Consumer Search with Observational Learning

Transcription

1 Consmer Search with Observational Learning Very Preliminary and Incomplete. Please Do Not Circlate. Daniel Garcia Sandro Shelegia May 2, 2014 Abstract This paper stdies social learning in a search environment. We model consmers who observe other consmers prchasing decisions before embarking on their own search for the best-fitting prodct. This form of social learning has two distinct effects on consmer search and firm pricing. First, consmers emlate others in the sense that they always make the first visit to the firm where their predecessor has prchased. Second, consmers free-ride on their predecessor s information and search less intensively than in the standard search model. Emlation encorages price competition becase firms fight for consmer visits, while free-riding has the opposite effect de to redced search. Both effects increase in the search cost, bt emlation is shown to dominate for most commonly sed distribtions of consmer preferences, therefore prices (eventally) fall as search cost increases, and may even go down to the marginal cost. We show that the reslts derived in a static dopoly model remain valid with a large nmber of firms and can be extended to a dynamic framework. JEL Classification: D11, D83, L13 Keywords: Consmer Search; Observational Learning, Emlation We thank Maarten Janssen for insightfl comments on an earlier version. Garcia grateflly acknowledges financial spport by the Hardegg Fondation. All errors are to be attribted to the athors only. Department of Economics, University of Vienna. daniel.garcia@nivie.ac.at Department of Economics, University of Vienna. sandro.shelegia@nivie.ac.at 1

2 1 Introdction Observational learning has been the object of stdy of an increasingly large literatre in economics since the seminal contribtions of Banerjee (1992) and Bikhchandani et al. (1992). In the classical model, a seqence of individals faces a simple decision problem nder ncertainty and where each individal observes the history of decisions of her predecessors. As arged by Banerjee (1992), this simple environment closely resembles the problem faced by consmers in many markets, where previos cstomers choices may be informative abot the relative vale of different prodcts. Search markets, where consmers have to actively engage in costly activities to gather information abot different options available to them, seem to be prominent examples of sch environments. Intitively, new consmers may free ride on the search effort of others and follow their advice. Importantly, this wold sbstantially change the elasticity of demand and, hence, eqilibrim prices. Ths, observational learning in search markets may have important implications on search behavior and ltimately on prices. To the best of or knowledge, no paper has stdied this important isse. We attempt to bridge this gap by analyzing a simple dopoly model of search with heterogenos prodcts in the spirit of Wolinsky (1986). In the model, a large nmber of consmers derive tility from a given good, that comes in two varieties sold by two different firms. Consmers are initially ninformed abot their valation for each variety or the price charged by each firm, bt learn them after engaging in costly seqential search. These valations are randomly drawn from some variety-specific distribtion. Firms simltaneosly choose prices so as to maximize expected profits taking the behavior of consmers as given. To this fairly standard setp borrowed from Wolinsky (1986) and Anderson and Renalt (1999) (henceforth ARW ) we add observational learning by informing individal consmers of the prchasing decision of their predecessor whose valation is positively correlated with consmer s own valation. Importantly, consmers do not observe whether their predecessors searched actively nor the price or tility derived. Ths, this is a model of (limited) observational learning. Strikingly, this change in the information strctre of consmers radically changes eqilibrim otcomes. In the ARW model, consmers first visit is directed to each firm with eqal probability (which is indeed optimal since the reslting eqilibrim prices are eqal). In or framework, becase tilities are positively correlated, consmers decide to search first the variety prchased by their predecessor. As conseqence, the proportion of first visits to a firm coincides (in the long-rn) with her market share, and since consmers are more likely to by the variety they sample first, this leads to a social mltiplier of demand. To nderstand the effect of this change, we first assme that the correlation between consmers valation is positive bt negligibly small, so that in a symmetric eqilibrim consmers learn nothing from their predecessors bt still follow them for their 2

3 first visit (which is still optimal). We show that in the reslting eqilibrim, prices are lower than in the ARW model, and for all commonly sed distribtions prices decrease in search cost once search cost is sfficiently large. The difference between or and Wolinsky-Anderson-Renalt model stems from the behavior of consmers who always by the variety they sample first (becase their valations for both prodcts exceed their reservation tility). In the ARW model, the firm who is visited by these consmers first is effectively a monopolist and, ths, as the proportion of sch consmers increase, as it does when search cost rises, prices tend to increase. In or model, however, these consmers are allocated according to the share of searchers who visit both stores. Since firms cannot price-discriminate between the two grops, as the proportion of consmers who stop at the first store increases, firms engage in increasingly fiercer competition for searchers who determine market shares, leading to Bertrand-like competition and eventally to prices that can be as low as the marginal cost. We then introdce differences in the distribtions of valations of each variety. More precisely, we assme that the valations that consmers derive from the variety offered by each firm are drawn from one of two distribtions, and one of those distribtions (High) dominates the other (Low) in the First Order Stochastic sense. Assming that neither firms nor consmers observe which distribtion realized, we can focs on the effect of learning in consmer behavior and prices. Note that a previos prchase of a given variety leads to an pwards pdate of its distribtion of valations and a downwards pdate in the rival s distribtion. Hence, consmers are now willing to accept a lower srpls from their first visit, ths redcing search effort. We term this the free-riding effect of observational learning. The effects on eqilibrim prices are, however, not straightforward. On the one hand, as explained above, less search leads to higher competition becase the social mltiplier becomes more important. On the other hand, a price deviation triggers a change of beliefs that may overcome this effect. If a firm deviates to a higher price, the proportion of sales in each demand state changes, ths changing the beliefs that incoming consmers have. Since the elasticity of demand is lower for the firm with a higher distribtion of valations, consmers become more pessimistic abot its rival s distribtion the higher is the price of the store they visit. Hence, the srpls (net of the price) they demand for bying right-away decreases in the price, redcing the elasticity and increasing prices. We show that for some distribtions, sch as niform and normal, the second effect is stronger for relatively small search costs while eventally the second effect dominates and prices may be lower than in the model withot learning. We are able to show that, jst as in the model with emlation bt no learning, in the model with learning, nder a reglarity condition, prices eventally decline in search cost and fall all the way down to marginal cost. The isse hinges on whether the likelihood ratio is sfficiently informative in the extreme case where almost no consmers search 3

