Dynamic Certification and Reputation for Quality

Transcription

1 Dynamic Cetification and Reputation fo Quality Iván Mainovic Andzej Skzypacz Felipe Vaas Febuay 21, 2017 Abstact We study fim s incentives to build and maintain eputation fo quality, when quality is pesistent and can be cetified at a cost. We chaacteize all eputation-dependent MPEs. They vay in fequency of cetification and payoffs. Low payoffs aise in equilibia because of ove-cetification taps. We contast the MPEs with the highest-payoff equilibia. Industy cetification standads can help fims coodinate on such good equilibia. The optimal equilibia allow fims to maintain high quality foeve, once it is eached fo the fist time. They ae eithe lenient o hash - endowing fims with multiple o one chance to impove and cetify quality. JEL Classification: C73, D82, D83, D84. Keywods: Voluntay Disclosue, Cetification, Dynamic Games, Optimal Stopping. 1 Intoduction Fims can affect the quality of thei poducts by investing in physical o human capital, eseach and development, o oganizational design. Customes often do not diectly obseve We would like to thank Simon Boad, Tim Baldenius (discussant), Ilan Guttman, Ginge Jin, Eik Madsen, Lay Samuelson, Segey Voontsov and wokshop paticipants at the Univesity of Minnesota, Zuich and Stanfod fo helpful comments. Stanfod Univesity, GSB. imvial@stanfod.edu Stanfod Univesity, GSB. skz@stanfod.edu Duke Univesity, Fuqua School of Business. felipe.vaas@duke.edu 1

2 these investments o thei esults, giving ise to a moal hazad poblem that leads to the unde-povision of quality. That poblem can be mitigated if the fim can invest to build a eputation fo quality. Howeve, fo the eputation to be cedible, customes need to obseve signals of quality. These ae often povided by the fim via voluntay, costly disclosues. To be cedible, such disclosues often ae cetified by a thid paty. Examples ange fom health cae (fo example, acceditation of HMOs by NCQA, descibed below), child cae (fo example, acceditation povided by the National Association fo the Education of Young Childen), and supplie elationships in B2B contacting (fo example, ISO 9000 cetification with ove one million oganizations independently cetified woldwide). 1 In this pape, we study the ole that an industy standad fo voluntay cetification plays in mitigating the unde povision of quality and in avoiding ove-cetification tap. Such self-egulation by incumbents has been citicized as a way to incease baies to enty (see fo example Lott (1987)). We ask if it can also be efficiency-enhancing by allowing fims to coodinate on equilibia that povide bette incentives to invest in quality and stonge eputations at a lowe cost of cetification. To this end, we analyze two types of equilibia. The fist class is Makov-Pefect equilibia in which the fim s cetification and investment stategies depend only on cuent eputation, which we define as the maket belief about cuent quality. We intepet these equilibia as plausible outcomes when the industy does not self-egulate to coodinate on a cetification standad. The second class we study ae optimal pefect Bayesian equilibia (hencefoth, best equilibia) in which the maket expectation of fims cetification (and investment) stategy can be a function of the whole histoy of the game and not just cuent eputation. Fo example, industy egulation can pevent fims fom e-cetifying too soon since the last successful o failed attempt to cetify. We adopt a capital-theoetic appoach to modeling both quality and eputation, as in Boad and Meye-te-Vehn (2013). The fim continuously and pivately chooses quality investment. Quality is pesistent, changing stochastically between two states, high and low, with the tansition ates depending on the instantaneous investment flows, so that cuent quality eflects all past investments. Reputation difts up if the fim is believed to be investing and difts down if not. Pofit flows depend on fim s eputation, which is defined as maket s belief about its quality. 2 This setting seems ealistic fo many makets. Fo 1 Othe souces of infomation about poduct quality include mandatoy disclosue (such as nutitional facts), thid-paty initiated eviews (such as eviews on Cnet.com), and consume epots (wod of mouth o consume epots on Amazon.com). See a suvey by Danove and Jin (2010). 2 Pofits can incease in peceived quality eithe because good eputation leads to a bigge demand fo 2

3 example, in the health-cae industy, HMOs invest in pocesses and pesonnel to povide high-quality sevices, quality is pesistent since human capital and oganizational capital ae pesistent but maintaining quality equies continuous investment to attact and etain talent, and to eact to changes in medical pactice o technology. Moeove, quality is had to obseve by individual customes and an impotant souce of infomation is the National Committee of Quality Assuance (NCQA) that since 1991 offes HMOs voluntay cetification pogam. The cetificates expie in thee yeas and total costs (diect fees and indiect costs) of pepaing acceditation ange fom $30, 000 to $100, 000 depending on the size of an HMO (and othe chaacteistics; see Jin (2005) fo a detailed desciption of the NCQA pogam). Quality is known pivately by the fim but at any time it can be cedibly evealed/cetified to the maket. We model cetification as a costly disclosue that allows the fim to cedibly and pefectly convey its cuent and patially pesistent quality to the maket. This is simila to the analysis of cetification in Jovanovic (1982) and Veecchia (1983), with the main diffeences being that in ou model quality is endogenous and disclosue is dynamic athe than static. Though we do not model the souce of this disclosue cost, we intepet it as epesenting the fee chaged by a cetifie in exchange fo its cetification and dissemination sevices (in the spiit of Lizzei (1999)), plus any costs necessay to allow the cetifie veify the fim s quality. Since the fim is pivately infomed about its quality, the maket leans about quality not only fom cetification but also fom the failue to cetify. This leads to multiplicity of equilibia that diffe in tems of the fequency of cetification. The diffeence in the two classes of equilibia we study is how maket expectations change in esponse to histoy. In the Makov-pefect equilibia maket expectations ae stationay - they depend only on the cuent eputation. In the optimal equilibia, the expected fequency of futue cetification can depend on past behavio. Fo example, if a high quality fim fails to maintain quality and e-cetify, the maket can expect a moe fequent cetification and less investment in the futue. We offe two sets of esults. Fist, we chaacteize Makov-pefect equilibia. When cetification costs ae low, thee is a ange of MPE equilibia with diffeent fequencies of cetification. In paticula, thee exist equilibia with a high fequency of cetification in the poduct o because it allows the fim to chage a highe pice, o both. Fo empiical evidence that cetification inceases demand, see fo example Xiao (2007) in the context of voluntay acceditation of child cae centes, and othe examples in Danove and Jin (2010). 3

