Prices and Heterogeneous Search Costs

Transcription

1 Pries and Heterogeneous Searh Costs José L. Moraga-González Zsolt Sándor Matthijs R. Wildenbeest First draft: July 2014 Revised: June 2015 Abstrat We study prie formation in a model of onsumer searh for differentiated produts when onsumers have heterogeneous marginal searh osts. We provide onditions under whih a symmetri Nash equilibrium exists and is unique. Searh osts affet two margins the intensive searh margin (or searh intensity) and the extensive searh margin (or the deision to searh rather than to not searh at all). These two margins affet the elastiity of demand in opposite diretions and whether lower searh osts result in higher or lower pries depends on the properties of the searh ost density. When the searh ost density has the inreasing likelihood ratio property (ILRP), the effet of lowering searh osts on the intensive searh margin has a dominating influene and pries derease. By ontrast, when the searh ost density has the dereasing likelihood ratio property (DLRP), the effet on the extensive searh margin is dominant and lower searh osts result in higher pries. We ompare these results with those obtained when onsumers have heterogeneous fixed searh osts. Keywords: sequential searh, searh ost heterogeneity, differentiated produts, monotone likelihood ratio JEL Classifiation: D43, D83, L13 Earlier versions of this paper were irulated under the title Do higher searh osts make markets less ompetitive? We would like to thank Simon Anderson, Guillermo Caruana, Alex de Cornière, Knut-Eri Joslin, Silvana Krasteva, Theo Dijkstra, Dmitry Lubensky, Andras Niedermayer, Marielle Non, Vaiva Petrikaitė, Régis Renault, Mariano Tappata, Makoto Watanabe, and Jidong Zhou for their useful remarks. Speial thanks go to Paulo K. Monteiro for his omments, help and useful referenes. The paper has also benefited from presentations at University Carlos III Madrid, University of Copenhagen, CEMFI (Madrid), the Netherlands Bureau for Eonomi Poliy Analysis (CPB), Paris Shool of Eonomis, Sauder Business Shool at UBC, University of East Anglia, University of Mannheim, University of Oxford, the Fifth Workshop on Searh and Swithing Costs (Bloomington, IN), the IIOC 2014 Conferene (Chiago), the 2014 MaCCI Workshop on Consumer Searh in Bad Homburg, and the SaM 2015 Workshop at the University of Bristol. Finanial support from the Marie Curie Exellene Grant MEXT-CT , grant ECO of the Spanish Ministry of Eonomy and Competitiveness, and grant PN-II-ID-PCE of the Romanian Ministry of National Eduation, CNCS-UEFISCDI, is gratefully aknowledged. Address for orrespondene: VU University Amsterdam, Department of Eonomis, De Boelelaan 1105, 1081HV Amsterdam, The Netherlands. j.l.moragagonzalez@vu.nl. Moraga is also affiliated with the University of Groningen, the Tinbergen Institute, the CEPR, and the Publi-Private Setor Researh Center (IESE, Barelona). Sapientia University Mierurea Ciu, Romania, and Affiliate Fellow at CERGE-EI, Prague. zsosan@gmail.om. Kelley Shool of Business, Indiana University, USA. mwildenb@indiana.edu.

2 1 Introdution Throughout history, tehnologial improvements have made it possible for onsumers to partiipate in markets that were previously beyond their reah. A reent example of suh a tehnologial development is the Internet. The introdution and sustained growth of e-ommere has signifiantly lowered the transation osts onsumers experiene while shopping. This has not only made it easier for onsumers to searh for and ompare produts but it has also made it easier for onsumers to get aess to new markets and hard-to-find produts. This has also led to the Internet s Long Tail phenomenon: the Internet allows onsumers to aess a muh larger produt seletion, inluding nihe produts that were previously diffiult to find, thereby reating a long tail in the sales distribution (see, e.g., Brynjolfsson, Hu, and Simester, 2011). 1 How are tehnologial advanes that make it easier for onsumers to searh expeted to affet market ompetitiveness? The onventional answer is that a redution in searh osts inreases the elastiity of demand and as suh redues pries. However, as illustrated by the example, this is not the omplete story. Lower searh osts also allow onsumers to searh for produts that previously were not part of their onsideration sets. Sine these new onsumers are likely to have higher searh osts (beause otherwise they would have been atively searhing before), this leads to ompositional hanges in the onsumer population that may potentially derease the elastiity of demand. More onsumer partiipation as a result of a tehnology-driven redution in searh osts may end up being the dominating fator, whih means firms will raise pries, even when searh osts go down for all onsumers. Hortaçsu and Syverson (2004) present empirial evidene that is onsistent with this partiular mehanism in their study of the US mutual fund industry during the late nineties. Not that long ago investors had to go through signifiant effort to searh for adequate investment opportunities while managing their finanial investments. However, the rise of Internet banking and online brokerages during the late nineties dereased transation osts and made it easier for individuals to searh for new investment opportunities. Hortaçsu and Syverson (2004) find that during this period investment fund fees went up and that searh osts dereased at the lower perentiles of the searh ost distribution while inreased at the upper perentiles. Aording to the authors, this surprising finding (i.e., a seond-order stohasti dominane derease in searh osts leading to a prie rise) is explained by the inrease in the number of households 1 The expansion of the railroad network in the United States in the 19th entury and the liberalization of airlines in the 20th entury are events that also led to signifiantly lower transportation osts and greater market integration. As with the Internet, lower transportation osts were not only benefiial to onsumers beause of redued transation osts but also beause they allowed onsumers to be ative in new markets. 2

