Costly quality, moral hazard and two-sided markets

PDF

Download

Open document

Flip pages

Download a page range

Download transcript

Copy asset link

Request this asset

Transcript (if available)

Content COSTLY QUALITY, MORAL HAZARD AND TWO-SIDED MARKETS by Guillaume Roger A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Ful¯llment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ECONOMICS) August 2008 Copyright 2008 Guillaume Roger ii Acknowledgements No dissertation is written without a dedicated supervisor. I therefore must ¯rst and foremost thank Guofu Tan for his active and intelligent support, and above all for teaching me the research process. Guofu always had this uncanny ability to makeme feel he wasbarely one step ahead ofme everystep of the way. It hastaken me a while to catch up to the reality of the situation, but now I know better. Of course I have also greatly bene¯ted from the advice and encouragements of Juan Carrillo and Simon Wilkie. Simon even put his money where his mouth was by o®ering me a generous fellowship with the Center for Communication, Law and Policy at USC. Thank you Simon. ThankyoutotheDepartmentofEconomicsatUSC,too. Theyhaveprovidedthe indispensable ¯nancial support graduate studies require. Special thanks to Caroline Betts for steering me on the right path early on, and to Young Miller for saving the day at least once a week. If there were angels they would be like Young. Last, but not least, many thanks to Guillaume Vandenbroucke (the other Guil- laume) for helping me getting LaTex under some kind of control. iii Table of Contents Acknowledgments ii Abstract iv Chapter 1: Non-linear pricing in a two-sided monopoly with costly quality 1 1.1 Introduction 1 1.2 Literature 3 1.3 Model 5 1.4 Linear pricing 6 1.5 Non-linear pricing 19 1.6 Conclusion 31 Chapter 2: Media competition: a two-sided duopoly with costly di®erentiation 3 2.1 Introduction 37 2.2 Literature 37 2.3 Introducing the model 39 2.4 Equilibrium characterization 41 2.5 The role of externalities 54 2.6 Conclusion 61 2.7 Appendix: proofs and additional information 63 Figure 1: Best reply functions and unique equilibrium 49 Table 1: Numerical comparisons: competition promotes quality 60 Chapter 3: Moral hazard and reputation on a two-sided platform 85 3.1 Introduction 86 3.2 Literature 90 3.3 Model 91 3.4 A simple background case 93 3.5 Moral hazard, reputation and linear fees 95 3.6 Registration and transaction fees 104 3.7 Incomplete replacement and the cleansing e®ect 113 3.8 Conclusion 118 3.9 Appendix: proofs 119 References 129 iv Abstract The object of this dissertation is to further the study of two-sided markets by departing from the standard setting of price competition alone. Speci¯cally the ¯rst two chapters introduce costly di®erentiation and in doing so contribute to the two-sidedmarketliteraturebyestablishingagenericdownwarddistortioninquality. Thisresultisrobusttodi®erentspeci¯cationsinthemonopolycase(Chapter1)and arises again in a duopoly (Chapter 2). In the latter, whether a Nash equilibrium exists on one side depends on the size of the pro¯ts to be extracted on the other. When competing platforms play in mixed strategies one of them may be inactive ex post. This work also extends a well-established model (Shaked and Sutton (1982)) in the Industrial Organization literature, which speaks to the role of quality as a source of endogenous di®erentiation. The last chapter allows for moral hazard on a trading platform. It contributes to the two-sided market literature by showing that opportunistic behavior on the part of sellers leads to lower transaction fees on both sides to 1) compensate buyers and 2) provide sellers with incentives to cooperate. Furthermore it breaks an equivalence result between transaction fees and lump-sum paymentestablishedbyRochetandTirole(2005). Herethesetwoformsofpayment plays a di®erent role: lump-sum fees are used to extract the sellers' surplus, while the linear prices are distorted downward to generate the incentives to cooperate. In this case moral hazard can be completely overcome. Unlike in the standard, one- sided moral hazard problem it is not obvious that this result should obtain. In this two-sided market problem, the incentives on the sellers' side depend on the buyers' participation,whilethebuyers'participationdependsonthesellers'incentives. The platform solves these problems simultaneously. 1 1 Chapter1: Non-linearpricinginatwo-sidedmonopoly with costly quality 1.1 Introduction A multiproduct monopolist is able to account for the in°uence of its commodities' consumption on each other. For example, a ¯rm that manufactures razors and blades internalizes the complementarity between these two products. This is an easy exercise because that ¯rms sells both products to the same consumers. From a di®erent angle, an agent purchasing a commodity from a retailer (an intermediary) cares not about the distribution of surplus between the retailer and the manufac- turer, but only about the total price she pays. In this paper we study a case that departs from these two basic tenets of economics. We consider a situation in which one group of consumers exerts a(n) (positive) externality on another set of agents, butneitherisabletotakeitintoaccount. Inthiscaseanintermediarycanintervene and internalize said externality by distorting prices appropriately. As a result, both the price level inclusive of markups, but also the price structure (who pays how much) a®ect the consumption decision. This setup has come to be referred to as a two-sided market, ¯rst pioneered by Rochet and Tirole (2003) and Caillaud and Jullien(2003). Hereweextenditbyintroducinga)aqualitydimension,inthesense of vertical di®erentiation, and b) non-linear pricing. The example to keep in mind, and that we carry throughout the paper, is that of media: a newspaper, say, can select the level of investment in quality (e.g. the size of the newsroom) and o®er a two-parts tari® that implements the optimal non-linear price. We claim two main results. With log-concave distribution functions assigned to the population of agents on both sides (e.g. consumers and advertisers) and general utility functions, the quality of the consumer good generically di®ers from the standard, one-sided case. It is (optimally) set at a lower level. This is true 2 under both linear and non-linear pricing. The intuition under linear pricing is that the positive externality exerted by consumers on advertisers implies a discount on the consumer side in order to increase coverage. With lower prices, the platform can then economize on costly quality. Another perspective suggests that quality is a means of extracting surplus, at the cost of market coverage (price increases in quality). But a loss of coverage is costly in this model, as it implies a loss of advertising revenue as well. This creates an incentive to lower quality. Under non-linear pricing the notion of discount makes little sense: up to the information rent, consumers are completely extracted. Rather, quality is distorted downwards to encourage participation. Thus less is extracted from consumers, more consumers participate and more can be extracted from the advertising side. Thesecondresultisrelevanttothenon-linearpricingcaseandpertainstotheex- act form of the advertisers surplus function. Consider two functions v(a;q A (e)) and v(a;q A ;e){thelatterbeingcompletelyseparable{oftype,quantityandexternality (e). In the ¯rst case, the object of the quality distortion (on the consumer side) is only to promote participation advertisers. In the second one, it is both to enhance participation and to increase the value of advertising to each participating adver- tiser. Thereasonisthatinthe¯rstinstancethenet(social)surplusfromadvertising v(a;q A (e))¡c(q A (e)) is unchanged at the margin by the externality, while in the second one (v(a;q A ;e)¡c(q A )) it clearly is. This di®erence in the surplus functions can be grounded in some reality. The case of v(a;q A (e)) corresponds to one where more consumers lead to a higher demand for advertising. This can be associated with informative advertising: more consumers need to be told about some product, or with an advertising technology that displays constant returns to scale. The sec- ond case is one where a greater consumer coverage leads to a higher valuation of advertising. This is typically the case under strategic advertising, and/or when the advertising technology exhibits increasing returns to scale, as in broadcasting. In equilibrium the optimal quality schedule µ(¯) di®ers depending on the return func- 3 tion, which implies that the two non-linear pricing schemes are not observationally equivalent. The present work di®ers from the literature on mechanism design with externali- ties (for example, Carrillo (1998), Brocas (2004)) in that we consider cross-market externalities as opposed to within market externalities. Technically it implies that all agents report their valuations independently to the mechanism designer, who then selects transfer and output (or quality) schedules to maximize her (private) payo®s. There is no strategic interaction between any agent. The mechanism re- mainsverysimpleanddistortionsarisenotfromtheagents'reportingstrategiesbut from the complementarity of the agents' consumption decisions. Up to the introduction of some advertising costs in the non-linear pricing case, the model we construct can be applied to both the linear and non-linear pricing problems. On the consumer side, the platform (the medium) selects quality and price, and price only on the advertising side. This quality is costly, which is an important detail, and should understood as a sunk cost. 1 Under complete informa- tion the advertisers can completely compute the market coverage, which they use to decide how much advertising to purchase { it can be thought of as quality in the eyes of the advertisers. The next section reviews the literature relevant to this problem. We then lay out the model in a general enough fashion in Section 3.3 and proceed ¯rst with the linear pricing case. Section 1.5 is concerned with the mechanism design problem and is the heart to the paper. 1.2 Literature This paper lies at the intersection of the literature on two-sided market ¯rst opened up by Rochet and Tirole (2002,2003), (now RT) and Caillaud and Jullien (2003), andthewell-establishedresearchonnon-linearpricingpioneeredbyMaskin andRi- 1 In the absence of such cost no trade-o® can arise. 4 ley (1984). Armstrong (2006) is also an early contributor and showed that the RT basicmodelcanwrittenintermsoflump-sumpaymentsinsteadoflinearprices. We depart from RT or Armstrong by letting outcomes on one side a®ect they other's payo®s directly and by allowing the monopolist to also choose a quality dimension. This paper di®ers from Maskin and Riley in that it introduces a cross-market ex- ternality, which modi¯es not the agents' revelation strategies but the monopolist's behavior. The present work also contributes to the study of the provision of quality by a monopolist (Tirole 1988, Chapter 2). With a quasi-linear utility function, Tirole shows that the monopolist underprovides quality as compared to the social planner because the former cares about the valuation of the marginal consumer while the lattertakesintoaccountthevaluationofthe average consumer{whichishigher. It is easy to verify that the social planner's solution is invariant with the externality in this model: the planner would set the advertising price equal to the marginal cost, thus yielding zero advertising pro¯ts. When optimizing with respect to the variablesontheconsumerside(priceandquality),theobjectivefunctionsarethere- foreidenticalwithandwithoutexternality. Herethemonopolistfurther(optimally) downgrades the quality level because consumers receive a discount { thanks to the externality. Therefore the valuation of the marginal consumer is lower than would be absent the advertising externality. Consumers are valuable not because they are willing to pay more but because a third party wishes to reach them. The literature on quality and two-sided markets mostly consists of Hotelling duopoly models. Gabszewicz, Laussel and Sonnac (2001) characterize pure-strategy equilibria in a Hotelling model with multi-homing on the advertising side and en- dogenouslocationsontheconsumerside. They¯ndtwokindsofequilibria: maximal di®erentiationwhenadvertisingisnotverylucrativeandco-locationotherwise. That is, distortions may arise and if they do, they are very stark. In contrast, Ferrando, Gabszewicz, Laussel and Sonnac (2003) take the locations as ¯xed; only prices can 5 be distorted. Gabszewicz and Wauthy (2006) do consider endogenous costless qual- ity, however in a rational expectation model with simultaneous price-setting in the consumer and advertising market and with the option of multi-homing. Because quality is costless, distortion has no bearing. 1.3 Model Consumers have an idiosyncratic valuation b for the information good (in the common-language sense of the word). The bene¯t b is distributed on an interval ¯ = £ ¯;¯ ¤ according to the distribution F (¯) with everywhere positive density f(¯), restricted to be log-concave. There is no ambiguity for consumers as to what quality is, when it is available. In other words, this is a model of vertical di®erentiation. Let µ2 £ = £ µ;µ ¤ denote the quality level chosen by the platform. When facing a price p C , where the superscript C stands for `Consumer', an agent's net utility is expressed as U =u(b;µ)¡p C the function u(:;:) is continuous and increasing in both b and µ. The variable µ is interpreted as quality. Using subscripts to denote partial derivatives, we impose Assumption 1 u(b;µ) follows u ¯ (:) > 0; u µ (:) > 0; u µµ (:)· 0 with u(0) = 0 and u µ (0)<1. which will be useful and called on throughout the paper. For ease of exposition we assume that any one reader has purchased the medium: there is no free viewing. Advertisers choose to purchase some space in quantity q A each because they derive an idiosyncratic bene¯t a for each consumer seeing the quantity q A of ads. This can be conceived of as a the marginal bene¯t of advertising. An advertiser's payo® is given by V = v(a;q A (e(D)))¡p A q A , with a 2 A = [®;®] ½R following a log-concave distribution G(®) with density g(:) > 0. The return function v(:;:) 6 satis¯es Assumption 2 Thefunctionv(:)iscompletelymonotonicinq A , i.e. : itisn-times di®erentiable and (¡1) n¡1 d n v(:) d[q A ] n ¸0; n=1;2::: and follows Assumption 3 v ® >0; v ®q A(®q A )>0; v ®q A q A ¸0; v ®®q A ·0. The term e(D) parametrises the externality arising from the consumer market, and it is quite natural to let e 0 > 0; e 00 · 0. The more consumers an advertiser can reach, the more an ad is valuable. Given e(D), an advertiser with type a purchases a quantity q A (a) such that v q A =p A . Platforms face no constraints on advertising space (the medium can always print one more page, for example), and it is assumed that advertising does not a®ect readership. 2 Thereisnocosttorunningadverts,andthemarginalcostofproducing the consumer content is also nil. Quality however is costly to provide: it takes the form of a sunk cost denoted k(µ) with k 0 (µ) > 0 and k 00 (µ) > 0. A medium collects revenues from both readers and advertisers, with monies from either side perfectly substitutable. Its instruments are consumer and advertising prices and the quality variable µ. 1.4 Linear pricing Theanalysisbeginswiththissimpleproblem,howevercaseinageneralformulation. For exposition purposes we ¯rst restrict ourselves to linear demands (which arise fromarestrictionofunitdemandforeachagent), andextendtheanalysistoamore general problem in Section 15. 2 Although we duly note restrictions on advertising time for free-to-air television, as well as an obvious distaste for advertising on TV by viewers 7 1.4.1 Special case: linear demands Webeginwithaspecialcasethatdepartsslightlyfromthegeneralframeworkunder consideration. Whilethisparticularformwillnot¯ndanequivalentintheforthcom- ing non-linear pricing problem, it remains an informative benchmark to consider. Model modi¯cations Consumers' net utility the simpler linear form u(b;p C ;µ)=µb¡p C which induces demand Pr ¡ µ¯¸p C ¢ =D C ¡ p C ;µ ¢ =1¡F µ p C µ ¶ on the part of readers. For ease of exposition we assume that any one reader has purchased the medium: there is no free viewing. Note that the sign of the cross- partial derivative D C p C µ (p C ;µ)= f 0 (:) p C µ +f(:) µ 2 is a priori ambiguous. Advertisers purchase at most one unit of space as long as eD C a¡p A ¸0, given some price p A and a consumer demand D C . Keeping in line with the main model, a is a random variable distributed according to G(:), with g(:) > 0 everywhere. The more readers the advertiser can reach, the more valuable is an ad and e can be interpreted as a scaling parameter. This generates the measure Pr µ a¸ p A eD C ¶ =q A (p A )=1¡G µ p A eD C ¶ 8 Analternativespeci¯cationmayconsistoflettingafollowaconditionaldistribution G(ajeD C ). Then demand would be de¯ned as Pr(a¸p A jeD C )=q A (p A ;eD C )=1¡G(p A jeD C ) In this case, it would be quite natural to have @q A @D C = ¡G D C ¡ p A jeD C ¢ > 0: the more readers the advertiser can reach, the larger the mass of advertiser bene¯ting from an ad. Platforms behave as before. A medium collects revenues from both readers and advertisers, with monies from either side perfectly substitutable. Its instruments are consumer and advertising prices and the quality variable µ. The pro¯t function writes M =D C (p C ;µ)p C +q A ¡ p A ;eD C (p C ) ¢ p A ¡k(µ) (1) The reader will notice that the term eD C enters the platform's pro¯t function through the quantity q A (:). Although advertisers cannot in°uence D C , and there- foretreatitasanexternality,itgeneratescomplementaritiesfortheplatform,which cantakeadvantageofthem. Thebenchmarkcaseistrivial. Giventheoptimalprice ¡ D(p C ;µ) D p C the platform selects quality to maximize ¼(µ)=¡ D 2 D p C ¡k(µ) (2) with su±cient ¯rst-order condition (FOC) ¡ D ¡ 2D µ D p C ¡DD p C µ ¢ (D p C) 2 =k 0 (µ) Log-concavity of the distribution function guarantees quasi-concavity of the ob- jective function. Denote the solution to this equation by µ M , and note dp C dµ = 9 ¡ D µ D p C ¡DD p C µ (D p C ) 2 > 0. Introducing externalities In a further departure from the main text, we use a linear speci¯cation, so the relevant advertising demand is q A (p A ;eD C ) = 1¡ G ³ p A eD C ´ . Fixing the quality level and consumer price, the su±cient ¯rst-order condition (by log-concavity of G(:)) with respect to p A yields the obvious q A =¡p A @q A @p A = p A eD C g µ p A eD C ¶ Using this expression (1) can be restated M(p C ;µ)=D C (p C ;µ)p C + ³ 1¡G ³ p A eD C ´´ 2 g ³ p A eD C ´ eD C ¡k(µ) (3) Thesecondterm µ 1¡G µ p A eD C ¶¶ 2 g ³ p A eD C ´ eD C isthemonopolyadvertisingpro¯t,apportioned by the `e®ective' consumer market coverage (eD C ). It is obviously positive, so that for the same quality level µ, if p C = p; M(p C ;µ) > ¼(p;µ). This formulation paves the way to a ¯rst result. Proposition 1 The optimal consumer price entails a discount. The platform behaves like a multi-product monopolist selling complementary prod- ucts { albeit with a single externality a®ecting its portfolio, here from consumers to advertising. It is able to internalize the impact of one product's sales on the other, and adjust its prices accordingly. However this platform di®ers from the more fa- miliar multi-product monopolist in that 1) it serves two markets, as opposed to selling two products to the same consumers; and 2) the services it provides these two markets with do not interact directly. 3 3 For instance, the advertisers do not use the consumer good. 10 Proof: The (su±cient) FOC D C +p C D C p C +eD C p C (1¡G(:)) 2 g(:) + 2(1¡G(:))D C p C [g(:)] 2 p A D C +g 0 (:)D C p C p A D C (1¡G(:)) 2 g(:) 2 =0 can be re-arranged more compactly. Using the de¯nition of the advertising price D C +D C p C ½ p C + · (1¡G(:)) 2 g(:) µ e+ 2[g(:)] 2 +g 0 (:)(1¡G(:)) [g(:)] 2 D C ¶¸¾ =0 (4) Withanincreasinghazardrate(whichfollowsfromlog-concavity),theterm2[g(:)] 2 + g 0 (:)(1¡G(:))ispositive. Suppose ^ p C ¸p C , then ^ D C ·D C , sotheFOC(4)implies that 0> ^ D C p C j ^ p C >D C p C j p C. Re-arranging, this leads directly to a contradiction. So the term in square brackets in (4) represents a discount o®ered to consumers. With this de¯nition, the pro¯t function (3) can again be recast as M(µ) =¡ [D C ] 2 D p C ¡D C h (1¡G(:)) 2 g(:) ³ e+ 2[g(:)] 2 +g 0 (:)(1¡G(:)) [g(:)] 2 D C ´i + (1¡G(:)) 2 g(:) eD C ¡k(µ) =¡ [D C ] 2 D p C ¡ h (1¡G(:)) 2 g(:) ³ 2+ g 0 (:)(1¡G(:)) [g(:)] 2 ´i ¡k(µ) (5) This is the marginal pro¯t attributable to the marginal consumer. According to (5) the monopolist chooses its quality variable so as to maximizes a payo® func- tion, which can only lie below that of the externality-free case (equation (2)), as (1¡G(:)) 2 g(:) ³ 2+ g 0 (:)(1¡G(:)) [g(:)] 2 ´ > 0. This seems to contradict the earlier observation that M(~ p; ~ µ)>¼(~ p; ~ µ). In fact, the platform internalizes the fact that it (optimally) o®ers a discount to consumers, and therefore need not o®er as high a quality level as it would absent advertising. Let ^ µ be the maximizer of (5), then Proposition 2 A su±cient condition for ^ µ · µ M is that the inverse hazard rate ( 1¡G g ) of the distribution G(a) be weakly convex. 11 The necessary (and su±cient) condition is marginally milder in that ^ µ < µ M , @ @µ µ (1¡G(:)) 2 g(:) ¶ @ @µ ³ g 0 (:)(1¡G(:)) [g(:)] 2 ´ > ¡ (1¡G(:)) 2 g(:) 2[g(:)] 2 +g 0 (:)(1¡G(:)) { the right-hand side being negative. The denominatormaybenegative{thatis, theinversehazardratemaybeconcave, but itsabsolute valuehas to be largeenough. However ¯ ¯ ¯ @ @µ ³ g 0 (:)(1¡G(:)) [g(:)] 2 ´¯ ¯ ¯ `large'implies that 1¡G ³ p A eD C ´ { the equilibrium volume of advertising { be `small'. One possible interpretation of this result goes as follows: for any quality level µ, the optimal price entails a discount, ^ p C < p M . When the platform increases its quality by dµ at cost k 0 (µ) to win a marginal consumer, it generates only ^ p C dD C < p M dD C in additional revenue from the consumer market. Hence, when quality is costly, it is optimal to restrict it to a lower level in presence of the externality. By way of analogy, this resembles the logic of subgame perfection in a strategic game. Notice also that the result essentially depends on the properties of the distribution G(:). Compared to the externality-free case, what matters is the behavior of the implicitdiscount(thesecondtermin(5))thattheplatforminternalizes. Whetherit is non-decreasing depends only on G(:), given the behavior of D(p C ;µ). When that is the case, the pro¯t function (5) reaches its maximum `earlier' (at a lower value of µ) than (2). Proof: We can write M(µ)=¼(µ)¡ h (1¡G(:)) 2 g(:) ³ 2+ g 0 (:)(1¡G(:)) [g(:)] 2 ´i , so @M(µ) @µ =0, @¼(µ) @µ ¡ @ @µ · (1¡G(:)) 2 g(:) µ 2+ g 0 (:)(1¡G(:)) [g(:)] 2 ¶¸ =0: Immediately, if the second term of the second equality is non-positive, we must have @¼(µ) @µ ¸ 0, hence ^ µ · µ M . The converse is obvious. To show su±ciency of the condition, observe that the term (1¡G(:)) 2 g(:) is necessarily increasing in µ so it is enough that ³ 2+ g 0 (:)(1¡G(:)) [g(:)] 2 ´ be increasing as well. That is, it is su±cient to have @ @µ µ 2+ g 0 (:)(1¡G(:)) [g(:)] 2 ¶ = @ @µ µ g 0 (:)(1¡G(:)) [g(:)] 2 ¶ ¸0: 12 Let º = ¸ ¡1 = 1¡G(x) g(x) denote the inverse of the hazard rate, with the property that @º @x = ¡ [g(:)] 2 +g 0 (:)(1¡G(:)) [g(:)] 2 = ¡ h 1+ g 0 (:)(1¡G(:)) [g(:)] 2 i < 0 because @¸ @x > 0 (by log- concavity). So @ @µ µ g 0 (:)(1¡G(:)) [g(:)] 2 ¶ =¡ @ 2 º @x 2 @x @µ ¸0, @ 2 º @x 2 ¸0; since @x @µ =¡ p A e[D c ] 2 D µ <0. With a uniform distribution, the inverse hazard rate is decreasing but at a con- stant rate { hence weakly convex, therefore ^ µ < µ M . Similarly if G(:) were expo- nential, with a constant hazard rate. 1.4.2 General case Thepreviousanalysisisinformative,butitcannothaveanequivalentinanon-linear pricingmodel. Thereforewecontinuewiththemoregeneralformulationofthemodel that can readily be converted into a mechanism design problem. Benchmark The analysis begins with the simple monopoly case absent advertising revenue, which will be used as a benchmark in the sequel. For any quality level µ, there exists a threshold b ¤ (p C ;µ) such that u(b ¤ ;µ)=p C . Hence, given Assumption 4 (Single-crossing condition) u bµ ¸0 we can show Claim 1 Properties of the threshold b ¤ (p C ;µ): ² db ¤ dµ <0; ² db ¤ dp C >0; d 2 b ¤ d[p C ] 2 =¡ u bb (b ¤ ;µ) u b (b ¤ ;µ) db ¤ dp C ¸(<)0 according as u bb (b ¤ ;µ)·(>)0 and ² d 2 b ¤ dp C dµ ·0 13 Proof:Fixp C ; u b db ¤ dµ +u µ =0. Similarly,¯xµ; u b (b ¤ ;µ) db ¤ dp C ¡1=0andu bb (b ¤ ;µ) db ¤ dp C + u b (b ¤ ;µ) d 2 b ¤ d[p C ] 2 = 0. Last, u bµ db ¤ dp C +u b d 2 b ¤ dp C dµ = 0. Re-arranging each of these yields the Claim. The consumers' behaviour induces demand Pr(¯¸b ¤ )=D C ¡ p C ;µ ¢ =1¡F ¡ b ¤ (p C ;µ) ¢ on the part of readers. In light of Claim 1 and Assumption 1, Claim 2 Properties of the consumers' demand: given p C , ² @D C @µ >0; ² @D C @p C <0 and ² @ 2 D C @p C @µ ¸0 which follows directly from Claim 1. The monopolist payo® function is then simply ¼(p C ;µ)=p C £ 1¡F(b ¤ (p C ;µ)) ¤ ¡k(µ) (6) with necessary and su±cient ¯rst-order conditions ~ p C (µ)= 1¡F ¡ b ¤ (p C ;µ) ¢ f(b ¤ (p C ;µ)) db ¤ dp C (7) where db ¤ dp C = 1 u p C (b;µ) >0. Therefore (6) rewrites ¼(µ)= £ 1¡F(b ¤ (p C ;µ)) ¤ 2 f(b ¤ (p C ;µ)) db ¤ dp C ¡k(µ) 14 Notice from (7) and Claim 1 that dp C dµ > 0 because the inverse hazard rate of F(:) is decreasing and db ¤ dµ <0. The necessary optimality conditions are then ¡[1¡F(b ¤ )] 2 6 4 db ¤ dµ db ¤ dp C £ 2[f(:)] 2 +[1¡F(:)]f 0 (:) ¤ +f(:) d 2 b ¤ dp C dµ h f(b ¤ ) db ¤ dp C i 2 3 7 5 =k 0 (µ) (8) The second term on the left-hand side of (8) is (almost) the derivative of the inverse of the hazard rate of F(:). By Claim 1 and the condition on the hazard rate, it is necessarily negative. Since 1¡F(:) is monotonic in b and the term in brackets has a constant sign (by the log-concavity assumption on F(:)), the LHS of (8) is also monotonic. Furthermore, 1¡F(b(p C ;µ))j µ=µ = 0, that is, the LHS originates at zero in the positive orthan. Last, d dµ ¡ 1¡F(b(p C ;µ)) ¢ j µ=µ > 0, while k 0 (0) = 0. Therefore the LHS and the RHS of (8) cross at most once for (strictly) positive values of µ. Denote this solution µ M . Introducing externalities Given e(D C ), an advertiser with type a purchases a quantity q A (a;e(D C )) such that v q A = p A ; that is, for each a 2 [®;®]; q A ´ q A (a;p A ;e(D C )). With our assumptions on v(:), an advertiser's demand q A is necessarily decreasing in p A and increasing in a. Market demand can then be obtained by integrating each function q A (a;e(D C )) over the set [®;®]. More precisely, letting Q A denote total demand, Lemma 1 Market demand is de¯ned as Q A = Z ® ^ ® q A ¡ x;p A ;e(D) ¢ dG(x) where ^ ®=maxf®;® 0 g and ® 0 is determined by the condition v q A ¡ ® 0 ;q A ¢ j q A =0 =p A : When the marginal bene¯t of advertising is unbounded at zero even the advertiser valuing it the least purchase some positive quantity { then ^ ® = ®. Otherwise, 15 depending on the level of p A , there may exist a positive measure of ¯rms electing to not advertise (® 0 >®). Proof: Fixa;p A ,sincev(:)iscontinuousinq A ,thereexistsapointwherev q A =p A . GivenAssumption3thiscanbemadetoholdcontinuouslyforeacha2[®;®]. When v q A ¡ ® 0 ;q A ¢ j q A =0 < Z < 1, there exists a price p A and a type ® 0 (p A ) such that v q A ¡ ® 0 ;q A ¢ j q A =0 = p A . This de¯nes ^ ®. The function q A (:;:;:) is a mapping from A£R£R7!R hence it is integrable. Furthermore Lemma 2 Properties of the market demand ² dQ A dD C ¸0; ² d 2 Q A d(D C ) 2 ·0 and ² d 2 Q A dp A de ·0 Proof: Given p A , for each a; 9 q A such that v q A ¡ a;q A (e(D C )) ¢ ¡ p A = 0, so di®erentiating with respect to D C v q A q Aq A e(D C ) e 0 + dq A de(D C ) e 0 =0 and once more v q A q A q A ³ q A e(D C ) e 0 ´ 2 +v q A q Aq A e(D C )e(D C ) (e 0 ) 2 + d 2 q A d[e(D C )] 2 (e 0 ) 2 +e 00 µ v q A q Aq A e(D C ) + dq A de(D C ) ¶ = 0: Re-arranging yields the obvious dq A dD C ¸ 0 and d 2 q A d(D C ) 2 · 0, thanks to Assumption 2 ande 0 >0. Similarly,di®erentiatingwithrespecttop A givesusv q A q A(:) dq A dp A ¡1=0, so that dq A dp A <0. Di®erentiating once more v q A q A q A(:) dq A de dq A dp A +v q A q A(:) d 2 q A dedp A =0; 16 therefore d 2 q A dedp A <0. Furthermore,di®erentiatingwithrespecttoa; v q A a +v q A q A dq A da = 0, so that by Assumption 3, q A (:) is increasing in the type ®. Recalling that the operator R dx is linear, the exercise in di®erentiation carries to Q A and the lemma obtains. With these preliminaries we can state the platform's objective function M =D C (p C ;µ)p C +Q A ¡ p A ;eD C (p C ) ¢ p A ¡k(µ) (9) where it can noticed that the outcome in the consumer market enters the revenues from the advertising market. For a meaningful analysis of this monopoly problem we impose Assumption 5 There exists 0 < Z < 1 such that v q A(a;q A )j q A =0 < Z for some a2(®;®) whichguaranteesthatsome¯rmswillnotconsumeinequilibrium(referLemma18). Maximizing (9) with respect to advertising prices yields the usual p A =¡ Q A @Q A @p A , that is, p A =¡ R ® ® 0 q A dG(z) R ® ® 0 @q A @p A dG(z)¡q A g(® 0 ) d® 0 dp A using Leibnitz' rule of di®erentiation, and where d® 0 dp A = 1 v ®q A j ® 0 >0 by Assumption 3. Increasing the price p A decreases total demand two-ways: the demand of infra- marginal advertiser drops and the marginal ¯rm ® 0 also shifts up the type space. Using this expression we can rewrite (9) as M(p C ;µ)=D C (p C ;µ)p C ¡ h R ® ® 0 q A dG(z) i 2 R ® ® 0 @q A @p A dG(z)¡q A g(® 0 ) d® 0 dp A ¡k(µ) (10) with q A ´ q A ¡ a;p A ;e(D C (p C ;µ)) ¢ . Our ¯rst result pertains to the consumer price level, for which we impose 17 Assumption 6 u ¯¯ (b ¤ ;µ)·0 which is quite natural but not necessary. (It can be construed as a su±cient condi- tion.) Then we have the analog to Proposition 1. Proposition 3 The optimal consumer price entails a discount. Proof: Using subscripts for partial derivatives, the ¯rst-order condition is D C + D C p C · p C ¡e 0 2Q A Q A e Q A p A ¡[Q A ] 2 Q A p A e [Q A p A ] 2 ¸ =0. Lettingp C ¤ denotethesolutiontothisequa- tion and re-arranging, p C ¤ =¡ D C D C p C +e 0 2Q A Q A e Q A p A ¡[Q A ] 2 Q A p A e [Q A p A ] 2 With Q A p A e > 0 given our assumptions on v(:), the second term on the RHS is always negative. The ¯rst one, ¡ D C D C p C is in fact the inverse hazard rate 1¡F(b ¤ ) f(b ¤ ) db ¤ dp C of the distribution F(b) evaluated at p C ¤ . Given our assumption of log-concavity of F(:), this is decreasing in p C , since b ¤ (p C ) is increasing. Proceeding by con- tradiction, suppose p C ¤ ¸ ^ p C , then 1¡F(b ¤ (p C ¤ )) f(b ¤ (p C ¤ )) db ¤ dp C · 1¡F(b ¤ (^ p C )) f(b ¤ (^ p C )) db ¤ dp C necessarily. Since e 0 2Q A Q A e Q A p A ¡[Q A ] 2 Q A p A e [Q A p A ] 2 < 0, we arrive at an obvious contradiction. Therefore p C ¤ < ^ p C . The pro¯t function then rewrites M(µ)=¡ [D C ] 2 D C p C +e 0 2Q A Q A e Q A p A ¡[Q A ] 2 Q A p A e [Q A p A ] 2 D C ¡ [Q A ] 2 Q A p A ¡k(µ) (11) Notice that the advertising pro¯t ¼ A = ¡ [Q A ] 2 Q A p A is positive, while the new term e 0 2Q A Q A e Q A p A ¡[Q A ] 2 Q A p A e [Q A p A ] 2 D C is negative: taking advantage of the externality from the consumer market to the advertising market is costly to the platform. This cost is the consumer discount highlighted in Proposition 3, which makes a dent in the platform's pro¯t. Thus we should expect the platform to internalize it in its choice of quality µ. Let µ ¤ denote the maximizer of (11) when it exists. We ¯rst claim 18 Lemma 3 There exists a unique maximizer µ ¤ of (11). Proof: Compared to ¼(µ), expression (11) only varies by e 0 2Q A Q A e Q A p A ¡[Q A ] 2 Q A p A e [Q A p A ] 2 D C ¡ [Q A ] 2 Q A p A ; which we rewrite ¡¼ A p C D C D C p C +¼ A . We know that ¼(µ) is quasi-concave in µ. Since Q A is concave in D C , we can show that @ @D C Ã ¡ Q A Q A p A ! =¡ Q D CQ A p A ¡Q A Q A p A D C [Q A p A ] 2 ¸ (<)0 according as ¡Q D CQ A p A ¡Q A Q A p A D C ¸ (<)0 Substituting, this holds if and only if vq A q A dq A de 1 v q A q A ¸(<)¡q A v q A q A q A dq A de dq A dp A v q A q A ; which is immediate. Further di®erentiation shows that @ 2 @[D C ] 2 µ ¡ Q A Q A p A ¶ · 0; that is, the price function p A is concave in the consumer demand D C . Since ¼ A is the product of two concave functions, it is concave in D C as well by Theorem 5 of Kantrowitz and Neumann (2005). Therefore ¡ ¼ A p C D C is decreasing in D C and consequentlytheterm¡¼ A p C D C D C p C isdecreasinginD C aswell. Sinceconsumerdemand D C is monotonic in µ; ¡¼ A p C D C D C p C decreases in µ, and the pro¯t function ¼ A is a concave transformation of consumer demand D C . With k(µ) convex, the problem represented by expression (11) is globally concave. The maximizer µ ¤ of (11) departs from µ M only if the derivative of this term di®ers from zero (when evaluated at µ M ). This derivative becomes so cumbersome that it fails to be informative. However we can proceed di®erently and still show the main result for this section. 