Competitive Pressure in a Fixed Price Market for Credence Goods

Transcription

1 Cometitive Pressure in a Fixed Price Market for Credence Goods Fridolin E. Marty University of Berne August 1999 Abstract We consider a market for credence goods. There are two tyes of exerts: ersons who never cheat and oortunistic exerts who take advantage of the information asymmetry. The rejection strategy of the clients and the nonfraudulent behavior of the honest colleagues may revent the cheating exerts from always recommending a high rice service. We comare rice cometition versus xed rices. Intriguingly, the xed rice regime may erform better concerning the information roblem than rice cometition. Therefore, a fraction of honest exerts is more owerful in discilining oortunistic exerts than rice cometition. Keywords: Credence Goods, Exert, Fraud, Perfect Bayesian Equilibrium. Journal of Economic Literature Classi cation Numbers: C72, D82, I10 Deartment of Economics, University of Berne, Gesellschaftsstrasse 49, CH-3012 Berne, Switzerland; fridolin.marty@vwi.unibe.ch, homeage: htt://www-vwi.unibe.ch/theory/leb_fm.html. I would like to Winand Emons and Gabrielle Wanzenried for helful discussions and suggestions. Financial suort from the Schweizerischer Nationalfonds is gratefully acknowledged. 1

2 1 Introduction In Euroe health markets are characterized by a small degree of cometition. Esecially, markets for hysicians services lack cometitive ressure, let alone rice cometition. Prices are xed in a tari which arises from a bargaining rocess between government and lobbyists. Quality cometition is not very vital either, since the rofessions reresentatives romote an overall high quality standard which, at least in some countries, is mostly met. Nevertheless, a moderate amount of cometition for clients is resent which naturally includes some kind of quality cometition as well. 1 In this article, neither do we address the reason for this shortage, nor do we ask whether and how one should increase cometition. We are rather interested in how this lack of rice cometition in uences the market outcome and whether other forms of cometition may mitigate the information roblem. In order to achieve this goal, we model weak cometition for clients in a regulated credence goods market and comare our ndings with a similar set u in an unregulated market with rice cometition. As exected, rice cometition better revents the exerts from cheating than a xed rice setting without honest exerts. Intriguingly, by introducing a fraction of honest exerts, however, the xed rice regime is referable in terms of handling the information roblem. Accordingly, a fraction of honest exerts is more owerful in discilining oortunistic exerts than intense cometition. Honest exerts can, therefore, mitigate the information roblem even with oor cometition. We consider a two eriod game between sellers and consumers within a regulated market where rices are xed. The strategic interaction between the exert and the client lasts at most two eriods. The client wants an exert to solve her roblem not knowing about the severity of her roblem and the tye of exert she is visiting. The exert sets u a diagnosis and rooses a roblem solving strategy which can be a treatment or an advice. Given the exert s roosal, the consumer decides to accet or to reject it. An acceted roosal leads to the transaction, whereas in case of rejection, the client stays in the market for another eriod. In her second eriod, the consumer, bearing the switching costs, consults another exert. We restrict our analysis to one ossible rejection such that a consumer wants the roblem to be solved 1 In Switzerland, for instance, cometition for clients is substantially restricted, because advertising is not allowed. 2

3 after two eriods. 2 Accordingly, the client always accets the advice in her second eriod exert. We assume heterogeneity among exerts: The rst tye of exert fully acts in the interest of his clients. The second tye, however, maximizes ro ts regardless of the consumer demands. That is, he cheats whenever this is ro table for him. Honest exerts are athologically honest, since they refuse to cheat even when it is ossible to do so. In contrast, dishonest exerts otentially cheat, because they are sometimes honest. 3 The assumtion of heterogeneous exerts can be justi ed by the current debate about hysicians salaries. Although there is no unequivocal evidence of the existence of two tyes of hysicians, the minority of hysicians who just want to make money is often blamed for soiling the reutation of the whole rofession. 4 Accordingly, the bargaining artners for the tari assume that there are two tyes of exerts. Furthermore, emirical evidence reveals at least strong di erences between exerts concerning their recommendation ractice (see, e.g., Marty (1998)). We comare our setu with Wolinsky s (1995) who resents an analogous model with rice cometition. Wolinsky (1995) also considers a two eriod exert-customer game, however, he assumes endogenous rice setting. The consumers o er rices which can be turned down by the exerts. Wolinsky (1995) identi es unique and multile equilibria deending on the search costs. In case of high search costs, the exerts always reject a lower rice. Consequently, all customers are served at a rice equal to the marginal cost of the major roblem in their rst eriod. This unique equilibrium in ure strategies involves maximal fraud, since the consumers ay the exerts as if they only had major roblems. For low search costs, two tyes of equilibria exist. The revious equilibrium in ure strategies still occurs. As a distinction to the former case, there also exist two interior equilibria. Here, the exert mixes between acceting and rejecting a lower rice o er, whereas the customer always sees another exert in case the exert rejects. This tye of equilibrium is not unique, since there are two robabilities of rejecting which satisfy the equilibrium conditions. Accordingly, 2 This assumtion is not as severe as it looks like in the rst lace: the seed of convergence of the udated robability of having the minor roblem is very high (see aendix). 3 See Ja e and Rusell (1976) for an analogous assumtion in a credit market setting. 4 Citation of a hysicians reresentative in italics, e.g., Tages-Anzeiger 15/5/98. Moreover, a reresentative of the health insurances conjectures that about 10 % to 15 % are black shee (Sonntagszeitung 13/12/98). 3

