Oligopolisti Markets with Sequential Searh and Asymmetri Information Maarten Janssen Paul Pihler Simon Weidenholzer 11th February 2010 Abstrat A large variety of markets, suh as retail markets for gasoline or mortgage markets, are haraterized by firms offering a fairly homogenous good produed at virtually the same ost. The present paper provides a theoretial examination of this type of market by developing a sequential searh model with inomplete information where onsumers are uninformed about the underlying prodution ost. We haraterize a perfet Bayesian equilibrium in whih onsumers follow reservation prie strategies and provide a suffiient ondition for suh an equilibrium to exist. Firms strategially exploit onsumers being uninformed about their prodution ost, and set on average higher pries ompared to the standard omplete information model. The importane of asymmetri information vanishes, however, when the number of firms beomes very large. Further, we find that expeted pries and onsumer welfare might be non monotoni in the number of firms. JEL Classifiation: D40; D83; L13 Keywords: Sequential Searh, Prie Dispersion, Asymmetri Information We are grateful to Benny Moldovanu, José Luis Moraga-González, Manfred Nermuth, Regis Renault, Sandro Shelegia, and Chris Wilson and to audienes at the EEA-ESEM 2009 in Barelona, the Workshop Searh and Swithing Costs in Groningen, the Theoretisher Ausshuß des Vereins für Soialpolitik, the Vienna University of Eonomis and Business, the University of Vienna, the Institute for Advaned Studies in Vienna, and the Meeting of Austrian Competition Authorities for helpful omments. An earlier version of this paper was irulated under the title Sequential Searh with Inompletely Informed Consumers: Theory and Evidene from Retail gasoline Markets; see Janssen, Pihler and Weidenholzer (2009). That working paper version ontains some preliminary empirial analysis onfirming some of the theoretial preditions we present here. Department of Eonomis, University of Vienna. Email: maarten.janssen@univie.a.at Department of Eonomis, University of Vienna. Email: paul.pihler@univie.a.at Department of Eonomis, University of Vienna. Email: simon.weidenholzer@univie.a.at 1
1 Introdution Consider a onsumer who observes the prie of gasoline at a gas station. Knowing that pries between different stations may vary onsiderably, the onsumer must deide whether to buy at the observed prie or searh for a better deal elsewhere. When making this deision, she must estimate how muh of the observed prie is due to ommon fators affeting all gasoline stations in a similar way, e.g. the prie of rude oil, and how muh is due to idiosynrati fators affeting the partiular seller being visited. If the onsumer believes that ommon fators are more relevant in determining the prie, she might onsider searhing for a heaper gas station not worthwhile and hene buy at the observed prie. Conversely, if she believes that the station harges a partiularly high prie ompared to other stations, she will probably find it optimal to look for a better deal. A key feature of this problem is that the onsumer must take her deision under inomplete information: she is unertain about the gas station s input (prodution) ost. Moreover, information is asymmetri, sine gasoline retailers are obviously aware of this ost. 1 In this paper we study how asymmetri information between firms and onsumers affets equilibrium in a market like the one desribed above. To this end, we introdue this feature into the standard sequential searh model with homogeneous goods, as developed by Stahl (1989). In our model, finitely many firms sell a homogenous produt on an oligopolisti market, with all firms faing the same stohasti prodution ost and being aware of its realization. Consumers have inelasti demand and engage in sequential searh for low pries. Unlike Stahl s sequential searh model and most searh literature, onsumers do not observe the firms prodution ost realization. Instead, they hold prior beliefs about the distribution of prodution osts and update these beliefs as they observe pries. 2 In this environment we examine the properties of a perfet Bayesian equilibrium satisfying a reservation pries property (PBERP). In suh an equilibrium, firms use mixed strategies and sample their pries from an optimal distribution that has no mass points; onsumers employ a (non-stationary) reservation prie rule, i.e., observing a ertain prie the onsumer buys if this prie is below her urrent reservation prie, and searhes for a lower prie otherwise. At the relevant reservation prie the onsumer is indifferent between buying and searhing for a better deal. Notie that a onsumer who observes a prie in the first round and ontinues to 1 This feature is not only found in gasoline markets, but haraterizes many environments suh as insurane or mortgage markets. 2 In our working paper, Janssen, Pihler, and Weidenholzer (2009), we provide empirial evidene for the model under investigation. In partiular, we show that the prie dispersion observed in the gasoline retail market in Vienna, Austria, is onsistent with a model of onsumer searh and diffiult to reonile with a model of horizontal produt differentiation. 2
searh will update her beliefs about the firms underlying prodution ost on the basis of her first prie observation, and therefore she will (generially) have a different reservation prie in the seond searh round. A onsequene of inomplete information within a sequential searh framework thus is that, if there exists an equilibrium where optimal searh behavior in eah round is haraterized by a reservation prie, the reservation prie must depend on the history of prie observations. This history-dependene renders the analysis onsiderably more ompliated ompared to the omplete information ase, whih may have lead the literature to restrit attention to other, often less plausible, searh protools when studying information asymmetry in onsumer searh models. 