on doctors, mechanics, and of credence goods...dulleck and kerschbamer: on doctors, mechanics, and...

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∗ Dulleck: Johannes Kepler University Linz andUniversity of Vienna. Kerschbamer: University of Innsbruckand CEPR. Previous versions have benefitted from com-ments by and discussions with Heski Bar-Isaac, GerhardClemenz, Winand Emons, Roger Gordon, Ulrich Kamecke,Gilat Levy, John McMillan, Dennis Mueller, GerhardOrosel, Carolyn Pitchik, Larry Samuelson, AndreasWagener, Elmar Wolfstetter, Asher Wolinsky, and threeanonymous referees. The first author gratefully acknowl-edges support by the DFG through SFB 373 at HumboldtUniversity Berlin and SFB 303 at the University of Bonn.

1. Introduction

“If a mechanic tells you that he has toreplace a part in your car, don’t forget

to ask him to put the replaced part into thetrunk of your car.” This is an example of day-to-day advice regarding the services of carmechanics. One hopes to avoid two types offraud by following this advice. On the onehand, the mechanic has to change a part and

cannot only claim (and charge for) thereplacement of a part he or she did not actu-ally change. On the other hand, the cus-tomer might be able to verify that themechanic did not provide an unnecessaryrepair by inspecting the defect of theexchanged part.

This example illustrates a situation wherean expert knows more about the type of goodor service the consumer needs than the con-sumer himself. The expert seller is able toidentify the quality that fits a customer’s needbest by performing a diagnosis. He can thenprovide the right quality and charge for it, orhe can exploit the informational asymmetryby defrauding the customer.

Goods and services where an expertknows more about the quality a consumerneeds than the consumer himself are called

Journal of Economic LiteratureVol. XLIV (March 2006), pp. 5-42

On Doctors, Mechanics, andComputer Specialists: The Economics

of Credence Goods

UWE DULLECK AND RUDOLF KERSCHBAMER∗

Most of us need the services of an expert when our apartment’s heating or our wash-ing machine breaks down, or when our car starts to make strange noises. And for mostof us, commissioning an expert to solve the problem causes concern. This concern doesnot disappear even after repair and payment of the bill. On the contrary, one worriesabout paying for a service that was not provided or receiving some unnecessarytreatment. This article studies the economics underlying these worries. Under whichconditions do experts have an incentive to exploit the informational problems associ-ated with markets for diagnosis and treatment? What types of fraud exist? What arethe methods and institutions for dealing with these informational problems? Underwhich conditions does the market provide incentives to deter fraudulent behavior?And what happens if all or some of those conditions are violated?

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6 Journal of Economic Literature, Vol. XLIV (March 2006)

credence goods. Michael R. Darby and EdiKarni (1973) added this type of good toPhillip Nelson’s (1970) classification of ordi-nary, search, and experience goods. Theymention provision of repair services as atypical example. Other examples includetaxicab rides, where a stranger in a city can-not be sure that the driver takes the short-est route to his destination, and medicaltreatments, where doctors may providewrong diagnosis to sell the most profitabletreatment or where they can try to chargefor a more expensive treatment than pro-vided. There are countless other examplesof situations were consumers face similarinformation problems and where theyworry about the two basic problemssketched out by the mechanic example: topay for goods or services they did notreceive or to receive goods or services theydid not need in the first place.

Consumers’ concerns about beingdefrauded by experts seem not to beunfounded: Winand Emons (1997) cites aSwiss study reporting that the averageperson’s probability of receiving one ofseven major surgical interventions is onethird above that of a physician or a mem-ber of a physician’s family. Asher Wolinsky(1993, 1995) refers to a survey conductedby the Department of Transportation esti-mating that more than half of auto repairsare unnecessary. He also mentions a studyby the Federal Trade Commission thatdocuments the tendency of optometriststo prescribe unnecessary treatment.These examples reveal that the informa-tional asymmetry matters. Monetaryincentives play a role too. For the case ofthe health industry, Jon Gruber, JohnKim, and Dina Mayzlin (1999) empiricallyshow that the frequencies of cesareandeliveries compared to normal child birthsreact to the fee differentials of healthinsurance programs. A similar observationhas been made by David Hughes andBrian Yule (1992), who document that thenumber of cervical cytology treatments is

correlated with the fee for this treatment.That treatment can also be affected bysupply of expert services is shown byVictor R. Fuchs (1978): “Other thingsequal, a 10 percent higher surgeon/popu-lation ratio results in about a 3 percentincrease in the number of operations . . . ”(p. 54).

These observations suggest that weneed a more profound understanding ofthe specific problems associated with mar-kets for diagnosis and treatment. Underwhich conditions do experts have anincentive to exploit the informationalproblems associated with markets fordiagnosis and treatment? What types offraud exist? What are the methods andinstitutions for dealing with these infor-mational problems? Under which condi-tions does the market provide incentivesto deter fraudulent behavior? And whathappens if all or some of those conditionsare violated?

The existing literature on credence goodsdoes not help much to answer these ques-tions. The studies performed up to nowcover very different and quite special situa-tions and in total they yield no clear pictureregarding the inefficiencies arising from theinformational asymmetry. Rather, theresults seem to depend sensitively on thespecific conditions of the settings consid-ered and the specific assumptions of themodels adopted in the analysis. More dis-turbingly, apparently similar models oftenlead to contradicting results (see section 2for details).

The present article presents a model ofcredence goods that highlights the informa-tion problems in markets for experts servic-es and clarifies the variety of results in thisarea. The model provides a unifying frame-work for the analysis of the effects of differ-ent market institutions and informationstructures on efficiency, and for the identi-fication of the forces driving the variousinefficiency results in the literature. Inaddition, the model allows us to generalize

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Dulleck and Kerschbamer: On Doctors, Mechanics, and Computer Specialists 7

some of the existing results and to show thelimits of other ones.

The key feature of credence goods is thatconsumers do not know which quality of agood or service they need. The provision ofa too low quality compared to the neededone is insufficient and the provision of a toohigh quality does not add extra value. Withrespect to the auto repair example, if a carbreaks down and only an adjustment to themotor is needed to put the car back on theroad, then the replacement of the entireengine does not add to the utility the cus-tomer derives from the repair. He does noteven perceive the difference. That is, thecustomer can ex post only observe (butmight be unable to verify) whether the prob-lem still exists. If the problem no longerexists, he can not tell whether he got theright or a too high treatment quality.Furthermore, he may not even be able toobserve whether a suggested treatmentquality was actually provided or not.

Credence goods can give rise to the fol-lowing two types of problem: (1) The con-sumer requires a sophisticated, complex,typically expensive intervention butreceives a simple, inexpensive treatmentand thus forgoes the benefits of the sophis-ticated intervention. We refer to this kind ofinefficiency as undertreatment. A secondtype of inefficiency arises if the consumerrequires an inexpensive treatment butreceives a sophisticated one. The additionalbenefits to the consumer from the sophisti-cated intervention are less than the addi-tional costs. We refer to this kind ofinefficient treatment problem as overtreat-ment. (2) In addition to these inefficienttreatment problems surrounding credencegoods, credence goods can also be associat-ed with pure transfers from consumers toproducers, when a consumer requires andreceives an inexpensive treatment but ischarged for an expensive one. In the longrun, such overcharging also leads to an inef-ficiency if consumers postpone car repairsor medical check-ups and the like because

1 There are, of course, a number of other potentialproblems (and therewith a number of other sources offraud) in the credence goods market. For example, expertsmay differ in their ability to perform a diagnosis or atreatment, and this ability may be their own private infor-mation. Or, the effort bestowed by an expert in the diag-nosis stage may be unobservable. We ignore theseproblems here, not because we regard them as less impor-tant, but rather because they seem to be less specific tothe credence goods market. For analyses of situationswhere effort is needed to diagnose the customer andwhere an expert’s effort investment is unobservable, seeWolfgang Pesendorfer and Wolinsky (2003) and Dulleckand Kerschbamer (2005a). The former contribution focus-es on the effect that an additional diagnosis (by a differentexpert) has on the consumer’s evaluation of a givenexpert’s effort and the latter paper studies competitionbetween experts and discounters when customers can freeride on experts’ advice.

of the high prices they have paid for theseservices in the past.1

A first step in the understanding of cre-dence goods is to identify the conditions thatdetermine which of these two basic prob-lems associated with markets for diagnosisand treatment is the more demanding one.To get a first impression, let us assume weown a garage and our customer cannotobserve the repair we perform, but that hewill only pay us in case his car is functioningwell after the repair (thus undertreatment isruled out). If this is the case and if we planto defraud him, we will claim that an expen-sive and difficult repair is needed even whena minor repair fixes the problem. If the cus-tomer authorizes us to perform the repair,we will provide the minor treatment andcharge for a more expensive one.Overtreatment is a strictly dominated strate-gy in this setting because the higher cost ofthe expensive repair does not affect the pay-ment of the customer. Next assume this cus-tomer follows the advice to ask for replacedparts and can thus verify the type of servicewe perform (but he is still technically notvery adept and therefore not able to deter-mine whether the provided service was actu-ally needed or not). In this case, we cannotovercharge the customer but we will consid-er overtreating him whenever we make ahigher profit selling an expensive instead of

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a cheap repair. In other words, overtreat-ment is only an issue if overcharging is notfeasible due to the specific situation.

In general, the legal framework as well astechnicalities and individual education andabilities determine whether, and if yes,which of the mentioned problems appears.Overcharging is only possible if the cus-tomer cannot control the service provided bythe expert. Asking the mechanic for replacedparts is a way to avoid overcharging and, ifone is able to inspect the exchanged part, adevice to avoid overtreatment too. If a cus-tomer is able to perform a diagnosis himself,there might be no credence goods problemat all. For instance, a patient with a medicaleducation might be able to determinewhether the treatment proposed by a physi-cian is really necessary. As the evidence citedabove reveals, these differences in abilitiesaffect the treatment prescribed in medicalenvironments.

Existing institutions address the informa-tional problems associated with markets fordiagnosis and treatment. The problem ofundertreatment is most famously controlledfor by the Hippocratic Oath of a physicianand its counterpart in the law. Warrantiesprovide similar incentives. The separation ofdoctors and pharmacies is an institution toavoid overtreatment by disentangling theincentives to prescribe drugs from the profitmade selling them. The fixed part of a taxiride tariff provides incentives for the driverto serve many individual customers andtherewith not to take longer routes than nec-essary. To avoid overcharging as well asovertreatment, in many repair industries thechamber of commerce issues standard work-times for some repairs to allow customers tobetter check upon a workman’s bill—by pro-viding a comparison to usual hours for theordered repair.

We will see below that the market mecha-nism itself might discipline experts. In theunifying model that we present, experts havean incentive to commit to prices that inducenonfraudulent behavior and full revelation

of their private information if a small num-ber of critical assumptions are satisfied.These conditions are (i) expert sellers face homogeneous customers; (ii) largeeconomies of scope exist between diagnosisand treatment so that expert and consumerare in effect committed to proceed with anintervention once a diagnosis has beenmade; and (iii) either the type of treatment(the quality of the good) is verifiable or a lia-bility rule is in effect protecting consumersfrom obtaining an inappropriate inexpensivetreatment (or verifiability and liability bothhold). In section 3 we will describe theseassumptions in more detail and in section 6we will discuss and review them in the lightof the examples given.

These three basic assumptions are centralin our simple, unifying model. We show thatby relaxing them one by one, a large part ofthe existing results on inefficiencies in thecredence goods market are obtained. Thus,the analysis of our simple model sheds lighton the general structure of credence goodsmarkets. This understanding of the generalstructure can help to make a next step, fromthe fractional analysis of certain situationsand certain institutions (as done in the exist-ing literature) toward a more complete pic-ture of the issues involved. Given thisunderstanding, we are then in a position tojudge other situations and other institutionsand to assist in the design of new policies andinstitutions that remove the inefficienciessurrounding credence goods.

The rest of the paper is organized as fol-lows. The next section briefly reviews theexisting literature on credence goods. In sec-tion 3 we introduce our model. Section 4shows that market institutions solve thefraudulent expert problem at no cost whenconditions (i), (ii), and (iii) hold. In the sub-sequent section we characterize the ineffi-ciencies that arise if at least one of theseconditions is violated. Section 6 builds a linkto real world situations by returning to thekey motivating examples mentioned in thetext and discussing the plausibility of each of

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these assumptions in each case. This sectionalso contains a table where the existing liter-ature is categorized according to ourassumptions. Section 7 concludes. Someproofs are relegated to the appendix.

2. On the Existing Literature

Before recapitulating the main contribu-tions of the literature within our model, wewant to review briefly the existing literaturewith respect to assumptions and results.

Models of credence goods markets needto make assumptions on at least three inter-related characteristics of the market consid-ered. They have to specify (1) the technologyof the suppliers, i.e., the costs for providingdifferent treatments; (2) the degree of com-petition together with the organization(“microstructure”) of the market; and (3) theinformation structure of consumers andcourts, i.e., whether consumers are able toobserve the quality provided and whetherthe quality provided and the result of treat-ment can be proven to the courts. We nowturn to each of these points and discuss whatthe literature assumes.

(1) There are several differences in theconsidered technologies. Some authorsassume that experts can serve arbitrari-ly many customers at constant(Wolinsky 1993 and 1995, Curtis R.Taylor 1995, Jacob Glazer and ThomasG. McGuire 1996, Dulleck andKerschbamer 2005a and 2005b, IngelaAlger and François Salanié 2003, Yuk-Fai Fong 2005) or increasing (Darbyand Karni 1973) marginal cost, in othercontributions experts are capacity con-strained (Emons 1997 and 2001, HughRichardson 1999). Also, some authorsconsider models where the right treat-ment fixes the problem for sure(Carolyn Pitchik and Andrew Schotter1987 and 1993, Wolinsky 1993 and1995, Taylor 1995, Dulleck andKerschbamer 2005a and 2005b, Fong2005), others focus on frameworks

2 Alger and Salanié (2003) focus on an intermediatecase where some but not all inputs needed for an inter-vention are observable and verifiable.

where success is a stochastic functionof service input (Darby and Karni 1973,Glazer and McGuire 1996, Emons1997 and 2001, Richardson 1999).

(2) The analyzed market conditions rangefrom experts having some degree ofmarket power (Pitchik and Schotter1987, Richardson 1999, Emons 2001,Dulleck and Kerschbamer 2005b,Fong 2005) to competitive frameworks(Wolinsky 1993 and 1995, Taylor 1995,Glazer and McGuire 1996, Emons1997, Alger and Salanié 2003, Dulleckand Kerschbamer 2005a). Also, insome contributions experts are able tocommit ex ante to take-it-or-leave-itprices (Wolinsky 1993, Taylor 1995,Glazer and McGuire 1996, Emons1997 and 2001, Dulleck andKerschbamer 2005a and 2005b, Algerand Salanié 2003, Fong 2005), in othersprices are determined ex post in a bilat-eral bargaining process (Wolinsky 1995,Richardson 1999), still others considermodels where prices are exogenouslygiven (Darby and Karni 1973, Pitchikand Schotter 1987 and 1993, Kai Sülzleand Achim Wambach 2005).