19 Proposition 4 µ ¤ <µ M Proof: The consumer price p C (µ) is continuous in µ and dp C dµ > 0. Therefore the reciprocal dµ dp C is well de¯ned, continuous and dµ dp C > 0 as well. Since p C ¤ < ~ p C the result follows. Henceadistortiongenericallyobtains. Thereasonisthat, givenquality, theplat- form is compelled to o®er a consumer discount to expand coverage, which increases advertising revenue. Since consumers pay a lower price, they are also prepared to acceptalesserproduct. Thatis,ittakesalowerqualityleveltoinducethemarginal consumer to purchase. 1.5 Non-linear pricing The model developed so far is adapted to be solved as a mechanism design problem involving an uninformed principal and continua of consumers on one side, and of advertisers on the other. The Spence-Mirrlees condition is assumed to hold and we also seek concavity and supermodularity of the virtual pro¯t function. More precisely we call on Assumption 4 and further impose Assumption 7 u ¯µµ ¸0; u ¯¯µ ·0 which are both standard in contract theory. Since u µ (0)<1 it may be optimal to shut down a positive measure of types. Let ¯ 0 ¸¯ the cut-o® such that only types ¯¸¯ 0 participate in the mechanism (this cut-o® is determined endogenously) and denote the measure of participating consumers by D = R ¯ ¯ 0 dF(:); D is decreasing in the threshold ¯ 0 . Advertisers and platforms behave as in Section 1.4. The principal solves max µ;p C ;q A ;p A M =E B £ p C ¡k(µ) ¤ +E A £ p A ¡c(q A ) ¤ (12) where k(µ) and c(q A ) are convex cost functions. 20 1.5.1 Externality-free benchmark In the consumer market a mechanism is a pair of functions hµ(¯);p C (¯)i mapping anannouncement ~ ¯2B intothespaceofallocationsandtransfers££P C ½R£R. This message ~ ¯ has to be optimal for the agent, that is u µ (¯;µ) _ µ¡ _ p C =0; 8 ~ ¯;¯ u µµ (¯;:)( _ µ) 2 +u µ (¯;:) Ä µ¡ Ä p C ·0 where ¯ is the true value and ~ ¯ the announcement. In particular, these conditions have to be satis¯ed when the agent reports the truth. Di®erentiating the agent's payo® when ~ ¯ =¯, this implies the incentive constraint _ U =u ¯ (¯;µ) (13) since the remaining terms are naught by the ¯rst-order condition. For the local second-order condition to be everywhere satis¯ed, the monotonicity condition u ¯µ (¯;µ) _ µ¸0 (14) obtainedbydi®erentiatingtheFOCwithrespecttothetruthandusingthefactthat the second-order condition (SOC) is non-positive, must hold. Finally the individual rationality constraint is the standard u(¯;µ)¡p C ¸0; 8¯¸¯ 0 (15) This standard problem has a familiar characterisation. Lemma 4 Suppose the principal's objective function is E B £ p C ¡k(µ) ¤ , subject to 21 (13)-(15). The optimal (second-best) quality level is determined by u µ (¯;µ SB (¯))=k 0 (µ SB (¯))+ 1¡F(¯) f(¯) u ¯µ (¯;µ SB (¯)) 8¯¸¯ 0 ; µ =0; 8¯ <¯ 0 . with the shutdown condition u(¯ 0 ;µ(¯ 0 ))=k(µ(¯ 0 ))+ 1¡F(¯ 0 ) f(¯ 0 ) u ¯ (¯ 0 ;µ(¯ 0 )) (16) and rents U = Z ¯ ¯ 0 u ¯ (x;µ(x))dx¸0; 8¯¸¯ 0 Proof: Standard and therefore omitted. Refer La®ont and Martimort (2002) for an exposition. Given our assumptions on u(¯;µ) and that inverse hazard rate 1¡F(¯) f(¯) is decreasing in ¯ (for F(:) is log-concave, whence its hazard rate is increasing), the solutionµ SB (¯)canbeshowntobeincreasinginthetype. Hencetheconstraint(14) is satis¯ed. This is the standard rent extraction-e±ciency trade-o®. Except at ¯ =¯, quality isdistortedcomparedtothe¯rst-bestu µ =k 0 . Ifthereexistsatype¯ 0 2Bsuchthat the principal would like to shut down all consumers below said type, it is identi¯ed by the rent binding at ¯ 0 (Condition (16)). For later reference, Lemma 5 8¯ 0 ¸¯; d¯ 0 dµ >0. Proof: Thethreshold¯ 0 isdeterminedeitherbytheexogenouslowerbound¯ when theFOCholdsforall¯2B,orendogenouslybytheshutdowncondition. Inthe¯rst case it is obvious that ¯ remains invariant. Di®erentiating the ¯rst-order condition with respect to ¯ yields _ µ(¯) ¸ 0; 8¯ ¸ ¯ 0 , whence its reciprocal d¯ dµ ¸ 0. By continuity of the schedule µ(¯), this extends to the case ¯ 0 =¯ when the shutdown condition is just binding at ¯. 22 In the optimal mechanism higher quality is associated with higher types, and this applies to the marginal agent. 1.5.2 Introducing externalities Although an advertiser's utility depends on the measure D of participating con- sumers, neither side is able to internalize its in°uence on the other one. That is, each advertiser selects its revelation strategy for some level D. Therefore the con- straint set for advertisers can be characterized in the same fashion as that applying to consumers, where a mechanism is a pair hq A (®;e(D));p A (®)i from the message space A into the space of allocations and transfers Q A £P A ½R£R. Hence we must have v q A ¡ ®;q A (:;e(D)) ¢ _ q A (~ ®;e(D))¡ _ p A (~ ®)=0; 8®;~ ® v q A q A(:;:)(_ q A (~ ®;:)) 2 +v q A(:;:)Ä q A (~ ®;:)¡ Ä p A (~ ®)·0 where ® is the true type. The incentive constraint becomes _ V =v ® ¡ ®;q A (:;e(D)) ¢ (17) by di®erentiating V(®) at ~ ®=®, and the monotonicity constraint v ®q A_ q A ¸0 (18) whereitisunderstoodthat _ q A = @q A (®;e(D)) @® . Last,allparticipatingadvertisersmust receive non-negative payo®s V =v ¡ ®;q A (e(D)) ¢ ¡p A ¸0; 8®¸® 0 (19) The constraints (17)-(19) inform us that the structure of the problem on the adver- tising side remains standard. Whatever distortion may be caused by the externality 23 e(D) arises not from the characterization of the feasible, incentive compatible set, butfromthetheplatform'sbehavior. Thisisbecausebothsidesreporttheirmessage to the principal independently of the other side's private information. Continuing on, the principal maximizes his objective function (12), subject to the constraints (13)-(15) and (17)-(19). In the general form Problem 1 max p C ;p A ;µ;q A M =¼ C (u(¯;:);U(¯;:))+¼ A ¡ v(®;q A (e(D)));V(®;q A (e(D))) ¢ s.t. (13)¡(15) and (17)¡(19) Optimization of M with respect to p A ;q A reduces to maximizing ¼ A pointwise, and yields, in particular, the standard conditions @¼ A @q A =0 which will be expanded later. However it is obvious from the de¯nition of the measure D that optimization with respect to p C ;µ may not be neutral on ¼ A . From now on we assume that ¯ 0 > ¯: absent any externality (Lemma 4), some consumers are shut down. As in Section 15, this may be achieved by bounding the consumers' marginal utility of quality at zero. Thus Assumption 8 There exists some Z 0 2R ++ such that u µ (¯;0)<Z 0 <1. Let¯ e 0 denotethemarginalparticipatingconsumerundertheexternalityregime. Us- ing standard techniques (Maskin and Riley (1984), La®ont and Martimort (2002)), the pro¯t function ¼ A = Z ® ® 0 £ v(z;q A (z;:))¡V(z;q A (z;:))¡c(q A (z;:)) ¤ dG(z) 24 obtained by substituting the de¯nition of the rent, can be turned into ¼ A = Z ® ® 0 · v ¡ z;q A (z;:) ¢ ¡ 1¡G(z) g(z) v ® (z;q A (z;:)¡c(q A (z;:))) ¸ dG(z) (20) (V(:;q A (z)) = R ® ® 0 v ® (z;q A )dz, simply substitute in ¼ A and integrate by parts.) Similarly, ¼ C = Z ¯ ¯ 0 · u(x;µ(x))¡ 1¡F(x) f(x) u ¯ (x;µ(x))¡k(µ(x)) ¸ dF(x) (21) Solving Problem 2 with these speci¯cations, we claim Proposition 5 Suppose V =v ¡ ®;q A (e(D)) ¢ , then 1. quality on the consumer side is uniformly distorted, i.e. µ e (¯)<µ SB (¯); and 2. more consumers participate, i.e. ¯ e 0 <¯ 0 Obviously there can be no distortion if all consumers already participate (¯ e 0 = ¯), which is why Assumption 8 is called upon. The proof consists in characterizing the mechanism, whence the claim is immediate. Proof: The problem remains separable when dealing with the advertising market, so maximizing the function M(µ;q A ) with respect to q A amounts to pointwise opti- mization of (20) only. Therefore, given ¯ 0 , the standard result v q A¡ 1¡G(®) g(®) v ®q A =c 0 (q A (®)) (22) with rents V(®) = R ® ® 0 v ® (z;q A )dz on the advertising side { by integrating _ V with V(® 0 )=0by(19). GiventheassumptionsonthedistributionG(:),(37)alsosatis¯es the monotonicity condition (18). The participation threshold ® 0 (¯ 0 ) is determined as in Lemma 4, viz. v(® 0 ;q(® 0 ))¡c(q(® 0 ))= 1¡G(® 0 ) f(® 0 ) v ® (® 0 ;q(® 0 )) (23) 25 for some ¯ 0 . We proceed in the same fashion for the consumer market, however using Leibnitz rule @M @µ =0 yields u µ ¡ 1¡F(¯) f(¯) v ¯µ ¡k 0 = R ® ® 0 @q A @e e 0 f(¯ 0 ) d¯ 0 dµ h v q A¡ 1¡G(®) g(®) v ®q A¡c 0 i dG(z) ¡ h v ¡ ®;q A ¢ ¡ 1¡G(®) g(®) v ® (®;q A )¡c(q A ) i g(® 0 (¯ 0 )) d® 0 d¯ 0 d¯ 0 dµ (24) for ¼ A is not independent of µ. The participation condition for the marginal con- sumer takes the same form as (16) in Lemma 4. The ¯rst term on the right-hand side of (24) is zero by FOC (37), which holds for all ® ¸ ® 0 . Thus all hinges on the sign of d® 0 d¯ 0 . By design, q(®) is a continuous, monotonic function and dq d® ¸ 0 (Condition (18)), hence the reciprocal exists and d® dq ¸0 as well. Thus we can write d® 0 d¯ 0 =¡ d® dq dq de e 0 f(¯ 0 ) < 0, whence the second term on the RHS is non-negative and µ e < µ SB . By Lemma 5 and Condition (16), the threshold ¯ e 0 < ¯ 0 and if u(:) is su±ciently concave, ¯ e 0 >¯. According to Proposition 5, µ e (¯) < µ(¯) 8¯ 2 £ ¯ e 0 ;¯ ¤ . That is, thanks to the externality, thereis a distortion at the top, evidenced by(24). The optimalschedule µ e (¯) is both shifted down and tilted, so that Condition (16) binds at a lower level of ¯. A natural Corollary can be appended. Corollary 1 For any consumer participating absent the advertising externality the information rent is lower under the externality regime, i.e. 8¯ 2 £ ¯ 0 ;¯ ¤ ; U e = R ¯ ¯ 0 u ¯ (¯;µ e (x))dx· R ¯ ¯ 0 u ¯ (¯;µ SB (x))dx Proof: Directly from the monotonicity condition u ¯µ (:)¸0 and Proposition 5. For each ¯, the rent increases in quality µ, therefore u ¯ (¯;µ e ) · u ¯ (¯;µ SB ) and the integrals, which are de¯ned on the same bounds, follow the same ranking. Thus some consumers bene¯t from the externality (those between ¯ e 0 and ¯ 0 ) because they can participate only under this regime, while others are hurt by it. However notice that the loss of rent on the part of the consumers is not transferred totheprincipal;rathertheyaredissipated. Becausequalityislower,soisthesurplus 26 extracted from consumers, but the platform recoups this loss from the advertising side. Proposition5happenstobeverysensitivetotheexactroleoftheexternalitye(D) in the advertisers' payo® function v(:). To make the point we alter the de¯nition of V(®;q A ) to V =v ¡ ®;q A ;e(D) ¢ , so that the quantity q A is not a function of the externality e(D), but the surplus function v is. Assumption 3 is also extended to include Assumption 9 8®2A; v e >0;v e® >0 and v ee® >0 When the rent V is separable in q A and e(D), the incentive constraint remains _ V =v ® (25) as does the monotonicity condition v q A ® _ q A ¸0 (26) Here too we obtain a distortion of the quality variable, that is Proposition 6 Suppose V = v(®;q A ;e(D)). There exists an optimal mechanism denoted h ^ µ e (¯);^ q A (®)i such that 1. quality on the consumer side is distorted, i.e. ^ µ e (¯)<µ SB (¯); and 2. more consumers participate, i.e. ^ ¯ e 0 <¯ 0 . However, with this new payo® function, the solution to Problem 2 takes a di®erent shape. Proof: The platform's problem on the advertising side is still characterized by the FOC (37) and u µ ¡ 1¡F(¯) f(¯) u ¯µ ¡k 0 = Z ® ® 0 e 0 f(¯ e 0 ) d¯ 0 dµ · v e ¡ 1¡G(z) g(z) v ®e ¸ dG(z) (27) 27 on the consumer side. The shutdown conditions remain as (16) and (23). But the optimal schedule q(®) is no longer a function of e(D), therefore d® 0 d¯ 0 = 0. Still, underAssumption9thederivativeofthevirtualsurplusv e ¡ 1¡G(z) g(z) v ®e iseverywhere positive. This proves the ¯rst claim. The second follows from Lemma 5 and the shutdown condition (16). In both cases the non-linear schedules hµ e (¯);q A (®)i and h ^ µ e (¯);^ q A (®)i can be implemented as a two-part tari®. Let º G ´¸ ¡1 G ´ 1¡G(®) g(®) denote the inverse hazard rate of the distribution G(®). Collecting Propositions 5 and 6, we can also show Proposition 7 The optimal schedules and equilibrium participation thresholds on the advertising side are such that 1. 8®¸ ^ ® 0 ; ^ q A (®)¸q A (®) and 2. ^ ® 0 ·(>)® 0 according as v ® ³ 1¡ dº G d® ´ ¸(<)v ®® º G Proof: Recall the ¯rst-order condition (37): v q A = c q A + 1¡G g v ®q A, the solution of which is an optimal schedule q A (®). Suppose ¯rst that V = v ¡ ®;q A (e(D)) ¢ and di®erentiate with respect to the externality e to obtain q A e µ v q A q A¡ 1¡G g v ®q A q A¡c q A q A ¶ =0,q A e =0 because the term in brackets is the second-order condition, which we know to be negative. However if V =v(®;q;e(D)), the same operation yields dq A de µ v q A q A¡ 1¡G g v ®q A q A¡c q A q A ¶ +v q A e ¡ 1¡G g v ®q A e =0 , that is, dq A de = 1¡G g v ®q A e ¡v q A e v q A q A¡ 1¡G g v ®q A q A¡c q A q A > · 0, 1¡G g v ®q A e ¡v q A e < ¸ 0 28 Therefore dq A de > 0, that is, 8® 2 [maxf^ ® e 0 ;® e 0 g;®];^ q A (®) ¸ q A (®), which proves the ¯rst claim. The second one does not directly follow, but with this in hand, consider the principal's net surplus function for a participating type ® : ^ ¼ A (®) = v(®;q A (®))¡ 1¡G g v ® (®;q A (®))¡c(q A (®)). It is increasing in ® and crosses zero for ®=® 0 (Condition (23)). Di®erentiate totally with respect to the schedule q A v ® d® dq A +v q A¡ h³ v ®q A +v ®® d® dq A ´ º G +v ® dº G d® d® dq A i ¡c q A = d® dq A h v ® ³ 1¡ dº G d® ´ ¡v ®® º G i +v q A¡º G v ®q A¡c q A | {z } =0 = d® dq A h v ® ³ 1¡ dº G d® ´ ¡v ®® º G i Since d® dq A >, all is determined by the sign of the expression in brackets. When it is zero, ^ ¼ A (®) = ¼ A (®): it takes the same value in both cases for any ® 2 A. Therefore ^ ¼ A (®) crosses the zero line at the same value ® 0 as ¼ A (®). When it is positive, ^ ¼ A (®)>¼ A (®)8®, so ^ ® 0 <® 0 and conversely. Item 1 of Proposition 7 mirrors the claims that µ e < µ SB and ^ µ e < µ SB (Propo- sitions 5 and 6), however in the opposite direction. While we cannot speak of a `distortion at the top', the schedule ^ q A (®) is everywhere above the schedule q A (®). This implies also that the rents accruing to the participating advertisers are larger, as in Corollary 1: Z ® maxf^ ® 0 ;® 0 g v ® (:;^ q A )dz¸ Z ® maxf^ ® 0 ;® 0 g v ® (:;q A )dz: In addition to a shift, the slope of the surplus function ¼(®) may be altered for each ®. The combination of these two changes may shift the point ® 0 at which it reaches zero. It fails to do so for a precise condition only. Notice in particular Corollary 2 If v(®;q A ;e) is weakly concave in ® (i:e: v ®® ·0), ^ ® 0 <® 0 necessar- ily. 29 which requires no proof. When the advertisers' return function is concave in the type, the marginal bene¯t is obviously decreasing. That is, as the platform lowers the participation threshold, additional advertisers are increasingly willing to pay. Conversely, if the return function is su±ciently convex, it prefers restricting partic- ipation to high-valuation agents only. This is because the optimal schedule ^ q A (®) becomes too steep and it would overprovide low-valuation advertisers. 1.5.3 Discussion When the second term on the right-hand side of FOC (24) is positive, the net marginal bene¯t of quality u µ ¡k 0 is larger than would be under the second-best { that is, quality is too low compared to the benchmark. In this case, for all participatingconsumerstheprincipalfailstoextractthewhole(second-best)surplus from this market. But she gets more consumers on board. Recall that Lemma 5 tells us that increasing quality µ reduces participation. Notice that distortions are independent of advertising costs. The di®erence be- tween Propositions 5 and 6 rests on the separability of the surplus function v(:) in q A and e(D). Proposition 5 tells us that the platform bene¯ts from capturing more consumers because it can expand participation on the advertising side. More consumers attract more advertisers. More precisely, when q A ´q A (®;e(D)), chang- ing D does not alter the marginal condition already given by (37). The economic intuition is that adding consumers a®ects quantities demanded by advertisers, but also the cost c(q A ) in the same magnitude, hence adding a marginal consumer does a®ectthequantitydemandedbytheinframarginaladvertisers. Therentextraction- e±ciency trade-o® (given by (37)) is set at its optimum already for any measure of consumers. However more participating consumers lure advertisers who would otherwise sit out: it lowers the participation threshold ® 0 . Hence a distortion away from the second-best µ SB to attract advertisers at the margin. The optimality con- 30 dition exactly balances the foregone pro¯t from the marginal consumer with the additional surplus that can extracted from the marginal advertiser. Remark 1 Suppose V = v ¡ ®q A (e(D)) ¢ : the type ® interacts with the quantity q(e(D)). The incentive constraint now reads _ V =v ® q A (®;e(D)) and the monotonic- ity condition becomes v ®q A®q_ q A ¸0. The integrand of (20) becomes v¡ 1¡G g v ® q A ¡ c(q A ) and the ¯rst-order condition (37) is altered to read v q A¡ 1¡G g ¡ v ®q Aq A +v ® ¢ ¡ c q A =0. The RHS of the optimality condition (24) becomes Z ® ® 0 @q A @e e 0 f(¯ e 0 ) d¯ e 0 dµ z · v q A¡ 1¡G(z) g(z) ¡ v ®q Aq A +v ® ¢ ¡c q A ¸ dG(z) | {z } =0 ¡ h v¡ 1¡G(®) g(®) v ® q A ¡c(q A ) i g(® 0 (¯ 0 )) d® 0 d¯ 0 d¯ 0 dµ When v(:) is separable in q A ;e(D), the participation level in the consumer mar- ket alters not the quantity q A (®) purchased directly, but the return v(®;q A ;e) for any level of q A . Therefore a marginal change in D a®ects the willingness to pay v(:;:;:) but not the cost c(q A (®)) directly. In addition, there is no connection be- tween the participation threshold ® 0 and the externality e(D). As a result µ cannot a®ect the advertisers' participation level. Rather, the optimal schedule is distorted away from the second-best µ SB to increase the inframarginal advertisers' valuation. The platform uses quality to control (here, expand) consumer coverage and extract more surplus from all participating advertisers. The optimality condition exactly balances the foregone pro¯t from the marginal consumer with the marginal surplus thatcan be extracted from all advertiserswhen one moreconsumer participates. In both cases a distortion at the top obtains, which stands in contrast to the standard literature. The reason is that the monopolist wants to expand consumer coverage (¯ e 0 ; ^ ¯ e 0 < ¯ 0 ), but this is costly (¯ 0 > ¯). In response it decreases quality for all consumers who participate. Proposition 7 suggests there is no observational equiv- alence between the two forms of distortion. Although in both cases the threshold 31 ¯ 0 is lowered, the implications are not the same on the advertising side. In partic- ular, the optimal schedule q A (®) is di®erent in the two cases, and the participation threshold may be as well. Directly from the characterization result, Corollary 3 µ e ·(>)µ SB ,e 0 ¸(<)0 and 1¡F(¯ e 0 )¸(<)1¡F(¯ 0 ),e 0 ¸(<)0 and ^ µ e ·(>)µ SB ,e 0 ¸(<)0 and 1¡F( ^ ¯ e 0 )¸(<)1¡F(¯ 0 ),e 0 ¸(<)0 Proof: Immediately from the FOC (24),(27) and the shutdown condition (16). That is, the direction of the distortion (on the consumer side) depends on the directionoftheexternality,whichisquiteintuitive. Inamorecomprehensivemodel, thissuggeststhatifexpandingconsumercoveragewereharmfultoadvertisers{say, because it dilutes the brand { the platform would increase quality and price on the consumer side to curtail consumer demand. 1.6 Conclusion This paper studies a monopoly problem with cross-market externality. The lan- guage of reference is that of the media industry, thus we speak of consumers and advertisers. As in much of the literature interested in complementarities { included that on two-sided markets { it highlights a price distortion on the part of the plat- form. Our innovation is two-fold: we allow for quality and extend the standard model to a non-linear pricing problem. It is established that quality is generically distorted and we provide a su±cient condition for this result in a simpli¯ed (linear pricing) case. Under non-linear pricing, the exact form and motivation for the qual- ity distortion depends on the underlying technology. We show that said distortion 32 may arise because it helps capturing advertisers at the margin only, or it may be motivated by an increase in valuation on the part of inframarginal advertisers as well. The mechanism remains very simple thanks to complete information (across markets) and because the advertisers' purchasing decision is orthogonal to that of the consumers. As a result their respective reporting strategies remain independent and only total consumer coverage matters to advertiser (not who is reached). A natural extension would consist in relaxing this assumption. 33 2 Chapter 2: Media competition: a two-sided duopoly with costly di®erentiation 2.1 Introduction \The only thing advertisers care about is circulation, circulation, circu- lation." Edward J. Atorino, analyst Fulcrum Global Partners, New York June 17, 2004 (The Boston Globe). The standard modus operandi for the media requires of them to satisfy two con- stituencies: consumers on one side and advertisers on the other. Typically adver- tisers also prefer reaching as large an audience as possible, but direct transfers to consumers are e®ectively impossible to implement. Unlike a more classical multi- product ¯rm problem where consumers internalize the bene¯ts of each product in their consumption decision of the other one(s), here each side fails to so. That is, therearecomplementaritiesthatpartiescannottakeadvantageof. Fromthissimple observation they can be construed as platforms competing in a two-sided market. But unlike e-Bay, say, whose only purpose is to facilitate transactions between buy- ers and sellers, a medium also provides an information (or entertainment) good to attract consumers. In this paper we develop a model of media competition in which a) platforms select the quality of the consumer good, where quality is costly; and b) competition cannot be reduced to the sole problem of attracting consumers. That is, media compete non-trivially in both the consumer and the advertising market. We compute the unique pure-strategy equilibrium of this game for a restricted set of parameters. Quite naturally it is asymmetric { like Shaked and Suton (1982). More generally a mixed-strategy equilibrium always exists and the distributions over the (subset of) actions are characterized and are symmetric. Beyond the posi- tive analysis, consumer prices are distorted downward { as in any two-sided market 34 problem { in order to take advantage of the complementarity they exert on adver- tising demand. The main result is that, when a pure-strategy equilibrium exists, the optimal quality level of the top ¯rm is lower than in the corresponding Shaked and Sutton (1982) benchmark. Pro¯t maximization alone is su±cient for this phe- nomenon to arise; that is, interference is not necessary. Quality and advertising become substitutes in the platforms' problem. Indeed, without advertising revenue, a high quality is a means of extracting consumer surplus, at the cost of giving away market share to the competition. With ancillary revenue, every consumer becomes more valuable because the platform can extract surplus from advertisers, hence the substitution phenomenon. It is further established that quality is declining in the magnitude of the advertising revenue. As the value of advertising increases, the discount o®ered to consumers deepens. Therefore the quality level required to in- duce the marginal consumer to purchase from the high-quality platform decreases. Beyond a well-de¯ned threshold, the quality-adjusted price of the top ¯rm is so low that the consumer market is preempted. However this fails to be an equilibrium as theexcluded¯rmsimplyhastoo®eramarginallyhigherqualityleveltomonopolize the market. Then platforms play in mixed strategies. Ex post either both media operate or both markets are monopolized (by the same ¯rm), depending on their exact play. This latter phenomenon arises not because there is too little surplus to extract but thanks to a large advertising market, which induces more competition. More formally, the model calls on Gabszewicz and Thisse's (1979) (also Shaked andSutton's(1982))verticaldi®erentiationconstructontheconsumerside,however adding a convex, sunk cost of quality. Advertisers are not strategic: their payo®s do no depend on what other advertisers do, but they do derive some idiosyncratic bene¯t from advertising. This ¯ts a framework of informative advertising, or one where commodity producers do not compete in this dimension, as in Anderson and Coate (2005). 4 Because advertisers care only about the market coverage of 4 In fact competition between producers is entirely side-stepped. 35 each medium { and not to whom they advertise, vertical di®erentiation emerges endogenouslyonthatside. Thegameisplayedinthreestages: ¯rstmediaplatforms simultaneously select their quality level. In the second stage, having observed each other's quality, they set consumer prices. Last, knowing the consumer market's con¯guration (their marketshare) they choose advertising prices. Consumers and advertisers purchase at most one unit of the good of interest { this is called the single-homing assumption. This importantdetailwill bediscussedatgreaterlength later; in particular it implies that price competition is still vivid in the advertising market, unlike in Gabszewicz, Laussel and Sonnac (2001). The media sector carries a signi¯cant weight in industrialized countries, with worldwide advertising expenditure estimated at US$ 600 billion in 2006 { approx- imately half of which were incurred in the United States. Print media is still a healthy subset of it wit $60Bn in advertising revenue in the US, a ¯gure that is comparable to that of broadcast television. The industry tends to be concentrated with few large companies such as Time Warner (with $44.2 Bn in 2006 revenue) Disney (34.3 Bn, 2006) or Tribune company ($5.5 Bn 2006 revenue) and Google ($12 Bn, 2007). The signi¯cance of media extends beyond easily measurable gains from trade: agents also have to invest leisure time in its consumption. On average an American consumers spends 4 hours a day watching TV, 40 minutes a day read- ing print media and, for those participating, 60 minutes a day on the internet. Yet unlike other large sectors of the economy (such as automobile) or industries fraught with externalities (airlines or telecommunications), it has not been the object of much academic research. According to Simon Wilkie, former chief economist of the the FCC, regulators su®er from this lack of interest on the part of economists. And inspiteofthislackofinformationthissectoroftheeconomyisoneofthemostregu- lated one: ownership, access (cable, internet interconnection), reach (broadcasting) and even content are the object of edicts of the FCC. 36 While this paper does not claim to be a policy prescription, it may inform policy makers and actors in the industry. In considering applications, the results seem to matchtheobservationthatinmostUScitieseitherasinglenewspapersurvives,ora verylargeonedominatesa(orafringeof)smalloutlet(s). 5 Thislone(ordominant) newspaper is nonetheless quite inexpensive. Although the static model we present cannot capture the dynamic evolution of print media competition, its predictions correspond to these observations. Two scenarii may be drawn. In the ¯rst one, a single newspaper remains at the end of a competitive process that eventually drives out other players. Alternatively, an exogenous shock a®ecting advertising pro¯ts disrupts the pure strategy equilibrium and leads to exit, as observed in the recent past across the US. Thus some players may be driven out not because of an exogenous market contraction, but because of a(n) (advertising) market expansion. The model analyzed herein is too stark to claim being a faithful description of the media industry, however it does contribute to its study by departing from much of the literature in three important ways. First quality is costly, which ¯ts much of theindustry: better,moreaccurateinformationrequiresmoreinvestmenttoretrieve andverifyit,andbettershowsdocostmoretoproduce. Second,itignoreswhatever disutility consumers may su®er from advertising. This choice can be debated, how- everwhatshouldnotbeisthata(commonlyused)convexdisutilityfunction(±(q A )) reduces the value of the marginal advertiser from the perspective of the platform. 6 In other words, it modi¯es the rate of substitution between surplus extraction from consumersand fromadvertisers. Introducingsuchdisutilitywouldextend therange of parameters on which the pure-strategy equilibrium can be sustained, as it re- duces the value of advertising to the platform. It otherwise does not modify the results qualitatively. Last, the model presumes of unit demand, and therefore of 5 Source: Poynter Online, circulation rankings as of November 6, 2002. Only New York City has more than one signi¯cant daily paper. 6 Inparticular,suchadisutilityfunctionmaynotbeconsistentwiththeframeworkofinformative advertising, in which the latter is necessary for consumer to discover goods. 37 single-homing on both sides, which can be rationalized through a budget constraint imposedonconsumersandaliquidityconstraintonadvertisers. Thisconstrainthas non-trivial implications. It de¯nes a proper subgame in the last stage, which would not arise in its absence; that is, there is meaningful price competition between plat- forms in the advertising market. In turn this hardens competition for consumers, in that a marginal increase in consumer coverage yields di®erent marginal bene¯ts in the advertising market, depending on the ranking of the platform in the consumer market. It also renders the pro¯t function bi-modal (hence not quasi-concave) at the consumer pricing stage, with associated equilibrium existence concerns. These features are generated not by the single-homing assumption but by the fact that platforms compete in both markets. They would also arise if advertisers could mul- tihome but made subject to a liquidity constraint. In that case platforms would competeforthemarginalunit, hencethemodelcanbedirectlycastintermsofunit demand. 2.2 Literature Rochet and Tirole (2002, 2003), Armstrong (2006) and Caillaud and Jullien (2003) are the seminal references when it comes to studying two-sided markets. In this paper, consumers' utility is not directly dependent on the number of advertisers. It may be a®ected indirectly by the price system. That is, the externality is one-sided only. TheworksclosesttooursarethoseofGabszewicz, LausselandSonnac(2001), hereafterGLS,Ferrando,Gabszewicz,LausselandSonnac(FGLS2003),Gabszewicz and Wauthy (2002) and Anderson and Coate (2005). GLS (2001) characterise pure- strategy equilibria in a Hotelling model with multi-homing on the advertising side and endogenous locations. In contrast, FGLS (2003) take the locations as ¯xed. Gabszewicz and Wauthy (2002) do consider endogenous costless quality, however in a rational expectation model with simultaneous price-setting in the consumer and advertising market and with the option of multi-homing. Anderson and Coate 38 (2005) conduct a welfare analysis of the broadcasting market; advertising may be underprovided, depending on its nuisance cost and its expected bene¯t to advertis- ers. Thereisnodirectcompetitionbetweenbroadcastersfortheadvertisersbusiness. OurresultsalsocontrastthesimplemodeldevelopedbyThorsonet al (2003),which she summarizes as Newsroom investment!Quality!Circulation!Revenue and according to which a quality investment uniformly improves revenue. Ellman and Germano (2004) disregard editorial independence and show that competing newspapersmayoptimallyselectwhattoreportinordertoalleviateadvertisers'dis- comfort with the news content. 