STRATEGIC LIQUIDITY SUPPLY AND SECURITY DESIGN *

Transcription

1 STRATEGIC LIQUIDITY SUPPLY AND SECURITY DESIGN * by Bruno Biais Université de Toulouse (GREMAQ-IDEI-CRG), and CEPR and Thomas Mariotti London School o Economics and Political Science Contents: Abstract 1. Introduction 2. The Basic Model 3. Liquidity Supply 4. Security Design 5. Robustness 6. Conclusion Appendi Reerences The Suntory Centre Suntory and Toyota International Centres or Economics and Related Disciplines London School o Economics and Political Science Discussion Paper Houghton Street No.TE/03/445 London WC2A 2AE January 2003 Tel.: * We would like to thank Mike ishman, Martin Hellwig, Roman Inderst, Bruno Jullien, Nobu Kiyotaki, Jean-Jacques Laont, David Martimort, Benny Moldovanu, John Moore, Patrick Rey, Jean-Charles Rochet, Ernst Ludwig von Thaden, and Jean Tirole, as well as seminar participants at Mannheim University, the London School o Economics and Political Science, Northwestern University, Lausanne University and Toulouse University, or insightul comments.

2 Abstract We study how securities and trading mechanisms can be designed to optimally mitigate the adverse impact o market imperections on liquidity. Asset owners seek to obtain liquidity by selling their claims on uture cash-lows, on which they have private inormation. Our analysis encompasses both the cases o competitive and monopolistic liquidity supply. In the optimal trading mechanism associated to an arbitrary given security, issuers with low cash-lows sell their entire holdings o the security, while issuers with larger cash-lows are typically ecluded rom trade. By designing the security optimally, issuers can eshew eclusion altogether. The optimal security is debt. Because o its low inormational sensitivity, debt mitigates the adverse selection problem. urthermore, by pooling all issuers with high cashlows, debt also reduces the ability o a monopolistic liquidity supplier to eclude them rom trade in order to better etract rents rom issuers with low cash-lows. Keywords: Security design, liquidity, mechanism design, adverse selection, inancial markets imperections. JEL Nos.: G32, L14. by the authors. All rights reserved. Short sections o tet, not to eceed two paragraphs, may be quoted without eplicit permission, provided that ull credit, including notice, is given to the source. Contact address: Dr Thomas Mariotti, Department o Economics, London School o Economics and Political Science, Houghton Street, London, WC2A 2AE, UK. t.mariotti@lse.ac.uk

3 1. Introduction While corporate inance oers insights in the design o optimal securities (see, e.g., Townsend (1979), Gale and Hellwig (1985), Allen and Gale (1988), Harris and Raviv (1989)), market microstructure analyzes how dierent trading mechanisms can oer variable degrees o liquidity, emphasizing the consequences o adverse selection and strategic behavior. We borrow rom these two approaches to study the interaction between security design and market mechanisms. Our objective is to study how the design o securities and markets can mitigate imperections and thus enhance the liquidity and eiciency o the issuance and trading processes. To motivate our analysis, consider an entrepreneur who owns a project yielding random cash-lows in the uture. Because o impatience or liquidity needs, she would like to sell today claims on these cash-lows. Brealey and Myers (2000, Chapter 15, page 419) describe the issuance process as ollows: Suppose that your company is likely to need up to $200 million o new long term debt over the net year or so. It can ile a shel registration or that amount. It then has prior approval [rom the SEC] to issue up to $200 million o debt, but it isn t obligated to issue a penny. [...] Now you can sit back and issue debt as needed, in bits and pieces i you like. Suppose Merrill Lynch comes across an insurance company with $10 million ready to invest in corporate bonds. Your phone rings. It s Merrill Lynch oering tobuy$10millionoyourbonds,pricedtoyield,say,81/2percent. I you think that s a good price, you say OK and the deal is done [...] Here is another possible deal. Suppose that you see a window o opportunity in which interest rates are temporarily low. You invite bids or $100 million o bonds. Some bids may come rom large investment banks acting alone; others may come rom ad hoc syndicates. But that s not your problem; i the price is right, you just take the best deal oered. This description underscores the process by which a security is designed and then marketed. It points at the role o the buyers in posting price oers, which the issuer can accept or not. The security can be purchased and priced by a single inancial institution, MerrillLynchintheeample, oritcanbeoered to several possible buyers, and sold to the highest bidder. Our model relects these stylized acts, and, in particular, encompasses the case where the inancial institution has market power. As a matter o act, the underwriting industry is highly concentrated. Brealey and Myers (2000, Chapter 15, page 415) report that the si largest underwriters (Merrill Lynch, Salomon Smith Barney, Morgan Stanley, Goldman Sachs, Lehman Brothers, and JP Morgan) manage the majority o the securities issues. Along with the major role o large banks, Benveniste, Busaba and Wilhelm (2002) point at the niche occupied by 1

