eferrals in Searc Markets Maria Arbatskaya and Hideo Konisi June 29, 2010 Abstract Tis paper compares te equilibrium outcomes in searc markets wit and witout referrals. Altoug it seems clear tat consumers would benefit from referrals, it is not at all clear weter firms would unilaterally provide information about competing offers since suc information could encourage consumers to purcase te product elsewere. In a model of a orizontally differentiated product and sequential consumer searc, we sow tat valuable referrals can arise in te equilibrium: a firm will give referrals to consumers wose ideal product is sufficiently far from te firm s offering. It is found tat te equilibrium prices are iger in markets wit referrals. Altoug referrals can make consumers worse off, referrals lead to a Pareto improvement as long as te searc cost is not too low relative to product eterogeneity. Similar results are obtained in te presence of referral fees and in te case were firms can price-discriminate among consumers and consumers can misrepresent teir tastes. Keywords: orizontal referrals, consumer searc, information, matcing, referral fees. JEL numbers: C7,D4,D8,L1. We tank Eric asmusen, egis enault, participants of seminars at Boston College, Emory University, Georgia Institute of Tecnology, IEB at Kobe University, and participants at te International Industrial Organization Conference, te Soutern Economic Association Meetings, and te Midwest Economic Teory Meetings. Corresponding Autor: Department of Economics, Emory University marbats@emory.edu Department of Economics, Boston College ideo.konisi@bc.edu 1
1 Introduction In a number of industries, consumers purcase products and services only infrequently, or else te products caracteristics cange often and are tus difficult to assess. Examples include professional services in areas of law, accounting, real estate, and ealt care, as well as specialized services and ig-tec electronic products. As a result, in tese industries a typical consumer needs to conduct a costly searc in order to find te products and services tat matc er tastes. In contrast, firms know te industry well and witout muc effort can gater information on products and services provided by teir competitors. Tey obtain useful information simply by engaging in teir everyday business and participating in business groups. Troug conversations wit teir customers, firms can also ascertain te customers preferences and inform tem about were tey can find te product or service tey are looking for. Suc referrals can reduce consumer searc costs tremendously. However, a referring firm faces an incentive problem in coosing between serving a consumer itself and referring te consumer to anoter seller. Under wat circumstances would firms inform consumers about products offered by oter firms? On te one and, directing consumers to oter firms tat matc consumers tastes would intuitively result in a loss of business. On te oter and, a service provider may feel tat a potential client s job does not matc is expertise. Altoug te temptation to keep customers by lowering te price exists, te provider would be wise to decline some of te requests for service based on te initial review. Clients may naturally ask for a referral to te professional wo is rigt for te job, and te service provider may ten refer tem to one of teir business contacts. Altoug referral fees would elevate te incentive problem by encouraging firms to refer, referrals can be made even witout suc fees. eferrals can be provided to customers purely out of goodwill. 1 It may also be a way of building referral relationsips wit business contacts. Te reciprocity in referral relationsips can be furter supported by professional 1 In retail markets, informal referrals occur wen a consumer searcing for a suitable product (a digital camera, a piece of furniture, or a collectable item) does not find it in a store, asks for referral, and is referred to a store tat sells it. One may argue tat te store s sales clerk cooses to provide a valuable referral to te consumer purely out of goodwill. However, if being elpful to customers in tat way is a norm for sellers, ten we would like to know te impact of suc a norm on te market outcome. 2
groups and associations tat promote networking between members of a profession for te purpose of excanging customer leads. eferral clubs aim specifically to provide a structured and positive environment for businesses to pass referrals to one anoter (e.g. Gold Star eferral Clubs and eferral Key). Clearly, te existence of referral networks can improve matcing outcomes and save consumer searc costs, especially if a business group is large and eterogeneous, and its members abide by an onor code regulating referral activity. 2 In tis paper, we focus on te effects of referrals on equilibrium prices, profits, and consumer welfare. In particular, we ask te following questions. Wic referral policies can exist in equilibrium? Are referral practices beneficial for buyers? Are tey beneficial for sellers? Do referrals increase or decrease equilibrium market prices? To analyze te role of referrals in searc markets we introduce referrals into a model of competition between firms producing a orizontally differentiated product. Te product space is a unit circle, wit firms evenly and consumers uniformly distributed over it. Consumer utility from a product decreases as a mismatc increases between a firm s product and te consumer s ideal product. A consumer can engage in sequential searc by randomly sampling products at a constant marginal searc cost. Upon a visit to a firm, te consumer learns te features of te product offered by te firm. Wolinsky (1983) analyzed symmetric oligopolistic price equilibrium in suc a model. In tis paper, we allow firms to refer consumers wo visit tem to oter sellers. A firm does not ave to provide consumers wit referrals if te firm would rater sell tem its own product. However, wen a firm knows tat a consumer would not purcase its product, it does not mind referring er to anoter firm. In tis paper, we assume tat wen indifferent, a firm always refers a consumer to te best-matcing product. In te bencmark model we derive and compare equilibria wit and witout referrals in te case of no referral fees, no price discrimination, and no misrepresentation of location by consumers; later, we relax tese assumptions. 2 Professional organizations recognize tat referral fees could potentially bias referrals and undermine te trust people ave in referrals made by professionals. To avoid conflicts of interest, professional associations in law, accounting, and real estate ave establised codes of onor tat regulate referral activity. For instance, te Code of Professional esponsibility of te American Bar Association prescribes te division of fees by lawyers in proportion to te actual services performed or responsibility assumed. In particular, te payment of referral fees is not allowed, except tat lawyers may pay te reasonable cost of advertising. 3
