ENTRY AND WELFARE IN SEARCH MARKETS*

Transcription

1 The Eonomi Journal, 128 (February), Doi: /eoj Published by John Wiley & Sons, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA. ENTRY AND WELFARE IN SEARCH MARKETS* Yongmin Chen and Tianle Zhang The welfare effets of entry are studied in a model of onsumer searh. Potential entrants differ in quality, with high-quality sellers being more likely to meet onsumer needs. Contrary to the standard view in eonomis that more entry benefits onsumers, we find that free entry is exessive for both onsumer welfare and total welfare when entry ost is relatively low, and onsumer welfare has an inverted-u relationship with entry ost. We explain why these results may arise naturally in searh markets due to the searh variety and searh quality effets of entry, and disuss their business and poliy impliations. Entry is of entral importane to ompetition and market performane. While it has long been known that free entry is effiient under perfet ompetition, eonomists have more reently reognised that the impat of unenumbered entry on total welfare is ambiguous when firms possess market power, due to onsumers gain and ompetitors loss that the entrant does not internalise (Von Weizsaker, 1980; Mankiw and Whinston, 1986; Cabral, 2004). The standard view in eonomis, however, is still that more entry will boost onsumer welfare. In homogeneous-produt markets, industry output under Cournot ompetition generally expands with entry (Seade, 1980). 1 Even in markets with differentiated produts, where it has been argued that prie-inreasing entry is theoretially unexeptional, the onsumer gain from greater produt variety will usually dominate any potential adverse prie effet (Chen and Riordan, 2008). This artile onduts a new analysis of entry and welfare in an important lass of markets those with onsumer searh, fousing espeially on how entry affets onsumer welfare, measured by aggregate onsumer surplus. Our interest in searh markets is partly motivated by the refletion that, despite the substantial progress in the eonomis of searh, 2 little attention has been paid to the effets of hanges in entry onditions, yet tehnologial progress suh as the Internet has drastially redued entry osts in many searh markets. We fous on onsumer welfare beause, as * Corresponding author: Yongmin Chen, Department of Eonomis, University of Colorado, Boulder, CO 80309, USA. yongmin.hen@olorado.edu. We thank two referees for their helpful omments. We also thank Mark Armstrong, Heski Bar-Isaa, Joe Farrell, Maarten Janssen, Louis Kaplow, Andrew Rhodes, David Sappington, Marius Shwartz, Guofu Tan, Mariano Tappata, Ralph Winter, partiipants in the 4th Workshop on Searh (Mosow), the 1st Tsinghua Conferene on Theoretial and Behavioral Eonomis (Beijing), the Fifth Annual Searle Conferene on Internet Searh and Innovation (Chiago), the Hal White Antitrust Conferene (Washington, DC), and in seminars at Aademia Sinia, Shanghai University of Finane and Eonomis, University of Arkansas, University of British Columbia, University of Colorado Boulder, University of South Carolina and Zhejiang University for useful disussions and omments. 1 Amir and Lambson (2000) demonstrate that prie an inrease in the number of firms under Cournot ompetition. Nevertheless, as the authors point out, the assumptions needed for suh an outome, whih involves an unstable equilibrium in a ertain sense, are restritive. 2 Starting from the seminal work of Stigler (1961), the literature has advaned in the diretions of searh for the best prie among ompeting homogeneous sellers (Stahl, 1989) and of searh for the best value among ompeting differentiated sellers (Wolinsky, 1986). [ 55 ]

2 56 THE ECONOMIC JOURNAL [ FEBRUARY we demonstrate, the ommon belief that unfettered entry benefits onsumers is atually misguided. This will have ramifiations for business praties as well as for antitrust and regulation poliies. We onsider a model where potential entrants differ in quality the probability that a seller s produt will math a onsumer s need. This probability is larger for a highquality firm, whose produt thus, has a high expeted value to onsumers. Eah firm privately learns its quality, and an hoose to enter the market by inurring an entry ost. After firms simultaneously make their entry deisions, those who have entered the market simultaneously hoose pries, whereas eah onsumer, observing the number of firms in the market, an ondut a sequential searh to find out the prie and produt value of one or more sellers. A onsumer reeives zero utility from a nonmathed produt, while her utility from a mathed produt is a random draw from a known distribution and is idential for all her mathes. 3 The model is thus a dynami game of inomplete information and the type-ontingent nature of the entry deision makes the model different from an otherwise standard two-stage entry game (Mankiw and Whinston, 1986). Under ertain onditions and for a given entry ost, the model has a unique symmetri (perfet Bayesian) equilibrium, where every potential entrant will hoose to enter the market if and only if its quality exeeds some threshold and, similar to Diamond (1971), the equilibrium prie is invariant with respet to the number of sellers. 4 We are interested in two related welfare questions at this equilibrium. First, given an entry ost, how will the expeted number of entrants under free entry ompare to those that maximise onsumer or total welfare? Seond, how will an exogenous hange in entry onditions, suh as the entry ost, affet welfare in the free entry equilibrium? We find that, holding everything else onstant, free entry leads to an exessive number of firms for onsumer welfare and, hene, also for total welfare, 5 when entry ost is below some ritial value; whereas entry is defiient for onsumer welfare when entry ost is above this ritial value. More strikingly, we find that onsumer welfare is an inverted-u funtion of entry ost, first inreasing and then dereasing, maximised at the ritial entry ost. We obtain these results in our model by first identifying two externalities of entry on onsumer searh, whih we term as the searh variety and searh quality effets: the entry of a firm expands the searh options available to eah onsumer, whih is the positive searh variety effet that the entrant does not internalise. But a marginal entrant also lowers searh quality beause it redues the expeted quality of sellers in the market and makes a searh less likely to produe a math. This negative searh quality effet is also not internalised by an entrant. 6 3 This formulation follows several reent papers on onsumer searh (Athey and Ellison, 2011; Chen and He, 2011). 4 In searh markets, more sellers an ause prie to rise (Satterthwaite, 1979; Stahl, 1989), to fall (Wolinsky, 1986), or to either inrease, derease or unhange (Janssen and Moraga-Gonzalez, 2004). Our model provides a useful baseline ase, making it transparent that the mehanism through whih entry affets onsumer welfare in our setting differs from the usual prie effet. 5 Sine entry also has the business-stealing effet (Mankiw and Whinston, 1986), if it is exessive for onsumer welfare, it must also be for total welfare. 6 More generally, entry an also affet onsumer welfare through a prie effet. We will study a variant of our model, where the prie effet is present, to hek the robustness of our welfare results.