7 Along the same vein, StrÄ omberg (2004) shows that pro¯t-maximizing media report information relevant to larger audiences, thereby providing incentives to political competitors to distort their messages to please this subset of the population. Crampes, Harichabalet and Jullien (2005) analyse the problem of entry in the media market using a model where platforms derive rev- enues from both consumers and advertising. This work is also related to an older strand of the industrial organization liter- ature. Building on the work of Gabszewicz and Thisse (1979), Shaked and Sutton (1982)showthatwhen¯rmscompeteinaverticaldi®erentiationmodel,theirpro¯ts, prices and market shares are ranked according to their quality choices. The equi- librium is unique and duopolists exhibit maximum di®erentiation to soften price competition. This model is slightly modi¯ed to re°ect ours, and used as a bench- mark when comparing quality levels. All proofs are collected in the Appendix, which also contains the derivations of the benchmark we contrast our results against and a supplementary section on symmetric equilibria. 7 The recent feud between the LA Times and Chrysler does lend some credence to this thesis. 39 2.3 Introducing the model There are two platforms, identi¯ed with the subscripts 1 and 2, and a continuum of consumer of mass 1 with private valuation b for information (in the common- language understanding of the word). The bene¯t b is distributed on an interval £ ¯;¯ ¤ following the function F (¯) and everywhere positive density f(¯). All con- sumers value quality in the sense of vertical di®erentiation { there is no ambiguity forconsumersastowhatqualityis. Letµ2£= £ µ;µ ¤ denotethequalityparameter of each good. For simplicity, it is assumed that one must purchase the medium to consume it: there is no free viewing. A consumer's net utility function is expressed as u(b;µ i ;p R i )=µ i b¡p R i ; i=1;2 when facing a price p R i , where the superscript R stands for `reader'. Let p R = ¡ p R 1 ;p R 2 ¢ ;µ = (µ 1 ;µ 2 ), and consumers buy at most one medium. Suppose further µ 1 >µ 2 without loss of generality. This induces the measure D R 1 ¡ p R ;µ ¢ ´Pr ¡ µ 1 ¯¡p R 1 ¸max © 0;µ 2 ¯¡p R 2 ª¢ (28) on the part of readers; D R 1 ¡ p R ;µ ¢ is simply the number of subscribers. Without loss of generality, consumers will purchase from provider 1 over provider 2 as long as ¯¸max n ^ ¯´ p R 1 ¡p R 2 µ 1 ¡µ 2 ; ~ ¯´ p R 1 µ 1 o and have demand D R 1 ¡ p R ;µ ¢ =min ½ 1¡F µ p R 1 µ 1 ¶ ;1¡F µ p R 1 ¡p R 2 µ 1 ¡µ 2 ¶¾ while the demand for information good 2 is determined for values of the parameter ¯2 h p R 2 µ 2 ; ^ ¯ i , or zero, and expressed as D R 2 ¡ p R ;µ ¢ =max ½ 0;F µ p R 1 ¡p R 2 µ 1 ¡µ 2 ¶ ¡F µ p R 2 µ 2 ¶¾ : 40 Advertisers have a pro¯t function A(y;x) separable in x;y; x 2 f0;1g denotes advertising consumption and y is a (vector of) variable(s). These include any other action a platform may undertake, such as hiring sales personnel, R&D and so forth. Let e i denote the quality of platform i as perceived by the advertisers. This plays the same role as µ i on the consumer side, but pertains to the advertisers decision. For any ^ y, they may choose to purchase one unit of space at most at price p A i if e i (A(^ y;1)¡A(^ y;0))¡p A i =e i a¡p A i ¸0; i=1;2 that is, they derive an increase in (expected) pro¯t a. In practice, while producers routinely place their messages on di®erent media, it is also true that they are cash- constrained { say because of risk aversion or credit markets imperfections. The one unit limit can be interpreted as a tight liquidity constraint. Announcers may value the bene¯t from advertising di®erently according to the parameter a, which is considered private and distributed following G(®) on [®;®] with mass 1. It is quite natural to let e i ´ e ¡ D R i ¢ , with @e(D R i ) @D R i > 0 and @ 2 e(D R i ) (@D R i ) 2 · 0, and of course, e ¡ D R i ¢ = 0 for D R i = 0: the more consumers the advertiser can reach, the more they value an ad, but this bene¯t is (weakly) concave. For example, following Shaked and Sutton (1982), if one thinks of the parameter taste b as disposable income, thevalueofthemarginalconsumertoadvertisersisclearlydecreasing. The distribution G(®) is restricted to having a monotonically increasing hazard rate. Like consumers, potential advertisers act as price takers and there is no strategic interactionbetweenthem,norbetweenadvertisersandplatforms. Therankingofthe platforms'marketsharesintheconsumermarketde¯nestheirrelativequalityinthe advertisingmarket. Givenpricesp A = ¡ p A 1 ;p A 2 ¢ , aproducerpurchasesfromchannel 1 over channel 2 (without loss of generality), only if e 1 ®¡p A 1 ¸max © 0;e 2 ®¡p A 2 ª . 41 This decision rule generates the measure Pr ¡ e 1 ®¡p A 1 ¸max © 0;e 2 ®¡p A 2 ª¢ ´q A 1 ¡ p A ;e ¢ where e=(e 1 ;e 2 ), whence we derive demands q A 1 ¡ p A ;e ¢ =min ½ 1¡G µ p A 1 e 1 ¶ ;1¡G µ p A 1 ¡p A 2 e 1 ¡e 2 ¶¾ and q A 2 ¡ p A ;e ¢ =max ½ 0;G µ p A 1 ¡p A 2 e 1 ¡e 2 ¶ ¡G µ p A 2 e 2 ¶¾ We assume neither constraint on advertising space (the medium can always print one more page, for example), nor that advertising a®ects readership. 8 The cost of running adverts is set at zero. Quality however is costly to provide and is modeled as an investment according to kµ 2 i . Platforms(say,magazineortelevisionchannels)¯rstchooseaqualitylevel. Given this quality, set prices ¯rst to consumers and, in a third stage, to advertisers. Upon observing these prices, they choose whether to purchase. The three-stage game is denoted ¡. A platform collect revenues from both readers and advertisers, with monies from either side perfectly substitutable. Ceteris paribus, a provider simply cares about total revenue. For any medium i=1;2, the objective function takes the form ¦ i =D R i ¡ p R ;µ ¢ p R i ¡kµ 2 i +q A i ¡ p A ;e ¢ p A i (29) 2.4 Equilibrium characterisation Section2.7.4oftheAppendixpresentsananalysisofsymmetricequilibriaingeneral form, which arise when platforms select symmetric quality (either because they are 8 Although we duly note restrictions on advertising time for free-to-air television, as well as an obvious distaste for advertising on TV by viewers, for example. 42 constrained by design, or as an equilibrium outcome), or when consumer prices are ¯xed. While the latter situation occurs quite naturally, say by mandate or simply because consumption cannot be monitored (as in the case of broadcasting), there is noreasonforplatformstolimitthemselvestosymmetricquality. Movingawayfrom symmetrysoftenspricecompetitionandleadstopositivepro¯ts(ShakedandSutton (1982,1983),Tirole(1988)). Inlinewiththeliterature,weimposesomestructureby assuming a uniform distribution on the bounded supports of the private parameters ¯and®intheconsumermarketandtheadvertisingmarket,respectively. Toprevent exogenous market preemption (in the consumer market) we impose Assumption 10 ¯¡2¯ >0 This rules out the trivial case in which the low-quality platform necessarily faces zero demand in the price game. Let e ¡ D R i ¢ = e£D R i for (non-trivial) simplicity, whichletsusinterpreteasascalingparameter. Fortheanalysistoremaintractable werestricttherangeofequilibriumoutcomestothosewerefull(advertising)market coverage arises, that is Assumption 11 e·1 and ® ® 2 µ 2; 2D R i ¡D R j e(D R i ¡D R j ) ¶ whereD R i >D R j andthedemandsareevaluatedatequilibrium. Thisisnotwithout loss of generality, as shown by Wauthy (1996): coverage is an equilibrium outcome. Wauthy (1996) shows it is optimal only in this range of parameter values. The condition on e guarantees the interval µ 2; 2D R i ¡D R j e(D R i ¡D R j ) ¶ to be non-trivial and well de- ¯ned. AsAssumption10,italsopreventsexogenouspreemption. Furtherdiscussion is postponed until after the analysis. Also, the parameter k needs to be su±ciently large for the cost function to have some bite; speci¯cally, Assumption 12 k > (2¯¡¯) 2 18µ 43 to guarantee an interior solution in the benchmark case. 9 Taken together, Assump- tions10and12guaranteethattheconsumermarketiscoveredinequilibrium,which greatly simpli¯es the analysis. To see why, suppose there is no externality and de- note the equilibrium quality levels are given by µ 0 1 = 1 2k µ 2¯¡¯ 3 ¶ 2 > µ 0 2 = µ. 10 The condition for a covered market is ¯¡2¯ 3 (µ 0 1 ¡ µ 0 2 ) · ¯µ 0 2 (see for example, Tirole, (1988)). Substitutingforthevaluesofµ 0 1 ;µ 0 2 andre-arranging, themarketiscovered for k ¸ 1 2µ µ 2¯¡¯ 3 ¶ 2 µ ¯¡2¯ ¯¡¯ ¶ , which is necessarily satis¯ed by Assumption 12. It follows that both ¯rms operate and the relevant demand functions in the consumer market are the competitive ones. It will be obvious that it is satis¯ed in an equilib- riumofthisgame. Weproceedintwosteps, ¯rstfocusingonagents'behaviorinthe advertising market, which is not directly a®ected by quality choices. Let ^ ®= ^ p A 1 ¡^ p A 2 ¢e and e¢D R = e(D R 1 ¡ D R 2 ) denote the di®erence in the platforms' quality. The following lemma re°ects these speci¯cations. Lemma 6 Suppose D R 1 ¸ D R 2 w.l.o.g. There may be three pure strategy equilibria in the advertising market. When D R 1 >D R 2 >0, the pro¯t functions write: ¦ A 1 =e¢D R µ 2®¡® 3 ¶ 2 ; ¦ A 2 =e¢D R 2 µ ®¡2® 3 ¶ 2 When D R 1 >D R 2 =0, platform 1 is a monopolist its pro¯ts are: ¦ AM 1 =eD R 1 µ ® 2 ¶ 2 For D R 1 = D R 2 , the Bertrand outcome prevails and platforms have zero advertising pro¯ts. Up to the endogenous quality in the advertising market, this is exactly the result of the classical analysis of vertical di®erentiation. Note that an equilibrium resting on 9 SaidbenchmarkisdevelopedinSection2.5. Intheabsenceofasu±cientlylargeparameterkthe Shaked and Sutton boundary result prevails, which renders across-model comparisons meaningless. 10 Refer section 2.5 for details. 44 D R i >D R j =0mayariseasanon-trivialequilibriumintheadvertisingmarket: take for example µ i =µ j and p R i <p R j , which may, or may not, be an out-of-equilibrium outcome in the consumer market. Proof:Theproofisstandardandthereforeomitted(itfollowstheproofofLemma15, Section 2.7.4). The low-quality ¯rm's survival requires su±cient heterogeneity among buyers, which is imposed by Assumption 11. Thanks to Assumption 10, consumer demand for the information good simply writes D R i =¯¡ p R i ¡p R j ¢µ ; D R j = p R i ¡p R j ¢µ ¡¯ for µ i > µ j and with ¢µ = µ i ¡µ j . When choosing its strategy in the consumer market, each ¯rm knows what to expect in the announcers' market: the platform thatholdsthelargermarketshareintheconsumermarketwillenjoyhighadvertising pro¯ts and conversely. Following Lemma 6 the pro¯t function (29) rewrites as ¦ i =p R i D R i (p R ;µ)¡k(µ i )+¦ A i ¡ e¢D R (p R ;µ) ¢ (30) on the equilibrium path. It is useful to bear in mind that when ¯rm 1 is dominant in the consumer market, ¢D R (p R ;µ) = D R 1 (p R ;µ)¡D R 2 (p R ;µ), and conversely if ¯rm 2 dominates. 2.4.1 Consumer price subgame From Lemma 6 three distinct con¯gurations may arise on the equilibrium path. In the ¯rst case, platform i dominates the consumer market, in the second one both share the consumer market equally and in the last one it is dominated by ¯rm j. 45 The pro¯t function (30) of each ¯rm i=1;2 writes ¦ i =p R i D R i (p R ;µ)¡kµ 2 i + 8 > > > > < > > > > : ¦ A i ; if D R i >D R j ; 0; if D R i =D R j ; ¦ A i ; if D R i <D R j . (31) Thisfunctioniscontinuous,butnotquasi-concaveandhasakinkatthepro¯le~ p R of consumerpricessuchthatD R i =D R ¡i . ThustheconditionsofTheorem2ofDasgupta and Maskin's paper (1986a) are not met. Their Theorem 5 is of limited use as it pertains to mixed strategies. In characterizing the equilibrium of this subgame we face discontinuous best-response correspondences owing the lack of quasi-concavity, which follows from the externality generated by advertising revenue. Proceeding by construction it is nonetheless possible to show that at least one equilibrium in pure strategies always exists. Denote ¢µ = µ 1 ¡µ 2 ; A = ³ 2®¡® 3 ´ 2 and A = ³ ®¡2® 3 ´ 2 . The space P R i £P R ¡i of action pro¯les (prices) can be divided into three regions: region I such that D R i > D R ¡i , region II such that D R i < D R ¡i and region III such that D R i =D R i . We begin with De¯nition 1 For i = 1;2, the platforms' `quasi-best responses' are de¯ned as the solution to the problem max p R i ¦ i ³ p R i ;D R i (p R ;µ);¦ A i (D R i ;D R j ) ´ , where the pro¯t function is de¯ned by (31). Therefore, letting µ 1 >µ 2 w.l.o.g, p R 1 ¡ p R 2 ¢ = 8 > > > > < > > > > : p R 1 ¡ p R 2 ¢ = 1 2 ¡ p R 2 +¢µ¯¡2eA ¢ ; if D R 1 >D R 2 ; 1 2 ¡ p R 2 +¢µ¯ ¢ ; if D R 1 =D R 2 ; p R 1 ¡ p R 2 ¢ = 1 2 ¡ p R 2 +¢µ¯+2eA ¢ ; if D R 1 <D R 2 ; and p R 2 ¡ p R 1 ¢ = 8 > > > > < > > > > : p R 2 ¡ p R 1 ¢ = 1 2 ¡ p R 2 ¡¢µ¯¡2eA ¢ ; if D R 1 <D R 2 ; 1 2 ¡ p R 1 ¡¢µ¯ ¢ ; if D R 1 =D R 2 ; p R 2 ¡ p R 1 ¢ = 1 2 ¡ p R 2 ¡¢µ¯+2eA ¢ ; if D R 1 >D R 2 ; 46 While it is always possible to ¯nd some point where `quasi-best responses' intersect (e.g. such that both play as if D R 1 < D R 2 ), it by no means necessarily de¯nes an equilibrium. Doing so assumes that in some sense platforms coordinate on a particular market con¯guration, say, such that D R 1 < D R 2 , which may not been immune from unilateral deviation. After all this is a non-cooperative game. To ¯nd the equilibrium, if it exists, we ¯rst need to pin down the ¯rms' true best replies. Lemma 7 Let µ 1 >µ 2 w.l.o.g. There exists a pair of actions (^ p 1 ;^ p 2 ) such that the best response correspondences are de¯ned as p R 1 ¡ p R 2 ¢ = 8 > < > : p R 1 ¡ p R 2 ¢ ; for p 2 ¸ ^ p 2 ; p R 1 ¡ p R 2 ¢ ; for p 2 < ^ p 2 ; (32) and p R 2 ¡ p R 1 ¢ = 8 > < > : p R 2 ¡ p R 1 ¢ ; for p 1 < ^ p 1 ; p R 2 ¡ p R 1 ¢ ; for p 1 ¸ ^ p 1 ; (33) Lemma 7 thus de¯nes the `true' best-response of each player. It says that platform 1, for example, prefers responding with p R 1 ¡ p R 2 ¢ for any prices p 2 ¸ ^ p 2 and switches to p R 1 ¡ p R 2 ¢ otherwise. Recall that consumer pro¯ts and advertising pro¯ts are sub- stitutes for the platforms. The best reply correspondence is discontinuous at that pointwhereplatformsareindi®erentbetweenbeingthedominantplatformandnot, thatis, betweenthecombinationofprices ³ p R i (p R j );p A i (p R i ) ´ and ³ p R i (p R j );p A i (p R i ) ´ . Anequilibriumhasa°avorofrationalexpectations, inthatplatformsmustselect actions that are consistent with each other (for example, both must play as if D R 1 > D R 2 ), as well as compatible with an equilibrium. Such a rationality requirement however is not necessary, as will soon be obvious; instead we let media play a standardNashequilibrium. Hencebyde¯nition, foreach¯rm, itsactionmustbean element of the best reply correspondence and these correspondences must intersect. We de¯ne a condition that captures both these features, and will show next that 47 it is both necessary and su±cient for an equilibrium to exist. From the `quasi-best responses', an equilibrium candidate is a pair of prices such that ¡ p ¤R 1 ;p ¤R 2 ¢ = 8 > < > : p R 1 ¡ p R 2 ¢ \p R 2 ¡ p R 1 ¢ ; if D R 1 >D R 2 or; p R 1 ¡ p R 2 ¢ \p R 2 ¡ p R 1 ¢ ; if D R 1 <D R 2 ; These actions may form an equilibrium only if their intersections are non-empty. Together, the de¯nitions of a best-response pro¯le (relations (32) and (33)) and of an equilibrium candidate sum to Condition 1 Either ^ p R 1 ¸p ¤R 1 and ^ p R 2 ·p ¤R 2 or ^ p R 1 ·p ¤R 1 and ^ p R 2 ¸p ¤R 2 or both. Consider an action pro¯le p ¤R satisfying this condition; from Lemma 7 each p ¤R i is an element of i's best response. Now, for it to be an equilibrium, players must choose reaction functions that intersect. This is exactly what Condition 1 requires. For example, the ¯rst pair of inequalities tells us that player 1's optimal action has to be low enough and simultaneously that of 2 must be high enough. When they hold, player2'sreactioncorrespondenceisnecessarilycontinuousuntil1reachesthe maximizer p ¤R 1 , and similarly for 1's best reply. Then Lemma 8 Condition1isnecessaryandsu±cientforatleastoneequilibriump ¤R = ¡ p ¤R 1 ;p ¤R 2 ¢ to exist. When both inequalities are satis¯ed, the game admits two equi- libria. When Condition 1 holds, the best-reply correspondences intersect in at least one subset of the action pro¯le space P R 1 £P R 2 . In this case, the Nash correspondence 48 p R 1 (p R 2 )£p R 2 (p R 1 ) has a closed graph and standard theorems apply. The potential multiplicityofequilibriaowestothediscontinuityofthebest-replycorrespondences. Let C = £ 2e ¡ A+A ¢¤ 2 = h 2e ³ ( 2®¡® 3 ) 2 +( ®¡2® 3 ) 2 ´i 2 . Condition 1 provides us with a pair of easy-to-verify conditions in terms of prices. Thus we can establish Lemma 9 (Existence) An equilibrium in pure strategies of the consumer price subgame always exists. It is unique and located in region I. There cannot exist a pair of alternative consumer prices ¡ p ¤¤R 1 ;p ¤¤R 2 ¢ compatible with a pair of quality choices (µ ¤ 1 ;µ ¤ 2 ) solving the platforms' problem: they fail the necessary condition laid out in Lemma 8. Sketch of a proof: Candidate equilibria can be constructed from the `quasi- reaction correspondences' of De¯nition 1. By Lemma 8, it is enough to verify that these candidates satisfy Condition 1 for them to form a Nash equilibrium. One of them always does, while the other one never can. The lack of quasi-concavity of the payo® functions induces discontinuity of the best-reply correspondences. To paraphrase Dasgupta and Maskin (1986a) however, this discontinuity is essential. Given sunk quality the con¯guration hµ 1 > µ 2 ;D 1 < D 2 i corresponding to the second line of Condition 1 can be interpreted as a need for advertising pro¯ts to be large enough for a second equilibrium to exist. However it entails playing a weakly dominated strategy for player 2: if she ¯nds it attractive to reduce her price so much, so must player 1. Thus the discontinuity set is not a trivial one { it has certainly not measure zero. It follows that mixed strategies cannot restore this second candidate equilibrium (Dasgupta and Maskin (1986a), Theorem5). Furthermoreitimpliesthatweneednotcallontherationalexpectation framework. Elimination of weakly dominated strategies is su±cient to rule it out and play a less strenuous Nash equilibrium. This is depicted in Figure 1. Now we arereadytocollecttheseresultsandtocomputeequilibriumconsumerpricesinthis subgame. 49 p 1 p 1 p 2 p 2 ˆ p2 ˆ p1 p 2 p 1 Figure 1: Best reply functions and unique equilibrium Proposition 8 (Consumer prices) Let µ 1 >µ 2 w.l.o.g. There may be two possi- ble con¯gurations arising in the consumer price subgame. For each, there exists a unique Nash equilibrium in pure strategies characterized as ² For ¢µ > p C ¯¡2¯ p R¤ 1 = 1 3 £ ¢µ ¡ 2¯¡¯ ¢ +2e ¡ A¡2A ¢¤ p R¤ 2 = 1 3 £ ¢µ ¡ ¯¡2¯ ¢ +2e ¡ 2A¡A ¢¤ ² If ¢µ· p C ¯¡2¯ p R¤ 1 = ¢µ¯ 2 ¡2eA p R¤ 2 =0 Consumerpricesthusincludea`discount'asplatformsengageincross-subsidization. The lure of advertising revenue intensi¯es the competition for consumers because theybecomemorevaluablethanjustfortheirwillingnesstopayfortheinformation good. Buttheintuitivereasoningwherebythelow-quality¯rmmay¯nditpro¯table to behave very aggressively in order to access large advertising revenue does not hold true (Lemma 9). Moreover, unlike in the Shaked and Sutton (1982) model, ¯¡2¯ > 0 is not su±cient to a®ord the low-quality ¯rm some positive demand: 50 ¢µ, de¯ned in the ¯rst stage, may be too narrow to sustain two ¯rms. That is, the high-quality platform may choose to act so as to exclude ¯rm 2. 2.4.2 First-stage actions In the ¯rst stage, platforms face the pro¯t function (31), which they each maximise by choice of their quality variable µ i . That is, each of them solves Problem 2 max µ i 2[µ;µ] p R¤ i D R i ¡ µ;p R¤ ¢ +¦ A i ¡ e;¢D R (p R¤ ;µ) ¢ ¡kµ 2 i subject to ^ ¯ = p R i ¡p R j µ i ¡µ j 2 £ ¯;¯ ¤ (34) where p R¤ is the relevant equilibrium price pro¯le characterized in Proposition 8, and p R¤ i the corresponding price chosen by platform i. £ N ¡i denotes the set of best responses of player i's opponents. The second constraint does not limit quality choices per se but is a natural restriction guaranteeing that platforms' demands remains bounded by the market size. 11 It can be rearranged as a pair of inequali- ties: ¢µ ¡ 2¯¡¯ ¢ + p C ¸ 0 and ¢µ ¡ ¯¡2¯ ¢ ¡ p C ¸ 0. Only the second one is constraining. On the equilibrium path the objective function of Problem 2 reads ¦ 1 = 8 > < > : 1 9 ¡ ¢µ(2¯¡¯) 2 +B 1 + C ¢µ ¢ ¡kµ 2 1 ; if ¢µ > p C ¯¡2¯ ; 1 9 ³ ¢µ(2¯¡¯) 2 +B 1 + p C(¯¡2¯) ´ ¡kµ 2 1 ; if ¢µ· p C ¯¡2¯ (35) where B 1 =(2¯¡¯)2e ¡ 2A¡A ¢ +3e ¡ ¯+¯ ¢ A (is constant in µ). The second line of the de¯nition of ¦ 1 rules out the arti¯cial case of ¯rm 1 facing a demand larger than the whole market (pro¯ts are bounded). It is derived by taking C ¢µ as ¯xed at 11 Observe that µ i !µ j ) ^ ¯!1. 51 its lowest value, that is, where ¢µ = p C ¯¡2¯ . For platform 2, pro¯ts are ¦ 2 = 8 > > > > < > > > > : 1 9 ¡ ¢µ(¯¡2¯) 2 +B 2 + C ¢µ ¢ ¡kµ 2 2 ; if ¢µ(¯¡2¯)> p C; 0; ¢µ(¯¡2¯)· p C and µ 2 =0; ¡kµ 2 2 ; ¢µ(¯¡2¯)· p C and µ 2 >0; (36) with B 2 = (¯¡ 2¯)2e ¡ A¡2A ¢ + 3e ¡ ¯+¯ ¢ A. A major di±culty may arise in solving this problem. Both platforms' pro¯t functions are the di®erence of two convex functions, which may be concave or convex. Section 2.7.2 of the Appendix studies the pro¯t function ¦ 1 (µ 1 ;µ 2 ) in the details necessary to support our results. In particular it identi¯es a threshold C f ´ · (2¯¡¯) 2 27k ¡µ ¸ 2 µ (2¯¡¯) 2 3 ¶ such that the function remains well behaved if C does not exceed C f . Otherwise, the necessary ¯rst-order condition fails to hold entirely. For C ¸ C f , the high-quality medium wouldliketopick ~ µ(e)´µ+ p C ¯¡2¯ ,where¦ 1 (:;:)reachesismaximum. Atthatpoint its rival is excluded (¢µ is low enough), and it still extracts as much surplus from consumers as it can without losing its status as monopolist. But then ¯rm 2 can `leap'overitandbecomethemonopolistatanegligibleincrementalcost. Intuitively, when advertising returns are large enough every consumer becomes very valuable to both platforms. Consequently the pro¯t function ¦ 1 (:;:) is linearly increasing up to µ 1 = ~ µ(e). For C <C f the function ¦ 1 (:;:) remains increasing, but now concave, on the portion beyond µ 1 = ~ µ(e) as well, where it admits a maximizer. To overcome this problem, Assumption 11 is strengthened and turned into Assumption 13 e< ¹ e´min ½ 1; µ (2¯¡¯) 2 27k ¡µ ¶ ¯¡2¯ 2(A+A) ¾ . 12 in order to study the optimal action pro¯le in pure strategies. Assumption 13 guarantees that when ^ µ 1 solves the ¯rst-order condition, ( ^ µ 1 ¡µ)(¯¡2¯)> p C, so that both media operate. If this restriction is met we can claim 12 This arises from the condition ( ^ µ1¡µ)(¯¡2¯)>(µ f 1 ¡µ)(¯¡2¯)¸ p C, where µ f 1 = (2¯¡¯) 2 27k is de¯ned in Section 2.7.2. 52 Lemma 10 Let µ 1 > µ 2 w.l.o.g. and Assumption 13 hold. Optimal actions consist of µ ¤ 2 =µ and µ ¤ 1 = ^ µ 1 , where ^ µ 1 uniquely solves (2¯¡¯) 2 =18kµ 1 + C (¢µ) 2 (37) Both platforms operate. We label the term C (¢µ) 2 the `market share e®ect': it acts as an incentive to reduce qualityandissimilartothatarisinginstandardBertrandcompetition. Incondition (37), ¯rm 1 trades o® the marginal bene¯t of quality (the left-hand side) not only with its marginal investment cost but also with the marginal advertising revenue thatitmustforegobecauseofhigherconsumerpricesinducedbyhigherquality(the RHS). For a positive k the low-quality ¯rm cannot deviate by `leaping' over its rival and o®ering a slightly higher-quality good. Given that it markets a lesser good, platform 2 selects µ to mitigate the price war. This is the Di®erentiation Principle at work, but here it is subsumed by the `market share e®ect'. Collecting the results from Lemmata 6 and 10 and Proposition 8, and letting platform 1 be the high-quality medium w.l.o.g., we can ¯nally state Proposition 9 (Equilibrium characterization) Suppose Assumption 13 holds. Thegame¡admitsexactlyoneequilibriuminpurestrategiesinwhichbothplatforms operate and choose di®erent qualities. It is characterized by the triplet of pro¯les ¡ µ ¤ ;p ¤R ;p ¤A ¢ de¯ned by Lemma 10, Proposition 8 and Lemma 6, respectively. Proposition 9 can be appended with the obvious Corollary 4 If a Nash equilibrium of the game ¡ exists, platforms may also play in non-trivial mixed strategies. for which the intuition is that of the Battle of the Sexes. Further we note Remark 2 Unlike the situation where C ¸ C f , in which platforms should select symmetric qualities with probability zero, the case µ 1 = µ 2 may arise with positive 53 probability following Corollary 4. 13 Recall that when a pure-strategy equilibrium exists, ¯rms randomize over the set fµ i ;µ ¤ i g instead of a continuous support. Each event carries a mass-point. Thus Proposition 17 (in the Appendix) is relevant to the case of endogenous quality, however only when a pure-strategy equilibrium exists. That aside, when C ¸ C f (Lemma 11 in the Appendix, Section 2.7.1), it is not immediate that the game admits a mixed strategy equilibrium, for the payo® corre- spondencesarenotupper-hemicontinuousandtheirsumisnotnecessarilysoeither. Nonetheless it is possible to show that Proposition 10 When ® ® 2 µ 2; 2D R i ¡D R j e(D R i ¡D R j ) ¶ a mixed-strategy equilibrium of the game ¡ always exists. The proof and discussion of this statement can be found in the Appendix. The conditions of the Proposition guarantee that the market is covered. Let H i (µ i ) be the probability distribution over i's play, £ N i the relevant support and µ c i the upper bound of the support. Let also R i (µ i ;µ j ) denote the revenue accruing to i. We can claim Proposition 11 There exists are unique pair of symmetric distributions H 1 ;H 2 satisfying H i (µ i )R j (µ i ;µ j )+ Z µ c i ~ µ i R j (s;µ j )dH i (s)=k(µ c j ) 2 with H i (s) 8 > < > : 2(0;1); s=µ i ; =1; s=µ c i . and h(s)=0; s2 ³ µ i ; ~ µ i ´ and µ c i de¯ned in Lemma 14. 13 Refer Section 2.7.1 in the Appendix. 54 Noticeably £ N j ½£ j . Indeed it is obvious from the pro¯t function (36) that playing any µ j 2 ³ µ; ~ µ(e) ´ is strictly dominated by selecting the lower bound. Furthermore, platforms place some mass at the lower bound µ. This is because ¦ i (µ i ;µ j ) > 0 for µ j > ~ µ(e): if j plays anything in the support £ N j but µ j ; i necessarily derives positive pro¯ts. Havingcharacterizedtheequilibriaofthegame,wewanttounderstandtheimpact oftheexternalitye(:)onplayers'behaviorandonthebreakdownoftheequilibrium. This discussion is the object of the next section. 2.5 The role of externalities The results of the preceding analysis are ¯rst contrasted with earlier ones estab- lished in the literature. We then investigate the impact of advertising pro¯ts on the behavior of quality by computing some comparative statics. Last we compare the level of quality provided in our duopoly to that arising in a monopoly, with and without externality. 2.5.1 Quality distortion and advertising revenue An important goal of this paper is to understand the behavior of quality when information providers have access to a substitute source of revenue. To study this problem it is useful to ¯rst establish a benchmark. The major reference in the vertical di®erentiation literature is Shaked and Sutton's 1982 paper, where however costs are completely ignored. It is immediate to adapt their model, and easy to show below that this second source of revenue acts as a substitute for quality for the platforms. Indeed Proposition 12 (Quality distortion) Inanypure-strategyequilibriumofthegame ¡, quality is lower than it would be absent advertising. Proof: Section 2.7.3 presents both the benchmark and the Proof. 55 Advertising income puts emphasis back on market share { while the introduction of di®erentiation had the opposite e®ect. This leads to more intense price compe- tition for consumers, and the discount extended by the platforms increases in the advertising pro¯ts (as shown in Propositions 8 and 13). Lower consumer prices uniformly relax the need to provide costly quality: at lower prices, the marginal consumer demands a lesser product to make a purchase. An intuition for the exact trade-o®canbesketchedasfollows: givenanyquality,inthesecondstage¯rmsmust o®eradiscounttoconsumers{qualityissunkbythen. Theextentofthatdiscount, given¯xedquality,isdeterminedbypro¯tstobecollectedontheadvertisingmarket. In the quality-setting stage, the high-quality ¯rm can further increase this discount by lowering quality: its consumer price is p ¤ 1 = 1 3 £ ¢µ(2¯¡¯)+2e(A¡2A) ¤ , while that of its rival is p ¤ 2 = 1 3 £ ¢µ(¯¡2¯)+2e(2A¡A) ¤ . Taking µ 2 ¯xed, for each dµ 1 ¯rm 1 can decrease its consumer price by more than ¯rm 2 can. If quality were interpreted as accuracy of reporting, this model implies that it is optimal for newsmedia to under-invest in said accuracy. Thus a contraction of the quality spread need not result from exogenous constraints, nor from outsiders' intervention (interference), but from players' pro¯t-maximizing behavior. This phe- nomenonresemblesthatobservedinotherindustriessuchassoftwareorgamedevel- opment: a widely used operating system need not provide the same intrinsic quality as a more marginal one because it supports so many applications. Proposition12suggeststhat¯rms'behaviordependsveryessentiallyontheexter- nality e(:). Things break down when C(e), the sum of marginal advertising pro¯ts per unit of quality, becomes too large { recall the condition (µ ¤ 1 ¡µ)(¯¡2¯)> p C. Using the de¯nitions of µ 0 1 , µ f 1 and C f , we can rearrange µ f 1 = 2 3 µ 0 1 . Following Assumption 13, there exists some threshold e < ¹ e and µ ¤ 1 2 ¡ 2 3 µ 0 1 ;µ 0 1 ¢ . Thus with advertisingexternalitiesapurestrategyequilibriumrequiresmoreconsumerhetero- geneity (a larger di®erence ¯¡¯) to exist than in the classical Shaked and Sutton (1982) case. In summary, 56 Corollary 5 (Taxonomy) Let µ 1 >µ 2 w.l.o.g. and ® ® 2 µ 2; 2D R i ¡D R j e(D R i ¡D R j ) ¶ , For e=0; k >0 The equilibrium is that of Problem 3 (adapted from Shaked and Sutton (1982)) with both ¯rms operating; For e>0; k =0 Maximum di®erentiation obtains with both ¯rms having positive demand; For ¹ e>e>0; k >0 The equilibrium is characterised by Proposition 9; For e> ¹ e; k >0 Proposition 11 applies. Note that the case e > 0;k = 0 yields the same di®erentiation result as Shaked and Sutton (1982) and Gabszewicz, Laussel and Sonnac (2001 and 2002). A pos- itive externality, single-homing and costly quality are necessary to Proposition 12 and its Corollary5. Intuitively, with free quality, why choose anything but the one that allows the largest surplus extraction from consumers? When the externality is powerful (e> ¹ e)no equilibrium (in pure strategies) exists. 2.5.2 The behavior of quality Since e(D R i ) = e£ D R i we continue to parametrize the value of the advertising market for platforms by e, as in Assumption 13. Thus the magnitude of the term C ´ (2e(A + A)) 2 is governed by e only, which can be construed as a measure of market size for the advertisers, for example. When a pure strategy equilibrium exists, we have Proposition 13 (Comparative statics) Let µ 1 > µ 2 w.l.o.g. At an equilibrium ¡ µ ¤ ;p ¤R ;p ¤A ¢ a. dµ 2 de =0, but dµ 1 de <0 and d 2 µ 1 de 2 <0 b. dp A 1 de > dp A 2 de >0 and dp R 1 de < dp R 2 de <0 57 c. dD R 1 de =¡ dD R 2 de >0 d. d¦ 1 de >0 and d 2 ¦ 1 de 2 <0 The e®ect of e on ¦ 2 is ambiguous. Proof: The proof is somewhat lengthy, but straightforward and relegated to the Appendix, Section 2.7.1. It is necessary to ¯rst characterize the behavior of µ 1 with respect to the parameter e, on which all other results depends. The presence of a second source of revenue not only depresses the quality of the consumer good, it does increasingly so as the advertising market becomes more valuable. Price competition is correspondingly more intense in the consumer mar- ket, but less in the advertising market { where di®erentiation is endogenous. As advertising revenues' weight increases, every consumer becomes more valuable to both platforms, so the consumer discount deepens { and does so faster for the high- quality ¯rm. It ¯nds it easier to enlarge its market share and therefore to become an increasingly better platform for advertisers. 2.5.3 Properties of the mixed-strategy equilibrium Although the distributions H 1 ;H 2 do not lend themselves to easy interpretation, more can be said about the nature of the equilibrium. Here we claim Proposition 14 Suppose e> ¹ e. When no platform plays at the lower bound µ, the market is necessarily monopolized ex post. Otherwise both operate. Thus the dominated ¯rm loses its investment kµ 2 i . The reason is that the length of theintervalµ c ¡ ~ µisnotsu±cienttoaccommodatetwo¯rms(because¢µ < 2e(A+A) ¯¡2¯ ). That is, if not trying to be the high-quality medium, the platform should seek maximal di®erentiation. Of course, whether it obtains depends on its opponent's exact play. 58 Interestingly either monopolization or the competitive situation may be an ex post outcome, which ¯ts industry patterns. 