4 which all the benefits of eputation fo the high quality fims ae dissipated by excessive cetification, an effect we call an ove-cetification tap. Moeove, we show that unde ou assumptions, the Makov-pefect equilibia do not ceate any value fo fims that stat at low quality. That is, even though in some Makov-pefect equilibia the fim invests in quality and eventually manages to cetify it, fo all positive costs of cetification, the equilibium yields the same payoff to the low-quality fim, as if quality could neve be impoved. 3 Moeove, in MPEs with on-path investment in quality, quality is tansitoy: even though the fim has the technology to maintain quality foeve, on path expected quality slowly dops afte cetification. The countepoductive effect of cetification in MPEs stesses the notion that cetification can be a double-edged swod: on one hand it allows fims to eap benefits of investments in quality, on the othe hand, it can ceate an (ove) cetification tap, if the maket expects the fim to e-cetify fequently. Paadoxically, high-quality fims caught in such a tap ean lowe pofits than if no cetification wee possible - this happens even in the MPE with the highest investment level. The intuition fo the low payoffs in any MPE is as follows. Fist, if cetification takes place only afte beliefs dop below some level, the fim cannot be investing in quality above that theshold since othewise maket beliefs would neve each it (ecall that in ou model, expected quality impoves when the fim invests and deteioates if it does not). Hence, it is not possible to foeve maintain high quality in any MPE and payoffs of a high-quality fim ae bounded away fom fist-best. Second, the fim with the lowest eputation cannot have stict incentives to invest in quality eithe. If it did, the fim would also have stict incentives to invest befoe it fails to cetify and maket beliefs would neve each the cetification theshold. As the cost of cetification goes down, the fim cetifies moe and moe often and all the savings ae dissipated by excessively fequent cetification. It may be at fist counte-intuitive that less-fequent cetification impoves incentives to invest in quality. The intuition is that with less-fequent cetification, the total expected continuation pofits fom cetifying high quality ae highe since less esouces ae spent on cetifying. Moeove, thee is a positive feedback effect: highe payoffs fom high quality incease incentives fo investment, and that inceases payoffs even futhe and so on. The second set of esults is a chaacteization of the best equilibia. The best equilibium not only delives highe payoffs than any MPE, but also diffes qualitatively fom all MPEs. 3 This stak esult depends on the assumption that if the fim invests maximally quality neve dops. Howeve, as we discuss late, the intuition fo ove-cetification tap and the coesponding benefit of coodination on bette equilibia is obust. 4

5 Fo low cetification costs we show that in the best equilibium the ex-ante payoff of the low-quality fim is stictly highe and inceases as cost of cetification goes down, conveging to the fist-best payoff when the cost of cetification declines to zeo. Moeove, once the fim eaches high quality, it is maintained foeve on the equilibium path in contast with all MPEs. In summay, the analysis implies that an industy standad fo voluntay cetification could allow fims to ceate and eap benefits fom building and maintaining eputation and avoid the ove-cetification tap. An impotant featue of such a system is that it keeps tack of the time since last cetification and sets the duation (i.e. the time the high quality fim is expected to e-cetify) optimally: 4 a shot duation induces excessive costs of cetification that by educing the value of eputation educes the incentives to invest; a long duation makes just-cetified fims est on thei lauels and shik since today s investments have small effect on long-tem quality. Finally, the best equilibium can be implemented by a system that keeps tack of the time since last cetification and a binay indicato whethe the fim is still in the system o not (a punishment can be implemented by emoving the fim fom the industy cetification pogam and letting it to its own devices). To limit cetification costs, the best equilibium takes one of two foms, hash o lenient. The diffeence between them is what happens when the fim stats at low quality. In the hash equilibium, the low quality fim has to wait a long time till cetification, so it passes it with a high pobability, but failue is hashly punished (the punishment can be intepeted as the fim being excluded fom the industy cetification pogam while maintaining the option to cetify independently accoding to one of the MPEs we descibed fist). In the lenient equilibium, the fim gets a shote time to fist cetification, but failue is not punished (beyond updating the eputation to the lowest level) the equilibium simply estats. In 4 These featues chaacteize many eal wold cetification pogams. Fo example, the pogam efeed to as Docto Boad Cetification, povides voluntay cetification fo doctos acoss 24 specialties (see This cetification pogam, administeed by the Ameican Boad of Medical Specialties (ABMS), which goes back to the ealy twentieth centuy, stated pescibing e-cetification evey 10 yeas in Despite its cost, almost 75% of doctos in the U.S. ae boad cetified because cetification is widely peceived as a signal of quality (see Bennan et al. (2004)). Howeve, this pogam is not exempt of contovesy. In 2014, the ABMS decided to incease ecetification fequency to 2-5 yeas, intoducing a gowing numbe of maintenance of ecetification equiements MOC which significantly inceased the cetification costs doctos bea (the pogam takes five to 20 hous a yea and costs $1,940 ove 10 yeas, including the exam. See Doctos Upset Ove Skill Reviews, WSJ, July 2104). This change motivated doctos acoss disciplines to potest, aguing that the ABMS became a monopoly that contols who can pactice medicine and use this powe to compel compliance and chage exobitant fees. Moe than doctos signed a petition (see to etun to the 10 yeas ecetification system (see Stop Wasting Docto s Time, NYT, Dec 15th, 2014). 5