(3) Without exception all contributions tothe credence goods literature implicit-ly or explicitly impose our condition(iii). That is, either the type of treat-ment (the input) is assumed to beobservable and verifiable (our verifia-bility assumption) or the result (theoutput) is supposed to be verifiable anda liability rule is assumed to be in effectprotecting consumers from obtainingan inappropriate inexpensive treatment(our liability assumption).2 Which ofthese two conditions is implicitly orexplicitly imposed (none of the contri-butions imposes both) unambiguouslydetermines the problem on which the

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3 In this context, “some” means that one subgroup ofcustomers is treated efficiently while another subgroupreceives one and the same treatment independently of theoutcome of the diagnosis. By contrast, “all” means thateach treated customer gets the same treatment (again,independently of the type of problem he has). In bothcases, some consumers might decide to remain untreated.

respective author(s) focus(es). Emons(1997 and 2001), Richardson (1999),and Dulleck and Kerschbamer (2005aand 2005b) explicitly impose the verifi-ability assumption and study the prob-lems of over- and undertreatment.Pitchik and Schotter (1987 and 1993),Wolinsky (1993 and 1995), Fong(2005), and Sülzle and Wambach(2005) analyze experts’ temptation toovercharge customers, implicitlyassuming the liability assumption tohold and verifiability to be violated.Taylor (1995) imposes these assump-tions explicitly. And in Darby and Karni(1973), the implicit assumption regard-ing verifiability varies according towhether capacity exceeds demand orvice versa. For the former case, theyanalyze experts’ incentive to overtreatcustomers, implicitly assuming verifia-bility to hold. For the latter case, theydiscuss the incentive to charge fortreatments not provided, implicitlyassuming verifiability to be violated.

Given the wide range of problems ana-lyzed in the literature, it is not surprisingthat the proposed solutions exhibit a broadrange of different equilibrium behavior:there are pure strategy equilibria in whichexperts “mistreat” (i.e., under- or overtreat)some (Dulleck and Kerschbamer 2005b) orall (Darby and Karni 1973, Richardson1999, Alger and Salanié 2003) consumers,and pure strategy equilibria which give riseto excessive search and diagnosis costs(Wolinsky 1993, Glazer and McGuire1996).3 There are pure and mixed strategyequilibria where the market outcomeinvolves fraud in the form of overchargingof consumers (Pitchik and Schotter 1987

4 Pitchik and Schotter (1993) claim that their formalframework encompasses both the verifiability and the non-verifiability case (p. 818). In our view, their assumptions onthe payoff structure (the profit of an expert authorized bya client to perform an expensive treatment is higher if theclient needs a cheap rather than the expensive treatment)hardly allow the verifiability interpretation, however.

and 1993, Wolinsky 1995, Fong 2005, Sülzleand Wambach 2005). And there are alsoequilibria where the only inefficiencies inthe credence goods market are experts’inefficient capacity levels (Emons 1997).Thus, the overall picture is rather blurred.At least for the nonexpert reader it is fairlydifficult to judge which set of conditionsdrives the presented results.

The unifying model that we present facili-tates the identification of the criticalassumptions that lead to the striking diversi-ty of results. In addition, it helps to removesome ambiguities in the literature. Forinstance, Emons (2001) attributes the maindifference between his own work and thepapers by Pitchik and Schotter (1987 and1993) and Wolinsky (1993) to the fact that“they all (implicitly) assume unnecessaryrepairs to be costless whereas our expertneeds resources for unnecessary treatments.This implies that overtreatment is alwaysprofitable in their set-up. In contrast, theprofitability of overtreatment in our modeldepends on demand conditions and is deter-mined endogenously” (p. 4). Our analysisbelow suggests that differences in two of thethree conditions specified in the previous(and further discussed in the next) sectionare more important: First, Pitchik andSchotter (1987 and 1993) and Wolinsky(1993) implicitly assume verifiability to beviolated and liability to hold whereas Emons(2001) considers the opposite constellation.4

Thus, the former papers focus on the prob-lem of overcharging, while the latter studiesexperts’ incentive to over- or undertreat cus-tomers. Secondly, Pitchik and Schotter(1987 and 1993) and Wolinsky (1993)assume that the efficiency loss of diagnosinga consumer more than once is rather low

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5 Many of results can also be obtained in an extendedmodel that allows for more than two types of problem andmore than two types of treatment (see Dulleck andKerschbamer 2005b for such a model). However, since ourgoal is to provide a simple unifying framework, we stick toa binary model.

6 For convenience, both the type of treatment and theassociated cost is denoted by c.

(low economies of scope between diagnosisand treatment), while Emons (2001)assumes this loss to be high (profoundeconomies of scope between diagnosis andtreatment).

Below we will reproduce the main resultsof these studies in our framework and discusswhich assumptions are crucial to get them.

3. A Basic Model of Credence Goods

To model credence goods, we assumethat each customer (he) has either a (minor)problem requiring a cheap treatment c– or a(major) problem requiring an expensivetreatment c–.5 The customer knows that hehas a problem but does not know howsevere it is. He only knows that he has an exante probability of h that he has the majorproblem and a probability of (1 − h) that hehas the minor one. An expert (she), on theother hand, is able to detect the severity ofthe problem by performing a diagnosis. Shecan then provide the appropriate treatmentand charge for it or she can exploit theinformation asymmetry by defrauding thecustomer.

The cost of the expensive treatment is c–and the cost of the cheap treatment is c–, withc– > c–.6 The expensive treatment fixes either

7 Of course, not all credence goods have such a simplepayoff structure. For instance, in the medical example thepayoff for an appropriately treated major disease mightdiffer from that of an appropriately treated minor disease.Similarly, the payoff for a correctly treated minor diseasemight differ from that of an overtreated minor disease.Introducing such differences would burden the analysiswith additional notation, without changing any of theresults, however.

problem while the cheap one is only good forthe minor problem.

Table 1 represents the gross utility of aconsumer given the type of treatment heneeds and the type he gets. If the type oftreatment is sufficient, a consumer gets util-ity v. Otherwise he gets 0. To motivate thispayoff structure, consider a car with either aminor problem (car needs oil in the engine)or a major problem (car needs new engine),with the outcomes being “car works” (ifappropriately treated or overtreated) and“car does not work” (if undertreated).7

An important characteristic of credencegoods is that the customer is satisfied in threeout of four cases. In general, he is satisfiedwhenever he gets a treatment quality at leastas good as the needed one. Only in one case,where he has the major problem but gets thecheap treatment, will he discover ex postwhat he needed and what he got.

Consumers may differ in their valuationfor a successful intervention or in their prob-ability of needing different treatments. Weintroduce heterogeneous customers in sec-tion 5. In section 4 we follow the main bodyof the literature on credence goods inassuming that customers are identical. Werefer to this as the homogeneity assumption(Assumption H).

TABLE 1UTILITY FROM A CREDENCE GOOD

Customer’s utility Customer needsc−

−c

Customer c− v 0

gets −c v v

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8 Since provision of treatment without diagnosis isassumed to be impossible, an increase in diagnosis costmeans that consulting more than one expert becomes lessattractive.

ASSUMPTION H (HOMOGENEITY):All consumers have the same probability hof having the major problem and the samevaluation v.

An expert needs to perform a diagnosisbefore providing a treatment. Regarding themagnitude of economies of scope betweendiagnosis and treatment, there are two dif-ferent scenarios to consider. If theseeconomies are small, separation of diagnosisand treatment or consultation of severalexperts may become attractive. With pro-found economies of scope, on the otherhand, expert and customer are in effect tiedtogether once the diagnosis is made. In sec-tion 5, we take the diagnosis cost d as ameasure of the magnitude of economies ofscope between diagnosis and treatment anddiscuss conditions under which expert andconsumer are in effect tied together.8 Insection 4, we work with the following short-cut assumption which we refer to as thecommitment assumption (Assumption C).

ASSUMPTION C (COMMITMENT):Once a recommendation is made, the cus-tomer is committed to undergo a treatmentby the expert.

As mentioned before, the focus of thecredence goods literature has beentwofold: inefficient treatment, eitherunder- or overtreatment, and overcharging.The inefficiency of treatment can bedescribed by referring to table 1. The caseof undertreatment is the upper right cornerof the table, the case of overtreatment isthe lower left corner. Note that overtreat-ment is not detected by the customer(v = v) and hence cannot be ruled out byinstitutional arrangements. This is not thecase with undertreatment; it is detected bythe customer (0 � v) and might even beverifiable. If this is the case, and if a legal

9 An undercharging incentive only exists if the price ofthe expensive treatment is such that customers reject a −crecommendation, and if the price of the cheap treat-ment exceeds the cost of the expensive one. Such pricecombinations are not observed in equilibrium.

rule is in effect that makes an expert liablefor the provision of inappropriate low qual-ity, we say that the liability assumption(Assumption L) holds.

ASSUMPTION L (LIABILITY): Anexpert cannot provide the cheap treatment c–if the expensive treatment c– is needed.

Referring again to table 1, the secondpotential problem is that the customer mightnever receive a signal that discriminatesbetween the upper left and the lower rightcell of the table. If this is the case, an expertwho discovers that the customer has theminor problem can diagnose the major oneso that the customer might authorize andpay for the expensive treatment althoughonly the cheap one is provided. This over-charging is ruled out if the customer is ableto observe and verify the delivered quality(he knows and can prove whether he is in thetop or the bottom row of the table), and werefer to situations in which consumers havethis ability as cases where the verifiabilityassumption (Assumption V) holds.9

ASSUMPTION V (VERIFIABILITY):An expert cannot charge for the expensivetreatment c– if she has provided the cheaptreatment c–.

Let us now describe the market environ-ment. There is a finite population of n ≥ 1identical risk-neutral experts in the cre-dence goods market. Each expert can servearbitrarily many customers. The expertssimultaneously post take-it-or-leave-itprices. Let p−i (p−

i, respectively) denote theprice posted by expert i ∈ {1, . . . ,n} for theexpensive (cheap) treatment c– (c–). Anexpert’s profit is the sum of revenues minuscosts over the customers she treated. Byassumption, an expert provides the appro-priate treatment if she is indifferentbetween providing the appropriate and

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10 Introducing some guilt disutility associated with pro-viding the wrong treatment would yield the same qualita-tive results as this common knowledge assumptionprovided the effect is small enough to not outweigh thepecuniary incentives.

11 The diagnosis cost d is assumed to include the timeand effort cost incurred by the consumer in visiting a doc-tor, taking the car to a mechanic, etc. It is also assumed toinclude a fair diagnosis fee paid to the expert to cover heropportunity cost. A more elaborate model would distin-guish between a search and diagnosis cost dc borne direct-ly by the consumer, a diagnosis cost de borne by the expert(where d = dc + de) and a diagnosis fee p charged by theexpert. Since one of our goals is to reproduce many of theexisting results on inefficiencies and fraud in credencegoods markets in a simple framework and since the bulk ofthe literature takes diagnosis fees as exogenously given (tothe best of our knowledge the only exceptions arePesendorfer and Wolinsky 2003, Alger and Salanié 2003,and Dulleck and Kerschbamer 2005a), we do not endoge-nize the price for diagnosis (but rather assume that p = de)for easier comparison.

12 Here, the implicit assumption is that the outsideoption is not to be treated at all. Again, the car exampleprovides a good illustration. A car may be inoperable for aminor or a major reason, with the lack of treatment givingthe same outcome (‘car does not work’) as undertreatment.The medical example behaves differently. For instance,letting a cancerous growth go untreated is much differentthan letting a benign growth go untreated. See footnote 7above, however.

providing the wrong treatment, and this factis common knowledge among all players.10

There is a continuum with mass one ofrisk-neutral consumers in the market. Eachconsumer incurs a diagnosis cost d perexpert he visits independently of whether heis actually treated or not.11 That is, a con-sumer who resorts to r experts for consulta-tion bears a total diagnosis cost of rd. Thenet payoff of a consumer who has been treat-ed is his gross valuation as depicted in table1 minus the price paid for the treatmentminus total diagnosis cost. The payoff of aconsumer who has not been treated is equalto his reservation payoff, which we normal-ize to zero, minus total diagnosis cost.12 Byassumption, it is always (i.e., even ex post)efficient that a consumer is treated when hehas a problem. That is, v − c– − d > 0. Also, byassumption, if a consumer is indifferentbetween visiting an expert and not visiting an expert, he decides for a visit, and if a

13 That consumers know experts’ costs of providing dif-ferent treatments is obviously a very strong assumption.Since it is explicitly or implicitly imposed by all authorswho work with the verifiability assumption and since weneed it to replicate their results, we introduce it here.

14 Here note that, from a game-theoretic point of view,there is no difference between a model in which naturedetermines the severity of the problem at the outset andour model where this move occurs after the consumer hasconsulted an expert (but before the expert has performedthe diagnosis).

15 We take the convention that each consumer can visita given expert at most once.

16 Throughout we assume that an expert’s agreement toperform the diagnosis means a commitment to provide atreatment even if treatment-provision is not profitable forthe expert. This assumption is not important for our resultsand we mention in footnotes what changes if the expert isfree to send off the customer after having conducted thediagnosis.

customer who decides for a visit is indifferentbetween two or more experts he randomizes(with equal probability) among them.

Figure 1 shows the game tree for the spe-cial case where a single expert (n = 1) courts asingle consumer. The variables v, h, c–, and c–are assumed to be common knowledge.13 Atthe outset, the expert posts prices p− and p− forc– and c– respectively. The consumer observesthese prices and then decides whether to visitthe expert or not. If he decides against thevisit, he remains untreated yielding a payoff ofzero for both players. If he visits the expert, arandom move of nature determines the sever-ity of his problem.14 Now the expert diagnosesthe consumer. In the course of her diagnosisshe learns the customer’s problem and recom-mends either the cheap or the expensivetreatment. Next the customer decideswhether to accept or reject the recommenda-tion. If he rejects, his payoff is − d, while theexpert’s payoff is zero.15 Under the commit-ment assumption (Assumption C), this deci-sion node of the consumer is missing. If theconsumer is committed or if he accepts underthe noncommitment assumption, then theexpert provides some kind of treatment andcharges for the recommended one.16 Underthe verifiability assumption (Assumption V),this decision node is degenerate: the expertsimply provides the recommended treatment.

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Figure 1. Game Tree for the Credence Goods Problems

overprovisionovercharging

underprovision

(p,p)

out in

c

c c

c cc c

r r

r

ra a

a

a

expert postsprices

consumer decideswhether to visit theexpert or not

nature deter-mines severityof problem

expert recom-mends (andcharges ifaccepted)

consumeraccepts or rejects

expert provides

c

c

c ccc

c

Under the liability assumption (AssumptionL), this decision node is degenerate wheneverthe customer has the major problem: then theexpert must provide the expensive treatment.The game ends with payoffs determined inthe obvious way.

The game tree for the model with manyconsumers and a single expert (n = 1) can bethought of as simply having many of thesesingle-consumer games going on simultane-ously, with the fraction of consumers withthe major problem in the market being h.With more than one expert (n > 1), theexperts simultaneously post prices p−

i and p−i

(i ∈ {1, . . . ,n}) for c– and c– respectively.17

17 In the efficiency results of section 4, the distinctionbetween the setting where there is only one expert and theone where there are many experts is simply a matter ofdetermining whether the surplus in the market is trans-ferred to consumers or experts. For the results in subsec-tion 5.1, experts need to have market power. In our simplemodel, market power corresponds to the single expert case.In subsection 5.2, we discuss the case of multiple diagnosis,a frequent phenomenon in markets for expert services. Toanalyze this phenomenon, we need more than one expert.

Consumers observe these prices and thendecide whether to undergo a diagnosis by anexpert or not and, if yes, by which expert.Thus, with n > 1 a consumer’s decisionagainst visiting a given expert (the “out”decision in the game tree) doesn’t mean thathe remains untreated: he might simply visita different expert. Similarly, a consumer’spayoff if he rejects a given expert’s treatmentrecommendation depends on whether hevisits a different expert or not.

With commitment, the game justdescribed is a multistage game withobserved actions and complete information;see Drew Fudenberg and Jean Tirole (1991,chapter 3).18 The natural solution conceptfor such a game is subgame-perfect equilib-rium and we will resort to it in section 4.Subgame-perfection loses much of its bite inthe noncommitment case where the less-informed customer has to decide whether to

18 Tirole (1990, chapter 11) refers to such games as“games of almost perfect information.”

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stay or to leave without knowing whether thebetter-informed expert has recommendedthe right or the wrong treatment. To extendthe spirit of subgame-perfection to this gameof incomplete information, we require thatstrategies yield a Bayes–Nash equilibriumnot only for each proper subgame, but alsofor continuation games that are not propersubgames (because they do not stem from asingleton information set). That is, we focuson perfect Bayesian equilibria in the non-commitment case.