3 2018] ENTRY AND WELFARE IN SEARCH MARKETS 57 The existene of the searh variety and quality effets of entry suggest that free entry may not maximise onsumer welfare. But this by itself does not tell us whether on balane entry will be too muh or too little for onsumers. While the analysis to establish our onsumer welfare results is rather involved, the intuition behind it is simple. When entry ost is low, the expeted number of entrants is large while the marginal entrant s quality is well below the average quality. Hene, the positive searh variety effet of entry is small but the negative searh quality effet is large. Consequently, free entry, under whih an entrant does not internalise these two externalities, is exessive for onsumer welfare. Moreover, for a small inrease in the entry ost, the inrease in searh quality is signifiant but the derease in searh variety is not, and thus onsumer welfare rises. Conversely, when entry ost is high, the positive variety effet is substantial but the negative quality effet is negligible, so that free entry is defiient for onsumers and a marginal redution in entry ost will inrease onsumer welfare. Remarkably, in our model, the trade-off between these two effets of entry varies smoothly so that onsumer welfare is a single peaked funtion of entry ost. 7 Our result on how entry affets onsumer welfare, while unonventional, is quite natural for searh markets. In fat, we an interpret searh markets very broadly, to inlude any market where firms have private information about produt quality and onsumers an obtain ostly quality information before purhase. Consider, for example, Akerlof s (1970) lassi model of used-ar market, where, under adverse seletion, low-quality sellers drive out high-quality sellers and the market may shut down ompletely. One may view our artile as taking Akerlof s model a step further by adding onsumer searh to it, so that a buyer an inur a searh (inspetion) ost to find out, possibly with the help of an auto mehani, whether a ar has a defet. 8 A high-quality seller, whose ar is less likely to be defetive, then has a higher probability to sueed in trading and, hene, more inentive to inur the (entry) ost to list its ar for sale. The buyers ability to detet a ar s flaw through ostly searh may thus mitigate the adverse seletion problem. 9 But if entry ost is very low, it will not prevent low-quality sellers from entering the market; buyers searh effiieny will then be too low and the market is likely to perform poorly. On the other hand, if entry ost is too high, very few sellers will enter the market, and even if their expeted quality is high, it will be hard for buyers with heterogeneous preferenes to find a math under the very limited searh opportunities. This, in essene, is the trade-off between the searh variety and searh quality effets of entry, as our analysis unovers In the literature on information ongestion in two-sided markets (Anderson and de Palma, 2009), externalities reated by both information senders and reeivers an also lead to defiient or exessive entries. By analysing a different trade-off in a novel model, we offer new insights on how entry affets onsumer welfare and total welfare in searh markets. 8 This, together with the onsumer s idiosynrati taste, may then determine whether the ar will meet her need. 9 There are related studies of searh and produt quality in the literature. For example, in Wolinsky (1983), pries are observable before onsumer searh and may serve as signals of produt quality that is privately known by firms. Dranove and Satterhwaite (1992) onsider a searh model where onsumers an imperfetly observe pries and qualities after inurring searh osts. They find that an improvement of prie or quality information may either inrease or derease welfare. 10 Importantly, the searh quality effet arises only beause asymmetri information on produt quality: if onsumers had perfet quality information, the entry of more firms, even of those with low qualities, would not redue searh effiieny beause onsumers ould always hoose to searh high-quality sellers first.

4 58 THE ECONOMIC JOURNAL [ FEBRUARY In searh markets, therefore, it will not be unusual for entry restritions to benefit onsumers. This an shed light on many business pratises. Consider, for instane, the market of apps for iphones and ipads. Apple learly has the inentive to inrease onsumer surplus in this market, whih would boost its profits from the sale of iphones and ipads. Although the entry of more app developers will offer users more hoies, the entry of low-quality sellers an redue searh quality and make it harder for onsumers to find a desired app. Apple appears to balane this trade-off by both inreasing entry ost and maintaining a minimum quality standard: it harges a fixed fee to eah entrant ($99/year) and the entrant s produt has to go through a stringent review proess. Only produts that are approved by Apple an be offered for sale to onsumers. In addition to entry barriers reated by private entities (as we shall disuss further in the onluding setion), government poliies an also limit entry, as for example, with a minimum quality requirement. A liene fee that ats as a transfer payment an also positively impat both onsumer and total welfare by raising the quality of the marginal entrant. On the other hand, an entry barrier that adds to physial ost of entry (suh as transation ost) might benefit onsumers but redue total welfare. We desribe our model in Setion 1 and haraterise its equilibrium in Setion 2. In Setion 3, we establish our main results on how the free entry outomes ompare with those that maximise onsumer or total welfare and how welfare may vary with entry onditions. In Setion 4, we analyse a variant of the main model, in whih a onsumer s value for eah math is an independent random draw and hene the mathed sellers of any onsumer are horizontally differentiated. Entry then also has a prie effet, whih ompliates the analysis; nevertheless, in the numerial examples that we onsider, the welfare results of our main model ontinue to hold. 11 In addition to serving as a robustness hek, this Setion also ontains a result of independent interest: we find that equilibrium market prie dereases in the expeted quality of sellers in the market. Setion 5 offers onluding remarks. Proofs that are more tehnial in nature are relegated to Appendix A. 1. The Model The market ontains a unit mass of onsumers, eah demanding one unit of a produt. There are N 2 potential entrants who an hoose to beome ative sellers, and the entry ost for eah seller is k > 0. Eah onsumer is ex ante unertain about whether a partiular firm offers a produt that she desires and how muh she is willing to pay for suh a produt. Speifially, with probability b i, potential entrant i s produt, i = 1, 2,..., N, meets a onsumer s need. The onsumer derives utility u from onsuming the produt of all her mathed sellers, and u is an independent 11 This variant of our model is losely related to Anderson and Renault (1999), who study a standard onsumer searh model with horizontally differentiated produts (i.e. b j ¼ 1 for all j in their model) and find that market entry is always exessive for total welfare. We explain in Setion 4 why the presene of vertial differentiation (i.e. b j \ 1 and different firms differ in their b j ) leads to very different results and new insights.

