Gilles Chemla

Christopher A Hennessy
email: chennessy@london.edu

Andres Almazan

Sudipto Bhattacharya

Patrick Bolton

James Dow

Sergei Guriev

Michel Habib

Yolande Hiriart

David Martimort

Sebastien Pouget

Jean Charles Rochet

Jean Tirole

Sheridan Titman

Skin in the Game and Moral Hazard

Keywords:

What determines equilibrium securitization levels, and should they be regulated? To address these questions we develop a model where originators can exert unobservable e¤ort to increase asset quality, subsequently having private information regarding quality when selling ABS to rational investors. In equilibrium, all originators have low/zero retentions if they are …nancially constrained and/or prices are su¢ ciently informative. Asymmetric information lowers e¤ort incentives in all equilibria. E¤ort is promoted by junior retentions, investor sophistication, and informative prices. Optimal regulation promotes e¤ort while accounting for investor-level externalities. It entails either a menu of junior retentions or a single junior retention with size decreasing in price informativeness. Mandated market opacity is only optimal amongst regulations failing to induce originator e¤ort.

Over the past two decades securitization markets have been an important source of funding for …nancial and non-…nancial corporations. As shown in Table 1, mortgage-related and non-mortgagerelated asset-backed securities (ABS below) accounted for over 30% of U.S. bond market issuance each year from 1996 to 2011, with the percentage exceeding 50% from 2002 to 2005. As shown in Table 2, non-mortgage-related ABS cover a diverse range of assets outside the housing sector: equipment; auto loans and leases; credit card debt; student loans; and trade receivables. Securitization markets collapsed in 2008, with issuance falling by 44% from 2007 levels. The majority of the decline is accounted for by the virtual disappearance of non-agency mortgage-backed securities.

However, it is apparent that weakness extends beyond the housing sector. For example, issuance of credit card and student loan ABS has also fallen signi…cantly in recent years. [START_REF] Gorton | Slapped in the Face by the Invisible Hand: The Panic of 2007[END_REF] argues that concern over asymmetric information regarding true asset values accounts for the collapse of ABS markets, and disputes the existence of moral hazard, e.g. the alleged failure of originators to carefully screen borrowers. In contrast, [START_REF] Mishkin | On Leveraged Losses: Lessons from the Mortgage Meltdown[END_REF] and [START_REF] Stiglitz | Freefall: America, Free Markets, and the Sinking of the World Economy[END_REF] argue that low originator retentions created moral hazard. In their behavioral narrative, unwary investors had simply overlooked moral hazard pre-crisis. Indeed, implicit in much discussion surrounding the crisis is the notion that ABS featuring low originator retentions are indicative of market irrationality. Moreover, implicit in the recently-passed Dodd-Frank Act is the view that government-mandated retentions will increase social welfare.

Understanding equilibrium in ABS markets and the formulation of optimal regulation have been hindered by the absence of a comprehensive theoretical framework allowing one to answer some fundamental questions. First, what levels of securitization should one expect to observe in unregulated ABS markets? Clearly, addressing this question is necessary before reaching any conclusion regarding whether observed structures are rational. Second, are there market failures and, if so, can a regulator improve upon unregulated market outcomes? Third, what are the policy options and conditions under which each dominates?

This paper develops a tractable, yet comprehensive, framework to address the positive and normative questions posed above. Although the primary focus is ABS, the economic setting is more general: Ex ante, an agent ("the originator" below) considers exerting costly unobservable e¤ort to increase the probability of producing a high quality asset. This e¤ort decision is made anticipating subsequent issuance of claims backed by the asset to fund a scalable investment with positive NPV. 1 The issuer privately observes the true asset quality (high or low) but investors do not.

There are three categories of investors: a speculator; competitive uninformed marketmakers; and rational uninformed investors with hedging motives. The originator can permit (block) speculator information production by choosing transparency (opacity).

The model delivers a rich set of predictions regarding how issuers will behave in unregulated ABS markets. We …rst investigate what securities will be marketed and retained by privately informed issuers. One possible equilibrium is a separating equilibrium in which high types separate from low types by retaining the minimal size junior tranche needed to deter mimicry by low types who fully securitize. In addition to this separating equilibrium, there may exist equilibria in which all originators pool and adopt identical structures. Pooling equilibria exist if both originator types are weakly better o¤ than at the separating equilibrium.

We show that if any pooling equilibrium can be sustained, a pooling equilibrium with full securitization can also be sustained. 2 In this sense, the originate-to-distribute business model (OTD below), which features zero issuer retentions, should not be viewed as an anomaly. However, we also show that pooling at full securitization can only be sustained as an equilibrium if prices are su¢ ciently informative and the originator's project NPV is su¢ ciently high. Intuitively, a high type will be willing to pool provided informed speculation drives prices su¢ ciently close to fundamentals. We also show some observed practices are hard to reconcile with notions of rational equilibrium. For example, a deliberate attempt by issuers to preclude speculative information production via opacity is shown to be inconsistent with investor sophistication. A high type should defect from opacity and sophisticated investors should know this.

We next evaluate ex ante e¤ort incentives of originators who anticipate such marketing of securities under asymmetric information. Since e¤ort increases the probability of developing a high quality asset, incentives are increasing in the size of the anticipated wedge between payo¤s accruing to owners of high and low quality assets. Critically, asymmetric information at the time of securitization reduces the size of this wedge, lowering e¤ort incentives. In this way, the model shows that the asymmetric information view of [START_REF] Gorton | Slapped in the Face by the Invisible Hand: The Panic of 2007[END_REF] and the moral hazard view of [START_REF] Mishkin | On Leveraged Losses: Lessons from the Mortgage Meltdown[END_REF] and [START_REF] Stiglitz | Freefall: America, Free Markets, and the Sinking of the World Economy[END_REF] are not competitors. Rather, if there is indeed asymmetric information between originator-distributors and investors at the time of security issuance, then in all possible equilibria, e¤ort incentives are lower than under observable types. In the separating equilibrium incentives are diminished since high types bear signaling costs. In pooling equilibria, incentives are diminished by price noise. In the extreme case of opacity and zero retentions, there is zero e¤ort incentive.

The analysis of unregulated ABS markets reveals two welfare arguments for government-mandated retentions. First, privately optimal retentions can be socially suboptimal since originators do not internalize e¤ects on investor welfare. When the high type credibly signals via junior retentions he bene…ts directly from his own marketed securities being priced at fundamentals on the issuance date. But he does not internalize the bene…t accruing to investors who can now e¢ ciently share risks being symmetrically informed. The second argument favoring regulation is that the payo¤ di¤erential between high and low types at the (interim) securitization stage may be insu¢ cient to induce originator e¤ort. In order to encourage e¤ort, low types should get low payo¤s. But if retentions are not mandated, a low type can always achieve his …rst-best payo¤ by admitting he is a low type and proceeding to securitize the entire asset. Government-mandated retentions o¤er a commitment device against markets implementing such incentive-reducing equilibria. As re ‡ected in the model, equilibria with low e¤ort incentives are especially problematic inasmuch as poor performance of a given originator's asset can trigger reductions in the value of other assets. For example, Campbell, Giglio, and Pathak (2011) and [START_REF] Gerardi | Foreclosure Externalities: Some New Evidence[END_REF] document negative externalities associated with foreclosed and/or distressed real estate.

A socially optimal mandatory retention scheme promotes e¤ort by increasing the spread between payo¤s to high and low types at the securitization stage, while accounting for costs imposed on investors as well as originators. There are two regulatory options. In a separating regulation issuers must choose from a menu of retentions. The menu is designed so that the chosen retention reveals the issuer's private information. In a pooling regulation all issuers must retain the same claim.

In the optimal separating regulation, originators choose from a menu of strictly positive junior tranche retentions of di¤ering size. Although menus featuring other retained claims (e.g. fractions of total cash ‡ow) can also induce truthful revelation of private information, junior tranche retentions minimize the cost of underinvestment by originators. In contrast to the separating equilibrium of unregulated markets, the separating regulation forces even the low type to retain a junior claim. This regulation achieves e¢ cient risk-sharing across investors since the originator's chosen retention reveals his private information, thus insulating investors from adverse selection.

In the optimal pooling regulation, issuers are forced to hold identical junior tranches. Intuitively, the gap between the interim payo¤s of high and low types is maximized if originators hold a junior claim. The size of the mandated retention is lower when price informativeness is high. That is, junior retentions and market discipline are substitutes in terms of e¤ort incentives. Thus, the exact details of the optimal pooling regulation requires taking a view on informational e¢ ciency. The disadvantage of the pooling regulation is that it entails costly speculator e¤ort and distortions in risk-sharing across investors. However, the pooling scheme imposes lower underinvestment costs on originators if prices are su¢ ciently informative.

Our model is most similar to those of [START_REF] Leland | Information Asymmetries, Financial Structure, and Financial Intermediation[END_REF] and [START_REF] Myers | Corporate Financing and Investment When Firms Have Information Shareholders Do Not Have[END_REF] in that we consider equilibrium security issuance by a privately informed originator. We depart from canonical signaling models in three ways. First, we consider that there is an e¤ort decision to be made before the securitization stage, with costly e¤ort increasing the probability of obtaining a high value asset. Second, at the securitization stage, securities are traded by an endogenously informed speculator. Third, rational uninformed investors with hedging needs also trade securities. The …rst model element allows us to address how the anticipation of interim-stage asymmetric information a¤ects ex ante e¤ort incentives. The second model element permits assessment of the role of price discipline. The third model element admits a proper analysis of social welfare and the e¢ ciency of risk-sharing in light of potential adverse selection facing uninformed investors.

Dang, Gorton and Holmström (2011) also analyze information production and social welfare, but ignore originator moral hazard. Similar to their analysis, we show opacity combined with full securitization maximizes interim-stage social welfare. 3 However, we show opacity is only socially optimal amongst regulatory schemes failing to induce originator e¤ort. Intuitively, opaque markets fail to provide price discipline. Thus, the choice between opacity and transparency must weigh interim-e¢ cient risk-sharing against ex ante moral hazard.

