Product Di erentiation and Quality Diversity in Markets for Experience Goods

Transcription

1 Product Di erentiation and Quality Diversity in Markets for Experience Goods Gerhard O. Orosel y and Klaus G. Zauner z January 2008 Abstract We analyze vertical product di erentiation in a model where a good s quality is unobservable to customers before purchase, a continuum of quality levels is technologically feasible, and minimum quality is supplied by a competitive fringe of rms. After purchase the true quality of the good is revealed with positive probability. To provide rms with incentives to actually deliver promised quality, prices must exceed unit variable costs. We derive su cient conditions for these incentive constraints to determine equilibrium prices, and show that under certain conditions only one or both of the extreme levels of quality, minimum and maximum quality, are available in the market. Jel Classification numbers: D82, L11, L15. Keywords: Moral Hazard. Product Di erentiation, Product Quality, Asymmetric Information, y Acknowledgements to be added. Department of Economics, University of Vienna, Hohenstaufengasse 9, A-1010 Vienna, Austria. gerhard.orosel@univie.ac.at. z Department of Economics, School of Social Sciences, City University London, Northampton Square, London EC1V 0HB, UK klaus.zauner@city.ac.uk (corresponding author).

2 1. Introduction In a seminal paper, Klein and Le er (1981) show that in order to provide incentives for a competitive rm to supply a high quality product, its price, the quality-assuring price, has to be above average production cost and include an informational rent. The threat of losing the rent from future purchases of customers enforces high quality provision and the associated quality-assuring price. 1 In this paper, we apply Klein and Le er s idea of qualityassuring prices to a market with price-setting rms and free but costly entry, and analyze which quality levels out of a continuum of technologically feasible levels are o ered in equilibrium. This study is therefore primarily related to the literature on the incentive approach to high quality provision. 2 However, rather than focusing on the quality-assuring price of a given exogenous quality, as Klein and Le er (1981), we study which quality levels are actually o ered in equilibrium when rms compete simultaneously in prices and quality. That is, we examine the consequences of Klein and Le er s insights about quality-assuring prices for the diversity of quality levels in the market. Our results can explain the casual observation that in some markets with asymmetric information there is little variety in available levels of quality, obtainable quality is generally low, and high quality is either impossible to nd or expensive. In fact, we show that under certain conditions at most two di erent levels of quality are o ered in equilibrium one of them being the lowest and the other one the highest technologically feasible level. Also, if high quality is available at all, then only for a price that is high relative to production cost. The lack of high quality products in a particular market may simply be the consequence of customer preferences. Customers may just not be willing to pay the higher production cost associated with better quality. In such a case complaints about the lack of high quality are irrelevant from the point of view of market performance. However, our case is di erent. Our results are not based on any unwillingness of the customers to pay the higher production cost of better quality. In our study, we consider a market for a good where (i) customers cannot observe its 1 This repeat-purchase contract enforcing mechanism is an example of a self-enforcing contract mechanism that plays an important role in contract theory. See, e.g., Crawford (1985), Levin (2003), MacLeod (2006). 2 This literature was pioneered by Klein and Le er (1981) and followed up by Shapiro (1983). See Bagwell and Riordan (1991), Bester (1998), Maksimovic and Titman (1991), Neeman and Orosel (2004), Riordan (1986), Rob and Sekiguchi (2004), Rogerson (1988), Wolinsky (1983), among others. 1

3 quality before deciding whether or not to buy, (ii) customers detect with positive probability the quality of the respective product after purchase, and (iii) contracts cannot be conditioned on quality. This type of good di ers somewhat from a regular experience good, as introduced by Nelson (1970), in that even after purchase customers may nd out the true quality not with certainty but only with positive probability. Examples of such goods can be found, e.g., among information services and products (Sarvary and Parker 1997), that is, services and products of legal, nancial, medical or technical consultants or advisors, and the provision of nancial and accounting audits. An illustration in case are retail nancial services. Retail nancial services rms collect and produce information and sell this information to investors, directly in the form of newsletters or reports, or indirectly through the set-up of a fund and the sale of shares on this fund that exploits this information. 3 Prospective customers of retail nancial services rms nd it hard to assess the quality of individual products and suppliers (Spencer 2000, p. 24). With physical goods, unobservable quality is frequently associated with the good s genetic, organic, and chemical properties but may also relate to the respective production processes (e.g., as regards child labor, environmental implications or animal welfare). Most of the time consumers are not able to detect these attributes by their own experience. However, checks and tests by public agencies, consumer organizations or watchdogs can nd out. Therefore, in our analysis we want to include goods where a customer s own experience will never reveal the true quality, but checks by certain agencies will, e.g., tests by health authorities will detect hazardous substances contained in a given product. Thus, checks and tests of such organizations are part of the contract enforcement mechanism. Since in any given period only few goods are tested, for a producer such tests are random events. Accordingly, we include in our model the case where customers cannot observe the true quality of the respective product even after consumption, but where after purchase the true quality of the good will become publicly known with positive probability. We are not interested in the situation where insu cient competition, e.g., a monopoly, is the cause of meagre quality diversity. Rather, we want to investigate the e ects of asymmetric information on quality diversity in equilibrium without limiting competition a priori. Therefore, we employ a model where market entry, and thus the intensity of competition, is endogenously determined by the interplay of a positive entry cost and market demand. 3 See, e.g., Admati and P eiderer (1986, 1990), Fishman and Hagerty (1995), Mahajan and Sweeney (2001). 2