3 Our paper attempts to fill this gap in the literature by examining a sequential searh framework with asymmetri information. Within this framework, we fous on three questions. What are the harateristis of a PBERP? Under whih onditions does suh an equilibrium exist? And what is the role of asymmetri information within a sequential searh framework? Conerning the first question, we first show that in a PBERP no firm will set a prie larger than the onsumers first round reservation prie. To gain intuition for this finding, assume a firm would harge the upper bound, and that this upper bound is higher than the onsumers first-round reservation prie. This firm would obviously not have any sales in the first searh round; moreover, in later searh rounds potential ustomers visiting the firm already have lower prie observations in their pokets 4, and thus the firm would not have any sales in later searh rounds either. This behavior annot be optimal. The entral step in haraterizing equilibrium thus turns out to be the haraterization of the onsumers first-round reservation prie. This is not a straightforward task, however, beause onsumers may in priniple searh more than one, leading to multiple instanes of updating of their prodution ost beliefs. However, as we show in this paper, this annot be the ase in a PBERP. Importantly, a onsumer who observes the first-round reservation prie in a PBERP and, being indifferent, ontinues to searh has a stritly higher reservation prie in round two than in round one. This implies that the firstround reservation prie is the prie at whih a onsumer is indifferent between buying now and ontinuing to searh exatly one more round, similar to the omplete information model. Conerning our seond researh question, we show that existene of a PBERP is not trivially 3 For example, Dana (1994) onsiders a model of asymmetri information with newspaper searh, i.e., uninformed onsumers an learn all pries at one by paying a searh ost. He emphasizes, however, that the... assumption of newspaper searh is learly restritive and is not as realisti as the assumption that searh is sequential (...). However, analyzing sequential searh equilibrium under inomplete information is extremely diffiult sine onsumers ould in priniple searh more than one and hene more than one instane of Bayesian updating ould our. (p. 747). 4 Reall that the prie distribution in a PBERP has no mass points. 3
guaranteed. This finding is losely related to Rothshild (1974), who shows that an optimal onsumer searh strategy does not need to satisfy a reservation pries property if the prie distribution is unknown to the onsumer, and thus obtaining prie observations also has an informational value as onsumers update their beliefs about the true prie distribution from whih pries are sampled. In an environment where the prie distribution is exogenously given, Rothshild shows that the onsumer searh strategy satisfies a reservation prie property if for any two pries the differene between these two pries is smaller than the differene in their informational ontent. Using similar ideas in an environment where the prie distribution is endogenously determined, and using speifi properties the equilibrium prie distribution has to satisfy, we speify a suffiient ondition for the existene of a PBERP. In partiular, we show that a PBERP exists in markets with either (i) a small support of the prodution ost distribution, (ii) relatively large searh osts, (iii) relatively many firms, or (iv) relatively few shoppers. If a PBERP exists, we show that it is neessarily unique. We extend Rothshild s logi in our framework by arguing that our ondition is also "neessary" in the sense that, if the ondition fails to hold, then one an always find distributions of the prodution ost suh that a PBERP does not exist. Moreover, we provide an example showing that even when the prodution ost is uniformly distributed, a reservation prie equilibrium fails to exist if our ondition is not satisfied. Thus, inomplete information introdues signifiant hanges to the sequential searh model with respet to the existene of reservation prie equilibria. Conerning our third question, we have the following observations. First, in the omplete information benhmark (whih is essentially the Stahl model with unit demand), we show that the expeted minimum prie in the market is independent of the number of firms, while the expeted prie is inreasing. Interestingly, when onsumers are unertain about the underlying ost level, the expeted prie and total onsumer surplus over informed and uninformed onsumers an be non-monotoni in the number of firms. Seond, examining equilibrium prie strategies used by firms, we show that the lower bound of the prie distribution is inreasing in the ost level while its upper bound is independent of the ost level. 5 Thus, the extent of equilibrium prie dispersion under inomplete information dereases as the ost level rises. This onstitutes an important and interesting differene to the omplete information setting where the extent of prie dispersion is independent of the prodution ost realization. Third, studying the welfare effets of inomplete information, we show that, from an ex-ante perspetive, 5 In a PBERP, the upper bound must be equal to the first round reservation prie of onsumers. Sine the latter annot depend of the ost realization whih is unknown to onsumers, this property arries over to the upper bound. 4
onsumer welfare is unambiguously lower under inomplete information and profits are unambiguously higher as ompared to the environment with omplete information. Both informed and uninformed onsumers pay in expeted terms higher pries. However, the ex-ante unertainty onerning the prie to be paid is typially muh higher under omplete information. Finally, when the number of firms in the industry beomes very large, the importane of asymmetri information vanishes and the expeted prie and the expeted minimum prie in the two models onverges to eah other. The main reason is that, with many firms, pries lose to the reservation prie do not onvey information to onsumers about the underlying ost of firms. This paper ontributes to a large and growing literature on equilibrium onsumer searh models starting from seminal ontributions by Reinganum (1979), Varian (1980), Burdett and Judd (1983), and Stahl (1989). In terms of the researh question being addressed, the papers most losely related to our paper are Benabou and Gertner (1993) and Dana (1994). 