Paper_Optimal Certification Design

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NES 20th Anniversary Conference, Dec 13-16, 2012
Article "Optimal Certification Design" presented by Sergei Kovbasyuk at the NES 20th Anniversary Conference.
Author: Sergei Kovbasyuk

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Paper_Optimal Certification Design

  1. 1. Optimal Certification Design Sergei Kovbasyuk∗ December 20, 2012 Abstract We study the ”seller-pays” business model of certification. In this model a pub- lic rating of a product is issued by a certifier who receives payments from a seller and partially internalizes buyers’ surplus. A feasible certification compatible with Bayesian updating by buyers is characterized when the payment to the certifier is: 1) fixed, 2) contingent on the rating and publicly known, 3) contingent on the rating and privately known to the seller and the certifier. We show that a feasible certi- fication under a fixed payment leads to uninformative rating and low welfare. In contrast, under a contingent public payment precise certification is feasible. The welfare is high. Finally, a contingent private payment implies imprecise ratings and can lead to rating inflation: the highest rating is issued for a wide range of qualities. The welfare is low. The welfare loss is caused by imprecise ratings and happens even though the buyers are fully rational. A desirable regulation of certifiers requires public payments.IntroductionA widespread business model for certifiers is the seller-pays paradigm, in which buyersaccess the ratings for free. A well-known example is the three biggest credit rating agencies ∗ EIEF (skovbasyuk@gmail.com). I am thankful to Jean Tirole for his advice. This paper benefitedfrom helpful comments by Jonathan Berk, Bruno Biais, Catherine Casamatta, Jacques Cr´mer, Larry eEpstein, Jana Friedrichsen, Mikhail Golosov, Bruno Jullien, Jean-Paul L’Huillier, Tymofiy Mylovanov,Marco Ottaviani, Nikita Roketskiy, Ilya Strebulaev, Daniele Terlizzese, Nicholas Trachter and seminarparticipants at Toulouse School of Economics, CREST, Einaudi Institute for Economics and Finance,Stanford GSB, Boston University, New Economic School, HEC Lausanne, University Pompeu Fabra,University of Zurich, Paris School of Economics, University of Mannheim, European University Institute,University of Copenhagen, CSEF University of Naples. The remaining errors are mine. 1
  2. 2. (Moody’s, Standard & Poor’s and Fitch), which provide public ratings of financial productsto investors for free. The agencies then charge the issuers a rating’s fee which is typicallyproportional to the volume of the issue. According to Partnoy (2006), around 90 percent ofrating agencies’ revenue is given by the fees paid by the issuers.1 The recent economic crisisraised concerns over the credibility of the seller-pays business model due to the conflict ofinterest it creates. Credit rating agencies were blamed for catering to issuers and inflatingthe ratings for asset backed structured products that were at the heart of the crisis.2 Ouranalysis demonstrates that this can happen even if per se the certifying agency cares aboutthe buyers’ and the buyers are fully rational. This paper examines the seller-pays certification assuming fully rational agents and noexternal forces that would make certain ratings highly valuable.3 Given that the rationalbuyers correctly infer the informational content of any rating, is there any flaw in the seller-pays business model? If yes, what is the nature of the flaw? Can a regulation mandating afixed certifier’s fee improve the situation? Shall a regulation require a transparent certifier’sfees or, as some credit rating industry participants put it, all relevant parties already knowthe fees and transparency is redundant? We answer the above questions in the following set-up. The seller of a product ofunknown quality applies for certification and commits to pay a fee t(r) to the certifier,possibly contingent on a rating r. The certifier issues a public rating which is free forthe buyers. The buyers update their beliefs about the product’s quality, the seller setsthe product’s price p(r) and the buyers purchase the desired quantity q(r). All agentsare Bayesian. The certifier partially internalizes the buyer’s welfare, due to liability orreputational concerns. He trades off the monetary payment from the seller against thebuyers’ surplus. We characterize feasible and optimal certification, and welfare in threedifferent environments 1) fixed payment, 2) payment contingent on the rating and publiclyknown, 3) payment contingent on the rating and private to the seller and the certifier. We document four findings. First, a certifier which internalizes the buyers’ surplus has 1 The fees correspond to 3-4 basis points of the volume of the issue for corporate bonds and go up to10 basis points for structured finance products. Similarly, investment banks when underwrite an initialpublic offering of shares advertise the offering among investors and charge the issuing firm a substantialfee. According to Ljungqvist et al. (2003), the fee constitutes around 7% of the volume of the issue forthe leading investment banks in the US. 2 For instance, of all asset-backed security collateralized debt obligations issued in 2005-2007 and ratedAAA by Standard & Poor’s, only 10% maintained AAA rating in 2009, while more than 60% weredowngraded below the investment grade International Monetary Fund (2009). Furthermore, Griffin andTang (2012) employ a model used by one of the main rating agencies and document that just before 2007the AAA tranches of CDOs were larger than what the rating agency model would deliver. 3 The conflict of interest in credit rating agencies may have been exacerbated by regulation based onratings Opp et al. (2012) or by seller’s forum shopping and trusting nature of some buyers Skreta andVeldkamp (2009), Bolton et al. (2012). The recent Dodd-Franck Act aims at reducing these tendencies. 2
  3. 3. a downward intrinsic bias: if the buyers were to take the certifier’s ratings at a face valuethe certifier would understate the true quality. Suppose the payment is fixed (t = const)and, hence, does not affect the certifier’s choice of a rating. A public rating affects theequilibrium price p(r), which increases with ratings, and the equilibrium quantity q(r). Inturn p(r) and q(r) affect the buyers’ surplus. The marginal effect of q on the buyers’ surplusis zero because q is chosen by the buyers to maximize their surplus. At the same time,the marginal effect of p on the buyers’ surplus is significant and negative. If ratings wereperfect, for a small understatement of quality the buyers’ surplus gain due to a decrease inprice would dominate the surplus loss due to a distortion in q(r). The certifier cares aboutthe buyers and has an incentive to downplay the quality in an attempt to lower the price. Second, we find that under a fixed payment only the uninformative certification is fea-sible. If ratings were perfect, then under a fixed payment the certifier would downplaythe true quality. But this is not consistent with rationality: in a Bayesian equilibriumthe ratings must be correct on average and cannot be systematically lower than the truequality. Therefore, perfect ratings are not feasible under a fixed payment. For a widerange of parameters we show than only the uninformative certification is feasible and thewelfare is low.4 Third, we find that the seller-pays business model performs well when the parties canfreely decide on payments provided that payments are publicly disclosed. If the certifierreceives a high payment for a high rating, then a rational buyer who observes the paymentshould infer that the rating might have been issued for money and interpret the ratingaccordingly. This in turn reduces the seller’s temptation to increase the payment for ahigh rating. We show that in the public payment environment a perfectly informativecertification is feasible. We solve for an optimal certification from the point of view ofthe seller. It results in a coarse rating and no payment for product qualities below somethreshold (which one can interpret as rejections) and a perfectly informative ratings withpositive and increasing payments above the threshold. We show that, contrary to conven-tional wisdom, an optimal certification under a public contingent payment leads to a moreprecise certification and to a higher welfare than a certification under a fixed payment.This is because a deliberately chosen contingent payment balances the certifier’s intrinsicbias and improves information revelation. Fourth, we find that rating inflation can occur when payments to the certifier are con-tingent on ratings and are not publicly disclosed. If buyers do not observe the payment,their perception of the ratings does not depend on actual payments. This creates an in- 4 In principle, in the sender-receiver cheap talk games inspired by Crawford and Sobel (1982) partiallyinformative equilibria may happen if the sender’s bias is small. We discuss this possibility in section 5.4 3