14 In addition ex post pro¯ts in mixed- strategycasearenotmonotonicallyranked: considertheplayhµ 1 ;µ c 2 i,whichimplies ¦ 1 >¦ 2 =0althoughµ 2 >µ 1 . Thisresultalsocomparesfavorablytotheindustry's idiosyncrasies, where the higher-quality publications or shows do not necessarily yield higher pro¯ts. 2.5.4 Gabszewicz et al. (2001 and 2002) Our results stand in contrast to those obtained by these authors. In their horizon- tal di®erentiation setup, Gabszewicz et al. (2001) ¯nd (mutually exclusive) pure strategy equilibria in which platforms either play at the extrema or converge to the center. Identical qualities in the present model leads to a mixed-strategy equilib- rium, with one ¯rm exiting ex post (by Proposition 17 in the Appendix). A pure strategy equilibrium always exists in their setup, thanks to a well-behaved pro¯t function. This rests on both costless quality and `multi-homing'. The latter results in no price competition in the advertising market, hence each platform acts as a monopolist on its audience { it is a bottleneck to advertisers. Technically no proper subgame is de¯ned at the advertising pricing stage, so the payo® function remains quasi-concave (advertising pro¯ts are simply increasing in consumer market share). Therefore the best-response correspondence in the consumer pricing subgame re- mainscontinuous. Incontrast, `single-homing'enhancescompetitionforconsumers: there is a higher premium to being the dominant platform. Not only does one ad- ditional consumer bring more revenue from both sides, it does so at a higher rate for all infra-marginal consumers as well. Had we allowed for `multi-homing' in this game, the unique equilibrium would also be symmetric. Costly quality necessarily leads to an interior solution and introduces a smooth trade-o® between surplus ex- traction from consumers and extraction from advertisers. For high values of e, this 14 Only New York City and Los Angeles have more than one signi¯cant newspaper, for example. 59 trade-o® is extreme, and combined with the asymmetric nature of the equilibrium, leadstothelow-quality¯rmbeingexcluded. Intheir2002papertheauthorsimpose single-homing in a similar model, however keeping di®erentiation ¯xed. Unlike in ourconstruct,multiple(asymmetric)equilibriamayariseintheirs. Whenplatforms aresymmetric at theprice-setting stage, playingeither of p R 2 ¡ p R 1 ¢ orp R 2 ¡ p R 1 ¢ isper- fectly reasonable and does not involve dominated strategies, unlike in the present paper. 2.5.5 Comparison to the monopoly case A question of interest is whether competition fosters the provision of quality in this economy. In the externality-free environment, the answer to this question is unam- biguouslypositive: withauniformdistributionitisstraightforwardtoshowthatthe monopolist selects price p M = µ¯ 2 and quality µ M = 1 2k ³ ¯ 2 ´ 2 , and it is immediate to verify that µ M <µ 0 1 when Assumption 10 holds. Given that quality µ ¤ 1 decreases in equilibrium when the externality e is introduced, it is not a priori obvious that this statement remains true in our duopoly. When the medium is a monopolist the pro¯t function for the advertising market writes ¦ A = eD ¡ ® 2 ¢ 2 (directly from Lemma 6). Total pro¯ts are ¦ M = p R ³ ¯¡ p R µ ´ +e ³ ¯¡ p R µ ´ ¡ ® 2 ¢ 2 ¡kµ 2 , whence p R¤ = 1 2 ¡ µ¯¡e( ® 2 ) 2 ¢ < p M . Substituting we can ¯nally write the monopolist's pro¯t function as ¦ M = 1 4µ £ µ¯+e ¡ ® 2 ¢¤ 2 ¡kµ 2 , with ¯rst-order condition µ ¯ 2 ¶ 2 ¡ ³ e 2µ ´ 2 µ ® 2 ¶ 4 =2kµ (38) which may not necessarily hold for all values of e. Suppose it does, we can de- ¯ne the function e(µ) ´ 2µ ½ ¡ 2 ® ¢ 4 · ³ ¯ 2 ´ 2 ¡2kµ ¸¾1 2 . The maximizer µ e of this function must satisfy @e(:) @µ = 0 (the SOC can easily be veri¯ed), and takes value µ e = ³ ¯ 2 ´ 2 ( 2 ® ) 4 k h 1+2( 2 ® ) 4 i . Thus the largest value of e such that the ¯rst-order condition 60 Parameters µ ¤¤ µ M µ ¤ 1 µ 0 1 e=0 n/a 9 8 n/a 2 ¡ 5 3 ¢ 2 e=1=4 1.12 n/a 1.37 n/a e=1=2 1.09 1.3 e=3=4 1.04 1.15 e=8=10 1.03 1.08 Table 1: Numerical comparisons: competition promotes quality. holds is e M = 2 ¡ 2 ® ¢ 2 ³ ¯ 2 ´ 2 ( 2 ® ) 4 k h 1+2( 2 ® ) 4 i ( ³ ¯ 2 ´ 2 ¡2 ³ ¯ 2 ´ 2 ( 2 ® ) 4 h 1+2( 2 ® ) 4 i )1 2 , which is always positive. From now on we suppose that e· e M . The optimality condition (38) admits two solutions; rewrite it as ³ ¯ 2 ´ 2 = '(µ) with ' 0 (µ) = 2k¡ ¡ ® 2 ¢ 4 ³ e 2 2µ 3 ´ , which may be positive or negative. Since ' 0 (µ) ¸ 0 for the SOC to be satis¯ed, the maximizer is unique and identi¯ed by the larger of the two roots of the ¯rst-order condition. Let µ ¤¤ =argmax¦ M (µ); µ ¤¤ <µ M immediately from (38). That is, (exogenously) reducing advertising opportunities would increase quality. This claim departs from Anderson (2004), whose (monopoly) broadcasting model shows that a binding cap on advertising quantity leads to lower (costly) quality. That is, increasing adver- tising volumes helps improve quality. In that model, advertising is the only source of revenue: no surplus is extracted from consumers. In the present paper, con- sumer revenue and advertising revenue are substitutes. If advertising were capped, the platform would o®er consumers a smaller discount and would therefore have to increase its quality. To evaluate quality across all market structures, we have to call on numerical estimations of the optimality conditions (37) and (38). With the parameter values ® = ¯ = 3=2;® = ¯ = 1=2; k = 1=4 they yield the results laid out in Table 1. On the proviso that an equilibrium in pure strategies exists, competition uniformly improves quality for a small enough externality. More precisely, Proposition 15 Suppose both (37) and (38) hold. There exists some e ¤ such that for e<e ¤ ; µ ¤¤ <µ M <µ ¤ 1 <µ 0 1 and for e>e ¤ ; µ ¤¤ <µ ¤ 1 <µ M <µ 0 1 . 61 When e becomes large enough the lure of advertising pro¯ts leads the high-quality platform in the competitive environment to drop its quality below the monopoly levelµ M . Thustheintroductionof advertisingrevenuecanreduce qualitybelowthe level provided even in the most distortionary environment. 2.5.6 Market coverage Under Assumption 13, both ¯rms operate and the market is fully covered. More precisely, e < ¹ e not only ensures that both platforms are viable, but also that full market coverage is the unique equilibrium outcome in the advertising market. Given market coverage in advertising, Assumption 12 guarantees market coverage in the consumer market. Under these restrictions, the formulation of advertising pro¯ts (Lemma 6) and of the demand functions D R 1 ;D R 2 are valid and represent the only equilibrium con¯guration that can arise in this game. This is no longer true absent those constraints, as established by Wauthy (1996). He extends the now standard Gabszewicz and Thisse (1979) model (or Shaked and Sutton (1982)), and shows market coverage to be an endogenous choice on the part of the ¯rm. With a broader parameter space, the low-quality ¯rm may optimally choose to not cover the market, that is, to not select the lowest quality possible { and some consumers do not purchase in this case. The di±culty we face in relaxing Assumption 13 is twofold. First, itimplies that thepro¯tfunctions de¯nedby Lemma6 are no longer the correct ones to use; others (well de¯ned by Wauthy (1996)) have to be called for. In other words, the solution (µ ¤ 1 ;µ ¤ 2 ) computed in this paper likely bears no resemblance to the new one. Second, the reaction functions p R i (p R j ) become fourth- order polynomials and it is impossible to compute the cut-o®s ^ p R i (p R j ). 2.6 Conclusion This paper has developed an analysis of platform competition when the production ofagoodisnecessarytoenticeoneofthepartiesontotheplatform,andwhere¯rms 62 compete on both sides of the market. Speci¯cally, players compete for consumers by choice of quality (in the sense of vertical di®erentiation) and prices. Using the language of the media industry, given an audience that determines their perceived quality by advertisers, they compete in prices in the advertising market. That is, vertical di®erentiation arises endogenously in the advertising market. Because of the assumption of inelastic demand, audience size not only induces a ranking in the advertising market, but also a premium to being the better platform for advertisers (as in the standard vertical di®erentiation models). This exacerbates the competition for consumers. In equilibrium maximum di®erentiation does not obtain, in departure from the standard literature. It is hampered because too costly in terms of market share. Indeed, the opportunity for additional revenue from each consumer renders them more valuable. Platforms therefore engage in cross-subsidization, and lower prices relaxtheneedforbetterqualitytoinduceaconsumertopurchasethemoreexpensive good. Qualities can come so close to each other that the low-quality platform becomes strictly dominated, at which point the equilibrium breaks down. It would face zero demand, not because of exogenous parameters, but because of ¯rm 1's actions. Three additional ingredients are necessary to these results: costly quality, here modeled as an investment, an externality from one side of the platform to the other and `single-homing'. In particular, the externality itself is not su±cient and would leadtomaximaldi®erentiation, howeverwithadi®erentdistributionofpro¯ts. The model lends support to popular claims of reduction in newsroom investment follow- ing the takeover of media by professional management. No ingerence is required for this phenomenon to arise: pro¯t-maximizing behavior is su±cient, for it turns quality and advertising ¯rst into substitutes. 63 Theresultswereportoweinparttothesimplestructurechosen,andinparticular to the assumption of complete market coverage. When it is no longer assumed that markets are covered, lower prices resulting from the cross-subsidization contribute to expanding market size; that is, they improve trade. In this case welfare analysis becomes more sensible. Importantly, media operate in conglomerates and strive to segment consumer markets (using real or perceived correlation between media and commodity consumption) to better serve their advertisers. These important characteristics are so far left out. 2.7 Appendix: proofs and additional information The Appendix contains the proofs of the Lemmata and Propositions developed in the main text, as well as two Propositions that rest on some exogenous restrictions on the players' behavior. 2.7.1 Proofs Proof of Lemma 7: First notice that playing a pro¯le ~ p R such that D R 1 = D R 2 can never be a best reply. When D R 1 = D R 2 advertising pro¯ts ¦ A i are nil for both platforms. So both players have a deviation strategy p R i +" in either direction since ¦ A i > ¦ A i > 0; i = 1;2 as soon as D R i 6= D R ¡i . Maximizing the pro¯t function (31) taking p ¡i as ¯xed leaves us with two `quasi-reaction correspondences', for each competitor, depending on whether D R 1 > D R 2 or the converse. Player i's pro¯t function can be rewritten ¦ i ¡ p R 1 (p R 2 );p R 2 ;¦ A i ¢ . Depending on ¯rm 2's decision, platform 1's pro¯t is either ¦ 1 = 8 > < > : ¦ 1 ³ p R 1 (p R 2 );p R 2 ;¦ A i ´ =¦ 1 ¡ 1 2 ¡ p R 2 +¢µ¯¡2eA ¢ ;p R 2 ;¦ A i ¢ ; or; ¦ 1 ¡ p R 1 (p R 2 );p R 2 ;¦ A i ¢ =¦ 1 ¡ 1 2 ¡ p R 2 +¢µ¯+2eA ¢ ;p R 2 ;¦ A i ¢ : 64 De¯ne g 1 (p R 2 ) ´ ¦ 1 ¡ p R 1 (p R 2 );p R 2 ;¦ A i ¢ ¡¦ 1 ³ p R 1 (p R 2 );p R 2 ;¦ A i ´ . This quantity is the di®erence in pro¯ts generated by ¯rm 1 when it chooses one `quasi-best response' over the other, as a function of the consumer price set by ¯rm 2. For p R 2 su±ciently low, g 1 (:) > 0. This function is continuous and a.e di®erentiable, for it is the sum of two continuous, di®erentiable functions. Using the de¯nitions of equilibrium advertising pro¯ts (in Lemma 6), it is immediate to compute dg 1 dp R 2 = d¦ A 1 (p R 1 ;p R 2 ) dp R 2 ¡ d¦ A 1 (p R 1 ;p R 2 ) dp R 2 <0,and d 2 g 1 d(p R 2 ) 2 =0,whencethereexistsapoint ^ p R 2 suchthatg 1 (^ p R 2 )=0. At ^ p R 2 ; ¦ i ³ p R 1 (^ p R 2 );^ p R 2 ´ = ¦ i ¡ p R 1 (^ p R 2 );^ p R 2 ¢ and platform 1 is indi®erent between these two pro¯t functions, that is between either best response p R 1 (^ p R 2 ) or p R 1 (^ p R 2 ). The same follows for platform 2, which de¯nes ^ p R 1 . Computing the pro¯t functions, it is immediate that ¦ 1 ³ p R 1 (p R 2 );p R 2 ;¦ A i ´ ¸¦ 1 ¡ p R 1 (p R 2 );p R 2 ;¦ A i ¢ ,p R 2 ¸ ^ p R 2 ´¡ ¡ ¢µ¯+e(A¡A) ¢ and ¦ 2 ³ p R 1 ;p R 2 (p R 1 );¦ A i ´ ¸¦ 2 ¡ p R 1 ;p R 2 (p R 1 );¦ A i ¢ ,p R 1 ¸ ^ p R 1 ´¢µ¯¡e(A¡A) Proof of Lemma 8: The pro¯t function ¦ i i=1;2 is strictly concave in p R i and thereforequasi-concave. Sinceplayeri'sactionsetisP R i µR,itiscompactandcon- vex. For each platform this can be partitioned into two subsets P R i = h p R;min i ;^ p R i i andP R i = h ^ p R i ;p R;max i i ,onwhichthebest-responsecorrespondencesde¯nedby(32) and (33) are continuous for each platform i. Consider any equilibrium candidate ¡ p ¤R 1 ;p ¤R 2 ¢ . By construction it is de¯ned as the intersection of the `quasi-best re- sponses', which is not necessarily an equilibrium. But when Condition 1 holds, following the de¯nitions given by equations (32) and (33), either p ¤R 1 2p R 1 (p R 2 ) and p ¤R 2 2 p R 2 (p R 1 ), or p ¤R 1 2 p R 1 (p R 2 ) and p ¤R 2 2 p R 2 (p R 1 ) (or both, if two equilibria ex- 65 ist). Thus at the point ¡ p ¤R 1 ;p ¤R 2 ¢ the reaction correspondences necessarily intersect at least once, whence the Nash correspondence has a closed graph and the Kaku- tani ¯xed-point theorem applies. To show necessity, suppose a pair ¡ p ¤R 1 ;p ¤R 2 ¢ is a Nash equilibrium. By de¯nition, p R 2 ¡ p R 1 ¢ \p R 1 ¡ p R 2 ¢ 6=, and by Lemma 7, either ¡ p ¤R 1 ;p ¤R 2 ¢ = p R 1 ¡ p R 2 ¢ \p R 2 ¡ p R 1 ¢ or ¡ p ¤R 1 ;p ¤R 2 ¢ = p R 1 ¡ p R 2 ¢ \p R 2 ¡ p R 1 ¢ , or both if two equilibria exist. For the ¯rst equality to hold, the ¯rst line of Condition 1 must hold, and for the second one, the second line of Condition 1 must be satis¯ed. Proof of Lemma 9: First construct a candidate equilibrium as follows. Suppose that platform maximize ¦ H 1 =p R 1 D R 1 (p R ;µ)¡kµ 2 1 +¦ A 1 and ¦ H 2 =p R 2 D R 2 (p R ;µ)¡ kµ 2 2 +¦ A 2 , respectively. Solving for the ¯rst-order conditions laid out in De¯nition 1 yields p ¤R 1 = 1 3 £ ¢µ ¡ 2¯¡¯ ¢ +2e ¡ A¡2A ¢¤ p ¤R 2 = 1 3 £ ¢µ ¡ ¯¡2¯ ¢ +2e ¡ 2A¡A ¢¤ From equilibrium prices it is straightforward to compute consumer demand: D R 1 = 1 3 h (2¯¡¯) 2 + p C i and D R 2 = 1 3 h (¯¡2¯) 2 ¡ p C i , hence the restriction D R 2 > 0 provided ¢µ > p C ¯¡2¯ and p ¤R 1 = ¢µ¯ 2 ¡e2A p ¤R 2 =0 otherwise. When D R i = 0 i = 1;2, p R j is determined by platform j's reaction corre- spondence only. Thus it easy to verify that the ¯rst line of Condition 1 is satis¯ed and that ¡ p ¤R 1 ;p ¤R 2 ¢ indeed constitutes an equilibrium by Lemma 8. This equilib- rium always exists because ^ p R 1 ¸ p ¤R 1 and ^ p R 2 · p ¤R 2 are always satis¯ed. Indeed, either both hold when both platforms are active, for ¢µ ¡ ¯+¯ ¢ +e ¡ A+A ¢ ¸0 is always true, or p ¤R 2 = 0> ^ p R 2 and ^ p R 1 >p ¤R 1 can be immediately veri¯ed when only ¯rm 1 is active. Another candidate equilibrium ¡ p ¤¤R 1 ;p ¤¤R 2 ¢ can be constructed by letting platform 1 play as if ¦ L 1 = p R 1 D R 1 (p R ;µ)¡kµ 2 1 +¦ A 1 and platform 2 as if 66 ¦ L 2 =p R 2 D R 2 (p R ;µ)¡kµ 2 2 +¦ A 2 , whence p ¤¤R 1 = 1 3 £ ¢µ ¡ 2¯¡¯ ¢ +2e ¡ 2A¡A ¢¤ p ¤¤R 2 = 1 3 £ ¢µ ¡ ¯¡2¯ ¢ +2e ¡ A¡2A ¢¤ with D R 1 = 1 3 h (2¯¡¯) 2 ¡ p C i and D R 2 = 1 3 h (¯¡2¯) 2 + p C i , therefore D R 1 > 0 if ¢µ > p C 2¯¡¯ . Notice that an equilibrium such that p ¤R 1 =0 p ¤R 2 =¡ ¢µ¯ 2 ¡e2A cannot exist, for these prices are not best response to each other. At the price setting stage the cost of quality is sunk, so for µ 1 > µ 2 there always exists some price p R 1 ¸ p R 2 such that consumers prefer purchasing from platform 1. Then when both ¯rms are active Condition 1 holds as long as ¢µ ¡ ¯+¯ ¢ ¡ e ¡ A+A ¢ · 0. Given that ¢µ¸ p C 2¯¡¯ , take the lower bound and substitute into the second line of Condition 1. Recalling p C =2e(A+A), e(A+A) Ã 2(¯+¯) ¯¡2¯ ¡1 ! >0; 8¯¸0 which violates the second pair of inequalities of the necessary Condition 1. So the second candidate can never be an equilibrium. For completeness, Condition 1 is also su±cient to rule out deviations from the pairs ¡ p ¤R 1 ;p ¤R 2 ¢ and ¡ p ¤¤R 1 ;p ¤¤R 2 ¢ . The SOC of the pro¯t function (31) is satis¯ed at prices p ¤R i and p ¤¤R i 8i;8p R ¡i , there cannot be any local deviation. Consider now deviations involving inconsistent actions,thatis,suchthatbothplatformsmaximizeeitherp R i D R i (p R ;µ)¡kµ 2 i +¦ A i or p R i D R i (p R ;µ)¡kµ 2 i +¦ A i . Since ¡ p ¤R 1 ;p ¤R 2 ¢ alwaysexists, the ¯rst line of Condition 1 always holds. It immediately follows from (32) and (33) that p R 1 ¡ p R 2 ¢ \p R 2 ¡ p R 1 ¢ = and p R 1 ¡ p R 2 ¢ \p R 2 ¡ p R 1 ¢ = as well. 67 Proof of Proposition 8: Directly from Lemma 9, which establishes existence and uniqueness of this equilibrium. In particular no such alternative equilibrium can exist when ¢µ < p C 2¯¡¯ . Consider such a situation, then the prices p R 1 = ¢µ¯ 2 ¡2eA p R 2 =0 do form an equilibrium for they satis¯es Condition 1. But the pair p R 1 =0 p R 2 =¡ ¢µ¯ 2 ¡2eA cannot be best responses to each other. At the price-setting stage, the cost of quality is sunk. So with µ 1 > µ 2 , there always exists some price p R 1 ¸ p R 2 such that consumers prefer purchasing from platform 1. Proof of Lemma 10: First o® the following simpli¯es the analysis and lets us focus on platform 1's problem. Claim 3 In any pure-strategy Nash equilibrium (µ ¤ 1 ;µ ¤ 2 ) such that µ ¤ 1 > µ ¤ 2 , µ ¤ 2 = µ necessarily. Proof: Assume the FOC (41) binds so that µ ¤ 1 = ^ µ 1 . Computing the slope of the pro¯t function ¦ 2 yields d¦ 2 dµ 2 = 8 > < > : ¡(¯¡2¯) 2 + C (¢µ) 2 ¡2kµ 2 <¡2kµ 2 ; if ¢µ(¯¡2¯)> p C; ¡2kµ 2 ; if ¢µ(¯¡2¯)· p C. whence it is immediate that d¦ 2 dµ 2 j µ 2 >µ < d¦ 2 dµ 2 j µ <0. Next delineate an impossibility. When C is said to be `large' the pro¯t function ¦ 1 (:;:) is no longer well behaved, as shown in Section 2.7.2. This leads to 68 Lemma 11 Let µ 1 >µ 2 w.l.o.g. and C¸C f ´ · (2¯¡¯) 2 27k ¡µ 2 ¸ 2 µ (2¯¡¯) 2 3 ¶ , a Nash equilibrium in pure strategies cannot exist. Proof:FollowsdirectlyfromClaims3and5inSection2.7.2. Anypair µ µ 2 + p C ¯¡2¯ ;µ 2 ¶ cannot be an equilibrium because ¯rm 2 can `jump' and assume the monopolist's role at incremental cost k" 2 . In line with the previous section of the Appendix, ¯rm 1's ¯rst-order condition reads ¡ 2¯¡¯ ¢ 2 ¡ C (¢µ) 2 ¡18kµ 1 =0andadmitsauniquemaximizer ^ µ 1 . Thisanalysis does not yet identify an equilibrium of this game but only platform 1's behavior, taking that of ¯rm 2 ¯xed. Suppose ¯rm 1 plays ^ µ 1 ; by Claim 3, platform 2 cannot increase its quality to any µ 2 2 ³ µ; ^ µ 1 ´ . So the pair ³ ^ µ 1 ;µ ´ is an equilibrium as long as ¯rm 2 cannot `jump' over ¯rm 1 and become the high-quality ¯rm. It will necessarily do so if platform 1 turns out to be a monopolist. To guarantee ¯rm 2 operates we need ( ^ µ 1 ¡µ)(¯¡2¯) > p C { Assumption 13 must holds. When ¯rm 2 does operate, the smallest `leap' it can undertake is such that ~ µ 2 ¸ ^ µ 1 +". Hence the no-deviation condition is ¦ 2 ³ ^ µ 1 ;µ ´ ¸¦ 2 ³ ^ µ 1 ; ^ µ 1 +" ´ , or ( ^ µ 1 ¡µ)(¯¡2¯) 2 +B 2 + C ( ^ µ 1 ¡µ) ¸ B 1 + p C(¯¡2¯) ¡9k( ^ µ 1 +") 2 ( ^ µ 1 ¡µ) £ (¯¡2¯) 2 +(2¯¡¯) 2 ¤ ¡18k ^ µ 2 1 +B 2 ¸ B 1 + p C(¯¡2¯) ¡9k( ^ µ 1 +") 2 ( ^ µ 1 ¡µ) £ (¯¡2¯) 2 +(2¯¡¯) 2 ¤ ¡9k ^ µ 2 1 +B 2 ¸ B 1 + p C(¯¡2¯) using the FOC (2¯¡¯) 2 ¡18k ^ µ 1 ¡ C ( ^ µ 1 ¡µ) 2 = 0 and the fact that k ^ µ 1 µ = kµ 2 = 0 (by assumption). Noting ^ µ 1 ¡µ > p C ¯¡2¯ , this condition is generically satis¯ed. Proof of Proposition 9: Suppose µ 1 >µ 2 . Since each proper subgame admits a uniqueNashequilibriumbyLemma10,Proposition8andLemma6,theequilibrium of the game ¡ must be unique. 69 DiscussionandproofofProposition10: TheassertionofProposition10holds triviallyby Corollary 4 whenAssumption 13 holds. The balance focuses on the case where it fails. Denote ~ µ =µ+ p C ¯¡2¯ fromnowon. Asbrie°yalludedtointhemaintext, itisnot immediate that the game ¡ admits a mixed strategy equilibrium, for the payo®s are not everywhere continuous. To see why, ¯rst de¯ne by µ c 1 the threshold such that ¦ 1 (µ c 1 ;µ)=0 when µ 1 >µ 2 . This point exists and exceeds ~ µ 1 because d¦ 1 dµ 1 j µ 1 > ~ µ 1 <0 and the cost function is convex. Obviously neither platform will want to exceed that threshold, so we may as well restrict the set of pure actions over which ¯rms randomize to be [µ;µ c i ]µ £ i ; i = 1;2. Next observe that any distribution over this setmustassignzeromasstoanyµ i 2(µ; ~ µ)byClaim3: anyactioninthisintervalis dominated by either µ or ~ µ. For [ ~ µ;µ c i ] large enough (and µ 2 ¸ ~ µ) there may be out- comessuchthat¢µ > p C ¯¡2¯ , inwhichcasebothplatformsareactive, or¢µ· p C ¯¡2¯ , in which case only the high-quality ¯rm operates. Take µ 1 > µ 2 > µ and suppose ¢µ > p C ¯¡2¯ and ¦ 1 > ¦ 2 > 0. Let µ 2 increase, both ¦ 1 and ¦ 2 vary smoothly. But while lim µ n 2 "µ 1 ¦ 1 =¦ 1 >0, lim µ n 2 #µ 1 ¦ 1 =¡kµ 2 1 , and similarly for ¯rm 2. Both payo®functionsarediscontinuousatthepoint µ 1 =µ 2 . Inthiscaseneitherthepay- o®s nor their sum are even upper-hemicontinous. Following Dasgupta and Maskin's (1986a) Theorem 5, it is ¯rst necessary to characterize the discontinuity set. If it has Lebesgue measure zero, a mixed strategy equilibrium does exist. Consider the case where µ 1 ¸ µ 2 w.l.o.g. and de¯ne ¨ 0 = n (µ 1 ;µ 2 )jµ 1 =µ 2 ;µ i 2[ ~ µ i ;µ c i ]8i o , the set on which the payo®s are discontinuous. Further de¯ne the probability measure ¹(µ 1 ;µ 2 ) over the set £ N = fµ 1 g[[ ~ µ 1 ;µ c 1 ]£fµ 2 g[[ ~ µ 2 ;µ c 2 ]. It is immediate that ¨ 0 has Lebesgue measure zero, so that Pr((µ 1 ;µ 2 )2¨ 0 ) = 0. Note that excluding the set ¨ 0 is remarkably convenient, for we do not know whether an equilibrium of the price subgame even exists (refer Section 2.7.4). Through this construct we can side-step this problem entirely. Next we claim 70 Lemma 12 Suppose µ 1 = µ 2 = µ, an equilibrium in mixed strategies exists in the consumer price subgame. The reader may recall that existence of a pure strategy Nash equilibrium is ruled out by Proposition 17 when µ 1 = µ 2 . In addition, a mixed strategy equilibrium in prices is not guaranteed to exist for all values of µ 1 = µ 2 . Here it holds because µ 1 = µ 2 = µ implies no ex post loss for either party. As each platform's payo®s are bounded below at zero and only one of them can operate (except at p R 1 =p R 2 ), their sum is almost everywhere continuous, except for the set of pairs (p R 1 = p R 2 ), which has measure zero. Proof: Let µ 1 = µ 2 = µ. The sum of pro¯ts ¦ = ¦ 1 +¦ 2 is almost everywhere continuous. Either ¦ = ¦ 1 > 0 8p R 1 < p R 2 , or ¦ = ¦ 2 > 0 8p R 1 > p R 2 , both of which are continuous except at p R 1 = p R 2 , where ¦ = ¦ 1 + ¦ 2 = 0. But the set ª = © (p R 1 ;p R 2 )jp R 1 =p R 2 ;(p R 1 ;p R 2 )2R 2 ª has Lebesgue measure zero. Theorem 5 of Dasgupta and Maskin (1986a) directly applies and guarantees existence of an equilibrium in mixed strategies. Therefore the pair µ 1 =µ 2 =µ may be part of an equilibrium of the overall game. Then Proposition 10 asserts that a mixed strategy equilibrium of the game ¡ exists. Proof of the Proposition: We only need showing that the payo® functions ¦ i i = 1;2 are lower-hemicontinuous in their own argument µ i . Without loss of generality, ¯x µ 1 > µ 2 . We know that ¦ 1 is continuous for any µ 1 > µ 2 (refer Section 2.7.2). From Claim 3 it is immediate that ¦ 2 is continuous for µ 1 > µ 2 . Last, for i=1;2 ¦ i = 8 > < > : 0; if µ 1 =µ 2 =µ; ¡kµ 2 i ; if µ 1 =µ 2 >µ. that is, ¦ i ; i = 1;2 is l.h.c. Since (µ 2 ;µ 1 )s:t µ 2 =µ 1 2 ¨ 0 , Theorem 5 in Dasgupta andMaskin(1986a)canbeapplied, whenceanequilibriuminmixedstrategiesmust exist. 71 Proof of Proposition 11: Let µ c i denote the upper bound of the support of the distributions, a precise de¯nition of which we will provide later. While Sharkey and Sibley (1993) provide an appealing approach to characterize mixed strategies in a problem of entry with sunk costs, it does not quite apply here. Indeed there is no proper entry stage and the payo®s depend not just on the ranking of the ¯rms' decisions (µ 1 ;µ 2 ), but on the di®erence µ 1 ¡µ 2 . This implies, in particular, that playing µ i = µ cannot be interpreted as a decision to not enter the market because ¦ i (µ i ;µ j ) > 0 for µ j such that ¢µ > 2e(A+A) ¯¡2¯ . Let H i (µ i ) be the distribution over ¡i's payo® realizations for µ i 2 h ~ µ;µ c i [fµg. For any play µ 2 , write the expected pro¯t of ¯rm 2 as E µ 1 [¦ 2 (µ 1 ;µ 2 )]= H(µ 1 )¦ 2 (µ 1 ;µ 2 )+ R µ 0 1 =µ 2 ~ µ 1 ¦ 2 (s;µ 2 )dH 1 (s)+ R µ c 1 µ 0 1 =µ 2 ¦ 2 (s;µ 2 )dH 1 (s) (39) with an atom at µ 1 . With probability R µ 0 1 =µ 2 ~ µ 1 dH 1 (s) it plays µ 1 such that medium 2 is the dominant ¯rm (µ 2 ¸µ 1 ). With probability R µ c 1 µ 0 1 =µ 2 dH 1 (s) it is the dominant ¯rm (the second integral). In this latter case, ¦ 2 (µ 1 ;µ 2 )=¡kµ 2 2 <0. We ¯rst claim Lemma 13 There is a mass point at µ i . More precisely, 8 i; H i (µ i )2(0;1) Proof: Suppose H 1 (µ 1 ) = 1, then argmaxE µ 1 [¦ 2 (µ 1 ;µ 2 )] = ~ µ 2 , so H 2 (µ 2 ) = 0 and H 2 (µ 2 )assignsfullmassat ~ µ 2 . Butthen¯rm1shouldplaysomeµ 1 > ~ µ 2 andbecome the monopolist. If H 1 (µ 1 ) = 0 it necessarily plays on h ~ µ 1 ;µ c 1 i and playing ~ µ 2 is a dominated strategy for ¯rm 2. It therefore assigns no mass at this point. But then 8 µ 2 2 ³ ~ µ 2 ;µ c 2 i ; ¦ 1 (µ 1 ;µ 2 )>0 and platform 1 should shift some mass to µ 1 . 72 The equilibrium conditions write8µ i 2£ N i ; E µ j [¦ i (µ i ;µ j )]=¦ i (µ i ; ~ µ j ) ¦ i (µ i ; ~ µ j )=0 (40) The ¯rst line asserts that the expected payo® cannot be worse than if not investing forsureandthesecondonethatifnotinvestingforsure, aplatformcanonlyexpect zero pro¯ts. Thus expected pro¯ts in the mixed-strategy equilibrium must be zero. We next need to determine the upper bound µ c i of the support of H i (µ i ) for each platform i=1;2. As a consequence of Lemma 13 it solves either ¦ i (µ j ;µ c i )=0 or ¦ i ( ~ µ j ;µ c i )=0 hence Lemma 14 µ c i =max n µ 0 i j¦ i (µ j ;µ 0 i )=0;¦ i ( ~ µ j ;µ 0 i )=0 o Proof: Let µ 0 i solve ¦ i (µ j ;µ 0 i )=0 and µ 00 i solve ¦ i ( ~ µ j ;µ 00 i )=0. Suppose µ 0 i <µ 00 i and µ c i = µ 0 i : there is a measure µ 00 i ¡µ 0 i on which i places zero weight. Then j should shift at least some weight to µ 0 i +²; ² > 0 and small, to obtain E ^ H(µ i ) [¦ j ] > 0 = E µ i [¦ j (µ i ;µ j )] (where ^ H(:) is an alternative distribution). Clearly this extends to any µ i 2[µ 0 i ;µ 00 i ). Rewriting the equilibrium condition (40) 8 µ j 2£ N j ; H i (µ i )R j (µ i ;µ j )+ Z µ 0 i =µ j ~ µ i R j (s;µ j )dH i (s)=kµ 2 j where R i (µ i ;µ j ) stands for platform i's revenue (gross of costs). Hence Proposition 11, the proof of which we complete below. 73 Proof of the Proposition: Existence is established by Proposition (10). For any play µ i , total revenue R j (µ i ;µ j is decreasing in µ j 2£ M j { refer Conditions (35) and (36). Thus for any distribution H i (µ i ) the LHS is decreasing while the RHS is strictly increasing. This complete the proof. ProofofProposition13: Uniquenessofthesubgame-perfectequilibriumrenders thecomparativestaticsexercisevalid. Forthe¯rstline,recallthatµ ¤ 2 =µisastrictly dominant strategy when an equilibrium exists, whence µ ¤ 2 is independent of e. Item (b) is stated and proven in Section 2.7.2. To show concavity, di®erentiate dµ ¤ 1 de once more and rearrange to ¯nd d 2 µ ¤ 1 de 2 = 8 ¡ A+A ¢ [¡(¢µ) 3 Á 0 ] 2 · dµ ¤ 1 de e ¡ (¢µ) 3 18k+2C ¢ ¡¢µ ¡ Á 0 +4C ¢ ¸ Since Á 0 ¸0 it is immediate that d 2 µ ¤ 1 de 2 <0. Next (c) obtains from the de¯nitions of equilibriumadvertisingpricestogetherwithitems(b): dp A 1 de = µ (¯+¯)+4e ¢µ¡e dµ ¤ 1 de (¢µ) 2 ¶ p A> µ (¯+¯)+4e ¢µ¡e dµ ¤ 1 de (¢µ) 2 ¶ p A = dp A 2 de > 0. Ascertaining the behaviour of consumer prices(d)isequallysimple: dp R 1 de = 1 3 h dµ ¤ 1 de (2¯¡¯)+2(A¡2A) i < 1 3 h dµ ¤ 1 de (¯¡2¯)+2(2A¡A) i = dp R 1 de < 0. For (e), call on the de¯nitions of consumer demands; in particular, we can compute dD R 1 de = 2 3¢µ (A+A) µ 1¡ e dµ ¤ 1 de ¢µ ¶ > 0, and the rest of the statement is obvious. To show (f), di®erentiate the pro¯t function, which yields d¦ 1 de = 1 9 " dµ ¤ 1 de (2¯¡¯) 2 + dB 1 de + 8e(A+A) 2 ¢µ¡C dµ ¤ 1 de (¢µ) 2 # ¡2kµ ¤ 1 dµ ¤ 1 de Collecting the terms and recalling the de¯nition of the FOC of (35), this rewrites d¦ 1 de = dB 1 de + 8e(A+A) 2 ¢µ >0. That ¦ 1 is concave is obvious, too. Proof of Proposition 14: When e is large enough platform 1 (the high-quality ¯rm) prefers playing such that ¢µ = 2e(A+A) ¯¡2¯ ´z(e) for any µ 2 (and µ 1 not so large as to induce negative pro¯ts). Its payo®s when ¢µ· z(e) are given by the second 74 line of (35), where B 1 (e)=2e(2¯¡¯)(2A¡A). This can re-arranged as ¼ 1 (e;µ)= 1 9 h ¢µ ¡ 2¯¡¯ ¢ 2 +2e[A(5¯¡4¯)¡A(¯+¯)] i ¡kµ 2 1 for ¢µ·z(e) and ¼ 1 (e;µ)= 1 9 · ¢µ ¡ 2¯¡¯ ¢ 2 +B 1 (e)+ [2e(A+A)] 2 ¢µ ¸ ¡kµ 2 1 if ¢µ > z(e). Let ¼ 1 (e;µ) = max¼ 1 (e;µ) for any pair µ 1 > µ 2 such that ¢µ = z(e). This is an upper bound on ¯rm 1's pro¯ts for any play by ¯rm 2. Clearly ¼ 1 (e;µ) is maximized for µ 2 =µ. Recall that we denote the corresponding value of µ 1 by ~ µ 1 . For any e and µ 2 ; @¼ 1 (e;µ) @µ 1 > 0 when ¢µ < z(e) and @¼ 1 (e;µ) @µ 1 < 0 when ¢µ = z(e) and µ 2 > µ. Therefore ¼ 1 (e;µ) reaches zero for some value µ 0 1 · µ c 1 . Thus no ¯rm will play out of these bounds. More precisely, @¼ 1 (e;µ) @µ 1 = 2¯¡¯ 9 ¡2kµ 1 >0; when ¢µ <z(e) and @¼ 1 (e;µ) @µ 1 = 2¯¡¯ 9 ¡2kµ 1 <0; for ¢µ =z(e); µ 2 >µ. with max @¼ 1 (e;µ) @µ 1 reached for µ 2 =µ. Since argmax ¦ 1 (µ 1 ;µ 2 )> ~ µ 1 when µ 2 >µ, it follows that @¼ 1 (e;µ) @µ 1 <j @¼ 1 (e;µ) @µ 1 j and thereforej ~ µ 1 ¡µ c 1 j<z(e). Proof of Corollary 5: For lines 1 and 3 the proof follows directly from Propo- sitions 9 and 12, as well as the analysis of ¦ 1 (:;:) in Section 2.7.2. When k = 0, because quality is a sunk cost in the original model, nothing is altered until plat- forms' have to choose their quality variable. That is, the analysis of the third and 75 second stages remains valid. In the ¯rst stage, they now face pro¯t functions ¦ 1 = 8 > > < > > : 1 9 £ ¢µ(2¯¡¯) 2 +B 1 + C ¢µ ¤ ; if ¢µ > p C ¯¡2¯ and ; 1 9 " ¢µ(2¯¡¯) 2 +B 1 + C p C ¯¡2¯ # ; if ¢µ· p C ¯¡2¯ . and ¦ 2 = 8 > > < > > : 1 9 £ ¢µ(¯¡2¯) 2 +B 2 + C ¢µ ¤ ; if ¢µ > p C ¯¡2¯ and ; 1 9 " ¢µ(¯¡2¯) 2 +B 2 + C p C ¯¡2¯ # ; if ¢µ· p C ¯¡2¯ . The FOC of the ¯rst line of ¦ 1 identi¯es a minimiser of ¦ 1 . That is, ¯rms will necessarily play the second line. Then we are back to Shaked and Sutton's model of maximum di®erentiation. 2.7.2 Analysis of the high-quality ¯rm's pro¯t function In the sequel µ 1 > µ 2 without loss of generality. The pro¯t function ¦ 1 (:;:) is obviously continuous for µ 1 < µ + p C ¯¡2¯ or the converse. Furthermore, assume e<1, then Claim 4 The function ¦ 1 is continuous for ¢µ = p C ¯¡2¯ Proof: Foreaseofnotation,let¦ 1 =¦ L 1 forall¢µ¸ p C ¯¡2¯ and¦ 1 =¦ R 1 otherwise. These are the de¯nitions of ¦ 1 (µ 1 ;µ) to the left and the right of the point such ¢µ = p C ¯¡2¯ foranypair(µ 1 ;µ 2 ). Totheleftplatform1isamonopolistwhosepro¯ts ¦ L 1 are necessarily bounded. The function is de¯ned as ¦ L 1 : £ 1 ££ 2 µR 2 7!R, thereforeTheorem4.5inHaaserandSullivanapplies: amappingfromametricspace into another metric space is continuous if and only if the domain is closed when the range is closed. So ¦ L 1 (µ 1 ;µ 2 ) is continuous at ¢µ = p C ¯¡2¯ , and is necessary the left-hand limit of the same function ¦ L 1 . Now consider a sequence µ n 1 such that ¢µ > p C ¯¡2¯ converging to p C ¯¡2¯ from above for some ¯xed µ 2 . This sequence exists and always converges for £ 1 µR is complete. As e < 1 and A and A are 76 necessarily bounded, C is ¯nite so there is some n and some arbitrarily small ± such that ¦ R 1 (µ n 1 ;µ 2 )¡¦ L 1 (µ 2 + p C ¯¡2¯ ;µ 2 ) < ±. That is, lim µ n 1 !µ 2 + p C ¯¡2¯ ¦ R 1 (µ n 1 ) = ¦ L 1 (µ 2 + p C ¯¡2¯ ;µ 2 ). Hence ¦ 1 is continuous for ¢µ = p C ¯¡2¯ . ¦ 1 (:;:) being the di®erence of two convex functions its exact shape is a®ected by that of these two primitives. Indeed, when C becomes large enough, it is no longer well behaved. Claim 5 There exists some C f such that ¦ 1 (:;:) admits a binding ¯rst-order con- dition for C · C f only. When C > C f , its maximum is reached at the kink: µ 1 =µ+ p C ¯¡2¯ . Proof: Seeking ¯rst-order conditions of ¦ 1 (:;:) with respect to µ 1 yields @¦ 1 @µ 1 = 8 > > > > > > > < > > > > > > > : µ 2¯¡¯ 3 ¶ 2 ¡2kµ 1 =0; for ¢µ· p C ¯¡2¯ ; µ 2¯¡¯ 3 ¶ 2 ¡ C (3¢µ) 2 ¡2kµ 1 =0; for ¢µ > p C ¯¡2¯ and C·C f ; µ 2¯¡¯ 3 ¶ 2 ¡ C (3¢µ) 2 ¡2kµ 1 <0; for ¢µ > p C ¯¡2¯ and C >C f ; (41) Whenbinding,thesecondlineofsystem(41)canberearrangedasÁ(µ 1 )= ¡ 2¯¡¯ ¢ 2 , with slope Á 0 (µ 1 )=18k¡ 2C (¢µ) 3 . Since ¢µ >0, this FOC has at most two solutions: one where Á 0 (µ 1 ) < 0 and the other with Á 0 (µ 1 ) > 0. The SOC requires Á 0 (µ 1 )¸ 0 forthe FOCto identifya maximizer, so thereexistsauniquelocalmaximizer of¦ 1 , denoted ^ µ 1 . Let µ 0 1 be the (unique) maximizer of the ¯rst line of system (41). It is immediate that ^ µ 1 < µ 0 1 and consequently µ 0 1 ¡µ 2 · p C ¯¡2¯ ; µ 1 2 BR 1 (µ 2 ) can never be true. That is, the two statements of the ¯rst line of (41) cannot be simultane- ously satis¯ed: ¯rm 1 would not play the ¯rst line of (35), but the second one. We rewrite: @¦ 1 @µ 1 = Ã 2¯¡¯ 3 ! 