Monopoly with Resale

Transcription

1 Monopoly with Resale Giacomo Calzolari Alessandro Pavan July 2004 Abstract his paper illustrates the intricacies associated with the design of revenue-maximizing mechanisms for a monopolist who expects her buyers to resell in a secondary market. We consider two modes of resale: the first is to a third party who does not participate in the primary market; the second is inter-bidders resale, where the winner in the primary market resells to the losers. he main contribution is in showing how revenue-maximizing mechanisms can be designed investigating the optimal informational linkage with the secondary market. o control the price in the resale game, the monopolist must design an allocation rule and a disclosure policy that optimally fashion the beliefs of the participants in the resale market. We show that it is generically impossible to maximize revenue through deterministic selling procedures and disclosing only the decision to trade with a particular buyer. o create the optimal informational linkage, the monopolist may need to induce stochastic allocations and disclose also the price paid in the primary market. he optimal allocation rule and disclosure policy maximize the expected sum of the bidders resale-augmented virtual valuations under the constraints imposed by the sequential rationality of the bidders offers in the resale game. Keywords: optimal information linkage between primary and secondary markets, disclosure policy, stochastic allocations, resale-augmented virtual valuations. Journal of Economic Literature Classification Numbers: D44, D82. We are grateful for their helpful comments to Larry Ausubel, Peter Cramton, Leonardo Felli, Ignatius Horstmann, enny Moldovanu, Mike Peters, William Rogerson, Vasiliki kreta, Jean irole, Daniel Vincent, Charles Zheng and seminar participants at Asset Meeting, occoni University, Fondazione Eni-Enrico Mattei, University of Maryland, University of Milan-icocca, Northwestern University, University of oronto, University of Venice, WZ erlin and the North American Econometric ociety ummer Meetings. We are also thankful to Maria Goltsman for valuable research assistance. Department of Economics, University of ologna, Italy. calzolari@economia.unibo.it Corresponding author. Department of Economics, Northwestern University, 2001 heridan Road, Andersen Hall Room 3239, Evanston, IL , UA. Phone: (847) Fax: (847) alepavan@northwestern.edu 1

2 2 G. Calzolari and A. Pavan 1 Introduction Durable goods are typically traded both in primary and secondary markets. Indeed, auctions for real estates, artwork and antiques are often followed by resale. he same is true for licenses, patents, reasury bills, emission and spectrum rights. imilarly, IPOs and privatizations generate ownership structures which change over time as a consequence of active trading in secondary markets. Resale may have different explanations. First, it may be a consequence of the fact that not all potential buyers participate in the primary market. 1 For example, a buyer may value a good only if it goes first into the hands of another agent. Intermediation is typical for instance in IPOs and reasury bills markets and is often believed to be value-enhancing. Alternatively, participation only in secondary markets may be due to a change in the environment: At the time the government decides to sell spectrum rights, a company may not bid in the auction because at that point it does not formally exist yet or it attaches a low value to the rights, possibly because of its current position in the market, or because of the business strategy of its management. After a merger, a privatization, or a successful takeover, the same company may develop interest in possessing the rights and decide to buy them from the winner in the primary market. 2 Furthermore, participation only in secondary markets may also be strategic as indicated in McMillan (1994) and Jehiel and Moldovanu (1996). econd, resale may be a direct consequence of misallocations in the primary market. As shown first in Myerson (1981), optimal auctions are typically inefficient when the distributions of the bidders valuations are asymmetric. y committing to a policy that misplaces the good into the hands of a buyer who does not value it the most, a seller can induce more aggressive bidding and raise higher expected revenues. When resale is possible, bidders may thus attempt to correct misallocations in the auction by further trading in a secondary market. 3 As indicated in Haile (1999, 2003), with resale, buyers willingness to pay in the primary market is endogenous for it incorporates the surplus expected in the secondary market which in turn depends on the information filtered by the monopolist through her selling procedure. In this paper, we examine the intricacies associated with the design of revenue-maximizing mechanisms for a monopolist who expects her buyers to resell. 4 he revenue-maximizing mechanisms are obtained by investigating the optimal informational linkage with the secondary market and showing how this linkage can be implemented through the design of a proper allocation rule and the adoption of an optimal disclosure policy. We analyze a simple two-stage game of incom- 1 ikhchandani and Huang (1989), Haile (1999), and Milgrom (1987) consider auctions followed by resale where the set of bidders in the primary market does not include all potential buyers. 2 Haile (2003) and chwarz and onin (2001) consider models where bidders valuations change over time. 3 ee also Gupta and Lebrun (1999) for an analysis of first-price asymmetric sealed bid auctions followed by resale where trade in the secondary market is motivated by the inefficiency of the allocation in the primary market. 4 Revenue-maximizing mechanisms without resale have been examined, among others, by ulow and Roberts (1989), Harris and Raviv (1981), Maskin and Riley (1984), Myerson (1981), Riley and amuelson (1981).