4 Our model is one of moral hazard and incentives rather than one of adverse selection and signaling. Since in the recent literature the term reputation is used predominantly, though not exclusively, in connection with adverse selection, we avoid that term in this paper. 4 our analysis the problem is not rms reputation with respect to (given) types but customers trust regarding rms behavior. Speci cally, we assume that customers trust rms whenever conditional on this trust a rm has no incentive to cheat, i.e., to provide lower quality than promised. This assumption concerns o -equilibrium beliefs, and it excludes equilibria where rms never produce higher than minimum quality because customers stubbornly believe that quality is always at its lowest feasible level. Whenever customers understand rms incentives su ciently well this assumption is justi ed. The results of our model seem to be, by and large, consistent with empirical work on product di erentiation. For example, Bresnahan (1981, p. 217) provides evidence for the US automobile market that products are much more tightly bunched at the low-quality end of the product spectrum, which seems to suggest that close to the minimum a relatively large number of basically identical qualities is available in the market whereas only a few high qualities are o ered. In addition, Bresnahan (1981, 1987) demonstrates that price-cost margins increase with higher quality. This observation, which is also supported by Kwoka (1992) and Feenstra and Levinsohn (1995), concurs with quality-assuring prices. 5 The rest of the paper is organized as follows. First, we discuss some related work in Section 2. In Section 3, we present the model. Then, in Section 4, we derive the qualityassuring prices from the incentive compatibility constraints for rms to provide high quality. In Section 5, we show that due to these incentive compatibility constraints only low quality may be available in the market even though preferences for high quality may be strong relative to production costs. In Section 6, we demonstrate that under certain conditions equilibrium prices are determined by incentive compatibility constraints, whereas customer preferences and the distribution of customer types only determine the quantities demanded, 4 Examples of papers on reputation which include adverse selection as an essential element are, among others, Holmström (1999), Tadelis (1999), Mailath and Samuelson (1998), (2001), Hörner (2002), Cripps et al. (2004). For our analysis of the moral hazard problem we need not be concerned with the questions of how reputation can be acquired, how it is lost, and how it can be used strategically. 5 Although the automobile market is a ected by many factors not included in our model, it is roughly consistent with our basic assumptions. In particular, it is an experience good market and unexpected low quality is punished by demand reduction. In 3

5 given equilibrium prices. In Section 7, we analyze the case where customers willingness to pay for quality is convex with respect to quality, and in Section 8 the case where it is concave. Finally, we conclude in Section 9. All mathematical proofs are relegated to the Appendix. 2. Related Work As pointed out above, our analysis is based on Klein and Le er (1981). In addition, it is also related to the literature on the analysis of Bertrand equilibria in markets with vertical product di erentiation (see, e.g., Gabszewicz and Thisse 1979, 1980, and Shaked and Sutton 1982, 1983). 6 However, it di ers from this literature (and the related analysis of endogenous sunk cost and the associated niteness result; see, e.g., Sutton 1986, 1991; Shaked and Sutton 1983, 1987; Anderson et al.1992, , Berry and Waldfogel 2006) in three important respects. 7 First, quality cannot be observed by customers before purchase and therefore, due to incentive reasons, high qualities must have high ( quality-assuring ) prices relative to production costs. That is, costs must also include an incentive component. Second, the quoted literature on Bertrand equilibria in markets with vertical product di erentiation assumes basically that the cost increase associated with an increase in quality is negligible in the sense that the cost increase is always below customers willingness to pay for that increase in quality. In contrast, we do not use the corresponding assumption in our analysis, which would be that customers willingness to pay for higher quality is always larger than the increase in production-cum-incentive cost. Third, while the respective literature concentrates on the case where rms decide rst on quality and then, given the quality choices, on prices, we consider the case where rms decide simultaneously about quality and price (i.e., 6 For a Cournot analysis of vertical product di erentiation see, e.g., Gal-Or (1983). Gabszewicz and Grilo (1992) analyze Bertrand equilibria of a vertically di erentiated duopoly where consumers have exogenous beliefs about which of the two rms sells high quality and which sells low quality. Another strain of literature, based on Mussa and Rosen (1978), deals with quality provision by a monopolist (see, e.g., Gabszewicz and Wauthy 2002 and the references therein). For an analysis of vertical product di erentiation with high and low qualities under monopolistic or competitive conditions see Carlton and Dana (2004). However, due to di erences in focus the conclusions of this literature are not readily comparable to our results. 7 A fourth respect in which our model di ers is our assumption that minimum quality is o ered at marginal cost by a competitive fringe. However, although such an assumption would a ect the equilibrium prices, the main results of the papers quoted above would remain essentially unchanged, since under the assumptions of these models the competitive fringe would have no customers. 4