6 Both papers onsider, however, simplified searh protools and do not onsider a sequential searh protool. Benabou and Gertner analyze a duopoly market where half the onsumers observe one prie and the other half observes the other prie at no ost. The only deision onsumers have to make is whether to also observe the prie of the firm they have not yet observed at a searh ost. Dana onsiders a model with two types of onsumers (informed and uninformed) where the uninformed onsumers are engaged in newspaper searh. These onsumers get a first prie quote for free and, on the basis of this prie, they deide whether or not to beome fully informed about all pries by paying a searh ost. Papers by Fershtman and Fishman (1992) and Fishman (1996) and the reent ontributions by Yang and Ye (2008) and Tappata (2008) use frameworks similar to Dana (1994), but extend them to a dynami setting. In suh environments, these papers study asymmetri prie adjustment to ost shoks, the so-alled rokets-and-feathers pattern. To the best of our knowledge, our paper is the first to introdue inomplete information into a sequential onsumer searh model. In a broader sense the urrent paper is related to reent work whih elaborates on the role of information gathering and information proessing in onsumer searh. 7 This literature fouses on obfusation (Ellison and Wolitzky (2009), Ellison and Ellison (2009)), boundedly rational agents (see, e.g., Spiegler (2006)), or information gatekeepers on the internet (see, e.g., Baye and Morgan (2001)). Another strand of the literature makes progress on the poliy impliations of the onsumer searh literature on onsumer protetion poliies (see, e.g., Armstrong, Vik- 6 Earlier work by Diamond (1971) and Rothshild (1974) has analyzed optimal searh behavior in a world where the prie distribution is unknown, but exogenously given. Reently, Gershkov and Moldovanu (2009) unover some formal relations between optimal stopping rules in the onsumer searh literature and the problem of ensuring monotone alloation rules in dynami alloation problems. 7 Baye, Morgan, and Sholten (2006) surveys a wide range of onsumer searh models. 5
ers, and Zhou (2009)) or on the empirial implementation of onsumer searh models (see, e.g., Lah (2007), Hortaçsu and Syverson (2004) and Moraga-González and Wildenbeest (2008)). The remainder of this paper is organized in the following way. In Setion 2 we briefly disuss the standard sequential searh model with ompletely informed onsumers, establishing a theoretial benhmark for omparison of our inomplete information model. This setion also ontains some new results (on the expeted prie of fully informed onsumers) and some new haraterizations making it easier to work with the model. In Setion 3 we develop our model with inompletely informed onsumers, define a perfet Bayesian equilibrium and haraterize its properties. In Setion 4 we summarize the effets of inomplete information on onsumer and produer welfare and ompare the models when the number of firms beomes large. Finally, in Setion 5 we onlude and disuss diretions for future researh. Proofs are provided in the Appendix. 2 Sequential searh with ompletely informed onsumers We start our analysis by desribing a sequential searh model with ompletely informed onsumers along the lines of Stahl (1989). This model will, at a later stage, serve as our benhmark to assess the impliations of inomplete (asymmetri) information within the sequential searh framework. Essentially, we modify Stahl s model along only two dimensions. First, to simplify the analysis we onsider a model of inelasti demand as in Janssen, Moraga-Gonzalez, and Wildenbeest (2005) and, seond, do not normalize marginal osts to zero. Solving the model for positive marginal osts is inevitable for our purposes, beause later on we want to analyze and ompare situations under different marginal ost levels. 2.1 Model We onsider an oligopolisti market where N 2 firms sell a homogenous good and ompete in pries. Eah firm n {1,..., N} faes the same prodution tehnology and the same marginal prodution ost, denoted by. Without loss of generality, we normalize fixed osts to zero. Eah firms objetive is to maximize profits, taking the pries harged by other firms and the onsumers behavior as given. On the demand side of the market we have a ontinuum of risk-neutral onsumers with idential preferenes. Eah onsumer j [0, 1] has inelasti demand normalized to one unit, and holds the same onstant evaluation v > 0 for the good. Observing a prie below v, onsumers will thus either buy one unit of the good or searh for a lower prie. In the latter ase, they have 6
to pay a searh ost s to obtain one additional prie quote, i.e. searh is sequential. A fration λ [0,1] of onsumers, the shoppers, have zero searh ost. These onsumers sample all pries and buy at the lowest prie. The remaining fration of 1 λ onsumers the non-shoppers have positive searh osts s > 0. These onsumers fae a non-trivial problem when searhing for low pries, as they have to trade off the searh ost with the (expeted) benefit from searh. Consumers an always ome bak to previously visited firms inurring no additional ost, i.e. we are onsidering a model of ostless reall. 8 We assume that v is large relative to and s so that v is not binding. In this setion onsumers are informed about the ost realization. 2.2 Equilibrium In this model, there exists a unique symmetri Nash equilibrium where onsumer behavior satisfies a reservation prie property. Moreover, Kohn and Shavell (1974) and Stahl (1989) argue that the reservation pries are stationary. That is, the onsumers reservation pries are independent from the history of prie observations and the number of firms left to be sampled (provided there is still at least one firm left) and an therefore simply be denoted by ρ k (). 9 To haraterize this equilibrium it is useful to introdue some more notation: we denote, for a given prodution ost, the distribution of pries harged by firms by F k (p ), its density by f k (p ), and the lower- and upper- bound of its support by p k () and by p k (), respetively. It is well-known that the presene of both shoppers and non shoppers, λ (0, 1), implies that there does not exist an equilibrium in pure strategies and that there are no mass points in the equilibrium prie distribution. The main reason behind this observation is that firms fae a tradeoff between setting low pries to ater to the shoppers and setting high pries to extrat profits from the non shoppers. Also, the upper bound of the equilibrium prie distribution must satisfy p k () = ρ k (), i.e., in a symmetri equilibrium no firm will set a prie higher than the reservation prie ρ k (). Given these two observations, the equilibrium prie distribution an be haraterized by Proposition 2.1 For λ (0, 1), the equilibrium prie distribution for the ost realization is given by ( 1 λ F k (p ) = 1 λn p k ) N 1 () p 1 p 8 Janssen and Parakhonyak (2007) analyze the ase where this assumption is replaed by ostly reall. 9 We use the supersript k to indiate variables and parameters of the model with ompletely informed onsumers who know the prodution ost realization. (1) 7