4 Wolinsky (1995) obtains a trile equilibrium for low switching costs. He identi es a mark-u over cost embodied in the rices of the small service desite rice cometition. The main roblem of Wolinsky s model is the multilicity of the equilibria. Moreover, his equilibrium in ure strategies with maximal fraud is valid for the whole range of switching costs. The latter imlies that this equilibrium can be sustained for each arameter constellation. Prices are xed in our setting. The cometitive ressure arise from cometition for clients. By introducing a fraction of honest exerts, the ro t maximizing exerts can lose clients to their honest colleagues. This setu allows to overcome two weaknesses of Wolinsky s model. First, the equilibrium in ure strategies does not exist anymore for low switching costs. That is, the equilibrium in which exerts always cheat and consumers always reject cannot be sustained for low switching costs. Second, our equilibria are unique, if the fraction of honest exerts is large enough. Intriguingly, the cometitive ressure in a xed rice setu is not necessarily weaker than with rice cometition. Accordingly, the existence of honest exerts may substantially strengthen the cometitive ressure in the sense that the fraudulent exerts cheat less frequently. Recall that the equilibrium with maximal fraud no longer exists for low switching costs. This feature is due to the heterogeneity of exerts, which revents the fraudulent exerts from cheating all the time. Since the honest exerts never cheat, the fraudulent exerts cannot always lay a tacit collusion strategy of ermanent cheating. As long as changing the exert is not too exensive, it is never otimal for the clients to accet continuously, because the existence of honest exerts makes it ro table to reject occasionally. Customers hoe that they will end u with an honest exert in the second eriod. Consequently, a xed rice regime with a fraction of honest exerts may be better at solving the information roblem than ure rice cometition. In our setu, we are also able to analyze the exert-client relationshi in general and overcharging in articular. Seci cally, we identify switching costs and rices as crucial variables for studying the information roblem. Furthermore, we investigate the interaction between these variables, having regulated markets in mind. The aer is organized as follows: In section 2 we resent the model and its solution. Section 3 comares cometition for clients versus rice cometition by contrasting our 4

5 results with Wolinsky s. The last section concludes. All roofs are relegated to the aendix. 2 The Model 2.1 The xed rice setu We consider a market a for credence good. In such a market, sellers acting as exerts determine the customer needs. Accordingly, customers never know ex ost which extent of the good was needed. This holds true even if the success of the good is observable. Time is divided into discrete eriods. There is neither a beginning nor an end of time, i.e., the time goes from 1 to 1: At the beginning of each eriod a cohort of consumers enters the market. This continuum of consumers with measure 1 join the consumers left from the revious eriod. They have either a major or a minor roblem. An exogenous fraction w 2 (0; 1) of each cohort su ers from a major roblem whereas afraction(1 w) has a minor one. Customers know that they have a roblem but they do not know how serious it is. When their roblem is solved they obtain utility of B: A treatment of the major roblem also solves the minor roblem but not vice versa. The suly side of market consists of a continuum of exerts with measure 1, who diagnose and reair roblems. Exerts belong to two grous. A fraction g 2 (0; 1) of exerts (tye g) fully act in the interest of their clients, so they will never cheat on their customers. In contrast, a fraction (1 g) of exerts (tye b) behave oortunistically, i.e., they may sell a major service to customers who only su er from a minor roblem. We cature this strategic decision by x 2 [0; 1] which denotes the robability of recommending a major service to customers who actually need a minor one. Accordingly, the goal of a tye b exert is to maximize his ro ts regardless of the customer s needs. Following Wolinsky (1995), we assume that the existence of a roblem is both observable and veri able, but the tye of service rovided by the exert is not observable to customers. That is, consumers neither know which service they need nor do they observe which treatment is actually erformed. This means that ayments can be conditioned on the resolution of a roblem but not on the tye of treatment. In addition, 5