6 knowing neither price nor quality chosen by competitors). This approach is natural in our context where quality is unobservable and rms have an incentive to cheat customers by producing and selling unobservably low quality for a high price. Bester (1998) considers a standard Hotelling model of spatial competition for a duopoly where the horizontally di erentiated good is an experience rather than an inspection good. The good s quality is either high or low and, as in our model, the price of high quality includes an incentive component. This puts a oor on the price for high quality that is above the production cost and that the rms cannot undercut. Consequently, price competition is mitigated and because of this the equilibrium outcome may (but need not) be minimum di erentiation rather than maximum di erentiation. Our model of only vertical product di erentiation di ers from Bester s in many respects (in particular, we consider a continuum of qualities, no spatial competition and endogenous entry, whereas Bester considers a continuum of locations, only two quality levels and an exogenously given duopoly) and thus a direct comparison is not possible. However, the two models share the property that asymmetric information about quality mitigates price competition. In fact, our result of low quality diversity may be interpreted as corresponding, in some sense, to Bester s result of minimum di erentiation, i.e., low spatial diversity. 8 Within the literature on vertical product di erentiation there is a branch that investigates how customers informational di erences with respect to di erent brands may constitute a barrier to entry (see, e.g., Schmalensee 1982, Bagwell 1990). The informational di erences are in fact di erences with respect to supplier reputation. Whereas the incumbent has reputation (the quality of his product is known ), a new entrant has no reputation (the quality of her product is not known ). In this literature, rms are of di erent types, e.g., high and low quality producers. Thus, customers face an adverse selection problem, and reputation is about rms (unalterable) types. In contrast, in our paper rms are not distinguished by types, and consequently, there is no adverse selection. Rather, we analyze the moral hazard problem associated with rms that can choose the quality they provide. Our paper shares some aspects with repeated moral hazard models which assume that, in each repetition, players cannot observe the opponents actions perfectly but only an imperfect 8 Models of horizontal and vertical di erentiation are not as di erent as frequently perceived. For the case of observable quality Cremer and Thisse (1991) have shown that the Hotelling model of horizontal di erentiation is actually a special case of the standard model of vertical di erentiation. 5

7 stochastic signal thereof (see, e.g., Abreu et al. 1986, 1990, Fudenberg et al. 1994, Green and Porter 1984, Rob and Sekiguchi 2004, among others). In contrast to these models, the unobservable actions (i.e., the actual quality choices) of opponents do not in uence a rm s stage-game pro ts in our model. However, quality monitoring at the end of each stage provides incentives, in our case, against a rm s repeated moral hazard problem of not providing the announced quality. 3. The Model We consider a market for a good that is homogeneous except for quality. That is, in our model there is (potentially) vertical product di erentiation but no horizontal product differentiation. Time is measured in discrete periods t 2 f1; 2; :::g. There is a pool of N (N possibly in nite) rms that are capable to produce each quality v 2 [0; 1] of the good considered at constant marginal cost c (v) > 0; where the function c : [0; 1]! R ++ is strictly increasing. That is, marginal cost is independent of quantity but increasing with respect to quality. Moreover, since c (v) is strictly increasing, quality can be measured without loss of generality in such a way that cost is linear in quality with positive slope > 0, 9 c (v) = c (0) + v: (3.1) Put di erently, we use production cost to measure quality. In addition to the N rms that are capable to produce each of the technologically feasible quality levels, there are in nitely many rms that are capable of producing minimum quality v = 0 at cost c (0) : Although a rm may be capable of producing multiple quality levels, we assume, in agreement with most of the literature, 10 that in each period it can produce and o er only one particular level of quality. The number N of potential producers of all quality levels v 2 [0; 1] should be interpreted to be large. Thus, the good can be supplied in many di erent quality levels. In equilibrium not all rms will be active in the market and produce. Firms that are active in the market are distinguished between brand names and no names. No names produce 9 Let V 2 [V min ; V max ] be any measure of quality with associated marginal cost C (V ) > 0 that increases in V but is constant in quantity. De ning v [C (V ) C (V min )] = [C (V max ) C (V min )] 2 [0; 1] gives the required normalized measurement of quality. For any quality level ^V with normalized equivalent ^v we get the marginal cost c (^v) = C ^V = C (V min ) + [C (V max ) C (V min )] ^v = c (0) + ^v; where c(0) C (V min ) and C (V max ) C (V min ). 10 See, e.g., Shaked and Sutton 1982, 1983, Gabszewicz and Thisse 1979, 1980, Klein and Le er

8 only minimum quality v = 0; whereas brand names may produce any quality level v 2 [0; 1] : Any rm can be active as a no name, and each of the N of potential producers of all quality levels can be active as a brand name. Even if a no name is capable of producing positive quality, it will not do so because customers believe that all no names provide only minimum quality (Assumption 3 below). A brand name has to announce the quality of its product in some appropriate way and we call this the announced quality. These announcements may take di erent forms, such as inscriptions on the product or its package, TV or newspaper ads, billboard advertising, promotion campaigns, etc. 11 It is also possible that the price itself serves as quality announcement. In fact, we will show below that in a stationary equilibrium quality can be inferred from the price, making explicit quality announcements dispensable. Actual quality is private information of the rm and may or may not coincide with the announced quality. Between periods brand names can change their announced and actual quality levels, respectively, at no cost. In the model we assume that brand names have to announce the quality in each period, but the absence of a fresh announcement may count as implicit renewal of the last one. We assume that no names have no entry cost and thus form a competitive fringe that sells minimum quality at marginal cost c (0) and has zero payo. In contrast, each brand name has to incur a positive entry cost > 0, identical for all rms, in order to establish the respective brand together with an associated distribution channel. A brand name that voluntarily leaves the market may enter again, either simultaneously or later, but it has to pay the entry cost for each entry. Brand names choose their respective prices, together with their respective quality levels, simultaneously at the beginning of each period. That is, we assume Bertrand competition. In each period there is an atomless continuum of customers of (Lebesgue) measure 1: Customers total expenditures are non-negative and uniformly bounded. Customers live 11 Whenever they are manipulable, public quality reports may play the role of quality announcements. Examples for this are report card programs in medical-care markets in the US (see Cutler et al. 2004) and performance indicators introduced by the National Health Service in the U.K. to report publicly on the quality of health care provided by hospitals and physicians. A variety of measures, such as limiting access to high-risk patients, allow health care providers to manipulate quality reports and thus misrepresent quality (for references see, e.g., Cutler et al. 2004, and Dranove et al. 2003). Similarly, in nancial markets public disclosure regulation is aiming at reporting publicly on the quality of retail nancial products (see, for example, Financial Services Authority 2003). Again, quality may be misrepresented by the respective retail nancial rms. Of course, rational customers will understand that these reports are manipulable. 7

