The Journal of Political Economy 99, 4 (1991), 689{ 721. [24] Dixit, A. Lawlessness and Economics : Alternative Modes of Governance. Princeton Univ. Press, 2004. [25] Djankov, S., McLiesh, C., and Shleifer, A. Private Credit in 129 Coun- tries. Journal of Financial Economics 84, 2 (2007), 299 { 329. 219 [26] Fabbri, D., and Klapper, L. Market Power and the Matching of Trade Credit Terms. Policy Research Working Paper Series 4754, The World Bank, Oct. 2008. [27] Fafchamps, M. The Enforcement of Commercial Contracts in Ghana. World Development 24, 3 (March 1996), 427{448. [28] Fafchamps, M. Trade Credit in Zimbabwean Manufacturing. World Devel- opment 25, 5 (May 1997), 795{815. [29] Fafchamps, M. Returns to Social Network Capital Among Traders. Oxford Economic Papers 54, 2 (April 2002), 173{206. [30] Fafchamps, M., Gunning, J. W., and Oostendorp, R. Inventories and Risk in African Manufacturing. Economic Journal 110, 466 (October 2000), 861{93. [31] Fafchamps, M., and Lund, S. Risk-sharing Networks in Rural Philippines. Journal of Development Economics 71, 2 (August 2003), 261{287. [32] Fafchamps, M., and Minten, B. Relationships and Traders in Madagascar. Tech. Rep. 35, 1999. [33] Fafchamps, M., and Minten, B. Social Capital and Agricultural Trade. American Journal of Agricultural Economics 83, 3 (August 2001), 680{85. [34] Farrell, J., and Klemperer, P. Coordination and Lock-In: Competition with Switching Costs and Network E ects, vol. 3 of Handbook of Industrial Organization. Elsevier, 2007, ch. 31, pp. 1967{2072. [35] Farrell, J., and Shapiro, C. Dynamic competition with switching costs. RAND Journal of Economics 19, 1 (Spring 1988), 123{137. [36] Friedman, J. W. A non-cooperative equilibrium for supergames. Review of Economic Studies 38, 113 (January 1971), 1{12. [37] Fudenberg, D., and Tirole, J. Customer poaching and brand switching. RAND Journal of Economics 31, 4 (Winter 2000), 634{657. [38] Fudenbgerg, D., and Villas-Boas, J. M. Behavior-Based Price Discrim- ination and Customer Recognition, vol. 1. Elsevier B.V., 2006. [39] Gal-Or, E. Information Sharing in Oligopoly. Econometrica 53, 2 (March 1985), 329{43. [40] Gallindo, A., and Miller, M. Can Credit Registries Reduce Credit Con- straints? Empirical Evidence on the Role of Credit Registries in Firm Invest- ment Decisions. 2001. 220 [41] Gehrig, T., and Stenbacka, R. Introductory o ers in a model of strategic competition. CEPR Discussion Papers 3189, C.E.P.R. Discussion Papers, Feb. 2002. [42] Gin e, X., Goldberg, J., and Yang, D. Identi cation Strategy: Field Experimental Evidence on Borrower Responses to Fingerprinting for Loan En- forcement. [43] Greif, A. Reputation and Coalitions in Medieval Trade: Evidence on the Maghribi Traders. The Journal of Economic History 49, 04 (December 1989), 857{882. [44] Greif, A. Institutions and the Path to the Modern Economy: Lessons from Medieval Trade. Cambridge: Cambridge Univ. Press, 2006. [45] Greif, A., Milgrom, P., and Weingast, B. R. Coordination, Commit- ment, and Enforcement: The Case of the Merchant Guild. The Journal of Political Economy 102, 4 (1994), 745{776. [46] Grozeva, V. Dynamic Competition with Customer Recognition and Switch- ing Costs in Overlapping Generations Framework. March 2009. [47] Holmes, T. J. The e ects of third-degree price discrimination in oligopoly. American Economic Review 79, 1 (March 1989), 244{50. [48] Jappelli, T., and Pagano, M. Information Sharing, Lending and Defaults: Cross-country Evidence. Journal of Banking & Finance 26, 10 (October 2002), 2017{2045. [49] Jorge, P. A. Revisiting Dynamic Duopoly with Consumer Switching Costs. Journal of Economic Theory 67, 2 (December 1995), 520{530. [50] Kallberg, J., and Udell, G. The value of private sector business credit information sharing: The US case. Journal of Banking and Finance 27 (March 2003), 449{469(21). [51] Kandori, M. Social Norms and Community Enforcement. The Review of Economic Studies 59, 1 (1992), 63{80. [52] Kim, B.-C., and Choi, J. P. Customer Infomation Sharing: Strategic In- centives and New Implications. Working Papers 07-27, NET Institute, Sept. 2007. [53] Klein, D. B. Promise Keeping in the Great Society: A Model of Credit Information Sharing. Economics and Politics, Vol. 4, No. 2, pp. 117-136, July 1992 . [54] Landa, J. T. A Theory of the Ethnically Homogeneous Middleman Group: An Institutional Alternative to Contract Law. The Journal of Legal Studies 10, 2 (1981), 349. 