3 Monopoly with Resale 3 plete information where in the first stage, a monopolist sells a durable and indivisible good in a primary market, whereas in the second stage, the primary buyer resells the good in a secondary market. rade in the resale game is the result of a simple ultimatum bargaining procedure in which players make take-it-or-leave-it offers with a probability distribution that reflects their relative bargaining abilities. Although stylized, the model illustrates the dependence of the resale surplus on the information disclosed in the primary market and is sufficiently tractable to allow for a complete characterization of the optimal allocation rule and disclosure policy from the monopolist s viewpoint. he first part of the paper considers the case where resale is to a third party who participates only in the secondary market. In this case, the monopolist uses the primary buyer as an intermediary to extract surplus also from those buyers who are not willing, or able, to contract directly with her in the primary market. he study of this simple environment illustrates the main intricacies associated with the design of revenue-maximizing mechanisms. wo results are highlighted in the analysis. First, the monopolist may find it optimal to sustain stochastic allocations in the primary market, for example, using lotteries to allocate the good and/or offering menus that induce the buyer to follow mixed strategies. y selling with different probability to different types, the monopolist uses the decision to trade as a signal of the buyer s valuation to induce the third party to offer a higher price in the resale game. Contrary to deterministic allocations, stochastic selling procedures give the monopolist a better control over the beliefs of the participants in the secondary market and do not require to exclude completely those buyers with a lower willingness to pay. o illustrate, suppose the buyer in the primary market has either a high or low valuation and assume the third party s prior beliefs are unfavorable to the buyer in the sense she is expected to offer a low price in the event she learns nothing about the value the latter attaches to the good. hen, if the monopolist uses a deterministic mechanism that sells to either type with certainty, the third party does not learn anything from the decision to trade and thus offers a low price. If, on the other hand, the monopolist sells only to the high type, the third party perfectly learns the value the buyer attaches to the good and offers a price equal to his high valuation, which again leaves no surplus to the buyer. In contrast, with a stochastic mechanism, the monopolist can sell to the high type with certainty and to the low type with probability positive but low enough to induce the third party to offer a high resale price, increasing the surplus the low type expects from resale and thus his willingness to pay in the primary market. econd, the optimal mechanism may require the adoption of a disclosure policy richer than the simple announcement of the decision to trade. In the example above, the monopolist could disclose two signals, the first with a higher probability when the buyer reports a high valuation, the second with a higher probability when he reports a low valuation. y making the two signals sufficiently informative (but not perfect), the monopolist may thus induce the third party to offer a high resale

4 4 G. Calzolari and A. Pavan price also to the low type. he advantage of disclosing information in addition to the decision to trade comes from the possibility to increase the level of trade without necessarily reduce the price in the secondary market. In the limit, if the monopolist knew the buyer s valuation, she could sell with certainty to either type and use only the stochastic disclosure policy to sustain a high resale price. hings are however more complicated when the buyer s valuation is not known to the initial seller. In this case, disclosure enhances the level of trade but does not allow the monopolist to sell with certainty, for the combination of a certain allocation rule and a stochastic disclosure policy is not incentive compatible. Indeed, if trade were certain, the high type would always select the contract with the lowest price, irrespective of the associated disclosure policy. ut then the low type would have an incentive to mimic the high type, paying the same price and inducing the monopolist to disclose the most favorable signal with a higher probability. he only way the monopolist can sort the two types and at the same time disclose informative signals to the third party is by making the high type pay a higher price than the low type, which is possible only if a higher price is associated with a lottery that delivers the good with a higher probability. We show how the optimal disclosure policy can be obtained as part of a direct revelation mechanism in which the seller commits to disclose abstract signals with a distribution that depends on the information revealed in the primary market. After characterizing the optimal mechanism, we propose a natural implementation in which these additional signals are simply the prices the buyer pays for lotteries that deliver the good with different probabilities. he optimal menu of price-lottery pairs may either perfectly separate the buyer s types or induce the latter to play a mixed strategy with type-dependent randomizations. In the first case, the monopolist discloses only the decision to trade, whereas in the second also the price paid. he analysis of the single-buyer case is extended in two directions. First, we allow for multiple bidders in the primary market, but maintain the assumption resale is to a third party who participates only in the secondary market. In this case, resale increases the willingness to pay and reduces the differences across types and hence the informational rents the monopolist must leave to the buyers to induce truthful information revelation. On the other hand, resale also creates new incentives for low types to misrepresent their valuations for the purpose of obtaining a higher price in the secondary market, which in turn may interfere with the monopolist s ability to sustain the same allocations as in the absence of resale. However, we show that by committing not to disclose any information in addition to the identity of the winner, it is always possible to implement exactly the same allocation rule as in a Myerson optimal auction in which bidders do not have the option to resell. It follows that a monopolist always benefits from resale when she can not contract with all potential buyers and is able to prevent the winner in the primary market from reselling to the losers. Finally, in the last part of the paper we examine revenue-maximizing mechanisms for a mo-

