OPTIMAL CERTIFICATION POLICY, ENTRY, AND INVESTMENT IN THE PRESENCE OF PUBLIC SIGNALS

Transcription

1 OPTIMAL CERTIFICATION POLICY, ENTRY, AND INVESTMENT IN THE PRESENCE OF PUBLIC SIGNALS JAY PIL CHOI AND ARIJIT MUKHERJEE A. We explore the optimal dislosure poliy of a ertifiation intermediary in an environment where (i) the seller s deision on entry and investment in produt quality are endogenous and (ii) the buyers observe an additional publi signal on quality. The intermediary mutes the seller s entry inentives but enhanes investment inentives following entry, and the optimal poliy maximizes rent extration from the seller in the fae of this trade-off. We identify onditions under whih full, partial or no dislosure an be optimal. The intermediary s report beomes noisier as the publi signal gets more preise, but if the publi signal beomes too preise, the intermediary resorts to full dislosure. In the presene of an intermediary, a more preise publi signal may also lead to lower soial welfare. JEL Classifiation: D4, L12, L4, L43, L51, L52 Keywords: Certifiation intermediaries, optimal dislosure poliy, investment and entry inentives, publi signal. 1. I Certifiation intermediaries are a ommon feature in many markets where the onsumers may not be able to readily assess the quality of the sellers produt. For example, redit rating agenies ertify finanial instruments, auditors ertify the finanial standings of organizations, numerous professional groups ertify the qualifiations and skills of their members, and a large numbers of agenies and laboratories offer ertifiation servie for produt safety. It is also interesting to note that in suh markets, the ertifiation intermediaries are often not the only soure of information as the onsumer may have aess to some information that is publily available. For example, in the U.S., the investors in a publily traded ompany not only onsider the firm s redit rating but also its filings with the Seurities and Exhange Commission as they are informative about the firm s overall finanial strengths. Similarly, though the firm may get its produt ertified by an intermediary, the ratings given by some non-profit independent ageny suh as Consumer Report an be an additional soure of imperfet publi signal on produt quality. There is a vast literature on ertifiation intermediaries that studies how their presene might improve trade effi ienies by alleviating the adverse seletion problem in the marketplae through the provision of relevant information (Lizzeri, 1991). The redution in Date: January 11, For their helpful omments and suggestions, we would like to thank Navin Kartik, Andrei Shevhenko, and seminar audienes at Mihigan State University. We also thank Cody Orr for his exellent researh assistane. Department of Eonomis, Mihigan State University, 110 Marshall Adams Hall, East Lansing, MI hoijay@msu.edu. Department of Eonomis, Mihigan State University, 110 Marshall Adams Hall, East Lansing, MI arijit@msu.edu.

2 2 CHOI AND MUKHERJEE information asymmetry between the trading parties an also enhane the sellers inentives to invest in quality and improves alloational effi ieny (Albano and Lizzeri, 2001). But the presene of an intermediary also has redistributive onsequenes. Clearly, the seller s investment inentives get muted as the intermediary aptures a share of the resulting gains. Moreover, the seller an be worse off in the presene of an intermediary if the intermediary an extrat more surplus than what gets reated due to the inreased investment inentives. The extant literature has not fully explored the interplay between suh redistributive and the effi ieny effets as the extent of entry by sellers is typially assumed to be exogenously fixed. In this paper we onsider a setting where the level of entry by the sellers is endogenous. In suh an environment, the presene of the intermediary gives rise to a novel trade-off: it enhanes the seller s inentives to invest in quality following entry but mutes his inentive to enter at the first plae. The goal of the paper is to explore the intermediary s optimal ertifiation poliy in the fae of this trade-off. In the presene of a publi signal on quality, we analyze how the informativeness of the intermediary s signal interats with the preision of the publi signal and draw out its welfare impliations. In order to explore the aforementioned trade-off between entry and investment inentives, we onsider a model of ertifiation intermediary with the following key features. The intermediary first ommits to a poliy that speifies a ertifiation fee and a dislosure poliy (we will elaborate on this shortly). Upon observing the ertifiation poliy, the seller deides whether to enter by inurring a fixed ost, and following entry, whether to invest to improve its produt quality. The investment inreases the likelihood of produing a high quality produt and the ost of investment depends on the seller s private type. After the quality is realized, the seller deides whether to use the intermediary s servie. The onsumers annot diretly observe the quality but an obtain relevant information from two soures: a publi signal whose preision is exogenously fixed, and the intermediary s signal (i.e., whether the seller has ertified his produt, and if so, what signal the intermediary has released). The onsumers offer a prie that is driven by their belief about the produt quality given the available information. As in Lizzeri (1999), we define a dislosure poliy as probability distribution over a set of signals onditional on the quality of the seller s produt. Suh a speifiation aommodates the extreme ases of full and no dislosure as well as the more generi ase of partial dislosure where the intermediary reveals a garbled information about the underlying produt quality. Also, in order to highlight the key trade-off between the entry and investment inentives in the most transparent way, we restrit parameters suh that entry is always effi ient but investment may be ineffi ient in the absene of the intermediary. We derive three key results. First, we haraterize a full-dislosure benhmark (Proposition 2) where the intermediary is required to reveal the quality without any noise should the seller opt to use its servie. In suh a setting, the extent of effi ieny in entry and investment ritially depends on the size of the entry ost. In partiular, we find that when the entry ost is below a utoff, the entry is effi ient. On the other hand the investment in quality, though improves ompared to the senario where there is no intermediary, remains ineffi - iently low ompared to the first-best (i.e., when the produt quality is perfetly observed by the buyers). In ontrast, if the entry ost exeeds the utoff, then both entry and investment remain ineffi iently low.