  4. 4. centive for the seller to elicit the highest rating by offering a generous compensation. Thecertifier, being offered a generous compensation for the highest rating, is willing to issuethis rating even when the actual quality of the product is not the highest. As a result,the highest rating is issued for a wide range of qualities (”pooling”) and is very imprecise.This phenomenon is akin to “rating inflation”, even though rational buyers correctly inferthe underlying quality of a product. We show that contingent private payments can bedetrimental for welfare even when the buyers are fully rational. The loss of welfare is notbecause the buyers are deceived (which does not happen under rationality) but becauseratings are imprecise and information is lost. The analysis points to possible improvements in regulation. It shows that public con-tingent payment causes no harm per se, while mandating an upfront fixed payment canreduce welfare. Instead a desired regulation would require disclosure of the certifier’s fee.For instance, currently credit rating agencies do not disclose their fees for individual deals.Indirect evidence suggests that these fees, especially the fees charged for new structuredfinance products, are substantial. Pagano and Volpin (2010) documents that Moody’sEBIT closely followed the popularity of new structured finance products: it increased from$541 million in the first quarter of 2003 to $1,439 million in the third quarter of 2007, andthen dropped to $683 million in the third quarter of 2009. Following the financial crisis theDodd-Frank reform increased the liability of credit rating agencies and required the dis-closure of their methodologies, yet it didn’t impose any regulation of compensations. Thisis striking given that the role of credit ratings in marketing financial products is almostas important as the role of underwriters, and underwriters are required to disclose theircommissions under the 1933 Securities Act. Buyers of securities can check underwriters’commissions in the prospectus published on the SEC website (www.sec.gov). In principle,similar requirements can be imposed on credit rating agencies. Our paper is related to Inderst and Ottaviani (2012), which analyses hidden payments toan information intermediary. Unlike our optimal contracting approach, their intermediarycannot provide buyers with precise information about the product via multiple ratings,but is bound to use pass-fail ratings: he can simply recommend the purchase or not.The intermediary panders to customers and advises them to buy the product they wouldlike, even when its production is inefficiently costly from a social welfare perspective. Bycontrast, in our model the intermediary cares about the net surplus of the buyers and isreluctant to give a high rating which leads to a high price. In Inderst and Ottaviani (2012)the kickbacks paid by the sellers can tilt the intermediary’s recommendations towards thecost-efficient product and improve social welfare. Authors conclude that caps and disclo-sure requirements on commissions may lower welfare. Conversely, our model with multiple 4
  5. 5. ratings stresses the adverse effect private payments have on information production. Toour knowledge, our paper is the first to study the endogenous partition of information inmultiple ratings under private contracts. On the methodological side our paper advances the literature on contracting for informa-tion by extending it to games among multiple receivers. First, the certifier’s communicationwith two strategic parties, the buyers and the seller, relates our paper to the literature oncheap talk with multiple audiences. In Farrell and Gibbons (1989) a sender S is informedabout the true state θ and sends a message m to two uninformed receivers: P and Q, whotake actions p and q respectively. Parties’ payoffs are represented in the Figure 1. Farrelland Gibbons show that the informativeness of communication between the sender S andreceiver Q can be limited if receiver P also receives the sender’s messages (subversion).In their model receivers do not interact: receiver P ’s action does not affect receiver Q’spayoff and vice versa. We expand Farrell and Gibbons (1989) in two ways. First, thetwo receivers interact strategically. Second, we allow one of the receivers (the seller) tocontract with the sender. Figure 1: Payoff structure Farrell-Gibbons This paper Krishna-Morgan sender S: m(θ) US (θ, q, p) US (θ, q, p) + t US (θ, p) + t (certifier) t(m) t(m) receiver P: p(m) UP (θ, p) UP (θ, q, p) − t UP (θ, p) − t (seller) game receiver Q: q(m) UQ (θ, q) UQ (θ, q, p) n.a. (buyers) As in Krishna and Morgan (2008), we allow public contingent contracts. Krishna andMorgan show that contingent contracts facilitate communication in the standard, single-receiver Crawford and Sobel (1982) model. By contrast, our two receivers play a gameafter receiving the message. Parties’ payoffs are recovered from primitives in a marketinteraction and are inherently interdependent. This makes communication nontrivial evenwhen the sender’s preferences are perfectly aligned with those of one receiver (the buyers).Differently from Krishna and Morgan (2008), the communication is not perfect becausethe second receiver (the seller) reacts to a sender’s announcement by choosing a priceand affects the first receiver’s (the buyers’) payoff. Furthermore, our study of privatecontracts between the seller and the certifier is also novel, as does not arise in a standardprincipal-agent (receiver-sender) set-up with only two parties. Our paper is also related to other contributions. Most of the literature on certification 5
  6. 6. takes one of two main approaches. Some contributions assume that certifiers can com-mit to a disclosure rule and ignore the issue of credibility. These papers are silent abouthow information production, credibility and welfare are affected when the certifier facesa contingent contract. Lizzeri (1999) finds that a monopolistic certifier who charges afixed fee for ratings discloses minimum information to ensure efficient market exchange.Similarly Farhi et al. (2012) argue that certifiers have no incentive to disclose rejections.They conclude that increasing transparency (requiring certifiers to reveal rejections) ben-efits sellers. We also advocate for transparency showing that public disclosure of seller’spayments to certifiers is welfare improving. The focus of our study is different, as we lookat the certifier’s ability to communicate information given the incentives he faces. Lerner and Tirole (2006)’s approach is closer to ours. They analyze forum shopping whencertifiers intrinsically care about product buyers and sellers can make costly concessions.In their main model buyers are homogeneous and the product is profitable only if buyersadopt it, which makes pass-fail examination optimal. They consider certifiers’ incentivesas given and focus on forum shopping by the sellers, finding that weak applicants go totougher certifiers and make more concessions. We consider heterogeneous buyers and allowthe seller to offer contingent payments to the certifier which enables us to analyze endoge-nous information production by the certifier. We study public, private and noncontingentcontracts and show that pass-fail ratings in general are not optimal. We also shed somelight on the issue of forum shopping for ratings which was the main objective of Lernerand Tirole (2006), of Skreta and Veldkamp (2009), and of Bolton et al. (2012). The paper is organized as follows. In section 1 we introduce a general environment. Insection 2 we, firstly prove that any feasible certification partitions the information availableto the certifier into potentially coarse grades (modified revelation principle). Secondly,we characterize feasible and optimal certification when the payment from the seller tothe certifier is public. In section 3 we proceed with the analysis of private payments andcharacterize feasible and optimal certification in this case. Section 4 is dedicated to welfareanalysis. Section 5 presents several extensions of the basic model: the case of the informedseller, forum shopping by the seller, the case when optimal certification is negotiated bythe seller and the certifier and, finally, an extended analysis of certification under a fixedpayment. Section 6 concludes.1 EnvironmentThere is a unit mass of potential buyers of a product of unknown quality θ sold by amonopolistic seller who has zero marginal cost of production. The quality is drawn from 6