respetively f k (p ) = 1 N 1 p k () (p ) 2 with support on [p k (), p k ()] with p k () = ( ) 1 1 λ λn N 1 ( p k () p p ) 2 N N 1 λn λn+1 λ + 1 λ λn+1 λ pk () and p k () = ρ k (). The proof follows essentially Stahl (1989) and is therefore omitted. Having haraterized the Nash equilibrium prie distribution onditional on the reservation prie ρ k (), we turn to optimal onsumer behavior. Given a distribution of pries F k (p ) and an observed prie p, it is straightforward to argue that the reservation prie ρ k () is impliitly determined by ρ k v ρ k () () = v s p f k (p )d p. p k () Using the result that the equilibrium prie distribution satisfies p k () = ρ k (), this ondition boils down to (2) ρ k () = s + E k (p ). (3) As the next lemma shows, this relation between reservation prie and expeted prie allows us to derive a simple formula for the expeted prie onditional on the ost realization, E k (p ). Lemma 2.1 The expeted prie onditional on the ost realization, E k (p ), is given by E k (p ) = + α s, (4) 1 α where α = 1 1 0 dz [0,1). 1+ λn zn 1 1 λ Janssen and Moraga-González (2004) have shown that α is inreasing in N. It follows that the expeted prie is inreasing in N as well. Note that (3) and (4) imply the following simple expression for the reservation prie, ρ k () = p k () = + s 1 α. (5) The reservation prie is thus a onstant markup over the ost, with the size of the markup being determined by the model s parameters. Note further that, by Proposition 2.1, p k () is a weighted average of and ρ k (). Consequently, it immediately follows that, provided s > 0, the lower bound satisfies p k () >. Thus firms make positive profits when harging pries aording to F k (p ). Furthermore, the following result obtains: Corollary 2.1 The equilibrium prie spread, i.e. the differene between the upper bound and the lower bound of the prie distribution, is independent of the realized ost level and given by p k () p k λn s () = λn + 1 λ 1 α. (6) 8
Figure 1: Prie distributions 1 0.9 0.8 0.7 =0 =0.25 =0.5 =0.75 =1 0.6 F k (p ) 0.5 0.4 0.3 0.2 0.1 0 1 1.2 1.4 1.6 1.8 2 p Parameters: N = 3, s = 0.01, λ = 0.01, U(0,1) The proof follows from (5) and Proposition 2.1. What is interesting about Proposition 2.1 is that a hange in leads to a one to one shift in the prie distribution, leaving the extent of prie dispersion unaffeted. In Figure 1 we highlight this point by plotting prie distributions for various realizations of ost. Note at this stage that, onditional on the ost, the average prie paid by a fration 1 λ of onsumers, i.e. the non-shoppers, is equal to E k (p ) as given in (4). This is, however, not the (average) prie paid by the λ shoppers who observe all pries in the market and buy at the heapest firm. This latter prie is given by E k (p l ), with p l = min{p 1, p 2,..., p N }. As firms hoose pries randomly and independent from eah other, it follows that the distribution of p l is given by F k l (p l ) = 1 [1 F k (p )] N. (7) Stahl (1989), and reently Morgan, Orzen, and Sefton (2006), and Waldek (2008), show that in their models the shoppers observing all pries are better off with entry. Surprisingly, it turns out that with unit demand and an endogenous reservation prie the expeted minimum prie is independent of N: Proposition 2.2 The expeted minimum-prie onditional on the ost realization, E k (p l ), is given by E k (p l ) = + 1 λ s, (8) λ 9
and therefore independent on N. In our model, entry has three different effets on the expeted minimum prie. First, the reservation prie is inreasing in the number of firms. As this implies that pries tend to be higher with more firms, this effet unambiguously makes shoppers worse off. A seond effet is that shoppers sample more firms when there are more firms available in the market. This has the unambiguous effet that for a given prie distribution shoppers are better off. Finally, prie dispersion inreases with more firms in the industry, implying that firms onentrate more on lower and higher pries at the expense of moderate pries. In fat, the lower bound of the prie distribution dereases in the number of firms. Surprisingly, the Proposition shows that the net result of these three effets is zero. The differene with earlier results an be explained as follows. Stahl (1989) onsiders a model with downward sloping (instead of unit) demand, while Morgan, Orzen, and Sefton (2006) onsider Varian s (1980) model where the onsumer reservation prie is exogenously given by the willingness to pay and onsumers have unit demand. 10 The two papers together may suggest that when we onsider the ase of unit demand and the reservation prie of non-shoppers being endogenously determined, the shoppers are also better off with entry. Our result shows that this is not the ase. Finally, we ompute the unonditionally expeted pries E k (p) = E k (p )d and E k (p l ) = E k (p l )d. These expeted pries will later on be important to assess onsumer welfare in the eonomy, as v E k (p l ) is the expeted equilibrium onsumer surplus attained by shoppers whereas v E k (p) is expeted equilibrium surplus of the 1 λ non-shoppers. 11 Formally, we obtain: Corollary 2.2 The unonditionally expeted pries E k (p) and E k (p l ) are given by where E() = g()d. E k (p) = E() + α s, 1 α and (9) E k (p l ) = E() + 1 λ s, λ (10) The proof follows trivially from Lemma 2.1 and Proposition 2.2. Note that both ex-ante expeted pries take the form of a markup over the ex-ante expeted ost, with the size of the respetive markup being determined by the model s parameters, where it is interesting to note that the mark-up for the expeted minimum prie is independent of N, and the mark-up of the expeted prie is inreasing in N. 10 Stritly speaking, Stahl (1989) only proves this result for the ase of infinitely many firms. For a finite number of firms he provides some numerial results. 11 We will ompute expeted onsumer surplus in the very beginning of the game, i.e. before the ost level is drawn from g(). 10
