Reputation and Competition

By JOHANNES HO¨ RNER*

This paper shows how competition generates reputation-building behavior in repeated interactions when the product quality observed by consumers is a noisy signal of firms’ effort level. There are two types of firms and “good” firms try to distinguish themselves from “bad” firms. Although consumers get convinced that firms which are repeatedly successful in providing high quality are good firms, competition endogenously generates the outside option inducing disappointed consumers to leave firms. This threat of exit induces good firms to choose high effort, allowing good reputations to be valuable, but its uncompromising execution forces good firms out of the market. (JEL C7, D8)

We are what we repeatedly do. Excellence, Examples of such “experience good” markets then, is not an art, but a habit. include nondurables such as wine, durables —Aristotle, Nicomachean Ethics such as appliances and cars, and most service providers such as lawyers, mechanics, and air- In traditional competitive theory, economists lines.1 In these settings, a consumer’s experi- assume that market participants have complete ence with a particular product becomes a knowledge of all relevant factors. This assump- precious guide—providing imperfect information has long been criticized as limiting the tion about a combination of the seller’s efforts, applicability of the theory, especially when ability, and luck. In these markets a seller’s competition is thought of as a dynamic process. reputation for quality therefore becomes a valu- (See, e.g., Friedrich A. Hayek, 1946; Joseph A. able asset. Schumpeter, 1950.) Indeed, the shortcomings of Since the seminal work of Benjamin Klein this approach are particularly clear when we and Keith B. Leffler (1981), several authors acknowledge that competition is in a large mea- have shown formally how firms in such markets sure competition for reputation and consumer may be induced to exert costly effort when the good will. The costly and apparently unwaver- fear of losing their reputation exceeds the tem- ing efforts many firms make in order to estab- porary advantage of cheating their customers. lish and maintain a reputation for excellence are (See Carl Shapiro, 1983; Russell Cooper and difficult to account for in the traditional Thomas W. Ross, 1984; Bengt Holmstrom, framework. 1999.) These theories of rational reputation Consider for instance the markets in which building must contend with two fundamental customers can only assess the quality of a sell- problems, as described by Joseph E. Stiglitz er’s product by purchasing and consuming it. (1989). First, a consumer’s refusal to purchase from a firm that has sold her a low-quality good must also be rational. In particular, it must be * Managerial Economics and Decision Sciences, optimal for a consumer to end a long relation- Kellogg School of Management, Northwestern University, ship with a firm she had considered trustworthy 2001 Sheridan Road, Evanston, IL 60201. This is a revised version of Chapter 1 of my Ph.D. dissertation, and I would after perhaps just a short string of bad experi- like to thank my adviser, George Mailath, for his invaluable ences. Second, as a reputation is only valuable if help and support. I would also like to thank Steven Mat- success brings profits, how could those possibly be thews, Andy Postlewaite, and Larry Samuelson for their driven down to zero in a competitive environment? encouragement and helpful advice, and particularly Dan Silverman, Steven Tadelis, and two referees for detailed comments. Any remaining mistakes are mine. The author gratefully acknowledges financial support from an Alfred P. 1 Phillip Nelson (1974) introduces the concept of an Sloan dissertation fellowship. experience good. 644 VOL. 92 NO. 3 HÖRNER: REPUTATION AND COMPETITION 645

However, the most formidable obstacle that a and this proposed equilibrium with high effort theory of rational reputation building faces re- unravels. mains the argument developed by Holmstrom Under persistent competition, however, firms’ (1999), in the framework of a monopoly, which revenues do not vary continuously with con- will be examined shortly. His reasoning shows sumers’ expectations. Because any consumer why there exists no equilibrium in which a can break off her relationship with a firm and single firm repeatedly exerts high effort. take some other preferred option, it does not This paper suggests how these three prob- matter how good a firm is thought to be, but lems may be simultaneously overcome and the- rather whether it is thought to be better than ories of competition and reputation reconciled. its rivals. This outside option, endogenously It offers a model in which many consumers and generated by competition, is precisely what is firms repeatedly trade. In each period, consum- required for consumers’ behavior to exert effec- ers pay up front for a good the expected quality tive discipline over sellers. of which increases with the effort exerted by the This crucial outside option will exist as long firm. Some firms, however, are inherently lazy as operating rivals with worse, similar, or better (inept) and always exert low effort. Consumers reputations charge appropriate prices. In view cannot distinguish inept firms from opportunis- of the price dispersion that these conditions tic firms, do not observe the effort level exerted, imply, it may seem that a firm could profitably and are thus confronted with both adverse se- attract consumers with a slight price decrease. lection and moral hazard problems. On the other This need not be so, however, if consumers hand, consumers do observe the prices posted believe that such a firm is bound to exert low by firms and (the existence of) their customer effort as a result. Indeed this consumer concern bases. Moreover, for those trades in which they is justified if the sequence of equilibrium prices were personally involved, consumers of course posted by a firm over time is so low that any realize the quality obtained (but since the out- further decrease in price necessarily violates its come reflects both the luck and the effort of the incentives to exert high effort. firm, this monitoring is imperfect). Consumers In this uncertain environment consumers are may freely switch firms at any time. able to identify preferred options because they I show in this setting how competition sup- observe both the prices and the customer bases ports the existence of equilibria in which oppor- of other firms. Importantly, it is the behavior of tunistic firms always exert high effort—a result the other consumers that ensures that the infor- that contrasts sharply with the results obtained mation conveyed by prices is dependable. In for a monopoly. To illustrate the importance particular, consumer knowledge of customer of competition for maintaining effort, consider bases prevents firms from raising their prices in the following version of Holmstrom’s argu- order to mimic rivals with better reputations. If ment (see also George J. Mailath and Larry a firm tried to do so, the flight of its old cus- Samuelson, 1998a). Suppose that in equilibrium tomers would effectively deter any potential a monopolistic (opportunistic) firm chose to ex- new ones. ert high effort always, and that its revenues The model sketched above has many equilib- varied continuously with consumers’ expecta- ria distinguished by the degree of patience tions about the efforts of the firm. As the firm’s shown by consumers when they experience low average product quality converges almost quality. In some equilibria, consumers may be surely to the expected quality associated with willing to stick with a firm in spite of a low- high effort, consumers become convinced quality experience, provided that the price de- that this firm is opportunistic rather than inept. creases sufficiently. This paper focuses instead Since this type of firm is always supposed to on the simpler, more dramatic case in which exert high effort, over time consumers’ expec- consumers are so exacting that they leave a firm tations—and hence the firm’s revenues— as soon as it disappoints them with a low- become inelastic to a bad outcome. In these quality experience. In these equilibria, the firms circumstances, low-quality outcomes will sim- that retain loyal customers are therefore those ply be attributed to bad luck. But then, of that always provide high quality; and the repu- course, the firm has an incentive to slack off tations of these firms increase with their age. 646 THE AMERICAN ECONOMIC REVIEW JUNE 2002