  7. 7. F (.) supported on [0, 1]. The buyers do not know actual quality. When buyers buy q units of a quality θ productat price p they receive a net surplusAssumption 1. S(θ, q, p) = θq − q γ /γ − pq, γ ≥ 2. The seller gets a profit π(θ, q, p) = pq − t net of the payment t given to the certifier. Forsimplicity the seller is also uninformed about θ and always applies to a certifier (we relaxthis assumption in section 5.1).Assumption 2. The product’s quality θ is private to the certifier. The certifier learns θ at no cost and issues a public rating m from an arbitrary set M .The rating does not affect the certifier per se, instead the certifier puts weight λ > 0 onthe buyers’ surplus S(θ, q, p) and receives monetary payments t(m) ≥ 0 for ratings fromthe seller.5Assumption 3. U (θ, q, p, m) = λS(θ, q, p) + t(m). In the first part of the analysis we abstract from the question who defines M, t andcharacterize possible equilibria taking M, t as given. Later on we turn to the optimaldesign of M, t. The timing is as follows. A set of ratings M and payments t(.) are defined. The sellerapplies for certification. The certifier observes θ and publishes a rating m ∈ M : hisstrategy [0, 1] → ∆(M ) assigns to each quality θ a probability distribution over possible ´ratings M with a density function σ(m|θ), σ(m|θ)dm = 1. Having observed m the Mseller sets the price p, her strategy is p : [0, 1] × M → R+ . The buyers and the sellerform beliefs M → ∆([0, 1]) that specify for each rating m a probability distribution over ´1the possible qualities θ ∈ [0, 1] with a density function µ(θ|m), µ(θ|m)dθ = 1. Having 0observed p and m the buyers decide on quantity q, their decisions generate a marketdemand q : R+ × M → R+ . Finally payoffs are realized. We assume that under a public payment all aspects of the game except θ are commonknowledge. Under a private payment, before applying for certification, the seller privatelyproposes payment t(.) to the certifier, so both θ and t(.) are not known to the buyers. Assumption 1 allows us to obtain closed form solutions. An attempt to perform theanalysis with a general surplus function is technically challenging. Assumption 2 simplifiesthe analysis, it is relaxed in section 5.1. 5 If the certifier cares about the seller’s gross profit as well U (b, m) = λS(θ, q, p) + ηpq + t(m) =λ(θq − q γ /γ) − (λ − η)pq + t(m), η > 0 the qualitative results are the same. Note that instead of adding ηinto the certifier’s utility we can introduce the shadow price of money η = λ−η < 1 in the buyers’ utility λS(θ, q, p) = θq − q γ /γ − η pq and get a similar problem (for details see remark 1). 7
  8. 8. Assumption 3 is crucial, because a certifier with λ = 0 does not care about the truequality θ and simply picks the highest transfer. Also if discovering the true θ required aneffort, then λ > 0 would be necessary to induce the certifier to bear the effort cost. Asimple extension of the model, where some buyers eventually learn the true θ and sue thecertifier in case of misreporting (m = θ), would justify U . Indeed, if the certifier’s costof litigating and settling with the irritated buyers were proportional to their surplus lossλ[S(θ, q(θ), p(θ)) − S(θ, q(m), p(m))], then dropping the constant S(θ, q(θ), p(θ)) we wouldget assumption 3. The parameter λ stands for various reasons why certifiers care aboutbuyers, from potential litigation costs and buyers’ interests being explicitly representedwithin certifiers’ decision bodies to reputation concerns and explicit monetary incentives.62 Public paymentsFirst, we characterize a potentially feasible certification and then analyse the optimalcertification. Intuitively, a certification feasible when t(.) is private should also be feasiblewhen t(.) is public, therefore we start with a less demanding case of public payments. LetM and t(.) be publicly known and consider a Perfect Bayesian Equilibrium.Modified revelation principleFirst, we prove a property of a feasible certification that greatly simplifies the analysis.The property states that an uninformed party needs to infer from a rating only the qualityrange (grade) for which the rating is issued. It echoes a modified revelation principleintroduced in Krishna and Morgan (2008) and Bester and Strausz (2001).Definition 1. A set of ratings G is a grading if each rating r ∈ G corresponds to a convexset of qualities (grade) gr and r = EF [θ|θ ∈ gr ]. Grades satisfy gr ∩ gr = ∅ for any r = rand ∪ gr = [0, 1]. r∈G For example, a completely uninformative grading has a single rating r = EF [θ] anda single grade gr = [0, 1]. A perfectly informative grading, on the other hand, has acontinuum of perfect ratings and grades r = gr = θ ∈ [0, 1]. In the cheap-talk literaturepartition equilibria with messages corresponding to intervals are common. Grading isa generalization of the partition concept, which permits intervals of perfect revelationalongside coarse intervals. 6 In governmental bodies like the Food and Drug Administration it is stated clearly that their aim isto serve the interests of consumers. Committees of Standard Setting Organizations that decide on newtechnological standards, on the other hand, are often composed of potential technology buyers. 8
  9. 9. Lemma 1. For any equilibrium under any {M, t(.)} there exists an outcome-equivalentequilibrium in pure strategies under a grading {G, t(.)} in which the certifier announcesrating r = EF [θ|θ ∈ gr ] whenever θ is in grade gr . All omitted proofs are in the appendix. The proof builds on two intuitive results.First, due to the convexity of parties’ payoff each rating gives rise to a unique marketoutcome q(m), p(m). Second, the single-crossing condition establishes that in equilibriumthe quantity q(θ) = q(m(θ)) is weakly increasing in quality. The monotone function q(θ)has at most a countable number of jumps and naturally defines grades {gr }r∈G over theinterval [0, 1], so that q(θ) and p(θ) are constant within each grade gr . The buyers perceive all products within a grade to be of the same quality (otherwisep and q would differ within a grade). In other words they do not have more preciseinformation than the grade itself. They also cannot have less precise information thanthe grade since they know q(θ) changes with grades. Therefore under any certification, inequilibrium, the buyers learn merely the grade, and considering gradings is without lossof generality.7 Considering gradings makes the analysis of a cheap-talk game simple andintuitive: the beliefs are naturally fixed by a given grading and it is enough to considerpure reporting strategies.2.1 Feasible certification under public paymentsA feasible certification is essentially a grading G and a payment t(.) that are compatiblewith the parties’ incentives: conditions (1), (2), (3), (4) and (5) listed below hold. Given a rating r ∈ G the buyers believe the quality to be in the grade gr , therefore fora price p they must buy a quantity q which maximizes their expected surplus: q(r, p) = arg max EF [θq − q γ /γ − pq|θ ∈ gr ], ∀r ∈ G. (1) q≥0For any r ∈ G the seller must set a price p that maximizes her profit taking into accountthe buyers’ reaction: p(r) ∈ arg max pq(r, p), ∀r ∈ G. (2) p≥0Given actual quality θ the certifier issues his preferred rating r ∈ G, his incentive compat- 7 One might wonder what happens if the certifier instead of reporting from G would report somethingelse a ∈ G or would not report at all. For these announcements one can set t(a) = 0 and buyers’ beliefs /such that E[θ|a] = EF [θ|θ ∈ gr ] for some r ∈ G. Then for any θ ∈ [0, 1] rating r dominates any a ∈ G /because λS(θ, q(r), p(r))+t(r) ≥ λS(θ, q(r), p(r)). Alternatively, we could dispose with the limited liabilityassumption and assume that the certifier is very risk averse, his utility from money is t for t ≥ 0 and −∞for t < 0. Then it suffices to specify t(a) < 0 for a ∈ G. / 9
  10. 10. ibility constraint is r(θ) ∈ arg max [λS(θ, q(r, p(r)), p(r)) + t(r)], ∀θ ∈ [0, 1]. (3) r∈GThe reports must be consistent with the grading: the certifier must truthfully report r ifand only if the true quality is in the appropriate grade gr r(θ) = r, ∀θ ∈ gr , ∀r ∈ G. (4)The certifier has limited liability: the payments are non-negative.8 Non-negative paymentsalso guarantee the participation constraint for the certifier. t(r) ≥ 0, ∀r ∈ G. (5)Certifier’s intrinsic biasThe buyers’ and the seller’s equilibrium strategies (1), (2) are characterized by correspond-ing first order conditions: 1 (γ − 1)r r γ−1 p(r) = , q(r) = . (6) γ γConsequently, in any feasible certification a rating r ∈ G pins down the market outcomep(r), q(r). The market reaction to a rating is natural: both equilibrium price and equilib-rium quantity increase with a rating. One may think that the certifier’s strategy (3) can also be easily described. For instance,one may conjecture that under a fixed payment (t = const) the certifier reports his infor-mation perfectly r = θ ∈ [0, 1]. This conjecture is false. The certifier’s reporting incentivesare not trivial. To see why perfect reports conflict the certifier’s incentives, imagine for asecond that the buyers were not Bayesian: the buyers were to take the certifier’s reportr ∈ [0, 1] at a face value, that is their posterior would be r.Definition 2. Suppose the buyers were to take a report r ∈ [0, 1] at a face value. If for agiven θ the certifier would report r < θ he is downward biased, if he would report r > θhe is upward biased.Proposition 1. Under a fixed payment the certifier is downward biased for any θ ∈ (0, 1]. 8 Negative payments are not plausible as they may provoke bogus applications. 10