5 Monopoly with Resale 5 nopolist who expects inter-bidders resale. o the best of our knowledge, this problem has been examined only by Ausubel and Cramton (1999) and Zheng (2002). Ausubel and Cramton assume perfect resale markets and show that if all gains from trade are exhausted through resale, then it is strictly optimal for the monopolist to implement an efficient allocation directly in the primary market. he case of perfect resale markets, although a benchmark, abstracts from important elements of resale. First, when bidders trade under asymmetric information, misallocations are not necessarily corrected in secondary markets (Myerson and atterthwaite (1983)). econd, and more important, efficiency in the secondary market is endogenous as it depends on the information revealed in the primary market which is optimally fashioned by the monopolist through her choice of an allocation rule and a disclosure policy. Zheng assumes it is always the winner in the auction who offers the price in the secondary market and suggests a mechanism that, under a few assumptions on the distributions of the bidders valuations, gives the monopolist the same expected revenue as in a standard optimal auction where resale is prohibited. Instead of selling to the bidder with the highest virtual valuation, the monopolist sells to the bidder who is most likely to implement in the secondary market the same final allocation as in the optimal auction of Myerson (1981). Although an important contribution, Zheng s result crucially relies on the assumption it is always the seller who offers the resale price. his assumption, justified on the grounds of symmetry between primary and secondary markets, has a lot of bite since it implies the monopolist can perfectly control the distribution of bargaining power in the secondary market through the allocation of the good in the primary market. However, that the original seller has full bargaining power rarely implies that any seller of the same good will necessarily have full bargaining power when contracting with prospective buyers. In general, the distribution of bargaining power is likely to be a function of the allocation of the good, but also of the individual characteristics of the players, such as their personal bargaining abilities. What is more, assuming it is always the seller who offers the resale price eliminates strategic effects which are important for the analysis of the design of revenue-maximizing mechanisms. First, when the monopolist can not perfectly control the distribution of bargaining power in the secondary market, it is generically impossible to achieve Myerson s expected revenue without prohibiting resale. econd, to create the optimal informational linkage the monopolist can not limit attention to deterministic selling procedures: the revenue-maximizing mechanism in the primary market may require the use of stochastic allocations and possibly the adoption of a disclosure policy richer than the announcement of the identity of the winner. We show how the optimal allocation rule and disclosure policy can be designed maximizing the expected sum of the bidders resale-augmented virtual valuations, taking into account the constraints on the sequential rationality of the bidders offers in the secondary market. As anticipated, auctions followed by an opportunity for resale have been analyzed also by Haile (1999, 2003) who studies equilibria in standard auction formats such as English, first-price, and

6 6 G. Calzolari and A. Pavan second-price sealed bid. His results well illustrate how the option to resell creates endogenous valuations and induces signaling incentives that may revert the revenue ranking obtained in the literature that assumes no resale. Our analysis builds on some of his insights, but differs from his in that we assume the monopolist is not constrained to use any specific auction format. Furthermore, the focus is on the design of the optimal informational linkage between primary and secondary markets from the monopolist s viewpoint and on the possibility to implement it through the adoption of an appropriate disclosure policy. Finally, another strand of literature related to our work considers bidding behavior in auctions followed by product market competition. Jehiel and Moldovanu (2000) focus on the effect of positive and negative externalities stemming from subsequent market interaction on equilibrium bidding strategies in second-price sealed bid auctions. hey also study the effects of reserve prices and entry fees on the seller s revenue and on welfare. More recently, Katzman and Rhodes-Kropf (2002) and Zhong (2002) study the effect of different bid announcement policies on the seller s expected revenue in standard auctions followed by ertrand and Cournot competition. Our paper differentiates from this literature for the analysis of resale instead of imperfect product market competition and for the use of mechanism design to derive the monopolist s optimal selling procedure and disclosure policy. o the best of our knowledge, this is new to the literature. he rest of the paper is organized as follows. ection 2 studies the case where the monopolist sells to a buyer in the primary market who then resells to a third party in a secondary market; this section illustrates the main intricacies of optimal mechanism design for a monopolist who expects resale. ection 3 extends the analysis to the case where the monopolist can contract with all potential buyers but can not prohibit the winner in the primary market from reselling to the losers. Finally, ection 4 concludes. All proofs are relegated to the Appendix. 2 Resale to hird Parties 2.1 he environment Consider an environment where in the primary market a monopolistic seller ( hereafter) trades a durable and indivisible good with a (representative) buyer,. If receives the good from, he can either keep it for himself, or resell it to a (representative) third party,, in a secondary market. 5 can not sell directly to, nor can or communicate with before they trade. For example, may represent a prospective future buyer that does not exist at the time needs to sell (such as in the case of a new firm that is expected to be created as a result of a merger or a privatization). Alternatively, may value the good only if it goes first into the hands of, as with 5 We adopt the convention of using masculine pronouns for and feminine pronouns for and.

7 Monopoly with Resale 7 an intermediate product that needs to be processed before it can be used also by.inthiscase, is likely to lack the necessary bargaining power to extract money from without selling anything to her, as indicated in Milgrom (1987). Finally, there can be legal or political impediments that prevent from contracting with, such as in the case of a privatization in which the government is constrained to sell only to domestic firms. Let x i {0, 1} represent the decision to trade between and player i, with i =,. When x i =1, the good changes hands. For example, for i =, x =1means that obtains the good from. imilarly, for i =, x =1means that obtains the good from. On the contrary, if x i =0, there is no trade between and player i. A trade outcome x i ª,ti, consists of the allocation of the good x i and a monetary transfer ti R between and player i. We denote with θ i the value i =, attaches to the good and with θ := (θ,θ ) Θ := Θ Θ aprofile of valuations for and. We also assume the value to is common knowledge and equal to zero. All players have quasi-linear preferences, respectively equal to u = t, u = θ x (1 x ) t + t, u = θ x x t. We make the following assumptions on valuations. A1: For i {,}, Θ i = ª θ i,θ i with θi := θ i θ i 0, θ i > 0, and Pr(θ i )=p i. A2: For any θ Θ, Pr(θ) =Pr(θ ) Pr(θ ). A3: is the only player who knows θ and is the only player who knows θ. A4: θ θ and θ θ. Assumptions A1-A4 identify two markets in which (i) agents have discrete independent private values, (ii) trade occurs under asymmetric information, and (iii) there are gains from trade in either market. Assumption A4 leads to two possible cases: A4.1: θ θ θ θ, A4.2: θ θ θ θ. In all other cases, the outcome in the resale bargaining game does not depend on the beliefs either or have about the rival s valuation. Assuming two types is a limitation, but is needed to keep the analysis tractable. he problem with a continuum of types stems from the difficulty to optimize over the space of all possible posterior beliefs for without imposing ad hoc restrictions. A similar difficulty arises in the literature of dynamic contracting where a principal needs to control for the belies of his future selves in the downstream interaction with the agent; although a complete characterization is available in thetwo-typecase(laffont and irole 1988, 1990), it is well known that the generalization to the continuum is hard to settle. he same problem also emerges in Zheng (2002) if one relaxes the assumptions on the distributions and on the specific bargaining game which are necessary for the optimality of his winner-selection rule.