4 specialized inancial institutions. or eample, they report that, out o iteen truckingindustry IPOs completed between 1990 and 1994, nine were managed by one bank (Ale. Brown). They urther argue that issues are oten priced by a single inancial intermediary. Empirically, the market power o inancial intermediaries and market makers has been evidenced in several contets (see Christie and Schultz (1994), or Chen and Ritter (2000)). Theoretically, it has been traced back to institutional constraints and private inormation. Thus the banking literature emphasizes that market power may arise because o capital adequacy requirements that limit entry, or because banks with preeisting relationships with irms are in a privileged position to lend to them (see reias and Rochet (1997)). In addition to market power, a second imperection is likely to hamper the eiciency o the security issuance process: irms issuing securities are oten likely to have private inormation about their uture cash-lows. As shown by Leland and Pyle (1977) and Myers and Majlu (1984), this creates an adverse selection problem, reducing the liquidity o the market and the gains rom trade that could be reaped by the issuers. The goal o the present paper is to analyse the security design and issuance process in presence o market power and adverse selection. In particular, we endeavor to shed light on the ollowing issues: (i) Through what channels does market power aect liquidity? How does it eacerbate the lemons problem induced by adverse selection? (ii) How do issuers react to the market power o inancial intermediaries? How can they mitigate the illiquidity it induces? Does this alter qualitatively the type o security they issue? Our analysis is in line with the insightul recent paper by DeMarzo and Duie (1999). In both papers, while the security is designed under homogeneous inormation, it is then traded ater the issuer has observed a private signal on the proitability o her assets. The preerences and inormation sets in our paper are similar to theirs. There are two major dierences with their approach. irst, we analyse the consequences o the market power o inancial intermediaries. In contrast, they study competitive liquidity suppliers earning zero epected proits in a signaling game similar to Kyle s (1985). Second, we take a mechanism design approach to analyze the trading process. Thus, in our analysis both the design o the security and the design o the issuance mechanism are endogenous. More precisely, in our model, liquidity suppliers oer trading mechanisms, to which the issuer reacts. In this screening game, the liquidity suppliers place transer schedules, speciying the prices at which they are willing to buy variable quantities. The issuer then selects rom this menu o oers the trade size maimizing her epected utility. To highlight the consequences o market power we analyse two etreme cases, namely the 2

5 case o a single monopolistic liquidity supplier, and the case o competitive liquidity suppliers. The set o trading mechanisms we analyze encompasses the simple case where the inancial intermediaries post a single price, as in the above quote rom Brealey and Myers. It is also consistent with the IPO process where investors place oers to buy a variable number o shares at certain prices, either through indications o interest in the book building process or through bids in IPO auctions. 1 or a given security design, the outcome o the trading interaction between the issuer and the liquidity suppliers has the ollowing characteristics. The worse the private signal o the issuer, the more she is eager to sell the security, and the greater her trade. As in Akerlo (1970), the good types, i.e., the issuers with large uture cash-lows, are those who suer the most rom the adverse selection problem. Because o the linearity o the problem, there is a bang-bang solution, relecting partial market break-down. Issuers with cash-lows above a certain threshold are entirely ecluded rom trade. In contrast, issuers with cash-lows below this threshold sell 100% o their holdings o the security. It ollows that the optimal trading mechanism can be implemented in a very simple way. Each inancial intermediary oers to buy the security by posting a linear price schedule. The issuer is then ree to accept any o these oers. Note that this eactly its Brealey and Myers description o the issuance process as quoted above. Relecting the adverse selection problem, and the nature o the screening game we analyse, the endogenous cost unction o the liquidity suppliers takes the orm o lowertail conditional epectations, as in Glosten (1994) and Biais, Martimort and Rochet (2000). This implies that, as long as a non-empty set o issuers are ecluded rom the market, the price at which the liquidity suppliers are willing to purchase any amount o the security is strictly lower than the unconditional epectation o the value o the security. This is analogous to the small trade spread arising in screening models o market microstructure. While the qualitative eatures o the market outcome are the same in the monopolistic and competitive cases, the spread, and correspondingly the raction o issuer types ecluded rom trade, are greater with a monopolistic liquidity supplier. Very much in line with the classical IO paradigm, the monopolist preers to reduce the volume o trade by ecluding more types, in order to etract greater rents rom the types who remain in the market. Our results contrast with DeMarzo and Duie (1999), where (i) ininitesimal trades have an ininitesimal impact on prices, (ii) issuers sell a raction o the security (which is interpreted as collateralization or tranching), and (iii) the good types are not entirely ecluded rom the market. The dierence in results is due to the dierence in trading mechanisms. Note that the separating equilibrium allocation obtained by DeMarzo and 1 Hanley and Wilhelm (1995) or Cornelli and Goldreich (1998) document empirically the placement o orders in the book building process. Biais and augeron-crouzet (2002) discuss IPO auctions. Benveniste, Wilhelm and Yu (1999) present empirical evidence on the determination o quantities sold by issuing irms ater bids have been placed. 3

6 Duie (1999) is implementable in our mechanism. In the competitive case, however, the e-ante eiciency o the allocation we characterize is greater than that o any equilibrium o their trading game. More generally, the competitive screening model we analyze allows to characterize the upper bound on the gains o trade that can be achieved given the agents preerences and inormation. While, as discussed above, adverse selection and market power induce ineiciencies, the issuer designs the security to mitigate these imperections and increase the gains rom trade. In line with the stylized act that debt is the major source o outside inancing (see, e.g., Grindblatt and Titman (1998, page 5)), we ind that the optimal security is a debt contract. This result relects two phenomena. irst, as in Myers and Majlu (1984) and DeMarzo and Duie (1999), debt mitigates the adverse selection problem, by making the payo o the security less sensitive to the high cash-low realizations. Second, and this is a distinctive contribution o our analysis, debt mitigates the adverse consequences o market power on the gains rom trade. To maimize proits, the monopolistic liquidity supplier seeks to reduce the rents earned by the agents with low cash lows, by making it costly or them to mimick the good types. When the payo o the security increases smoothly with the cash-low rom the project (as with equity), this is achieved by ecluding the best types rom trade. This partial market break-down is avoided with debt, as long as the ace value is not too high. Indeed, with debt contracts, the payo othesecurityisthesameorallissuerswithuture cash-lows above the debt service. Hence the liquidity supplier must either include them all, or eclude them all rom the market. Since the latter would be quite costly, as it would imply loosing a large raction o the most proitable customers, he preers to design his schedule so that all issuers participate to the market. Hence, the optimal design o the security enables to entirely avoid eclusion rom the market. In contrast with the signaling model o DeMarzo and Duie (1999), all issuers types sell 100% o their security holdings. Correspondingly, there is no inormational content o trades: the epectation o the value o the security given a sale is equal to its unconditional value. This low inormation content o the sale o debt securities in our model is in line with the results o several empirical studies (see, e.g., Dann and Mikkelson (1984), Eckbo (1986), and Mikkelson and Partch (1986)). One could argue that there are three limitations to our analysis. irst, we assume that the issuer initially designs one security, and is then restricted to that security. 2 Second, or technical reasons, we require that both the security payo and the residual claim o the agent be increasing in the inal cash-low generated by the asset. This rules out mechanisms which have been shown to be optimal in other security design analyses, such as the live-or-die contract obtained by Innes (1990) in a moral hazard contet. 2 This is in line with DeMarzo and Duie (1999), but diers rom Nachman and Noe (1994), where the security is designed by the agent ater she has observed her private inormation, and thus conveys a signal o the proitability o her assets. 4