Te results are somewat surprising. Te norm of referring consumers to te bestmatcing competitors tends to increase prices and is generally preferred by sellers. Tis result can be understood as follows. Wen a firm refers consumers to te rival firm offering tem te igest utility, te referring firm is actually weakening te incentives of te rival firm to cut prices. eferrals make te rival s demand more inelastic and tus tey function as a pro-collusive device. Tis suggests tat altoug referrals provide consumers wit valuable information tat decreases searc costs and improves te product matc, consumers may be worse off in te presence of referrals. 3 Tis appens wen te benefits from referrals (i.e. lower searc costs and better product matc) are outweiged by te loss to consumers from te equilibrium price increase. We sow tat for sufficiently low searc costs relative to degree of product eterogeneity, consumers prefer markets witout referrals. At te same time, referrals lead to a Pareto improvement in markets wit relatively ig searc costs (Proposition 4). We ten ask ow te analysis canges if consumers can misrepresent teir locations and/or firms can adopt discriminatory prices. We find tat referral equilibria can arise even if we allow firms to price-discriminate among consumers based on location and allow consumers to misrepresent teir location. In tis case, we sow tat firms would contract teir referral regions and give imperfect referrals to consumers wo misrepresent teir location. Importantly, consumer misrepresentation counteracts firms efforts to make additional profits troug price discrimination (Proposition 6). Tat is, even toug firms are allowed to pricediscriminate, tey would not be able to do so because consumers would always buy at te lowest price offered by a firm to any customer. An introduction of positive referral fees and commissions does not cange equilibrium referral beavior; it only increases equilibrium price and reduces consumer market participation. Given differences in opinion about referral policies and teir prominence in many industries, it is peraps surprising tat tere is not muc economic teory on te topic. One exception is a study by Garicano and Santos (2004), wic examines referrals between vertically differentiated firms (vertical referrals). Due to complementarity between te values of opportunities and firms skills, efficient matcing involves assigning more valuable oppor- 3 Suc a negative externality on consumers from improved information was first demonstrated by Anderson and enault (2000) in an economy wit orizontally differentiated searc goods. 4
tunities to ig-quality firms. Te autors sow tat flat referral fees can support efficient referrals from ig-quality to low-quality firms but not vice versa. Te moral azard problem arises because a low-quality firm as an incentive to keep te best opportunities to itself rater tan refer tem to a ig-quality firm. 4 However, te model does not allow te firms clients to conduct searces on teir own. In contrast, tis paper introduces referrals into a sequential consumer searc model of Wolinsky (1983, 1984, 1986), in wic consumers can always randomly visit a firm and at a cost learn about te firm s product and price. Firms are orizontally differentiated and terefore referrals are orizontal rater tan vertical. Te rest of te paper is organized as follows. Section 2 outlines te fundamental features of our bencmark model. Section 3 analyzes te model of price competition between firms selling a orizontally differentiated product in searc markets wit and witout referrals, and compares te referral and random searc equilibria tat arise in suc markets. In Section 4, we examine ow te results in Section 3 cange if we relax te assumptions of no consumer misrepresentation, no price discrimination, and no referral fees. Section 5 offers concluding comments. Te Appendix contains all te proofs. 2 Consumer Searc and eferrals We model competition between firms tat produce a orizontally differentiated product following Wolinsky (1983, 1984, 1986). We assume tat tere is a very large number of firms, n, located symmetrically on a circle of unit circumference and producing te product of te location at a zero marginal cost. Tere is unit mass of consumers per firm, and eac consumer as a unit demand and is caracterized by a valuation of te product (willingness-to-pay) and preferences over te orizontally differentiated products. Consumers ideal positions are uniformly distributed over te unit circle. Independently of spatial preferences, eac consumer as a value v for er ideal product (te product tat is a perfect matc wit er tastes). We assume tat v is uniformly distributed over an interval [0, 1], and tus te equilibrium consumer participation is endogenously determined in te model. consumer s ideal position and valuation are independently distributed. Eac 4 For an early study on te effects of fee splitting on referrals in ealt care see Pauly (1979). An empirical study by Spurr (1990) on referral practices among lawyers examines te proportion of cases referred between lawyers, as dependent on te value and nature of a claim, advertising activity, and oter factors. 5