8 8 G. Calzolari and A. Pavan Primary Market In the primary market, designs a mechanism which consists in a trading procedure along with a disclosure policy. As proved in Pavan and Calzolari (2002), there is no loss of generality in restricting attention to direct revelation mechanisms φ : Θ R 4({0, 1} Z) in which reports a message θ Θ, pays a transfer t (θ ) R andwithprobabilityφ (1,z θ ) receives the good and information z Z is disclosed to in the secondary market. ince players have quasi-linear payoffs, it is without loss of generality to restrict attention to mechanisms φ : Θ R 4({0, 1} Z) instead of φ : Θ 4(R {0, 1} Z), for t (θ ) can always be read as the expected transfer between and. Note that we allow tochooseanymenu(indexedby Θ ) of joint distributions over {0, 1} Z. Imposing restrictions on the correlation between the (marginal) distribution over x and that over z (for example, assuming the lottery over the decision to trade is independent from that used to disclose information to ) may preclude the possibility to characterize the optimal mechanism. We do not assign any precise meaning to the set Z at this point, but we assume it is sufficiently rich to generate any desired posterior beliefs in the secondary market. As we prove in Lemma 1, with a finite number of players and types, Z can be restricted to be a finite set. his abstract representation of information transmission between the two markets allows to replicate with a direct revelation mechanism fairly general disclosure policies. he implementation of the optimal mechanism at the end of the section will suggest a natural interpretation of Z as the set of possible prices paid by in the primary market. Note that the disclosure policy is stochastic for two reasons. First, trade between and may be subject to uncertainty which may be reflected into the signal z when this is correlated with the decision to trade. econd, it may well be in the interest of to commit not to fully disclose to all the information that has been revealed in the primary market. Indeed, we assume is not exogenously compelled to release any particular information and can commit to any disclosure policy of her choosing (the case where can not commit to her disclosure policy is discussed at the end of ection 2.4). Furthermore, can not charge for the information disclosed, which is consistent with the assumption that can not contract with. We relax this assumption in ection 3, where we allow to extract money also from secondary buyers that do not obtain the good in the primary market but receive valuable information about rivals valuations. Finally, can not make the price contingent on the outcome in the secondary market, nor can she design the resale game, for example by assigning bargaining power to one of the two bidders. If this were the case, could also indirectly control the final allocation and the analysis of the constraints imposed by resale would be uninteresting. econdary Market

9 Monopoly with Resale 9 he resale price is assumed to be the result of a random ultimatum bargaining procedure: With probability λ, makes a take-it-or-leave-it offer to, whereas with the complementary probability λ =1 λ, makes a take-it-or-leave-it offer to. Restricting the two players to make take-it-or-leave-it offers that consist of a single price instead of more general mechanisms is without loss of generality in this environment with quasi-linear preferences, private values and finite types (Maskin and irole, 1990, Prop. 11). Hence, one can interpret λ also as the probability a player designs the resale mechanism in the secondary market. his bargaining procedure is stylized, yet common in the contract theory literature. It has the advantage of being particularly tractable and at the same time it allows for an explicit analysis of the dependence of the resale outcome on the information disclosed by the monopolist in the primary market, which is the interest of our analysis. Note that λ =1corresponds to the bargaining game examined in Zheng (2002), where it is always the seller who makes the offer in the secondary market. We believe this assumption may not be appropriate for two reasons. First, it implies the initial monopolist can perfectly control the distribution of bargaining power in the secondary market which in turn eliminates important strategic effects associated with resale (For example, disclosure becomes irrelevant when it is always who makes the price in the resale game). econd, we do not find the justification of symmetry with the primary market fully convincing: hat the initial seller has full bargaining power rarely impliesthatanysellerofthesamegoodwillnecessarily have full bargaining power when contracting with prospective buyers. In general, the distribution of bargaining power is likely to be a function of the allocation of the good, but also of the individual characteristics of the players. y assuming the distribution of bargaining power is entirely identity-dependent (and parametrized by the exogenous probability each bidder offers the resale price), we isolate in this paper strategic effects that are new to the literature and show how they affect the design of optimal selling procedures. 6 iming At t =1,publicly announces a selling mechanism φ Φ,whereΦ is the set of all possible feasible (direct revelation) mechanisms. 7 If refuses to participate in φ, the game ends and all players enjoy their reservation payoffs which are equal to zero. If accepts, he reports θ, pays an expected transfer t (θ ) andwithprobabilityφ (1,z θ ) receives the good and information z Z is disclosed to in the secondary market. 6 In addition to Zheng s representation of the resale game, the other modelizations considered in the literature are the following: (a) resale is the result of any possible bargaining procedure which leads to valuations in the primary market satisfying a set of reasonable conditions (cfr Haile, 1999); (b) the resale game is exogenous, but assumed to lead to efficient allocations with certainty (Ausubel and Cramton, 1999). 7 Pavan and Calzolari (2002) formally prove that assuming publicly announces her mechanism corresponds to the most favorable case for the initial seller. It is important to note that although can observe the direct mechanism φ, she does not necessarily observe its realization, i.e. the announcement θ, the decision to trade, x, or the transfer t.