Rajan, Seru and Vig (2010) analyze a setting most similar to ours in that they too consider a bank that can exert unobservable e¤ort prior to entering into securitization contracts. However, they assume each loan is fully securitized with an exogenous probability. In contrast, we …rst characterize the full set of equilibrium ABS structures and then assess the e¤ect of each on e¤ort incentives. Their model does not allow for informed trading and they do not analyze optimal regulation. [START_REF] Gorton | Banks and Loan Sales: Marketing Nonmarketable Assets[END_REF], [START_REF] Parlour | Loan Sales and Relationship Banking[END_REF], and Plantin (2011) consider a di¤erent agency setting in which contracting occurs before a bank chooses e¤ort. The respective agency problems are di¤erent. The pre-contracting e¤ort we consider is akin to screening of loan applicants while the post-contractual e¤ort they consider is akin to monitoring of loan recipients. These papers do not analyze speculative information production or optimal regulation. Plantin (2011) shares one of our predictions: securitization rates should be higher if banks place high value on immediate funding. Hartman-Glaser, Piskorski and Tchistyi (2012) also analyze optimal contracting before unobservable e¤ort. Their optimal dynamic contract features a single positive transfer to the agent made only after a su¢ cient time with no defaults. Their privately optimal contract is socially optimal and there is no case for regulation.

The role of price informativeness in alleviating moral hazard has been analyzed in other contexts. [START_REF] Holmström | Market Liquidity and Performance Monitoring[END_REF], [START_REF] Maug | Large Shareholders as Monitors: Is There a Trade-o¤ Between Liquidity and Control?[END_REF], [START_REF] Kahn | Ownership Structure, Speculation and Shareholder Intervention[END_REF], [START_REF] Aghion | Exit Options in Corporate Finance: Liquidity versus Incentives[END_REF] and Faure-Grimaud and Gromb (2004) show price informativeness promotes insider e¤ort.

Each of these papers assumes pure noise trading, precluding social welfare analysis. These papers do not analyze socially optimal mandatory retention regulations.

The remainder of the paper is as follows. Section I describes the model. Section II analyzes the …nal continuation game in which market-makers set prices. Section III analyzes the subgame in which the privately informed originator chooses retentions. Section IV analyzes originator e¤ort incentives. Section V contains an analysis of the sources of welfare losses in unregulated market equilibria, followed by an analysis of socially optimal mandatory retention regulations.

I. The Model

This section describes the production technology, endowments, investor preferences and the timing of events. Figure 1 provides an overview of the time-line.

A. Production Technology, Endowments and Preferences

There is a single storable consumption good and four periods: 1, 2, 3 and 4. Agents consume in periods 3 and 4 and consumption must be non-negative. The originator (denoted O) has one unit of endowment in period 1 which he can use to fund a veri…able investment in an asset generating a veri…able cash ‡ow in period 4. O has no other endowment and is risk neutral with von Neumann-Morgenstern (vNM) utility function C 3 + C 4 : At the time of the initial investment, O has the option to exert unobservable e¤ort which increases expected cash ‡ow. In particular, by exerting e¤ort O increases the probability of the asset being of high quality from to ; where 0 < < < 1. A high quality asset generates cash ‡ow H with probability q and L with probability 1 q: A low quality asset generates H with probability q and L with probability 1 q: It is assumed: 0 < q < q < 1; L 2 (0; H); and qH + (1 q)L > 1: This last assumption implies O always …nds it optimal to invest his initial endowment in the asset.

The originator e¤ort cost is denoted c, and this cost is non-pecuniary. We assume the e¤ort cost is less than the expected increase in cash ‡ow that it produces.

A1 : 0 < c < ( )(q q)(H L):
Assumption A1 implies the originator would exert e¤ort if he planned to retain all claims to future cash ‡ow.

At the start of the interim period (period 2), Nature draws q: Then O privately observes q, with q being labeled the asset type below. Outside investors do not have access to the same information as O at this time and cannot observe q. For example, an originator may have superior granular information regarding local real estate market conditions, permitting a superior forecast of the terminal asset payo¤.

At the start of period 3, O gets exclusive access to a scalable linear investment technology providing a private bene…t > 1: The private bene…t is not observable or veri…able and cannot be transferred to other agents. Since the private bene…t cannot be sold to other agents, any funding for the new investment must come from marketing some portion of the cash ‡ows coming from the original underlying asset. 4There are three categories of investors. First, there is a continuum of deep-pocketed risk neutral market-makers (MM below

Pr[q = qjs = s] = Pr[q = qjs = s] = :
The …nal set of investors is a measure-one continuum of agents who have no information regarding the asset type, labeled uninformed investors (UI). The UI are identical ex ante aside from idiosyncratic di¤erences in risk-aversion parameters ( ) discussed below. UI are risk-neutral over period 3 consumption and risk-averse over period 4 consumption.

An extant literature treats uninformed trading as exogenous. Although such noise trading frameworks are a bit simpler, they su¤er from two weaknesses in terms of policy analysis. First, noise trading models preclude analysis of total social welfare. Second, by treating uninformed investors as price-insensitive, such models fail to capture deadweight losses arising from portfolio distortions in response to perceived security mispricing. In light of these weaknesses, we depart from the standard noise trading setup. Instead, we model the UI as choosing portfolios optimally given well-de…ned utility functions described below.

Prior to the trading of securities in period 3 each UI privately learns whether he is invulnerable or vulnerable to preference and endowment shocks. The utility function of an arbitrary UI, privately observable to them, is:5 The proportion of vulnerable UI is an independent random variable v 2 fv; vg; with each possible v equiprobable and v < v. Whether an agent is vulnerable or not is not observable or veri…able to others, nor is their realized endowment. This prevents writing insurance contracts directly on individual endowments. Further, the realized v is not observable or veri…able. Thus, securities markets are necessarily incomplete.

B. Securitization Stage

The Securitization Stage takes place in period 2. This stage approximates a shelf registration of securities whereby a prospective issuer of a set of securities registers them in advance and is then free to pull securities "o¤ the shelf" over some time interval without further …ling requirements.

Shelf registrations are commonly used for ABS. 7 Applying a result of [START_REF] Maskin | The Principal-Agent Relationship with an Informed Principal, II: Common Values[END_REF] for signaling games in general, [START_REF] Tirole | The Theory of Corporate Finance[END_REF] shows allowing an issuer to …rst register a set of claims and then choose "from the menu" can improve their payo¤ by restricting the set of equilibria.

The Securitization Stage begins with O registering two securitization structures, ( ; ) with the number of structures equal to the number of possible types. 8 Each structure speci…es the amounts (M L ; M H ) that will paid to outside investors in the respective cash ‡ow states should the originator choose to issue it. Investors are assumed to have limited liability, so payments to them must be non-negative.

Next O, selects one of the registered structures to bring to the market, committing to retain the residual cash ‡ow rights. The payo¤ vector on the retained security is denoted (R L ; R H ): Since the originator has no outside endowment other than the asset, both R L and R H must be non-negative.

The cash ‡ow rights retained by the originator are assumed to be a legally veri…able contractual commitment, consistent with the mandatory disclosure rules of Regulation AB of the Securities Exchange Act. It is worth noting that the owner of a high quality asset stands to bene…t from such a retention commitment as it allows him to credibly signal positive information.

Total state-contingent payo¤s on retained and marketed securities are equal to the cash ‡ow generated by the underlying asset:

R L + M L = L R H + M H = H:
Notice, the preceding payo¤ identities assume the originator invests all funds raised from investors in the new investment paying him the private bene…t : That is, it is assumed the originator does not place any of the funds raised from investors into risk-free storage. Doing so would simply allow the originator to raise promised payments to investors by one dollar for each additional dollar raised and stored. This would have no e¤ect on any agent's utility since the originator's level of new investment would be unchanged, while outside investors would be no better or worse o¤ relative to storing the funds on their own accounts.

Voluntary disclosure of additional information is also possible at the Securitization Stage. In particular, the originator has the option to disclose in the prospectus additional information about the underlying asset (transparency) or not (opacity). This additional information can be used by the speculator to acquire a signal of the asset quality.

C. Trading Stage

The Trading Stage of the model takes place in period 3. For simplicity, it is assumed all securities trading takes place in this period, just after investors observe their private information. Thus, informational asymmetries across investors are treated as an inherent feature of securities markets. 9There are two securities markets: a market for risk-free bonds and a market for an Arrow security paying 1 if the realized cash ‡ow is L. These two securities span the only two veri…able states (L and H), and the introduction of markets for redundant securities would have no e¤ect on the equilibrium set.

At the start of the Trading Stage, S chooses whether to pay e to acquire a signal of the asset type. Recall, signal acquisition is only possible if the originator opted for transparency at the Securitization Stage. Next, each UI privately observes whether he is vulnerable to shocks. Next, each agent other than the MM submits his market order. 10 Each MM observes the aggregate buy order and aggregate sell order. The MM then compete à la Bertrand to clear markets. The originator provides the promised supply of payo¤s (M L ; M H ) according to the securitization structure but then engages in no further trading. Clandestine sales by the originator at this stage would contradict the retentions disclosed in the Securitization Stage. 11 The payo¤ pair (M L ; M H ) is sold on the two markets as M H units of risk-free bonds and (M H M L ) units of the L-state Arrow security.

Equivalently, one can think of the MM as pricing the claim to (M L ; M H ) using the equilibrium Arrow security price.

As is standard in the literature on General Equilibrium with Incomplete markets, short-selling is possible in both securities markets, but courts will impose an arbitrarily high utility penalty on any agent who fails to deliver promised payments to securities market counterparties, thus ruling out reneging on short sales. 12 The model is solved by backward induction. As in [START_REF] Maskin | The Principal-Agent Relationship with an Informed Principal, II: Common Values[END_REF], the equilibrium concept is pure strategy perfect Bayesian equilibrium (PBE).

D. Benchmark: Observable Types

Before characterizing equilibrium under asymmetric information, it is useful to analyze outcomes if the asset type was observable. This benchmark setting is particularly useful in framing our argument that interim-state asymmetric information regarding asset quality can be understood as a root cause of ex ante moral hazard.

If q was observable, O would market all cash ‡ow and receive securitization proceeds equal to the true expected cash ‡ow qH + (1 q)L: Full securitization would occur since > 1 implies there are gains from trade, and these would be fully exploited under symmetric information. Therefore, if types were observable, the maximum e¤ort cost the originator would be willing to incur is times the expected increase in cash ‡ow arising from e¤ort:

b c obq = ( )(q q)(H L): (1)
Assumption A1 implies that the originator would pay the cost c if types were observable. Since the originator would exert e¤ort with observable types, a failure of the originator to exert e¤ort in the full model can be understood as arising from asymmetric information at the Securitization Stage regarding asset quality.