5 2018] ENTRY AND WELFARE IN SEARCH MARKETS 59 draw from distribution F with density f on support ½u; uš, where u [ u With probability 1 b i, i s produt does not meet the onsumer s need, in whih ase the onsumer utility from the produt is normalised to zero. 13 Thus, we onsider b i as a measure of i s quality. 14 Potential sellers differ in their quality. In partiular, we assume that b i draws from umulative distribution funtion G with density funtion g > 0 on support [0, 1]. Finally, the prodution ost of eah seller is normalised to zero. 15 The timing of the model is as follows. First, b i is realised and is known privately by i. Seond, potential entrants simultaneously hoose either to enter the market or to stay out. Third, the market struture is determined, with n entrants as sellers. Consumers are informed of the number of sellers. Although n = 0 is always a possibility, our analysis will fous on situations where n 1, and we assume that k is relatively small so that a potential entrant with a suffiiently high b i will enter the market. Fourth, sellers simultaneously and independently set their pries, after whih eah onsumer, without knowing whether any partiular seller is a math, her value u for the math, and the seller s prie, hooses whether and how to ondut sequential searh. Eah searh will enable the onsumer to disover the aforementioned information from a seller, with searh ost s. We study symmetri perfet Bayesian equilibrium of this game. Throughout the artile, we maintain the assumption that 1 F(u) and 1 G(b) are log-onave. Let p m ¼ arg maxfp½1 F ðpþšg; p m ¼ p m ½1 F ðp m ÞŠ: p Then, p m exists uniquely and is interior, due to the log-onavity of 1 F. The logonavity of 1 G will be used in Lemma 1 to show that the equilibrium profit of an entrant is inreasing in its quality type. 2. Market Equilibrium Suppose for a moment that, given k, a potential entrant will enter the market if and only if its quality exeeds some threshold t. We first study equilibrium for any given threshold t. We then show that in equilibrium the expeted profit of potential entrant i indeed inreases in b i ; thereby onfirming the optimality of the threshold-based entry strategy for eah potential entrant. The equilibrium threshold t f is then determined, whih is shown to inrease in k. 12 Note that b i is firm- but its realisation is onsumer- (in the sense that a math for one onsumer does not neessarily imply a math for another). Also, u is onsumer-speifi but, for eah onsumer, it is equal aross her mathed sellers. 13 For example, a onsumer may have a speifi requirement for a produt, suh as a ertain quality feature for a ar, and a high-quality seller is more likely to meet the requirement. Or, a onsumer may need to improve a produt s performane (suh as the energy effiieny of a house) and a high-quality firm is more likely to find the right solution to the problem. It ould also be that the onsumers are input purhasers on an intermediate-good market and a high-quality supplier is more likely to meet eah buyer s quality standard. R 14 u In fat, the expeted value of seller i s produt to a onsumer is simply b i u udf ðuþ, whih inreases in b i. Thus, a high-quality seller is more likely to offer onsumers a high-value produt. 15 More preisely, the marginal ost of prodution is equal to the onsumer utility from a non-mathed produt, whih has been assumed to be zero.

6 60 THE ECONOMIC JOURNAL [ FEBRUARY For any given t, the expeted quality of an entrant is: Z 1 xg ðxþ dx t ðtþ ¼ 1 GðtÞ ; (1) where > t for all t 2 [0, 1) sine R 1 t xg ðxþ dx [ t½1 GðtÞŠ. First, onsider the sellers prie strategy and onsumers searh strategy. If there is only one seller (n = 1), its equilibrium prie will be p m and onsumers will searh if: Z u ðu p m Þf ðuþ du s 0: (2) p m Condition (2) is satisfied if s is not too large, whih we assume throughout the artile. With n 2 sellers, from standard arguments (Diamond, 1971; Chen and He, 2011), there is a unique equilibrium where eah seller sets p ¼ p m ; eah onsumer will searh sequentially and will purhase from the first math, provided that u p m. The onsumer will exit the market without purhase if u \ p m or if she has searhed all n sellers without finding a math. Thus, in equilibrium, seller i s expeted profit for any given t when there are n entrants in the market (inluding i) is: ðb i Þ¼b i p m / n ; (3) where: / n ¼ 1 n X n 1 j¼0 ð1 Þ j ¼ 1 ð1 Þn n is the expeted number of onsumers who visit seller i when n firms (n 1 rivals) enter the market. We next determine the endogenous number of sellers. Consider a potential seller s entry deision. From (3), a seller s expeted profit, when there are n entrants, is inreasing in b i. To determine the equilibrium t, we onsider the deision of i with b i. The post-entry expeted profit for i is: (4) where: Eðpjb i Þ¼ XN d n ðtþ ðb i Þ; (5) d n ðtþ ¼ N 1 n 1 ½1 GðtÞŠ n 1 GðtÞ N n (6) is the probability that n 1 other potential entrants enter and ðb i Þ is the expeted profit for i if it hooses entry simultaneously as the n 1 others. Our analysis will utilise Lemma 1 below, whih states that: (i) an inrease in the marginal entrant s quality will raise the average quality of all entrants in the market; but (ii) the marginal inreases relatively more than the average.