Firmsknowtepositions ofalloterfirms in te industry, wereas consumers do not know wic firms offer wic products. Wen a consumer visits a firm, se learns of te firm s position and its price, wile te firm learns of te consumer s ideal product (tastes). Te firm remains uncertain about te consumer s willingness-to-pay even after er visit. Tis assumption captures te idea tat firms may find it easier to extract information about a consumer s ideal product tan about te consumer s willingness-to-pay for te ideal product. Consider a consumer wose product valuation is v and wo learns tat te firm s product is located at distance x from er ideal position. Let te consumer s utility for te firm s product (gross of price and searc costs) be u(x, v) =v x, were>0is te taste parameter tat denotes te degree of product eterogeneity and x represents te disutility a consumer receives from consuming a product located at a distance x in te product space from te consumer s ideal product. For example, a ig corresponds to markets were consumers feel strongly about product caracteristics, e.g. color, weigt, or design. Te case of =0corresponds to a omogeneous product market. Consumer searc is sequential, wit a constant marginal cost of searc c>0, common across consumers. Eac consumer cooses weter to initiate searc, and a consumer wo decides to searc can always coose not to buy te product of te firm sampled and continue to searc for anoter product. Consumer searc is random, witout replacement and wit perfect and costless recall. Tat is, at eac step of te sequential searc process, a consumer samples firms products randomly, but se does not sample te same product more tan once, and se can always recall previously sampled products. We assume tat wen indifferent between searcing or not, consumers do not searc. A firm s referral policy can be conditional on consumer location. At te same time, firms are assumed to set nondiscriminatory prices in te bencmark model wit or witout referrals. Wen a firm refers a customer to anoter firm, te customer can costlessly follow te referral. Tis is consistent wit te idea tat searc costs come mainly in te form of learning about product caracteristics, and terefore referrals substantially lower consumers searc costs. Te assumption is not crucial, toug. We also assume tat wenever a firm knows tat it cannot sell its own product to a consumer given its own price and oter firms prices, and tus is indifferent weter to provide a referral or not and were to refer 6
te consumer, it refers te consumer to te best-matcing firm. Tis assumption of onest referrals resolves firms indifference and is in te spirit of te onor codes adopted in markets for professional services. Note tat onest referrals are in fact incentive compatible in te sense tat firms do not gain by deviating unilaterally from te equilibrium in wic onest referrals are supported. 5 3 Te Main Analysis We start in Section 3.1 wit te analysis of price competition in a random searc market witout referrals. We assume tat firms set nondiscriminatory prices, and we ten derive te equilibrium price p tat caracterizes te unique symmetric (pure-strategy) equilibrium wit consumer participation, wic we call te random searc equilibrium. Ten, in Section 3.2 we study te searc model wit referrals and no referral fees. In te model, firms simultaneously set uniform prices and coose referral policies. Our focus is again on te symmetric pure-strategy equilibria in wic all firms use te same price and referral policy. We will sow tat since a firm refers a consumer if and only if te consumer would oterwise leave te firm to engage in random searc, firm i s referral rule is caracterized by te same critical distance as te consumers stopping rule. Hence, a symmetric referral equilibrium is described by a common price p and a referral rule x, suc tat if te distance between a consumer s ideal product and a firm is greater tan x,tefirm refers te consumer to te best-matcing firm. We derive te unique symmetric equilibrium wit consumer participation, (p, x ), wic we call te referral equilibrium. 3.1 andom Searc In tis section, we assume tat firms do not make referrals, and so te only way for consumers to receive information about a product s location and price is to engage in sequential random searc. We will support te equilibrium wit a symmetric price p by following Wolinsky s (1983, 1986) tecniques. To find te equilibrium in te random searc model, we assume 5 Suppose a firm were to deviate and refer a departing consumer to a poorly matcing firm. Te consumer would follow te referral because se expects it to be valuable in te equilibrium. Ten, se would eiter purcase te referred product, or if se learns tat te product is of poor matc, se would engage in additional random searc. Tus, te consumer would not purcase te disonest firm s product anyway. 7
tat all firms except firm i carge price p and tat consumers rationally expect te common price level, and we ten derive te best-response price for firm i inamarketwitanarbitrary large number of firms. 6 By setting te best-response price equal to p, weobtainasymmetric equilibrium in te monopolistically competitive market. To allow for te possibility of an equilibrium wit consumer participation, we assume c, and later we will sow tat tis condition is necessary for te existence of a symmetric 4 equilibrium wit consumer participation (Proposition 1). 7 Wen consumers participate in te market, firm i s profit depends on te optimal stopping rule for a consumer engaged in random sequential searc. Lemma 1 derives te optimal stopping rule. Lemma 1. Assume tat c 4 and te number of firms n is very large. Wen all firms except firm i set a common price p and firm i sets a price p i, ten te optimal stopping rule foraconsumerengagedinrandomsearcistostopsearcingatfirm i if and only if firm i s product is witin distance µ p x(p i,p)=x pi + (1) from er ideal product, were x = p c is te critical distance in te symmetric equilibrium. In a symmetric equilibrium, te probability tat a consumer stops er searc at firm i is 2x =2 p c 1 because te optimal stopping rule tells er to stop searcing once se samples a product witin distance x on bot sides of er ideal product on te circumference. For example, if firm i were to set a different price, p i 6= p, itcouldtenaffect consumers searc beavior somewat. If firm i were to surprise consumers by carging a iger price, firm i would prompt more consumers to continue random searces. Note tat in a symmetric random searc equilibrium, it is strictly dominated for firm i to set a price p i suc tat x(p i,p) > 1 2 or x(p i,p) < 0. Incasex(p i,p) > 1 2,wicoccurswenp i 6 We could ave taken te finiteness of n seriouslybyanalyzingann-firm searc market, but tis would greatly complicate te analysis because te distribution of unsampled products canges as searc proceeds. Following Wolinsky (1983, 1984), we assume away tis possibility by letting n be very large (see footnote 2 on page 276 in Wolinsky (1983) for a justification of te approac). For very large n we can also ignore te possibility of a consumer not being able to find a firm located witin te critical distance x. 7 Note tat tere is always a symmetric equilibrium witout consumer participation in te market. In suc an equilibrium, all firms carge a proibitively ig price, and consumers do not searc correctly expecting te market price to be very ig. Tis is an equilibrium because consumers only receive price information by engaging in searc, and if a firm were to deviate by reducing its price it would not be able to convey te price information to consumers. Hence, no deviation is profitable. 8