Consider …nally equilibrium risk-bearing with observable types. If q was observable, the speculator would not pay e > 0. The MM would set the price of the L-state Arrow security to 1 q. At that actuarially fair price, all vulnerable UI would fully insure against negative endowment shocks, buying units of the L-state Arrow security. 13

II. The Trading Stage

This section determines UI security demand, speculator e¤ort and price setting by the MM.

Given that we con…ne attention to pure strategies, there are two possible information con…gurations at the start of the Trading Stage: all agents know the asset's type or all agents other than the originator are uninformed regarding the type. We consider these two cases in turn.

A. Asset Type Common Knowledge

Competition between MM ensures risk-free bonds trade at price 1 per unit of face value. Let P denote the price the L-state Arrow security. If the type (q) is known to all agents at the start of the Trading Stage, the MM set P = 1 q: With the type known, the speculator has no incentive to incur e¤ort costs, and any trading by the speculator is of no consequence for any agent's expected utility, including her own.

Since securities markets span the veri…able cash ‡ow states, an uninformed investor's portfolio problem can be framed as a choice of state-contingent period 4 portfolio payo¤s, here denoted

(x L ; x H ):
With common knowledge of type, an optimal UI portfolio solves:

max (x L ;x H ) y ui 3 x H q x L (1 q) + q minfC H 4 ; 0g + (1 q) minfC L 4 ; 0g (2)
subject to

C L 4 = (1 ) + x L ; C H 4 = + x H ; C L 4 0; C H 4 0:
As shown in the appendix, one …nds the following optimal UI portfolios under common knowledge of q. Vulnerable UI purchase units of the L-state Arrow security while invulnerable UI sell units of the risk-free bond. The sharing of risks under common knowledge of type is socially e¢ cient ex post, with all vulnerable UI buying from the MM fairly priced insurance against costly consumption shortfalls.

B. Asset Type not Common Knowledge

Consider the remaining case when asset type is not common knowledge at the start of the Trading max

(x L ;x H ) y ui 3 x H [1 E(P j )] x L E(P j ) (3)
+[ q + (1 )q] minfC H 4 ; 0g + [1 ( q + (1 )q)] minfC L 4 ; 0g subject to C L 4 = (1 ) + x L ; C H 4 = + x H ; C L 4 0; C H 4 0:
As shown in the appendix, the solution of the preceding program implies the following optimal portfolios. Invulnerable UI sell units of the risk-free bond. Vulnerable UI buy units of the L-state Arrow security if b , where:

b

E[P j = 1] (1 q) + (1 )(1 q) : (4)
The remaining vulnerable UI do not trade. Under optimal portfolios, vulnerable UI with b achieve the critical consumption level C 4 = regardless of the realized asset payo¤. In contrast, vulnerable UI with < b consume in state H but only 0 in state L. Intuitively, all vulnerable UI have an incentive to insure against consumption shortfalls by purchasing units of L-state payo¤s.

However, if they expect the Arrow security price to exceed its expected payo¤ ( b > 1), they forego insurance provided their personal loss from consumption shortfalls is su¢ ciently low.

Consider then Trading Stage outcomes if the originator had chosen opacity in the Securitization

Stage. In this case, the MM know order ‡ow cannot possibly contain any information regarding the asset type. Consequently, regardless of the observed order ‡ow, the MM set the price of the Arrow security equal to 1 q (1 )q: It follows from equation ( 4) that in this case b = 1: Thus, under opacity risk-sharing is e¢ cient since all vulnerable UI purchase units of the L-state Arrow security.

Consider next Trading Stage outcomes under transparency, conjecturing the speculator will indeed …nd it optimal to acquire the noisy signal of the asset type provided e is su¢ ciently low.

Integrating over uninformed investors' optimal demands (x L ), the aggregate UI demand for the

X U I L 1 Z 1 x L ( )f ( )d = v [1 F ( b tran )] 8 v 2 fv; vg: (5)
Consider next the speculator's trading strategy in the market for the L-state Arrow security. The speculator cannot make trading gains by shorting, since the MM will justi…ably impute any shortselling to the speculator. So she will place buy orders for the L-state Arrow claim when she observes the negative signal s. In order for the speculator to make positive expected trading gains, she must choose a buy order size such that the MM cannot infer s with probability one. This can only be achieved by choosing an order size for the L-state Arrow claim such that MM cannot distinguish between: speculator buying (based upon signal s) cum low UI demand (v) versus speculator not buying (based upon signal s) and high UI demand (v). Using the aggregate demand expression from equation ( 5), we obtain the following condition pinning down the buy order size (X S L ) that masks the speculator across the states (s; v) and (s; v):

X S L + v [1 F ( b )] = v [1 F ( b )] ) X S L = (v v) [1 F ( b )]: (6)
For brevity, let:

[1 F ( b )]: (7)
P (X AG L ) = 1 q + (q q) Pr[ q = qjX AG L (8) Pr[q = qjX AG L = (2v v) ] = (1 ) + 2 Pr[q = qjX AG L = v ] = 1 Pr[q = qjX AG L = v ] = 1 + 1 + 2 :
Since each UI has measure zero, any order ‡ow con…guration o¤ the equilibrium path must arise from a deviation by the speculator. O¤ the equilibrium path, MM form worst-case beliefs from the speculator's perspective. An aggregate buy (sell) order o¤ the equilibrium path is imputed to her observing s (s). Given such beliefs, no deviation generates a positive gross trading gain for the speculator.

Having pinned down the speculator's optimal signal-contingent trading strategy, we consider next the conditions under which she will pay the e¤ort cost e. If the speculator acquires the signal, her equilibrium expected gross trading gain G as computed from Table 3 is:

G = 1 2 (1 )(q q)(2 1)(v v) [1 F ( b )]: (9)
It is readily veri…ed that if the speculator were to instead remain uninformed, her optimal strategy is to abstain from trading. 13 Returning to Table 3 we …nd that under transparency vulnerable UI form the following price expectation:

E[P j = 1] = (1 q) + (1 )(1 q) + (1 )(q q)(2 1) v v v + v ( 10
)
) b tran = 1 + (1 )(q q)(2 1) v v v+v (1 q) + (1 )(1 q) :
The remainder of the analysis assumes the …xed cost of speculator e¤ort satis…es the following assumption, which implies the speculator exerts e¤ort in the Trading Stage provided the originator chose transparency at the Securitization Stage.

A2 : e 1 2 (1 )(q q)(2 1)(v v) 2 4 1 F 0 @ 1 + (1 )(q q)(2 1) v v v+v (1 q) + (1 )(1 q) 1 A 3 5 :
And it then follows from equation ( 10) that risk-sharing will be distorted under transparency since a subset of the vulnerable UI fail to insure given that the expected L-state Arrow security price is above its expected payo¤.

The following proposition summarizes the continuation equilibrium at the Trading Stage.

Proposition 1 [Trading Stage] If the type is common knowledge: the L-state Arrow security price is P = 1 q and all vulnerable investors insure against shocks. Under opacity: P = (1 q) + ( 1)(1 q) and all vulnerable investors insure against shocks. Under transparency: the speculator acquires the costly signal; P is set according to equation [START_REF] Gorton | Slapped in the Face by the Invisible Hand: The Panic of 2007[END_REF]; and vulnerable investors only insure against shocks if b tran as de…ned in equation [START_REF] Hartman-Glaser | Optimal Securitization with Moral Hazard[END_REF].

III. The Securitization Stage

Continuing the backward induction, this section describes the set of continuation equilibria at the Securitization Stage. This subgame begins with Nature drawing the type q 2 fq; qg, which is then privately observed by the originator. The other players have a common prior for the probability of the type being q: This -contingent subgame may be of independent interest as it resembles a standard corporate …nance signaling game where the equilibrium set is predicated upon investor priors. For simplicity, we borrow terminology from Tirole (2006) when possible.

Recall, at the Securitization Stage the originator performs a shelf-registration of two securitization structures and then chooses one from the menu. A separating menu contains two di¤erent structures such that each type prefers a di¤erent structure. If such a menu is registered, the subsequent choice of structure reveals the type to all agents, so the type becomes common knowledge at the start of the Trading Stage. A pooling menu contains only one securitization structure so there is no possibility of the type being revealed by the choice of structure, so the type is not common knowledge at the start of the Trading Stage. 14 We …rst characterize the least-cost separating (LCS) allocations which maximize the utility of each originator type within the set of separating menus. We conjecture and then verify the high type will not mimic the low type. The LCS allocations allow the low type to fully securitize his asset since this raises his payo¤ and relaxes the non-mimicry (NM) constraint. The high type's LCS retention solves:

max (R L 0;R H 0) qR H + (1 q)R L + [q(H R H ) + (1 q)(L R L )] (11)
subject to the following NM constraint:

[qH + (1 q)L] qR H + (1 q)R L + [q(H R H ) + (1 q)(L R L )]:
Solving the above program yields the following lemma.

Lemma 1 The Least-Cost Separating Allocations entail zero retention by the low type while the high type signals by retaining a junior security with payo¤ s R L = 0 and R H = (q q)(H L)=( q q):

The respective continuation utilities for the low and high type are:

U lcs = [qH + (1 q)L] (12)
U lcs = [qH + (1 q)L] ( 1)
q (q q)(H L) ( q q) :

In an LCS allocation, the low type receives his perfect information payo¤. The high type receives his perfect information payo¤ minus foregone NPV due to signaling via retention of a claim paying zero if the realized cash ‡ow is L: Throughout the analysis we refer to such claims as junior in that their payo¤ is equal to that of a junior claim when there is a senior debt claim with face value between L and H: The next lemma is parallel to a general result from [START_REF] Maskin | The Principal-Agent Relationship with an Informed Principal, II: Common Values[END_REF],

showing that the LCS payo¤s constitute a lower bound. In light of the preceding lemma, the PBE in which the LCS allocations are proposed will be denoted as the Least-Cost Separating Equilibrium (LCSE). Note, the lemma implies there can be no other separating equilibrium. Thus, we turn next to determination of pooling equilibria. Before doing so, it is worth emphasizing that the lemma shows the equilibrium set consists of pooling structures that Pareto-improve upon the LCSE from the perspective of originators. These structures do not have to be Pareto-improving for all agents in order to be PBE. To the contrary, it is apparent that any pooling cum transparency makes the UI worse o¤ by exposing them to adverse selection in securities trading.