3 OPTIMAL CERTIFICATION POLICY 3 The intuition is as follows. With mandated full dislosure, the intermediary s problem is similar to that of a monopolist by harging a higher ertifiation fee it an extrat more rents from the seller (i.e., the buyer of its ertifiation servie) but also redues the likelihood that the seller would purhase the ertifiation servie. Note that with full dislosure, only a seller with high quality produt opts to ertify. Thus, in order to ensure that the intermediary faes a robust demand for its servie, it must protet both the entry and the investment inentives of the seller. In other words, the intermediary hooses the ertifiation prie so as to trade-off the gains from rent extration with the losses from both the entry and the investment distortions. When the entry ost is low, the optimal ertifiation fee does not affet the entry inentives and trades off rent extration with distortions in investments only. But as the entry ost inreases, the entry gets distorted and the intermediary needs to redue its ertifiation prie to indue entry. However, if the entry ost beomes suffi iently large, aommodating entry for all types of the seller beomes too ostly. In response, the intermediary hooses a prie suh that entry is viable only if the seller also invests in quality (following entry), and all seller types with relatively high investment ost are forelosed from the market. Hene, under full dislosure, when the entry osts are relatively low the intermediary leads to more effi ient investment (ompared to the ase when there is no intermediary) and also maintains effi ieny in entry. But, if the entry ost is relatively high, the presene of the intermediary leads to a trade-off as it enhanes the effi ieny in investment but distorts the effi ieny in entry. Next, we haraterize the optimal dislosure poliy for the intermediary and explore the interplay between the informativeness of the intermediary and the preision of the publi signal (Propositions 4 and 5). If the publi signal is suffi iently preise, the optimal poliy alls for full dislosure irrespetive of the ost of entry. Otherwise, the dislosure poliy is more nuaned. If the ost of entry remains either too low or too high, full dislosure is still optimum. But for a moderate ost of entry the intermediary resorts to partial dislosure where, with some probability, the low quality produt reeives the same ertifiation that the high quality produt gets. Moreover, as the preision of the publi signal inreases, the intermediary s report beomes inreasingly more noisy and eventually absolutely uninformative. But one the publi signal beomes suffi iently preise, the intermediary reverts to full dislosure. To see the argument, reall that under full dislosure the intermediary has exatly one instrument, namely the ertifiation fee, to trade off the gains from rent extration with the losses from diminished inentives for both entry and investment. For low entry ost the optimal fee does not distort entry, but as the ost inreases the trade-off beomes more aute as the ertifiation fees not only mutes investment inentives of the seller but also his inentives to enter the market at the first plae. The intermediary must signifiantly lower its ertifiation fee in order to inentivize entry of all types, and as a result, ends up leaving large amount rents to a high quality seller. A noisy dislosure an attenuate this trade-off as it allows the intermediary to extrat more rents from the seller while the resulting damage to entry inentives is partly restored through the garbling of information. By partly pooling the low and high quality seller, the intermediary an inrease its payoff by ensuring effi ient entry and induing the seller use its servie irrespetive of the realized quality level. This is due to the fat that under partial dislosure, even a low quality seller expets to be pooled with the high quality and feth a

4 4 CHOI AND MUKHERJEE better prie from the onsumers, whereas the onsumers may believe the seller to be of low quality if he does not use the intermediary. However, when the ost of entry is too high, even if the intermediary resorts to partial dislosure effi ient entry would still all for a signifiant redution of the ertifiation fee. And at this point it may be more profitable for the intermediary to swith bak to full dislosure and harge a higher fee in order to extrat rents from the high quality sellers only. This implies that partial dislosure an be an useful strategy when the entry ost is intermediate and the trade-off between the rent extration and seller s entry and investment inentives are most aute under full dislosure. The argument for the optimality of partial dislosure is also useful in understanding how the intermediary responds to the hanges in the preision of the publi signal. The optimal partial dislosure poliy stipulates a ertain amount of spread between the expeted pries that a high and a low quality seller gets from the ustomer. Reall that the prie that the onsumers offers for a produt is tied to their posterior belief about the produt s quality. Hene, given the preision of the publi signal, the intermediary garbles its own report so as to attain the optimal spread in the onsumer s belief. But notie that as the intermediary s signal is (weakly) informative, its ability to influene the onsumer s posterior belief is onstrained by the preision of the publi signal. The spread in the onsumers posterior beliefs that a low and a high quality seller expets to emerge following the intermediary s report is at least as large as the spread that would have been obtained when the onsumers an aess the publi signal only. Until this onstraint is binding, the intermediary adds more noise in its signals as the publi signal beomes more preise. Thus, the informational ontent provided by the publi signal and the ertifier interats as substitutes. But one the onstraint beomes binding, it is optimal for the intermediary not to dislose any further information. Finally, as disussed earlier, when the publi signal beomes very preise, the intermediary swithes from partial dislosure to full dislosure; i.e., the intermediary s and publi signal beome omplements. Our third key result analyzes how the soial welfare under the intermediary s optimal poliy varies with the preision of the publi signal. We show that in the presene of the intermediary, the preision of the publi signal affets the soial welfare only if the preision is at a moderate range. Also, the welfare under intermediary need not be monotone with the publi signal s preision. And finally, the value of the intermediary i.e., the hange in soial welfare due to the intermediary s presene dereases in the preision of the publi signal. Reall that while the intermediary may improve the effi ieny in investment, it an indue ineffi iently low entry, partiularly when the entry ost is relatively high and the intermediary resorts to full dislosure. The findings above stem from the fat that an inrease in the preision of the publi signal an lead to a regime hange in the intermediary s dislosure poliy where it swithes from partial dislosure that indues effi ient entry to a full dislosure poliy where entry is restrited. The resulting welfare loss due to the entry ineffi ieny an outweigh the gains due to stronger investment inentives, and an lead to an overall drop in the soial welfare. In ontrast, as the publi signal beomes more preise, welfare in the absene of the intermediary always inreases as it provides more inentives to invest in quality. This implies that when the publi signal is suffi iently preise, the intermediary s presene is detrimental to soial welfare.