Along with the zero-profit assumption, the I. The Basic Model conditions described above uniquely identify the equilibrium prices and market shares. Prices This section outlines a model which isolates are shown to rise with a firm’s reputation, a the role of competition in providing incentives. result that accords with the findings of the em- For simplicity, it does not allow for entry of pirical literature on reputations (see, for in- firms after the beginning of the game and pos- stance, Allen T. Craswell et al., 1995). In tulates an exit behavior for firms that is justified addition, prices are initially low enough to deter in the richer model of Section II. fly-by-night attempts, and eventually reflect a premium for quality. A. The Market In the related literature, the exit option mechanism of this paper is close to that of Albert O. Consider a market in which firms and con- Hirschman (1970), though its role is rather com- sumers repeatedly trade. Time is discrete, in- plementary. Hirschman discusses how consum- dexed by t ϭ 0, 1, . . . and the horizon is ers’ exit may constitute a mechanism of infinite. In each period, firms and consumers recuperation for firms that accidentally fell be- may trade. In this event, a consumer pays the hind. This paper shows instead why the exit firm up front and enjoys either a good outcome option keeps a firm “on its toes” in the first or a bad outcome, depending on the unobserved place. The literature on the market for reputa- effort level exerted by her chosen firm. Firms tions which focuses on the economics of name are of two types, which is private information. trading and on the incentives such markets may Good firms choose either high or low effort, generate is also germane (see David M. Kreps, while bad firms only exert low effort. Normal- 1990; Mailath and Samuelson, 1998b; Steven ize the cost of high and low effort to c ʦ (0, 1) Tadelis, 1999, 2000). Research in credit mar- and 0 respectively. High effort generates a good kets is also related (see Douglas W. Diamond, outcome, or success, with probability ␣ which is 1989; Harold L. Cole et al., 1995; Mitchell A. larger than the corresponding probability ␤ gen- Petersen and Raghuram G. Rajan, 1995; Cesar erated by low effort. Specifically, 0 Ͻ ␤ Ͻ Martinelli, 1997). Diamond (1989), for in- ␣ Ͻ 1, so that a bad outcome, or failure, may stance, investigates reputation formation and occur even under high effort (all the results also the evolution of reputation’s mitigating effects hold for ␣ ϭ 1). Firms maximize their payoff, on the conflicts of interest between borrowers that is, the expected discounted sum of profits. and lenders. Diamond shows that, with suffi- In the initial period, the measure of firms is one, cient adverse selection, reputation does not ini- and good firms account for a fraction ␸0 ʦ (0, tially provide adequate incentives to borrowers 1) of the market. with short credit histories. Over time, however, Consumers (also referred to as clients or cus- when a good reputation is acquired, reputation tomers) are infinitely-lived, and their measure is provides improved incentives. one. In each period, every firm can serve a In Section I, I develop a basic model in which continuum of consumers, while each consumer the role of competition in easing the problem of may pick only a single firm.2 All the consumers preserving reputations appears very clearly. of a given firm experience the same outcome. This simple game focuses on the mechanisms of (This simplifying assumption is not necessary the exit option, their generality and importance, for the results, as discussed in Section III). and on the relationship between prices, incen- Consumers are Bayesian: they have beliefs over tive constraints, and the zero-profit condition. firms’ types and use all the available informa- Section II then develops a richer framework, tion to update their beliefs according to Bayes’ building on the basic model to show how its rule. They know ␸0, but do not know the type insights extend to a world with both entry and of a particular firm. A consumer maximizes the exit of firms, and in which consumers may choose among firms with diverse reputations. Section III discusses the role of the assumptions 2 Specifically, if the unit measure of consumers was and the robustness of the results. Section IV divided equally among the unit measure of firms, each firm concludes. would serve a unit measure of consumers. VOL. 92 NO. 3 HÖRNER: REPUTATION AND COMPETITION 647

FIGURE 1. TIMING OF MOVES IN A PERIOD.

expected discounted sum of utilities, and prefers previously quit observe the price distribution set good outcomes to bad ones. The utilities asso- by firms. Third, loyal consumers observe their ciated with a good and a bad outcome are nor- own firm’s new price. Fourth, firms observe malized to 1 and 0, respectively. In each period, whether they have consumers willing to trade. consumers may either trade or take their outside Fifth, consumers do not observe the effort level option, which provides a utility of ␥ ʦ [␤, ␣]. exerted by firms and only observe the outcome The outside option can be thought of as the generated by the firm they chose. value of their next best alternative. For instance, To complete the description of the informa- ␥ ϭ ␤ is the outside option if the alternative tion structure, the following assumptions are consists of a separate competitive sector com- made, but do not affect the results. They are posed only of bad firms. At the end of each chosen both because they are simple and mini- period, consumers may freely leave (or quit) mal, and because they make consumers’ switch- their firm and switch to another one, as ex- ing less attractive, and therefore reputations plained below. The discount rate, ␦ ʦ [0, 1), is harder to preserve. common to both firms and consumers. A client of firm j knows the sequence of The game proceeds as follows. In the initial prices and outcomes of this firm since she began period, consumers observe prices which have trading with it. However, she does not observe been simultaneously set by firms and either take prices or outcomes of any other firm during that their outside option or trade with a firm ran- relationship. If she quits firm j, she is hence- domly selected from the firms posting the price forth unable to distinguish it from other firms. they choose. Firms then simultaneously and in- That is, while she recalls all the prices she paid dependently exert an effort level which gener- and the outcomes she experienced, she only ates an outcome for each firm. Consumers then distinguishes between two kinds of firms at the decide whether to stay or leave. More generally, end of a period: the one with which she just at the end of period t ϭ 0, 1, ... , after trading traded, if any, and all the others for which, in with a firm, consumers decide whether to be the event she quits, she only observes the posted loyal and remain with their firm, or to quit and prices.3 This means that a switching consumer become a switching consumer. Quitting is cost- only observes the price distribution and chooses less. At the beginning of the next period, firms a price rather than a particular firm. It is ines- simultaneously set prices or exit. Consumers sential but convenient to assume that a firm then simultaneously decide whether to trade or posting such a price has then a probability of to take their outside option. The loyal consum- obtaining her patronage proportional to its actual ers only consider trading with the firm previously chosen, while consumers who quit may 3 also choose at that point with which firm to Assuming instead that loyal consumers and switching consumers have symmetric access to information, for in- trade. Finally, firms simultaneously exert an stance, would not affect the results, although consumers effort level which generates an outcome. The who stay would play a weekly dominated strategy in that timing of moves is summarized in Figure 1. case. Here instead, consumers who decide whether to quit The following five assumptions about the in- face a trade-off between potentially valuable information and the opportunity to choose among all the firms, as formation structure are important. First, firms identified through their price. In general, factors which observe whether they have loyal consumers be- make quitting more attractive only reinforce the competitive fore choosing a price. Second, consumers who pressures described later. 648 THE AMERICAN ECONOMIC REVIEW JUNE 2002 size. That is, the probability of obtaining her point in time differ at most by sets of consumers patronage is proportional to the mass of its per firm of measure zero. Equilibrium strategies loyal consumers, as a fraction of the total size of are symmetric, that is, consumers choose iden- firms posting such a price. Finally, a consumer tical strategies, as do firms of a given type. who chose her outside option in a period starts Strategies are formally defined and discussed in the next one as a switching consumer would. Appendix A. Firms observe their own outcomes as well as The focus is on establishing conditions under the price distribution, but not the outcomes of which high-effort equilibria exist; i.e., equilibria other firms. Furthermore, firms observe the ac- in which good firms always exert high effort on tual measure of their loyal consumers and of the equilibrium path. Of particular interest are those consumers who agree to trade with them. nonrevealing high-effort equilibria, which are Formal definitions of histories are given in Ap- defined to be high-effort equilibria in which all pendix A. operating firms’ strategies specify identical It is further assumed, in this section only, prices in each period on the equilibrium path that firms exit if and only if all their consum- (but that price may of course vary over time), ers quit (that is, if and only if the measure of and consumers always trade rather than use remaining consumers is zero). While it is innoc- their outside option (on the equilibrium path). uous to assume that a firm does not exit otherwise Although focusing on such symmetric strategies (after all, it may always choose low effort and set is certainly restrictive at this point, it permits a positive prices, which guarantees nonnegative simple illustration of the impact of competition profits), forcing a firm to exit as soon as all its on incentives. consumers quit is stringent at this point. It cap- tures the intuition, formalized as a result in B. From Competition to Selection Section II, that such a firm, having no customer base, is indistinguishable from entrants, and Consider then a nonrevealing, high-effort therefore realizes zero profits in a competitive equilibrium. At the end of each period, partition equilibrium. the consumers into groups according to the his- An equilibrium of this game refers to a Per- tory of outcomes and quitting decisions which fect Bayesian Equilibrium in symmetric, pure, they have experienced. Since a consumer may Markovian strategies. Markovian strategies are follow the strategy of switching in every period strategies in which consumers’ decisions only tЈ м t, her total equilibrium payoff at the end of depend on firms’ prices and on their own beliefs period t is at least the weighted average of the about the firms’ types, while firms decisions groups’ payoffs. This simply means that, by only depend on consumers’ beliefs and willing- switching in every period, a consumer can en- ness to trade. These strategies only depend on sure herself a payoff of: payoff-relevant variables and are therefore particularly simple. In each period, consumers de- ϱ cide whether to trade (and possibly with which s Ϫ t i (1) ␦ usdi firm to trade) depending on the prices they ͸ ͵ sϭt ϩ 1 ͓0,1͔ observe and the beliefs that they associate with those prices. Their belief about their particular ϱ firm is then updated according to the outcome ϭ ␦s Ϫ tui di which they observe, given the equilibrium strat- ͵ ͸ s egies (as are their beliefs about other firms, on ͓0,1͔ s ϭ tϩ1 the basis of which they decide whether to stay i or quit). Similarly, a firm sets its price as a where us is the utility of consumer i in period s. function of the measure of its consumers and This implies that all consumers have the their beliefs. Their updated beliefs and measure same total expected continuation payoff at the then determine the effort level chosen by a good end of each period, regardless of her history, firm. It is assumed that the equilibrium actions for otherwise a consumer of the least favored of each firm are constant on histories in which group would gain from deviating. This in the set of consumers taking a decision at each turn implies that all consumers have the same VOL. 92 NO. 3 HÖRNER: REPUTATION AND COMPETITION 649 expected utility per period, and therefore the rium prices, the worst punishment that can be same beliefs about their current firm. In a inflicted upon disappointing firms is the con- high-effort equilibrium, customers of firms sumers’ best alternative, and the patronage of that have been repeatedly successful certainly switching consumers represents the best possi- do not leave (these firms would not exert high ble reward for successful firms. As time passes, effort otherwise). Therefore, all consumers the dynamics of competition generate the out- must have such a maximal belief about their side option required for consumers to optimally firm, and thus they must quit as soon as they quit after a bad outcome, but not after a good one. experience a bad outcome. This establishes: While best for incentives, these equilibria leave little hope of survival for a particular firm, PROPOSITION 1: In any nonrevealing, high- whether good or bad. Since high effort gener- effort equilibrium, the only firms operating in a ates a bad outcome with positive probability, a given period are those firms which have not constant fraction of good firms is compelled to experienced a single bad outcome up to then. exit in every period. This is true even though consumers know that, eventually, almost all Observe that the argument, and therefore the firms which operate are of the good type. From result, do not require strategies to be Markovian this perspective, it may seem that modeling or symmetric, and all a consumer needs to ob- firms as a continuum is more than a technically serve are the outcomes she experiences. It is convenient assumption. The model of Section II certainly not surprising that it is a (symmetric) shows why this concern is unfounded, and also equilibrium strategy for consumers to quit as shows that the unbounded features which this soon as they are disappointed, that is, as soon as basic model exhibits are not necessary for the they experience a bad outcome. It may be more argument to hold. However, the reasoning and surprising that there are no other outcomes in- the result break down if there are switching duced by equilibrium strategies. Indeed, if some costs or if the average belief over the effort level consumers were willing to show more patience of the remaining firms is noisy. This is because, with disappointing firms, one might expect for any given cost, there is a time at which that others would optimally choose to do beliefs of consumers are so close to one that the so too, since they may prefer the potentially second best outcome does not induce them to valuable information gathered during their rela- quit any more, since posteriors do not change in tionship to the uncertainty associated with an response to different outcomes. unknown firm.4 A “market for lemons” logic The focus on nonrevealing equilibria is es- is at work here: among those consumers who sential for Proposition 1 to hold. More sophis- are supposed to be loyal, those with the ticated equilibria exist, in which firms with most pessimistic beliefs only gain from quitting different histories charge different prices and instead, so that only the most optimistic ones consumers are willing to forgive a firm for a bad remain loyal. In particular, if there were more outcome. This issue is tackled in Section III. than two outcomes, and more desirable out- First however, it remains to be established un- comes were also more likely to be generated der which conditions nonrevealing, high-effort by high effort, then the operating firms would equilibria do indeed exist, and what the features be precisely those which have produced noth- of equilibrium prices are. ing but the best possible outcome in every period. Notice that in such an equilibrium, by pursu- ing their own myopic interest, consumers pro- C. The Equilibrium with Reputation vide firms with the strongest possible incentives to exert high effort. That is, given the equilib- In any nonrevealing, high-effort equilibrium, operating firms are precisely those which have been repeatedly successful. Therefore, the prob- 4 At this point, it should be clear that giving less infor- ability assigned by consumer i in period t that mation to consumers about their current firm or more infor- i mation about other firms (as would be the case if her current firm is good, consumer i’s belief ␸t, information was symmetric between loyal and switching is obtained by t successive applications of consumers) would only reinforce the conclusions. Bayes’ rule, conditional on the occurrence of a 650 THE AMERICAN ECONOMIC REVIEW JUNE 2002 good outcome, given that good firms exert high initial period if the discounted continuation effort, starting with ␸0. This number is denoted value per consumer exceeds the cost of effort by ␸t. More precisely, [as can be seen from equation (4) with t ϭ 0]. As a consequence, the initial price p0 must be (2) ␸t ϭ ␣␸tϪ1/͓␣␸tϪ1 ϩ ␤͑1 Ϫ ␸tϪ1͔͒ t м 1. negative if good firms realize zero profits overall. Intuitively, this negative price prevents the Because only a fraction ␣␸t Ϫ 1 ϩ ␤(1Ϫ ␸t Ϫ 1) outbreak of “fly-by-night”firms that could make a of firms operating in period t Ϫ 1 still operates profit by entering the market, exerting low ef- in period t, the mass of consumers per operating fort, and leaving immediately (as in Shapiro, firm in period t, or market size, nt, satisfies: 1983). With a positive initial price and costless low effort, bad firms would clearly realize zero ͑3͒ nt ϭ nt Ϫ 1 /͑␣␸tϪ1 ϩ ␤͑1 Ϫ ␸t Ϫ 1 ͒͒ profits, as they could always choose positive prices and low effort thereafter. Since good t м 1 n0 ϭ 1. firms exert high effort persistently in a high- effort equilibrium and charge a negative price Three conditions are necessary for a nonreveal- initially, some prices must necessarily exceed ing, high-effort equilibrium to exist. First, the cost later on, reflecting a premium for quality. equilibrium value of a good firm per consumer In particular, these premia must keep on being in period t, Vt, must exceed the payoff from a collected frequently enough. one-shot deviation to low effort. That is, It is instructive to further restrict attention to the special case in which good firms are pre- (4) Vt ϭ pt Ϫ c ϩ ␣␦nt ϩ 1 Vt ϩ 1 /nt cisely indifferent between high and low effort in every period. This state of affairs can be thought