4. Efficiency and Honesty with Credence Goods

We start by stating our efficiency result:PROPOSITION 1: Under Assumptions H

(Homogeneity), C (Commitment), and eitherL (Liability) or V (Verifiability) or both,market institutions solve the fraudulentexpert problem at no cost.

The proof for this result, as well as theintuition behind it, relies on three observa-tions that are reported as Lemma 1–3 below.Lemma 1 discusses the result under the ver-ifiability assumption. With verifiability alone(i.e., without liability), experts find it optimalto charge a uniform margin over all treat-ments sold and to serve customers honestlyas the following result shows.

LEMMA 1: Suppose that Assumptions H(Homogeneity), C (Commitment), and V(Verifiability) hold, and that Assumption L(Liability) is violated. Then, in any sub-game-perfect equilibrium, all consumers areefficiently served under equal markupprices. The equilibrium prices satisfy p− −c– = p− − c– = v − d − c– − h(c– − c–) if a singleexpert provides the good (n = 1), and p− −c– = p− − c– = 0 if there is competition in the

credence goods market (n ≥ 2).PROOF: First note that with verifiability

each expert will always charge for the treat-ment she provided. The treatment qualityprovided depends upon the type of price vec-tor (tariff) under which the customer isserved. There are three classes of tariffs to

19 The assumption that it is common knowledge amongplayers that experts provide the appropriate treatmentwhenever they are indifferent plays an important role inLemma 1 in generating a unique subgame-perfect equilib-rium outcome. Without this assumption, there exist othersubgame-perfect equilibria which are supported by thebelief that all experts who post equal markup prices—or,that experts who post equal markup prices that are too low(in the monopoly case: too high)—deliberately mistreattheir customers. We regard such equilibria as implausible(see footnote 10 above) and have therefore introduced thecommon knowledge assumption which acts as a restrictionon consumers’ beliefs.

consider, tariffs where the markup for themajor treatment exceeds that for the minorone (p− − c– > p− − c–), tariffs where the markupof the minor treatment exceeds that for themajor one (p− − c– < p− − c–), and equal markuptariffs (p− − c– = p− − c–). A consumer under a tar-iff in the first class will always get the major, aconsumer under a contract in the secondclass always the minor interventions. Onlyunder contracts in the last class is the expertindifferent between the two types of treat-ment and, therefore, behaves honestly.19

Consumers infer experts’ incentives fromtreatment prices. Thus, if a single expert pop-ulates the market (n = 1), the maximal profitper customer the monopolist can realize withequal markup prices is v − d − c– − h(c– − c–);the maximal obtainable profit with tariffs sat-isfying p− − c– > p− − c– is v − d − c–; and the max-imal profit with tariffs satisfying p− − c– < p− − c–is (1 − h)v − d − c–. Thus, since v > c– − c–, theexpert will post the proposed equal markuptariff and serve customers honestly. Next sup-pose that n > 1. Further suppose that at leastone expert attracts some customer(s) under atariff (p̂−, p̂−) that violates the equal markuprule. Then she can increase her profit byswitching to an equal markup tariff (p−,p−) satisfying either p− − c– = p− − c– = p̂− − c– + (1 − h)(c– − c–) (if the old tariff had p̂− − c– >p̂− − c–), or p− − c– = p− − c– = p̂− − c+ h(v − c– + c–) (inthe opposite case). Thus, only equal markupprices can attract customers in equilibrium.The result then follows from the observationthat p− − c– = p− − c– = 0 is the only equal markupconsistent with Bertrand competition.�

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Figure 2. Consumers’ Expected Utility with Verifiability

)( cchpdv

)( ccpdv

pdvh)1(

0 cp cp

u( p, p)~

Figure 2 illustrates the equal markupresult graphically. In this figure, theterm tu(p−,p−) denotes the expected net payoffof a consumer who resorts to an expert withposted price-vector (p−,p−). First notice thatverifiability solves the problem of overcharg-ing; that is, the seller cannot claim to havesupplied the expensive treatment when sheactually has provided the cheap one. Thereremains the incentive to provide the wrongtreatment. Such an incentive exists if theintervention prices specified by the tariff aresuch that providing one of the treatments ismore profitable than providing the other. So,if we fix the markup for the cheaper inter-vention at p− − c– and increase the markup forthe expensive intervention from 0 (as it isdone in figure 2), then the expert’s incen-tives remain unchanged over the interval(0, p− − c–): she will always recommend andprovide the cheap treatment at the price p−.Consequently, a consumer’s expected utilityis constant in this interval at (1 − h)v − d − p−and there is an efficiency loss from under-treatment of size h(v − c– + c–). Similarly, if westart at p− − c– and increase p− − c–, then theexpert will always recommend and provide

the expensive treatment at the price p−, so thatthe consumer’s utility is v − p− − d implying anefficiency loss from overtreatment of size(1 − h)(c– − c–). Only at the single point wherethe markup is the same for both treatments(p− − c– = p− − c– = ∆, say), will the expert per-form a serious diagnosis and recommend theappropriate treatment. At this point there isno efficiency loss and consumer’s expectedutility jumps discontinuously upward to v − p− − h(c– − c–) − d.

Consumers infer the experts’ incentivesfrom the intervention prices. So, expertscannot gain by cheating. Consequently, thebest they can do is to post equal markup tariffs and to behave honestly. In themonopoly case (n = 1), the expert has all themarket power. Hence, posted prices aresuch that the entire surplus goes to theexpert. With two or more experts, on theother hand, Bertrand competition drivesprofits down to zero and consumers appro-priate the entire surplus as experts are notcapacity constrained in our model.

The verifiability assumption is likely to besatisfied in important credence goods mar-kets, including dental services, automobile

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and equipment repair, and pest control. Formore sophisticated repairs, where the cus-tomer is usually not physically present dur-ing the treatment, verifiability is oftensecured indirectly through the provision ofex post evidence. In the automobile repairmarket, for instance, it is quite common thatbroken parts are handed over to the cus-tomer to substantiate the claim that replace-ment, and not only repair, has beenperformed. Similarly, in the historic carrestoration market the type of treatment isusually documented step by step in pictures.

Is there any evidence of equal markups inthese markets? Many suppliers in the autorepair and historic car restoration market setstandard job-completion times and thencharge a uniform hourly rate. This might beinterpreted as evidence for uniform marginsover all treatments sold.

Equal markup prices are also plausible incases with expert sellers. Computer storesare an obvious example. Customers can con-trol which quality they receive. Other exam-ples of uniform margins are the pricingschemes of insurance brokers and travelagents. The markup insurance brokers andtravel agent charge (the margin plus anybonuses to the agent offered by theprovider) are similar for all products. Oftenalso a fixed fee is involved, independent ofwhat insurance is bought or which bookingis made.

Lemma 1 facilitates the interpretation ofthe Emons (1997) result on the provisionand pricing strategy of capacity constrainedexperts. Emons considers a model in whicheach of a finite number of identical poten-tial experts has a fixed capacity. Shebecomes an active expert by irreversiblydevoting this capacity to the credence goodsmarket. Once she has done this, she can useher capacity to provide two types of treat-ment at zero marginal cost up to the capac-ity constraint. One type of treatmentconsumes more units of capacity than thesecond. Total capacity over all potentialexperts exceeds the amount necessary to

serve all customers honestly. Emons propos-es a symmetric equilibrium in which eachpotential expert’s entry decision is strictlymixed. Thus, active experts may either haveto ration their clientele due to insufficientcapacity or they may end up with idle capac-ity. In the former case, they charge pricessuch that (1) all the surplus goes to theexperts and (2) the price for the morecapacity-consuming treatment exceeds theprice of the second treatment by such anamount that the profit per unit of capacityconsumed is the same for both types oftreatment. In the latter case, all expertscharge a price of zero for both types oftreatment. In both cases experts serve cus-tomers honestly. This is exactly what ourLemma 1 would predict: the situationwhere demand exceeds total capacity corre-sponds in our model to a situation wheren = 1, since experts have all the marketpower in that case and since n is our param-eter for market power. With all the marketpower, experts appropriate the entire sur-plus and consumers get only their reserva-tion payoff of zero. Furthermore, withinsufficient capacity, equal markup pricesimply a higher price for the more capacity-consuming treatment since the opportunitycost in terms of units of capacity used ishigher. By contrast, if capacity exceedsdemand, the opportunity cost is zero forboth types of treatment. Thus, the price hasto be the same for both treatments to yieldequal markups. Furthermore, with idlecapacity, experts have no market power(which corresponds in our model to a situa-tion where n ≥ 2). Without market power,competition drives prices down to opportu-nity costs and customers appropriate theentire surplus.

To summarize, our analysis shows thatmany specific assumptions made by Emons(e.g., that there is a continuum of cus-tomers, that capacity is needed to providetreatments, that success is a stochasticfunction of the type of treatment provided,etc.) are not important for his efficiency

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20 The only inefficiency that remains in the Emonsmodel is the suboptimal amount of capacity provided inequilibrium. This inefficiency has nothing to do with thecredence goods problem, however. It is rather a coordina-tion failure type of inefficiency similar to that arising in thesymmetric mixed strategy equilibrium of the grab-the-dollar game prominent in Industrial Organization.

21 In the Emons model, the inefficiencies described insubsection 5.1 arise for any n since experts have marketpower whenever capacity is insufficient to cover demand.

22 Assumption C is not important for the Emons resultas Lemma 8 below shows.

23 If the expert is not obliged to treat the customer afterhaving conducted the diagnosis, the price charged in equi-librium changes to tp = −c +�(v − d − −c), where � = 1 forn = 1 and � = 0 otherwise. The rest of Lemma 2 remainsunaffected.

result.20 What is important, however, is thatconsumers are homogeneous (see Proposition2 below)21 and that the type of treatment isverifiable (see Proposition 4, below).22

Let us turn now to the liability case.Under the liability assumption, expertscharge a uniform price for both types oftreatment and serve customers honestly asthe following result shows.

LEMMA 2: Suppose that Assumptions H(Homogeneity), C (Commitment), and L(Liability) hold, and that Assumption V(Verifiability) is violated. Then, in any sub-game-perfect equilibrium, each expertcharges a constant price for both types oftreatment and efficiently serves her cus-tomers. The price charged in equilibrium isgiven by tp = v − d if a single expert providesthe good (n = 1), and by tp = c– + h(c– − c–) ifthere is competition in the credence goodsmarket (n ≥ 2).23

PROOF: Obvious from the discussionbelow and therefore omitted.�

Figure 3 depicts the expected utility of aconsumer under the conditions of Lemma 2.In the setting under consideration, the cus-tomer cannot control the service provided bythe expert. In this case, overtreatment is astrictly dominated strategy because the high-er cost of the expensive repair does not affectthe payment of the customer. Also, under-treatment is no problem because of the lia-bility assumption. So each expert efficiently

24 As a referee pointed out, this may be a too naive viewon HMOs. In section 6, we discuss the plausibility ofAssumptions L in this context.

serves her customers. There remains theincentive to overcharge, that is, to alwaysclaim that an expensive and difficult repair isneeded even when a minor treatment fixesthe problem. Such an incentive exists when-ever p− ≠ p−. Consumers know this and expectthe expert to charge the higher price andthen to provide the cheapest sufficienttreatment. Thus, their behavior depends ontp = max{p−, p−} only. The rest is trivial. In themonopoly case (n = 1), the expert has all themarket power. Thus, she charges tp = v − d.With n > 1, the price-posting game is a stan-dard Bertrand game. Thus the pricecharged in equilibrium is tp = c– + h(c– − c–) bythe usual price-cutting argument.

Lemma 2 offers an explanation for the fre-quently observed fixed prices for expertservices in environments where experts haveto provide reliable quality because otherwisethey are punished by law or by bad reputa-tion. One example is health maintenanceorganizations (HMOs) providing medicalservice to members at an individualized,constant price per customer. Our analysissuggests that the schemes offered by theHMOs are cheaper than a health insurancesystem. Under insurance, the customer doesnot care about the price after having paid theinsurance premium. With the HMO, thecompany will make sure that the cheapestsufficient quality will be provided.24

Lemma 2 facilitates the identification ofthe driving forces behind Taylor’s (1995)efficiency results. Taylor studies an elaboratemodel of a durable credence good whichmay be in one of three states: health, dis-ease, or failure. The good begins in the stateof health, passes over to the state of disease,and from there either to the state of failureor, if efficiently treated, back to the state ofhealth. The amount of time spent in each ofthe first two states is governed by an expo-nential distribution. When the good enters

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Figure 3. Consumers’ Expected Utility with Liability

0

pdv

p p

ppdv ,max

),(~ ppu

the state of failure it remains there forever.The main problem for the owner of the goodis that he never discovers whether the goodis healthy or diseased. An expert, on theother hand, can observe this by performing adiagnostic check. If the check reveals thegood to be diseased, she can perform thenecessary treatment. Taylor shows that, ifexperts post treatment prices ex ante, thenthey essentially offer fair insurance againstthe presence of disease by charging a fixedprice equal to the expected treatment cost,as our Lemma 2 would predict. However, inthe Taylor model, this insurance solution isnot efficient since the good also needs main-tenance by the owner for survival. If theowner opts for low maintenance, then themaintenance cost per unit of time is low butthe good is later more expensive to treatwhen it becomes diseased. Obviously, in thisset up fixed prices are inefficient since theyprovide poor incentives for owners to per-form maintenance. Taylor proposes twoalternative ways to solve this problem: expost contracts where experts commit totreatment prices only after having learnedthe owner’s level of care, and multiperiodcontracts for settings with repeated interac-tions between an owner and an expert.

These solutions would dominate the simplefixed price rule in our setup too if weassumed that the good needs costly mainte-nance and that the owner’s level of care isrevealed to the expert in the diagnosticcheck as is assumed by Taylor.

Our framework shows that many detailsof the more sophisticated Taylor model arenot important for the efficiency results.What is important, however, is that a liabili-ty rule protects consumers from obtaininginsufficient treatment (without liabilityexperts would never cure the good, exceptfor the prospect of repeat business) and thatthe type of treatment is not verifiable (oth-erwise equal markup prices would provideincentives for maintenance).

Let us now consider the case where boththe liability and the verifiability assumptionhold. In this case, a lower markup for themajor than for the minor intervention is suf-ficient to induce nonfraudulent behavior asthe following result shows.

LEMMA 3: Suppose that Assumptions H (Homogeneity), C (Commitment), L(Liability), and V (Verifiability) hold. Then, in any subgame-perfect equilibrium, eachexpert posts and charges prices yielding alower (≤) markup for the more expensive

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25 Obviously, if an expert is not obligated to treat a cus-tomer after having conducted the diagnosis, equilibriumprices will also satisfy the condition −p ≥ −c.

treatment and efficiently serves her cus-tomers. Posted (and charged) equilibriumprices satisfy p− + h(p− − p−) = v − d and p− −c– ≤ p− − c– if a single expert provides the good(n = 1), and p− + h(p− − p−) = c– + h(c– − c–) and p− − c– ≤ p− − c– if there is competition in the credence goods market (n ≥ 2).25

PROOF: The proof is similar to that ofLemma 1, the only difference being that theliability rule prevents the expert from profit-ing by providing the cheap treatment whenthe expensive one is needed.�

Under the conditions of Lemma 3, verifi-ability prevents an expert from claiming tohave supplied the major treatment when sheactually has provided the minor one (i.e.,overcharging is ruled out). In addition, lia-bility prevents her from providing the minorintervention when her customer needs themajor one (i.e., undertreatment is ruled out,too). There remains the incentive to providea major treatment when the consumer onlyneeds the minor one. This overtreatmentincentive disappears if prices are such thatthe markup for the minor treatment exceedsthat of the major treatment.

Lemmas 1–3 discussed possible idealenvironments that imply that the pricemechanism solves the credence goods prob-lem at no cost. This no-fraud result suggeststhat asymmetric information alone cannotexplain experts’ cheating.