is below p 1 x, all consumers wo visit firm i buy tere since i s offer is better tan 2 engaging in furter random searc, and a firm could increase its price sligtly witout losing any customers. In case x(p i,p) < 0, wic occurs wen p i is above p + x,noconsumer would purcase firm i s product, but firm i could sligtly reduce its price to generate positive sales and profits. Tus, we can restrict our attention to p i p ( 1 2 x ),p+ x. To derive firm i demandwenitsetsapricep i and all oter firms carge p, we need to first find te probability tat a consumer samples firm i on a kt random draw, stops searc to buy firm i s product, and ten sum over all k, 1 k n. Te probability of drawing firm i as te kt firm is 1 for a consumer, since all firms are symmetric and at n eac step in te sequential searc, te next store is cosen randomly wit equal probability and witout correlation. 8 Te probability tat a consumer searces at least k times is (1 2x ) k 1 (wic, in accordance wit te optimal stopping rule, is te probability tat all previous k 1 draws were farter tan distance x = p c on bot sides from er ideal position). If firm i carges p i, te probability tat a consumer purcases at firm i upon te visit is 2x(p i,p). As a result, wit a very large n, firm i s demand per searcer is ed i n (p i,p) = 1 nx (1 2x ) k 1 2x(p i,p) (2) n k=1 = 1 µ 1 (1 2x ) n 2x(p n 1 (1 2x i,p). ) To finis te derivation of firm i s demand function, we need to find te measure of searcers. Let v denote te critical valuation for market participation, wic is te valuation of te consumer indifferent between initiating searc and staying outside te market. Only consumers wit valuations v [v, 1] enter te market. Since tere is unit mass of consumers per firm, te total measure of searcers is n(1 v ),andfirm i s demand function is µ 1 (1 2x Di n (p i,p)=(1 v ) n ) 2x(p 1 (1 2x i,p). (3) ) For zero marginal costs, te profit function of firm i is ten π n i (p i,p)=p i D n i (p i,p) in te 8 Wit n firms, firm i is selected first wit probability 1/n. It is selected on a second draw if it is not selected first (probability 1 1/n), and if it is selected as te second firm among te remaining n 1 firms (probability 1/(n 1)). Tus, te probability of firm i being selected on te second draw is also 1/n. Similarly, we can sow tat firm i is selected on te kt draw wit probability 1/n (see Wolinsky, 1986). 9
n-firm market. In te limit as n, firm i s profit function becomes π i (p i,p) = lim n π n i (p i,p)=(1 v ) µ p i 1+ p p i. (4) c A firm s price does not affect consumers market participation decisions and tus does not affect te number of consumers wo visit te firm. Tis is because consumers decisions to engage in searc are based on price expectations, not on te actual prices set by firms. Terefore, te measure of searcers does not depend on p i,andfirm i cooses p i to maximize te expression in te square brackets in (4). Te properties of te unique random searc equilibrium and comparative statics results are stated in Proposition 1. Proposition 1. Consider te limit case of n. Tere exists a unique symmetric random searc equilibrium if and only if c min 1, 4 4ª.Teequilibriumpriceisp = c, critical valuation for market participation is v =2 c, andprofits are π =(1 v )p. Te equilibrium price increases and consumer market participation decreases in searc cost, c, and product eterogeneity,. Profits can increase or decrease in c and. Condition c is necessary for sustaining sequential consumer searc in a symmetric 4 equilibrium. 9 Te oter condition c 1 is needed to guarantee tat te equilibrium price is 4 not proibitively ig. If tis condition is violated, ten v =2 c > 1, andallconsumers expect a negative payoff andstayawayfromtemarket. Te comparative statics analysis of Proposition 1 sows tat, as expected, te equilibrium price in te random searc equilibrium increases wit searc cost and product eterogeneity. Interestingly, te equilibrium profits may increase or decrease wit c and depending on teir levels. Te equilibrium profits increase in c and for sufficiently low levels of te parameters, c < 1. For oter values of c and, te decline in market participation is 16 not compensated for by te iger price associated wit larger c and. In oter words, for 9 Wen c> 4, tere is no symmetric pure-strategy equilibrium wit consumer participation in te random searc model. Wen c> 4, te optimal stopping rule in a symmetric equilibrium is for a consumer to always stop searcing regardless of te product found since for suc parameter values x = p c > 1 2. However, witout sequential searc, firms ave an incentive to raise prices above te expected price level, and terefore a symmetric equilibrium wit consumer participation does not exist wen c> 4.Tistypeof non-participation equilibrium is described in Stiglitz (1979) and is also reminiscent of te Diamond paradox (Diamond, 1971). However, suc an equilibrium is rater uninteresting, and we will restrict parameter values to focus on equilibria wit consumer participation. 10