м pt ϩ ␤␦nt ϩ 1 Vt ϩ 1 /nt @t м 0 of as the outcome of Bertrand competition which drives prices down to a level below which it is common knowledge that incentives where pt is the common price charged in period cannot be maintained if market operations keep t. Second, consumers must prefer trade to the on taking place according to the equilibrium outside option: outcome. More precisely, such a sequence satisfies the property that no price can be lowered without violating some incentive constraint of (5) ut ϭ ␣␸t ϩ ␤͑1 Ϫ ␸t͒ Ϫ pt м ␥ @t м 0. the good firm, holding both its other prices and market sizes fixed at their equilibrium levels. It Finally, in any period, the equilibrium price is true that a lower price could be interpreted by must be optimal for the firm given consumers’ sophisticated consumers, in the spirit of forward beliefs, and in particular, given how consumers induction, as a signal from a firm that its future interpret deviations to out-of equilibrium prices. prices and market sizes will also differ from If any such deviation is perceived as certainly their equilibrium values, possibly in a way such coming from a bad firm, then there may be that high effort was and remains persistently many equilibrium price sequences, but all these optimal. On the other hand, in view of a price sequences and associated belief functions need cut, consumers’ skepticism may never be more not be equally reasonable. legitimate than when prices are supposed to be Equilibria in which all firms, whether good or at the minimal level compatible with the good bad, realize zero profits overall are of particular firms’ incentives. This motivates the following interest, both because they satisfy a standard definition. assumption of competitive theory that could certainly be derived as a result in a larger game allowing for free entry, and because this condi- Definition 1: A competitive equilibrium with tion is rarely compatible with the emergence of reputation, or competitive equilibrium, is a non- reputations in the earlier literature (with the revealing equilibrium in which good firms exert notable exception of Shapiro, 1983). Now good high effort on the equilibrium path, the incen- firms are only willing to exert high effort in the tive constraint of equation (4) holds with equal- VOL. 92 NO. 3 HÖRNER: REPUTATION AND COMPETITION 651 ity for any t м 0, and firms’ payoffs are zero in in Appendix B, states that this willingness is the initial period. also sufficient for existence.