6 othe wods, the fim is given multiple chances to impove and cetify its quality no matte how many times it has failed befoe. Intuitively, the hash equilibium povides stonge incentives and hence can economize on cetification costs, but it also sometimes tigges inefficient punishment on the equilibium path (false-positive when the fim is unlucky in achieving high quality by the deadline despite appopiate investment). If cetification costs ae small, the best equilibium is lenient. On the othe hand, if cetification costs ae lage and quality impoves sufficiently easily (both in tems of cost of investment and aival ate of impovements), the optimal equilibium is hash. The best equilibium does not allow fims with an expied cetificate to cetify as soon as thei quality impoves. At fist blush, this might seem inefficient but it s not: since maket beliefs ae coect on aveage, fom the ex-ante point of view, the fim would not benefit in tems of evenues fom ealy cetification but would only incu the cetification costs moe often. This is a limitation of time-contingent cetification pogams that implement a fixed cetificate duation, but allow fims with expied cetificates to e-cetify as soon as thei quality impoves. The analysis of this class of equilibia is povided in Appendix A. While we intepet the diffeence between the MPEs and the best equilibia as a potential benefit of having an industy standad to coodinate maket beliefs, in pactice fims can affect maket expectations about the fequency of cetification (and hence ty to coodinate on bette equilibia) in othe ways too. Fo example, they sometimes esot to thid paties to ceate cetification with a pe-announced duation. 5 Theefoe, ou analysis can be intepeted moe boadly as showing in an equilibium setup fist the potential costs of ove-cetification, and second the benefits of managing maket expectations about timing of cetification. We assume the eputational benefit of voluntay cetification is the only way customes ewad fims fo poviding high quality. In some industies, thee ae othe moe impotant mechanisms. Fo example, waanties ae a common way to educe the moal hazad poblem, as is the theat of losing epeated customes of expeience goods. Moeove, thee ae othe souces of infomation that affect the fim s eputation. In seveal impotant industies voluntay cetification plays a fist-ode ole (as the examples in the beginning of the into- 5 Deviating fims could be eithe denied by the thid-paty cetifie woied about ceating a pecedent in the industy and educing the value of the cetification pogam, o punished by expectations that once they cetify soone than expected, the maket would expect them to cetify even moe often in the futue. Such concens fo eputation fo eticence o not evealing infomation too often ae well known to manages in aeas beyond cetification. See fo example Houston Lev and Tucke (2010) fo voluntay eanings guidance by fims. 6

7 duction suggest). One of the easons is that veifying in cout custome satisfaction may be expensive o impossible in such makets, so that waanties ae impactical (as they appea to be in the makets fo HMOs, child cae and many examples of supplie elationships). Anothe eason is that many customes have eithe one-off o ae tansactions with the fim in such makets, so that dynamic theats of losing business if quality tuns out to be low offe low-poweed incentives. The co-existence of infomation coming fom cetification and thid paties (e.g., wod-of-mouth o eviews) seems to be moe elevant to these makets. While we think that many of the economic effects identified in this pape ae impotant also in a model with both cetification and thid-paty infomation, a pope analysis of such a model is beyond the scope of this pape. 1.1 Related Liteatue As we mentioned above, ou pape can be viewed as a dynamic vesion of Jovanovic (1982); Veecchia (1983) with endogenous quality. Ou model of quality and intepetation of eputation is as in Boad and Meye-te Vehn (2013). 6 Simila papes that conside incentives to invest in quality with exogenous public news include Dilme (2016); Halac and Pat (2016). Thee ae two main diffeences between ou pape and this liteatue. Fist is how we model infomation: in ou model it is geneated endogenously by the fim, while in thei models the maket obseves exogenous signals about the quality. Second, these pevious models study only Makov-Pefect equilibia, and ou model contasts MPEs with the optimal equilibia. The contast between what can be achieved in each class is the main esult of ou pape. An implication of these esults (that we do not to emphasize) is that focusing on MPEs in eputation models can ule out ealistic behavio. 7. A stand of the liteatue studies cetification, focusing on the behavio of a monopoly cetifie who can commit in advance to both a cetification fee and a disclosue ule (see e.g., Lizzei (1999), Albano and Lizzei (2001)). In this pape we take the cetification technology as exogenous and focus instead on fim s investment behavio, but we believe ou model could be also used to study pofit-maximizing cetifies. Ou model suggests that an optimal stategy of a cetifie would involve a non-tivial decision about pice as well as the duation of cetification. Fo example, in ou model longe duation can actually esult in moe cetification since it could povide stonge incentives to maintain quality (and only 6 See Mailath and Samuelson (2015) fo a ecent suvey on the eputation liteatue. 7 In some eputation models all equilibia ae Makov, as shown in Feingold and Sannikov (2011) o Bohen (2016), but as we show hee, focusing on MPEs sometimes leads to paadoxical esults 7