Table 2 recapitulates the role our fourassumptions play for the efficiency result. Ifconsumers are homogeneous, price discrimi-nation is no issue. As we will see below, pricediscrimination in expert markets proceedsalong the dimension of quality of adviceoffered.

The commitment assumption preventsconsumers from visiting more than oneexpert. This assumption is important forefficiency if liability holds while verifiabilityis violated. Why? Because then a constant

26 As one of the referees pointed out, a second diagno-sis is not necessary in many settings. Our assumption thatprovision of treatment without diagnosis is impossibleshould not be taken literally, however, but rather as a shortcut for situations in which there exist economies of scopebetween diagnosis and treatment. As Darby and Karni(1973, footnote 2) have put it, “ . . . it is easier to repair anydamage while the transmission or belly is open to see whatis wrong, than to put everything back together and goelsewhere to repeat the process for the actual repair.”

price across treatments is necessary to pre-vent experts from overcharging (see Lemma2). A constant price across treatmentsimplies that consumers with minor prob-lems subsidize those with major ones. Ifconsumers are not committed to undergo atreatment once a diagnosis has been madeand if the cost of getting a second opinion islow, such cross-subsidization becomes infea-sible because it invites cream skimming by adeviating expert who sells the minor inter-vention only at a lower price. Nondeviatingexperts who charge a constant price acrosstreatments will realize that only consumerswho need a major intervention visit themand they will adjust their prices accordingly.The result is specialization, some expertssell only the minor, others only the majorintervention. Specialization is inefficientbecause it implies a duplication of searchand diagnosis costs.26

The verifiability assumption preventsovercharging, while the liability assumptionprevents undertreatment. If none of theseassumptions holds, each expert will alwaysprovide the minor and charge for the majorintervention. Thus undertreatment of con-sumers who need the major treatmentresults. Consumers anticipate this and iftheir valuation for a successful treatment istoo low they leave the market.

In the next section, we will discuss these(and other) inefficiencies in more detail.

5. Various Degrees of Inefficiencies andFraud in the Credence Goods Market

In this section, we characterize the ineffi-ciencies that might arise if at least one of the

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TABLE 2ASSUMPTIONS AND THEIR IMPORTANCE FOR THE EFFICIENCY RESULT

Assumption Effect

Homogeneity prevents price discrimination

Commitment prevents consumers from visiting more than one expert

Liability prevents undertreatment

Verifiability prevents overcharging

conditions of Proposition 1 is violated. Webegin with the homogeneity assumption(Assumption H).

5.1 Heterogeneous Consumers: InefficientRationing and Inefficient Treatment ofsome Consumer Groups

Dropping the homogeneity assumptioncan give rise to two different kinds of ineffi-ciency: inefficient rationing of consumersand/or inefficient treatment of some con-sumer groups. To show this, we consider asetting in which a single expert (n = 1) sellsverifiable treatments (Assumption V holds)to heterogeneous consumers (Assumption His violated). To keep things simple, we con-centrate on the case where consumers onlydiffer in their expected cost of efficient treat-ment. More precisely, we assume that a con-sumer of type h has the major problem withprobability h and the minor problem withprobability (1 − h). Since this assumptionimplies that higher type consumers have ahigher expected cost of efficient treatment,we refer to them as high cost consumers.Consumers’ types (i.e., the probabilities h)are drawn independently from the sameconcave cumulative distribution functionF(.), with differentiable strictly positive den-sity f(.) on [0,1]. F(.) is common knowledge,but a consumer’s type is the consumer’s pri-vate information. If a consumer gets a treat-ment that fixes his problem, he obtains atype-independent gross utility of v, and if notone of zero, exactly as in our basic model.

Let us start with a setting in which theexpert cannot price discriminate among

consumers. Without price discrimination,the expert chooses equal markup prices.Why? Because consumers infer the expert’sincentives from the intervention prices.Thus, the expert cannot gain from cheating.Consequently, the best she can do is to postan equal markup tariff and to provide seriousdiagnosis and appropriate treatment. Withan equal markup tariff, the monopolisticexpert is interested in two variables only, inthe magnitude of the markup and in thenumber of visiting customers. If consumersare very similar, the expert finds it profitableto serve all consumers. Otherwise, prices aresuch that some consumers do not consult hereven though serving them would be effi-cient. This is nothing but the familiarmonopoly-pricing inefficiency. We record itin Lemma 4.

LEMMA 4: Suppose that Assumptions C(Commitment) and V (Verifiability) hold,and that Assumptions H (Homogeneity)and L (Liability) are violated. Further sup-pose that a single expert (n = 1) who cannotprice-discriminate among customers servesthe market, and that consumers differ intheir expected cost of efficient treatmentonly. Then, in the unique subgame-perfectequilibrium, the expert posts and chargesequal markup prices (p− − c– = p− − c–). If thedifference in expected cost between the bestand the worst type is large relative to theefficiency gain of treating the worst type(c– − c– > (v − d − c–)f(1)), then prices aresuch that (i) high cost consumers decide toremain untreated (p− > v − d) and (ii) allother types visit the expert (p− < v − d) and

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27 That the condition −c − −c > (v − d − −c)f (1) is necessaryand sufficient for an interior solution is easily verified bychecking that the expert’s profit is an increasing function of∆ at ∆ = v − −c − d if and only if this condition is satisfied.

get serious diagnosis and appropriate treat-ment. Otherwise all consumers are effi-ciently served under equal markup prices(p− ≤ v − d).

PROOF: From the proof of Lemma 1 weknow that, for given net utilities for the con-sumers, the monopolist’s profit is highestwith an equal markup tariff. Thus, themonopolist will choose such a tariff and shewill provide the appropriate treatment to allof her customers. With an equal markup tar-iff, the monopolistic expert is interested intwo variables only, in the magnitude of themarkup (p− − c– = p− − c– = ∆, say) and in thenumber of visiting consumers. The resultthen follows from the observation that theexpert’s problem is nothing but the familiarmonopoly pricing problem for demandcurve D(p−) = F[(v −p− − d)/(c– − c–)] and netrevenue per customer ∆ = p− − c–.27

�

For our next result, we allow the expert topost more than one tariff. Since consumersdiffer in their expected cost of efficienttreatment, the monopolist might want to tar-get a specific tariff for each consumer or atleast different tariffs for different consumergroups. However, in the absence of informa-tion about the identity of each consumer(the expert only knows the aggregate distri-bution of probabilities of needing differenttreatments), the expert must make sure thatconsumers indeed choose the tariff designedfor them and not the tariff designed forother consumers. This puts self-selectionconstraints on the set of tariffs offered by themonopolist. As our next result shows, themonopolist uses the quality of advice offeredas a self selection device.

LEMMA 5: Suppose that the generalconditions of Lemma 4 hold except that theexpert can now price discriminate amongconsumers (rather than being restricted toa single price vector only). Then, in any

subgame-perfect equilibrium, the expertposts two tariffs, one with equal markupsand one where the markup for the majortreatment exceeds that for the minor one.28

Both tariffs attract customers and in totalall consumers are served. Low cost con-sumers are served under the former tariffand always get serious diagnosis andappropriate treatment; high cost con-sumers are served under the latter andalways get the major treatment, sometimesinefficiently.

PROOF: The proof parallels Dulleckand Kerschbamer’s (2005b) proof of (their)Proposition 2. We reproduce it here in theappendix to this paper.�

When price discrimination is possible,the expert finds it optimal to provide high-quality diagnosis and appropriatetreatment to low-cost consumers only.High-cost consumers are potentiallyovertreated; that is, they are induced todemand a high-quality intervention withoutan honest diagnosis.

Why is such a policy optimal? The reasonis simple. Under perfect price discrim-ination, the monopolist would providehigh-quality diagnosis and appropriatetreatment to all consumers and gain theentire surplus by charging customer specif-ic prices. In the absence of informationabout the identity of consumers, perfectdiscrimination is infeasible. With imperfectdiscrimination, the expert sells seriousdiagnosis and appropriate treatment at arelative high price to low-cost consumersonly. For the rest of the market, this policyis unattractive since the expected price islarger than the valuation of a successfulintervention. Offering efficient diagnosisand treatment at a lower expected price tothe residual demand is impossible becausethis would induce low-cost consumers toswitch to the cheaper policy. So, the expert

28 The menu may contain some redundant price vectorstoo, i.e., some tariffs that attract no consumers.

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offers, in addition to the expensive efficientdiagnosis and treatment policy, a cheaperbut also less efficient one. Her difficulty indesigning this second policy is to preventthe low-cost segment from choosing thecheaper policy. The solution is to potential-ly overtreat the residual demand consistingof high-cost consumers; that is, to inducethem to demand a high-quality interven-tion (a new engine) without a serious diag-nosis. For low-cost consumers, this policy isunattractive since their problem is mostlikely to be a minor one, implying that buy-ing the expensive efficient diagnosis andtreatment policy still entails a lower expect-ed cost than buying the high quality inter-vention (without an honest diagnosis) at abargain price.

While the equilibrium behavior outlinedin Lemma 5 is obviously an abstraction andit is probably impossible to point out anindustry that features exactly this kind ofprice discrimination. the result identifiesan element that may be present in the con-duct of some industries. The IT industry,for example, features some degree of sec-ond degree price discrimination: there arePC manufacturers who distribute theirequipment through IT warehouses thatoffer only selected qualities of equipmentat a relatively low price and through spe-cialized dealers that offer the entire assort-ment as well as advice on choosing the rightquality; some consumers (presumably theless profitable ones) buy from warehousesellers, others consult an expert seller andget serious diagnosis and appropriateequipment.

The equilibrium outlined in Lemma 5 isessentially the overtreatment equilibrium ofDulleck and Kerschbamer (2005b). Dulleckand Kerschbamer go on to show that theresult changes dramatically when con-sumers differ in their valuation for a suc-cessful intervention (rather than in theirexpected cost of efficient treatment). In thiscase, no overtreatment will be observed butundertreatment appears.

29 To the best of our knowledge there are only threefurther contributions with heterogeneous consumers, oneof them is the more verbal paper by Darby and Karni(1973), the second is the 1993 article by Pitchik andSchotter, and the third is Fong (2005). In the former twoarticles, heterogeneity is only used to purify a mixed strat-egy equilibrium; in the third contribution, consumers’ lackof commitment power plays an important role for theresult—we therefore relegate the discussion of that articleto the next subsection.

Dulleck and Kerschbamer’s undertreat-ment result stands in sharp contrast to thefindings in another credence goods paperthat has the verifiability assumption andheterogeneous consumers.29 In a modelwith capacity constrained experts who pro-vide procedures to consumers who differ intheir valuation for a successful interven-tion, Richardson (1999) finds that all treat-ed consumers are overtreated; that is, theyget a high-quality intervention independ-ently of the outcome of the diagnosis. Acloser look at his paper reveals differentdriving forces behind the Richardsonovertreatment and the Dulleck andKerschbamer (over- and undertreatment)results. The Dulleck and Kerschbamerresults are driven by a combination of mar-ket power and the ability to price discrimi-nate among heterogeneous consumers.Obviously, the type of treatment must alsobe verifiable (otherwise overtreating wouldbe strictly dominated by overcharging). Bycontrast, Richardson’s findings result froma lack of commitment power. Consider anexpert who can ex ante precommit only tothe price for a low quality basic interven-tion. At the diagnosis stage, the expert cantell the consumer that the basic interven-tion is insufficient to cure his problem andinform him about the additional amount hewould have to pay if he accepts an upgradeto a more advanced procedure. The con-sumer knows that he might have a seriousproblem and that the basic interventionfails in this case. He is therefore preparedto pay some additional amount for astronger intervention. If this amountexceeds the difference in treatment costs

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30 Here, remember that equal markup prices are nec-essary to induce an expert to perform serious diagnosis andto provide the appropriate treatment (see the discussion ofLemma 1).

(as is the case under Richardson’s assump-tions), then the expert has an incentive toalways recommend a stronger treatmenteven if the basic intervention would havebeen sufficient to cure the problem.30

Before proceeding further, notice thatthe inefficiencies of lemmas 4 and 5 disap-pear if experts have no market power. Withn > 1, price-competing experts provide bothtypes of treatment at marginal cost leavingno leeway for inefficiencies of any kind.Note here that the relevant condition is notn = 1, but rather that experts have marketpower in providing treatments. In a modelin which capacity is required to serve cus-tomers (cf. e.g., Emons 1997 and 2001 orRichardson 1999), experts have marketpower (independently of n) whenever tightcapacity constraints hamper competition.Similarly, consumer loyalty, travel coststogether with location, search costs, collu-sion and many, many other factors mightgive rise to market power.

We summarize the results of this subsec-tion in the following proposition.

PROPOSITION 2: Subgame-perfect equi-libria exhibiting inefficient rationing and/orinefficient treatment of some consumergroups can exist if Assumption H(Homogeneity) is violated and experts havemarket power.

5.2 No Commitment: Overcharging andDuplication of Search and DiagnosisCosts

In this subsection, we drop the commit-ment assumption. Under certain condi-tions, this gives rise to two different typesof equilibria—overcharging equilibria andspecialization equilibria. In both, the cre-dence goods problem manifests itself ininefficiently high search and diagnosis costsas some consumers end up visiting morethan one expert and being diagnosed more

31 If experts are not committed to treat their customersafter having conducted a diagnosis, then a lower bound forthe price of the expensive treatment (−p ≥ −c) emergesendogenously. So only the upper bound for prices (−p ≤ −c + d)is required in this case to prove part (i) of the lemma.

than once. We begin with the overchargingscenario. To get the overcharging resultstudied in the literature, firms’ price settingpower needs to be restricted. In part (i) ofLemma 6, this is captured by referring to a(legal) rule requiring experts to choose theprice for the expensive treatment from agiven range of cost-covering prices. Such arule is, of course, unrealistic. Part (ii) of thelemma shows that the overcharging equi-librium ceases to exist if prices are fullyflexible.

LEMMA 6: (i) Suppose that AssumptionsH (Homogeneity) and L (Liability) hold, andthat Assumptions C (Commitment) and V(Verifiability) are violated. Further supposethat there is some competition in the cre-dence goods market (n ≥ 2), that economiesof scope between diagnosis and treatmentare relatively low (d < (c– − c–)(1 − h)), andthat a (legal) rule is in effect requiringexperts to choose the price for the expensivetreatment from a given range of cost-cover-ing prices (p− ∈ [c–, c– + d]).31 Then thereexists a symmetric weak perfect Bayesianequilibrium in which experts overchargecustomers (with strictly positive probabili-ty). In this equilibrium, experts post pricessatisfying p− = c– + ∆ and p− = c– > c– + ∆ .Experts always recommend the expensivetreatment if the customer has the majorproblem, and they recommend c– with proba-bility ω ∈ (0,1) if the customer has the minorproblem. Consumers make at least one, atmost two visits. Consumers at their first visitalways accept a c– recommendation, and theyaccept a c– recommendation with probability� ∈ (0,1) and reject it with probability (1 −�). Consumers who reject, visit a second(different) expert, and on that visit theyaccept both recommendations with certain-ty. A customer who accepts to be treatedalways gets the appropriate treatment.

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32 For experts to be indifferent between recommend-ing the minor and recommending the major interventionwhen the customer has the minor problem, the probabili-ties � and ω, and the markup ∆ must satisfy ∆ [1 + ω(1 −�)] = (−c − −c)[� + ω (1 − �)]. To understand this equation,note that recommending the cheap treatment guarantees aprofit of −p − −c = ∆ > 0 for sure, while recommending theexpensive treatment when only the cheap one is requiredis like playing in a lottery yielding a payoff of −p − −c > ∆ withprobability [� + ω (1 − �)] / [1 + ω (1 − �)], and zero oth-erwise. This probability takes into account that a fraction1/[1 + ω (1 − �)] of customers are on their first visit (andhence are accepting the −c recommendation with probabil-ity �), while the remaining fraction ω (1 − �)/[1 + ω (1 − �)]are on their second visit (accepting the −c recommendationfor sure).