3 that partiipated in the mutual fund market for the first time. These new investors arguably had higher searh osts and were less savvy. Hortaçsu and Syverson argue that the entry of these new investors hanged the omposition of the investor population and made demand more inelasti, explaining the inrease in mutual fund fees. The main objetive of our paper is to understand under whih onditions a mehanism like the one desribed above ours. In partiular, our goal is to derive general onditions under whih a first- or seond-order stohasti dominane derease in searh osts triggers suffiient onsumer entry suh that the elastiity of demand is lowered and equilibrium pries go up. In order to do so, we develop a model of onsumer searh for differentiated produts in whih onsumers have heterogeneous searh osts and the deision to partiipate in the market is endogenous. Our model, whih is presented in Setion 2, builds on the seminal model of onsumer searh for differentiated produts introdued by Wolinsky (1986), and further studied by Anderson and Renault (1999). 2 We extend this model by allowing for arbitrary searh ost densities. This is ruial for studying the seond-order stohasti dominane (SOSD) dereases in searh osts observed in the mutual fund industry by Hortaçsu and Syverson (2004). Setion 3 deals with marginal searh ost heterogeneity and haraterizes the symmetri Nash equilibrium of the model. Our first ontribution is the study of the existene and uniqueness of a symmetri Nash equilibrium in pure strategies. Beause a diret verifiation of the seond order onditions fails to deliver lear-ut results we proeed by studying the onditions under whih the profit funtion of a typial firm is quasi-onave. We show that when the density of searh osts is suffiiently log-onave, an equilibrium exists and is unique. In order to prove our existene result we use an aggregation theorem due to Prékopa (1973) that shows that integration preserves log-onavity (see also Caplin and Nalebuff, 1991, p.46). Our seond ontribution is the study of the omparative statis effets of lower searh osts. In Setion 4 we show that a hange in searh osts affets two margins, namely, the intensity with whih onsumers searh (whih we all the intensive searh margin) and the share of onsumers who hoose to searh for a good deal in the first plae (whih we refer to as the extensive searh margin). By assuming that all onsumers searh at least one, the literature has typially foused on the effets of the intensive searh margin on prie determination and 2 Arguably Wolinsky s framework has beome the workhorse model of onsumer searh for differentiated produts. Reent work that builds on this artile inlude Bar-Isaa, Caruana, and Cuñat (2012), who extend the model to the ase of quality-differentiated firms and study design differentiation, Armstrong, Vikers, and Zhou (2009), who study searh and priing behavior in the presene of a prominent firm, Haan and Moraga-González (2011a), who study the emergene and the prie effets of prominene, Moraga-González and Petrikaitė (2013), who examine the effet of searh osts on mergers, and Zhou (2014), who studies multi-produt searh. 3

4 has thereby negleted the role of the extensive searh margin. 3 Reognizing that pries do not only affet the intensive searh margin but may also affet the extensive searh margin turns out to be ritial for a omplete understanding of the funtioning of onsumer searh markets. We show that if the extensive margin does not play a role, any derease in searh osts in the sense of first-order stohasti dominane (FOSD) results in a lower equilibrium prie. This would happen when the range of searh osts is suffiiently small and all onsumers hoose to searh in market equilibrium. Consumers, faing lower searh osts, all hoose to searh more and demand produts that are more attrative. Firms, antiipating that average demand beomes more elasti, respond by reduing their pries. Under ertain onditions on the density of math values (satisfied by for example the uniform distribution), a derease in searh osts in the sense of seond-order stohasti dominane (SOSD) also results in lower pries. If the extensive searh margin does play a role, we get very different results. Speifially, we show that a shok that redues searh osts leads to more searh but also brings new onsumers into the market with relatively high searh osts. We establish that for searh ost densities with the dereasing likelihood ratio property (DLRP), a derease in searh osts in the sense of FOSD or SOSD results in a higher equilibrium prie. That is, for densities with the DLRP a derease in searh osts has a bigger effet on the extensive searh margin than on the intensive searh margin, and as a result pries go up. This is exatly what seems to have ourred in Hortaçsu and Syverson s (2004) study of the mutual fund industry: easier searh led to entry of new onsumers and this resulted in a more inelasti omposition of demand so pries went up in spite of the inreasing popularity of Internet banking and online trading. For searh ost densities with the inreasing likelihood ratio property (ILRP) we find the opposite: a derease in searh osts leads to lower pries. For this type of densities the effet of a fall in searh osts on searh intensity dominates the effet on demand omposition. We believe that this mehanism may also be important for understanding other existing empirial evidene regarding the effets of the Internet on retail prie ompetition. The ommonly held belief is that the Internet improves onsumer searh tehnology and this should naturally lead to lower pries online than offline. Despite this belief, the empirial evidene is mixed: although some studies indeed find lower pries online than offline (Brynjolfsson and Smith, 2000; Sott Morton, Zettelmeyer, Silva-Risso, 2001), others find no effets or even the opposite (Clay, 3 This point is also the entral tenet in Anderson and Renault (2006), who, by allowing for arbitrary searh osts, are able to reonile the empirial observation that muh of the advertising we observe in arguably searh environments does impart only math information and not prie information. 4

5 Krishnan, and Wolff, 2001; Clemons, Hann, and Hitt, 2002; Goolsbee, 2001; Lee, 1998). The entry of new buyers as a result of a redution in transation osts due to the Internet may very well be part of the explanation for why some studies have found online pries to be higher than offline pries. A third ontribution of our paper is more tehnial in nature and relates to the properties of trunated densities. Speifially, as part of our proofs for our findings on the omparative statis effets of lower searh osts we show that for any two densities that an be ranked aording to the ILRP, the orresponding distributions trunated from above an be ranked aording to the FOSD riterion, irrespetive of the loation of the trunation point. Conversely, if the densities an be ranked aording to the DLRP, then the orresponding distributions trunated from above an be ranked aording to the reverse FOSD. To prove this result we use a tool from the statistis literature by Ahlswede and Daykin (1978) that has also been used by for instane Athey (2002), although for a different appliation. Finally, in Setion 5 we study what happens to pries when there is a hange in the distribution of fixed searh osts rather than marginal searh osts. We show that when onsumers differ in their fixed osts of searh (but fae a similar marginal ost of searh) we obtain quite different results beause a derease in the fixed osts of searh results in an inrease in demand without altering the elastiity of demand. As a result, pries do not hange. This illustrates an important point: hanges in the size of the market do not always lead to hanges in demand elastiity. The mehanism by whih pries are affeted by hanges in the extensive searh margin goes via the hanges in the gains from searh of the average onsumer who hooses to searh. Changes in the market that do not affet the gains from searh of the average onsumer (as happens when fixed searh osts vary while marginal searh osts are fixed) do not have a bearing on the elastiity of demand, whih implies that pries do not hange. Related literature Most of the previous literature has proeeded under the restritive assumption that searh osts are required to be low enough, de fato implying that all onsumers hoose to searh at least one in equilibrium rather than to not searh at all (e.g., Stahl, 1989; Burdett and Judd, 1983; Wolinsky, 1986). As pointed out by Stiglitz (1979), the alternative assumption that searh osts are large may ause the market to ollapse. 4 As we have disussed before, this unravelling of 4 For example, in the setting of Diamond (1971), the only prie that an be part of a market equilibrium is the monopoly prie (the well-known Diamond paradox ). If the searh ost is relatively high, the surplus onsumers derive at the monopoly prie may be insuffiient to over the ost of the first searh, in whih ase onsumers rather do not searh at all and the market fails to exist. 5