3 Sequential searh with inompletely informed onsumers We now turn to the analysis of the inomplete information model and modify the model presented in Setion 2 by postulating that onsumers are uninformed about the firms prodution ost. Let nature randomly draw from a ontinuous distribution g() with ompat support on [, ]. Consumers do not know the ost realization and they all hold orret prior beliefs about the prodution ost distribution and update their beliefs aording to Bayes rule as they observe pries. 3.1 On out-of-equilibrium beliefs In the model with inompletely informed onsumers, the exat speifiation of out-of-equilibrium beliefs plays an important role in determining reservation pries. To see this point, assume that onsumers hold out-of-equilibrium beliefs that are suh that, if a prie above their reservation prie is observed, they think that the lowest ost level has been realized with probability one and therefore ontinue to searh. In suh a ase, in equilibrium no firm would set a prie above the reservation prie (more details will be given shortly) and therefore suh a prie observation is learly an out-of-equilibrium event. Note that under these partiular beliefs, one an support a (first round) reservation prie with the property that a onsumer who observes it will stritly prefer to buy instead of atually being indifferent between buying and searhing for a lower prie. In a omplete information setting, this ould never be a reservation prie as onsumers would then also be willing to buy at a slightly higher prie. However, in the inomplete information ase under these partiular out-of-equilibrium beliefs, a onsumer would prefer to searh for lower pries thinking that the lowest ost level has been realized and thus that pries should be low. Consequently, there would be a disontinuity in the willingness of onsumers to buy around this reservation prie. However, we think that suh a disontinuity is diffiult to defend in a onsumer searh model and we ertainly do not want the omparison between the omplete and inomplete information settings to depend on the arbitrary hoie of out-of-equilibrium beliefs. We therefore insist that, if at a reservation prie onsumers stritly prefer to buy, out-of-equilibrium beliefs should be suh that onsumers also should buy at a slightly higher prie. This effetively defines the first round reservation prie as the prie at whih the onsumer is indifferent between buying and ontinuing to searh, in a way similar to the familiar omplete information searh model. In the following, we limit attention to equilibria satisfying suh a reservation pries property. 11
3.2 Equilibria with reservation pries property We start by providing a formal definition of what we mean by equilibria satisfying a reservation pries property. This requires first to introdue some more notation. In partiular, we denote by ρ t (p 1,..., p t 1 ) the reservation prie of a onsumer in searh round t who has observed pries p 1,..., p t 1 in the t 1 previous searh rounds. Note that unlike the omplete information model, any reservation prie ρ t (p 1,..., p t 1 ) held by onsumers has to be independent of the prodution ost and that the reservation prie ρ 1 in the first round is not onditional on any prie observation, and we write ρ 1 = ρ. We have: 12 Definition 3.1 A perfet Bayesian equilibrium satisfying a reservation pries property (PBERP) is haraterized by: 1) eah firm n {1,...,N} uses a prie strategy that maximizes its (expeted) profit, given the ompeting firms prie strategies and the searh behavior of onsumers; 2) given the (possibly degenerate) distribution of firms pries, onsumers searh optimally given their beliefs, and update their beliefs given their prie observations (if possible) and formulate out-of-equilibrium beliefs whenever they observe a non-equilibrium prie; moreover, optimal onsumer searh is of the following form: i) after observing p t = ρ t (p 1,..., p t 1 ) in round t and p 1,..., p t 1 in previous rounds, the onsumer is indifferent between buying and ontinuing to searh; ii) after observing any p t < ρ t (p 1,..., p t 1 ) in round t and p 1,..., p t 1 in previous rounds, the onsumer buys. In what follows, we onentrate on the haraterization of this type of equilibrium and determine onditions for existene. 3.3 Properties of PBERP We first examine the properties of a PBERP, assuming that suh an equilibrium exists. In the next subsetion, we onsider the existene question. A first observation is that in a PBERP, the upper bound of the prie distribution has to be equal to the reservation prie of onsumers in the first searh round, i.e. p() = p = ρ for all [, ]. Suppose this was not the ase and that for some, p() > ρ. If a firm harges p(), it will not sell to shoppers in any PBERP, as p() 12 This definition is an adaptation of the reservation prie equilibrium defined by Dana (1994) to the ase of sequential searh. 12
does not have positive probability and therefore shoppers observe lower pries with probability one. Furthermore, a firm setting p() will not sell to non-shoppers either, as these onsumers will ontinue to searh after observing p in the first searh round, and will then find a lower prie in a subsequent searh round with probability one. On the other hand, it an also not be the ase that for some, p() < ρ sine firms ould profitably deviate to a prie equal to ρ beause non-shoppers would ontinue to buy. Under asymmetri information firms fae virtually the same maximization problem as in the omplete information benhmark. The only major differene is that upper bound of the prie distribution is now onstant at p = ρ for all realizations of the ost. Formally: Proposition 3.1 In any PBERP the equilibrium prie distribution for the ost realization is given by respetively ( ) 1 1 λ p p N 1 F(p ) = 1 λn p f (p ) = 1 p N 1 (p ) 2 with support on [p(), p] with p() = ( ) 1 1 λ λn N 1 ( p p p ) 2 N N 1 λn λn+1 λ + 1 λ λn+1 λ p and p = ρ. The proof is omitted, as it is a straightforward extension of the proof of Proposition 2.1. Inspetion of (11) reveals that F(p ) first order stohastially dominates (FOSD) F(p ) whenever >. Furthermore, we have that p() is inreasing in, implying that onsumers who observe pries below p() an rule out ertain (high) ost realizations. Employing the tehniques used in 2.1 we an write E(p ) as (11) (12) E(p ) = (1 α) + α p. (13) Figure 2 visualizes all these observations by plotting the prie distributions F(p ) for different realizations of the prodution ost. This figure points to the fat that the prie spread is dereasing in as stated in the following orollary: Corollary 3.1 If onsumers are uninformed about the firms ost realization, the prie spread in a PBERP is equal to λn p p() = ( p ) λn + 1 λ and therefore is dereasing in the ost level. Corollary 3.1 establishes a sharp ontrast to the omplete information model in whih the prie spread is independent of the realized ost level. 13