6 it imlies that an exert might be induced to misreresent a minor service as a major one. The minor roblem can be solved with a single service, namely service L. L denotes the rice of this service. Its marginal costs are normalized to zero so that L also denotes the mark-u of the minor service. In order to solve the major roblem, the exert sells a set of services and earns a mark-u of t + L > 0: Accordingly, t can be interreted as the mark-u of cheating. In the following, we call this set of services H: Naturally, exerts never cheat if t =0: The asymmetric information roblem between client and exert therefore only arises if t is strictly ositive. Due to the strong informational osition of the exerts, however, it is imossible for the government to assure that t =0: In determine the set of services H, for instance, the exert enjoys a degree of discretionary in uence to acquire additional ro ts. We do not distinguish di erent cases concerning the observability. 5 Fraudulent behavior may involve overtreatment or overcharging according as the services erformed are observable or not. Observability makes cheating less attractive since the cheating exert has to bear higher marginal costs (c H ). As a result, fraudulent behavior is less likely. The structure of equilibria are, however, the same for both kinds of consumer information (see Marty (1999a)). For the ease of resentation, we assume that cheating does not entail additional costs to the exert. In doing so, the marku of cheating equals the rice di erential of the two services. 6 Furthermore, we assume that the mark-u for the diagnosis is zero to exclude diagnosis from strategic considerations (see Emons (1997, 1999) for the interaction of diagnosis and reair). We consider a two eriod game concerning the strategic interaction between exert and client. In the rst eriod, a customer visits an exert who is either of tye g or of tye b: The customers have the ossibility to reject a service. They never reject an L-service because this is the cheaest way to obtain utility B: Since the customers do not know the tye of exert, they randomly reject a high service o er in eriod 1 with robability y 2 [0; 1]: In case of rejection, they visit another exert which leads to switching costs of k for the consumers. Although they still do not know their own 5 See Marty (1999a) for the di erent e ciency roerties of observable and nonobservable services, resectively. 6 This requires that the services are not observable or c H = c L =0. Otherwise, the consumer has to ay L + t + c H for service H; not just L + t as it is assumed in the following. 6

7 tye of roblem, they always follow the advice of an exert in the second eriod, since the game ends for the customers in the second eriod and we assume B>k+ H : The latter assures that the consumer maximizes her utility by acceting in the second eriod. The assumtion of the game ending after two eriods revents the customer from rejecting twice hoing to receive an L-advice in the third eriod. Alternatively, a two eriod game can be modeled by resuming that the switching costs k for a third oinion are large enough. 7 Contrary to Marty (1999a), we assume that the exert cannot recognize a second eriod customer, so he has to form beliefs about the clients age. The time structure of the model is summarized as follows: t =0: the consumer identi es a roblem not knowing if it is a minor or a major one, and sees an exert t =1:1 : theexertrooseseitheraminororamajortreatment. t =1:2 : the consumer decides whether to accet the advice or to see another exert. t =2:1 : in case of rejection, the second oinion exert rooses a treatment. t =2:2 : the consumer accets. The remainder of this section is organized as follows.in the rst subsection, we outline the strategy sace. The second subsection resents the equilibrium analysis. We rst comute the erfect Bayesian equilibrium in mixed strategies, followed by the erfect Bayesian equilibrium in ure strategies. The section concludes with the benchmark case of only dishonest exerts (tye b). 2.2 Strategies The exerts have the strategies e H=H; e L=H; e L=L, and e H=L in both eriods. The strategy e L=H is to be read as follows: recommend service L to a customer who actually needs the service H: L=H e is never chosen, because the roblem is both observable and veri able. Accordingly, all exerts recommend a major service to a customer who has a major roblem (strategy e H=H). In addition, an exert of tye g always chooses 7 Customers comare the switching costs with the bene t from rejecting. This bene t, however, becomes smaller for each new eriod since the robability of having a minor roblem given a major advice converges to zero as long as there are honest exerts in the market. Accordingly, restricting the analysis to two eriod resuoses that this convergence is fast enough. See the aendix for comarative statics on this convergence. 7