1 c 1, sellers prefer to operate in searc markets wit lower searc costs and lower 16 4 product eterogeneity. Intuitively, if searc cost c goes up, ten te profits may go down despite a iger equilibrium price p because te price increase reduces equilibrium consumer participation. Janssen et al. (2005) and Konisi (2005) similarly find tat te equilibrium profits are influenced by tese two effects. [Figure 1 ere] Te equilibrium consumer decisions to engage in searc, buy at a firm located at a distance x, or engage in sequential searc are illustrated in Figure 1. Only consumers wose product valuations are at least v =2 c engage in searc. Consumers visiting a firm located witin distance x = p c from teir ideal positions buy te product, wile oters continue to searc. Figure 1 also depicts te expected utility of a consumer wo first draws a product located at distance x from te consumer. 3.2 Searc wit eferrals In tis section, we allow firms to provide referrals to consumers. We assume tat tere are no referral fees and tat a firm (salesperson) can observe a customer s ideal position x from aving a conversation wit er, but cannot observe er willingness-to-pay v. Terefore, te referral tat a firm offers its customer can only be conditioned on te observed consumer s ideal product. We continue to assume in te bencmark model tat firms carge nondiscriminatory prices, but we will relax tis assumption in Section 4. In te model, firms simultaneously set nondiscriminatory prices and referral policies, wile consumers old rational expectations and select te optimal stopping rule for teir sequential searc wit costless and perfect recall of products and referrals. In a searc market wit referrals, a consumer cooses weter to start a random searc given er product valuation v; after eac draw te consumer learns te firm s product and decides weter to follow te firm s referral if it is given, continue random searc, purcase te best examined item, or leave te market witout purcasing any product. Our focus is again on symmetric 11
pure-strategy equilibria (i.e., equilibria in wic firms coose te same price and referral policy). As discussed in Section 2, we assume tat firms referrals are onest in te sense tat wenever a firm knows tat it cannot sell its own product to a consumer at te given prices, it refers er to te best-matcing firm. A firm would ten refer a consumer if and only if te consumer s ideal position is sufficiently far away, so tat te consumer prefers engaging in furter random searc to buying at te firm. Te optimal stopping rule in sequential searc wit referrals is for a consumer visiting firm i to stop searcing at firm i if and only if firm i s product is witin distance x (p i,p) from er ideal product. Since a firm refers a consumer if and only if te consumer would oterwise leave te firm to engage in additional random searc, firm i s referral rule is caracterized by te same critical distance as te consumer stopping rule. Te symmetric equilibrium referral rule states tat if te distance between a customer s position and a firm is more tan x = x (p, p), tefirm refers te consumer to er best-matcing firm. Hence, a symmetric referral equilibrium can be caracterized by a pair (p, x ). To find te equilibrium (p, x ), consider firm i coosing an arbitrary price p i,and suppose tat all oter firms set a common price p and referral rule x.teoptimalreferral tresold x (p i,p) for firm i is based on te location of a consumer wo is indifferent between buying from firm i at price p i andcontinuingtosearcatrandomamongfirms carging p. Lemma 2 derives te equilibrium referral rule x and optimal stopping rule x (p i,p) for a consumer engaged in searc wit referrals. Lemma 2. Assume tat c and te number of firms n is very large. Suppose firm i 4 sets a price p i and oter firms set a common price p and use referral rule x. Te optimal stopping rule for a consumer engaged in searc wit referrals is toµ stop searcing at firm i p if and only if firm i s product is witin distance x (p i,p)=x pi + from er ideal product, were x = 1 1 2 2p 1 4(c/) is te equilibrium referral rule. [Figure 2 ere] 12
Figure 2 illustrates te optimal consumer searc and firm i s referral policy, assuming tat firm i sets a price p i and oter firms set a common price p and use referral rule x.a consumer randomly cooses te first firm to visit, say, firm i, located at distance x from er ideal position. Wen te distance between te consumer and firm i is greater tan x (p i,p), as in te case of consumer B, te consumer is referred by firm i to te best-matcing firm (firm j), and se as to decide weter to buy at firm i, follow te referral, continue random searc, or leave te market. Wen firm i is located witin distance x (p i,p) from te consumer s ideal position, as in te case of consumer A, te consumer does not receive a referral and as to eiter buy at firm i, continue random searc, or leave te market. In te symmetric equilibrium, a consumer follows te referral and buys at firm j wen firm i is located farter away tan x = x (p, p), and se buys at firm i immediately oterwise. In case c>, te optimal stopping rule in a symmetric equilibrium is for a consumer 4 to always stop searcing regardless of te product found. Tis is te same as in te case of no referrals. Tus, consumers searc at most one firm, and since referrals are given only to consumers wo would oterwise engage in additional random searc, consumers do not receive referrals. Hence, just as in te model witout referrals, a symmetric equilibrium wit consumer participation does not exist in searc markets wit referrals wen c>/4. Next, we calculate te demand function of firm i assuming tat oter firmscoosea symmetric price p and give referrals to customers located farter tan x. We can restrict our attention to p i p + x 1 2, p + x because it is strictly dominated for firm i to price outside tis interval. In case p i <p+x 1 2,weavex (p i,p) > 1 2,andallconsumers wo visit firm i buy tere since i s offer is better tan engaging in furter random searc or following te referral. But ten firm i would rater set a sligtly iger price. In case p i >p+ x,weavex (p i,p) < 0, and no consumer visiting firm i purcases its product. Moreover, v p i <v p x olds for any v, and even consumers wo are referred to firm i do not purcase from it. Tus, demand for firm i in tis case is zero, and firm i would benefit byreducingitspricetogeneratepositivedemandandprofits. For p i p + x 1, p + 2 x, consumers located at x x (p i,p) [0, 1 ] buy from 2 firm i. Assuming firm i follows te optimal referral policy x (p i,p), itsdemand per searcer 13