(ii) The strategy profile sketched in part (i)of this lemma ceases to form part of a weakperfect Bayesian equilibrium, if experts arecompletely free in choosing prices.

PROOF: See the appendix.�

In the setting under consideration, verifi-ability is violated. In this case, overtreat-ment is a strictly dominated strategybecause the higher cost of the expensiverepair does not affect the payment of thecustomer. Also, undertreatment is no issuebecause of the liability assumption. So eachexpert efficiently serves her customers.There remains the incentive to overcharge,that is, to claim that a major intervention isneeded when a minor one fixes the prob-lem. Such an incentive exists in the equilib-rium characterized in Lemma 6 since −p > −p.In the equilibrium, experts do not over-charge their customers all the time (butonly with strictly positive probability)because recommending the cheap treat-ment guarantees a positive profit of −p − c−= ∆ > 0 for sure, while recommending theexpensive treatment when only the cheapone is required is like playing in a lotteryyielding a payoff of −p − −c > ∆ if the con-sumer accepts, and zero, otherwise. By con-struction, the expected payoff under thelottery equals ∆, making experts exactlyindifferent between recommending honest-ly and overcharging.32

The overcharging equilibrium sketched inpart (i) of Lemma 6 is essentially the equi-librium outlined by Pitchik and Schotter

33 Sülzle and Wambach’s (2005) paper is essentially acomparative statics study on the overcharging equilibriumof Lemma 6. The main focus is on the effect of insurancearrangements on the (mixed strategy) equilibrium behaviorof consumers and experts.

34 See Andreu Mas-Colell, Michael Whinston, andJerry R. Green (1995) for the definition of weak perfectBayesian equilibrium. Roger B. Myerson (1991) calls thesame concept a weak sequential equilibrium.

(1987)—and further studied by Sülzle andWambach (2005)—for a setting with exoge-nously given payoffs.33 Wolinsky (1993)argues that the equilibrium might also existwith flexible prices if sufficiently few expertscompete for customers. Lemma 6 shows thatthe overcharging configuration of part (i)continues to form part of a weak perfectBayesian equilibrium if experts have somefreedom to choose prices, but that it ceasesto form part of such an equilibrium if pricesare fully flexible.34

The reason for why overcharging equilib-ria with the essential features as outlined inpart (i) of the lemma cease to exist if expertsare completely free in choosing prices is thatthe markup −p − −c = ∆, which is necessary tosupport the described behavior as a weakperfect Bayesian equilibrium, invites priceundercutting by a deviating expert who spe-cializes in the minor intervention. How can adeviating expert specialize in the minorintervention? In the present model whereeach expert’s decision variables are her post-ed prices and her recommendation policy,specialization on the minor interventionoccurs via the commitment to a tariff whichinduces customers who are diagnosed torequire the major intervention to rejecttreatment and to visit another expert. Howdoes such a specialization tariff look? In theequilibrium of Lemma 6, all experts post (−p, −p) = (−c + ∆, −c). Thus, by posting a tariffthat has −p > −c + d, a deviating expert cancredibly signal that she will provide reliablediagnosis and only the minor treatment. Thepoint is that, when the customer has themajor problem, the expert cannot sell theminor intervention because of liability. Thus,she will reveal his true condition. And, if the

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customer has the minor problem, the expertwill not recommend the major treatmentbecause the customer would reject it and hewould visit a nondeviating expert (if heaccepts he pays −p > −c + d, if he rejects andvisits a nondeviating expert his cost is −c + d).Under the conditions of part (i) of Lemma 6,such specialization tariffs are infeasible. Ifthey become feasible because prices areflexible (as in part (ii) of the lemma) theovercharging configuration ceases to formpart of a weak Bayesian equilibrium.

The “equilibrium with fraud” discussedby Darby and Karni (1973) resembles apurified version of the overcharging equilib-rium of Lemma 6. In their analysis “ . . .increasing the amount of services pre-scribed on the basis of the diagnosis,increases the probability of entering thecustomer’s critical regions for going else-where [ . . . ] Taking this consideration intoaccount, the firm will take fraud up to thepoint where the expected marginal profit iszero” (p. 73). Adapting Lemma 6 to an envi-ronment where consumers are heteroge-neous with respect to their search costwould yield an equilibrium with these prop-erties (see Pitchik and Schotter 1993). Theanalogy is not perfect, however, as the prob-lem discussed by Darby and Karni is that ofovertreating and not that of overchargingcustomers. Overtreatment can only poseproblems if the customer can observe andverify the type of treatment he gets (ourAssumption V), since, if he cannot, over-charging is always more profitable. But, withverifiability and flexible prices there are noequilibria exhibiting fraud, as Lemma 8below shows. In other words, to support theDarby and Karni (1973) configuration withfraud as an equilibrium, payoffs again needto be exogenously fixed as they are in theirenvironment.

A result that resembles a degenerate ver-sion of the overcharging equilibrium ofLemma 6 is the “no-cheating result” ofFong (2005). In Fong’s basic model, there is a single expert who offers two types of

35 Since in Fong’s model there is a single expert,Assumption C is not required to get this result.

36 With a fixed price below −c, the expert turns downconsumers with a major problem; with a fixed price above−c, no consumer will ever consult the expert (as −c is abovev − d by Fong’s first departing assumption).

37 If the c− recommendation is accepted for sure, theexpert will always recommend c−; if only c− recommenda-tions are accepted, she will always recommend c−. Thus, inan equilibrium in pure strategies a price vector with p− ≠ p−is equivalent to a fixed price contract.

treatment to a continuum of homogeneousconsumers in an environment in which lia-bility holds while verifiability is violated.Since the expert is completely free in choos-ing prices, our model would predict that shesets a single price for both treatments at theconsumers’ net valuation for a successfulintervention (that is, −p = −p = v − d; seeLemma 2).35 However, this solution is infea-sible in Fong’s setup due to a combinationof two assumptions: First, consumers’ valu-ation for a successful intervention isassumed to be small in comparison to treat-ment costs. Translated to the present con-text, Fong’s assumption amounts tov − d < −c < v, while we assume −c < v − d. Thismodification alone would not change any-thing provided that it is ex ante efficient tovisit the expert (i.e., v − d − −c − h(−c − −c) ≥ 0):The experts would still offer to fix bothtypes of problem at the price v − d cross-subsidizing the losses made on selling c− bythe gains made on selling −c. Fong’s seconddeparture from our assumptions—that theexpert has the option to reject a customerafter performing a diagnosis—implies thattreatment prices cannot be lower than treat-ment costs and thereby prevents this (andany other) fixed price solution.36 Also, purestrategy equilibria in which the expert poststwo different prices cannot exist because insuch an equilibrium it must be commonknowledge that only one of the prices willbe charged by the expert.37 It remains, asthe only possibility, an equilibrium in mixedstrategies. In such an equilibrium, theexpert posts two prices (−p = v − d / (1 − h),−p = v) and nevertheless behaves honestly.

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38 In Fong’s model, where consumers who reject do notreenter the market, for the expert to be indifferentbetween recommending the minor and recommendingthe major treatment when the customer has the minorproblem, the probability � must satisfy � = (_p − _c)/(

_p − _c)

= [_v − _d/(1 − _h) − _c]/(_v − _c).

The intuition behind this “no-cheatingresult” parallels that for Lemma 6. Theexpert does not recommend the majorintervention when the minor one fixes theproblem because customers accept themajor treatment with a low enough proba-bility while they accept the minor one forsure.38 The only difference to the over-charging equilibrium in Lemma 6 is thatconsumers who get a −c recommendation arenot kept indifferent between accepting andrejecting by the expert’s cheating behaviorbut rather by the fact that the price chargedfor the major treatment equals exactly theirvaluation for a successful intervention (−p = v). Although there is no duplication ofsearch and diagnosis costs in Fong’s model,the equilibrium is still inefficient because afraction of consumers who need the majortreatment remain untreated to keep theexpert indifferent.

Based on this basic model, Fong derivestwo overcharging equilibria, both based onidentifiable heterogeneity among con-sumers. To see how they work, assume thatthere exist two groups of consumers—onewith valuation −v, the other with valuation −v > −v, where −v − d < −c—and that a con-sumer’s valuation is observable to theexpert. Obviously, from the expert’s pointof view it would be ideal if she could pricediscriminate between the two groups. Thisis assumed to be impossible; that is, theexpert is constrained to post a single pricevector only. With a single vector, she canimplement the no-cheating result for a sin-gle customer group only. If the fraction of−v consumers is large, she will implementthe no-cheating result for those consumersand ignore the rest of the market. By con-trast, if there are many −v consumers, theexpert tailors the no-cheating tariff to their

39 One difference between the overcharging equilibri-um of Lemma 6 and Fong’s overcharging equilibriumdeserves mentioning: In the equilibrium of Lemma 6, con-sumers’ incentive to reject a c− recommendation increasesin ω because a higher ω implies that the consumer has ahigher probability to pay less for the treatment on his sec-ond visit. By contrast, in Fong’s model consumers have noopportunity to visit a second expert. Thus, an additionalassumption is needed to keep consumers indifferent;namely that the loss born by the consumer if his problemis left untreated is greater if c− rather than −c is required.

40 In the Wolinsky (1993) model, where experts committo an unobservable recommendation policy at the price-posting stage of the game, an additional restriction onbeliefs is required to prove the result. In the presentmodel, that requirement is implied by the notion of perfectBayesian equilibrium.

valuation (−p = −v − d /(1 − h), −p = −v) and shewill sometimes overcharge −v consumers tokeep them indifferent between acceptingand rejecting a −c recommendation (exactlyas in our Lemma 6).39

To summarize, our analysis shows thatmany specific assumptions made by Fong(e.g., that there is no diagnosis cost or thatthe treatments for the two problems are notsubstitutable) are not important for his find-ings. What is important, however, is that lia-bility holds, while verifiability is violated,that consumers are not committed, that theexpert is not committed either, and that con-sumers’ valuation for successful treatment israther low.

Overcharging equilibria with the essentialfeatures as outlined in Lemma 6 cease toexist if there is more than one expert and if experts are free in choosing prices. With fully flexible prices, a continuum of experts, and low economies of scopebetween diagnosis and treatment, the onlyperfect Bayesian equilibria that survivewhen Assumptions H and L hold whileAssumptions C and V are violated are spe-cialization equilibria similar to the one out-lined in the next lemma. This has beenshown by Wolinsky (1993).40

LEMMA 7: Suppose that Assumptions H(Homogeneity) and L (Liability) hold andthat Assumptions C (Commitment) and V(Verifiability) are violated. Further suppose

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41 If experts are not committed to treat their customersafter having conducted the diagnosis, the condition d <(−c − −c)(1 − h) changes to d < (−c − −c)(1 − h)/h.

that there is enough competition in the cre-dence goods market (n ≥ 4) and thateconomies of scope between diagnosis andtreatment are relatively low (d < (−c − −c)(1 − h)).41 Then there exists a perfectBayesian equilibrium exhibiting specializa-tion. In this equilibrium, some experts (atleast two) post prices given by −p = −c and −p > −c + d (we call such experts “cheap experts”)and some other experts (again at least two,“expensive experts”) post prices given by p− ≤ −p = −c. Cheap experts always recommendthe appropriate treatment, expensive expertsalways the expensive one. Consumers firstvisit a cheap expert. If this expert recom-mends −c, the customer agrees and gets −c. Ifthe cheap expert recommends −c, the cus-tomer rejects and visits an expensive expertwho treats him efficiently.

PROOF: See the appendix.�

In the specialization equilibrium ofLemma 7, liability (again) solves the prob-lem of undertreatment and the cost differ-ential −c − −c that of overtreatment. Theincentive to overcharge customers is elimi-nated because cheap experts lose their cus-tomers if they recommend the expensivetreatment (see the discussion above).

Is there any supporting evidence for verti-cal specialization as outlined in Lemma 7 inthe market place? Wolinsky (1993) mentionsthe automobile repair industry as an exam-ple of a market featuring some degree ofspecialization of this kind: there are back-yard garages who specialize in minor repairsand there are certified dealer garages whooffer the whole spectrum of treatments;some consumers (presumably those with alower opportunity cost of time) first visit abackyard garage and, if the problem turnsout to be a major one, they turn to a certifieddealer.

Note that the condition for the strategiesdescribed in Lemma 1 to form part of a

42 Glazer and McGuire (1996) characterize a similarequilibrium for a setting in which there are (by assump-tion) two types of experts, safe ones who can successfullyserve all consumers, and risky ones whose treatmentmight fail. The focus of their work is on optimal referralfrom risky to safe experts after risky experts’ diagnosis of aconsumer’s problem.

43 Wolinsky (1993) doesn’t explicitly impose the liabilityassumption. This assumption is implicit in his specificationof consumer payoffs, however.

perfect Bayesian equilibrium even if thecommitment assumption (Assumption C) isnot imposed and even if n ≥ 4 is exactly thatthe restriction imposed by Lemma 7 on d isviolated; that is, the diagnosis cost d mustexceed (1 − h)(−c − −c). To verify this, noticethat a deviation that might jeopardize theequilibrium of Lemma 1 is a specializationtariff that has −p < −c + h(−c − −c) and −p ≥ −c + h(−c − −c) + d. The expected cost to aconsumer who visits the deviator first and, ifrecommended the expensive treatment,resorts to a nondeviating expert is d + (1 − h)

−p + h[−c + h(−c − −c) + d]. Consulting only anondeviating expert, on the other hand,costs d + −c + h(−c − −c). Thus, to attract cus-tomers, the deviator must post a price vectorwith a −p such that d + −c + h(−c − −c) ≥ d +(1 − h) −p + h[−c + h(−c − −c) + d], which isequivalent to −p ≤ −c + h[(−c − −c) − d /(1 − h)].But, if d > (1 − h)(−c − −c), then such a pricevector doesn’t cover cost, and no deviation isprofitable.

The equilibrium outlined in Lemma 7 isessentially the specialization equilibrium ofWolinsky (1993), the only difference beingthat experts can refuse to provide a treatmentafter having conducted the diagnosis in theWolinsky model while they cannot refuse inour present framework.42 What drives theWolinsky result is the combination of twoassumptions, the assumption that consumersare neither able to observe the type of treat-ment they need nor the type they get (ourAssumption V is violated), and the assumptionthat experts are liable for providing the cheaptreatment when the expensive one is needed(our Assumption L holds).43 Specialization

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44 Since the uniform price charged by the monopolisticexpert under the conditions of Lemma 1 does not exceedconsumers’ gross utility v, they will not quit even if notcommitted, for their only alternative is to remain withoutany treatment.

equilibria cease to exist if Assumption L is violated and they also cease to exist ifAssumption V holds. We postpone the discus-sion of the case where neither liability norverifiability holds to the next subsection andrecord the rest of the result as Lemma 8.

LEMMA 8: Suppose that prices are flexi-ble, that consumers are homogeneous(Assumption H) and that the type of treat-ment is verifiable (Assumption V). Then theequilibria summarized in Lemma 1 (for thecase where Assumption L is violated) andLemma 3 (for the case where Assumption Lholds) remain the only perfect Bayesianequilibria even if Assumption C is violated.

PROOF: The proof is similar to that ofLemma 1 and therefore omitted.�

Why is it that specialization equilibriainvolving costly double advice exist if liabilityholds while verifiability is violated, while theycease to exist if verifiability holds? The pointis that verifiability allows experts to set treat-ment prices close to marginal cost. Withprices close to marginal cost, consumers haveno incentive to search for a second opinion.

For obvious reasons, perfect Bayesianequilibria exhibiting specialization and per-fect Bayesian equilibria in which expertsovercharge customers also cease to exist if asingle expert serves the market.44

We summarize the results of this subsec-tion in the following proposition.

PROPOSITION 3: Perfect Bayesian equi-libria exhibiting specialization and perfectBayesian equilibria in which experts over-charge their customers might exist ifAssumption C (Commitment) is violated.