7 2018] ENTRY AND WELFARE IN SEARCH MARKETS 61 Part (i) is straightforward, and while (ii) is also intuitive, it relies on the log-onavity of 1 G. LEMMA 1. For all t 2 [0, 1): ðiþ d dt ¼ ðiiþ dðt=þ dt g ðtþ ð tþ [ 0; 1 GðtÞ g ðtþtð tþ=½1 GðtÞŠ ¼ 2 [ 0: (7) By Lemma 1, the proof of whih is ontained in Appendix A: ðtþ ¼p m t 1 ð1 Þ n (8) n inreases in t. That is, given n, the expeted profit for the marginal entrant is higher if it has a higher quality. It an also be verified that ðtþ dereases in n. Lemma 2, whih is also proved in Appendix A, establishes that the expeted post-entry profit for the marginal entrant is inreasing in its quality: LEMMA 2. E(p t) inreases in t. Notie that the marginal entrant will earn zero if it has b i ¼ 0, and will earn p m if it has b i ¼ 1. Therefore, for any given k 2½0; p m Þ, there exists a unique threshold t f t f ðkþ 2½0; 1Þ that satisfies: Eðpjt f Þ¼k; (9) and t f ¼ t f ðkþ inreases in k, with t f ¼ 0 for k = 0 and t f! 1ask! p m. We have thus shown that there exists a symmetri equilibrium where eah potential entrant will enter if and only if its quality reahes the threshold t f, and t f monotonially inreases in k. Moreover, it is straightforward to hek that there an be no other symmetri equilibrium. Summarising the above disussion, we have: PROPOSITION 1. For any given k 2ð0; p m Þ, there exists a unique symmetri equilibrium where: (i) potential entrant i, i = 1, 2,..., N, will enter the market if and only if b i t f, with t f 2ð0; 1Þ, defined in (9), being an inreasing funtion of k, and eah seller will harge prie p m ; (ii) eah onsumer will searh sequentially in random order, purhase from the first math if u p m and make no purhase if either she finds no math or u \ p m. 3. Welfare Analysis In our model, the number of entrants (n) is unertain, depending on the number of potential entrants (N), the realisations of b i and entry ost (k). Hene, a proper measure of entry is the expeted number of entrants, whih is determined by t, the

8 62 THE ECONOMIC JOURNAL [ FEBRUARY minimum possible quality of atual entrants. A lower t orresponds to a higher expeted number of sellers in the market. In equilibrium, through the dependene of t f on k, the expeted number of sellers in turn will be determined by k. We an then ompare it with the number that maximises onsumer or total welfare and explore how welfare may vary with entry onditions. 3.1 Consumer Welfare For a given t, onsumer welfare, measured by expeted aggregate onsumer surplus (net of searh ost) is: where: k n ðtþ ¼ V ¼ XN N n k n ðtþv n ð; p m Þ; (10) ½1 GðtÞŠ n GðtÞ N n (11) is the probability that exatly n sellers have entered, and: V n ð; p m Þ¼ Xn i¼1 Z u ð1 Þ i 1 ðu p m Þf ðuþdu Xn ð1 Þ i 1 is ð1 Þ n ns (12) p m i¼1 is the onsumer welfare with n 1 sellers when their expeted quality is. InV n above, the first term is the (weighted) sum of benefit when a onsumer has searhed and purhased from the i th seller, while the seond and the third terms are the expeted searh ost when the onsumer ends up with and without purhase respetively. We define: Z u U ¼ ðu p m Þf ðuþ du; MðtÞ ¼1 ½1 GðtÞŠ; (13) p m where Φ is a onsumer s expeted surplus from a math and M(t) indiatesthe probability that a potential entrant will not be a math when the entry threshold is t. LEMMA 3. Consumer welfare V V(t) an be expressed as: V ¼½1 MðtÞ N Š U s : (14) Equation (14) has an intuitive interpretation. The probability that a onsumer will (eventually) find a math is 1 M ðtþ N : Sine Φ is the expeted surplus to a onsumer from a math and s/ is the searh ost adjusted by the expeted math probability per seller, Φ s/ reflets the expeted net benefit from a searh that yields a math. With a unit mass of onsumers, onsumer welfare is the onsumer s expeted net benefit from the entry of firms under threshold t.

9 2018] ENTRY AND WELFARE IN SEARCH MARKETS 63 Note that given the distribution of u, searh ost s, and the number of potential entrants N, V is entirely determined by t through = (t) and M(t). Totally differentiating (14) with respet to t, we have: dv dt ¼ NMðtÞN 1 dm dt U s fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} searh variety effet þ½1 M ðtþ N Š s d 2 : (15) dt fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} searh quality effet Thus, the impat of inreased entry (i.e. a derease in t) on onsumer welfare an be deomposed into two parts: a searh variety effet and a searh quality effet. The first term in (15), the variety effet, is the hange in V due to dm/dt: a derease in t raises the expeted number of entrants, providing onsumers with more searh opportunities to obtain the expeted net benefit, Φ s/. From (13) and by Lemma 1: dmðtþ dt ¼ d ½1 GðtÞŠ þ g ðtþ ¼g ðtþt [ 0: (16) dt The seond term, the quality effet, is the hange in V due to d/dt. Note from (7): d g ðtþð tþ ¼ [ 0; (17) dt 1 GðtÞ and hene more entry has a negative quality effet: a derease in t redues the average math probability of sellers in the market, lowering onsumer searh effiieny. The hange in onsumer welfare from a marginal entrant depends on the balane of these two opposing effets. Sine t f ðkþ is monotonially inreasing, a redution in k has the same two effets as a redution in t f in equilibrium. Define V f V ðt f Þ as the onsumer welfare in the free-entry equilibrium. We an now state our main result onerning free entry and onsumer welfare. Its proof first establishes that V(t) is single-peaked and then uses the fat that t f monotonially inreases in k. THEOREM 1. There exists some k 2ð0; p m Þ suh that, relative to what maximises onsumer welfare, the expeted number of entrants under free entry is too high when k \ k and too low when k [ k. Furthermore, onsumer welfare is an inverted-u funtion of k, first inreasing and then dereasing, maximised at k. Proof. From (15), utilising (16) and (17), and noting 1/[1 G(t)] = /[1 M(t)], we have: dv dt h i ¼ 1 M ðtþ N s g ðtþ 2 1 GðtÞ ð tþ NM ðtþn 1 g ðtþt U s " 1 M ðtþn s ¼ g ðtþ 1 MðtÞ ð tþ NMðtÞN 1 t U s # : (18)