4 except those with the lowest valations for one of the prodcts. We are able to show that distribtions with finite lower bond always satisfy the condition, and so ore prediction on the relationship between search cost and eqilibrim prices is rather robst. In the reminder of the paper we consider extensions to oligopoly and dynamic pricing. First we stdy the effects of competition on eqilibrim prices in the model withot learning. We show that prices decrease in search cost and converge to marginal costs as the proportion of movers vanish, for any nmber of firms. We frther show that prices decrease in the nmber of firms for any search cost. Of special interest is the limit price when the nmber of firms grows large. In this case, whether there is correlation across individals or not, there is nothing to be learnt abot rival firms qalities from the realization of sampled varieties. Armstrong and Chen (2009) shows that, as long as it is exogenos, the order of visits does not change prices in the ARW model since the elasticity of demand is the same for all firms. In or model, however, prices are lower than in the ARW model and have a U-shape relationship with search costs. If search costs are very low and there are many firms, Bertrand competition for byers obtains. On the other hand, if search costs are very high, we know that firms fiercely compete for first visits and prices also converge to marginal costs. For intermediate search costs, however, firms charge positive prices. Regarding dynamic pricing we show that if firms can adjst prices after the introdctory period, and that in sbseqent periods consmers make first visits proportional to first-period market shares, prices are frther redced in the introdctory period with observational learning, and then go p to the levels of the ARW model. Ths observational learning mechanism is complementary to the dynamic market share bildp mechanism and together they lead to a very low introdctory prices and then price hikes. Literatre Review Or model of observational learning is inspired by recent evidence in economics and marketing. Mobis et al. (2005) present reslts of a field experiment designed to nderstand individal demand for different prodcts for which individals receive information. They show that those sbjects who are connected throgh social links to others who are informed abot a particlar good, vale the goods more. Importantly, they find that this effect is stronger for gadgets than for services, which are more likely to be sbject to direct observation. In another field experiment, Cai et al. (2009) show that restarantgoers are more likely to order those goods that are presented to them as more poplar. Finally, Moretti (2011) analyzes the movie market where an nexpected increase in the first-week s box office has a persistent a significant effect in ftre attendance. Two recent papers have analyzed consmer search with observational learning, bt both assme that prices are fixed exogenosly. Kircher and Postlewaite (2008) stdy 4

5 consmers who differ in their willingness to search and firms differ in qality. Althogh prices are fixed, firms may decide to offer a valable service to any consmer who visit their store. They show that eqilibria may arise where high-qality firms offer service to those consmers who search more actively and those who search less actively follow their advice. Hendricks et al. (2012) presents a model of observational learning with mltiple types in the spirit of Smith and Sørensen (2000) where each consmer has to decide between acqiring a costly signal abot qality of a single good, bying it right-away or not bying. The model is cast in a more traditional herding framework where consmer receive a signal before deciding whether to engage in costly search. They focs on the long-rn dynamics of sales for high and low qality prodcts and the possibility of bad herds arising. Or model lacks pre-search information, therefore we cannot stdy bad herds, bt instead focs on pricing. In the consmer search literatre with price competition, closest papers to ors are Armstrong et al. (2009) and Armstrong and Zho (2011), who present a model of prominence in consmer search where one firm is sampled first by all cstomers. In Armstrong et al. (2009) a given firm is made prominent exogenosly. In the reslting eqilibrim, the prominent firm charges a lower price than her rivals becase her share of retrning cstomers (who are typically less responsive to prices) is lower. One may view or framework as that of endogenos prominence, where share of first visits depends on price. Armstrong et al. (2009) show that as the nmber of firms grows, (exogenos) prominence becomes irrelevant while in or model, competition becomes even fiercer and prices decrease in the nmber of firms. Armstrong and Zho (2011) stdy several different models that endogenize firm s prominence. One sch model is based on observable price competition where consmers rationally search the lowest-pricing firm first. Becase demand is discontinos in prices, the reslting eqilibrim involves mixed strategies and has a property that higher search cost leads to (stochastically) lower prices. One may view observability of prices as an extreme example of observational learning where consmers observe market shares, and ths prices. We show that imprecise information abot prices reslts in a pre strategy eqilibrim featring inverse relationship between search cost and prices. Another closely related model can be fond in Haan and Moraga-Gonzalez (2011), where prominence depends on advertising efforts and firm profits may decrease in search cost (althogh prices increase and consmer srpls decreases). Or paper is also related to the literatre that stdies effects of social learning on monopoly pricing. Campbell (2013) and Chhay (2010) analyze the impact of word-ofmoth commnication on the monopoly price and prodct design. Perhaps closest to or work are two contribtions by Bose et al. (2006) and Bose et al. (2008) who stdy dynamic interaction between a monopolist and a seqence of consmers with common valation who observe each other prchasing decisions. While most of this literatre has stdies monopoly, an exception is Kovac and Schmidt (2014) who stdy a dynamic market where 5