5.3 Neither Liability Nor Verifiability: TheCredence Goods Market Breaks Down

In this subsection, we consider an envi-ronment resembling a hidden action version

45 Proposition 4 relies on the assumption that con-sumers are able to verify whether some kind of treatmenthas been provided or not. If this is not the case, eachexpert always has an incentive to provide no treatment atall and the credence goods market ceases to exist for anyvalue of v (see also the next footnote).

of George A. Akerlof’s (1970) lemons model:consumers can neither observe the type oftreatment they get (Assumption V is violat-ed), nor can they punish the expert if theyrealize ex post that the type of treatmentthey received is not sufficient to solve theirproblem (Assumption L is violated too).Under these adverse conditions, theexpert(s) always provide(s) the cheap (andcharge(s) for the expensive) treatment.Consumers anticipate this and consult anexpert only if the price of the expensivetreatment is such that getting the minorintervention at this price for sure increasestheir expected utility. If there is no −p ≥ −c thatattracts customers, no trade takes place andthe credence goods market ceases to exist.45

PROPOSITION 4: Suppose that AssumptionH (Homogeneity) holds and that AssumptionsV (Verifiability) and L (Liability) are violat-ed. Then there is no perfect Bayesian equilib-rium in which experts serve customersefficiently. If consumers valuation v is suffi-ciently high (−v ≥ (−c + d)/(1 − −h)), then eachexpert charges a constant price (given by tp =(1 − h)v − d if a single expert provides thegood, and by tp = −c if there is competition inthe credence goods market) and always pro-vides the cheap treatment. If the consumers’valuation is too low, then the credence goodsmarket ceases to exist.

PROOF: Obvious and therefore omitted.�

The assumption that one treatment ismore expensive than the second one isimportant for the negative result inProposition 4. Without this assumption(that is, if −c = −c), expert(s) have no incentiveto mistreat customers and therefore behavehonestly. This helps to explain the Emons(2001) results. In his 2001 contribution,Emons investigates the same basic model

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as in the 1997 paper (see our discussion insection 4). That is, again capacity isrequired to provide treatments to homo-geneous consumers, and once a givencapacity level has been installed bothtypes of treatment can be provided at zeromarginal cost up to the capacity con-straint. Again, one type of treatment con-sumes more units of capacity than thesecond. Also, there is no liability rule ineffect that protects consumers from get-ting an inappropriate cheap treatment.The major difference between the twoEmons contributions is (1) that the 2001article considers a credence goodsmonopolist while the earlier paper isabout competing experts, and (2) that the2001 article distinguishes between thecases of verifiable and unverifiable treat-ment, and between observable and unob-servable capacity while the earlier articledeals only with the case of verifiable treat-ment together with observable capacity.The major result of Emons (2001) is thatfor three out of the four possible constel-lations the monopolist always behaveshonestly. Only when capacity and treat-ment are both unobservable no tradetakes place.

Efficiency with homogeneous consumersand verifiable treatments is exactly whatone would expect given our Lemma 8.What is more intriguing is that Emonsobtains efficiency even in one of the non-verifiability cases. The intuition for thisresult is as follows. With observable capaci-ties, the expert can publicly precommit to atechnology (i.e., to a capacity level) thatallows her to provide the right treatment toall consumers at zero marginal cost (c− = c−= 0). Consumers observe capacity and,since they know that there is nothing themonopolist can do with her technology butto provide honest services, they trust theexpert and get honest treatment. By con-trast, with unobservable capacities, theexpert has no precommitment technologyat hand and the credence goods market

46 In the Emons paper, the credence good marketbreaks down even if consumers’ gross valuation v is high.The reason is the Emons assumption that not only the typeof treatment is unobservable, but also whether treatmenthas been provided at all or not. Under this assumption,expert’s capacity investment is zero irrespective of con-sumers’ valuation v. If only the type of treatment wereunobservable, as is the case in the present paper, and if vwere high enough, then the expert would install a capacitylevel that allows her to sell c− to all consumers, exactly asone would expect from our Proposition 4.

47 Wolinsky (1993) and In-Uck Park (2005) study cre-dence goods models where the reputation mechanismincreases efficiency. That the possibility of reputationbuilding may also go in the opposite direction has beenshown by Jeffrey C. Ely and Juuso Välimäki (2003) and byEly, Fudenberg, and David K. Levine (2005) in theirpapers on “bad reputation.”

breaks down.46 To summarize, our analysisshows that many specific assumptions madeby Emons (e.g., that capacity is needed toprovide treatments, that success is a stochas-tic function of the type of treatment provid-ed, etc.) are not important for his findings.What is important, however, for the positivepart of his result is that consumers are homo-geneous; with heterogeneous customers theeffects described in subsection 5.1 abovewould emerge and inefficiency would appear.

The equilibrium of Proposition 4 is ratherextreme and one is tempted to argue infavor of public intervention, e.g., the intro-duction of a legal rule that makes the expertliable for providing an inappropriately inex-pensive treatment. In reality, liability rulesare far from being perfect mechanisms,however (see the discussion in section 6). Isthere a way out? In practice, experts mightbe kept honest by their need to maintaintheir reputation (if bad reputation spreads,reputation considerations might mitigate theproblem even if consumers are expected tobuy only once), or by their desire to retaincustomers who are expected to frequentlyneed a treatment.47

6. Discussion: Theory and Real WorldExamples

We started our analysis by showing thatexperts have an incentive to commit to

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prices that induce nonfraudulent behaviorand full revelation of their private informa-tion if a small set of critical assumptionshold. These conditions are (1) expert sellersface homogeneous customers (AssumptionH); (2) large economies of scope existbetween diagnosis and treatment so thatexpert and consumer are in effect commit-ted to continue with a treatment once adiagnosis has been made (Assumption C);and (3) either the type of treatment (thequality of the good) is verifiable(Assumption V) or a liability rule is in effectprotecting consumers from obtaining aninappropriately inexpensive treatment(Assumption L). This positive result temptsone into thinking that problems in credencegoods markets are not very prevalent.However, some of our organizing assump-tions sound more innocent than they reallyare. In this section, we make the link to realworld situations by returning to the keymotivating examples mentioned in theintroduction and discussing the plausibilityof Assumptions C, V, and L in each case(strictly speaking Assumption H is neversatisfied in practice). This section also pro-vides a table where the existing literature iscategorized according to our assumptions.

Let us start by considering the example ofcar repairs and other repair services. Withrepair services, strict liability is difficult toimpose in practice. For instance, an insuffi-ciently repaired item may work for sometime and problems may develop later on. Tomitigate the undertreatment problem insuch a situation, the liability needs to cover alonger period. But during this longer periodthe item may stop working for reasons unre-lated to the expert’s behavior. Also, anextended liability period introduces a moralhazard problem on the consumers’ side insituations where the eventual performanceof the product depends not only on theexpert’s but also on the consumer’s behavior:If the consumer is fully compensated for abreakdown during the extended liabilityperiod, then he has no incentive to take care

48 Casual observation suggests that seemingly less tech-nically able customers face a higher risk of becoming vic-tim of overtreatment than seemingly more able customers.See, for example, the article in the Los Angeles Times fromthe 23rd of February 2000, “Your Wheels, More WomenAre Beginning To Take A Peek Under The Hood” statingthat women are more likely to be defrauded by mechanics.

of the product (see Taylor 1995 for a modelcapturing this feature).

The verifiability assumption poses similarproblems. For instance, the advice that opensthe article urging customers of repair servicesto ask the expert for replaced parts is soundonly if the consumer can verify that the partsactually came from his product and if he hasenough knowledge to identify the parts asthose he is charged for. The latter problem ispronounced by the fact that most real-worldrepair settings feature a large number of pos-sible treatments (Is the presented part an oilor an air filter? Did my brakes get their drumsground and new brake shoes or just newbrake shoes?). Some customers may haveenough expertise to secure verifiability, othersdo not. Thus, technical expertise, or expert’sexpectation of its existence on the consumer’sside, may affect market outcomes.48 Theexisting literature has ignored consumers’heterogeneity in expertise so far. We thinkthis is an interesting area for future research.

Next consider the commitment assump-tion. This assumption may be a reasonableapproximation to reality for complicatedproblems where repair is more or less a by-product of diagnosis and where an addition-al diagnosis adds nothing to the informationthat is revealed during the repair processanyway (for example, if the problem is a bro-ken part in the gear box of a car). However,it may be violated in many other cases.

With respect to taxicab rides, the taxime-ter, registering time and distance travelled,ensures verifiability and paying on reachingyour destination ensures liability. Thisimplies that the only problem to worry aboutis overtreatment, that is, the driver’s incen-tive to take a circuitous route. Obviously,such an incentive only exists if the driver

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feels that her passenger is a stranger in thecity (and it is therefore often helpful todemonstrate one’s knowledge of the city bygiving the driver details on which route totake). Thus, again heterogeneity in con-sumers’ knowledge—some consumers needrecommendation and treatment; others canself-diagnose the problem and need only theexpert’s treatment—is an important featureof the market. For market segments withmany strangers, our model would predictthat drivers commit to prices where themarkups for shorter routes exceed those forlonger ones (Lemma 3). The usual two-parttariff consisting of a fixed fee and a meterapproximates such a tariff. The approxima-tion is not perfect, however. An importantaspect of the taxicab-rides market is that thecost of selling −c instead of −c varies with thetime of the day: during peak times theopportunity cost for the driver to use a cir-cuitous route is high, while it approximateszero during off-peak times. To incorporatesuch variations in opportunity cost into atime-invariable two part tariff, the variablepart of the tariff has to be set equal to zero sothat consumers essentially pay a fixed price.A fixed price over all destination is infeasibleof course, given the large heterogeneity inrides. However, for more homogeneousmarket segments fixed price contracts maywell be optimal. And in some segments theyindeed exist; for instance, in many cities taxi-cabs charge a fixed price for trips to andfrom the airport and from one city to thenext, independently of the exact pick-uplocation and/or the concrete destination.

Finally consider the commitment assump-tion. In the case of taxicab rides, this assump-tion is implicitly enforced by not asking thedriver for the route he plans to take.

Medical treatments offer the most compli-cated and maybe the most important envi-ronment. Strict liability is difficult to imposein this context. With many sicknesses there isno sufficient treatment, with others successis only random. Thus, a failing treatment isno perfect signal of undertreatment. Also,

49 We thank a referee for suggesting this discussion.He/she mentions federal laws—ERISA—that preemptstate suits; and state and (proposed) federal limits on med-ical malpractice awards. A (possibly too naive) alternativeexplanation for the undertreatment complaints againstHMOs is that they result from HMOs’ incentive to providean inexpensive treatment whenever it is sufficient. Bydoing so, they prescribe the inexpensive treatment moreoften than it is prescribed under an insurance scheme.

treatment success is often impossible or verycostly to measure for a court, while stillbeing observed by the consumer (how canone prove the presence/absence of pain, forinstance). In such a situation, a patient maymisreport treatment success to sue thephysician for compensation. Similarly, thephysician may claim that the treatment wassuccessful, even if she knows it was not. Withindividual physicians, the Hippocratic Oath(and/or intrinsic motivation) may work rea-sonable well as a substitute for formal liabil-ity. With profit seeking institutions, thissolution does not work. This may explain thecomplaints against HMOs in the UnitedStates that they often provide inexpensivetreatment where expensive ones are needed.The undertreatment problem is worsenedby the fact that HMOs have sought, andsometimes received, relief from liabilitythrough government intervention.49

Verifiability is also a too demandingassumption with medical treatments.Patients usually are either physically not ableto observe treatment—as during an opera-tion—or simply lack the education to verifythe treatment delivered by a physician (did Iget a root canal and a crown or just a crownfrom the dentist?).

The commitment assumption (as a short-cut for situations where there are largeeconomies of scope between diagnosis andtreatment) seems reasonable for some butnot for all cases. For instance, in surgery theassumption of profound economies of scopeis reasonable, whereas for the prescriptionand preparation of drugs it is not.

Insurance coverage seems to be a peculi-arity of the market for medical treatments

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and asks for some attention. On the onehand, patients have less incentive to proper-ly take care of themselves because healthrisks are covered by insurance. On the otherhand, they are less worried about overtreat-ment as they do not carry the full socialcosts. For treatments with positive sideeffects (for example, vitamin treatments ormassages), patients may even actively ask forovertreatment. But, given that overtreat-ment is often associated with negative sideeffects (for example, a chemo therapy in thecase of cancer), patients’ interests are at leastin some cases realigned with parsimonioustreatment. On the supply side, insurancecompanies may have an interest to colludewith physicians in expensive cases such thatsome patients get inefficiently undertreatedas long as this cannot be proven. All theseindirect incentive effects are not addressedin our model.

Table 3 provides a summary of our find-ings and categorizes the literature. First sup-pose that the homogeneity and thecommitment assumption hold. Then themarket provides incentives to deter fraudu-lent behaviour provided either the verifiabil-ity or the liability assumption is satisfied. Ifa liability rule protects consumers fromobtaining insufficient treatment while thetype of intervention is not verifiable thenexperts are tempted to overcharge cus-tomers, while to overtreat customers is astrictly dominated strategy because the high-er cost of the major intervention does notaffect the payment of the customer. In thiscase, a fixed price agreement is an efficientway to solve the credence goods problem(Case 1). Similarly, if verifiability holds sothat overcharging is no problem while liabil-ity is violated (so that experts might not onlybe tempted to over- but also to undertreatconsumers) then experts’ commitment toequal markup prices—i.e., to tariffs wherethe differences in intervention prices reflectthe differences in treatment costs—providesincentives for nonfraudulent behaviour(Case 2). If consumers can observe and ver-

ify the type of treatment they get (verifiabil-ity holds) and punish the expert if they real-ize ex post that the type of intervention theyreceived was insufficient to solve their prob-lem (liability holds too), then the onlyremaining problem is overtreatment. In thiscase, experts’ commitment to prices wherethe markup for the minor interventionexceeds (weakly) that for the major oneinduces truthful behaviour (Case 3).

If the homogeneity assumption is dropped,different consumers might be treated differ-ently: High quality advice and appropriatetreatment might be provided to the mostprofitable market segment only. Less prof-itable consumers might be induced todemand either unnecessary or insufficientprocedures (Case 4).