10 64 THE ECONOMIC JOURNAL [ FEBRUARY Therefore, for t 2 (0, 1), dv/dt = 0 if: " # t ¼ 1 þ NM ðtþn 1 1 MðtÞ U s 1 1 M ðtþ N s : (19) If t = 0, the LHS of (19) < the RHS of (19); if t? 1, the LHS of (19) > the RHS of (19). Furthermore, from Lemma 1, the LHS of (19) monotonially inreases in t. Sine dm(t)/dt 0, d/dt 0, and: d½m N 1 ð1 M Þ=ð1 M N ÞŠ dm M N 2 ¼ ð1 M N Þ 2 ðn NM þ M N 1Þ ¼ M N 2 ð1 M Þ ð1 M N Þ 2 N XN 1 M! j 0; the RHS of (19) dereases in t. Therefore, there exists a unique t 2ð0; 1Þ that solves (19), with dv/dt > 0ift \ t and dv/dt < 0ift [ t. Furthermore, sine t f ¼ tðkþ is monotonially inreasing and dv f =dk ¼ðdV f =dt f Þtf 0 ðkþ, V f first inreases and then dereases in k, maximised at some k 2ð0; p m Þ. Finally, sine the entry threshold assoiated with the maximum onsumer welfare is t ¼ t f ðk Þ, relative to what maximises onsumer welfare, free entry is exessive when k \ k but defiient when k [ k. As t, or entry ost k, dereases, more potential entrants hoose to enter the market but the marginal entrant has a lower quality. When k is high, entry is defiient and a derease in k benefits onsumers, both beause the opportunity to searh an additional entrant is highly valuable when the expeted number of entrants is small and beause the margin entrant has a relatively high quality, so that the positive variety effet dominates. 16 When k is relatively low, entry is exessive and an inrease in k benefits onsumers beause in this ase, the negative quality effet of entry dominates. 17 We note that searh ost is ruial for our onsumer welfare results in Theorem 1. As searh ost approahes zero, from (15), the searh quality effet vanishes so that dv/dt < 0; onsequently both t and k approah 0, and more entrants will (almost) always benefit onsumers beause of the positive variety effet. Then, free entry will (almost) always be defiient from onsumers perspetive and our novel result that there is too muh equilibrium entry for onsumers in searh markets when entry ost is small, beomes irrelevant. 18 The result below shows more generally how searh ost and the number of potential entrants affet k, the entry ost that maximises onsumer welfare. j¼0 16 When t? 1, t and are lose. So, the first term in the square braket of (18) approahes zero, while the seond term is positive and inreasing in t. Hene, dv/dt < 0 when t is high. 17 When t? 0, the first term in the square braket of (18) approahes some positive onstant, while the seond term approahes zero. Hene, dv/dt > 0 when t is low. 18 Our analysis for Theorem 1 remains valid for arbitrarily small s, as long as it is stritly positive. If s = 0, there would be no pure-strategy prie equilibrium beause firms would want to underut eah other to ompete for onsumers with multiple mathes but they would also want to raise the prie to onsumers with only one math. The analysis would then be very different from ours. Thus, again, ostly searh plays a ruial role in our model.

11 2018] ENTRY AND WELFARE IN SEARCH MARKETS 65 COROLLARY 1. k,ort, inreases in searh ost (s) and in the number of potential entrants (N). Proof. Sine t ¼ tðk Þ, it suffies to show that t inreases in s and in N. Sine the LHS of (19) inreases in t and is independent of s while the RHS dereases in t and inreases in s, t inreases in s. Moreover, sine M < 1, dðn ln M M N þ 1Þ=dM ¼ N =M NM N 1 [ 0, and N ln 1 1 N þ 1 ¼ 0, we have: d½nm N 1 ð1 M Þ=ð1 M N ÞŠ dn Therefore, t inreases in N. M N 1 ¼ ð1 M N Þ 2 ð1 MÞðN ln M M N þ 1Þ \ 0: The entry ost (or the quality threshold) that maximises onsumer welfare inreases in searh ost and in the number of potential entrants. Intuitively, with a high searh ost, it is more ostly for onsumers to searh more varieties. It follows that fewer sellers with higher quality tend to be better for onsumers and, hene, k (or t ) is higher. Also, when the number of potential sellers is high, the variety effet is less signifiant beause for a given k, the expeted number of entrants is large and, hene, an inrease in t tends to be more benefiial to onsumers. Therefore, t also inreases in the number of potential entrants. 3.2 Total Welfare We next onsider total welfare. For given k and t, the (expeted) industry profit is: P ¼ XN n¼0 k n ðtþn½ ðþ kš; (20) where ðþ k is the expeted profit for a seller of quality in a market with n sellers. 19 The result below is proved in Appendix A. LEMMA 4. For any given t, industry profit is: h i PðtÞ ¼p m 1 MðtÞ N kn ½1 GðtÞŠ; (21) and the free-entry equilibrium industry profit is: P f ¼ 1 t h f 1 M ðt f Þ N i p m : (22) Note that ½1 M ðt f Þ N Šp m is the expeted industry revenue when at least one seller s produt mathes a onsumer s need. Sine the marginal entrant with t f earns zero profit, 1 t f = reflets the expeted profit margin of eah entrant. 19 Note that in our model, in the free-entry equilibrium, only the marginal entrant earns zero profit, while the other entrants earn positive profits.