Lemma (2) provides a simple algorithm for assessing whether a pooling structure is in the set of PBE. One must simply compute expected utilities across the two originator types and compare them with the respective LCSE utilities. Originator utility in the event of pooling is equal to times expected securitization revenues, plus the expected payo¤ on the retained security, with both expectations computed conditional upon the privately known type. Under opacity, expected securitization revenues are equal across types, i.e. there is no market discipline. Thus, under opacity, the following two inequalities must be satis…ed by any pooling equilibrium:

U op = (qM H + (1 q)M L ) + (1 )(qM H + (1 q)M L ) +qR H +(1 q)R L U lcs ( 13
)
U op = (qM H + (1 q)M L ) + (1 )(qM H + (1 q)M L ) +qR H +(1 q)R L U lcs :
Under transparency, informed trading drives prices closer to fundamentals and securitization revenues vary across originator types. The following two inequalities must be satis…ed by any pooling equilibrium featuring transparency:

U tran z (qM H + (1 q)M L ) + (1 z)(qM H + (1 q)M L ) +qR H +(1 q)R L U lcs (14) U tran z (qM H + (1 q)M L ) + (1 z)(qM H + (1 q)M L ) +qR H +(1 q)R L U lcs where z( ) 1 2 2 1 + 2 + (1 ) 2 + 2 + z( ) 1 (1 z):
The endogenous variable z plays a critical role in the model, measuring the informational e¢ ciency of prices. For example, in the hypothetical case where z = 1; there is no mispricing. In fact, z is increasing in ; with z 2 ( ; (1 + )=2]: Intuitively, if the speculator has a more precise signal, the expected wedge between price and true value is lower.

Exploiting equations ( 13) and ( 14), the following proposition characterizes the set of continuation equilibria featuring pooling.

Proposition 2

The set of pooling perfect Bayesian equilibrium marketed cash ‡ows with opacity (transparency) is the convex set de…ned by equations 13 [START_REF] Leland | Information Asymmetries, Financial Structure, and Financial Intermediation[END_REF]. In any pooling equilibrium, marketed high state payo¤ s (M H ) are strictly greater than L: If there is a pooling equilibrium with marketed payo¤ s (M 0 L ; M 0 H ); then for all M 0 H 2 (M 0 H ; H] there is a pooling equilibrium with marketed payo¤ s

(M 0 L ; M 0 H ).
If there is a pooling equilibrium with partial securitization, there is a pooling equilibrium with full securitization. Under transparency, a necessary and su¢ cient condition for a pooling equilibrium with full securitization is

q q q q 1 2 2 1 + 2 + ( 1
) 2 + 2 + ( z( )) :
Under opacity, a necessary and su¢ cient condition for a pooling equilibrium with full securitization is q q q q :

The intuition for Proposition 2 is as follows. In order for a pooling equilibrium to be supported, both types must be weakly better o¤ than at the LCSE. And the low type is able to fully securitize his asset in the LCSE. In order to improve upon this, M H must be su¢ ciently high to ensure the marketed claim is risky, as stated in the second sentence of the proposition. Moreover, the second sentence in the proposition implies total marketed payo¤s must increase with cash ‡ow (M H > M L ) in any equilibrium. It is also worth noting that it is possible for originators to pool at structures in which the retained claim is not junior (R L > 0). In fact, it is possible for originators to pool at structures in which the payo¤ on the retained claim is decreases with cash ‡ow (R L > R H ), thus creating a reward for poor performance.

The fourth statement of the proposition demonstrates that uniformly high levels of securitization do not constitute prima facie evidence of market irrationality. To the contrary, a necessary condition for the existence of any pooling equilibrium is the existence of a pooling equilibrium featuring full securitization. The …nal two statements in the proposition show that price informativeness and funding value are substitutes in supporting pooling equilibria, with the required informational e¢ ciency threshold for pooling (z) decreasing in : Thus, pooling can be an equilibrium if and only if informational e¢ ciency is high or originators attach very high value to immediate funding.

It follows from the …nal two statements of the proposition that if pooling at opacity can be sustained as an equilibrium, then pooling at transparency can also be sustained as an equilibrium.

Intuitively, the high type is more willing to pool if prices are closer to fundamentals as is the case under transparency. Finally, pooling at opacity is easier to sustain as a continuation equilibrium at than under : Intuitively, the high type is more willing to pool at opacity if investors have more favorable prior beliefs, a standard result in signaling models.

Further intuition regarding the set of pooling equilibria is provided by Figures 3A and3B. Using equation [START_REF] Leland | Information Asymmetries, Financial Structure, and Financial Intermediation[END_REF], each …gure plots pairs of marketed cash ‡ows (M L ; M H ) in pooling equilibria that just pin the two originator types to their respective LCSE payo¤s. The better-than set is the region above the respective indi¤erence curves. 15 curve is above that of the low type, re ‡ecting his reluctance to pool at opacity given that securities will be priced far from fundamentals. Thus, under opacity the high type's indi¤erence curve is the relevant boundary for the set of pooling equilibria. Comparing across the …gures it is apparent that the level of securitization sustainable as a pooling equilibrium is in ‡uenced by price informativeness.

Post-crisis there has been much debate about whether observed securitization levels and the opacity of structures constitute evidence of investor irrationality. Using the perfect Bayesian equilibrium concept, one cannot argue full securitization and/or opacity are inconsistent with rationality.

After all, one implication of Proposition 2 is that full securitization cum opacity can be sustained as a rational market equilibrium if is su¢ ciently high. However, it can be argued that the PBE concept constitutes a weak test of rationality inasmuch as it can admit o¤-equilibrium beliefs that seem unreasonable. The following proposition identi…es structures satisfying the Intuitive Criterion, which imposes restrictions on o¤-equilibrium beliefs.

Proposition 3 A necessary and su¢ cient condition for a perfect Bayesian equilibrium to satisfy the Intuitive Criterion is that interim type-contingent utilities for the originator (U ; U ) satisfy

( q q)U ( 1)qU (q q)[qH + (1 q)L]:
The Least-Cost Separating Equilibrium satis…es the Intuitive Criterion. Opacity never satis…es the Intuitive Criterion. Pooling with transparency and partial securitization satis…es the Intuitive Criterion if and only if

[( 1)q(z z) (1 z)(q q)][M H M L ] 1 ( 1)(L M L ):
Pooling with transparency and full securitization satis…es the Intuitive Criterion if and only if z z 1 z q q q q : Proposition 3 shows a PBE only satis…es the Intuitive Criterion if there is a su¢ ciently large spread between the high and low type interim utilities. Pooling at opacity violates the Intuitive Criterion since all originators get paid the same price for marketed securities. A more "sophisticated" market would infer that only low types prefer opacity. Proposition 3 also shows full securitization can satisfy the Intuitive Criterion. However, comparing inequalities across Proposition 2 and Proposition 23 equilibrium. From equation ( 15) and Lemma 1 it follows that originator willingness-to-pay in the LCSE is b c lcs = [ ( )(q q)(H L)] q q q q : (17

)
The …rst square bracketed term in the expression for b c lcs is the maximum e¤ort cost the originator would pay under observable types. The second bracketed term is a number less than one. Intuitively, at the LCSE the high type bears the underinvestment cost of signaling while the low type gets his symmetric information payo¤. Consequently, there is less incentive to put in e¤ort aimed at becoming a high type.

Consider next e¤ort incentives if the continuation equilibrium entails pooling. Here we must distinguish between pooling cum transparency versus pooling cum opacity. The respective willingnessto-pay expressions are:

b c tran = ( )(q q)[R H R L + (M H M L ) (z z)] (18)
b c op = ( )(q q)[R H R L ]:

Notice that under transparency there are two sources of e¤ort incentives: the retained claim and market discipline, with high types expecting a higher price for their marketed security (since Proposition 2 shows M H > M L in any pooling equilibrium). In contrast, under opacity the only source of e¤ort incentive is the retained claim. It is apparent from the preceding equation that for the same level of retentions, e¤ort incentives are higher under transparency than opacity. Further, under transparency originator e¤ort incentives are higher for higher assumed values of since the wedge between z and z is increasing in .

Equation ( 18) also shows that under opacity and zero retentions, there is zero e¤ort incentive.

Under transparency, e¤ort incentives exist even with zero retentions, with the implied willingnessto-pay equal to:

b c otd tran [ ( )(q q)(H L)][z z]: (19)
The …rst square-bracketed term in the expression for b c otd tran is the cuto¤ cost that would obtain under symmetric information regarding asset type. It can be veri…ed that the second bracketed term is number less than one half.

B. Equilibrium E¤ort

This subsection completes the backward induction to determine the equilibrium set. For the purpose of this analysis, it is useful note that Lemma The …rst statement of the proposition follows from the fact that it is always possible to construct an equilibrium based on the LCSE as a continuation path. The demonstration of the rest of the proposition is as follows. Under the …rst inequality, it is impossible to sustain an equilibrium with no-e¤ort since no-e¤ort is inconsistent with any continuation equilibrium that could possibly follow from no-e¤ort. Under the second inequality, it is impossible to sustain an equilibrium with e¤ort, since e¤ort is inconsistent with any continuation equilibrium that could possibly follow from e¤ort. In the remaining case, it is possible to support an equilibrium with e¤ort by positing the continuation equilibrium corresponding to b c max and it is possible to support an equilibrium with no-e¤ort by positing the continuation equilibrium corresponding to b c min :

The importance of the preceding proposition is to highlight the possibility of multiple equilibrium e¤ort levels. Of course, since continuation payo¤s determine the originator's willingness-to-pay (equation ( 15)), the possibility of multiple equilibrium e¤ort levels is a natural consequence of the fact that there are potentially multiple Securitization Stage continuation equilibria. To take an example, suppose (q q)=( q q) and [z( ) z( )] 1: The …rst inequality implies full securitization combined with either opacity or transparency falls within the set of continuation equilibria. And we have seen that if the continuation equilibrium entails opacity, the originator will not exert e¤ort regardless of the required cost c. In contrast, if the continuation equilibrium entails transparency, the second inequality (combined with Assumption A1) implies the originator will exert e¤ort for each possible c. Therefore, in this example, for each possible c value, no-e¤ort and e¤ort can both be sustained as equilibrium decisions at the origination stage.