7 Third, while we analyse the competitive case as an inormation-constrained Pareto optimum, one might wonder i it could emerge rom an actual trading game, where liquidity suppliers would post competing schedules. We show that our analysis is robust to these three limitations. irst, we analyze, in the competitive case, the situation where, instead o one security, the issuer initially designs a menu o securities, among which she will be able to choose at the trading stage. We ind that the equilibrium allocations arising in this more general setting are eactly the same as those arising in our basic model. urthermore, no monotonicity assumption is needed to obtain the optimality o debt. We also study the case where several liquidity suppliers post non-eclusive competing transer schedules. We show that the trades arising in the competitive case are a Nash equilibrium o this oligopolistic liquidity supply game. The paper is organized as ollows. The model is described in Section 2. In Section 3, we analyse the design o the trading mechanism. In Section 4, we turn to the security design problem. Etensions o our analysis are presented in Section 5. Section 6concludes. 2. The Basic Model Our model is in line with DeMarzo and Duie s (1999). We etend their paper along two dimensions. irst, we consider arbitrary trading mechanisms. Second, we allow or market power on the part o the liquidity supplier The Etensive-orm o the Game Agents. There are two agents, an issuer and a liquidity supplier. The issuer owns assets that generate uture cash-lows drawn rom an absolutely continuous c.d.. G with positive density g over a compact interval =[, ] R ++. Both agents are risk-neutral and the market interest rate is normalized to zero. The issuer has an incentive to raise cash by selling part o her assets. The reason or this is that she is more impatient than the liquidity supplier. We accordingly denote by (0, 1) the issuer s discount actor, while the liquidity supplier s is normalized to one. The e-ante private value o the assets to the issuer is thus lower than the value they have or the liquidity supplier. There are thereore gains rom trade rom transerring the assets rom the issuer to the liquidity supplier. Security Design. In order to raise cash, the issuer designs a limited-liability security backed by her assets. In general, the payo o this security may depend on any e-post contractible inormation. or simplicity, we assume that the payo o this security canonlybemadecontingentontherealizedcash-low, i.e., there eists a measurable mapping ϕ : R + such that = ϕ(). As Harris and Raviv (1989) or Nachman and Noe (1994), we shall in a irst step restrict the set o admissible securities by imposing the ollowing limited liability and monotonicity conditions: 5

8 (LL) ϕ() [0,]orall ; (M) ϕ is non-decreasing on ; (MR) Id ϕ is non-decreasing on, where Id is the identity unction on. Conditions (M)-(MR) require that both the payments to the liquidity supplier and to the issuer be non-decreasing in the realized cash-low. Together with (LL), they imply that the set Φ o admissible securities payments is a subset o Lipschitz unctions on. Wedenoteby =[, ] the interval o easible payments associated to an admissible security. Timing and Inormation Structure. There are two periods, and ive stages. The sequence o events in the irstperiodisasollows: (i) irst, the agent designs the security ; (ii) Net, a trading mechanism T :[0, 1] R is designed or the sale o any raction q [0, 1] o the securitized asset; (iii) The issuer privately learns the realization o the cash-lows ; (iv) I the issuer accepts the trading mechanism T, she trades a volume q o the security, or which she obtains a transer T (q). inally, in the second period, (v) The value o the cash-lows is publicly revealed and any remaining consumption takes place. We shall return in more detail to the question o who designs the trading mechanism at stage (ii). Meanwhile, two eatures o this etensive orm are worth emphasizing. irst, the issuer has perect advance knowledge o the cash-lows at the interim stage (iii). This diers rom DeMarzo and Duie (1999), who allow or noisy private signals. Second, the issuer designs her security beore she learns the realization o the cashlows. Thereore, unlike in Nachman and Noe (1994), the choice o a security cannot be used by the issuer as a signal o the proitability o her assets. The assumption that, at stage (i), the issuer designs a single security, rather than a menu o securities, will be relaed in Section 5. This etensive orm o the game is in line with the description o the issuance process oered by Brealey and Myers (2000) and quoted in our introduction. Anticipating that it will need unds in the uture, the irm designs the security and goes through the shelregistration process. This corresponds to stage (i) in our game. Once the security is shel-registered, it can be issued some time later, or eample one year later. This corresponds to stage (iv) in our game. 6