10 10 G. Calzolari and A. Pavan At t =2, if x =1, bargaining between and takes place according to the procedure described above. Otherwise, the game is over. Figure 1 summarizes the trading environment. [Insert Figure 1 here] he game described above can be solved by backward induction examining first how the price in the secondary market is influenced by the outcome in the primary market and then choosing the selling procedure that maximizes revenue taking into account the effect of disclosure on the resale surplus. 2.2 he outcome in the secondary market Consider first the case where offers the price in the resale game. Given information z Z, updates her beliefs about θ using ayes rule and the fact that received the good in the primary market so that Pr( θ x φ =1,z)= (1,z θ )p φ (1,z θ )p + φ (1,z θ )(1 p ). Note that, even if does not directly observe x,shealwaysmakesheroffer contingent on the event x =1: indeed, trade between and in the secondary market is possible only if received the good from. Foranyz Z and θ Θ, then offers a price 8 t (θ,z)= ( θ if Pr( θ x =1,z) θ θ θ, θ if Pr( θ x =1,z) < θ θ θ. which is obtained comparing the surplus θ θ with the surplus (θ θ ) 1 Pr( θ x =1,z) that can achieve by offering a lower price at the risk of ending up without buying. It follows that the surplus obtains from reselling to when it is the latter to offer the price is r ( θ,θ z) =0for ( any θ Θ and z Z. θ if t r (θ,θ z) = (θ,z)= θ, 0 otherwise. 8 In the case is just indifferent between offering a high and a low price, we assume she offersahighprice. In addition, we assume sells to when he is indifferent between accepting s offer and retaining the good. hese assumptions are not needed and are introduced just to simplify the exposition.

11 Monopoly with Resale 11 Letting r (θ z) :=E θ [r (θ,θ z)], we then have that the difference in the surplus expected by the two types is r (z) :=[r (θ z) r (θ z)] [ θ, 0], with r (z) = p θ 0 if Pr( θ x =1,z) < θ θ θ, if Pr( θ x =1,z) h θ θ θ, θ if Pr( θ x =1,z) θ θ θ. θ θ θ, Next, consider the case where asks the price. When his personal value for the good is θ, asks a resale price ( θ t if p > θ θ, θ (θ )= θ θ if p θ θ. θ θ which is obtained comparing the surplus θ θ that can guarantee himself asking a low price with the expected surplus p ( θ θ ), that can be obtained by asking a higher price θ.itfollows that the surplus expects from resale when he makes the offer is θ θ if t (θ )= θ, and θ = θ, s (θ,θ )= θ θ if t (θ )=θ, 0 otherwise. Letting s (θ ):=E θ [s (θ,θ )], we then have that s := [s (θ ) s (θ )] [ θ, 0], with θ if p θ θ, ³ θ θ i s = θ θ p (θ θ ) (θ θ ) if p, θ θ, θ θ θ θ p θ if p > θ θ. θ θ Note that both s and r are negative: resale (to third parties) not only increases the value the primary buyer attaches to the good, but also reduces the differences between types. As illustrated below, this affects the monopolist s ability to extract surplus in the primary market as well as the structure of the optimal mechanism. 2.3 he optimal mechanism in the primary market At t =1, both and have correct expectations on how the resale outcome depends on the mechanism adopted in the primary market. aking these expectations into account, the monopolist then chooses a mechanism φ Φ which maximizes her expected revenue subject to the buyer s individual rationality and incentive compatibility constraints U (θ ):= X z Z φ (1,z θ ) {θ + λ s (θ )+λ r (θ z)} t (θ ) 0, (IR(θ ))

12 12 G. Calzolari and A. Pavan U (θ ) X z Z φ (1,z ˆθ ) {θ + λ s (θ )+λ r (θ z)} t (ˆθ ), for any ˆθ Θ. (IC(θ )) Formally, the program for the optimal mechanism can be written as ( max E θ t (θ ) P : φ Φ subject to IR(θ ) and IC (θ ) for any θ Θ. Given the price formation process in the resale (sub)game, there is no loss of generality in assuming chooses a mechanism that discloses only three signal, say z l with l =1, 2, 3, such that t (θ,z 1 )= θ for any θ, t (θ,z 2 )=θ for any θ, t (θ,z 3 )= θ if θ = θ and t (θ,z 3 )=θ otherwise. In words, signals z 1 and z 2 correspond to information that induces to offer respectively a high and a low price independently of her valuation, whereas signal z 3 induces θ to offer a high price and θ a low price. Indeed, if φ were to disclose more than three signals, then at least two signals would induce the same offer from in the secondary market. ut then could simply replace φ with another mechanism φ 0 in which these two signals are replaced with a single one, preserving the outcome in the resale game and hence generating the same revenue in the primary market. he formal argument is in Lemma 1. Lemma 1 (Cardinality of the signal space) Let # φ Z := # {z Z : φ (1,z θ ) > 0 for some θ Θ }. For any mechanism φ such that # φ Z>3, there exists another mechanism φ 0 such that # φ 0 Z 3 which is payoff-equivalent for all players. Also note that since the offer t (θ,z 1 ) is always contingent on the event trade occurred in the primary market, the information discloses to when x =0has no value. Hence, with a slight abuse of notation, in what follows we will assume that does not disclose any signal when x =0. his also implies that for each message θ Θ, P 3 l=1 φ (1,z l θ )=1 φ (0 θ ), where φ (0 θ ) is the probability of no trade. As in standard screening models, since the high type can always guarantee himself at least the same payoff as the low type by announcing b θ = θ, at the optimum the two constraints IC(θ ) and IR(θ ) necessarily bind, which implies that 9 t (θ )= X 3 φ l=1 (1,z l θ ) {θ + λ s (θ )+λ r (θ z)}, 9 he proof is in the Appendix Lemma 2.