Figure 2: Prie distributions 1 0.9 0.8 0.7 =0 =0.25 =0.5 =0.75 =1 0.6 F u (p ) 0.5 0.4 0.3 0.2 0.1 0 1.5 1.51 1.52 1.53 1.54 1.55 p Parameters: N = 3, s = 0.01, λ = 0.01, U(0,1) Let us now turn the fous on onsumers searh behavior. Assuming onsumers have orret prior beliefs about the prodution ost and use Bayesian updating after observing a prie p, let δ( p) be the (posterior) probability density funtion of the prodution ost onditional on a prie observation p. By Bayes rule, we have that: g() f (p ) δ( p) = g( ) f (p )d. (14) It remains to speify onsumers out-of-equilibrium beliefs, i.e. beliefs on the ost level for prie observations not in the support of the equilibrium prie distribution. As argued above, we want to avoid out-of-equilibrium beliefs that reate a disontinuity in the willingness of onsumers to buy around the reservation prie. To this end, we assume that for a prie observation above the upper bound of the prie distribution onsumers hold the same beliefs on the ost level as if they had observed the upper bound, i.e. δ( p) = δ( p) for p > p. 13 In the following, we derive several lemmas that will prove useful to examine the properties of PBERP. First, we identify an important feature of Bayesian updating in our framework that plays a key role in our main results: a onsumer who has observed a prie p [p(), p] will put more probability mass on higher realization of the prodution ost and less mass on lower realizations of the prodution ost than under the prior distribution g(). 13 As at pries below the lower bound of the prie distribution in the lowest ost senario onsumers buy regardless of their beliefs on the realized ost level, beliefs in these states are irrelevant. 14
Lemma 3.1 For any p [p(), p], the posterior distribution of ost levels δ( p) first order stohastially dominates the prior distribution g(). Lemma 3.1 implies that non-shoppers who have observed any p [p(), p] expet a higher ost level E( p) than if they hadn t observed any prie, i.e. E( p) = δ( p)d > g()d = E(). Moreover, we find that the higher the prie observed in the interval p [p(), p], the more optimisti the onsumer is about the possibility of finding low pries if she ontinues searhing. Formally, Lemma 3.2 For all p [p(), p), there is a unique ost level ĉ suh that δ( p) p > 0 if < ĉ = 0 if = ĉ < 0 if > ĉ. Consequently, for all p, p [p(), p] with p > p, the posterior distribution of ost levels δ( p) FOSD δ( p ). Lemma 3.2 appears puzzling at first sight. However, the intuition behind it is readily seen: in the interval [p(), p], the ratio of densities f (p )/ f (p ) is inreasing in p for any pair, with <. This implies that higher pries in [p(), p] are relatively more likely under low osts than under high osts, whih in turn explains why higher prie observations in p [p(), p] lead the onsumer to beome more optimisti about the ost realization. We now move on to the haraterization of reservation pries under inomplete information. Reall that in the omplete information model, the reservation prie is defined as the prie at whih the onsumer is indifferent between buying now and ontinuing to searh. In the present ontext, this would translate into defining the (first round) reservation prie ρ by the indifferene ondition v ρ = v s E(p ρ), with E(p ρ) = E(p )δ( ρ)d. The reservation prie ρ would thus be impliitly given by ρ = s + E(p )δ( ρ)d. (15) However, under inomplete information reservation pries are not stationary and do dependent on the searh history, due to Baysian updating of beliefs. The arguments used above may 15
hene not be valid, and it is not obvious that the first round reservation prie should satisfy (15). In what follows, we however prove that (15) still provides a proper haraterization of the reservation prie. The intuition is the following: if a onsumer in searh round one observes the round one reservation prie and deides to ontinue searhing, she will find a prie stritly below her round two reservation prie with probability one, and thus neessarily buys in round two. Thus, the reservation prie in round one is the prie at whih the onsumer is indifferent between buying and searhing one more firm. Lemma 3.3 establishes this result. Lemma 3.3 In any PBERP, after observing the upper bound of the prie distribution in the first searh round, a onsumer s reservation prie in the seond searh round satisfies ρ 2 (ρ) > ρ. It follows that there does not exist a PBERP where onsumers follow a stationary reservation prie searh rule. This is an important differene with the omplete information model. Using equation (13) and ρ = p gives us p = s + = s + (1 α) ( ) (1 α) + α p δ( p)d δ( p)d + α p δ( p)d. Sine δ( p)d = 1 and δ( p)d = E( p), we further have that the reservation prie is impliitly defined by ρ = p = E( p) + s 1 α. (16) Substituting (16) into (13), we arrive at the following result onerning the onditional expeted prie. The laim onerning the onditional expeted minimum prie is a little more diffiult to arrive at, but essentially follows from the fat that (1 λ)(e(p ) ) + λ(e(p l ) ) = (1 λ)(ρ ), (17) i.e., the sum of the expeted industry profits over shoppers and non-shoppers has to be equal to N times the expeted individual profit, whih equals (1 λ)(ρ ). Proposition 3.2 In any PBERP the onditionally expeted pries E(p ) and E(p l ) in a PBERP are given by E(p ) = + α s + α [E( p) ], (18) 1 α (1 λ) E(p l ) = + (s + (1 α)[e( p) ]. (19) λ 16