is µ µ 1 n 1 ed i(p n i,p) = 2x (p i,p)+ n n = 1 µ 1+ 2(p p i) n 1 n 1 (1 2x ), (5) were x = 1 2 1 2p 1 4(c/) and te number of firms n is very large. Te first term represents demand from consumers wo visit firm i first, wile te second term represents demand from consumers wo visit oter firms first and are ten referred to firm i. Let v denote te critical valuation for market participation, wic is te valuation of te consumer wo is indifferent between initiating sequential searc and staying outside te market in te referral equilibrium. Since tere is unit mass of consumers per firm, te total measure of participating consumers is n(1 v ), andfirm i s demand function is µ Di(p n i,p)=(1 v) 1+ 2(p p i). (6) Hence, wen firm i sets a price p i and all firms are expected to set price p and use referral rule x, firm i s profit function in te limit as n is µ π i (p i,p)= limp i D n n i(p i,p)=(1 v) p i 1+ 2(p p i). (7) As before, a firm s price does not affect consumers market participation decisions and, terefore, (1 v ) does not depend on p i. Given tis, firm i cooses p i to maximize te expression in te square brackets in (7). In Proposition 2 we describe te unique pure-strategy symmetric equilibrium wit consumer participation, wic we call te referral equilibrium. Proposition 2. Consider te limit case of n. Tere exists a unique symmetric referral equilibrium if and only if c and eiter 1 or c 2 3/4 1/. Te 4 equilibrium price is p = 2,referralruleisx = 1 1 2 2p 1 4(c/), critical valuation for market participation is v = p + x,andprofits are π = p (1 v ). eferral intensity decreases wit consumer searc cost and increases wit product eterogeneity. Te equilibrium price is perfectly insensitive to searc cost, wereas it increases as product eterogeneity increases. Consumers market participation and profits decrease wit searc cost, c, and tey can increase or decrease wit product eterogeneity,. 14
According to Proposition 2, te symmetric referral equilibrium exists if and only if c 4 and eiter 1 or c 2 3/4 1/. Te inequality c 4 is necessary for sustaining sequential consumer searc in a symmetric equilibrium. Te second condition obtains from 1 2p 1 4(c/), wic is equivalent to v 1, werev = p + x = 2p 1 4(c/) is te critical valuation for market participation. Figure 3 illustrates te equilibrium consumer decisions and te expected consumer utility for different realizations of v and x. Only tose consumers wose product valuations are at least v engage in searc. Consumers visiting a firmlocatedwitindistancex buy te product, wile oters follow te firm s referral and buy from te referred seller. [Figure 3 ere] Wy does searc cost c not matter in te determination of price p in tis case? By lowering its price, firm i can increase its sales only troug an increase in te retention rate 2x (p i,p)=2x 2(p i p)/ of consumers wo visit firm i first. However, a cange in te retention rate, wic equals 2/, isnotaffectedbysearccostc since sequential searc never takes place in te equilibrium. Tus, in searc markets wit referrals te equilibrium price is determined only by eterogeneity parameter. We can see an analogy wit te Diamond paradox (Diamond 1971): in bot cases, sequential searc does not occur, and te equilibrium price is independent of te level of searc cost (as long as it is positive). In te current model, owever, tere is still competition among firms trying to retain initial customers, and te monopoly price does not prevail as te equilibrium price. 3.3 Comparison of te andom Searc and eferral Equilibria So far, we ave described te random searc and referral equilibria. Proposition 3 below compares te regions of teir existence, wile Proposition 4 compares te properties of te equilibria, and sows tat unless searc cost c is very low relative to eterogeneity parameter, te referral equilibrium Pareto dominates te random searc equilibrium. Proposition 3. If 1, ten bot te random searc and referral equilibria exist if and only if c 4. However, if 1 5 3 ten te referral equilibrium exists for a larger range 15
of searc cost c, wereas if 5, ten te random searc equilibrium exists for a larger 3 range of searc cost c tan te referral equilibrium (in particular, te referral equilibrium cannot exist for >2). Figure 4 illustrates Proposition 3 by sowing te regions of parameter values of and c for wic te random searc and referral equilibria exist. Te random searc equilibrium exists for any level of product eterogeneity, provided searc cost is sufficiently low. contrast, a referral equilibrium may exist only for a sufficiently low eterogeneity parameter, 2. As product eterogeneity approaces te critical level of 2, te market collapses because te equilibrium price p = approaces unity and tus becomes proibitively ig 2 for consumers, regardless of te magnitude of searc cost. In [Figure 4 ere] Intuitively, wen te product is igly eterogeneous, firms ave greater market power to set ig prices in te random searc and referral equilibria, wic discourages consumers market participation. Even for a very ig, a random searc market would not close down for a sufficiently low searc cost, since te low searc cost imposes competitive pressure on firms in te market. In te referral equilibrium, prices are not affected by searc costs, and te searc market wit referrals does not open for >2. We next compare te random searc and referral equilibria in te region of parameter values for wic bot of tem exist. Consumer surplus is directly related to te value of te marginal consumer te consumer wo is indifferent between entering and staying out of te market. 10 We use te critical values for market participation v and v to sow tat v v, and terefore all consumer types are better off and market participation is larger in te referral equilibrium, as long as searc cost is not very low relative to product eterogeneity. We furter sow tat for relatively ig searc cost, te referral equilibrium 10 To see tis, suppose a critical valuation for market participation in a searc market is bv. Te expected utility of a consumer wo values te product at v bv is EU(v) =v bv, and consumer welfare is 1 v EU(v)dv = 1 2 (1 bv)2. Terefore, bv fully determines te expected utility of any consumer and te total consumer welfare. Higer consumer market participation is equivalent to iger consumer benefits. 16