The preceding proposition o¤ers two alternative interpretations of the apparent decline in lending standards in the run-up to the credit crisis of 2007-2008, with di¤ering implications for regulation.

One interpretation is that no-e¤ort was inevitable in any unregulated market equilibrium. This

V. Social Welfare and Optimal Mandatory Retentions

Up to this point attention has been con…ned to a positive analysis of potential equilibria in unregulated markets. This section addresses three normative questions. First, what are the market failures and sources of welfare losses in the various unregulated market equilibria? Second, can mandatory retentions increase social welfare? And …nally, which form of mandatory retention scheme maximizes social welfare? Anticipating, the alternative policy options will create winners and losers so Pareto improvements are not generally available. For example, opacity or policies that induce issuers to reveal their private information bene…t uninformed investors as they are insulated from adverse selection. At the same time, such policies make the speculator worse o¤ by preventing her from making trading gains. Below we evaluate alternatives taking the perspective of a utilitarian social planner placing equal weight on the utility of each agent. Social welfare is then the sum of expected utilities, accounting for all private bene…ts and externalities.

A. Welfare in Unregulated Markets

This subsection considers the welfare losses implicit in the various unregulated market equilibria.

To understand the source of welfare losses, it is useful to recall outcomes if the asset type was observable. As discussed above, with observable q, the originator would …nd it optimal to exert e¤ort given that the expected output increase exceeds e¤ort costs (Assumption A1). At the Securitization Stage the entire asset would be marketed since the originator's investment has positive NPV. And it was shown in Section II that with known q each vulnerable UI would fully insure against consumption shortfalls by purchasing units of the L-state Arrow security at an actuarially fair price of 1 q.

Invulnerable UI would borrow in period 3 against their future endowment windfall in order to shift consumption forward as desired. Finally, the speculator would not exert costly e¤ort to acquire information and would simply consume his endowment. The implied social welfare with observable types is equal to the sum of the expected utilities of the originator, uninformed investor and speculator:

W obq = [( (1 q) + (1 )(1 q))L + ( q + (1 )q)H] c ( 20
)
+y ui 3 1 2 ( + )[ (1 q) + (1 )(1 q)] + 1 + 2 + y s 3 :
Consider …rst the welfare gap if the unregulated market equilibrium entails pooling at opacity cum full securitization of the underlying asset ("OTD"). Such an equilibrium has a number of bene…ts in terms of social welfare. The speculator does not exert costly e¤ort. And with symmetric ignorance, e¢ cient risk-sharing is achieved, with each vulnerable UI buying fairly priced insurance against consumption shortfalls (Proposition 1). Finally, with full securitization, there are no underinvestment costs. In fact, the only social cost of such an equilibrium is that it provides zero e¤ort incentive (b c = 0), as shown in Section IV. So here the welfare gap is equal to the net social value of originator e¤ort. Using the social welfare formulae in the appendix, this social value of originator e¤ort is equal to:

W obq W otd op = ( )(q q) (H L) + 1 2 ( + ) c: (21)
The …rst term in the square brackets in the preceding equation is the expected increase in the asset's cash ‡ow resulting from originator e¤ort, which is scaled up by his funding value : The second term in square brackets is the expected increase in the endowment of uninformed investors resulting from originator e¤ort. Recall, a low realized cash ‡ow results in an endowment loss of units for each vulnerable UI, with the aggregate measure of the vulnerable UI being an equiprobable random variable 2 f ; g: Essentially, the second term captures the social value of reductions in externalities arising from distressed or foreclosed properties. The failure of lenders to account for such externalities at the time of loan origination is a …rst market failure.

As shown below, the net social value of originator e¤ort is actually a key welfare loss associated with any unregulated market equilibrium failing to induce e¤ort. And at this point it is worth addressing the following question: Why does an unregulated securitization market admit equilibria failing to induce originator e¤ort? Essentially, the unregulated market admits as equilibria securitization structures achieving a su¢ ciently high payo¤ to originators post-e¤ ort, as shown in Lemma 2.

For example, as shown in Proposition 2, if is very high originators may pool at OTD cum opacity.

And such an outcome actually maximizes interim-stage social welfare. However, such an anticipated outcome results in zero ex ante e¤ort incentive. Tension between interim-e¢ ciency and moral hazard is common to many agency settings (see e.g. [START_REF] Fudenberg | Moral Hazard and Renegotiation in Agency Contracts[END_REF]). Anticipating, the tension between ex ante and interim e¢ ciency provides one potential rationale for government intervention. Regulation can commit issuers not to implement some structures, even some with a high level of interim-e¢ ciency (e.g. opaque OTD), with the goal of restoring e¤ort incentives.

Consider next the welfare gap if the unregulated market equilibrium entails pooling at opacity and partial securitization. In this case, the social welfare gap is increased by an amount equal to the foregone project NPV due to originator retentions. However, the net social value of originator e¤ort is potentially recaptured since the retained claim increases e¤ort incentives provided R H > R L ; as shown in equation [START_REF] Myers | Corporate Financing and Investment When Firms Have Information Shareholders Do Not Have[END_REF]. We have the following welfare gap.

W obq W op = ( 1) ( op (1 q) + (1 op )(1 q))R L + ( op q + (1 op )q)R H (22)
+ op ( )(q q) (H L) + 1 2 ( + ) c :
Consider next the welfare gap if the unregulated market equilibrium entails pooling at transparency. In this case, there are four sources of welfare losses. First, there is a welfare loss equal to the foregone NPV from investment due to originator retentions. Second, the net social value of originator e¤ort is lost if b c tran < c: Third, under transparency the speculator exerts costly e¤ort gathering information. Fourth, as shown in Proposition 1, the existence of an informed speculator distorts risk-sharing in that a subset of vulnerable UI forego insurance against consumption shortfalls, fearing adverse selection. As shown in Lemma 2, in their own decisionmaking, originators do not account for the negative externality associated with pooling, a more subtle market failure. The implied total welfare loss under a transparent pooling equilibrium is:

W obq W tran = ( 1) ( tran (1 q) + (1 tran )(1 q))R L + ( tran q + (1 tran )q)R H (23) + tran ( )(q q) (H L) + 1 2 ( + ) c +e + 1 2 ( + ) " Z b 1 ( 1)f ( )d # [ tran (1 q) + (1 tran )(1 q)]:
Consider …nally the welfare gap if the LCSE is the unregulated market equilibrium. The LCSE has a number of bene…ts. In the LCSE the private information of the originator is credibly signaled at the Securitization Stage so there is common knowledge of the asset type at the Trading Stage.

As shown in Proposition 1, it follows that the speculator does not exert e¤ort. And with the type revealed, each vulnerable UI purchases a fairly priced Arrow security to insure against consumption shortfalls, so that risk-sharing is e¢ cient. Thus, there are only two sources of welfare loss in the LCSE. First, high type retentions result in foregone project NPV. Second, the net social value of originator e¤ort is lost if b c lcs < c: We have the following welfare gap in the LCSE:

W obq W lcs = ( 1) lcs q (q q)(H L) q q (24)

+ lcs ( )(q q) (H L) + 1 2 ( + ) c :
The next two subsections consider socially optimal mandatory retention schemes that serve to induce speculator e¤ort. In contrast, the following proposition singles out the optimal regulation inducing no-e¤ort.

Proposition 6 Mandating opacity and zero originator retentions is is socially optimal amongst regulations failing to induce originator e¤ ort.

The preceding proposition shows that an optimal regulation need not mandate retentions or more transparent disclosure. To the contrary, if the regulator is content to tolerate originator moral hazard and forego the net social value of originator e¤ort (equation ( 21)), then the government should actually mandate zero retentions, with the goal of maximizing originator funding. And if e¤ort incentives are not a concern, there is no need for market discipline, so opacity is optimal.

Proposition 6 is consistent with the arguments in Dang, Gorton and Holmström (2011) regarding the bene…ts of opacity. Opacity conserves speculator e¤ort costs and promotes e¢ cient risk-sharing.

With this in mind, it follows that a high degree of investor sophistication is not necessarily bene…cial in terms of social welfare. In particular, if investor beliefs are "sophisticated" in the sense of satisfying the Intuitive Criterion, then pooling at opacity cannot be sustained as an unregulated market equilibrium (Proposition 3).

B. Motivating E¤ort via Separating Regulations

From Proposition 6 it follows that inducing e¤ort is a necessary condition for some regulation other than mandatory opaque OTD to be socially optimal. Therefore, the remainder of the analysis is devoted to determining socially optimal methods for inducing originator e¤ort. This subsection determines the optimal mandatory retention scheme amongst those inducing originators to exert e¤ort, as well as compelling them to credibly reveal the true asset type to investors. From a social perspective, all schemes meeting these two objectives result in the same expected utility for the speculator, who consumes her endowment, and the UI, who fully insure against negative shocks.

Therefore, the socially optimal separating regulation maximizes the expected utility of the originator subject to appropriate incentive constraints.

We begin by noting that if e¤ort is incentive compatible (IC below) in the LCSE, there is no socially preferable separating scheme. Consider then the socially optimal separating regulation when the IC constraint is violated at the LCSE (c > b c lcs ). Let (M L ; M H ) and (M L ; M H ) denote the cash ‡ows to be marketed by low and high types, respectively. The planner's problem is to maximize the expected utility of the originator subject to IC, non-mimicry by the low type (with the high type's non-mimicry constraint being slack), and limited liability for the originator. We solve the following relaxed program which ignores some limited liability constraints and then verify the neglected constraints are slack:

max M L ;M H ;M L ;M H q(H M H ) + (1 q)(L M L ) + [qM H + (1 q)M L ] (25)
+(1 )fq(H M H ) + (1 q)(L M L ) + [qM H + (1 q)M L ]g s:t: IC : q(H M H ) + (1 q)(L M L ) + [qM H + (1 q)M L ] c = q(H M H ) + (1 q)(L M L ) + [qM H + (1 q)M L ] N M : q(H M H ) + (1 q)(L M L ) + [qM H + (1 q)M L ] q(H M H ) + (1 q)(L M L ) + [qM H + (1 q)M L ] LL : M L L; M L L
NM must bind in the relaxed program otherwise the objective function could be increased by raising M H by an in…nitesimal amount while still meeting all constraints. Substituting the binding NM constraint into the objective function and IC constraint allows one to rewrite the relaxed

The separating regulation described in Proposition 7 accomplishes two distinct tasks: provision of ex ante e¤ort incentives by increasing the wedge between high and low type Securitization Stage continuation utilities and revelation of the originator's private information. This last e¤ect is socially valuable since it eliminates the speculator's incentive to pay costs to acquire information and also serves to insulate uninformed investors from adverse selection, facilitating e¢ cient risk-sharing.