12 66 THE ECONOMIC JOURNAL [ FEBRUARY From (14) and (22), total welfare at the free entry equilibrium is: h i W f ¼ 1 M ðt f Þ N U s þ 1 t f p m : (23) PROPOSITION 2. In equilibrium: (i) industry profit dereases in k; and (ii) total welfare dereases in k when s is suffiiently small or k is suffiiently high. Proof. From (22) and (23), sine k affets P f and W f only through t f, and sine t f inreases in k, it suffies to show that the stated relationships for k hold for t f : (i) Reall from (7) and (16) that d(t/)/dt > 0 and dm/dt 0. Thus dp f =dt f \ 0. (ii) From Proposition 1, onsumer welfare dereases in t when t is high. Thus, sine dp f =dt f \ 0, W f ¼ V f þ P f must derease in t f when t f is suffiiently high. Furthermore: dw f ¼ NMðt f Þ N 1 g ðt f Þt f U s þ 1 t f p m dt f h i þ 1 Mðt f Þ N s d 2 dðt f =Þ p m : dt f dt f Reall that dðt f =Þ=dt f [ 0. Hene, dw f =dt f \ 0ifs? 0. A marginal inrease in entry ost raises t f, whih redues the expeted number of sellers and, hene, the probability of sales. Additionally, a higher entry ost redues an inframarginal seller s profit margin. Consequently, industry profit is redued with a higher entry ost. On total welfare, a higher k will inrease onsumer welfare by raising t f when k \ k, whih an potentially outweigh the profit effet. But when k is large, profit and onsumer welfare move in the same diretion and, hene, W islowerwithanevenhigherk. Also,whens is small, the low searh ost an largely offset the redution in sellers quality to provide searh inentives, so that the profit hange will dominate and hene an inrease in entry ost will lower total welfare. Example 1 illustrates how the equilibrium onsumer welfare, industry profit and total welfare vary with entry ost k. EXAMPLE 1. Suppose that N = 3, s = 0.05, with b i and u being uniformly distributed on [0, 1]. Then, from (1) and (13), = (1 + t)/2, M ¼ð1 þ t 2 Þ=2, Φ = 1/8 and p m ¼ 1=4. From (5), t f solves EðpjtÞ ¼tð4t 2 þ t 4 þ 7Þ=48 ¼ k. From (14), we have V f ¼ð5t f þ 1Þ ð1 t f Þð4t 2 f þ t 4 f þ 7Þ=320, t ¼ 0:497, and k ¼ 0:083. Moreover, from (22), P f ¼ ð1 t f Þ 2 ð4t 2 f þ t 4 f þ 7Þ=32, and thus W f ¼ð11 5t f Þð1 t f Þ ð4t 2 f þ t 4 f þ 7Þ=320. In Figure 1, onsumer welfare is the inverted-u urve, while both industry profit and total welfare derease with k.

13 2018] ENTRY AND WELFARE IN SEARCH MARKETS Consumer Welfare Industry Profit Total Welfare k Fig. 1. Welfare and Entry Cost Now onsider the soially optimal t, denoted as t o t o ðkþ, for whih we do not impose the free-entry ondition E(p t) = k. From (21) and (14), for any given t, total welfare is given by: W ðtþ ¼ U s h i þ pm 1 MðtÞ N kn ½1 GðtÞŠ: (24) Thus: dw dt ¼ s d h i 2 1 M ðtþ N dt U s þ pm NM ðtþ N 1 tgðtþþkng ðtþ: (25) At the free entry equilibrium, sine the marginal entrant has zero net profit due to Eðpjt f Þ¼k, the marginal entrant must redue industry profit due to the business-stealing effet. From Proposition 1, for t f \ t t f ðk Þ, free entry is exessive for onsumer welfare. Therefore, when k k (or t f t ), free entry must be soially exessive, with t o [ t f. When k [ k, entry is defiient for onsumer welfare but it an still be soially exessive when the negative profit effet is onsidered. However, when k is large, the profit effet is small relative to the effet on onsumers, and entry is soially defiient, as we establish in the result below. PROPOSITION 3. Free entry is soially exessive (i.e. t o [ t f ) when k k, and it is soially defiient (i.e. t o \ t f ) when k is suffiiently large (but still smaller than p m ). Proof. We have already argued t o [ t f when k k. It remains to show t o \ t f when k (\p m Þ is suffiiently large. From the proof of Lemma 2, any t t f satisfies k ðt=þp m ½1 M ðtþ N Š=fN ½1 GðtÞŠg. Substituting this into (25), we have:

14 68 THE ECONOMIC JOURNAL [ FEBRUARY dw ( 1 MðtÞN sð tþ dt t tf 1 GðtÞ 2 þ t pm U s ) þ pm NM ðtþ N 1 t g ðtþ: Thus, when k! p m, t? 1,? 1,M(t)? 1 and, from (16), lim t!1 ½1 MðtÞ N Š= ½1 GðtÞŠ ¼ lim t!1 ½ N M ðtþ N 1 g ðtþtš=½ g ðtþš ¼ N. Hene, the right-hand side of the above inequality approahes: ðu sþng ð1þ\0: Therefore, when k is suffiiently large (but still smaller than p m ), dw =dtj t tf \ 0, so that free entry is soially defiient (i.e.t o \ t f Þ. As in the ase of onsumer welfare, searh ost also plays an important role for our total welfare results. As s? 0 and hene k! 0, the result in Proposition 3 that free entry is exessive for total welfare when k k, while still valid, is virtually irrelevant. More entry will then inrease onsumer welfare due to the searh variety effet but redue industry profit due to business stealing, with the net impat on total welfare potentially ambiguous. When k (\p m Þ is suffiiently high, however, Proposition 3 applies and the equilibrium number of entrants will be too small relative to the soial optimum. 4. Differentiation Among Mathed Sellers So far, we have assumed that a onsumer has the same value (u) from all of her mathed sellers, even though u is ex ante unertain to the onsumer. As we mentioned earlier, one advantage of this formulation is that equilibrium prie will then be invariant to the number of sellers, whih substantially simplifies the analysis. We now onsider an alternative setting where a onsumer has heterogeneous values for sellers who math her need. Speifially, as in Wolinsky (1986), we assume that a onsumer s value for eah mathed seller i, u i, is independently drawn from distribution F on support ½0; uš, with density f. 20 Thus, there is horizontal differentiation among mathed sellers. 21 Everything else is the same as in the main model. A key aspet in whih this variant differs from the main model is that entry will now also affet market prie. Our analysis in this Setion proeeds as follows: first, we haraterise the equilibrium priing strategy given the number of ative sellers (n) and their average quality (). Next, we show that the equilibrium market prie ( Þ dereases in. This additional prie effet introdues a ompliation to the expeted profit for a seller. In partiular, unlike in the main model, it is no longer lear that a potential entrant s expeted profit will inrease in t beause a higher t, whih results in a higher average quality, now also leads to a lower equilibrium prie. After presenting the equilibrium 20 That is, in ontrast to our main model in whih the values of a onsumer s mathed sellers are perfetly dependent, this formulation onsiders the other polar ase where these values are independent. More realistially, the values of a onsumer s mathed sellers may be neither perfetly dependent nor independent; but, like others in the literature, we fous on these two polar ases for analytial tratability. 21 Searh models with horizontally differentiated sellers following Wolinsky (1986) inlude, for example, Anderson and Renault (1999), Armstrong et al. (2009), Eliaz and Spiegler (2011), Haan and Moraga- Gonzalez (2011) and Bar-Isaa et al. (2012).