8 el=l by assumtion, i.e., she recommends a minor service to a customer with an L roblem in both eriods. A tye b exert, on the contrary, chooses with robability x the strategy H=L; e i.e., he tries to sell with robability x a major service to a customer with an L roblem, since the mark-u for the service H is higher than for the L-service. As mentioned earlier, customers always accet a minor service recommendation since it is the cheaest way to solve their roblem. In the rst eriod, a major service is rejected with robability y. The incentive to reject an H-service in the rst eriod is the rosect of getting an L-service recommendation in the second eriod. That is, some consumers are unlucky enough to end u with a tye b exert who lays the strategy H=L e aying for the exensive service. We consider the erfect Bayesian equilibria of this game. 8 Tye b exerts maximize their ro ts and the clients maximize utility, given all information available to them. Seci cally, the layers udate information by alying Bayes Rule and act otimally given their beliefs. Tye g exerts roose to solve the correct roblem irresectively of their ro ts. 2.3 Equilibrium analysis Perfect Bayesian equilibrium in mixed strategies We rst consider the mixed strategy equilibrium of this game. In the rst eriod, the customers reject a major service with robability y inordertosetatyeb exert indi erent between cheating and telling the truth when facing a client with a minor roblem. [1 y (1 b c )] ( L + t) = L (1) The ro t of tye b exerts telling the truth is L. By cheating, an exert may obtain L + t. t is the additional mark-u which is earned for erforming the major service. The ro t of the exert, however, deends on the rejecting strategy (y) of his client. b c is the tye b exerts belief about the fraction of consumers who are in the second eriod and accet for sure. This belief is borne out in the equilibrium, i.e., b c =[(1 g) x y ]=[1 + (1 g) x y ]. The value of b c is obtained by considering the 8 For a formal descrition of this equilibrium concet see Fudenberg and Tirole (1991).325 8

9 consumers with the minor roblem. The fraction (1 g) x y of the consumers who start the same eriod ends u in the second eriod. Another cohort of consumers joins these second eriod consumers building together the consumer oulation that is in the market. Therefore, a fraction of b c of the consumers is in their second eriod. According to equation (1), the customer randomizes over rejecting (y) and acceting (1 y) with y = t L + t (1 (1 g) x ) (2) The tye b exert chooses to roose the wrong service to a client who needs a minor roblem with robability x such that the customer is indi erent between acceting and rejecting. k =(g +(1 g)(1 x)) rob(lj e H) t (3) where rob(ljh)= e (1 w)(1 g) x (4) (1 w)(1 g) x + w. Accordingly, the left hand side of (3) reresents the cost of rejecting a service, whereas the right hand side denotes its bene t. rob(ljh) e is the robability of su ering from a minor roblem desite receiving an H-advice in the rst eriod. (g +(1 g)(1 x)) is the robability for receiving an honest recommendation. When the customer always accets the recommendation of the exert then she takes the risk to be cheated in the rst eriod. She avoids, however, the switching costs k and the risk of obtaining an H-advice again in the second eriod desite having already rejected it in the rst eriod. The latter is haening either because she ends u with atyeb exert who cheats or because she actually has a major roblem. A customer who always rejects the advice in the rst eriod has to incur switching costs but is cheated less frequently, since there exists a chance of meeting an honest exert in the second eriod. It follows from (3) that the tye b exert randomizes between cheating (x) and telling the truth (1 x) according to x 1;2 = ½ (t k)= [2t(1 g)] + A= [2t(1 g)(1 w)] (t k)= [2t(1 g)] A= [2t(1 g)(1 w)] ; (5) 9

10 where A =(1 w) (t k) 2 w(t + k) 2 : The two robabilities y 2 [0; 1] 9 and x 2 [0; 1] constitute a erfect Bayesian equilibrium in mixed strategies when at least one robability is strictly lower than 1: In order to establish a mixed equilibrium, A has to be ositive, i.e., k<[t (1 + w 2 w)]=(1 w) =k 1 or k>[t (1+w+2 w)]=(1 w) =k 2 : For k>k 2 the otimal mixing robability of x would be negative. Accordingly, only the rst inequality is a candidate for a erfect Bayesian equilibrium in mixed strategies. It deends on the arameter constellation whether an equilibrium exists and whether it is unique or not. We can distinguish two cases deending on k 1 Q (2g 1) t. In the rst case, k 1 < (2g 1) t; we obtain a unique equilibrium in mixed strategies for k <[g (1 g)(1 w) t]=(1 g+wg) = b k: In the second case, k1 > (2g 1) t, the whole range of k<k 1 involves Bayesian equilibria. They are, however, unique only for k< b k: Proosition 1 There exists a unique erfect Bayesian equilibrium in mixed strategies. for k< g (1 g)(1 w) t (1 g + wg) The tye b exerts choose strategy e H=L with robability x 2 H-advice in the rst eriod with robability y 2.10 = b k: (6) and the clients reject an We observe a ositive amount of fraud in the mixed strategy Nash equilibrium. Due to the switching costs k>0 the equilibrium is ine cient Perfect Bayesian equilibrium in ure strategies We only consider equilibria for k>0: As long as k<k 1 ; all equilibria involve mixed strategies. In contrast, equilibria in ure strategies arise for k> b k: For the secial case b k<k<k1, it deends on whether k Q (2g 1) t: A trile equilibria is obtained for 9 y 1;2 = [2(1 w) t]=[(1 w)(2 L + t + k) A] : 10 See the aendix for the case k = bk: 10