6 the market will not arise when onsumer searh osts are heterogeneous. Two related papers (Janssen, Moraga-González, and Wildenbeest, 2005; Rauh, 2004) also allow for endogenous onsumer partiipation, although in both papers the type of searh ost heterogeneity that is assumed is less general than in our model. Janssen, Moraga-González, and Wildenbeest (2005) employ Stahl s (1989) setting, with the muh used shoppers and non-shoppers setup. They show that when the extensive searh margin plays a role, pries will always inrease if the searh ost dereases. Our paper shows instead that pries an go both diretions and provides onditions under whih one or the other ours. This means that by not restriting the extent of onsumer heterogeneity in the market, our model may result in qualitatively very different outomes than a model that uses the standard shoppers and non-shoppers setup. Moreover, the shoppers and non-shoppers setup does not allow for studying the omparative statis effets of SOSD dereases in searh osts, suh as those that ourred in the US mutual fund industry. In Rauh (2004) onsumer partiipation is endogenous and proves to be important for the welfare analysis of prie eilings; however, he assumes searh osts are uniformly distributed and does not examine the omparative statis effets of higher searh osts. A few reent papers have put forward situations in whih lower searh osts do not neessarily lead to lower pries, although these findings rely on very different mehanisms. For instane, Chen and Zhang (2011) enrih Stahl s (1989) setting by adding loyal onsumers, and show that in suh an environment a redution in the searh ost sometimes leads to higher equilibrium pries. In a different framework where searh is prie-direted, Armstrong and Zhou (2011) show that lower searh osts lead to higher pries. A similar result is obtained in Haan and Moraga-González (2011b), who extend the Anderson and Renault (1999) model by making pries observable before searh. In a model in whih onsumers searh for various produts from multi-produt firms, Zhou (2014) demonstrates that produt externalities an lead firms to raise their pries when searh osts go down. Lastly, in a model with vertial relations, Janssen and Shelegia (2015) enounter situations where retail pries inrease as searh osts derease. Finally, our paper relates to the reent literature on obfusation, whih points out that firms have inentives to obfusate their produts by raising the osts onsumers have to inur to inspet their offers (Ellison and Wolitzky, 2012; Wilson, 2010). Our result tells that under ertain onditions firms may benefit from doing exatly the opposite, that is, by lowering searh osts. 6

7 2 Sequential searh for differentiated produts Our model builds on the seminal ontribution of Wolinsky (1986); speifially we examine a ompetitive version of his model and we allow onsumers to have heterogeneous searh osts. 5 Consider a market with infinitely many onsumers and firms and, without loss of generality, normalize the number of onsumers per firm to 1. Firms produe horizontally differentiated produts using the same onstant returns to sale tehnology of prodution; let r be the marginal ost of the firms. Aiming at maximizing their expeted profits, firms hoose their pries simultaneously. We fous on symmetri Nash equilibria (SNE); let p denote an SNE prie. A onsumer l has tastes for a produt i desribed by the following indiret utility funtion: { εil p u il = i if she buys produt i at prie p i ; 0 otherwise. The parameter ε il is a math value between onsumer l and produt i. We assume that the math value ε il is the realization of a random variable distributed on the interval [0, ε] aording to a umulative distribution funtion (CDF) denoted by F. Let f be the probability density funtion (PDF) of F. We assume that f is differentiable and that 1 F is log-onave. Math values ε il are independently distributed aross onsumers and produts. private information of onsumers so personalized priing is not possible. sequentially in order to maximize their expeted utility. Moreover, they are Consumers searh While searhing, they have orret beliefs about the equilibrium prie. The total ost of searh of a onsumer l who searhes n times is k l + n l, where k l is the onsumer s fixed ost of searh and l indiates her marginal ost of searh. In the following setions we examine the role of searh ost heterogeneity in our model. We start with the ase in whih onsumers have heterogeneous marginal searh osts. In Setion 5 we study what happens when onsumers have heterogeneous fixed searh osts. 5 In the Appendix we present numerial results showing that the main insights of our paper also hold when the number of firms is finite. With a finite number of firms, however, proving existene and uniqueness of equilibrium is a hallenge beause the demand of a typial firm onsists of a sum of two non-neessarily quasi-onave funtions of its own prie (see Anderson and Renault, 1999). 7