6 two firms offer a homogenos prodct and consmers learn prices from others. Since or focs is on competition and we abstract from dynamic isses 1, we view or work as complementary to theirs. Finally, a nmber of papers have stdied the relation between crrent market shares and ftre demand. Becker (1991) directly introdces aggregate demand into the individal tility fnction. Caminal and Vives (1996) stdies a dynamic signaling game where new cohorts of consmers observe past market shares of an experience good, bt not prices and try to infer qality from this information. Firms se prices to maniplate market shares to attract consmers. While their setting is different in that consmers do not search, we also find that firms se prices to attract consmers, bt here consmers free-ride on their predecessors efforts and de to search cost, herds form, while in Caminal and Vives these effects are mte. The next section introdces the general model with observational learning with correlated preferences and endogenos prices. Section 3 solve the model in the special case of no correlation between preferences of consmers. Section 4 stdies the model with correlated preferences and establishes comparative statics reslts inclding with respect to the nmber of firms. Section 5 introdces an extension to dynamic pricing and Section 6 concldes. 2 Model Consider a market poplated by a large nmber of consmers i = 1, 2,..N, interested in prchasing a single nit of a differentiated good that comes in two varieties, each sold by one firm, 1 and 2. Consmers are initially ncertain abot their valation of each firm s prodct, bt may acqire this information throgh seqential search. We assme that the first visit is free bt the second has a cost of c in tility nits and all consmers can recall their previosly sampled varieties at no additional cost. Following Anderson and Renalt (1999) we make the simplifying assmption that all consmers will by one of the two varieties. 2 We describe now the demographics of the model. In every period, a cohort of consmers arrive to the market. Every consmer makes all his decisions in that one period. Let 1 t i T be the period in which consmer i arrives. The tility that every consmer i derives from variety j depends on his type k {1, 2,.., K}, and in each cohort there is a single individal of each type, so that there are N = K T consmers. Consmers do not know t i and hold a common prior ν(t) over it. We introdce observational learning by informing every consmer arriving in a cohort t > 1 of the prchasing decision of the previos consmer of his type. Importantly, they do not observe the information that has 1 See Section 4 2 This amonts to assming that their otside option is sfficiently bad. 6

7 led to the prchase or whether their predecessor sampled more than one variety. We introdce correlation between preferences of individals of a given type in the following fashion. All consmers of a type draw their valations for each firm s prodct from one of two potential distribtions, a High distribtion (denoted by G H ()) and a Low distribtion (G L ()). Both distribtions are eqally likely to realize for each type, and these realizations are independent across types and firms. We assme that both distribtions have the same (possibly infinite) spport [, ū] and that G H First Order Stochastically Dominates (FOSD) G L. Let g H and g L be the corresponding densities. As is sal in the economics literatre, we assme that the Monotone Likelihood Ratio property holds so that g H(x) g g L is an increasing fnction of x. Let λ() = H () (x) g H ()+g L be the () conditional probability of H given. Let G() = 1G 2 H() + 1G 2 L() be the nconditional distribtion of valations of a random consmer. We shall assme that G() is logconcave. 3 Finally let (S 1, S 2 ) k {H, L} 2 be the realized state for a given type. To smmarize, all consmers of the same type draw their tility for a firm from the same distribtion. E.g. consmers of type k may draw their tility from distribtion H for firm 1 and L for firm 2. Even thogh consmers cannot know the tility of the consmer they observed, the fact that that consmer bys from a firm is informative abot which state (S 1, S 2 ) k has realized. In order to disentangle changes in information from changes in the distribtion of valations, it is convenient to define two axiliary distribtions G 1 () and G 2 () sch that G H () = (1 r)g() + rg 1 () while G L () = (1 r)g() + rg L () and sch that G() = 1 2 G 1() G 1(), where r [0, 1] measres the degree of correlation across consmers of a given type. This specification garantees that for any r the nconditional distribtion (withot knowing state S) for a firm is G(), while r controls how close G L is to G 1 and G H is to G 2. If r = 0 valations are independent, and drawn from G(), while for r > 0 valations of any two members of the same type are positively correlated. Conveniently, for every r the nconditional distribtion of valations remains constant bt the amont of information contained in the prchasing decision of a predecessor may vary significantly (as allowed by the disparity between G 1 and G 2 ). Ths r proxies correlation of valations among members of a type. On the spply side, there are two firms spplying any qantity of their variety at a constant nit cost normalized to zero. Importantly, firms commit to serve all incoming cstomers at a single price p j set simltaneosly before the state of the market is realized in order to maximize expected profits. We shall devote most of or attention to a symmetric, pre-strategy eqilibrim where both firms charge the same price p. 3 Log-concavity of G H and G L does not garantee Log-concavity of G. 7

8 2.1 Consmer Behavior Consmers in the first cohort, lacking any information to discriminate firms, make their first visit randomly and by there if and only if i p i > ŵ p, where ŵ solves ū ŵ ( ŵ)g()d = c, (1) where g() = g H()+g L () 2 and is the density of G(). ŵ is the familiar reservation tility from Wolinsky (1986). It is fond by eqalizing expected benefit from search (with free recall) to the search cost. Unsrprisingly, the first consmer of any type is no different in or model as compared to the ARW model. As is well known, in ARW there is no search for c > ū g()d, adn in the version of the model where the otside option is infinitely bad, prices are infinitely large. To avoid this, henceforth we assme Assmption 1. The search cost satisfies c < ū g()d c. All remaining consmers observe a predecessor bying from firm j and expect the same price p in both stores before embarking on their first search. Since valations are positively correlated, the expected srpls from firm visiting firmj is (weakly) larger than that of firm j and so the consmer shold visit that store. Upon visit consmer learns his tility realization for good j, ij and the price that the firm j charges. Consmer will search if and only if ij p j < w(p j ) p, where w(p j ) solves ū w p j +p ( w + p j p )(q(w; p j )g H () + (1 q(w; p j ))g L ())d = c (2) where q(; p) is the (posterior) probability that at the other store the valations are drawn from the high distribtion. This probability will, in general, depend on the valation drawn from firm j (becase preferences are correlated) and the price of firm j becase it affects the probability consmers by at that firm, and ths conditional distribtion of the other store. In principle, q(; p) is a very complicated object, since each consmer may infer from not only how likely it is that a given distribtion realized bt also his cohort (which is potentially informative abot the informational content of the prchasing decision of the predecessor). In particlar, the probability that a consmer arriving in cohort t bys from firm 1 if the state is (SS ) {H, L} 2 is x t SS (p 1, p 2 ) = x t 1 SS (p 1, p 2 )(1 G S (w(p 1 ))) + (1 x t 1 SS (p 1, p 2 ))G S (w(p 2 ))(1 G S (w(p 1 ))) + w(p1 ) p 1 +p 2 G S ( p 1 + p 2 )g S ()d which increases in x t 1 SS, leading to a link between past and crrent market shares. It 8