8 high-quality fims e-cetify). Ou model of cetification as a costly infomation disclosue with timing chosen by the fim is simila to that in Schaa and Zhang (2015). In that pape quality is fixed so the fim cetifies at most once and the focus of that pape is not on incentives to invest in quality but on the inteplay between exogenous public news and endogenous cetification. Ou pape is also somewhat elated to the ecent liteatue on eputation with infomation acquisition. (see e.g., Liu (2011)), whee it is the buyes who can acquie infomation about the fim. The main diffeence is that in ou model quality is endogenous and pesistent, and it is the fim that incus costs to povide infomation. Ou model shaes some featues with the statistical discimination liteatue initiated by Aow (1973). 8 The undeinvestment poblem descibed in this pape is diven by the unobsevability of quality and investment choices. The etun to investment depends on the pofits that the fim can assue by cetifying high quality. In tun, these pofits ae detemined by the buyes expectation about past investments. In some sense, investment, cetification, and buyes beliefs ae stategic complements, so that undeinvestment becomes a self-fulfilling pophecy and an industy standad can help the fims and customes coodinate on equilibia with stonge incentives to invest. The emainde of the pape is oganized as follows. In Section 2 we descibe the model. In Section 3, we study equilibia when the fim chooses when to cetify based on its cuent eputation. We contast this case with the optimal pefect Bayesian equilibia in Section 4 and discuss the implications fo the optimal pattens of cetification, investment and eputation. 2 Model Thee is one fim and a competitive maket of identical consumes, sometimes efeed to as the maket. Time t [0, ) is continuous. At evey time t, the fim chooses pivately investment in quality, makes a decision about cetification, and sells a poduct, when the maket s demand depends on peceived quality (fim s eputation). We boow the model of investment in quality developed by Boad and Meye-te Vehn (2013). In paticula, at time t the fim s poduct quality is denoted by θ t {L, H} whee we nomalize L = 0 and H = 1. Initial quality is commonly known to be low, θ 0 = L, but subsequent quality depends on investment and unobsevable technology shocks. Shocks ae geneated accoding to a Poisson pocess with aival ate λ > 0. Quality θ t is constant 8 See Aow (1998) fo a eview of this liteatue. 8

9 between shocks and is detemined by the fim s investment at the most ecent technology shock s t that is, θ t = θ s and P(θ s = H) = a s. The fim obseves poduct quality and chooses an investment plan a = {a t } t 0, a t [0, 1] which is pedictable with espect to the filtation geneated by θ = {θ t } t 0. Investment has a maginal flow cost k > 0. Consumes obseve neithe quality no investment. We denote thei conjectue about the fim s investment by ã = {ã t } t 0. This specification implies that, given an investment policy a, quality jumps fom L to H at an exponential time with ate λa t and jumps fom H to L at a ate λ(1 a t ). As a consequence, investment has a pesistent effect on poduct quality, as in the case when investment efes to employee taining. 9 Since λ measues the likelihood of shocks, a highe λ can be intepeted as captuing the instability of the fim s economic envionment. On the technical side, note that since we assume a t [0, 1], in the absence of investment, poduct quality can only expeience negative shocks, and when investment is maximal, poduct quality can only expeience positive shocks. To focus on the ole of cetification in eputation, and unlike Boad and Meye-te Vehn (2013), we assume thee ae no public signals about fim quality. Instead, the fim has access to an extenal (unmodeled) paty e.g., a cetifie who can cedibly cetify the cuent quality of the fim fo a fee c. Poduct quality becomes public infomation at the time of cetification. We denote the fim s cetification stategy by d t {0, 1} and the maket s conjectue about the fim s cetification stategy by d. The fim is isk neutal and discounts futue payoffs at ate > 0. We model the maket in a educed fom by assuming that the fim s pofit flow is a linea function of its eputation, p t, whee p t = d[θ Eã, t Ft d ] and Ft d is the infomation geneated by the fim s obseved cetification choices. Thee ae multiple ways to intepet this specification of pofits. Fo example, as in Boad and Meye-te Vehn (2013), the fim may be selling a limited amount of the poduct pe peiod and the customes compete fo poduct in a Betand fashion which leads to pices being equal to the expected value of the poduct flow. Altenatively, the pice may be fixed and the demand fo the poduct may be popotional to the fim s eputation. Given the fim s investment and cetification stategy (a, d) and the maket s conjectue 9 Also a etention and selection policy fo employees has pesistent effects on the quality of the wokfoce of a fim. 9

10 about them (ã, d) the fim s expected pesent value equals E a,d,θ 0 0 [ e t( p t a t k ) dt ] e t c d t t 0 The conjectued investment and cetification pocess (ã, d) detemine the fim s pofit flow fo a given histoy while the actual stategy (a, d) detemines the distibution ove quality and histoies. Befoe studying the equilibium, note that in the absence of disclosue the evolution of eputation is given by the odinay diffeential equation ṗ t = λ ( ã t p t ). (1) When ã t = 0, the eputation p t difts downwad and when ã t = 1 it difts upwad. Thoughout the pape we assume that k is sufficiently small, k < λ. This implies that λ+ a t = 1 is the fist best investment, namely the investment the fim would choose if eithe quality o investment wee obseved by the maket. Definition 1. An equilibium is a pai of stategies (a, d) and conjectues (ã, d) such that given the maket conjectues, the fim s stategy is optimal and conjectues ae coect on the equilibium path. Thoughout the pape we focus on pue stategy equilibia in which the fim s cetification stategy, d, is pue. Thee ae seveal possible histoies off-the-equilibium path: the fim may cetify soone than expected, in which case we assume consumes believe the cetification is tuthful (so that beliefs ae eset to p t = 1). Moeove the fim may fail to cetify even if it is believed to have maintained high quality by investing a t = 1. In that case the beliefs ae not esticted by Bayes ule. In what follows, we study two classes of equilibia. Fist, in Section 3, we conside belief-contingent (Makov pefect, MPE) equilibia in which the investment and cetification stategies depend on eputation and quality. Late, in Section 4, we conside non stationay equilibia in which the stategies depend on the complete histoy. 10