13 Monopoly with Resale 13 t (θ ) = P 3 l=1 φ (1,z l θ ) θ + λ s θ + λ r (θ z) ª + P 3 l=1 φ (1,z l θ )[ θ + λ s + λ r (z l )]. In other words, θ must be given a price discount (informational rent) 3X U (θ )= φ (1,z l θ )[ θ + λ s + λ r (z l )] l=1 to truthfully reveal his type. Note that θ + λ s + λ r (z l ) is positive since both s and r (z l ) are clearly smaller than θ. Other way stated, resale reduces the differences in valuations, but does not invert the ranking across types. Also note that contrary to standard screening mechanisms without resale, the informational rent U (θ ) depends not only on the probability of receiving the good when announcing b θ = θ, but also on the information disclosed by the monopolist to when reports a low type. ubstituting t (θ ) and t (θ ) into U := E θ t (θ ), the revenue can be written in terms of resale-augmented virtual valuations, whichare defined as the sum of the standard virtual valuations as in Myerson (1981), that is M(θ ):= θ and M(θ ):=θ p 1 p θ, augmented by the virtual surplus each type expects from resale. he latter consists of the real surplus λ s (θ )+λ r (θ z), adjusted by the effect of resale on the informational rent for the high type. Definition 1 (Resale-augmented virtual valuations) Let V (θ z l ):=M(θ )+λ s (θ )+λ r (θ z l ), V (θ z l ):=M(θ )+λ s (θ ) + λ r (θ z l ) p 1 p s p r (z l ), 1 p be the resale-augmented virtual valuation forabuyerwithprivatevalueθ, conditional on disclosing information z l to in the secondary market, with l =1,...,3. Note that since r (z 1 )= θ, r (z 2 )=0, and r (z 3 )= p θ, we have that r (θ z l ) p r (z l )= 1 p θ + p 1 p θ if z l = z 1, 0 i if z l = z 2, p h θ + p 1 p θ if z l = z 3, which implies that the virtual valuations of the low type can be ranked as follows V (θ z 1 ) V (θ z 3 ) V (θ z 2 ). On the other hand, the high type does not expect any surplus from resale when it is who offers the price in secondary market, and hence V (θ z l )=θ + λ s (θ ) for any z l Note that by substituting the expressions for M(θ ),s (θ ) and r (θ z l ) into V (θ z l ), the virtual valuations can be read as functions of the exogenous variables only.

14 14 G. Calzolari and A. Pavan Using Definition 1, the optimal selling mechanism for can thus be obtained by choosing an allocation rule and a disclosure policy that solve the following (reduced) program. Lemma 2 (eliefs-constrained reduced program) he optimal mechanism in the primary market φ maximizes h P3 U = E θ l=1 V (θ z l )φ (1,z l θ )i subject to and P 3 l=1 φ 1,zl θ [ θ + λ s + λ r (z l )] P 3 l=1 φ (1,z l θ )[ θ + λ s + λ r (z l )] θ Pr( θ x =1,z 1) θ θ, (1) Pr( θ x =1,z 2) θ, (2) hθ θ i Pr( θ x =1,z 3) θ θ, θ θ θ θ. (3) (M) Constraints (1) (3) guarantee that, given the mechanism φ and the information z l, it is indeed sequentially optimal for to offer the equilibrium price in the secondary market. Constraint (M) is a disclosure-augmented monotonicity condition which guarantees that the low type does not gain from mimicking the high type. Note that (M) reduces to the standard monotonicity condition according to which trade must occur with a higher probability when the buyer reports a higher type only if λ =1, in which case the information disclosed in the primary market has no effect on the resale price. he remaining constraints, (IC), (IR) and (IR), have already been embedded into the reduced program via the resale-augmented virtual valuations. In what follows, we first construct a solution to the program of Lemma 2. Next, we discuss the implications for the optimality of stochastic allocations and richer disclosure policies (Proposition 1). Finally, we propose a simple implementation where the signals are the prices pays in the primary market (Proposition 2). he solution to the reduced program of Lemma 2 depends on the sign of the virtual valuations V (θ z l ), which in turn depends on the severity of the adverse selection problem as well as on s prior beliefs about the value attaches to the good. For simplicity, consider case A4.1 (overlapping supports). he results for A4.2 are similar and are omitted for brevity. Note that under A4.1, there are no signals z 1 that can induce to offer a high price when she has a low valuation and hence necessarily φ (1,z 1 θ )=0for any θ. On the other hand, θ may be willing to offer a high price, but only if she believes assigns a high value to the good in which case an offer at a low price is likely to be rejected. In what follows, we will say that s prior beliefs are favorable to the buyer when θ is expected to offer a high price even in the event she learns nothing from the outcome in the primary market. his is clearly the most favorable case also for the monopolist who does not