15 2018] ENTRY AND WELFARE IN SEARCH MARKETS 69 analysis for a given n, we turn to numerial analysis to show that the welfare results of the main model still hold under additional funtional and parameter onditions. Suppose first that there are n N sellers in the market who all set prie. 22 Following Kohn and Shavell (1974) and Wolinsky (1986), onsumers optimal searh strategy is to sample sellers sequentially, with reservation value a() from mathed seller i that satisfies: Z u a ðu i aþf ðu i Þ du i ¼ s: (26) Note that the market is ative only when sellers are expeted to harge a. A onsumer stops searhing when she finds a math with u i a; if no suh produt is found after she searhes all sellers, she buys the produt from the mathed seller with the highest u i, and she buys nothing if no math is found or if u i \ for all mathes. Sine u i is independently and identially distributed for eah of a onsumer s mathed sellers, for onveniene we shall drop the subsript i for the rest of the Setion. Totally differentiating of (26) with respet to and rearranging terms, ¼ Z u a ðu aþf ðuþ du s Z u ¼ f ðuþ du 2 [ 0: (27) ½1 F ðaþš a Hene, a inreases with. That is, the benefit of searh is larger if the expeted quality of sellers is higher. We assume that s is suffiiently small suh that onsumers will indeed searh in equilibrium. Next, we haraterise the ondition for the equilibrium where sellers harge the same prie despite differenes in math probabilities. 23 If there is only one seller (n = 1), then it optimally harges p 1 ¼ p m. So, suppose that n 2. If other sellers harge in equilibrium, given the searh strategy by onsumers, a seller with b i harges p to maximise: i ðp; Þ¼pfb i ½1 F ðp þ a ÞŠu n þ b i R n ðp; Þg; (28) where: u n ¼ 1 X n 1 X j j ð1 Þ h ½F ðaþš j h 1 ½1 þfðaþšn ¼ (29) n h n½1 F ðaþš j¼0 h¼0 is the number of onsumers who ome to seller i for the first time after sampling j 2 {0, 1,..., n 1} other sellers and finding no math or the valuation is below a, and: 22 Even though sellers differ in math probabilities, a seller is either a math or no math after the searh by a onsumer and the seller ompetes in pries only with other mathed sellers for the onsumer. Sine all mathed sellers of a onsumer are horizontally differentiated and are ex ante idential to the onsumer, it is appropriate to onsider a symmetri prie equilibrium. Alternatively, one may onsider a possible asymmetri prie equilibrium where firms with higher math probabilities hange higher pries but it does not appear to be analytially tratable in the model here. 23 Intuitively, all mathed sellers of a onsumer are horizontally differentiated as in Wolinsky (1986), and thus the equilibrium has a similar struture.

16 70 THE ECONOMIC JOURNAL [ FEBRUARY Z ( pþa pn X n 1 n 1 ) R n ðp; Þ¼ ð1 Þ n 1 j ½F ðu p þ ÞŠ j f ðuþdu j ¼ p Z pþa pn p j¼0 ½1 þ F ðu p þ ÞŠ n 1 f ðuþ du (30) is the number of returning onsumers who have sampled all sellers and have not found any value above a, while seller i is a math that gives the highest valuation. It follows that: Z a R n ð ; Þ¼ ½1 þ F ðuþš n 1 f ðuþ du: (31) From the first-order ondition of (28), at an equilibrium with p i ¼ for all i = 1,..., n, the equilibrium satisfies p 0 =@pj p¼pn ¼ 0: ½1 F ðaþšu n þ Z a ½1 þ F ðuþš n 1 df ðuþ ff ðaþu n Z a ½1 þ F ðuþš n 1 df ðuþg ¼ 0:g (32) If ¼ 0, the LHS of (32) is positive. If ¼ a, the LHS of (32) beomes f½1 F ðaþš af ðaþgu n, whih is negative beause a ½1 FðaÞŠ=f ðaþ [ p 1 ½1 F ðp 1 ÞŠ=f ðp 1 Þ¼0, where the inequality holds due to p 1 \ a and ½1 F ðaþš= f ðaþ \ ½1 F ðp 1 ÞŠ=f ðp 1 Þ. Thus, there exists some 2ð0; aþ that solves (32), where is given by: Z a ½1 F ðaþšu n þ ½1 þ F ðuþš n 1 f ðuþ du p ¼ Z n a : (33) f ðaþu n ½1 þ F ðuþš n 1 f 0 ðuþ du Lemma 5 below, whih is proved in Appendix A, states that sellers will harge at any symmetri prie equilibrium and provides a suffiient ondition for the unique existene of suh an equilibrium. 24 LEMMA 5. At any symmetri prie equilibrium of the alternative model, eah seller sets aording to (33) and onsumers searh with reservation value a() that satisfies (26). If F follows a uniform distribution, then the symmetri prie equilibrium exists and is unique. We next state a result on how equilibrium prie may vary with the average quality of sellers in the market. The proof is also ontained in Appendix A. PROPOSITION 4. In the alternative model where eah onsumer s value is independent for every math, given the number of sellers (n), an inrease in leads to a derease in. 24 Searh models generally also have a trivial equilibrium where firms are expeted to and indeed harge very high pries, and no onsumer engages in searh. As in the literature, we do not onsider suh trivial ases.