9 is straightforward to see that this mapping has a fixed point where market shares are stationary (x SS (p 1, p 2 )). In Appendix 1 we show that, provided that the nmber of consmers of each type is sfficiently large, consmers optimal search strategy is arbitrarily close to the one compted for a stationary market share distribtion. In this case, dropping the time sbscript, we can write a firm s market share in state SS when it charges p while the other firm charges the eqilibrim price as x SS (p, p ) = x SS (p, p )(1 G S (w(p))) + (1 x SS (p, p ))G S (w(p))(1 G S (w(p))) w(p) p+p + G S ( p + p )g S ()d. We can solve for x SS (p, p ) from the above x SS (p, p ) = p +p+w p +p+w g S ()G S (p p+) d+(1 G S ( p +p+w ))G S (w ) g S ()G S (p p+) d.(3) w(p) g S()G S (p p+) d+(1 G S ( p +p+w ))G S (w )+G S (w(p)) For stationary market shares, q(; p) satisfies q (; p) = x HH (p)λ() + x LH (p)(1 λ()) (x HH (p) + x HL (p))λ() + (x LH (p) + x LL (p))(1 λ()) (4) The above formla ses market shares in every state and weights the conditional probability of H given by these market shares. As expected, if in all states both firms have eqal market share (x SS = 1/2), which given or assmption abot G H and G L imply G L = G H, the first visit is not informative and q (; p) = 1/2. Notice that q (; p) is increasing in becase λ is an increasing fnction and the market shares satisfy x HH x LH and x HL x LL. Intitively, a higher leads the consmer to pdate pwards the probability he attaches to the crrent firm having a high distribtion (S = H). Importantly, this also increases the belief he holds abot the other firm s distribtion, since it redces the informativeness of the predecessor s prchase at this firm regarding the other firm s distribtion. This completes the characterization of the consmer search rle. In particlar, w(p) is implicitly defined in (2) where q(; p) is given by (4) where x SS is defined in (3). In the reminder of the paper we will se stationary market share distribtions. One may think of a redced-form model where consmers observed a predecessor, proportion of first visits matches firm s market share, and search and firm strategies are consistent with the above. 2.2 Eqilibrim Conditions In a symmetric eqilibrim, prices are eqal and, ths, do not affect search behavior. The consmers search rle can be rewritten as 9

10 ū ŵ ( ŵ)(q(ŵ)g H () + (1 q(ŵ))g L ())d = c (5) where q(ŵ) = q(ŵ; p ) is the eqilibrim probability that the rival firm has the High distribtion given the reservation tility. Notice that q() q(ū) 1 since observing a 2 previos consmer bying in a store is bad news abot the prospects in the rival store. This effect is only flly mitigated if the consmer learns that the crrent firm offers high valations, in which case it learns nothing abot his prospects in the rival firm. Hence, we have the following trivial observation. Proposition 1. Fix c > 0, in a niqe symmetric eqilibrim, ŵ < w, and, hence, consmers free-ride on others effort. Firms profits depend on the search rle sed by consmers both on and off-theeqilibrim path. In particlar, let w (p) be the implicit derivative of the search ctoff with respect to p. In the ARW model model withot learning, w (p) = 1 so that an increase in price is compensated with a higher reqired tility. This is no longer the case with observational learning. Consmers visiting a firm with a different price adjst their beliefs abot the distribtion of valations in the rival firm, while keeping their beliefs abot its price constant. 4 In particlar, the higher the price, the less likely it is that a consmer ends p in that firm if its rival has a High realization of valations. Ths, in general, w (p) < 1. Given this search behavior, consmers are split into for grops in the ( 1, 2 ) space (plot). As is standard in Wolinsky-type models, those whose valation for both varieties is lower than the reservation tility ŵ search independently of the variety they sample first and then by from the highest-srpls offering firm. Those whose tility profile satisfies j < ŵ < k only search if they visit firm j and always by from k. Finally, those whose valations are higher than their reservation tility by from the firm they visit first. This grop has low-demand elasticity and is typically assigned randomly across firms. In or model, these consmers are assigned to each firm with probabilities eqal to market shares and, ths, become qite elastic. We term this the emlation effect of observational learning. Expected demand for a firm that charges p while its competitor charges p and consmers se reservation tility fnction w(p) is the average of demands over all possible states. Becase states between firms are independent, and eqally likely, the demand is given by: x(p, p ) = 1 x SS (p, p ). (6) 4 SS Here x(p, p ) implicitly depends on w(p) and ths on consmer search rle. 4 The assmption of passive beliefs is common in the literatre. For a discssion see Janssen and Shelegia (2014) 10