The proposition shows that in order to restore e¤ort incentives the high type is forced to hold a larger junior tranche than in the LCSE. Examination of the low type contract reveals a stark contrast between the LCSE and the socially optimal separating regulation inducing e¤ort. In the LCSE, a low type fully securitizes his asset and achieves his symmetric information payo¤. In contrast, the optimal regulation mandates that the low type must also retain a junior claim, albeit of smaller size than that of the high type. Such a mandated retention increases the wedge between high and low type continuation payo¤s, restoring e¤ort incentives.

It is apparent that in terms of continuation utilities both originator types are worse o¤ than at the LCSE. Lemma 2 shows an unregulated market would never implement such an outcome since it is interim-ine¢ cient from the perspective of originators. The role of the government regulation here is to serve as a commitment device to implement interim-ine¢ cient equilibria in order to restore ex ante e¤ort incentives. Finally, the optimality of forcing the originator to hold a junior claim in the context of the separating regulation is a consequence of the fact that in the present setting a standard single-crossing condition is satis…ed, with high types placing a higher relative valuation on high state payo¤s. This fact makes the retention of a junior claim a less costly signaling device.

That is, other retained claims might su¢ ce to separate types and restore e¤ort incentives, but they would generate larger underinvestment costs. This signaling argument is distinct from the traditional moral hazard argument that calls for risk-neutral agents, such as our originator, to be residual claimants (see e.g. [START_REF] Innes | Limited Liability and Incentive Contracting with Ex Ante Action Choices[END_REF]).

C. Motivating E¤ort via Pooling Regulations

Consider next the socially optimal means of inducing originator e¤ort using some form of pooling regulation such that all originators are forced to retain the same claim. A pooling regulation can be used in combination with either mandated transparency or opacity.

Consider …rst the optimal pooling regulation combined with mandated transparency. The socially optimal regulation maximizes the weighted average of originator utilities subject to the appropriate IC constraint, since the expected utility of all other agents is the same across all transparent pooling regulations. The social planner's program is:

max M L L;M H H U + (1 )U = L + (H L)[ q + (1 )q] + ( 1)[M L + (M H M L )( q + (1 )q)]
subject to

IC : U U = (q q)[(H L) (M H M L )(1 (z z))] c :
If the IC constraint is slack then the solution to the above program is full securitization. Consider then the remaining case in which the IC constraint binds (c > b c otd tran ). Substituting the IC constraint into the objective function it follows that the optimal pooling contract cum transparency calls for the originator to market the following bundle of cash ‡ows:

M L = L (30)
M H = L + ( )(q q)(H L) c ( )(q q)[1 (z z)] = H (H L)(c b c otd tran ) + ( )(q q)(H L) b c otd :
Consider next the optimal pooling regulation when combined with mandated opacity. Again, the socially optimal regulation maximizes the weighted average of originator utilities. The social planner's program is:

max M L L;M H H U + (1 )U = L + (H L)[ q + (1 )q] + ( 1)[M L + (M H M L )( q + (1 )q)]
subject to

IC : U U = (q q)[(H L) (M H M L )] c :
Here the IC constraint must bind since otherwise the optimum would entail full securitization, but this would necessarily violate the IC constraint. Substituting the IC constraint into the objective function it follows that the optimal pooling regulation cum opacity calls for the originator to market the following bundle of cash ‡ows:

M L = L (31)
M H = H c ( )(q q) :
We have established the following proposition.

Proposition 8 Socially optimal pooling regulations for inducing e¤ ort call for originators to retain junior claims paying zero in state L. If the regulation mandates transparency, the retained claim has state H payo¤ equal to:

R tpool H = (H L)(c b c otd tran ) + ( )(q q)(H L) b c otd tran :
If the regulation mandates opacity, the retained claim has state H payo¤ equal to:

R opool H = c ( )(q q) :
Proposition 8 shows that if the regulatory intent is for originators to pool at a common structure, the socially optimal means of providing e¤ort incentives is for the originator to retain a junior claim such that R L = 0. Intuitively, reductions in R L serve to relax the respective IC constraints as well as increasing the level of originator fundraising. The proposition also shows that originators must be forced to hold larger junior claims if the regulation mandates opacity. After all, under opacity market discipline is absent at the time of securitization, so all e¤ort incentives must come from the retained claim. Finally, it is readily veri…ed that R tpool H decreases with the informational e¢ ciency of markets, as measured by z z: Intuitively, under the pooling regulation, originator retentions and market discipline are substitute mechanisms for providing e¤ort incentives. Thus, the optimal pooling regulation cum transparency requires making a judgement about informational e¢ ciency.

D. Welfare Comparisons Across Regulations

Having characterized the optimal regulations within each category in the preceding two subsections, we can now determine the optimal e¤ort-inducing regulation. Appendix B contains the social welfare equations. For brevity, this section compares social welfare losses across the alternatives. : Essentially, the opaque pooling regulation imposes the same high level of retentions that the separating regulation reserves for the high type, with low types being permitted to retain smaller claims. We have:

( 1) qR sep H + (1 )qR sep H < ( 1)[ q + (1 )q]R opool H ) W sep > W opool : (32)
It follows from the preceding argument that the optimal e¤ort-inducing regulation is either the separating regulation or a pooling regulation with mandatory transparency. Qualitatively, the two regulations di¤er along the following lines. The separating scheme conserves speculator e¤ort costs and achieves …rst-best risk-sharing. Again, this is due to the fact that the separating scheme restores symmetric information across investors by compelling originators to credibly signal positive information via higher retentions. In contrast, no such signal is sent under the transparent pooling scheme. To the contrary, transparency allows the speculator to exert costly e¤ort in order to gain an informational advantage over other investors. Risk-sharing is then distorted as a subset of uninformed investors fail to insure against endowment shocks fearing adverse selection (Proposition

1) qR sep H (1 )qR sep H ( 1)[ q + (1 )q]R tpool H + e+ 1 2 ( + ) [ (1 q) + (1 )(1 q)] h R b 1 ( 1)f ( )d i : (
The …rst term on either side of the preceding inequality measures expected underinvestment costs under the two regulatory schemes. Of course, this will be the case only if prices are su¢ ciently informative. Therefore, determination of the optimal regulation for inducing e¤ort requires taking a view on the informational e¢ ciency of markets. If it is low, the separating regulation is preferred. It it is high, then the pooling regulation has the potential to yield higher social welfare.

Once the optimal e¤ort-inducing regulation has been determined, its respective social welfare loss should be compared with the social value of originator e¤ort in equation ( 21). If it is smaller, then the optimal regulation induces e¤ort. Otherwise, it is optimal to forego e¤ort incentives and instead mandate opacity and full securitization (Proposition 6).

Conclusions

This paper revisits a canonical problem in corporate …nance, security issuance and retention when the issuer has private information. The model departs from prior literature in three ways.

First, we address how the anticipated issuance-stage equilibrium a¤ects ex ante e¤ort incentives.

Second, we analyze how the prospect of informed trading alters equilibrium retentions and e¤ort incentives. Finally, we consider how informed trading e¤ects the e¢ ciency of risk-sharing when rational uninformed investors shift portfolios in response to adverse selection. The primary focus is on ABS markets, where moral hazard and adverse selection problems appear acute, but the setup considered approximates many other real-world settings. For example, owner-managers of private …rms choose e¤ort anticipating a subsequent sale of securities under asymmetric information. Mature conglomerates expend resources to improve the quality of business units prior to carve-outs.

We consider …rst potential equilibria in unregulated markets. One possible equilibrium is a separating equilibrium in which high types separate from low types by retaining the minimum junior tranche needed deter mimicry by low types who fully securitize. In addition, pooling equilibria exist if both originator types are weakly better o¤ than at the separating equilibrium. We show that if any pooling equilibrium can be sustained, a pooling equilibrium with full securitization can also be sustained. In this sense, full securitization should not be viewed as an anomaly. Unobservability of types at the securitization stage reduces e¤ort incentives in all unregulated market equilibria. This is because asymmetric information at the time of security issuance reduces the payo¤ di¤erential between owners of high and low quality assets. Transparency and sophisticated investor beliefs were shown to increase originator e¤ort incentives. Finally, there can be multiple self-ful…lling levels of originator e¤ort in unregulated markets.

Privately optimal retentions can be socially suboptimal since originators do not internalize e¤ects on investor welfare. In particular, when the high type credibly signals via large junior retentions he bene…ts directly from his own marketed securities being priced at fundamentals at the time of issuance. But he does not internalize the bene…t accruing to investors who can now e¢ ciently share risks being symmetrically informed. Further, the anticipation of asymmetric information at the time of securitization reduces originator e¤ort incentives. We show mandated retentions can raise social welfare by increasing e¤ort incentives in an e¢ cient way, accounting for investor-level externalities.

The …rst policy implication to emerge from the model is that originators should be required to hold junior tranches. The underlying logic for this prescription depends on the nature of the regulation. In a pooling regulation, retention of a junior claim increases the spread between payo¤s accruing to high and low types. In a separating regulation, retention of a junior claim allows issuers to signal with minimal reduction in their investment. Second, in contrast to standard signaling results, it is optimal to impose mandatory retentions on even the low type, since this increases ef-fort incentives e¢ ciently. Third, in the optimal pooling regulation, the size of the mandated junior retention is decreasing in informational e¢ ciency. Fourth, a necessary condition for the pooling regulation to dominate is su¢ cient informational e¢ ciency. Fifth, the separating (pooling) regulation generally maximizes welfare if e¢ cient risk-sharing (originator investment) is the dominant concern.

Finally, if the net social value of originator e¤ort is low, then it is optimal to forego e¤ort incentives altogether and instead maximize investment and the e¢ ciency of risk-sharing. This is achieved by mandating opacity and zero retentions.