8 3 Marginal searh ost heterogeneity Assume that onsumers differ in their marginal osts of searh. 6 Speifially, assume that a buyer l s marginal searh ost l is drawn independently from a differentiable umulative distribution funtion G with support [, ]. Let g be the density of G. We refer to the differene between the upper and lower bound of the searh ost distribution as the range of searh osts. We require the lower bound of the searh ost distribution to be suffiiently low beause otherwise no onsumer would searh and the market would ollapse. In this setion, we assume that onsumers do not need to inur any fixed ost of searh; the role of fixed searh osts is relegated to Setion 5. Existene and uniqueness of an SNE prie We now haraterize the SNE prie. In order to do so, we derive the payoff of a firm, say i, that deviates from the SNE prie p by harging a prie p i p. Next, we ompute the first order ondition (FOC), apply the symmetry ondition p i = p, and study the existene and uniqueness of the SNE. For later use, we define the monopoly prie as p m = arg max p (p r)(1 F (p)). Consider the (expeted) payoff to a firm i that deviates from the equilibrium by harging a prie p i. In order to ompute firm i s demand, we first need to haraterize onsumer searh behavior. Sine onsumers do not observe deviations before searhing, we an rely on Kohn and Shavell (1974), who study the searh problem of a onsumer who faes a set of independently and identially distributed options with a known distribution. Kohn and Shavell show that the optimal searh rule is stati in nature and has the stationary reservation utility property. Aordingly, onsider a onsumer with searh ost and denote the solution to h(x) ˆ ε x (ε x)f(ε)dε = (1) in x by ˆx(). The left-hand-side (LHS) of equation (1) is the expeted benefit in symmetri equilibrium from searhing one more time for a onsumer whose best option so far is x. Its righthand-side (RHS) is the onsumer s ost of searh. Hene ˆx() represents the threshold math value above whih a onsumer with searh ost will optimally deide not to ontinue searhing for another produt. The funtion h is monotonially dereasing. Moreover, h(0) = E[ε] and h(ε) = 0. It is readily seen that for any [, min{, E[ε]}], there exists a unique ˆx() that 6 Most models in the literature abstrat from onsumer searh ost heterogeneity or only allow for speial forms of heterogeneity, typially with some onsumers having a searh ost equal to zero (usually referred to as the shoppers ), while the rest faes a positive and idential searh ost (the non-shoppers ). There are some exeptions to this in the literature (e.g., Bénabou, 1993; Rob, 1985; Rauh, 2009; Stahl, 1996; Tappata, 2009) but in these papers searh osts are assumed to be suffiiently low so that all onsumers searh at least one. We will later show that this assumption is restritive. 8

9 solves equation (1). Differentiating equation (1) suessively, we obtain ˆx 1 () = 1 F (ˆx()) < 0; ˆx () = f(ˆx()) [ˆx ()] 2 1 F (ˆx()) whih implies that ˆx() is a dereasing and onvex funtion of on [, min{, E[ε]}], with ˆx(E[ε]) = 0 and ˆx() ε. 7 In order to ompute firm i s demand, onsider a onsumer with searh ost who shows up at firm i to inspet its produt after possibly having inspeted other produts. Let ε i p i denote the utility the onsumer derives from firm i s produt. Obviously, if alternative i is not the best one so far, the onsumer will disard it and searh again. Therefore, only when the deal offered by firm i happens to be the best so far, the onsumer will onsider to stop searhing and buy the produt right away. For this deision, the onsumer ompares the gains from an additional searh with the osts of suh a searh. In this omparison, the onsumer holds orret expetations about the equilibrium prie so she expets the other firms to harge p. The expeted gains from searhing one more firm, say firm j, are equal to ε ε i p i +p [ε j (ε i p i +p )]f(ε j )dε j. Comparing this to equation (1), it follows that, onditional on having arrived at firm i, the probability that buyer stops searhing at firm i is equal to Pr[ε i p i > ˆx() p ] = 1 F (ˆx() + p i p ). With the remaining probability, the onsumer finds the produt of firm i not suffiiently satisfatory and ontinues searhing; with infinitely many firms, suh a onsumer will surely buy at another firm. Beause a onsumer with searh ost may visit firm i after having visited no, one, two, three, et. other firms, the unonditional probability she stops searhing and buys at firm i is > 0, 1 F (ˆx() + p i p ). (2) 1 F (ˆx()) To obtain the payoff of firm i we need to integrate expression (2) over the onsumers who deide to searh the market for a satisfatory produt; in other words, we need to integrate over those onsumers who derive expeted positive surplus from partiipation. To ompute the surplus a onsumer with searh ost obtains from partiipation, we note that she will stop and buy at the first firm she visits whenever the math value she derives at that firm is greater than ˆx(); otherwise she will drop the first option and ontinue searhing. In the latter ase she will enounter herself exatly in the same situation as before beause, onditional on partiipating, she will ontinue searhing until she finds a produt satisfatory enough to buy it and leave the 7 Consumers with searh ost > E[ε], if any, will automatially drop from the market and an therefore be ignored right away. Therefore ˆx() is well-defined for every onsumer that matters for priing. 9

10 market. Denoting by CS() her onsumer surplus, we have that: 8 CS() = ˆx() p. Setting this surplus equal to zero, and using equation (1), we obtain the ritial searh ost value above whih onsumers will refrain from partiipating in the market: (p ) = ˆ ε p (ε p )f(ε)dε. (3) Depending on how large the range of searh osts is, more or less onsumers will hoose to searh the market for a satisfatory deal. Correspondingly, we define 0 (p ) min{, (p )}. We refer to the deision to searh as the extensive searh margin. Beause (p ) is dereasing in p, if onsumers expet a higher equilibrium prie then fewer of them will hoose to searh the market for an aeptable produt. 9 The payoff to the deviant firm i is then: π(p i ; p ) = (p i r)d(p i, p ), (4) where the demand firm i reeives D(p i, p ) is ˆ 0 (p ) 1 F (ˆx() + p i p ) g()d if p i < p, D(p i, p 1 F (ˆx()) ) = ˆ 0 (p ) 1 F (ˆx () + p i p ) g () d if p i p, 1 F (ˆx ()) max{,ĉ(p i )} (5) where ĉ (p i ) is the solution of the equation ˆx () + p i p = ε in. The FOC is given by (for p i < p ): ˆ 0 (p ) 1 F (ˆx() + p i p ˆ ) 0 (p ) g()d (p i r) 1 F (ˆx()) Applying symmetry, i.e., p i = p, we an rewrite the FOC as: 10 p = r + 0 (p ) f(ˆx() + p i p ) g()d = 0. 1 F (ˆx()) G( 0 (p )) (6) f( x()) 1 F (ˆx()) g()d. 8 For a formal derivation of this expression, see the Appendix. 9 The standard assumption in the searh ost literature has been that 0(p ) =, whih implies that all onsumers searh at least one. This kind of fully-overed market assumption (in the sense that all onsumers searh) is learly restritive and, as we will show later, it an lead to wrong onlusions about the effets of higher searh osts on equilibrium pries. 10 Note that the first-order neessary ondition for a symmetri equilibrium is the same when we assume that the deviating prie p p. 10