11 In eqilibrim, symmetric eqilibrim price satisfies the following condition x SS (p, p ) + p x SS (p, p ) p SS SS = 0 and frther SS x SS (p, p ) = 1 2 to Eqations for x SS by symmetry. So the eqilibrim pricing rle simplifies x SS (p 1, p ) p = 0 (7) p SS 1 are non-trivial and depend on a complex way on the search rle consmers se. Becase of this, while we derive some reslts for any G and r > 0, before proceeding to analyzing the general case we trn to the special case where r = 0. 3 Emlating Consmers To better nderstand these two effects, we start by isolating emlation from free-riding. We do so by assming that r 0 so that valations across consmers are nearly independent, while consmers still follow others prchasing decisions for their first visits. This is the limit case of the general model when distribtions become increasingly similar, and since the model is ex-ante symmetric, making the first visit to the firm where the predecessor has prchased remains optimal (inclding when r = 0). More generally, one can view this model as a behavioral modification of Anderson and Renalt (1999) where consmers first visits are allocated according to market shares rather than randomly. While the modification is rather small, it has dramatic effect on prices. Becase the visit of the predecessor contains no information, the reservation tility is compted as in the ARW model and is given by (1). As for the firms, let x denote market share of firm 1. Becase first visits also follow prchases, the share of firm 1 s first visits is also x. Ths its demand can be written as x(p 1, p ) = x(p 1, p )(1 G(ŵ + p 1 p )) + (1 x(p 1, p ))G(ŵ)(1 G(ŵ + p 1 p )) ŵ+p1 p + G( p 1 + p )g()d From the above, it is straightforward to solve for x(p 1, p ) and find firm 1 s pricing eqation. Proposition 2. Sppose r = 0. In a symmetric eqilibrim with emlating consmers, the reservation tility is compted as in (5) and the price is p = 2 ŵ (2 G(ŵ))G(ŵ) g2 ()d + (1 G(ŵ))g(ŵ) (8) 11

12 while in the ARW model ˆp = 2 ŵ 1 g2 ()d + (1 G(ŵ))g(ŵ) (9) Ths, p p, with a strict ineqality for w < ū. The intition for this reslt is rather simple. In the ARW model, price competition is restricted to the small sbset of consmers whose valations for both firms are close and lower than the reservation tility. In the model with observational learning, those consmers whose valation for both varieties is larger than the reservation tility become sensitive to price changes becase they follow the lead of those with relatively low valations. This is the social mltiplier effect on demand. As the proportion of consmers who do not search beyond the first firm grows, price competition for searching consmers who bring all others to the store intensifies leading to lower prices. Recall that c = ū g()d is the average tility at a given store. Using the argment above, we can state: Proposition 3. For any log-concave G with bonded spport, for any lim c c ˆp = 0. Also, if G has nbonded spport bt lim G()/g() = 0, then lim c ˆp = 0. The first part of the proposition has been explained. The second part pts a restriction on the ratio of the cmlative to the density at the lower bond that is satisfied by all distribtions whose spport is bonded below as well as for most log-concave distribtions whose spport is nbonded (e.g. Normal). This condition is not satisfied for the Logistic distribtion, bt it can also be readily verified that prices decrease in search costs for this distribtion too. 4 Eqilibrim with Learning The effects of learning on prices is ex-ante ambigos becase learning adds two competing effects. First, as highlighted by Proposition 1, for a given c and a symmetric price p, ŵ is lower the bigger the difference between distribtions. Consmers are then willing to accept lower srpls the higher the price (w (p) < 1), which pshes prices pwards. On the other hand, in the presence of emlation, lower ŵ may lead to lower prices throgh fiercer competition for increasingly rare searchers. This effects can be seen in pictres... Notice that, eventally, free-riding joins forces with emlation and prices decrease faster and converge to marginal cost for lower search costs than withot learning. In particlar, the maximm search costs at which a pre-strategy eqilibrim exists eqals the expected valation of G L. This is becase, in the example, the likelihood ratio grows nbondedly at the lower bond. Hence, a consmer who arrives at a store offering sfficiently low tility is almost sre that the crrent firm s distribtion of valations is 12

13 Low, which implies that with arbitrarily high probability the rival offers Low valations too, for otherwise the probability that my predecessor ended p here is negligible. To see this, notice that lim x SS (p, p ) = c c G S (ŵ) G S (ŵ) + G S (w(p)) If λ() = 1, x L,H (p, p ) = 0 and ths, q() = 0. In sch a case, search is nattractive and the consmer stays. Define c L = ū g L()d. Since prices approach marginal costs as the share of movers vanishes, we have the following conterpart of Proposition 3: Proposition 4. Sppose G has a finite spport. As c c L, we have ˆp 0. Prices converge to zero as the search cost converges to the expected valation of G L. The following Corollary sggests that learning itself may decrease prices by letting the ex-ante distribtion of valations constant. Let ˆp(r) be the price if the distribtion is governed by r [0, 1] and recall that higher r correspond to higher correlation and, therefore, higher informativeness of the prchasing decision of a predecessor. We have the following Corollary Corollary 1. Take a distribtion G. For c ( c L, c), ˆp(1) = 0 < ˆp(0). That is, as the informativeness of the prchasing decision increases, the eqilibrim price elasticity increases and, ths, prices decrease. (10) This is becase those consmers whose valation for the variety they sample first is low become increasingly pessimistic abot their prospects in the other store the higher is the correlation across consmers, and, therefore, the less inclined they are to search. The following figres illstrate or reslts for a niform distribtion G on [0, 1]. In this example G 1 is a trianglar distribtion on [0, 1] with mode 0 while G 2 is a trianglar distribtion on [0, 1] with model 1. As reqired, the mixtre of the two is niform on [0, 1], G H FOSD G L and the likelihood ration is monotone for any r > 0. Figres 1 and 2 show prices and reservation tilities in varios models. As expected, prices are always lower in the model with pre emlation (black) than in the ARW model. As predicted by theory, in ARW model price is increasing in s and reaches 1 at c = 1/2. This is where even a consmer who draws 0 at the first store refses to search frther (after this, prices are infinite becase otside option is ). In or model with learning (r = 1), as predicted by Proposition 4, becase = 0 >, price is zero at c = 1/3, which is the average of G 1. In or model withot learning (r = 0), price converges to zero at c = 1/2, as implied by Corollary 2. Intition for both is simple. When c = 1/3 in the model with learning, a consmer who is sre that tilities from the other store are drawn from G L will not search. Bt this is what happens in eqilibrim when a consmer draws tility close to - she reasons that becase distribtion in the crrent firm is almost srely G L, the fact that she came here indicates that in the other 13