9 one period and are distinguished by types s 2 S = [s min ; s max ] R according to their willingness to pay for quality. Their preferences are speci ed by the following assumption. Assumption 1 (Customer Preferences). Each customer buys at most one unit of the good. The payo from not buying the good is normalized to zero. For customers of type s 2 S the payo from buying one unit of the good of quality v for the price p is given by U (v; p; s) = R (v; s) p: For all v 2 (0; 1] and s 2 S, the function R (; ) is continuous and strictly increasing in quality v. The function R (; ) gives the willingness to pay of type s for quality v: Notice that the shape of this function, in particular, whether it is convex or concave, depends on the way quality is measured. The normalization of (3.1) implies that we use (constant) marginal cost to measure quality, and the function R (; ) is de ned using this measurement. In this study we consider the case where at the time of purchase a product s quality is private information of the producer and cannot be observed by other agents; nor can contracts be conditioned on quality. In such a situation, a rm could save on cost by producing lower quality than announced, and in the following we refer to such behavior as cheating. There are many di erent aspects of quality. Some of these a customer can observe immediately after purchase (like whether a fruit tastes good), some she can detect after a relatively short period of use (like the mileage of a car), some she can discern only after a long time (like the durability of a good), and some she will never know with certainty (like some features of product reliability, such as the probability of a breakdown). Sometimes, a customer can learn the quality of a product, for example the quality of insurance or health care, only after exogenous random events, as the occurrence of an accident or illness (see Israel 2005). Similarly, quality di erences in various information services, like nancial auditing, may become apparent only in particular circumstances, such as bankruptcy or other disasters. Moreover, some aspects, like the content of certain hazardous elements in food or toys, can be detected by an appropriate monitoring agency but not by the ordinary customer herself. We model the customers learning of the respective good s quality in a way that is consistent with all these cases (but not restricted to them). Speci cally, Assumption 2 below covers the case where in each period the true quality is revealed to the public with some probability ' 2 (0; 1] ; and where with complementary probability 1 ' there is no information about a product s quality. Notice that we do not exclude the case ' = 1 (perfect monitoring), which following Klein and Le er (1981) is generally assumed in the 8

10 literature on experience goods. Our results allow, but do not require, that monitoring is imperfect. On the other hand, strategic aspects of imperfect monitoring are not relevant for our investigation and therefore we model imperfect monitoring as simple as possible. For the same reason we ignore quality certi cation. 12 In some cases, the information that a certain product does not have the announced quality may spread slowly and the rm which sells it in the market may be able to exploit the trust among customers it has acquired in the past for a relatively long period after it has started to cheat. In other cases, this information may become public almost instantaneously and force the rm to close down. For example, a watchdog agency may randomly test brand names products, and if the test shows that the true quality is below the announced quality the respective rm may have to close down, e.g., because it loses its customers or is forced by a legal authority to exit the market. The relevant point is not how fast or to what extent the respective rm s business is reduced, but that a cheating rm risks that it will be punished either by the public or by a legal authority. The expected punishment gives rise to an incentive compatibility constraint that, if satis ed, induces the rm to provide the announced quality. Regardless of the details of the model, this constraint necessarily implies that the price of a good above minimum quality must be su ciently above its marginal cost. Otherwise the rm would cheat and produce only minimum quality. Since customers cannot immediately observe quality and contracts cannot be conditioned on quality, a rm that produces high quality must earn an informational rent. The threat of losing this rent provides the incentive for the rm to actually produce the announced quality. The previous discussion motivates the following assumption, which among other cases captures the situation where watchdog agencies perform random tests of the brand names products with some probability ' 2 (0; 1], and rms caught cheating have to exit the market and receive a payo of zero from that moment onwards Quality certi cation may alleviate, at a cost, the moral hazard problem (see, e.g., Biglaiser and Friedman 1994, and Albano and Lizzeri 2001). However, it also introduces strategic play by certi ers (for a recent reference see Strausz 2005). While accounting for it would complicate our analysis and detract from our main issues, it would presumably not alter our results signi cantly (notice that rms do not cheat in equilibrium). 13 The assumption that the payo drops to zero is a simpli cation. A signi cant reduction (to a still positive value) is su cient. Empirically, such reductions can be observed. For example, with respect to food quality Coyle (2002, pp ) points out: A company involved in the spread of a foodborne pathogen can... face costs imposed by courts or government agencies, including nes, product recalls, and temporary 9