15 Monopoly with Resale 15 have any incentives in changing s behavior in the resale game. At the optimum, sells to either type if V (θ z 3 ) 0 andonlyto θ otherwise. Indeed, when V (θ z 3 ) < 0, the rent must leave to θ in case she sells also to θ is so high that she is better off excluding the low type from trade. hings are more difficult for when s prior beliefs are unfavorable, inwhichcaseθ is expected to offer a low price when her posterior beliefs are close to her prior. he monopolist may then attempt to change s behavior in the resale game by disclosing a signal z 3 with sufficiently higher probability when announces b θ = θ than when he reports b θ = θ so as to induce θ to offer a high price conditional on receiving information z 3. Disclosure has however a cost, for it precludes the possibility to sell with certainty to either type. o see this, suppose trade occurs with probability one both when reports b θ = θ and b θ = θ (i.e. φ (1,z 2 θ )+φ (1,z 3 θ )=1 for any θ ). From IC, must give θ a price discount (informational rent) 11 U (θ ) U (θ )+ θ + λ s + λ ( p θ ) φ (1,z 3 θ ). ut then if the low type pretends he has a high valuation and reports b θ = θ,hegets U (θ ) θ + λ s + λ ( p θ ) φ 1,z3 θ = = U (θ )+λ p θ φ 1,z3 θ φ (1,z 3 θ ) >U (θ ) since necessarily φ 1,z3 θ >φ (1,z 3 θ ). In other words, if the menu offered by the monopolist is such that trade is certain whatever the buyer s choice, the high type, who is not interested in the information disclosed to, would always select the contract with the lowest price. ut then the low type would have an incentive to mimic the high type, paying the same price and inducing the monopolist to disclose the most favorable signal, z 3, with a higher probability. he only way the monopolist can sort the two types when she discloses information is by making the high type pay a higher price than the low type, which is possible only if a higher price is associated with a lottery that delivers the good with a higher probability. It follows that to be incentive compatible, any mechanism which discloses both signals z 2 and z 3, must necessarily be associated with a stochastic allocation rule. With unfavorable beliefs, thus faces a trade-off between selling with certainty but inducing a low resale price, or sustaining a higher resale price but at the cost of not being able to trade with certainty. o solve this trade off, first note that θ is willing to offer a high price conditional on z 3 ifandonlyifpr( θ x =1,z 3) θ, or equivalently φ θ θ (1,z 3 θ ) J(θ )φ 1,z3 θ, where J(θ ):= p (θ θ ) (1 p ) θ < 1 in case of unfavorable beliefs. ince U is increasing in φ 1,z3 θ and since a higher φ 1,z3 θ allows to disclose signal z3 with a higher probability also when reports θ, whenever at the optimum decides to disclose signal z 3, then necessarily φ 1,z3 θ = 1 and φ (1,z 3 θ )=J(θ ). ubstituting φ 1,z3 θ and φ (1,z 3 θ ) into the monotonicity 11 Recall that r (z 2 )=0and r (z 3 )= p θ.

16 16 G. Calzolari and A. Pavan condition (M), we then have that the upper bound on φ (1,z 2 θ ) is given by φ (1,z 2 θ )= 1 J(θ )/K < 1 φ (1,z 3 θ ), where K [0, 1) is a "discount" factor which captures the cost of not being able to sell with certainty to θ when disclosing both signals z 2 and z It follows that the (virtual) value of trading with θ and disclosing both signals is φ (1,z 3 θ )V (θ z 3 )+φ (1,z 2 θ )V (θ z 2 )=J(θ )V (θ z 3 )+ 1 J(θ )/K V (θ z 2 ). o determine the optimal mechanism, thus compares the above with the value V (θ z 2 ) of selling to θ with certainty and disclosing only the less favorable signal z 2. It follows that when V (θ z 2 ) >KV(θ z 3 ) > 0, finds it optimal to favor trade over a higher resale price and at the optimum φ 1,z2 θ = φ (1,z 2 θ )=1. When instead V (θ z 2 ) is positive but less than KV(θ z 3 ), the optimal mechanism consists in selling with certainty to the high type and with probability φ (1,z 2 θ )+φ (1,z 3 θ ) < 1 to the low type. Finally, when V (θ z 2 ) < 0, but V (θ z 3 ) 0, the rent must leave to the high type in case she sells to θ and θ offers a low price is so high that does not find it profitabletodisclosesignalz 2, so that at the optimum φ (1,z 2 θ )=0and φ (1,z 3 θ )=J(θ ). In this case, the allocation rule is stochastic, but the optimal disclosure policy is deterministic and consists in disclosing only signal z 3 whatever the buyer s type. Wesummarizetheaboveresultsinthe following Lemma and in Figure 2. Lemma 3 (Optimal mechanism) Assume A1-A4.1. When beliefs are favorable, the optimal allocation rule and disclosure policy are deterministic: discloses only signal z 3 and sells to the low type if and only if V (θ z 3 ) > 0. When instead beliefs are unfavorable, discloses only signal z 3 when V (θ z 2 ) < 0, both z 2 and z 3 when V (θ z 2 ) [0,K V(θ z 3 )], and only z 2 when V (θ z 2 ) >K V(θ z 3 ) > 0. Furthermore, excludes the low type when V (θ z 3 ) < 0, and sells to θ 1. with probability J(θ ) when V (θ z 2 ) < 0 <V(θ z 3 ), 2. with probability J(θ )+[1 J(θ )/K] when V (θ z 2 ) [0,K V(θ z 3 )], 3. with probability one when V (θ z 2 ) >K V(θ z 3 ) > 0. In all cases, the high type always receives the good with certainty. [Insert Figure 2 here] 12 Note that K = the initial prior of. [ θ +λ s ]J(θ ) [ θ +λ s ]J(θ )+[1 J(θ )]λ p θ is increasing in J(θ ), that is, it is higher the more favorable