9 Our model is also consistent with the case where the security is irst designed and sold to an intermediary at stage (i), and then resold by the intermediary at stage (iv). Consider or instance a manuacturing irm that will generate cash-lows in the uture, and currently needs cash. It initially sells to a inancier a security with payo contingent on these cash-lows (this corresponds to stage (i) in our game). The inancier can be a bank providing a loan, a supplier providing trade credit, or a venture capitalist purchasing convertible bonds. Ater this initial echange, the inancier naturally receives inormation about the project (this corresponds to stage (iii) in our game). The inancier may be itsel subject to a liquidity shock, and thus led to demand liquidity rom the market. 3 To obtain liquidity, it sells the security it holds in its portolio (this corresponds to stage (iv) in our game). The price at which it initially purchases the security rom the manuacturing irm relects its rational epectations about uture market liquidity. The security is initially designed to maimize market liquidity, and correspondingly the initial sale price Comparison with DeMarzo and Duie (1999) The irst main dierence between our model and the setting considered by DeMarzo and Duie (1999) is that we take an alternative approachtomodelingthetradinggame. They consider a signaling game, whereby the issuer, ater observing her signal, chooses the size o her trade, and the liquidity suppliers react to this quantity by quoting prices. In contrast, we take a mechanism design approach. The trading mechanism is a menu o pairs {q, T(q)} q [0,1], rom which the inormed agent selects her optimal trade. This menu o trades, designed beore the private signal is observed, can be interpreted as a screening mechanism. I the transer schedule T is concave, it amounts to a sequence o limit orders, as in Biais, Martimort and Rochet (2000). The allocation arising in the separating equilibrium considered by DeMarzo and Duie (1999) is implementable in the trading mechanism. It is not the optimal allocation, however, as established in the net section. This is because the screening mechanism yields more commitment power, as the liquidity supplier can commit to a menu o trades beore the quantity q is observed. 4 The second dierence is that, while in the contet o the signaling model, competitive liquidity supply is warranted, we allow or strategic liquidity supply. We consider 3 In the case o a bank, the need or liquidity can be due to prudential rules (see Dewatripont and Tirole (1994)). In the case o a trade creditor it can stem rom a transient cash-low gap, or an investment opportunity, combined with credit rationing constraints. In the case o a venture capitalist it can relect an opportunity to invest in new projects combined with constraints on raising new unds. 4 This commitment power makes it possible to engineer cross-subsidization between the issuer s types. Indeed, in the equilibrium o our screening model, the liquidity supplier will earn proits when trading with issuers whose cash-lows are high, while he will make losses when trading with issuers whose cash-lows are low. This cross-subsidization o the bad types by the good types also takes place in the screening games analyzed by Glosten (1994) and Biais, Martimort and Rochet (2000). 7

10 two polar cases. In the monopolistic case, the trading mechanism is designed at stage (ii) by the liquidity supplier, to maimize his epected proit, under the incentive and participation constraints o the inormed agent. The latter constraint requires that the inormed agent accepts to participate in the trading mechanism at stage (iv). In the alternative case, reerred to as the competitive case, the trading mechanism is designed at stage (ii) by the issuer to maimize her epected utility, subject to the participation constraint o the liquidity suppliers. By comparing the allocations arising in the monopolistic case and in the competitive case, we shed some light on the consequences o market power or market liquidity Incentive Compatibility Conditions Given a security design, and a transer schedule T, the issuer selects what raction q o the security to sell to the liquidity supplier, conditional on her private inormation about uture cash-lows. At this interim stage, since the issuer has perect advance knowledge o the cash-lows, and since the security s payo is only contingent on these, she also perectly knows the realization = ϕ() o. Her utility is T (q)+( q), while the proit o the liquidity supplier is q T (q). Thus, the type o the issuer is entirely summarized by, and the set o possible types is. The Lemons Problem. An issuer with type inds it attractive to trade q rather than not to trade at all and consume i and only i: T (q) q. This condition holds i the security payo is low enough, the unit price T (q) o the q security is high enough, and is low enough so that the issuer is suiciently impatient. Overall, the willingness to trade reveals a relatively low type. This underscores the nature o the adverse selection problem arising in our model, which is very much in line with Akerlo s (1970) lemons problem. Implementable Mechanisms. Under adverse selection, the trading mechanism must be incentive compatible and satisy the issuer s individual rationality constraint or all realizations o. There is no loss o generality in applying the revelation principle (Myerson (1979)). I.e., any implementable allocation achieved via a transer schedule T can also be achieved via a truthul direct mechanism (τ,q): R [0, 1] that stipulates a transer and a trading volume as a unction o the issuer s report o her type. Incentive compatibility requires that: arg ma ˆ τ( ˆ) q( ˆ);. (1) 8

11 We denote by U the corresponding inormational rent: U () =sup ˆ τ( ˆ) q( ˆ). (2) U is analogous to the inormational rent o a regulated irm with privately observed marginal cost, as in Baron and Myerson (1982). We take the dual approach and characterize the set o pairs (U,q) that correspond to an incentive compatible mechanism. This set is characterized in the ollowing lemma. Lemma 1 Apair(U,q) is implementable i and only i: (i) U is conve on ; (ii) or almost every, U () = q(). Lemma1simplyrelects the act that U is the upper enveloppe o a amily o aine and decreasing unctions o. Conveity o U together with U = q implies the ollowing important property. Lemma 2 In any implementable allocation, the volume o trade q is non-increasing in the security payo, and consequently in the cash-low. The intuition is in the line o Akerlo (1970). As discussed above, issuers with relatively largeuturecash-lows are relatively less eager to trade at a given price than issuers with lower uture cash-lows. That issuers with low cash-lows are always ready to trade depresses the price, which makes issuers with high cash-lows even less eager to trade. In the limit this can lead to a market break-down, where the issuers with the largest cash-lows obtain zero gains rom trade. Lemmas 1 and 2, and their intuition are similar to Proposition 1 in DeMarzo and Duie (1999). E-Post Rationality Constraints. In addition to the above incentive compatibility constraint, a easible trade mechanism must also satisy the issuer s e-post participation constraint. Speciically, since the issuer has always the option not to trade, and since in this case she cannot be compelled to pay anything to the liquidity supplier, the issuer s inormational rent U must always be non-negative: Since U is non-increasing by Lemma 1, this simpliies to: U () 0;. (3) U () 0. (4) 9