Notice that, since the incentive constraint of the initial period binds in a competitive equilib- THEOREM 1: A competitive equilibrium ex- rium, bad firms realize zero profits overall as ists if and only if long as good firms do, so that one of those constraints is redundant. More importantly, (7) min͕␣␸t ϩ ␤͑1 Ϫ ␸t ͒ Ϫ p*t ͒} м ␥. prices are uniquely determined. t

This condition is obviously satisfied when- PROPOSITION 2: If a competitive equilibrium ever the cost of effort is low enough and the ϱ exists, the equilibrium prices {p*t}tϭ0 are given outside option ␥ is hardly better than the utility by ␤ generated by low effort. On the other hand, it is violated if firms are very impatient (small ␦) ␣c or if monitoring is of bad quality [small (␣ Ϫ (6) p* ϭϪ 0 ͑␣ Ϫ ␤͒ ␤)/␣]. This condition is clearly necessary for existence, as consumers would use their outside c ␸t Ϫ 1 option at some point otherwise. If this condition p*t ϭ ␣ Ϫ ␤␦ @t м 1. is satisfied, consumers are willing to trade, and ␦͑␣ Ϫ ␤͒ ͩ ␸t ͪ as by construction good firms prefer to exert high effort, it only remains to check that firms They are strictly increasing over time and con- choose to post the equilibrium prices, given verge to consumers’ beliefs. Lower prices are unattrac- tive for firms if such price cuts are interpreted as ͑1 Ϫ ␦͒/␦ p*ϱ ϭ 1 ϩ c. certainly coming from bad firms, an interpreta- ͩ ͑␣ Ϫ ␤͒/␣ͪ tion which has been motivated before. Higher prices are hardly attractive either, provided that The proof is in Appendix B. As expected, the consumers do not believe that such higher initial price is negative, to ensure that firms prices are more likely to come from a good firm realize zero profits overall. Prices gradually rise, than the equilibrium price is. slowly early on if the fraction ␸0 is low enough, Among the unsatisfactory features of this and eventually converge to a level which re- simple model, notice that the equilibrium flects a premium for quality. The eventual pre- market size (per firm) increases at a geometric mium, (p*ϱ Ϫ c), is simply the cost times the rate. As a consequence, the total value of a ratio of the interest rate (1Ϫ␦)/␦ over the qual- firm, ntVt ϭ ntc/␦(␣ Ϫ ␤), diverges over time. ity of monitoring, as measured by (␣Ϫ␤)/␣. Moreover, the whole argument seems to rely Perhaps more surprising, prices increase with on the persistence of competition among firms the initial fraction of good firms ␸0. If this which are repeatedly successful. These are fraction is high, fewer firms fail in each period, inexorable consequences of a model with exit and thus successful firms attract fewer addi- but no entry, but they do not challenge the tional consumers, reducing their incentives to robustness of the main insights of the paper, exert high effort, which must therefore be con- as the next section shows. veyed by higher prices. In every period, a fraction of good firms is forced to exit, even though II. A Richer Model and its Stationary it is common knowledge that the fraction of bad Equilibrium firms vanishes over time. However, good firms expect to operate longer than bad firms, so that This section extends the previous model to an the future higher prices provide incentives to economy in which both entry and exit of firms good firms without generating rents for bad ones. occur. As before, on the equilibrium path, good Of course, consumers must be willing to firms always exert high effort because of their trade at those prices. The next theorem, proved reputation, and consumers quit a firm as soon as 652 THE AMERICAN ECONOMIC REVIEW JUNE 2002

FIGURE 2. TIMING OF MOVES IN A PERIOD. they experience a bad outcome. Moreover, the made to the basic model. The timing of moves firms’ decision to exit after a bad outcome is is summarized in Figure 2. now voluntary, and consumers freely choose Although this market cannot be formally among the offerings of firms with various rep- modeled as a game, Markovian strategies are utations. This global competition determines the still well defined. A nonrevealing, high-effort distribution of market sizes, in addition to the equilibrium is then a family of Markovian strat- equilibrium prices. egies and of consumers’ beliefs such that beliefs The previous framework is enriched as fol- are correct, strategies are mutual best responses, lows. Time extends into the infinite past as well and (i) consumers (mixed) strategies are identi- as into the infinite future. The total mass of cal, as is the restriction to prices of firms’ pure (doubly infinitely lived) consumers is normal- strategies, and, on the equilibrium path, (ii) ized to one, as is the mass of firms entering the good firms always exert high effort, (iii) con- market in every period. Because in the station- sumers always trade and quit as soon as they ary model analyzed below, an equal mass of experience a bad outcome, and (iv) firms opti- firms exits, the measure of operating firms re- mally exit when no loyal consumer remains. mains constant over time. A fraction ␸0 ʦ (0, In such an equilibrium, firms of a given age, 1) of entrants are of the good type. that is, firms that have been operating for a The assumption previously made about exit given number of periods, post the same price. is dropped. Instead, firms freely decide whether However, firms of different ages may charge to exit. However, the timing and information different prices. Hence, a variety of alternatives structure are modified as follows. As before, is available to consumers, and it is therefore consumers decide whether to stay or to go at the convenient to allow them to independently ran- end of a period, on the basis of which firms then domize, both when they decide whether to stay choose a price or exit. Loyal consumers, that is, or to quit, and when they choose among the consumers who chose to stay decide then different offerings. (The measurability issues whether to trade. Those (loyal) consumers who raised by mixed strategies in such a context, agree to trade form the customer base. Switch- discussed in Robert J. Aumann, 1964, for in- ing consumers, that is, consumers who quit pre- stance, could be avoided at the expense of ex- viously, observe the joint distribution of prices positional simplicity.) Observe that, unless and of the existence of a customer base, and consumers leave with probability one even after decide then with which firm to trade, if with a good outcome in some circumstances, suc- any. Hence, rather than only observing prices, cessful firms always retain some loyal consum- they also observe whether a given firm, as iden- ers. The logic underlying Proposition 1 applies tified by its price, has a customer base or not. here too: if any kind of firm of age i retains (Assuming instead that they observe the actual loyal consumers, it must be those firms which size of the customer base would only reinforce have been repeatedly successful, and consum- the results.) This enables switching consumers ers, who could as well trade with the successful to make inferences from the price about a firm’s ones, must therefore have quit the other unlucky reputation on the basis of possibly dependable firms from the same generation.5 information. In view of the well-known attempts made by various suppliers to appear 5 popular, there is little doubt that such informa- As a referee pointed out, because the equilibria of interest are stationary ones, the formal analysis should be tion, if available, is indeed a determining factor restricted to a generic subset of [0,1]2 for the probabilities ␣ in consumers’ decisions. No other change is and ␤, because, in equilibrium, the exit decision of a VOL. 92 NO. 3 HÖRNER: REPUTATION AND COMPETITION 653

Notice that, for consumers to be indifferent and the mass of firms of age i, ␭i, satisfies among alternatives, older firms, which in equilibrium have better reputations, also charge higher (9) ␭i ϭ ͓␣␸iϪ1 ϩ ␤͑1 Ϫ ␸iϪ1͔͒␭iϪ1 prices. Hence, the prices set by a firm increase with its age. Suppose furthermore that entrants i м 1 ␭0 ϭ 1 realize zero profits. First, as before, it implies that the initial price must be negative, since the value while the market size of a firm of age i, per consumer of a firm of age one must exceed the ni , satisfies cost of effort. Second, as a firm whose consumers have all quit is indistinguishable from an entrant, c such a firm must also realize zero profits and may (10) ni ϭ 2 niϪ1 ␦͑␣ Ϫ ␤͒ ͑␸i Ϫ ␸0͒ then just as well exit.

Attention is focused, for the same reasons as ϱ before, on a particular class of equilibria. м ␭ ϭ i 1 ͸ ni i 1. iϭ0 Definition 2: Astationarycompetitiveequilib- rium with reputation, or stationary equilibrium, is As expected, prices rise over time, from an anonrevealing,high-effortequilibriuminwhich initial negative level to an asymptotic level the payoff of firms entering the market is zero, and which is larger than the cost of effort under the good firms are always indifferent between high conditions given below guaranteeing existence and low effort on the equilibrium path. of a stationary equilibrium. It may appear coun- terintuitive that prices decrease with the cost of effort, as well as, for i large enough, with the In a stationary equilibrium, in each period, a prior ␸0. If the cost of effort increases, for measure ␭i of firms of age i charge price pi and instance, the continuation value per consumer each serves a mass ni of consumers. Their value of a good firm of age one must increase, so as to per consumer, Vi , does not depend on their preserve incentives of entrants. This increase is type, and therefore satisfies V0 ϭ 0, as well as exacerbated by the imperfect monitoring, as ͑ni ϩ 1/ni͒Vi ϩ 1 ϭ c/␦͑␣ Ϫ ␤), all i. Moreover, measured by (␣ Ϫ ␤)/␣, and therefore exceeds since consumers must be indifferent among the the increase in cost, so that, for entrants to offerings of firms of different ages, both the realize zero profits, the initial price must de- equilibrium sequence of prices and market sizes crease. But older firms have then to decrease are determinate. Recall that ␸i is simply the their price so as to remain competitive. The probability that a firm is good, given that good growth rate of a firm of age i, ͑ni ϩ 1 Ϫ ni͒/ni, firms exert high effort and a string of i good decreases with its age, but increases with the outcomes occurs. prior ␸0 for i sufficiently large. Indeed, when the fraction of good firms among entrants increases, the utility derived by consumers from PROPOSITION 3: In any stationary equilib- trading with those firms grows, as their price rium, prices pi are given by does not depend on ␸0. Older firms must then lower their price. Since incentives are conveyed ␤ either through future premia or through growth (8) p ϭ ͑␣ Ϫ ␤͒͑␸ Ϫ ␸ ͒ Ϫ ciм 0 i i 0 ␣ Ϫ ␤ perspectives, lower prices then imply higher growth rates. This is not true on the other hand for younger firms if ␸0 is sufficiently low, for consumer only depends on her belief about her own firm then an increase of the prior actually enhances and not on her beliefs about other firms, as beliefs about the reputational advantage of young firms over other firms are stationary. Hence, it is necessary that the entrants. belief generated by a string of good outcomes followed by a bad one be different from the belief generated by any It is then easy to understand why consumers string composed only of good outcomes. Enlarging slightly may randomize their decisions, to ensure that the the consumers’ state space would do as well. prescribed equilibrium distribution of consumers 654 THE AMERICAN ECONOMIC REVIEW JUNE 2002 across firms obtains. As a consequence, the measure of consumers who quit exceeds the size of the customer bases of unlucky firms. Two conditions are necessary for the existence of a stationary equilibrium. First, prices must be low enough so that consumers agree to trade. Second, as the measure of consumers is finite, the eventual growth rate of a firm must be accordingly bounded. Specifically, for any ϱ choice of (positive) n0, the series ¥i ϭ 0 ni␭i must converge, where the terms are recursively 12345678910 defined as in Proposition 3. These conditions are also sufficient. FIGURE 3. INDUSTRY DYNAMICS