5 OPTIMAL CERTIFICATION POLICY 5 Literature review: There is by now a large literature on the role of ertifiation intermediaries in markets plagued by asymmetri information (some early ontributions inlude, among others, Biglaiser, 1993; Biglaiser and Friedman, 1994; Lizzeri, 1999; and Albano and Lizzeri, 2001; also see Dranove and Jin, 2010, for a survey). In relation to this literature, our paper is losest to Lizzeri (1999) and Albano and Lizzeri (2001). Lizzeri analyzes the role of ertifiation intermediaries in a model of adverse seletion, and shows that the optimal hoie for the intermediaries often entails no dislosure or partial losure in the form of minimum quality ertifiation. While Lizzeri assumes that the seller s produt quality is exogenously fixed, Albano and Lizzeri (2001) extends Lizzeri s model to endogenize the quality. They analyze the issue of the optimal degree of information revelation and show that the presene of intermediary enhanes effi ieny by inreasing the sellers inentives to provide high quality (though the full information first-best alloation remains infeasible). 1 But in ontrast to our setup, these models abstrat away from the question of the seller s entry inentives and the interplay between the publi signal on quality and the intermediaries dislosure poliy (i.e., they assume that the seller is already in the market and the intermediary is the only soure of information for the buyers). A reent paper that explores how ertifiation intermediaries may influene the sellers entry deision is Harbaugh and Rasmusen (2018). They onsider a pure adverse seletion model without investment and show that exat grading is never optimal. As in our model, oarse grading i.e., partial dislosure is used to indue more partiipation by senders; in their model with a ontinuum of types, the middle types have more inentives to partiipate in the ertifiation proess under a pass-fail sheme than an exat grading sheme as they would like to distinguish themselves from the low types but pool with the high types. They assume a non-profit ertifier whose objetive is to enhane the amount of information available to the buyers while the seller inurs an exogenous ertifiation ost. Our model, however, assumes a profit-maximizing ertifier with an endogenous ertifiation prie and the ost of garbling information omes from redued investment inentives rather than the loss of information per se. In a related work, Dubey and Geanakoplos (2010) onsider the optimal grading sheme in induing students efforts, that is, they fous on the optimal dislosure poliy for investments (but without onsidering partiipation onstraints). They show that oarse grading (i.e., partial dislosure) often motivates students to work harder when the students are about their status (relative rank in the lass). 2 While these papers present theoretial analyses of the intermediary s impat on the agents behavior, Hui et al. (2018) provide an empirial analysis on the effets of ertifiation poliies on the evolution of markets. By using a hange in the ertifiation poliy of ebay as a natural experiment, they investigate how the stringeny of ertifiation poliy affets the types of entrants and seller behavior. However, the ertifiation poliy in their model is exogenous and the servie is unilaterally provided by a platform owner (i.e., ebay) without any prie. In ontrast, our fous is on haraterizing the optimal ertifiation poliy by a profit-maximizing ertifier. Our paper is also related to the literature on the optimal design of information struture (Ostrovsky and Shwarz, 2010; Kamenia and Gentzkow, 2011; Rayo and Segal, 2010). Kamenia and Gentzkow analyze the general Bayesian persuasion problem in whih a sender 1 In a related paper, Belleflamme and Peitz (2014), assume that onsumers observe the investment (but not the realization of the produt quality) and show that the firm overinvest in quality ompared to the full informatioin benhmark. 2 Similar issues are also disussed by Costrell (1994) and Boleslavsky and Cotton (2015).

6 6 CHOI AND MUKHERJEE hooses the optimal information struture for a signal to be revealed to a reeiver, and derive general onditions under whih the ability to ontrol the informational environment is benefiial to the sender. 3 As in the Bayesian persuasion literature, we also assume that the intermediary an preommit to a partiular dislosure poliy. However, in our setting the intermediary s problem annot be modeled as a standard sender s problem beause the distribution of the produt quality (i.e., the underlying state of the world that the sender reveals information on) is endogenous to the sender s dislosure poliy. In a more reent paper, Boleslavsky and Kim (2018) develop a framework to analyze the optimal design of information struture in the presene of moral hazard. In partiular, they onsider a three-player Bayesian persuasion game whih inludes an agent, in addition to the sender and the reeiver, whose private effort determines the distribution of an unobservable state. Our setup is similar to Boleslavsky and Kim in that the distribution of the state of the world, i.e., the produt quality is endogenously affeted by the seller s entry and investment deisions. Nonetheless, our framework does not onform to the Boleslavsky and Kim s setup beause we also have an element of adverse seletion with heterogeneous agent types and additional onstraint on the entry whereas their model fouses only on the investment margin with only one type of agent. 4 Finally, it is worth noting that our analysis on the interplay between the informativeness of the intermediary and the preision of the publi signal is related to the literature on the adverse onsequenes of transpareny (if we interpret more preise publi signal as more transpareny in an ageny relationship). Prat (2005), for instane, shows that an agent with areer onerns may ignore his signal, to the detriment of the prinipal, and behave as a onformist when his ation is observed (i.e., transparent). Levy (2007) onsiders the effet of transpareny on ommittee deisions and identifies irumstanes under whih a seretive ommittee that uses a partiular voting rule makes better deisions on average. Their analyses, however, are in ompletely different ontexts and rely on areer-based reputation effets. Also, these papers only onsider a binary hoie between no transpareny and full transpareny. In ontrast, we onsider a ontinuum of preision levels in the publi signal, whih enables us to ondut a omparative stati analysis and derive the non-monotoniity result. The remainder of the paper is organized as follows. In Setion 2, we set up our basi model of adverse seletion with investments and endogenous entry in the presene of intermediary. In Setion 3, we haraterize the market equilibrium in the absene of intermediary as a benhmark. Setion 4 explores the role of intermediary in our set-up, but onsider an intermediary who is onstrained to fully dislose the quality. In Setion 5, we present a general analysis of optimal dislosure poliy in whih the intermediary an hoose both the ertifiation prie and the dislosure poliy. We identify onditions under whih the intermediary may adopt partial dislosure by garbling information. Setion 6 disusses the impliations of the optimal poliy, its welfare onsequenes, and presents a onlusion. All proofs are given in the Appendix. 3 The Bayesian persuasion literature has been extended to many different diretions to address the possibilities of multiple senders (Au and Keiihi, 2017; Gentzkow and Kamenia, 2017; Li and Norman, 2015), multiple reeivers (Chan et al. 2017), and a privately informed reeiver (Kolotilin et al. 2017), for instane. The framework has also been applied to a variety of ontexts inluding voting (Alonso and Camara, 2016), monopoly priing (Roesler and Szentes, 2017), prie disrimination (Bergemann et al. 2015), and areer onerns (Rodina, 2017). 4 See Rosar (2017) for an analysis of a Bayesian persuasion game with voluntary partiipation.