  11. 11. Intuitively a public rating r affects both equilibrium quantity q(r) and price p(r), that inturn affect the buyers’ surplus. The marginal effect of quantity on buyers’ surplus is zerosince the buyers choose the quantity; consequently only the effect of price is relevant. Aprice set by a seller increases with a rating which causes a loss of buyers’ welfare. Thereforeunder a fixed fee the buyer-protective certifier would underreport quality: this would leadto a first order gain due to reduced price and only to a second order loss due to a distorted ∂[λS(θ,q(r),p(r))+t]quantity. Formally, given θ the certifier’s report must satisfy ∂r = 0. Suppose ∂S(θ,q(r),p(r))he reports r = θ, from the buyers’ problem (1) we have ∂q |r=θ = 0. For a ∂[λS(.)+t]constant fee ∂t ∂r = 0 and we get ∂r |r=θ = λ ∂S(.) ∂p |r=θ = ∂p ∂r −λqp = −λ γ−1 q < 0, that is γthe certifier prefers not to report r = θ. One can show that he reports r < θ for θ ∈ (0, 1]. The publicity of the ratings is key for this result. As was pointed out in Farrell andGibbons (1989) in cheap-talk models with multiple audiences (the buyers and the seller)the private communication with one of the audiences (buyers) may be easier than the publiccommunication.9 Certain certifiers do advise their clients privately; investment banks orconsultants are among the examples. The analysis of private certifiers is out of the scopeof this paper.Remark 1. If the certifier were to internalize profits U = λS(θ, q, p) + ηpq + t he would be γ−1downward biased for θ ∈ (0, 1] if η < η ∗ = γ λ. Interestingly, if the certifier cared about the utilitarian social welfare (η = λ) he wouldbe upward biased. A price p > 0 is a transfer between the buyers and the seller and doesnot affect the utilitarian social welfare, but it discourages the buyers from consuming the 1socially optimal quantity. Indeed if the certifier reports r = θ then q(r) = (r/γ) γ−1 < 1q ∗ (θ) = arg max [θq − q γ /γ] = θ γ−1 . Therefore, the utilitarian certifier would reportr = γθ > θ in an attempt to restore efficiency. We let η = 0 because the goal of thepaper is to study how a general payment t, of which t = ηpq is a special case, affectsthe certifier’s incentives. We find that in general a payment t = ηpq is not a part of theoptimal certification. From now on the buyers are Bayesian, therefore in a feasible certification the ratingsmust be correct on average. It is easy to see that the uninformative certification with asingle rating EF [θ] and no payment t = 0 is always feasible. Indeed, the quantity is given 1 γ−1by q = (EF [θ]/γ) γ−1 , the price is p = γ EF [θ], hence (1) and (2) are satisfied. Conditions(3),(4) and (5) are also trivially satisfied, and the certification is feasible. This naturalbenchmark corresponds to a market with no certification. In order to solve for the model explicitly, we introduce an assumption common to the 9 ∂p If the certifier privately reports to buyers the seller cannot react to a report ∂r = 0, hence a fullyinformative private communication may be feasible under a fixed payment. 11
  12. 12. cheap-talk literatureAssumption 4. The distribution of quality is uniform: F (.) = U [0, 1].Proposition 2. (Monotonicity of payments.) In a feasible certification with at leasttwo ratings the payments strictly increase with ratings: for any r > r one has t(r) > t(r ). The idea is that the certifier’s intrinsic bias in favor of a lower rating is so strong, that itis necessary to pay him an extra amount for a higher rating in order to encourage him toissue the higher rating. Monotonicity of payments is a very handy property, it implies thatthe certifier’s limited liability constraint may bind only for the lowest rating, this greatlysimplifies the analysis.Corollary 1. Under a fixed payment only the uninformative certification is feasible. Intuitively, the certifier’s bias is powerful enough to prevent any informative certifica-tion under a fixed payment, which echoes the Crawford and Sobel’s babbling equilibriumwhen the sender’s bias is extreme. For an informative certification to emerge, the cer-tifier’s intrinsic downward bias must be compensated by an increasing payment, so thathis reporting incentives are relatively balanced.10 Under a contingent public payment thecertifier’s bias can be undone altogether with an appropriate choice of payments.Remark 2. A fully informative certification G = [0, 1] is feasible under a contingent publicpayment. 1 Indeed a payment t(0) ≥ 0, t (r) = λq(r)p (r) = λ γ−1 (r/γ) γ−1 , ∀r = θ ∈ [0, 1] in- γduces perfect certification. Thus a contingent payment can improve the informativeness ofcertification with respect to a fixed payment.2.2 Optimal certification under public paymentsHaving described a feasible certification G, t under a contingent public payment we char-acterize a certification maximizing the seller’s expected profit. In section 5.3 we let theseller and the certifier negotiate G, t and find similar results. The certification optimal forthe seller is a natural benchmark for the analysis. If the certifier uses the same grading Gand payment schedule t(.) to rate a representative sample of sellers, then the certificationwe solve for delivers the highest profit to a representative seller. This certification provides 10 This finding hinges on the product’s price being close to the expected quality p = γ−1 r, γ ≥ 2. It γturns out that for γ < γ ∗ ≈ 1.74 an informative certification is feasible under a fixed payment. Howeverthis implies a product price of less than 50% of the product’s expected quality p = γ−1 < 0.43. Prices of r γfinancial products should be close to the expected value: the ratio should be close to one and we shouldhave γ 2. 12
  13. 13. the best incentives for innovation for a seller who has not yet developed a product and doesnot know its quality. Finally, if the seller markets a set of products of different qualities,then she must push the certifier to adopt a certification that maximizes her profit fromselling a representative set of products. An optimal certification must be feasible and maximize the seller’s expected profit: max E[pq − t|G, t(.)], s.t.(1), (2), (3), (4), (5). (7) {G,t(.)}Under a fixed payment only the uninformative certification is feasible and, a fortiori, isoptimal. When payments can depend on ratings one obtainsProposition 3. If λ < λ∗ an optimal certification under a public payment commands asingle imprecise rating with no payment for low qualities θ < θ∗ (λ) and perfect ratings withpositive payments for high qualities θ ≥ θ∗ (λ), if λ ≥ λ∗ the uninformative certification is γ2λ γ(γ−1)2γ/(γ−1) −(γ 2 +γ−1) γ(γ−1)2γ/(γ−1) −2γ 2 +γoptimal. Here θ∗ (λ) = γ(γ−1)+λ(γ 2 +γ−1) γ(γ−1)2γ/(γ−1) −γ(2γ−1) , λ∗ = γ 2 +γ−1−(γ−1)2γ/(γ−1) . Under the optimal certification, low quality applicants with θ < θ∗ (λ) pay nothing andreceive low, imprecise quality assessments that can be interpreted as rejections. Appli-cants above some minimal threshold θ ≥ θ∗ (λ) are perfectly certified and pay for ratings.Optimal certification is intuitive. In a feasible certification payments increase with ratings(proposition 2). In the absence of a limited liability constraint the seller would implementperfect certification with a zero expected payment by asking the certifier to pay a certainamount upfront. Under limited liability a net payment from the seller to the certifiercannot be negative, hence perfect ratings become expensive for the seller. The paymentsincrease additively with ratings, therefore the best way to economize on expected paymentsis to pay nothing for the lowest rating and increase the likelihood θ∗ (λ) of this rating beingissued. An optimal certification is weakly less informative when the certifier is more protectiveof the buyer. The threshold θ∗ (λ) increases with λ and the interval of perfect certification[θ∗ (λ), 1] shrinks when λ < λ∗ . Intuitively when λ increases, the certifier internalizes thebuyers’ surplus to a greater extent and a larger compensation is necessary to balance hisintrinsic downward bias. Perfect ratings become more expensive and the seller prefers acertification where a large set of qualities is associated with the cheap imprecise rating.Interpreting the lowest rating as rejections we conclude that a tougher certifier λ > λrejects a higher fraction of applicants θ∗ (λ ) > θ∗ (λ). Contrary to conventional wisdom, a contingent payment, supposedly feeding the certi-fier’s conflict of interest, causes no apparent harm to the buyers. Quite the opposite infact; a public contingent payment can implement a perfect certification while a fixed pay- 13
  14. 14. ment leads to uninformative certification and information loss. Somewhat surprisingly ,under a public payment optimal certification becomes more informative when the certifiercares little about the buyers: λ → 0. The subsequent analysis shows that this findingdoes not hold if the seller pays the certifier privately. In the latter case λ → 0 is notoptimal. Another reason for λ > 0 may be the certifier’s moral hazard: a costly effort maybe necessary to learn the quality θ. Clearly, only a certifier with λ > 0 may choose to bearthe effort cost. A rough assessment of λ for real certifiers can be done based on the information aboutthe fees. The model predicts t(r) ≈ λp(r)q(r), where p(r)q(r) is the gross revenue of theseller. Rating agencies charge around 3 basis points of the volume of the issue as a feefor rating corporate bonds, this implies λ ≈ 0.0003. Investment banks when underwritean initial public offering of shares charge a fee of 7% of the offering volume, this givesλ ≈ 0.07.3 Private paymentsCertain certifiers refuse to disclose the compensation they receive from sellers, for instancemain credit rating agencies do not reveal the issuers’ payments for ratings. Essentially, thepayments are private: they are not observed by the buyers. The privacy of payments mayprompt the seller to secretly skew the certifier’s compensation for high ratings (looselyspeaking the seller may bribe the certifier). For simplicity we still assume that the set offeasible ratings G is publicly known. This assumption can be easily relaxed. In real worldthe number of grades and ratings typically is observable, while the exact borders betweengrades are not. Any certification feasible under a private contract is also feasible under apublic contract, hence the modified revelation principle applies. However, a certificationfeasible under a public contract may be prone to manipulations under privacy, thereforethere must an additional constraint. The timing is as follows. Given a publicly known grading G the seller privately proposespayments t(.) to the certifier. The certifier learns actual θ and reports r ∈ G. Given therating r the buyers and the seller believe the quality to be in the corresponding grade gr ,the seller sets a price p and buyers buy a quantity q.3.1 Feasible certification under private paymentsCorollary 1 fully characterizes certification under a fixed private payment. Here we considera contingent private payment t(.). When the seller proposes t(.) she takes into account the 14