Note: ␣ ϭ 0.7, ␤ ϭ 0.35, ␦ ϭ 0.99, c ϭ 0.1, ␾0 ϭ 0.05. THEOREM 2: A stationary equilibrium exists if and only if Observe that, since equilibrium existence requires that the share of business allotted to old ␣ Ϫ ␤ 2 firms vanishes with their age, and prices converge, (i) c Ͻ ␣␦ (1Ϫ␸ ) ͩ ␣ ͪ 0 the total equilibrium value niVi of a firm converges too. This contrasts with the results obtained in the and basic model. On the other hand, the market size may or may not converge depending on the pa- ␤ rameters, as long as it does not grow at a rate (ii) ␥ Ϲ ␣␸ ϩ ␤(1 Ϫ ␸ ) ϩ c . larger than ␣,therateatwhichgoodfirms exit. 0 0 ␣ Ϫ ␤ Depending on the specificcharacteristicsofthe market and the technology, firms lucky enough to Condition (ii), which ensures that consumers survive may then either grow without bound or are willing to trade, is satisfied whenever the experience the sequence of youth, maturity and outside option ␥ is sufficiently close to ␤. Con- declining vigor, which Alfred Marshall (1961) dition (i), which is equivalent to the conver- stressed. Be that as it may, the value of a firm gence of the series aforementioned, gives an remains bounded, and more importantly, the logic upper bound on the cost of effort, equal to the of competition implies that “success brings credit product of the effective discount rate of good and credit brings success; credit and success help firms (that is, the product of the discount and the to retain old customers and bring new ones” (Mar- hazard rates), the (square of) quality of moni- shall, 1961 p. 315). An illustration of industry toring (␣ Ϫ ␤)/␣, and the initial fraction of bad dynamics for a generation of firms is given in firms. It is intuitive that reputations are easier to Figure 3. preserve when success is statistically more in- Notice finally that the argument developed formative, or good firms more patient. With here does not rely on the cardinality of the set of many entrants of the good type, prices of older firms. An older firm with an excellent reputation firms must be set low enough to remain com- does not only face rivals with similar good- petitive, and the resulting rents from success wills, but also younger firms offering cheaper may be too low to provide incentives. Notice services, whose competition alone suffices to also that this condition implies that c Ͻ ␣ Ϫ ␤. exert effective discipline over the older firm. That is, a stationary equilibrium exists only if high effort is the efficient action. III. Discussion Theorem 2 is proved, and the equilibrium strategies described, in Appendix B. These A. Pricing strategies are the straightforward analogues of those detailed in the previous section and are In the equilibria examined so far, consumers therefore not discussed here. have been extremely alert, quitting a firm as soon VOL. 92 NO. 3 HÖRNER: REPUTATION AND COMPETITION 655 as that firm failed to keep up with the highest outcome is relaxed. While some industries dis- standards. As Proposition 1 shows, consumers’ play such characteristics (i.e., arts, entertain- behavior is necessarily exacting in an equilibrium ment, etc.), in many others, consumers’ latest with reputation, as long as firms of a given age set experiences are only imperfectly correlated. identical prices in spite of possibly different his- Thus, suppose that, although the fraction of tories. There is however no reason to focus exclu- consumers enjoying a good outcome increases sively on such pricing strategies. Firms may with the effort level, satisfied and dissatisfied compensate consumers for a loss in reputation by consumers necessarily coexist, in sizes deter- an appropriate reduction in prices, so as to retain at mined by the effort level in a deterministic least some of their clients. The pay of lawyers, fashion. Specifically, under high (respectively physicians, consultants, or even professional ath- low) effort, a fraction ␣ (respectively ␤) of letes, for instance, typically reflects their recent consumers experience a good outcome (of performances. In such markets, the restriction to course, a consumer only observes her own re- nonrevealing equilibria is importantly limiting. As alization of the effort level). Assume, to ensure consumers’ exit becomes less threatening, the finiteness of the mass of firms operating, that scope for competitive equilibria with reputation is firms may be exogenously forced to exit in any reduced for two reasons. First, the punishment for period with probability ␪ Ͼ 0 which can be failing to provide high quality is alleviated, since arbitrarily small. In such a modified framework, unsuccessful firms may keep on operating profit- a stationary high-effort equilibrium in which ably. Second, since some consumers remain loyal entrants realize zero profits exists under condi- to such firms, successful ones fail to attract as tions very similar to those of the equilibrium many additional consumers as they would other- derived in the previous section. On the equilib- wise, reducing accordingly the reward from pro- rium path, consumers remain loyal if and only if viding high quality. As a consequence, reputations they experience a good outcome. Firms of age i are harder to preserve, and equilibria with reputa- set price pi ϭ (␣ Ϫ ␤)(␸i Ϫ ␸0) Ϫ p0, leav- tion only exist for very low values of the cost of ing consumers with belief ␸i indifferent be- effort and of the outside option. It remains never- tween staying and switching to an entrant. All theless possible to analyze, if not to solve explic- firms keep on operating, as some consumers itly for, such a stationary equilibrium, in which necessarily stay. Thus, the fraction of good prices of unlucky firms reflect their previous per- firms among firms of a given generation re- formances, along the lines of the previous section. mains constant over time, at level ␸0, but the Prices are pinned down by consumers’ indiffer- relative market share of good firms increases. ence among offerings, so that firms only exit if Accordingly, loyal consumers believe that their they experience histories such that their consum- firm is good with probability ␸i , and given the ers’ beliefs fall below the prior ␸0,astheprofits of equilibrium prices, optimally quit as soon as a firms with such beliefs must be negative, and bad outcome occurs. Notice that the quitting market sizes are then determined by the binding decision does not depend here on out-of- incentive compatibility constraints (to preserve equilibrium beliefs. Good firms find it optimal stationarity, it is however necessary to introduce to exert high effort in period i as long as the an arbitrarily small exogenous hazard rate, as value per consumer in period i ϩ 1 satisfies there would be otherwise a positive probability that a good firm never exits in equilibrium, while ni ϩ 1 bad firms eventually exit for sure). No addi- V м c/[(1 Ϫ ␪)␦(␣ Ϫ ␤)] n iϩ1 tional insights would be gained from such a i model, and the degree of sophistication de- manded from consumers in such an equilibrium for all i м 0. In a stationary equilibrium then, would be tremendous. these incentive constraints bind and are used to determine market sizes, and the zero-profit con- B. Idiosyncratic Outcomes dition pins down the initial price. Obviously, for ␪ small enough, these conditions are similar to Related issues arise if the assumption that all the ones of the richer model, and equilibrium the clients of a given firm observe the same exists under analogous restrictions. 656 THE AMERICAN ECONOMIC REVIEW JUNE 2002