14 1.0 p 0.8 ARW r =1 r = c Figre 1: Prices in or model with r = 1 (ble), r = 0 (black) and ARW model (red) for trianglar distribtions. store the distribtion is also G L, or otherwise almost all qees wold end p at the other store. Ths even thogh close to is bad news abot the crrent store, it is also bad news abot the next store. One interesting aspect revealed by Figre 1 is that prices in the model with information may be lower than withot information. This seems to contradict with Proposition 1 (and also Figre 2) that states that with information (r > 0) there is free-riding, ths prices shold be higher. This pzzle is resolved by noting that even thogh it is tre that reservation tility is lower in the model with information, which on its own wold reslt in higher prices, lower w also reslts in more elastic demand for firms (becase very few consmers actally search), leading to lower prices. Figre 3 shows w (p) as a fnction of c. For c 0, the derivative of reservation tility approaches 1. This is becase when almost all consmers search, price contains almost no information and so reservation tility matches it one for one. For all c > 0, w (p) is less than 1, reflecting the informational content of price deviations. Finally, Figre 4 illstrates eqilibrim demands of a given type for varios states as fnctions of c. In symmetric states (LL and SS), demand for both firms is eqal to 0.5 and is independent of search cost. Matters are more interesting in asymmetric states. As the search cost increases, demand for a firm with low distribtion facing a firm with high distribtion shrinks (the opposite is tre for high distribtion vs low distribtion). Natrally, when search cost is small some consmers by at the store where tility is drawn from G L, and so the same proportion of consmers visits this store first. As c 1/3, search vanishes and almost all consmers are steered first to the firm with high distribtion, and becase ŵ approaches 0, almost all stay there. Ths, a herd forms where all consmers make the first visit to the store with high distribtion, and 14

15 r =1 w ARW and r = c Figre 2: Reservation tilities in or model with r = 1 (ble) and or model with r = 0 and also ARW model (red) for trianglar distribtions. then correctly avoid searching frther. 5 The interesting aspect of sch optimal collective behavior is that it coincides with zero prices. Note thogh that prices are zero not becase of optimal collective behavior in eqilibrim, bt rather for the opposite reason ot of eqilibrim. Namely, if a firm were to deviate and charge a higher price, even thogh for a qarter of types the state is HL, almost no consmers of these types wold visit it. This means that pon price deviations bad herds form where small price differences reslt in qarter of consmer incorrectly visiting a firm with worse distribtion and slightly better price. 4.1 The Effects of Competition We have assmed throghot that there are only two firms in the market. This assmption ensres that both decisions and information sets are binary and renders the analysis of consmer learning and behavior tractable. As the nmber of firms increases, the nmber of states increases exponentially and so do the complexity of the information strctre. Ths, to analyze the effects of competition we go back to the simple model of consmers emlation where the learning channel is mted (i.e. r = 0). Notice, however, that as the nmber of firms becomes larger, and learning becomes more complex, the amont of information abot other firms distribtion contained in the prchasing decision of a predecessor also decays exponentially. Intitively, as the nmber of firms grows large, the market share of a given firm is, approximately, independent of the realization of the state in any other firm (mean-field approximation). Ths, while a prchasing decision by a 5 The word herd here is sed in contrast to the economics literatre. In or model consmers do not possess pre-search information, and have to obtain it actively, ths herds in traditional sense cannot arise. 15

16 w 0 (p) Figre 3: Derivative of reservation tility with respect to p in or model for r = 1 and trianglar distribtions. c predecessor remains informative abot the distribtion of valations for the variety sold in the crrent firm, it is irrelevant in determining the distribtion of valations for any other variety. Hence, assme that r = 0 and let n be the nmber of firms in the market. The stationary market share of a given firm choosing a price p if all its rivals choose p is x(p, p ) = x(p, p )(1 G(ŵ p + p)) + 1 x(p, p ) n 1 G(ŵ) j (1 G(ŵ p + p)) n 1 + ŵ p +p G( p + p ) n 1 g( p + p )d. Notice that market shares only determine the nmber of first visits to each firm, bt not the sbseqent searches. Let p n be the price in a symmetric eqilibrim with n firms. For an arbitrary distribtion G, we have j=1 p n = ng(ŵ) G(ŵ) n n 1 g(ŵ)( n 2 i=0 G(ŵ)i +(n 1)G(ŵ) n 1 ) (n 1)n ŵ G()n 2 g() 2 d. (11) Recall that ŵ is still defined by (5). This can be contrasted with the eqilibrim price in the ARW model given by ˆp n = 1 g(ŵ)( n 2 i=0 G(ŵ)i (n 1)G(ŵ) n 1 )+(n 1)n ŵ G()n 2 g() 2 d. (12) Proposition 5. Assme r = 0. For every n and bonded G, the price with emlating 16