11 We now show that a andidate symmetri equilibrium prie exists and is unique. First onsider the ase in whih searh osts are suffiiently low, i.e. < (p ), hene G( 0 (p )) = 1. In terms of exogenous parameters, this ours under the following restrition: Condition 1. ˆ ε 1 r+ f( x()) 1 F ( x()) g()d ε r 1 f( x()) 1 F ( x()) g()d f(ε)dε. Beause under Condition 1 0 (p ) =, the expression (6) gives the andidate equilibrium prie expliitly. In this ase, obviously, there exists a unique andidate equilibrium prie. When searh osts are not restrited to be suffiiently small and Condition 1 is violated then > (p ) and orrespondingly G( 0 (p )) < 1. In this ase the equilibrium prie is given impliitly by the solution to equation (6). We now show that equation (6) has a unique solution in this situation as well. For this we define the funtion L(p) G( (p)) (p r) ˆ (p) f( x()) g () d 1 F ( x()) for p [r, p m ], where p m as before denotes the monopoly prie. Note that Also observe that L(p m ) an be written as L (p m ) = ˆ (p m ) L (r) = G( (r)) > 0. 1 F (ˆx()) (p m r)f(ˆx()) g () d. (7) 1 F (ˆx()) The sign of this expression depends on the sign of the numerator of the fration in the integrand. We now argue that L(p m ) < 0 beause 1 F (ˆx()) (p m r)f(ˆx()) 0 for all [, (p m )]. In fat, note that by log-onavity of f, beause ˆx() dereases in, it follows that f(ˆx())/ [1 F (ˆx())] dereases in, whih implies that 1 F (ˆx()) (p m r)f(ˆx()) inreases in. Beause ˆx((p)) = p, if we set = (p m ) in the expression 1 F (ˆx()) (p m r)f(ˆx()), we get the monopoly priing rule 1 F (p m ) (p m r)f(p m ) = 0. We an now onlude that L (p m ) < 0 beause the expression 1 F (ˆx()) (p m r)f(ˆx()) is inreasing in and takes on value zero when we ompute it at the upper bound of the integral. Taken together, L (r) > 0 and L (p m ) < 0 imply that a andidate equilibrium prie p [r, p m ] exists. We finally note that dl (p) dp = g ( (p)) = d (p) dp d (p) (p r) f (p) g ( (p)) dp 1 F (p) (p r) f (p) d (p) g ( (p)) 1 F (p) dp ˆ (p) ˆ (p) f(ˆx())g () d f(ˆx())g () d (8) 11

12 is negative for any p [r, p m ], whih implies that the andidate equilibrium prie is unique. This follows from the fat that 1 F (p) (p r) f (p) 0 (beause it is the first order derivative of the monopoly payoff (p r) (1 F (p)), whih is log-onave) and d (p) /dp < 0 (beause, from equation (3), is dereasing in p). In partiular, at the andidate equilibrium prie p we must have dl(p )/dp < 0. Our next result provides onditions for existene of a symmetri equilibrium. Theorem 1. There may be two types of SNE in the model of sequential searh for differentiated produts with heterogeneous marginal osts of searh: (A) An SNE where all onsumers ondut at least a first searh, in whih ase Condition 1 holds so that (p ). In this type of equilibrium firms harge a prie given by expression (6) after setting 0 (p ) =, and where ˆx() solves equation (1). (B) An SNE where only some of the onsumers searh for a satisfatory produt, in whih ase Condition 1 is violated so that > (p ). In this type of equilibrium firms harge a prie given by expression (6) after setting 0 (p ) = ε p (ε p )f(ε)dε; moreover, the fration of onsumers G ( ) ε p (ε p )f(ε)dε onduts at least a first searh while the rest of the onsumers do not searh at all and leave the market. Let ˆx 1 (t) denote the inverse funtion of ˆx(). When the funtion g(ˆx 1 (t)) is log-onave in t, then the SNE exists and is unique. The proof, whose details are in the Appendix, is as follows. Beause a diret verifiation of the seond order onditions does not deliver lear-ut results, we proeed by showing that the demand funtion of an individual firm is a log-onave funtion of its own prie under the assumption that g(ˆx 1 (t)) is log-onave in t. One this is proven, we know that the firm profit funtion (4) is quasi-onave in its own prie so that the unique andidate equilibrium prie given by expression (6) is indeed an equilibrium. In order to prove that the demand funtion of a firm is log-onave in the firm s own prie, we first show that the demand from a single onsumer is log-onave both in prie and in onsumer reservation values; after this we make use of Theorem 6 in Prékopa (1973), whih shows that integration over onsumer reservation values preserves log-onavity. 4 The omparative statis effets of lower searh osts We now proeed to study the impat of lower marginal searh osts on the equilibrium pries given by Theorem 1. In order to address this question, we parametrize the CDF of searh 12

13 osts by a positive parameter β that shifts the distribution. Speifially, let G(; β) denote the parametrized CDF of searh osts; let g(; β) denote the parametrized PDF of searh osts with support [, ]. 11 We study FOSD and SOSD dereases in searh osts. Conretely, we say that searh osts derease in the sense of FOSD whenever a derease in β implies an inrease in the searh ost distribution for all, i.e., when G(; β)/ β 0 for all (and < 0 for some ). Alternatively, we say that searh osts derease in the sense of SOSD whenever ˆ G(x; β) dx 0 for all (and < 0 for some ). β Note that SOSD is implied by FOSD (not vie-versa). Let p (β) be the orresponding SNE prie for a given β. In order to develop some intuition, it is useful to reall that, as usual with monopolisti ompetition, we an write the equilibrium prie as: where p (β) = r 1 1 ɛ(p ;β) ɛ(p ; β) p D (p ; β) D(p ; β) represents the prie elastiity of demand. We are interested in the behavior of p (β) with respet to β. 4.1 Low searh osts Consider first the ase of Theorem 1A. When the upper bound of the searh ost distribution is suffiiently low and all onsumers searh, a hange in searh osts only affets the intensive searh margin. The equilibrium prie is equal to p (β) = r + (9) 1 (10) f( x()) 1 F (ˆx()) g(; β)d. Note that the denominator of the fration on the RHS of equation (10) is nothing else than the expetation of the hazard rate f(ˆx())/(1 F (ˆx())). By log-onavity of 1 F, note that the mentioned hazard rate inreases in ˆx() and dereases in. From Theorem 1 in Hadar and Russell (1971), a FOSD derease in searh osts then immediately implies that the expeted value of the hazard rate goes up. It follows, then, that 11 To be as general as possible we allow the support of the density of searh osts to depend on β. We ome bak to this issue later. 13