17 Monopoly with Resale 17 wo important insights can be derived from Lemma 3. First, the presence of a secondary market significantly changes the properties of revenue-maximizing allocations in the primary market. In particular, contrary to standard mechanisms that do not explicitly account for resale and for which trade is deterministic, a monopolist who expects her buyers to resell may need to induce stochastic allocations in the primary market (either through lotteries, or mixed strategies). his follows from the fact that the decision to trade is an informative signal of the buyer s valuation; as a consequence, deterministic mechanisms in which the monopolist sells with certainty to a subset of types and excludes the others, typically fail to induce the desired beliefs in the resale game. o see how stochastic allocations may lead to higher revenues than deterministic ones, consider area EFD in Figure 2, where for simplicity we restricted attention to λ =1, which is the most interesting case for the analysis of the manipulation of beliefs in the secondary market. Note that in this case, s prior beliefs are unfavorable J(θ ) < 1 andv(θ z 2 ) < 0 <V(θ z 3 ), implying that finds it optimal to trade with θ if and only if θ offers a high resale price. With deterministic mechanisms, then neither, nor, would benefit from resale. Indeed, if were to sell to both types, then would learn nothing about s valuation and since her prior beliefs are unfavorable, she would offer a low price in the resale game. If, on the other hand, were to sell only to the high type, then the third party would offer a price equal to the buyer s high valuation, once again leaving no surplus to the buyer. In contrast, with a stochastic allocation, can trade with positive probability also with the low type and at the same time induce to offer a high resale price in the secondary market. econd, to increase the level of trade in the primary market without reducing the probability of a high resale price, the monopolist may need to disclose information in addition to the decision to trade. o see this, consider area HDI in Figure 2 in which beliefs are unfavorable and V (θ z 2 ) > 0. If were to sustain a high resale price disclosing only the decision to trade, then the best she could do is selling to the high type with certainty and to the low type with probability J(θ ). In contrast, by adopting a richer disclosure policy, can increase the level of trade with θ above J(θ ) and at the same time guarantee that θ continues to offer a high resale price with probability J(θ ). It follows that when trade with the low type is profitableevenintheevent offers a low price, that is when V (θ z 2 ) > 0, combining lotteries with a stochastic disclosure policy leads to a higher revenue. 13 We conclude that Proposition 1 (tochastic allocations and disclosure policy) Inthepresenceofresale,it is generically impossible to maximize revenue with deterministic allocations. Furthermore, to optimally fashion the beliefs of the participants in the secondary market, a monopolist may need to adopt 13 Recall that a stochastic disclosure policy is incentive compatible only if it is associated with a stochastic allocation rule.

18 18 G. Calzolari and A. Pavan a disclosure policy richer than the simple announcement of the decision to trade with a particular buyer. We now turn attention to the implementation of the optimal mechanism. o create the desired informational linkage, the monopolist has two natural instruments: (1) she can sell to different types with different probabilities so that the decision to trade is itself informative of the buyer s valuation; (2) she can disclose the price pays in the primary market as an additional signal. he next proposition indicates a simple implementation based on these two instruments. We limit attention to the interesting case in which in the direct mechanism discloses both signals z 2 and z 3. In all other cases, z does not convey any information and the optimal disclosure policy is simply the announcement of the decision to trade. Proposition 2 (Prices as signals) When the optimal informational linkage can not be implemented disclosing only the decision to trade, the monopolist uses the price as an additional signal and offers a menu of lotteries which induce the buyer to follow a mixed strategy. Let the menu consists of two price-lottery pairs: if pays τ H he receives the good with certainty, whereas if he pays τ L <τ H, he receives the good with probability positive, but less than one. he menu is designed so as to induce the high type to pay τ H and the low type to randomize choosing τ H with probability J(θ ) and τ L with probability 1 J(θ ).Given s mixed strategy, it is optimal for θ to offer a high price when she observes τ H and a low price when she observes τ L, so that the signals z 2 and z 3 in the direct mechanism are simply the prices pays in the primary market. Note that needs to combine lotteries with mixed strategies. o see this, suppose tries to induce to randomize over τ L and τ H without using lotteries, i.e. by selling with certainty whatever the price. hen, the high type who does not care about the information disclosed to would always pay the lowest price τ L. Anticipating that will never offer a high price if she observes τ H, the low type would also pay τ L. o sort the two types and create the desired informational linkage, must thus associate τ L with a lottery that delivers the good with probability strictly less than one. Next, suppose tries to implement the optimal mechanism with lotteries, but without making play a mixed strategy. If perfectly separates the two types and discloses the price, she fully informs about s valuation, which is clearly never optimal. If, on the other hand, she separates the two types and keeps the price secret, then the only thing can do to sustain a high resale price is assign τ L to a lottery that gives the good with probability at most equal to J(θ ). On the contrary, by inducing the buyer to play a mixed strategy and using the price as a signal, the monopolist can increase the level of trade with the low type and induce to offer a high resale price with the same probability.