2 ¡2kµ 1 >0; for ¢µ· p C ¯¡2¯ 77 Recall that the pro¯t function is continuous, so it does not jump anywhere. Be- cause ¦ 1 is monotonically increasing below ^ µ 1 and the SOC is monotonic beyond ^ µ 1 , it is concave for C · C f and ^ µ 1 is a global maximizer. The binding ¯rst- order condition de¯nes a function C(µ 1 ;µ 2 ) ´ (¢µ) 2 £ (2¯¡¯) 2 ¡18kµ 1 ¤ , whence dC(:) dµ 1 = 0 , µ f 1 = (2¯¡¯) 2 27k . Substituting back into C(µ 1 ;µ 2 ) gives the cut-o® value C f ´ · (2¯¡¯) 2 27k ¡µ 2 ¸ 2 µ (2¯¡¯) 2 3 ¶ . When C > C f , the ¯rst-order condition (41) is everywhere negative, hence d¦ 1 dµ 1 j µ 1 <µ+ p C ¯¡2¯ >0 d¦ 1 dµ 1 j µ 1 >µ+ p C ¯¡2¯ <0 Whilethispro¯tfunctionisnotdi®erentiablefor¢µ = p C ¯¡2¯ ,ithasbeenestablished that it is nonetheless continuous for any such pair (µ 1 ;µ 2 ). It is monotonic on either side of ¢µ = p C ¯¡2¯ , so that ^ µ 1 such that ¢µ = p C ¯¡2¯ , is the unique maximum of ¦ 1 (µ 1 ;µ 2 ) given some ¯xed µ 2 . Last in this section we examine the behavior of the quality variable µ 1 when the ¯rst-order condition (41) does bind. Claim 6 Let ^ µ 1 solve ¡ 2¯¡¯ ¢ 2 ¡ C (¢µ) 2 ¡18kµ 1 =0, then d ^ µ 1 de <0 and d ^ µ 1 dk <0. Proof: Di®erentiatethe¯rst-ordercondition(41); aftersomemanipulationswecan write dµ ¤ 1 de = 8¢µe(A+A) 2 ¡ 2e(A+A) ¢ 2 ¡18k(¢µ) 3 dµ ¤ 1 de ¸ (·)0 , 2C ¡ 18k(¢µ) 3 = ¡(¢µ) 3 Á 0 (µ 1 )j µ 1 =µ ¤ 1 ¸ (·)0 so that dµ ¤ 1 de < 0 (assuming the SOC holding strictly at µ ¤ 1 , otherwise dµ ¤ 1 de is not de¯ned and we need to consider the left derivative). The second statement is similar: di®erentiate the ¯rst-order condition of (35) to ¯nd 2C(¢µ) ¡3dµ 1 dk ¡ 18µ 1 ¡ 18k dµ 1 dk = 0, which is rearranged as dµ 1 dk = 18µ 1 (¢µ) 3 2C¡18k(¢µ) 3 . The denominator is exactly the SOC of (35), which we know to hold, multiplied by (¢µ) 3 . 78 2.7.3 Elements of Proof of Proposition 12 { unique subgame perfect equilibrium of the Shaked and Sutton model In the Shaked and Sutton (1982) model there exists a unique equilibrium in the price subgame. In the ¯rst stage of the game, ¯rms solve Problem 3 max µ i 2£ i p ¤ i D i ¡kµ 2 i for i = 1;2 and with demand D 1 = 1 3 ¡ 2¯¡¯ ¢ , D 2 = 1 3 ¡ ¯¡2¯ ¢ and prices p 1 = ¢µ 3 ¡ 2¯¡¯ ¢ ,p 2 = ¢µ 3 ¡ ¯¡2¯ ¢ , respectively. Thisproblemisconcave8i, and, given equilibrium prices p ¤ i 8i, has obvious maximizers µ 0 2 =µ and µ 0 1 = 1 2k µ 2¯¡¯ 3 ¶ 2 with µ 0 1 < µ thanks to k > (2¯¡¯) 2 18µ . These individually optimal maximizers also form a Nash equilibrium, for although ¦ 1 ¡ µ 0 1 ;µ 0 2 ¢ > ¦ 2 ¡ µ 0 1 ;µ 0 2 ¢ 8k > 0 15 , it is also true that Claim 7 ~ µ 2 >µ 0 1 such that ¦ 2 ³ µ 0 1 ; ~ µ 2 ´ ¸¦ 2 ¡ µ 0 1 ;µ 0 2 ¢ . Proof: Consider a deviation ~ µ 2 =µ 0 1 +², ² arbitrarily small. We can compute ¯rm 2 pro¯t from this deviation as ¦ 2 ³ µ 0 1 ; ~ µ 2 ´ =² µ 2¯¡¯ 3 ¶ 2 ¡k ~ µ 2 2 <0 and the marginal pro¯t µ 2¯¡¯ 3 ¶ 2 ¡2k(µ 0 1 +²)<0. This completes the equilibrium characterization of the benchmark model. Proof of Proposition 12: In the Shaked and Sutton (1982) model the game admits a unique equilibrium, as shown in the Appendix, Section 2.7.3. In the ¯rst stage playersselectµ 0 2 =µandµ 0 1 = 1 2k µ 2¯¡¯ 3 ¶ 2 <µ. Firm1's¯rst-orderconditioninthis problem reads µ 2¯¡¯ 3 ¶ 2 ¡2kµ 0 1 = 0 while that of Problem 2 is µ 2¯¡¯ 3 ¶ 2 ¡2k ^ µ 1 = C (3¢µ) 2 >0. Therefore ^ µ 1 <µ 0 1 . 15 We can readily compute these pro¯ts with closed form solutions: ¦ 1 > ¦ 2 , ³ 2¯¡¯ 3 ´ 2 · 1 2k ³ 2¯¡¯ 3 ´ 2 ¡µ ¸ > ³ ¯¡2¯ 3 ´ 2 · 1 2k ³ 2¯¡¯ 3 ´ 2 ¡µ ¸ , which can be re-arranged as µ 0 1 · 1 2 + 2 3 ¯¯ ¯ 2 ¡¯ 2 ¸ >µ, and always holds 79 2.7.4 Analysis and presentation of symmetric equilibria In this section we study a constrained version of the problem, that of symmetric equilibria. Giventhepro¯tfunction(29),thefollowingresultwillbeusefulthrough- out. Suppose without loss of generality that ¯rm 1 is the high-quality platform for advertisers, that is e 1 >e 2 . Lemma 15 For any quality pro¯le (µ 1 ;µ 2 ) and any pair of action (p R 1 ;p R 2 ), there are three pure strategy equilibria in the advertising market. When e 2 >0, p A 1 (µ;p R )=c+ 1¡G(^ ®) g(^ ®) ¢e>p A 2 (µ;p R )=c+ G(^ ®)¡G( p A 2 e 2 ) e 2 g(^ ®)¡g( p A 2 e 2 )¢e e 2 ¢e When e 1 > e 2 = 0, platform 1 is a monopolist. The equilibrium is a pair of prices such that p AM 1 ¡c p AM 1 = 1 ´ A >p A 2 =0 The third equilibrium entails e 1 =e 2 , whence p A 1 (µ;p R )=p A 2 (µ;p R )=c Equilibrium prices, quantities and pro¯ts are functions of the actions chosen in the consumer market, which are summarized by the externality variable e(D R i ). In the ¯rst equilibrium quantities are q A 1 (µ;p R )=1¡G(^ ®)>q A 2 (µ;p R )=G(^ ®)¡G ³ p A 2 e 2 ´ and necessarily advertising pro¯ts ¦ A 1 (µ;p R ) > ¦ A 2 (µ;p R ) > 0. In the monopoly case, they read q AM 1 (µ;p R ) = 1¡G ³ p AM 1 e 1 ´ > q A 2 = 0, and obviously ¦ AM 1 (µ;p R ) > ¦ A 2 (µ;p R ) = 0. In the symmetric equilibrium, q A 1 = q A 2 = 1 2 ¡ 1¡G ¡ c e ¢¢ and neces- sarily ¦ A 1 (µ;p R )=¦ A 2 (µ;p R )=0. The number of equilibria in this subgame follows the number of permutations of the leadership role. Proof:ThesymmetricequilibriumobtainsdirectlyfromBertrandcompetitionwhen e 1 = e 2 . Suppose e 1 > e 2 = 0 { for whatever reason. Platform 2 has no customers 80 in the advertising market since v 2 · 0 and therefore q A 2 = 0. Consequently ¯rm 1 optimally prices in the advertising market using the monopoly pricing rule p A 1 ¡c p A 1 = 1 ´ A where´ A denotethepriceelasticityofdemand. Thereverseobtainswhene 2 >e 1 = 0. In an equilibrium where both ¯rms are active and e 1 > e 2 > 0 we necessarily have p A 1 >p A 2 . If not, ®e 1 ¡p A 1 >®e 2 ¡p A 2 for any ® and q A 2 =0. We also have Claim 8 In equilibrium, 1¡G(^ ®)>G(^ ®)¡G ³ p A 2 e 2 ´ . Proof: By contradiction. Suppose that in equilibrium (at prices ^ p A i ; i = 1;2), 1¡G(^ ®) · G(^ ®)¡G ³ ^ p A 2 e 2 ´ . The necessary ¯rst-order conditions for both ¯rms imply g(^ ®) ¢e (^ p A 1 ¡c)· 0 @ g(^ ®) ¢e ¡ g ³ ^ p A 2 e 2 ´ e 2 1 A (^ p A 2 ¡c) which is obviously impossible when ^ p A 1 > ^ p A 2 . The fact that ¦ A 1 > ¦ A 2 follows directly. Furthermore, ¦ A 2 > 0 since ^ p A 2 > c and ¯rm 2 faces strictly positive demand. This concludes the proof of the Lemma. Symmetric equilibria This section presents a fairly intuitive result when platforms are constrained in their actions. Such constraints may owe to technological limita- tions, as in Proposition 16, below or be somewhat more arbitrary (but nonetheless plausible), as in Proposition 17, further. Both are impossibility results, the ¯rst one resting on quite a familiar empirical observation. Proposition 16 Fix p R 1 = p R 2 = 0. A pure strategy equilibrium does not exist. If a mixed-strategy equilibrium exists, platforms set their advertising price as if each were a monopolist. The conditions of this proposition and the results are a stylized version of the prob- lem faced by free-to-air broadcasters, where access to the good cannot be controlled 81 bytheproviders. Proposition16helpsrationalisingthegivingawayoffreegoodsby media platforms that are naturally constrained in their pricing (for example local radio stations). Proof: Given p R 1 =p R 2 =0, consumer demand is given by D R i = 8 > > > > < > > > > : 1; if µ i >µ j ; 1 2 ; if µ i =µ j ; and 0; if µ i <µ j . for i6=j; i=1;2, whence platform i faces payo®s ¦ i = 8 > > > > > > > < > > > > > > > : ¦ AM ¡k(µ i ); if µ i >µ j ; ¡k(µ i ); if µ i =µ j ; 0; if 0=µ i <µ j ; or ¡k(µ i ); if µ <µ i <µ j . following directly from Lemma 15, and where ¦ AM denotes monopoly pro¯ts in the advertising market when demand in the consumer market is D R i (0;µ i ) = 1. In particular, suppose µ i =µ j such that ¦ A i (e;µ i )¸0 and one of the platforms charges k(µ i ), j can o®er k(µ i )¡² and become a monopolist in advertising. Thus Claim 9 When consumer prices are identical a pure strategy equilibrium cannot exist. Proof: Given p R i = p R j = 0, µ i > µ j implies that i is a monopolist and j faces negative payo®s for µ j >µ. Suppose µ i =µ j ; then e i =e j and any price advertising price k(µ i ) leading to ¦ A i (e;µ i )¸0 is dominated by k(µ i )¡², and j is a monopolist. There exists some quality level ~ µ, such that ¦ AM i ( ~ µ i ;p A i ) = 0. At this point ¯rm j should not enter, but then i deviate slightly to ~ µ i ¡² and make a positive pro¯t. Hence no pure strategy is deviation proof. 82 A symmetric equilibrium in mixed strategies of this game is a pro¯le ¡ ¾ A ;¾ µ ¢ of distributions over price and quality parameter ¡ p A i ;µ i ¢ i=1;2 . Since no dominated strategy can be part of a mixed strategy equilibrium, the set of pure strategies over which platforms randomise can be restricted to the subsets h µ; ~ µ i , where ~ µ = k ¡1 ¡ ¦ AM ¢ , and p A i as de¯ned in Lemma 15. It is easier to ¯rst deal with the advertising market. Claim 10 Irrelevance. In any symmetric, mixed-strategy equilibrium platforms will behave as strict monopolists in the advertising market. Proof: The proof follows directly from the fact that ¯rms necessarily play a mixed strategy equilibrium. Since µ i 2 h µ; ~ µ i i 8i, each event has probability zero and one platform will be a monopolist in the advertising market with certainty. It will therefore not face competition in this market and behave regardless of what its opponent does. Thus¾ A is necessarily degenerate and the only source of uncertainty is the choice of quality. It is not the object of this paper to characterize ¾ µ , the existence of which is not certain. Given symmetric consumer prices, platforms can only choose symmetric strate- giesintheremainingvariables: theyloseaninstrumentto compete and mustfollow the Bertrand logic of competition, however in quality. But this costly quality is a sunk cost by the time the pricing decision must be made in the advertising market. Therefore marginal-cost pricing in advertising cannot include the cost of producing the information good, which they also fail to recover from consumers. Symmetric qualities necessarily lead to the Bertrand result in advertising, whence the quality investment in the consumer market is necessarily too costly. However, if an equi- librium can be sustained in mixed strategies, players can foresee that only one of them will remain. So they select their distributions anticipating the monopoly rent in the advertising market. The behavior of the opponent becomes irrelevant; it is 83 as if there were no competition in the advertising market 16 . More precisely, the restrictions imposed on the game necessarily lead players to a subgame where only one of them will survive. Next we lay out a proposition analyzing the price subgame when quality µ i is (arbitrarily) restricted to be symmetric. This kills di®erentiation and leads to an outcome that somewhat mirrors Bertrand competition. Here it takes an extreme form owing to the positive externality a®orded by the advertising market: ¯rms' competition is intensi¯ed in the consumer market. Duopolists each maximize the pro¯t function (29) by choice of prices © p R i ;p A i ª i=1;2 , in the last two stages, given some ¯xed quality level. We can think of this situation as a degenerate version of the game without the quality-setting stage, or as some case where quality turns out to be symmetric in the ¯rst stage. Proposition 17 Fixµ i =µ j . Anequilibriuminpurestrategiesinthepricesubgame does not exist. If a mixed-strategy equilibrium exists, platforms set prices as if each were a monopolist in the advertising market. If platforms are symmetric in the consumer market (identical quality and prices), they necessarily are so in the advertising market as well. But with costly quality, there is a limit to incurring losses in the consumer market: they maynot be covered by advertising pro¯ts. Thanks to randomization, it is optimal for ¯rms to price as if they were a monopolist in the advertising market: indeed, either they are a monopolist, or they are absent altogether. Irrelevance strikes again: competition in the advertising market is ignored at the consumer price-setting stage. 16 This result is extremely sensitive to the choice of cost function. With marginal cost increasing in quality { as opposed to an investment cost{ the sunk-cost problem vanishes. Instead, marginal- costpricing in advertising caninclude the marginal cost of quality. Thus it is as if platforms where setting prices to marginal cost on both sides of the market. The externality is then neutralized and we refer to this phenomenon as independence. 84 Proof: UsingtheresultofLemma15,givenapro¯le(e 1 ;e 2 )andassumingacovered market, the payo®s in the advertising market read (¦ A 1 ;¦ A 2 )= 8 > > > > > > > > > > < > > > > > > > > > > : (¦ A 1 ;¦ A 2 ); if e 1 >e 2 ; (¦ A 1 ;¦ A 2 ); if e 1 <e 2 ; (0;0); if e 1 =e 2 ; (¦ AM 1 ;0); if e 1 >e 2 =0; (0;¦ AM 2 ); if 0=e 1 <e 2 . where ¦ AM i denotes monopoly pro¯ts when ¯rm j reaches no consumer. This can arise when µ i =µ j but p R i <p R j , for then e j =0. For any ¯rm i, total pro¯ts are ¦ 1 =p R i D R i ¡ µ;p R ¢ ¡k(µ i )+ 8 > > > > > > > < > > > > > > > : ¦ A i ; ¦ A i ; 0; ¦ AM i : Whoever ends up with a larger market share in the consumer market necessarily dominates in the advertising market. Claim 11 When µ 1 =µ 2 , a pure strategy equilibrium cannot exist. Proof: Note that for any µ 1 =µ 2 in the consumer market any ¯rm playing p R j >p R i surrenders a monopoly position in the advertising market. It is obvious that any price p R i ¸ k(µ i ) 1 2 D R i ; 8i is dominated by k(µ i ) 1 2 D R i ¡², for then j becomes a monopolist in theadvertisingmarketatthecost ². Supposeµ 1 =µ 2 =µ. Bothengageinthisform of Bertrand competition for the monopoly privilege in advertising until reaching some price p R i such that ¦ 1 = p R i D R i (µ;p R )+¦ AM i = 08i. At this point, j should play some p R j > p R i and stay out, which is costless. But then p R i = p R j ¡² > p R i becomesabestresponse. Thusthereisnopure-strategyequilibrium. Ifµ 1 =µ 2 >µ, 9 p R i 3 ¦ 1 =p R i D R i (µ;p R )¡k(µ)+¦ AM i =08i as well. But at that point, either 85 p R i <p R j and j realizes ¦ j =¡k(µ)<0, or p R i =p R j and ¦ i =¦ j =¡k(µ)<0 since Bertrand competition prevails in advertising as well. So j has a strict incentive to plays p R j < p R i . Following this logic, 9 · p R i 3 ¦ 1 = · p R i D R i (µ;p R )¡ k(µ) + ¦ AM i = ¡k(µ) 8i. If j plays p R j = · p R i , ¦ i = ¦ j < ¡k(µ) while for any p R j > · p R i , ¦ i (p R j ;· p R i )=¦ j (p R j ;· p R i )=¡k(µ). Butthenagainp R i =p R j ¡²> · p R i becomesabest response. Therefore no pure-strategy can exist. Irrelevance also works here, of course. An equilibrium in mixed strategies of this game is a pro¯le ¡ ¾ R ;¾ A ¢ of distributions over prices ¡ p R i ;p A i ¢ i=1;2 , where ¾ A is degenerate. The only source of uncertainty a®ecting the payo®s is the choice of consumer price p R i . 86 3 Chapter 3: Moral hazard and reputation on a two- sided trading platform 3.1 Introduction Marketplaces, stock exchanges and, more recently, internet-based trading platforms bring together sellers and buyers. These markets are fraught with externalities and as a result have attracted the attention of economists, who have developed a con- venient analytical framework to handle them, dubbed two-sided markets. Typically the actors involved have a limited set of instruments to collect revenues, whether optimally by design or as a result of history. It follows that attempts to internalize the externalities arising between trading parties entails pricing distortions. In this paperweextendaconvenientandwell-acceptedmodelbyintroducinganadditional source of distortion: we let one side (the sellers') have the choice of taking a good or a bad action. That is, we allow for moral hazard. After all, trading platforms (such as e-Bay, for example) exert little to no control on the price nor the quality of the commodities sold. They do not observe any of the many items traded, and do not even vet their members { unlike say NASDAQ, whose members are licensed. Opportunisticbehaviorshouldreasonablybeexpected. Tocapturethisissue, welet the seller's action directly a®ects the buyer's payo®s and taking the wrong action is sociallycostly. Ourfocusisnot onthepriceoftheitemtrading,whichforallintents and purposes we take as exogenous. Rather we concern ourselves for the fees the platform can charge for intermediation in the face of such a problem. The research question is con¯ned to ¯nding optimal prices in the presence of moral hazard. We claim two important results. First, with linear prices only, the platforms optimally adjusts both the buyers' fee and the sellers' fee downwards. Thus we do not observe the so-called \topsy-turvy" e®ect that usually obtains on two-sided markets. On the buyers' side, moral hazard depresses the expected value of each 87 trade, and therefore the willingness to pay for the match. Absent a response from the platform, this can be quite severe and because buyer demand for transactions and seller demand are complementary in this model, it may substantially a®ect its pro¯t. To alleviate the impact of a drop in buyers' willingness to pay, the intermediary provide the sellers with incentives to take the high action. It does so by increasing the value of trade each period, that is, by decreasing the sellers' fee. Thus we have a legitimate price decrease on the buyers' side and what we call a mitigatingprice distortion onthesellers'(Section3.5.3). Inequilibriumtherestillis a measure of sellers who do deviate, however this can be construed as a second-best outcome. Thisresultisshowntoberobusttoalterationsinthedistributionofsellers resulting from progressive exclusion of deviating sellers sellers (Section 3.7). The secondresultstatesthatwithupfrontpayments(tojointheplatform,say)andlinear fees, the platform can overcome the moral hazard problem entirely. That is, it can fully replicate the benchmark allocation, in which no seller takes the bad action. Introducing lump sum fees drastically modi¯es the sellers' incentives. It can be conceived of as a bond that the sellers pay in advance, and recover through future trade. Mathematically, it turns the participation decision into an intertemporal decision, as is the incentive constraint. This di®ers from the case of linear fees where the participation decision is a one-period problem only. With appropriate payments, the critical value de¯ning the measure of participating and deviating sellers can be made to exactly coincide, hence no one deviates in equilibrium. With this in hand, we attract attention to the facts that (1) registration and transaction fees are no longer equivalent { unlike in Rochet and Tirole (2004) and Armstrong (2006), and (2) subsidies need to be carefully targeted: paying sellers to play, in this model, is damaging { in contrast to Caillaud and Jullien (2003). This work also contributes to the discipline's understanding of moral hazard. While the ¯rst- best can always be achieved in a standard principal-agent model with appropriate transfers, here the participation of buyers depends on the action of sellers and the 88 incentivesofsellersdependsontheparticipationofbuyers. Thatis, whetheraseller can be induced to take the high action depends not just on his transfers, but also on those implemented on the other side. At the heart of this chapter is the following. Buyers can form an \opinion" of the platform, which corresponds to the probability that a seller they are matched with will take the good action (a). This opinion really is the reputation of the platform, which is not based on a theory of information. Instead, reputation is to beunderstoodasanequilibriumoutcome. Theplatform'spricesgenerateabehavior onthepartofsellers,whichbuyersanticipateandoptimallyrespondto. Conversely, the prices also induce a behavior from buyers, that sellers optimally reply to. In equilibrium these behaviors have to be consistent. This de¯nes the equilibrium reputationoftheplatform. Agoodreputationishelpfultotheplatform, inthatthe buyers' expected value of a trade and the social surplus increase with reputation. The platform can therefore charge them a higher fee for each trade. We consider a set of long-lived agents (platform and sellers) and call on intertemporal incentives to discipline sellers, and steer the platform's reputation. On the one hand a seller taking the low action (a) is detected (ex post) with some probability (less than one) and can be excluded. On the other, if not deviating, a seller can be guaranteed to have access to the platform the following period. This is akin to a grim-trigger strategy in a repeated game. This speci¯c punishment may be extreme, and it may even not be the optimal punishment, but it is quite a natural one. For example, e-Bay reserves the right to exclude participants if they misbehave. As brie°y mentioned, price formation for the good traded is abstracted from; that is, the model we thoroughly analyze entails two-sidedness by construction (it followsRochetandTirole(2004)). Thisisakintosayingthatthecommodity'sprice is exogenous. The following two examples justify this approach. 89 Example 1 Posted prices. Consider a website such as half.com (now owned by e-Bay). Sellers compete by posting prices for homogenous goods. Take therefore p to be the competitive price level and let ¯(a) denote a buyer's gross valuation with ¯(a)» ~ F(¯(a)), log-concave. Fix a and de¯ne b(a)´maxf0;¯(a)¡pg { this is the idiosyncratic bene¯t from trading on the platform. Facing t b a consumer will derive a positive net surplus as long as b(a) ¸ t b , hence buyers' demand for transaction is D b ´ Pr ¡ b(a)¸t b ¢ = 1¡F(t b ) { a shift of ~ F(:). Similarly, let ¾(a)» ~ G(¾(a)) be a seller's valuation (say, a reservation value). The gross surplus is then s(a)´ maxf0;p¡¾(a)g and sellers' demand is de¯ned as D s ´Pr(s(a)¸t s )=1¡G(t b ). This is exactly the framework of our simple model. Example 2 Creditcardswith no-surchargeruleThisisasimpli¯edexposition of the case analyzed by Rochet and Tirole (2003). Consumers receive a private bene¯t b from using a credit card over another form of payment, for which they pay t b . Merchants may also derive an idiosyncratic bene¯t s and pay t s to their ¯nancial institution for each transaction. In essence the no-surcharge rule prevents retailers from passing on the cost t s to consumers. Hence each side's demand, D b ´ Pr(b¸ t b ) and D s ´Pr(s¸t s ), fails to internalize the total cost (t s +t b ). This too ¯ts our simple model. More generally, the manner in which trading parties determine the commodity's price a®ects whether two-sidedness arise. For example, in Caillaud and Jullien (2003), the intermediary is not a two-sided one without registration fees. In such a case the moral hazard problem is exacerbated. Indeed, when two-sidedness fails the platform e®ectively uses a single price, hence it cannot implement the distortion de- scribedearlier. Ifupfrontpaymentsareallowed,ourresultstands: moralhazardcan be entirely overcome. Intuitively, lump-sum payments re-introduce two-sidedness, hence it is natural that the result holds. More precisely, at most three prices are required to deal with moral hazard. 90 The next section reviews the relevant literature. We then lay out the model and present an educational case. Section 3.5 introduce the benchmark and analyzes the moral hazard problem with linear fees only. The next one introduces upfront payments and Section 3.7 presents a robustness check. Last we conclude. All the proofs are in the Appendix. 3.2 Literature This paper lies as the crossroad of the newly developed literature on two-sided mar- kets, and some older work on the role of intermediaries. The early two-sided market literature (Rochetand Tirole, (2002,2003)), Armstrong (2005), Caillaud and Jullien (2003)havefocusedoncharacterizingtheoptimalpricingruleofamonopolistunder di®erent governance models, on showing the equivalence of up-front fees and linear prices and have introduced competition between platforms, respectively. Caillaud and Jullien (2003) introduce the notion of \divide-and-conquer strategies", consist- ing of subsidising one side and extracting from the other one. We build on the Rochet-Tirole model, which o®ers the most °exibility, and stay with the monopoly problem, but introduce moral hazard. Biglaiser (1993) and Biglaiser and Friedman (1992) investigate the role of intermediaries. This approach endows said interme- diary with a superior technology (either innate or acquired at a cost) to inspect goods, which provide him with private information . The intermediary is long-lived and therefore wants to build a reputation for reliability. In contrast, our platform does not inspect commodities, nor does it take ownership. Therefore it does not set prices for the commodities, nor does it hold information it could signal. Instead its monitoring technology informs it ex post, and it can only alter the fees it charges its users. Yet it is able to play a socially bene¯cial role, both by matching agents and by disciplining the potential deviating one. 91 3.3 Model There is a single platform that o®ers intermediation services. In each period, there is also a continuum of buyers and sellers needing intermediation { they cannot ¯nd each other by themselves. Conditional on patronizing the platform, buyers and sellers match every period with exogenous probability ¹ 2 (0;1) { the matching technology. The common discount factor is ± 2 (0;1) and where required, time is indexed by ¿. Whenever a buyer and a seller matched through the platform complete a trade, they must pay transaction fees t b and t s , respectively. These fees are set in advance by the platform. Agents are heterogenous; s2 S = [s;s], s» F(:) denotes sellers' type. If matched, a seller can choose an action a 2 fa;ag; a seller of type s who takes action a values a transaction according as v(s;a)= 8 > < > : s+d; a=a; s; a=a. Action a is costly to the seller as compared to action a. The net payo® per transaction to a seller (taking into account transaction fee) is then v(s;a)¡t s . The buyers' type follows b» G(:) on a bounded support B = £ b;b ¤ . A buyer of type b values a transaction with a seller taking action a followingu(b;a)= 8 > < > : b; a=a; b+h; a=a. Thus u(b;a) > u(b;a) for all b and u(b;a) is (strictly) increasing in b for all a 2 fa;ag. The net payo® per transaction to a buyer is u(b;a)¡t b . Throughout we impose Assumption 14 h>d. whenceu(b;a)+v(s;a)>u(b;a)+v(s;a)forallsandb. Inanytransaction,thehigh action (high-cost action) a is socially bene¯cial. It is assumed that the distributions F(:) (with density f(:)) and G(:) (with density g(:)) are log-concave, atomless and with everywhere positive densities. Information: In the model we analyze it is irrelevant whether the seller observes the buyer's type because we abstract from the price at which they trade the com- 92 modity. A further discussion on this point is postponed to the end of this paper. However, the fact that the buyer does not observe the type of the seller is relevant because the seller's choice of action a may depend on his type. Thus assuming that the buyer does not observe the seller's action before the transaction is completed a®ords a role for reputation. Without this information the buyer's valuation of a transaction (i.e., the buyer's valuation before the transaction takes place) depends on his expectations about the seller's action. The platform also does not observe sellers and buyers' types, however it does have a(n) (imperfect) monitoring tech- nology that a®ords it ex post information. Speci¯cally, whenever a seller deviates by choosing the low action a, the platform observes it with some probability. The platform never receives a wrong signal of deviation: no signal is sent if the seller does not deviate. Weconsiderasituationwhereactionsarenotveri¯able,orwherecontractingcosts are prohibitive compared to the gains from trade. Therefore contracts specifying trade contingent on the seller's action a are not feasible. If such contracts were feasible, the fact that the seller's action is not observed by the buyer before the transaction takes places becomes irrelevant, and so does reputation. Rewards and Punishment: Since contracting has been ruled out by design, we callonintertemporalincentivestosustainanyhigh-actionequilibrium. Weconsider a natural form of punishment. If detected, deviating sellers are excluded from the platform. That is, they are prevented from trading through the platform forever after. Trading platforms such as e-Bay, for example, do reserve the right to exclude sellers whose behavior they deem inappropriate. For ease of exposition we postpone a precise de¯nition of the equilibrium concept until it becomes necessary. Inthissetupwheretheplatformusessellersexclusiontomitigatethemoralhazard problem, we formalize the idea of equilibrium reputation of the platform, analyze how fees a®ect that reputation, and address the issue of optimal fees. 93 3.4 A simple background case Beforeexposingtheanalysisofthemodelundermoralhazardwepresentasimpli¯ed version for expositional purposes. We characterize optimal prices when sellers can take only one action, which is observed by the platform. Assume for concreteness that this action is a in the ensuing exposition. Given a match between a buyer of type b and a seller of type s and transaction fees ht s ;t b i, the buyer accepts to transact if and only if b + h¡ t b ¸ 0 and the seller if and only if s¡ t s ¸ 0. As a consequence, D b (t b ) ´ 1¡ G(t b ¡ h) and D s (t s )´1¡F(t s ) correspond, respectively, to the proportion of buyers and to the proportion of sellers who accept to trade given a match. Thus, given transaction feesht s ;t b i, the platform's pro¯t is given by ¦(t s ;t b )= 1 X ¿=0 ± ¿ ¹D b (t b )D s (t s )[t s +t b ]. Since F(:) and G(:) are time invariant, the platform maximizes its discounted (present)valueofpro¯tsbychoosingthetransactionfeesthatmaximizetheperiod- pro¯t. Thus, the platform solves max t s ;t b ¼(t s ;t b )=¹D b (t b )D s (t s )[t s +t b ]. Re-arranging the ¯rst-order conditions, we obtain that the optimal fees t s and t b must satisfy t s +t b =¡ D b (t b ) D 0 b (t b ) =¡ D s (t s ) D 0 s (t s ) , (42) which are the Rochet and Tirole (2003) optimality conditions. Equivalently the second condition in (42) writes t b " D b ;t b = t s " D s ;t s , 94 where " D j ;t j ´ D 0 j £ t j D j represents the transaction-fee t j elasticity of D j , j = s;b. UnderourstandardassumptionsaboutthedistributionsF(:)andG(:),(42)becomes t s +t b = 1¡G(t b ¡h) g(t b ¡h) = 1¡F(t s ) f(t s ) (43) and admits a unique solution since the inverse hazard rate of a log-concave function is decreasing. As exposed by Rochet and Tirole (2003), the optimality conditions equalize the markups extracted from each side of the market. For completeness, if instead the only action that the seller can take is the low action a, the fees that maximise the platform's pro¯ts satisfy t s +t b = 1¡G(t b ) g(t b ) = 1¡F(t s ¡d) f(t s ¡d) . (44) With this in hand we can claim Proposition 18 The optimal fees satisfy 1. t s <t s ; 2. t b >t b and 3. t s +t b >t s +t b . Thesumofpricesforeachtransactionincreasesbecausethehighaction aissocially valuable. As the social surplus increases, the platform's share (for each transaction) alsoincreases. Thereaderwillalsonoticethatthispriceincreaseisnotuniform: the buyers' price is higher if sellers take the good action, while the sellers' price drops. Proposition 18 can be appended with the following corollary. Corollary 6 ¦(t s ;t b )>¦(t s ;t b ). Thus the platform bene¯ts from the sellers taking the high action a and would obviously prefer implementing it. 95 3.5 Moral hazard, platform reputation and linear fees Let now sellers have the choice of a 2 fa;ag. Under the punishment regime, a seller who takes a low action may be excluded with probability ®. In what follows, assume that the distribution of sellers F(:) and that of buyers G(:) remains time invariant,in spite of exclusion. Thatis,whenevertheplatformexcludessomesellers for choosing action a, they are replaced by new identical sellers. In Section 3.7 we analyze and discuss the main implications of relaxing this requirement. Let r denote reputation of the platform in the eyes of the buyers. This is the proportion of sellers among those that transact on the platform, who are expected to take action a. Our focus is on the platform's reputation, not that of individual sellers. 