11 k >(2g 1) t whereas a unique equilibrium in ure strategy arises for k <(2g 1) t: In all equilibria in ure strategies, the tye b exerts always recommend the major service (x =1)and all clients always accet this recommendation (y =0). The switching cost k are then too high in order to revent fraudulent behavior. Proosition 2 For (2g 1) t< b k<k k 1, a trile equilibrium exists. The tye b exerts choose strategy H=L e with robability x 1 ;x ; or 2 x, and the clients reject an H-advice in the rst eriod accordingly (y 1;2; ): -insert gure 1 and 2 about here- Figures 1 and 2 resent the equilibria by dislaying the otimal mixing strategy of the tye b exert. In gure 1, we see the otimal mixing robability of the fraudulent exert as a function of the switching costs. The equilibrium is unique, because k 1 < (2g 1) t or g>(1 w)=(1 w): In gure 2, we obtain a corresondence between the mixing robability and the switching costs, since the equilibrium is not always unique Perfect Bayesian equilibrium with only tye b exerts In order to build the benchmark case, let us consider the situation with ro t maximizing exerts only. We can see form inequality (6) that the equilibria in mixed strategies are no longer unique since b k =0for g =0: We obtain a trile equilibrium for g =0 and k<k 1. Two equilibria are in mixed strategies, and one equilibrium is in ure strategies. In the equilibrium in ure strategies, tye b exertsalwaysrecommenda major service which the clients accet in the rst eriod. The reason for this nding is a kind of collusion among the exerts which makes the search for a second oinion useless. The ro t maximizing exerts do not fear losing clients to the tye g s. As a result, the clients have no longer any incentive to reject as long as the exerts are always cheating. This contrasts with the case of a ositive fraction of tye g exerts. Here, for k< b k; the customers sometimes reject although the tye b exerts always cheat. 11

12 -insert gure 3 about here- In gure 3, the mixing robability of the exerts in equilibrium is a corresondence of the switching costs. For k<k 1 ; we identify a trile equilibrium. 3 Comarison with rice cometition In order to comare our ndings with Wolinsky s, we set t = H L. In Wolinsky s setu, there is a single service H with rice H : Themarkuofcheatingwiththis service is H L ; due to the assumed nonobservability of the service reformed. This allows overcharging. In Wolinsky s model, only ro t maximizing exerts, who behave oortunistically, are in the market. They face rice cometition since rices are endogenously determined. Wolinsky obtains similar results like those resented in gure 3. He also nds equilibria in ure and in mixed strategies, resectively. Furthermore, the ure strategy equilibrium arises for the whole range of switching costs like in our case without honest exerts. For low and medium values of the switching costs, Wolinsky also obtains equilibria in mixed strategies. In contrast to our ndings, the mixing strategy of the clients is always trivial with rice cometition, because the clients constantly reject a major service. The exerts, on the other hand, are mixing between cheating and telling the truth with a robability strictly between zero and one. Nevertheless, Wolinsky identi es a trile equilibrium as well, since there exists two di erent, otimal mixing robabilities for the exerts. Like Wolinsky, we observe that the clients fully su er from the information asymmetry for high switching costs. In this case, no kind of cometition can revent the oortunistic exerts from always recommending the high rice service. Furthermore, we identify equilibria in mixed strategies for low values of k too. The cut-o value of k, however, is lower in our setu. Accordingly, the lack of rice cometition excludes the equilibria in mixed strategies for medium values of k: -insert gure 4 about here- Figure 4 shows that the switching costs can be higher with rice cometition than with xed rices in order to achieve an equilibrium in mixed strategies. For switching 12

13 costs higher than k 1 ; only unique equilibria in ure strategies arise. k indicates the analogous threshold for rice cometition. The di erence between the two cuto values with and without rice cometition deends on the ex ante robability of su ering from a major roblem. 11 Therefore, rice cometition is necessary in order to achieve the equilibria in mixed strategies in the sickle-shaed area between the two curves k 1 and k. The second di erence between the two tyes of cometition concerns the behavior of the clients in the equilibrium in mixed strategies. Exogenous rice setting may reduce the number of rejections. The su cient condition L > (1 g) t ensures that the clients reject with robability less than one. Remember, that the clients reject each major service recommendation with endogenous rices. Since the switching costs are the only cause of ine ciency in a setu with overcharging, a lower rejection rate results in a more e cient outcome. Accordingly, a xed rice setting may be more e cient than rice cometition in this resect. By introducing honest exerts, the equilibrium in ure strategies no longer exists for low values of k: That is, the collusion strategy of the oortunistic exerts does not work anymore. Moreover, we obtain a unique equilibrium for a wide range of arameter constellations. The only excetion is a arameter environment such that (2g 1) t< b k<k<k 1 ; where we observe a trile solution as if honest exerts were not resent. -insert gure 5 and 6 about here- In gure5,wedislay b k; k 1 and k as functions of the fraction of tye g exerts. We reverse the usual order of the axis in order to comare this grah with gures 1 to 3. k 1 and k are constant functions of the fraction of tye g exerts, whereas b k is a nonmonotonic function of it. Recall that b k indicates the threshold value for which a rejection is ro table for the clients, although all tye g exerts are constantly cheating. That is, switching costs lower than b k make sure that the collusion equilibrium cannot be sustained anymore. There is a value g such that b k is maximized. 12 This value 11 This ga is maximized for w =0:13: The maximum is 0:3 t: 12 g = 1 w 1 w : Notice that lim w!0 [g ]=0:5: 13