221 [55] Li, L. Cournot Oligopoly with Information Sharing. The RAND Journal of Economics 16, 4 (1985), 521{536. [56] Liu, Q., and Serfes, K. Customer Information Sharing among Rival Firms. European Economic Review 50, 6 (2006), 1571 { 1600. [57] Love, I., and Mylenko, N. Credit Reporting and Financing Constraints. SSRN eLibrary (2003). [58] Luoto, J., McIntosh, C., and Wydick, B. Credit Information Systems in Less Developed Countries: A Test with Micro nance in Guatemala. Economic Development and Cultural Change 55 (2007), 313{334. [59] Marquez, R. Competition, Adverse Selection, and Information Dispersion in the Banking Industry. Review of Financial Studies 15, 3 (2002), 901{926. [60] McGee, J. S. Predatory price cutting: The standard oil (n. j.) case. Journal of Law and Economics 1 (1958), 137{169. [61] McMillan, J., and Woodruff, C. Inter rm Relationships And Informal Credit In Vietnam. The Quarterly Journal of Economics 114, 4 (November 1999), 1285{1320. [62] McMillan, J. N., and Woodruff, C. Private Order under Dysfunctional Public Order. SSRN eLibrary (2000). [63] Nilssen, T. Two kinds of consumer switching costs. RAND Journal of Eco- nomics 23, 4 (Winter 1992), 579{589. [64] Novshek, W., and Sonnenschein, H. Ful lled Expectations Cournot Duopoly with Information Acquisition and Release. Bell Journal of Economics 13, 1 (Spring 1982), 214{218. [65] Okuno-Fujiwara, M., and Postlewaite, A. Social Norms and Random Matching Games. Games and Economic Behavior 9, 1 (1995), 79 { 109. [66] Olegario, R. Credit reporting agencies. In Credit Reporting Systems and the International Economy, M. J. Miller, Ed. Boston: MIT Press, 2003, ch. 3. [67] Padilla, A. J., and Pagano, M. Endogenous Communication Among Lenders and Entrepreneurial Incentives. The Review of Financial Studies 10, 1 (1997), 205{236. [68] Padilla, A. J., and Pagano, M. Sharing default information as a borrower discipline device. European Economic Review 44, 10 (2000), 1951 { 1980. [69] Pagano, M., and Jappelli, T. Information Sharing in Credit Markets. The Journal of Finance 48, 5 (1993), 1693{1718. 222 [70] Pazgal, A., and Soberman, D. Behavior-Based Discrimination: Is It a Winning Play, and If So, When? MARKETING SCIENCE 27, 6 (2008), 977{ 994. [71] Raith, M. A General Model of Information Sharing in Oligopoly. Journal of Economic Theory 71, 1 (October 1996), 260{288. [72] Safavian, M., and Wimpey, J. When Do Enterprises Prefer Informal Credit? SSRN eLibrary (2007). [73] Schneider, F. G. The Size and Development of the Shadow Economies of 22 Transition and 21 OECD Countries. SSRN eLibrary (2002). [74] Staiger, D., and Stock, J. H. Instrumental Variables Regression with Weak Instruments. Econometrica 65, 3 (1997), 557{586. [75] Stock, J. H., and Yogo, M. Testing for weak instruments in iv regression. In Identi cation and Inference for Econometric Models: Essays in Honor of Thomas Rothenberg, D. Andrews and J. Stock, Eds. Cambride University Press, 2005, ch. 5. [76] Taylor, C. R. Supplier sur ng: Competition and consumer behavior in subscription markets. RAND Journal of Economics 34, 2 (Summer 2003), 223{46. [77] Telser, L. G. Cutthroat competition and the long purse. Journal of Law and Economics 9 (1966), 259{277. [78] To, T. Multi-period competition with switching costs: An overlapping gen- erations formulation. Journal of Industrial Economics 44, 1 (March 1996), 81{87. [79] Townsend, R. M. Financial Systems in Northern Thai Villages. The Quar- terly Journal of Economics 110, 4 (November 1995), 1011{46. [80] Van Horen, N. Customer Market Power and the Provision of Trade Credit : Evidence from Eastern Europe and Central Asia. Policy Research Working Paper Series 4284, The World Bank, July 2007. [81] Vercammen, J. A. Credit bureau policy and sustainable reputation e ects in credit markets. Economica 62, 248 (1995), 461{78. [82] Villas-Boas, J. M. Dynamic competition with customer recognition. RAND Journal of Economics 30, 4 (Winter 1999), 604{631. [83] Villas-Boas, J. M. Price cycles in markets with customer recognition. RAND Journal of Economics 35, 3 (Autumn 2004), 486{501. [84] Vinogradova, E. Working Around the State: Contract Enforcement in the Russian Context. Socioecon Rev 4, 3 (2006), 447{482. 223 [85] Vives, X. Duopoly Information Equilibrium: Cournot and Bertrand. Journal of Economic Theory 34, 1 (October 1984), 71{94. [86] Vives, X. Information Sharing Among Firms. October 2006. 224