12 2.4. The Epected Utilities o the Agents Given a security and a schedule T, the epected proit o the liquidity supplier is: (q() T (q()) dg ϕ () (5) where G ϕ is the c.d.. o the random variable = ϕ(). Similarly, the e-ante epected inormational rent o the issuer is : (T (q()) q()) dg ϕ (). (6) Adding the epected proits o the liquidity supplier and the epected rent o the issuer, we obtain the total gains rom trade: (1 ) q() dg ϕ (). (7) Thus, the gains rom trade are an increasing unction o the dierence between the discount rate o the liquidity suppliers and that o the issuer, and o the amount o cash-lows transerred rom the second period to the irst E-Ante Eiciency As a benchmark, we irst consider the case where a benevolent social planner chooses a trading mechanism so as to maimize social welare. ollowing Holmström and Myerson (1983), eiciency is deined at an e-ante stage, i.e., beore the issuer learns the value o the uture cash-lows. Thus an e-ante optimal mechanism solves: (T (q()) q()) dg ϕ () sup (T,q) subject to the liquidity supplier s participation constraint: (q() T (q()) dg ϕ () π or some π 0. Solving this program is immediate. The participation constraint o the liquidity supplier is binding, and the optimal trading volume is q =1. Itollows then rom (7) that an equity contract maimizes the epected gains rom trade. 3. Liquidity Supply In this section, we analyze the optimal price-quantity schedule or a given security design. We consider the two polar cases o competitive and monopolistic liquidity supply. In the competitive case, the schedule T is designed by the issuer, who etends a take-it-or-leave it oer to the liquidity supplier; the situation is reversed in the monopolistic case. 10

13 The Competitive Case. We irst consider the case where the issuer has all the bargaining power. Given a security, the issuer s problem is to design the transer schedule T to maimize her epected rent (6), subject to her incentive compatibility condition (1), her e-post individual rationality condition (3), and the participation constraint o the liquidity supplier, that his epected proit be non-negative. Recall that the epected rent o the issuer is equal to the epected total gains rom trade minus the epected proit o the liquidity supplier: (1 )q() dg ϕ () (q() T (q())) dg ϕ (). To maimize her rent, the issuer designs the schedule so as to saturate the participation constraint o the liquidity supplier and set his epected proit to zero. The liquidity supplier s zero-proit conditionsimpliies the program o the issuer to the choice o a trading volume q which maimizes the overall epected gains rom trade (7) under her incentive compatibility condition, characterized in Lemma 1, and her e-post participation constraint (4). The only dierence between this problem and the design o the e-ante eicient allocation is the e-post participation constraint, since the e-ante eicient trading proile is incentive compatible. The Monopolistic Case. Now turn to the case where the liquidity supplier has all the bargaining power. The liquidity supplier s task is to choose a transer schedule T in order to maimize his epected proit (5), subject to the incentive compatibility condition (1) and the e-post individual rationality condition (3). Recall that the epected proit o the liquidity supplier is equal to the epected total gains rom trade minus the epected inormational rent o the issuer: (1 )q() dg ϕ () (T (q()) q()) dg ϕ (). The relevant constraints are again the incentive compatibility conditions, characterized in Lemma 1, and the e-post participation contraint (4). Since the inormational rent is non-increasing, the participation constraint o the issuer must be binding at the upper end o the support. The Optimal Trading Mechanism. The menus (τ c,q c )and(τ m,q m )oered respectively by the issuer and by the liquidity supplier are characterized in the ollowing proposition. Proposition 1 There eist c m and τ c 0 τ m 0 =0such that, or all, and or i {c, m}, (i) τ i () =τ i 0 + i whenever i and τ i () =τ i 0 otherwise; (ii) q i () =1whenever i and q i () =0otherwise. 11

14 Moreover, τ c 0 =0whenever c <. In both the competitive and the monopolistic cases, issuers with cash-lows below a the threshold i sell 100% o the security, while issuers above this threshold do not trade at all. Correspondingly, issuers with small uture cash-lows obtain large gains rom trade, while issuers with large uture cash-lows can ace a market break-down, and obtain no gains rom trade. This bang-bang solution diers markedly rom the signaling equilibrium analyzed by DeMarzo and Duie (1999), where the trade smoothly decreases with the issuer s type. As in the monopoly pricing model o Riley and eckhauser (1983), it arises because o the combined eect o the linearity o the preerences and the screening nature o the trading game. In order to saturate the liquidity supplier s break-even constraint in the competitive case, it can be necessary to allow or a lump-sum transer τ0 c given to the issuer independently o her trade. This can however only arise when no type o the issuer is ecluded rom trade. Indeed, when some types are ecluded rom trade, it is preerable to increase the price o the security, in order to make trading more attractive or the good types and thus minimize the etent o the market break-down, rather than giving a lump-sum transer. As the threshold value o the cash-low above which the issuer eits the market is greater in the competitive than in the monopolistic case, more gains rom trade are achieved in the ormer than in the latter. This bears some analogy with credit rationing models such as Bolton and Scharstein (1990), or market microstructure models such as Biais, Martimort and Rochet (2000). The intuition is that the monopolistic liquidity supplier trades o the beneits o a high volume o trade against the incentive costs o inducing the issuer to reveal truthully low realizations o the cash-lows. This renteiciency trade-o is less acute when the issuer designs the trading mechanism, since the rent etraction motive is not present. Implementation. The optimal transer schedule can be implemented with a limit order to buy, or bid price, posted by the liquidity supplier, at which he stands ready to buy up to one unit o the security. 5 Saturating the participation constraint o the liquidity supplier and, or simplicity, neglecting the lump-sum ta τ0, i we obtain the price at which the competitive liquidity supplier purchases the security: c dg ϕ () G ϕ ( c ) = E( c ). 5 In line with the analogy drawn in the market microstructure literature between limit orders and options (Copeland and Galai (1983)), we can also interpret this arrangement as the option, or the issuer, to sell her securities at a predetermined price. 12