7 OPTIMAL CERTIFICATION POLICY 7 2. M P: We onsider an environment with three types of players: a seller, a ertifiation intermediary, and a set of idential buyers. A : The seller deides whether to enter a market to sell his produt to a set of idential buyers. The buyers valuation of the seller s produt is based on its quality. The produt quality ould be either high or low and generates a value v {0, 1} for the buyer, where v = 0 if the quality is low and v = 1 if it is high. If the seller deides to enter the market, he inurs an entry ost of k. Upon entry he an undertake an investment in quality in order to inrease the likelihood of produing a high quality produt. Let I {0, 1} denote the seller s investment deision where I = 1 if he invests in quality and I = 0 otherwise. We have (1) Pr (v = 1 I = 1) = α > 1 2 = Pr (v = 1 I = 0). The ost of this investment is determined by the seller s type θ [0, 1], whih is assumed to be uniformly distributed on [0, 1]. For a seller of type θ, the ost of investment is /θ for θ 0 (and C > 1 for θ = 0). 5 The seller s type is his private information and known to him before he makes his entry deision. The produt quality is also privately observed by the seller, leading to an information asymmetry in the produt market. But the buyers an obtain information on quality through two hannels. First, the seller an hire a ertifiation intermediary who an verify the quality and dislose additional information. The intermediary is assumed to be a monopolist in the market for ertifiation servies. At the beginning of the game, the intermediary ommits to a ertifiation prie p and a dislosure poliy D that speifies what it may dislose to the buyers, given the underlying quality of the produt. However, the intermediary may not fully reveal the quality of the produt and an potentially garble its report. In order to allow for suh a noisy dislosure, we define a dislosure poliy as a mapping D : {0, 1} X where X is a pre-speified signal spae (and hene, a part of the poliy). That is, the dislosure poliy sends a signal x X that is drawn aording to a given probability distribution, onditional on the true quality of the produt. We assume that the intermediary does not inur any ost to evaluate the produt quality. Seond, in addition to the intermediary s signal, the buyers observe a publi signal z {0, 1} that provides noisy information about the produt quality, where Pr (z = j v = j) = π [1/2, 1), for j = 0, 1. The parameter π represents the preision of the publi signal; when π = 1 the pubi signal 2 is ompletely uninformative. The seller makes his entry deision after observing the intermediary s offer (p, D). Also, the seller deides on whether to hire the intermediary after the quality realization but before 5 For simpliity we normalize the seller s probability of produing a high quality produt without any investment to be 1 2. Also, if Pr (v = 1 I = 0) = α 0 < 1 2 < α 1, we may not have a unique equilibrium even in the absene of the intermediary, and the haraterization of the optimal poliy beomes analytially intratable.

8 8 CHOI AND MUKHERJEE the publi signal is realized. Observing the available signals on the quality, the buyers simultaneously bid for the produt and the produt is sold at the highest bid. All players are assumed to be risk neutral. This implies that the produt is sold at the expeted value of the produt given the information available to the onsumers. T: The following time line summarizes the game. Stage 1: The intermediary ommits to his ertifiation poliy (p, D). Stage 2: The seller observes his type θ and the intermediary s poliy and deides whether to enter. (If there is no entry, the game ends.) Stage 3: If the seller enters, he deides whether to investment on quality, and observing his produt quality, deides on whether to hire the ertifiation intermediary. Stage 4: The intermediary reveals its signal x on the produt quality. Stage 5: The publi signal z on quality is revealed. Observing x (if available) and z, the buyers bid for the produt and the produt is sold at the highest bid. S : The strategies of the players are as follows: the intermediary s strategy is to hoose a ertifiation poliy (p, D). The seller s strategy has three omponents: (i) entry and (ii) investment deisions given his type and the intermediary s poliy, and (iii) deision on hiring the intermediary given his produt quality and the intermediary s poliy. Finally, the buyers strategy is to hoose a bid given the available information (i.e., the publi signal, the intermediary s ertifiation poliy, whether the intermediary was hired or not, and if hired, the intermediary s report). We use (pure strategy) perfet Bayesian Equilibrium as the solution onept. The optimal dislosure poliy is defined as the (p, D) pair that indues the highest feasible equilibrium payoff for the intermediary. Below, we maintain the following parametri restritions to streamline our analysis. Let := α 1 denote the soial value of investment in quality. 2 Assumption 1. (i) α < ; (ii) < k < Observe that Assumption 1 (i) implies <, i.e., at least for some types of the seller, it is soially effi ient to invest in quality. We maintain a stronger assumption (than simply requiring < ) so as to rule out ertain orner solutions that lutter our analysis without offering any important eonomi insight. Assumption 1 (ii) plays a similar role by stipulating a moderate range of entry ost where the interplay of entry and investment deisions is nontrivial and leads to a rih set of equilibrium harateristis. 3. B: F- In this setion, we onsider two benhmark ases that inform our subsequent analysis of the intermediary s dislosure poliy. First, onsider the soial first-best the entry and investment rule that maximizes the aggregate surplus of the seller and the buyers. Note that the surplus reated by a seller of type θ with investment I {0, 1} is Pr (v = 1 I) I/θ k. Hene, the first-best requires