11 Assumption 2 (Quality Monitoring). If in any period t 2 f1; 2; :::g the true quality of a product is below the quality announced by its producer, the respective brand name has to exit the market with positive probability ' 2 (0; 1] at the end of this period and receives a payo of zero from the next period onwards. With complementary probability 1 ' providing lower than announced quality in any period has no e ect for the respective brand name. The assumption that cheating results in market exit with probability ' need not be taken literally. The model covers all cases where in expectation a cheating brand name loses ' percent of its future payo. For example, rather than having to leave the market with probability '; a brand name that cheats may lose ' percent of its customers for sure. Or it may lose '= percent of its customers with probability 2 (0; 1) : Out-of-equilibrium beliefs may lead to (implausible) equilibria where brand names cannot exist. To avoid such equilibria we assume, like Bester (1998, pp ), that customers trust rms whenever conditional on this trust the respective rm has no incentive to cheat. Assumption 3 (Customer Beliefs). Customers cannot observe the true quality of a product. They believe that the true quality is the announced quality unless given these beliefs it is optimal for a rm to provide lower quality. Otherwise, they believe that the true quality is the minimum quality v = 0: No names are believed always to provide minimum quality. Since no names have no entry cost, they always o er quality v = 0 for the price p (0) = c (0) under perfectly competitive conditions. Therefore, we restrict the terms entry and incumbent to brand names. Moreover, because no names do not behave strategically, we do not treat them explicitly as players in the game. In contrast, potential and actual brand names act strategically. In their decisions they take the competitive fringe of no names into account, as well as the strategies of all players. We assume that all rms have the same discount rate > 0: The associated discount factor is denoted by 1 2 (0; 1). A rm s payo is the sum of its discounted pro ts 1+ minus its entry cost, if it has entered the market, and zero otherwise. If more than one brand name o ers the same quality v 2 (0; 1] for the lowest price in the market, all those brand names share the respective demand equally. Our analysis is based on the following game with imperfect information. The set of players or permanent plant closures as well as large liability settlements and associated legal costs. Potential market and liability losses are strong incentives for food rms to ensure the food supply is as safe as possible. 10

12 consists of brand names, customers, and nature. Brand names know the distribution of customer preferences, but cannot observe the individual types. The cost function (3.1) and the rest of the model are common knowledge. In each period t 2 f1; 2; :::g, the game proceeds in four phases. In the rst phase, at the beginning of the period, brand names decide simultaneously about exit and entry. That is, brand names that are in the market decide whether to exit, and brand names that are not in the market (or not any more) decide whether to enter. In the second phase, still at the beginning of the period, all brand names in the market observe the moves made in the rst phase and choose simultaneously announced quality, actual quality, and price (each for the respective period). Customers and rms observe the announced qualities and prices, but actual qualities are private information, unobservable to customers and rival rms. In the third phase, which takes place at the end of the period, customers decide whether and from which supplier to buy one unit of the good. These decisions are executed and the period s payo s accrue. Finally, in phase four, nature moves and each brand name that had provided some quality below the announced quality has to leave the market forever with probability ' > 0. A brand name that has to leave the market receives no further payo (but keeps the payo s received so far). Since we are not interested in collusion among rms, we want to rule out folk theorem type results. In standard models this can be done by considering only those equilibria of the dynamic game that consist of playing a particular equilibrium of the stage game in every period. In our model the situation is somewhat di erent because the only reason why rms do not cheat are their future rents, and therefore the game that incumbent rms play in each period is not the relevant stage game. In any one-period stage game (as well as in any nite version of the dynamic game) incumbents would always cheat, and since customers would anticipate this and thus not buy the respective product, no brand name would enter the market and no positive quality would be available in equilibrium. In our model, the analog of an equilibrium (in pure strategies) of the dynamic game that consists of playing an equilibrium of the stage game in every period, is what we call an equilibrium in stationary strategies or, for short, a stationary equilibrium. We de ne a brand name s strategy to be stationary, if it satis es the following two conditions: (i) the rm either enters the market in the rst period t = 1 or not at all; (ii) if the rm had entered, its announced quality, actual quality, and price are constant in time and independent of the history of actions. Accordingly, an equilibrium (in pure strategies) is stationary, if all equilibrium strategies are 11

13 stationary. The requirement that equilibrium strategies are independent of the history of actions makes collusion unattainable. In a stationary equilibrium, equilibrium strategies are not only constant along the equilibrium path but also at o -equilibrium nodes. However, this holds only for equilibrium strategies. Deviating strategies are not constrained to be stationary. With the exception of Section 5, our equilibrium concept throughout the paper is the one of a stationary equilibrium in pure strategies. For the results of Section 5 collusion plays no role and therefore we consider all Nash equilibria in pure strategies Incentive Compatibility Constraints In this section, we derive the incentive (compatibility) constraint for a rm to provide a given quality v 2 (0; 1] in a stationary equilibrium. Consider a putative stationary equilibrium, where in some period t a rm o ers a product of some announced and actual quality v 2 (0; 1] for some price p; sells x units in this period, and considers to do the same in every future period. An alternative, deviating strategy is to cheat and provide only minimum quality for K 1 periods, risking involuntary exit; and to produce again quality v thereafter, conditional on still being in the market. Actually we need to consider only the case K = 1, because whenever some deviating strategy with K > 1 generates a higher expected payo than the putative equilibrium strategy, this holds as well for some one-period deviation. 15 For K = 1 the rm s incentive constraint to actually provide quality v is [p c (0)] x + (1 ') [p c (v)] x 1 [p c (v)] x 1 where the left hand side is the payo from always producing quality v and the right hand side is the expected payo from producing quality v = 0 in the next period and, if undetected, quality v thereafter. Equivalently, the bene t from cheating, which is [c (v) c (0)] x = vx; must not exceed the cost of cheating, which is ' [p c (v)] x = ' [p c (v)] x: Both 1 14 Since actual qualities are private information, the only proper subgames are the ones starting at the second phase of the rst period, after market entry. It is easy to see that all equilibria are subgame perfect. 15 See the one-stage deviation principle in Fudenberg and Tirole (1991, Theorem 4.2, p. 110). Although the theorem is not directly applicable because it assumes observed actions, the proof holds analogously for our game. Alternatively, direct calculation shows that considering any deviation for K > 1 periods leads to an incentive compatibility constraint that is identical to the one below. ; 12