17 Asusualintheanalysisoftwosidedmarkets,westartbystudyingbuyers'optimal decisions given the behavior on the seller side and sellers' optimal decisions given the behavior on the buyer side. Next, these two are put together to obtain platform equilibrium transactions and reputation. Buyers' transaction decisions Fix seller participation D s and reputation r. If matched the buyer faces the choice of whether to accept to trade. She does not observethe seller's action or type. Conditional on a match, her expected valuefrom the transaction is ru(b;a)+(1¡r)u(b;a)¡t b , which is equal to b+rh¡t b . Hence the buyer accepts the transaction if and only if b+rh¡t b ¸0. We can then de¯ne D b (t b ;r)´ Z b¸t b ¡rh g(b)db=1¡G(t b ¡rh). (45) 17 Arguably, one may attempt to construct a platform's reputation from the sellers', along the lines of Tirole (1996). 96 D b (t b ;r)correspondstotheproportionofbuyerswhoaccepttotradeontheplatform given the transaction fee t b and reputation r; it represents the buyers' demand for transactions. Following the assumptions on the buyer's distribution G(:), D b (t b ;r) is continuous, decreasing in t b and increasing in r. Therefore reputation is valuable to the platform. Sellers' transaction decisions Fix buyer participation D b . If matched a seller faces two choices: whether to accept the trade and which action a2fa;ag to take. Both decisions depend on the seller's payo® from the current transaction, and on the continuation value of future trades through the platform. For a seller of type s who is allowed to trade on the platform, let V(s) denote the present value of the expected future payo® at the beginning of a period before he knows whether he will be matched to a buyer in that period. Because the problem under consideration is stationary, V(s) satis¯es the following (Bellman) equation V(s) = ¹D b maxf0+±V(s);s+d¡t s +(1¡®)±V(s);s¡t s +±V(s)g +(1¡¹D b )±V(s). (46) With probability ¹D b the seller is matched with a buyer who wishes to trade. In that case, his faces three options. First, the seller may decide to not trade and gets 0;heissuretoremainontheplatformthefollowingperiod,inwhichhewillreceived V(s) suitably discounted by ±. Second, the seller may decide to trade and to choose action a, collect s+d¡t s , and to remain on the platform only with probability 1¡®. Third, the seller may decide to trade and select action a, in which case he receives s¡t s and is sure to remain on the platform. Last, with probability 1¡¹D b the seller is not matched with a buyer willing to trade and therefore can only wait until next period. Let us ¯rst analyze a seller's decision whether to trade upon a match { the participation decision. 97 Lemma 16 When matched with a buyer, a seller of type s accepts to trade if and only if s+d¸t s . According to Lemma 16, given transaction fee t s , only sellers with a high-enough valuation trade whereas the others do not. Speci¯cally, all the sellers to whom the value of a transaction when they choose the low action a is positive will trade on the platform. We can then de¯ne D s (t s )´ Z s¸t s ¡d f(s)ds=1¡F(t s ¡d). D s (t s )representstheproportionofsellersacceptingatradewhenmatchedgiventhe transaction fee t s . In our framework it is interpreted as the demand for transaction servicesbysellers. Followingtheassumptionsontheseller'sdistributionF(:),D s (t s ) is continuous and decreasing in t s . For future convenience, let s ¤ (t s )´minfs2S :s¸t s ¡dg: Thus, s ¤ (t s )correspondstothecut-o®valueof s abovewhichsellers acceptto trade on the platform and we can write D s (t s )=1¡F(s ¤ (t s )). Consider now a seller facing the choice of actions a or a { this is the incentive problem. A seller takes the socially bene¯cial action a if and only if the immediate gainfromcheatingonthecurrenttransactionv(s;a)¡v(s;a)=ddoesnotexceedthe expected value of the loss that is associated with being excluded from the platform. That is, a seller of type s chooses action a if and only if d·®±V(s). (47) Thus a seller's optimal action depends on his value of trading on the platform. The morehevaluesit, thelesslikelyheistochoosethelowaction; beingexcludedworks 98 as a stronger punishment. Lemma 17 characterizes of the seller's optimal action as a function of the transaction fee t s and the buyers' participation D b . Lemma 17 When matched with a buyer, a seller of type s who accepts to trade (and expects buyers' participation to be D b in the subsequent periods) chooses the high action a if and only if d·® ± 1¡± ¹D b [s¡t s ]. (48) Whencondition(48)holds,aselleroftypesascribeasu±cientlyhighvalueV(s)to trading on the platform so that choosing the high action is optimal. From condition (48)itisimmediatethatsellers'incentivestochoosethehighactioninatransaction depend not only on t s but also on the buyers' participation in the platform D b . Thus, the behavior of the sellers is a®ected by the buyers' participation. These cross e®ects between di®erent sides of a market are a central feature of two-sided markets. 18 However this is di®erent from Rochet and Tirole's (2003) or Armstrong (2006) models, where each side is essentially myopic and only responsive to prices. Let s ¤¤ (t s ;D b )´min ½ s2S :d·® ± 1¡± ¹D b [s¡t s ] ¾ . Because the right-hand-side of (48) is increasing in s, s ¤¤ (t s ;D b ) represents the cut- o® value of s above which sellers choose the high action a, given transaction fee t s and buyer participation D b . It is clear that if (48) is satis¯ed then s¸t s , which in turn implies that s ¸ t s ¡d. Thus for all t s and D b , s ¤¤ (t s ;D b ) ¸ s ¤ (t s ). (When 0<D s (t s )<1,thisinequalityisstrict.) Wecanthenconcludethatgiven t s andD b for: s < s ¤ (t s ) sellers do not trade; s2 [s ¤ (t s );s ¤¤ (t s ;D b )) sellers trade and choose a; and s¸s ¤¤ (t s ;D b ) sellers trade and choose a. 18 For example, in Caillaud and Jullien (2003), the sellers' decision to register in the platform depends on the buyers' participation in the platform and vice-versa. In our case, it is the seller's incentive to choose the high action in a transaction that depends on the buyer's participation in the platform. 99 Giventransactionfeet s andbuyer'sparticipationD b ,wecanderivetheproportion of sellers that choose the high action among those that trade on the platform. Throughout we denote this proportion by k(t s ;D b ). Formally, k(t s ;D b )´ 1¡F(s ¤¤ (t s ;D b )) D s (t s ) , where 1¡F(s ¤¤ (t s ;D b )) corresponds to the measure of non-deviating sellers. 3.5.1 Benchmark case The proper benchmark to consider is not the result developed in Proposition 18, where either none or all sellers cooperate. Instead we need to allow for a subset of them to do so, depending on the exact problem they face { that is, on their type. Here we conduct a thought experiment as follows. Suppose that the platform can select the reputation r it wants, as well as the transaction fees (t s ;t b ) it charges. In thisproblemtheconstructionofthesellers'demandisdi®erentbecausenoincentive is necessary to induce cooperative behavior. By setting its reputation r, it is as if the platform were able to perfectly monitor traders and decide how many to allow to deviate. That is, a seller who accepts a trade will have to agree to what the platform wants him to do { deviate or not. But for any r2 (0;1) he cannot know in advance what action is expected of him. Therefore his expected utility is v(s;r;a)=rs+(1¡r)(s+d)=s+(1¡r)d andthenetpayo®isv(s;r;s)¡t s ,sothemarginalsellerisgivenbys ¤ =t s ¡(1¡r)d. Ex post some sellers will have negative utility { just like some buyers may. Then sellers' demand is D s =1¡F (t s ¡(1¡r)d) and the platform solves max t s ;t b ;r ¦(t s ;t b ;r)=¹ h 1¡G(t b ¡rh) i [1¡F(t s ¡(1¡r)d)](t s +t b ) 100 Therefore we can claim Proposition 19 Suppose the platform can select the triple ht s ;t b ;ri; then r = 1 and prices satisfy t s =t s ; t b =t b as de¯ned in Proposition 18. Although sellers' demand D s is decreasing in r (because the surplus v(s;a;r) is decreasing in r), a higher reputation is bene¯cial for the platform. This hinges on Assumption 14: the bene¯t of the good action a accruing to the buyers exceeds the cost to the sellers. As the social surplus increases, so does its appropriation by the buyers. Hence it is possible to extract more of it from said buyers (in the form of higher transaction fees) and to redistribute it to sellers (in the form of lower transaction fees) so as to secure their participation. The optimality conditions (43) and (44) exactly balance this trade o®. Forthcoming results will be contrasted against this benchmark. 3.5.2 Equilibrium reputation and transactions Withthesepreliminaryresultswecanturntotheissueoftheplatform'sequilibrium reputation and transactions (i.e., participation) given transaction fees t b and t s . De¯nition 2 (Equilibrium transactions and reputation) Given transactions fees t b and t s , an equilibrium is a triple hD ¤ b ;D ¤ s ;r ¤ i such that: 1. D ¤ b =D b (t b ;r ¤ ), 2. D ¤ s =D s (t s ) and 3. r ¤ =k(t s ;D ¤ b ) whenever D s (t s )>0. Equilibrium transaction volumes and reputation are then characterized by three conditions: the buyers' transaction decisions are optimal given transaction fee t b andplatformreputation; thesellers'participationdecisionsareoptimalgiventrans- actionfeet s ;andplatformreputationisconsistentwiththesellers'optimalbehavior regarding action a given transaction fee t s and buyer participation. 101 The determination of equilibria with a positive volume of transactions consists essentially of the determination of the equilibrium reputation. Note that given a platform's equilibrium reputation, the equilibrium volume of transactions on the platform can be immediately pinned down. We therefore focus on the existence and characterization of equilibrium reputation in the remainder of this section. From De¯nition 2 it follows that r ¤ is an equilibrium reputation of the platform when transaction fees are t b and t s and D s (t s )>0 if and only if r ¤ =k(t s ;D b (t b ;r ¤ )). (49) Lemma 18 provides conditions under which an equilibrium reputation exists. Let ¨=ft s :D s (t s )>0g. Lemma 18 For all t s 2 ¨ there exists a(n equilibrium) reputation satisfying (49). Furthermore if t s 2 ¨ is such that D s (t s ) < 1, then there exists a(n equilibrium) reputation r ¤ 2[0;1) satisfying (49). Ingeneraltheremayexistmultipleequilibriumreputationsforanygiventransaction fees t s and t b . In the remainder of the paper, we focus on the highest of them and denote it by r ¤ (t b ;t s ). The highest equilibrium reputation for given transaction fees is the equilibrium with the highest volume of transactions on the platform and also the equilibrium with the highest platform's pro¯ts. Hereafter we also refer to r ¤ (t b ;t s ) as the platform equilibrium reputation. Next we analyze the impact of transaction fees t s and t b on the equilibrium reputation r ¤ (t b ;t s ). Proposition 20 For all t s 2¨, r ¤ (t b ;t s ) is non-increasing in t b and t s . A higher transaction fee t b for buyers implies that fewer buyers trade given a match with a seller. For any seller, this decreases the expected future value of being able to trade through the platform, which reduces the e®ectiveness of exclusion from the platform as a punishment device. The incentive to choose the more costly action a 102 is therefore less powerful, and in equilibrium reputation of the platform drops. The impact of increasing the transaction fee t s for sellers is slightly more intricate. A highertransactionfeeforsellersa®ectsnotonlythesellersdecisiontotradethrough theplatformbutalsotheirincentivestochooseactiona. Inparticular, higherprices reduce the sellers' participation D s . But a higher transaction fee also depresses the sellers' expected future value of being able to trade to the platform, reducing their incentives to choose a. As a result, fewer sellers { only those with high valuation { choose action a. Thus a higher t s implies that less sellers choose action a, but alsolesssellerswithlowvaluationthatwouldchooseaction aelecttotradethrough the platform. Whether increasing t s has a positive or negative e®ect on reputation depends on the interplay between those two e®ects. In turn, these depend on the sensitivityofthesellers'valuationv(s;a)onsandonthedistributionofsellersF(s). With our speci¯cations the result obtains. 3.5.3 Optimal linear fees The platform's problem is max t s ;t b ¦(t b ;t s )= X ¿¸0 ± ¿ ¹D b (t b ;r ¤ (t b ;t s ))D s (t s )[t s +t b ]. (50) In the presence of moral hazard, (50) tells us that the choice of the transactions fees a®ects the platform's pro¯t through a new channel: they enter equilibrium reputation, which in°uences the demand for services by the buyers. This e®ect is important for the determination of optimal transaction fees. Stationarity implies thatasinthebenchmarkcaseofSection3.5.1,theplatformmaximizesitsdiscounted (present)valueofpro¯tsbychoosingthetransactionfeesthatmaximizetheperiod- pro¯t ¼(t s ;t b )=¹ h 1¡G(t b ¡r ¤ h) i [1¡F(t s ¡d)] ³ t s +t b ´ . 103 If the pro¯t function ¼ is di®erentiable at the solution to the platform's pro¯t prob- lem, the optimal fees b t s and b t b satisfy the following ¯rst-order conditions (ignoring functions' arguments and assuming positive transaction volumes D s and D b ) b t s + b t b = 1¡G(t b ¡r ¤ h) g(:) ¡ 1¡ @r ¤ @t b ¢ = [1¡F(t s ¡d)][1¡G(t b ¡r ¤ h)] f(:)[1¡G(:)]¡g(:) @r ¤ @t s [1¡F(:)] , (51) where @r ¤ =@t b · 0 and @r ¤ =@t s · 0 from Proposition 20. Condition (51) can be written in terms of elasticities to obtain b t b " D b ;t b +" D b ;r [" r ¤ ;t b +" r ¤ ;t s t b ts ] = b t s " D s ;t s , where" D b ;r ´(@D b =@r)(r=D b )representsthereputationelasticityofD b ,and" r ¤ ;t j ´ (@r ¤ =@t j )(t j =r), j = b;s, represents the transaction-fee t j elasticity of the equilib- rium reputation of the platform. Evidently if reputation does not a®ect buyers' demand (i.e. @D b =@r = 0) or transaction fees do not a®ect the equilibrium repu- tation of the platform (i.e. @r ¤ =@t s = 0 and @r ¤ =@t b = 0) then condition (51) is identical to (42): this is the standard case. More broadly, not only there is a new term capturing the impact of reputation on demand enters each of the ¯rst-order conditions,thisreputationcannotbeperfectlycontrolled. Thisisthesourceofprice distortions. Our ¯rst main result follows. It mirrors Proposition 18 and provides us with a ranking of prices in these two environments. Proposition 21 ^ t s <t s and ^ t b <t b . Moral hazard depresses both prices for the platform. To explore the exact mecha- nism, ¯x t s for now. Knowing that some sellers will systematically deviate (Lem- mata 17 and 18), it is obvious that the buyers revise their expected value of a trade downwards. The platform responds by lowering the transaction fee to buyers. On thesellers'side,keepingfeesatthelevelt s depressesintertemporalincentives,which 104 we know induces reputation r ¤ to drop (Proposition 20). Too low a reputation may leadtoaverylowtransactionfeet b ,oraverylowbuyerdemand,bothofwhichmay be excessively costly because of the complementarity of buyer and seller demand in (50). But the platform can improve on this situation by distorting the sellers' price downwards; then its reputation improves, which alleviates the loss of revenue on the buyers' side. The optimality balances these two e®ects exactly. It requires (1) a price decrease on the buyers' side in the face of moral hazard; and (2) a price distortion on the sellers' side to limit the buyers' price drop. This departs from standard optimality conditions in two-sided markets in that prices decrease on both sides. Clearly the benchmark allocation, in which pro¯ts are the highest, is out of reach with linear fees alone { simply because s ¤¤ (t s ;D b ) > s ¤ (t s ). The platform cannot prevent a measure of sellers, however small, from deviating. 3.6 Registration and transaction fees As evidenced in the preceding analysis, linear fees are not su±cient to mitigate moral hazard. In this section we consider the case in which sellers have to register in order to access the matching technology of the platform. Let T s and T b be the registration fees for sellers and buyers, respectively. These are paid only once by a seller and a buyer. For simplicity we assume throughout that a buyer is in the market for only one period. So at the end of the period the buyer leaves the market regardless of whether she has traded through the platform, and is replaced by an identical buyer. 19 Buyers' registration and transaction decisions A buyer decides to register withtheplatformiftheexpectedvalueoftradeexceedstheregistrationfeerequired by the platform. Given the platform's reputation r, transaction fee t b and seller 19 This assumption simpli¯es the exposition of the results but is immaterial in terms of the analysis. The absence of moral hazard on the buyers' side implies that a buyer's problem is always a static one. 105 participation D s , the expected value of trade for a buyer of type b is U(b)=¹D s [r(b+h)+(1¡r)b¡t b ]=¹D s [b+rh¡t b ]. A buyer of type b registers in the platform if and only if U(b)¡T b ¸0. Let e b ¤ (t b ;T b ;r;D s )´min n b2B :¹D s [b+rh¡t b ]¡T b ¸0 o . Because U(b) is increasing in b, buyers of type b¸ e b ¤ (t b ;T b ;r;D s ) register whereas buyers of type b < e b ¤ (t b ;T b ;r;D s ) do not. Then buyers' registration with the platform is de¯ned as R b (t b ;T b ;r;D s )´1¡G( e b ¤ (t b ;T b ;r;D s )). Sellers(eitherregisteredorconsideringtodoso)careonlyaboutbuyerswhoregister and, given transaction fees, accept to trade. A buyer registers and trades through the platform if and only if b¸maxft b ¡rh; e b ¤ (t b ;T b ;r;D s )g. We can then rede¯ne buyers' participation (those who register and accept to trade given a match) as D b (t b ;T b ;r;D s )´1¡G(maxft b ¡rh; e b ¤ (t b ;T b ;r;D s )g). Sellers'registrationandtransactiondecisions Asellerfacesadi®erentprob- lem here too. Because the registration fee is paid at the outset and only once, it does not a®ect directly sellers' decisions whether to accept a trade and which action a2fa;ag to select. The upfront payment only in°uences the registration decision, not whether to trade any period thereafter. Therefore a seller's value of being able to trade through the platform is also not a®ected directly by the registration fee T s . 20 As before, given transaction fee t s and buyers' participation D b , sellers of 20 It may have an in°uence through the equilibrium reputation of the platform, which at this point we take as given. 106 type s<s ¤ (t s ) never trade through the platform { even if registered; sellers of type s such that s ¤ (t s ) · s < s ¤¤ (t s ;D b ) trade and choose action a; and sellers of type s¸ s ¤¤ (t s ;D b ) trade and choose action a. Using this, the solution to the Bellman equation (46) and the value of trade to a seller of type s reads V(s)= 8 > > > > < > > > > : 0 if s<s ¤ (t s ) ¹D b 1¡±[1¡®¹D b ] [v(s;a)¡t s ] if s ¤ (t s )·s<s ¤¤ (t s ;D b ) ¹D b 1¡± [v(s;a)¡t s ] if s ¤¤ (t s ;D b )·s. V(s) is continuous and increasing in s. Naturally a seller of type s registers if and only if V(s)¡T s ¸0. (52) Let e s ¤ (t s ;T s ;D b )´minfs2S :V(s)¡T s ¸0g. BecauseV(s)isincreasingins, sellersoftypes¸e s ¤ (t s ;T s ;D b )paytheregistration fee and others do not. Let R s (t s ;T s ;D b )´1¡F(e s ¤ (t s ;T s ;D b )) denote the measure of registered sellers. From the buyers' perspective the relevant sellers are those that are registered and trade, not those only registered. Given fees t s and T s and buyers' participation D b , a seller does so if and only if his type s¸ maxfs ¤ (t s );e s ¤ (t s ;T s ;D b )g. Hence the sellers' participation (those who register and trade) D s is rede¯ned as D s (t s ;T s ;D b )´1¡F(maxfs ¤ (t s );e s ¤ (t s ;T s ;D b )g). 107 Byde¯nitionD s (t s ;T s ;D b )·R s (t s ;T s ;D b ). Furthermore,sinceonlysellersoftype s>s ¤ (t s ) have a strictly positive value of being able to trade through the platform, D s (t s ;T s ;D b )=R s (t s ;T s ;D b ) whenever T s >0. Asbefore, wecande¯netheproportionofsellerswhodonotdeviateamongthose that participate. This is the proportion of sellers that choose action a among those that register and trade in the platform. Let k(t s ;T s ;D b )´ 8 > < > : 1¡F(s ¤¤ (t s ;D b )) Ds(t s ;T s ;D b ) if e s ¤ (t s ;T s ;D b )<s ¤¤ (t s ;D b ) 1 if e s ¤ (t s ;T s ;D b )¸s ¤¤ (t s ;D b ). Because of registration, this proportion now depends on the registration fee T s . 3.6.1 Platform's equilibrium reputation and transactions The major di®erence to the case where only transaction fees were considered is that now sellers' participation decisions depend on buyer's participation and conversely. This is because participation by an agent requires registration, and the value of being registered depends on the measure of agent's on the other side of the market. It follows that the de¯nition of equilibrium registrations, transactions volumes and reputation needs to be slightly adjusted. De¯nition 3 (Equilibrium with registration) Given transaction fees t b and t s and registration fees T s and T b , an equilibrium is a tuple hR ¤ b ;R ¤ s ;D ¤ b ;D ¤ s ;r ¤ i such that: 1. R ¤ b =R b (t b ;T b ;r ¤ ;D ¤ s ), 2. R ¤ s =R s (t s ;T s ;D ¤ b ), 3. D ¤ b =D b (t b ;T b ;r ¤ ;D ¤ s ), 4. D ¤ s =D s (t s ;T s ;D ¤ b ) and 108 5. r ¤ =k(t s ;T s ;D ¤ b ) whenever D s (t s ;T s ;D ¤ b )>0. When registration becomes necessary to trade on the platform, the de¯nition of an equilibrium also re°ects the fact that agents' registrations decisions are optimal. In particular the de¯nition of platform reputation remains as before, with only a necessary technical adjustment. Now sellers' participation corresponds to the proportion of sellers who register and accept to trade. Then the reputation of the platform is the proportion of sellers who choose action a, among those that register and trade on the platform (participate). Beforeproceedingtotheanalysisoftheequilibriumreputation(givenregistration and transaction fees), we outline an equivalence result which will be useful through- out. We say that two equilibria are outcome equivalent if the transactions volumes D s and D b , the platform reputation r an the platform's pro¯ts are identical in both equilibria. Lemma 19 Foreachequilibriumwhentransactionandregistrationfeesareht b 0 ;T b 0 > 0;t s 0 ;T s 0 i, there exists an outcome equivalent equilibrium when transaction and reg- istration fees are ht b 1 ;T b 1 =0;t s 0 ;T s 0 i for some t b 1 . RochetandTirole(2005)¯rstestablishedequivalenceofregistrationandtransaction feesinasimplesettingthatisexactlyoursonthebuyersside. 21 Aswewillseebelow, here it will not always be true. Remark 3 In Lemma 19 we consider only equilibria when the registration fee T b is strictly positive. This is without loss of generality, since it is never optimal for the platform to set a negative registration fee T b . That would mean to subsidize registration of buyers that do not trade, which does not increase participation. Thanks to Lemma 19, we can restrict our analysis to the case in which the platform charges registration fees only to sellers, or equivalently T b = 0. Therefore in the 21 In Rochet and Tirole (2005) and Armstrong (2004) as in the buyers' side in our model there is no moral hazard problem. 109 sequel we consider only the case of transaction fees t b , t s and registration fee T s . When buyers do not have to register to trade, the buyers' problem { to trade or not given a match with a seller { depends only on transaction fee t b and reputation r. So, the decision rules are identical to those presented in Section 3.5. Buyers' participation is then given by D b (t b ;r) as de¯ned by (45). IntheremainderofthissectionourapproachmirrorsthatofSection3.5.2. With- out buyer registration, the determination of equilibrium again consists essentially of the determination of the equilibrium reputation. Once equilibrium reputation is characterized,buyerparticipationcanbeimmediatelyobtainedand,withthesetwo, sellers' registration and participation. Given fees t b , t s and T s , the determination of equilibrium reputation corresponds to ¯nding some r ¤ that satis¯es the following condition r ¤ =k(t s ;T s ;D b (t b ;r ¤ )). (53) ThisistheanalogtoCondition(49). Themaindi®erenceisthatnowtheright-hand side of (53) can be 1 if T s is high enough; in such a case, sellers who register have a valuation that is so high that they optimally choose action a whenever matched with a buyer. Also, because now the sellers' participation depends on the buyers' participation, t b and r also enter the denominator of k(t s ;T s ;D b (t b ;r)). Lemma 20 Suppose that t b , t s and T s are such that D s (t s ;T s ;D b (t b ;r)) > 0 for all r2[0;1]. Then, there exists a(n equilibrium) reputation satisfying (53). InthecomplementarycasewhereD s (t s ;T s ;D b (t b ;r))=0forsomer2[0;1],thefact that D s (t s ;T s ;D b (t b ;r)) is increasing in D b and D b (t b ;r) increases in r implies that D s (t s ;T s ;D b (t b ;0)) = 0. Hence there exists an obvious equilibrium in which r = 0 and no trade occurs. Next we turn to the behavior of the equilibrium reputation of a platform with respect to the variables t s ;t b and T s . As in Section 3.5.2, we focus on the highest-reputation equilibrium, which we naturally denote by r ¤ (t b ;t s ;T s ). Again, let b ¨=f(t b ;t s ;T s ):D s (t s ;T s ;D b (t b ;r))>0;8r2[0;1]g, then 110 Proposition 22 8 (t b ;t s ;T s )2 b ¨, 1. r ¤ (t b ;t s ;T s ) is non-decreasing in T s ; 2. r ¤ (t b ;t s ;T s ) is non-increasing in t s ; 3. r ¤ (t b ;t s ;T s ) is non-increasing in t b if d¸®±¹(1¡± 2 )=[(1¡±)+®±¹]. Transaction fees t b and t s a®ect the equilibrium reputation of the platform in two ways. First, they in°uence the sellers' participation decision. Second, they have an impact on the incentive of the sellers who participate to choose the high action a. While the ¯rst e®ect is positive, the second one is negative. Thus, the aggregate e®ect of transaction fees t b and t s on the equilibrium reputation of the platform de- pends on the magnitude of these two e®ects. Under the assumptions in Proposition 22, the second e®ect is dominant. When it comes to the registration fee T s , only the ¯rst e®ect is present. While the registration fee a®ects (negatively) the sellers' participation by reducing the participation of sellers with low valuation, it does not a®ect sellers incentives regarding the action a. So by increasing registrations fees a platform can increase its reputation. Next we study how this bears on the determination of the platforms registration and transaction fees. 3.6.2 On the role of registration and transaction fees So far we have established that we could ignore registration fees on the buyers' side { registration fees and transaction fees are equivalent to the risk-neutral, un- constrained buyer. On the sellers' side, these two prices play very di®erent roles. Indeed, our next important result below implies that Lemma 19 does not have an equivalent on the sellers' side. Proposition 23 Consider the benchmark, i.e. ht s 0 = t s ;t b 0 = t b ;r = 1i. There exists a set of fees ht s 1 ;T s 1 ;t b 1 i that generates the same allocation. 111 That is, the tripletht s 1 ;T s 1 ;t b 1 i generates the same buyer demand D b , the same seller demand D s and the same pro¯ts as can be achieved in the benchmark. So registra- tion fees can be so helpful as to implement the best-possible allocation de¯ned in Section 3.5.1. The reason turns out to be remarkably simple and widely applicable. With linear fees only, cooperation is supported by intertemporal incentives while the participation decision is relevant for the current period only. In contrast, the lump-sum payment force sellers to consider the present discounted value of partici- pation. Althoughitisaonce-onlydecision, italsoisan intertemporal problem. The appropriatemixoftransactionandregistrationfeeensuresthatthemarginalpartic- ipating seller never deviates, and may be made to correspond to the marginal seller in the absence of moral hazard. Buyers behave correspondingly and the measure of participating buyers is exactly the one obtained in the benchmark case. 3.6.3 Optimal fees Given that the set ht s ;t b ;r = 1i is one that maximises pro¯ts, the corresponding allocation is the one preferred by the platform. Therefore Proposition 24 Withregistrationfees,theplatformimplementthetripletht b 1 ;t s 1 ;T s 1 i such that t b 1 =t b , T s 1 = d ®± and t s 1 =s ¤ 0 ¡ (1¡±)d ±¹D 0 b <t s . Notice that T s = d ®± ; so it is as if participating sellers commit to not deviating by foregoing the expected present value of the deviation gain d upfront. They are then rewarded with lower transaction fees forever after. 3.6.4 Discussion The lump-sum payment T s > 0 acts as a bond that sellers deposit with the plat- form, which can then distort the sellers' price downward to some t s < ^ t s to provide incentives to recover its value through a higher period-surplus. Absent the bond, there are too many sellers compared to the benchmark since ^ t s < t s . So the plat- 112 form introduces T s and decreases t s to replicate the allocation ht s ;t b ;r ¤ = 1i. As in Lemma 19, t s and T s are substitutes in both the participation condition and the incentive constraint { they just do not substitute at the same rate in each instance. Increasing T s beyond d ®± no longer alters the incentive constraint, but it does re- duce participation further. The platform can only use the registration fee T s to increase ~ s ¤ (t s ;T s )uptos ¤ : increasingt s insteadwouldcreateanewwedgebetween s ¤ (t s ;T s ) and s ¤¤ (t s ;T s ). AccordingtoProposition23transactionandregistrationfeesarenolongergener- ically interchangeable, in contrast to the buyers' side (Lemma 19). This departs from Rochet and Tirole's (2005) canonical model, where they show equivalence of both kinds of payment. The moral hazard problem voids the equivalence result of Lemma 19 for the simple reason that transaction fees and registration fees play a di®erent role. The former enters both the incentive constraint and the participation decision, while the latter is neutral on incentives. In the absence of moral hazard of course there is no incentive constraint to satisfy, so that these two prices can substitute perfectly in the participation decision. The platform can never reach the ¯rst-best with transaction fees alone: moral hazard can be completely addressed only if introducing registration fees. On another point, T s >0 necessarily: any T s <0 worsens the incentive problem bydrivingawedgebetweentheparticipationcut-o® ~ s ¤ (t s ;T s )andtheincentivecut- o® ~ s ¤¤ (t s ;T s ). As a result, `divide-and-conquer' strategies 22 advocated by Caillaud and Julien (2003) and Hagiu (2006), albeit in a model of price competition, need to be more carefully considered. Subsidizing the side subject to moral hazard likely does more harm than good. 22 In a model of competition between platforms, DC strategies consist in subsidizing one side to attract it on the platform and extracting surplus from the other 113 3.7 Incomplete replacement and the cleansing e®ect Theanalysisconductedsofarreliesheavilyonthestationarityoftheproblem,which owes to the assumption of full replacement of excluded sellers. In what follows we relax it and brie°y characterize the impact of incomplete seller replacement on the equilibrium reputation. Incomplete replacement induces a progressive truncation of the distribution of sellers remaining. That is, we forego stationarity. To keep the analysis simple, we limit ourselves in this section to a three-period model. To characterize the impact of incomplete seller replacement, our strategy is to ¯rst de- velopa¯nite-horizonbenchmarkwithfullreplacementandthenassessthedi®erence between said benchmark and its equivalent with incomplete replacement. 