14 the equilibrium prie (10) unambiguously dereases when β dereases. The intuition is straightforward: when searh osts go down, onsumers reservation values rise; this inreases searh ativity and demand beomes more elasti. In fat, when all onsumers searh D(p ; β) = 1 and the elastiity of demand simply equals ɛ(p ; β) = p ˆ f(ˆx()) g(, β)d. (11) 1 F (ˆx()) As we have argued above, a fall in searh osts inreases the value of the integral in equation (11) and thereby the elastiity of demand goes up. The equilibrium prie must then derease. With additional onditions on the math value density we an state what happens when searh osts derease in the sense of SOSD (see Theorem 2 of Hadar and Russell, 1971). Proposition 1. Let G(; β) be a parametrized searh ost CDF with positive density on [, ]. Then: (A) An FOSD derease in searh osts results in a fall in the equilibrium prie given by Theorem 1A. (B) An SOSD derease in searh osts results in a fall in the equilibrium prie given by Theorem 1A provided that f [1 F ] 2 + 4ff [1 F ] + 3f 3 0 (whih for example holds when f is uniform). The proof of Proposition 1B an be found in the Appendix. 4.2 Suffiiently high searh osts We now move to the ase of Theorem 1B. In this ase not all onsumers hoose to searh so a hange in the distribution of searh osts affets both the intensive and the extensive searh margins. The equilibrium prie is given by the solution to equation ˆ (p L(p ; β) G( (p ); β) (p ) f(ˆx()) r) g(, β)d = 0. 1 F (ˆx()) It turns out to be quite useful to rewrite this, dividing by G( (p ); β), as follows: where ˆ (p L(p; β) 1 (p ) f(ˆx()) r) g(, β)d = 0, (12) 1 F (ˆx()) g(, β; (p )) g(, β) G( (p ); β) denotes the searh ost density trunated from above by (p ). 14

15 By the impliit funtion theorem, the impat of a (small) derease in β on the equilibrium prie in equation (12) is given by the sign of dp (β) dβ L(p ;β) β = L(p ;β). (13) p We have already argued above that the funtion L(p ; β) is dereasing in p, and so is L(p ; β), so the denominator of equation (13) is negative. Therefore the sign of dp (β)/dβ depends on the sign of L( )/ β. Note that equation (12) is very similar to equation (10). In fat, the only differene is that the integral in equation (12) is the expeted value of the hazard rate of the onsumers who hoose to searh, and that is why the integral involves the trunated density. Therefore, the answer to the question what happens to the prie when searh osts go down boils down to finding out what happens to the mentioned expeted value. Denoting the distribution of searh osts trunated above by (p ) by G(, β; (p )) ˆ g(x, β; (p )))dx, we an state diretly (and without proof) the following extension of Proposition 1. Proposition 2. Let G(; β) be a parametrized searh ost CDF with positive density on [, ]. Then: (A) An FOSD derease in searh osts results in a fall (inrease) in the equilibrium prie given by Theorem 1B if G(, β)/ β ( )0 for all (and < (>)0 for some ). (B) Assume that f [1 F ] 2 + 4ff [1 F ] + 3f 3 0. Then an SOSD derease in searh osts results in a fall (inrease) in the equilibrium prie given by Theorem 1B if ˆ G(x; β, (p )) dx 0 for all (and < 0 for some ). β This result emphasizes that what really matters for the relationship between the equilibrium prie and β is the way in whih the (trunated) searh ost distribution of the partiipating onsumers is affeted by hanges in β. If a FOSD derease in searh osts results in a FOSD derease in the distribution of searh osts onditional on partiipation, then the equilibrium prie will surely derease. Otherwise, the equilibrium prie will inrease. This result is similar for SOSD dereases in searh osts. The intuition behind this result is as follows. When not all onsumers searh the elastiity of demand is ɛ(p ; β) = p ˆ (p ) f(ˆx()) g(, β)d (14) 1 F (ˆx()) 15

16 A fall in searh osts makes onsumers more piky and they searh more. Beause of this effet, firms have an inentive to lower their pries. However, a fall in searh osts brings new onsumers into the market. These onsumers turn out to have relatively large searh osts beause otherwise they would have been searhing previously. Beause of this effet, firms have an inentive to raise their pries. If the seond effet is weak and the distribution of searh osts onditional on searh has lower searh osts then the prie will fall. If the seond effet is suffiiently strong, then the prie will rise. The question is then, under whih onditions a (FOSD or SOSD) derease in searh osts results in a (FOSD or SOSD) derease in the trunated distribution of searh osts? We now present a well-known result from the statistis literature that enables us to address this question: 12 Lemma 1. [Ahlswede and Daykin, 1978]. Let f 1, f 2, f 3, f 4 be non-negative funtions on [a, b] suh that f 1 (x) f 2 (y) f 3 (max {x, y}) f 4 (min {x, y}) for all x, y [a, b]. Then ˆ b a f 1 (x) dx ˆ b a f 2 (x) dx ˆ b a f 3 (x) dx ˆ b a f 4 (x) dx. Athey (2002) uses this result to show that log-supermodularity is preserved by integration. Our use of this Lemma is different; as far as we know, it has not been used before to prove the following result. Proposition 3. Assume that g(; β) has the inreasing (dereasing) likelihood ratio property, that is, g (; γ) g (d; β) ( )g (; β) g (d; γ) for any d and β γ. Then, the distribution of searh osts trunated from above by (p ) orresponding to β (γ), first-order stohastially dominates the distribution of searh ost trunated from above by (p ) orresponding to γ (β), that is, G (; β (p )) ( )G (; γ (p )) for all and (p ). The proof of this result is in the Appendix. Note that the ILRP (DLRP) is equivalent to log-supermodularity (log-submodularity) (f. Athey, 2002). By ombining Propositions 2 and 3 we an state suffiient onditions under whih a redution in searh osts will derease or inrease the equilibrium prie. Namely, when the searh ost 12 We are indebted to Paulo Monteiro for alerting us about this result. 16