9 OPTIMAL CERTIFICATION POLICY 9 all types of the seller to enter (reall from (1) and Assumption 1 (ii), Pr (v = 1 0) k > 0) and all types θ θ F B I to invest, where: α θ F B I = 1 B, i.e., θfi = 2. As a seond benhmark, onsider the investment and entry inentives of the seller in absene of the intermediary where the buyers only soure of information is the publi signal z. Let Θ E [0, 1] and Θ I Θ E be the set of the seller s types that enter the market and invest in quality, respetively. Sine the buyers ompetitively bid for the produt, the equilibrium is defined as follows: (i) The seller obtains a prie of E (v z, Θ E, Θ I ), i.e., the expeted valuation of the produt given the realized publi signal and set of seller types that enter and invest. (ii) Given the seller s expeted prie (prior to the realization of the publi signal), only the types in Θ E enters, and among the entrants, only the types in Θ I invests. Denote v i (Θ E, Θ I ) = E (v z; Θ E, Θ I ) Pr (z I = i), z {0,1} whih is the expeted prie the seller reeives when his investment deision is I and the buyers believe that the set of types that enter is Θ E and the set of types that invest is Θ I. Now, for type θ, the expeted payoff from entry and investment deision I is: V (I; θ, Θ E, Θ I ) = { v1 (Θ E, Θ I ) k θ if I = 1 v 0 (Θ E, Θ I ) k if I = 0. The following proposition haraterizes the equilibrium. Proposition 1. (Equilibrium without intermediary) If there is no intermediary in the market, the equilibrium has the following features: (i) Entry is always effi ient all types of the seller enter (i.e., Θ E = [0, 1]). (ii) Investment may be ineffi ient. There exists a threshold > 0 suh that if > no type invests. Otherwise, there exists a unique utoff θ NI I (> θ F B I ) suh that all types θ θ NI I invest. Moreover, θ NI I is dereasing in π. The proposition above has a simple intuition. Note that even if the seller does not invest, he earns at least 1 2 k > 0 by entering the market irrespetive of his own type (v 0 (Θ E, Θ I ) 1 2 for all Θ E and Θ I ). Hene, all types enter. Moreover, as the seller s payoff from investment is monotonially inreasing in his type, the investment deision must follow a utoff strategy: unless the ost of investment is too high, all types above a threshold invest. However, there is under-investment ompared to the first-best beause the seller annot reeive a fair return from his investment due to asymmetri information on the produt quality. But the investment ineffi ieny an be mitigated via the availability of a publi signal, and the extent of suh ineffi ieny dereases with the signal s preision. The more preise is the publi signal the higher is the return from investment, and the stronger investment

10 10 CHOI AND MUKHERJEE inentive moves the type threshold loser to the first-best. In the limit, if the publi signal is perfetly informative (i.e., π = 1), there is no information asymmetry and we restore the first-best (i.e., θ NI I = θ F B I ). In ontrast, if the publi signal is ompletely uninformative (i.e., π = 1 ), there is no inentive to invest for the seller regardless of his type (i.e., θni 2 I = 1). 4. I F D Given the benhmark analyses above, we now explore the role of intermediary in our model. We first onsider a simple ase where the intermediary, if hired, is obligated to fully dislose the quality. That is, under (mandatory) full dislosure, the intermediary an only set the prie p 0 for its ertifiation servie. The analysis of the full dislosure ase learly illustrates the trade-offs that the intermediary faes while hoosing its ertifiation fee, and it is also instrutive in understanding when and how a partial dislosure poliy may be optimal. Let µ (x, z) = Pr (v = 1 x, z) be the buyers (posterior) belief on quality given the intermediary s report x and the publi signal z. Note that under full dislosure, x {0, 1, } where, with a slight abuse of notation, we denote x = when the seller does not use the intermediary. (Clearly, as the seller deides on hiring the intermediary after observing his produt quality, the buyers would update their beliefs even when the seller deides not to hire the intermediary.) In what follows, we analyze the optimal PBE for the intermediary, i.e., we derive the equilibrium ertifiation prie p and the assoiated belief µ that maximize the intermediary s profit given that the beliefs are onsistent and the seller s entry and investment deisions are sequentially rational (onditional on his type). Lemma 1. In the optimal PBE under full dislosure, only the high quality seller hires the intermediary. The argument is straightforward (hene, we omit the formal proof): Notie that in any PBE with p > 0, it annot be the ase that the seller hires the intermediary irrespetive of his produt quality. Under full dislosure, the low quality seller is sure to get a prie of 0 from the buyers if he hires the intermediary, and he an save on the ertifiation fee by not hiring the intermediary at the first plae. Likewise, in the optimal PBE, it annot be the ase that the seller never ertifies irrespetive of his produt quality. If suh an equilibrium exists, the intermediary makes 0 whereas (at the time of the ertifiation stage) the high quality seller an only expet from the buyers a bid that is stritly less than 1 (as the publi signal is noisy). Any suh equilibrium, even if it exists, gives lower payoff to the intermediary ompared to an equilibrium where the intermediary harges a suffi iently low prie ε > 0 (with full dislosure), and µ (, z) = 0 for all z {0, 1} (i.e., the buyers take a skeptial posture à la Milgrom and Roberts (1986) and believe that the produt is of low quality if the seller does not hire the intermediary). For suh a prie, the high quality seller would find it profitable to use the intermediary, and the intermediary would earn a stritly positive payoff. In light of the above lemma, we fous on the fully separating PBE where only the high quality seller hires the intermediary. If suh a PBE exists, notie three salient features of suh an equilibrium: First, the publi signal does not affet beliefs. Seond, for any prie of ertifiation, p, a seller of type θ enters the market only if