The model delivers a broader message. It is commonly argued that the decline in lending standards prior to the subprime crisis of 2007/8 was inevitable given the preceding shift from relationship banking to the OTD business model. And in fact, existing theoretical models of the ABS market support the notion that lender laxity is a necessary consequence of OTD. After all, so the argument goes, an originator has no incentive to screen if he is going to sell all claims on cash ‡ow at an unconditional price re ‡ecting only the average ABS quality. And indeed, in existing ABS models prices are uninformative precisely because traders are assumed to be incapable of generating information about asset quality. In reality, traders can generate useful information and securities prices can be informative. Viewed from this perspective, the problem of lender laxity must be understood as a failure of the price mechanism. Moreover, for those believing in the importance of market discipline, praise of and calls for increased opacity seem exactly the wrong policy response to the subprime crisis unless one is willing to accept lender laxity as a fact of life.

The model presented gives a stylized overview of the agency problems and policy tradeo¤s in ABS markets. However, it surely fails to capture some features of speci…c ABS markets. Given the size of each of the respective ABS markets, it would be useful for future models to take a more granular perspective in order to better capture the institutional details, risk characteristics, and agency problems inherent in the di¤erent ABS classes. Next-generation models should also pursue a richer speci…cation of underlying stochastic processes and better capture dynamics.

set since such a contract would lower at least one type's payo¤. Further, it follows a necessary condition for a pooling menu to be in the equilibrium set is that both types are weakly better o¤ than at the LCS. We next establish su¢ ciency. To this end, consider any conjectured equilibrium in which both types receive at least their LCS payo¤. Deviations to a separating contract cannot be pro…table for either type since no separating contract improves upon the LCS payo¤s. Consider next deviations to a pooling menu. We need only identify and stipulate o¤-equilibrium beliefs su¢ cient to deter deviation.

Consider …rst deviations with total marketed cash ‡ows such that M H M L : Such deviations are assumed to be imputed to the low type. The low type payo¤ to such a deviation is:

q(H M H ) + (1 q)(L M L ) + [qM H + (1 q)M L ] U lcs :

And the high type payo¤ to deviating is:

q(H M H ) + (1 q)(L M L ) + [qM H + (1 q)M L ]

q(H M H ) + [qM H + (1 q)L] < U lcs :

Consider …nally a deviation to a pooling contract with M H < M L : Such deviations are assumed to be imputed to the high type. The high type payo¤ to deviating is then:

q(H M H ) + (1 q)(L M L ) + [qM H + (1 q)M L ] = qH + (1 q)L + ( 1)[qM H + (1 q)M L ]
qH + (1 q)L + ( 1)L < U lcs :

And the payo¤ to the low type from such a deviation is:

q(H M H ) + (1 q)(L M L ) + [qM H + (1 q)M L ] = qH + (1 q)L + ( q q)M H + [ (1 q) (1 q)]M L < qH + (1 q)L + ( q q)M L + [ (1 q) (1 q)]M L = qH + (1 q)L + ( 1)M L U lcs :

This is a contradiction.N

Lemma. M H > L in any pooling equilibrium.

Proof: Suppose to the contrary there exists a pair of marketed payo¤s (M 0 L ; M 0 H ) in the equilibrium set while M 0 H < L: Since K L (q; z) > 0 and K H (q; z) > 0 (preceding lemma) we know K L (q; z)M 0 L + K H (q; z)M 0 H + qH + (1 q)L < K L (q; z)L + K H (q; z)L + qH + (1 q)L = ( 1)L + qH + (1 q)L < U lcs :

This is a contradiction.N

The third statement in the proposition follows from the fact that K H > 0 for both types if there is a pooling equilibrium. To prove the fourth statement in the proposition, assume there is a pooling equilibrium at the marketed pair (M 0 L ; M 0 H ): Since K L (q; z) > 0 and K H (q; z) > 0, we know the high type's payo¤ at full securitization is K L (q; z)L + K H (q; z)H + qH + (1 q)L K L (q; z)M 0 L + K H (q; z)M 0 L + qH + (1 q)L U lcs

And the low type is always better o¤ when pooling at full securitization than under his LCS allocation since K L (q; z)L + K H (q; z)H + qH + (1 q)L > [qH + (1 q)L] = U lcs :

Finally, to establish the existence of a pooling equilibrium at full securitization we need only check the condition under which the high type is better o¤ than at the LCS (since the low type is necessarily better o¤). We have: K L (q; z)L + K H (q; z)H + qH + (1 q)L [qH + (1 q)L] ( 1)q (q q)(H L) ( q q) m z (q q)=( q q): Proposition 3: The Intuitive Criterion

We begin by recalling that with two types (t; t 0 ), a PBE fails to satisfy the Intuitive Criterion if there exists: an unsent menu proposal m 0 ; a type t 0 who is strictly better o¤ than at the posited PBE by proposing m 0 for all best responses with beliefs focused on t 0 ; and a type t who is strictly better at the posited PBE than at m 0 for all best responses for all beliefs in response to m 0 .

With this de…nition in mind a few lemmas are immediate. First, a PBE will never be pruned via a low type deviation (imputed to him) since the associated payo¤ is weakly less than his LCS payo¤.

Second, no separating menu can prune the PBE set since any separating contract yields either type weakly less than his LCS payo¤. Third, any pruning high type pooling contract deviation must feature M H > L since a deviation to M H L imputed to him yields strictly less than his LCS payo¤. Thus, without loss of generality in pruning the set of PBE attention can be con…ned to high type deviations to pooling contracts entailing M H > L M L : The following lemma further narrows the set of relevant deviations.

Lemma: If a deviation to (M 0 L ; M 0 H ) prunes a PBE, so too does a deviation to (L; M 1 H ) where ) U = ( q q) M H + [ (1 q) (1 q)] M L ) U = [( q q)(1 q)=q + ( (1 q) (1 q))](L M 0 L ) < 0:N Lemma: A necessary and su¢ cient condition for a perfect Bayesian equilibrium to satisfy the Intuitive Criterion is that the associated type-contingent interim utilities (U ; U ) satisfy (q q)[qH + (1 q)L] ( q q)U ( 1)qU :

Proof: From the preceding lemma, a necessary and su¢ cient condition to prune a PBE is to …nd an M H such that q(H M H ) + [qM H + (1 q)L] < U q(H M H ) + [qM H + (1 q)L] > U

The …rst inequality immediately above implies an upper bound M up H < H and the second implies a lower bound M low H > L: Thus, there exists a feasible pruning deviation i¤ M up H > M low H : The inequality stated in the lemma is necessary and su¢ cient to ensure M low H M up H so that no pruning deviation exists.N

The LCSE utilities satisfy the necessary and su¢ cient condition stated in the preceding lemma.

We turn now to proving a …nal lemma.

Lemma: A PBE survives the Intuitive Criterion if and only if

( 1)q(z z) (1 z)(q q) (M H M L ) ( 1) (L M L ) :

Proof: Let REV and REV denote the expected revenues of high and low types, respectively. We established above the following necessary and su¢ cient condition to satisfy the Intuitive Criterion:

q s v 0 v v 2 q s v 0 v v 2 q s v (v v) v (2v v) (1 ) 2 q s v (v v) v v (1 ) 2 q s v (v v) v (2v v) (1 ) 2 q s v (v v) v v (1 ) 2 q s v 0 v v (1 )(1 ) 2 q s v 0 v v (1 )(1 ) 2

1 2

 2 See Dubey, Geanakoplos and Shubik (2005) for GEI with …nite penalties to reneging and endogenous default.

Lemma 2

 2 The set of equilibrium menus at the Securitization Stage consists of the Least-Cost Separating Allocations and any pooling menus giving each originator type at least his respective Least Cost Separating Allocation payo¤ .

  Figure 3A depicts transparency and Figure 3B depicts opacity. To isolate the role of price informativeness, model parameters are held …xed across the two …gures. Consider …rst Figure 3A. With transparency, the low type's indi¤erence curve is above that of the high type. Thus, the low type's indi¤erence curve is the relevant boundary for the set of pooling equilibria. Intuitively, the low type is more reluctant to pool than the high type if prices are close to fundamental value. Consider next Figure 3B. With opacity, the high type's indi¤erence

Proposition 5

 5 2 and Proposition 2 imply the set of Securitization Stage continuation equilibria depends upon investors' prior belief ( ) regarding the probability of the issuer being a high type. Intuitively, once the Securitization Stage subgame is reached, we have a standard signaling game where the equilibrium set naturally depends upon prior beliefs regarding the distribution of types. With this in mind, let E SS denote the set of possible Securitization Stage continuation equilibria resulting from = and let E SS denote the set of possible continuation equilibria resulting from = : Each set includes the LCSE and the set of -contingent pooling equilibria described in Proposition 2. Let denote a generic member of either set and b c( ) denote the corresponding willingness-to-pay using the formulae in the preceding subsection. Next let:E SS f 2 E SS : b c( ) cg E SS f 2 E SS : b c( ) cg:Notice, for each of the sets de…ned above, the willingness-to-pay is consistent with the posited continuation path, e.g. b c( ) c if is a continuation equilibrium arising when no e¤ort has been exerted ( That is, b c min measures the minimum willingness-to-pay computed over the set of continuation equilibria resulting from = : Conversely, b c max measures the maximum willingness-to-pay computed over the set of continuation equilibria resulting from = : Since both sets include the LCSE (Lemma 2), it follows b c min b c max : We have the following proposition. The set of pure strategy perfect Bayesian equilibria for the full game is the nonempty set (No-E¤ ort, E SS ) [ (E¤ ort, E SS ): If c < b c min ; the originator exerts e¤ ort in any equilibrium. If c > b c max ; the originator does not exert e¤ ort in any equilibrium. If c 2 [b c min ; b c max ], both e¤ ort and no-e¤ ort can be equilibrium outcomes.

  interpretation corresponds to c > b c max . An alternative interpretation is that unregulated markets were simply trapped in an equilibrium with low e¤ort incentives. This interpretation corresponds to c 2 [b c min ; b c max ], with unregulated markets happening to implement an equilibrium with a low b c: In this case, a su¢ cient remedy for lender laxity is light-touch regulation selecting an e¤ort-inducing equilibrium from the set of potential unregulated market equilibria, e.g. mandating transparency when opacity was otherwise viable as an equilibrium.The prior analysis also suggests a potentially critical role for investor sophistication in alleviating originator moral hazard by way of eliminating Securitization Stage continuation equilibria generating low e¤ort incentives. For example, the Intuitive Criterion precludes pooling at full securitization cum opacity, an outcome that destroys e¤ort incentives. More generally, from Proposition 3 it follows the Intuitive Criterion demands that the gap between type-contingent interim utilities be su¢ ciently large, which is precisely what is needed to promote originator e¤ort ex ante, as shown in equation[START_REF] Maskin | The Principal-Agent Relationship with an Informed Principal, II: Common Values[END_REF].