The main insights of the paper are thus robust ni ϩ 1 (11) ␦ V to more sophisticated pricing schemes, as well n i ϩ 1 as to a relaxation of the correlation between i quality levels experienced by clients. Reputa- ck Ϫ cj cj Ϫ ck tions can be preserved whenever success guar- ʦ max ,min ␣ Ϫ ␣ ␣ Ϫ ␣ antees sufficiently high future revenues, both ͫ 0 Ϲ j Ϲ k Ϫ 1 k j k ϩ 1 Ϲ j Ϲ n Ϫ 1 j kͬ with respect to current expenditures and to future revenues in case of failure. The rewards of all i м 0 success may be conveyed either through price increases, precisely sustainable under competition because of the reputation’s improvement, where the interval is nonempty because action k or by a widening of the customer base, which maximizes ␣k Ϫ ck by definition of efficiency. presupposes however that potential customers If in a competitive equilibrium with reputation, may be able to identify such a firm. This in turn this discounted continuation value is defined to requires that consumers have reliable informa- be equal to the lower extremity of the interval, tion about a firm’s reputation, either through the as suggested by the earlier definitions, good direct evidence of a firm’s popularity or indi- firms will optimally be choosing the efficient rectly through the advice of acquaintances. The action in equilibrium. availability of such information also alleviates The limited applicability of the analysis the moral hazard problem because it backs up should of course be borne in mind. Although the threat of exit by dissatisfied consumers. The competition may help preserve reputations, more a consumer knows about the reputation of severe restrictions have been imposed on pa- her firm’s rivals, the more effective the exit rameters to guarantee the existence of equi- option will be. Even if loyal consumers have libria with reputation. When consumers’ “insider” information at their disposal, and do information is poor and the control of firms not observe other market activity during their over the quality of the product is limited and relationship, as is assumed in the model developed costly, competition per se is unlikely to make in this paper, even then, the uncompromising adifference. behavior of consumers may disclose precisely the information required for this hard line to remain optimal later on. IV. Conclusion

This paper explains why competition helps C. Efficiency preserve reputations. Without competition, a firm cannot always work hard to provide a It has been already pointed out in the liter- high-quality product, when the clients’ mon- ature with a single firm that equilibria with itoring is imperfect and their up-front pay- reputation need not build on the efficient ac- ment continuously depend on the firm’s tion when more than two actions are allowed reputation, because the firm’stemptationto (see Mailath and Samuelson, 1998a; Holm- shirk becomes irresistible as its reputation strom, 1999). In presence of competition, it improves. Competition, on the other hand, depends on the generalization of a competi- endogenously generates the outside option for tive equilibrium which is adopted, and in par- consumers that is necessary to keep firms on ticular on the determination of equilibrium their toes, as it gives consumers the power of prices. Suppose for instance that firms have n choosing between the offerings of rival sup- actions available, generating a good outcome pliers whose prices adjust to their reputation. with probability ␣j at cost cj , j ϭ 0, ... , This threat of exit provides incentives for ... n Ϫ 1, where 0 Ͻ ␣0 Ͻ ␣nϪ1 Ͻ 1, 0 Ϲ firms to try their best to keep up with the ... c0 Ͻ Ͻ cnϪ1 Ϲ 1, and consider implementing standards of the industry, but its uncompro- the efficient action k, where for simplicity mising execution forces able but unlucky 0 Ͻ k Ͻ n Ϫ 1. Incentive compatibility then firms out of the market. As put by Andrew Car- requires that: negie, “while the law [of competition] may be VOL. 92 NO. 3 HÖRNER: REPUTATION AND COMPETITION 657 sometimes hard for the individual, it is best for after trading once with a firm [observing first a the race, because it ensures the survival of the price distribution F, then, in case she accepts to fittest in every department.” This competitive trade, choosing a price in the support of the pressure works best if the information about a price distribution, Supp F, observing then an firm’s reputation transmitted through prices is outcome, either good (G) or bad (B), and de- reliable, as is the case if it is backed up by the ciding then to quit (Q)]. An element of RL (␴) visible behavior or advice of an experienced describes the information collected by a con- customer base. sumer during ␴ ϩ 2 periods, from the point at Now competition per se does not guarantee which she initially chose a firm as a switching that reputations are valued and moral hazard consumer, until she quits that particular firm or problems thereby solved. On the contrary, the chooses her outside option, after having been analysis emphasizes the conditions which are loyal for ␴ ϩ 1 consecutive periods (T refers to prerequisites for such a healthy competition. her decision to accept to trade after observing a Customer competence, for instance, which af- price in ޒ, while S refers to her decision to stay fords many problems of policy and implemen- at the end of a period). To formalize duration, tation, is an important ingredient. The main define l(⅐) by l(rS) ϭ 1 @rS ʦ RS, l(rL) ϭ point is rather that a theory of reputations which ␴ ϩ 2 @rL ʦ RL (␴). Define finally fails to take competition into account neglects L an important aspect of economic life. To para- I͑R ͒͑␴͒ ϭ ͓ ഫFʦ F ϫ Supp F ϫ ͕G,B͖ phrase Schumpeter (1950), “even if correct in F logic as well as in fact, it is like Hamlet without ϫ ͕S͖] ϫ ͓ޒ ϫ ͕T͖ ϫ ͕G,B͖ ϫ ͕S͖͔␴ the Danish prince.” as the collection of elements describing the information collected during a relationship which APPENDIX A has not been “ended” after ␴ ϩ 1 periods and L L L let I(R ) ϭ ഫ␴м0I(R )(␴), and for i(R ) ʦ This Appendix defines formally histories and I(RL), l(i(RL)) ϭ ␴ ϩ 1 whenever i (RL) ʦ strategies for the basic model. I (RL)(␴). Define then C ϭ Histories H0 A and, @t м 1, Let denote the set of Borel measures on ޒ, with totalF masses smaller than or equal to 1. n C ϭ L S Ͻ Define: Ht ͑n͒ ͹ Ei͉Ei ʦ R ഫ R , @i n, ͭ i ϭ 1 S n R ϭ ഫFʦ ͓F ϫ ͕͑O͖ ഫ ͑Supp F F L S S ϭ En ʦ R ഫ R ഫ I͑R ͒, ͸ l͑Ei ͒ t i ϭ 1 ͮ ϫ ͕G,B͖ ϫ ͕Q͖))] C C C C L Ht ϭ ഫn Ht ͑n͒ H ϭ ഫt Ht . R ͑␴͒ ϭ ͓ ഫFʦ F ϫ Supp F ϫ ͕G,B͖ ϫ ͕S͖͔ F ϫ ޒ ϫ ϫ ϫ ␴ C ͓ ͕T͖ ͕G,B͖ ͕S͖͔ Hence, a history ht for a consumer at the beginning of period t, or history of length t, con- ϫ ͓ޒ ϫ ͕͑O͖͒ഫ͕͑T͖ ϫ ͕G,B͖ ϫ ͕Q͖͒͒] sists of a succession of n “relationships,” for some n, with firms, some to which she may L L R ϭ ͕R ͑␴͉͒␴ м 0͖. have been loyal for a while, others which she immediately quit after joining them (and some An element of RS describes the information “relationships” in which she preferred to use her collected during one period by a switching con- outside option), and her “current” relationship, sumer who leaves in the following period, either the nth one, which may not be over yet. Define C C C because she chose her outside option ({O}), or Ht ͑Q͒ as the set of histories ht ʦ Ht ͑n͒, for 658 THE AMERICAN ECONOMIC REVIEW JUNE 2002