14 of g is most owerful in discilining the oortunistic exerts. For values lower than g, every increase in the fraction of tye g exerts reduces the area for which the equilibrium with the maximum amount of fraud exists. We call this e ect the direct e ect: collusion is less ossible the more honest exerts are in the market. For values above g, however, the reverse holds true: a unique equilibrium in mixed strategies becomes less likely whereas the equilibrium with the maximum amount of fraud is more likely. We call this e ect the indirect e ect: collusion becomes easier the more honest exerts are in the market. That is, cheating exerts can exloit the fact that they are a minority comared to the honest exerts. Note that the maximum amount of fraud is always de ned relative to the fraction of tye g exerts. Naturally, an increase in g never increases the total amount of fraud since a tye g exert can be relaced by a constantly cheating exert at most. This, however, cannot increase the absolute amount of fraud. Additionally, b k de nes the threshold for unique equilibria in mixed strategies. Switching costs higher than b k give rise to a unique equilibrium in ure strategies whereas switching costs between b k and k 1 may cause multile equilibria. We obtain a unique equilibrium if the fraction of tye g exerts is higher than g ; i.e., when the indirect e ect dominates. Otherwise, a trile equilibrium will arise in this interval. Figure 6 analyses the threshold value g which is deendent on the ex ante robability w of the major roblem. The lower w; the larger g which indicates that the direct e ect becomes more imortant the smaller the fraction of customer su ering from a major service is. Accordingly, a major service recommendation is a stronger signal for being with an oortunistic exert when the major roblem is not likely. The value g cannot be lower than 0:5 since the indirect e ect may only redominate when the tye b exerts are the minority. 4 Discussion By setting rices exogenously it is ossible to reroduce the outcome of rice cometition among exerts. For a medium size of the switching costs, however, rice cometition is better at solving the information roblem. For low values of the switching costs, on the other hand, a fraction of athologically honest exerts are more 14

15 e cient in solving the credence good roblem than rice cometition. This is our main nding: a fraction of honest exerts may create cometitive ressure in credence good markets that even rice cometition is not able to generate. Concerning the equilibria, the results with honest exerts are more aealing in terms of uniqueness and behavior of the clients. Recall, that the equilibrium in ure strategies, which features the maximum amount of fraud, no longer exists for low switching costs. In addition, the clients do not always reject a major service advice which is more e cient. It must be said that the nice results for the xed rice setu resuoses otimal rice setting. Otimal rice setting, however, needs a lot of information on the technology that is di cult to obtain. This additional information roblem is not analyzed in our model. Price cometition reveals information about the cost structure. Therefore, rice cometition would erform better if the whole information roblem was considered. By roviding adequate incentives, exerts behave honestly. In health markets, for instance, hysicians working for an HMO face roer incentives. Accordingly, the fraction of honest exerts could be increased when olicy succeeds in romoting HMOs. Collecting a second oinion is a driving force behind discilining oortunistic exerts. This holds true for all tyes of cometition. Accordingly, an imortant tool for weakening the credence goods roblem is to lower the switching costs. In ractice, some health insurance comanies rovide second oinion diagnoses for free for certain oerations which makes sense in view of our analysis. 5 Conclusion The aim of this aer is to show how the tye of cometition in uences the market outcome in credence goods markets. Intriguingly, cometition for clients within a xed rice setu does not necessarily erform worse with resect to the information roblem than rice cometition. Moreover, a fraction of honest exerts in the market are much more owerful in discilining oortunistic exerts than intense cometition. That is, for low switching costs, the equilibrium with a maximum amount of fraud does not exist anymore. Accordingly, a fraction of honest exerts creates cometitive ressure that even rice cometition is not able to generate. 15