17 1.0 x x HL (p,p ) x HH (p,p )=x LL (p,p ) x LH (p,p ) c Figre 4: Eqilibrim demand of a given type for varios states for trianglar distribtions. consmers is lower than the price in the ARW model and goes to zero with c. Frther, ˆp n bt also p n/ˆp n decreases in n. As with the ARW model prices go to zero with n in or model. Important distinction is that as the nmber of firms grows, the relative gap between prices in or and ARW model also grows, ths competition componds the emlation effect. The comparison becomes more clear when we take the limit as the nmber of firms grows. In this case, thee share of retrning cstomers vanishes and the price converges to p = (1 G(ŵ))G(ŵ) g(ŵ) (13) so that lim c 0 p = lim c c p = 0. On the other hand, as shown in Anderson and Renalt (1999), the price in the ARW model (eq. 12) with an infinite nmber of firms coincides the classical Perloff-Salop formla ˆp = 1 G(ŵ). (14) g(ŵ) Armstrong et al. (2009) shows that this price is independent of the way consmers decide their search paths, as long as this order is exogenos. This shows that the effect of prominence is qalitatively different from the effect of emlation, particlarly when the nmber of firms is large. The next figre illstrates eqilibrim prices for different n. As shown in Proposition 5, prices decrease in n. 17

18 1.0 p Figre 5: Eqilibrim prices as fnction of search cost for n = 2 (prple), n = 4 (red) and n = (ble) for or model (solid) and ARW (dashed). 5 Dynamics and Lock-in Effects In this Section we explore the connection between or model and the Switching Cost literatre and, in so doing, introdce dynamic considerations. We consider the following extension of the Benchmark model. The economy has two dates τ = 1, 2. In τ = 1 the market strctre is as presented above. Firms set prices for the first date and consmers arrive in different cohorts and observe their predecessors prchasing decision. In τ = 2, firms observe their sales and simltaneosly choose prices for the second date. In order to introdce a simple dynamic link we assme that consmers visit first the store they prchased from in the first period bt their tilities are drawn anew. As in the standard switching cost model, the market in the first date is more competitive than in the second. In this case, the difference in competition comes from the observational learning that does not occr in the second Again, for simplicity, we assme that r = 0, so that no learning abot demand occrs in eqilibrim. assme that valations are niformly distribted in [0, 1]. Frther, we follow Armstrong et al. (2009) and In the second periods, firms choose prices to maximize expected profits given their nmber of first visit. In particlar, given a share of first-date market, her second-date market share is c x 2 (p, p 2 (x 1 )) = x 1 (1 (w p + p 2 (x 1 ))) + (1 x 1 )(w p 2 (x 1 ) + p 1 (x 1 ))(1 (w p + p 2 (x 1 ))) + w p+p 2 (x 1 ) ( w + p p 2 (x 1 ))d Let = p 1 (x 1 ) p 2 (x 1 ) be the expected difference in prices in the second period given 18

19 some market shares and let = p p 2 (x 1 ) be the realized difference. We have x 2 ( ) = x 1 (1 (w )) + (1 x 1 )(w )(1 (w )) + w ( w + )d From Haan and Moraga-Gonzalez (2011) we know that (x 1 ) is increasing in x 1 and ( 1) = 0. If a symmetric eqilibrim exists 2 = 0 and p 2 (x 1) = p 2 ( 1 ) = ˆp. Now, let 2 π(x 1 ) be the expected profit in the second date given market share x 1. π(x 1 ) is clearly increasing in x 1. Firm 1 solves then max p 1 p 1 x 1 (p 1, p ) + π(x 1 (p 1, p )) (15) Clearly p 1 p and in any symmetric p 2 (x 1) > p 1 so that prices increase over time. 6 Natre of observational learning In or model consmers observe the prchasing decision of a single predecessor. While this assmption is rather crde, it is perhaps more plasible than assming that consmers observe the whole seqence of prchasing decisions or the tre market shares. In any case, this assmption greatly simplifies the analysis since the nmber of possible information sets a given consmer may end p in grows exponentially in the nmber of predecessors. More importantly, in or model all consmers have the same reservation tility strategy. This wold not be the case if they observe a larger set of consmers. On the other hand, if consmers observed market shares withot noise, the eqilibrim wold converge to the mixed-strategy pricing eqilibrim presented in Armstrong and Zho (2011) for the case in which consmers observe prices. The reason for this is that even a small deviation in price wold reslt in all consmers first visiting the deviating firm, ths profit wold jmp p discretely. This is the familiar Bertrand pressre on prices. Bt prices cannot be zero becase for relatively small search cost a firm can still earn profit even if it is visited last. Therefore, there has to be a mixed strategy eqilibrim. 6 This shows that when consmers posses a lot of information abot their predecessors prchases, eqilibrim reslts in mixed strategies. We analyze the other extreme where consmers have very limited information. What happens between these two extremes is extremely hard to analyze, bt we believe that for relatively limited observational learning eqilibrim is qalitatively similar to ors, ntil when a lot of learning reslts in mixed strategies. 6 This is in contrast to or reslts (Propositions 3 and 4) that say that prices go down to zero.there search costs are so high that almost no one searches beyond the first firm, ths not matching the other firm s low price reslts in zero profits. 19