14 arguments give the incentive constraint ' [p c (v)] x vx. Re-arranging, we get p c (v) + v: (4.1) ' In a stationary equilibrium, this inequality has to hold for all quality levels v 2 (0; 1] that are actually produced. It cannot be that some quality v is sold at a lower price because at a lower price the respective rm would cheat and this would be anticipated by its customers. Let ^p (v) denote the minimal incentive compatible price for quality v 2 [0; 1] in a stationary equilibrium, i.e., ^p (v) c (v) + v; v 2 [0; 1] : (4.2) ' These prices include the incentive cost v in addition to the production cost c (v) : This ' incentive cost is proportional to the discount rate and vanishes when converges to 0: 16 In any stationary equilibrium the price p (v) must satisfy p (v) ^p (v) (4.3) for each quality v 2 [0; 1] that is sold in the market. 17 Because of (4.3), prices exceed marginal costs for positive levels of quality and this has signi cant consequences for rms incentives to di erentiate. If quality were immediately observable to customers, Bertrand competition would imply that whenever two or more rms o er the same quality v the price p (v) must equal marginal cost c (v) ; and consequently pro ts are zero. Therefore, rms have a strong incentive to di erentiate from each other. In contrast, this is not true in our case of unobservable quality and the reason is (4.3). Since equilibrium prices for positive quality must always exceed marginal costs, Bertrand competition does not any more imply that pro ts are zero when two or more rms o er the same quality. This reduces, and may even eliminate, the incentive of rms to di erentiate from each other. As a consequence, little variety of quality may result. 16 Thus, ^p (v) approaches zero pro t pricing for! 0: This result is not due to the way we model (imperfect) monitoring, but follows from the assumption that the punishment for cheating consists in forced market exit. If in a stationary equilibrium the (constant) pro t per period and unit sold is positive, say > 0; its capitalized value and therefore the punishment of forced market exit goes to in nity as the discount rate goes to zero. This exploding punishment explains why the required rent per period, and with it the incentive cost, converges to zero for! 0: 17 There is a related problem in the regulation of banks: deposit-rate ceilings can be used as incentives for banks to invest in safe rather than in ine ciently risky assets ( gambling assets ) because deposit-rate ceilings increase banks pro ts per period and thus their charter values (Hellmann et al. 2000, Repullo 2004). 13

15 5. Minimum Quality Only Although the incentive constraints (4.3) have been derived for stationary equilibria, this section shows that they have some relevance for all equilibria. We say that some given quality v 2 [0; 1] is available in the market in some period t, if in period t some rm produces this quality v and o ers its product at a price which at least some customers are willing to pay, i.e., at a price that results in sales. Recall that by normalization every customer s payo is zero when she does not buy the good, and that minimum quality v = 0 is always o ered for the price p (0) = c (0) : Assume that some given quality v > 0 is o ered for some price p (v) ^p (v) : If R (v; s) ^p (v) < max [R (0; s) c (0) ; 0] for all s 2 S; customers will not buy quality v > 0 for the price p (v) : Each customer prefers either to buy minimum quality for the price c(0) or not to buy quality v at all. Consequently, quality v > 0 will not be available in a stationary equilibrium. If the inequality R (v; s) ^p (v) < max [R (0; s) c (0) ; 0] holds for every customer s 2 S for all positive levels of quality v 2 (0; 1], then in any stationary equilibrium either only minimum quality is available or the good is not available at all. Speci cally, the inequality implies that minimum quality v = 0 is available in the market in each period if R (0; s) > c (0) for some s 2 S; whereas if R (0; s) < c (0) for all s 2 S the respective good is never available (due to lack of demand). Proposition 1 generalizes this observation to all equilibria. Proposition 1. If for no positive level of quality v 2 (0; 1] customers are willing to pay the price ^p (v), i.e., if R (v; s) ^p (v) < max [R (0; s) c (0) ; 0] for all s 2 S and all v 2 (0; 1] ; then every Nash equilibrium in pure strategies has the property that positive quality is never available in the market. Depending on preferences, the respective good is either available only in minimum quality or, if R (0; s) < c (0) for all s 2 S, is not available at all. The intuition of Proposition 1 is as follows. By the proposition s assumption customers are not willing to pay the price ^p (v) that is necessary to provide the incentive to actually produce quality v > 0 when pro ts per period are constant in time. Since positive quality can be sold only for some price below ^p (v), an equilibrium requires that along the equilibrium path pro ts of a rm that sells v > 0 in some period t increase in the future. As pro ts can be positive in the future only if the respective rm provides again positive quality, the argument feeds on itself and pro ts have to increase ever more. In fact, it can be shown that with t! 1 pro ts must diverge. Since customers expenses are uniformly bounded, this is 14