16 Our model with two tyes of exerts is characterized by a unique equilibrium for most arameter constellations. This feature is frequently missing in other credence goods models. Therefore, it imroves some of them substantially. Seci cally, we are able to restrict the trivial equilibrium in which all oortunistic exerts cheat and all clients accet to a lausible range of arameter constellations. 16

17 6 Aendix Proosition 1 and 2 x 1;2 = ( (t k) + A 2t(1 g) 2t(1 g)(1 w) (t k) 2t(1 g) A 2t(1 g)(1 w) y 1;2 = ( 2(1 w) t (1 w)(2l+t+k) A 2(1 w) t (1 w)(2l+t+k)+ A For establishing an equilibrium in mixed strategies x i 2 (0; 1) and y i 2 (0; 1); i =1; In order to avoid comlex solutions, we need A =(1 w)[(t k) 2 w(t + k) 2 ] 0: That is, k t (1+w 2 w) (1 w) = k 1 or k t (1+w+2 w) = k (1 w) 2 : The second inequality imlies a negative x,since 1) if k>k 2 then k>t; because (1+w+2 w) (1 w) 2) a) if k>tthen x 1 < 0; > 1, 1+w +2 w>1 w, 2 w> 2w because (1 w)(k t) > A ) (1 w) 2 (k t) 2 > (1 w)[(t k) 2 w(t + k) 2 ] ) (1 w) 2 (k t) 2 > (1 w)[(t k) 2 w(t + k) 2 ] ) 4wkt (w 1) < 0: b) if k>tthen x 2 < 0; because t k A<0: Therefore, only the rst inequality matters. Here, x is always ositive, since 1) if k<k 1 then k<t;because (1+w 2 w) (1 w) = (t k+wt+wk) A 2(1 g)t A < 0 3) x 1(k =0)= 1 1 g > 0; and x 2(k =0)=0 4) x 1(k = k 1 )=x 2(k = k 1 )= w w (1 g)(1 w) > 0: But sometimes, x is greater than 1: 1) x 1 < 1 ) (t k)(1 w)+ A 2t(1 g)(1 w) < 1: < 1, 1+w 2 w<1 w, 2w <2 w = t k+wt+wk A 2(1 g)t A 13 We neglect boundary cases like for examle x 1 =0and y 1 2 (0; 1): > 0 17

18 1.case: k<(2g 1) t = e k; then x 1 > 1 for all arameters. 2.case: k>(2g 1) t = e k; then x 1 < 1 if k> g (1 g)(1 w) t (1 g+wg) = b k 2) x 2 < 1 ) (t k)(1 w) A 2t(1 g)(1 w) < 1: 1. case: k<(2g 1) t = e k; then x 2 < 1 if k< g (1 g)(1 w) t (1 g+wg) = b k 2. case: k>(2g 1) t = e k; then x 2 < 1 for all arameters. It remains to show that k 1 = t (1+w 2 w) (1 w) > b k = tg(1 g)(1 w) (1 g+wg) Therefore, we maximize b k subject to g : FOC: (1 w)(1 2g+g2 wg 2 ) =0) g = 1 w (1 g+wg) 2 1 w SOC: 2(1 w)w (1 g+wg) 3 < 0: We obtain b k (g )= (1+w 2 w) (1 w) : That is, the maximized value of b k subject to g is k 1 : Furthermore, e k (g )= (1+w 2 w) (1 w) : e k (g ") < b k and e k (g + ") > k 1 ; () (1 w)(1 2g+g2 wg 2 ) (1 g+wg) 2 t<2t () w (1 + 2g 2g 2 + wg 2 )+(1 g) 2 > 0 As a result, e k never lies between k 1 and b k: Accordingly, if e k<k 1 ; we obtain an x 2 (0; 1) for b 1 k<k<k 1,andanx 2 (0; 1) for 0 <k<k 2 1: In contrast, if e k>k 1 ; x =2 (0; 1); whereas 1 x 2 (0; 1) for k<b 2 k (< k 1 ). Concluding, we obtain a unique solution for k< b k and we obtain a double solution if e k<k 1 and b k<k k 1 : For b k<k= k 1 ; the double solution coincide. Now, we show that if e k<k 1,theng< 1 w : 1 w e k =(2g 1) t< t (1+w 2 w) = k (1 w) 1 ) (1 w)(2g 1) < (1 + w 2 w) ) 2g(1 w) < 2 2 w ) g< 1 w 1 w : Secial case k = b k 18