Dropping the commitment assumption(and assuming low economies of scopebetween diagnosis and treatment) opens upproblems of multiple diagnoses. Customerswho are not happy with the repair price pro-posed by the first expert they visit can searchfor a second opinion. This search for a secondopinion makes fraudulent behaviour by thefirst expert less attractive (she knows that shewill lose the business of some consumers shediagnoses as requiring a major intervention),but it also leads to duplication of search anddiagnosis costs. As table 3 reveals, droppingthe commitment assumption (and assuminglow economies of scope) alone is not enoughto get equilibria characterized by multiplediagnosis. The type of treatment has also tobe nonverifiable. Why is it that specializationequilibria involving costly double advice arisewhen Assumptions C and V are violated (as inCase 5 in the table) while efficiency prevailswhen Assumption C alone or Assumptions Cand L are violated (Cases 2 and 3)? Undernonverifiability an expert who has recom-mended the major intervention to a con-sumer with a minor problem will not have toprovide it anyway. Consequently, overtreat-ment is not a problem in this setting. Thereremains the incentive to overcharge. In anefficient equilibrium, this incentive is

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removed by experts’ commitment to charge aconstant price across treatments (Case 1 inthe table). The reason why this solution is notattainable with low economies of scope is thatthe constant price across treatments impliesthat consumers with minor losses subsidizethose with major losses. If consumers are notcommitted and if the cost of getting a secondopinion is low (low economies of scope) thiscross-subsidization invites ex ante creamskimming by a deviating expert who special-izes in the minor intervention (by committingto prices that induce the customer to rejectthe expensive recommendation with certain-ty, see Case 5 in the table). By contrast, if the

type of intervention is verifiable, treatmentprices can be set close enough to marginalcost to deter cream skimming (Case 2).Dropping assumptions C and V might alsolead to overcharging equilibria (Case 6 in thetable). In such an equilibrium, liabilityremoves the under-, the cost differentialbetween the interventions the overtreatmentincentive. So, only experts’ temptation toovercharge consumers remains. In equilibri-um experts do not overcharge their customersall the time (but only with strictly positiveprobability) because recommending thecheap treatment guarantees a small positiveprofit (−p − −c > 0) for sure (because customers

TABLE 3CREDENCE GOODS: ASSUMPTIONS, PREDICTIONS, AND LITERATURE

Assumptions∗ Equilibrium Equilibrium Case treated inCase dropped characterized by prices Section (S), Lemma (L), Prop. (P) Literature

1 V Full Efficiency and −p = p− S 4, L 2 Emons (1997)

2 L; L and C Truthful Revelation −p − −c = p− − c− S 4, L 1; S 5.2, L 8 Taylor (1995)

3 none; C of Private information −p − −c ≤ p− − c− S 4, L 3; S 5.2, L 8

Price Discrimination:

4 H and LOver- and/or −p − −c ≠ p− − c− S 5.1, L 5

Dulleck &

Undertreatment for some customers Kerschbamer (2005b)

(exp. having market power)

Specialization:some exp.: p− = c−, −p = �

Wolinsky (1993)

5 C and V Duplication of Searchother exp.: p− ≤ −p = −c

S 5.2, L 7 Glazer&McGuire (1996)

and Diagnosis Cost

Overcharging:

Duplication of search −p = −c > p− > c− Pitchik&Schotter (1989)

and diagnosis cost Wolinsky (1993)

6 C and V (when prices inflexible); S 5.2, L 6 Sülzle&Wambach (2003)

Some Consumers Untreated

(when valuation low and −p = v > p− = v − −d(1−−h)− Fong (2005)

monopolistic expert not committed)

Lemons Problem: Akerlof (1970)

7 L and V+ Undertreatment/ constant price S 5.3, P 4 Emons (2001)

MarketBreak Down

∗ - H: Homogenity - all customers have the same v and h- C: Commitment - customers are commited to undergo treatment once a diagnosis has been performed- V: Verifiability - customers observe and are able to verify to courts the quality of treatment received (rules out overcharging)- L: Liability - experts cannot provide −c when −c is needed (rules out undertreatment)

+- and the following combinations of Assumptions dropped: C, L&V; H, L&V; C, H, L&V

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accept such a recommendation with certain-ty), while recommending the expensive treat-ment when only the cheap one is required islike playing in a lottery yielding a high payoff(−p − −c > −p − −c) if the consumer accepts, andzero, otherwise. By construction, the expect-ed payoff under the lottery equals −p − −c, mak-ing the expert exactly indifferent betweenrecommending honestly and overcharging.As we have shown in subsection 5.2 this over-charging equilibrium ceases to exist if expertsare completely free in choosing prices. Thena deviating expert can again specialize on theminor intervention and thereby attract allconsumers on their first visit.

Whenever neither verifiability nor liabilityholds, an expert has an incentive to providelow quality but to charge for high quality. Inthis situation we have Akerlof’s (1970) marketfor lemons problem and the market maybreak down all together (Case 7 in table 3).

7. Conclusions

Information asymmetries in expert–cus-tomer relationships are an everyday life phe-nomenon. Education and experience giveexperts the ability to diagnose the exactneeds of customers who themselves are onlyable to detect a need but not the most effi-cient way to satisfy it. The information prob-lems in markets for diagnosis and treatmentsuggest that experts may be tempted todefraud customers. The present article hasshown that the price mechanism alone is suf-ficient to solve the fraudulent expert prob-lem if a small set of critical assumptions aresatisfied and that most existing results oninefficiencies and fraud in credence goodsmarkets can be reproduced by dropping oneof those assumptions.

Our systematic approach is new.Previous work has fostered the impressionthat the equilibrium behavior of expertsand consumers in the credence goods mar-ket delicately depends on the details of themodel. By contrast, the present paper hasshown that the results for the majority of

the specific models can be reproduced in asimple unifying framework.

Our analysis suggests that market institu-tions solve the fraudulent expert problem atno cost if (1) expert sellers face homoge-neous customers, (2) large economies ofscope exist between diagnosis and treatmentso that expert and consumer are in effectcommitted to continue with a treatmentonce a diagnosis has been made, and (3)either the type of treatment is verifiable or aliability rule is in effect protecting con-sumers from obtaining an inappropriateinexpensive treatment.

We have shown that inefficient rationingand inefficient treatment of some consumergroups may arise if condition (i) fails to hold,that equilibria involving overcharging of customers or specialization of experts—bothleading to a duplication of search and diag-nosis costs—may result if condition (ii) isviolated, and that the credence goods mar-ket may break down altogether if condition(iii) doesn’t hold.

Our model might be considered restric-tive in several respects. It rests on theassumption that there are only two possibletypes of problem, that only two types oftreatment exist, that treatment costs areobservable, that posted prices are take-it-or-leave-it prices, that experts can diagnose aproblem perfectly, and so on. This is certain-ly a justified criticism. Nevertheless, oursimple model is sufficient to derive mostresults of that class of models that have beenthe focus of research in the credence goodsliterature.50 Thus, our simple model pro-vides a useful benchmark for the develop-ment of more general frameworks whichallow for an assessment of the robustness ofthe results.

50 Our model is insufficient, however, to reproduce theWolinsky (1995) result, which relies on the assumptionthat posted prices are bargaining prices. It is also insuffi-cient to provide Taylor’s (1995) theoretical micro-founda-tions of several features observed in markets for diagnosisand treatment.

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51 We thank an anonymous referee for suggesting thisdiscussion.

With respect to robustness, a comparisonof common experience with the results ofthis paper suggests that something is missingfrom existing models of credence goods.51

As discussed earlier, one common problemin such markets is that of overtreatment, thatis, that a more expensive treatment is pro-vided than is needed. Suppose for example,that our car needs either a new fuse or a newengine. A common fear is that the garagewill sell us an engine, even when the fusewould be sufficient, because it is more prof-itable to replace the engine. Some of uswould suspect this will be the case even if wecan tell whether the car is fixed when thegarage is done (and we refuse payment if thecar is not fixed) and even though we can seewhether the engine has been replaced, i.e.,even if liability and verifiability are satisfied.Proposition 1 of this article tells us that suchfraud cannot occur if liability and verifiabili-ty, as well as homogeneity and commitment,are satisfied; whereas common experiencesuggests the opposite. What accounts for thisdissonance? The answer appears to be that,in such situations, prices should adjust sothat the firm can make just as much profit onselling us the fuse (when we only need thefuse) as on selling us the engine. What doesthis mean? For experts facing tight capacityconstraints (for example, work time) therequirement is that the markup on theengine must exceed the markup on the fuseby such an amount that the profit per unit ofcapacity consumed is the same for bothtypes of intervention. Such markups are like-ly to be feasible. By contrast, if experts holdidle capacities then the absolute markup hasto be the same for both types of treatment.But, are consumers really prepared to paythe same absolute markup for a fuse as foran engine? Common experience tells us thatthey are not, to the contrary, most of uswould punish an expert who charges a highabsolute margin on a fuse with malicious

gossip. Is there a way out? For some of us afeasible solution might be to follow the day-to-day advice mentioned in the beginning ofthe article: to ask the mechanic to put thereplaced part in the back of the car and toinspect the defect of this part. It remains tohope that future research comes up withsolutions for this problem that are both moreefficient (in the sense that consumers do nothave to investigate exchanged parts) and fea-sible also for the less technically adept. Forthe meantime, a valuable advice might be toconsult an expert featuring a lengthy queueof customers waiting for service.

Appendix

Proofs of lemmas 5, 6, and 7 follow.PROOF OF LEMMA 5: The proof pro-

ceeds in four steps. In Step 1 we show thatany arbitrary menu of tariffs partitions thetype-set into (at most) three subintervalsdelimited by cut-off values h10, h12, and h02,with 0 ≤ h10 ≤ h12 ≤ h02 ≤ 1 and either h12 = h02

or h02 = 1 (or both) such that (i) the optimalstrategy of types in [0,h10) is to choose a tar-iff where the markup for the minor treat-ment exceeds that for the major one, (ii) theoptimal strategy of types in (h10,h12) is todecide for an equal markup tariff, and (iii)the optimal strategy of types in (h12,1) iseither to choose a tariff where the markupfor the major treatment exceeds that for theminor one (h02 = 1], or to remain untreated(h12 = h02).

52 Our strategy is then to show inStep 2 that an optimal price-discriminatingmenu cannot have h10 = h12 (that is, there

52 The borderline types h10 and h12 are indifferentbetween the strategies of the types in the adjacent inter-vals (whenever such intervals exist). Here note that weallow for h12 = 1 (all consumers are served and no con-sumer chooses a tariff where the markup for the majortreatment exceeds that for the minor one), for h10 = h12 (noconsumer is attracted by an equal markup tariff), and forh10 = 0 (no consumer is attracted by a tariff where themarkup for the minor treatment exceeds that for the majorone). Price discrimination requires, however, that at leasttwo of the three relations hold as strict inequalities.

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53 An immediate implication of this observation is thatsuccessful price discrimination requires that some typesare mistreated with strictly positive probability. Why?Since at least two tariffs must attract a positive measure ofconsumers and since only one of them can be an equalmarkup tariff.

54 Under a tariff where the markup for the major treat-ment exceeds that for the minor one, neither the con-sumer nor the expert cares about the associated −p. Alltariffs in this group that have the same ∆02 can thereforebe thought off as being a single tariff without any loss ingenerality. The argument for tariffs where the markup forthe minor treatment exceeds that for the major one issymmetric.

must be an equal markup tariff which attractsa strictly positive measure of types), to shownext (in Step 3) that h10 = 0 whenever h10 < h12

(that is, the expert has never an incentive topost a menu where both an equal markuptariff and a tariff with a higher markup forthe cheap treatment attract types), and toshow in the end (Step 4) that the expert hasindeed always a strict incentive to cover astrictly positive interval by a tariff where themarkup for the major treatment exceeds thatfor the minor one (h12 < h02 = 1).

Step 1: First note that any arbitrary menuof tariffs can be represented by (at most)three variables, by the lowest ∆02 ≡ −p − −cfrom the class of tariffs where the markupfor the major treatment exceeds that for theminor one (we denote the lowest ∆02 in thisclass by ∆ l

02), by the lowest ∆10 ≡ −p − −c fromthe class of tariffs where the markup for theminor treatment exceeds that for the majorone (we denote the lowest ∆10 in this class by∆l

10), and by the lowest equal markup∆12 ≡ −p − −c = −p − −c from the class of allequal markup tariffs in the menu (denotedby ∆l

12). 53 To see this, note that each possi-

ble tariff is member of exactly one of thesethree classes and that a customer whodecides for a tariff in a given class willalways decide for the cheapest one.54 Animmediate implication is that each menu oftariffs partitions the type-set into the abovementioned three subintervals. This followsfrom the fact that the expected utility underthe ∆ l

02 tariff is type-independent (implying

that either h12 = h02, or h02 = 1, or both), whilethe expected utility under both, the ∆l

12 tariffand the tariff where the markup for theminor treatment exceeds that for the majorone, is strictly decreasing in h, and from v > −c − −c (implying that the ∆l

10-function issteeper than the ∆l

12-function).Step 2: To see that h10 < h12, suppose to the

contrary that h10 = h12. Then h10 > 0, sinceh10 = h12 = 0 is incompatible with price-dis-crimination (and since—by Lemma 4—anon-price-discriminating expert will alwaysdecide for an equal markup tariff). But sucha menu is strictly dominated since the ∆l

10

tariff can always be replaced by a tariff withequal markups of ∆12 = ∆l

10 + h10(v − −c + −c);the latter attracts exactly the same types asthe replaced one and yields a strictly higherprofit.

Step 3: To see that h10 = 0 wheneverh10 < h12, suppose to the contrary that0 < h10 < h12. Then the expert’s profit is strict-ly increased by removing all tariffs where themarkup for the minor treatment exceeds thatfor the major one from the menu. This fol-lows from the observation that (by themonotonicity of the expected utility—in h—under equal markup tariffs) all types in[0,h10) switch to ∆l

12 when all tariffs wherethe markup for the minor treatment exceedsthat for the major one are removed from themenu, and from the fact that the expectedprofit per customer is strictly higher under∆l

12 than under ∆l10 whenever 0 < h10 < h12,

since ∆l12 ≤ ∆l

10 is incompatible with theshape of expected utilities (∆l

12 ≤ ∆l10 implies

that v − d − −c − h(−c − −c) − ∆l12 > (1 − h)v − d −

−c − ∆l10 for all h > 0 contradicting h10 > 0).

Thus, h10 = 0 < h12 ≤ h02 ≤ 1. So, if price dis-crimination is observed in equilibrium, it isperformed via a menu that contains two tar-iffs, one with equal markups and a tariffwhere the markup for the major treatmentexceeds that for the minor one.55

55 The menu might contain some redundant vectorstoo, which can safely be ignored, however.

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Step 4: We now show that the expert hasalways a strict incentive to post such a menu.Consider the equal markup tariff posted bythe expert under the conditions of Lemma 4.The markup in this vector is at least ∆12 = v − d − −c, in an interior solution evenhigher. First suppose that the monopolist’smaximization problem under the conditionsof Lemma 4 yields an interior solution (i.e.,∆12 > v − d − −c). Then the expert can increaseher profit by posting a menu consisting oftwo tariffs, the one chosen under the condi-tions of Lemma 4, and a second tariff, wherethe markup for the major treatment exceedsthat for the minor one, with −p = v − d. Thelatter vector guarantees each type an expect-ed utility equal to the reservation utility of 0.Thus, all types that remain untreated underthe conditions of Lemma 4 will opt for itsince they are indifferent. Also, all typesserved under the conditions of Lemma 4 stillchoose the equal markup vector since v − d −−c − h(−c − −c) is strictly decreasing in h. Hence,since v − d > −c, and since all types in [0,1]have strictly positive probability, the expert’sexpected profit is increased.56 Now supposethat the monopolist’s maximization problemunder the conditions of Lemma 4 yields thecorner solution ∆12 = v − d − −c. Then again themonopolist can increase her profit by postinga menu consisting of two tariffs, a tariff wherethe markup for the major treatment exceedsthat for the minor one, with −p = v − d, and an equal markup tariff that maximizesπ(∆12) = ∆12F[(v − d − c− − ∆12)/(−c − −c)] + (v −d − −c)(1 − F[(v − d −c− − ∆12)/(−c − −c)]). Sinceπ(∆12) is strictly increasing in ∆12 at ∆12 = v −d − −c an interior solution is guaranteed.�

56 Here note that the expert can do even better byincreasing ∆ l

12. This follows from the observation that theexpert’s trade-off under the conditions of Lemma 4 isbetween increasing the markup charged from the types inthe segment of served customers and losing some types tothe unprofitable segment of not served consumers, whilethe trade-off here is between increasing the markupcharged from the types in the segment of customers servedunder the more profitable equal markup vector and losingsome types to the segment of customers served under theless profitable ∆ l

12 tariff.

PROOF OF LEMMA 6 Part (i): Firstnotice that, under the conditions of Lemma6, liability prevents undertreatment, and thecost differential (−c − −c) prevents overtreat-ment. So, each expert’s provision policy istrivial and we will therefore focus in the restof the proof on experts’ price-posting andrecommendation policy and on consumers’visiting and acceptance decisions. For therecommendation and acceptance behaviordescribed in part (i) of the lemma to poten-tially form part of a (weak) Perfect BayesianEquilibrium (henceforth PBE), the equilib-rium probabilities ω ∗ and �∗, and themarkup ∆∗ must satisfy

(1)

and

(2)

The first of these two equations guaran-tees that consumers who get a c− recommen-dation are indifferent between accepting and rejecting,57 the second equation guar-antees that experts are indifferent betweenrecommending −c and recommending c− when the consumer has the minor problem.58

∆∗∗ ∗ ∗

∗ ∗= −+ −( )

+ −( )( ) .c c� �

�

ωω

11 1

d c ch

h h= − −

−( ) −( )+ −( )

∗∗ ∗

∗( )∆1 1

1ω ω

ω

57 If a consumer rejects, he incurs an additional diagno-sis cost of d for sure. His benefit is to pay less for the treat-ment on his second visit with probability [(1 − ω∗)ω∗

(1 − h)]/[h + (1 − h)ω∗] because he has the minor problemwith probability [ω∗(1 − h)]/[h + (1 − h)ω∗] given that theexpert has recommended −c.