  15. 15. market reaction (1), (2) the certifier’s incentive compatibility constraint (3) and the limitedliability constraint (5).11 However, she ignores the certifier’s truth-telling constraint (4).With a private payment the seller may want to deceive the buyers and manipulate thecertifier’s actual reports away from the implied grading G, her maximization problem is ˆ 1 max E[pq − t|G, θ]dθ, s.t.(1), (2), (3), (5). (8) {t(.)} 0 Note the difference with the problem (7), where the truth-telling constraint (4) waspresent. Of course Bayesian buyers can’t be deceived and in an equilibrium the truth-telling constraint (4) must hold. That is, the private payment solving (8) must induce thecertifier’s reports consistent with (4). In other words, the grading G must be robust tohidden payment manipulations by the seller. Note that (8) implies (1), (2), (3), (5), therefore (4) and (8) are necessary and sufficientfor a certification to be feasible under a private payment. The uninformative certificationwith zero transfers is always feasible. Clearly, if the certifier can only issue a rating thatconfirms the ex ante expected quality of the product, the seller cannot manipulate the cer-tifier’s reports. She also has no reason to pay the certifier: (8) implies t = 0. The followingproposition provides a necessary and sufficient conditions for a feasible certification undera private payment in a general case.Proposition 4. A feasible certification G, t under a private payment has no intervals ofperfect revelation: G contains at most a countable number of coarse ratings. At any borderpoint between any two ratings both ratings deliver to the seller the same virtual profitπ (θ, r) = p(r)q(r) + λS(θ, q(r), p(r)) − λq(r)(1 − θ):˜ π (θi−1 , ri−1 ) = π (θi−1 , ri ), i = 2, ..., N. ˜ ˜ (9) The detailed proof is in the appendix. To illustrate the idea behind the proof we reformu-late the problem (8) using the certifier’s indirect utility U (θ) = max[λS(θ, q(r), p(r))+t(r)]. r∈GUsing envelope theorem dU (θ) = Sθ (θ, q(r), p(r)) = λq(r) ≥ 0 we express the seller’s ex- dθ ´1pected profit as Π = π (θ, r)dθ − U (0). The seller’s revenue p(r)q(r) increases with ˜ 0ratings, thus for each border point θi , i = 1, ..., N − 1 per se the seller prefers a rating ri 11 One should not take our model literally, where a single seller contracts with the certifier. Insteadone can think of a representative sample of sellers, each of whom is uncertain about her product andattempts to tilt the certifier in her favour. The sum of these efforts would result in something close to theproblem we analyse. This is coherent with the observation in Partnoy (2006): “Because rating agenciesrate thousands of bond issues, they do not depend on any particular issuer, so the concern about conflictsis more systemic than individualized.“ 15
  16. 16. over a rating ri−1 . If she attempts to marginally increase the chance of receiving ri insteadof ri−1 and inflates the payment t(ri ) she will have to raise payments for all ratings aboveri to maintain incentive compatibility, which is costly. This additional cost is captured bythe term λS(θ, q(r), p(r)) − λq(r)(1 − θ) in the virtual profit. Note that the lover the θithe higher is the associated cost of payment inflation. For the seller not to be wiling tomanipulate the payments she must perceive the same virtual profit from the two borderingratings (9). In order to see that a perfect certification is not possible, suppose there was an ininterval of perfect certification r = θ ∈ (a, b). Then maximization of Π with respect to 1 2 ∂ π (θ,r) ˜r(θ) would require a necessary condition ∂r = ( γ ) γ−1 1 − λ γ γ+γ−1 + r 2 −1 λ 2θ−1 (γ−1) r = 0 2θ−1for r = θ ∈ (a, b). The ratio θ is strictly increasing, the necessary condition would beviolated for some θ ∈ (a, b) and, hence, no interval of perfect certification is possible.Certifier’s bias under a private paymentA private payment may be skewed in favour of high ratings. Consequently, the certifiermay be endogenously upward ”biased” for high ratings. Unfortunately it is impossibleto measure the certifier’s bias as it depends on a particular payment t(.) (for instance ift = 0 the certifier has no bias caused by payments). Nevertheless, some properties of afeasible certification can be established. We explicitly characterize a feasible certificationand illustrate its properties in case γ = 2, which gives rise to the buyers’ utility functionU = qθ − q 2 /2 − pq. For brevity we consider λ ∈ (0, 2] as degree to which the certifier mayinternalize the buyers’ surplus.Corollary 2. If γ = 2 a feasible certification G, t under a private payment depends on λ. 2 1) If λ ∈ (0, 11 ] then G, t admits N = 1 rating with borders θ0 = 0, θ1 = 1. 2 2 2) If λ ∈ ( 11 , 3 ] then G, t admits N = 1, 2 ratings. If N = 2 the ratings’ borders are 11λ−2θ0 = 0, θ1 = 10λ+4 , θ2 = 1. 3) If λ ∈ ( 2 , 2] then G, t admits N = 1, ..., ∞ ratings. If N ≥ 2 the ratings’ borders 3 i (8−4λ+8λDN )DN +1−j +(8λ+(8−4λ)DN )Djare θ0 = 0, θN = 1, θi = (3λ−2)(1−D2N )(1−D) , i = 1, ..., N − 1, D = j=1 √5λ+2−4 λ(λ+2) 3λ−2 . The proof is a solution of (9) for γ = 2, it is in the appendix. We illustrate the resultwith a figure. Given λ we depict the boundaries of ratings for a certification with the ∗highest number of ratings feasible. In the figure, θ1 (λ) is the upper boundary of the low 16
  17. 17. ∗rating (the thick solid line) and θN −1 (λ) is the lower boundary of the high rating (the thickdashed line). Figure 2: The certification with the highest feasible number of ratings given λ. λ 2 ∗ ∗ θ1 (λ) θN −1 (λ) low rating high rating 1 2 3 2 11 θ 0 0.2 0.4 0.6 0.8 1 2 If the certifier is lax λ ≤ 11 only the uninformative certification is feasible. Indeed, if thecertifier were to use more than one rating then, given that λ is small, the seller would offera private payment which would induce the certifier to report a high rating even when thequality is low. Such a manipulation would be profitable because a small premium wouldbias the certifier’s reports away from what is implied by the grading. In equilibrium thetruth-telling constraint requires reports to be consistent with the grading. Therefore, aninformative certification is impossible. 2 When λ ∈ ( 11 , 2 ] the certifier is sufficiently tough to implement an informative certifica- 3tion with two ratings. The high rating is issued for a wide range of qualities and involves ∗more pooling than the low rating: θ1 (λ) < 1/2. This is because the certifier’s reports areaffected by a generous payment offered for the high rating. For the same reason a finergrading with more than two ratings in not possible: if the certifier were to use more thantwo ratings the seller would offer a large payment for the highest rating and would inducethe certifier to misreport. 2 The most interesting case is λ ∈ ( 3 , 2], when infinitely many ratings are feasible. As canbe seen on the picture the low and the high ratings are very imprecise, while intermediateratings are relatively precise. It appears as if the certifier is downward ”biased” for lowqualities and upward ”biased” for high qualities. To understand the intuition, notice thatthe seller may want to manipulate the payment for two reasons. First, the seller does 17
  18. 18. not like to pay for ratings per se and, therefore, she prefers low payments. Second, theseller prefers high ratings over low ratings and she may want to increase payments for highratings. The seller has a strong incentive to increase t(rN ), which lowers the threshold θN −1and increases the likelihood of obtaining the highest rating. An incentive to increase thepayment for the second highest rating is not as strong, because an increase in t(rN −1 ) notonly lowers the threshold θN −2 but also raises the threshold θN −1 . The second effect isunwelcome since the likelihood of rating rN −1 increases at the expense of the likelihoodof the highest rating rN . In order to counterbalance the second effect t(rN ) has to goup. By similar reasoning, incentives to inflate t(rN −2 ) are even lower than incentives toinflate t(rN −1 ), and so on. As a result, under a private payment the seller offers a highcompensation for high ratings, which in turn makes the certifier endogenously ”upward”biased for high ratings and less upward ”biased” for low ratings. In fact, for low ratings thecertifier’s upward bias because of the payments becomes smaller than the certifier’s intrinsicdownward bias (proposition 1). Overall, therefore, the certifier is downward ”biased” forlow ratings. One can interpret the imprecise highest rating as a manifestation of so called ”ratinginflation”. In our framework all agents are rational and nobody is deceived, yet ”ratinginflation” can be harmful as the informational value of the highest rating is limited. Theinformational value of the rating is low not because buyers fail to interpret it, but becausea high payment from the seller induces the certifier to give the highest rating even to theproducts of a quality below the highest.3.2 Optimal certification under private paymentsAn ex ante optimal certification under a private payment must be feasible and it mustmaximize the seller’s profit max E[pq − t|G, t(.)], s.t.(4), (8). (10) {G,t(.)}Proposition 5. Under a private payment the uninformative certification with zero pay-ments is optimal. Formal proof is in the appendix. The result is striking because the seller is informationloving per se and is eager to buy an informative certification. However, in a feasiblecertification under a private payment, the payment rapidly increases with ratings. As aresult, the seller’s profit net of the payment becomes a concave function of quality: under 18