15 This is reminiscent o the result obtained by Glosten (1994) in a screening model with competitive market makers, where the bid is equal to the lower tail epectation o the inal value o the security. In the competitive case, the threshold c above which issuers opt out rom trading, and the bid price are pinned down by combining this lower tail epectation and the e-post rationality condition o the issuer: c = E( c ). In line with basic price theory, the valuation o the marginal issuer or the security is equated with the security s price. The dierence between the bid price and the unconditional epectation o the value o the security is similar to the bid-ask spread. The greater the probability mass corresponding to low cash-low realizations, the lower the bid price, the wider the spread, and, consequently, the greater the mass o high cash-lowsissuerswhoare deterred rom trading. This is similar to the result obtained in screening models o market microstructure (Glosten (1989, 1994), Biais, Martimort and Rochet (2000)) that the small trade spread maps into the set o investors types who are ecluded rom trade. 4. Security Design In both the competitive and the monopolistic environments, the issuer s problem is to choose a security, or equivalently a unction ϕ Φ, in order to maimize her epected rent, anticipating the equilibrium price at which she will be able to sell the securities. or simplicity we assume hereater that: µ d G() 1 ;. (8) d g() This condition is slightly stronger than the standard assumption o log-concavity o the density g. It ensures that one may neglect the constraint that U be conve when solving or the optimal transer schedule. In other terms, it enables us to ocus on the irst-order conditions o the mechanism design problem, while warranting that the second-order conditions hold Debt and Equity To build some intuition about the security design problem, we irst compare liquidity supply with debt and with equity. Equity. I the issuer designs a pure equity contract, i.e., ϕ =Id, the optimal schedules (τe,q c E)and(τ c E m,qe m ) are as stated in the net proposition. Proposition 2 I the issuer designs an equity contract, then: 13

16 (i) In the competitive case, c E =min{, c } where c is the largest such that: = φ g(φ) dφ. G() (ii) In the monopolistic case, m E =min{, m },where m is the largest such that: 1 G() g() 0. When E c or E m is equal to, all issuer types achieve gains rom trade. Otherwise, issuerswithhighcash-lows are ecluded rom the market. To determine i issuers with type should be ecluded rom the market, the monopolistic liquidity supplier compares the gains rom trade (1 ) g() that can be achieved with these agents, with the rent G() they must be let. This rent increases with the cumulative distribution o types up to, since, as incentive compatible rents are decreasing with types, rents let to type must be let to all types below. Debt. I the issuer designs a debt contract with ace value d, i.e.,ϕ =min{id,d}, the optimal schedules (τd,q c D)and(τ c D m,qd) m are as stated in the net proposition. 6 Proposition 3 I the issuer designs a debt contract with ace value d, then: (i) In the competitive case, c D = d i d d c and c D = c E otherwise, where d c is the largest d such that: d g() d +(1 G(d))d = d. (ii) In the monopolistic case, D m = d i d d m and D m = E m the largest d such that: otherwise, where d m is d g() d +(1 G(d))d d = m E ( m E ) g() d. 6 In DeMarzo and Duie (1999), the interpretation o min{id,d} as a standard debt contract requires the assumption that the unsold raction o the security is not held on the balance sheet o the issuer at the time o deault. This does not arise in our model, since, when the issuer trades, the security is entirely transerred to the liquidity supplier. 14

17 Theintuitionistheollowing. In the competitive case, d c is the largest ace value such that the liquidity supplier s participation constraint and the issuer e-post rationality condition are consistent. I the ace value o the debt is too high, d>d c,market equilibrium then requires that the highest types, with cash-low between E c and d, be ecluded rom the market. This eectively converts the debt contract into an equity contract, since or all the issuers who participate to the market, ϕ() =. Ontheother hand, i d d c then the participation constraint o the liquidity supplier is consistent with the e-post rationality condition o all issuer types. In that case, all issuers sell their security, and thus reap gains rom trade. Now turn to the monopolistic case. When the issuer designs a debt contract with ace value d, the liquidity supplier has the option to shut-down the upper tail o the payo distribution by setting the price at which the issuer can sell his security to D m < d. Id is larger than dm, the highest ace value o debt such that the liquidity supplier obtains the same epected proit than under an equity contract, it is indeed optimal or the monopolistic liquidity supplier to do so. The shut-down threshold is then optimally set at E m. Just as the issuer in the competitive case, the liquidity supplier is eectively converting a debt contract into an equity one. On the other hand, i d d m, then all issuer types ind it preerable to sell their security. Debt Versus Equity. Since by assumption the support o the cash-low distribution is bounded, equity is just a particular case o debt with a ace value equal to. Hencethe optimal debt contract always weakly dominates a pure equity contract rom the issuer s viewpoint. It is nevertheless interesting to determine eactly in which circumstances debt strictly dominates equity. The same argument applies or both the competitive and monopolistic cases, i {c, m}. Wedeined d i as the maimum ace value such that a debt contract with ace value d [E,d i i ] can always be entirely traded between the two agents. Note that a debt contract with d = E i is equivalent, rom the issuer s point o view, to a pure equity contract. With the ormer, all issuers entirely sell the security, but the rent o issuers with cash-lows above d is 0. With the latter, only issuers with cash-lows up to E i sell the security. Moreover, di >E i whenever E i <. Since the e-ante utility o the issuer is clearly increasing in d, wehavetheollowingresult. Proposition 4 or any i {c, m}, i i E <, then the issuer is always strictly better o designing an optimal debt contract with ace value d i than an equity contract. Technically, designing a debt contract is equivalent or the issuer to creating an atom at d in the distribution o types. Since the liquidity supplier is making gains on the high types, he will not restrict the volume o trades provided that d is not high enough to jeopardize the issuer s incentives. The latter is thus able to get a better price d or her shares and thereby to increase her level o utility. 15