20 7 Discssion Or aim in this paper was to introdce simple observational learning into a standard consmer search model with horizontally differentiated prodcts. We did so and showed that consmer search models change qalitatively with sch learning. To achieve this goal, we have made several important assmptions that can be relaxed and or modified in ftre work. First, we followed Anderson and Renalt (1999) in assming that all consmers by. This assmption greatly increases the tractability of the search model and simplifies learning across individals. In its defense, it shold be noted that an extensive margin of demand shold psh prices downwards. Since or main reslts concern srprisingly low prices, an active extensive margin wold reinforce these reslts. Moreover, it allows s to concentrate on the bsiness-stealing effect of information, since the amont of prchases is kept constant. Second, we have introdced correlation across consmers valations in the simplest feasible way. If we went one step frther in the direction of simplicity, and assmed that all consmers of the same type have the same tility, the eqilibrim search rle wold not have standard reservation tility property. This is becase, in a ptative ctoff-strategy eqilibrim, the prchasing decision of a predecessor is more informative for tilities jst below the ctoff and so search is less valable there. Therefore, consmers wold want to stop for tilities below the ctoff. As a reslt, the eqilibrim wold have to featre an interval of tilities where probability of searching frther transitions smoothly from one to zero. Althogh very interesting, this search rle renders the fll model less tractable. Moving in the direction of more general correlation of tilities within types is also hard to model since consmers learning abot the distribtion in other stores is highly non-linear and their beliefs are formed by mixtres of trncated distribtions. Hence, or model is a simple, yet appealing, compromise. We believe that pshing in either direction wold be important for or frther nderstanding of observational learning in search markets. Finally, we have for the most part abstracted away from dynamic pricing considerations. It is highly intitive that firms wold take advantage of consmers learning and change their prices accordingly. While we have addressed this isse in a somewhat rdimentary fashion by introdcing a period where learning has been completed, price changes dring the learning process may be interesting, bt are fairly complicated to handle. 20

21 A Appendix 1 The following two lemmata prove that the stationary eqilibrim we stdy can be reached as the limit of the dynamic economy of the model as N grows. Lemma 1. Sppose there is a niqe ctoff ŵ(p i ). Then, for every ɛ > 0, there exists a T < sch that for all states and prices w(p i ) ŵ(p i ) < ɛ. Proof. The idea for the argment follows the ideas of Lemma 2 in Thomas and Cripps (2014), althogh or model is mch simpler. Let M 1 (p 1, p ) and M 2 (p 1, p ) represent the probabilities that a consmer following ŵ bys from firm 1 if he visits this firm first and second respectively. These probabilities are independent of t. The evoltion of market shares can be readily compted as x t x t 1 = x t 1 x t 2 (M 1 (p 1, p 2 ) M 2 (p 1, p 2 )) (16) Notice that that M 1 M 2 max j,s (1 G S (w(p j )). If min j,s G S (w(p j )) = 0, then market share for firm 1 is either 0 or 1 (one of the firms is an absorbing state). In sch a case, the reslt holds trivially. Otherwise, 7 or where x W x t x t 1 {max j,s (1 G S(w(p j ))} t x 1 x 0 (17) x t x t 1 {max S (1 G S(w(p j ))} t x W 1 2 (18) are the shares of the wolinsky consmer (i.e. the first consmer). Clearly, this Fixed Point converges to the stationary market shares by Blackwell s Sfficiency Conditions and the parameter of convergence is {max j,s (1 G S (w(p j ))} which is strictly less than one. Since λ and x are continos fnctions, for every ɛ > 0, there exists ɛ T 1 < sch that T 1 = ln( ) and, therefore q max j,s (1 G S (w(0)) T (; p) q (; p) < ɛ. Let 2 T = LT 1, for L large enogh we have that sp T t=1 1 T q t(, p) < (L 1)ɛ + 2 2L < ɛ (19) Becase (2) defines a contraction mapping between q and ŵ(p), the reslt follows. Lemma 2. If λ () < 1 (gh () g L ())d, there exists a niqe w(p i) satisfying eqation (2). Proof. Taking derivatives in (2) we get q () ( w+p p )(g H () g L ()) (1 q()g H (w) (1 q()g L (w))) > 0 (20) w p+p 7 In eqilibrim, G S (w(p )) > 0 if c < c L = dg L (). 21

22 where q () < λ (). Rewriting we get λ () < 1 (q()g H(w) + (1 q())g L (w)) w p+p ( w + p p ) (21) RHS is a decreasing fnction of w, so the spremm is attained at w =. This condition is satisfied for all trianglar distribtions and for normal distribtions with small enogh different in means. TBC Proof of Proposition 2 Proof. First, in the case where consmers follow market shares, we need to compte demand as a fixed point of the market flow eqation. In particlar, given a conjectre price p for the opponent firm and a reservation tility w(p) = ŵ + p p, the market share of a firm charging p if the rival sticks to eqilibrim price is G(w(p)) ŵ x = 1 G( p + p )dg() G(w(p)) + G(ŵ)(1 G(w(p))) ŵ G( p + p )dg() + w(p ) G( p + p )dg() taking derivatives with respect to p, eqating p = p sbstitting w (p) = 1 and, therefore, ŵ = w(p) and x (p) = 2 ŵ g()2 d + (1 G(ŵ))G (ŵ) G(ŵ)(2 G(ŵ)) (22) Since, in a symmetric eqilibrim, x = 1, the pricing eqation 2 x (p)p + x(p) = 0 (23) is solved by p = 2 ŵ G(ŵ)(2 G(ŵ)) g()2 d + (1 G(ŵ))G (ŵ) It is easy to see that the Second Order Condition is satisfied for log-concave distribtions (show). The price for the ARW model was derived in Anderson and Renalt (1999). Simple inspection shows that the price is higher in the ARW model, for any ŵ. Proof of Proposition 3 Proof. Form (2), it is obvios that w is continos and decreasing in c. Hence, it sffices to show that p () < p (ū). To see this, notice that (24) p (ū) = p() = 1 2 ū g2 ()d (25) 22