11 3 Makov Pefect Equilibia: Cetification Taps In this section, we conside (pue stategy) Makov pefect equilibia. That is, we study equilibia in which the fim stategy (a, d) is a function of its cuent quality θ and eputation p, and not the full histoy of the game; in paticula, it does not depend on the fim s actions befoe the last cetification, since evey cetification e-sets beliefs to p t = 1 (Recall that thoughout the pape we estict attention to pue cetification stategies). Maket conjectues about the fim s stategies ae hence a function only of eputation p. Wheneve the fim is expected to cetify ( d(p) = 1) the continuation value, V θ (p), satisfies V H (p) = max { V H (1) c, V H (0) }. (2) on the othe hand, when the fim is not expected to cetify ( d(p) = 0), the continuation value satisfies the HJB equation: 0 = max p ak + λ(ã(p) p)v L(p) + λad(p) V L (p) (3) a [0,1] { 0 = max max p ak + λ(ã(p) p)v H(p) λ(1 a)d(p) V H (p), (4) a [0,1] } V H (1) c V H (p), whee, following Boad and Meye-te Vehn (2013), we efe to D(p) V H (p) V L (p) as the value of quality namely the capital gain the fim expeiences when its quality impoves, given its eputation p. The fist step is to analyze the cetification stategy. Wheneve the maket expects the high quality fim to cetify, eputation dops to zeo, if the fim fails to do so. Hence, the fim has two options: (i) cetify and get a continuation value V H (1) c, (ii) do not cetify and get a continuation value V H (0). Equation (2) says that the continuation value is the maximum between these two altenatives. On the othe hand, wheneve the fim is not expected to cetify, beliefs evolve accoding to Equation (1). If the fim cetifies, its net gain (loss) is V H (1) c V H (p); hence, the fim has incentives to cetify if and only if V H (p) V H (1) c. 11

12 In othe wods, the fim cetifies wheneve the gain caused by cetification outweighs the (lumpy) cetification cost. Wheneve V H (p) > V H (1) c, the fim does not cetify and the continuation value satisfies the diffeential equation V H (p) = max a [0,1] p ak + λ(ã(p) p)v H(p) λ(1 a)d(p) (5) The economic intuition behind Equation (5) is the following: the flow continuation value, V H (p) has thee pats: i) the cuent pofit flow, ii) the capital gains fom changes in maket beliefs (that affect futue pofit flows) and iii) the potential capital gains o losses fom changes in pivately known quality. The next step is to analyze the fim s investment decision. Inspection of the HJB equation, eveals that the fim s optimal investment policy is: 0 if λd(p) < k a(p) = 1 if λd(p) > k, and any a is optimal when λd(p) = k, because the net pesent value of the investment is zeo at that point. Note that due to the poductivity of investment being symmetic acoss states, the fim s investment incentives ae independent of the state θ: investment inceases the pobability of a positive shock when the state is low and educes the pobability of a negative shock when the state is low, but in both cases the maginal benefit of investment is the same. This symmety allows us to wite the equilibium investment stategy as a function of maket beliefs alone, a(p). Tivially, if the fim could not communicate its quality to the maket the value of quality would be zeo, D(p) = 0, leading to zeo investment, a = 0. By contast, if the infomation about quality wee public, the fim would fully intenalize the benefit of investment, leading to fist best levels (i.e., a = 1). So unlike standad disclosue models (such as Dye (1985); Jovanovic (1982)) hee infomation allows the fim to sustain investment and maintain a high level of quality. One might thus think that cetification should play a positive ole, as it does in many static settings. Fo example, Albano and Lizzei (2001) demonstate that cetification plays a positive ole, even when the cetifie has monopoly powe. We next show that this esult does not hold in ou (dynamic) setting even when the cetification cost is abitaily small, at least as long as cetification is based on cuent eputation. To undestand the link between cetification and investment incentives, obseve that the 12

13 value of quality when the fim is not cetifying evolves as follows: D(p) = λ(ã(p) p)d (p) λd(p). (6) Let p c = sup{p 0 : d(p, H) = 1} be the highest eputation at which the high type decides to cetify and let τ c = inf{t > 0 : p t = p c, p 0 = 1} be the time that it takes to each this eputation. Since ṗ t = λ ( ) ã t p t, we can integate (6) ove time to get that fo any t [0, τ c ], o equivalently fo any p [p c, 1], the value of quality at time t is: D(p t ) = e (+λ)(τc t) D(p c ). (7) So the value of quality deteioates following the last cetification. Cetification has long lasting effects on eputation because quality is pesistent. In tun, the fim has the weakest incentive to invest ight afte it cetifies high quality. Futhemoe, at the time/eputation the fim cetifies, the value of quality is: D(p c ) = V H (p c ) V L (p c ) = V H (1) c V L (p c ). Natually, if the fim does not cetify at time t = τ c, then the maket infes that quality is low θ τc = L, and, as a consequence, eputation dops to zeo and emain at that level until the fim e-cetifies. Theefoe, V L (p c ) = V L (0). Ou fist lemma, shows that any equilibium with positive cetification can be chaacteized by two thesholds p a and p c such that the fim neve invests befoe the cetification time. 10 Lemma 1. Any pue stategy Makov pefect equilibium is equivalent to an equilibium defined by two thesholds p a and p c such that: p a p c, a(p) = 0 if p > p a and d(p, θ) = 1 {p pc,θ=h}. This is a stak esult. Fist, it implies that in any equilibium whee the cetification stategy is contingent on eputation, the fim eithe neve invests in quality o only invests when eputation is at the lowest level. Second, it implies that the fim neve invests in quality while its eputation is above the cetification theshold. This, combined with the maket s Bayesian updating implies that the fim invests, if at all, only when the maket knows with cetainty that quality is low. 10 Fomally, we say that two equilibia (â, ˆd) and (a, d) ae equivalent if (â t, ˆd t, ˆθ t ) = (a t, d t, θ t ) a.s., each t, whee ˆθ and θ ae the quality pocesses induced by the investment stategies â and a, espectively. 13