11 (2) max OPTIMAL CERTIFICATION POLICY 11 { α (1 p) } θ k, 1 (1 p) k 2 0. Hene, in any equilibrium, we must have: p p := 1 k + α, as otherwise even the most effi ient type (θ = 1) would hoose to stay out. Third, all types of the seller enter if 1 (1 p) k > 0, i.e., 2 p < p E := 1 2k. Note that for p < p E, the seller gets a stritly positive (expeted) payoff by entering the market even if he does not invest following entry. Also, for any p [p E, p) entry is profitable only if the seller also invests in quality (p E < p by Assumption 1 (ii)). Based on the above observations, we an now haraterize the optimal ertifiation prie for the intermediary. In any PBE the set of seller s types that enter (Θ E ) and invest (Θ I ) are pinned down by their respetive type utoffs, θ E and θ I (say), where Θ E = [θ E, 1] and Θ I = [θ I, 1]. (The argument is the same as in the benhmark ases disussed above, and follows from the fat that the ost of investment is dereasing in the seller s type). Now, the intermediary an adopt one of two possible types of priing strategies: (i) harge p p E and indue full entry, i.e., Θ E = [0, 1], or (ii) harge p (p E, p] suh that low types stays out and all types that enter also invests in quality, i.e., Θ I = Θ E. Thus, in the most profitable PBE for the intermediary, he harges a prie that solves the following problem: max p Π (p) := p Pr (v = 1 θ E, θ I ) = p [ 1 (θ 2 I θ E ) + α (1 θ I )] ] { ΠE (p) := p ( 1 = θ 2 I + α (1 θ I ) ) if p p E Π I (p) := pα (1 θ I ) if p [p E, p), where θ E and θ I are the marginal types of the seller for whom it is profitable to enter and profitable to invest (respetively), given the intermediary s prie. (Notie that θ I θ E is the mass of entrant types who do not invest whereas 1 θ I is the mass of types who invest following entry.) The lemma below states the solution to the above problem. Lemma 2. There exist three entry ost utoffs k E < k I < k suh that the intermediary s payoff at the most profitable PBE is: where Π = { Π E if k < k Π I otherwise, (3) Π E = max p p E Π E (p) = { ΠE := α + 2 α if k < k E Π E := α (1 2k) ( ), 1 otherwise 2kα

12 12 CHOI AND MUKHERJEE and (4) Π I = max p>p E Π I (p) = { ΠI := α + k 2 (α k) if k k I Π I := α (1 2k) ( ) 1 otherwise. The intuition for the above result is as follows. By raising the ertifiation prie the intermediary an extrat more rents from the seller but also redues the likelihood that the seller would hire the intermediary at the first plae. The latter effet on the intermediary s demand stems from the fat that the ertifiation prie distorts both the investment and the entry inentives of the seller. Reall that under full dislosure only the high-quality seller hires the intermediary, and the seller s produt quality is more likely to be high if he invests in quality. But the larger is the ertifiation prie the weaker is the seller s inentive to invest as the intermediary keeps a larger share of the trade surplus. And if there is not enough surplus left for the seller to over his entry ost, he may not enter the market. Consider the ertifiation prie p (say) that the intermediary would have harged if the seller were already on the market. In suh a setting, the optimal prie trades off the gains from the rent extration with the losses that emanates only due to weakened investment inentives of the seller. Notie that for suffi iently low entry ost (k < k E ), p < p E. So, even in our setting the intermediary would harge p as suh a prie would not affet entry, and all types of the seller would be on the market. But as the entry ost inreases beyond the threshold k E, we have p > p E, and there would be ineffi iently low entry at the prie p the seller may stay out if his type is suffi iently low (i.e., ost of investment is suffi iently high). As long as the entry ost is moderately low (i.e., when k E < k < k ) the intermediary is better off by lowering its prie to p E. By harging a lower ertifiation prie, the intermediary forgoes a part of the rent that it ould have extrated from the seller. But suh a loss is more than offset by the gains from inreased likelihood of having a high-quality seller as the redution in the ertifiation prie restores entry effi ieny and also strengthens the seller s investment inentives. However, when the entry ost is signifiantly large, aommodation of entry would onsiderably hurt the intermediary as it would require a large redution in the ertifiation prie. As a result, when the entry ost rosses a threshold (k ) the intermediary finds it optimal to restrit entry by raising his ertifiation prie suh that all entrants types have inentive to invest in quality. In light of the above lemma, we an haraterize how the intermediary s optimal priing poliy (under full dislosure) affets the entry and investment effi ienies in our setting. 2k Proposition 2. (Effi ieny impliations of intermediary under full dislosure) If k < k, there is effi ient entry, i.e., all types enter, but investment remains ineffi ient as only the types θ θ f I invest, where { } α θ f I := min,. 2k But for k > k, both entry and investment remain ineffi ient as only the types θ θ i I enter and invest where