From Proposition 3.2 it immediately follows that the unonditionally expeted pries E(p) and E(p l ) are given by 3.4 Existene of PBERP E(p) = E() + α s + α [E( p) E()], (20) 1 α (1 λ) E(p l ) = E() + (s + (1 α)[e( p) E()]). (21) λ Having established some properties any PBERP should satisfy, we now move to the existene question. Note that we have so far impliitly assumed that non-shoppers would like to buy at all pries below ρ. While this is straightforward to establish in the framework with omplete information, it is not obvious under inomplete information sine onsumers update their beliefs about the true ost as they observe pries. In partiular, after observing a prie p < p() a onsumer may suddenly think that the ost is very low and thus may deide to ontinue searhing. Moreover, we need to verify that for all ost realizations firms find it optimal to set the pries impliitly speified above. In partiular, we need that p() > for all values of. Again, under asymmetri information this ondition is not automatially satisfied as the reservation prie (and thereby the upper bound of the prie distribution) is independent of the ost realization. The next Proposition establishes a suffiient ondition for the existene of PBERP with non-stationary reservation pries. Rothshild (1974) already observed that if onsumers sample from an unknown distribution, it may happen that they prefer to buy at high pries, whereas they ontinue to searh (and do not buy) at lower pries. He also provides a suffiient ondition under whih it is optimal for onsumer to atually follow a reservation prie strategy. The suffiient ondition he states is that the prie differene between any two pries should be smaller than the differene in informational ontent of these pries. Rothshild fouses, however, on the onsumer searh problem for a given (but unknown) prie distribution. We show that these onsiderations also arise in onsumer searh models where firms are strategially hoosing pries. Importantly, our suffiient ondition is in terms of the exogenous parameters of the model. Proposition 3.3 If then a unique PBERP exists. ( ) λn s, (22) λn + 1 λ 1 α The proof is based on the following onsiderations. We first show that observing a prie p with p() < p < ρ, an uninformed onsumer prefers to buy instead of ontinuing to searh and 17
buy in a later round. Note that at these pries, onsumers assign positive density to any ost realization and by Lemma 3.1 beome more pessimisti about the possibility of finding lower pries when ontinuing to searh. We then examine lower prie observations p < p(), where onsumers an rule out ertain high ost realizations. For a PBERP to exist, onsumers must still find it optimal to buy at suh pries. This, in turn, requires that onsumers do not infer from observing a prie p < p() that the ost is low enough so that ontinued searh pays off. To rule out this ase, we exploit the idea that a onsumer who finds it optimal to buy at a prie p if he knows the ost realization is, i.e. p ρ k (), ertainly has to find it optimal to buy in the unknown ost ase at the same prie. Consequently, by imposing p( ) ρ k () we an guarantee that a onsumer will find it optimal to buy at all prie observations smaller than the reservation prie ρ. This ondition translates into inequality (22) haraterizing the existene of a PBERP. Furthermore, inequality (22) also is suffiient to ensure that firms will set pries as speified above, i.e. p() > holds for all [, ]. It is interesting to see when the ondition in Proposition 3.3 holds. Clearly, this is the ase when the support of the ost distribution is small or s is large. More interestingly, it is also the ase when N is large enough (for any given values of the other parameters). To see this, note that both λn λn+1 λ and α approah one as N approahes infinity; the RHS of inequality (22) therefore approahes infinity as well. Finally, note that when + Ns, a PBERP exists also for small values of λ. To arrive at this observation, we evaluate λn λn + 1 λ 1 1 1 1 0 dz 1+ λn zn 1 1 λ when λ is lose to zero. Applying l Hopital s Rule, we get that in a neighborhood of λ = 0 N (λn+1 λ) 2 10 Nz N 1 /(1 λ) 2 (1+ λn 1 λ zn 1 ) 2 dz = N 10 Nz N 1 dz = 1 10 z N 1 dz = N. For λ lose to 0, the right hand side of our inequality is thus approximately equal to + Ns. We summarize our findings regarding the existene of PBERP in the following orollary: Corollary 3.2 A PBERP exists in environments with (i) a suffiiently small support of the ost distribution,, and/or (ii) suffiiently large searh osts s, and/or (iii) suffiiently many firms N, and/or (iv) a suffiiently small fration of shoppers λ, provided that + Ns holds. 18
For general funtions g() the ondition established in Proposition 3.3 is almost neessary in the following sense. If p( ) > ρ k (), then one an onstrut a density funtion of the ost parameter, g(), that is onentrated on values lose to the two extremes and (see Figure 3) suh that, after observing a prie smaller than p( ), onsumers suddenly onsider it extremely likely that the ost is lose to. In partiular, if a prie observation p is in the interval (ρ k (), p( )) onsumers will then prefer to searh. Figure 3: A ost distribution onentrated around the two extremes. g() One may then wonder whether, if we restrit the prior ost distribution, existene of a PBERP may always be guaranteed. Considering a uniform distribution of prodution osts, the next example demonstrates that this is not the ase. Figure 4 displays the net benefits of searh in a duopoly market with searh osts equal to s = 0.00675, a shopper-share equal to λ = 0.025, and prodution osts drawn from the uniform distribution U(0, 1). As an easily be seen, for this parameter onstellation no PBERP exists: the onsumer does not prefer to buy for all pries below the potential reservation prie defined by equation (16). While for pries between ρ = 1.0260 and p( ) = 1.0248 the onsumer stritly prefers to buy, when observing pries slightly below p( ), the net benefits of searh are inreasing rapidly. Indeed, the net searh benefits beome positive for an interval of pries a bit below p( ). The reason is that when the onsumer observes pries just below p( ), she infers that the expeted prodution ost is relatively low and so is the expeted prie. When she would observe even lower pries, the searh benefits inrease rapidly and it beomes profitable not to buy at the observed prie but to searh for a lower prie, even though the onsumer would have bought had she observed 19