16 not feasible and the proposition follows. Even when customers are not willing to pay the prices ^p (v) for v 2 (0; 1] ; they may be willing to pay the production cost c (v) < ^p (v) : In particular, it may be the case that every single customer prefers the highest feasible quality to all other alternatives when prices equal marginal cost, i.e., max [R (v; s) c (v) ; 0] < R (1; s) c (1) may hold for all v < 1 and all s 2 S. If this is the case, only the highest feasible quality would be consumed in the rst-best, whereas due to asymmetric information either only minimum quality is available in equilibrium or the good is not available at all. Proposition 1 holds analogously for the more general case where the payo function U (v; p; s) is any function U : [0; 1] R + S! R that is strictly decreasing in the price p: 18 The proof of Proposition 1 shows that if U [v; ^p (v) ; s] < max fu [0; c (0) ; s] ; 0g for all v 2 (0; 1] and all s 2 S; then in every Nash equilibrium in pure strategies positive quality is never available in the market. Moreover, Proposition 1 does not depend on the details of the game, such as simultaneous entry of rms, and extends to equilibria in mixed strategies. 6. Equilibrium Prices and Incentive Constraints For the reasons explained in Section 3, we examine stationary equilibria (in pure strategies) in the rest of the paper. Along any stationary equilibrium path, all incumbents actions are constant. However, deviating strategies are not constrained to be stationary. If we restrict for each rm the strategy set to stationary strategies, we get a new game, which we call the restricted game. Although even for a stationary equilibrium of the original game all strategies have to be considered as possible deviations, the following lemma shows that it is actually su cient to look only at stationary strategies. Lemma 1. A strategy pro le that constitutes a Nash equilibrium in pure strategies of the restricted game is also a Nash equilibrium in pure strategies of the original game. 19 In a stationary equilibrium of our game nothing can be learned from the past and the strategic situation is exactly the same in every period. It follows that equilibrium strategies 18 If the parameter s is interpreted as income thought of as a Hicksian composite commodity, the payo becomes U (v; p; s) = u (v; s p), which frequently is further simpli ed to U (v; p; s) = w (v) (s p) ; as, e.g., in Gabszewicz and Thisse (1979, 1980) and Shaked and Sutton (1982, 1983). 19 Obviously, the converse also holds: if a strategy pro le of stationary strategies constitutes a Nash equilibrium of the original game, it also constitutes a Nash equilibrium of the restricted game. 15

17 are sequentially rational and that a stationary equilibrium is a perfect Bayesian equilibrium. Obviously, in a stationary equilibrium incumbents have positive pro ts, do not cheat and do not exit. For each quality v 2 [0; 1] that is available in the market there can be only one price p (v) and it must hold that p (v) ^p (v) ; where ^p (v) is given by (4.2). Otherwise the respective rm(s) would cheat. Below we will derive a condition that implies the equality p (v) = ^p (v) for all intermediate quality levels v 2 (0; 1). Consequently, under this (su cient but not necessary) condition equilibrium prices of intermediate quality levels are completely determined by the incentive constraints (4.3), whereas customer preferences and the distribution of customer types only determine the quantities demanded, given (predetermined) equilibrium prices. In addition, the price for minimum quality v = 0 is given by its cost c (0) : For maximum quality v = 1 the incentive constraint determines the price p (1) = ^p (1) in two di erent circumstances. One case occurs when the incentive cost is su ciently ' large to make the price ^p (1) = c (1) + optimal (in the set fp j p ^p (1)g of incentive ' compatible prices) even for a rm that is the sole brand name in the market. The other case, explained next, consists of the situation where two or more brand names o er quality v = 1: If at least two brand names o er the same quality v > 0; Bertrand competition will drive the price p (v) to the lowest possible value. In the case of observable quality this lowest possible value is the marginal cost c (v) : In contrast, when quality is unobservable, the lowest possible value is ^p (v) in any stationary equilibrium, since a brand name that had announced quality v > 0 and charges a price p (v) < ^p (v) will cheat. Because of Assumption 3 (Customer Beliefs), customers will buy its product for a price below ^p (v) only if the price is c (0) : Thus, if the rm charges a price p (v) < ^p (v) ; its pro t is zero, whereas it is positive if p (v) = ^p (v). Consequently, whenever two or more brand names o er the same quality v it must hold that p (v) = ^p (v) in any stationary equilibrium. 20; 21 Although it 20 As Klein and Le er (1981, p. 625) put it... the quality-assuring price is, in e ect, a minimum price constraint enforced by rational consumers. 21 Since brand names have no xed cost of production, every brand name that has incurred the entry cost can stay in the market and guarantee itself a non-negative pro t per period. Consequently, it is impossible that a brand name drives another brand name out of the market by undercutting ^p (v). In the case where (contrary to our model) brand names have a xed cost of production, a di erent argument gives the same result. With xed costs, undercutting will trigger a war of attrition, and in wars of attrition the most plausible equilibria are those in mixed strategies. Since this implies expected payo s of zero, whereas cheating gives strictly positive payo s, the respective brand names will cheat. Because customers will recognize this, undercutting is not pro table. 16