3.7.1 Finite horizon with full replacement This section develops a three-period model with ¿ = 0;1;2. It reproduces some of the results of the in¯nite-horizon model tobe usedas a benchmark against which to compare the impact of incomplete replacement. To isolate the impact of cleansing on the equilibrium reputation and to allows us to compare these results to those in the in¯nite horizon case studied so far, we assume the following. Assumption 15 The platform can commit to a sequence of prices © t s ¿ ;t b ¿ ª 2 ¿=0 . Assumption 16 The platform uses constant prices, viz. t i ¿ = t i ; i = s;b; ¿ = 0;1;2. Assumption 15 implies that the platform can commit to prices at the outset that may be suboptimal ex post (i.e., in periods 1 and 2). Under assumption 16 the platform's prices are the same in every period. The buyers' demand for transactions is the same as before, so we turn directly to thesellers'decisions. Becauset s ¿ =t s for¿ =0;1;2(Assumption16), ineachperiod ¿ a seller of type s accepts to trade if and only if v(s;a)¸t s . Thus, for each period 114 ¿, the seller participation cuto® is s ¤ (t s ). As a consequence, sellers' participation is the same in all periods and given by D s (t s )=1¡F(s ¤ (t s )). Consider now the sellers' choice of action a. Unlike in the in¯nite-horizon model the platform faces three di®erent incentive problems, one in each period. Hence there are three relevantthresholds s ¤¤ ¿ , ¿ =0;1;2. It is obviousthatall sellers select the low action a in the last period: intertemporal incentives only apply to periods 0 and 1. In period ¿ =0;1 the incentive constraint reads d=v(s;a)¡v(s;a)·®±V ¿+1 (s), (54) where V ¿ (s) denotes seller s discounted value, at the beginning of period ¿, when he is allowed to trade on the platform. That is, conditional on a match, a seller of type s does not deviate if the gain from deviation is no greater than the expected valueofthepunishment{foregoingV ¿+1 (s). Accordingto(54), theincentivetonot deviate today is provided through next periods' payo®s. Therefore the relevant cut- o®pointcharacterizingthemarginalsellerdependsonnext periods'transactionfees and reputation not today's. While is also true in the in¯nite horizon, the problem remains stationary so that today's action and expected value of future payo®s are the same as tomorrow's. In this ¯nite-horizon model, the sequence of equilibrium transactions and repu- tation given transaction fees can be obtained recursively starting in the last period. Because in period 2 sellers have no incentive to take the more costly action, the platform's equilibrium reputation in that period, r ¤ 2 , is zero. Thus, given transac- tionfeest b andt s , equilibriumtransactionsandreputationinperiodtwoare D s (t s ), D b (t b ;0) and r ¤ 2 = 0. To obtain equilibrium transactions and reputation in period one we can proceed in the following way. Given that buyer participation in period two is D b (t b ;0), V 2 (s)=¹D b (t b ;0)maxf0;v(s;a)¡t s g. (55) 115 Combining this value and the incentive constraint (54), we obtain the period-one thresholds ¤¤ 1 ,abovewhichsellerschoosethehighaction. Sincesellers'participation is the same in every period and is given by D s (t s ), determining the cuto® s ¤¤ 1 gives us the period-one equilibrium reputation r ¤ 1 immediately. Equilibrium transactions in period one are then D s (t s ) and D b (t b ;r ¤ 1 ). Repeating the process, one ¯nds the equilibriumtransactionsandreputationinperiodzero. Speci¯cally, given V 2 (s)and buyer participation in period one D b (t b ;r ¤ 1 ), we have V 1 (s) = ¹D b maxf0+±V 2 (s);s+d¡t s +(1¡®)±V 2 (s);s¡t s +±V 2 (s)g + h 1¡¹D b (t b ;r ¤ 1 ) i ±V 2 (s). (56) Again, combining (56) this and (54) determines the equilibrium threshold s ¤¤ 0 in periodzeroandconsequentlytheequilibriumreputation r ¤ 0 sincesellerparticipation is constant at D s (t s ). Buyer participation in period zero is then D b (t b ;r ¤ 0 ). With this information we can readily claim the following. Lemma 21 Under a ¯nite horizon ¿ = 0;1;2, constant transaction fees and with full replacement s ¤¤ 0 ·s ¤¤ 1 . Furthermore r ¤ 0 ¸r ¤ 1 ¸r ¤ 2 =0. Evidently no seller can be expected to take the high-cost action in the last pe- riod. Total payo®s are decreasing over time, so a seller who does not deviate in the penultimate period necessarily does not deviate in the ¯rst one either. Under the assumptions of Proposition 20 and Assumption 16, the equilibrium reputation r ¤ ¿ ; ¿ =0;1;2 displays exactly the same properties as r ¤ stated in Proposition 20. 3.7.2 Incomplete replacement: the cleansing e®ect In this section we relax the assumption of complete population replacement by adopting the perspective that only a measure of the excluded sellers in any given period is replaced. This phenomenon is assumed to be independent across agents 116 and across periods; that is, it is not predictive of the average lifetime of an agent. While we keep imposing time-invariant prices, the di®erence here is that for any pairht b ;t s i, in each period ¿ a fraction of the population of the sellers with type in [s ¤ ¿ ;s ¤¤ ¿ ] disappears. Thus the distribution of sellers who can trade on the platform may change every period, directly a®ecting the platform's equilibrium reputation. With complete replacement and a ¯nite horizon, the platform's reputation varies over time because sellers' incentives weaken as the last period approaches. With incomplete replacement, the platform reputation may vary over time also because thedistributionofsellersthatareallowedtotradeonitmaychangeovertime. That is, the denominator D s of the ratio 1¡F(s ¤¤ ¿ )=D s may vary. The incentive constraints (54) remain valid, although the de¯nition of the sellers' demand and of reputation need be altered to re°ect the progressive truncation of the original distribution of sellers F. Before proceeding, some useful notation is introduced. Let F denote the measure induced by F on the set of sellers' types [s;s], and F ¿ denote that measure in period ¿, ¿ = 0;1;2. Naturally, F 0 = F. We capture the idea of incomplete replacement of excluded sellers by assuming the following. Assumption 17 (Incomplete seller replacement) For ¿ = 0;1, given cuto® points s ¤ ¿ and s ¤¤ ¿ , 1. F ¿+1 (A)·F ¿ (A) if Aµ[s ¤ ¿ ;s ¤¤ ¿ ]; and 2. F ¿+1 (B)=F ¿ (B), if Bµ[s;s] and B\[s ¤ ¿ ;s ¤¤ ¿ ]=. Under Assumption 17, a portion of those sellers who in a given period are better o® trading and choosing the low action is removed from the next period's pool of sellers. This is the \cleansing e®ect". As in the case of full replacement the seller participation cuto® remains the same in each period and is given by s ¤ (t s ). Still, seller participation may change from one period to the other since the mass of sellers on the platform may change due 117 to cleansing. Speci¯cally, seller participation in period ¿ is given by D ¿ s (t s ) = F ¿ ([s ¤ (t s );s]). Conceptually,ourde¯nitionofplatformreputationremainsthesame. Given cuto® s ¤¤ ¿ , the platform reputation in period ¿ is given by r ¿ = F ¿ ([s ¤¤ ¿ ;s]) D ¿ s (t s ) , whereF ¿ ([s ¤¤ ¿ ;s])correspondstothemeasureofnon-deviatingsellersinthatperiod. To obtain the equilibrium reputations, which we denote by r c¤ ¿ ;¿ =0;1;2 (where the superscript c stands for \cleansing"), we proceed as follows. As before, sellers have no incentive to choose the high action in the last period and therefore r c¤ 2 =0. This determines V 2 (s), which is identical to that obtained in full replacement case. From this and the incentive compatibility constraint (54) of period one, we obtain s ¤¤ 1 , which is also identical to that in the case of full replacement. Unlike before, s ¤¤ 1 is not su±cient to determine r c¤ 1 . The period-one equilibrium reputation is a®ected by the measure F 1 of sellers that are on the platform. Under the transition law implicit in Assumption 17, this measure depends on s ¤¤ 0 and so does reputation in period one. The trouble is that reputation in period one a®ects V 1 (s), which in turn a®ects s ¤¤ 0 . Thus, the equilibrium cuto® s ¤¤ 0 and reputations r c¤ 0 and r c¤ 1 are simultaneously determined by the conditions d = v(s ¤¤ 0 ;a)¡v(s ¤¤ 0 ;a)=®±V 1 (s ¤¤ 0 ), r ¤c 0 = F([s ¤¤ 0 ;s]) D 0 s (t s ) r c¤ 1 = F 1 ([s ¤¤ 1 ;s]) D 1 s (t s ) . (57) Let r ¤ ¿ denote the equilibrium reputation in period ¿ when there is full replacement. The next proposition compares the equilibrium reputations with and without the cleansing e®ect. Proposition 25 For the same prices t s and t b , r c¤ ¿ ¸r ¤ ¿ for ¿ =0;1;2. 118 That is, holding prices constant across the two regimes, cleansing improves repu- tation. The cleansing e®ect alters the equilibrium reputation in two ways. First, it directly reduces the population of lower-type sellers on the platform, who are likely to deviate in future transactions. Second, it in°uences reputation indirectly by providing higher incentives for sellers not to deviate. This is because the sell- ers' continuation value V ¿ (s) increases in reputation, as the buyers' participation increases in reputation as well. 3.8 Conclusion This paper studies simple pricing strategies available to a monopoly platform in the face moral hazard on one side of the (two-sided) market. It does so by developing a notionofequilibriumrestingonrationalexpectationsonthepartofbothsidesofthe market. Thisequilibriumreputationisshowntoexistandtobeinterior. Compared to a moral hazard-free the platform can only use linear prices, it must decrease the fee it charges to buyers to make up for a lower expected value of trade. On the othersideitdistortsthefeeitleviesfromsellerstoincreaseintertemporalincentives and promote cooperation. If upfront payments are possible, the platform uniformly improves the outcomes. In the absence of frictions the socially desirable action can alwaysbeimplemented, withpossiblylargelump-sumpaymentsandnegativelinear fees. Weconductsatisfactoryrobustnesschecks. Anobviousextensionofthispaper is one in which platforms compete. A less evident, but critical one, consists in introducing a proper price-formation mechanism for the good sold, which we so far have abstracted from. 119 3.9 Appendix: proofs 3.9.1 Proofs of the propositions and corollaries Proof of Proposition 18: We begin by proving point 1. Let º i (x) denote the inverse hazard rate of the distribution i = F(x);G(x). Suppose t s ¸ t s , then by Conditions (43) and (44), º F (s ¤ )<º F (s ¤ ) (since it is decreasing), so that t s +t b < t s +t b . Itthenfollowsthatt b <t b ,whenceº G (b ¤ )<º G (b ¤ ). Butthent s +t b >t s +t b by Conditions (43) and (44) as G(b) also has a decreasing inverse hazard rate. Thus we are led to a contradiction and t s < t s necessarily. Next onto item 3. Suppose that t s +t b · t s +t b . Then, by Conditions (43) and (44) and the fact that º F (:) and º G (:) are decreasing, t b ·t b ¡h (58) and t s ¡d·t s . (59) Addinginequalities(58)and(59)andre-arrangingoneobtainsthatt b +t s +(h¡d)· t b +t s . Since by assumption h > d, we are led to a contradiction. Point 2 follows immediately from points 1 and 3. Proof of Corollary 6: From point 3 in Proposition 18, conditions (43) and (44), and the fact that the inverse hazard rate of the distributions F and G is decreasing, itfollowsthatt s ¡d>t s andt b >t b ¡h. Thisimpliesthat1¡F(t s )¸1¡F(t s ¡d) and 1¡ G(t b ¡ h) ¸ 1¡ G(t b ). Thus, D s (s ¤ ) > D s (s ¤ ); D b (b ¤ ) > D b (b ¤ ) and t s +t b >t s +t b , Corollary 6 obtains. Proof of Proposition 19: The ¯rst-order conditions of the problem max t s ;t b ;r ¦(t s ;t b ;r)=¹ h 1¡G(t b ¡rh) i [1¡F(t s ¡(1¡r)d)](t s +t b ) 120 read t b +t s = 1¡G(t b ¡rh) g(t b ¡rh) and t b +t s = 1¡F(t s ¡(1¡r)d) f(t s ¡(1¡r)d) with g(:)h[1¡F(t s ¡(1¡r)d)]¡df(:) h 1¡G(t b ¡rh) i =0 as a necessary condition for r <1, so h 1¡F(t s ¡(1¡r)d) f(:) ¡d 1¡G(t b ¡rh) g(:) =0 that is, (t s +t b )(h¡d)=0 which is impossible since h > d. Instead r = 1 necessarily. The characterisation of optimal prices follows immediately. Proof of Proposition 20: We start by the e®ect of t b on r ¤ (t b ;t s ). Take (t s ;t b 0 ) and (t s ;t b 1 ) in ¨ such that t b 0 · t b 1 . Because D b (t b ;r) is non-increasing in t b and s ¤¤ (t s ;D b ) is non-increasing in D b , k(t s ;D b (t b ;r)) is non-increasing in t b . Thus, k(t s ;D b (t b 0 ;r))¸ k(t s ;D b (t b 1 ;r)) for all r2 [0;1]. Suppose that r ¤ (t b 0 ;t s ) < 1. Since by de¯nition r ¤ (t b 0 ;t s ) is the highest value of r that satis¯es k(t s ;D b (t b 0 ;r)) = r, k(t s ;D b (t b 0 ;r))<rforallr2(r ¤ (t b 0 ;t s );1]. Hence,k(t s ;D b (t b 1 ;r))·k(t s ;D b (t b 0 ;r))< r for all r2 (r ¤ (t b 0 ;t s );1], which implies that r ¤ (t b 1 ;t s )· r ¤ (t b 0 ;t s ). The case where r ¤ (t b 0 ;t s ) = 1 is trivial, since by de¯nition r ¤ (t b ;t s )· 1 for all t s and t b . We focus now on the e®ect of t s on r ¤ (t b ;t s ). We only show here that in ¨, k(t s ;D b (t b ;r)) is non-increasing in t s . The remainder of the argument is analogous to that pre- sented above associated with the e®ect of t b on r ¤ (t b ;t s ). Fix r. Suppose ¯rst that (t s ;t b ) 2 ¨ is such that k(t s ;D b (t b ;r)) = 0. From the fact that s ¤¤ (t s ;D b ) is 121 non-decreasing in t s , it follows that k(t s 1 ;D b (t b ;r)) = 0 for all for all (t s 1 ;t b ) 2 ¨ with t s 1 ¸ t s . Suppose now that (t s ;t b ) 2 ¨ is such that 0 < k(t s ;D b (t b ;r)) < 1. Di®erentiating k(t s ;D b (t b ;r)) with respect to t s , we obtain that the derivative is non-positive if and only if 1¡F(s ¤¤ ) f(s ¤¤ ) f(s ¤ ) 1¡F(s ¤ ) · @s ¤¤ (t s ;D b )=@t s @s ¤ (t s )=@t s . (60) The hazard rate f(s)=(1¡F(s)) is increasing in s since F(s) is log-concave. When 0<k(t s ;D b (t b ;r))<1, s ¤ (t s )< s ¤¤ (t s ;D b ). Therefore, the left-hand-side of (60) is less than one, and (60) is satis¯ed since @s ¤ (t s )=@t s =@s ¤¤ (t s ;D b )=@t s =1. Proof of Proposition 21: Suppose ¯rst that ^ t b + ^ t s ¸ t s +t b . Then, ^ t s ¸ t s or ^ t b ¸ t b . In either case this implies ^ t b + ^ t s < t s +t b , for the right-hand side of (51) (the hazard rates º F (y), º G (y)) are both decreasing and both @r ¤ =@t b ·0 and @r ¤ =@t s · 0. Thus we arrive at a contradiction. But this does not rule out ^ t s ¸ t s or ^ t b ¸ t b and ^ t b + ^ t s · t s + t b . However, suppose that ^ t b ¸ t b ; as we know it implies ^ t b + ^ t s < t s +t b therefore we must have ^ t s < t s . Hence D b ( ^ t b ;r ¤ ) < D b (t b ) by Condition (??) and D s ( ^ t s ) > D s (t s ) by Lemma 16. Since both D b ( ^ t b ;r ¤ ) < D b (t b ) · 1 and 1 ¸ D s ( ^ t s ) > D s (t s ), D b ( ^ t b ;r ¤ )D s ( ^ t s ) < D b (t b )D s (t s ). Therefore ¼( ^ t s ; ^ t b ;r)<¼(t s ;t b ) and there exists a pairht b ;t s i that dominatesh ^ t b ; ^ t s i. Suppose now that ^ t s ¸ t s ; since ^ t s + ^ t b < t s +t b ; ^ t b < t b necessarily. So ¸ G ( ^ t b ) · ¸ G (t b ), and ¤ G ( ^ t s ) · ¤ G (t b ), hence D b ( ^ t b ;r ¤ ) < D b (t b ;r ¤ ). Simultaneously, since ^ t s ¸ t s ; D s ( ^ t s )·D s (t s ). Again ¼( ^ t s ; ^ t b ;r)<¼(t s ;t b ) and there exists a pairht b ;t s i that dominatesh ^ t b ; ^ t s i. ProofofProposition22: Let¡(r;t s ;t b ;T s )´k(t s ;T s ;D b (t b ;r)). Westartbythe e®ect of T s on r ¤ (t b ;t s ;T s ). Take (t s ;t b ;T s 0 ) and (t s ;t b ;T s 1 ) in ¨ such that T s 0 ·T s 1 . Because ¡(r;t s ;t b ;T s ) is non-decreasing in T s , ¡(r;t s ;t b ;T s 1 )¸¡(r;t s ;t b ;T s 0 ) for all r2 [0;1]. Suppose ¯rst that r ¤ (t b ;t s ;T s 0 ) = 1. This implies that ¡(1;t s ;t b ;T s 0 ) = 1, 122 whichinturnimpliesthat¡(1;t s ;t b ;T s 1 )=1. Thus,r ¤ (t b ;t s ;T s 1 )=1=r ¤ (t b ;t s ;T s 0 ). Suppose now that r ¤ (t b ;t s ;T s 0 )<1. Then ¡(r ¤ (t b ;t s ;T s 0 );t s ;t b ;T s 1 )¸¡(r ¤ (t b ;t s ;T s 0 );t s ;t b ;T s 0 )=r ¤ (t b ;t s ;T s 0 ). From continuity of ¡(r;t s ;t b ;T s ) in r (see proof of Lemma 20) and the fact that r ¤ (t b ;t s ;T s ) is by de¯nition the highest of the solutions to ¡(r;t s ;t b ;T s ) = r , it follows that r ¤ (t b ;t s ;T s 1 ) ¸ r ¤ (t b ;t s ;T s 0 ). The case in which r ¤ (t b ;t s ;T s 0 ) = 0 is trivial. We next show the e®ect of t s on r ¤ (t b ;t s ;T s ). Suppose ¯rst that T s · 0. Then, D s (t s ;T s ;D b ) = 1¡F(s ¤ (t s )), in which case the result in Proposition 20 applies. Suppose now that T s > 0. In this case, D s (t s ;T s ;D b ) = 1¡F(e s ¤ (t s ;T s ;D b )). We only show here that in ¨ the function ¡(r;t s ;t b ;T s ) is non-increasing in t s . The remainder of the argument is analogous to that presented above associated with the e®ect of T s on r ¤ (t b ;t s ;T s ). Di®erentiating ¡(r;t s ;t b ;T s ) with respect to t s when 0<¡(r;t s ;t b ;T s )<1, we obtain that the derivative is non-positive if and only if f(e s ¤ ) 1¡F(e s ¤ ) @e s ¤ (t s ;T s ;D b ) @t s ¡ f(s ¤¤ ) 1¡F(s ¤¤ ) @s ¤¤ (t s ;T s ;D b ) @t s ·0. (61) Because (i) @e s ¤ (t s ;T s ;D b )=@t s = @s ¤¤ (t s ;T s ;D b )=@t s = 1, (ii) e s ¤ (t s ;T s ;D b ) < s ¤¤ (t s ;T s ;D b ) when ¡(r;t s ;t b ;T s ) < 1 and (iii) the hazard rate f(s)=(1¡ F(s)) is increasing in s, condition (61) holds. Also note that when ¡(r;t s ;t b ;T s ) = 0, s ¤¤ (t s ;T s ;D b ) = s and that s ¤¤ (t s ;T s ;D b ) is non-decreasing in t s . This establishes point 2. We now show the e®ect of t b on r ¤ (t b ;t s ;T s ). When T s ·0 the sellers demand is given by D s (t s ;T s ;D b ) = 1¡F(s ¤ (t s )), in which case the result in Proposition 20 applies. If T s >0, then D s (t s ;T s ;D b )=1¡F(e s ¤ (t s ;T s ;D b )). Again, we only show here that in ¨ the function ¡(r;t s ;t b ;T s ) is non-increasing in t b . Di®erentiating 123 ¡(r;t s ;t b ;T s ) with respect to t b when 0<¡(r;t s ;t b ;T s )<1 and using the fact that @D b (t b ;r)=@t b ·0, we obtain that the derivative is non-positive if and only if f(e s ¤ ) 1¡F(e s ¤ ) @e s ¤ (t s ;T s ;D b (r;t b )) @D b ¡ f(s ¤¤ ) 1¡F(s ¤¤ ) @s ¤¤ (t s ;D b (r;t b )) @D b ¸0. (62) Because the hazard rate f(s)=(1¡ F(s)) is increasing in s and e s ¤ (t s ;T s ;D b ) < s ¤¤ (t s ;T s ;D b )when¡(r;t s ;t b ;T s )<1,condition(62)holdsif@e s ¤ (t s ;T s ;D b )=@D b ¸ @s ¤¤ (t s ;T s ;D b )=@D b . Note that @e s ¤ (t s ;T s ;D b )=@D b =¡ 1+± [1¡±(1¡®¹D b )]D b (63) and @s ¤¤ (t s ;D b )=@D b =¡ v(s ¤¤ ;a)¡t s D b . (64) Comparing(63)and(64)andusingthefactsthatats ¤¤ (48)issatis¯edwithequality andbyconstructionD b ·1,weobtainthat@e s ¤ (t s ;T s ;D b )=@D b ¸@s ¤¤ (t s ;D b )=@D b if d ¸ (1¡ ± 2 )®±¹=[(1¡ ±) + ®±¹]. Finally, note that when ¡(r;t s ;t b ;T s ) = 0, s ¤¤ (t s ;T s ;D b )=sandthats ¤¤ (t s ;D b (r;t b ))isnon-decreasingint b . Thisestablishes point 3. ProofofProposition23: Letthesolutiontothebenchmarkcasebeht s 0 ;t b 0 ;r =1i. Let s ¤ 0 denote the corresponding threshold type of seller above which sellers trade. Because r = 1, only sellers who take the high action trade, which implies that s ¤ 0 =t s 0 . Thus seller participation is given by D 0 s =1¡F(t s 0 ). Buyer participation is givenbyD 0 b =1¡G(t b 0 ¡h). Thediscountedvalueoftheplatform'spro¯tassociated with the solution of the benchmark case is ¦ 0 = 1 1¡± ¹D 0 s D 0 b (t b 0 +t s 0 ) 124 Under moral hazard, this allocation can be implemented as follows. Choose the buyers' transaction fee t b 1 = t b 0 . Choose the sellers' transaction fee such that sellers of type above s ¤ 0 prefer to choose the high action. Using the no-cheating condition (48), this transaction fee, which we denote t s 1 , satis¯es d=® ± 1¡± ¹D 0 b [s ¤ 0 ¡t s 1 ] (65) and therefore s ¤¤ (t s ;D 0 b ) = s ¤ 0 . Now select the registration fee T s 1 such that only sellers of type above s ¤ 0 trade. That is, choose T s 1 =V(s ¤ 0 )= 1 1¡± ¹D 0 b [s ¤ 0 ¡t s 1 ] (66) in which case only sellers of type above s ¤ 0 register (and trade). Thus seller par- ticipation D 1 s = D 0 s . All those who trade choose the high action, which implies a reputation of 1. Since t b 1 = t b 0 buyer participation also equals the benchmark's: D 1 b = D 0 b . All that remains to show is that the platform's pro¯t is the same as in the solution of the benchmark. It consists of the revenues associated with the transaction fees and the revenues associated with the registration fees in the ¯rst period. Speci¯cally, ¦ 1 = 1 1¡± ¹D 1 s D 1 b (t b 0 +t s 1 )+T s 1 D 1 s = 1 1¡± ¹D 0 s D 0 b (t b 0 +t s 1 )+T s 1 D 0 s = 1 1¡± ¹D 0 s D 0 b (t b 0 +t s 1 )+ 1 1¡± ¹D 0 b [t s 0 ¡t s 1 ]D 0 s = 1 1¡± ¹D 0 s D 0 b (t b 0 +t s 0 ) = ¦ 0 where the second equality follows from the fact that D 1 s = D 0 s and D 1 b = D 0 b ; the third equality follows by substituting T s 1 by its value and noting that s ¤ 0 =t s 0 . 125 ProofofProposition24: t b 1 =t b fromtheproofofProposition23. Condition(66) and (65) implies that T s 1 = d ®± . Last, Condition (65) delivers the third line directly. Proof of Proposition 25: The result is trivial for ¿ = 2. To obtain that r c¤ 1 ¸ r ¤ 1 , note that s ¤¤ 1 and s ¤ 1 are the same with full replacement and with cleansing. Moreover, note that as in the case of full replacement s ¤¤ 0 <s ¤¤ 1 . By Assumption 17, this implies thatF 1 ([s ¤¤ 1 ;s])=F([s ¤¤ 1 ;s]). Thus, r c¤ 1 = F([s ¤¤ 1 ;s]) D 1 s (t s ) ¸ F([s ¤¤ 1 ;s]) D 0 s (t s ) =r ¤ 1 , wheretheinequalityfollowsfromthefactthatD 0 s (t s )=F 0 ([s ¤ (t s );s])¸F 1 ([s ¤ (t s );s])= D 1 s (t s ). Finally, to obtain that r c¤ 0 ¸r ¤ 0 , note that r c¤ 1 ¸r ¤ 1 implies that equilibrium V 1 (s) with cleansing is bigger than equilibrium V 1 (s) with full replacement. This implies s ¤¤ 0 with cleansing is smaller than s ¤¤ 0 with full replacement. SinceF 0 =F, this implies that r c¤ 0 ¸r ¤ 0 . 3.9.2 Proofs of the lemmata Proof of Lemma 16: First the only if part. Suppose that s +d < t s . This implies that also s < t s . Since V(s) is non-negative because the seller can always choose not to trade at any given match, it is clear from direct inspection of each of the seller's payo®s inside the curly brackets on the right-hand side of (46) that nottradingdominatesbothtradingandchoosingaction aandtradingandchoosing action a. Now the if part. Suppose that s+d¸ t s . If it is also the case that s¸ t s , then regardlessofV(s)amatchedselleroftypesisbettero®tradingandchoosingaction a than not trading at all{the third term inside the curly brackets in (46) is greater than the ¯rst. So in this case trading and choosing action a clearly dominates not trading. If instead s < t s , the opposite happens, i.e., to trade and choose 126 action a is dominated by not to trade at all. As a consequence, (46) collapses into V(s) = ¹D b maxf0+±V(s);s+d¡t s +(1¡®)±V(s)g+(1¡¹D b )±V(s). When s+d¡t s ¸0,thesolutiontothisequationisV(s)=¹D b [s+d¡t s ]=[1¡±(1¡®¹D b )]. Thisimpliesthat±V(s)·s+d¡t s +(1¡®)±V(s),meaningthattradingandchoosing action a is better than not trading. Proof of Lemma 17: As a preliminary fact note that the right-hand side of (48) can be written as ®±V ND (s), where V ND (s) = [1=(1¡±)]¹D b [s¡t s ] corresponds to the present value of the expected payo®s to a seller of type s who trades on the platform and never deviates. We now prove the result in the lemma. First the only if part. We prove the equivalent statement that if condition (48) is not satis¯ed then the seller's optimal decision when matched is not to choose action a. Suppose that condition (48) is not satis¯ed. This means that the seller is better o® deviating once and then, if not excluded from the platform, choosing the high action forever after. Since the problem is stationary, the seller's optimal decision given a match must be always the same. Thus (always) choosing action a given a match is not optimal. Now the if part. Suppose that (48) holds. Since the right-hand side of (48) is given by ®±V ND (s) and by de¯nition V ND (s)·V(s), this implies that d·®±V ND (s)·®±V(s). That is, condition (47) is satis¯ed and therefore deviating cannot be optimal to the seller. ProofofLemma18: Considerthetransactionfeet s 2¨. Notethatk(t s ;D b (t b ;r)) is continuous in r at every r in [0;1], since D b (t b ;r) is continuous in r; s ¤¤ (t s ;D b ) is continuous in D b and F(s) is continuous in s. Because k(t s ;D b (t b ;r))2[0;1] for all r 2 [0;1] and is continuous in [0;1], it follows by Brouwer's Fixed Point Theorem 127 that it has a ¯xed point in [0;1]. This establishes existence. If 0<D s (t s )<1, then s ¤¤ (t s ;D b ) > s ¤ (t s ) necessarily. In this case, k(t s ;D b (t b ;r)) < 1 for all r 2 [0;1], which implies that r ¤ =1 cannot be a solution to (49). ProofofLemma19: Consideranequilibriumwhentransactionandregistrations fees are ht b 0 ;T b 0 > 0;t s 0 ;T s 0 i. Let D 0 s > 0, D 0 b > 0 and r 0 denote, respectively, the sellers' participation, buyers' participation and reputation in that equilibrium. Now consider the following transaction and registration fees ht b 1 ;T b 1 = 0;t s 0 ;T s 0 i, where t b 1 =t b 0 +T b 0 =¹D 0 s . We start by showing that D 0 s , D 0 b and r 0 also constitutes an equilibrium with these transaction and registration fees. From the de¯nition of e b ¤ it follows directly that e b ¤ (t b 1 ;T b 1 = 0;r 0 ;D 0 s ) = e b ¤ (t b 0 ;T b 0 ;r 0 ;D 0 s ). Moreover, since T b 0 > 0 and T b 1 = 0, it follows that b ¤ (t b 0 ;r 0 ) < e b ¤ (t b 0 ;T b 0 ;r 0 ;D 0 s ) and that b ¤ (t b 1 ;r 0 )= e b ¤ (t b 1 ;T b 1 ;r 0 ;D 0 s ). ThisimpliesthatD b (t b 1 ;0;r 0 ;D 0 s )=D b (t b 0 ;T b 0 ;r 0 ;D 0 s ). Thus, since by de¯nition of equilibrium D 0 b =D b (t b 0 ;T b 0 ;r 0 ;D 0 s ), we obtain that D 0 b =D b (t b 1 ;0;r 0 ;D 0 s ). (67) Also by de¯nition of equilibrium, D 0 s =D s (t s 0 ;T s 0 ;D 0 b ) (68) and r 0 =k(t s 0 ;T s 0 ;D 0 b ). (69) Because (67)-(69) hold simultaneously, D 0 s , D 0 b and r 0 constitute an equilibrium when fees areht b 1 ;T b 1 =0;t s 0 ;T s 0 i. It remains to show that the platform's pro¯ts are thesameinbothequilibria. Sellers'registrationdecisionsarethesameinbothequi- libria and given by R s (t s 0 ;T s 0 ;D 0 b ). Since e b ¤ (t b 1 ;T b 1 = 0;r 0 ;D 0 s ) = e b ¤ (t b 0 ;T b 0 ;r 0 ;D 0 s ) buyers' registration decisions are also the same in both equilibria. They are given by R b (t b 0 ;T b 0 ;r 0 ;D 0 s ) = R b (t b 1 ;0;r 0 ;D 0 s ). Denoting the platform's (expected) period 128 pro¯t in the equilibrium with fees ht b 0 ;T b 0 > 0;t s 0 ;T s 0 i by ¼ 0 and in the equilibrium with feesht b 1 ;0;t s 0 ;T s 0 i by ¼ 1 , we obtain ¼ 1 = ¹D 0 s D 0 b h t s 0 +t b 1 i +R s T s 0 = ¹D 0 s D 0 b [t s 0 +t b 0 + T b 0 ¹D 0 s ]+R s T s 0 = ¹D 0 s D 0 b [t s 0 +t b 0 ]+D 0 b T b 0 +R s T s 0 = ¹D 0 s D 0 b [t s 0 +t b 0 ]+R b T b 0 +R s T s 0 =¼ 0 ; where the penultimate equality follows from the fact that whenever T b > 0 (and D s >0), D b (t b ;T b ;r;D s )=R b (t b ;T b ;r;D s ). This concludes the proof. ProofofLemma20: Whent b ,t s andT s aresuchthatD s (t s ;T s ;D b (t b ;r))>0for allr2[0;1],itisclearthatk(t s ;T s ;D b (t b ;r))isde¯nedforallr2[0;1]. Fixt s ;t b ;T s satisfying the above condition. By construction k(t s ;T s ;D b (t b ;r)) 2 [0;1] for all r2 [0;1]. Moreover, k(t s ;T s ;D b (t b ;r)) is continuous in r at every r in [0;1]. Note that D b (t b ;r) is a continuous function of r, s ¤¤ (t s ;D b ) is continuous in D b and F(s) iscontinuousins,whichimpliesthat1¡F(s ¤¤ (t s ;D b (t b ;r)))iscontinuousinr. Also notethate s ¤ (t s ;T s ;D b )iscontinuousinD b implyingthatmaxfs ¤ (t s );e s ¤ (t s ;T s ;D b )g is continuous in D b and consequently that D s (t s ;T s ;D b (t b ;r)) is continuous in r. Finally, note that by construction k(t s ;T s ;D b (t b ;r)) is continuous at every point r where k(t s ;T s ;D b (t b ;r)) = 1. Because k(t s ;T s ;D b (t b ;r)) 2 [0;1] for all r 2 [0;1] and is continuous in [0;1], it follows by Brouwer's Fixed Point Theorem that it has a ¯xed point in [0;1]. Proof of Lemma 21: From direct inspection of (55) and (56), and the fact that D b (t b ;r) is increasing in r and r ¤ 1 ¸ 0; V 1 (s) ¸ V 2 (s);8 s 2 [s;s]. This, together with the fact that V ¿ (s) is increasing in s and [v(s;a)¡v(s;a)] is non-increasing in s implies that s ¤¤ 0 ·s ¤¤ 1 . The second part of the Lemma follows from the ¯rst part and the fact that sellers' participation is the same in every period. 129 References [1] Alesina, A. and H. Rosenthal, 1995, Partisan Politics, Divided Government and the Economy, Cambridge University Press, Cambridge. [2] Aldrich, J., 1993, Rational choice and turnout. American Journal of Political Science, 37, 246-78. [3] AndersonS.,2004,Regulationoftelevisionadvertising.Mimeo,TheUniversity of Virginia. [4] Anderson S. and Steven Coate, 2005, Market provision of Broadcasting: a welfare analysis. Review of Economic Studies, 72, 947-972. [5] AndersonS.andJohnMcLaren, 2004, Mediamergersandmediabias. Mimeo, The University of Virginia. [6] Armstrong, M., 2006, Competing in two-sided markets. Rand Journal of Eco- nomics, 37, 668-691. [7] Biglaiser, G., 1993, Middlemen as Experts. Rand Journal of Economics, 24, 212-223. [8] Biglaiser, G. and J. Friedmand, 1992, Middlemen as guarantors of quality. Mimeo, Department of Economics, University of North Carolina [9] Brocas, I., 2007, Auctions with type-dependent and negative externalities. Mimeo, Department of Economics, University of Southern California. [10] Caillaud, B. and Bruno Jullien, 2003, Chicken and egg: Competition among intermediation service providers. Rand Journal of Economics, 34, 309-328. [11] Carrillo, J., 1998, Coordination and Externality. Journal of Economic Theory, 78, 103-129. [12] Crampes, C., Carole Harichabalet and Bruno Jullien, 2005, Competing with advertising revenue. Mimeo, Universit¶ e de Toulouse. [13] Dasgupta, P. and Eric Maskin, 1986, The existence of equilibrium in discon- tinuouseconomicgames,I:Theory.TheReviewofEconomicStudies,53,1-26. 130 [14] Dasgupta, P. and Eric Maskin, 1986, The existence of equilibrium in discon- tinuous economic games, II: Applications. The Review of Economic Studies, 53, 27-41. [15] Ellman, M. and Fabrizio Germano, 2004, What do newspapers sell? working paper 800, Universitat Pompeu Fabra. [16] Ferrando J., Jean-J. Gabszewicz, Didier Laussel and Nathalie Sonnac, 2003, Intermarket network externalities and competition: an application to the me- dia industry. Mimeo, CORE, Universit¶ e Catholique de Louvain. [17] GabszewiczJ.-J., DidierLausselandNathalieSonnac, 2001, Pressadvertising and the ascent of of the pens¶ ee unique. European Economics Review, 45, 234- 251 [18] Gabszewicz J.-J., Didier Laussel and Nathalie Sonnac, 2002, Concentration in the press industry and the theory of the circulation spiral. Mimeo, CORE, Universit¶ e Catholique de Louvain. [19] Gabszewicz J.-J.and Jean-Fran» cois Thisse, 1979, Price competition, quality and income disparity. Journal of Economics Theory, 20, 340-359 [20] Gabszewicz J.-J.and Xavier Wauthy, 2002, Two-sided markets and price com- petitionwithmulti-homing.Mimeo,CORE,Universit¶ eCatholiquedeLouvain. [21] Hagiu A., 2006, Pricing and commitment by two-sided platforms. Rand Jour- nal of Economics, 37, 720-737. [22] Hagiu A., 2007, Merchant or two-sided platform? Rewiew of Network Eco- nomics, 6, 115-133. [23] Haaser N. and Joseph Sullivan, 1991, Real analysis. Dover Publications, New York. [24] Kantrowitz R. and Michael Neumann, 2005, Optimization for products of concave functions. Rend. Circ. Mat. Palermo (2), 54, 291-302. [25] La®ont J.-J., Scott Marcus, Patrick Rey and Jean Tirole, 2003, The o®-net cost pricing principle. Rand Journal of Economics, 34, 370-390. 131 [26] La®ont J.-J. and David Martimort, 2002, The theory of incentives. Princeton University Press. [27] MaskinE.andJohnRiley,1984,Monopolywithincompleteinformation.Rand Journal of Economics, 15, 171-196. [28] MantralaM.,PrasadA.Naik,ShrihariSridharandEstherThorson,2007,Up- hill or downhill? Locating the ¯rm on a pro¯t function. Journal of Marketing, 70, 25-42. [29] Owen B. and Michael Spence, 1977, Television programming, monopolistic competition and welfare. The Quarterly Journal of Economics, 91, 1-23. [30] Rochet J.-C. and Jean Tirole, 2002, Cooperation among competitors: some economicsofpaymentcardassociations.RandJournalofEconomics,33,1-22. [31] Rochet J.-C. and Jean Tirole, 2003, Platform competition in two-sided mar- kets. Journal of the European Economic Association, 1, 990-1029. [32] Rochet J.-C. and Jean Tirole, 2004, Two-sided markets: an overview. Mimeo, IDEI, Universit¶ e de Toulouse. [33] Rochet J.-C. and Jean Tirole, 2006, Two-sided markets: a progress report. Rand Journal of Economics, 37, 645-667. [34] ShakedA.andJohnSutton,1982,Relaxingpricecompetitionthroughproduct di®erentiation. The Review of Economic Studies, 49, 3-13. [35] ShakedA.andJohnSutton,1983,Naturaloligopolies.Econometrica,51,1469- 1483. [36] ShakedA.andJohnSutton,1987,Productdi®erentiationandindustrialstruc- ture. The Journal of Industrial Economics, 36, 131-146. [37] Sharkey W. and David Sibley, 1993, A Bertrand model of pricing and entry. Economics Letters, 41, 199-206. [38] StrÄ ombergD.,2004,Massmediacompetition,politicalcompetitionandpublic policy. The Review of Economic Studies, 71, 265-284. [39] Tirole J., 1988, The Theory of Industrial Organisation, MIT Press. 132 [40] Tirole J., 1996, A theory of collection reputations (with applications to the persistence of corruption and to ¯rm quality). The Review of Economic Stud- ies, 63, 1-22. [41] Wauthy X., 1996, Quality choice in models of vertical di®erentiation. The Journal of Industrial Economics, XLIV, 345-353.