18 4.2. Debt as the Optimal Security To begin with, we establish some useul properties o an optimal design that hold independently o the considered environment. The optimality o debt then requires a separate argument in the competitive and the monopolistic case. Preliminaries. irst, note that risk-ree cash-lows are not subject to adverse selection problems. Hence, in the line o Myers and Majlu (1984), it is always optimal to sell these, in order to maimize trade and thus the gains rom trade. Consequently, it is optimal to design the security to yield at least the worst possible realization o the cash-low. This yields the ollowing lemma. Lemma 3 I is an optimal security, then ϕ() =. Our net proposition is key to our results. The argument generalizes the result obtained when comparing debt and equity. Let i {c, m}, and consider a security with payo ϕ such that issuers above a certain threshold i do not trade. What this proposition asserts is that the issuer could strictly gain by oering instead an alternative security, with payo capped at a level slightly above the shut-down level i. That alternative security ε would have payo ϕ ε =min{ϕ, i + ε}, and would be such that all issuers would trade. Rationally anticipating the participation o all issuer types, including the better ones, the liquidity supplier would be ready to pay a slightly better price, ( i + ε), than or,aslongasε is not high enough to make the issuer s incentives prohibitively epensive. At that price, issuers with high cash-lows would be willing to sell the security, given that its payo is capped just above i. The increase in price implies that the security ε strictly dominates the original security rom the issuer s point o view. Given our assumptions on,thisimpliesthatitisnotoptimalorthe issuer to design a security involving shut-down or good types. Thus, we can state the ollowing proposition. Proposition 5 The optimality o security requires that all issuer types entirely sell their holdings to the liquidity supplier. This result, which obtains in the contet o the optimal trading mechanism characterized in the previous section, underscores its dierence with the signaling models o Leland and Pyle (1977) and DeMarzo and Duie (1999). In these models, an inormed agent can credibly signal the quality o a project only by retaining part o the cashlows generated by this project. or an arbitrarily chosen security, the analogue o this phenomenon in our screening model is the possibility o market break-down. rom the issuer s point o view, this way o signaling the quality o her assets is however very costly. Hence she is better o designing her security to avoid market break-down altogether. As a consequence, the market or an optimal security will be very liquid. or instance, in the competitive case, the price at which such a security will be traded will just be equal to the unconditional epectation R dgϕ () othevalueothis security, thereby eliminating the bid-ask spread. 16

19 The Competitive Case. The program o the issuer is to maimize the total gains rom trade, subject to her own incentive compatibility and e-post participation constraints at the trading stage, and to the zero-proit constraint o the liquidity suppliers. The analysis o Section 3 implies that these constraints simpliy to a bang-bang trading structure, whereby issuers with types above a certain threshold do not trade, while those below entirely sell their security at a price equal to a lower tail epectation. Proposition 5 simpliies the situation urther by mandating to concentrate only on securities such that there is no shut-down. Thus we can restate the issuer s problem o choosing an optimal security as an ininite-dimensional linear programming problem: sup (1 ) ϕ() g() d ϕ Φ subject to the no shut-down condition: ϕ() g() d ϕ(). This inequality can alternatively be seen as an e-post participation constraint or the issuer, requiring that the price R ϕ() g() d o the security be greater than its present value or all issuer types, even or the issuer with the greatest possible cashlow, i.e., ϕ(). Note that in ormulating the issuer s problem, we have already taken into account the liquidity supplier s break-even constraint, which must be saturated at the optimum. The issuer s security design problem can then be analyzed as ollows. Let us orm the Lagrangian: µ L(ϕ, λ) =(1 ) ϕ() g() d + λ ϕ() g() d ϕ(), where λ is the Lagrange multiplier o the issuer s e-post participation constraint. By (LL) (M) (MR), any ϕ Φ is absolutely continuous, the derivative ϕ is a.e. welldeined with 0 ϕ 1, and ϕ() = R ϕ(ξ) dξ or all. Hence, integrating by parts, we get: L(ϕ, λ) = (1 + λ ) ϕ() G() d +(1 )(1 + λ) ϕ(). The maimization o L(ϕ, λ) with respect to ϕ can thus be treated as a standard optimal control problem. We then have the ollowing result. Proposition 6 Suppose that (8) holds. Then the debt contract with ace value d c is an optimal security rom the issuer s point o view. 17