L S i some n, with En ʦ R ഫ R , that is, as the set Thus, the mapping ␰t specifies whether con- of histories of length t such that the consumer sumer i agrees to trade in period t, and possibly i starts period t as a switching consumer, and at which price, while ␳t specifies whether the C C C .Ht ͑Q͒ as the set of histo- consumer accepts to stay or quitsگ define Ht ͑S͒ϭHt ries of length t such that the consumers starts A strategy for a good firm j consists of two period t as a loyal consumer. Histories of firms mappings, which for each t м 0 and each j F are simpler. Define: history ht ʦ Ht specify which price to post and whether to exert high effort. That is, F H0 ϭ A j F pt : Ht 3 ޒ, and F Ht ϭ ͓ ഫFʦ F ϫ Supp F ϫ ޒ ϩ F j F ␶t : Ht ϫ ഫFʦ ͑F ϫ Supp F͒ ϫ ޒϩ 3 ͕H,L͖. F ϫ ϫ ϫ ޒ* t ͕H,L͖ ͕G,B͖ ϩ ] j A strategy for a bad firm j is a function pt , for F F F each t м 0, mapping H into reals. H ϭ ഫtм0 Ht . t Markovian Strategies In every period in which a firm operates, a firm observes the price distribution and the price it The belief ␪i of consumer i about the propor- posts itself, the mass of consumers which tion of good firms and her belief ␸i about her accept to trade, the effort level exerted, the current firm, if any, are the only variables which outcome obtained, and the mass of consumers are directly payoff-relevant to her, along with staying. As a firm exits as soon as all con- the prices she may observe. The state of the sumers quit, histories in which the mass of game for consumer i is defined to be her belief i i i consumers staying is zero are ignored. Elements ␻ ʦ ⍀i about the distribution of (␸ ,␪ )iʦ[0,1]. C C C of Ht (Ht ͑Q͒ and Ht ͑S͒) for consumer i are Markovian strategies for consumer i are de- i i denoted by ht͑Q͒ and ht͑S͒, respectively, and fined as follows. If ⍀i is induced by a history F i i similarly elements of Ht for firm j are denoted ht ʦ Ht͑Q͒, for some t, then consumer i de- j by ht . cides whether and with which firm to trade according to the mapping ␰i(Q): Strategies i ␰ ͑Q͒: ⍀i ϫ 3 ޒ ഫ ͕O͖ A strategy for consumer i specifies for each F i C i i t м 0, and each history ht ʦ Ht , two decisions: with ␰t͑Q͒͑␻ ,F͒ ʦ Supp F ഫ ͕O͖ whether to trade or not, and possibly with which i i firm, and at the end of the period, whether to while if ⍀i is induced by a history ht ʦ Ht͑S͒, stay or quit, in case she has not just chosen the she decides whether to trade according to the outside option. That is, it consists of two map- mapping ␰i(S): t t pings, ␰i͑X͒, ␳i͑X͒, X ʦ ͕S,Q͖: i ␰ ͑S͒:⍀i ϫ ޒ 3 ͕T,O͖. ␰i͑Q͒: HC͑Q͒ ϫ 3 ޒ ഫ ͕O͖ t t F Her decision to stay or switch is given by: i i with ␰t͑Q͒ϫ͑ht͑Q͒,F͒ ʦ Supp F ഫ ͕O͖ i ␳ :⍀i 3 ͕S,Q͖. i C ␰t͑S͒: Ht ͑S͒ ϫ ޒ 3 ͕T,O͖ The state of the game for firm j is similarly i C j ␳t͑Q͒: Ht ͑Q͒ ϫ ഫFʦ ͑F ϫ Supp F͒ defined as its belief ␻ ʦ ⍀j about the distri- F i i bution of (␸ ,␪ )iʦ[0,1]. Markovian strategies ϫ ͕G,B͖ 3 ͕S,Q͖ for (good) firm j are given by mappings:

i C j j ␳t͑S͒: Ht ͑S͒ ϫ ޒ ϫ ͕T͖ ϫ ͕G,B͖ 3 ͕S,Q͖. p : ޒ*ϩ ϫ ⍀j 3 ޒ ␶ : ޒ ϩ ϫ ⍀j 3 ͕H,L͖. VOL. 92 NO. 3 HÖRNER: REPUTATION AND COMPETITION 659

The pricing rule p j specifies a price as a func- It is clear from the last expression of equation ϱ tion of the mass of loyal consumers and of j’s (B2) that ͕ p*t ͖t ϭ 1 is an increasing sequence, and beliefs. The effort decision ␶ j specifies the ef- converges to: fort level as a function of the mass of consumers trading and of j’s beliefs. Similarly, a Mark- c p* ϭ ͓͑␣ Ϫ ␤͒ ϩ ␤͑1 Ϫ ␦͔͒ ovian strategy for a bad firm j is a pricing rule ϱ ␦ ␣ ␤ j ͑ Ϫ ͒ p :ޒ*ϩ ϫ⍀j 3 ޒ. In the equilibria defined below, however, ͑1 Ϫ ␦͒/␦ consumers strategies only depend on prices and ϭ 1 ϩ c. on their beliefs about their firm and about the ͩ ͑␣ Ϫ ␤͒/␣ͪ proportion of good firms (as these strategies are symmetric, it is straightforward to extend their As for p*0 ϭϪ͓␤/͑␣ Ϫ ␤͔͒c,itfollowsfrom domain to the state space ⍀i). The notation used V0 ϭ 0, given that hereafter is accordingly simpler. n1 c V ϭ . APPENDIX B 1 1 ͩ ␦͑␣ Ϫ ␤͒ͪ

PROOF OF PROPOSITION 2: Direct inspection yields p Ͻ p . Recall equation (4): 0 1

nt ϩ 1 PROOF OF THEOREM 1: Vt ϭ pt Ϫ c ϩ ␣␦ Vt ϩ 1 nt As by construction, good ﬁrms are willing to nt ϩ 1 exert high effort and consumers are willing to м pt ϩ ␤␦ Vt ϩ 1 , @t м 0, nt trade on the equilibrium path under the assumption of the theorem, it remains to show that as well as equation (3): ﬁrms do not want to deviate from equilibrium pricing and to complete the description of strat-

ntϪ1 egies and beliefs. n ϭ t м 1 n ϭ 1. Define consumer i’s beliefs as follows. At the t ␣␸ ϩ ␤͑1 Ϫ ␸ ͒ 0 tϪ1 tϪ1 end of period t Ϫ 1, consumer i’s belief about the proportion of good firms remaining in pe- Using equation (4), one gets that i riod t, ␪ , equals ␸t [defined by equation (2)]. Define then, for F ʦ , and ␪i ʦ [0,1], nt c F V ϭ . ⌽␪i[F]: Supp F 3 [0,1] by ⌽␪i[F](p) ϭ ␸t t ϩ 1 i n ϩ ͩ ␦͑␣ Ϫ ␤͒ͪ i t 1 if p ϭ pt and ␪ ϭ ␸t and ⌽␪ [F](p) ϭ 0other- wise. This mapping is used by consumer i, after Substituting back into equation (4), both for Vt i C a history ht ʦ Ht ͑Q͒, to update her beliefs, and Vt ϩ 1, one obtains: given her prior beliefs and the observed price distribution, and also determines her belief after c nt Ϫ 1 (B1) p*ϭ Ϫ ␤␦ her own choice (to trade and with which firm to t ␦͑␣ Ϫ ␤͒ ͩ n ͪ trade). After a history hi ʦ HC͑S͒, consumer i t t t uses the rule ⌽␪i:ޒ 3 [0,1] defined by i i ␸ ␪ ␸ i and upon using equation (3), ⌽␪ (p)ϭ t if p ϭ pt and ϭ t,and⌽␪ (p) ϭ 0 otherwise, to update her beliefs given the price she observes. After updating her belief c ␸t Ϫ 1 (B2) p*t ϭ ␣ Ϫ ␤␦ either way, she revises them again, in the ␦͑␣ Ϫ␤͒ ͩ ␸t ͪ event she agrees to trade, using Bayes’ rule ␾:[0,1]ϫ {G,B} 3 [0,1], given that good c firms exert high effort and that either a good ϭ ͓͑␣ Ϫ ␤͒␸ ϩ ␤͑1 Ϫ ␦͔͒ . ␦͑␣ Ϫ ␤͒ t Ϫ 1 outcome (G) or a bad one (B) is observed. 660 THE AMERICAN ECONOMIC REVIEW JUNE 2002