Consider …rst a comparison

  of social welfare under the separating regulation versus the opaque pooling regulation. Both regulatory schemes have the bene…t of conserving speculator e¤ort costs and achieving …rst-best risk-sharing. In the separating regulation, symmetric information across investors is restored by inducing issuers to credibly signal their type. Under the opaque pooling regulation, investors have symmetric ignorance. Under both schemes the only deadweight welfare loss is expected underinvestment by originators. Here it is readily veri…ed that expected underinvestment costs are higher under the opaque pooling regulation since R sep H < R sep H = R opool H

1 ) 0 :

 10 . Finally, as shown in Proposition 7, the separating scheme necessarily generates underinvestment costs since R sep H > R sep H In contrast, as shown in Proposition 8, the transparent pooling regulation only generates underinvestment costs if b c otd tran < c: The separating scheme yields higher social welfare if:

  construction the high type achieves the same payo¤ deviating to (L; M 1 H ) as opposed to (M 0 L ; M 0 H ): Further, since the high type gains from both deviations it must be that M 1 H > L and M 0 H > L M 0 L : Thus, for either deviation the most favorable belief is that it is being made by the high type. Given such beliefs the low type must have been worse o¤ deviating to (M 0 L ; M 0 H ): Relative to that deviation payo¤, the low type is even worse o¤ deviating to (L; M 1 H ) with the change in utility (for beliefs focused on the high type) computed viaU = q(H M H ) + (1 q)(L M L ) + [qM H + (1 q)M L ]

  ). Each has a vNM utility function C 3 + C 4 . Second, there is a risk neutral speculator S with vNM utility function C 3 + C 4 . Her period 3 endowment is y s

	3	H, so she
	has su¢ cient wealth to buy the entire asset. S is unique in having the requisite skill to learn about
	asset quality. The possibility of the speculator learning about asset quality depends on whether
	there is opacity or transparency. Under opacity the speculator does not receive any signal. Under
	transparency the speculator can receive an informative signal, but this requires her to incur a …xed
	non-pecuniary e¤ort cost e 0. If the speculator exerts e¤ort, her signal is s 2 fs; sg with:

Table 3

 3

depicts potential orders confronting the MM on the equilibrium path. The MM infer s = s when the state is (s; v) and s = s when the state is (s; v). However, MM cannot infer s in the states (s; v) and (s; v): Using Bayes'rule the MM revise beliefs and set price as follows based upon aggregate demand for the L-state Arrow security (X AG L ):

  The rest of the terms capture e¤ort costs and welfare losses from foregone insurance under the transparent pooling regulation. Since the transparent pooling regulation results in speculator e¤ort costs and distorted risk-sharing, it is clearly dominated if it also creates higher underinvestment costs. Here it is worth noting that R tpool

			H	converges to R	sep H as
	approaches one-half. Since R sep H < R	sep H it follows that a su¢ cient condition for the separating
	regulation to be superior is for to be su¢ ciently low. Conversely, the pooling regulation imposes no
	underinvestment costs at all if b c otd tran	c and may be superior accounting for underinvestment costs.

Table 3 :

 3 Aggregate Demand Outcomes

	Type Signal	% Uninformed	Informed	Uninformed	Aggregate	Probability
		Vulnerable	Demand	Demand	Demand

The fact that securities are written on an asset in place, excluding the new investment, departs from some corporate …nance settings and models.

Here we refer only to the continuation equilibrium.

Pagano and Volpin (2010) also develop a model of tradeo¤s associated with primary market opacity.

This setup is equivalent to an alternative setup without new investment but with O being impatient and having vNM utility function C3 + C4:

Smooth utility functions could be assumed at the cost of more complex aggregate demands.

The characterization of equilibrium securitization structures and originator e¤ort incentives are unchanged if there is negative correlation between UI endowments and cash ‡ow.

Some proposals call for mandatory retentions as a requirement for using the shelf registration procedure. See "SEC Proposes Asset-Backed Securities Reform," 12 April 2010, Harvard Law School Forum on Corporate Governance and Financial Regulation.

That the number of structures should correspond to the number of types is shown by[START_REF] Maskin | The Principal-Agent Relationship with an Informed Principal, II: Common Values[END_REF].

If one were to attempt trading prior to information revelation, all agents would have incentives to accelerate information gathering.1 0 The characterization of equilibrium retentions is unchanged if one instead considers limit orders.1 1 Regulations AB and M of the Securities Exchange Act prohibit originators from clandestine trading.

[START_REF] Dow | Arbitrage, Hedging, and Financial Innovation[END_REF] Trading based upon a completely uninformative signal generates a loss.

[START_REF] Dubey | Default and Punishment in General Equilibrium[END_REF] Another class of pooling menus subsumed in this case is when the menu contains two distinct structures but both types would choose the same structure from it.

[START_REF] Faure-Grimaud | Public Trading and Private Incentives[END_REF] High type indi¤erence curves always have negative slope while low type indi¤erence curves can have positive or negative slope. These properties follow from signing (KL; KH ) in the appendix.

higher degree of price informativeness.

IV. Originator E¤ort

As the last step in the backward induction to characterize the equilibrium set, this section considers the originator's e¤ort decision in period 1.

A. Originator Willingness-to-Pay

Let b c denote the maximum cost the originator would be willing to incur in order to increase the high type probability from to : For each pair of type-contingent originator utilities (U ; U ) in the set of Securitization Stage continuation equilibria, the willingness-to-pay (b c) is:

The preceding equation delivers a simple message: ex ante e¤ort incentives are increasing in the wedge between type-contingent continuation utilities.

Before considering e¤ort incentives under any speci…c continuation equilibrium, we prove that asymmetric information results diminished e¤ort incentives. In any PBE:

The …rst inequality in equation ( 16) follows from the fact that the low type receives at least his symmetric information payo¤ in any PBE, as shown in Lemma 2. The last inequality follows from the fact that the high type gets less than his symmetric information payo¤ in any PBE. In the LCSE, the high type underinvests in order to signal positive information. And in any pooling equilibrium the high type's securities are underpriced. We have established the following key result.

Proposition 4 In all unregulated market equilibria, originator e¤ ort incentives are less than under symmetric information regarding types.

program as:

Substituting the right side of IC 0 into the objective function we …nd it is increasing in M L from which it follows the socially optimal separating contract entails:

Next, we substitute (M L ; M H ) into the NM constraint to compute the low type's utility under the socially optimal separating contract:

Any pair (M L ; M H ) giving the low type the correct utility level su¢ ces. For example, set:

We have established the following proposition.

Proposition 7

The socially optimal separating regulation for inducing originator e¤ ort calls for both types to retain junior claims paying zero in state L. In state H, the retained claims of the high and low types have respective payo¤ s:

Uninformed Investor Portfolios

Consider …rst portfolio choice under common knowledge of type. Each vulnerable UI solves the following program:

Utility is increasing in x L for all x L 2 (0; ) and is decreasing in x L for all x L : Utility is decreasing in x H for all x H 0 and increasing in x H for all x H 2 ( ; 0): Consider next portfolio choice for each invulnerable UI. They solve the following program:

For an invulnerable UI utility is decreasing in x L and x H on the relevant interval so their optimal portfolio payo¤ is ( ; ).

Consider next UI portfolio choice when the type is not common knowledge. A vulnerable UI solves the following program:

We conjecture (and verify):

Under the stated conjecture, utility is increasing in x L for all x L 2 (0; ) i¤ b , and is otherwise decreasing. Utility is decreasing in x L for all x L : Utility is decreasing in x H for all x H 0 and increasing in x H for all x H 2 ( ; 0):

The optimal portfolio for an invulnerable UI solves:

For an invulnerable UI utility is decreasing in x L and x H on the relevant interval so their optimal portfolio payo¤ is ( ; ).

Lemma 1: LCS Allocations

The program can be written as max

subject to

We solve a relaxed program ignoring the last constraint and then verify the neglected constraint is slack. In this relaxed program the nonmimicry constraint must bind since otherwise the objective function could be increased by raising M H by an in…nitesimal amount. From the binding nonmimicry constraint M H can be expressed as:

Substituting M H (M L ) into the objective function and ignoring constants, the relaxed program can now be expressed as:

This objective function is strictly increasing in M L , implying optimality of M L = L. Substituting this value into M H (M L ) and verifying the neglected constraint is slack, it follows an LCS allocation entails:

Lemma 2: Set of Equilibrium Payo¤ s

Each type can guarantee himself at least his LCS payo¤ (in any sequential equilibrium) by proposing the LCS retention scheme. It follows that no other separating contract is in the equilibrium Proposition 2: Characterization of Pooling Equilibria

For brevity, we express the constraints on pooling equilibria as follows.

The case of opacity is subsumed in the prior equations by setting z = z = :

We begin by proving a few useful lemmas.

Lemma. M H > M L in any pooling equilibrium.

Proof: Suppose to the contrary there exists a pair of marketed payo¤s

The low type's pooling payo¤ would be

with the second inequality following from K H (q; z) > 0 and the last line following from

Lemma. If there is a pooling equilibrium, then K H (q; z) > 0.

Proof: Suppose to the contrary there exists a pair of marketed payo¤s (M 0 L ; M 0 H ) in the equilibrium set while K H (q; z) 0: Since M H > M L in any pooling equilibrium we then know the high type's equilibrium payo¤ is:

The full securitization condition follows immediately from the preceding lemma. And …nally, no opaque structuring (z = z) satis…es the condition stated in the lemma.

Appendix B: Social Welfare

This appendix presents expressions for total social welfare under both unregulated and regulated market equilibria. 1)(1 q)] + 1 + 2

( 1)[ q + (1 )q]R opool H :