Figure 4: Net benefits of searh 2 x 10 3 0 2 4 6 8 10 12 1 1.005 1.01 1.015 1.02 1.025 p Parameters: N = 2, s = 0.00675, λ = 0.025, U(0,1) a slightly higher prie. In ase a PBERP does not exist, it is important to know what type of equilibrium does exist. Unfortunately, it turns out this is a very diffiult issue to resolve. For example, it an be shown that allowing for reservation pries whih are more general than the ones in Definition 3.1 does not overome the non-existene problem. Note that a more general reservation prie strategy is a strategy aording to whih onsumers buy if, and only if, they observe a prie at or below a ertain ut-off prie, but where onsumers are not indifferent between buying and ontinuing searhing at the reservation prie. As explained before, suh reservation pries ould be supported in our framework by speifying out-of-equilibrium beliefs in suh a way that after observing pries higher than the upper bound of the prie support (the reservation prie), onsumers believe that the underlying osts are very low and therefore stritly prefer to ontinue searhing. The next result establishes that there are parameter values for whih equilibria where onsumers follow suh strategies do not exist. Proposition 3.4 If s is relatively small or is relatively large and g() has a relatively high probability mass lose to, then an equilibrium where onsumers follow a reservation prie strategy does not exist. Together with the obvious fat that there annot be a hole in the pries at whih the nonshoppers deide to buy with probability one, Proposition 3.4 implies that for some parameter 20
values onsumers have to follow a mixed strategy in equilibrium. We leave it for further researh to fully haraterize these equilibria. 4 Welfare impliations and the impat of entry We are now ready to ompare the two models more formally. The most important basis for this omparison is the examination of the split of welfare between uninformed onsumers, informed onsumers and firms generated in the two models and how this split of welfare depends on the number of firms in the industry. First, we assess the impat of inomplete information on (i) the ex ante expeted prie, (ii) the ex ante expeted lowest prie, and (iii) the ex ante expeted profit of firms by omparing the two omplete and inomplete information senarios. Proposition 4.1 In the PBERP of the sequential searh model with inomplete information, the ex-ante expeted prie paid by non-shoppers, E(p), the ex-ante expeted prie paid by shoppers, E(p l ), and the ex-ante expeted profit made by firms, are higher than in the omplete information model. Consequently, onsumer surplus is lower and produer surplus is higher. Proposition 4.1 illustrates that onsumer welfare is higher when the onsumers are informed about the firms prodution ost suggesting that poliy interventions induing observability of prodution ost benefit onsumers. The main reason for this result is the following. As uninformed onsumers update their beliefs about the underlying ost level upon observing a prie, the reservation prie in the unknown ost ase is larger than the unonditional a priori expeted ost (whih is relevant in the known ost ase), i.e., the expressions in equations (20) and (21) are larger than the ones in (9) and (10). It is, however, important to note that the ex ante variation of pries is muh larger in the known ost ase than in the unknown ost ase, as a quik inspetion of Figures 1 and 2 reveals. This is beause the upper bound of the prie distribution in the known ost ase shifts with the ost parameter, while this is not the ase in the unknown ost ase. For the welfare omparison in our model, this is of no relevane as onsumers are supposed to be risk neutral. However, in ase onsumers would be risk averse, it may be that the welfare omparison seriously depends on how onsumers value the ex ante prie variation. Further, we have that : Proposition 4.2 The onditionally expeted profits of firms are dereasing in the ost level when onsumers are uninformed about the ost realization, whereas these profits are independent of when onsumers are perfetly informed. In partiular, onditionally expeted profits 21
under inomplete information are higher (lower) for low (high) ost realizations ompared to when onsumers are perfetly informed. Again, the fat that in the known ost ase the reservation prie shifts with the underlying osts is responsible for this result. Finally, we study the effet of hanges in the number of firms on the welfare evaluation of the different groups in the industry, and espeially how the differenes between the two models depends on N. For the known ost ase, these results are already summarized at the end of Setion 2. It is diffiult to evaluate the relevant expressions for the unknown ost ost as it is diffiult to evaluate the impat of N on the reservation prie. We are able, however, to asertain the following two results. First, when the number of firms in the industry is large, the differene between the two models beomes negligibly small, i.e., the omplete information model provides a good approximation of the more ompliated inomplete information model. 14 This is the ontent of the next Proposition. 15 Proposition 4.3 If the number of firms inreases without bound we have ( ) lim ρ E(ρ k ()) = 0 N ( ) lim E(p) E k (p) = 0 N ( ) E(p l ) E k (p l ) = 0. lim N The proof of this proposition exploits the fat that around the upper bound of the prie distribution (whih is the reservation prie), the prie distributions in the unknown ost ase are almost idential for different ost realizations. Bayesian updating yields therefore not more information and the reservation prie is simply based on the ex ante expeted ost. Thus, for large N the reservation prie in the unknown ost ase is very lose to the ex ante expeted reservation prie in the known ost ase. It is striking that the differene between the two models beomes negligible in ompetitive markets where N is large. This is in ontrast to results in? where, in a ontext where pries signal quality, the differene between the omplete and inomplete information models remains for large N. A seond result is that the effets of inreased ompetition in the unknown ost ase may be non monotoni. As an be verified from Figure 5, there are numerial examples where the expeted prie, E(p), is first dereasing in the number of firms and starts to inrease only above a ertain number of firms in the market. The fat that the expeted prie E(p) may be 14 We already know from Setion 3.4 that the existene ondition for PBERP is satisfied for large N. 15 Note that we assume that the willingness to pay is large enough suh that the onstraint that pries must be below v is not binding. 22