18 is implausible that two or more brand names o er the same intermediate quality v 2 (0; 1) ; our analysis will show that for maximum quality v = 1 this may well be the case. For the rest of the paper we assume that R (; ) is twice continuously di erentiable and that R vs (; ) > 0 for v > 0, where subscripts of R (; ) always denote the respective partial derivatives. The assumption R vs (; ) > 0 is a single crossing condition and means that customer types can be de ned in such a way that higher types have a higher willingness to pay for additional quality, i.e., have a higher marginal willingness to pay for quality. Assumption 4 (Increasing Di erences). Higher customer types have a higher willingness to pay for additional quality, i.e., R vs (; ) > 0 for v > 0: Because of Assumption 4, the di erence in willingness to pay between any two quality levels v 0 and v 00 > v 0, R (v 00 ; s) R (v 0 ; s) = R v 00 R v (v; s) dv; is strictly increasing in s for any v 0 such pair v 0 ; v 00 2 [0; 1] : 22 Speci cally, the additional amount of money that a customer of type s is willing to pay when the good is of highest rather than of lowest feasible quality, r (s) R (1; s) R (0; s), is strictly increasing. For convenience we identify customer types by this di erence r (s), i.e., without loss of generality we assume s R (1; s) R (0; s). Although in a stationary equilibrium the price for any quality level v cannot be below the minimal incentive compatible price ^p (v) ; it can, of course, exceed that price. In the following we provide a su cient condition for this not to be the case for quality levels v below the maximum. Speci cally, we show that whenever it holds for all types s 2 S that R v (v; s) > for some v 2 (0; 1), then p (v) = ^p (v) : Consequently, if R v (v; s) > for all v 2 (0; 1) and all s 2 S; then p (v) = ^p (v) for all v 2 [0; 1) since p (0) = ^p (0) = c (0) holds trivially. Substituting for ^p (v) gives p (v) = c (v) + 1 v = c (0) + + v = c (0) + v; where ' ' is de ned as 1 + = '+ > 1: Because of R ' ' vs (; ) > 0, the condition R v (v; s) > for all v 2 (0; 1) and all s 2 S is equivalent to R v (v; s min ) > for all v 2 (0; 1) : Moreover, it is also equivalent to the condition that R (v; s) c (v) = R (v; s) c (0) v increases in v for all s 2 S: As a consequence, all agents would prefer the highest feasible quality v = 1 to any other quality (though not necessarily to abstention) at marginal cost pricing p (v) = c (v) Except for points of in ection the converse also holds. Whenever for all v 00 ; v 0 2 [0; 1] ; v 00 > v 0 ; the di erence R (v 00 ; s) R (v 0 ; s) = R v 00 R v 0 v (v; s) dv; is strictly increasing in s for all s 2 S, then R vs (; ) > 0 ;s) except for isolated points ~v 2 [0; 1]. This follows R(v 0 = R v 00 R v 0 vs (v; s) dv: 23 If R (; ) is strictly concave in v (i.e., if R vv (; ) < 0), such a preference for the highest feasible quality is even equivalent to R v (; ) > because then R v (v; ) for some v 2 (0; 1) implies R v (v; ) < for 17

19 Proposition 2. Assume that R v (v; ) > for some intermediate quality level v 2 (0; 1) ; i.e., all customers marginal willingness to pay for quality v exceeds the marginal cost of quality. Then, if there exists a stationary equilibrium where quality v is o ered, its price p (v) is the minimal incentive compatible price, i.e., p (v) = ^p (v) : If the assumption R v (v; ) > holds for all intermediate quality levels v 2 (0; 1), then this result p (v) = ^p (v) holds for each quality level v 2 [0; 1) that is below the highest feasible level. 24 Because of Proposition 2, if customers have a su ciently strong preference (relative to production cost) for high quality, then prices for all quality levels v 2 [0; 1) below 1 are determined by the sum of production and incentive cost. The intensity of competition plays no role. Customer preferences, as long as they satisfy R v (; ) > for v > 0; and the distribution of customer types determine the quantities demanded but not the equilibrium prices (which follow from the incentive constraints only). In addition, whenever two or more rms o er the highest feasible quality v = 1; the price for maximum quality v = 1 is also determined by the incentive constraint, i.e., p (1) = ^p (1) ; because of Bertrand competition. Provided R v (v; ) > for all v 2 (0; 1), Proposition 2 implies that in a stationary equilibrium quality v is a function of the price p. Speci cally, v = 1 [p c (0)] if c (0) p < c (0) + ; and v = 1 if p c (0) + : If beliefs infer v = 0 from p < c (0), prices themselves can serve as quality announcements and no explicit quality announcements are needed. It is useful to compare Proposition 2 with corresponding results of standard oligopoly models of vertical product di erentiation, where quality is observable. In these models, it all v 2 (v; 1) and thus R (v; ) c (v) > R (v; ) c (v) for all v 2 (v; 1) : Notice that when R (; ) is strictly concave in v; R (v; s) c (v) = R (v; s) c (0) v achieves an interior maximum where R v (v; s) = ; provided such an interior maximum exists. Consequently, if R (; ) is strictly concave in v and all customers prefer the highest feasible quality v = 1 to any other quality at prices p (v) = c (v), then it must hold that R v (v; ) > for all v 2 (0; 1) : 24 For any given v 2 (0; 1) it is su cient that R v (v; s) > holds for the (unique) type s (v) that solves R (v; s) R (0; s) = v (thus, s (v) is the indi erent type de ned by R (v; s) ^p (v) = R (0; s) c (0)), whenever such a type exists. Because of R vs (v; s) > 0; this implies R v (v; s) > for all higher types s > s (v) ; but for lower types s < s (v) it may hold that R v (v; s) : If a type s (v) does not exist, either R (v; s) R (v; s) R (0; s) < v for all s 2 S and no type will demand quality v at the price ^p (v) ; or R (0; s) > v for all s 2 S and only then quality v can be available in the market and the assumption R v (v; ) > is needed. If for each v 2 (0; 1) the type s (v) exists and satis es R v (v; s) >, then p (v) = ^p (v) for all v 2 (0; 1) : Moreover, the proposition can be extended to the case where the strict inequality in R v (v; s) > is replaced by the weak inequality. 18