20 The intuition is that a debt contract trades-o in an optimal way two conlicting objectives. On the one hand, it is eicienttotranserasmuchcash-lows rom the second period to the irst. On the other hand, the lemons problem limits the etent to which this can be done. By imposing a cap on the security payo, a debt contract minimizes this adverse selection cost, in support o Myers and Majlu s (1984) peckingorder hypothesis. The Monopolistic Case. In the monopolistic case, the issuer designs the security to maimize her epected gain rom trade, anticipating the optimal response o the monopolistic liquidity supplier, and her own reaction, relected in her incentive compatibility and e-post participation constraints. The issuer anticipates that the liquidity supplier will design his schedule to maimize his epected proit. She designs the optimal security to mitigate the adverse consequences o this rent etraction strategy on the gains rom trade. rom the previous section, we know that the transer schedule optimally designed by the monopolistic liquidity supplier is a simple take-it-or-leaveit oer to buy all the security at a given price. In designing this oer the liquidity supplier trades-o the beneit rom a large market, rom which no issuer would be ecluded, with the beneits o a smaller market, ecluding issuers with high cash-lows, but etracting more rents rom the others. Proposition 5 implies that with the optimal security there is no shut-down. Thus we can re-state the issuer s problem o choosing an optimal security as an ininite-dimensional linear programming problem: sup ϕ Φ subject to the no shut-down condition: (ϕ() ϕ()) g() d (ϕ() ϕ()) g() d, (ϕ() ϕ( )) g() d;. This can be interpreted by comparing the security design problem to a principal-agent problem with moral hazard. The principal is the issuer, who designs the security, while the agent is the liquidity supplier, and the moral-hazard variable is the decision by the agent to shut-down the market or not. The issuer s security design problem can then be analyzed as ollows. Let us orm the Lagrangian: L(ϕ, Λ) = (ϕ() ϕ()) g() d µ + (ϕ() ϕ()) g() d (ϕ() ϕ( )) g() d dλ( ), where Λ is the Lagrange multiplier associated to the no shut-down condition. It is a distribution unction on, i.e., a non-decreasing, right-continuous unction such that 18

21 Λ() = 0. The ollowing lemma provides a suicient condition or ϕ Φ to be an optimal security (see, e.g., Luenberger (1969, 8.4, Theorem 1)). Lemma 4 Let ϕ Φ, andλ be a distribution unction on such that: µ (ϕ() ϕ()) g() d (ϕ() ϕ( )) g() d dλ( ) =0 and: L(ϕ, Λ) L( ϕ, Λ); ϕ Φ. Then ϕ is an optimal security in Φ. To prove the optimality o debt, we proceed as ollows. Suppose that (8) holds. Then, by Proposition 3, the optimal debt contract rom the issuer s viewpoint has ace value d m. Given this contract, the only point at which the liquidity supplier s shut-down constraint is binding is at the level E m. This suggests taking as a Lagrange multiplier Λ apoint-massate m, i.e., a mapping o the orm Λ λ() =λχ { m E } or some λ > 0. or this choice o Λ, the Lagrangian can be re-written as: L(ϕ, Λ λ )=(1 λ) (ϕ() ϕ()) g() d à + λ m E (ϕ( m E ) ϕ()) g() d +(1 ) m E! ϕ() g() d. Proceeding as in the competitive case, this epression can be urther simpliied to: L(ϕ, Λ λ )=(1 λ) ϕ() G() d à + λ m E ϕ() G() d +(1 ) m E! ϕ() g() d. It is clear rom this epression that the second term on the right-hand side is maimized by setting ϕ =Id. The same is true or the irst term i λ 1. Overall, a pure equity contract maimizes the Lagrangian i λ 1. But then, the no shut-down condition would not be binding, and Lemma 4 would not apply. This means that, in order to derive the optimality o debt, we must select λ > 1. Intuitively, the shadow cost o the no shut-down condition must be high enough or debt to be an optimal security. Lemma 5 There eists λ > 1 such that ϕ =min{id,d m } maimizes L(ϕ, Λ λ ) with respect to ϕ Φ. 19

22 Since or the debt contract ϕ =min{id,d m }, m E (ϕ() ϕ()) g() d (ϕ() ϕ(e m )) g() d =0, the ollowing result is an immediate consequence o Lemmas 4 and 5. Proposition 7 Suppose that (8) holds. Then the debt contract with ace value d m is an optimal security rom the issuer s point o view. Just as in the competitive case, a debt contract is optimal rom the issuer s point o view. The act that the ace value o debt is smaller than in the competitive case relects the liquidity supplier s market power. It is interesting to note that, in both cases, the optimal security is risky debt. That this optimal security is inormationally sensitive, stands in stark contrast with the results o DeMarzo and Duie (1999). In their model, i the issuer observes a perectly inormative signal about the realization o the cash-lows, there eists an optimal security whose payo does not depend on her private inormation and is identically equal to the lowest possible realization o the cash-lows, ϕ =. Thisdierence between this result o and ours relect both the dierence between our screening trading mechanism and their signaling game, and, in the monopolistic case, the act that the liquidity supplier would be able to etract all the rent i the security was not inormationally sensitive. 5. Robustness We now investigate the robustness o our results to some o the assumptions underlying our basic model Menus o Securities So ar, we have assumed that the choice o a security is made e-ante. We now rela this assumption, by allowing the issuer to design e-ante a menu o securities, rom which she will select which to trade at the interim stage. 7 A menu o securities is then amapping(, ˆ) 7 ψ(, ˆ) such that ψ(, ˆ) [0,] or all (, ˆ) 2. or eample, this includes the case where, i ˆ is in a certain set, then the security is a debt contract, while i ˆ is in the complementary set, then the security is an equity contract. Note that we do not impose any monotonicity condition on the menu o securities. By the revelation principle, there is no loss o generality in ocusing on truthul direct mechanisms (τ,q) : R [0, 1] that stipulate a transer and a trading 7 Similar results would obtain i the menu o securities was designed instead by the inanciers supplying liquidity to the issuer. 20