13 OPTIMAL CERTIFICATION POLICY 13 θ i I := α k. All other types stay out. Π θ I Π α θ I Π I θ i I Π E θ F B I θ f I 0 2 k E k 1 k k E k 1 k 2 Figure 1. The intermediary s payoff (Π ) and assoiated utoff type for investment (θ I ) as a funtion of entry ost k. Figure 1 depits the intermediary s equilibrium payoff as a funtion of entry ost k and the orresponding utoff types for investment (as given by Lemma 2 and Proposition 2). Observe that θ f I is (weakly) dereasing whereas θi I is inreasing in the entry ost. The intuition is simple and, again, an be seen from Lemma 2: When k < k E, entry is effi ient and the intermediary s optimal prie is independent of k it is the same prie that he would have harged if the seller were already on the market. So, the seller s share of the trade surplus is also independent of the entry ost, and, onsequently, the seller s type utoff for investment (θ f I ) remains onstant. But when k E < k < k, the intermediary aommodates entry by setting p = p E = 1 2k. Thus, higher k alls for lower ertifiation prie and sine the seller gets to keep a larger share of the trade surplus, his inentive to invest inreases. As a result θ f I is dereasing in k. In ontrast, for k > k, entry is profitable only if the seller invests. So with larger entry ost only the more effi ient types enter; i.e., θ i I is inreasing in k. To onlude our analysis of the full dislosure ase, we ompare the entry and investment inentives with and without the intermediary (as given by Proposition 1 and 2 above). 6 Proposition 3. (Comparing full- and no-dislosure) In the presene of the intermediary (under full dislosure) entry is ineffi ient iff k k. The effi ieny in investment is affeted as follows: 6 We omit the proof of Proposition 3 as it readily follows from the fats that θ f I is dereasing and θi I is inreasing in k (as given in Proposition 2), and θ NI I is dereasing in π (as given in Proposition 1).

14 14 CHOI AND MUKHERJEE (i) If (i.e., when the intermediary is absent, all types enter but none invests), the intermediary always improves effi ieny in investment. (ii) If < (i.e., when the intermediary is absent, all types enter and all types θ > θ NI I invest), for all k, there exists a utoff π k suh that the intermediary improves investment inentives iff π < π k. The threshold π k is inreasing in k for k < k but dereasing otherwise. 5. O D P In this setion, we turn to the question of optimal dislosure poliy where the intermediary may hoose both his ertifiation prie as well as his dislosure poliy. In partiular, we allow for partial dislosure where the intermediary may (partially) pool the low and high quality sellers by using a stohasti dislosure rule. To simplify the analysis, we adopt the diret revelation approah. As the buyers bid ompetitively for the produt, the equilibrium bid is simply the expeted quality of the produt given the intermediary s dislosure poliy, the set of types Θ E and Θ I who enter and invest, and the realized signals (x, z). For a given dislosure poliy, let t 0 and t 1 be the ex-ante expeted bids (or transfers from the buyers) for a low and high quality seller (respetively) when he uses the intermediary but before he learns the realization of the signals (x, z). That is, t i = E (x,z) v=i E v [v x, z, Θ E, Θ I ], i {0, 1}. With this diret revelation approah, the intermediary s ertifiation poliy ertifiation prie p and dislosure poliy D an be represented by the triplet (p, t 0, t 1 ). Under a stritly partial dislosure poliy we have 0 < t 0 < t 1 < 1 whereas under full dislosure, we have t 0 = 0 and t 1 = 1. Unfortunately, the diret revelation approah to solving the intermediary s optimal poliy still laks algebrai tratability. The ompliation primarily stems from the fat that both entry and the hiring of intermediary are disrete hoies of the seller, and the intermediary s payoff need not be ontinuous in his ertifiation poliy (p, t 0, t 1 ). To irumvent this problem, we proeed as follows: First, we note that if a partial dislosure poliy is optimal then we must have full market overage where all types (θ) of the seller enter and the seller always hires the intermediary irrespetive of his produt quality. Next, we haraterize the optimal partial dislosure poliy under full market overage i.e., we impose the onstraint that the poliy must indue all seller types to enter and ertify (regardless of the quality). Finally, we derive the optimal poliy by omparing the intermediary s maximal payoff under the onstrained partial dislosure poliy (as defined above) and its full dislosure ounterpart. We begin our analysis with the following lemma. Lemma 3. If a partial dislosure poliy is optimal, it must entail full market overage : in equilibrium, (i) all types (θ) of the seller enter the market, and (ii) the sellers for both produt qualities use the intermediary.

15 OPTIMAL CERTIFICATION POLICY 15 To see the intuition behind the above result, onsider the latter part first. If the intermediary plans to sell its servie only to the high quality seller, it is best to maximize the differene in pries that a low and a high quality seller would reeive from the buyer. Clearly, this is attained under full dislosure as it removes all information asymmetries regarding the produt quality. Thus, a partial dislosure poliy an be optimal only when the intermediary intends to indue both the high and low quality sellers to ertify their produts. A similar logi applies to the first part of the lemma. In any equilibrium, if it is the ase that only some types of the seller enter, it also must be the ase that those types who enter also invest. (Sine the entry ost is the same for all types, if its profitable for some type θ to enter even though it would not invest following entry, then entry must be profitable for all types of the seller.) But if the intermediary only intends to indue entry of those types who would invest as well, it an be argued that it is (weakly) optimal to adopt the full dislosure poliy. Thus, we an onlude that partial dislosure is adopted by the intermediary only when it intends to engage in full market overage. In light of Lemma 3, we an now formulate the intermediary s optimal partial dislosure poliy. We maintain the off-equilibrium belief µ (, z) = 0 as it is the most favorable to the the intermediary, i.e., if the seller hooses not to use the intermediary, the buyers offer a prie of 0. Thus, for all types of the seller to enter and ertify regardless of the realized produt quality, the ertifiation prie p should satisfy the following two onditions: (IR E ) and 1 2 (t 0 + t 1 ) k p 0, (IR C ) t 0 p 0. The first ondition above (IR E ) is the individual rationality ondition for entry and states that it is profitable for all types (θ) of the seller to enter the market even if he deides not to invest in quality. Otherwise, a seller with low θ will not enter the market as his ost of investment may be prohibitively high; onsequently, we would not have full market overage. The seond ondition (IR C ) states that the seller has inentives to ertify even when his produt quality is low. Obviously, under full dislosure we have t 0 = 0, and (IR C ) annot be satisfied with any positive ertifiation prie. Hene, under full market overage, the intermediary needs to garble information if it were to reeive a higher ertifiation prie. So, for any dislosure poliy (t 0, t 1 ) with full market overage, the intermediary s payoff is: { Π (p) = p = min t 0, 1 } 2 (t 0 + t 1 ) k. As before, we an argue that the seller s investment deision follows a utoff strategy as the ost of investment is monotonially dereasing in the seller s type. Let v (θ I ) be the prior expeted value of the produt i.e., the probability that the seller s produt is of high quality (so v = 1) given that all types enter and all types above θ I invest, but without any information on the signals (x, z). That is,