17 density has the ILRP with respet to parameter β, then a derease in β will ause a FOSD derease of the trunated searh ost distribution, thereby leading to a prie derease. By ontrast, when the searh ost density has the DLRP, a derease in β will ause a FOSD inrease of the trunated searh ost distribution and this will result in a higher prie. The final observation we make is that we need to ensure that ILRP and DLRP are ompatible with FOSD and SOSD. Obviously, ILRP is ompatible with FOSD (and SOSD by impliation) beause the latter is implied by the former. For DLRP to be ompatible with FOSD (and hene SOSD) we just need the (upper bound of the) support of searh ost to inrease with β (below we present a few examples). To summarize, ombining Propositions 1-3, we have obtained the following omparative statis results. Theorem 2. Let G(; β) be a parametrized searh ost CDF with positive density on [, ]. Then: (A) If a derease in β signifies a FOSD derease of searh osts, then the equilibrium prie in Theorem 1A will derease. If it signifies a SOSD derease in searh osts then the prie will derease provided that f [1 F ] 2 + 4ff [1 F ] + 3f 3 0. (B) If a derease in β signifies a FOSD or a SOSD derease in searh osts and if g(; β) has the ILRP with respet to β, then the equilibrium prie in Theorem 1B will derease. If a derease in β signifies a FOSD or a SOSD derease in searh osts and if g(; β) has the dereasing likelihood ratio property with respet to β (in whih ase we must have d/dβ > 0), then the equilibrium prie in Theorem 1B will inrease. The intuition behind the role played by the inreasing or dereasing likelihood ratio property of the searh ost density in Theorem 2 an better be onveyed by means of a ouple of instrutive examples. Suppose that searh osts are distributed on a fixed interval and that the density is dereasing. Moreover, suppose that the searh ost density satisfies the ILRP, that is, for γ β, the relative frequeny g(; γ)/g(; β) is inreasing in. The ILRP implies that g(; β) must derease more rapidly than g(; γ). Beause both are densities and their integrals must equal 1, it must be the ase that g(; β) is higher than g(; γ) for low searh osts and otherwise for high searh osts. This in turn implies that for any trunation point, the average onsumer under g(; β) has a lower searh ost than that under g(; γ). For densities with the ILRP, a derease in searh osts has a relatively stronger effet on the intensive searh margin than on the extensive searh margin and this auses the average onsumer to beome less elasti after searh osts go down. This leads to a derease in the equilibrium prie. A similar argument 17

18 an be made for inreasing densities. In that ase, g(; β) must inrease less rapidly than g(; γ) whih implies that the frequeny of low searh ost onsumers under g(; β) is higher than under g(; γ). An example, whih is further elaborated in the Appendix, follows. Example 1: ILRP Assume that math values are distributed on [0, 1] aording to the uniform distribution and that searh osts are distributed on [0, β] aording to the density funtion g() = β(2 + δ) 2 β 2, with δ 0. (1 + δ) The CDF of g is G() = (β(2 + δ) ) β 2 (1 + δ) and note that an inrease in the parameters β or δ signify a FOSD shift of the searh ost CDF. When β hanges, the support hanges but when δ hanges, the support remains the same. Moreover, we observe that g has the ILRP with respet to parameters β and δ. For this example, we an then state that an equilibrium exists and is unique and that an inrease in either β or δ inreases the equilibrium prie given by Theorem 1A and inreases the equilibrium prie given by Theorem 1B. The effet of lower searh osts when the extensive margin plays a role is illustrated in Figure 1. The red urve in graph 1(a) is the density of searh osts for δ = 2; the blue urve for δ = 1. In this graph we set β = 1. The move from the red to the blue density represents a FOSD derease of searh osts, as an be seen in graph 1(). The vertial dashed line indiates the threshold searh ost above whih onsumer hoose not to searh the market for a satisfatory produt. The orresponding trunated densities and distributions are given in graphs 1(b) and 1(d). What we see is that a FOSD derease of searh osts translates into a FOSD derease of the searh osts of partiipating onsumers. With densities that satisfy the DLRP, g(; γ)/g(; β) is dereasing in and we orrespondingly have the opposite. The following example illustrates the situation when the searh ost density satisfies the DLRP. Example 2: DLRP Assume that math values are distributed on [0, 1] aording to the uniform distribution and that the searh ost density is [ g() = 1 + δ ( ) ] δ 1 +, with 0 < δ 1/2. β(2 + δ) β The CDF of g is G() = [ 1 + δ + β(2 + δ) ( ) ] δ β 18

19 g g 1.0 g g g g p p 1.0 (a) Densities (b) Trunated densities G 1.0 G G G 1, p G G 2, p p p () Distributions (d) Trunated distributions Figure 1: ILRP and the effet of higher searh osts on (trunated) densities and distributions. and note that an inrease in parameter β implies a FOSD shift of the searh ost CDF. Moreover, g has the DLRP with respet to parameter β. (For DLRP to be onsistent with FOSD, as mentioned in Theorem 2, the support has to vary when we move β, whih is the ase here.) For this example we an then state that an equilibrium exists and is unique and that a derease in searh osts dereases the equilibrium prie given by Theorem 1A but inreases the equilibrium prie given by Proposition 1B. Example 2 is illustrated in Figure 2. The blue urves represent the densities and distributions, as well as the trunated ounterparts when β = 1.5. The red urves show the ase of β = 1. The parameter δ = 1/2. The move from the blue urves to the red ones represent a FOSD fall in searh osts, as an be seen in graph 2(). What we an see is that a derease in searh ost results in a pool of onsumers partiipating that have higher searh osts than before (graphs 2(b) and 2(d)). As a result, pries will inrease as searh osts fall. 5 Fixed searh ost heterogeneity Sometimes onsumers also fae a fixed searh ost before their atual searh begins, for instane when they first have to start up a omputer, log onto a website, or drive to a shopping mall. 19