Asset Metadata

Creator Roger, Guillaume (author)

Core Title Costly quality, moral hazard and two-sided markets

Contributor Electronically uploaded by the author (provenance)

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Economics

Publication Date 08/04/2008

Defense Date 04/30/2008

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag differentiation,,media,moral hazard,OAI-PMH Harvest,trading platforms,two-sided markets

Language English

Advisor Tan, Guofu (committee chair), Carrillo, Juan D. (committee member), Wilburn, Kenneth (committee member), Wilkie, Simon J. (committee member)

Creator Email guillaume_the_bum@yahoo.com.au,roger@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m1529

Unique identifier UC174987

Identifier etd-Roger-2257 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-114871 (legacy record id),usctheses-m1529 (legacy record id)

Legacy Identifier etd-Roger-2257-0.pdf

Dmrecord 114871

Document Type Dissertation

Rights Roger, Guillaume

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email uscdl@usc.edu

Abstract (if available)

Abstract The object of this dissertation is to further the study of two-sided markets by departing from the standard setting of price competition alone. Specifically the first two chapters introduce costly differentiation and in doing so contribute to the two-sided market literature by establishing a generic downward distortion in quality. This result is robust to different specifications in the monopoly case (Chapter 1) and arises again in a duopoly (Chapter 2). In the latter, whether a Nash equilibrium exists on one side depends on the size of the profits to be extracted on the other. When competing platforms play in mixed strategies one of them may be inactive ex post. This work also extends a well-established model (Shaked and Sutton (1982)) in the Industrial Organization literature, which speaks to the role of quality as a source of endogenous differentiation. The last chapter allows for moral hazard on a trading platform. It contributes to the two-sided market literature by showing that opportunistic behavior on the part of sellers leads to lower transaction fees on both sides to 1) compensate buyers and 2) provide sellers with incentives to cooperate. Furthermore it breaks an equivalence result between transaction fees and lump-sum payment established by Rochet and Tirole (2005).