  19. 19. a private payment the seller becomes information averse. In other words, if the seller hasan opportunity to privately pay the certifier, then from an ex ante perspective she prefersnot to have an informative certifier around. If such a certifier was present then the sellerwould not resist a temptation to offer a high payments for high ratings. Consequently,in expectation the seller would end up paying the certifier more than the extra profit shewould gain from an informative certification.4 Welfare analysisFor any feasible certification G, t denote the ex ante expected buyers’ surplus as S(G),the seller’s gross expected profit as P (G) and the expected payment to the certifier asT (G, t) = E[t|G]. The seller’s net expected profit is Π = P (G) − T (G, t) and the certifier’sexpected payoff is U = λS(G)+T (G, t). We define the social welfare as a sum of the buyers’surplus, the seller’s profit and the certifier’s payoff W = S + Π + U = (1 + λ)S(G) + P (G). A social planner with the objective functions W is information loving. Intuitively in-formation is beneficial for the buyers because high quality products have higher valueand should be purchased in larger quantities than the low quality products. At the sametime, the seller also benefits from information because higher quality products have higherequilibrium prices and are sold in larger quantities. Formally substitute for q(r), p(r) in γ −1 γ−1 γ−1 γ−1S(θ, q, p) and for any r ∈ G get E[S(θ, q, p)|θ ∈ gr ] = γ2 r γ and E[pq|θ ∈ gr ] = γ −1 −1 γγ−1 γ r γ−1 γ γ−1 . Denote a constant ω = (γ+1+λ) γ−1 γ γ2 γ−1 then W = ω r(θ) γ−1 Pr(θ ∈ gr ). r∈G γFunction r γ−1 is convex, hence welfare increases when G’s precision improves. For instance, γthe uninformative certification delivers WU = ω( 1 ) γ−1 while the fully informative certifi- 2 ´ γ 1 γcation delivers WF = ω x γ−1 dx = ω 2γ−1 > WU because 2γ−1 < 2 γ−1 for γ ∈ (1, ∞). γ−1 γ−1 0Remark 3. If the social planner with the utilitarian objective function W were to choosecertification, she would induce a fully informative certification. In reality, a regulator can hardly impose a particular certification. Nevertheless, theregulator may be able to impose restrictions on contracts between the parties. For instance,require a fixed payment for certification or mandate full disclosure of payments from theseller to the certifier. We proceed assuming that the social planner can choose betweenthree contracting regimes: private contingent payments, public contingent payments andfixed payments; we also assume that under any regime the certification optimal for the sellertakes place. Corollary 1 and proposition 5 state that the uninformative certification prevailsunder a fixed-fee and under a private contingent payment. According to proposition 3, aninformative certification emerges under a public contingent payment iff λ < λ∗ , thus we 19
  20. 20. haveProposition 6. If λ < λ∗ then a a public contingent payment strictly dominates a fixed-feeor a private contingent payment from the social welfare perspective. If λ ≥ λ∗ all threeregimes lead to the uninformative certification and deliver the same social welfare. Generally an optimal certification preferred by the seller is not fully informative andthe information is under-provided in an equilibrium compared to the social optimum. Theutilitarian social planer does not care about cost of implementing the fully informativecertification because the payment to the certifier is a redistribution which does not affectthe total welfare. The seller, on the other hand, bears all the cost of certification butdoes not internalize a surplus which accrues to the buyers, hence she prefers not the fullyinformative certification. One way to get closer to the social optimum is to ask buyersto pay the certifier for his services. The free-rider problem and lack of commitment onthe buyers’ side make such a scheme dubious. An alternative regulation, which mightappear controversial at first glance, calls for subsidizing the seller’s certification expenses.Indeed, under a public contract if the seller were to receive a subsidy for each dollar spenton certification, she would induce a more informative certification and the social welfarewould improve.5 ExtensionsIn this section we present several extensions of the basic model. First, we generalize ouranalysis to the case of informed seller. Second, we let the seller choose among multiple cer-tifiers. Then, we let the seller and the certifier negotiate an optimal certification. Finally,we extend our analysis to markets with a moderate price reaction to ratings and find thatan informative certification can be feasible and optimal under a fixed payment.5.1 Informed sellerHere we generalize our results to the case when the seller is informed. We rule out otherways to signal quality than certification.12 Suppose the certification maximizing the ex anteexpected profit of an uninformed seller G, t is established as a common business practice.The seller learns her quality before applying for certification. The buyers hold pessimisticbeliefs: if the seller does not apply they perceive her product to be of quality r1 = min G.The rest of the game proceeds as before. 12 Other signalling ways like warranties or advertising may not be credible in the context of financialproducts. For instance an issuer of bonds can’t issue a meaningful warranty on her own bonds. 20
  21. 21. A certification optimal under a fixed payment and under a private contingent paymentrequires zero payments and is uninformative. When λ ≥ λ∗ the optimal certification undera public contingent payment is also uninformative. It follows that in these cases the optimalcertification can be sustained as an equilibrium when the seller is informed. Indeed theseller is indifferent about certification and it is an equilibrium strategy to apply.Proposition 7. If λ < λ∗ a certification G, t optimal for an uninformed seller under apublic contingent payment can be sustained as an equilibrium when the seller is informed. The proof is in the appendix. Intuitively under an optimal certification the seller’s profitincreases with ratings. A seller who has not applied for certification G, t is perceived tohave a low quality product. Given that the payment for the lowest rating is zero it is aweakly dominant strategy for the seller to apply: in the worst case she will get the lowestrating and will not pay anything. If she will get a higher rating her profit will be higher,so it is an equilibrium strategy for her to apply. The signalling equilibrium presented hereis not unique. For instance, if a seller who has not applied for a certification is perceivedto have a quality E[θ] = 1 , while a seller who has applied is perceived as r1 < E[θ] = 2 , 2 1then no seller would apply. In this case, very sceptical beliefs about applicants are outof equilibrium path, hence they can happen in equilibrium. The equilibrium described inproposition 7 delivers the highest profit to an average seller, and can be thought of as abenchmark.5.2 Forum shoppingIn the original set-up with the uninformed seller we allow the seller to choose a certifierout of a continuum of certifiers with different degrees of buyer-protectiveness λ ∈ [0, ∞)known to all parties. This environment can be seen as an approximation of a competitivecertification industry. Our previous results suggest that if the payments between the sellerand the certifier are either private or fixed, then the optimal certification for the seller isuninformative independently of λ. In these cases the analysis is trivial. Consider public contingent payments. Assume each of the certifiers, depending on hisλ, proposes a certification preferred by the seller G(λ), t(λ). The seller has to choose acertifier with the best offer G, t.Proposition 8. Under a public contingent payment the seller hires the least buyer-protectivecertifier: λ = 0. The proposition follows from proposition 3, low λ leads to low θ∗ (λ) and high informationproduction. The seller is information loving, therefore she chooses a certifier with thesmallest λ. 21