20 g 1.4 g g Β 1 g Β g Β 1.5 g Β p p (a) Densities (b) Trunated densities G 1.0 G G Β 1 G Β G Β 1.5, p G Β 1, p p p () Distributions (d) Trunated distributions Figure 2: DLRP and the effet of higher searh osts on (trunated) densities and distributions. In this setion, we study the effet of an inrease in fixed searh osts on equilibrium pries. The main objetive is to show that beause fixed osts do not have a bearing on the gains from searh, inreases in fixed searh osts, even if they result in a fall in demand, do not have an impat on the equilibrium prie. Suppose that onsumers have to pay a fixed searh ost upfront, denoted k, in order to start searhing. Suppose further that onsumers are heterogeneous in their fixed searh osts. Let K(k) be the distribution of fixed searh osts, with support [k, k]. After paying this ost, onsumers proeed as in the model of Setion 2, and fae a ommon marginal searh ost equal to. Sine fixed searh osts are sunk after the first attempt to find a satisfatory produt, the searh rule given by the solution to equation (1) remains the same. Correspondingly, the surplus of a onsumer with fixed searh ost k is equal to CS(k) = ˆx() p k. For a onsumer to enter the market this surplus must be stritly positive. Setting CS(k) = 0 and solving for k gives a ritial fixed searh ost, denoted k(p ), above whih onsumers will abstain from searhing for } an aeptable produt. Defining k 0 (p ) min { k, k(p ) the payoff analogous to the payoff of 20

21 the deviant firm i in equation (4) is given by and zero otherwise. integrating we obtain ˆ k0 π(p i ; p (p ) ) = (p i r) k 1 F (ˆx() + p i p ) dk for p i < ε ˆx() + p 1 F (ˆx()) Beause fixed searh osts do not affet the reservation value ˆx(), by π(p i ; p ) = K( k 0 (p ))(p i r) 1 F (ˆx() + p i p ). (15) 1 F (ˆx()) Inspetion of equation (15) immediately reveals that a hange in the distribution of the fixed searh osts has no bearing on equilibrium priing beause, even though it affets demand, it does it in a way that does not modify the elastiity of demand. The main distintion between heterogeneity in the fixed searh ost and heterogeneity in the marginal searh ost is that omposition-of-demand effets do not matter for priing in the former ase, but do matter in the latter ase. Beause the marginal benefits from searh for the onsumers who hoose to searh are not affeted by an inrease in fixed searh osts, the elastiity of demand is onstant no matter the pool of onsumers who hoose to searh. With heterogeneity in the marginal osts of searh, this is quite different beause the elastiity of demand intimately depends on the pool of onsumers who hoose to searh. We onlude by emphasizing that the mehanism we have identified in this paper does not just arise for any hange in the omposition of demand. In fat, for the elastiity of average demand to inrease or derease as searh osts inrease, onsumers marginal benefits from searh need to hange as well. 6 Conlusions Most existing onsumer searh models are analyzed under the assumption that all onsumers searh at least one in equilibrium. In this paper we have argued that it is hard to reonile this type of fully-overed-market assumption with the general idea that distint onsumers have different searh osts and that, in many markets, some onsumers may simply find it not worthy to start searhing for a satisfatory produt. Sine the deision to searh depends on the equilibrium prie, the existing literature has negleted an important role of the prie mehanism, namely, that pries ought to affet the share of onsumers who hoose to searh for a produt in the first plae. We have studied how pries are determined in a model of searh for differentiated produts while allowing for arbitrary searh ost heterogeneity. Our first ontribution has been to provide 21

22 results on the existene and uniqueness of equilibrium. We have shown that when the searh ost density is suffiiently log-onave, a symmetri Nash equilibrium exists and is unique. In proving this result, we have exploited a mathematial result due to Prékopa (1973) whih shows that log-onavity is preserved by integration. We have also revisited the question how a derease in searh osts affets the level of pries. A derease in searh osts affets not only how muh onsumers searh but also who hooses to searh. Reognizing that the prie mehanism affets the searh omposition of demand turns out to be ritial for understanding how pries and profits are affeted by lower searh osts. When searh osts derease there are two effets operating in opposite diretions: the average onsumer searhes more but the identity of the average onsumer hanges beause of the inflow of onsumers with relatively high searh osts. We have referred to the first effet as the effet on the intensive searh margin and to the seond one as the effet on the extensive searh margin. Our seond ontribution has been on identifying suffiient and intuitive onditions on searh ost densities under whih a derease in searh osts in the sense of FOSD or SOSD has an unambiguous effet on the equilibrium prie. We have singled out a ritial property of searh ost densities that plays a deisive role, namely, log-supermodularity (or monotone inreasing likelihood ratio property). When the searh ost density is log-supermodular, a derease in searh osts has a relatively stronger impat on the extensive searh margin than on the intensive searh margin. The intuition is that with log-supermodular densities, a derease in searh osts inreases the relative frequeny of onsumers with low searh osts. As a result, the average onsumer who hooses to searh has a lower searh ost and is orrespondingly more elasti. This auses the equilibrium prie to fall. By ontrast, when the searh ost density is log-submodular (or has the monotone dereasing likelihood ratio property), a derease in searh fritions raises the relative frequeny of onsumers with high searh osts. Correspondingly, the average (over the onsumers who ontinue hoosing to searh for an aeptable produt) demand beomes less elasti and thereby pries inrease. This result is in line with Hortaçsu and Syverson s (2004) empirial observation that pries went up in the US mutual fund industry during the late nineties in spite of the SOSD derease in searh osts that they estimate. 22