17 2018] ENTRY AND WELFARE IN SEARCH MARKETS 71 It may seem surprising that a higher average quality would lead to a lower market prie but in a searh market this result is quite natural, for the following reason. An inrease in the average quality of sellers in the market indues a higher onsumer reservation value in their searh deision beause the expeted benefit from another searh is higher. This fores sellers to lower pries in order to indue onsumers to purhase without further searh. 25 For a given entry ost, when the equilibrium prie is given by in (33), there exists a free-entry equilibrium that is similar to the one in the main model, with the marginal entrant s quality, t f, now defined by (34) below. PROPOSITION 5. In the alternative model, suppose that the equilibrium prie is given by in (33). Then, for any k 2ð0; p m Þ, there exists an equilibrium for the entire model where: (i) potential entrant i will enter the market if and only if b i t f, eah entrant will harge, and t f satisfies: X N 1 ½1 þfð ÞŠ n d n ðt f Þt f ¼ k; (34) n (ii) onsumers will searh sequentially with reservation value a that satisfies (26). Proof. For a given t and thus, from (28) and (A.1), in the symmetri equilibrium with n sellers the profit for seller i is: ( Z ) 1 ½1 þf a ðaþšn i ¼ b i ½1 F ðaþš þ b n½1 F ðaþš i ½1 þ F ðuþš n 1 f ðuþ du 1 ½1 þfð ÞŠ n ¼ b i : n Thus, the expeted post-entry profit for entrant i is Eðpjb i Þ¼ P N d nðtþi, whih inreases in b i. For the seller with math probability t, its expeted profit from entry is: EðpjtÞ ¼ XN 1 ½1 þfð ÞŠ n d n ðtþt ; n whih is a ontinuous funtion of t. Sine the marginal entrant with t = 0 has zero profit, and the marginal entrant with t = 1 has profit p m, for any k 2ð0; p m Þ,E(p 0) < k and E(p 1) > k. Therefore, there exists some t f 2½0; 1Þ suh that Eðpjt f Þ¼k. That is, given k, there exists some t f suh that potential entrants with b i t f will enter. Finally, from Proposition 5, the priing strategy and onsumer searh behaviour are optimal when there are n sellers. Different from the main model, here, we have not proven that t f is an inreasing funtion of k. The ompliation is that, as t inreases, equilibrium prie dereases and 25 Anderson and Renault (1999) study the effet of onsumer taste for diversity on equilibrium prie in a searh model. They show that, when the preferene for diversity is low, equilibrium prie may derease as the preferene for diversity inreases due to the inreased onsumer searh.

18 72 THE ECONOMIC JOURNAL [ FEBRUARY thus the impat on the expeted profit of the marginal seller with quality t is unlear. 26 For the rest of this Setion, we assume that: (i) N = 3 and (ii) F and G are both uniform distributions on [0, 1]. Then, it an be verified numerially that t f inreases in k for various values of s. Furthermore, onsumer welfare initially inreases but eventually dereases in t. The intuition is similar as in the main model: a lower t f leads to a higher expeted number of sellers in the market but to a lower sellers average quality. The inrease in variety benefits onsumers by expanding their searh opportunities, whereas the derease in quality harms onsumers by reduing their searh effiieny. However, here prie is also affeted, in two opposing diretions: greater variety ats to redue equilibrium pries, whereas lower quality works in the opposite diretion as onsumers searh less due to the lower searh benefit. Nevertheless, as in the main model, when t f is high, and thus the number of ative sellers is low, the variety effet tends to dominate, so that a further inrease in t f results in lower onsumer welfare. On the other hand, when t f is low, the quality effet tends to dominate, so that an inrease in t f results in higher onsumer welfare. Sine t f inreases in k, it follows that onsumer welfare also first inreases and then dereases in k. Figure 2 shows how onsumer welfare varies with k for three different values of searh ost. For a given s, let k be the entry ost that maximises onsumer welfare. Then, same as in the main model (Theorem 1), from the standpoint of onsumer welfare, entry is V k s = 0.05 s = 0.04 s = 0.03 Fig. 2. Consumer Welfare and Entry Cost 26 Reall that in the main model, equilibrium prie is independent of.

19 2018] ENTRY AND WELFARE IN SEARCH MARKETS 73 exessive when k k but defiient when k [ k. Notie from Figure 2 that for a higher s, k (or t Þ is higher, also same as in the main model (Corollary 1). This is beause with a higher searh ost, the option to searh more varieties is less valuable while the lower quality of sellers (or lower searh effiieny) is more detrimental to onsumers. Consider next total welfare, W. Same as in the main model (Proposition 3), when k k, entry is soially exessive, again beause of the overentry for onsumers and the additional negative effet on industry profit of the marginal entrant. Furthermore, when k is suffiiently large (but still less than p m Þ, entry an be soially defiient (as in the main model, too). For instane, if s = 0.03 and k = 0.2, we an ompute that the quality ut-off is t f ¼ 0:9 under free entry but t o ¼ 0:823 for the soial optimum, and the intuition is also similar to that in the main model: sine k ¼ 0:2 [ k 0:045, entry is too low from onsumers perspetive, and although the marginal entrant has the negative externality on industry profit, the onsumer effet dominates the profit effet when k is large, despite the prie effet of entry in this ase. Therefore, the results of our main model ontinue to hold in this alternative setting, under additional funtional and parameter restritions. 27 While our most novel result is about the effets of entry on onsumer welfare, our finding about the total welfare effets of entry also differs from those in several losely related papers. In partiular, Wolinsky (1984) studies the optimality of entry in a irle model with onsumer searh. Similar to our main model, entry has no prie effet in his model, where for simpliity, he assumes that the prie is exogenously given. He reports an overentry result: the market will offer exessive variety from the standpoint of total welfare when entry ost is suffiiently low. This is beause in his model, the soially optimal variety is bounded: when the number of varieties is suffiiently high (or the entry ost is suffiiently low), onsumers will find a brand satisfying the sequential searh stopping rule and not searh further. In this ase, an extra entrant will not benefit onsumers and, onsequently, will redue total welfare due to the negative externality on other sellers profits. Our models also predit that entry is soially exessive when the entry ost is suffiiently small but, in our ase, entry an also be insuffiient when the entry ost is relatively large. In both of our models, overentry in terms of total welfare ours under low entry osts beause the marginal entrant redues the average quality of sellers in the market, leading to lower searh effiieny (reenforing the negative externality on profits); whereas defiient entry an arise under high entry osts beause onsumers benefit from more searh opportunities to find a math, whih an overome the negative externality on profits. Our model with horizontal differentiation in this Setion is more losely related to Anderson and Renault (1999), whih studies a standard onsumer searh model with differentiated produts. They find that market entry is always exessive beause in their model, entry is exessive in the limiting ase of zero searh ost and the presene of positive searh osts exaerbates the distortion: as searh ost rises, 27 We have also studied the ase where N = 2, with F and G being standard uniform. The results are qualitatively the same.