Pareto-dominates te random searc equilibrium. Tat is, te ex ante expected payoffs of eac consumer and eac firmintereferralequilibriumareigerforsomeagentsandat least as ig for oter agents tan in te random searc equilibrium. Proposition 4. Wenever bot random searc and referral equilibria exist, we ave p p and x x.allconsumersarebetteroff in te referral equilibrium if and only if c 0.09, i.e. searc cost is not very low relative to product eterogeneity. Tus, c 0.09 guarantees tat firms earn more profits in te referral equilibrium, and te referral equilibrium Paretodominates te random searc equilibrium. Tese results are not surprising. If searc cost c is very low, consumers surely prefer te random searc equilibrium, since te random searc equilibrium price p is low wile te referral equilibrium price p is insensitive to c. If searc cost is not very low, te referral equilibrium is favored by consumers; and since te equilibrium price is iger in te referral equilibrium tan in te random searc equilibrium, firms favor tis equilibrium as well. Te referral equilibrium Pareto-dominates te random searc equilibrium wen te gain to consumers due to iger matcing quality and lower searc costs is iger tan te loss due to increased price. From te welfare perspective, referrals andle te information problem of matcing buyers and sellers but lead to iger prices in searc markets. Te reason te equilibrium prices are iger under referrals is tat in te referral equilibrium, a fraction 1 2x = p 1 4(c/) of consumers are referred to teir best-matcing products. Not surprisingly, te firm faces a more inelastic demand. 11 Te fact tat prices are iger in te referral equilibrium tan in te random searc equilibrium indicates tat firms benefit more tan consumers from te presence of referrals. Tat is, if consumer welfare is iger under referrals, ten firms find tem beneficial as well, but te reverse is not true. It could be te case tat firms find te referral equilibrium more profitable, wile te referrals make consumers worse off. 12 Since firms usually enjoy iger 11 ecall tat firm i s demand is proportional to 1+2 p p ³ i and 1+ p p i c in searc markets wit and witout referrals, respectively. Hence, for c 4, firm i s demand is more inelastic in a searc market wit referrals. 12 Altoug firms usually would benefit from establising onest referrals, tere are some parameter values for wic profits are lower in te referral equilibrium tan in te random searc equilibrium. For tese parameter values, it must be tat consumer market participation is lower (and consumers are worse off) 17
profits in te referral equilibrium, it sould be selected over te random searc equilibrium by firms and professional associations wenever bot equilibria exist. We next study ow social welfare depends on te parameters of te searc model. We define social welfare as a sum of consumer welfare and profits. According to footnote 10, consumer welfare is 1 EU(v)dv = 1 (v bv) dv = 1 (1 v v 2 bv)2, wic is only a function of te critical valuation for market participation bv. Te social welfare in te searc model wit and witout referrals is ten a function of price p and bv: W (p, bv) =(1 bv) p + 1 2 (1 bv)2. (8) Proposition 5 summarizes te comparative statics results for social welfare in te random searc and te referral equilibria. Proposition 5. Social welfare in te random searc and referral equilibria is iger wen searc cost is lower. Te effect of product eterogeneity on social welfare is negative in te random searc equilibrium, and it is ambiguous in te referral equilibrium. Te comparative statics results for te random searc equilibrium indicate tat social welfare is iger in markets wit low searc costs and low product eterogeneity. Similarly, for te referral equilibrium, social welfare is iger in markets wit low searc costs, but te effect of product eterogeneity on social welfare is ambiguous. Te results are quite intuitive. An increase in searc cost does not affect te equilibrium price, but it reduces referral intensity and market participation. Terefore, iger searc costs are detrimental from te point of view of bot te consumer and social welfare. In contrast, iger product eterogeneity stimulates referral activity and can improve social welfare despite a price increase. 4 Extensions In tis section, we consider modifications of te bencmark model. We start in Section 4.1 by examining te case were consumers can misrepresent teir location. Ten, in Section 4.2 we allow firms to price-discriminate based on consumer location. Finally, in Section 4.3 we briefly state wat appens wen referral fees or commissions are introduced. because te referral equilibrium price is iger. Terefore, te referral equilibrium can be Pareto dominated by te random searc equilibrium. 18
4.1 Misrepresentation by Consumers In te main analysis, we assumed tat firms can correctly identify consumers ideal locations. However, a consumer wose true location is outside a firm s referral region but close to te referral border x (say, x, were > 0 is small) as an incentive to report er position to be x +ε so tat se can receive a referral, were ε>0is arbitrarily small. Altoug suc manipulation would result in an imperfect referral, it is beneficial as long as x >. To capture te possibility of consumer manipulation, we will assume tat (1) as before, firms know oter firms positions, but tey cannot observe te location of a consumer wo visits tem, and tat (2) altoug consumers cannot tell firms positions prior to teir visits, tey can immediately identify a firm s position upon a visit to te firm. Maintaining te assumption of nondiscriminatory pricing, we obtain results similar to tose of te bencmark model. Figure 5 illustrates te equilibrium referrals obtained by consumers depending on teir distance from a sampled firm (firm i). In te symmetric equilibrium firms provide referrals for anybody outside of [ x,x ]. Consumers located farter tan x from firm i trutfully report teir location and obtain perfect referrals. Consumers located in te region [ x, x ) ( x 2 2,x ] ave an incentive to misrepresent teir position in order to receive an imperfect referral. Tese consumers report teir location to be x + ε (were ε>0 is arbitrary small) in order to receive a referral to a firm located at x + ε, wic is less tan x 2 to consumers in region [ x, x 2 2 ]. from teir ideal position. Firm i can sell its product only [Figures 5 and 6 ere] Figure 6 illustrates consumer decisions in te referral equilibrium wit consumer misrepresentation and te equilibrium utility enjoyed by consumers, as dependent on te distance from te first firm tey visit. Lemma 2 can be modified by replacing te equilibrium referral rule x = 1 2 1 2p 1 4c/ wit x = 1 1 2 2p 1 8c/. We refer to tis statement as Lemma 3 and use it to prove Proposition 6. 19