16 16 CHOI AND MUKHERJEE v (θ I ) = 1 2 θ I + α (1 θ I ) = (1 θ I). Any dislosure poliy (t 0, t 1 ) must satisfy the following four onditions: First, a Bayes rationality (BR) ondition that requires the expeted posterior mean quality must be equal to the prior mean quality: (BR) v (θ I ) t 1 + (1 v (θ I )) t 0 = v (θ I ). Seond, as the intermediary s signal is (weakly) informative, we must have the following bounds: (L 1 ) t 1 t 1 (θ I ) := E z v=1 E v [v z, Θ E = [0, 1], Θ I = [θ I, 1]]. and (U 0 ) t 0 t 0 (θ I ) := E z v=0 E v [v z, Θ E = [0, 1], Θ I = [θ I, 1]]. Finally, as disussed earlier in the ase of full dislosure, if it is inentive ompatible for all types θ θ I to invest, we must have: (IC) { } θ I = min (t 1 t 0 ), 1. Hene, the intermediary s optimal partial dislosure poliy solves the following program: P : max t0,t 1,θ I [0,1] min { 1 2 (t 0 + t 1 ) k, t 0 } s.t. (BR), (L 1 ), (U 0 ), and (IC). In order to solve the above program we first onsider a relaxed problem that ignores the boundary onditions (L 1 ) and (U 0 ). If the optimal utoff for the investment type, say ˆθ I, in the relaxed problem satisfies the boundary onditions, then ˆθ I is also the solution to the original program P. Otherwise, we need to adjust the investment type utoff in order to aount for the boundary onditions. As shown in the proof of Lemma 4, lowering the investment utoff level θ I (i.e., induing investment by more types) relaxes both boundary onditions. Moreover, both of these onditions beome stritly non-binding as θ I beomes suffi iently small, as stronger inentives for investment alls for more preise information revelation; i.e., a larger spread between t 1 and t 0. Therefore, there is a positive utoff level θi suh that t 1 t 1 (θ I ) and t 0 t 0 (θ I ) if and only if θ I θ I. When the boundary onditions are not satisfied, the solution to the original problem thus requires a redution of the utoff level from ˆθ I to θ I. Lemma 4. (Optimal partial dislosure poliy under full market overage) The solution to P, ( t 0, t 1, θ I ), is haraterized as follows:

17 OPTIMAL CERTIFICATION POLICY 17 (i) The optimal utoff above whih all types of the seller invest in quality is given by θ I = min{ˆθ I (k), θ I (π)}, where ˆθ I is independent of π but dereasing in k, and θ I is independent of k but dereasing in π. (ii) There exist two thresholds, π and π, where 0 < π < π < 1, suh that for any given k, θi = ˆθ I (k) if π < π, and θ I = θ I (π) if π > π. Otherwise, i.e., if π [π, π], there exists a threshold k π [, ] suh that θ 4α 4 I = θ I (π) if k < k π and θ I = ˆθ I (π) otherwise. Moreover, k π is inreasing in π. (iii) The seller s pries t 0 and t 1 solve (BR) and (IC) evaluated at θ I. As the publi signal beomes more preise, the boundary onditions (L 1 ) and (U 0 ) beome tighter and there is less room for the intermediary to garble information. When the onditions start binding, the optimal partial dislosure poliy implements θ I = θ I (π). Note that when the boundary onditions hold with equality (i.e, when they are binding), the intermediary s (garbled) signal is pure noise as it does not ontain any information on the produt quality. This implies that the investment threshold in this ase is the same as the one under no intermediary, that is, θ I (π) = θ NI I (π). We an now haraterize the intermediary s optimal poliy by omparing his payoff under full dislosure and partial dislosure (with full market overage). Proposition 4. (Optimal dislosure poliy) There exists a threshold π F D (π F D > π, where π is as defined in Lemma 4) suh that: (i) If π > π F D, full dislosure is optimal for all k. (ii) Otherwise, there exists an interval (k 1 (π), k 2 (π)), where k 1 (π) k k 2 (π), suh that partial dislosure is stritly optimal if and only if k (k 1 (π), k 2 (π)). Moreover, (k 1 (π), k 2 (π)) = (, 1 4 2) for π < π, and the interval shrinks with π for π [π, π F D ]. Proposition 4 illustrates how the optimal dislosure poliy varies with the entry ost and the preision of the publi signal. When the publi signal is relatively impreise, partial dislosure is optimal only if the entry ost is suffi iently large. Otherwise, partial dislosure beomes optimal only for a moderate range of entry ost if the ost is too high or too low, the intermediary resorts to full dislosure. Moreover, this range of entry ost (i.e., where partial dislosure is optimal) gets smaller as the publi signal beomes more informative; when the publi signal is suffi iently preise (i.e., π > π F D ) full dislosure is optimal regardless of the ost of entry. To see the intuition for the result, reall the trade-off with ertifiation fee under full dislosure. A larger fee extrats more rents from the seller who uses the intermediary but redues the likelihood that the seller would use the intermediary at the first plae. The redution in the demand for the intermediary s servie stems from the fat that when the intermediary extrats a larger share of the trade surplus, both the investment and the entry inentives of the seller beome muted. But this trade-off ould be softened with partial dislosure. By partially pooling the low quality seller with the high quality one, the intermediary an ensure that an entrant gets a relatively high prie from the buyer even if he ends up with a low quality produt. Thus, suh