19 Proosition 3 For k = g (1 g)(1 w) = b k; we distinguish two cases: (1 g+wg) As long as (i) k 1 < (2g 1) t and L > (1 g) t or (ii) k 1 > (2g 1) t and L (1 g) t, a erfect Bayesian equilibrium in mixed strategies and one in ure strategies exist. As long as k 1 > (2g 1) t and L > (1 g) t, two di erent erfect Bayesian equilibria in mixed strategies exist. For k 1 < (2g 1) t and L (1 g) t; a unique erfect Bayesian equilibrium in ure strategies is observed. x 1 Notice that y (x =1)= We distinguish two cases: t : L+gt (i) k 1 < (2g 1) t =2 (0; 1) but x 2 =1with y 2 2 (0; 1), L > (1 g) t (ii) k 1 > (2g 1) t x 2 2 (0; 1) and x 1 =1with y 1 2 (0; 1), L > (1 g) t x 2 2 (0; 1) but x 1 =1with y 1 =2 (0; 1), L (1 g) t Case with only ro t maximizing exerts (g =0) x 1;2 = ( (t k) 2t + A 2t(1 w) (t k) 2t A 2t(1 w) y 1;2 = ( 2(1 w) t (1 w)(2l+t+k) A 2(1 w) t (1 w)(2l+t+k)+ A In order to establish an equilibrium A =(1 w)[(t k) 2 w(t + k) 2 ] 0 That is, k t (1+w 2 w) (1 w) = k 1 or k t (1+w+2 w) = k (1 w) 2 : The second inequality imlies a negative x. Proof analogous to the case g>0: Therefore, only the rst inequality matters. Here, x 1;2 2 (0; 1); because 1) if k<k 1 then k<t;because (1+w 2 w) (1 w) = (t k+wt+wk) A 2t A < 0 < 1, 1+w 2 w<1 w, 2w <2 w = t k+wt+wk A 2t A 19 > 0

20 3) x 1(k =0)=1> 0; and x 2(k =0)=0 4) x 1(k = k 1 )=x 2(k = k 1 )= w w (1 w) 2 (0; 1): Comarative statics on the convergence of rob h L= H e i h rob L= H e i h after one rejection: rob L=1H e i (1 w)(1 g) x = (1 w)(1 g) x +w h rob L= H e i h after two rejections: rob L=2H e i (1 w)(1 g) = 2 (x ) 2 (x ) 2 [1 2g+g 2 w+2wg wg 2 ]+w h rob L= H e i h after three rejections: rob L=3H e i (1 w)(1 g) = 3 (x ) 3 (x ) 3 [1 3g+3g 2 g 3 w+3wg 3wg 2 +wg 3 ]+w L=3H e i =@g < 0; i.e., the convergence is faster, the more tye g exerts are in the market. L=3H e i =@w < 0; i.e., the convergence is faster, the higher the robability for a major roblem. L=3H e i =@x > 0; i.e., the convergence is faster, the smaller the robability of cheating. Numeric examle: h 1 w =rob[l] g x rob L= H e i h rob L=2H e i h rob L=3H e i Since the twice udated robability (of having the minor roblem) is very low, the assumtion that the game ends after two eriod is not severe. 20

21 References [1] DARBY, M. R. and KARNI, E., Free Cometition and the Otimal Amount of Fraud, Journal of Law and Economics 16 (1973), [2] EMONS, W., Credence Goods and Fraudulent Exerts, Rand Journal of Economics 28 (1997), [3] EMONS, W., Credence Goods Monoolists, International Journal of Industrial Organization (1999), forthcoming. [4] FUDENBERG, D., and TIROLE, J., Game Theory, MIT Press [5] JAFFEE, D.M., and RUSSELL, T., Imerfect Information, Uncertainty, and Credit Rationing, Quarterly Journal of Economics 90 (1976) [6] MARTY, F. E., The Exert-Client Information Problem, Mimeo (1999a). [7] MARTY, F. E., Caacity as a Determinant of the Suly for Physicians Services, Diskussionsschriften Universität Bern 98-5 (1998). [8] PITCHIK, C. and SCHOTTER, A., Honesty in a Model of Strategic Information Transmission, American Economic Review 77 (1987), [9] WOLINSKY, A., Cometition in Markets for Credence Goods, Journal of Institutional and Theoretical Economics 151 (1995),

22 x 2, x 1 x * x 2 * 0 kˆ k 1 k Figure 1: unique equilibrium for k < (2g 1) t 1

23 x 1, x 2, x 1 x * x 1 * x 2 * 0 kˆ k 1 k Figure 2: unique, double or trile equilibrium for k > (2g 1) t 1

24 x 1, x 2, x x * =1 x 1 * x 2 * 0 k given k1 k Figure 3: trile equilibrium for g=0 and k < k 1

25 k (w) t = H L k 0 k 1 1 w Figure 4: cut-off value of the switching costs as a function of the fraction of clients suffering from a major roblem

26 tye g 1 g* 0 kˆ Figure 5: kˆ as a function of the fraction of tye g exerts k 1 k k

27 g* Figure 6: g* as a function of w 1 w