i C After a history ht ʦ Ht ͑Q͒, consumer i de- ޒϩ 3 {H,L} as mapping firm j’s beliefs and cides whether to trade and with which firm to the measure of its consumers into the effort i trade according to the mapping ␰␪i ͑Q͒ defined level which maximizes payoff. Obviously, high i as follows. Given ⌽␪i[F], define ␰␪i ͑Q͒: effort is specified by this mapping on the equi- i F 3 Supp F ഫ ͕0͖by ␰␪i ͑Q͒͑F͒ϭp where p librium path. maximizes ␣⌽ i[F](p) ϩ ␤(1 Ϫ⌽ [F](p)) Ϫ p ␪ ␪i over p ʦ Supp F if this maximum exceeds ␥, i and let ␰␪i ͑Q͒͑F͒ϭO otherwise. Hence, PROOF OF PROPOSITION 3: Notice that the i i ␰␪i͑Q͒͑F͒ϭp ϭ minpʦSupp F p,or␰␪i ͑Q͒͑F͒ϭpt, binding incentive constraints imply that: i or ␰␪i ͑Q͒͑F͒ϭO. Notice in particular that i ␰␪i͑Q͒͑F͒ϭp only if p Ͻ 0. After a history c ntϪ1 i C (B3) pt ϭ Ϫ ␤␦ ht ʦ Ht ͑S͒,consumeri decides whether to trade ␦͑␣ Ϫ ␤͒ ͩ n ͪ i t according to the mapping ␰␪ i ͑S͒: ޒ ϫ [0,1] de- i i i i fined by ␰␪i ͑S͒͑p,␸ ͒ϭT if ␣␸ ϩ ␤(1 Ϫ ␸ ) Ϫ i i p м ␥ and by ␰␪i ͑S͒͑p,␸ ͒ϭO otherwise, and p0 ϭϪ͓␤ /͑␣ Ϫ ␤)]c. However, consum- where ␸i is her belief after observing price p (in ers must be indifferent among offerings, or for i equilibrium, of course, ␸ ϭ⌽␪i(p)). that matter, indifferent between what a firm of In the event in which consumer i traded in age i Ͼ 0 offers and what an entrant offers. period t, her switching decision is then taken That is, according to ␳␪ i :[0,1]3 {S,Q}, defined by i i i ␳␪i(␸ ) ϭ S if and only if ␸ м ␾(␪ ,G): she ␣␸i ϩ ␤͑1 Ϫ ␸i͒ Ϫ pi ϭ ␣␸0 ϩ ␤͑1 Ϫ ␸0͒ Ϫ p0 stays if and only if her belief updated using Bayes’ rule, ␸i, exceeds or equals her belief from which it immediately follows that revision about the averaging operating firm, ␾(␪i,G), given that only firms having experi- ␤ (B4) p ϭ ͑␣ Ϫ ␤͒͑␸ Ϫ ␸ ͒ Ϫ c. enced successes still operate in period t ϩ 1. i i 0 ␣ Ϫ ␤ Observe at this point that if firm j does not set price pt in period t,itsconsumersleaveattheend Combining equations (B3) and (B4), one gets of the period, and will have traded in period t only j if the price posted, p ,wassufficiently negative. c Hence, firm j cannot realize positive profits by (B5) ni ϭ 2 niϪ1 @i Ͼ 0. deviating from the equilibrium price. ␦͑␣ Ϫ ␤͒ ͑␸i Ϫ ␸0 ͒ Firm j’s belief about the distribution of beliefs of consumers about its own type and about As before, since all firms which produce a fail- the fraction of good firms remaining in the ure exit, it must be that economy, f j ʦ F j and g j ʦ G j, respectively, is defined in the obvious way: firm j believes (B6) ␭i ϭ ͑␣␸iϪ1 ϩ ␤͑1 Ϫ ␸iϪ1͒͒␭iϪ1 i м 1 that all consumers have common belief ␪ ϭ ␸t at the beginning of period t. If it has any loyal and ␭0 ϭ 1 by normalization. By normalization ϱ consumer, it also assigns probability one that also, ¥iϭ0ni␭i ϭ 1, provided that there exists they all believe that firm j is good with proba- n0 Ͼ 0 such that the series defined by the bility ␸t. These beliefs are then updated in left-hand side coverage, which is the case when- accordance with the belief functions of consum- ever an equilibrium exists, as shown below. ers and their description is omitted. Define then, both for good and bad firms, the pricing rule p j: j j F ϫ G ϫ ޒϩ 3 ޒ as mapping its beliefs and PROOF OF THEOREM 2: the measure of its consumers into the price that n maximizes payoffs given consumers’ strategies. First, consider the partial sums ¥i ϭ 0ni␭i , j j In equilibrium, f and g are degenerate and defined for n0 Ͼ 0, ␭0 ϭ 1, and where ni, ␭i, assign probability one to the distribution assign- i Ͼ 0 are defined by induction in equation ing probability one to the belief ␸t in period t. (B5) and (B6) for n Ͼ 0. Notice that the j j j Similarly, for good firms, define ␶ : F ϫ G ϫ (positive) ratio ni ϩ 1␭i ϩ 1/ni␭i converges to VOL. 92 NO. 3 HÖRNER: REPUTATION AND COMPETITION 661

2 ␣c/␦͑␣ Ϫ ␤͒ ͑1 Ϫ ␸0͒ as i 3 ϱ. Hence, the such that ni Ϲ ni ϩ 1 @i Ϲ i* Ϫ 1, and ni м 2 ϱ series converge for c Ͻ ␦͑␣ Ϫ ␤͒ ͑1 Ϫ ␸0͒/␣ and ni ϩ 1 @i м i*, where the sequence ͕ni͖i ϭ 0 2 diverge for c Ͼ ␦͑␣ Ϫ ␤͒ ͑1 Ϫ ␸0͒/␣.Suppose is given by Proposition 3 (as ni ϩ 1/ni decreases 2 then that c ϭ ␦͑␣ Ϫ ␤͒ ͑1 Ϫ ␸0͒/␣.Inthatcase, in i, i* is uniquely determined). Also, let observe that now be the set of (joint) distributions overF ޒ ϫ {Y,N }, with generic element F, where Y (N) refers to the presence (absence) of a cus- ni ϩ 1␭i ϩ 1 i Ϫ 1 3 tomer base, and let ␴1 be the first coordinate k C ͩ ni␭i ͪ mapping. For histories ht ʦ Ht ͑Q͒, and for any Fʦ , define then the updating rule ⌽ i F ␤ ␤ Ϫ ␣ ϩ ␣␸0 ϩ ␤͑1 Ϫ ␸0͒ [F]: Supp F 3 [0,1] as follows: i 3 0 ͩ ␣ͪ ␣␸0 ⌽͓F͔͑p,Y͒ ϭ ␸i if p ϭ pi , and the series diverge by Raabe’s test. Hence ⌽͓F͔͑p,N͒ ϭ ␸ if p ϭ p , the series converge if and only if 0 0 ⌽͓F͔͑z͒ ϭ 0 otherwise. 2 ␦(␣ Ϫ ␤) (1 Ϫ ␸0) c Ͻ ␣ For any F ʦ , given ⌽[F], define the function ␰k(Q): SuppF F ഫ {O} 3 [0,1] which and, as these (positive) series are homogeneous maps choices (firm or outside option) into prob- of degree 1 in n0, one may always pick n0 so abilities. Specifically, if ( pi ,Y) ʦ Supp F @i м ϱ that ¥i ϭ 0ni␭i ϭ 1, which was the desired 1, (p0,N) ʦ Supp F and if they all achieve normalization. maxzʦSuppF ␣⌽[F](z) ϩ ␤ (1 Ϫ⌽[F](z)) Ϫ Second, observe that consumers are willing to ␴1( z), larger than ␥, then let: trade on the equilibrium path as long as the utility ␥ ␭ ͑n Ϫ n ͒ from trade exceeds ,butastheutilityfromtrade k i ϩ 1 i ϩ 1 i does not depend on the age of the firm by con- ␰ ͑Q͒͑pi ϩ 1 ,Y͒ ϭ i* Ϫ 1 ¥l ϭϪ1␭l ϩ 1 ͑nl ϩ 1 Ϫ n1 ͒ struction, it is sufficient that trade with an entrant is preferred to the outside option. That is, for 0 Ϲ i Ϲ i* Ϫ 1, ␤ ␭0 ͑n0 Ϫ n Ϫ 1 ͒ ␥ Ϲ ␣␸0 ϩ ␤͑1 Ϫ ␸0 ͒ ϩ c k ␣ Ϫ ␤ ␰ ͑Q͒͑p0 ,N͒ ϭ i* ϭϪ1 , ¥l ϭϪ1 ␭l ϩ 1 ͑nl ϩ 1 Ϫ nl ͒ which is precisely condition (B2). and ␰k͑Q͒͑z͒ ϭ 0 It remains, as for Theorem 2, to construct an equilibrium by defining strategies, belief func- for any other z ʦ Supp F ഫ ͕O͖. tions and by verifying that strategies are best responses, given beliefs. As the model has no Otherwise, if the maximum defined above is initial node, histories are complicated objects larger than ␥, let ␰k(Q) assign equal weight to whose definition is therefore omitted. To avoid all the maximizers and no weight to other ele- confusion, consumers are indexed by the letter ments of Supp F ഫ {O}. Finally, if the maxi- k, firms by j, and i refers to the age of a firm. As mum is below ␥, let ␰k(Q)(O) ϭ 1. k C before, it is useful to distinguish sets of histories For histories, ht ʦ Ht ͑S͒,define ⌽:[0,1] ϫ C C Ht ͑Q͒ and Ht ͑S͒, corresponding to sets of ޒ 3 [0,1] by ⌽(␸i, p) ϭ ␸i if p ϭ pi (for some pi histories up to period t along which consumer k of the sequence defined in Proposition 3) and k has quit her firm or used her outside option at consumer k’spriorbelief␸ ϭ ␸i,andby⌽(␸, p) ϭ the end of period t Ϫ 1 and to sets of histories 0otherwise.Define then ␰k(S): [0,1] ϫ ޒ 3 along which she stayed with her firm, respec- {T,O}by␰k (S)(␸, p) ϭ T if ␣␸ ϩ ␤(1 Ϫ ␸) Ϫ k tively. Let nϪ1 ϭ 0 and define i* ʦ ގ ഫ {ϱ} p м ␥,andby␰ (S)(␸, p) ϭ O otherwise. 662 THE AMERICAN ECONOMIC REVIEW JUNE 2002

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