14 We povide a detailed poof in the Appendix, but hee is the economic intuition. Suppose the fim has just cetified so p = 1. If the fim is expected to invest in quality at some belief p a, befoe the belief eaches p c (i.e. if p a > p c ), then the maket belief would neve coss p a (ecall that ṗ t = λ ( ã t p t ) ). But if so, the maket belief would neve dop to the cetification theshold and we get a contadiction, since a fim that is neve expected to cetify, has no incentives to invest at all. 11 With this esult at hand we can futhe chaacteize the equilibia. Since V L (0) equals the discounted expected gain deived fom a positive quality shock, net of both the investment costs equied to enable such a shock, and the cetification expense equied to communicate to the maket that quality inceased, we have V L (0) = λa(0)(v H(1) c) a(0)k. (8) + λa(0) If p c > 0 (so that thee is cetification in equilibium) then, since failing to cetify at p c makes the maket update that the quality is low, V H (p c ) = V H (0) = V H (1) c. Theefoe, the value of quality at p = p c is D(p c ) = D(0) = ( V H (1) c ) + a(0)k. + λa(0) This expession allows us to fully chaacteize the set of MPE. Lemma 1 implies that, in any equilibium, the fim has at most weak incentives to invest. Hence, in any equilibium with positive investment we have D (p c ) = D(0) = k λ. Because the fim is indiffeent about the level of investment, the continuation value at p = 0 can be computed assuming that a = 0. This yields the bounday condition V L (0) = V L (p c ) = 0. (9) Similaly, we can also compute the continuation value assuming that a(0) = 1. If we combine Equations (8) and (9) we find that V H (p c ) = V H (1) c = k λ. (10) 11 As we show in the poof, even if the fim at p a chooses an inteio level of investment by (7) at slightly lowe beliefs it would have stict incentives to put full investment, leading to the same contadiction 14

15 Using these bounday conditions, we can solve fo the continuation value in the no-disclosue egion (p c, 1] and detemine the disclosue theshold p c. The next poposition chaacteizes the equilibium. Poposition 1. In any Makov Pefect Equilibium, (i) Thee is investment only if p t = 0. (ii) The payoff of a low quality fim is zeo when p t = 0. That is, V L (0) = 0. (iii) The payoff of a high quality fim when p t = 1 is lowe than the payoff if cetification is unavailable. That is, V H (1) 1/( + λ). In paticula, the set of pue stategy Makov pefect equilibia is chaacteized as follows: (i) If c < 1 k, then, thee is an inteval P +λ λ c thesholds. The lowe theshold is given by = [p c, p + c ] of equilibium cetification p c [ 1 c 1 +λ k λ ] λ +λ, and the uppe theshold is the unique equilibium theshold in which the zeo pofit condition V H (1) = c holds. In any equilibium with p c > p c the fim neve invest, that is a(p t ) = 0. On the othe hand, when p c = p c we have that fo any a [0, 1], thee is an equilibium in which the high quality fim cetifies wheneve p t p c and invests a(p t ) = a 1 {pt=0}. The fim s payoffs ae the same in all the equilibia with positive investment and ae given by V L (p c ) = 0 and V H (1) = k λ + c. (ii) If 1 +λ k λ c 1 +λ, then the fim neve invests and thee is an inteval P c = [p c, p + c ] such that fo any p c P c thee is an equilibium such that a high quality fim cetifies wheneve p t p c. The equilibium with p c = p + c zeo pofit condition V H (1) = c holds, while p c = p c the smooth pasting condition V H (p c) = 0 holds. 15 is the unique equilibium in which the is the unique equilibium in which

16 (iii) If c > 1 +λ thee is a unique equilibium in which the fim neithe invests no cetifies. The equilibium taxonomy depends on the cost of cetification. Natually, fo vey high values of c, the equilibium entails no disclosue hence zeo investment. When the cost is intemediate, thee is some cetification, but no investment can be suppoted. The most inteesting case aises when the cost is low; then, some investment can be suppoted. In the following, we assume that c is low enough so that positive investment can be suppoted. Specifically, we assume that c < 1 +λ k λ. Pehaps the most supising obsevation in Poposition 1 is that, in any MPE, cetification is essentially unable to mitigate the fim s unde-investment poblem. Even in the equilibia that have the most investment, the etun to investment is at best zeo (i.e., when the fim invests, it is indiffeent between positive investment and zeo investment). The intuition fo this esult is as follows. As agued in Lemma 1, in equilibium the fim is only willing to invest when its eputation is at the bottom, p = 0. But why is the etun to investment zeo at that point? The eason is that if the fim had stict incentives to invest in quality at p = 0, then by continuity it would also have stict incentives to invest befoe eaching p c (since D(p c ) = D(0) and D(p) is continuous in p fo p > p c ). But then we would get the same contadiction as in Lemma 1: eputation would neve each the cetification theshold and the fim would actually have no incentive to invest. Second, this indiffeence implies V L (0) = 0: since the fim has at most weak incentives to invest in quality at p = 0, its equilibium payoff can be computed using the stategy of neve investing. 12 The existence of MPE with vey high fequency of cetification, no investment, and vey low payoff (as low as V H (1) = c) which we efe to to as an ove-cetification tap, appeas vey obust. It extends to a model with additional public news and a moe geneal quality tansition pocess. The intuition is that as long as the fim knows its quality if the maket expects it to e-cetify fequently, the fim may find it vey difficult to convince buyes that it delays cetification because it wants to get out of the tap and not because it has failed to maintain high quality. A high enough cetification fequency can be chosen to dissipate most of the gains fom eputation and theeby educe o fully emove investment incentives. 12 This helps explain two stak consequences of Poposition 1 fo equilibia with positive investment. The ex-ante payoff of the high-quality fim is inceasing in the cetification costs and costs of investment, k. The high-quality fim is bette off when the cetification is moe expensive and investment is moe costly! The intuition is as follows. The fequency of cetification must be high enough to dissipate enough pofits so that V H (1) is low enough that the L type is indiffeent between investing and not investing at p = 0. The highe c o k, the less attactive is investment to the low type, so the cetification needs to be less fequent to keep it indiffeent (notice that p c deceases in k). That helps the high type. 16