Lemma 3. Assume tat c and te number of firms n is very large. Suppose firm i sets 8 apricep i and oter firms set a price p and use referral rule x. Te optimal stopping rule for a consumer engaged in searc wit referrals is toµ stop searcing at firm i if and only if p firm i s product is witin distance x B (p i,p)= 1 pi 2 x + from er ideal product, were x = 1 1 2 2p 1 8(c/) is te symmetric equilibrium referral rule. Note tat unlike in Lemma 2, consumers purcase a firm s product in te equilibrium only if tey are located witin distance x B (p, p) = 1 2 x from te firm. Altoug tis appears to be a minor modification, te demand curve is kinked in tis case. Tis is coming from te imperfect referral demand. On te one and, if p i > p,tensomeimperfect referral consumers may leave firm i to engage in furter searc. On te oter and, if p i <p, ten all imperfect referral consumers simply purcase te firm s product. Since te measure of consumers arriving at a firm depends only on expected rater tan actual price, tis means tat price cutting does not generate additional demand from consumers arriving troug imperfect referrals. Te kinked demand function generates a continuum of symmetric equilibrium prices, as described in Proposition 6. Proposition 6. Suppose tat consumers can misrepresent teir location. Consider te limit ³ 1 3 4 case of n. Wen c, and eiter 4 or c 8 3 8 a continuum of symmetric referral equilibria wit prices p p 1 8(c/), critical valuation for market participation v 1 2 1 2 π 2,tereexists [, ], referral rule x 4 2 = = p + 1 2 x,andprofits = p (1 v ). eferral intensity is lower in te case of consumer misrepresentation, and it decreases wit consumer searc costs and increases wit product eterogeneity. Proposition 6 sows tat our results obtained in Section 3 are qualitatively robust even if we allow consumers to misrepresent teir location. Te main differences are tat te total referralactivityislower,somereferralsareimperfect,andtereisarangeofpricessupported in te referral equilibrium. Figure 7 illustrates ow te region of parameter values for wic te referral equilibrium exists would contract in te presence of manipulative consumers. [Figure 7 ere] 20
4.2 Price Discrimination by Firms Next, we consider te possibility of price discrimination by firms. In te main analysis, we assumed tat firms must carge te same (uniform) price to all consumers, even toug firms can observe a consumer s preferred location and refer te consumer to te most suitable product. Te reader may wonder weter firms would try to use teir information about consumer location to price-discriminate between consumers based on te distance, x, between te consumer s ideal position and te firm s product. We will address tis possibility by assuming tat firms can engage in suc price discrimination. Tat is, we let eac firm i coose a price function p i (x) instead of te uniform price p i and ten examine ow our results are affected by tis modification. We find tat as long as consumers can misrepresent teir location, it does not matter weter or not firms are allowed to price-discriminate between consumers based on location. Under price discrimination, consumers can misrepresent teir position in two ways: to obtain a referral and to obtain te lowest price from te referred firm. Let p be te lowest price a firm carges any consumer reporting to be located in region x x x. By misrepresenting er location, a consumer is able to always purcase at te lowest price p. Tus, te situation is exactly te same as in te uniform pricing case, and te referral equilibria (p,x ) are as in Proposition 6. Proposition7 sows tat te result is drastically different if consumers cannot misrepresent teir location. Proposition 7. If firms can price-discriminate based on consumer location wile consumers cannot misrepresent teir location, ten tere is no symmetric pure-strategy equilibrium wit consumer participation in searc markets wit or witout referrals. Te non-existence of te equilibrium is due to te presence of a positive searc cost and tefacttatanoptimalpricescedulewouldexactlycompensateconsumersfortedisutulity of mismatc. Tis omogenizes te product and leads to te well-known paradoxical result tat an introduction of a small searc cost dramatically canges te equilibrium prices in te sequential searc markets (from te marginal cost to te monopoly price level) and possibly leads to market closure (Diamond, 1971, and Stiglitz, 1979). Te reason is tat in te symmetric equilibrium, tere is always an incentive to raise prices, and terefore a typical 21
equilibrium involves a proibitive price suc as p =1. Tis is an equilibrium because as long as all firms carge te same price, consumers ave no incentive to searc more tan once, and as long as no consumer conducts sequential searc, firms would set te monopoly price. Our analysis sows tat te uniform price assumption may be important for ensuring te existence of non-trivial equilibria in searc markets witout consumer misrepresentation. 4.3 eferral Fees and Commissions Te activities of real estate agents can illustrate te role of referrals in searc markets and motivate te introduction of commissions into te model. A ome buyer contacts a real state agent in a realtor s office. Te real state agent obtains a full commission if se can sell a property andled by te realtor s office as a seller broker (usually about 6% of te ome sale price in te U.S. real estate industry). However, if te agent as no properties suitable for te buyer, te agent will refer te buyer to a property andled by an agent at anoter realtor s office to obtain a alf-commission as a buyer broker. Te agent obviously prefers to sell a property as a seller broker, but if se knows tat a buyer will not purcase any of te properties se andles, se works ard to find a property suitable for er buyer-client for a alf-commission. Witout tis system of referrals, a buyer would need to visit many realtor s offices before finding a suitable property. Tus tis referral practice would seem to reduce buyers searc costs substantially. However, it also affects consumer demand and market prices. Suppose tat, as in te real estate market, tere is a common commission level. As in te bencmark model, we assume tat firms cannot price-discriminate between consumers, and we look for a symmetric equilibrium. A firm s profit functionintiscaseisasumof profits from consumers buying at te firm on teir first visit, on teir visit by referral, and tepaymentsbyoterfirms for te referrals te firm makes. Since all firms offer te same commission, a referring firm cannot do better tan to refer a consumer to te best-matcing firm. We can sow tat tere exists a symmetric referral equilibrium wen te commission is small enoug to guarantee consumer participation. We obtain te same type of referral equilibrium wit modified symmetric equilibrium price and consumer participation. Te 22