58 Recommending the cheap treatment guarantees aprofit of ∆∗ > 0 for sure, while recommending the expen-sive treatment when only the cheap one is required is likeplaying in a lottery yielding a payoff of −p − c− > ∆∗ withprobability [α∗ + ω∗(1 − α∗)]/[1 + ω∗(1 − α∗)] , and zerootherwise. This probability takes into account that a frac-tion 1/[1 + ω∗(1 − α∗)] of customers are on their first visit(and hence, are accepting the −c recommendation withprobability α∗), while the remaining fraction ω∗(1 − α∗)/[1 + ω∗(1 − α∗)] are on their second visit (accepting the −crecommendation for sure). Here notice that a consumerwho is indifferent between accepting and rejecting a −c rec-ommendation on his first visit will accept the −c recommen-dation on his second visit because the probability ofneeding the minor treatment is lower on the second thanon the first visit.

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Obviously, consumers who get a −c recom-mendation will accept with certainty. Also,since −p < −c, and since a −c recommendation isalways accepted, experts will always recom-mend c− when the consumer has the majorproblem. So, if prices are exogenously fixed at(−p, −p) = (−c + ∆∗, −c) and if ∆∗ is such that theprobabilities �∗ and ω∗ satisfying equations (1)and (2) above are in (0,1), then the behaviordescribed in the lemma forms part of a PBE.59

To show that this behavior continues toform part of a weak PBE even if experts arefree to choose −p while −p can vary within therange specified in part (i) of the lemma, wehave to specify out-of-equilibrium beliefsand strategies that support the equilibrium.Each expert’s strategy consists of a price vec-tor (−p, −p) and a recommendation policy forall possible price vectors. Each consumer’sstrategy consists of a visiting strategy for eachset of posted price vectors and an acceptancebehavior for each price vector. We first spec-ify consumers’ beliefs and acceptance behav-ior and experts’ recommendation policy foreach subgame beginning at the moment aconsumer visits an expert characterized byher price vector (−p, −p). Then we specify con-sumers’ visiting and experts’ price postingstrategy. In the end we verify that we haveindeed identified a weak PBE.

Let ω i denote the probability that theexpert recommends treatment −c given that consumer’s problem is diagnosed to bei ∈ {m, M}, where m stands for minor, andM for major. Also, let −µ (−µ) be the proba-bility a consumer assigns to the event thathe has the major problem given that theexpert has recommended treatment −c (−c).Similarly, let −� (−�) be the probability that aconsumer accepts the recommendation c− (−c). Finally, let k denote a consumer’sexpected cost if he follows the proposedequilibrium strategy (that is, k = (1 − h)(1 − ω ∗)(−c + ∆∗) + [h + (1 − h)ω ∗]−c + d =

59 Here note that for any (−c, c−, d, h) with d < (−c − c−)(1 − h) there always exists a ∆∗ such that both α∗ and ω∗

are in (0,1).

(−c + ∆∗) + (−c − −c − ∆∗)[h + (1 − h)ω ∗] + d)and let π denote experts’ profit per customer if they follow the proposed equi-librium strategy (that is, π = (1 − h)[1 + ω∗(1 − �∗)]∆∗). Suppose that con-sumers’ beliefs are correct at the proposedprice vector and that consumers’ beliefs atother price vectors are given by (i) −µ(−p, p−)= 1 and −µ(−p, −p) = 0 iff p ≤ d + (1 − ω ∗)(−c + ∆∗) + ω∗−c and −p ∈ [−c,−c + d); and (ii)−µ(−p, −p) = h and −µ(−p, −p) = 0 in any othercase. Further suppose that consumers’acceptance decisions are given by (i) −�(−p,−p) = 1 iff −p ≤ d + (1 − ω ∗)(−c + ∆∗) + ω∗−c,and −�(−p, −p) = 0 otherwise; and (ii) −�(−p, −p) = 1 iff either −p ≤ d + (1 − ω ∗)(−c + ∆∗) + ω∗

−c and −p ≤ −c + d, or p−> d + (1 − ω ∗)(−c +∆∗)+ω∗c− and −p ≤ k, and −�(−p, −p) = 0otherwise. Also suppose that deviatingexperts’ recommendation policy is given byωm(−p, −p) = ωM(−p, −p) = 1.60 Finally supposethat consumers’ visiting strategy prescribesnot to visit a deviating expert, and thatexperts’ price-posting strategy prescribesnot to deviate to a price vector differentfrom the proposed one.

Now we verify that we have indeed identi-fied a weak PBE. First observe that con-sumers’ acceptance strategies are indeedoptimal given their beliefs: If a single expertdeviates, the proposed price vector is stillavailable since at least two experts post thisvector in equilibrium. The expected cost tothe consumer under that vector is k withprior beliefs, it reduces to d + (1 − ω∗)(−c + ∆∗) + ω∗−c if the consumer believes toneed the cheap treatment for sure, and itincreases to −c + d if the consumer believes toneed the expensive treatment for sure.Given this, consumers’ acceptance strategies

60 Here note that for out-of-equilibrium price-vectorssatisfying −p ≤ d + (1 − ω∗)(c− + ∆∗) + ω∗c− and p− � [−c, c− + d],the proposed beliefs are not consistent with the proposedequilibrium strategies. Sequential rationality and full consis-tency of beliefs at all information sets (full PBE) wouldrequire that for all out-of-equilibrium price-vectors withthese properties experts and consumers play mixed strategiessimilar to the ones specified in Lemma 6.

Mar06_Art1 2/20/06 6:35 PM Page 39


are indeed optimal given their beliefs. Nextobserve that ωm(−p, −p) = ωM(−p, −p) = 1 isindeed optimal for a deviating expert, eitherbecause −�(−p, −p) = 1 and −p ≥ −c, or because

−�(−p, −p) = −�(−p, −p) = 0. Given deviatingexperts’ recommendation policy and −p ≥ −c,consumers obviously prefer to ignore anydeviating expert. This establishes that no devi-ation to an alternative price vector will attractany customers and so (and since π > 0) nodeviation will occur.

Part (ii): To verify part (ii) of the lemma itsuffices to specify a profitable deviation.Consider a deviation to a price vector (−p, −p)with −p ∈ (−c,d + (1 − ω∗)(−c + ∆∗) + ω∗ −c) and−p > −c + d. First observe that, for a consumerserved under such a price vector, it is optimal to set −�(−p, −p) = 1 and −�(−p, −p) = 0 forany belief that he might have.61 Given con-sumers’ acceptance behavior, sequentialrationality for the expert requires that shechooses ωm = 0 and ωM = 1.62 Given deviat-ing experts’ recommendation- and con-sumers’ acceptance-behavior, the expectedcost to a first-time consumer who plans tovisit the deviator is k̂ = d + (1 − h)−p + h(d + −c).63 So, if k̂ < k then the deviatorattracts all first-time consumers and servesthem at an expected profit of π̂ = (1 −h)(−p −−c) per customer. The inequality k̂ < k isequivalent to −p < −c + ∆∗ + ω∗(−c − −c − ∆∗) −dh/(1−h), which (using the indifferenceequations (1) and (2) above) is again equiva-lent to −p < −c + ∆∗ + dω ∗/[(1 − ω ∗)(1 − h)].Now suppose the deviating expert sets −p =−c + ∆∗ + dω ∗/[(1 − ω ∗)(1 − h)] − �, with �

61 This follows from the fact that the only alternativeprice vector available if a single expert deviates is (−p, −p) =(c− + ∆∗, c−) and that the cost to the consumer under thatvector is d + (1 − ω∗)(c− + ∆∗) + ω∗c− if the consumerbelieves to need the cheap treatment for sure while it isc− + d if the consumer believes to need the expensivetreatment for sure.

62 This follows from −p > c− (so that it is indeed optimal torecommend c− to a consumer with the minor problem) and

−p < c− (so that it is never optimal to recommend c− to a con-sumer who needs c−), where the latter inequality is impliedby consumers’ indifference between accepting and reject-ing a c− recommendation under (−p, −p) = (c− + ∆∗, c−) and by

−p < d + (1 − ω∗)(c− + ∆∗) + ω∗c−.

strictly positive but smaller than dω∗/[(1 − ω∗)(1 − h)]. Then she attracts all consumers and,thus, makes an expected profit π̂ > (1 − h)∆∗.The expected profit per expert in the pro-posed symmetric overcharging equilibriumis π /n = (1 − h)[1 + (1 − �∗)ω∗]∆∗/n which isstrictly less than (1 − h)2∆∗/n implying thatπ̂ < π/n even for n = 2. This establishes thatthe weak PBE sketched in part (i) of thelemma ceases to exist if experts are complete-ly free in choosing prices since a profitabledeviation always exists.�

PROOF OF LEMMA 7: Again, liabilityprevents undertreatment, and the cost dif-ferential (−c − −c) prevents overtreatment. Soeach expert’s provision policy is again trivialand we will therefore again focus on experts’price-posting and recommendation policyand on consumers’ visiting and acceptancedecisions. To show that the proposed strate-gies are part of a (full) PBE we have to spec-ify reasonable out-of-equilibrium beliefs andstrategies that support the equilibrium.Using the notation introduced in the proofof Lemma 6, we first specify consumers’beliefs and acceptance behavior and experts’recommendation policy for each subgamebeginning at the moment a consumer visitsan expert characterized by her price vector(−p, −p). Then we specify consumers’ visitingand experts’ price posting strategy. In theend we verify that we have indeed identifieda PBE.

Let k denote the expected cost to the cus-tomer if he follows the proposed equilibriumstrategy; that is, k = d + (1 − h)−c + h(−c + d).Suppose that consumers’ beliefs are correctat the proposed price vectors and that con-sumers’ beliefs at other price vectors aregiven by (i) −µ(−p, −p) = 0 and −µ(−p, −p) = 1 ifeither −p = −p ≤ −c + d, or −p > −c + d and

−p < −c; (ii) −µ(−p, −p) = h and −µ(−p, −p) = 1 if −p > −c + d and −p > −c; (iii) −µ(−p, −p) = 0 and

63 With probability (1 − h) the consumer has the minorproblem and gets treated for the price −p by the deviator;with probability h he has the major problem, the deviatorrecommends c−, the consumer rejects, visits a nondeviatorand gets the right treatment for the price c−.

Mar06_Art1 2/20/06 6:35 PM Page 40


−µ(−p, −p) is such that a customer who gets a −c recommendation is indifferent betweenaccepting and rejecting whenever −p ∈(k, −c + d) and −p < −c + d; and (iv) −µ(−p, −p) = 0 and −µ(−p, −p) = h in any other case.Also suppose that new consumers’ (first timevisitors’) acceptance decisions are given by(i) −�(−p, −p) =1 for −p ≤ k and −�(−p, −p) = 0 for

−p > k whenever −p > −c + d and −p > −c; and

−�(−p, −p) =1 for −p ≤ −c + d and −�(−p, −p) = 0 for

−p > −c + d whenever either −p ≤ −c + d or

−p ≤ −c; and (ii) −�(−p, −p) = 1 whenever −p < k; −�(−p, −p) is such that an expert who observesthat the customer has the minor problem isexactly indifferent between recommending c−and recommending −c whenever −p ∈(k, −c + d)and −p < −c + d; and −�(−p, −p) = 0 for any otherprice vector. Further suppose that theacceptance decisions of consumers on theirsecond visit are given by (i) −�(−p, −p) =1 for

−p ≤ k and −�(−p, −p) = 0 for −p > k whenever −p > −c + d and −p > −c; and −�(−p, −p) =1 for

−p ≤ −c + d and −�(−p, −p) = 0 for −p > −c + dwhenever either −p ≤ −c + d or −p > −c; and (ii) −�(−p, −p) = 1 whenever −p ≤ −c + d and −�(−p, −p) = 0 otherwise. Also suppose thatexperts recommend in accordance with con-sumers’ beliefs. Finally suppose that con-sumers visiting strategy prescribes not todeviate from the proposed visiting behaviorprovided no deviating expert offers either

−p < −c and −p > −c + d, or −p < −c, and thatexperts’ price-posting strategy prescribes notto deviate to price vectors different from theproposed ones.

Now we verify that we have indeed identi-fied a PBE. First observe that customersbeliefs reflect experts’ incentives: The expertrecommends honestly (i.e., ωm = 0 andωM = 1) if −p < −c + d and −p = −p, and if −p > −c + dand −p < −c, either since −�(−p, −p) = −�(−p, −p) = 1and −p = −p (in the former case) or since

−�(−p, −p) = 1, −�(−p, −p) = 0 and −p < −c (in the lattercase); the expert always recommends

−c (i.e., ωm = 0 and ωM = 0) if −p > −c + d and

−p > −c, since −�(−p, −p) = 0 and −p > −c ; the expertrecommends −c if the consumer has themajor problem and she randomizes between

recommending −c and recommending c− if hehas the minor problem (i.e., ωm ∈(0,1) andωM = 1) whenever −p ∈ (k,−c + d) and −p ∈(−c, −c + d) since −�(−p, −p) =1 and −�(−p, −p) is suchthat she is exactly indifferent between bothrecommendations; and the expert recom-mends the expensive treatment (i.e., ωm = 1and ωM = 1) in all other cases, either because�−(−p, −p) = 1 and −p > −p, or because −�(−p, −p) =−�(−p, −p) = 0. Next observe that customers’strategies are optimal given their beliefs.First consider consumers’ acceptance strate-gies. If a single expert deviates, the proposedequilibrium offers are still available since atleast two experts make each offer. Thus, theabove described acceptance strategies areoptimal given consumers’ beliefs. Next con-sider new consumers’ visiting strategy. If noexpert deviates, the relevant alternatives are(i) to visit a cheap expert first and to rejectthe −c recommendation as proposed, or (ii) tovisit an expensive expert first and to acceptthe −c recommendation. The former strategyhas an expected cost of d + (1 − h)−c +h(−c + d), the latter a cost of −c + d, while thebenefit is the same. Since (−c − −c)(1 − h)/h > (−c − −c)(1 − h) > d, the former cost isstrictly lower and customers’ visiting strategyis optimal if no expert deviates. Given thatthe equilibrium offers are still available if asingle expert deviates, and given the abovespecified beliefs, no customer has an incen-tive to visit a deviating expert if her postedprices do not satisfy either −p < −c and −p > −c + d, or −p < −c. Finally observe that noexpert has an incentive to deviate.Deviations to prices satisfying −p < −c and −p > −c + d are unattractive since only the

−c recommendation is accepted. Devia-tions to price vectors where −p > −c and −p ≥ −c are unattractive since they attract no customers. Price vectors where −p < −c are unprofitable too, if they only attractcustomers who first visit a cheap expert, and,if recommended the expensive treatment,resort to the deviator. The expected cost to the customer of this latter strategy is d + −c + h(p̂ + d − −c) where p̂ is the price

Mar06_Art1 2/20/06 6:35 PM Page 41


posted by the deviator for the expensivetreatment. Going directly to the deviator andaccepting her recommendation, on the otherhand, costs at most p̂ + d. Thus, in order to avoid being visited only by consumerswith the major problem, the deviation mustsatisfy d + −c + h(p̂ + d − −c) > d + p̂, which is equivalent to p̂ < −c + dh /(1 − h). To cover expected treatment cost the price p̂ must also satisfy p̂ ≥ −c + h(−c − −c). But,

−c + h(−c − −c) > −c + dh /(1 − h) since (1 − h)(−c − −c) > d. This proves that no deviation byan expert is profitable.�

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