17 As we show in the next section, while the existence of low-payoff-no-investment MPEs appea quite obust even fo low costs of cetification, thee exist equilibia with investment and high payoffs. Theefoe, an industy standad o othe ways to coodinate on bette equilibia can be vey effective in impoving the outcome of a cetification pogam. Remak. The esult that all MPEs have no investment until the eputation dops to zeo depends on ou assumption that quality can only impove if the fim chooses full investment. Fo example, if instead quality jumped fom H to L at a ate λ(1 a t (1 ɛ)) fo some small ɛ, then fo small costs of cetification thee would exist MPEs with investment fo all t. Roughly, in such an MPE, ight afte successful cetification, eputation deteioates slowly fom p 0 = 1 despite the belief that the fim chooses a t = 1. It is then possible to pick p c in a way to economize on cetification costs while still maintaining incentives fo a t = 1. Such equilibia ae vey simila to the time-based equilibia that we discuss in Section 5. One can also use ou chaacteization of equilibia to evisit the natual question of picing of cetification. Conside the equilibia with the most efficient investment. Fom the point of view of the fim, cheape cetification is offset by the equilibium effect that the maket expects it to cetify moe often. The latte effect dominates, making the fim wose off as c deceases. A pofit-maximizing cetifie faces a downwad-sloping demand cuve: lowe c leads to moe fequent cetification. If the maginal cost of the cetifie is close to zeo (the cost of poviding additional cetification), we expect the optimal pice to be vey low. To see this, conside the exteme case of zeo maginal cost. Then, as c goes down, cetification and hence investment ae moe fequent. Since paying c is just a tansfe, the oveall efficiency inceases. At the same time, the pofits of the fim go down, which implies that the pofit of the cetifie goes up as well. Hence the cetifie pofits go up as c deceases towads zeo (the limit evenues ae positive since the fequency of cetification goes to infinity). This tendency to set low fees to benefit fom moe fequent cetification adds a new consideation to ou standad intuition fom the static model in Lizzei (1999). In ou dynamic context, the cetification inefficiency is exacebated as the cost of cetification vanishes. Indeed, the pesent value of expected cetification expenses inceases as the cetification cost vanishes because the fequency of cetification inceases as well. A pioi, one could hope that the best MPE conveges to fist best when c goes to zeo, as in static settings. As we have shown, this is not the case and one of the easons is that the fequency of cetification inceases faste than the eduction in the cost; hence, the pesent value of futue cetification costs does not go to zeo. Howeve, this is not the only eason why the 17

18 limit is not efficient. Even if the cost whee just a tansfe that doesn t affect oveall welfae, the equilibium would not convege to fist best. The eason is that, even in the limit, investment is highly inefficient. While in the fist best thee is constant full investment in any MPE with investment, a high quality fim neve invests and a low quality fim only invests when it is known to be low quality. In the limit when c goes to zeo, quality is known by the maket effectively at evey instant, but investment emains inefficient. We summaize this discussion in the following coollay: Coollay 1. In the limit when c 0 the equilibium outcome conveges to p t = θ t and a t > 0 if and only if θ t = L. Poof. The esult follows fom the chaacteization of the equilibium in Poposition 1 and the obsevation that the disclosue theshold p c conveges to 1 when c goes to zeo so the set of disclosue times in the limit is dense in R +. 4 Escaping the Tap: Best Equilibium and Industy Standad As mentioned in the Intoduction, the dynamic eputation liteatue often chaacteizes voluntay disclosue without commitment by focusing on MPE. We intepet the esults of the pevious section as suggesting that without a coodination device, such as industy standads o othe thid-paty coodination, fims may be unable to eap benefits fom voluntay cetification, o that most o even all the value of eputation may dissipate via excessive cetification. In fact, the pevious section showed that voluntay cetification without (implicit o explicit) commitment to coodinate consume expectations and fim actions, esults in too much cetification, too little investment, and no net benefits fo low-quality fims enteing the maket. To model an industy standad that coodinates fims and custome expectations we now look at non-makov equilibia. In this section, we study the best Pefect Bayesian Equilibia of ou game. We show that even if the industy standad cannot impose fines o bonuses upon cetification, and can only announce a time schedule fo expected cetifications and e-cetifications of high-quality fims, it can esult in vastly supeio outcomes fo the fims. We also povide insights about the featues of optimal industy standads, showing that not only highe payoffs can be achieved, but also that the optimal standad (the stategy in the optimal equilibium) has quite natual and ealistic featues. 18

19 We exploit the ecusive natue of the poblem to analyze the set of equilibium payoffs. Since in ou game the fim has pivate infomation about its type, which changes ove time, this is not a epeated game. Yet, because cetification pefectly eveals high type, thee ae no extenal signals about quality, and we look at equilibia in pue cetification stategies, we can use the times of cetification on the equilibium path to define a egeneative pocess. We can then use this egeneative pocess to factoize the equilibium payoffs using a pocedue analogous to that in Abeu, Peace and Stacchetti (1990) (heeafte, APS). We begin by intoducing some notation. Let d t (H) {0, 1} be the equilibium cetification decision at time t conditional on θ t = H. Define the sequence of times T n = inf{t > T n 1 : d t (H) = 1}, T 0 = 0 ecusively (T n+1 can depend on the public histoy up to T n ). In equilibium, a high quality fim cetifies at time T n so p Tn = 1 if θ Tn = H. A low quality fim does not cetify at this time and this is intepeted as pefect evidence the fim has low quality, i.e., p Tn = 0 if θ Tn = L. Accodingly, on the equilibium path thee is a common belief about the fim quality at each T n. This means that the set of continuation payoffs at time T n, n 0, only depends on θ Tn and not the whole histoy of the game. Hence, with the addition of a public andomization device, the set of continuation equilibia is the same at evey T n. 13 Theefoe, in ode to chaacteize the equilibium payoff set we can use the tools fom APS and decompose any equilibium into cuent stategies and continuation values afte public signals geneated by cetification (which in ou setting is the only souce of public signals). To poceed with this ecusive chaacteization, it is convenient to measue the time elapsed since T n 1. Hence, fo any date s [T n 1, T n ], we let t = s T n 1 and τ = T n T n 1. The continuation value at time t is denoted by U θt (t θ 0 ) (it depends on the quality at the last cetification date, θ 0, and the cuent θ t known by the fim). Adapting the APS appoach, we factoize the fim s payoff using the time τ when a high quality fim cetifies fo the fist time, the investment stategy up to time τ, and the continuation value given the cetification decision at time τ. Let s denote the wost and best equilibium payoffs of a type θ 0 at t = 0 (that is, at the date T n 1 ) by U θ0 and U θ0, espectively. The wost payoffs have to be individually ational fo the fim, and we can use the Makov equilibia in Poposition 1 to detemine the wost payoff fo eithe type. In paticula, the wost Makov pefect equilibia minimax the fim 13 The andomization device is needed fo this claim since othewise past outcomes could be used to coodinate on continuation play. As we show late, the optimal equilibia we constuct do not use the andomization device. 19