  22. 22. Intuitively, a certifier who cares little about the buyers can be cheaply compensated forhis intrinsic bias and induced to perfectly certify the product. As a result, in the absenceof other concerns, such as private payments or certifier’s moral hazard in informationacquisition, the seller picks the certifier which cares about the buyers the least.5.3 Optimal negotiated certificationIn sections 2.2 and 3.2 we have analysed certification optimal for the seller. In real world thecertifier often decides on the grading and on the associated payments. To accommodatethis possibility we let the certification G, t be jointly decided by the certifier and theseller before any party learns the actual quality (both parties are uninformed). Given afeasible certification G, t the seller’s ex ante expected profit is equal to the gross expectedprofit net of the expected payment Π = P (G) − T (G, t). In a similar way, the certifier’sexpected payoff is a sum of a weighted buyers’ expected surplus and the expected paymentU = λS(G)+T (G, t). If the parties fail to agree on G, t the uninformative certification with 1 1t = 0 obtains; and the parties get their reservation utilities Π0 = P ( 2 ) and U0 = λS( 2 ).To model negotiations we use a generalized Nash bargaining solution where the seller’sbargaining power is ν ∈ [0, 1]: max (Π − Π0 )ν (U − U0 )1−ν , s.t. Π ≥ Π0 , U ≥ U0 . {G,t f easible} It is easy to see that an optimal negotiated certification under a fixed payment or undera private payment is uninformative and requires no payments. Under a fixed payment onlythe uninformative certification is feasible hence it is optimal. Under a private payment inany feasible informative certification the seller gets a profit Π < Π0 (for details see theproof of proposition 5), therefore the negotiation leads to the uninformative certificationwith no payments. The same is true for a certification under a public contingent paymentif λ ≥ λ∗ .Proposition 9. If λ < λ∗ the negotiated certification under a public contingent paymentis as follows. 1) If the certifier is not very buyer-protective λ ≤ λ∆ and the seller has a low bargainingpower ν ≤ ν ∗ the optimal certification is fully informative. 2) If λ > λ∆ or ν > ν ∗ the optimal certification leads to a coarse rating and zero paymentfor θ ∈ [0, θ∗ (ν, λ)], and perfect ratings with positive payments for θ ∈ (θ∗ (ν, λ), 1]. Thethreshold θ∗ (ν, λ) is increasing in ν. θ∗ (ν, λ), λ∆ and ν ∗ are characterized in the appendix. The detailed proof can be found in the appendix. The idea is the following. For a 22
  23. 23. given total expected transfer T (G, t) the seller and the certifier have congruent preferencesabout certification. As a result, an optimal negotiated certification is qualitatively similarto the certification maximizing the seller’s expected profit (which obtains for ν = 1): athreshold θ∗ (ν, λ) separates a low imprecise rating from high perfect ratings. The differenceis that the threshold θ∗ (ν, λ) depends on the bargaining power of the seller ν. Only ratingsabove the threshold θ∗ (ν, λ) command positive payments to the certifier. If the seller hasa low bargaining power ν, the certifier gets a high expected payment T (G, t). This inturn implies a wide interval of perfect certification with positive payments and, hence, alow threshold θ∗ (ν, λ). If the certifier’s bargaining power is high enough ν ≤ ν ∗ a fullyinformative certification takes place.5.4 Informative certification under fixed public paymentsThroughout the analysis we have assumed γ ≥ 2 which guaranteed a significant price γ−1reaction to a rating p = γ . We showed that in this case no informative certification wasfeasible under a fixed payment. The assumption γ ≥ 2 is reasonable in a context of financialmarkets, where competition among investors (potential buyers) drives the product’s priceclose to it’s expected value p ≈ r. Nevertheless, in consumer goods markets one mayexpect a less competitive demand and a lower reaction of price to quality changes. Herewe let γ ∈ (1, 2) and analyse a certification under a fixed public payment. We do notanalyse contingent payments because of technical difficulties. A certification is feasiblewhenever (1), (2), (3), (4) and t(r) = t ≥ 0, ∀r ∈ G hold.Proposition 10. A feasible certification G, t under a fixed payment is a partition {θi }i=0,...,Nof interval [0, 1] into grades gr = [θi−1 , θi ), i = 1, ..., N , such that θ0 = 0, θN = 1, and S(θi−1 , q(ri )p(ri )) = S(θi−1 , q(ri−1 ), p(ri )), i = 2, ..., N, (11) i) the number of ratings is not greater than N (γ) < ∞, N (γ) = 1 if γ ≥ γ ∗ , N (γ) ≥ 2if γ < γ ∗ , γ ∗ ≈ 1.74. ii) if N ≥ 2 high ratings are more precise than low ratings : θi − θi−1 < θi−1 − θi−2 ,i = 1, ..., N . The detailed proof can be found in the appendix. Observe that γ determines the mark-up γ−1p= γ r. When γ ≥ γ ∗ ≈ 1.74 the mark-up is high and the certifier fails to communicateproduct quality; in this case only the uninformative certification is feasible. An optimal certification under a fixed public payment maximizes the seller’s expectedprofit. The seller’s problem is (7) with an additional constraint t(r) = t, for any r ∈ G. 23
  24. 24. Proposition 11. An optimal certification under a fixed public payment is a partition{θi }i=0,...,N (γ) with the maximum feasible number of ratings N (γ) and zero payments. Formal proof is in the appendix. The result is intuitive. A fixed fee does not influencethe certifier’s reports and the seller sets a zero fee. Equilibrium prices and quantities arehigher when a high quality is certified, hence the seller’s profit is a convex function ofquality. The seller is information loving and prefers the finest feasible certification. The findings of this section accord with practices of some real certifiers. Often the cer-tifiers that charge fixed fees issue few coarse ratings. For instance the Food and DrugAdministration sets standard application fees and publishes pass-fail quality assessments.Similarly certifiers of consumer goods such as the European New Car Assessment Pro-gramme or the Michelin Guide typically charge fixed fees and evaluate products on five-and three-star scales. The analysis stresses that a moderate price reaction to ratings, whichis common to consumer goods, is the prerequisite of an informative certification under afixed payment. In case of a financial product, on the other hand, the price is very close tothe perceived product’s value and is very sensitive to a rating change. Our theory predictsthat in the latter case an informative certification under a fixed payment is problematic.6 ConclusionThis paper has studied the issuer-pays business model of certification assuming that theratings issued by the certifier are cheap talk; as Moodys’ website (www.moodys.com)explains the ratings are ”statements of opinion but not statements of fact”. We develop a model with fully rational agents and characterize feasible and optimalcertification when the payment from the seller to the certifier is: 1) fixed and publiclyknown, 2) contingent on the rating and publicly known, and 3) contingent on the ratingand privately known to the seller and the certifier. We show that if the payments between the seller and the certifier are publicly knownthen requiring the certifier to charge a fixed fee reduces the informativeness of ratingsand lowers welfare. Fundamentally, there is no harm from contingent payments as longas payments are public. If a rational buyer observes that a seller has paid an unusuallyhigh fee for a rating, then his perception of the product’s quality goes down. This in turnprevents the seller from paying an unusually high fee and improves the precision of theratings. By contrast, a phenomenon akin to “rating inflation” can occur when payments are pri-vate for the seller and the certifier. If buyers do not observe the payment, their perceptionof the ratings does not depend on actual payments. This creates an incentive for the seller 24
  25. 25. to elicit the highest rating by offering a generous compensation. As a result, the highestrating is issued for a wide range of qualities and has limited information value. Welfare islow. The welfare loss is a result of imprecise ratings and happens even when the buyersare fully rational. Our analysis points to possible improvements in regulation. A regulation requiring fixedfees is not desirable. A desirable regulation requires certifiers to fully disclose all thepayments they receive for a ratings.7 AppendixProof of proposition 1. Under any contract (M, t(.)) in a Perfect Bayesian equilibrium the buyers buy their ´1preferred quantity given the price p and realized beliefs µ(θ|m): q(p, m) = arg max (θ − q≥0 0p)q − q γ /γ)µ(θ|m)dθ. The seller maximizes her profit given the buyers’ demand q(p, m) ´1and realized beliefs µ(θ|m): p(m) = arg max q(p, m)pµ(θ|m)dθ. p≥0 0 ´1 Denote E[θ|m] = θµ(θ|m)dθ. The buyers’ problem has a unique solution q(p, m) = 0 1(E[θ|m] − p) γ−1 if p ≤ E[θ|m] and q(p, m) = 0 otherwise. If E[θ|m] > 0, the solution γ−1to the seller’s problem is unique p(m) = γ E[θ|m]. The equilibrium quantity is q(m) = 1(E[θ|m]/γ) γ−1 . If E[θ|m] = 0 any p ∈ [0, ∞) can happen in equilibrium. In this case q = 0and all equilibria deliver the same payoffs to the parties; we set p(m) = q(m) = 0. Itfollows that for each m ∈ M the beliefs function µ(θ|.) induces a unique market reactionp(m), q(m) and considering pure strategies of the seller and of the buyers is without lossof generality. The certifier maximizes his payoff given θ, p(m), q(p, m) and the buyers’ beliefs functionµ(θ|.). Given θ the certifier issues a rating m with positive probability σ(m|θ) ≥ 0 ifm(θ) ∈ arg max [λS(θ, q(p(m), m), p(m)) + t(m)], otherwise σ(m|θ) = 0. Bayes’ rule, when m∈M ´1applicable, requires µ(θ|m) σ(m|θ )dF (θ ) = σ(m|θ)f (θ). 0 In equilibrium the quantity is weakly increasing in quality. For θ2 > θ1 take anym2 : σ(m2 |θ2 ) > 0 and any m1 : σ(m1 |θ1 ) > 0. The certifier’s revealed preferencewrites λS(θ2 , q(m2 ), p(m2 ))+t(m2 ) ≥ λS(θ2 , q(m1 ), p(m1 ))+t(m1 ), λS(θ1 , q(m1 ), p(m1 ))+t(m1 ) ≥ λS(θ1 , q(m2 ), p(m2 )) + t(m2 ). Combining inequalities and substituting for S weget λ(q(m2 ) − q(m1 ))(θ2 − θ1 ) ≥ 0 and q(m2 ) ≥ q(m1 ). For every θ ∈ [0, 1] the set of quantities implemented with positive probability q(θ) = 25