20 payoffs, so that U H = c and U L = By the standad bang-bang popety, we can focus attention on equilibia with continuation payoffs that andomize at τ ove {U θ0, U θ0 } based on the fim s cetification choice at time τ. In pinciple, thee ae two such andomizations to conside: when the fim cetifies and when it does not. When the fim cetifies, continuing with the best equilibium is good fo both on-path expected payoffs and fo incentives to invest. So the equilibium with the highest ex-ante payoff must continue to U H when the fim cetifies. Theefoe, to descibe continuation stategies fo the best equilibium if we stat with type θ, we only need to specify the pobability β of tansitioning to U L (a punishment phase coesponding to the wost equilibium) if the fim fails to cetify at τ. The fim s incentives to invest at t ae detemined by the value of quality given, as befoe, by D(t θ 0 ) U H (t θ 0 ) U L (t θ 0 ). Fo any t [0, τ), the continuation values satisfy HJB equations analogous to the Makovian case: 0 = max a [0,1] 0 = max a [0,1] p θ 0 t ak + U L (t θ 0 ) + λad(t θ 0 ) U L (t θ 0 ) (11) p θ 0 t ak + U H (t θ 0 ) λ(1 a)d(t θ 0 ) U H (t θ 0 ), (12) whee p θ 0 t is the eputation p t given p 0 = θ 0. As we did in the analysis of the Makov pefect equilibium, we can integate these HJB equations between time t and τ to get D(t θ 0 ) = e (+λ)(τ t) D(τ θ 0 ). (13) A diect consequence of equation (13) is that incentives to invest ae inceasing in time. The fim s optimal investment policy is to invest as soon as D(t θ 0 ) k/λ, this means that investment stategy is fully chaacteized by the time τ a at which this incentive compatibility constaint is satisfied, and can be witten as a t = 1 t>τa. That the investment stategy is completely detemined by D(τ θ 0 ) tuns out to be quite useful. Given (τ θ0, β θ0, U θ0, U θ0 ), the fim s optimal investment stategy (descibed by τ a ) depends deteministically on D(τ θ 0 ) which equals: D(τ θ0 θ 0 ) = U H c ( β θ0 U L + (1 β θ0 )U L ) = U H c (1 β θ0 )U L. 14 At t = 0 the high-quality fim just incued cost c to cetify. Hence, its continuation payoff has to be at least c since othewise it would deviate at T n 1. 20

21 The pevious equation shows that, fo a given set of continuation payoffs and fo a given stating type θ 0, once we specify τ and β, the fim s investment policy is uniquely detemined by the incentive compatibility constaints and so is the total payoff fom this equilibium. In othe wods, given (U θ0, U θ0 ), the best equilibium is fully chaacteized by two pais (τl, β L ), (τ H, β H ) that ae the times to next cetification oppotunity and the punishment pobability at that time that depend on the maket belief about fim quality at the last time of possible cetification (o the beginning of the game). Theefoe, to find the optimal equilibium, we only need to optimize ove (τ θ, β θ ). We do this by fist computing the fim s payoff as: U θ0 (τ, β) τ 0 e t (p θ 0 t 1 t τa k)dt + e τ ( p θ 0 τ (U H c) + (1 p θ 0 τ )(1 β)u L ). Thus, we have educed the poblem of finding the best equilibium to solving the following optimization poblem (fo a given set of continuation payoffs): U θ0 = max τ 0,β [0,1] U θ0 (τ, β). (14) Now, stictly speaking, this is a elaxed poblem because thee ae two incentive compatibility constaints that we have ignoed so fa: (1) a high quality fim does not cetify befoe time τ, and (2) a high quality fim does not skip the oppotunity to cetify at time τ. We can ignoe (1) because we can always attach continuation payoff U H = c if the fim cetifies when it is not supposed to do so (so, befoe it spends c fo cetification it gets payoff 0). We ignoe (2) fo the moment and veify late on (in the poof of Poposition 2) that it is not optimal fo a high quality fim to delay cetification at time τ. The next step in ou analysis is to show that the optimal βθ is eithe zeo o one, so that the optimal equilibium/best industy standad does not andomize when the fim fails to cetify. Lemma 2. In the best equilibium the pobability β of tiggeing a punishment when the fim fails to cetify at τ is eithe zeo o one. This esult holds whethe the best equilibium implements full effot o not. When βl = 0 we call the equilibium lenient since failing to cetify does not tigge punishment and the fim is given multiple oppotunities to cetify till it finally gets a success. When βl = 1 we call the equilibium hash since afte failing to cetify the fist time, the low-quality fim neve cetifies again, being essentially shut-down. The poof of the lemma 21