  26. 26. {q(m) : σ(m, θ) > 0} contains at most three distinct elements. Suppose for some θ the ˜ ˜set q(θ) contains four elements q < q1 < q2 < q. For any m1 : q(m1 ) = q1 one must have ˜ ˜ ˜σ(m1 , θ ) = 0 if θ = θ. Indeed if σ(m1 , θ) > 0 for some θ < θ one would have q(θ) > q ∈ q(θ) ˜which contradicts the weakly increasing quantity. If σ(m1 , θ) > 0 for some θ > θ one would ˜ ˜have q(θ) < q ∈ q(θ) also a contradiction. Therefore σ(m1 , θ) > 0 iff θ = θ, the Bayes’ 1 ˜ ˜rule implies E(θ|m1 ) = θ and q1 = q(m1 ) = (θ/γ) γ−1 . By analogous reasoning for any 1 ˜m2 : q(m2 ) = q2 one obtains q2 = q(m2 ) = (θ/γ) γ−1 = q1 , which contradicts q2 > q1 .Hence q(θ) contains at most three elements. For almost every θ ∈ [0, 1] the set of q(θ) is single valued, that is randomization couldhappen at points that have measure zero. Indeed in equilibrium quantity weakly increaseswith θ, it follows that q(θ) = sup{q(θ)} is monotone. Since q(θ) contains at most threeelements q ∈ q(θ). At any point θ where q(θ) is not single valued function q(θ) is dis-continuous (for any θ < θ and θ > θ one has q(θ ) ≤ inf{q(θ)} < q(θ) ≤ inf{q(θ )}).A monotone q(θ) can be discontinuous only at a countable number of points: for almostevery point q(θ) is single valued. In an equilibrium denote by Θd the set of θs where q(θ) and q(θ) differ. For each θ ∈ Θdfind m such that σ(m, θ) > 0 and q(m) = q(θ), set σ (m, θ) = 1. Set σ (m, θ) = σ(m, θ)for θ ∈ Θd so that the modified reporting strategy would implement q(θ) for any θ ∈ [0, 1].σ (m, θ) is consistent with the certifier’s incentives if the meanings of messages do notchange, that is for any m such that σ(θ|m) > 0 for some θ ∈ Θd we have E(θ|m) =´1 ´1 θµ (θ|m)dθ = θµ(θ|m)dθ.0 0 If m is issued for a single point θ ∈ Θd , that is σ(θ, m) > 0 and σ(x, m) = 0 for ´1 ´1x = θ, then xµ(x|m)dx = θ. For all such m we set beliefs so that xµ (x|m)dx = θ, 0 0these beliefs are consistent with σ . If m is issued for θ ∈ Θd and some θ = θ, thenq = q(m) is implemented for an interval of θs because q(θ) is weakly monotone. Using ´1the notation Θq = {θ : q(θ) = q} one can show xµ(x|m)dx = EF [x|x ∈ Θq ]. It implies 0that if we change the beliefs for a single point θ ∈ Θd the expectation would not change:´1 ´1 xµ (x|m)dx = xµ(x|m)dx. It follows that for any equilibrium which implements q(θ)0 0we can construct an equilibrium which implements q(θ). Let Q be the set of qs such that q(θ) = q for some θ ∈ [0, 1]. For every q ∈ Q define a setof qualities that induce the same q(θ), that is Θq = {θ : q(θ) = q}. Since q(θ) is monotoneeach Θq ⊆ [0, 1] is either an interval or a point, moreover ∪ Θq = [0, 1] and Θq ∩ Θq = ∅ q∈Qif q = q. For each q ∈ Q define rq = EF [θ|θ ∈ Θq ], then G = {rq }q∈Q is a grading byconstruction. For each q ∈ Q pick a message mq which induces q in equilibrium: q(mq ) = q 26
  27. 27. and let t(rq ) = t(mq ). It is immediate to see that for an equilibrium under {M, t(.)} which implements q(θ)there exists an equilibrium in pure strategies under {G, t(.)} which also implements q(θ). The market reaction conforms to relation p = (γ−1)q γ−1 . Therefore, given θ the functionq(θ) pins down the parties’ payoffs S(θ, q, p) = θq − q γ /γ − pq and π(q, p) = pq. In anyequilibrium q(θ) and q(θ) coincide for almost every point. Thus for any equilibrium underM, t there exists an outcome equivalent equilibrium in pure strategies under G, t QED. Proof of proposition 2. Take two consecutive ratings ri−1 = EF [θ|θ ∈ [θi−2 , θi−1 )] andri = EF [θ|θ ∈ [θi−1 , θi )] (considering intervals is without loss of generality). The incentivecompatibility and truthful reports (3), (4) require λS(θi−1 , ri−1 ) + t(ri−1 ) = λS(θi−1 , ri ) + 1 −1 γ 2 −γ+1t(ri ). Using (6) we express S(θ, r) = (θ − γ 2 r))r γ−1 γ γ−1 and we get t(ri ) − t(ri−1 ) = γ γ 1 1 2 −γ+1 −1 −1 γλγ γ2 (ri γ−1 γ−1 − ri−1 )γ γ−1 − λθi−1 (ri γ−1 γ−1 − ri−1 )γ γ−1 , i = 2, ..., N. Observe that ri γ−1 − γ 1 1 1 1 1 1ri−1 γ−1 = (ri + ri−1 )(ri γ−1 − ri−1 γ−1 ) + ri ri−1 γ−1 − ri−1 ri γ−1 ≥ (ri + ri−1 )(ri γ−1 − ri−1 γ−1 ) 1 1 2 −γ+1for γ ≥ 2 since ri−1 < ri . Thus we get t(ri ) − t(ri−1 ) ≥ λ(ri γ−1 − ri−1 γ−1 )( γ γ2 (ri + −1 1 1 −1 γ 2 −γ+1ri−1 ) − θi−1 )γ γ−1 ≥ λ(ri γ−1 − ri−1 γ−1 )( γ2 ri−1 − γ−1 θi−1 )γ γ2 γ−1 since ri ≥ θi−1 . Moreover θi−1 +θi−2for F (.) = U [0, 1] (4) implies ri−1 = 2 therefore (γ 2 − γ + 1)ri−1 − (γ − 1)θi−1 ≥γ 2 −3γ+3 2 θi−1 > 0 since γ ≥ 2. Thus t(ri ) > t(ri−1 ) QED. Proof of proposition 3. Using the certifier’s indirect utility U (θ) = max[λS(θ, q(r), p(r))+ r∈G dU (θ)t(r)] and the envelope theorem = Sθ (θ, q(r), p(r)) = λq(r) ≥ 0 the seller’s ex- dθ ´1pected profit can be expressed as Π = π (θ, r))dF (θ) − U (0), π (θ, r) = p(r)q(r) + ˜ ˜ 0λS(θ, q(r), p(r)) − λq(r) 1−F (θ) . f (θ) Since dU (θ) dθ ≥ 0 the incentive constraint (3) holds. Due toproposition 2 the constraint (5) can be replaced with t(r1 ) ≥ 0 ⇔ U (0) ≥ λS(0, q(r1 ), p(r1 )).The seller’s problem is max Π, s.t. (1), (2), (4), U (0) ≥ λS(0, q(r1 ), p(r1 )). (12) {r(.), U (0)} First, solution has U (0) = λS(0, q(r1 ), p(r1 )) ⇔ t(r1 ) = 0. Second, F (.) = U [0, 1], (1),(2) for any r imply EF [˜ (θ, r), θ ∈ gr )] = π (r, r) = p(r)q(r)+λS(r, q(r), p(r))−λq(r)(1−r) π ˜ 2 +γ−1 1 −1 γ−1is convex for r ∈ (0, 1]. Indeed denote β = γ + λγ γ2 , then π (r, r) = (βr − λ)r γ−1 γ γ−1 ˜ 3−2γ −1 d2 π (r,r) ˜ 1 dr2 = (γβr − (2 − γ)λ)r γ−1 γ γ−1 (γ−1)2 > 0 for r ∈ (0, 1] since γ ≥ 2. Maximize ´Π= π (r, r)dF (θ) − λS(0, q(r1 ), p(r1 )) subject to (4). Given that π (r, r) is convex ˜ ˜ r∈G θ∈grthe solution to the last problem has at most one coarse rating r1 corresponding to [0, θ1 ].Indeed otherwise replacing an extra coarse rating r = r1 with an interval of perfect θ1revelation r = θ ∈ Gr would raise Π. (4) requires r1 = 2 , θ ∈ [0, θ1 ] and r = θ, 1θ ∈ (θ1 , 1]. Substitute S(0, q(r1 ), p(r1 )) = − γ q(r1 )γ − p(r1 )q(r1 ). Optimizing Π(θ1 ) = 27
  28. 28. ´1π ( θ21 , θ21 ))θ1 +˜ π (θ, θ)dF (θ) + λ γ q( θ21 )γ + λp( θ21 )q( θ21 ) over θ1 ∈ [0, 1] we obtain θ∗ = ˜ 1 θ1 λ γ(γ−1)2γ/(γ−1) −(γ 2 +γ−1)min β γ(γ−1)2γ/(γ−1) −γ(2γ−1) ,1 ≥ 0 since γ ≥ 2. Reporting r(θ) = θ for θ ∈ [0, θ∗ ] and r(θ) = θ for θ ∈ (θ∗ , 1] is a unique solution. ∗Of course there exists an equivalent solution with a pooling interval [0, θ∗ ); the expectedparties’ payoffs are the same in both cases and we refer to them as a single solution. ∗The corresponding grading is G = { θ2 ∪ (θ∗ , 1]}; condition (3) pins down the payment ´θ ∗ ∗t(r(θ)) = λq(r(x))dx − λS(θ, q(r(θ), p(r(θ))) + λS(0, q( θ2 ), p( θ2 )), θ ∈ [0, 1] QED. 0 Proof of proposition 4. Perfect ratings are not possible (see the argument in the text), therefore each ratingri ∈ G, i = 1, ..., N − 1 corresponds to a grade [θi−1 , θi ), θi−1 < θi , rN corresponds to[θN −1 , 1]. Denote r1 the lowest rating in G. Due to proposition 2 constraint (5) in problem(8) can be replaced with t(r1 ) ≥ 0 ⇔ U (0) ≥ λS(0, q(r1 ), p(r1 )). Problem (8) is equivalentto ˆ1 max π (θ, r)dθ − U (0), s.t. (1), (2), (3), U (0) ≥ λS(0, q(r1 ), p(r1 )). ˜ (13) {r(.), U (0)} 0Here π (θ, r) = p(r)q(r) + λS(θ, q(r), p(r)) − λq(r)(1 − θ). A unique solution to this ˜problem is U (0) = λS(0, q(r1 ), p(r1 )) ⇔ t(r1 ) = 0, and an increasing function r∗ (θ) = ∂2πarg max[˜ (θ, r)], ∀θ ∈ [0, 1] since π ˜ ∂θ∂r = 2λ ∂q > 0. Border points between any two ratings ∂r {r∈G} ˜ ∗ ˜ ∗issued with positive probability ri−1 , ri are given by π (θi−1 , ri−1 ) = π (θi−1 , ri ), i = 2, ..., N . ∗ Condition (4) requires θi−1 = θi−1 , i = 2, ..., N . Hence we obtain (9): π (θi−1 , ri−1 ) = ˜π (θi−1 , ri ), i = 2, ..., N .˜ ∂q Given that Uθr = ∂r > 0 and r∗ (θ) is increasing from U (θ) = max[λS(θ, q(r), p(r))+t(r)] r∈Gand U (0) = λS(0, q(r1 ), p(r1 )) ⇔ t(r1 ) = 0 we can find t(.) satisfying (3) QED. Proof of corollary 2. For γ = 2 and F (.) = U [0, 1] condition (9) becomes (3λ −2)θi − 2(5λ + 2)θi−1 + (3λ − 2)θi−2 = −8λ, i = 2, ..., N . Since θ0 = 0, θN = 1 a solution 2 11λ−2with N = 2 exists iff λ ∈ ( 11 , 6) and is characterized by θ1 = 10λ+4 . Suppose N ≥ 3and take the first difference (3λ − 2)dθi − 2(5λ + 2)dθi−1 + (3λ − 2)dθi−2 = 0. One 2must have dθi ≥ 0 for i = 1, .., N , which is not possible if λ ≤ 3 , therefore N < 3 2 2 5λ+2 8λfor λ ≤ 3 . Suppose λ > 3 , denote x = 3λ−2 , then 1 + x = 3λ−2 and we obtain adifference equation dθi − 2xdθi−1 + dθi−2 = 0, i = 3, ..., N . Characteristic polynomial √D2 − 2xD + 1 = 0 delivers D = x − x2 − 1 < 1 and D = 1/D. Solution to thedifference equation is dθi = ADi + A D−i , i = 1, ..., N . Conditions θ0 = 0, θN = 1 require(dθ2 + dθ1 ) − 2xdθ1 = −(1 + x) and 1 − 2x(1 − dθN ) + (1 − dθN − dθN −1 ) = −(1 + x) 28

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